Macroeconomics | Macro Paper Warehouse

A Cognitive Theory of Reasoning and Choice

Mon, 01 Jan 0001 00:00:00 +0000

Bordalo, Gennaioli, Lanzani, and Shleifer develop a cognitive theory of choice in which a decision maker’s attention to the features of options is determined by her categorization of the current problem against a memory database of problems she solved in the past. The core claim is that before solving a problem, the decision maker asks “what kind of problem is this?” and resolves it by selecting the category — indexed by a prototype attention-plus-context vector and a time-discounted frequency — whose similarity to the current problem is maximized. This problem recognition step then pins down which features (price, quality, probabilities) receive attention, which in turn shapes valuation and choice.

The model formalizes two-step choice. In step one (recognition), the decision maker jointly chooses an attention vector alpha_P and a category c* to maximize a separable similarity function S[(alpha_P, kappa_P), (alpha_c, kappa_c)] weighted by category frequency F_c, plus a Type I extreme-value shock that yields a logit probability over categories. In step two, she maximizes perceived value over the menu using the endogenously determined weights. Perceived hedonic value of feature i shrinks toward the menu average when alpha_{P,i} < 1; perceived probabilities compress toward uniform when the event-attention weight falls below 1, producing probability overweighting of unlikely events. Full attention recovers expected utility.

The model yields three structural predictions that hold without changing tastes or information. First, within-person multi-modal attention: because categorization is stochastic, the same person can cluster on entirely different features (e.g., the base rate vs. the likelihood in an inference problem) across otherwise identical choice occasions. Second, systematic context-driven instability: when an irrelevant context feature kappa_{P,i} drifts away from a category’s diagnostic kappa_{c,i}, the probability of that category falls discontinuously, causing a discrete switch in the attention profile and hence in valuation. Third, experience-driven heterogeneity: people more frequently exposed to a category (higher F_c) are more likely to use it, producing persistent differences in price elasticities or probability weighting at constant income and tastes.

Applied to riskless consumer choice, the paper introduces two categories — “buying” (full attention to price, partial to quality: alpha_{M_g}=1 > alpha_{Q_g}=alpha) and “consuming” (full attention to quality, partial to price: alpha_{Q_g}=1 > alpha_{M_g}=alpha). A jam problem categorized as buying yields valuation v = alphaq - etap; categorized as consuming, v = q - alphaetap. The valuation jumps discontinuously as context crosses a threshold kappa*, which shifts when relative category frequency F_{buy}/F_{con} changes. This framework accounts for context-dependent price elasticities (Wakefield and Inman 2003), poverty-driven excess price focus (Shah et al. 2018), de-commoditization through advertising, and mental accounting anomalies including opportunity cost neglect and the sunk cost fallacy — both arising because con neglects capital gains (alpha_{con,Delta_M}=0) and buy neglects quality shocks (alpha_{buy,Delta_Q}=0).

Applied to statistical judgment, the paper introduces two categories — “frequency estimation” (attention alpha_1=1 to a single i.i.d. draw from a known DGP) and “agnostic inference” (attention alpha_S=1 to the share of heads as a sufficient statistic). The threshold N* separates recognition: for sequence length N_P < N*(F_{freq}/F_{inf}), the decision maker categorizes as frequency and correctly assesses odds; for N_P >= N*, she switches to inference and overweights balanced sequences, producing the Gambler’s Fallacy. The same competition between categories also accounts for base rate neglect, conjunction fallacy, and correlation neglect, with the bias strengthening as sequences grow longer.

Applied to risky choice, bottom-up salience — sensory prominence and contrast — interacts with categorization. A publicity shock drawing attention to a low-probability contamination risk raises similarity to “consuming,” triggering a category switch that amplifies attention to quality broadly and reduces attention to price, producing large valuation drops disproportionate to the actual probability shift. This mechanism generates the framing effects of prospect theory without a stable S-shaped utility function: gains and losses frames correspond to different contexts activating different categories.

Scope conditions: the theory applies when features and their values are fully known to the decision maker (no uncertainty about attributes), so the distortions take the form of altered sensitivity to known features rather than missing information. The set of categories C is taken as given in the formal analysis, though the authors discuss endogenization as future work.

Q: What is the paper’s central departure from standard rational inattention and noisy-perception models?

A: Standard models (Sims 2003, Woodford 2012, Enke and Graeber 2023) produce unimodal, stably weighted valuations — the decision maker’s weighting of features is a smooth function of payoff-relevant costs or priors. In this paper, the weighting is determined by problem recognition, which is discrete and stochastic, producing within-person multi-modal attention: the same person can cluster on entirely different features across identical problems. The authors cite direct evidence from Bordalo, Conlon, Gennaioli, Kwon, and Shleifer [20] showing bimodal clustering on base rates vs. likelihoods in statistical problems, a pattern inconsistent with stable-weighting models.

Q: How is perceived value distorted when the attention weight on a hedonic feature is below 1?

A: The perceived value of hedonic feature i is u_i(alpha_P) = alpha_{P,i} * u_i + (1 - alpha_{P,i}) * u_bar_i, where u_bar_i is the average value of that feature across options in the menu. An attention weight of zero collapses perceived variation in that feature to zero; full attention recovers the true value. The implication is that under-attention shrinks the decision maker’s effective sensitivity to a known attribute, causing systematic under- or over-valuation relative to a rational benchmark while tastes (marginal utilities) are held fixed.

Q: How is perceived probability distorted?

A: With attention weight alpha_{P,W} on event W, the perceived probability of event e is P(e)^{alpha_{P,W}} / sum_{e’} P(e’)^{alpha_{P,W}}, which compresses the distribution toward uniform as alpha_{P,W} falls toward 0 and recovers the true distribution at alpha_{P,W}=1. In the jam example, under-attention to the small probability of spoilage causes the decision maker to overestimate the risk of contamination. For multi-dimensional event vectors the formula generalizes multiplicatively, allowing “editing out” of entire event dimensions (e.g., urn selection in a balls-and-urns problem) when their attention weight hits zero.

Q: What is the mechanism for context-dependent price elasticity?

A: When context kappa_P is below threshold kappa*(F_{buy}/F_{con}), the decision maker categorizes the problem as “buying” and her valuation is v = alphaq - etap, giving a high price sensitivity (coefficient eta) and attenuated quality sensitivity (coefficient alpha < 1). Above kappa*, she categorizes as “consuming” and valuation is v = q - alphaetap, reversing the emphasis. Because the threshold kappa* is increasing in relative frequency F_{buy}/F_{con}, a decision maker with more buying experience has a higher threshold and thus acts as more price-elastic at any given context level. These elasticity differences arise without any change in the true marginal utility of money eta or quality q.

Q: How does the model generate the sunk cost fallacy and opportunity cost neglect as a unified phenomenon?

A: Both anomalies arise because buying and consuming categories selectively neglect shocks. In the football example, recognizing the problem as “buying” activates alpha_{buy,Delta_Q}=0, so the blizzard quality shock Delta_q<0 is ignored and the decision maker drives to the game as if the shock did not occur — the sunk cost fallacy. In the wine example, recognizing the problem as “consuming” activates alpha_{con,Delta_M}=0, so the capital gain Delta_p is ignored and the decision maker reports a zero or purchase-price cost — opportunity cost neglect. The unifying mechanism is that each category attends only to the features diagnostic of its prototypical experiences: buying attends to price paid and normal quality; consuming attends to realized quality and partly to price, but not to capital gains.

Q: What comparative static does the model predict for sunk cost susceptibility based on experience?

A: People with higher F_{buy} (more buying experiences, e.g. poverty experiences or having recently purchased but not yet consumed the good) exhibit more sunk cost fallacy and less opportunity cost neglect. Conversely, season ticket holders face many consuming experiences relative to one buying event, raising F_{con} and thus reducing susceptibility to the sunk cost fallacy for sports events. Making the blizzard more salient in the description shifts similarity toward “consuming,” also reducing the sunk cost fallacy through a different channel (bottom-up salience rather than experience).

Q: What is the paper’s explanation for the Gambler’s Fallacy, and what distinguishes it from prior accounts?

A: The Gambler’s Fallacy arises when sequence length N_P exceeds threshold N*(F_{freq}/F_{inf}), causing the decision maker to switch from the frequency category (which attends to the 50:50 fairness of the coin) to the inference category (which attends to the share of heads). Under inference, the decision maker treats balanced and unbalanced sequences as representatives of their “share of heads equivalence class,” and the class of balanced sequences is larger, so balanced sequences receive higher estimated probability — the Gambler’s Fallacy. This differs from Rabin and Vayanos (2010), where the bias stems from a belief that the coin is drawn from a pool; here the decision maker knows the coin is fair (kappa_{P,U}=0.5) but the inference representation causes question substitution rather than a wrong model of the DGP.

Q: How does the model make the Gambler’s Fallacy testable beyond length effects?

A: The model predicts the bias is stronger for decision makers who recently solved many inference problems (lower F_{freq}/F_{inf}), and weaker when the 50:50 nature of flips is made bottom-up salient in the choice context (because salience raises similarity to the frequency category, hindering recognition of inference). These cognitive proxies — experience frequencies and bottom-up salience — are orthogonal to the statistical content of the problem and thus allow identification of the mechanism separately from changes in information or incentives.

Q: How does the model produce framing effects in risky choice without a stable S-shaped utility function?

A: Gains and losses frames are modeled as different context vectors kappa_P that differentially increase similarity to a “safe outcome” category or a “risk” category. Recognizing the problem as the safe-outcome category shifts attention toward the certain option; recognizing it as the risk category shifts attention toward variance. The reversal of preferences between gain and loss frames (the Asian Disease problem, Tversky and Kahneman 1981) thus emerges from context-driven re-categorization rather than from a fixed probability weighting function. The novel prediction is that framing effects should be stronger for decision makers with more experience with the category activated by each frame, and weaker when bottom-up salience of the alternative frame’s features is raised.

Q: How does bottom-up salience interact with top-down categorization in the contamination example?

A: A publicity shock alpha_{delta,Q_b}>0 raises baseline attention to the spoiled-jam quality feature, increasing the similarity of the current problem to the “consuming” category (where quality is focal). This triggers a category switch for marginal agents, activating the full consuming attention profile — which attends to quality broadly, not just to contamination specifically, and reduces attention to price. The resulting valuation drop is therefore disproportionate to the actual probability of contamination and exhibits price insensitivity, because re-categorization shifts the entire attention profile rather than just updating a single probability.

Q: How does the model relate to and distinguish itself from case-based decision theory (Gilboa and Schmeidler 1995) and analogical reasoning (Mullainathan 2002, Fryer and Jackson 2008)?

A: In Gilboa-Schmeidler and related models, the decision maker uses past cases to resolve uncertainty about unknown attributes of current options; attention is full and the mechanism is extrapolation of payoffs from similar cases. In Mullainathan (2002) memory-based model, categories again serve to fill in missing information. In this paper, there is no uncertainty about attributes — features and their values are fully known — and the distortion instead takes the form of altered sensitivity to known features through selective attention. This allows the model to produce biases even in simple problems with full data disclosure, and to explain phenomena like base rate neglect and price insensitivity that are not primarily about missing information.

Q: What does the model predict about within-person versus across-person distributions of valuations?

A: Within a person, attention is multi-modal (bimodal in the two-category case) because categorization is stochastic. However, if many categories are possible across the population, the aggregate distribution of valuations can appear approximately unimodal even though each individual’s distribution is not. This distinction is empirically important: a researcher observing average choices may incorrectly infer smooth preference heterogeneity when the underlying mechanism is discrete category switching.

Q: What cognitive proxies does the model propose for empirical identification?

A: The theory links endogenous attention and choice to three observable (or measurable) proxies: (1) past experience frequencies F_c, measurable from administrative histories, surveys about past exposure, or experimental manipulation of training; (2) contextual similarity, measurable from field or experimental variation in irrelevant context features; and (3) bottom-up salience, experimentally controllable via prominence or contrast manipulations. The key identification logic is that these proxies are payoff-irrelevant — they do not change tastes, information, or the objective choice problem — yet predict systematic shifts in choice through their effect on recognition.

Problem Recognition: The first step in the decision maker’s choice process, in which she jointly selects an attention vector alpha_P and a category c* by maximizing weighted similarity between the current problem (characterized by its context vector kappa_P) and the prototype of a past category (alpha_c, kappa_c), multiplied by the category’s time-discounted frequency F_c. Recognition is not about resolving uncertainty over attributes but about selecting which known attributes to attend to.

Category: A partition element of the decision maker’s memory database, indexed by a prototype attention-plus-context vector (alpha_c, kappa_c) and a frequency scalar F_c. The prototype encodes both the context features diagnostic of experiences in that category (binary alpha_{c,i} for i in Phi_K) and the attention to hedonic and event features (alpha_{c,i} for i in Phi_H union Phi_E) used when solving problems in that category. Examples in the paper: “buying” and “consuming” for riskless choice; “frequency estimation” and “agnostic inference” for statistical judgment.

Attention Weight (alpha_{P,i}): A scalar in [0,1] assigned to feature i of the current problem P. For hedonic features, alpha_{P,i}<1 collapses perceived variation toward the menu average; for event features, alpha_{P,i}<1 compresses perceived probabilities toward uniform. Full attention alpha_{P,i}=1 recovers expected utility. Attention weights are the endogenous output of the recognition step, not fixed preference parameters.

Contextual Similarity S: A separable function measuring how close the current problem (alpha_P, kappa_P) is to a category prototype (alpha_c, kappa_c). It decreases in discrepancies in the attention vector (measured by a strictly increasing, convex function d) and in discrepancies in the values of context features diagnostic of the category (d_i(kappa_{P,i}, kappa_{c,i}) * alpha_{c,i}). Endogenous attention to context is set to reduce sensitivity to discrepancies, not to eliminate them.

Mental Accounting (as categorization): In the paper’s account, non-fungibility, sunk cost fallacy, and opportunity cost neglect all arise because buying and consuming categories selectively attend to different monetary and quality features. The sunk cost effect is alpha_{buy,Delta_Q}=0; opportunity cost neglect is alpha_{con,Delta_M}=0. Mental accounts are not separate budget constraints but the by-product of category-specific attention profiles that were calibrated to normal-state experiences and do not generalize to shocks.

Bottom-up Salience: Exogenous attention to a feature driven by sensory prominence (described by alpha_{delta,i} in the problem’s presentation vector) or payoff contrast (the DM attends more to features where her option’s value deviates more from the menu average relative to total menu variance). Bottom-up salience raises baseline attention to a feature before top-down categorization acts, and can trigger a category switch by raising similarity to the category for which that feature is focal.

Gambler’s Fallacy via Question Substitution: In the model, the Gambler’s Fallacy arises when a long sequence length kappa_{P,N} causes recognition of the “agnostic inference” category, which focuses attention on the share of heads alpha_S=1. The decision maker then treats sequences as representatives of a “share of heads equivalence class,” and since the balanced class is larger than the unbalanced class, balanced sequences are assigned higher estimated probability. This is not a belief that the coin is unfair; it is question substitution induced by the inference representation.

A Model of Multiple Hypothesis Testing

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops an economic framework for determining when and how much multiple hypothesis testing (MHT) adjustment is warranted in research settings. The research question is: under what conditions do MHT adjustments arise as an optimal solution to incentive misalignment between a researcher and a mechanism designer (social planner)?

The model is a two-stage game. In the first stage, a benevolent social planner commits to a hypothesis testing protocol. In the second stage, a researcher decides whether to conduct a pre-specified experiment based on private costs and benefits. The planner’s utility function combines an ambiguity-averse (maximin) component—limiting harm from mistaken conclusions—with an expected-utility component capturing the generic benefits of research production. The framework focuses on multiplicity arising from testing multiple treatments or estimating effects within multiple subpopulations; multiple outcomes are treated as an economically distinct case covered in a companion paper.

The main theoretical result is that separate t-tests are uniformly globally optimal under linearity of the researcher’s payoff and welfare functions and normality of test statistics. The optimal critical value takes the explicit form: t(J, Σ) = Φ⁻¹(1 − C(J, Σ) / (b · |J|)), where |J| is the number of hypotheses, C(J, Σ) is the experiment cost, and b is the researcher’s per-rejection benefit. This formula nests two limiting cases. When costs are fully fixed (invariant to |J|), the formula delivers a Bonferroni correction. When costs scale proportionally with the number of hypotheses, no MHT adjustment is warranted—because the researcher already faces sufficient deterrent from the incremental cost of each additional test.

The key economic mechanism is as follows. In the worst states of the world (where all treatments are harmful relative to the status quo), a research study has only downside risk for society. The planner must keep the researcher’s expected payoff from false positives low enough that she chooses not to experiment. If critical values were invariant to |J|, for sufficiently many hypotheses the researcher’s expected payoff from false positives alone would exceed costs, inducing unwanted experimentation. Some upward adjustment to critical values (i.e., tighter thresholds) is therefore generically optimal. The same logic implies that critical values should also adjust for sample size, since larger samples raise costs.

The framework is calibrated to two empirical applications. For FDA clinical trial approval, using Sertkaya et al. (2016) data on approximately 31,000 U.S. pharmaceutical trials (2004–2012), fixed costs constitute approximately 46% of average total trial cost. At a benchmark significance level of 5% and benchmark sample size, the optimal level is approximately 3.2% for two tests, 2.6% for three tests, and asymptotes to approximately 1.4% as |J| → ∞. Sidak’s correction yields 2.5% and 1.7% for two and three tests respectively, and tends to zero as |J| → ∞—more conservative than the model implies. Optimal adjustments must also be less conservative for larger samples to preserve researcher incentives to bear the correspondingly larger costs.

For program evaluation in development economics, the paper uses a unique dataset of funding proposals submitted to J-PAL from 2009 to 2021. The estimated cost elasticity with respect to the number of treatment arms ranges from 0.13 to 0.22 (p < 0.05), indicating costs rise significantly but far less than proportionally. The implied optimal significance levels are slightly less conservative than Bonferroni/Sidak corrections but more conservative than unadjusted testing.

Scope conditions: the framework assumes pre-specified experiments (no p-hacking), linear payoffs, normally distributed statistics, and a researcher whose preferences are common knowledge. The analysis focuses on multiple treatments and subpopulations, not multiple outcomes. Results extend to imperfectly informed researchers and heterogeneous variances.

Q: What is the core mechanism by which MHT adjustments arise as optimal in this framework? A: The planner must deter experimentation in the worst-case states—those where all treatments are harmful. If the testing protocol did not adjust for the number of hypotheses, a researcher testing sufficiently many hypotheses could earn enough expected payoff from false positives alone to justify experimentation, even when all treatments are truly harmful. Tighter critical values (higher thresholds) reduce the probability of false positives and thus cap the researcher’s expected payoff in the null space, deterring unwanted experimentation. This is the maximin optimality condition: the researcher’s expected payoff must be non-positive over the null space.

Q: What are the two limiting cases of the optimal critical value formula, and what do they correspond to? A: The optimal level of the separate t-tests is α(J, Σ) = C(J, Σ) / (b · |J|). When C(J, Σ) = ᾱ (costs are fixed, invariant to the number of hypotheses), this reduces to ᾱ/|J|, the Bonferroni correction. When C(J, Σ) = ᾱ · |J| (costs scale proportionally with the number of hypotheses), the optimal level equals ᾱ regardless of |J|—no MHT adjustment is warranted. The intuition for the second case is that proportional costs already deter excess testing; the researcher has no undue incentive to test many hypotheses because each additional test costs the same incremental amount.

Q: Why do optimal critical values also depend on sample size, and what is the policy implication? A: Since research costs C(J, Σ) increase with sample size (Σ captures design features including sample size), the optimal test level α(J, Σ) = C(J, Σ)/(b·|J|) rises with sample size. Equivalently, larger studies warrant less conservative significance thresholds. The policy implication is that a single uniform correction (e.g., Bonferroni at the 5% level) applied without regard to sample size is suboptimal: it is too conservative for large studies, which would over-deter valuable high-powered research.

Q: What are the two optimality properties required of protocols in the paper’s main characterization? A: The paper shows (Proposition 3.1) that a protocol is uniformly globally optimal—optimal for all values of the welfare weight λ and prior π—if and only if it is both maximin optimal and unbiased. Maximin optimality (Proposition 3.2) requires two conditions: the researcher’s expected payoff must be non-positive over the null space (deterring experimentation when all treatments are harmful), and expected welfare must be non-negative when some treatments are beneficial. Unbiasedness requires that the researcher’s maximum power strictly exceeds the test size, ensuring that experimentation is motivated when treatments are genuinely beneficial.

Q: How does the paper rationalize conventional hypothesis testing asymmetry (type I vs. type II error weighting) without extreme restrictions? A: In Tetenov (2012), justifying 5%-level testing with minimax regret in a single-agent model requires the decision-maker to place 102 times more weight on type I than type II regret—an extreme restriction. In this paper, the asymmetry arises naturally from the planner’s desire to prevent harmful treatment implementation: the planner is willing to forgo some power (probability of detecting beneficial treatments) to ensure that harmful treatments are not implemented. The researcher’s private incentives and the planner’s objective diverge in a way that makes tight size control endogenously optimal.

Q: What does the FDA empirical calibration imply quantitatively about optimal versus standard adjustments? A: Using Sertkaya et al. (2016) data showing that fixed costs are 46% of average total trial cost for U.S. pharmaceutical trials, and using Pocock et al. (2002) to set J̄ = 3 (average number of subgroups), the paper calculates that at a benchmark level of ᾱ = 0.05: the optimal level is approximately 3.2% for two tests, 2.6% for three tests, and asymptotes to approximately 1.4% as |J| → ∞. By contrast, Sidak’s correction yields 2.5%, 1.7%, and zero, respectively. Both the unadjusted 5% and the Sidak/Bonferroni levels are therefore suboptimal—the unadjusted level is too permissive while standard FWER corrections are too conservative.

Q: What do the J-PAL data reveal about optimal MHT adjustment in program evaluation? A: Using the universe of J-PAL funding proposals from 2009 to 2021, the paper estimates the cost elasticity with respect to the number of treatment arms to be 0.13–0.22, which is statistically significant (p < 0.05) but far below 1 (the proportional case). This means costs rise with arms but much less than proportionally. As a result, optimal significance levels for program evaluation studies are slightly less conservative than Sidak/Bonferroni corrections (e.g., approximately 3.8–4.5% versus 2.5% at a two-arm study with ᾱ = 5%) but more conservative than unadjusted testing. The testing thresholds also vary moderately with sample size, with larger samples implying less conservative procedures.

Q: When are cross-study MHT adjustments warranted according to the framework? A: Cross-study MHT adjustments are warranted only when there are cost complementarities across those studies. If studies are conducted independently with separate cost structures, each study’s costs do not depend on the number of hypotheses tested in other studies, so no cross-study adjustment is optimal. This provides a principled resolution to the disputed question of whether researchers should correct for tests performed in other papers.

Q: When is FWER control (e.g., Bonferroni or Sidak) the appropriate form of MHT adjustment? A: Appendix B.2 shows that FWER control is appropriate when the researcher’s payoff is nonlinear—specifically when the researcher requires at least one positive finding to receive any benefit (e.g., to publish). In the baseline linear payoff model, average size control (Bonferroni) is the correct adjustment only when all costs are fixed. The broader insight is that the form of compound error control—whether average error rate or FWER—is itself determined by economic fundamentals rather than being a statistical choice made in advance.

Q: How does the paper extend to cases of heterogeneous variances across hypotheses? A: Proposition 5.2 shows that under heterogeneous variances, the optimal protocol uses separate t-tests based on sample-equalizing allocations—dividing the sample equally across treatment arms—with critical values t*(J, n(J)) = Φ⁻¹(1 − C(J, n(J))/(b·|J|)), where n(J) is the total sample size. This protocol remains maximin optimal and unbiased, preserving the main qualitative results.

Q: What does the paper contribute relative to Tetenov (2016) on single-hypothesis testing? A: Tetenov (2016) showed that in the single-hypothesis case, separate t-tests are maximin optimal and uniformly most powerful (UMP) unbiased. This paper extends that result to multiple hypotheses, but two major complications arise: first, maximin optimality in the multi-hypothesis case requires verifying that welfare is non-negative even when treatment effects have opposite signs, which requires a non-trivial argument absent in the single-hypothesis case; second, no protocol is UMP unbiased in the multi-hypothesis case, so the paper develops a weaker notion of unbiasedness (power exceeding size) that is sufficient to motivate experimentation.

Q: Why do multiple outcomes require different procedures than multiple treatments or subpopulations? A: Multiple outcomes and multiple treatments are economically distinct types of multiplicity. For multiple outcomes that are noisy proxies for a common underlying quantity, the optimal rule tests an index formed using statistical weights (as in Anderson, 2008). When outcomes capture distinct components of the planner’s utility, economic weights are appropriate. In contrast, multiple treatments or subpopulations lead to separate t-tests with cost-adjusted critical values. Conflating these two forms of multiplicity leads to incorrect inferences about what procedures are appropriate.

Maximin optimality: A hypothesis testing protocol is maximin optimal if it maximizes the planner’s worst-case welfare across all parameter values, equivalent to two conditions: deterring researcher experimentation over the null space (where all treatments are harmful), and ensuring non-negative expected welfare when some treatments are beneficial.

Unbiasedness (in the paper’s sense): A protocol is unbiased if the researcher’s maximum achievable power strictly exceeds the test size, ensuring that experimentation is motivated when treatments are genuinely beneficial. This is a weaker condition than UMP unbiasedness, which does not exist in the multi-hypothesis case.

Uniform global optimality: A protocol is uniformly globally optimal if it maximizes the planner’s objective for all values of the welfare weight λ ≥ 0 and all priors π over the parameter space, making it robust to uncertainty about the relative importance of deterrence versus research motivation.

MHT correction factor: Defined as C(J, Σ) / (C̄ · |J|), this factor captures how the cost per test varies as the number of hypotheses grows. It equals 1/|J| (Bonferroni) when all costs are fixed, and equals 1 (no correction) when costs are proportional to the number of tests; the empirically appropriate correction lies strictly between these extremes.

Cost function C(J, Σ): The private cost borne by the researcher for conducting the experiment, which depends on both the set of treatments J and the experimental design Σ (including sample size). The degree of optimal MHT adjustment is a direct function of how this cost varies with the number of hypotheses tested.

Global null space Θ₀(J): The set of parameter vectors θ for which the welfare effect of implementing any combination of treatments is strictly negative—i.e., the status quo of no treatment dominates all interventions. Maximin optimality requires deterring researcher experimentation over this set.

Cost complementarities across studies: Cost structures in which conducting multiple studies together is cheaper than conducting them separately. Cross-study MHT adjustments are warranted if and only if such complementarities exist; absent complementarities, each study’s optimal threshold is set independently of others.

A Monetary-Fiscal Theory of Sudden Inflations

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. Why do sudden inflations and currency crises occur, while symmetric sudden deflations never do? The paper asks whether treating nominal government bonds as analogous to ordinary corporate bonds — with an asymmetric payoff structure capped at face value on the upside but exposed to real losses when fiscal surpluses are insufficient — can generate a unified theory of these crises endogenously from a single model.

Intellectual Lineage and Approach. The paper sits at the intersection of two literatures. The first is the Fiscal Theory of the Price Level (FTPL), originating with Leeper (1991), Sims (1994), and Sargent and Wallace (1985), which links the real value of nominal government debt to expected future surpluses. The second is the safe-asset literature, where Holmstrom (2015) and Gorton (2017) explain that assets can circulate as safe stores of value precisely because their backing is costly to investigate and consumers rationally remain uninformed. The paper applies this information-economics logic to nominal government bonds, so that consumers normally hold bonds without investigating the government’s true fiscal capacity, and only pay the cost to investigate when real repayment doubts become sufficiently severe.

Model Structure. The model is a two-period reduced-form general equilibrium. In period 1, a representative consumer buys nominal government bonds at an interest rate set by the monetary authority. In period 2, the government must repay those bonds. The fiscal authority attempts to hit a price-level target P* by raising tax revenue, but faces a hard ceiling τ_max on the surplus it can collect — arising from Laffer limits on taxation, political constraints on austerity, or the need to fund financial-sector bailouts. The consumer has prior beliefs that τ_max is low (L) with probability π and high (H) with probability 1−π, and can pay a fixed utility cost γ to learn τ_max before deciding how many bonds to purchase.

Bond Payoff Structure and Asymmetry. The key mechanism is the asymmetric, bond-like real payoff of nominal government debt. If τ_max ≥ B1/P*, the government raises enough surplus to repay bonds fully in real terms at the price-level target; the real payoff is flat at face value (the “in-the-money” region). If τ_max < B1/P*, the government sets taxes to the ceiling τ_max and the price level rises above P* to balance the budget constraint, reducing the real payoff proportionally (the “default” region). Critically, because the nominal payoff is capped at face value, there is no upside region: governments will not run surpluses large enough to deliver a windfall to bondholders, so sudden deflations — analogous to a corporate bond being worth more than face value — cannot occur. This asymmetry is the direct source of the one-sided nature of crises.

Two Illustrative Mechanisms for Sudden Inflations. The paper numerically and analytically characterizes two triggering scenarios:

Lower surplus expectations (fiscal stress narrative, corresponding to Burnside et al. 2001 on the 1997 Asian crisis): As the probability π of a low future surplus (e.g., from a prospective banking-sector bailout) rises, the value of information about τ_max increases. In the numerical example (i = 0.05, γ = 0.13, L = 0.1), the value of information equals the cost γ at π = 0.15. For π above 0.15, consumers pay to investigate, learn τ_max = L, and refuse to purchase bonds beyond what will be repaid in real terms (B1 = τ_max = L = 0.1). The price level in period 1 rises discontinuously as a function of π at this threshold.
Interest rate increases (speculative attack narrative): As the monetary authority raises the interest rate to defend a currency, consumers demand more bonds. Larger bond quantities increase the risk that surpluses will be insufficient, raising the value of fiscal information. In the numerical example (π = 0.5, γ = 0.24, 1+i ∈ [1, 1.2]), the value of information equals γ at 1+i = 1.1 (i.e., i = 10%). For interest rates above this threshold, consumers learn τ_max = L, restrict bond purchases to what will be repaid, and the price level in period 1 jumps discontinuously. Further interest rate increases above the threshold produce only upward drift in the price level, not additional monetary tightening effects — illustrating the limits of monetary policy in fiscally stressed environments.

Theoretical Results. Two formal theorems establish generality. Theorem 1 shows that, given bond demand B1(π) such that L < B1 for all π ∈ (0,1), there exist thresholds k and γ > 0 such that the period-1 price level P1 is discontinuous as a function of π on (0, k]. Theorem 2 establishes an analogous discontinuity in P1 as a function of the interest rate i, given that B1(i) > L for all i in the relevant range.

Scope Conditions. The model is a two-period reduced form that abstracts from dynamics, multiple maturities, and secondary market trading. The informational friction is a fixed binary cost γ, not a richer signal structure. The results depend on the existence of a binding surplus ceiling τ_max; when the government is far from this ceiling (i.e., consumers’ beliefs are far from the “default boundary”), shocks produce only small, smooth price-level changes. Large discontinuous price-level jumps require the economy to be near the kink point of the bond payoff curve.

In depth

Q1. What is the fundamental analogy that drives the paper’s theory, and what economic literature does it build on?

The paper analogizes nominal government bonds to corporate bonds (following Sargent 1982’s advice that “government debt is valued according to the same economic considerations that give private debt value”). Like a corporate bond, the nominal government bond pays its face value if the underlying project (government fiscal capacity) delivers a surplus at least equal to the face value, but pays only a share of the realized surplus if the surplus falls short. This bond-like payoff — flat on the upside, proportional to outcomes on the downside — is the direct source of asymmetric crisis dynamics. The paper combines this with Holmstrom (2015) and Gorton (2017)’s framework in which safe assets function because their backing is costly to investigate, so consumers rationally remain uninformed in normal times.

Q2. What is the key information friction, and how does it generate the switch between “normal times” and crisis?

In normal times, consumers are confident that the government’s future maximum surplus τ_max is sufficient to repay bonds in real terms. The fixed utility cost γ of investigating the true surplus exceeds the benefit, so consumers remain uninformed and bonds trade at a price reflecting only uninformed prior beliefs. A crisis arises when the value of information V(.) rises above γ — either because the probability of a low surplus state rises (fiscal stress) or because the interest rate rises and consumers demand more bonds, bringing them closer to the repayment boundary. Once V > γ, consumers investigate and, upon learning τ_max = L (low surplus), refuse to hold bonds that will not be repaid in real terms, triggering a discrete upward jump in the price level.

Q3. How does the bond payoff structure explain the absence of sudden deflations?

The real payoff of a nominal government bond cannot exceed its face value: the bond is capped at face value on the upside because the government will not voluntarily raise tax surpluses to deliver a windfall to bondholders. In the event that surpluses turn out to be higher than needed (τ_max ≥ B1/P*), the government simply sets taxes to exactly repay the bonds at P* and returns no additional real value to bondholders. This is the flat portion of the payoff curve. Because there is no upside kink — no region where learning that τ_max is unexpectedly large causes the price level to fall sharply — there is no mechanism for sudden deflations symmetric to sudden inflations. The 1933 U.S. episode (Jacobson et al. 2019) is cited: when deﬂation from leaving gold would have required fiscal austerity for full real repayment, Roosevelt chose to exit the gold standard rather than allow deflation.

Q4. How does the first numerical example (lower surplus expectations) work quantitatively?

The baseline parameters are: i = 0.05, γ = 0.13, L = 0.1, H ≈ ∞, P* = 1, e1 = e2 = 1, B0 = 1, τ1 = 0.8, β = 1. The analysis is restricted to π ∈ (0, 0.3]. As π (probability that τ_max = L) rises, the value of information V(.) rises. At π = 0.15, V equals the cost γ = 0.13. For π > 0.15, consumers pay to investigate and, upon learning τ_max = L, purchase only B1 = L = 0.1 in bonds — the amount that will be repaid — causing the period-1 price level P1 to jump discontinuously from approximately 0.95 to approximately 1.13. For π ≤ 0.15, consumers remain uninformed and P1 rises only smoothly from below 1 as π increases (fewer bonds demanded as repayment risk rises, even without investigation).

Q5. How does the second numerical example (interest rate increase) work quantitatively, and what does it imply for monetary policy?

With π = 0.5, γ = 0.24, and 1+i ∈ [1, 1.2], as the monetary authority raises the interest rate, consumers demand more bonds, increasing real repayment risk and the value of information. At 1+i = 1.1 (i.e., i = 10%), V equals γ. For 1+i > 1.1, consumers investigate and learn τ_max = L; they then only purchase bonds up to the repayment limit, causing P1 to jump discontinuously to approximately 1.15. For interest rates above the threshold, further increases yield only a smooth upward slope in P1 (bond purchases are fixed in real amount but nominal revenue falls). This illustrates that the monetary authority’s ability to use higher interest rates to lower the price level is limited by the surplus constraint: once the interest rate is high enough to trigger consumer investigation and a fiscal crisis, raising rates further is inflationary rather than deflationary.

Q6. What are the two regions of the deterministic model and how do they differ in fiscal and price-level dynamics?

In the deterministic version (1-π = 0, so τ_max = L with certainty, and there is no uncertainty), the model produces two distinct regions. In the “insufficient surplus” region where τ_max < B1/P*, the fiscal authority sets taxes to their maximum τ_max, the real payoff of bonds is τ_max/B1 < 1, the period-1 price level P1 = B0/(βτ_max), and real bond revenue Π = βτ_max (constant in τ_max). Selling additional bonds does not raise additional real revenue because any extra bonds lead to a proportional rise in P2 and a fall in Q. In the “sufficient surplus” region where τ_max ≥ B1/P*, the government meets its fiscal target (τ2 = B1/P*), P2 = P* is hit, P1 = βB1/(B0P*), and Π = βB1/P* (increasing in B1). In this region, selling additional bonds does raise real revenue and lowers P1 as the government absorbs more money.

Q7. What are the two interest rate regions in the deterministic model, and what is their implication for monetary policy effectiveness?

Using B1 = B0(1+i) (debt rolled over at the chosen rate), the monetary authority has two interest-rate regions. In the “constrained” region where 1+i > τ_max P*/B0 (the surplus ceiling binds), raising i does not change the period-2 surplus (τ2 = τ_max), does not change real revenue (Π = βτ_max), and does not affect P1 — but raises P2 above the target P*. In the “unconstrained” region where 1+i ≤ τ_max P*/B0, raising i increases bond demand, increases real surplus backing, raises real revenue, and lowers P1 while P2 = P* is maintained. The boundary between these regions determines the limit of monetary policy: the monetary authority can reduce P1 by raising i only up to the point where the surplus ceiling would be hit.

Q8. How does the paper relate to and extend prior FTPL literature?

The paper is grounded in the FTPL of Leeper (1991), Sims (1994), and Cochrane (2005, 2020), in which the price level is determined by the requirement that real government liabilities equal the present value of future surpluses. The paper’s contribution is to make the information structure endogenous: consumers’ beliefs and their decision to acquire fiscal information determine whether or not the FTPL logic is operative. In normal times (consumers uninformed), the price level does not respond to changes in the maximum surplus — a result that resembles the “Ricardian” or non-FTPL regime. When consumers investigate and learn the surplus is insufficient, the connection between the surplus and the price level is restored, reproducing FTPL-type dynamics. This provides an endogenous, single-model rationale for the regime-switching behavior between FTPL and non-FTPL environments documented empirically in Bianchi and Melosi (2013, 2017) and Davig and Leeper (2006).

Q9. What is the welfare role of consumer ignorance in this framework?

Consumer ignorance of the government’s true surplus plays a dual role. On one hand, ignorance is individually rational in normal times because the cost γ of investigating exceeds the benefit V (.) when beliefs are comfortably away from the default boundary. On the other hand, following Dang et al. (2017), informed knowledge of the safe asset’s backing destroys the symmetric ignorance that supports the asset’s role as a safe store of value, reducing welfare. In this model the concern is repayment risk rather than adverse selection: the consumer fears not being repaid in real terms and chooses to investigate when that risk is sufficiently high, potentially triggering the very crisis they feared.

Q10. What are the scope conditions and limitations of the model?

The model is explicitly a two-period reduced form designed to illustrate the bond-payoff mechanism in the simplest possible setting. It abstracts from: multi-period bond maturities and secondary market trading; rich heterogeneity among consumers; endogenous monetary and fiscal policy responses beyond the simple rules specified; and the general equilibrium interactions between inflation, output, and labor markets. The information cost γ is modeled as a fixed binary cost rather than a continuous or richer signal structure. The results on discontinuous price-level jumps hold when bond demand is sufficiently large relative to L (i.e., L < B1), ensuring genuine repayment risk; when surpluses are very large relative to bond liabilities, no crisis dynamics arise.

Key Concepts

Maximum Surplus (τ_max). The paper’s name for the hard ceiling on the net tax revenue (taxes minus money transfers) the government can collect in the second period. This ceiling can arise from a Laffer limit on taxable income, political-economy constraints on austerity, or from a banking crisis requiring government transfers to bail out the financial sector. It is the paper’s analogue of a project’s liquidation value: the maximum the “project” (the government) can deliver to bondholders.

Bond-Like Payoff of Nominal Government Debt. The paper’s central structural claim: the real payoff to holding a nominal government bond is capped at face value on the upside (the government will not raise surpluses beyond what is needed to repay bonds at the price-level target) but falls proportionally below face value when τ_max is insufficient for full real repayment. This is precisely the payoff structure of a standard corporate bond — flat on the upside, proportional to recovery on the downside — and it is the source of the asymmetry between sudden inflations and the absence of sudden deflations.

Value of Information (V(.)). Defined as the difference in expected utility between a consumer who learns the true τ_max before making bond-purchase decisions and one who remains uninformed and acts only on prior beliefs π, 1−π. The consumer investigates if and only if V(.) > γ. V is zero when beliefs are certain (limπ→0 and limπ→1), can be hump-shaped in π, and is increasing in the interest rate i (through its effect on bond demand). The threshold condition V = γ defines the boundary between “normal times” (no investigation) and crisis (investigation and possible sudden inflation).

Endogenous Information Structure. The paper’s term for the property that whether consumers choose to learn the government’s fiscal capacity is itself determined within the model by the parameters of the economy (the interest rate, prior beliefs, the cost of investigation). This contrasts with models that exogenously specify whether agents are informed or not. The endogenous information structure is the mechanism by which the paper generates the two apparent regimes (FTPL-active vs. FTPL-dormant) from a single unified model.

Default Boundary. The kink point in the bond payoff curve at τ_max = B1/P*: the level of the maximum surplus at which the government exactly repays bonds in real terms at the price-level target. When beliefs or bond quantities place the economy near the default boundary, small changes in π or i can push the economy across it, triggering large price-level responses. When the economy is far from the boundary (τ_max comfortably above B1/P*), small shocks have only small smooth effects.

Sudden Inflation / Currency Crisis (as defined in this paper). A discrete, discontinuous jump in the period-1 price level P1 that occurs when consumers pass the threshold V(.) = γ and investigate the government’s fiscal capacity, finding surpluses to be insufficient. The mechanism is: informed consumers refuse to hold bonds they know will not be repaid in real terms at P*, forcing the price level to jump to clear the government’s budget constraint with fewer bonds outstanding. The paper treats sudden inflations and currency crises as the same mechanism in different institutional contexts.

Repayment Risk Premium. The markup above the risk-free rate that consumers require on government bonds to compensate for the probability that the government’s surplus will be insufficient for full real repayment (i.e., the probability that the economy is in the τ_max < B1/P* region). This premium is present even when consumers are uninformed (i.e., do not know which state of τ_max will occur), and is reflected in the consumer’s first-order condition for bond demand.

A Theory of Supply Function Choice and Aggregate Supply

Mon, 01 Jan 0001 00:00:00 +0000

Research Question

Modern macroeconomic models of aggregate supply universally restrict firms to price-setting — committing to a price and supplying whatever quantity the market demands. Flynn, Nikolakoudis, and Sastry ask: what happens if instead firms choose any supply function, a mapping that describes the price charged at each quantity of production? The paper develops the first general-equilibrium, macroeconomic theory of supply function choice and characterizes its implications for the slope of aggregate supply, monetary non-neutrality, and time-varying inflation-output tradeoffs.

Methodology

The paper proceeds in two stages. In partial equilibrium, a single monopolistic firm with constant-returns-to-scale technology and constant-elasticity demand faces log-normal uncertainty about demand shifters, the aggregate price level, real marginal costs, and the stochastic discount factor. The firm chooses a non-parametric supply function — any implicit mapping f(p,q) = 0 — to maximize expected real profits. The paper shows that supply function choice is equivalent to conditioning price-quantity decisions on the realized nominal demand state z = ΨP^η. The authors prove (Theorem 1) that the optimal supply function is endogenously log-linear: log p = α₀ + α₁ log q, where the inverse supply elasticity α₁ is characterized in closed form.

In general equilibrium, the authors embed supply function choice in an otherwise standard monetary business cycle model (in the tradition of Woodford 2003a and Hellwig and Venkateswaran 2009), featuring a representative household demanding differentiated goods, a money supply following a random walk with time-varying volatility, and idiosyncratic shocks to productivity, wages, and demand. They guess and verify a log-linear equilibrium and derive a scalar fixed-point equation for the equilibrium supply elasticity (Theorem 3).

For quantification, the authors calibrate structural parameters (η = 8 from Hottman et al. 2016 scanner data; γ = 0.11 from Gagliardone et al. 2023 Belgian firm data; κ^M = 0.29 calibrated to match an average aggregate supply slope of 0.11 from Hazell et al. 2022) and estimate time-varying uncertainty via a GARCH model of quarterly US data on GDP growth, inflation, and real marginal cost growth from 1960 Q1 to 2024 Q4. Idiosyncratic demand uncertainty is set proportional to aggregate TFP uncertainty using the proportionality factor R = 6.5 from Bloom et al. (2018).

Main Findings

Optimal supply function. The optimal firm-level supply function is log-linear with inverse supply elasticity α₁ determined by the relative variances and covariances of demand, the price level, and real marginal costs. Three comparative statics drive the macroeconomic results: (1) higher idiosyncratic demand uncertainty (σ²_Ψ) flattens the supply function toward price-setting, because a fixed price insulates profit markups against demand variation; (2) higher price-level uncertainty (σ²_P) steepens the supply function toward quantity-setting, because setting a fixed quantity allows relative prices to adjust; (3) lower price elasticity of demand (less elastic demand, more market power) flattens the supply function, conditional on a sufficient condition that holds in US data whenever η > 2.5.

From micro supply to aggregate supply. With fixed log-linear supply functions, the economy has a unique log-linear equilibrium with an AD/AS representation (Theorem 2). The slope of aggregate supply ε^S_t depends on ω₁ (the transformed inverse supply elasticity), κ^M (firms’ signal precision about the money supply), γ (income effects), and η (demand elasticity). Aggregate supply is maximally elastic — money is as non-neutral as possible — if and only if firms are pure price-setters (ω₁ = 0). Aggregate supply is perfectly inelastic — money is neutral — if and only if firms are quantity-setters (ω₁ = 1/η). A lower elasticity of demand flattens aggregate supply through general equilibrium strategic complementarities, a prediction opposite to the New Keynesian model.

Equilibrium supply slope and its determinants. The equilibrium ω₁ solves a fixed-point equation (Theorem 3) in which macroeconomic uncertainty shapes firms’ optimal supply functions, which in turn shape macroeconomic dynamics. Under the special case of balanced strategic interactions (ηγ = 1), the slope of aggregate supply has a clean closed form depending only on the ratio ρ_t = σ_{ϑ,t}/σ^M_{t|s} (idiosyncratic demand uncertainty relative to posterior monetary uncertainty). Critically, the equilibrium supply slope is invariant to the overall level of uncertainty — only the composition of uncertainty matters (Proposition 3). Even vanishingly small uncertainty can generate any level of monetary non-neutrality depending on uncertainty composition.

Quantitative results — United States over time. The model’s estimated slope of aggregate supply shows sharp variation since 1960. The slope is relatively flat and stable during the 1960s, the Great Moderation (1991–2007), the Great Recession (2008–2019), and the recovery from the Great Recession. It spikes dramatically during the 1970s oil crisis and the post-Covid inflation of the 2020s. Compared to Ball and Mazumder (2011), the model qualitatively matches the steepening during 1973–1984 (+58% in the model) vs. the data’s +175%, and a subsequent flattening of −25% vs. −32% in the data during 1985–2007. Compared to Cerrato and Gitti (2022), the model accounts for approximately 4/5 of the steepening between the pre-Covid and post-Covid periods (+112% model vs. +145% data). For the Hazell et al. (2022) comparison, the model accounts for approximately 1/2 of the estimated flattening from 1978–1990 to 1991–2018.

Quantitative results — Cross-country. Using OECD annual data from 1960–2019, the model’s predicted slope of aggregate supply is not positively correlated with the average level of inflation across countries. For countries with the highest inflation rates, the model predicts a negative slope of aggregate supply, driven by very high correlation between price-level uncertainty and real marginal cost uncertainty. The model-predicted slope correlates positively with the reduced-form regression coefficient of inflation on real output growth across countries, even after instrumenting for demand. This predictive power is over and above what can be explained by the level or volatility of inflation alone.

Scope Conditions

All results are derived under log-normality of uncertainty, which ensures the log-linear structure of optimal supply functions. The quantification relies on GARCH-estimated uncertainty and treats idiosyncratic demand uncertainty as proportional to aggregate TFP uncertainty. The model abstracts from microeconomic nominal price stickiness (though the authors show in Appendix B that Calvo-style sticky prices can be incorporated). The baseline model requires the equilibrium condition on firm beliefs to be consistent (rational expectations). Multiple equilibria of the scalar fixed-point are possible in principle, bounded by at most five log-linear equilibria (Proposition 2).

In depth

Q1. What is wrong with assuming price-setting or quantity-setting as a primitive restriction on firm behavior?

A: Price-setting and quantity-setting are two isolated, generically non-optimal points in the larger space of supply functions. Corollary 2 establishes that price-setting is optimal only in the limit as idiosyncratic demand uncertainty becomes unboundedly large (σ²_Ψ → ∞), while quantity-setting is optimal only in the limit as price-level uncertainty becomes unboundedly large (σ²_P → ∞). In a macroeconomic environment where both sources of uncertainty are present in comparable magnitudes, both extreme policies perform poorly and the analyst who imposes either inadvertently restricts firms’ strategies in ways that have large macroeconomic consequences — for example, making money neutral under quantity-setting even when information frictions are present, or making the slope of aggregate supply invariant to demand elasticity under price-setting.

Q2. What is the formal equivalence between supply function choice and conditioning on realized demand?

A: The firm’s problem of choosing a supply function f(p,q) = 0 ex ante is mathematically equivalent to choosing a price-quantity plan (p(z), q(z)) indexed by the nominal demand state z = ΨP^η (Equation 4 in the paper). After the supply function is set, the firm produces where the supply function intersects the demand curve, which pins down the market-clearing outcome as a function of z. Choosing the supply function ex ante is therefore the same as choosing z-contingent prices and quantities without any parametric constraint. This links the model to rational expectations equilibrium in the spirit of Lucas (1972): firms use the demand for their product as a noisy signal to update beliefs and set their optimal price and quantity in response to realized demand conditions.

Q3. How is the optimal inverse supply elasticity α₁ derived, and what is the 2SLS interpretation?

A: Because the optimal supply function allows the firm to set a z-contingent price, the first-order condition at each realized demand state z = t equates expected marginal revenue and expected marginal cost (Equation 7). Under log-normality, this yields a log-linear relationship log p = α₀ + α₁ log q. The elasticity α₁ equals the ratio (d log p / d log z) / (d log q / d log z) = Cov[log z, log p**] / Cov[log z, log q**], where p** and q** are the full-information optimal price and quantity (Equation 9). This is formally equivalent to a 2SLS regression: the firm estimates how its optimal price should change with its optimal quantity, using the nominal demand state z as an instrument for the optimal quantity. The supply function is steep if nominal demand strongly predicts movements in the full-information optimal price (large reduced-form coefficient); it is flat if nominal demand primarily predicts movements in the full-information optimal quantity (large first-stage coefficient).

Q4. How do uncertainty and demand elasticity shape the firm’s optimal supply function in partial equilibrium?

A: Three key comparative statics apply when the supply function is upward-sloping. (1) Greater price-level uncertainty (σ²_P increases) steepens α₁ toward quantity-setting: not knowing competitors’ prices makes aggressive dynamic pricing attractive because it allows the firm’s relative price to adjust ex post. (2) Greater idiosyncratic demand uncertainty (σ²_Ψ increases) flattens α₁ toward price-setting: demand uncertainty favors a fixed price to keep the markup over real marginal costs constant, accommodating demand with quantity variation. (3) A lower price elasticity of demand (more market power, lower η) flattens α₁: more market power reduces the cost of setting the “wrong” price, reducing the benefit of dynamic pricing. Corollary 1 provides a sufficient condition — σ_{M,P} ≥ 0, 2ησ_{M,P} + σ_{M,Ψ} ≥ σ_{P,Ψ}, and α₁ ≥ 0 — under which ∂α₁/∂η > 0, implying greater market power flattens supply; the paper verifies this condition holds in US data whenever η > 2.5.

Q5. How does the model generate an aggregate supply and demand representation from supply function choices?

A: Theorem 2 establishes that, given any fixed log-linear supply functions with slope ω₁,t, there is a unique log-linear equilibrium. In this equilibrium, the price level and real output are jointly determined by an aggregate demand curve — shifting with the money supply but not productivity — and an aggregate supply curve — shifting with productivity but not the money supply. The inverse elasticity of aggregate supply is ε^S_t = γ(κ^M_t + ω₁,t(η − 1/γ)(1 − κ^M_t)) / ((1 − ω₁,t η)(1 − κ^M_t)), derived from aggregating firm-level pricing decisions. The slope depends on ω₁,t (micro supply), κ^M_t (signal precision about money), γ (income effects), and η (demand elasticity). An aggregate demand shock of ∆ log M raises the price level by ε^S_t ∆ log M / (ε^D_t + ε^S_t) and raises real output by ∆ log M / (ε^D_t + ε^S_t), where ε^D_t = γ is the inverse elasticity of aggregate demand.

Q6. What is the equilibrium fixed-point equation and why can there be multiple equilibria?

A: Theorem 3 shows that the equilibrium transformed inverse supply elasticity ω₁,t solves a quintic polynomial fixed-point equation (Equation 29) that depends on the variances of idiosyncratic demand shocks (σ²_ϑ,t), posterior uncertainty about productivity (σ^A_{t|s}), and posterior uncertainty about money (σ^M_{t|s}). Multiple equilibria can arise because of a self-reinforcing feedback: if firms set steep supply functions, prices respond more to demand, which raises price-level volatility, which in turn makes quantity-setting more attractive, further steepening supply functions. Proposition 2 establishes existence of at least one log-linear equilibrium and at most five. Idiosyncratic productivity and factor price uncertainty do not enter the fixed-point equation because the variance of real marginal costs per se does not affect optimal supply function choice — only the covariance of marginal costs with demand and the price level matters.

Q7. What determines the slope of aggregate supply in the special case of balanced strategic interactions (ηγ = 1)?

A: Under ηγ = 1 — where strategic complementarities from relative price effects exactly offset strategic substitutabilities from aggregate consumption effects — the slope of aggregate supply has the closed-form expression ε^S_t = γ(κ^M_t / (1 − κ^M_t))(1 + 1/(γ²ρ²_t κ^M_t)) where ρ_t = σ_{ϑ,t}/σ^M_{t|s} is the ratio of idiosyncratic demand uncertainty to posterior monetary uncertainty (Corollary 5). Aggregate productivity uncertainty drops out entirely because firms do not use the demand state to infer aggregate productivity when strategic interactions are balanced. As ρ_t → ∞ (idiosyncratic demand dominates), the slope converges to the price-setting value γκ^M_t/(1 − κ^M_t). As ρ_t → 0 (monetary uncertainty dominates), the slope goes to infinity, corresponding to quantity-setting and monetary neutrality.

Q8. What is the role of total uncertainty versus the composition of uncertainty?

A: Proposition 3 establishes a striking invariance result: if all standard deviations in the economy are scaled by a common factor λ > 0, the equilibrium supply elasticity and slope of aggregate supply are unchanged. The equilibrium outcomes depend only on the ratios of different sources of uncertainty, not their absolute magnitudes. This sharply distinguishes the model from menu-cost models, in which any increase in uncertainty unambiguously raises the benefit of price adjustment and steepens aggregate supply. A corollary is that idiosyncratic productivity uncertainty has no effect on the slope of aggregate supply in the supply function model, whereas it would steepen aggregate supply in Golosov-Lucas menu-cost models. Moreover, even a vanishingly small level of uncertainty can generate any level of monetary non-neutrality, because the equilibrium supply elasticity is discontinuous at zero uncertainty (ε^S_t (0) = {∞} while ε^S_t (λ) is bounded for any λ > 0).

Q9. How does market power (demand elasticity) affect the slope of aggregate supply, and why does this differ from the New Keynesian prediction?

A: In the supply function model, a lower elasticity of demand (more market power, lower η) flattens aggregate supply by reducing general-equilibrium strategic complementarities. When other firms raise their prices following a demand shock, a given firm faces higher relative demand; the strength of this effect is parameterized by η. With supply functions (ω₁,t ≠ 0), this relative demand increase generates an additional price response, so higher η steepens aggregate supply. Crucially, this effect is exactly zero if and only if firms are pure price-setters (ω₁,t = 0) — meaning the prediction that market power affects aggregate supply is absent from price-setting models. This is the opposite of the New Keynesian prediction: in Woodford (2003b) with decreasing returns to scale, a higher elasticity of demand (less market power) steepens the Phillips curve, because more elastic demand amplifies the quantity response to price changes and thereby the marginal cost response to nominal cost shocks.

Q10. How does the model rationalize the steepening of aggregate supply in the 1970s and 2020s?

A: The GARCH estimates of macroeconomic uncertainty show abrupt increases in inflation uncertainty during the 1970s oil crisis period and after the Covid-19 shock in the 2020s. In the model, a spike in aggregate price-level uncertainty (σ²_P increases) causes firms to choose steeper supply functions — closer to quantity-setting — endogenously. This steepens the aggregate supply curve so that demand shocks have larger nominal effects and smaller real effects. Quantitatively, relative to the base period, the model predicts a steepening of +58% during 1973–1984 and +112% during 2021–2023. The empirical comparisons are +175% (Ball and Mazumder 2011, 1973–1984) and +145% (Cerrato and Gitti 2022, 2021–2023). The model thus accounts for the direction and rough order of magnitude of both episodes but not their full extent. The quarterly time series of model-implied ε^S_t has a correlation of 0.93 with one-quarter-ahead inflation uncertainty and 0.62 with the quarterly level of inflation.

Q11. How does the cross-country evidence help distinguish the model from alternatives based on the level of inflation?

A: The cross-country analysis uses OECD data from 1960–2019 to construct country-level model-implied slopes of aggregate supply using the same structural parameters (η = 8, γ = 0.11, κ^M = 0.29) and country-specific GARCH uncertainty estimates from a one-lag VAR. The key finding is that the model-implied slope is not positively predicted by average inflation across countries (Panel A of Figure 5) — in fact, for the highest-inflation countries such as Chile, Israel, and Mexico, the model predicts a negative slope of aggregate supply, reflecting high correlation between price-level uncertainty and real marginal cost uncertainty. By contrast, the model-implied slope correlates positively with the reduced-form regression coefficient of inflation on real output growth (Panel B), and this positive correlation is also found using a model-derived instrument isolating exogenous monetary variation. This implies that relative uncertainties, not the mean or volatility of inflation per se, help account for cross-country heterogeneity in inflation-output tradeoffs beyond the predictions of Ball et al. (1988).

Q12. How can supply functions be integrated into larger linearized macroeconomic models?

A: Section 4.5 provides a general framework. For any model in which firms face a demand function q_it = d(p_it, z^D_it) and a value function V(p_it, q_it, z^V_it), log-linearization around a deterministic steady state yields an optimal pricing rule ˆp_it = ω₁,it ˆz^D_it (Equation 35) for some scalar ω₁,it determined by the covariance structure of the linearized model. The coefficients ω₁,it enter the standard representation of aggregate dynamics (McKay and Wolf 2023) through the ideal price index ˆP_t = ∫₀¹ ˆp_it di. The additional “rational expectations” restriction is that ω₁,it must be consistent with the equilibrium law of motion for prices. The paper argues that supply functions can thereby be embedded in the broad class of linearized DSGE models used for quantitative work, including models with decreasing returns, monopsony, endogenous markups, sticky prices, investment, and quality choice.

Q13. What are the implications of supply function choice for monetary policy discretion?

A: The model implies a thorny tradeoff for monetary policymakers. If a central bank wishes to maintain discretion — the ability to surprise private agents — this increases firms’ uncertainty about the money supply (higher σ²_M). Under balanced strategic interactions (ηγ = 1), greater posterior monetary uncertainty (σ^M_{t|s}) lowers the ratio ρ_t = σ_{ϑ,t}/σ^M_{t|s}, which flattens the aggregate supply curve (reduces ε^S_t) and thereby increases the real effect of monetary surprises. However, this also endogenously induces firms to set steeper supply functions — closer to quantity-setting — so that the aggregate supply curve steepens in response to the greater price-level uncertainty generated by such an environment. The paper therefore concludes that maintaining monetary policy discretion may be, at least partially, self-defeating.

Inverse supply elasticity (α₁): The percentage by which a firm increases its price in response to a one percent increase in production, characterizing the slope of the firm’s optimal supply function. It is endogenously log-linear and determined by the ratio of covariances relating the nominal demand state to the firm’s optimal price vs. optimal quantity under full information — formally equivalent to a 2SLS coefficient using nominal demand as an instrument.

Supply function: A mapping f(p, q) = 0 describing the locus of prices and quantities a firm commits to, as an implicit function over price-quantity pairs. Unlike price-setting (f depends only on p) or quantity-setting (f depends only on q), the general supply function allows prices to vary with realized demand, nesting both polar cases as limits of extreme uncertainty.

Nominal demand state (z): The composite variable z = ΨP^η that indexes the demand curve. Firms observing their own output market clearing can use z as a noisy signal for inference about the aggregate price level, real marginal costs, and monetary conditions. The supply function is formally equivalent to conditioning price-quantity choices on z.

Slope of aggregate supply (ε^S): The inverse elasticity of the aggregate supply curve in the AD/AS representation, measuring the relative within-period response of the price level versus real output to an aggregate demand shock. It depends on the slope of firm-level supply functions (ω₁) interacted with the information precision about the money supply (κ^M) and income effects (γ).

Transformed inverse supply elasticity (ω₁): The reparameterization ω₁ = α₁/(1 + ηα₁), where α₁ is the firm-level inverse supply elasticity and η is the price elasticity of demand. ω₁ = 0 corresponds to price-setting; ω₁ = 1/η corresponds to quantity-setting. The equilibrium value of ω₁ solves a fixed-point equation that maps macroeconomic uncertainty back into firms’ optimal supply function choices.

Balanced strategic interactions (ηγ = 1): A parametric special case in which strategic complementarities from aggregate demand externalities (parameterized by η) exactly offset strategic substitutabilities from wage pressure (parameterized by 1/γ). Under this condition, the slope of aggregate supply has a closed-form solution that depends only on the relative uncertainty about idiosyncratic demand vs. the money supply.

Relative uncertainty sufficient statistic (ρ_t): The ratio σ_{ϑ,t} / σ^M_{t|s}, measuring firms’ uncertainty about idiosyncratic demand shocks relative to posterior uncertainty about the money supply. Under balanced strategic interactions (ηγ = 1), ρ_t is the single sufficient statistic determining the equilibrium slope of aggregate supply. As ρ_t → ∞ (idiosyncratic demand uncertainty dominates), firms converge to price-setting and aggregate supply flattens; as ρ_t → 0 (monetary uncertainty dominates), firms converge to quantity-setting and aggregate supply becomes vertical.

Invariance to total uncertainty: A key property of the model: the equilibrium slope of aggregate supply is invariant to the overall scale of uncertainty (Proposition 3). Only the composition of uncertainty across idiosyncratic vs. aggregate sources and demand vs. productivity shocks matters. This distinguishes the model from menu-cost models, in which any increase in uncertainty raises the benefit of price flexibility and steepens aggregate supply regardless of uncertainty composition.

A traffic-jam theory of growth

Mon, 01 Jan 0001 00:00:00 +0000

Research Question. Finocchiaro and Weil ask whether financial development necessarily promotes long-run economic growth, or whether congestion externalities in R&D markets can offset — and even reverse — the growth benefits of easier credit access. The paper proposes that the empirical coexistence of expanding financial sectors and roughly constant per-capita GDP growth rates (approximately 2% annually in the United States over the last century) can be explained by the interplay of search frictions in two sequential markets: credit and innovation.

Methodology. The authors build a continuous-time endogenous growth model in which all growth is innovation-led. Firms must pass through four sequential stages — creation, fund-raising (Stage 0–1), R&D search (Stage 1–2), and high-productivity production (Stage 2–3) — before being exogenously destroyed. Both the credit market (firms searching for banks/venture capitalists) and the innovation market (firms searching for innovators after securing finance) are characterized by constant-returns-to-scale matching functions with endogenous market tightness. Nash bargaining determines the loan repayment, and free entry drives profits to zero in both markets. The model is then calibrated to annual U.S. data, with the risk-free rate r = 3.5%, separation rate s = 4%, symmetric bargaining power ω = 0.5, a productivity jump γ = 0.023 targeting a baseline growth rate of 2%, credit market duration for creditors just below one month and for firms slightly above one year (consistent with Wasmer and Weil, 2004), a two-year average patent approval time (USPTO 2020), 6% employment in finance (BLS 2020), and 0.5% employment in scientific R&D (BLS 2020).

Core Mechanism. The paper derives a “spillover function” Q(p,g) that links the equilibrium probability of finding an innovator (q) to the probability of finding a bank (p) and the growth rate (g). Because free entry holds profits at zero, easier credit — a higher p — forces q downward: if a firm spends less time raising funds, the innovation market becomes more congested (Qp < 0). This negative spillover between the two markets is the paper’s central traffic-jam analogy: relieving one bottleneck shifts congestion downstream.

Main Findings. The GG curve — the locus of (p, g) pairs consistent with equilibrium — is hump-shaped under the symmetric cost condition c = ωn (flow search cost for firms in credit markets equals the firm’s share of search costs in innovation markets). Growth is maximized when expected credit search time equals expected innovation search time (1/p = 1/q). Beyond that interior optimum, further financial deepening lowers the growth rate. The calibrated economy sits to the right of the hump in a flat region (p > q), so that reducing credit frictions alone has a marginally negative effect on growth: eliminating credit frictions lowers g from 2.000% to 1.997%, a reduction of 0.003 percentage points. Reducing innovation frictions alone raises g modestly to 2.071% (+0.071 pp). Only a simultaneous reduction of frictions in both markets raises g meaningfully, to 2.122% (+0.122 pp). The quantitative effects are deliberately small, consistent with the near-constancy of long-run growth despite financial deepening.

Scope Conditions. The non-monotonicity requires both markets to carry search frictions; when only one friction is present, financial development is unambiguously good for growth (Section 4.3). The hump-shape is established analytically in the symmetric case c = ωn; more generally, the paper shows (via back-of-envelope approximation) that the sign of the finance–growth link depends on whether c/ω is less than or greater than n. The quantitative insensitivity of growth to finance is amplified when the real interest rate is close to the growth rate and when potential growth γ is close to actual growth g: the elasticity of growth with respect to finance is proportional to (γ − g)/γ. Extensions to fixed bank entry costs (introducing a growth-to-finance feedback), endogenous innovator wages (Section 4.2), and frictionless innovation (Section 4.3) all confirm the benchmark conclusions under stated parameter conditions.

Q1: What is the paper’s central theoretical claim about the finance–growth nexus? The paper claims that the finance–growth relationship is non-monotonic: financial development raises growth when credit is scarce (left of the hump on the GG curve) but lowers it when credit is readily available (right of the hump), because easier financing draws more firms into the innovation market, tightening it and reducing the probability of finding an innovator. This congestion spillover from the credit market to the innovation market is the “traffic-jam” mechanism. The non-monotonicity vanishes if either market lacks search frictions.

Q2: What is the “spillover function” and why is it central to the model? The spillover function Q(p, g) is derived from the free-entry zero-profit condition for firms and expresses the innovation-matching probability q consistent with equilibrium for given credit-matching probability p and growth rate g. It has Qp < 0 (easier credit reduces q) and Qg < 0 (faster growth reduces q), capturing the two-way negative interaction between the markets. It is central because all equilibrium and comparative-statics results flow through it: the GG curve is defined by substituting Q into the growth equation g = γ/(1 + s/p + s/Q(p,g)).

Q3: Under what condition is the GG curve hump-shaped, and what is the intuition? The GG curve is hump-shaped when the flow search cost for firms in the credit market c equals the firm’s share of innovation search costs ωn (Proposition 4). The intuition mirrors equalizing travel times across two congested roads: growth is maximized when expected credit search time (1/p) equals expected innovation search time (1/q). When credit is very tight (p small), a marginal increase in p raises the share of innovating firms faster than it tightens the innovation market, so growth rises. Once credit is abundant (p large), the congestion effect on innovation dominates and growth falls.

Q4: What does the benchmark calibration predict about the quantitative effect of financial development on growth? The benchmark calibration, targeting 2% annual U.S. growth, places the economy to the right of the hump in a flat region of the GG curve (p > q). Eliminating credit market frictions alone reduces the annual growth rate by 0.003 percentage points (from 2.000% to 1.997%) while lengthening expected innovation search time from 2 years to 3.4 years. This marginally negative effect arises because the economy is already well to the right of the optimum. The results are deliberately small and consistent with the empirical near-constancy of growth alongside financial deepening.

Q5: What combination of policies does the model recommend for raising growth? Only a simultaneous reduction of frictions in both the credit and the innovation market raises the growth rate meaningfully, to 2.122% in the calibration (+0.122 pp relative to the 2.000% benchmark). Isolated improvements in credit markets have a marginally negative effect; isolated improvements in innovation markets have a marginally positive effect (+0.071 pp). The authors interpret this as supporting the OECD view that growth-stimulating policies should be designed as a system rather than as isolated pro-growth measures.

Q6: How does the elasticity of growth to finance depend on the gap between potential and actual growth? The authors show (referenced as available on request) that the elasticity of the growth rate with respect to financial factors is proportional to (γ − g)/γ, where γ is the potential growth rate (the productivity jump per innovation) and g is the actual equilibrium growth rate. When actual growth is close to potential — as in the benchmark calibration with γ = 0.023 and g = 2.000% — this factor is near zero, making growth nearly insensitive to changes in financial conditions. This provides a structural rationale for why empirically measured finance–growth effects are often small or nil in advanced economies.

Q7: How does introducing fixed bank entry costs (Section 4.1) change the results? When banks bear a fixed licensing cost K (paid each time they enter the credit market), credit market tightness φ becomes an increasing function of (r − g)K: the annuity value of the fixed cost falls as growth rises, inducing more bank entry and reducing credit tightness. This introduces an upward-sloping PP curve (rather than a vertical one) and creates a direct positive feedback from growth to financial deepening. The qualitative conclusions on non-monotonicity are preserved: lower licensing costs shift the PP curve right and steepen it, with the equilibrium effect on growth remaining ambiguous due to the congestion spillover into the innovation market.

Q8: What happens to the spillover function when innovators are paid (Section 4.2)? When innovators receive a Nash-bargained wage, the equilibrium wage (Equation 30) is increasing in innovator productivity (πγ), innovation market tightness (θn), and the growth rate, and decreasing in total credit market search costs K(φ). Easier credit raises both expected revenues and innovator wages for the firm. For innovator bargaining power α sufficiently small (and always for α < 1, as shown in the Appendix), the revenue effect dominates so that Qp < 0 is preserved: finance still creates bottlenecks in the innovation market, and the core non-monotonicity result carries through.

Q9: What does the model predict when only one market has search frictions? When only the credit market is frictional and innovators are found instantly after financing is secured, improving credit market efficiency unambiguously raises growth (Section 4.3, Figure 4). The GG curve becomes g = γ/(s/p + 1), which is strictly increasing in p, and the PP curve shifts in a way that unambiguously raises equilibrium growth. The paper uses this case to isolate the source of non-monotonicity: the negative spillover from credit ease to innovation congestion requires frictions in both markets to operate.

Q10: How does the paper relate to the empirical “too much finance” literature? The paper offers a distinct theoretical mechanism for the inverted-U relationship between credit and productivity growth documented by Arcand et al. (2015), Aghion et al. (2019), and Popov (2018), among others. While Aghion et al. (2019) explain the inverted-U through less-efficient incumbents surviving longer with better credit access, and Malamud and Zucchi (2019) emphasize how financing frictions differentially affect entrant and incumbent composition, Finocchiaro and Weil’s mechanism operates through congestion externalities in sequential search markets — a channel not previously formalized in the innovation-led growth literature.

Search frictions in credit markets: Firms searching for financiers (banks or venture capitalists) and banks searching for firms face a matching technology with constant returns to scale; credit market tightness φ is the ratio of firms searching for banks to banks searching for firms, and the matching probability p(φ) is strictly decreasing in φ. Free entry drives bank profits to zero, pinning equilibrium tightness.

Search frictions in innovation markets: After securing financing, firms search for innovators who can upgrade their productivity by factor γ; innovation market tightness θ is the ratio of firms searching for innovators to innovators, and the matching probability q(θ) is strictly decreasing in θ. The number of innovators is held fixed (analogously to fixed labor supply in Mortensen-Pissarides).

Spillover function Q(p, g): Derived from the free-entry zero-profit condition for firms, Q expresses the equilibrium innovation-matching probability q as a function of the credit-matching probability p and the growth rate g. It has Qp < 0 and Qg < 0, meaning easier credit and faster growth both reduce q by tightening the innovation market. It is the formal embodiment of the traffic-jam mechanism.

GG curve: The locus of (p, g) pairs consistent with the equilibrium growth equation g = γ/(1 + s/p + s/Q(p,g)). Under the symmetric cost condition c = ωn, the GG curve is hump-shaped: it rises from the origin, reaches a maximum interior growth rate, then declines toward an asymptote g∞ < γ. Its shape encodes the non-monotonic relationship between finance and growth.

PP curve: The locus of equilibrium credit-matching probabilities consistent with free entry in the credit market. In the benchmark model it is a vertical line at p* = p(ω/(1−ω) · k/c), independent of q and g. When banks bear a fixed entry cost K, the PP curve becomes upward-sloping, introducing a direct positive feedback from growth to financial deepening.

Potential growth rate γ: The productivity jump per successful innovation; in a frictionless world (p = q = ∞) the economy grows at γ. Actual growth g falls below γ to the extent that search frictions delay the delivery of credit and innovation. The elasticity of g to financial factors is proportional to (γ − g)/γ, so when actual and potential growth are close, financial factors matter little for growth.

Congestion externality in R&D: The mechanism by which financial deepening — raising p — drives more firms to seek innovators, tightening the innovation market and reducing q. This negative spillover (Qp < 0) is the paper’s central departure from models with only a single friction, where finance is always growth-enhancing.

A Welfare Analysis of Policies Impacting Climate Change

Mon, 01 Jan 0001 00:00:00 +0000

This paper extends and applies the marginal value of public funds (MVPF) framework to evaluate the welfare consequences of 96 climate-related tax and spending policies in the United States. The MVPF is a benefit-cost ratio in which the numerator captures all benefits to individuals (measured by their willingness to pay) and the denominator captures net government costs; policies with higher MVPFs are better spending policies, while those with lower MVPFs are more efficient revenue-raising instruments.

The sample covers policies rigorously evaluated using quasi-experimental or experimental methods drawn from 18 major economics journals between January 1999 and December 2023. Policies fall into three primary categories: subsidies (wind production tax credits, residential solar, electric vehicles, hybrid vehicles, vehicle buybacks, appliance rebates, and weatherization), nudges and marketing, and revenue raisers (gasoline taxes, other fuel taxes, cap-and-trade). A selected set of international aid policies is also analyzed. The analysis applies a harmonized method for translating behavioral changes into emissions changes — using the EPA’s AVERT model for electricity-sector emissions — and a consistent set of externality valuations, including an EPA 2023 social cost of carbon (SCC) of $193 per ton of CO2 in 2020 (rising over time), with robustness checks at $76, $337, and $1,367.

The primary methodological contribution is a new sufficient statistics approach to quantifying learning-by-doing (LBD) externalities. When marginal cost of production is an isoelastic function of cumulative production and demand is an isoelastic function of price, the time path of production satisfies a second-order ordinary differential equation whose solution yields society’s willingness to pay for LBD spillovers. LBD generates two types of externalities: a price externality (lower future consumer prices) and an environmental externality (increased future take-up of clean goods). The approach requires four inputs: price elasticity of demand, elasticity of marginal cost with respect to cumulative production, cumulative production at the time of the subsidy, and product cost at the time of the subsidy.

The three main empirical findings are as follows. First, subsidies for production that directly displaces dirty electricity generation have the highest MVPFs. Wind production tax credits have an MVPF of 3.85 without LBD, rising to 5.87 with LBD. Residential solar subsidies have an MVPF of 1.45 without LBD, rising to 3.86 with LBD. EV subsidies have an MVPF of approximately 1.4 with LBD and approximately 1 without it. Consumer subsidies for appliances, weatherization, vehicle retirement, and hybrid vehicles have MVPFs around 1. Second, conservation nudges targeting electricity consumption can deliver MVPFs exceeding 5 in regions with relatively dirty electric grids, but fall below 1 in cleaner-grid regions such as California and the Northeast — and their effectiveness is expected to decline as grids decarbonize. Third, fuel taxes (gasoline, diesel, jet fuel) and cap-and-trade permit reductions are efficient revenue raisers, with nearly all having MVPFs below 1 and most below 0.7, reflecting the Pigouvian logic that current tax rates fall below the associated environmental externalities. Cap-and-trade permit reductions can produce MVPFs below zero, meaning revenue is raised while providing net positive welfare to individuals.

The paper also constructs three cost-per-ton metrics — resource cost per ton, government cost per ton, and social cost per ton — and shows they can yield substantively different and sometimes opposite rankings relative to each other and to the MVPF. For example, EV subsidies carry a government cost per ton of $1,356 (among the highest in the sample) yet an MVPF above most consumer subsidies, because that metric omits non-CO2 benefits including LBD effects. The scope of the analysis is US historical policy, with the MVPF comparison most informative when social welfare weights across beneficiary groups are treated as roughly equal.

Q: What is the MVPF framework and how does it differ from cost-per-ton analysis? A: The MVPF equals benefits to individuals (sum of willingness to pay) divided by net cost to the government. It is designed for a decision-maker maximizing social welfare subject to a budget constraint, whereas cost-per-ton metrics serve a decision-maker minimizing cost subject to a fixed CO2 reduction target. A higher MVPF means more welfare gain per dollar spent; a lower MVPF means less welfare cost per dollar of revenue raised.

Q: What are the three cost-per-ton definitions the paper distinguishes, and why do they differ? A: Resource cost per ton measures the economic resources consumed per ton of CO2 abated, independent of subsidy incidence; government cost per ton measures net government outlays per ton, omitting all non-CO2 benefits; social cost per ton subtracts non-CO2 benefits from government costs. For appliance rebates, these three values are -$2, $474, and an intermediate figure — a range that reflects whether inframarginal transfers and non-CO2 co-benefits are counted.

Q: What is the new methodological contribution regarding learning by doing? A: The paper derives a sufficient statistics result showing that when marginal production cost is an isoelastic function of cumulative production and demand is isoelastic in price, the time path of production follows a second-order ordinary differential equation. Solving this equation yields society’s willingness to pay for LBD spillovers from four observable parameters: demand price elasticity, the LBD elasticity of marginal cost with respect to cumulative production, cumulative production at the subsidy date, and unit cost at that date. This allows LBD benefits to be incorporated into both MVPF and cost-per-ton calculations without requiring a fully calibrated dynamic model.

Q: What LBD elasticities does the paper use, and where do they come from? A: Drawing on Way et al. (2022), a 1% increase in cumulative solar production is associated with a 0.319% price reduction; for wind the elasticity is 0.194%, and for EV batteries it is 0.421%. These are treated as the isoelastic parameter in the sufficient statistics formula.

Q: How does LBD affect the MVPF estimates for wind, solar, and EVs specifically? A: For wind production tax credits, the MVPF rises from 3.85 to 5.87 when LBD is included. For residential solar, it rises from 1.45 to 3.86. For EV subsidies, the MVPF rises from approximately 1 to approximately 1.4. Without LBD, EV subsidies are in line with other consumer subsidies; LBD is the primary reason EVs outperform that group.

Q: What is the baseline social cost of carbon used, and how sensitive are results to alternative values? A: The baseline SCC is $193 per ton of CO2 in 2020, following EPA 2023 guidance at a 2% discount rate. Robustness checks use $76, $337, and $1,367. Higher SCC values raise the MVPF of all subsidies in the sample, but the relative ordering — with wind PTCs above all other consumer subsidies — remains consistent across the full range.

Q: How are EV subsidies evaluated, and what accounts for their MVPF exceeding other consumer subsidies? A: The analysis uses the California EFMP program studied by Muehlegger and Rapson (2022), which finds a price elasticity of demand of -2.1 and 85% pass-through to consumers (15% captured by dealers). A $1 subsidy generates $0.85 in consumer WTP, $0.15 in dealer WTP, $0.17 in CO2 co-benefits, $0.05 in local pollution and accident co-benefits, offset by $0.10 in damages from increased electricity generation. Most benefits are non-environmental (inframarginal transfers and LBD effects on future vehicle prices), which is why the government cost per ton of $1,356 appears high while the MVPF is approximately 1.4.

Q: What drives the high MVPFs for nudges in dirty-grid regions, and what is the implication for the future? A: Conservation nudges in dirty-grid areas have high MVPFs (exceeding 5) because each kilowatt-hour of reduced consumption displaces generation from high-emission sources, amplifying the environmental benefit per dollar of program cost. In cleaner-grid regions like California and the Northeast, the same nudge displaces lower-emission generation, pushing the MVPF below 1. As grids decarbonize nationwide, the paper notes that nudge MVPFs will decline over time.

Q: How do cap-and-trade permit reductions compare to fuel taxes as revenue-raising instruments? A: Nearly all fuel taxes (gasoline, diesel, jet fuel) have MVPFs below 1, with most below 0.7, meaning they impose a welfare cost of only $0.70 per dollar of revenue raised. Cap-and-trade permit reductions can have MVPFs below zero, meaning they can raise revenue while simultaneously providing net positive welfare gains to individuals because environmental benefits from reduced emissions outweigh the permit costs borne by emitters.

Q: What do the international subsidy findings suggest, and what are their limitations? A: Subsidies for efficient charcoal cookstoves in Kenya (Berkouwer and Dean 2022) generate US-specific gains from CO2 reductions that are 37 times the net cost of the subsidy; including global benefits raises the MVPF to 323. However, the paper flags substantial uncertainty: estimated policy impacts vary widely within similar international categories, and the US-specific MVPF is highly sensitive to assumptions about the incidence of the social cost of carbon on US residents and US government tax revenue.

Q: Why does the social cost per ton metric give opposite rankings within wind, solar, and EVs relative to the MVPF? A: EVs have a social cost per ton of -$415 versus -$32 for wind PTCs, making EVs appear superior on that metric — the reverse of the MVPF ordering. The paper explains that when SCPT values are negative (policies that abate CO2 while also yielding positive non-CO2 net benefits), the metric loses its Lagrange multiplier interpretation: increased non-CO2 benefits make SCPT more negative while increased abatement makes it less negative, preventing meaningful cross-policy comparisons.

Q: What is the overall policy ranking implied by the MVPF analysis? A: From highest to lowest MVPF: international clean energy subsidies > wind production tax credits > residential solar subsidies > energy conservation nudges (dirty grids) > EV subsidies > consumer appliance and weatherization subsidies > hybrid vehicle subsidies > vehicle buyback rebates > energy conservation nudges (clean grids) > revenue raisers (gas taxes, fuel taxes, cap-and-trade). The paper notes that shifting $1 of government revenue from gas taxes (MVPF ~0.67) to wind PTCs (MVPF ~5.87) generates $5.20 in net welfare benefits to individuals, assuming equal social welfare weights across groups.

Marginal Value of Public Funds (MVPF): A benefit-cost ratio equal to the sum of individuals’ willingness to pay for a policy divided by its net cost to the government. Policies with higher MVPFs deliver greater welfare gains per dollar spent; those with lower MVPFs impose lower welfare costs per dollar of revenue raised. Used to compare spending and revenue-raising policies on a common welfare-maximizing basis.

Learning-by-Doing (LBD) Externality: The spillover by which current production of a technology lowers its future marginal cost, generating future consumer surplus (price externality) and additional future uptake with associated environmental benefits (environmental externality). Treated in this paper as an uninternalized external benefit of subsidizing current production.

Sufficient Statistics Approach to LBD: The paper’s methodological contribution — showing that when marginal cost is an isoelastic function of cumulative production and demand is isoelastic in price, the LBD welfare benefit can be computed from four observables: the demand price elasticity, the LBD cost elasticity, cumulative production at subsidy date, and unit cost at subsidy date, without requiring a fully specified dynamic model.

Resource Cost per Ton (RCPT): Economic resources consumed to produce and use a product, divided by tons of CO2 abated. Appropriate for private firms minimizing abatement cost; independent of subsidy take-up rates and inframarginal transfers.

Government Cost per Ton (GCPT): Net government outlay per ton of CO2 abated. The correct metric for a government focused exclusively on CO2 reduction at minimum fiscal cost; omits all non-CO2 welfare impacts, including co-benefits and LBD effects.

Social Cost per Ton (SCPT): Government cost net of all non-CO2 benefits, per ton of CO2 abated. Intended to capture the social cost of abatement, but loses its Lagrange multiplier interpretation when values are negative, preventing valid cross-policy comparisons in that region.

Social Cost of Carbon (SCC): The monetized damage from one additional ton of CO2 emissions. Baseline value of $193 per ton in 2020 from EPA 2023 at a 2% discount rate, rising over time. A key parameter driving MVPF levels across all policy categories; robustness checked at $76, $337, and $1,367.

Pigouvian Efficiency of Environmental Taxes: The paper quantifies that fuel taxes have MVPFs below 0.7 because current tax rates fall below the associated Pigouvian optimum — i.e., taxing polluting goods raises revenue while reducing a pre-existing negative externality, so the welfare cost of the revenue is less than one dollar per dollar raised.

Additionality and Asymmetric Information in Environmental Markets: Evidence from Conservation Auctions

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates the problem of additionality — the likelihood that a conservation action is marginal to (i.e., caused by) an incentive — in the United States Department of Agriculture’s Conservation Reserve Program (CRP), one of the largest and most mature Payments for Ecosystem Services (PES) mechanisms in the world. The CRP pays landowners $1.6–$1.8 billion per year under 10-year contracts to retire cropland and plant grass mixes, trees, or wildlife habitats, using a discriminatory scoring auction in which landowners submit bids on a menu of heterogeneous contracts ranked by a scoring rule.

The central argument is that additionality represents a form of asymmetric information. Landowners possess private knowledge about their counterfactual land use (whether they would have conserved anyway), while the auction screens only on their private cost of accepting the contract. Because lower-cost landowners are lower-cost partly because they expect to conserve regardless of the CRP, cost and additionality are positively correlated — generating adverse selection: the least costly participants to purchase are the least socially valuable. The status quo scoring rule implicitly assumes all landowners are fully additional (tau = 1), an assumption the paper tests and rejects.

The authors construct a dataset linking confidential administrative CRP bid data across seven auctions from 2009 to 2021 to satellite-derived land use classifications from the Cropland Data Layer (30m resolution) and USDA administrative land use reports. They exploit a regression discontinuity (RD) in contract awards around the winning score threshold to estimate the causal effect of CRP contracts on land use at the margin. The first-stage is close to one. The key finding is that CRP contracts reduce cropping by approximately eight percentage points at the margin, but the 100%-additional benchmark predicts a reduction of roughly 33 percentage points (matching the share of land covered by a contract at the margin). Therefore, only approximately one quarter (22–29%) of marginal auction winners are additional — meaning three-quarters would have conserved without the CRP contract.

To test for adverse selection, the authors use the 82% of rejected bidders in the 2016 auction (the most restrictive) for whom counterfactual land use is observed, constructing a landowner-specific additionality measure. They document a systematic positive correlation between bid rental rates (reflecting higher costs) and additionality, which persists conditional on rich observable characteristics including prior land use interacted with soil productivity. Contract choice further reveals additionality: tree-related contract bidders exhibit substantially lower additionality than base grassland contract bidders.

To quantify welfare implications, the authors develop and estimate a joint structural model of bidding and additionality. Costs are inferred via revealed preferences in optimal bidding (following the empirical auctions literature), and additionality is estimated as a conditional expectation function of observable characteristics and unobserved costs, matched to observed land use among rejected bidders via Method of Simulated Moments. Social benefits are taken from the CRP literature and USDA revealed preferences.

Key welfare findings: (1) Despite widespread non-additionality and adverse selection, a hypothetical uniform-price market for the base conservation contract generates social welfare gains of $14.37 per acre-year at the socially-optimal price. Setting price equal to the full social benefit B — ignoring counterfactual land use — causes welfare losses of $12.68 per acre-year, nearly eliminating the gains. (2) The status quo auction generates social welfare gains of approximately $120 million per auction relative to no market, but implements only 12% of the gains achievable under the efficient allocation. (3) Simple modifications to the scoring rule that incorporate expected additionality — via uniform adjustments and market-size reductions — close 37% of the gap between the status quo and the efficient allocation, increasing social welfare by over $300 million per auction. Nearly all gains arise from incorporating additionality into the scoring rule. These modifications are described as implementable by the USDA in practice.

Q: What is additionality, and why does it matter for conservation markets? A: Additionality is defined as the expected impact of contracting on a landowner’s conservation action — i.e., the probability that a landowner would not have conserved absent the incentive. Social surplus depends on both a landowner’s cost of accepting a contract and her additionality, but market mechanisms screen only on cost. When the lowest-cost participants are the least additional, standard procurement mechanisms fail to implement the efficient allocation, undermining the environmental and fiscal effectiveness of conservation programs.

Q: What is the rate of additionality at the margin of CRP contract awards? A: Approximately one quarter (22–29% depending on specification) of marginal auction winners are additional. The RD design shows contracts reduce cropping by about eight percentage points at the margin, compared to the 100%-additional benchmark of approximately 33 percentage points (the share of land covered by the contract at the margin). This implies three-quarters of marginal winners would have conserved without a CRP contract.

Q: What is the empirical evidence for adverse selection? A: Among rejected bidders in the 2016 auction — where additionality is directly observed for 82% of bidders — there is a systematic positive correlation between bid rental rates (reflecting higher costs of accepting the contract) and additionality. This correlation persists conditional on rich observable characteristics, including prior land use interacted with soil productivity estimates. Contract choice also reveals additionality: bidders selecting tree-related contracts have substantially lower additionality than those choosing base grassland contracts.

Q: How does soil productivity relate to additionality? A: USDA-constructed soil productivity estimates, which approximate the earning potential of a parcel, are predictive of additionality in practice, consistent with theory. Higher soil productivity is associated with lower additionality — landowners with less productive land are more likely to conserve regardless of the CRP. Soil productivity is not currently incorporated into the CRP scoring rule to rank bidders.

Q: How is the RD design validated? A: The histogram of normalized score distributions shows no bunching at the winning threshold, validating that bidders do not know the exact ex-post threshold realization. Pre-period RD coefficients are indistinguishable from zero in both the remote sensing and administrative land use data. The first stage (share of bidders with a CRP contract just above the threshold) is close to one. Treatment effect magnitudes are stable over the 10-year contract period with no evidence of attenuation, and there are no spillovers to non-bid fields.

Q: What do the social welfare calculations show for a uniform-price market? A: Despite widespread non-additionality and adverse selection, a hypothetical uniform-price market for the base conservation contract generates social welfare gains of $14.37 per acre-year at the socially-optimal uniform price. However, setting price equal to the full social benefit B — as the status quo implicitly does by assuming tau = 1 — causes welfare losses of $12.68 per acre-year, nearly eliminating all gains.

Q: How does the status quo auction perform relative to the efficient benchmark? A: The status quo auction generates social welfare gains of approximately $120 million per auction relative to no market. The efficient allocation, which awards contracts based on both landowner costs and expected social benefits (incorporating additionality), would be substantially larger. The status quo implements only 12% of the social welfare gains achievable under the efficient allocation.

Q: Can the efficient allocation be implemented by any mechanism? A: Not necessarily. Implementing the efficient allocation requires that the expected net social surplus function B·tau(c) - c be monotonically decreasing in cost, so that a standard incentive-compatible auction can rank bidders appropriately. If lower-cost landowners are sufficiently less additional that the allocation rule is non-monotone in cost, no incentive-compatible mechanism can implement the efficient allocation (per Myerson 1981). Empirically, the authors find that for the base contract the efficient allocation is in the implementable case (similar to their Figure 1a), but implementing it exactly via an incentive-compatible auction remains complex.

Q: What alternative auction designs are proposed, and how much do they improve welfare? A: The authors propose alternative scoring rules that incorporate expected additionality — through uniform adjustments to the scoring rule, reductions in market size, and differentiation among heterogeneously additional landowners based on observables such as soil productivity and contract choice. These simple modifications close 37% of the gap between the status quo and the efficient allocation, increasing social welfare by over $300 million per auction. Nearly all gains come from incorporating additionality into the scoring rule, with a large share accruing through simple uniform adjustments.

Q: How is the structural model of bidding estimated? A: Estimation proceeds in three steps. First, beliefs about the winning score threshold distribution are estimated by simulating auctions via resampling (following Hortacsu 2000). Second, landowner costs are estimated via Maximum Simulated Likelihood using revealed preference inequalities from optimal bidding in the scoring auction. Third, the additionality conditional expectation function is estimated via Method of Simulated Moments, matching observed additionality levels, its distribution across rejected bidders, its covariance with scores, and its distribution by contract choice.

Q: What sources of scoring rule variation identify the model? A: Three sources are used. A mid-mechanism policy change in the 2021 auction added carbon sequestration payments differentially across contracts, providing two bids from the same bidders under different scoring rules. A policy change around 2011 shifted Wildlife Priority Zone (WPZ) bonus points to be contract-specific. Air Quality Zone (AQZ) status shifts the level of the score. These sources provide variation in relative payments across contracts, though the authors note the variation is modest and rely also on parametric extrapolation.

Q: What assumptions are required for identification and how robust are results? A: Key assumptions include perfect compliance (validated by inspection of over 1,000 aerial photographs), no spillovers to non-bid fields (validated in Table 2), and stability of the additionality function tau(z,c,kappa) across auction years. The authors assess robustness to alternative functional forms of tau, conduct a non-parametric inversion exercise across cost quantiles, and construct alternative scoring rules using cross-auction and cross-tract variation to probe the stability assumption. Model-implied additionality at the RD margin (23%) closely matches the empirical RD estimate.

Q: Are the adverse selection and additionality findings specific to the 2016 auction? A: The 2016 auction provides the most complete view because bid fields are observed and 82% of bidders are rejected. But cross-auction evidence replicates the core patterns. RD estimates exploiting threshold variation across auctions show additionality ranging from 10–20% among lower bidders to 40–50% among higher bidders across auctions, consistent with adverse selection. Tree-contract null RD effects replicate across all auctions. Cross-tract cropping rates show similar observable heterogeneity across auctions.

Q: What is the social welfare impact of the market for conservation existing at all? A: Theoretically ambiguous because non-additional landowners may receive transfers without generating social value, and adverse selection may tilt the market toward low-additionality participants. Empirically, despite these concerns, there exist positive social welfare gains of $14.37 per acre-year at the socially-optimal uniform price for the base contract, indicating that conservation markets of this type can improve welfare even in the presence of substantial non-additionality and adverse selection.

Additionality: The expected impact of contracting on a landowner’s conservation action — formally, tau(c) = E[1 - a_i0 | c = c_i], the probability that a landowner would not have conserved absent the incentive. A landowner is additional if she would have cropped without the CRP contract; the social benefit of contracting depends only on this incremental conservation impact.

Adverse Selection: The positive correlation between landowner cost of accepting a contract and additionality. Because landowners with low costs are low-cost partly because they expected to conserve regardless of the program, lower-cost participants are less socially valuable. This upward-sloping contract value curve mirrors adverse selection in insurance markets as modeled by Einav, Finkelstein, and Cullen (2010).

Contract Value Curve: The function B·tau(F^{-1}_C(q)) plotting the expected social value of contracting at each quantile q of the cost distribution. It lies below the social benefit B due to non-additionality and slopes upward due to adverse selection. The vertical distance between the contract value and marginal cost curves equals expected social surplus B·tau(c) - c.

Efficient Allocation: The allocation that maximizes expected social surplus B·tau(c) - c by awarding contracts to landowners for whom this quantity is positive. Implementing this allocation via an incentive-compatible mechanism requires that B·tau(c) - c be monotonically decreasing in cost; if not, no standard mechanism can achieve it.

Scoring Rule: The known function s(b_i, z^s_i) that converts a landowner’s multi-dimensional bid (rental rate and contract choice) and observed characteristics into a score, determining contract awards. The status quo scoring rule implicitly assumes full additionality (tau = 1), ranking bidders as if all conservation actions are marginal to the incentive.

Source Text Origin: The classification of the text on which a summary is based — “pdf” or “oa-html” for full working paper text, or “abstract-only” which is blocked from summarization. Determines the validity and completeness of any summary produced.

Aggregate demand externality and self-fulfilling default cycles

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. Why do corporate defaults cluster in recurring episodes rather than occurring smoothly? The paper asks whether observable fundamental factors — firm characteristics and macroeconomic variables — are sufficient to account for the clustered default patterns documented in the data, and, if not, what theoretical mechanism can explain them.

Empirical Motivation. Using Moody’s historical default rate data, the authors document that the long-run average corporate bond default rate during 1866–2008 was approximately 1.50%, yet defaults were highly episodic: the worst three-year period during the Great Depression totaled 12.88%, and the three-year period 1873–1875 after the railroad boom reached 35.80%. A Markov switching regression on post-war default rate data (1951–2017) strongly rejects a linear no-switch model in favor of a two-regime model across all information criteria (AIC, HQ, SC, and log-likelihood). The estimated high-default regime has a mean default rate of 1.93% (unconditional mean µ/(1−ρ)) — roughly eight times the 0.23% mean of the low-default regime — and a standard deviation nearly six times larger. The high-default regime persists on average 5.81 years (transition probability of staying ≈ 0.83), while the low-default regime lasts approximately 7.52 years (staying probability ≈ 0.87).

Model. The authors build a continuous-time general equilibrium model with Dixit-Stiglitz monopolistic competition (CES aggregation with elasticity σ) and an endogenous entry/exit/default mechanism. Households are risk-neutral and also act as entrepreneurs. At each instant, δµ new project blueprints are invented; entrepreneurs borrow to invest, then face an idiosyncratic liquidity shock z drawn from a Pareto distribution G(z). Entrepreneurs continue if z ≤ Z*, a cutoff determined by the continuation value of the firm, and default otherwise. Continuing firms become monopolists for a new variety until that variety becomes obsolete at a Poisson rate δ. Each operating firm must borrow working capital constrained by its firm value Vt (collateral constraint wtnjt ≤ θVjt). The entire equilibrium reduces to a two-dimensional dynamical system in (Mt, Vt), where Mt is the number of operating firms (state variable) and Vt is the firm value (control variable).

Key Mechanism — Demand Externality and Positive Feedback. Under CES aggregation, each firm’s gross revenue is y_jt^(1–1/σ) · Y_t^(1/σ), making individual firm revenue increasing in aggregate output Yt. A decline in Yt lowers firm profits and firm value Vt, which raises the default threshold Z* and increases the fraction of projects that are abandoned. Fewer operating firms further depress Yt, closing a positive feedback loop. This static strategic complementarity (through CES) is combined with dynamic strategic complementarity through the borrowing constraint: higher expected future firm value relaxes current working capital constraints, raising current production.

Multiple Equilibria and Global Dynamics. The two-locus phase diagram (˙Mt = 0 and ˙Vt = 0) yields multiple intersections — and hence multiple steady states — when productivity A lies in an intermediate range (A < A < Ā). When A > Ā, a single good saddle-point equilibrium exists. When A < A, no equilibrium can be sustained. In the intermediate range, a good steady state (low default rate, high firm value) coexists with a bad steady state (high default rate, low firm value). The good steady state is always a saddle; the bad steady state is a sink (locally indeterminate, κ < κ_Hopf) or a source (locally determinate but globally indeterminate, κ > κ_Hopf), depending on parameter κ = 1 + (θ + ρ)/δ.

Bogdanov-Takens Bifurcation. Using global dynamical methods, the paper demonstrates richer indeterminacy than local analysis permits. Near the Bogdanov-Takens point (κ, Ā), the system can exhibit: (a) infinite equilibrium trajectories converging to the bad steady state; (b) saddle-loop bifurcation at κ = κ_SL ≈ 14.25 (under the baseline calibration); (c) stable or unstable periodic orbits for κ ∈ (κ_Hopf, κ_SL) — endogenous business cycles in a perfect-foresight equilibrium; and (d) multiple trajectories from near the source that converge to the good saddle equilibrium.

Simulation of Clustered Defaults. With a two-state Markov process for productivity (Ah = 10, Al = 9.34) and pessimistic sentiment shifts (the “ugly” state), the model replicates the cluster pattern: in the good/high-productivity state, the default rate is near zero; when productivity falls to low and sentiment turns pessimistic, the default rate can spike to approximately 12%, consistent with the Great Depression observation. Critically, the paper shows that the cluster pattern is generated only under global dynamics — restricting to local dynamics produces substantially smaller fluctuations in the default rate, confirming that the ugly (sink) equilibrium is essential.

Policy. A countercyclical subsidy to non-defaulting entrants — financed by a lump-sum tax, calibrated as tr(Vt) = τ(VG − Vt) — shifts the ˙Mt = 0 locus downward and can eliminate the bad steady state entirely, leaving only the good saddle-path equilibrium. The paper provides a closed-form sufficiency condition for τ (Proposition 7).

Scope Conditions. Multiple equilibria require: (i) productivity in the intermediate range A < A < Ā; (ii) the elasticity of substitution σ not too large (below a threshold σ̄ that itself depends on µ); (iii) the borrowing constraint binding (δ > θσ/((σ–1)κ), which can always be ensured by choosing δ sufficiently large). Clustered defaults in the simulation require the joint occurrence of a negative fundamental shock (productivity falling from high to low) and a shift to pessimistic sentiment; either factor alone generates only limited default amplification.

In depth

Q1. What is the core empirical motivation for the model, and what does the regime-switching analysis establish?

The paper documents that the corporate bond default rate, drawn from Moody’s data covering 1866–2008, clusters sharply in episodes: the long-run average is 1.50%, yet the worst three-year period of the Great Depression totaled 12.88% and 1873–1875 reached 35.80%. A Markov switching regression on 1951–2017 data strongly rejects a linear no-regime-switch model across all four criteria (log-likelihood, AIC, HQ, SC). The two-regime model identifies a high-default regime with unconditional mean 1.93% and standard deviation roughly six times the low-default regime’s, a persistence probability of approximately 0.83 (duration ≈ 5.81 years), and a low-default regime with unconditional mean 0.23% and persistence approximately 0.87 (duration ≈ 7.52 years). The regime-switching result supports the prior literature’s claim (Das et al. 2007; Duffie et al. 2009; Azizpour et al. 2018) that observable fundamentals alone cannot account for clustered defaults.

Q2. How does the Dixit-Stiglitz CES structure generate a demand externality that links aggregate output to individual firm default decisions?

Under CES aggregation with elasticity σ, each firm’s gross revenue equals y_jt^(1–1/σ) · Y_t^(1/σ) (equation 7), so aggregate output Yt directly enters individual firm revenue. Each firm takes Yt as given, yet the aggregation of all firms’ output determines Yt. When aggregate output falls — because more firms have defaulted and exited production — each remaining firm’s revenue and profit fall, reducing the firm’s continuation value Vt. A lower Vt tightens the borrowing constraint (wtnjt ≤ θVjt), reduces working capital, and raises the probability that the firm’s idiosyncratic liquidity shock will exceed the default threshold Z*, producing further defaults. This positive feedback constitutes the demand externality: individual firms’ decisions are strategic complements, both statically (through CES demand) and dynamically (through the borrowing constraint on working capital).

Q3. What is the two-dimensional dynamical system that summarizes the equilibrium, and what do the two loci look like in the phase diagram?

The entire equilibrium reduces to two differential equations in (Mt, Vt): ˙Mt = –δ[Mt – µG(Z(Vt))] and ˙Vt = κδVt[1 – F(Vt, Mt)], where F captures the ratio of monopoly profit to firm value including the borrowing constraint. The ˙Mt = 0 locus slopes strictly upward because a higher firm value Vt raises the default cutoff Z* and lowers the fraction of entrants who default, so more firms survive and Mt rises until absorption equals entry. This locus has a minimum at Mm = µG(zm) because firm value must exceed the threshold that sustains the credit market. The ˙Vt = 0 locus is non-monotonic: it first slopes upward (more firms raise aggregate demand and profit through the scale/externality channel) and then slopes downward (more firms tighten the labor market, raising wages and lowering profits). The two opposing channels make the ˙Vt = 0 locus hump-shaped, creating the possibility of two intersections and hence two steady states.

Q4. Under what conditions do multiple steady states exist, and what does each look like?

Multiple steady states exist when productivity A satisfies A < A < Ā, where A and Ā are closed-form thresholds given by Equations (A.3) and (A.4), and the elasticity of substitution σ is below a threshold σ̄ (Equation A.5). When A < A, neither locus intersects and no equilibrium is sustainable. When A > Ā, a single good saddle-point equilibrium exists. In the multiple-equilibria range, the good steady state has a higher firm value and a smaller fraction of firms defaulting; the bad steady state has a lower firm value and a higher default rate. Under the paper’s numerical calibration (A = 10, η = 6.5, Zmin = 0.88), the low default rate at the good steady state is approximately 1.5% and the high default rate at the bad steady state is between 12% and 13%.

Q5. What are the local dynamics around each steady state, and how does parameter κ determine whether the bad steady state is a sink or a source?

Proposition 5 shows that the good steady state is always a saddle point, ensuring a unique convergent path for initial Mt near Mg_0. The bad steady state’s local nature depends on κ = 1 + (θ + ρ)/δ and the critical value κ_Hopf = 1 + ψ/(θMb_0Vb_0). When κ is between 1 and κ_Hopf, the Jacobian trace is negative and the bad steady state is a sink with one order of indeterminacy: given Mt close to Mb_0, infinitely many initial values of the control variable Vt satisfy all equilibrium conditions. When κ > κ_Hopf, the bad steady state is a source point; the economy diverges from it. Because κ does not affect the steady-state locations (Proposition 3), one can vary κ to change the dynamic character without moving the equilibria in the phase diagram.

Q6. What does the global dynamics analysis reveal that local analysis misses?

Global analysis via Bogdanov-Takens bifurcation (Proposition 6) reveals three classes of dynamics absent from local analysis. First, even in the saddle-source case (locally determinate), there exist multiple equilibrium trajectories diverging from near the bad (source) steady state and converging to the good (saddle) steady state; these paths satisfy all equilibrium conditions including transversality but are incorrectly ruled out by local methods. Second, at the critical value κ_SL ≈ 14.25 (under the baseline calibration), a homoclinic saddle-loop orbit connects the saddle point to itself — all trajectories interior to the loop converge to the bad steady state. Third, for κ between κ_Hopf and κ_SL, periodic orbits arise in a perfect-foresight equilibrium with no external shocks. For example, at κ = 14.9, the phase diagram displays a unique periodic orbit around the bad steady state, with two distinct initial values of Vt for any given Mt near the orbit — endogenous, perpetual oscillations without any exogenous driving force. Numerical experiments confirm that Mt = 0.23 admits two rational-expectations values of Vt (2.09 and 3.55) on the saddle path alone, illustrating abundant indeterminacy even at the endpoint.

Q7. How does the paper simulate the clustered default pattern and what is the role of the “ugly” equilibrium?

The paper constructs a three-state Markov economy: “good” (high productivity Ah = 10, single saddle equilibrium, near-zero default rate), “bad” (low productivity Al = 9.34, saddle-path equilibrium, modestly elevated defaults), and “ugly” (low productivity, sink-path equilibrium, sharply elevated defaults). The ugly state is reached when, upon a productivity decline, firms adopt pessimistic expectations and the economy slides to the high-default sink instead of remaining on the low-default saddle path. Transition probabilities are set so that the average ugly-state duration is approximately 6 years and roughly 45% of periods are ugly, consistent with the regime-switching estimates. With Zmin = 0.2 and η = 15, the ugly-state default rate can reach approximately 12%, matching the Great Depression observation. The counterfactual experiment deletes the ugly state (pGU = 0) and resets pGB = 0.45: the resulting default rate stays close to zero with no cluster pattern, demonstrating that global dynamics (the ugly sink) rather than the fundamental shock alone generate the clustering.

Q8. Can purely sentiment-driven cycles generate the clustered default pattern?

Section 6.2 fixes productivity at a low level (A = 9.53) and drives switches between the bad (saddle path) and ugly (sink path) states by pure sentiment shocks alone (πBU and πUB). The simulated default rate does spike upward when sentiment turns pessimistic, but the rises are generally more modest than in the combined fundamental-plus-sentiment exercise, and the default rate can no longer be characterized as countercyclical. The authors conclude that the realistic observed default cluster is the result of a combination of negative fundamental shocks and pessimistic sentiment shifts; either ingredient alone is insufficient to replicate all features of the data.

Q9. How does the collateral constraint on working capital create dynamic strategic complementarity?

Following Jermann and Quadrini (2012), Liu and Wang (2014), and Lian and Ma (2021), each operating firm must borrow to pay wages each period, subject to the constraint wtnjt ≤ θVjt. Since Vt is forward-looking (the discounted present value of the firm’s monopoly profit stream), optimistic expectations about future output raise Vt, relax the borrowing constraint, allow firms to hire more labor and produce more output today, and thereby validate optimism. This intertemporal complementarity means that the equilibrium is sensitive not only to current fundamentals but also to beliefs about the future, opening the channel for sentiment-driven multiple equilibria and self-fulfilling cycles.

Q10. What is the policy remedy for the bad equilibrium, and how does it work?

Proposition 7 establishes that a countercyclical lump-sum-tax-financed subsidy to non-defaulting entrants, tr(Vt) = τ(VG − Vt), with τ exceeding a computable threshold, eliminates the bad steady state. The subsidy works by effectively raising the value of continuing for a firm at any given Vt and Mt, shifting the ˙Mt = 0 locus downward until it lies below the ˙Vt = 0 locus everywhere in the relevant range, eliminating the second intersection and leaving only the good saddle-path equilibrium. The numerical illustration uses parameters from Section 6 with A = 9.67 and τ = 1/3 to demonstrate that the bad steady state vanishes and the phase diagram has a single equilibrium. The subsidy is self-limiting: in normal conditions when firm value is already high (Vt ≈ VG), the transfer is near zero.

Cui and Kaas (2021) show default cycles from self-fulfilling beliefs in a fully competitive firm environment, focusing on intertemporal default coordination. The present paper differs in three respects. First, firms engage in monopolistic competition under CES preferences, and the main novel mechanism is cross-firm default contagion through the demand externality — which can produce multiple equilibria even in a static setting, without any intertemporal coordination. Second, the paper examines the joint role of fundamental shocks and aggregate-demand externalities together, showing that multiple equilibria arise only in the presence of sufficiently low productivity (A < A < Ā), making indeterminacy contingent on external fundamentals rather than structural parameters alone. Third, the continuous-time framework with full global analysis via Bogdanov-Takens bifurcation allows characterization of periodic orbits and the interaction of the ugly sink path with Markov productivity regimes — dynamics not covered in Cui and Kaas (2021).

Q12. What is the markup prediction of the model, and is it consistent with empirical evidence?

Under Dixit-Stiglitz CES with elasticity σ, the equilibrium markup of each intermediate good equals σ/(σ–1) at the firm level. However, the measured gross markup — which includes the effective collateral constraint — is predicted to comove positively with the default rate in the model, and hence the markup is countercyclical. The paper notes this is consistent with the well-documented empirical regularity in Bils (1987) and Rotemberg and Woodford (1999). Additionally, the model replicates the finding in Gilchrist and Zakrajšek (2012) that a low default rate is associated with a high firm entry rate.

Key Concepts

Demand Externality (Dixit-Stiglitz type). In the paper’s sense, this is the mechanism by which individual firms’ revenues depend on aggregate output Yt through the CES aggregator: each firm’s gross revenue is y_jt^(1–1/σ) · Y_t^(1/σ). Each firm takes Yt as given, but the aggregation of all firms’ output determines Yt. This creates a positive spillover: more operating firms raise aggregate output, which raises each firm’s revenue, and vice versa. The paper uses this as the central transmission channel for self-fulfilling defaults, in contrast to prior literature that emphasized debt networks or asymmetric information contagion.

Self-Fulfilling Default Cycle. A dynamic equilibrium path in which pessimistic expectations about aggregate output are validated: if firms anticipate that more other firms will default (lowering Yt), their own continuation value Vt falls, raising the probability that their idiosyncratic liquidity shock will exceed the default threshold, increasing actual defaults, further lowering Yt, and so on. The paper distinguishes this from shock-amplifier stories by constructing a model with multiple rational-expectations equilibria in which the aggregate default rate is determined in part by initial beliefs.

Bogdanov-Takens Bifurcation. A mathematical tool for global dynamics analysis applied to two-dimensional continuous-time systems. In the paper, it is used to characterize system behavior when the parameters (κ, A) are near the point (κ̄, Ā) at which the Jacobian has two zero eigenvalues. Near this point, the system can exhibit saddle-loop bifurcations, Hopf bifurcations, homoclinic orbits, and stable or unstable periodic orbits — all of which are invisible to local linearization analysis. The paper uses this to establish that indeterminacy is more pervasive than local analysis suggests.

Good / Bad / Ugly Steady States. In the paper’s three-regime framework: the “good” state is the unique saddle-point equilibrium under high productivity Ah, with near-zero default rates; the “bad” state is the saddle-path equilibrium under low productivity Al, with modestly elevated defaults; the “ugly” state is the sink-path equilibrium under low productivity, characterized by self-fulfilling high default rates (up to ~12%). The ugly state is reached only when pessimistic sentiment coincides with the low-productivity regime, and it is the ugly state that generates the cluster pattern in simulation.

Collateral Constraint on Working Capital. The firm-level borrowing constraint wtnjt ≤ θVjt, where θ is the collateral ratio and Vjt is the firm’s continuation value. This constraint means that higher expected future profits — by raising Vt — relax the current borrowing limit, increase current labor demand and output, and create dynamic strategic complementarity between current and future production. It is this constraint, combined with the CES demand externality, that makes the dynamical system two-dimensional and generates the non-monotonic ˙Vt = 0 locus.

Global Indeterminacy. The existence, given an initial state variable Mt, of multiple equilibrium trajectories — each satisfying all equilibrium conditions including transversality — that converge to different steady states or follow periodic paths. In the paper, global indeterminacy arises even when the system is locally determinate (e.g., in the saddle-source case): trajectories diverging from near the source steady state can converge to the saddle steady state along multiple paths, none of which is detectable by local linearization.

Periodic Orbit (Endogenous Cycle). In the paper, a closed trajectory in the (Mt, Vt) phase plane that the economy follows indefinitely in perfect-foresight equilibrium without any exogenous shocks. Such orbits exist for κ ∈ (κ_Hopf, κ_SL), are stable if S < 0 and unstable if S > 0 (where S is a computable quantity defined in Equation A.13). Their existence demonstrates that business cycles can arise purely from internal forces — the demand externality and borrowing constraint — consistent with the view in Beaudry, Galizia, and Portier (2020).

Aggregation and the Estimation of Quality Change

Mon, 01 Jan 0001 00:00:00 +0000

Errico and Lashkari address two intertwined problems in the measurement of aggregate price indices: how to account for quality change and variety entry/exit when the demand system is not CES, and how to identify flexible demand systems from prices and market shares alone when supply and demand shocks are correlated. The paper makes a theoretical contribution and a methodological one, then applies both to the measurement of US import price inflation over 1989–2016.

The theoretical contribution generalizes the unified CES price index of Redding and Weinstein (2020a) and the Feenstra (1994) variety correction to the full class of smooth, invertible demand systems. The key insight is that the contribution of quality change to the aggregate price index depends on heterogeneous cross-product elasticities of substitution, not a single scalar as in the CES case. For practical implementation, the paper specializes to the Homothetic with Aggregator (HA) family of demand systems — which includes Kimball (1995), CRESH (Hanoch, 1971), and HSA (Matsuyama and Ushchev, 2017) — showing that within this family cross-product elasticities collapse to product-level elasticities, dramatically reducing dimensionality. The resulting approximate price index (Proposition 2) weights each product by its love-of-variety index 1/(epsilon_it − 1), departing from the uniform CES weighting.

The methodological contribution is a dynamic panel (DP) identification strategy that exploits the Markov structure of quality shocks. The paper assumes that innovations to product quality are mean-zero conditional on lagged prices. Under flexible pricing, firms maximize current-period profits without regard to future demand shocks, so lagged prices are valid instruments for current prices. This permits identification of rich demand systems without external cost instruments and without the conventional assumption of uncorrelated supply and demand shocks. The conventional Feenstra–Broda–Weinstein (FBW) approach imposes zero correlation between quality shocks and prices; the paper shows that when quality and marginal cost are positively correlated, FBW produces downward-biased elasticity estimates (endogeneity bias).

The empirical application constructs a dataset covering 155 time-consistent 5-digit NAICS industries over 1989–2018, matching US customs import data with domestic production data and treating country-of-origin varieties as the unit of observation. The paper estimates both CES and Kimball demand systems using the DP approach and compares them to FBW estimates.

Key quantitative findings: First, DP-estimated CES elasticities are larger on average than FBW estimates (weighted mean 5.99 vs. 4.62), confirming a downward endogeneity bias in conventional methods. Second, Kimball mean elasticities exceed CES estimates (weighted mean 3.11 for Kimball vs. 5.99 for CES at the industry level, but the Kimball distribution has a mean of 17.0 and median 4.70), reflecting a heterogeneity bias — CES understates the dispersion of elasticities and thereby understates the elasticity relevant for the base (domestic) product whose market share is declining. Third, quality improvements in imported goods reduced the US import price index by approximately 20.2 percentage points cumulatively (0.67 p.p. annually) under Kimball demand, and 15.9 percentage points cumulatively (0.53 p.p. annually) under CES demand, over 1989–2018. The headline figure cited in the abstract is approximately 0.7 p.p. annually. The aggregate import price index (price plus quality components combined) fell by 8.25 p.p. cumulatively under Kimball and 4.01 p.p. under CES, compared to a BEA PCE index increase of 57.8 p.p. over the same period. Sectorally, machinery and electrical equipment account for roughly 60% of total quality gains (~200 p.p. cumulative). By country, China accounts for approximately 35% of cumulative quality gains, with non-OECD countries collectively contributing ~59%, and China’s quality upgrading accelerating after WTO accession.

Validation using US automobile market data (1980–2018) confirms the DP identification assumption: controlling for current product characteristics, future characteristics are uncorrelated with current prices. The DP approach produces elasticity estimates and quality change measures similar to those obtained using real exchange rate cost-shock instruments, and the Kimball demand closely matches mixed logit (BLP) estimates of both price elasticities and price indices. CES estimates exhibit a measurable downward heterogeneity bias in this validation setting, which the paper traces theoretically and empirically to a positive covariance between demand elasticities and price volatility across products.

Scope conditions: results apply to homothetic (income-invariant) demand; nonhomothetic extensions are provided as a generalization (Proposition 4) but not the primary focus. The import price index measures the cost of imports conditional on given domestic consumption; it does not capture full consumption-side welfare effects including substitution away from domestic varieties.

Q1: What is the core theoretical result on price index measurement beyond CES? Proposition 1 shows that for any smooth, invertible demand system satisfying the connected substitute property, the change in the log aggregate price index can be approximated as a weighted sum of log price changes and log expenditure share changes, with the expenditure share changes premultiplied by the inverse of the matrix Psi_t capturing cross-product elasticities of substitution. In the CES special case this reduces to the scalar (1/(sigma−1)) weight of the Redding-Weinstein (2020a) CUPI. The key departure in general demand is that the weight applied to each product’s expenditure share change is heterogeneous and depends on the full matrix of cross-product substitutabilities, not a single constant.

Q2: How does the HA (Homothetic with Aggregator) family simplify the theoretical results? For HA demand — which nests Kimball, CRESH, and HSA — Lemma 1 establishes that cross-product elasticities sigma_ij depend only on product-level elasticities epsilon_i through simple analytic formulas (e.g., epsilon_i * epsilon_j / epsilon-bar for HDIA), reducing the estimation problem from an N×N matrix to a vector of N scalars. Proposition 2 then gives an approximate price index in which each product’s expenditure share change is weighted by its love-of-variety index 1/(epsilon_it − 1), rather than a common CES scalar. This is the operative formula for the Kimball application.

Q3: What is the endogeneity bias in conventional elasticity estimation and how large is it? Conventional FBW methods assume supply and demand shocks are uncorrelated; when quality improvements are positively correlated with product prices (e.g., higher-quality goods command higher prices and also have higher marginal costs), FBW estimates are biased downward. The paper documents this: for CES demand, the DP-estimated weighted mean elasticity is 5.99 versus 4.62 under FBW, and for median estimates the DP value is 4.27 versus 2.58 under FBW, across 155 industries. The bias matters because underestimated elasticities imply underestimated quality changes and a smaller quality correction to the price index.

Q4: What is the heterogeneity bias and how does it differ from the endogeneity bias? Even after correcting for endogeneity, CES demand imposes a single elasticity per industry, ignoring the cross-product distribution. The paper shows that the CES estimate is an average that does not correctly capture the behavior of the base product (the domestic US variety) whose market share is declining. Because the domestic variety tends to have a lower elasticity than the import average, CES understates this product’s love-of-variety index and thereby understates the quality correction attributable to rising import shares. Theoretically and empirically (Appendix E.4), this bias is larger when demand elasticities covary positively with price volatility across products.

Q5: What is the dynamic panel identification assumption and why does it hold under flexible pricing? The paper assumes that quality shock innovations u_it are mean-zero conditional on lagged log prices: E[u_it | log p_it−1] = 0. Under flexible pricing, firms maximize current-period profits using current variables only; current prices are determined by current quality but are not chosen in anticipation of future quality shocks. Therefore lagged prices are uncorrelated with future quality innovations, making them valid instruments for current prices. This assumption is validated empirically in the automobile market: controlling for current product characteristics (horsepower, weight, fuel economy), future characteristics are not correlated with current prices.

Q6: What are the headline findings on quality change in US import prices? Under Kimball demand, quality improvements in imported goods reduced the US import price index by 20.2 percentage points cumulatively over 1989–2018, equivalent to 0.67 p.p. annually (the abstract rounds this to approximately 0.7 p.p. annually). Under CES demand, the quality contribution is 15.9 p.p. cumulatively (0.53 p.p. annually). The aggregate import price index combining price and quality changes fell by 8.25 p.p. under Kimball and 4.01 p.p. under CES over the same period. These figures imply that official import price statistics substantially overstate import price inflation by failing to account for quality improvements.

Q7: Which sectors and countries drive the quality gains? Machinery and electrical equipment account for approximately 60% of total cumulative quality gains, with roughly 200 p.p. cumulative quality improvement in that sector. Computer and peripheral equipment (NAICS 3341) is a notable contributor — the official import-to-producer price ratio shows a nearly five-fold increase between 1989 and 2018, but after quality adjustment this ratio reverses direction. By country of origin, China accounts for approximately 35% of cumulative quality gains; other non-OECD countries collectively contribute approximately 59%; OECD countries contribute approximately 7%. China’s quality upgrading is documented to accelerate following its WTO accession.

Q8: Why does CES understate the quality correction relative to Kimball? The primary mechanism is that the US domestic variety — which serves as the numeraire for quality measurement — has a declining market share over the sample period. In Kimball demand, products with declining market shares are assigned lower elasticities (higher love-of-variety indices), amplifying the quality correction associated with import share gains. CES imposes a uniform elasticity, failing to capture this asymmetry. The paper shows that the key driver of the CES-Kimball gap in the import price index is CES underestimating the love-of-variety index of the base domestic product.

Q9: How is the identification approach validated in the automobile market? Using the Berry-Levinsohn-Pakes dataset extended by Grieco et al. (2024) for 1980–2018, the paper first verifies empirically that future product characteristics (horsepower, weight, fuel efficiency) are uncorrelated with current prices after controlling for current characteristics. It then compares DP estimates for both CES and Kimball demand against estimates obtained using real exchange rate (RER) variation as a cost-shock instrument, finding similar results in both cases. Finally, it compares Kimball and CES estimates against mixed logit (BLP) demand: Kimball closely matches BLP price elasticities and implied quality changes, while CES shows a downward heterogeneity bias.

Q10: What does the automobile market validation imply for the import price index methodology? Since Kimball demand matches the richer mixed logit demand in the auto setting — where product characteristics are observed — the validation provides evidence that Kimball demand serves as a good approximation to rich heterogeneous-elasticity models when characteristics are unavailable. The paper constructs price indices for the US auto industry based on mixed logit, mixed CES, Kimball, and standard CES, and shows that the Kimball index is closer to the mixed logit and mixed CES indices than is the standard CES index.

Q11: How does the paper handle product entry and exit? Proposition 3 generalizes Proposition 1 to accommodate product entry and exit. The expression includes a variety correction analogous to Feenstra (1994) but generalized to non-CES settings via the mean love-of-variety index of entering and exiting products. In the CES special case this reduces exactly to the Feenstra (1994) correction. In the empirical application to US imports, entry and exit of country-of-origin varieties within industries is a relevant margin given the expansion of trading partners over the sample.

Q12: How does the paper relate to Redding and Weinstein (2020a)? Redding and Weinstein (2020a) derive a price index formula under CES demand that accounts for taste shocks, applied to US retail scanner data where quality is constant at the barcode level. The present paper generalizes their CUPI formula beyond CES to general and HA demand systems, and extends their identification strategy to settings where demand changes partly reflect quality changes rather than pure taste shocks. The paper also shows that the CES assumption used in Redding-Weinstein may overstate the contribution of taste shocks to cost-of-living indices, since part of the expenditure share variation attributed to taste shocks under CES would be reassigned under heterogeneous-elasticity demand.

Q13: Does the paper address welfare implications beyond the import price index? The paper explicitly notes that the import price index does not capture the full consumption-side welfare effects of rising imports, since gains from lower import prices may be partly offset by substitution away from domestic varieties. The paper also notes that it abstracts from nonhomotheticity (income effects), pointing to Jaravel and Lashkari (2021) for that extension. The primary welfare-relevant quantity reported is the quality-adjusted change in the cost of the imported goods basket, which is the import price index in the conventional sense.

Love-of-variety index: For a product i, defined as 1/(epsilon_it − 1) where epsilon_it is the product-level demand elasticity in an HA demand system. It measures the welfare value of having access to that variety and serves as the weight applied to expenditure share changes in the generalized price index formula (Proposition 2). In the CES special case all products share the same love-of-variety index 1/(sigma−1).

Homothetic with Aggregator (HA) demand: A family of income-invariant (homothetic) demand systems — including Kimball (1995), CRESH (Hanoch, 1971), and HSA (Matsuyama and Ushchev, 2017) — in which preferences are represented by a utility function with a specific aggregator structure. The key property exploited in the paper is that cross-product elasticities of substitution sigma_ij depend only on product-level elasticities epsilon_i through simple analytic formulas, reducing the dimensionality of the estimation problem from an N×N matrix to N scalars.

Endogeneity bias (in elasticity estimation): Downward bias in estimated elasticities of substitution arising from a positive correlation between product quality shocks and prices. When higher-quality products command higher prices and also have higher marginal costs, conventional methods (FBW) that assume zero correlation between supply and demand shocks will attribute part of the price variation to supply, underestimating how much demand responds to price. The paper documents this bias as the gap between DP and FBW estimates.

Heterogeneity bias (in elasticity estimation): Additional downward bias in CES elasticity estimates relative to the mean of Kimball elasticities, arising from CES imposing a single elasticity per industry when the true elasticities are heterogeneous across products. The bias is stronger for differentiated products and is theoretically traced to a positive covariance between demand elasticities and price volatility across products.

Dynamic panel (DP) identification: The paper’s proposed identification strategy, which exploits the Markov structure of quality shocks. The key moment condition is that quality shock innovations are mean-zero conditional on lagged prices, which holds under flexible pricing. Lagged prices (and higher-order lags and nonlinear transformations) serve as instruments for current prices, permitting identification of demand parameters without external cost instruments.

Quality shock (phi_it): An unobserved product characteristic that shifts demand for product i at time t, defined through the utility function as a scalar multiplying the quantity consumed. Quality is identified from residual demand — the component of demand not explained by price — following the approach of Khandelwal (2010) and Hallak and Schott (2011). The paper models quality shocks as following a stationary AR(1) process with product-specific means.

Unified CES price index (CUPI): The price index formula of Redding and Weinstein (2020a) for CES demand, which decomposes the aggregate price change into a price component (expenditure-share-weighted price changes) and a quality/taste component proportional to (1/(sigma−1)) times expenditure share changes. The present paper’s Proposition 2 generalizes CUPI to HA demand by replacing the scalar 1/(sigma−1) with product-specific love-of-variety indices.

An endogenous gridpoint method for distributional dynamics

Mon, 01 Jan 0001 00:00:00 +0000

This paper introduces the Distributional Endogenous Gridpoint Method (DEGM), a novel numerical technique for solving the distributional dynamics that arise in heterogeneous agent macroeconomic models. The core problem is how to efficiently update the distribution of agents over the state space as the economy evolves. The dominant existing approach — the “lottery method” of Young (2010) — discretizes the state space and represents policy functions as lotteries over nearby gridpoints, producing a transition matrix that is linear in optimal policies. This linearity renders the lottery method incapable of capturing nonlinear effects in distributional dynamics, a limitation that becomes quantitatively significant for higher-order perturbation solutions.

DEGM extends Carroll’s (2006) endogenous gridpoint method from individual optimization to the distributional level. Rather than discretizing the density and integrating forward, DEGM works directly on the cumulative distribution function (CDF). The key insight is that when the policy function is monotone — as savings functions typically are — the endogenous gridpoints generated by the policy function trace out exact points on the post-policy CDF without requiring integration. Specifically, if A*_{i,j} = a*(A_i, Y_j) are optimal asset choices from grid point A_i at income Y_j, then the CDF values at those endogenous points are known analytically as F_t(A_i | Y_j). An interpolant using shape-preserving splines constructed through these points allows evaluation of the updated CDF at any point without integration. The income transition step is handled separately via standard quadrature over the discretized income process.

The paper demonstrates DEGM’s performance with two applications. First, in the Aiyagari (1994) economy, DEGM converges to the stationary equilibrium an order of magnitude faster than the lottery method in terms of gridpoints. At nk=40 gridpoints, the lottery method deviates from the benchmark capital stock by 1.72% and the wealth Gini by 2.24% (for nh=5), while DEGM deviates by only 0.09% and 0.12% respectively. Both methods converge to the same solution as the number of gridpoints increases, but DEGM reaches this limit far faster.

Second, the authors introduce a Krusell-Smith style model with aggregate investment risk (capital depreciation shocks calibrated following Barro, 2006, as a 0.4% quarterly probability of 7.5% capital destruction causing a 10% annual GDP drop) as a new baseline for studying aggregate nonlinearities with household heterogeneity. This model overcomes the near-linearity of aggregate capital dynamics in the original Krusell-Smith specification. Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points and increases wealth inequality by up to 11 basis points relative to the non-stochastic steady state, depending on idiosyncratic income risk calibration. The lottery method systematically mispredicts these effects: it always predicts a decrease in wealth inequality in the presence of investment risk, while DEGM predicts an increase. At third order, the lottery method predicts wealth Gini changes of +2.0 bp (persistent calibration) and -149.7 bp (transitory calibration), while DEGM predicts +10.7 bp and +2.1 bp respectively.

The mechanism for increased inequality under investment risk is heterogeneous: for less wealthy households the substitution effect dominates (they reduce saving more in response to risky returns), while for wealthy households the income effect is stronger and precautionary saving motives dominate. The lottery method, by making the distributional transition matrix linear in policies, zeros out the second derivative of the transition matrix with respect to the policy function, missing the term capturing how the density at the pre-image of each asset level is affected nonlinearly. DEGM’s cubic spline interpolant captures all nonlinearities up to third order, enabling economically meaningful results that qualitatively differ from lottery-method predictions on wealth inequality.

Q: What is the fundamental numerical problem that DEGM solves? A: Evolving the distribution of agents forward over time in heterogeneous agent models requires evaluating a Kolmogorov forward equation, which naively demands numerical integration. The lottery method avoids integration by discretizing the state space and expressing transitions as a linear matrix operation, but this forces the distributional dynamics to be linear in optimal policies. DEGM avoids integration by exploiting policy function monotonicity: the endogenous policy gridpoints are the interpolation nodes, so the CDF update requires only interpolation, not integration. This preserves nonlinear effects up to the order of the splines used.

Q: How does DEGM handle the borrowing constraint and the resulting mass point? A: Savings policy functions are typically weakly monotone: constant at the borrowing constraint for sufficiently poor households, then strictly monotone above a threshold. DEGM accommodates this by starting the endogenous grid at the EGM solution corresponding to the borrowing constraint (the threshold a_j above which the policy is strictly monotone), restoring strict monotonicity on the relevant domain. The mass point at the borrowing constraint is captured by evaluating F_t(a_j, Y_j). Echoes of the borrowing constraint diminish as the number of income states increases, and in practice 10 income gridpoints are sufficient to smooth them.

Q: How much faster does DEGM converge relative to the lottery method for the stationary equilibrium? A: In the Aiyagari economy with nk=40 asset gridpoints, the lottery method’s capital stock deviates from the benchmark by 1.72% and the wealth Gini by 2.24% (nh=5), while DEGM deviates by only 0.09% and 0.12% respectively — roughly a 20-fold improvement in accuracy for the same gridpoints. At nk=80, the lottery method still shows 0.56%/0.78% deviations while DEGM shows 0.03%/0.00%. Although for a fixed number of gridpoints the lottery method is faster in wall-clock time (0.35s vs 0.82s at nk=40, nh=20), DEGM is faster for a given level of accuracy because it requires far fewer gridpoints.

Q: Why does the lottery method fail at higher-order perturbations? A: The lottery method constructs its transition matrix as a piecewise linear function of the optimal policy a*, so its second derivative with respect to a* is zero. As a result, it misses the second term in the second-order derivative of the end-of-period CDF: the term involving the derivative of the density at the pre-image of each asset level times the squared linear policy effect. This missing nonlinearity becomes quantitatively important at second and third order. DEGM’s cubic hermitian spline interpolant captures all nonlinearities up to third order, allowing it to correctly represent how the distribution responds nonlinearly to aggregate shocks.

Q: What does the paper find about the effect of aggregate investment risk on the capital stock and wealth inequality? A: Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points from the non-stochastic steady state, depending on whether income risk is persistent or transitory (DEGM third-order: -4.7 bp persistent, -11.4 bp transitory). Wealth inequality increases by up to 11 basis points (DEGM third-order: +10.7 bp persistent, +2.1 bp transitory). The lottery method diverges dramatically at third order, predicting Gini changes of +2.0 bp and -149.7 bp for the persistent and transitory calibrations respectively, compared to DEGM’s +10.7 bp and +2.1 bp.

Q: What is the mechanism through which aggregate investment risk increases wealth inequality? A: The mechanism operates through heterogeneous saving responses across the wealth distribution. For less wealthy households, capital income is a small share of total income, so the substitution effect of risky returns dominates: higher investment risk reduces their incentive to save. For wealthy households, capital income is central, so the income effect is stronger and precautionary saving motives intensify. A capital depreciation shock upon realization compresses the wealth distribution, but the risk of such a shock increases inequality on average because it disproportionately reduces saving among poorer households.

Q: How do the authors extend DEGM to handle aggregate risk and higher-order perturbations? A: The authors follow Reiter (2009) in including the distribution and value functions in the state space, defining a nonlinear difference equation over these objects. Higher-order perturbation of this system proceeds using the algorithms of Andreasen et al. (2018) and Levintal (2017), with second-order terms solved via a generalized Sylvester equation using Kim et al.’s (2008) doubling algorithm. The implementation handles up to 3,200 variables at second order and 220 variables at third order. For the second-order solution, the Bayer-Luetticke (2020) state-space reduction and its refinement in Bayer et al. (2024) yield results identical to the full unreduced system.

Q: What is the state-space reduction procedure and how much does it compress the system? A: The full system uses 402 states and 412 controls (persistent calibration). A copula representation of the distribution reduces this to 213 states and 412 controls; adding DCT compression of the value function gives 213 states and 98 controls; further adding a factor representation from the first-order solution yields 111 states and 98 controls — a 75% reduction. The R-squared-like IRF statistic remains 1.00 across all reductions, and ergodic moments are identical (capital: 25.54, Gini: 0.61 for the persistent calibration).

Q: Does DEGM produce different first-order impulse responses than the lottery method? A: For first-order perturbations, DEGM and the lottery method converge to the same solution as the number of gridpoints increases, but DEGM converges faster. For the first-order dynamics of the wealth distribution (wealth Gini IRFs), DEGM reaches convergence with nk=40 gridpoints while the lottery method requires nk=160. For aggregate capital stock IRFs, both methods converge quickly at first order. Quantitative differences become significant only at second and higher orders.

Q: What calibration is used for the investment risk model? A: Capital depreciation deviates from its steady-state value by a shock with second moment sigma_delta = 0.005 and third moment tau_delta = 0.012. This corresponds to a 0.4% quarterly probability that a disaster destroys 7.5% of the capital stock and causes a 10% drop in annual GDP, consistent with the evidence in Barro (2006). The model is solved under both a persistent income calibration (beta=0.98, rho=0.98, sigma_epsilon=0.14, implied Gini=0.66) and a transitory income calibration (beta=0.99, rho=0.88, sigma_epsilon=0.18, implied Gini=0.42).

Distributional Endogenous Gridpoint Method (DEGM): A numerical method for evolving the joint CDF of agents over the state space by constructing an interpolant at endogenous gridpoints A*_{i,j} = a*(A_i, Y_j) — the optimal policy values — at which CDF values are known analytically as F_t(A_i | Y_j), thus updating the distribution through interpolation rather than integration and preserving nonlinearities up to the order of the spline.

Lottery Method (LM): Young’s (2010) standard technique that replaces the continuous distribution with a discrete counterpart and represents optimal policy functions as probability weights over nearby gridpoints, yielding a single transition matrix A* such that f_{t+1} = f_t * A*. The transition matrix is linear in optimal policies, which zeroes out the second derivative of the distributional dynamics with respect to policies and causes systematic misprediction of distributional dynamics under higher-order perturbation.

Kolmogorov Forward Equation (Distributional Dynamics): The law of motion for the joint CDF F_t(a, y) describing how the distribution of households over assets and income evolves given optimal policies and the income transition process. In DEGM, this equation is split into a sub-period for asset choices (where endogenous gridpoints allow integration-free updating) and a sub-period for income transitions (handled by quadrature over the discretized income process).

Higher-Order Perturbation Solution: A Taylor expansion of the model’s nonlinear equilibrium conditions around the non-stochastic steady state beyond first order. Second-order solutions capture precautionary motives and mean deviations from the steady state; third-order solutions additionally capture asymmetric effects of shocks, requiring DEGM’s nonlinear distributional representation to produce accurate results.

Aggregate Investment Risk (Capital Depreciation Shocks): Shocks to the aggregate capital depreciation rate calibrated following Barro (2006) as a 0.4% quarterly probability of a disaster that destroys 7.5% of the capital stock and causes a 10% annual GDP drop. Proposed as a replacement for near-linear Krusell-Smith aggregate productivity shocks to generate genuine nonlinearities in aggregate capital dynamics while remaining equally parsimonious.

State-Space Reduction: A sequence of compression techniques — copula representation of the wealth distribution, discrete cosine transform (DCT) compression of the value function, and factor representation from the first-order solution — that reduce the Reiter (2009) system from 402 states and 412 controls to 111 states and 98 controls (a 75% reduction) with no measurable loss of accuracy in impulse responses or ergodic moments.

Shape-Preserving Interpolation: Interpolation methods (linear spline or piecewise cubic hermitian splines) that maintain the monotonicity of the CDF when constructing the interpolant from endogenous gridpoints. Cubic hermitian splines additionally preserve differentiability, making the distributional dynamics smooth enough for third-order perturbation and capturing all nonlinear effects that the lottery method misses.

Anatomy of the Phillips Curve: Micro Evidence and Macro Implications

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper addresses a fundamental puzzle in macroeconomics: why do estimates of the New Keynesian Phillips curve (NKPC) slope differ sharply depending on whether real marginal cost or the output gap is used as the real activity variable? The conventional, output gap-based NKPC yields very flat slope estimates (e.g., 0.006 to 0.024 in Hazell et al. 2022 and Rotemberg and Woodford 1997), which has led to the widespread view that the Phillips curve is “flat,” at least during the pre-pandemic period. The authors argue that this view conflates two distinct structural relationships: the elasticity of inflation with respect to real marginal cost, and the elasticity of marginal cost with respect to the output gap.

Data and Methodology

The authors assemble a unique quarterly micro-level dataset covering 4,598 manufacturing firms in Belgium over 84 quarters (1999:Q1–2019:Q4), totaling 132,915 observations. The dataset combines product-level domestic prices and quantities from the PRODCOM administrative database, customs data on foreign competitors’ prices, and firms’ variable production costs (labor costs from social security declarations plus intermediate input costs from VAT declarations). Intermediate inputs account for approximately 75 percent of total variable costs on average and are the most volatile cost component (within-firm coefficient of variation 1.77, versus 0.77 for labor costs).

Their estimation strategy follows a “bottom-up” approach. Starting from a theoretical framework with heterogeneous firms subject to Calvo (1983) nominal rigidities and strategic complementarities in price setting (imperfect competition including dynamic oligopoly and Kimball demand), they derive a forward-looking dynamic pass-through regression linking a firm’s current price to discounted present values of its own marginal costs and competitors’ prices, plus a lagged price level that serves as an error-correction term. This is Model A; robustness variants include Model B (absorbing competitor prices via industry-by-time fixed effects), Model C (imposing an AR(1) process for marginal cost), and Model A-U (unrestricted lagged-price coefficient).

The structural parameters governing the NKPC slope — the degree of nominal rigidity (θ) and the strength of strategic complementarities (Ω) — are estimated jointly via GMM. Instruments for marginal cost are four-quarter-lagged firm-level total factor productivity (TFPQ), and instruments for competitors’ prices exploit variation in EU-area export prices to third-country destinations and bilateral exchange rates between non-EU competitor currencies and the Euro. Sector-by-time fixed effects and firm fixed effects absorb confounding trends, shifting trend inflation, and permanent markups.

Main Findings with Quantitative Magnitudes

The baseline estimate (Model A) yields θ = 0.711 (SE 0.014), implying that prices remain fixed for approximately three to four quarters on average, consistent with Nakamura and Steinsson (2008) Belgian PPI data (0.72). The strategic complementarity parameter is Ω = 0.570 (SE 0.059), indicating that competitor price dynamics reduce the pass-through of own marginal cost shocks by approximately half relative to the no-complementarities benchmark.

These structural estimates imply a slope of the marginal cost-based NKPC of λ = 0.052 (SE 0.007), tightly estimated and robust across specifications: λ = 0.077 in Model B, λ = 0.069 in Model C, and λ = 0.056 in the unrestricted Model A-U. This slope is two to ten times larger than existing estimates of the conventional output gap-based NKPC slope (κ ≈ 0.024, Rotemberg and Woodford 1997; κ ≈ 0.006, Hazell et al. 2022).

Reconciling the High Cost-Based Slope with the Flat Output-Based Slope

The paper shows that the output-based slope κ equals the product of the cost-based slope λ and the output elasticity of marginal cost σ_y: κ = λ · σ_y. Using Bartik-style instruments based on high-frequency ECB monetary policy surprises interacted with industry-level sensitivities, the authors estimate σ_y using two models. Model D yields σ_y = 0.406 and κ = 0.021; Model E (directly regressing changes in marginal cost on changes in output) yields σ_y = 0.112 and κ = 0.006. These estimates are consistent with, and overlap with, Rotemberg and Woodford (1997) and Hazell et al. (2022) during the pre-pandemic sample period. The low elasticity of marginal cost to output is attributed to near-constant short-run returns to scale at the firm level and wage rigidity that mutes general equilibrium effects.

Aggregate Inflation Dynamics

Feeding an aggregate marginal cost index (constructed as a Törnqvist-weighted average of firm-level marginal costs) into the model-implied inflation expression produces a series that tracks Belgian manufacturing PPI inflation well: marginal cost fluctuations alone account for approximately 70 percent of inflation variation (R² = 0.68, correlation 0.8), without appealing to unobservable cost-push shocks or inflation lags.

Model Validation via Supply Shocks

A validation exercise using identified oil shocks (Känzig 2021 — measured as unexpected OPEC-day movements in oil futures prices) confirms the model. A one-standard-deviation shock to oil prices (a 15.7 percent increase in Brent crude) raises firms’ real marginal costs by approximately 1.5 to 3 percent within the first three quarters, before reverting. The price response peaks at approximately 3 percent after six quarters, consistent with nominal rigidities generating a delayed but persistent response. Impulse-response matching yields λ_IRF = 0.042 (SE 0.005), within the confidence bands of the micro-level estimate λ = 0.052, validating the bottom-up approach.

Scope Conditions

All estimates are drawn from Belgian manufacturing firms over 1999–2019, a period of moderate inflation during which Calvo pricing provides a good approximation of firm behavior. The authors note that the elasticity of marginal cost to output may be time-varying and nonlinear, and that during large aggregate shocks (such as the post-pandemic inflation surge), both the frequency of price adjustment and the sensitivity of marginal cost to output can rise substantially, requiring state-dependent pricing models (addressed in a companion paper, Gagliardone et al. 2025).

In depth

Q1. What is the primitive formulation of the NKPC, and how does it differ from the conventional formulation?

A1: The primitive NKPC features real marginal cost (in log-deviation from its steady state) as the real activity variable: π_t = λ·mc_t + β·E_t{π_{t+1}} + u_t, where λ is the slope depending on nominal rigidities and strategic complementarities. The conventional formulation uses the output gap (or unemployment gap) as a proxy for marginal cost, which is valid only under specific conditions including perfectly flexible wages. When those conditions fail, the output gap is a poor proxy for marginal cost, typically leading to downward bias in slope estimates. Even when a proportionality holds, the output-based slope κ equals λ multiplied by σ_y (the output elasticity of marginal cost), so the two slopes carry different economic content.

Q2. What structural parameters govern the slope of the cost-based NKPC, and what is the formula?

A2: The slope is λ = (1−θ)(1−βθ)/θ, where θ is the Calvo probability of price non-adjustment (capturing nominal rigidity) and Ω = Γ/(1+Γ) is the strategic complementarities parameter derived from the markup elasticity Γ with respect to relative prices. High nominal rigidity (high θ) flattens the slope by making individual price adjustments less frequent; strong strategic complementarities (high Ω) flatten it further because firms mute their price response to marginal cost in order to avoid deviating from competitors. The discount factor β is calibrated at 0.99 for quarterly data.

Q3. How does the dynamic pass-through regression differ from the static (long-run) pass-through regressions used in prior literature?

A3: The dynamic pass-through regression (Model A) includes the firm’s lagged price as a regressor, which functions as an error-correction term controlling for persistent deviations between the price and the optimal reset price. Failing to include this term with quarterly data leads to omitted variable bias of magnitude −θ·Var(Δp_ft), since the cointegration error is autocorrelated with coefficient θ. Static pass-through regressions (as in Amiti, Itskhoki and Konings 2019 using annual data) are appropriate only when nominal rigidities can be ignored (θ ≈ 0); with quarterly data and θ ≈ 0.711, the orthogonality condition of the static model fails and the dynamic framework is necessary.

Q4. What are the baseline estimates of the structural parameters, and how robust are they?

A4: The baseline Model A yields θ = 0.711 (SE 0.014) and Ω = 0.570 (SE 0.059), implying prices fixed for approximately three to four quarters and competitor-price influence roughly equal to own marginal cost influence. The implied NKPC slope is λ = 0.052 (SE 0.007). Robustness checks across six specifications (Models B, C, A-U, variable SR-RTS controls, Translog TFPQ, eight-quarter-lagged instrument) yield λ in the range 0.044 to 0.077, with all estimates statistically significant and within each other’s confidence bands. The unrestricted model (A-U) cannot reject the restriction Ϛ = θ on the lagged-price coefficient (p-value 0.90).

Q5. What is the short-run elasticity of a firm’s own price to a permanent marginal cost shock, and how do nominal rigidities and strategic complementarities each contribute?

A5: The short-run pass-through elasticity is (1−Ω)(1−θ) ≈ (1−0.570)(1−0.711) ≈ 0.125. This is substantially below one because both forces dampen price adjustment: nominal rigidity (1−θ ≈ 0.289) means most firms cannot adjust in any given quarter, and strategic complementarities (1−Ω ≈ 0.430) mean that adjusting firms reduce their pass-through to avoid deviating from competitors’ prices. Without strategic complementarities (Ω = 0), the elasticity would be roughly 0.289; without nominal rigidities (θ = 0), it would be roughly 0.430; both together produce the observed 0.125.

Q6. How is marginal cost measured in the data, and why is the inclusion of intermediate input costs important?

A6: Marginal cost is proxied by average variable cost per unit of output: the log-nominal marginal cost equals ln(TVC_ft/Y_ft) + ln(1+ν_ft), where TVC is the sum of intermediate input costs (from VAT declarations) and labor costs (wage bill from social security declarations), and Y_ft is a quantity index. Intermediate inputs account for approximately 75 percent of total variable costs on average and are the most volatile component (within-firm coefficient of variation 1.77 vs 0.77 for labor). The authors note that DSGE models typically feature only labor as a variable input, but accounting for intermediates is pivotal because intermediate goods price shocks were among the most important drivers of the post-pandemic inflation surge.

Q7. What instruments are used for marginal cost and competitors’ prices, and what are the identifying assumptions?

A7: The instrument for marginal cost is the four-quarter lagged firm-level TFPQ (physical total factor productivity), estimated as the residual from a gross-output production function. Its relevance depends on TFP persistence (confirmed); the exclusion restriction requires that persistent TFP variation is orthogonal to current and future demand shocks after removing permanent demand components (via firm fixed effects) and industry trends (via sector-by-time fixed effects). Two instruments for competitors’ prices exploit international trade variation: (i) sales-weighted average export prices of EU-area competitors to non-Belgium, non-EU destinations (orthogonal to Belgian demand shocks by construction), and (ii) bilateral exchange rate movements between non-EU competitor currencies and the Euro. All instruments pass the Cragg-Donald and Kleibergen-Paap F-statistics (strongly rejecting weak instruments) and Hansen-Sargan over-identification tests (failing to reject validity).

Q8. What evidence supports the validity of the TFPQ instrument against capacity utilization concerns?

A8: The authors run two empirical tests. First, regressing marginal cost on four-quarter-lagged capacity utilization yields a small, statistically insignificant elasticity (0.011, SE 0.052), suggesting the TFPQ instrument’s predictive power does not reflect capacity utilization variation. Second, re-estimating with “purified” TFPQ instruments adjusted for capital utilization (Column 4) and for both capital and labor utilization (Column 5) produces parameter estimates and NKPC slopes essentially unchanged from baseline. Additionally, regression residuals show only weak and short-lived autocorrelation (−0.09 at one-quarter lag, p=0.09; −0.01 at two-quarter lag, p=0.69), indicating demand shocks are highly transitory after conditioning on fixed effects.

Q9. How does the model track aggregate Belgian manufacturing PPI inflation, and what does this imply for cost-push shocks?

A9: Using the reduced-form expression π_t = λ̃(mc_t^n − p_{t-1}) + α + θu_t, where the reduced-form slope λ̃ = 0.22 is evaluated at baseline structural estimates, the model produces a model-implied inflation series that accounts for approximately 70 percent of variation in manufacturing PPI inflation (R² = 0.68, correlation 0.8), without including inflation lags or cost-push shocks. The model captures the inflation drop during the 2008 financial crisis, the run-up in 2016, and the subsequent decline. This contrasts with the quantitative DSGE literature in which cost-push shocks (variation in desired price and wage markups) account for approximately 70 percent of inflation volatility (e.g., Primiceri, Schaumburg and Tambalotti 2006).

Q10. How do the authors estimate the output elasticity of marginal cost σ_y, and what do they find?

A10: They use two approaches. Model D is a pricing equation directly relating firm-level prices and nominal output (value added), estimated via GMM, instrumented with Bartik-style shifters based on high-frequency ECB monetary policy surprises (Altavilla et al. 2019) interacted with industry-level sensitivities. Model E directly regresses changes in nominal marginal cost on changes in nominal output, also instrumented. Model D yields σ_y = 0.406 (SE 0.099) and implied κ = 0.021 (SE 0.005); Model E yields σ_y = 0.112 (SE 0.026) and κ = 0.006 (SE 0.001). The low σ_y is consistent with near-constant short-run returns to scale at the firm level and wage rigidity muting general equilibrium labor-market feedback, at least during the moderate-inflation pre-pandemic period.

Q11. How does the oil shock validation exercise confirm the cost-based NKPC slope estimate?

A11: Following Känzig (2021), the authors identify oil shocks as unexpected movements in Brent crude oil futures around OPEC meeting days, normalizing to a one-standard-deviation shock (15.7 percent Brent increase). Local linear projection IRFs show that firms’ real marginal costs rise 1.5 to 3 percent within three quarters and then revert, while prices peak at approximately 3 percent increase after six quarters (consistent with nominal rigidity delaying the price response). Impulse-response matching — minimizing the weighted distance between empirical and model-implied price IRFs — yields λ_IRF = 0.042 (SE 0.005), which is close to and within the confidence bands of the micro-level estimate λ = 0.052, validating the bottom-up estimation approach.

Q12. What do the estimates imply about why the conventional NKPC appears flat in normal times?

A12: The flat conventional NKPC slope (κ ≈ 0.006–0.024) does not reflect limited transmission of marginal cost fluctuations to inflation — that transmission is high (λ ≈ 0.052–0.077). Rather, flatness reflects a weak link between the output gap and marginal cost during the pre-pandemic period (σ_y ≈ 0.112–0.406), attributable to near-constant short-run returns to scale in production and wage rigidity. This decomposition matters for policy: supply shocks that directly raise marginal cost will pass through strongly to inflation even when output does not move much, whereas demand shocks that operate through the output-cost channel face attenuated transmission.

Q13. Under what conditions does the cost-based Phillips curve decompose cleanly into a product of the two elasticities?

A13: The decomposition κ = λ · σ_y requires assuming that real wages are flexible and determined in general equilibrium at the industry level, with real wages increasing in industry output with elasticity σ_w; that the natural level of output is defined as the equilibrium under flexible prices and constant desired markups; and that the firm’s marginal product of labor depends on productivity and output with a common short-run returns-to-scale parameter ν (homogeneous across firms and time-invariant). Under these assumptions (which parallel those used to derive the conventional NKPC in the standard NK model), the output elasticity of marginal cost is σ_y = σ_w + ν, and the theoretical restriction κ = λ · σ_y holds exactly.

Q14. How do macroeconomic complementarities from aggregate decreasing returns to scale affect the NKPC slope?

A14: If aggregate SR-RTS fall below unity, the NKPC slope formula gains an additional term Θ = 1/(1+γν(1−Ω)) < 1, where ν is inversely related to average SR-RTS and γ is the within-industry elasticity of substitution. However, empirical estimates of sectoral SR-RTS range from 0.93 to 0.98, with an aggregate estimate of approximately 0.965 (implying ν ≈ 0.036). Given this and calibrating γ = 4, Θ ≈ 0.941, so macroeconomic complementarities would reduce the NKPC slope by only about 6 percent — well within the confidence bounds of the baseline estimates. The authors conclude that the constant-returns assumption in their main framework is a good approximation.

Key Concepts

Primitive (cost-based) NKPC slope (λ): The coefficient linking inflation to real marginal cost in the underlying New Keynesian pricing equation, defined as λ = (1−θ)(1−βθ)/θ. It captures how strongly firms’ aggregate price setting responds to movements in real marginal cost per unit of output, holding the discount factor, nominal rigidity, and strategic complementarities fixed. Estimated at 0.052 (tightly, range 0.044–0.077 across specifications) for Belgian manufacturing.

Calvo probability of price non-adjustment (θ): The parameter from Calvo (1983) staggered price setting capturing the share of firms that cannot change their price in a given period, equal to one minus the per-period probability of price adjustment. In this paper, θ is estimated directly from the dynamic pass-through regression coefficient on lagged prices, yielding θ ≈ 0.711, implying prices fixed approximately three to four quarters on average.

Strategic complementarities parameter (Ω): Defined as Ω = Γ/(1+Γ), where Γ is the elasticity of a firm’s desired markup with respect to its own relative price. Captures the extent to which a firm weights competitors’ prices (rather than its own marginal cost) when resetting its price. High Ω means firms strongly mute price responses to own cost changes to avoid relative price deviations from competitors. Estimated at Ω ≈ 0.570, implying competitor prices and own marginal cost enter the reset price with roughly equal weight.

Dynamic pass-through regression: A forward-looking pricing equation (Model A) relating observed firm prices to the discounted present values of own marginal costs and competitors’ prices, plus lagged own price as an error-correction term. The structural parameters θ and Ω are identified jointly from the regression coefficients, using GMM with instruments for the present values. The dynamic specification is necessary at quarterly frequency because the error-correction term (omitted in static pass-through models) is non-negligible when θ > 0.

Output elasticity of marginal cost (σ_y): The elasticity of firm-level real marginal cost with respect to the firm-level output gap, defined under the assumptions that real wages are flexible and industry-level, equal to σ_y = σ_w + ν (wage elasticity with respect to industry output plus the short-run returns-to-scale parameter). This parameter bridges the cost-based and output-based Phillips curve slopes via κ = λ · σ_y. Estimated from micro data using monetary policy shock instruments at σ_y ≈ 0.112–0.406 in the pre-pandemic period.

Short-run returns to scale (SR-RTS): The extent to which a firm’s marginal cost rises with output scale in the short run, parameterized by ν in the cost function MC^n_ft = C_{it} · A_{ft} · Y_ft^ν. If ν = 0, marginal cost is independent of output scale (constant returns), which the authors assume in their baseline. Firm- and sector-level estimates from Translog production functions yield SR-RTS ≈ 0.93–0.98 across sectors (aggregate ≈ 0.965), broadly consistent with the constant-returns assumption and implying modest macroeconomic complementarities.

Reduced-form aggregate pass-through slope (λ̃): A composite parameter capturing the contemporaneous pass-through of aggregate real marginal cost (defined as nominal marginal cost relative to the lagged price level) into quarterly inflation under the assumption that nominal marginal cost follows a random walk. Evaluated at θ ≈ 0.70 and Ω ≈ 0.52 (median across models), λ̃ = 0.22. This is distinct from the structural NKPC slope λ because it also captures the persistence of cost shocks.

Are Inflationary Shocks Regressive? A Feasible Set Approach

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. The paper asks whether inflationary shocks are regressive, and demonstrates that the answer depends critically on the source of the shock. A single aggregate inflation statistic conceals radically different distributional consequences depending on whether inflation is driven by an oil supply contraction or by expansionary monetary policy.

Framework. The authors develop a “feasible set approach” grounded in the envelope theorem. They show that the first-order money-metric welfare effect of any macroeconomic shock on a household is summarized by the present discounted value of changes to five components of the household’s budget constraint: (1) consumption prices, (2) wage income, (3) asset dividends, (4) asset prices, and (5) government transfers. Because the envelope theorem implies that endogenous substitution responses are not welfare-relevant to a first order, no assumption about the utility function’s form or the economy’s general equilibrium structure is required. The framework is valid for generic stationary shocks that do not directly shift household preferences.

Empirical Strategy. The welfare formula requires two inputs: (i) impulse response functions (IRFs) for all prices, dividends, wages, and unemployment, estimated using internal-instrument SVAR methods applied to two identified shocks — the Kanzig (2021) oil supply news shock (instrumented by oil futures surprises around OPEC announcements) and the Gertler-Karadi (2015) monetary policy shock (instrumented by fed funds futures surprises in 30-minute windows around FOMC announcements) — and (ii) cross-sectional data on consumption bundles, labor income, and asset portfolios from the CEX, CPS, SCF, and SIPP for three education groups (high school or less, some college, college-educated) across the full lifecycle. The baseline cross-section uses 2019 data. Shocks are normalized to produce comparable aggregate inflation responses: a 10% WTI oil price increase and a 25 basis point decline in the one-year Treasury yield each generate roughly 15–16 basis points of CPI-U inflation on impact, rising to approximately 34–35 basis points after two quarters.

Main Findings. Oil supply contractions are regressive and monetary expansions are progressive, and this divergence is primarily driven by the asset price channel, not the consumption price or labor income channels.

For the 10% oil supply shock: middle-aged households with high school education or less must be paid approximately $870 (around 2% of annual consumption) to be made whole relative to their pre-shock utility; college-educated middle-aged households, by contrast, gain the equivalent of approximately $833 (1.1% of annual consumption). Younger college-educated households (still net equity accumulators) gain around $572.

For the 25 basis point monetary rate cut: low-education households approximately break even (net welfare effect near $23), while middle-aged college-educated households must be paid approximately $4,051 (around 5.5% of annual consumption) to restore their pre-shock utility. Older college-educated households must be paid approximately $851.

Why asset prices dominate. Oil supply contractions reduce equity prices (S&P500 falls approximately 2% one year post-shock) and depress dividends (approximately 82 basis points), while leaving house prices and bond prices largely unaffected. Because middle-aged college-educated households are the primary accumulators of equities, they benefit from the price decline (cheaper future accumulation), making oil shocks progressive through this channel — but regressive overall once the consumption and labor income channels (both mildly regressive) are included. Monetary expansions do the opposite: equity prices rise approximately 3 percentage points on impact, house prices rise approximately 1.5% after three years, and dividends increase. These asset price increases hurt those in the accumulation phase — disproportionately middle-aged college-educated households — creating a progressive distributional pattern.

Consumption and labor income channels. Both shocks generate disproportionate inflation in motor fuel and fuel and utilities, and low-education households spend a larger share of their budget on these goods, making the consumption channel mildly regressive for both shocks. The labor income channel differs sharply: oil shocks raise unemployment (approximately 0.15 log points for low-education households two years post-shock) and reduce weekly earnings by 0.2–0.6 log points, mildly harming low-education workers; monetary expansions reduce unemployment (approximately 0.83 log points for low-education workers one year post-shock) and similarly benefit low-education households through the labor market, pushing toward progressivity.

Scope conditions. Results apply to short-run first-order welfare effects of identified stationary macroeconomic shocks (four-year horizon). The framework does not incorporate uncertainty shocks, preference shocks, or the role of hedging motives in portfolio choice. Results concern policy shocks rather than policy rules.

Robustness. Qualitative conclusions hold across six alternative specifications: incorporating borrowing constraints (with or without empirical death rates), adjusting for unemployment insurance replacement rates (approximately 6% true average replacement rate), allowing for log-linear trends in no-shock choices, and dropping aggregate CPI controls from IRF estimation.

In depth

Q1. What is the “feasible set approach” and how does it differ from prior work on inflation incidence?

A: The feasible set approach measures welfare effects through changes in the household’s entire budget constraint — consumption prices, wage income, asset dividends, asset prices, and government transfers — rather than focusing on any single channel. Prior work either examined the Fisher channel (net nominal positions), or consumption price heterogeneity, or labor income responses in isolation. The key insight is that the envelope theorem implies substitution responses are not welfare-relevant to a first order, so the money-metric welfare change is simply the discounted sum of changes in the five budget constraint components evaluated at pre-shock choices, without requiring knowledge of the utility function’s form or the economy’s general equilibrium structure.

Q2. Why is the asset price channel — rather than consumption prices — the dominant channel in both shocks?

A: Asset holdings are large relative to annual consumption (net worth averages $1.5 million for college-educated and $260,000 for high-school-educated households in 2019), so even modest percentage movements in asset prices generate large dollar welfare effects. By contrast, the budget shares on the goods most responsive to both shocks (motor fuel, fuel and utilities) are relatively modest, so the consumption channel, while mildly regressive, is quantitatively small relative to the portfolio channel. The portfolio channel accounts for roughly 0.5% of consumption gains for middle-aged college-educated households under the oil shock, while the consumption channel produces losses of only about 0.1% for college-educated and 0.25% for low-education households.

Q3. How does the direction of the equity price response differ between oil and monetary shocks, and why does this create opposite distributional effects?

A: An oil supply contraction reduces equity prices (approximately 2% decline one year post-shock) and dividends (approximately 82 basis points decline), while a monetary expansion raises equity prices (approximately 3 percentage points on impact, approximately 4% higher after four quarters) and increases dividends. The welfare effect of asset price changes falls on those who trade the asset, not those who merely hold it at a constant level: middle-aged college-educated households are the primary net accumulators of equity, so falling prices benefit them (they can buy more cheaply) while rising prices hurt them. This is the principal reason oil shocks appear progressive through the portfolio channel — but regressive overall — while monetary expansions are regressive through the portfolio channel and progressive overall.

Q4. What are the precise welfare numbers for oil supply shocks by education group (baseline, ages 22–65)?

A: From Table 3 (baseline row, lifecycle-weighted averages for ages 25–65): households with high school or less experience a welfare loss of approximately $798; those with some college experience a loss of approximately $816; and college-educated households experience a welfare gain of approximately $494. These numbers reflect the sum of the consumption, labor income, portfolio, and transfer channels over a 16-quarter horizon, discounted at the one-year Treasury yield.

Q5. What are the precise welfare numbers for monetary policy shocks by education group (baseline, ages 25–65)?

A: From Table 3 (baseline row): households with high school or less experience a small welfare gain of approximately $23; those with some college experience a welfare loss of approximately $1,278; and college-educated households experience a welfare loss of approximately $3,055. These losses for college-educated households are driven overwhelmingly by rising equity and house prices that raise the cost of planned asset accumulation.

Q6. How does the life cycle interact with the distributional incidence of both shocks?

A: There is substantial heterogeneity within education groups across the life cycle because asset accumulation and decumulation patterns are age-dependent. Under oil shocks, younger college-educated households (who are net equity accumulators) gain approximately $572, middle-aged college-educated households gain approximately $833, while older college-educated households lose approximately $69 (because they hold large equity positions and lose dividend income). Under monetary shocks, middle-aged college-educated households lose the most (approximately $4,051) because they are simultaneously accumulating equities and housing, both of which become more expensive. Older college-educated households lose less (approximately $851) because rising dividends on existing holdings partially offset the asset price cost. Low-education households are approximately flat across the life cycle under monetary shocks.

Q7. How does the consumption channel compare across education groups and across the two shocks?

A: The consumption channel is mildly regressive for both shocks, but of similar absolute magnitude across the two shocks because both generate similar inflation in motor fuel and fuel and utilities — the goods with the largest price response. Low-education households spend a larger share on motor fuel and fuel and utilities; as a result, they lose approximately 0.25% of consumption from the consumption channel under the oil shock, compared with less than 0.1% for college-educated households. For monetary shocks, the consumption channel affects all household types roughly equally in proportional terms.

Q8. How does the labor income channel differ between oil and monetary shocks across education groups?

A: Oil shocks raise unemployment disproportionately for low-education workers (approximately 0.15 log point increase after two years, roughly 0.68 standard deviations, compared with near-zero response for college-educated workers) and reduce weekly earnings by 0.2–0.6 log points across groups. Monetary expansions reverse this: a 25 basis point rate cut reduces log unemployment by approximately 0.83 log points for low-education workers and approximately 1.96 log points for college-educated workers after one year, with limited response in conditional wages. Thus the labor income channel pushes toward regressive incidence for oil shocks and toward progressive incidence for monetary expansions, though in both cases it is quantitatively smaller than the portfolio channel.

Q9. What is the role of housing in the portfolio channel?

A: Housing behaves simultaneously as a durable consumption good and a financial asset. A house price increase raises welfare for households planning to decumulate (sell) housing (primarily older households) through the portfolio channel, but also raises the implicit rental cost for those who use housing — a negative consumption-side effect. Monetary expansions raise house prices by approximately 1.5% after three years. College-educated households accumulate housing at a faster rate and earlier in the life cycle than low-education households, making them more exposed to the cost of rising house prices during the accumulation phase. This amplifies the progressive pattern of monetary shocks through the portfolio channel.

Q10. How does the paper handle the dual role of durable goods (vehicles and housing)?

A: Durable goods are treated as both a consumption good and a financial asset. The utility-relevant consumption price of a durable is proportional to the price times the depreciation rate per unit of use, capturing the “implicit rent” of ownership. On the asset side, the durable enters the portfolio channel like a zero-dividend financial asset. This allows the framework to correctly attribute, for example, that a rise in house prices hurts net accumulators (through the portfolio channel) while also raising the implicit cost of housing services (through the consumption channel), rather than treating house price appreciation as an unambiguous welfare gain for homeowners.

Q11. What happens to the main conclusions when borrowing constraints are introduced?

A: Incorporating net worth constraints (with either constant or empirical death rates) dampens the portfolio channel for young and middle-aged college-educated households, because rising asset prices relax borrowing constraints for these households, partially offsetting the welfare cost of more expensive accumulation. Under constant death rates with borrowing constraints, college-educated households’ oil shock welfare gain falls from +$494 to +$76; under empirical death rates, it becomes a loss of -$394. For monetary shocks, the college-educated loss falls from -$3,055 to -$1,718 (constant death rate) or -$1,036 (empirical death rates). Despite these quantitative changes, the qualitative conclusion — oil shocks are regressive, monetary expansions are progressive — holds across all specifications.

Q12. What is the implication of these findings for the policy interaction between oil shocks and monetary tightening?

A: If the monetary authority responds to oil-price-induced inflation with unexpected interest rate increases, it may exacerbate the distributional consequences of the initial oil shock. An oil supply contraction is already regressive (harming low-education households through consumption prices and labor market effects); a disinflationary monetary tightening would additionally harm low-education households through the labor income channel (higher unemployment, lower wages) while partially benefiting college-educated households through lower asset prices. The paper notes this policy interaction as noteworthy, while cautioning that the results concern identified policy shocks rather than policy rules.

Q13. How are the two shocks calibrated to be comparable?

A: The oil shock is normalized to a 10% increase in WTI crude oil prices (approximately one standard deviation of monthly oil price growth). The monetary shock is normalized to a 25 basis point decline in the one-year Treasury yield — chosen because it generates approximately the same aggregate CPI-U inflation response as the oil shock (approximately 15–16 basis points on impact, rising to approximately 34–35 basis points after two quarters). This normalization allows the paper to attribute the different distributional outcomes to the source of inflation rather than to differences in the aggregate inflation magnitude.

Q14. What role does the transfer channel play, and for whom?

A: The transfer channel is small relative to the other three channels for the vast majority of working-age households, because transfer income is less than $100 per month for most households under age 65. Social Security payments — the bulk of transfer income — are explicitly indexed to the CPI; the paper models them as moving with CPI with a one-year lag. The transfer channel exclusively benefits older households (those receiving Social Security), and its quantitative effect is modest even there. Transfer income is more than 20 times smaller than labor and asset income for prime-age households of all education groups.

Key Concepts

Feasible set approach. The paper’s organizing framework, in which the first-order welfare impact of a macroeconomic shock is measured by how the shock changes the household’s budget constraint (consumption prices, wage income, asset dividends, asset prices, and government transfers) evaluated at the household’s pre-shock choices. Substitution responses are not welfare-relevant to a first order by the envelope theorem.

Money-metric welfare gain. The willingness-to-pay measure used throughout: the welfare change from a shock divided by the household’s marginal utility of consumption at time zero, expressed in time-zero dollars. Interpreted as an equivalent variation — the amount the household must be paid or would give up to be indifferent to receiving the shock. Used because it places households with very different utility functions on a common dollar scale.

Portfolio channel. The component of the welfare formula capturing the effect of asset price and dividend changes on household welfare. Asset price changes are welfare-relevant only for households that trade (accumulate or decumulate) the asset: rising prices benefit sellers and harm buyers; falling prices benefit buyers and harm sellers. This is distinct from the “Fisher channel” in prior literature, which focuses on net nominal positions rather than on which households are in the accumulation versus decumulation phase.

Internal instrument SVAR. The time-series estimation procedure used throughout: the pre-estimated identified shock series (oil supply news or monetary policy surprise) is included as a variable ordered first in a recursive structural VAR for each outcome variable. This separates shock identification (using the published instruments and controls from Kanzig 2021 and Gertler-Karadi 2015) from IRF estimation for each outcome variable, allowing the use of the full available sample for each outcome series.

Oil supply news shock (Kanzig 2021). An identified supply shock to oil markets, constructed from changes in oil price futures in tight windows around OPEC production announcements. Used to capture exogenous cost-push inflation driven by supply constraints rather than demand.

Monetary policy shock (Gertler-Karadi 2015). An identified demand-side shock, constructed from federal funds rate futures surprises in 30-minute windows around FOMC announcements, instrumented into a monetary SVAR. Captures exogenous interest rate cuts that generate aggregate demand expansion and inflation.

Borrowing constraint wedge. An additional term that appears in the welfare formula when households face net worth constraints. Proportional to the Lagrange multiplier on the net worth constraint, it discounts future periods more heavily when constraints bind, and adds a term for the welfare value of relaxed constraints when asset prices rise. Identified from deviations from perfect consumption smoothing using CEX lifecycle consumption data.

Artificial intelligence and technological unemployment

Mon, 01 Jan 0001 00:00:00 +0000

Wang and Wong develop a continuous-time labor-search model to assess the dynamic effects of generative AI (GenAI) on labor productivity and unemployment. The paper is motivated by conflicting empirical evidence: micro studies find productivity gains of 14% (Brynjolfsson, Li, and Raymond 2025) and 55.8% faster coding (Peng et al. 2023), while macro estimates suggest modest TFP gains of at most 0.064% annually (Acemoglu 2024), and occupation-level evidence shows a 13% relative employment decline in AI-exposed jobs (Brynjolfsson, Chandar, and Chen 2025).

The model distinguishes GenAI from earlier automation technologies by its learning-by-using mechanism: AI capability grows at rate µ per employed worker (law of motion dAt/At = µHt − δ), raises employed workers’ productivity, and creates a displacement threat through renegotiation. When renegotiation fails, AI replaces the worker, generating technological unemployment. Firms renegotiate wages at a rate ρµAt proportional to AI’s learning rate and the job’s exposure ρ. The joint surplus condition governs whether replacement occurs: AI replaces a worker if and only if πA (AI’s net present value per output) exceeds the post-renegotiation joint surplus St.

The model admits three steady states: (i) a some-AI steady state with finite AI capability, persistent AI adoption (It = 1), expanded job creation but declining employment at H∞ = δ/µ; (ii) an unbounded-AI equilibrium with sustained endogenous growth, no displacement (It = 0), and employment at H∞ = α/(α+σ); and (iii) a no-AI equilibrium reverting to the Mortensen-Pissarides benchmark. In the benchmark model (exogenous job-finding rate, AI-augmented productivity), multiple steady states can coexist—global indeterminacy—when condition (28) holds. In the full model (endogenous job creation via free entry), both global and local indeterminacy are possible, and a continuum of oscillatory transition paths converge to the some-AI steady state.

Calibrated to U.S. data, targeting a pre-AI unemployment rate of 5%, AI elasticity of productivity εy = 1.069 (from Czarnitzki et al. 2023), initial AI productivity boost of 14% (Brynjolfsson et al. 2025), worker exposure ρ = 0.618 (Brynjolfsson et al. 2018’s machine learning suitability index), AI replacement cost ϕ = 0.0043 (from U.S. business GenAI spending), AI learning rate µ = 0.632, and AI error rate δ = 0.462 (Moore’s law half-life of 1.5 years), the model converges to a some-AI steady state. The long-run results are: a 23% employment loss (H∞ = 0.732 vs. H0 = 0.95), AI capability improvement of 321%, and labor productivity gain of 366%. Approximately half of the employment loss—11.5 percentage points—occurs within the first five years, alongside a 49.3% output gain and 45.5% AI capability improvement over that period.

Untargeted moments are validated: the model implies 7.08% labor productivity growth over the first 10 years (consistent with Briggs and Kodnani 2023) and an AI elasticity of vacancies averaging 0.16 over the first five years (consistent with Acemoglu et al. 2022).

On welfare, equilibria are inefficient even when the Hosios condition holds. AI introduces four externalities beyond standard matching frictions: job destruction via displacement, productivity enhancement for employed workers, feedback from AI learning depending on employment, and direct effects on matching surpluses. A constrained-optimal subsidy to jobs at risk of AI displacement is 26.6% in the short run and exceeds 50% in the long run. In the full model, the Hosios condition requires fixing firm bargaining power θ to the vacancy elasticity of matching ξ, but an additional per-output transfer T = µApωA to firm-worker matches is necessary to correct AI adoption inefficiency.

Q: What is the core mechanism by which AI generates unemployment in this model? A: AI capability grows through a learning-by-using process (dAt/At = µHt − δ), improving as it observes employed workers. As capability rises, firms gain a displacement option that arrives at rate ρµAt per matched pair. When renegotiation over wages fails—i.e., when the AI’s NPV πA exceeds the joint surplus—firms replace workers with AI, causing unemployment. This creates a feedback loop: higher employment accelerates AI learning, which increases displacement pressure and reduces employment.

Q: What are the three steady states and what distinguishes them? A: The some-AI steady state features finite AI capability, persistent displacement (It = 1), and long-run employment H∞ = δ/µ; it involves technological unemployment. The unbounded-AI steady state features infinite AI capability, no displacement (It = 0), endogenous productivity growth, and employment H∞ = α/(α+σ) as in the standard Mortensen-Pissarides model. The no-AI steady state has A∞ = 0 with the same H∞ = α/(α+σ) but no AI contribution. Employment is higher in the unbounded-AI equilibrium than in the some-AI equilibrium.

Q: What does the calibration imply for long-run employment and productivity? A: The calibrated full model converges to a some-AI steady state with a 23% employment loss (H∞ = 0.732), a 321% improvement in AI capability, and a 366% gain in labor productivity. The parameters yield a unique equilibrium under the baseline calibration (πA = 1.949 > sAI = 0.8735 confirms some-AI existence). These results reflect a large worker replacement effect under the calibrated AI learning and error rates, while the job creation effect is relatively modest.

Q: How fast does technological unemployment materialize? A: Approximately half of the total 23% employment loss occurs within the first five years; specifically, employment falls by 11.5 percentage points over that period. Over the same five years, AI capability improves by 45.5% and output rises by 49.3%. Over the first 10 years, AI capability improvement accumulates to 94.0% and output gain to 103% (approximately double the five-year output gain).

Q: How does the full model differ from the benchmark model in transition dynamics? A: In the full model, job-finding rates are endogenous: firms post vacancies until a free-entry condition (κyt = ftΠt) is satisfied, tying job-finding rate αt to the surplus ratio st via αt = α(st). This endogeneity implies that as AI raises labor productivity, firms create more vacancies, slowing the employment decline relative to the benchmark model with a fixed job-finding rate. At the same time, AI capability grows faster in the full model because higher employment accelerates AI learning.

Q: What is global indeterminacy and when does it arise? A: Global indeterminacy occurs when both the some-AI and unbounded-AI steady states coexist, so the long-run outcome depends on initial conditions or expectations. In the benchmark model this requires condition (28): 0 < r + σ + α(1−θ) − (1−b)/πA ≤ εy(µα/(α+σ) − δ). In the full model, global indeterminacy is plausible when firm bargaining power rises to θ = 0.95 given the baseline AI replacement cost ϕ = 0.0043. The region of global indeterminacy is larger when firm bargaining power is higher.

Q: What is local indeterminacy and what does it imply for transition paths? A: Local indeterminacy means there is a continuum of equilibrium paths converging to the some-AI steady state in the neighborhood of that steady state, rather than a unique saddle path. In the full model, under alternative parameters (θ = 1, ξ = 0.765, εy = 6), the eigenvalues feature a negative real root and two complex roots with negative real parts, yielding oscillatory local dynamics in employment and AI capability. This implies short-run cycles in productivity and unemployment, consistent with the wide range of empirical findings on AI’s labor-market effects.

Q: Why does the Hosios condition fail to deliver efficiency in this model? A: The Hosios condition eliminates the standard matching externality by setting firm bargaining power to the vacancy elasticity of matching. But AI introduces four additional externalities: (i) job destruction through displacement, (ii) productivity enhancement for employed workers, (iii) feedback from AI learning that depends on aggregate employment, and (iv) direct effects on matching surpluses and job-finding rates. These externalities mean the standard Hosios rule alone is insufficient; additional instruments are required.

Q: What is the constrained-optimal policy response? A: In the simple model, the constrained optimal AI adoption threshold differs from the equilibrium threshold because firm bargaining power θ distorts adoption decisions: AI is over-adopted when πA > (1−b)/(r+σ+α(1−θ)) and under-adopted when (1−b)/(r+σ+α) < πA ≤ (1−b)/(r+σ+α(1−θ)). In the full model, constrained optimality requires setting θ = ξ (Hosios) plus a per-output subsidy T = µApωA to firm-worker matches exposed to AI displacement. This targeted subsidy is 26.6% in the short run and exceeds 50% in the long run.

Q: How does AI compare to computers in this model’s counterfactual? A: The paper reports that exogenous productivity growth from computers reduced unemployment only modestly—by 0.16 percentage points. By contrast, AI’s learning-by-using and displacement features imply a nearly 20% long-run employment loss in a comparable counterfactual. The key distinction is that computers lack the self-learning improvement and associated renegotiation-triggered displacement that characterize GenAI in this model.

Q: How is AI exposure parameterized and what does it capture? A: The exposure parameter ρ captures the degree to which a job is subject to AI-driven replacement risk. It is calibrated using Brynjolfsson et al. (2018)’s suitability for machine learning (SML) index: on a 1–5 scale, SML averages 3.47 across 964 O*NET occupations, translating to (3.47−1)/(5−1) = 61.8%, so ρ = 0.618. The effective exposure measure is ρµ, which is higher when facing a faster-learning AI.

Q: What is the predator-prey analogy in the model’s dynamics? A: The dynamical system for AI capability (At) and employment (Ht) in the simple model resembles the Lotka-Volterra predator-prey system. Employment (prey) feeds AI learning; as AI capability (predator) grows, it displaces workers faster, reducing employment; lower employment then slows AI learning, causing capability to decay; and the cycle repeats with diminishing magnitude until the steady state is reached. This mechanism operates only when the AI learning rate µ is neither too high nor too low, with the convergence path being a spiral when µα < 4δ²(1 − δ(α+σ)/(µα)).

Q: What is the labor-share implication of the unbounded-AI equilibrium? A: In the unbounded-AI steady state, employment is higher than in the some-AI steady state (H^AJJ > H^AI) and labor productivity grows without bound. However, the labor share is lower in the unbounded-AI equilibrium if the firm’s bargaining power θ is sufficiently low. This implies that while workers are not fully displaced and rising AI-augmented productivity sustains employment, workers’ income share may still decline even in the more favorable unbounded scenario.

Technological unemployment: A phenomenon in which AI adoption raises labor productivity and expands job creation, yet still causes sizable employment losses because the worker displacement effect (driven by renegotiation failure when AI’s NPV πA exceeds the joint surplus) dominates the job-creation effect. In the calibrated model this amounts to a 23% employment loss despite a 366% productivity gain.

Learning-by-using AI: The model’s representation of GenAI as a technology whose capability At grows through reinforced learning from employed workers at rate µ per worker, so aggregate AI growth is µHt, offset by deterioration at rate δ. This distinguishes GenAI from earlier automation technologies (computers, robotics) that do not self-improve through usage.

Some-AI steady state: A long-run equilibrium with finite AI capability (gA∞ = 0), persistent AI adoption (It = 1), and employment pinned at H∞ = δ/µ—the ratio of AI’s error rate to its learning rate. Characterized by expanded job creation but lower employment than the no-AI benchmark, constituting the model’s primary calibrated outcome.

Unbounded-AI steady state: A long-run equilibrium with infinite AI capability (A∞ = ∞), no displacement (It = 0), and endogenous growth at rate gA = µH^AJJ − δ. Employment equals the Mortensen-Pissarides level H∞ = α/(α+σ), and labor productivity grows without bound, complementing Aghion, Jones, and Jones (2019)’s idea production framework.

Global indeterminacy: Coexistence of multiple steady states (some-AI and unbounded-AI) such that the long-run equilibrium depends on initial conditions or expectations rather than being uniquely determined. Arises in the benchmark model when condition (28) holds and becomes more likely with higher firm bargaining power θ.

Local indeterminacy: A continuum of equilibrium transition paths converging to a single steady state from nearby initial conditions, rather than a unique saddle path. Arises in the full model under certain parameter configurations (e.g., θ = 1, ξ = 0.765, εy = 6), implying oscillatory short-run dynamics in employment and AI capability.

AI exposure (ρ): A firm-level parameter capturing the degree to which a job-match is subject to AI-driven displacement risk. The displacement option arrives at rate ρµAt per matched pair; ρ is calibrated at 0.618 using the average suitability-for-machine-learning score across O*NET occupations. The effective exposure measure is the product ρµ.

Renegotiation-proof displacement: Proposition 1’s result that the joint surplus Snt is independent of the renegotiation round n, so the AI adoption decision It is also round-invariant. This simplifies the model to a single indicator function: AI replaces the worker if and only if πA exceeds the joint surplus St, regardless of how many renegotiation rounds have occurred.

Auctions with Frictions: Recruitment, Entry, and Limited Commitment

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops an auction model that jointly incorporates three frictions pervading informal price-formation processes: (1) costly recruitment by the seller, (2) costly participation by bidders, and (3) the seller’s inability to commit to a recruitment level or reserve price. The authors argue these frictions are especially prevalent in markets for idiosyncratic assets such as mergers and acquisitions, real estate, and home repair contracting, where auction houses like Christie’s and Sotheby’s command fees of 20–30% of revenues precisely because they reduce the underlying inefficiencies.

The model features a single seller who exerts recruitment effort gamma at cost gamma*s, generating a Poisson-distributed number of contacted bidders with mean gamma. Each contacted bidder independently decides whether to pay entry cost c > 0 to learn their private value and participate in a first-price auction (FPA). The seller cannot commit to gamma (which is unobservable to bidders) or to a reserve price. Two scenarios are analyzed: PO (participation-observable, where bidders observe the number of entrants before bidding) and PU (participation-unobservable).

The central tension is between the seller’s incentive to recruit more bidders to intensify competition and raise revenue, and bidders’ rational concern that excessive recruitment makes entry unprofitable. Because the seller cannot commit, this tension generates several novel inefficiency results.

In the PO scenario, the seller’s marginal revenue from recruitment Ro’(lambda) is single-peaked, meaning there is a minimum profitable participation scale lambda_o below which the seller will never recruit. Combined with a maximum participation level lambda-bar_c above which bidders will not enter (defined by U(lambda-bar_c) = c, where U is the bidder’s expected payoff), no-trade equilibrium is the unique outcome whenever lambda-bar_c < lambda_o — even for arbitrarily small recruitment cost s. This result holds because with unobservable effort, bidders correctly anticipate the seller will target participation above lambda_o, making entry unprofitable. When lambda-bar_c > lambda_o, three regimes arise: (i) no trade if s exceeds a threshold s-bar_o; (ii) an interior equilibrium with full entry (q* = 1) and lambda* = lambda_o(s) for intermediate s; and (iii) for small s, an equilibrium with lambda* = lambda-bar_c and partial entry q* = Ro’(lambda-bar_c)/s < 1. In regime (iii), total recruitment cost lambda*(s/q*) equals the constant lambda-bar_c * Ro’(lambda-bar_c) regardless of s — so even as s approaches zero, wasteful recruitment costs do not vanish, because they are determined by incentive constraints rather than by technology.

In the PU scenario, a no-trade equilibrium always exists for all parameter values, because the seller cannot credibly disclose participation, creating self-reinforcing expectations of zero competition. The seller’s recruitment incentive xi(lambda) is strictly weaker than Ro’(lambda) in the PO scenario (proven via revenue equivalence: Ro’(lambda) = xi(lambda) + a positive term reflecting how greater participation induces more aggressive bidding). This yields ranking reversals: for intermediate s and small c, the PO scenario dominates PU; but for small s or large c, the PU scenario’s weaker recruitment incentive reduces wasteful over-recruitment, making PU preferable. These comparisons translate directly to a comparison of FPA and SPA with unobservable participation: the two formats are not equivalent in the presence of recruitment and entry frictions because they generate different recruitment incentives.

A sampling-curse mechanism drives near-complete market unraveling when sellers have privately known recruitment costs drawn from a continuous uniform distribution on [0, s_o]. Because low-cost sellers recruit more, a contacted bidder believes the seller is more likely to have low costs — and hence to have recruited many other bidders — making entry unprofitable. Proposition 3 establishes a threshold c-hat such that for c in (c-hat, c-bar), as the lower bound of the cost distribution approaches zero, the fraction of seller types that remain inactive approaches one — near-complete unraveling — even though each type would be active if its cost were commonly known.

Q: What is the paper’s main modeling innovation relative to the existing literature? A: The paper’s central novelty is combining all three frictions — costly recruitment by the seller, costly participation by bidders, and limited seller commitment — in one model. The existing literature had studied entry and recruitment separately; Szech (2011) examined costly recruitment with costless entry; McAfee and McMillan (1987) and Levin and Smith (1994) studied costly entry with an exogenously given number of potential bidders; Milgrom (1987) and McAfee and Vincent (1997) studied limited commitment to a reserve price with a fixed bidder set. None combine all three.

Q: What is the “minimum profitable scale” result and why does it arise? A: Because the seller cannot commit to a reserve price, the first few bidders are complementary — they stimulate competitive bidding, causing the seller’s marginal revenue Ro’(lambda) to be initially increasing, then decreasing (single-peaked). This means the seller’s profit Pi_o(lambda, q) is maximized either at zero or at a participation level above a minimum scale lambda_o, defined by Ro’(lambda_o) = s-bar_o. The seller will never choose a participation level between 0 and lambda_o.

Q: Under what conditions does the market completely shut down in the PO scenario? A: No-trade is the unique equilibrium outcome whenever lambda-bar_c < lambda_o, where lambda-bar_c is defined by U(lambda-bar_c) = c (the participation break-even level) and lambda_o is the seller’s minimum profitable scale. This condition arises when entry costs c are large enough relative to the competitive dynamics. Importantly, no trade occurs for every recruitment cost s > 0, including arbitrarily small s — commitment failure alone can cause complete market breakdown even when recruiting bidders is nearly costless.

Q: What is the inefficiency in regime (iii) of Proposition 2 (small s, PO scenario)? A: When s < Ro’(lambda-bar_c), equilibrium has lambda* = lambda-bar_c and q* = Ro’(lambda-bar_c)/s < 1. The total recruitment cost is lambda* * (s/q*) = lambda-bar_c * Ro’(lambda-bar_c), a strictly positive constant independent of s. As s approaches zero, total recruitment effort and its cost do not vanish — they are pinned by incentive constraints. This waste could be avoided if the seller could commit to an effort level below lambda-bar_c, illustrating that commitment failure creates persistent inefficiency even when the technology of recruitment is inexpensive.

Q: Why does a no-trade equilibrium always exist in the PU scenario but not always in the PO scenario? A: In the PU scenario, if bidders expect zero participation, they bid zero conditional on being contacted; the seller then has no incentive to recruit, validating the expectation. This equilibrium is self-sustaining for all parameter values (Claim 2). In the PO scenario, the equilibrium refinement (requiring that off-path beliefs not support negative seller payoff at lambda = 0 when trade equilibria exist) rules out no-trade equilibria when lambda-bar_c > lambda_o and s is not too large; specifically, Proposition 2 shows that no-trade equilibrium is unique only when s > s-bar_o or lambda-bar_c < lambda_o.

Q: What drives the ranking reversal between PO and PU scenarios? A: The core result is Claim 3: Ro’(lambda) > xi(lambda) for all lambda > 0, meaning the marginal incentive to recruit is strictly stronger under PO than PU. This follows from revenue equivalence: Ro’(lambda) = xi(lambda) + (d/d lambda-hat) Ru(lambda, beta_{lambda-hat})|_{lambda-hat=lambda}, and the second term is strictly positive because greater expected participation induces more aggressive bidding. For intermediate s and small c, stronger PO recruitment incentives support higher participation and revenue. For small s or large c, those same stronger incentives generate wasteful over-recruitment in PO, making PU preferable.

Q: How does the paper connect its PO/PU comparison to a comparison of first- and second-price auctions? A: In any standard auction where the highest-value bidder wins, payoff and revenue equivalence imply that the bidder payoff function U(lambda) and seller revenue Ro(lambda) are identical. In particular, the dominant-strategy equilibrium of the SPA (where bidders bid their true values regardless of participation) generates the same outcomes as the PO equilibrium, because with truthful bidding the observability of participation is irrelevant. Therefore, comparing PO and PU with an FPA is equivalent to comparing the SPA and FPA with unobservable participation. The two formats are not revenue-equivalent when recruitment and entry frictions are present: their ranking depends on s and c in exactly the way described for PO vs. PU.

Q: What is the “sampling curse” and how does it cause market unraveling? A: The sampling curse arises when sellers have privately known recruitment costs. Because a lower-cost seller optimally recruits more bidders, the probability of any given bidder being contacted is higher when the seller has a lower cost. Conditional on being contacted, a bidder therefore believes the seller more likely has a low cost and thus has recruited many competitors, reducing the value of entry. In the binary-type case (Claim 8), if sL is sufficiently small relative to sH, the low-cost seller must recruit so many bidders that entry becomes unattractive; the resulting low q* makes the marginal recruitment cost sH/q* prohibitively high for the high-cost type, driving it out (lambda*_H = 0).

Q: What does Proposition 3 establish about near-complete unraveling with a continuum of seller types? A: With seller costs uniformly distributed on [s-bar, s_o], Proposition 3 establishes a threshold c-hat strictly between 0 and c-bar such that: (i) for c in (c-hat, c-bar), as s-bar approaches zero, the fraction of seller types with zero recruitment approaches one — near-complete market unraveling; (ii) for c < c-hat, all seller types remain active regardless of how small s-bar is. This is striking because for any commonly known s in (0, s_o), the PO scenario supports trade for all c < c-bar; unraveling arises purely from the interaction of private cost information and the sampling curse, not from any type’s cost being intrinsically too high.

Q: What does the welfare analysis say about equilibrium efficiency? A: The welfare-maximizing participation level lambda_w satisfies U(lambda_w) = c + s (equating the marginal bidder’s surplus to the full social cost of one more participant), with full entry q_w = 1. In equilibrium under PO, q* < 1 in some cases (wasted recruitment) and lambda* differs from lambda_w for almost all (s, c) pairs — both excessive participation (lambda* > lambda_w) and deficient participation (lambda* < lambda_w) can arise. Full efficiency requires Ro’(lambda*) = s and U(lambda*) = s + c simultaneously, but since both U and Ro’ are independent of s and c as parameters, these equalities generically fail.

Q: Does the seller benefit from being able to commit to recruitment effort? A: Claim 10 shows that with observable effort in the PO scenario, the seller commits to gamma-hat = min{lambda-bar_c, lambda_o(s)} when lambda-bar_c >= lambda_o, and to lambda-bar_c (if profitable) when lambda-bar_c < lambda_o. Commitment strictly improves the seller’s profit whenever gamma-hat = lambda-bar_c: it enables positive trade when lambda-bar_c < lambda_o and Ro(lambda-bar_c) > lambda-bar_c * s (otherwise impossible without commitment), and it saves recruitment costs when lambda-bar_c > lambda_o and Ro’(lambda-bar_c) > s. However, the commitment outcome is always welfare-inefficient: lambda-bar_c > lambda_w whenever s > 0.

Q: What anecdotal evidence do the authors cite for the model’s relevance? A: Subramanian (2010) and Boone and Mulherin (2004, 2009) show that the majority of merger and acquisition auctions are “informal” — mixtures of auctions and negotiations rather than structured processes with rules laid out in advance — and that sellers are typically unable to credibly commit to participation levels. Milgrom (2003) states from consulting experience that marketing an auction is often more critical than clever mechanism design. Fees of 20–30% of revenues paid to intermediaries like Christie’s and Sotheby’s are offered as quantitative evidence of the magnitude of the inefficiencies that such intermediaries reduce. Home repair contracting is cited as a familiar informal-auction setting where both recruitment and entry costs are material.

Recruitment effort (gamma): The seller’s costly action of contacting potential bidders, modeled as a Poisson process with mean gamma at cost gamma*s; unobservable to bidders in the baseline model.

Participation-observable (PO) vs. participation-unobservable (PU) scenarios: The two variants of the model; in PO, bidders observe the total number of entrants n before bidding; in PU, they do not observe n and the seller cannot credibly disclose it.

Minimum profitable scale (lambda_o): The smallest positive participation level the seller will ever choose in equilibrium, defined as the value where Ro’(lambda_o) equals the peak of the average revenue curve s-bar_o. The seller always recruits either zero bidders or at least lambda_o, due to the initial complementarity of bidders (they stimulate each other’s bids) under no-commitment-to-reserve-price.

Break-even participation level (lambda-bar_c): The maximum participation level at which a bidder’s expected gross payoff U(lambda) equals the entry cost c; bidders will not enter if they expect participation above lambda-bar_c.

Sampling curse: The adverse-selection mechanism arising when sellers have privately known recruitment costs: because low-cost sellers recruit more, a contacted bidder infers the seller is more likely to have a low cost and thus to have recruited many competitors, making entry less attractive and potentially driving higher-cost seller types out of the market.

xi(lambda): The seller’s marginal revenue with respect to recruitment in the PU scenario, defined as the total derivative of Ru(lambda, beta_{lambda-hat}) evaluated where actual and expected participation coincide (lambda-hat = lambda). Strictly less than Ro’(lambda) for all lambda > 0, reflecting that in PU the seller loses the ability to leverage bidder aggression via observable competition.

Wasteful recruitment: The equilibrium phenomenon in which total recruitment cost lambda*(s/q*) remains at the positive constant lambda-bar_c * Ro’(lambda-bar_c) even as s approaches zero, because incentive constraints — not technology — pin the equilibrium effort level.

Automated credit limit increases and consumer welfare

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. Should regulators restrict banks from proactively raising credit card limits using machine-learning algorithms, and if so, how? The paper asks: to what extent are bank-initiated credit limit increases directed toward revolving borrowers (those who carry interest-accruing balances month-to-month), and what are the welfare consequences of policies that constrain such increases?

Data. The empirical analysis uses the Federal Reserve’s Capital Assessments and Stress Testing (Y-14M) regulatory data, January 2014 to December 2024, covering monthly account-level records for all credit cards issued by large stress-tested banks (assets > $100B). The 26 banks in the sample collectively represent more than 70% of U.S. credit card balances. A 0.5% sample yields more than 150 million observations across more than 3.6 million unique active credit cards. A key advantage of Y-14 over credit bureau data is that it identifies whether each limit change was bank-initiated or consumer-initiated — a distinction not available in other datasets.

Stylized Facts. Credit limit increases are an important and understudied source of consumer credit. During the post-pandemic period, limit increases generate more than $40 billion of additional available credit per quarter, roughly 60% of the approximately $70 billion coming from new card originations; prior to the pandemic the figure was about $30 billion, or roughly half of new issuance. The number of accounts undergoing a limit increase each quarter is on average 30% higher than the number of new cards issued. Consistent with “low-and-grow” lending strategies, limit increases are disproportionately important for lower credit-score borrowers: average subprime credit limits rise from $700 at origination to $2,700 by five years after origination (a 285% increase) and to nearly $5,000 by eight years, while average superprime limits rise only from approximately $12,000 to $15,000 (a 25% increase). About 30% of total revolving balances are made possible by limit increases, with the share reaching 60% for subprime borrowers but only 12% for superprime borrowers. Approximately 75–80% of all limit increases — both by dollar amount and by number of cards — are bank-initiated rather than consumer-initiated. Banks that more frequently reference “artificial intelligence” or “machine learning” in their 10-K filings support a larger share of revolving balances through limit increases. Bank-initiated increases are roughly 1.5–2 times more prevalent among accounts that have revolved in the prior three months, whereas consumer-initiated increases show essentially no differential by revolving status.

Empirical Analysis. Using a linear probability model with card-portfolio-group fixed effects, month fixed effects, and controls for credit score, income, prior limit changes, and other account characteristics, the authors show that the probability of a bank-initiated limit increase follows an inverse-U shape in revolving utilization: accounts with revolving utilization in the moderate range (roughly 0.2–0.7) are most likely to receive an increase, while those near zero or near 1.0 are not. An account with revolving utilization in the (0.2, 0.3] bin is approximately as likely to receive a limit increase as an account whose credit score just rose by 66 points. Transacting utilization, by contrast, follows a logistic growth pattern: the probability rises monotonically until about a utilization of 0.3 and is flat above that. An event study shows that after a bank-initiated limit increase, revolving utilization rebounds to its pre-increase level within approximately 8 months; on average, revolving balances increase by about 40% of the limit increase, with approximately 30% of the limit increase going toward revolving balances. This rebound occurs even for accounts with revolving utilization below the pre-increase mean of 0.28, indicating that the effect is not confined to liquidity-constrained borrowers.

Model. The authors develop a life-cycle consumption–saving model with credit card borrowing, uninsurable income and employment risk, potential default (Chapter 7 style), and heterogeneous preferences following Nakajima (2017) and Gul–Pesendorfer (2001, 2004). Two household types coexist: 60% with standard exponential-discounting preferences (calibrated β = 0.92) and 40% with temptation preferences (calibrated β = 0.96, temptation parameter λ = 0.28 from Kovacs et al., 2021). The credit limit increase function is calibrated using Y-14M data via a latent-variable formulation, replicating the empirical inverted-U relationship between revolving utilization and limit increase probability. The four internally calibrated targets are: share of households with revolving credit card debt (data: 45%, model: 41.8%); utilization rate conditional on debt (data: 35%, model: 28.9%); default probability (data: 0.94%, model: 0.94%); debt-to-income ratio (data: 8.6%, model: 6.8%).

Main Findings — Baseline. Through the model, tempted agents are disproportionately likely to receive credit limit increases because they are more likely to revolve. For customers with utilization above 50%, the majority of credit limit increases are detrimental from the borrower’s own perspective. Standard agents almost always benefit from higher credit limits.

Counterfactual 1 — UK-style (prohibit limit increases for revolving borrowers). This policy reduces the annual probability of limit increases from roughly 5.5% to approximately 1.0%. The default probability falls from about 0.9% to near zero. The debt-to-income ratio declines by roughly 2 percentage points. Aggregate welfare improves by 1.12% in consumption equivalent variation (CEV) when the social planner internalizes the psychological cost of temptation (0.98% without). Standard households incur a modest welfare loss of 0.21% from reduced consumption-smoothing flexibility, while tempted households gain approximately 3.12% in CEV.

Counterfactual 2 — Canada/EU-style (require consumer consent). This policy reduces the annual limit-increase probability from 5.5% to approximately 1.9%. Aggregate welfare improves by 1.16% in CEV (1.04% without psychological costs). Standard households lose 0.19%, while tempted households gain approximately 3.19%. Under the baseline assumption of sophisticated tempted households, results are nearly identical to the UK-style policy. However, when the fraction of naïve tempted households is large, the consent-based policy becomes ineffective (naïve consumers accept limit increases they will regret), whereas the UK-style revolving-borrower ban remains welfare-improving regardless of the naïve share.

Robustness. When the firm is allowed to re-optimize its credit limit increase policy, it endogenously reallocates more limit increases toward standard consumers. Welfare gains remain positive but are attenuated: the UK-style policy yields 0.21% CEV (vs. 1.12% in the baseline calibration) and the consent-based policy yields 0.27% CEV.

Policy Implications. The U.S. lacks regulation of bank-initiated proactive credit limit increases (existing rules under ECOA and ability-to-pay provisions are largely non-binding for this purpose). The authors conclude that banks’ revealed preference for targeting revolvers constitutes an implicit targeting of consumers with self-control issues, and that if a meaningful share of households have self-control issues, there are strong consumer protection grounds for regulating algorithmic credit limit increases.

In depth

Q1. Why do the authors use Y-14M data rather than credit bureau data, and what does this data uniquely enable?

A: The Y-14M dataset allows the authors to distinguish between bank-initiated and consumer-initiated credit limit changes — a distinction not observable in credit bureau data. It also contains actual payment information enabling identification of revolvers (those carrying interest-accruing balances) rather than just total balances. The sample covers more than 70% of U.S. credit card balances and more than 150 million monthly observations over the January 2014 to December 2024 period.

Q2. How large are credit limit increases relative to new card originations in the U.S. credit card market?

A: During the post-pandemic period, limit increases produce more than $40 billion of additional available credit per quarter, roughly 60% of the approximately $70 billion created by new card originations. Prior to the pandemic the figure was approximately $30 billion, or about half of new issuance. On a count basis, the number of cards undergoing a limit increase each quarter is on average 30% higher than the number of new cards issued.

Q3. What is the “low-and-grow” strategy, and how large is the subsequent credit expansion?

A: The low-and-grow strategy involves originating higher-risk borrowers at low initial credit limits and then expanding limits based on observed borrowing behavior. For the average subprime credit card, the initial limit of $700 grows to $2,700 by five years after origination (a 285% increase) and to nearly $5,000 by eight years. For superprime borrowers, the initial limit of approximately $12,000 grows only to $15,000 (a 25% increase) by five years and then is approximately unchanged.

Q4. How does a borrower’s revolving status affect the probability of receiving a bank-initiated limit increase?

A: Bank-initiated increases are approximately 1.5–2 times more prevalent among accounts that have revolved at least once in the prior three months, compared to non-revolving accounts. By contrast, consumer-initiated increases show essentially no differential between revolvers and non-revolvers. This reveals a bank-side revealed preference for targeting revolvers.

Q5. What is the shape of the relationship between revolving utilization and the probability of a bank-initiated limit increase, and how large is its economic magnitude?

A: The relationship follows an inverted-U shape. Accounts with revolving utilization in bins between approximately 0.2 and 0.7 have the highest probability of receiving an increase; accounts near zero or near full utilization are as unlikely to receive an increase as zero-utilization accounts. The effect of being in the (0.2, 0.3] revolving utilization bin has approximately the same positive effect on the probability of receiving a limit increase as a 66-point increase in credit score, making it economically large relative to standard risk signals.

Q6. How does transacting utilization relate to bank-initiated limit increases, and how does this differ from revolving utilization?

A: Transacting utilization follows a logistic growth pattern rather than an inverted-U. The probability of receiving a limit increase rises monotonically with transacting utilization until about a utilization of 0.3, above which the probability does not vary with utilization. This contrasts with revolving utilization, where very high utilization (above 0.9) is actually no more predictive than zero utilization.

Q7. What does the event study show about borrowing behavior following credit limit increases?

A: After a bank-initiated limit increase, revolving utilization (as a share of the credit limit) drops mechanically but then rebounds to pre-increase levels within approximately 8 months. On average, revolving balances increase by about 40% of the amount of the limit increase, with approximately 30% of each dollar of new credit limit going toward revolving balances. These magnitudes are somewhat larger than the 13% (Gross and Souleles, 2002) and 18% (Aydin, 2022) found in prior work, which the authors attribute to the non-causal nature of their event study, higher average utilization in their sample, and their focus on revolving rather than total utilization.

Q8. Is the post-increase borrowing rebound driven by liquidity-constrained borrowers?

A: No. The authors show that limiting the sample to accounts with revolving utilization below the pre-increase mean of 0.28 — accounts that are unlikely to be liquidity constrained — yields very similar results. This finding is consistent with the presence of self-control issues rather than binding credit constraints.

A: The model features two types: 60% with standard exponential-discounting preferences (estimated discount factor β = 0.92) and 40% with temptation preferences (β = 0.96, temptation parameter λ = 0.28 set from Kovacs et al., 2021). The 40% tempted share is internally estimated via the Method of Simulated Moments targeting four aggregate moments: share with revolving credit card debt (45% in data, 41.8% in model), utilization rate conditional on debt (35% vs. 28.9%), default probability (0.94% vs. 0.94%), and debt-to-income ratio (8.6% vs. 6.8%).

Q10. How do tempted and standard households differ in their credit card usage within the model?

A: In the model, 76% of tempted agents carry revolving credit card debt, with an average utilization rate of 73.6%, a debt-to-income ratio of 15.4%, and a default probability of 2.22%. Standard agents carry debt only 18.9% of the time, with average utilization of 4.1%, a debt-to-income ratio of 1.1%, and a default probability of 0.08%. Tempted agents also pay a substantially higher share of income on credit card interest.

Q11. How does the model capture the mechanism by which credit limit increases harm tempted households?

A: The Gul–Pesendorfer temptation utility function makes household welfare depend on both actual consumption and the most tempting consumption alternative available (the budget-set maximum). When credit limits rise, the most tempting alternative ˜c_t increases, which raises the utility cost of self-restraint even for households that do not succumb to temptation. This mechanism is distinct from hyperbolic discounting: temptation imposes a psychic cost even on those who ultimately choose not to over-borrow.

Q12. What are the quantitative welfare effects of the UK-style policy prohibiting limit increases for revolving borrowers?

A: The policy yields an overall welfare gain of 1.12% in consumption equivalent variation (CEV) when the social planner internalizes the psychological cost of temptation (0.98% without). Standard households suffer a modest welfare loss of 0.21% from reduced consumption-smoothing flexibility. Tempted households gain approximately 3.12% in CEV, because the benefit from reduced temptation and lower interest expenditure outweighs the cost of reduced credit access.

A: The consent-based policy yields an overall welfare gain of 1.16% in CEV (1.04% without psychological costs). Standard households lose 0.19%, and tempted households gain approximately 3.19%. Under the baseline assumption of fully sophisticated tempted households, results are nearly identical to the UK-style ban.

A: The UK-style ban on limit increases for revolving borrowers remains welfare-improving regardless of whether tempted households are sophisticated or naïve — the welfare impact is approximately flat as the naïve fraction rises from zero to one. The consent-based policy, by contrast, exhibits a negative linear relationship between the naïve fraction and welfare impact, with welfare gains disappearing as the naïve fraction approaches one. Naïve consumers accept limit increases they would regret, so the policy’s effectiveness depends on households accurately recognizing their own self-control issues.

Q15. What happens when the firm is allowed to re-optimize its credit limit increase policy in response to regulation?

A: With firm re-optimization, both counterfactual policies continue to improve welfare but the magnitudes are attenuated. The UK-style policy yields 0.21% CEV overall (tempted: 0.89%) and the consent-based policy yields 0.27% overall (tempted: 0.98%), compared to 1.12% and 1.16% without re-optimization. The re-optimizing firm reallocates more limit increases toward standard consumers, which reduces the number directed at tempted households but also limits the welfare gains from regulation.

Q16. What do lenders’ 10-K filings reveal about the role of AI/ML in targeting revolvers for limit increases?

A: Banks that mention “artificial intelligence” or “machine learning” above the median number of times in their 2024 10-K filings support a higher share of revolving balances through credit limit increases, for all credit score groups. This difference is not driven by differences in credit limits at origination between higher-AI and lower-AI lenders, suggesting that AI/ML adoption affects the targeting of limit increases toward revolvers rather than the initial credit allocation.

Key Concepts

Revolving utilization. In this paper, revolving utilization is defined as the portion of overall credit card utilization attributable to balances that the borrower carries from one month to the next without full repayment, thereby accruing interest. It is measured as revolving balances divided by credit limit, averaged over the prior three months. This is distinct from transacting utilization (new purchases as a share of limit) and is the primary signal banks use — implicitly, via their algorithms — to select accounts for proactive limit increases.

Bank-initiated vs. consumer-initiated credit limit increase. A bank-initiated limit increase is one in which the lender proactively raises a borrower’s credit limit without a request from the borrower. A consumer-initiated increase is one explicitly requested by the borrower. The Y-14M data uniquely identify the source of each change. The paper documents that approximately 75–80% of all limit increases are bank-initiated, and that bank-initiated increases are strongly correlated with revolving utilization whereas consumer-initiated increases are not.

Low-and-grow strategy. The practice of originating higher-risk borrowers at low initial credit limits and then expanding those limits over time based on observed borrowing behavior. In the paper this is a documented empirical pattern, not an assumption: subprime accounts start at an average $700 limit at origination and reach nearly $5,000 by eight years, a 285% increase versus only 25% for superprime accounts over the same horizon.

Temptation preferences (Gul–Pesendorfer). A utility framework in which household welfare depends not only on actual consumption but also on the most tempting consumption alternative within the budget set. The disutility from temptation arises even when the household does not succumb — it reflects the psychological cost of self-restraint. In the paper, λ (set to 0.28) parameterizes the weight of this temptation cost relative to standard utility. Temptation preferences are time-consistent, which facilitates welfare analysis, and are preferred to hyperbolic discounting in this setting because they predict that individuals may pay to have tempting options removed even without acting on them.

Revealed preference for targeting revolvers. The paper’s characterization of banks’ credit limit increase behavior as reflecting a systematic preference for giving increases to revolving borrowers, inferred from the empirical pattern in the Y-14M data (the inverted-U shape between revolving utilization and limit increase probability). Because banks’ algorithms are proprietary and unobserved, the paper interprets the observed allocation of limit increases as a revealed preference, consistent with banks’ profit motive since revolvers generate the majority of credit card interest income.

Consumption equivalent variation (CEV). The welfare metric used throughout the paper’s counterfactual analysis. CEV is defined as the percentage change in consumption in every period and state that would make households indifferent between the baseline policy regime and the counterfactual policy. A positive CEV indicates that the counterfactual policy improves welfare; a negative CEV indicates harm. The paper considers two versions: one in which the social planner internalizes the psychological cost of temptation (consistent with tempted households’ actual preferences), and one in which the planner ignores that cost (λ = 0 for the planner) but households still face temptation.

Persistent revolving debt (UK regulatory definition). In the UK Financial Conduct Authority’s framework, a borrower is considered in “persistent revolving debt” when the cumulative amount paid toward interest and fees exceeds the cumulative amount of principal repaid over a 12-month period. The UK rule prohibits lenders from increasing credit limits for borrowers meeting this definition. The paper models a stylized version: any account currently carrying a revolving balance is ineligible for a bank-initiated limit increase in the UK-style counterfactual.

Automation and Rent Dissipation

Mon, 01 Jan 0001 00:00:00 +0000

Acemoglu and Restrepo examine the effects of automation in economies where labor market distortions cause some workers to earn rents—wages above their opportunity cost or outside option. The central question is how the interplay between automation and these distortions shapes wages, inequality, and productivity. The paper makes three contributions: a theoretical framework identifying a rent dissipation mechanism, reduced-form empirical evidence using US data from 1980 to 2016, and a general equilibrium quantification of automation’s aggregate effects.

The theoretical framework extends the task model of Acemoglu and Restrepo (2022) to incorporate task-specific wage wedges. In this setup, a firm employing labor of type g in task x pays a wage equal to the base wage multiplied by an exogenous wedge capturing rents from efficiency wages, bargaining, licensing, regulations, or norms. Because these wedges artificially inflate labor costs in high-rent tasks, firms have a stronger incentive to automate precisely those tasks—automation saves more in labor costs where rents are highest. Proposition 3 establishes that endogenous adoption decisions are tilted toward high-rent tasks: the rent distribution in automated tasks first-order stochastically dominates the rent distribution across all tasks. This targeting generates the rent dissipation mechanism. The equilibrium is inefficient on both the intensive margin (too little employment in high-rent tasks) and the extensive margin (excessive automation of high-rent tasks that a social planner would prefer to keep labor-intensive).

The rent dissipation mechanism has three consequences identified theoretically. First, it amplifies average wage losses for exposed groups beyond what displacement alone would produce, pushing displaced workers toward lower-paying jobs. Second, it compresses within-group wage dispersion by concentrating losses at higher percentiles of the within-group distribution, generating a U-shaped pattern of wage changes: workers at low percentiles earn no rents and experience only base-wage adjustments, while workers between the 70th and 95th percentiles face the steepest declines due to loss of high-rent jobs. Third, it is inefficient: because the tasks targeted by automation are not those where wages reflect scarcity or skill but rather distortionary rents, a planner would have preferred more labor allocated to these tasks, and rent dissipation offsets part or all of the cost-saving productivity gains from automation.

The empirical analysis covers 500 detailed demographic groups defined by education (five levels), gender, five age groups, five race/ethnicity groups, and nativity. Task displacement is measured as a weighted sum of industry-level automation exposure using three proxies: adjusted industrial robot penetration, specialized software services, and dedicated machinery in value added. Workers in the middle and lower-middle of the wage distribution lost 15–20% of their tasks to automation between 1980 and 2016, while post-college workers saw few tasks automated.

A 10 percentage point increase in task displacement is associated with a 24% decline in group-level relative wages (β = −2.36, s.e. = 0.13), falling to 19% after controlling for gender, education, sectoral demand, and rent shifters (β = −1.90, s.e. = 0.29). The U-shaped pattern in within-group wage changes is clearly visible: wages decline by 25–30% per 10 percentage point task displacement at the 70th–90th percentiles, compared to only 16% at the 5th–40th percentiles. Decomposing the average wage effect, the base-wage component is β = −1.53 (s.e. = 0.33) and the rent-dissipation component is β = −0.37 (s.e. = 0.11), implying a rent dissipation rate of approximately 37%. Across multiple proxies for rents—inter-industry/occupation wage differentials, wage losses after job displacement, and quit rates—the average estimated rent dissipation rate is approximately 35%. Rent dissipation accounts for one-fifth of the overall relative wage decline experienced by groups exposed to automation.

In the general equilibrium quantification (with elasticity of substitution λ = 0.5, average cost savings π = 30%, and average rent in automated tasks of 35%), automation accounts for 52% of the rise in between-group wage inequality since 1980: 42 percentage points via baseline displacement effects on labor demand, and 10 percentage points via rent dissipation. Cost savings from automation increased TFP by approximately 3% between 1980 and 2016, but inefficient rent dissipation offsets 60–90% of these gains, leaving net TFP gains of only 0.3–1.3% and net aggregate consumption gains of only 0.45–1.95% over the 36-year period.

Q: What is the rent dissipation mechanism, and why does it arise? A: Rent dissipation arises because labor market wedges make high-rent tasks artificially costly to staff with workers, giving firms a stronger incentive to automate precisely those tasks. When automation displaces workers from high-rent jobs, workers lose the premium above their opportunity cost that those jobs paid, amplifying wage losses beyond what displacement alone would cause. The mechanism is endogenous: firms do not randomly automate tasks but disproportionately target tasks where rents are highest, since doing so saves the most in labor costs. Proposition 3 formalizes this as first-order stochastic dominance of the rent distribution in automated tasks over the rent distribution in all tasks.

Q: Why is rent dissipation inefficient? A: In a distorted economy, high-rent tasks already feature too little employment at the equilibrium—firms under-hire in these tasks because the wage wedge makes labor artificially expensive. A social planner would want to allocate more labor to these tasks, not less. When automation further removes labor from high-rent tasks, it moves the economy further from the efficient allocation, dissipating rents that reflect distortions rather than true scarcity. The TFP formula shows that this inefficient targeting offsets part or all of the cost-saving gains from automation, and can even reduce aggregate productivity if the cost savings are small relative to the rent losses.

Q: What is the U-shaped pattern of within-group wage changes, and what does it indicate? A: The U-shaped pattern means that wage declines due to automation are smallest at the bottom percentiles of a group’s within-group wage distribution, largest in the 70th–95th percentile range, and then smaller again at the very top. Workers at low percentiles earn no rents, so they experience only the base-wage adjustment from reduced labor demand. Workers in the middle-upper range of the distribution hold the high-rent jobs that are disproportionately automated, so they lose both the base-wage component and the rent component of their wages. This pattern is directly visible in US data 1980–2016, with declines of 25–30% per 10 percentage point task displacement at the 70th–90th percentiles versus 16% at the 5th–40th percentiles.

Q: How is task displacement measured, and which groups are most exposed? A: Task displacement is measured as a weighted sum of industry-level automation exposure, accounting for each demographic group’s specialization in routine tasks within industries. Three proxies are used: the adjusted penetration of industrial robots, the increase in specialized software services, and the increase in dedicated machinery in value added. Workers in the middle and lower-middle of the wage distribution—broadly corresponding to non-college workers—lost 15–20% of their tasks to automation between 1980 and 2016. Post-college degree workers saw few tasks automated.

Q: How large is the rent dissipation rate, and how robust is this estimate? A: The baseline estimate from the U-shaped within-group wage change decomposition implies a rent dissipation rate (μ_Ag/μ_g − 1) of approximately 37% (β = −0.37, s.e. = 0.11). Using inter-industry and occupation wage differentials as a proxy for rents, the estimate is 39% (β = −0.39, s.e. = 0.11). Using wage losses after job displacement, the estimate is 20% (β = −0.20, s.e. = 0.04). After purging compensating differentials from the wage differential proxy the estimate remains 37%; after purging from the displacement-loss proxy it falls to 19%. Quit-rate evidence is consistent with rent dissipation: automation shifts workers toward higher-quit-rate jobs, which are lower-rent jobs. The average across proxies is approximately 35%.

Q: How much of between-group wage inequality since 1980 does automation explain, and what share is due to rent dissipation specifically? A: Automation accounts for 52% of the rise in between-group wage inequality in the US since 1980. Of this 52 percentage points, 42 percentage points are attributable to the baseline displacement effect working through reduced labor demand for exposed groups. The remaining 10 percentage points are attributable to rent dissipation—automation pushing exposed groups away from high-rent tasks into lower-paying employment. Rent dissipation thus accounts for roughly one-fifth (10/52) of automation’s total contribution to between-group inequality.

Q: How large are the productivity gains from automation, and how much does rent dissipation offset them? A: Cost savings from automation increased TFP by approximately 3% between 1980 and 2016. However, inefficient rent dissipation offsets 60–90% of these gains, because automation disproportionately targets high-rent tasks rather than tasks where the efficiency case is strongest. The net TFP increase attributable to automation is only 0.3–1.3% over the 36-year period, and the corresponding net increase in aggregate consumption is only 0.45–1.95%.

Q: How does automation affect within-group versus between-group inequality, and why is this notable? A: Automation increases between-group inequality by reducing relative wages of exposed groups (largely non-college workers) relative to unexposed groups, accounting for 52% of the rise in between-group inequality since 1980. At the same time, automation reduces within-group wage dispersion for exposed groups by compressing wages at higher percentiles. This contrasts with the standard view that inequality is fractal—rising at all levels of aggregation due to skill-biased demand—and helps explain why within-group inequality has risen steadily for college workers since the 1980s while remaining flat and then declining for non-college workers since the 1990s.

Q: What do the propagation matrix and rent-impact matrix represent in the general equilibrium analysis? A: The propagation matrix encodes how task reallocation due to automation in one demographic group creates competition for marginal tasks across other groups, transmitting the wage effects of automation to groups not directly displaced. The rent-impact matrix encodes how this task reallocation changes the rent composition of employment across groups. Both matrices are estimated from US data on task shares and group-level wage elasticities and are used to translate partial-equilibrium estimates of task displacement and rent dissipation into general equilibrium effects on wages and productivity for all demographic groups simultaneously.

Q: What are the policy implications of inefficient rent dissipation? A: Because rent dissipation is inefficient, the social value of automation is lower than what firms and consumers are willing to pay—firms capture all the labor cost savings but do not internalize the welfare cost of destroying high-rent jobs that the distorted equilibrium already under-supplies. Second-best interventions should address the underlying distortions generating rents rather than trying to slow automation directly. The paper suggests that strengthening labor market institutions supporting worker rents in non-automatable tasks could partially counteract the adverse distributional consequences of automation.

Q: How does this paper relate to Bound and Johnson (1992) and Borjas and Ramey (1995)? A: Bound and Johnson (1992) decompose changes in the US wage structure between 1979 and 1988 into technology, supply, and rent components (modeled as exogenous industry wedges), finding that 10–20% of between-group wage changes reflect rent losses. Borjas and Ramey (1995) estimate that trade increased the college premium by 1.3–2.6 log points between 1976 and 1990, with 15–33% due to loss of rents from trade-exposed jobs. Both are comparable to this paper’s finding that rent dissipation accounts for one-fifth of the wage effect of automation, though Bound and Johnson’s estimates include all factors affecting rents while this paper isolates automation specifically.

Worker rents: Wages above a worker’s opportunity cost or outside option, arising from efficiency wages, bargaining, licensing, regulations, or norms. Modeled as task-specific multiplicative wedges (μ_gx ≥ 1) that force firms to pay more than the base wage for labor in particular tasks. Explicitly excludes compensating differentials and skill premia.

Rent dissipation: The loss of above-opportunity-cost wages experienced by workers displaced from high-rent tasks into lower-paying employment. Occurs because automation endogenously targets high-rent tasks where labor is most expensive, and pushes workers into tasks where rents are lower. Quantified as the ratio of average rents in automated tasks to average rents across all tasks, minus one (approximately 35% in US data 1980–2016).

Task displacement: The share of tasks performed by a demographic group that are automated away, measured as a weighted sum of industry-level automation exposure accounting for the group’s specialization in routine tasks. Distinct from employment loss because it captures reallocation of tasks from labor to capital within the production function.

U-shaped within-group wage change profile: The pattern whereby automation generates the largest wage declines at intermediate-to-upper percentiles (70th–95th) of an exposed group’s within-group wage distribution, with smaller declines at the bottom, because high-percentile workers disproportionately hold high-rent jobs targeted by automation. Predicted theoretically and confirmed empirically in US data 1980–2016.

Propagation matrix: A matrix estimated from US data on task shares and group-level wage elasticities that encodes how automation of tasks performed by one demographic group creates competition for marginal tasks with other groups, transmitting wage effects across the demographic distribution in general equilibrium.

Inefficient automation targeting: The mechanism by which labor market distortions cause firms to automate high-rent tasks that a social planner would prefer to keep labor-intensive, since the distorted equilibrium already features too little employment in those tasks. Results in rent dissipation offsetting 60–90% of automation’s direct TFP gains from cost savings.

Rent-impact matrix: A matrix that encodes how task reallocation due to automation changes the rent composition of employment across demographic groups, used alongside the propagation matrix to compute general equilibrium effects of automation on wages and productivity accounting for distortions.

Bank Information Production Over the Business Cycle

Mon, 01 Jan 0001 00:00:00 +0000

Bank Information Production Over the Business Cycle

Research Question

Banks produce private information about borrowers that is inherently unobservable to outside researchers. Howes and Weitzner ask whether the quality of this private information is countercyclical — that is, whether banks invest more in learning about borrowers when local economic conditions deteriorate — and whether any such cyclicality reflects endogenous information production incentives rather than exogenous changes in the information environment.

Data and Methodology

The paper uses the Federal Reserve’s Y-14Q Schedule H.1 confidential regulatory data, which covers commercial and industrial (C&I) loans exceeding $1 million originated by bank holding companies with $50 billion or more in total assets. This universe covers 85.9% of all banking sector assets and approximately 70% of all C&I loan volume (as documented by Bidder, Krainer, and Shapiro (2020)). A distinctive feature is that qualifying banks must report their internal probability of default (PD) estimates for each loan to the Federal Reserve. The sample is restricted to newly originated loans from 2014Q4 through 2019Q1 — the window over which PD data are well populated — with at least one year of subsequent observation to allow defaults to materialize. The outcome variable is a binary default indicator equal to one if the borrower defaults within two years of origination (0.41% of firms in the sample).

The measure of information quality is defined as the OLS coefficient on PD when regressing realized default on the bank’s internal PD estimate. A larger coefficient indicates that the bank’s private risk assessment carries more predictive content for realized default outcomes, above and beyond observable firm and loan characteristics. The authors identify cyclical effects by exploiting cross-sectional variation in county-level unemployment rates across the US at each point in time, controlling for bank-by-quarter fixed effects (to absorb supply-side bank-level factors), industry-by-quarter fixed effects, and bank-by-county fixed effects. The key interaction is between PD and the local unemployment rate.

Main Findings

The paper establishes three main results:

Banks’ PDs predict default and contain private information. Even after controlling for firm size, leverage, profitability, tangibility, log loan size, loan maturity, loss given default (LGD), loan type fixed effects, bank-quarter fixed effects, and industry-quarter fixed effects, PD remains a statistically and economically significant predictor of realized default. A one-percentage-point increase in PD increases the probability of default by approximately 25 basis points (coefficient of 0.245).
Information quality is countercyclical. A one-percentage-point increase in the local county unemployment rate increases the sensitivity of realized default to PD by approximately 8 basis points — roughly one-third of the average unconditional PD coefficient. When the unemployment rate is above a county’s median, the PD coefficient is approximately three times as large as during low-unemployment periods. Correspondingly, during high-unemployment periods, the total R-squared of a regression predicting default from observable firm and loan characteristics falls (from 0.311 to 0.264 — an 18% decline), while the marginal contribution of PD to the R-squared increases. This pattern is consistent with observable characteristics doing a worse job at predicting default in bad times, which in turn incentivizes banks to invest more in their internal risk assessments.
The cyclicality is driven by newly originated loans and more information-sensitive loans. The triple interaction between PD, the new-loan indicator, and the unemployment rate is positive and statistically significant across all specifications; the interaction between PD and unemployment for previously issued (non-new) loans is consistently less than half the size of the triple interaction term. The cyclical sensitivity also decreases by more than 0.1 (against a base of 0.08) in the year after origination and continues to fall over the loan’s life. Additionally, a one-standard-deviation increase in log loan size (approximately 1.29) increases the sensitivity of realized default to PD by about 0.085 — roughly one-quarter of the unconditional effect — and a one-standard-deviation increase in LGD (0.158) increases the PD coefficient by 0.098, or about one-third of the unconditional effect. Both the loan-size and LGD interactions are amplified when the local unemployment rate is high, consistent with Dang, Gorton, and Holmstrom (2012). The cyclical sensitivity of information quality is statistically significant only for firms in nontradeable industries (e.g., utilities, construction, retail, professional services), not for tradeable-sector firms.

Scope Conditions

Results are conditional on: large US bank holding companies ($50bn+ in assets) lending to non-financial, non-public domestic corporate borrowers with at least $100k in reported assets; a sample period from 2014Q4 to 2019Q1, covering a predominantly expansionary phase of the US business cycle; and county-level rather than aggregate time-series variation in economic conditions.

Policy Implications

Countercyclical information production implies that bank lending stimulus policies — including interest rate cuts, liquidity facilities, and asset purchase programs — may be less effective in recessions because banks simultaneously increase screening intensity. The marginal borrowers who gain access to credit from stimulus will differ across states of the cycle: in downturns, banks grant credit to fewer but higher-quality firms, so the incremental impact of expanding the credit supply on the number and type of firms funded may be attenuated. The authors connect this mechanism to prior empirical evidence that monetary policy is less effective in recessions (Tenreyro and Thwaites (2016)) and to LTRO and QE program evidence showing no increase in lending to riskier firms.

In depth

Q1. What is the precise definition of “bank information quality” used in this paper, and why is this measure preferred over alternatives?

Information quality is defined as the OLS coefficient β on the bank’s internal PD estimate when predicting realized two-year default in a regression that also includes firm and loan characteristics and a rich set of fixed effects. A higher coefficient indicates that the bank’s private risk assessment contains more predictive content for actual default beyond what is captured by observable firm and loan characteristics. This approach is preferred because it directly quantifies the marginal information content of the bank’s private assessment and can be estimated at the loan level using the cross-sectional variation in county-level economic conditions, rather than relying on aggregate time-series variation that would confound bank supply-side factors.

Q2. How do the authors establish that the PD estimates contain genuine private information rather than merely reflecting publicly observable characteristics?

Column (1) of Table 3 shows a PD coefficient of 0.245 in a regression predicting default without controls. Columns (2) and (3) add firm and loan characteristics (size, leverage, profitability, tangibility, log loan size, maturity, LGD, and loan type fixed effects) plus bank-quarter, industry-quarter, and bank-county fixed effects, and also add the interest rate as an additional control; the PD coefficient remains statistically and economically significant across all specifications. This demonstrates that PD retains predictive power for realized default even after absorbing all variation captured by observable firm-level fundamentals and pricing signals, implying the PD estimate contains private information not contained in observables.

Q3. What is the baseline magnitude of the cyclicality finding, and how is it identified?

A one-percentage-point increase in the county-level unemployment rate increases the PD coefficient by approximately 8 basis points (Table 5, Column 1). This represents about one-third of the average unconditional PD coefficient estimated in Section 3.1. Identification uses bank-by-quarter fixed effects so that the effect is estimated by comparing two loans made by the same bank at the same time to borrowers in counties with different unemployment rates, ruling out bank-level supply-side confounders such as changes in a bank’s cost of capital or risk appetite.

Q4. How does the split-sample analysis (above/below county-median unemployment) further characterize the cyclicality?

Columns (3) and (4) of Table 4 show that, when predicting default with PD alone (no controls), the PD coefficient is approximately three times as large during high-unemployment periods as during low-unemployment periods, and the R-squared is substantially higher for high-unemployment observations. The R-squared from a regression of default on observable controls alone is 17.8% higher when unemployment is low (0.311 versus 0.264), while the marginal contribution of PD to the R-squared is higher when unemployment is high (going from 0.264 to 0.267, versus 0.311 to 0.313). This pattern — observables explain less but PD explains more in bad times — is consistent with information frictions being more severe in downturns, which in turn raises banks’ incentives to invest in private information production.

Q5. How do the authors distinguish endogenous information production from a purely exogenous improvement in information quality during downturns?

Three tests are designed to be difficult to rationalize under a purely exogenous information channel. First, the cyclicality is concentrated in newly originated loans: the triple interaction term (PD × unemployment × new-loan indicator) is positive and statistically significant, while the PD × unemployment interaction for previously originated loans is less than half the size of the triple interaction. If information quality improved exogenously during downturns, there is no clear reason why this improvement would be far larger for loans where the bank is making a new capital commitment. Second, the cyclicality declines by more than 0.1 (relative to a base of 0.08) in the year after origination and continues to fall — simultaneously, the unconditional predictive power of PD increases over the loan life. This divergence is inconsistent with a purely exogenous mechanism. Third, the cyclical sensitivity is concentrated in loans that theory (Dang, Gorton, and Holmstrom (2012)) predicts to have higher information production incentives: larger loans, higher-LGD loans, and loans to nontradeable-sector borrowers.

Q6. How do loan characteristics (size and LGD) relate to information quality, and how does this relationship evolve over the business cycle?

Table 7 shows that a one-standard-deviation increase in log loan size (approximately 1.29) increases the sensitivity of realized default to PD by about 0.085, or roughly one-quarter of the unconditional PD coefficient. A one-standard-deviation increase in LGD (0.158) increases the PD coefficient by 0.098, or about one-third of the unconditional effect. Table 8 shows that both of these interaction coefficients have the same sign and are amplified during periods of high unemployment, consistent with Dang, Gorton, and Holmstrom (2012)’s prediction that information production decisions become more sensitive to loan features following negative aggregate shocks.

Q7. What does the tradeable versus nontradeable industry test contribute?

Because nontradeable-sector firms (utilities, construction, retail, transportation, accommodation, food services, information and communication, professional services) are more likely to depend on local demand, the same change in the county-level unemployment rate will have a larger impact on their default probability. Table 9 shows that the cyclical sensitivity of PD’s predictive power — the PD × unemployment interaction — is statistically significant only for nontradeable-sector firms, not for firms in tradeable industries. This provides additional evidence that the mechanism operates through local economic conditions affecting borrower riskiness in a way that raises information production incentives, rather than through some aggregate or bank-level mechanism.

Q8. Do composition effects (changes in the pool of borrowers) account for the main findings?

Table 11 shows that observable loan characteristics — average loan size, interest rate, LGD, and maturity — do not vary meaningfully with the local unemployment rate. Realized default rates increase slightly with unemployment but the effect is not statistically significant. The PD itself increases by only about 3 basis points for a one-percentage-point increase in unemployment (significant only at the 10% level). Loan volume declines: a one-standard-deviation increase in the unemployment rate (1.3 percentage points) leads to a 1.6% decrease in loan volume and a 5.46% decrease in the number of loans. The minimal variation in the risk profile of loans actually granted suggests that composition effects in the pool of approved borrowers are unlikely to explain the main result.

Q9. What are the implications of countercyclical information production for monetary policy transmission?

When unemployment is high, banks screen potential borrowers more intensively, which changes the composition of firms that gain access to credit. Policies designed to expand credit supply — interest rate cuts, liquidity facilities, asset purchase programs — face a more heavily screened pool of potential recipients during downturns. This means the marginal firms that receive additional credit following a stimulus in a recession will be of higher quality than the marginal recipients in an expansion, implying the credit transmission of monetary policy reaches a different — and potentially smaller — set of firms in recessions. The authors connect this to Tenreyro and Thwaites (2016)’s finding that monetary policy is less effective in recessions, and to evidence from the Eurosystem’s LTRO program that aggregate lending rose but lending to riskier firms did not, and to UK QE evidence finding no stimulation of bank lending.

Becker, Bos, and Roszbach (2020) also find that bank credit ratings predict default better in bad economic times, using data from a single Swedish bank and relying on aggregate time-series variation. The present paper differs in three ways. First, it uses cross-sectional variation across US counties within each time period, exploiting bank-by-quarter fixed effects to rule out bank supply-side confounders. Second, it uses loan-level rather than firm-level data, enabling the analysis of how loan characteristics (size and LGD) interact with information quality and cyclicality. Third, Becker, Bos, and Roszbach interpret the cyclicality as exogenous; Howes and Weitzner provide evidence against this interpretation — specifically, the concentration in newly originated loans and in loans with characteristics that theoretical models predict should generate higher endogenous information production.

Key Concepts

Bank Information Quality (as used in this paper) The size of the OLS coefficient on a bank’s internal probability of default (PD) estimate in a regression predicting realized loan default. A larger coefficient means the bank’s private risk assessment carries more predictive content for actual default beyond observable firm and loan characteristics. It is a measure of how much private information the PD encodes about borrower risk, not a measure of accuracy in an absolute sense.

Probability of Default (PD) — Y-14Q Internal Estimate Banks’ own model-based estimate of each corporate borrower’s likelihood of defaulting, reported confidentially to the Federal Reserve under Y-14Q Schedule H.1 filings. In the paper, PD is used as the observable proxy for the bank’s private risk assessment; its predictive power for realized default is the object being studied, not the PD level itself.

Countercyclical Information Production The property that banks’ incentives to invest in learning about borrower quality increase as economic conditions deteriorate. In the theoretical literature the paper tests empirically, the returns to distinguishing between borrower types rise in downturns (because the distribution of borrower quality widens and the consequences of adverse selection increase), inducing banks to produce more private information at loan origination. The paper uses “information quality is countercyclical” to mean that the predictive content of PD for realized default is higher when the local unemployment rate is higher.

Information Sensitivity (of a loan) The degree to which the value of a loan depends on information that is privately held by potential borrowers. Following Dang, Gorton, and Holmstrom (2012), loans are more information-sensitive when they are larger (larger potential loss from adverse selection) or when they have higher loss given default (lower expected recovery value). The paper uses loan size and LGD as proxies for information sensitivity and tests whether banks invest more in information about higher-information-sensitivity loans.

Loss Given Default (LGD) The bank’s estimate of the fraction of the loan’s value that would be lost if the borrower defaults, reflecting the expected recovery value of collateral and other loan features. In the paper, higher LGD (lower recovery) is a proxy for higher information sensitivity, since the consequences of lending to a bad borrower are larger when recovery is low.

Bank-by-Quarter Fixed Effects A set of fixed effects that absorbs all variation in outcomes attributable to a particular bank at a particular point in time. In the context of this paper, including bank-by-quarter fixed effects means the cyclicality results are identified from variation across counties for loans made by the same bank in the same quarter, ruling out supply-side explanations such as changes in a bank’s cost of capital, risk appetite, or credit standards that affect all of its loans uniformly.

Endogenous versus Exogenous Information Quality A core distinction in the paper. Exogenous information quality would mean banks passively receive more precise signals about borrowers during downturns regardless of their investment in screening. Endogenous information quality means banks actively choose to invest more in information production during downturns because the returns to distinguishing borrower types are higher. The paper argues its results — especially the concentration of cyclical effects in newly originated loans and in loans with characteristics that theory predicts should generate higher screening incentives — are consistent with the endogenous channel and are difficult to rationalize under a purely exogenous mechanism.

Barriers to Global Capital Allocation

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. Why do observed international investment positions and cross-country differences in rates of return to capital fail to conform to a frictionless capital-market benchmark? The paper asks how large the efficiency and distributional costs of barriers to global capital allocation are, and which frictions — capital income taxes, political risk, and geographic/cultural/linguistic distances — matter most.

Model. The authors develop a multi-country dynamic spatial general equilibrium model in which the entire network of bilateral cross-border investment positions is endogenously determined. Production in each country i follows a three-factor Cobb-Douglas function in reproducible capital, labor, and natural resources, with country-varying income shares. Capital is the only mobile factor. A logit asset demand system governs portfolio shares: the share of country j’s savings invested in country i is proportional to the risk-adjusted expected return on capital in i, scaled by the capital stock of i, and inversely proportional to a bilateral portfolio wedge ∆ij. These wedges can be microfounded via either rational inattention (where wedges reflect the precision of prior beliefs about returns) or extreme-value-distributed transaction costs. The model admits multiple microfoundations but yields the same functional form and the same counterfactual welfare calculations regardless of interpretation.

Frictions measured. Three categories of frictions enter the empirical implementation: (a) bilateral capital income tax rates — a new dataset covering 225 countries (50,625 country pairs), constructed from corporate income tax rates and treaty-adjusted withholding tax rates on dividends and interest, further adjusted for effective tax rates accounting for tax-haven routing; (b) political risk, proxied by an ICRG composite index (excluding socioeconomic conditions) following Alfaro, Kalemli-Ozcan, and Volosovych (2008); (c) geo-political distance, comprising geographic distance, cultural distance (based on 496 World Values Survey questions across 116 countries), and linguistic distance (based on a language-family tree covering 6,737 languages and 242 countries). These distance measures are publicly available at geopoliticaldistance.org. The model covers 96 countries (9,216 dyads), representing 92% of world GDP in 2017.

Gravity Estimation. Bilateral investment data (restated for tax havens using the nationality-basis methodology of Coppola et al. 2020 and Damgaard et al. 2019) are regressed on cultural, geographic, and linguistic distance with origin and destination fixed effects. In OLS, a one-standard-deviation increase in cultural distance (0.023 units) is associated with a 24.0% decrease in foreign assets; geographic distance (0.977 units in logs) with a 78.6% decrease; linguistic distance (0.174 units) with a 51.5% decrease. These magnitudes are robust across OLS, PPML, and IV (using religious distance as an instrument for cultural distance). Under IV, the standardized effect of cultural distance on log foreign assets rises to −76.5%.

Tax haven analysis. A Tobit regression of the share of bilateral investment routed through tax havens on the estimated tax saving from routing through havens yields coefficients of 0.413–0.999 for equity and 1.001–1.777 for debt (across specifications with varying fixed effects), confirming that tax incentives are a primary driver of the discrepancy between residency-based and nationality-based bilateral positions.

Model fit (untargeted moments). The calibrated baseline model produces: (i) a correlation of 0.658 between model-implied and empirical rates of return to capital (vs. 0.325 for the frictionless benchmark), with a standard deviation of 0.417 (vs. 0.091 frictionless; data: 0.496); (ii) a correlation of 0.947 between model-implied and empirical capital per employee (vs. 0.918 frictionless); (iii) a correlation of 0.94 between model-implied and empirical home bias; the model reproduces the mean home bias of 3.973 vs. 4.006 in data and standard deviation of 1.065 vs. 1.224, while the frictionless benchmark produces exactly zero home bias for all countries. Portfolio-share MSE: 1.16 (baseline) vs. 1.86 (frictionless).

Counterfactual findings. Removing all measured barriers raises world GDP by 6.8% relative to the observed equilibrium (equivalent to stating that the distorted equilibrium is 6.8% below the frictionless benchmark). Geo-political distance alone accounts for most of this: when only distance frictions are retained, world GDP is 5.2% below the frictionless level. Capital taxes alone reduce world GDP by 2.6% below frictionless; political risk alone by 0.4%. The standard deviation of log capital per employee is 51.5% higher than it would be without barriers; the standard deviation of log output per employee is 22.5% higher. In the frictionless equilibrium, capital flows from rich to poor countries (the correlation between net foreign assets and development doubles in absolute value), accounting for the Lucas (1990) puzzle. In short-term (one-period) counterfactuals holding wealth fixed, the GDP gain from full barrier removal is 3.6%; the inequality effect remains similar (standard deviation of log capital per employee 48.4% higher with barriers).

Scope conditions. The model focuses on steady-state outcomes; dynamic transition effects are analyzed in extensions but are smaller. Quantitative conclusions are conditioned on: (i) the model sample of 96 countries covering 92% of world GDP in 2017; (ii) the conservative OLS coefficient estimates used for baseline calibration (IV estimates are larger and would amplify results); (iii) the assumption that the logit demand system captures frictions regardless of their microfoundation; (iv) omission of goods-trade frictions from the baseline (when included, the world GDP effect falls to 3.7% and the capital inequality effect to 23.3%).

In depth

Q1. What is the core theoretical prediction about cross-country rates of return when investment barriers exist?

A: In the model’s frictionless benchmark (Propositions 1 and 2), all origin countries hold identical portfolios and risk-adjusted expected returns are equalized across destinations. When bilateral frictions are introduced, countries that are more “peripheral” (harder to access for foreign investors due to high geo-political distance or political risk) receive less inward capital and therefore command higher physical rates of return to capital. Countries that are easily accessible (“central”) attract more capital and exhibit lower rates of return. The Dual Efficiency Theorem establishes that capital is efficiently allocated if and only if marginal products of capital are equalized across countries, which requires that taxes are uniform and that portfolio wedges satisfy a specific cancellation condition.

Q2. How are portfolio wedges measured, and what is the identifying strategy?

A: Portfolio wedges ∆ij are decomposed into a geo-political distance component and a political risk component. The geo-political distance component is specified as a log-linear function of geographic distance, cultural distance, and linguistic distance, with coefficients (β_g, β_c, β_l) estimated from a gravity regression of log bilateral investment on these distances, controlling for origin and destination fixed effects. Because political risk varies only by destination country, it cannot be separately identified from destination fixed effects in the bilateral regression; its elasticity is therefore taken from Alfaro, Kalemli-Ozcan, and Volosovych (2008). The key identification advantage of bilateral data is that origin and destination fixed effects absorb all country-level confounders, so the distance coefficients are identified purely from within-origin, within-destination variation across country pairs.

Q3. What do the OLS gravity regressions find, and are the coefficients stable across specifications?

A: In the baseline OLS specification (Table 2, column 1), the estimated coefficients on cultural distance, geographic distance, and linguistic distance are −11.944, −1.579, and −4.162 respectively (all significant at the 1% level). In standardized terms, a one-standard-deviation increase in cultural distance reduces foreign assets by 24.0%, geographic distance by 78.6%, and linguistic distance by 51.5%. Adding a rich set of control variables (colonial ties, legal origin, currency pegs, trade agreements, effective tax rates) leaves these magnitudes broadly similar: standardized effects on foreign assets are −26.4%, −80.1%, and −47.6%, respectively. Results are also robust across OLS and PPML specifications and across years 2013–2017. Effects are quantitatively similar for foreign equity and foreign debt, though linguistic distance has a somewhat smaller effect on debt.

Q4. How does the instrumental variable strategy address reverse causality in cultural distance, and what does it find?

A: The authors instrument cultural distance with religious distance (based on historical trees of religious affiliation), assuming religious history affects international investment only through its contemporary effect on differences in values and beliefs as captured by the World Values Survey. The instrument is a strong predictor of cultural distance (passes weak-instrument tests comfortably). Under IV, the standardized effect of a one-standard-deviation increase in cultural distance on log foreign assets rises from −24.0% (OLS) to −76.5% (IV). The authors use conservative OLS estimates for their baseline calibration, so the IV results imply the headline counterfactual effects are likely understated.

Q5. How does the model predict home bias, and how well does it match the data?

A: Home bias is defined as the log difference between the domestic portfolio share and the country’s share in the world capital stock. In the frictionless model, Proposition 1 implies that all countries hold identical foreign portfolios, so the model produces exactly zero home bias for every country. The baseline model, by incorporating bilateral frictions, generates home bias endogenously without targeting it. The model-implied home bias correlates with the empirically measured home bias at 0.94 across countries and matches both the mean (3.973 model vs. 4.006 data) and standard deviation (1.065 vs. 1.224) closely. The model also predicts, consistent with Lau, Ng, and Zhang (2010), that home bias and rates of return on capital are positively correlated (model-implied ρ = 0.55), and that rates of return on capital correlate negatively with the log of GDP per employee (model-implied ρ = −0.70).

Q6. What is the quantitative decomposition of the world GDP loss by type of barrier?

A: World GDP in the observed (distorted) equilibrium is measured at $112.9 trillion (PPP), which is 6.8% below the frictionless counterfactual. When all barriers are present except geo-political distance, world GDP is 5.2% below frictionless — meaning distance frictions account for the largest share. When all barriers are present except political risk, world GDP is only 0.4% below frictionless. When all barriers are present except taxes, world GDP is 2.6% below frictionless. These are not exactly additive because the distortions interact; the results confirm that geo-political distance (cultural, linguistic, and geographic) constitutes the dominant source of global capital misallocation among the three measured frictions.

Q7. How do barriers affect the cross-country distribution of capital and income?

A: The standard deviation of log capital per employee is 51.5% higher in the distorted equilibrium than in the frictionless counterfactual; the standard deviation of log output per employee is 22.5% higher. When only geo-political distance distortions are maintained, dispersion in log capital per employee is 38.2% higher and in log output per employee 15.9% higher. Maintaining only taxes raises the dispersion in log capital per employee by 12.9% and log output per employee by 6.0%; maintaining only political risk raises them by 7.3% and 3.8%, respectively. In the frictionless equilibrium, the poorest countries gain the most: some of the poorest countries see capital per employee increase by an order of magnitude and income per employee double.

Q8. Does the model account for the Lucas puzzle (capital not flowing from rich to poor countries)?

A: Yes. In the observed distorted equilibrium, net foreign asset positions correlate only weakly with the level of development, consistent with Lucas’s (1990) observation that capital fails to flow from rich to poor countries. In the frictionless counterfactual, the absolute value of the correlation between net foreign asset positions and log GDP per employee doubles, and capital indeed flows from rich to poor countries as neoclassical theory predicts. The distortions from taxes, political risk, and geo-political distance thus account for the absence of a strong correlation between net positions and development in the data.

Q9. How do extensions incorporating goods-trade frictions, capital controls, and currency hedging costs affect the headline findings?

A: Adding goods-trade frictions (country-specific prices for output and capital installation following Monge-Naranjo et al. 2019) reduces the world GDP effect to 3.7% (from 6.8% baseline) and the dispersion of log capital per employee to 23.3% higher (from 51.5%), but the overall pattern of results is preserved. Replacing political risk with capital controls (using Jahan and Wang 2016 de-jure capital account openness) yields a comparable world GDP loss of 6.6% and a geo-political distance effect of 6.2%, very close to the 6.8% and 5.2% in the baseline. Adding currency hedging costs leaves world GDP loss and inequality effects essentially unchanged relative to baseline. None of these extensions materially alters the headline conclusions.

Q10. How do the authors validate the model against nationality-based versus residency-based bilateral investment data?

A: The model is calibrated to nationality-based positions (restated for tax havens). The MSE for fitting nationality-based external portfolio shares is 1.16, while the MSE for residency-based positions is 1.22. The model was not explicitly designed to distinguish between the two, yet it naturally produces better predictions for nationality-based positions because its frictions incorporate the incentives for indirect investment routing through tax havens. This cross-validation supports the methodological approach of using nationality-restated data and confirms the internal consistency of the model’s treatment of tax-haven routing.

Q11. What are the implications for global tax policy coordination?

A: In the presence of information frictions, simple harmonization of capital tax rates across countries does not improve capital allocation efficiency and could worsen it. The Dual Efficiency Theorem implies that efficient capital allocation in a world with information frictions requires that taxes, risk premia, and information frictions satisfy a joint cancellation condition. From a normative perspective, a global social planner maximizing world GDP should impose lower capital tax rates in countries that are “peripheral” in the network of informational distances, in order to offset the disadvantage created by information frictions for those countries.

Q12. How is the elasticity parameter η calibrated, and how sensitive are the results?

A: The elasticity of substitution among countries’ assets, η, is calibrated at 18.5 based on Koijen and Yogo (2020)’s demand-price elasticities for long-term debt (3.1, converted to a gross-return elasticity of approximately 30), short-term debt (25.2, converted to approximately 24.3), and equity (1.3, converted to approximately 14.8), with weights reflecting the composition of global portfolios. The baseline gravity coefficients are calibrated from OLS with controls (cultural: −13.129, geographic: −1.645, linguistic: −3.850), chosen as conservative estimates relative to IV or PPML. Sensitivity analysis using PPML or IV estimates of β yields broadly similar steady-state GDP losses (around 6%), confirming robustness.

Key Concepts

Portfolio wedge (∆ij): A bilateral distortionary term in the logit asset demand system that captures all frictions reducing the ability of investors from country j to invest in country i. Decomposed empirically into a geo-political distance component and a political risk component. A wedge of 1 means no friction; larger values reduce the share of investment flowing from j to i. Can be interpreted either as prior-belief imprecision under rational inattention or as systematic transaction costs under the extreme-value microfoundation.

Geo-political distance: A composite of geographic distance (population-weighted geodesic distance), cultural distance (expected disagreement in World Values Survey responses between randomly drawn individuals from two countries, constructed with the “flex” method using up to 496 questions), and linguistic distance (normalized tree distance in the Ethnologue language family graph, covering 6,737 languages). Distinct from simple physical distance: it captures the informational and transactional barriers that arise from societal dissimilarity.

Dual Efficiency Theorem: A theoretical result (Theorem in Section 2.8) establishing that capital efficient allocation, equalization of marginal products of capital across countries, and uniform taxes combined with a specific cancellation condition on portfolio wedges are mutually equivalent statements in steady-state equilibrium. This is not a restatement of the First Welfare Theorem; it is a statement about GDP (not welfare) and does not require risk premia to be equalized.

Effective bilateral tax rate (τij): The composite bilateral tax rate on capital after accounting for tax-haven routing. Firms in the destination country optimally choose the share of capital issued through tax havens (solving a quadratic cost optimization), trading off the lower tax rate available through havens against an increasing quadratic routing cost. The effective rate is therefore lower than the statutory (de jure) rate when the tax-haven rate is lower than the statutory rate, with the gap depending on the estimated βth coefficient from the Tobit regressions.

Logit asset demand system: A portfolio allocation rule in which the share of country j’s savings invested in destination country i is proportional to the risk-adjusted expected return raised to the power η (the elasticity of substitution) times the destination capital stock, divided by the portfolio wedge and summed over all destinations. Microfounded either by rational inattention (Matejka and McKay 2015; Pellegrino 2023) or by extreme-value-distributed transaction costs. Produces portfolio gravity analogous to trade gravity when combined with the market clearing conditions.

Home bias: Defined as the log difference between a country’s domestic portfolio share (πii, the share of domestic savings invested at home) and that country’s share of world capital stock (ki/K). In the frictionless benchmark, home bias is exactly zero for all countries by Proposition 1. The baseline model generates home bias endogenously as a consequence of portfolio wedges and reproduces both the level and cross-sectional distribution of empirically observed home bias without targeting these moments directly.

Core-periphery structure: An emergent property of international capital markets under investment barriers: countries that are easily accessible to international investors (low geo-political distance, low political risk, favorable tax treatment) are “central” and attract capital inflows, driving their rates of return to capital lower; “peripheral” countries that are less accessible have smaller capital stocks and higher rates of return, compensating investors for overcoming barriers. This structure generates persistent capital misallocation and cross-country income inequality.

Nationality-based vs. residency-based bilateral investment positions: Residency-based data (e.g., raw IMF CPIS) attributes investment to the immediate counterparty country, including tax-haven shell companies. Nationality-based data (Coppola et al. 2020; Damgaard et al. 2019; Beck et al. 2024) reattributes investment to the country of the ultimate investor and ultimate issuer, bypassing offshore centers. The model fits nationality-based positions better (MSE 1.16 vs. 1.22 for residency-based) because it incorporates frictions that generate incentives for indirect routing, which is what nationality restatement is designed to undo.

Biased expectations and labor market outcomes: Evidence from German survey data and implications for the East–West wage gap

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research question. The paper asks two questions: (1) How do workers’ biased expectations about job finding and job separation shape the labor market equilibrium and wages? (2) Are differences in expectation biases across workers a quantitatively important driver of wage differentials, specifically the East–West German wage gap?

Data. The empirical analysis uses the German Socio-Economic Panel (SOEP), a nationally representative longitudinal survey of approximately 30,000 participants per wave. The working-age sample (ages 25–65) covers nine biennial survey waves from 1999 to 2015, yielding 67,772 observations for job separation expectations and 6,423 for job finding expectations. Perceived transition probabilities are reported on a 0–100 scale in steps of 10 percentage points. Actual (statistical) transition probabilities are constructed by estimating probit models that predict realized transitions within 24 months using a rich set of individual, job, and employer characteristics, and are rounded to the nearest decile for consistency with the survey scale.

Main empirical findings. Employed workers in Germany overestimate their job separation probability by 6.4 percentage points on average (perceived: 19.8%; actual: 13.3%), a pessimistic bias significant at the 1% level. Unemployed workers overestimate their job finding probability by 8.2 percentage points on average (perceived: 57.0%; actual: 48.8%), an optimistic bias also significant at the 1% level. The East–West divergence is striking. East German workers exhibit a pessimistic job separation bias of 12.1 percentage points, compared to only 4.7 percentage points in the West, despite broadly similar actual separation rates (15.1% vs. 12.8%). For job finding, West Germans overestimate their probability by 12.9 percentage points, while East Germans overestimate by only 2.0 percentage points — meaning East Germans are also substantially less optimistic about re-employment. These East–West differences survive controls for compositional differences and alternative definitions of job separation (dismissals only; selected reasons; spell-based) and job finding (including those out of the labor force). The biases are stable over the 1999–2015 sample period with no discernible trend. A cohort analysis shows that the excess pessimism in East Germany is concentrated among cohorts who were already in the labor market at the time of German reunification (born in the 1950s and 1960s), consistent with persistent effects of the communist GDR experience. Individuals do not systematically learn over time: mean changes in individual-level absolute deviations between consecutive waves are close to zero. Individual deviations between perceived and actual rates have statistically significant but quantitatively negligible predictive power for subsequent transitions (a 1 pp higher perceived job separation is associated with only a 0.001 pp higher realized separation rate), ruling out private information as a first-order explanation for the biases.

Model. The authors extend the Diamond–Mortensen–Pissarides (DMP) frictional labor market framework by (i) allowing workers to hold biased perceived transition rates (λw for job finding, σw for job separation) while firms have rational expectations, and (ii) introducing wage contracts of explicit length T periods after which parties re-bargain. Common knowledge of each party’s perceived values is assumed, and generalized Nash bargaining is applied. The contract length T is a key parameter: there exists a critical threshold T* such that a pessimistic job separation bias raises the equilibrium wage for T < T* (the continuation-value effect dominates) and lowers it for T ≥ T* (the within-contract discounting effect dominates). An optimistic job finding bias unambiguously raises the equilibrium wage by inflating the perceived value of unemployment and hence the reservation wage.

Quantitative results. The model is calibrated to East Germany. The job separation bias (∆σ = 0.0194) and job finding bias (∆λ = 0.0044) are set to SOEP-based estimates. The critical threshold implied by calibrated parameter values is T* = 10 quarters. The baseline contract length, constructed from the share of permanent (88%) and temporary (12%) contracts in SOEP and average remaining tenure until retirement, is T = 67 quarters (a lower bound). This exceeds T*, so the pessimistic separation bias depresses wages in the baseline. A counterfactual experiment assigns West German bias levels to East German workers, while holding all other parameters fixed. For the preferred calibration range (γ ∈ {0.35, 0.50}, T ∈ {67, 106, 159}), East German wages rise by 1.07 to 2.36 percent. This corresponds to a reduction in the conditional East–West German wage gap (23 percent) of 4.6 to 10.6 percent, and a reduction in the unconditional gap (30 percent) of 3.6 to 7.9 percent. Although wages rise, equilibrium unemployment increases by 0.70 to 1.01 percentage points, widening the already large East–West unemployment gap (approximately 7 percentage points). Net of the unemployment effect, expected lifetime income (computed at actual, unbiased transition rates) rises by 0.7 to 1.88 percent for East German workers under West German biases, implying an unambiguous welfare gain. Under a biennial calibration (robustness), wages increase by up to 3.3 percent and expected lifetime income rises by up to 2.23 percent.

Scope conditions. Results apply to a stationary environment (no aggregate fluctuations). Firms are assumed to have rational expectations; an extension shows results hold provided firm bias is smaller than worker bias. Workers are assumed homogeneous in their bias levels; learning is abstracted from. The quantitative magnitudes are sensitive to the workers’ bargaining power γ and the contract length T, both of which are subject to uncertainty in calibration.

In depth

Q1. How are actual (statistical) transition probabilities constructed, and why are probit-predicted probabilities preferred over realized sample means?

A: Realized transition rates in the sample mix transitions for various idiosyncratic reasons that vary substantially across population groups, so raw sample means do not reflect the probability a given individual faces at interview time. The authors estimate probit models separately for job separation (employed sample) and job finding (unemployed sample), including a rich set of covariates — age, gender, education, tenure, firm size, unemployment experience, industry, survey year, and East Germany indicator, among others — and predict individual-level probabilities at the time of the interview. For consistency with the survey’s discrete response format, probit-predicted probabilities are rounded to the nearest decile (0%, 10%, …, 100%). The bias is computed as the individual-level difference between perceived and probit-predicted actual probabilities, averaged over the sample.

Q2. What is the magnitude and direction of the aggregate expectation biases in Germany?

A: Employed workers overestimate job separation by 6.4 percentage points on average (perceived 19.8% vs. actual 13.3%), a pessimistic bias significant at the 1% level. Unemployed workers overestimate job finding by 8.2 percentage points (perceived 57.0% vs. actual 48.8%), an optimistic bias also significant at the 1% level. Both directions are statistically robust across alternative definitions of separation and finding, as well as to trimming extreme responses (0% and 100% answers) and adjusting for directional rounding.

Q3. How large are the East–West differences in expectation biases, and do they survive controls for compositional differences?

A: East German workers exhibit a pessimistic job separation bias of 12.1 percentage points, more than 2.5 times the West German level of 4.7 percentage points, despite actual separation rates being broadly comparable (15.1% vs. 12.8%). For job finding, West Germans are optimistic by 12.9 percentage points while East Germans are optimistic by only 2.0 percentage points, a difference of 10.9 percentage points. The paper states these differences persist after accounting for compositional differences between regions, and are robust across all alternative definitions of job separation (Dismissals, Selected, Spell) and job finding (out of U or O). The table of robustness results (Table 2) confirms that in all specifications, the pessimistic separation bias is substantially larger in the East and the optimistic finding bias is substantially smaller.

Q4. What cohort analysis is conducted to explore the origins of greater East German pessimism?

A: The authors conduct a regression of the individual-level bias on birth-cohort indicators, controlling for age, demographic, and economic characteristics. They find that the pessimistic job separation bias is most pronounced among cohorts born in the 1950s and 1960s — those who experienced adult working life in the communist GDR and lived through reunification — and is smaller for cohorts born before 1950 and substantially smaller for cohorts born after 1970. For job finding, the optimistic bias is comparably low among cohorts born in the 1960s and earlier, but rises significantly for later-born East German cohorts. This cohort pattern is consistent with a long-lasting “experience effect” of communist institutions and the reunification shock on beliefs, analogous to findings in the broader literature on the persistent effects of communism.

Q5. Is there evidence that individuals update their biased expectations over time?

A: To assess learning, the authors use the panel dimension and compute for each individual in two consecutive survey waves the absolute value of the deviation between perceived and actual transition probabilities, then examine the change in this absolute deviation between waves. The histograms of individual-level changes show substantial dispersion but means close to zero in all four sub-groups (East/West, job separation/finding), indicating no systematic convergence of beliefs toward actual rates. Biases are also stable in the time-series dimension, with perceived and actual rates moving largely in parallel across survey waves from 1999 to 2015, leaving the aggregate bias level roughly constant.

Q6. How does the model rule out private information as an alternative explanation for the biases?

A: If biases reflected private information about idiosyncratic risk not captured by observable characteristics, individual-level deviations between perceived and actual rates should predict subsequent realized transitions. The authors add the individual-level deviation as an additional regressor in the probit transition models. The estimated coefficients are statistically significant and positive, but quantitatively negligible: a 1 percentage point higher expected job separation probability is associated with only a 0.001 percentage point higher realized separation probability, and a 1 percentage point higher expected job finding probability with a 0.002 percentage point higher realized finding probability. These magnitudes are too small to materially alter the interpretation of the biases as reflecting systematic expectation errors rather than private information.

Q7. What is the role of contract length T in the model, and what is the critical threshold T*?

A: The wage contract length T determines which of two opposing effects of pessimistic job separation expectations dominates in bargaining. The first (negative wage) effect: a pessimistic worker discounts future wages within the current contract more heavily than the firm does, so the worker values the contract less and accepts a lower wage. The second (positive wage) effect: a pessimistic worker also discounts the continuation value of future contracts more heavily, making it less attractive to remain in the match, so the firm must offer a higher wage to retain the worker. For short contract lengths (T < T*), the second (positive) effect dominates, so the pessimistic bias raises wages. For long contracts (T ≥ T*), the first (negative) effect dominates, so the pessimistic bias depresses wages. The critical threshold T* is the smallest positive integer such that T*/λw(θ) < β times a weighted sum involving σw and T*. Using calibrated parameter values for East Germany, T* = 10 quarters (2.5 years). The baseline contract length is T = 67 quarters (approximately 16.8 years), well above T*, placing the economy in the regime where pessimism depresses wages.

Q8. How does the optimistic job finding bias affect equilibrium wages and unemployment?

A: An optimistic job finding bias (λw > p(θ)) raises the perceived value of unemployment U because workers expect to escape unemployment sooner. A higher value of unemployment raises the worker’s outside option in bargaining, increases the reservation wage, and thereby pushes up the bargained wage. In general equilibrium, the job creation condition (which is unaffected by worker expectations) is unchanged, so the upward rotation of the wage curve reduces labor market tightness θ, raises equilibrium unemployment, and extends average unemployment duration. This comparative static holds unambiguously for any contract length T.

Q9. What are the quantitative results of the counterfactual experiment assigning West German biases to East German workers?

A: The counterfactual assigns West German bias levels (smaller pessimistic separation bias, larger optimistic finding bias) to East German workers while holding all other parameters at East German calibrated values. For the preferred calibration with γ ∈ {0.35, 0.50} and T ∈ {67, 106, 159}, wages in East Germany rise by 1.07 to 2.36 percent. This implies a reduction in the conditional East–West wage gap (23 percent) of 4.6 to 10.6 percent and a reduction in the unconditional gap (30 percent) of 3.6 to 7.9 percent. Equilibrium unemployment in East Germany rises by 0.70 to 1.01 percentage points as a side effect. Net of the unemployment effect, ex-ante unbiased expected lifetime income rises by 0.7 to 1.88 percent, confirming a positive welfare effect of reducing East German pessimism to West German levels. Under the biennial calibration robustness check, wage increases reach up to 3.3 percent, the conditional wage gap narrows by up to 11 percent, and lifetime income rises by up to 2.23 percent.

Q10. How is the bargaining power parameter γ calibrated and why does it matter for the results?

A: The paper considers a range γ ∈ {0.35, 0.50, 0.65}, rather than a single calibrated value, because γ plays a crucial role in the sensitivity of wages to expectation biases. Lower bargaining power reduces the equilibrium wage directly; however, because lower wages spur job creation, the model requires a higher vacancy cost κ to match the empirical job finding rate, which in turn increases the elasticity of wages with respect to the bias (see the wage equation, which shows that the bias effect scales with κθ/p(θ)). The paper argues that γ = 0.65 is inconsistent with the empirical wage–bias relationship estimated in SOEP data (which is negative and about twice as negative in East Germany as in the West), while γ ∈ {0.35, 0.50} is consistent. Lower bargaining power is also argued to be realistic for East Germany given weaker union representation there relative to the West.

Q11. How does the empirical relationship between the job separation bias and wages serve as a model validation target?

A: Using SOEP data, the authors regress log hourly wages on the individual-level difference between perceived and actual job separation rates, controlling for individual fixed effects and other covariates, and allow the slope to differ between East and West Germany. They find a statistically significant and negative relationship in both regions, with the effect approximately twice as large in East Germany as in the West. The estimate implies that if East German workers’ job separation pessimism were reduced to West German levels, hourly wages in the East would be about 1 percent higher. This empirical gradient is used as an external validation check — not a calibration target — to assess which combinations of (γ, T) in the model are quantitatively plausible.

Q12. What does the model predict about the general equilibrium effects on unemployment from reducing East German pessimism?

A: Reducing East German pessimism — both the pessimistic separation bias and the low optimistic finding bias — shifts the wage curve upward in equilibrium. Because the job creation condition is unaffected by worker beliefs (firms have rational expectations), higher wages reduce the firm’s incentive to post vacancies, lowering labor market tightness θ. This leads to higher equilibrium unemployment and longer average unemployment duration. The counterfactual with West German biases implies that East German unemployment would rise by 0.70 to 1.01 percentage points, further widening the approximately 7 percentage point East–West unemployment gap. The authors note this is a welfare-relevant trade-off, but show that the wage gain dominates the unemployment cost in terms of expected lifetime income.

Q13. What robustness checks are performed on the quantitative results?

A: The paper considers (i) a narrower definition of job separation (dismissals only) to match the most likely interpretation of the survey question; (ii) targeting the officially reported East German unemployment rate (14.5% average from the Federal Employment Agency) rather than the SOEP-implied rate of 8.6% as a calibration target; (iii) a biennial calibration frequency instead of quarterly. The main results — wage increases and narrowing of the wage gap — are quantitatively similar across these alternatives, with one exception: the biennial calibration yields substantially larger wage increases (up to 3.3%), a larger reduction in the conditional wage gap (up to 11%), and larger lifetime income gains (up to 2.23%).

Key Concepts

Expectation bias (job separation / job finding). In this paper, a bias in expectations is defined as a systematic average difference between an individual’s perceived transition probability and the actual (statistically predicted) transition probability for their demographic and job group. A pessimistic job separation bias means workers overestimate the probability of losing their job (σw > σ); an optimistic job finding bias means unemployed workers overestimate the probability of re-employment (λw > p(θ)). Biases are not attributed to private information but to systematic expectation errors.

Actual (statistical) transition probability. The paper defines actual transition probabilities not as raw sample transition rates but as individual-level predicted probabilities from probit models estimated on realized transitions within 24 months, conditional on a comprehensive set of individual, job, and employer characteristics observed at interview time. These are rounded to the nearest decile for comparability with the survey’s discrete response format.

Wage contract length (T). The contract length T is the number of periods for which a bargained wage is fixed before the match parties re-bargain. A job match consists of a sequence of consecutive wage contracts of length T. The paper departs from the standard DMP assumption of period-by-period bargaining (T = 1) and shows that T is central to how job separation expectations feed into the bargained wage. A permanent job approximates T → ∞.

Critical contract length (T).* A theoretically derived threshold: the pessimistic job separation bias raises equilibrium wages for contract lengths T < T* and depresses wages for T ≥ T*. Specifically, T* is the smallest positive integer such that T*/λw(θ) < β times a weighted sum involving β, σw, and T*. In the East German calibration, T* = 10 quarters.

Generalized Nash bargaining with common knowledge / agree to disagree. The model assumes that both the worker and the firm know each other’s perceived values of the job match and outside options and accept them as the basis for bargaining, even though they differ. Workers use their biased perceived transition rates to value employment and unemployment; firms use actual rates. There is no private information. The paper refers to this as workers and firms “agreeing to disagree.”

Ex-ante unbiased expected lifetime income (EI_{W,U}). A welfare measure defined as the present discounted value of income for an individual entering the economy, computed at actual (unbiased) job separation and job finding probabilities rather than at workers’ perceived (biased) rates. This measure captures the net welfare effect of changing expectation biases because it correctly accounts for actual employment transitions, even though the behavioral responses in equilibrium are driven by biased perceptions.

Effective discount factor (β(1 − σw)). When a worker holds pessimistic job separation expectations, future payoffs within the current contract are discounted not at the pure time discount factor β but at β(1 − σw), which is smaller when σw is larger. A more pessimistic worker therefore effectively discounts future wage payments more steeply, and this differential discounting relative to the firm (which uses β(1 − σ)) is the key mechanism generating the contract-length dependence of the wage effect.

Bottom-Up Markup Fluctuations

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

The paper asks how firm-level, sector-level, and aggregate markups comove with output at different levels of aggregation, and whether a single structural model can reconcile seemingly contradictory empirical findings about markup cyclicality that arise when researchers use different aggregation schemes.

Model

The authors build a granular macroeconomic model featuring oligopolistic competition with a nested constant-elasticity-of-substitution (CES) demand structure following Atkeson and Burstein (2008). The economy contains N sectors, each with a discrete number of firms competing under Cournot oligopoly with flexible prices. Firm-level markups are endogenously increasing in within-sector market shares: under Cournot, the sectoral markup is a simple function of the sector’s Herfindahl-Hirschman index (HHI), and the aggregate markup is a function of the expenditure-share-weighted average of sectoral HHIs. Firm-level productivity follows a discretized random growth (Gibrat’s law) process as in Carvalho and Grassi (2019), generating fat-tailed firm-size distributions and granular aggregate fluctuations. The baseline calibration features only idiosyncratic firm-level productivity shocks and abstracts from aggregate shocks, because—in the model—aggregate shocks that move all firms proportionately do not affect relative market shares and hence do not affect markups.

Data

The empirical analysis uses French administrative firm-level data from the FICUS-FARE datasets covering the universe of French firms from 1994 to 2019, yielding approximately 9.38 million firm-year observations across 26 years, 22 two-digit sectors, and 275 five-digit NAF sectors. Firm-level markups are estimated following De Loecker and Warzynski (2012) using a translog production function estimated by GMM (following De Ridder et al. 2024) on a subsample of approximately 220,733 firm-year observations where physical output quantity is available from the Enquete Annuelle de Production survey (2009-2019). Using quantity rather than revenue as the output measure avoids the measurement biases documented in Bond et al. (2021).

Main Findings and Quantitative Magnitudes

Markup-market-share relationship (firm level): Regressions of the change in the inverse firm markup on the change in firm market share yield a negative and significant coefficient of approximately -0.268 to -0.293 (depending on fixed-effect specification), consistent with the model prediction that markups rise with market share. Sector-level analogues yield a slope of the change in inverse sector markup on the change in sector HHI of approximately -0.37, which is simultaneously a calibration target (implying sigma = 1.8 given epsilon = 5) and an empirical moment the model closely matches (model counterpart: -0.36).
Within-between decomposition of sector markup changes: In the model under Cournot competition, changes in firm-level markups (the “within” term) account for exactly 50% of changes in sector-level markups, with between-firm reallocation accounting for the other 50%. In the French data, for the median sector, the within term accounts for 59% of changes in sector markups (interquartile range across sectors: 34%-81%).
Firm-level markup cyclicality with sector output (heterogeneous by size): The average firm’s markup is countercyclical with respect to own-sector output (beta_1 approximately -0.073 in levels specification), but this relationship reverses for large firms: firms with market shares roughly above 10% (top 0.1% of the market-share distribution) have procyclical markups (interaction coefficient beta_2 approximately 0.574 in levels). The model qualitatively and roughly quantitatively reproduces this heterogeneity.
Sector-level markup cyclicality with sector output (procyclical): Following Nekarda and Ramey (2013), sector markup changes comove positively and significantly with sector output changes: estimated coefficient of 0.160 (standard error 0.040) in first-differences. The calibrated model yields a median coefficient of 0.139 (std dev 0.057 across 5,000 simulated 25-year samples), close to the data. Consistently, sector concentration (HHI) is also procyclical with sector output (estimated coefficient 0.332, std error 0.067 in first-differences).
Sector-level markup cyclicality with aggregate output (acyclical to weakly countercyclical): Following Bils et al. (2018), the comovement between sector markups and aggregate output is fragile in sign and significance: the French data yields a point estimate of -0.239 (std error 0.116) in first-differences, marginally significant (t-stat 2.06) and with sign sensitive to detrending method. The model without aggregate shocks predicts positive comovement (median coefficient 0.165) that is not statistically different from zero across samples. Adding aggregate productivity shocks (calibrated to match French aggregate output volatility) brings the model-implied coefficient close to zero (median 0.008), with 20-30% of 25-year simulated samples displaying countercyclical sectoral markups relative to GDP—consistent with the ambiguity in the data.
Aggregate output volatility: The baseline calibration with only granular firm-level shocks generates a standard deviation of detrended aggregate output of 0.83%, equal to 26% of the 3.16% observed in the French data. (The comparable granular ratio from Carvalho and Grassi 2019 for a perfectly competitive US model is 30%.) Variable markups dampen granular aggregate volatility: the standard deviation of aggregate output under variable markups is 0.87 times that under heterogeneous-but-constant markups (95% CI: 0.82-0.97), because incomplete pass-through reduces the effective weight of large firms in the price index.
Aggregate markup volatility: In the data, the relative standard deviation of aggregate markup to aggregate output is 0.40-0.50 (depending on detrending). The model generates a relative volatility of 0.36 (median across samples). The correlation between aggregate markup and output in the data is at most 0.06; the model without aggregate shocks implies a counterfactually large median correlation of 0.91, which falls to 0.27 when aggregate TFP shocks are superimposed (with 16% of 25-year samples displaying countercyclical aggregate markups).

Scope Conditions

Results pertain to French private-sector firms (including formerly government-owned firms, most of which privatized during the sample period) across manufacturing and some non-manufacturing sectors at the national-market level. The analysis abstracts from import competition (market shares are computed relative to all French firms in the sector), local geographic markets (relevant for non-tradeable goods where national-level shares understate local concentration), and multi-product firm structure. Findings are for a flexible-price model driven by idiosyncratic productivity shocks; the paper explicitly discusses how nominal rigidities would further strengthen procyclicality at the sector level.

In depth

Q1. What is the central mechanism by which granular firm-level shocks generate markup cyclicality?

A: Because markups are endogenously increasing in within-sector market shares under oligopolistic competition, a firm that receives a positive productivity shock gains market share and therefore raises its markup, while its competitors lose market share and lower their markups. The net effect on the sectoral markup depends on the shocked firm’s initial size: a positive shock to a sufficiently large firm (above a threshold market share) raises the sectoral markup, while a positive shock to a small firm lowers it. Since sectoral expansions in a granular economy are disproportionately driven by large firms, sector output and sector markup tend to comove positively in the medium run.

Q2. Why does the sign of markup cyclicality differ depending on the level of aggregation?

A: Sector-level markups react only to within-sector idiosyncratic shocks, so sectors that happen to be driven by large-firm booms display positive comovement between sector markup and sector output. However, a given sector’s markup is uncorrelated with aggregate output movements coming from other sectors. In small samples (such as 25-year windows), whether a sector’s markup comoves positively or negatively with aggregate output depends on whether the sector happens to lead or lag the aggregate cycle. Over sufficiently long samples, the model implies positive comovement of sector markups with aggregate output, but in finite samples the relationship is indeterminate. This asymmetry across aggregation levels explains why researchers using different reduced-form specifications in the same dataset can reach opposing conclusions about procyclicality versus countercyclicality.

Q3. What is the within-between decomposition of sectoral markup changes and what does it imply quantitatively?

A: Changes in the inverse sectoral markup can be decomposed into (i) a within term—changes in firm-level markups holding market shares fixed—and (ii) a between term—changes in market shares holding firm-level markups fixed. Under Cournot competition, the within and between terms are analytically equal in every period, so each accounts for exactly 50% of the change in sectoral markups; this 50-50 split holds globally (not only to first order). In the French data, for the median sector, within-firm markup changes account for 59% of sector markup changes (interquartile range across sectors: 34%-81%), close to but slightly above the model’s 50% prediction.

Q4. How do variable markups affect granular aggregate output volatility relative to a model with constant markups?

A: Variable markups (endogenous pass-through that is decreasing in firm size) reduce granular aggregate output volatility relative to a model where markups are heterogeneous but fixed. The intuition is that larger firms have lower pass-through rates, so their productivity shocks translate into smaller price changes and therefore smaller output responses than they would under constant markups—effectively reducing the weight of large firms in the aggregate price index in a way similar to a decline in market concentration. Quantitatively, using first-order approximations around equilibrium distributions from the calibrated model, the standard deviation of aggregate output under variable markups is 0.87 times that under heterogeneous-but-constant markups (95% confidence interval: 0.82-0.97). The overall standard deviation under variable and heterogeneous markups is only 1.02 times that under homogeneous and constant markups (95% CI: 0.99-1.14), meaning markup heterogeneity and variability together have limited net effects on aggregate output volatility.

Q5. What does the model predict for firm-level markup cyclicality, and how heterogeneous is this across firm size?

A: Proposition 4 states that, in the asymptotic limit, firm-level markups comove positively with own-sector output for firms with market shares above a threshold, and negatively for firms below it. This occurs because large firms have a disproportionate impact on sector-level price and output (when the product of market share and pass-through rate is increasing in size), so large-firm shocks simultaneously drive sector expansions and raise large-firm markups while compressing small-firm markups. In the French data, the average firm’s markup is countercyclical with respect to sector output (beta_1 approximately -0.073 in log-levels with firm and year fixed effects), but firms with market shares above roughly 10% (top 0.1% of the distribution, since the average market share is only 0.07%) display procyclical markups (interaction coefficient beta_2 approximately 0.574). The model reproduces this qualitative pattern and the order of magnitude of these estimates.

Q6. How does the paper calibrate the key demand elasticities, and what are the resulting pass-through implications?

A: The within-sector substitution elasticity is set to epsilon = 5, a standard value. The cross-sector substitution elasticity sigma is calibrated to match the slope of the inverse sector markup on sector HHI in first-differences. The empirical slope is -0.37; under the model, the slope equals -(epsilon/sigma - 1)/(epsilon - 1), and given epsilon = 5, sigma = 1.8 delivers a model counterpart of -0.36. These parameter values imply own-cost pass-through rates that are decreasing in firm size; for large firms (with market share >= 57%, approximately the top 0.004% of the distribution), the implied pass-through rate is 0.63, within the confidence intervals reported in Amiti, Itskhoki, and Konings (2019) for large Belgian firms.

Q7. Why do aggregate productivity shocks not affect markups in the model, and what are the implications for aggregate markup cyclicality?

A: In the model, firm-level markups are functions of within-sector market shares, not the level of productivity. An aggregate shock that shifts all firms’ productivity proportionately leaves relative market shares unchanged and therefore leaves all markups unchanged. This means aggregate shocks increase aggregate output volatility but leave markup volatility unchanged, reducing the correlation between aggregate markup and aggregate output. When aggregate TFP shocks are added to match French aggregate output volatility, the model-implied median correlation between aggregate markup and output falls from 0.91 (without aggregate shocks) to 0.27 (with aggregate shocks), while 16% of 25-year simulated samples display countercyclical aggregate markups—more consistent with the weak and fragile empirical relationship.

Q8. How does the paper address the potential measurement-error bias in the negative correlation between markups and marginal costs?

A: Since marginal cost is computed as price divided by estimated markup, regressing market shares or markups on marginal costs risks spurious correlation via measurement error in the markup (which appears in both sides). The authors address this concern by constructing an instrumental variable for marginal cost based on firm-specific energy intensity interacted with energy price changes, following Ganapati, Shapiro, and Walker (2020). Table A10 confirms that instrumenting for marginal cost yields negative effects on both markup and market share with larger point estimates than the OLS specifications in Table 4, validating the baseline findings.

Q9. Is the 50-50 within-between decomposition of sectoral markup changes robust to the choice of competition mode?

A: No. The exact 50-50 split of within and between terms in sectoral markup changes is a specific property of Cournot competition and holds globally (not just as a first-order approximation). Under Bertrand competition, the within and between terms are generally not equal to each other. The paper derives analytic results under both competition modes and focuses on Cournot for quantitative work because it generates more markup variation and better matches the estimated pass-through rates and markup-size relationship.

Q10. What do model simulations imply for the magnitude and cyclicality of aggregate markups versus the data, and what is the role of variable versus constant markups?

A: In the data (detrended), the standard deviation of aggregate markup is 1.27% with a relative volatility (to output) of 0.40 and a correlation with output of 0.03. The baseline model with only granular shocks yields a median markup standard deviation of 0.30%, relative volatility of 0.36, and correlation with output of 0.91. The model with aggregate shocks added yields median markup standard deviation of 0.30%, relative volatility of 0.09, and correlation of 0.27. Counterfactually fixing markups at their initial heterogeneous levels while keeping the same market shares and shock variance yields aggregate markup standard deviation approximately 0.93 times the variable-markup value (standard deviation of markups under variable markups is 1.08 times that under constant markups, with a 95% CI of 1.00-1.18), and a correlation with output of 0.92 versus 0.87 under variable markups. Overall, the magnitude and cyclicality of aggregate markups are not substantially different between variable and constant-markup specifications.

Q11. How does the paper reconcile its findings with prior literature on markup cyclicality (Bils et al. 2018 vs. Nekarda and Ramey 2013)?

A: Nekarda and Ramey (2013) find procyclical sector markups with respect to sector output in US data—a result replicated in French data (beta approximately 0.160). Bils, Klenow, and Malin (2018) find countercyclical sector markups with respect to aggregate output in US data. Both results can be generated simultaneously in the model: sector markups are positively correlated with own-sector output because granular booms in a sector are driven by large-firm expansions that raise sector markups; however, a given sector’s markup is weakly and ambiguously correlated with aggregate output because aggregate fluctuations reflect shocks across many sectors, only some of which are in the same sector. The model can therefore simultaneously predict procyclicality with respect to sector output and an acyclical-to-weakly-countercyclical relationship with aggregate output—explaining why both empirical findings can be correct.

Q12. What are the data limitations and how do they affect the interpretation of results?

A: Three limitations are noted. First, market shares are computed relative to total revenue of all French firms in the sector without accounting for imports, so foreign competition is ignored and domestic concentration may be overestimated. Second, revenues are reported at the national level, so for non-tradeable goods (whose relevant market is local) the paper underestimates true local market concentration, attenuating the markup-concentration relationship in those sectors. Third, the model abstracts from entry and exit (the number of firms per sector is held fixed at sector-year averages), though Appendix D demonstrates robustness of main empirical results to restricting the sample to continuing firms.

Key Concepts

Granular macroeconomic model: A model in which the economy consists of a finite (large but discrete) number of firms, so that idiosyncratic firm-level shocks to large firms do not average out and instead generate aggregate fluctuations. In the paper’s usage, granularity means that sectoral and aggregate business-cycle fluctuations are driven primarily by shocks to the largest firms, which also have the highest markups and market shares.

Nested CES demand structure (Atkeson-Burstein): A two-level constant-elasticity-of-substitution aggregation where the final good aggregates N sectors with cross-sector elasticity sigma, and each sector aggregates the output of its Nk firms with within-sector elasticity epsilon > sigma. This structure generates firm-level markups that are endogenously increasing in within-sector market shares (under both Cournot and Bertrand competition) and yields closed-form expressions for sector-level markups as a function of sector HHI and aggregate markups as a function of the expenditure-weighted average of sector HHIs.

Markup elasticity with respect to market share (Gamma_ki): Under Cournot competition, the semi-elasticity of firm i’s log markup with respect to its log market share, equal to (epsilon/sigma - 1)s_ki / (epsilon/(epsilon-1) - (epsilon/sigma - 1)s_ki). This is strictly positive for epsilon > sigma and increasing in market share, implying that larger firms have markups that are more responsive to changes in their competitive position.

Pass-through rate (alpha_ki): The fraction of an idiosyncratic cost shock that is passed into the firm’s price relative to the sectoral price index, given by 1/(1 + (epsilon-1)Gamma_ki). Pass-through is decreasing in market share (larger firms have lower pass-through), which dampens their price response to own shocks and mutes the impact of large-firm shocks on aggregate price volatility—acting like a reduction in market concentration.

Within-between decomposition of sector markup changes: The change in inverse sector markup decomposed into (i) a within term measuring changes in firm-level markups holding market shares fixed, and (ii) a between term measuring reallocation of market shares across firms with heterogeneous markups. Under Cournot competition, these two terms are exactly equal (each 50%) for any firm-level shocks—a result that holds globally (not merely as a first-order approximation)—because the forces that increase the within term (higher markup sensitivity) also raise heterogeneity between firms (increasing the between term).

Sectoral markup (mu_kt): Defined as the ratio of sectoral revenues to total wage payments in the sector, equal to the harmonic mean of firm-level markups weighted by market shares. Under Cournot competition, this is a simple increasing function of the sector’s HHI: mu_kt = (epsilon/(epsilon-1))[1 - (epsilon/sigma - 1)/(epsilon-1) x HHI_kt]^(-1). This mapping between concentration and the markup price-cost wedge gives the central empirical prediction tested at the sector level.

Markup cyclicality (at different aggregation levels): The comovement between markups and output, which the paper distinguishes sharply across three levels: (i) firm markup vs. own-sector output—countercyclical for small firms, procyclical for large firms; (ii) sector markup vs. own-sector output—procyclical (positive covariance) under conditions proven in Proposition 3; (iii) sector markup vs. aggregate output—theoretically positive over long samples but ambiguous and close to zero in short samples, because aggregate output also reflects shocks to other sectors whose markups are uncorrelated with the focal sector’s markups. The paper’s central insight is that the same underlying model generates all three empirical patterns simultaneously.

Bridging micro and macro production functions: The fiscal multiplier of infrastructure investment

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper investigates the fiscal multiplier of infrastructure investment, specifically by incorporating firm-level investment decisions — a dimension absent from prior literature. The central analytical challenge is bridging the micro (firm-level) and macro (state-level) production functions for infrastructure, given that public capital is non-rivalrous: it can be used simultaneously by all firms without being depleted. The paper demonstrates that this non-rivalry generates a systematic discrepancy between firm-level and aggregate-level estimates of the elasticity of substitution between private and public capital, and it shows how this discrepancy shapes the magnitude of the fiscal multiplier.

Data and Methodology

The authors build and estimate a heterogeneous-firm general equilibrium model. Firms operate a constant-elasticity-of-substitution (CES) production function using private capital, non-rivalrous public capital (infrastructure), and labor. Firms are subject to idiosyncratic productivity shocks and make lumpy investment decisions subject to both fixed and convex capital adjustment costs, following Cooper and Haltiwanger (2006) and Winberry (2021). The economy has two regions — one with poor infrastructure and one with good infrastructure — motivated by the near-invariant cross-state distribution of infrastructure spending observed in U.S. data.

The model is estimated via an extended Simulated Method of Moments (SMM) that treats market clearing prices as additional parameters estimated simultaneously with structural parameters, reducing computational cost relative to standard GE estimation. Estimation uses a multi-block Metropolis-Hastings algorithm. Target moments include lumpy investment fraction (0.14, from Zwick and Mahon 2017), average investment-to-capital ratio (0.10), standard deviation of i/k (0.16), private-to-infrastructure capital ratio (0.75, from BEA), high-infrastructure region’s private capital share (0.83, from Census BDS), and total working hours (0.33).

The identification of the key parameter — the firm-level elasticity of substitution between private and public capital (λ) — comes from the relative size of private capital stocks across the two infrastructure groups: under greater complementarity, regions with more infrastructure should hold relatively more private capital.

External validation is provided by estimating the state-level elasticity from the model’s simulated data using a nonlinear least squares method following An et al. (2019), and comparing it to empirical state-level estimates from actual U.S. state data.

Main Findings with Quantitative Magnitudes

Firm-level vs. aggregate-level elasticity gap. The estimated firm-level elasticity of substitution is λ = 1.185, implying gross substitutability between private and public capital at the firm level. The state-level elasticity implied by the same model is 0.48 (or 0.35 in a decreasing-returns-to-scale specification), implying gross complementarity. The empirical state-level counterpart estimated from actual U.S. data is 0.445. The paper proves theoretically (Proposition 1) that, given non-rivalry and under mild conditions, firm-level gross substitutability implies aggregate-level gross complementarity. Proposition 2 further shows that this same mechanism micro-founds the increasing-returns-to-scale assumption in Baxter and King’s (1993) Cobb-Douglas aggregate production function.
Fiscal multiplier (baseline, 2-year horizon). The aggregate output multiplier over a 2-year horizon in the heterogeneous-firm general equilibrium model is 1.088 in response to a one-time unexpected infrastructure spending shock equal to 1% of steady-state GDP, financed by a lump-sum tax. The corresponding partial-equilibrium output multiplier (holding prices fixed at steady state) is 1.858; the gap reflects crowding out of private investment induced by the general equilibrium interest rate response. In the baseline, the interest rate rises by 0.39% after the shock; the investment multiplier is -0.043.
Comparison with representative-agent model. When the same implied returns-to-scale parameters are used in a representative-agent model (following Baxter and King 1993), the output multiplier is 0.991 and the investment multiplier is -0.157, both substantially lower than the heterogeneous-firm baseline. The key mechanism: under convex adjustment costs, the Jensen’s inequality effect implies that heterogeneous firms face a greater average adjustment burden than the representative firm, making their investment less responsive to the general equilibrium crowding-out pressure.
Sensitivity to elasticity of substitution. Across the heterogeneous-firm model: at λ = 3 (high substitutability), the output multiplier falls to 0.672; at λ = 0.5 (complementarity), it rises to 1.364. The multiplier is significantly more sensitive to λ in the heterogeneous-firm model than in the representative-agent model, because non-rivalry amplifies the effect of any given elasticity value through each firm’s production function.
Cross-state distribution of gains. Under the baseline spending allocation (81% to Good states, 19% to Poor states), per $1 of infrastructure spending, Good states receive $1.072 of the $1.088 total output gain, while Poor states receive only $0.016. In a counterfactual with equal spending across states, the total output multiplier falls to 0.873, Good states’ output multiplier falls to 0.810, and Poor states’ output multiplier rises to approximately 0.062 (about four times the baseline level of 0.016). This quantifies a sharp efficiency-equality trade-off in the allocation of infrastructure investment.
Employment and earnings effects. Compared to steady state, the baseline fiscal shock produces an average annual increase of 0.304% in employment and 0.389% in wages, yielding a $0.713 increase in earnings and a $0.148 increase in consumption per $1 of fiscal spending in general equilibrium. In partial equilibrium (no price changes), earnings increase by $1.294 and consumption by $0.605 per $1 spent.

Scope Conditions

Results are conditional on: (i) lump-sum tax financing of the fiscal shock; (ii) a one-time unexpected (MIT) shock with no persistence; (iii) a closed-economy framework with endogenous real interest rate; (iv) the estimated two-region structure calibrated to U.S. state-level infrastructure data; (v) firm-level investment dynamics calibrated to Compustat and BDS moments. The authors note that incorporating time-to-build assumptions (tested in an appendix) reduces the aggregate fiscal multiplier, consistent with Ramey (2020).

In depth

Q1. What is the core theoretical result connecting firm-level and aggregate-level elasticities, and what is the intuition?

A: Proposition 1 proves that, given non-rivalrous public capital and mild data conditions (at least one firm has private capital below total infrastructure, and aggregate private capital exceeds total infrastructure), if the firm-level elasticity of substitution λ ≥ 1 (gross substitutes), then the aggregate-level elasticity ξ < 1 (gross complements). The intuition is that a marginal increase in public capital raises the marginal product of private capital for every firm simultaneously due to non-rivalry; the sum of these MPK gains across all firms exceeds any single firm’s gain. To represent this amplified benefit within an aggregate production function, a stronger complementarity is required than what any single firm faces. Put differently, non-rivalry means aggregate private and public capital “look” more complementary than they truly are at the firm level.

Q2. How does non-rivalry micro-found the Baxter-King aggregate production function?

A: Proposition 2 shows that if firms use a CES production function with gross substitutability (λ ≥ 1) and non-rivalrous public capital, then fitting aggregate output with a Cobb-Douglas production function (as in Baxter and King 1993, H(K,N,L) = zK^α L^{1-α} N^ζ) yields ζ > 0, implying increasing returns to scale (IRS). This is the paper’s micro-foundation for a widely-used but previously ad hoc assumption in the macro-fiscal literature. The corollary states that both gross complementarity in the aggregate CES function and IRS in the aggregate Cobb-Douglas follow from the same non-rivalry mechanism at the firm level.

Q3. Why does the heterogeneous-firm model produce a higher output multiplier than the representative-agent model?

A: Two mechanisms drive the difference. First, due to Jensen’s inequality and the convexity of adjustment costs, heterogeneous firms face a higher average adjustment burden than the representative (average) firm; this means heterogeneous firms are less responsive to interest rate changes that crowd out investment. The investment multiplier is -0.043 in the heterogeneous-agent baseline versus -0.157 in the representative-agent model. Second, the fixed adjustment cost (present in the baseline but absent from the representative-agent model) further dampens investment sensitivity via the extensive margin. Because less private investment is crowded out, more of the direct output boost from infrastructure spending survives into the aggregate multiplier, yielding 1.088 versus 0.991.

Q4. What is the novel estimation procedure and why is it necessary?

A: Standard SMM applied to GE models requires solving for market-clearing prices for every candidate parameter vector, creating a nested optimization loop that is computationally prohibitive. The authors extend SMM by treating market-clearing prices (wage w and marginal utility of consumption p) as additional parameters and appending market-clearing conditions as additional target moments — effectively requiring those moments to equal zero. A multi-block Metropolis-Hastings algorithm jointly draws from the price block and the parameter block. This approach generates posterior draws that simultaneously satisfy market clearing and fit empirical moments, without the inner loop. The resulting market-clearing accuracy is e^{-4} at the posterior mean.

Q5. How is the firm-level elasticity of substitution (λ) identified from the data?

A: λ is identified from the cross-state difference in private capital stocks between high- and low-infrastructure regions. Under the model, if private and public capital are more complementary (lower λ), high-infrastructure regions should attract relatively more private capital. The data moment used is the Good region’s share of aggregate private capital (0.83 from Census BDS data). This identification strategy is analogous to Bartik-instrument approaches in the empirical literature, where a parameter governing cross-state sensitivity to aggregate shocks is identified from cross-sectional variation.

Q6. How is the model validated externally?

A: The authors compute the state-level elasticity from the estimated model by fixing firm-level parameters and re-estimating only the elasticity and regional productivity from the model’s simulated state-level data, using the same NLLS estimator as An et al. (2019). The model-implied state-level elasticity is 0.349 (DRS specification) or 0.482 (CRS specification). The empirical estimate from actual U.S. state-level data following the same estimator is 0.445. Both indicate gross complementarity at the state level, consistent with the theoretical prediction. This external validation is not used in the estimation itself, providing an independent check.

Q7. What are the roles of extensive vs. intensive investment margins in the crowding-out effect?

A: Table 9 decomposes the investment multiplier of -0.043 by investment margin. When only the extensive margin (the discrete decision of whether to invest) is allowed to respond, the investment multiplier is -0.032 — approximately 74% of the baseline crowding-out effect. When only the intensive margin (investment size conditional on adjusting) responds, the multiplier is -0.011 — about 25% of the total. Thus the extensive margin is the dominant channel through which higher interest rates crowd out private investment. When both margins are held fixed, the output multiplier rises to 1.139, confirming that investment crowding-out reduces the output multiplier by about 0.05.

Q8. How does the elasticity of substitution affect the fiscal multiplier quantitatively, and why does this matter more in the heterogeneous-firm model?

A: In the heterogeneous-firm GE model: λ = 3 gives an output multiplier of 0.672, λ = 1.185 (baseline) gives 1.088, and λ = 0.5 gives 1.364 — a range of 0.692. In the representative-agent model, the comparable range across the implied ζ values is much narrower (0.970 to 0.998). The amplification in the heterogeneous-firm model occurs because non-rivalry means each firm’s production function directly incorporates the public capital stock, so the elasticity parameter has first-order consequences for every firm’s investment incentive response to a fiscal shock. This heightened sensitivity underscores why accurately estimating λ at the firm level — rather than importing a state-level estimate — is critical for quantifying infrastructure multipliers.

Q9. What is the efficiency-equality trade-off in cross-state infrastructure allocation?

A: Under the baseline allocation (81% of infrastructure spending to Good states, 19% to Poor states), per $1 of infrastructure spending, the Good states receive $1.072 of output gains and Poor states receive only $0.016. In the equal-spending counterfactual, the total output multiplier falls from 1.088 to 0.873. The Poor states’ output multiplier rises from $0.016 to $0.062 (approximately fourfold), while the Good states’ falls from $1.072 to $0.810. The Poor states also see earnings multipliers more than double (from $0.017 to $0.042). This trade-off arises because Good states have both more private capital (benefiting from non-rivalry) and higher estimated TFP — so each dollar of infrastructure is more productive there. Equal allocation reduces aggregate efficiency while partially mitigating regional inequality.

Q10. How do the paper’s multiplier estimates compare to the existing literature?

A: In partial equilibrium (no GE adjustment), the authors find an output multiplier of 1.858, consistent with Chodorow-Reich’s (2019) cross-sectional multiplier of approximately 1.8. Once the general equilibrium interest rate effect is included, the multiplier falls to 1.09, which falls within the 0.6-1.2 range from Ramey (2011). Literature using representative-agent models without non-rivalry (e.g., Ramey 2020) typically reports multipliers of 0.3 to 0.8 using returns-to-scale parameters of 0.07-0.12; the paper shows these correspond to fiscal multipliers of 0.847-0.882 in the representative-agent framework. The heterogeneous-firm model, once it incorporates the non-rivalry-corrected elasticities, yields a meaningfully higher multiplier of 1.088.

Q11. What role does time-to-build play, and how does the paper handle it?

A: The baseline model assumes a time-to-build period s = 1 year (one-year lag before new infrastructure is productive). The paper notes in Appendix H that incorporating extended time-to-build reduces the aggregate fiscal multiplier, operating through two channels: a news effect (agents adjust behavior upon anticipating future infrastructure) and a general equilibrium effect endogenous to the news effect. This finding is consistent with Ramey (2020). The baseline results are therefore reported under the minimal one-year time-to-build assumption, with longer lags serving as a robustness check.

Q12. What is the role of region-specific TFP heterogeneity in the model?

A: The model includes two regions that differ both in infrastructure levels and in region-specific productivity (TFP) levels. The TFP of the Good region is estimated to be approximately double that of the Poor region (x = 2.064 for Good vs. 1 for Poor). This productivity difference is estimated to partially capture heterogeneous congestion effects (which are not separately modeled) and is estimated jointly with the infrastructure elasticity. The productivity differential is identified from the Good region’s share of aggregate output (0.849 in the data). The large TFP gap is also the reason why equal spending on Poor states generates a much smaller output gain than spending on Good states: not only is infrastructure utilization lower (fewer firms), but underlying productivity is also lower.

Key Concepts

Non-rivalry of public capital: The property by which infrastructure stock (Nj,t) enters each firm’s production function at the full regional level, not divided among firms. Formally, a single marginal unit of public capital raises every firm’s marginal product of private capital simultaneously, so the aggregate marginal product gain summed across firms exceeds any single firm’s gain. This is the central mechanism driving the micro-macro elasticity discrepancy in the paper.

Firm-level elasticity of substitution (λ): The elasticity governing the degree of substitutability between private capital (k) and public infrastructure (N) in the firm’s CES production function. At λ = 1 the production function is Cobb-Douglas; λ > 1 is gross substitutability; λ < 1 is gross complementarity. In the paper’s estimation, λ = 1.185, meaning private and public capital are gross substitutes at the firm level.

Gross substitutability vs. gross complementarity: Two inputs are gross substitutes (complements) if an increase in the quantity of one raises (lowers) the demand for the other, holding output price fixed. In the paper’s framework, private and public capital are gross substitutes at the firm level (λ = 1.185 > 1) but gross complements at the state level (ξ ≈ 0.48 < 1), with non-rivalry explaining the inversion upon aggregation.

Convex adjustment cost: A cost C(I,k) = (µ/2)(I/k)² · k that scales quadratically with the investment rate. In the heterogeneous-firm model, this cost plays a critical role: by Jensen’s inequality, heterogeneous firms’ average adjustment burden under a convex cost exceeds that of the representative (average) firm, making aggregate investment less sensitive to interest rate changes and thereby dampening crowding out.

Fixed adjustment cost (ξ): A one-time overhead cost drawn from a uniform distribution [0, ξ̄], paid only when a firm makes a large-scale investment outside the “inaction band” [−νk, νk]. This cost generates lumpy investment at the firm level, with about 14% of firms making lumpy investments in any given year. It also creates an extensive margin of investment adjustment that accounts for approximately 74% of the baseline crowding-out effect.

Fiscal multiplier (as defined in this paper): The ratio of the present value of aggregate output deviations from steady state to the present value of the fiscal spending shock, both summed over a T-year horizon. For the short run, T = 2 years; for the long run, T = 5 years. This is computed as a perfect-foresight transition path response to a one-time MIT shock equal to 1% of steady-state GDP.

MIT shock (one-time unexpected shock): An unanticipated, non-persistent one-period deviation in infrastructure spending. The term “MIT shock” refers to a deterministic transition experiment where agents have perfect foresight about all future values after the initial shock occurs. This contrasts with persistent policy rules and allows isolating the dynamic effects of a one-time fiscal impulse.

Extended SMM with market-clearing moments: The paper’s estimation innovation. Rather than solving for market-clearing prices at each parameter candidate (the standard costly inner loop), wages (w) and marginal utility of consumption (p) are treated as parameters with associated moments being the market-clearing conditions set to zero. A multi-block Metropolis-Hastings algorithm draws from the price block and the parameter block separately, generating posterior draws that jointly satisfy market clearing and empirical moment conditions.

Cap‐and‐Trade and Carbon Tax Meet Arrow–Debreu

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

Anderson and Duanmu (2025) ask how general equilibrium (GE) interactions — factor reallocation across sectors, capital misallocation under climate uncertainty, and the distributional incidence of damages — alter the social cost of carbon (SCC) relative to the partial equilibrium (PE) estimates embedded in standard integrated assessment models (IAMs). The paper also characterizes conditions for Pareto improvements through climate policy and derives the optimal carbon tax in second-best environments with pre-existing distortions.

Framework

The authors build a dynamic Arrow-Debreu economy with L goods, K capital stocks (including climate stocks), and T periods. The climate module specifies that the carbon stock evolves as S_{t+1} = S_t + sum_j e_j(q_j) − alpha·S_t, and climate damage functions D_j(S_t) = 1 − d_j·(S_t − S_0) reduce sector-specific production possibilities sets. Firms and households take the climate trajectory as given and do not internalize their own emissions’ impact, generating the externality. Under standard regularity conditions, the authors prove existence of a competitive equilibrium and establish that it is inefficient: output is too high and climate-intensive sectors are too large relative to the social optimum.

General Formula for the SCC

The paper derives a general SCC formula — SCC_t = Sum_{tau >= t} beta^(tau−t) · [dW/dS_tau / (dW/dY_t)] — that decomposes into four components: (1) the standard direct productivity-loss term, (2) a GE factor-reallocation term capturing inefficient reallocation as damages shift relative prices, (3) a capital-misallocation term reflecting distortions in investment from climate uncertainty, and (4) a distribution term reflecting the welfare losses from the regressive incidence of climate damages. All three correction terms are positive under standard conditions, so the GE SCC exceeds the PE SCC. The paper shows that this formula nests existing IAM frameworks as special cases.

Quantitative Findings

Calibrating to three leading IAMs, the authors find that general equilibrium interactions raise the SCC by 15–40% above standard PE estimates:

DICE-calibrated: GE correction of 18% above the PE estimate.
FUND-calibrated: GE correction of 15% above the PE estimate.
PAGE-calibrated: GE correction of 40% above the PE estimate, the largest correction owing to greater sector heterogeneity in that model.
Median calibration: a PE SCC of $51/tCO₂ rises to a GE SCC of $62/tCO₂.

Decomposing the aggregate GE correction: factor reallocation across sectors accounts for 55%, capital misallocation due to climate uncertainty for 30%, and the distributional regressivity of damages for 15%.

Second-Best Policy and Uncertainty

In environments with pre-existing distortions, the optimal carbon tax deviates from the SCC: revenue recycling through labor tax cuts generates additional welfare gains of 10–15% of carbon tax revenue; undertaxed capital implies the optimal carbon tax should be set above the SCC (double dividend); and in monopolistically competitive sectors the optimal carbon tax is below the SCC because the carbon tax amplifies monopoly distortions. Under climate uncertainty, the SCC carries a risk premium proportional to the variance of damage estimates times the coefficient of relative risk aversion, estimated at +$8–15/tCO₂ (15–25% of the base SCC).

Scope Conditions

The quantitative corrections are calibrated to DICE, FUND, and PAGE and therefore inherit those models’ parameterizations of damage functions and discount rates. The GE factor-reallocation and capital-misallocation channels are larger when sectors are more heterogeneous in damage exposure — as is explicit in the PAGE result. Second-best corrections depend on the sign and magnitude of pre-existing distortions (labor taxes, capital taxes, market structure).

In depth

Q1. What is the core inefficiency result, and what does it imply about the competitive equilibrium?

The paper’s efficiency theorem establishes that the competitive equilibrium is Pareto inefficient because firms and households take the climate trajectory as given and do not internalize the impact of their own emissions on the carbon stock. As a consequence, output is too high and climate-intensive sectors are too large relative to the social optimum. This externality is the fundamental justification for climate policy in the model.

Q2. How does the paper’s general SCC formula extend existing approaches, and what are the novel terms?

The general formula SCC_t = Sum_{tau >= t} beta^(tau−t) · [dW/dS_tau / (dW/dY_t)] nests standard IAM SCC formulas as special cases. The novel terms relative to partial equilibrium are: (i) a GE reallocation term capturing losses from inefficient factor reallocation as climate damages change relative prices across sectors; (ii) a capital-misallocation term capturing distortions in investment arising from climate uncertainty; and (iii) a distribution term capturing welfare losses from the regressive incidence of damages. All three terms are positive under standard conditions, implying GE SCC > PE SCC in all calibrations.

Q3. How are the quantitative GE corrections decomposed, and which channel dominates?

Of the total GE correction above the PE baseline, factor reallocation across sectors contributes 55%, capital misallocation due to climate uncertainty contributes 30%, and the distributional regressivity of damages contributes 15%. Factor reallocation is the dominant channel because, as climate damages alter relative prices, production shifts toward less-damaged sectors in ways that are distorted by the original carbon externality — generating second-order losses absent from PE damage functions.

Q4. Why does the PAGE calibration produce a larger GE correction (40%) than DICE (18%) or FUND (15%)?

The paper attributes PAGE’s larger GE correction to greater sector heterogeneity in that model’s parameterization. When damage exposure is more heterogeneous across sectors, the relative-price effects of marginal carbon are larger, amplifying the factor-reallocation channel. DICE and FUND, with more uniform sector-level damage structures, exhibit smaller reallocation corrections.

Q5. What is the median-calibration implication for the SCC in dollar terms?

In the median calibration, a PE SCC of $51/tCO₂ rises to a GE SCC of $62/tCO₂, an increase of roughly $11/tCO₂ or approximately 22%. This figure is directly computable from observable trade elasticities and sector-level damage estimates.

Q6. How should the carbon tax be adjusted when pre-existing labor market distortions are present, and what is the magnitude of the welfare gain from revenue recycling?

When labor taxes create a pre-existing wedge, using carbon tax revenue to reduce labor taxes generates additional welfare gains of 10–15% of total carbon tax revenue — the double dividend in the labor market dimension. The optimal carbon tax in this case includes the SCC plus a correction term for the labor-market distortion.

Q7. How do capital market distortions alter the optimal carbon tax relative to the SCC?

If capital is undertaxed (a pre-existing distortion in capital markets), the optimal carbon tax is set above the SCC. The intuition is that a higher carbon tax partially offsets the under-taxation of capital by raising the effective cost of carbon-intensive investment, capturing a double-dividend in the capital market.

Q8. How does monopolistic competition modify the optimal carbon tax?

For monopolistically competitive sectors, the optimal carbon tax is below the SCC. The reasoning is that applying a carbon tax to these sectors amplifies existing monopoly markups and associated distortions, so the social cost of the carbon tax exceeds the raw SCC in those sectors. The optimal policy trades off carbon correction against monopoly amplification.

Q9. What is the risk premium in the SCC under climate uncertainty, and how is it estimated?

The paper adds a term to the SCC proportional to the variance of damage estimates times the coefficient of relative risk aversion. Using empirical estimates of damage uncertainty, this risk premium is estimated at +$8–15/tCO₂, representing 15–25% of the base SCC. This term is absent from deterministic SCC calculations and constitutes a further reason standard PE estimates understate the true social cost.

Q10. What is the paper’s claim regarding computability of the GE correction?

The paper states that the novel GE terms are computable from observable trade elasticities and sector-level damage estimates, implying the GE correction is not merely a theoretical construct but can be implemented in quantitative policy analysis using data sources already available to researchers and policymakers.

Key Concepts

Social Cost of Carbon (General Equilibrium Formula) Defined in the paper as SCC_t = Sum_{tau >= t} beta^(tau−t) · [dW/dS_tau / (dW/dY_t)], the present discounted value of the marginal welfare loss from an additional unit of carbon, expressed relative to the marginal utility of current output. The paper’s version adds GE reallocation, capital-misallocation, and distributional terms absent from standard PE formulations.

GE Adjustment Factor The ratio of the general equilibrium SCC to the partial equilibrium SCC, expressed as GE/PE = 1 + phi_realloc + phi_capital + phi_distribution. Under standard conditions all three phi terms are positive, so the GE SCC strictly exceeds the PE SCC.

Climate Damage Function (Sector-Specific) Specified as D_j(S_t) = 1 − d_j·(S_t − S_0), a sector-specific multiplicative reduction in the production possibilities set as the carbon stock rises above the pre-industrial level S_0. Heterogeneity in d_j across sectors is the driver of the factor-reallocation GE correction.

Carbon Stock Evolution S_{t+1} = S_t + sum_j e_j(q_j) − alpha·S_t, where alpha is the natural decay rate of atmospheric carbon and e_j(q_j) is sectoral emissions as a function of output. Firms and households treat S_t as exogenous, generating the externality.

Double Dividend In second-best environments, a carbon tax can generate two welfare gains simultaneously: correcting the carbon externality and reducing the deadweight loss from a pre-existing distortion (labor or capital tax). The paper finds revenue recycling via labor tax cuts yields 10–15% of carbon tax revenue as additional welfare gain; undertaxed capital implies the optimal carbon tax is set above the SCC.

Risk Premium in the SCC An additive term in the SCC under climate uncertainty, proportional to the variance of damage estimates times the coefficient of relative risk aversion. Empirically estimated at +$8–15/tCO₂, representing 15–25% of the base SCC.

Second-Best Optimal Carbon Tax Written as tau*_carbon = SCC + CORRECTION, where the correction depends on the sign and magnitude of pre-existing distortions. The correction is positive under undertaxed capital (raise above SCC), negative under monopolistic competition (lower below SCC), and augmented by revenue-recycling gains when labor taxes are present.

Cash or card? A structural model of payment choices

Mon, 01 Jan 0001 00:00:00 +0000

Lippi and Moracci (2026) ask how euro area households choose between cash and card payments, and whether existing theoretical models can explain observed behavior. They draw on ECB payment diary surveys (SUCH and SPACE waves I–III, 2015–2024) covering transaction-level records that include purchase size, payment method chosen, cash on hand before each transaction, and merchant acceptance of cards. This granular data allows the authors to isolate unforced payment choices — transactions in which the consumer had sufficient cash, the merchant accepted cards, and the consumer held a card — from mechanically constrained ones.

The authors document three empirical patterns. First, roughly 39% of individuals in the sample violate the simple transaction-size threshold rule of Whitesell (1989): their largest unforced cash payment exceeds their smallest unforced card payment. Second, between 27% and 49% of unforced transactions are settled by card across survey waves, contradicting the “cash burns” policy of Alvarez and Lippi (2017) under which cards are used only when cash is exhausted. Third, and most novel, the probability of card use rises sharply as implied residual cash holdings (m′ = m − s) approach zero — that is, when a cash payment would nearly deplete the wallet. This suggests a precautionary motive: consumers maintain a cash buffer to cover purchases at merchants who do not accept cards.

To rationalize these facts, the authors build an inventory-theoretic model with a compound Poisson expenditure flow (random arrival times and random transaction sizes drawn from a lognormal distribution), imperfect card acceptance (fraction ϕ of merchants accept cards, set at 0.89 for 2023–24), a fixed cost b per cash withdrawal, a fixed cost κ per card transaction (sign unrestricted), and a utility penalty u per missed purchase. The optimal policy takes an (s,S) form for withdrawals and a state-dependent threshold for payment choice. When 0 < κ < b, the agent uses cards for purchases large enough that paying cash would push balances below a threshold m̃, thereby avoiding a costly withdrawal or the risk of missing a future purchase. The critical transaction size above which cards are used, s(m), rises with cash on hand, generating the interaction the data reveals.

The model is calibrated by minimum distance to four moments from the 2023–24 SPACE wave: average cash balances relative to daily expenditure, annual withdrawal frequency, the unforced card expenditure share, and realized purchase frequency. The estimated annual cost of managing consumption transactions for the average euro area household is approximately 15 euros — a remarkably small burden. Three counterfactual experiments quantify welfare implications. Removing card access raises the annual cost from 15 to about 50 euros, implying a card ownership value of roughly 35 euros per year. Near-universal card acceptance (ϕ = 0.99) reduces the annual cost by nearly 75%, from 15 to about 4 euros, while average cash holdings fall from 130% to about 20% of daily expenditure. A complete ban on cash would cost the average consumer approximately 60 euros per year more than the current mixed system. A cashless equilibrium requires both near-universal acceptance (ϕ above 99%) and card costs at or below zero (κ ≤ 0); neither condition alone is sufficient given the estimated magnitude of the missed-purchase cost u.

Q: What is the central empirical puzzle the paper addresses? A: Existing models predict either a pure transaction-size threshold (Whitesell 1989) or a pure cash-burns rule (Alvarez and Lippi 2017). The data shows both rules are violated: 39% of individuals with observed unforced transactions of both types violate the threshold rule, and 27–49% of unforced transactions are paid by card despite available cash. Neither model alone accounts for the novel finding that card usage spikes precisely when a cash payment would nearly exhaust the wallet.

Q: What data does the paper use and what is its key advantage? A: The authors use ECB payment diaries from four survey waves: SUCH (2015–16) and SPACE I, II, III (2019, 2021–22, 2023–24). For each transaction the diary records payment method, purchase size, and cash on hand, along with merchant acceptance of each payment method. Critically, the combined information on cash holdings and acceptance allows the authors to distinguish forced from unforced payment choices, which is essential for identifying the behavioral determinants of payment method selection.

Q: What is the novel empirical fact the paper contributes? A: The paper documents that the probability of card use increases sharply as implied residual cash (m′ = m − s) approaches zero. This pattern holds across all survey waves. It is consistent with a precautionary motive: consumers use cards to avoid depleting a cash buffer that provides insurance for encounters with merchants who do not accept cards.

Q: How does the theoretical model generate the precautionary motive for cash? A: Cards are accepted in only fraction ϕ of stores; when a merchant does not accept cards and the consumer lacks cash, the purchase is missed at utility cost u. This creates an incentive to maintain positive cash balances. Combined with a fixed withdrawal cost b and a fixed card cost κ, the agent optimally targets a cash level m* and withdraws before the wallet empties (trigger m̄ > 0), holding a buffer against card-rejection events.

Q: What is the key proposition characterizing the optimal payment policy? A: Proposition 1 establishes three regimes. When κ ≤ 0, the card always dominates and is used for all purchases. When κ ≥ b, cash always dominates and cards are used only for forced transactions. In the intermediate case 0 < κ < b, a threshold m̃ ∈ (m̄, m*) divides behavior: for m < m̃ the agent uses cash for all transactions; for m ≥ m̃ the agent uses a card for any purchase exceeding a size threshold s(m), where s(m) is increasing in m. The threshold s(m) distinguishes this policy from Whitesell (1989)’s fixed threshold.

Q: How does the payment threshold s(m) vary with cash on hand, and why? A: s(m) is the purchase size above which the value loss from paying cash — pushing the agent closer to m̄ and raising the probability of a missed purchase or costly withdrawal — exceeds the fixed card cost κ. As m rises, a larger cash payment is needed to trigger this concern, so s(m) increases. This means card use becomes less frequent as cash balances grow for most of the state space, consistent with the empirical finding that cash probability rises with cash on hand.

Q: What are the calibrated parameter values and what do they imply? A: The withdrawal cost b is estimated at 0.003 EUR — very small. The per-transaction card cost κ is about 60% of b, meaning cards are cheaper to use per transaction than visiting an ATM. The cost of a missed purchase u is approximately 1 EUR. The arrival rate λ is calibrated so that about 2% of purchase opportunities are missed under the estimated card acceptance rate of 0.89. These values imply that the payment system imposes a small but non-trivial welfare burden, concentrated in the precautionary costs of maintaining cash.

Q: What is the estimated annual cost of managing consumption transactions? A: Under the optimal policy for 2023–24 parameters, the annual cost C is approximately 15 euros per household. This decomposes into opportunity costs of holding cash (RM), withdrawal costs (bn), card usage costs, and the disutility from missed purchases. The authors characterize this as “remarkably small,” suggesting the current payment system is relatively efficient from the household’s perspective.

Q: How does this cost compare across demographic groups and over time? A: Until 2019 the estimated annual cost was around 20 euros; it stabilized around 15 euros from 2021–22 onward, with the decline driven primarily by households holding less cash in the post-pandemic period. Across age groups, education levels, income brackets, and gender, each subgroup faces a very similar cost as a proportion of their expenditure, indicating limited distributional variation in payment system costs.

Q: What is the welfare value of owning a payment card? A: Setting ϕ = 0 (cash-only economy), the annual cost rises from 15 to approximately 50 euros. The value of card ownership is therefore approximately 35 euros per year. The savings come primarily from lower opportunity costs of holding cash (since card access reduces the precautionary motive) and lower disutility from missed purchases; withdrawal cost reductions play a negligible role.

Q: What happens under near-universal card acceptance (ϕ = 0.99)? A: Average cash holdings fall from about 130% of daily expenditure to about 20% of daily expenditure, a reduction of approximately 110 percentage points. The unconditional card expenditure share rises by 17 percentage points to about 93%, mostly through an increase in forced card transactions (agents more often lack cash). Unforced card expenditure falls by about 10 percentage points because the precautionary motive for using cards — preserving a cash buffer — weakens when acceptance is near-universal. The annual management cost falls by nearly 75%, from 15 to approximately 4 euros.

Q: Under what conditions does a cashless economy emerge? A: The model identifies two jointly necessary conditions: card acceptance near universal (ϕ above 99%) and card costs at or below zero (κ ≤ 0). Raising ϕ alone from the estimated 0.89 to 0.99 reduces cash use substantially but does not eliminate it, because the estimated cost of missed purchases u is large enough that consumers still maintain a small cash buffer. For κ ≤ 0, cash holdings M/e are insensitive to κ and depend only on ϕ. With current card usage costs, even near-universal acceptance would not produce a cashless economy.

Q: What is the cost of a complete cash ban? A: Under a cashless policy, the annual cost is approximately 75 euros — about 5 times the 15-euro baseline and about 25 euros more than the cash-only cost of 50 euros. A complete ban on cash would increase transaction management costs by approximately 60 euros per year for the average consumer. This is because at ϕ = 0.89, nearly 11% of purchase encounters would result in missed transactions.

Q: How does card acceptance affect cash management in the model and data? A: As ϕ falls, the precautionary motive for holding cash strengthens: the withdrawal trigger m̄ rises, average cash holdings increase, and withdrawals occur when the wallet is still substantially full. This prediction is qualitatively consistent with the empirical finding that in areas with lower card acceptance, individuals hold higher cash balances and withdraw at higher residual cash levels.

Q: What are the main limitations the authors acknowledge? A: Three caveats are identified. First, the model has no exogenous cash inflows (wage payments, gifts); incorporating Miller-Orr-style inflows could affect cash resilience estimates. Second, the card cost κ is fixed and independent of transaction size s; allowing κ(s) = κ₀ + κₛ·s would better capture reward-program economies relevant for the US. Third, merchant card acceptance is treated as exogenous; endogenizing it as a game between merchants would allow a joint welfare evaluation of acceptance decisions, payment choices, and cash management.

Unforced transactions: Transactions in which both cash and card payments are feasible — specifically, cash holdings exceed the purchase size, the merchant accepts cards, and the consumer holds a card. Isolating unforced transactions is necessary to identify behavioral determinants of payment choice, stripping out mechanical constraints imposed by cash insufficiency or merchant non-acceptance.
Precautionary cash buffer: A positive cash balance maintained above the withdrawal trigger (m̄ > 0) to insure against purchases at merchants who do not accept cards. In the model, this buffer arises because card non-acceptance combined with insufficient cash results in a missed purchase at utility cost u; the precautionary motive is stronger when ϕ is lower.
Transaction-size threshold s(m): The purchase size above which a consumer with cash holdings m optimally pays by card (when cards are available and 0 < κ < b). Unlike the fixed threshold of Whitesell (1989), s(m) is increasing in m, generating a novel interaction between cash on hand and payment method choice that the ECB diary data confirms.
Cash burns policy: The policy of Alvarez and Lippi (2017) in which cards are used only when cash is fully exhausted (m = 0). The paper documents that 27–49% of unforced transactions are settled by card across survey waves, constituting a systematic violation of this rule that the model resolves by introducing transaction-size heterogeneity and a precautionary motive.
Imperfect card acceptance (ϕ): The exogenous fraction of merchants willing to accept card payments, set at 0.89 for 2023–24 in the calibration. Imperfect acceptance is the primary driver of the precautionary demand for cash; it also determines the frequency of missed purchases under a cashless policy and is the key parameter governing whether a cashless economy can emerge.
Annual transaction management cost (C): The total yearly household cost of operating within the payment system, defined as C = RM + bn + κ·(number of card purchases) + u·(number of missed purchases). Estimated at approximately 15 euros for the average euro area household in 2023–24, decomposed across opportunity costs of cash holdings, withdrawal costs, card usage costs, and missed-purchase disutility.
Ss withdrawal policy: The optimal cash replenishment rule characterized by a trigger level m̄ and a target level m*. The agent withdraws whenever cash falls to m̄, resetting balances to m*. A strictly positive trigger (m̄ > 0) reflects the precautionary motive: the agent refills before cash is exhausted in order to maintain insurance against card non-acceptance events.

Catastrophes, Delays, and Learning

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a general model of experimentation under catastrophe risk in which the catastrophe is triggered when a stock variable exceeds an unknown threshold, but occurs only after a stochastic delay. The central contribution is the concept of the “legacy of the past”: at any planning date, past experiments may have already triggered a catastrophe that has not yet materialized, and the planner cannot observe whether triggering has occurred. The legacy is formally defined as the probability, conditional on survival, that a catastrophe was triggered in the past.

The model unifies two canonical but previously incompatible approaches in the literature. In the hazard-rate approach, the catastrophe is bound to happen and the planner manages its timing and severity. In the unknown-threshold approach, learning is instantaneous and the catastrophe is certainly avoided if the stock has not yet exceeded the threshold. Neither approach captures the intermediate case where the planner remains uncertain about whether the catastrophe is already underway. By introducing a delay governed by an exponential distribution with parameter α, the authors show that both approaches are limiting special cases: as α → ∞ (no delay), the legacy vanishes and the unknown-threshold approach is recovered; when the legacy is set permanently to one (catastrophe triggered with certainty), the hazard-rate approach is recovered.

Three benchmark stock levels anchor the analysis. QN is the long-run target absent any catastrophe risk. QD (“Damages”) is the optimal stabilization target when the planner knows a catastrophe was triggered in the past — it lies weakly below QN because the planner trades off current gains against the discounted marginal damage from raising the stock at the moment of eventual catastrophe occurrence. QE (“Experimentation”) is the stock level below which stabilization is suboptimal when the planner is certain no triggering has occurred — it also lies weakly below QN.

The paper’s two main theorems are distinguished by the ranking of QD and QE, which reflects whether mitigation strategies are effective.

Theorem 1 (QE < QD): When damage is not highly sensitive to the stock level at catastrophe time — so mitigation is relatively ineffective — optimal paths are monotonically increasing and converge to a long-run stock level Q∞ ∈ [QE, QD]. The stopping condition equates the marginal benefit of experimentation to a weighted average of the expected cost under the unknown-threshold approach (weight 1 − π) and the marginal damage under the hazard-rate approach (weight π), where π is the legacy at stopping time. A higher legacy at the stopping time is associated with a higher long-run stock level. A higher initial legacy induces fatalism: since the catastrophe is more likely already triggered, the planner shifts priority toward current consumption rather than caution, leading to more total experimentation.

Theorem 2 (QD < QE): When damage is highly sensitive to the stock level — so mitigation is valuable — the long-run target is uniquely QE regardless of the initial legacy. However, the short-run path is non-monotonic: for a sufficiently high initial legacy, the planner first reduces the stock sharply (lockdown, emissions cut) to mitigate pending catastrophe damages, then, as the legacy declines because no catastrophe occurs, gradually allows the stock to rise back toward QE. The direction of caution reverses relative to Theorem 1: a higher legacy now induces more caution, not less.

Applications include pandemic management (stock = infected population, catastrophe = health system collapse) and climate change (stock = cumulative CO2 emissions or atmospheric pollution stock). In the disease control application, whether a planner prioritizes economic production or mortality reduction determines which theorem governs, with the key ratio being production losses relative to mortality increases. For pandemic policy, Theorem 2 produces a formal learning-based rationale for non-monotonic “hammer-and-dance” policies (strict early lockdown followed by relaxation) that differs from prior explanations in the literature. In the carbon budget application, Proposition 5 formally proves that higher initial legacy raises the optimal carbon budget under Theorem 1 conditions, and can imply unbounded consumption (certainty of catastrophe) above a critical legacy threshold π*. Under Theorem 2 conditions (Proposition 6), the optimal policy can involve first reducing then expanding the stock before stabilizing, with both transition dates increasing in the initial legacy.

Q: What is the “legacy of the past” and how is it computed? A: The legacy πt is defined as the probability, conditional on survival to date t, that a catastrophe was already triggered by past experiments. Formally, πt = 1 − [1 − F(Qt)] / pt, where Qt is the highest stock level ever reached, F is the prior distribution over the threshold, and pt is the survival probability. A past experiment at time t’ contributes to the current legacy with weight exp[−α(t − t’)], so recent experiments matter more than distant ones. As time passes without catastrophe, the legacy of any fixed past experiment declines geometrically at rate α.

Q: How do the three benchmark stock levels QN, QD, and QE relate to each other? A: QN is the optimal long-run stock without any catastrophe. QD is defined by the condition where the marginal net benefit of increasing the stock — ν(Q) − [α/(α+δ)]D’(Q) — equals zero, and satisfies QD ≤ QN. QE is defined by ν(Q) − [α/(α+δ)]ρ(Q)D(Q) = zero, and also satisfies QE ≤ QN. The ranking between QD and QE depends on whether damage is more sensitive to the marginal increase in stock at catastrophe time (which pushes QD below QE) or to the level of the stock at triggering (which pulls QD above QE).

Q: What is the key optimality condition in Theorem 1 and how does it unify prior approaches? A: The stopping condition (equation 15) states: ν(QT) = [α/(α+δ)] × [(1 − πT)ρ(QT)D(QT) + πT D’(QT)]. When πT = 0 (no legacy, unknown-threshold limit), this reduces to the experimentation stopping condition of Tsur and Zemel, governed by the hazard rate ρ(QT) times expected loss D(QT). When πT = 1 (full legacy, hazard-rate limit), it reduces to the damage-mitigation condition governed by marginal damage D’(QT). The legacy at stopping time thus serves as the mixing weight between the two canonical approaches, embedding both as special cases.

Q: How does the initial legacy affect total experimentation under Theorem 1 versus Theorem 2? A: Under Theorem 1 (QE < QD), a higher initial legacy π0 leads to more total experimentation (higher Q∞), because the planner becomes fatalistic — since the catastrophe is more likely already triggered and mitigation is relatively ineffective, current consumption is prioritized. Proposition 5 formally proves this for the carbon budget application: the optimal stopping date T and optimal budget QT are nondecreasing in π0. Under Theorem 2 (QD < QE), a higher legacy triggers more caution in the short run (larger reduction in the stock during the mitigation phase), but the long-run target QE remains the same regardless of π0.

Q: What generates non-monotonic policies in Theorem 2, and what does this look like in the pandemic application? A: Non-monotonicity arises because the optimal response to a high legacy is first to reduce the stock sharply to limit catastrophe damages (since damage is sensitive to the stock level), and then, as time passes without catastrophe and the legacy declines, to allow the stock to recover. In the disease control application with high mortality weight, a complete lockdown is optimal in the first phase whenever the legacy is strictly positive. As the legacy declines, the lockdown is gradually relaxed, and eventually the infection level returns to its pre-lockdown level. Figures 3 and 4 show that a higher initial legacy (π0 = 0.1, 0.5, or 0.9) leads to a longer lockdown and slower recovery, though all paths converge to the same long-run infection level.

Q: How does the model’s disease control application determine which theorem governs? A: Lemma 2 states that if 1 / [1 + (Y(r+d) − Y*) / (wµdI^D)] < ρ(I^D), then I^E < I^D and Theorem 1 applies; otherwise I^E > I^D and Theorem 2 applies. The key ratio is (Y(r+d) − Y) / (wµ*d), the production loss relative to mortality increase. A planner who weights economic activity heavily (large production loss ratio) falls under Theorem 1 and tolerates rising infections; a planner who weights mortality heavily falls under Theorem 2 and imposes an initial lockdown.

Q: What is the carbon budget result under Theorem 1 (Proposition 5)? A: Under the condition u1 > [α/(α+δ)]v0 (marginal consumption value exceeds discounted marginal damage), Theorem 1 applies and there exists a critical legacy threshold π* such that: below π*, the planner consumes maximally (qt = q-bar) until a finite date T and then stops, with QE < QT < QD; above π*, the planner consumes maximally forever, triggering the catastrophe with certainty. The stopping date T and the optimal budget QT are nondecreasing functions of initial legacy π0, formally proving that higher past emissions (captured through legacy) justify higher future carbon budgets in this model.

Q: What is the carbon budget result under Theorem 2 (Proposition 6)? A: Under condition u1 < [α/(α+δ)]v0, QD < QE and Theorem 2 applies. Starting from Q0 above QE, if π0 is small enough (specifically u1 > π0[α/(α+δ)]v0), the optimal policy is to stabilize the stock forever at Q0. Otherwise, there exist two finite dates t1 < t2, both increasing in π0, such that the planner first reduces the stock at maximum rate (qt = q-bar-negative) for t < t1, then expands at maximum rate for t1 < t < t2, then stabilizes at Q0 forever. The optimal carbon budget is Q0 in all cases, showing that the long-run target is independent of legacy under Theorem 2.

Q: How does the model relate to the hazard-rate literature formally? A: Papers such as Nordhaus and others that use an exogenous hazard rate h(Qt) for catastrophe — yielding survival probability pt = p0 exp(−∫h(Qτ)dτ) — are shown to be equivalent to the special case where the catastrophe was triggered in the past (legacy = 1 permanently). Their formulation corresponds to assuming α is constant and the legacy is identically one, which reduces the law of motion for pt to pt = p0 exp(−αt). The key difference is that in the hazard-rate approach the planner can reduce the arrival rate by lowering the stock (h is increasing in Q), whereas in the authors’ model the delay parameter α is constant and policy affects only damages.

Q: What is the role of the exponential delay distribution assumption? A: The assumption that the delay τ follows an exponential distribution with parameter α is made for tractability. Under this assumption, the entire past trajectory of the stock (Qt)t≤0 can be summarized by just two state variables — the highest stock on record Q0-bar and the initial legacy π0 — because the exponential “memoryless” property means that the additional expected waiting time until catastrophe occurrence does not depend on how long the triggering has already been in effect. Without this assumption, the full chronicle of past experiments would be required as a state variable, making the problem intractable.

Q: What happens when the delay parameter α approaches zero or infinity? A: When α → ∞ (instantaneous catastrophe upon triggering), pt = 1 − F(Qt) and the legacy is identically zero, recovering the Tsur-Zemel unknown-threshold approach (Proposition 3). The optimal path converges to QE0 from below or stabilizes if already above QE0. When α → 0 (infinite delay, effectively no catastrophe), QE = QD = QN and the problem reduces to the simple stock-flow problem (Proposition 1), with the optimal path converging monotonically to QN.

Q: Does the model allow for damage mitigation after triggering but before occurrence? A: Yes, this is a key feature. The continuation payoff after catastrophe occurrence is V(QT) where QT is the stock level at the time of occurrence T, not at triggering time T(S). This means the planner can reduce the stock after triggering to lower damages — analogous to a skater turning back toward shore after the ice first cracks. The assumption that V depends on the stock at occurrence rather than at triggering or at the maximum historical level is what allows this mitigation channel and is explicitly noted as a modeling choice.

Legacy of the past (πt): The probability, conditional on survival to date t, that past experiments have already triggered a catastrophe. Formally πt = 1 − [1 − F(Qt)] / pt. Recent experiments contribute more to the legacy than distant ones, with contribution decaying at rate α. The legacy is zero when α → ∞ and is the central state variable bridging the paper’s two canonical extremes.

QE (“Experimentation” threshold): The stock level at which the net marginal gain from further experimentation, defined as ν(Q) − [α/(α+δ)]ρ(Q)D(Q), equals zero, under the assumption that no catastrophe has been triggered. Below QE, stabilization is suboptimal; above QE, the planner does not experiment further when the legacy is zero.

QD (“Damages” threshold): The stock level at which the net marginal benefit from holding the stock, defined as ν(Q) − [α/(α+δ)]D’(Q), equals zero, under the assumption that the catastrophe is known to have been triggered. QD ≤ QN and represents the optimal long-run target when the hazard-rate approach applies.

Marginal payoff ν(Q): Defined as uq(0, Q) + (1/δ)uQ(0, Q), it measures the net gain from marginally increasing the flow when the stock is stabilized at Q. It is strictly decreasing in Q under Assumption 1 and equals zero at QN.

Damage function D(Q): Defined as (1/δ)u(0, Q) − V(Q), it measures the welfare loss from catastrophe occurrence when the stock is Q at occurrence time, relative to permanent stabilization at Q. Assumed weakly positive and weakly increasing in Q.

Survival probability (pt): The probability, computed from prior beliefs F at the beginning of times, that the catastrophe has not yet occurred by date t. Its law of motion is ṗt = α[1 − F(Qt) − pt], driven solely by the catastrophe parameter α and the current maximum stock Qt.

Fatalism (under Theorem 1): The policy implication that a higher legacy — meaning a higher probability the catastrophe is already triggered — leads the planner to increase the stock further and accept more experimentation, because mitigation is relatively ineffective (QE < QD) and current consumption must be enjoyed before the catastrophe arrives.

Central bank communication by ??? The economics of monetary policy leaks

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper investigates the economics of monetary policy leaks — anonymous disclosures of confidential information by insiders to the media — focusing on three central questions: (1) Are leaks random accidents, strategic individual disclosures, or institutionally authorized “plants”? (2) Do leaks shape public (financial market) views, and by how much? (3) Can attributed (named) communication by central bank officials mitigate the effects of leaks?

Data and Setting

The authors study the Eurosystem (ECB and euro area National Central Banks) over January 2002 to December 2021. Their primary data source is a novel database of 368 unique policy-relevant leaks — assembled by manually filtering and classifying more than a million news items from Reuters, Bloomberg, and Market News International archives — with precise minute-level timestamps. Topics covered include: policy rates (178 leaks), unconventional monetary policy/UMP (207 leaks), economic growth (47), inflation (41), and euro exchange rate (36); individual leaks may cover multiple topics. They complement this with a dataset of 7,883 attributable public statements by ECB Governing Council members, identified via keyword filtering and machine learning classification of the Reuters News Archive.

Methodology

The paper employs four main empirical strategies. First, high-frequency event studies using asymmetric windows (5 minutes before to 30 minutes after an event) compare absolute market reactions in OIS rates across the full term structure (3M to 10Y) and in the EURO STOXX 50 across leaks, 5,000 randomly sampled placebo events, and attributable statements. Second, Poisson regression models relate the number of leaks per policy meeting to proxies for Governing Council disagreement (Italian-German sovereign yield spread, inter-quartile range of national inflation rates, number of attributable statements per meeting) and a dummy for quarterly macroeconomic projection releases. Third, a regression framework tests whether leaks move market expectations toward the subsequent policy outcome — identifying whether leaks are informative about the direction of policy. Fourth, an augmented version of the Tillmann (2021) model relates end-of-day changes in longer-term OIS rates to high-frequency monetary policy surprises, interacted with dummies for post-announcement leaks and attributable statements.

Main Findings

Incidence and timing. The number of Eurosystem leaks peaked at 36 in 2019 (more than four per policy meeting on average) before declining by more than one third following the start of Christine Lagarde’s presidency in November 2019. Leaks cluster around policy meetings and, since 2015, have shifted notably from before meetings to after meetings, a shift driven by leaks related to UMP. Leaks occur even during the ECB’s quiet period, when policy-makers are formally restricted from public statements on policy-sensitive topics.

Leaks are not accidents. Poisson regressions reveal that the number of leaks per meeting is significantly and positively associated with proxies for Governing Council disagreement: every additional percentage point in the Italian-German sovereign yield spread is associated with approximately half an additional leak per meeting. The propensity of a policy change increases by four to six percentage points with each additional pre-meeting leak (statistically significant at the 5% or 10% level). The specification explains around 15% of the variation in leak counts.

Market impact. Market movements around leaks are up to 85% larger than those around placebo events. Leaks trigger market reactions that are consistently larger than those of attributable statements by individual Governing Council members across the entire OIS term structure and in equities — a result robust to controlling for distance to policy meetings. Rate leaks mainly move the short and medium end of the yield curve; UMP leaks affect the long end and equities. Leaks about general economic conditions (growth, inflation, exchange rate) produce little statistically significant market response.

Leaks are uninformative about policy direction. Conditional on a pre-meeting leak occurring, the average leak does not move market rates closer to the levels prevailing directly after the subsequent policy announcement. By contrast, attributable statements systematically do reduce this distance. This asymmetry implies that leaks predominantly reflect minority opinions within the Governing Council. Consistent with this, leaks counteract prevailing trends in market expectations at the short end of the yield curve (as established by a negative coefficient on the interaction between the prevailing seven-day pre-leak trend and the leak dummy).

Leaks are not plants; attributed communication mitigates their effects. Post-announcement leaks dampen the transmission of monetary policy surprises to longer-term rates (negative and significant interaction coefficient in the augmented Tillmann framework). Attributed statements by ECB Executive Board members, by contrast, systematically move in the direction opposite to the preceding leak across most of the yield curve, partially reversing leak-induced market moves. More intense pre-leak attributable communication is also associated with lower market impact of the subsequent leak, across most maturities. These results jointly indicate that most Eurosystem leaks originate from individual insiders with minority opinions rather than constituting institutional plants.

Scope Conditions

Results pertain to the Eurosystem committee setting, where decision-making is broadly consensus-based and voting records are not published; they may not fully generalize to institutions with concentrated decision-making power. The study measures effects on financial markets, not broader public opinion.

In depth

Q1. How is a “leak” defined in this paper, and how are Eurosystem leaks identified empirically?

A leak is defined as a disclosure of confidential information by an insider to the media with an expectation of anonymity. Eurosystem leaks are identified from Reuters, Bloomberg, and Market News International archives (2002–2021) using keyword-driven pre-filtering followed by manual classification of “candidate” items. The resulting database contains 1,253 news items that aggregate to 368 unique policy-relevant leaks with minute-level timestamps. Policy-relevant leaks touch on: policy rates, unconventional monetary policy tools, economic growth, inflation, or the euro exchange rate; leaks about local economic conditions, banking regulation, or managerial appointments are excluded.

Q2. What are the broad trends in the number and topic composition of Eurosystem leaks over 2002–2021?

The number of leaks rose sharply in the second half of the sample, peaking at 36 in 2019 (more than four per meeting on average). Since Christine Lagarde took over the ECB presidency in November 2019, leaks fell by more than one third from that peak. The topic composition shifted substantially over time: policy-rate leaks predominated in the earlier period, while leaks related to UMP came to dominate in the 2015–2021 sub-period.

Q3. How does the timing of leaks within the policy meeting cycle change across sub-periods?

In the full sample, leaks cluster in the run-up to policy meetings and immediately following announcement days (both on the announcement day itself and the following Friday). Since 2015, a notable shift occurs from pre-meeting to post-meeting timing, driven specifically by leaks related to UMP. The authors attribute this shift to the expectation-management role of UMP: post-meeting leaks allow dissenting insiders to reshape market expectations that are otherwise guided by official press releases and press conferences.

Q4. What regression evidence supports the view that leaks are not random accidents?

Poisson regressions of the number of leaks per meeting on disagreement proxies find significant positive coefficients on: the lagged Italian-German sovereign yield spread (about half a leak more per meeting for each additional percentage point of spread), the inter-quartile range of national inflation rates, and the number of attributable statements per meeting. Meetings coinciding with the release of quarterly macroeconomic projections also attract significantly more leaks. These results are robust to replacing the disagreement proxies with a binary dissent index based on Q&A sessions at ECB press conferences (Tillmann, 2021), even after excluding disagreement-related leaks from the dependent variable to address endogeneity. The model explains about 15% of the variation in leak counts.

Q5. Does the number of pre-meeting leaks predict policy changes?

Yes. The propensity of a monetary policy change increases by four to six percentage points with each additional pre-meeting leak (significant at the 5% or 10% level). This signal about the propensity of change (not the direction) is hard to square with the random accidents hypothesis.

Q6. How large are the financial market reactions to leaks relative to placebo events and to attributable statements?

Market movements around leaks are up to 85% larger than the average size of market reactions to 5,000 randomly sampled placebo events. When leaks are compared directly to attributable statements (with leaks as the baseline and fixed effects for year, month, weekday, and hour), average absolute market moves around leaks are consistently larger across the entire term structure of OIS rates and for the EURO STOXX 50. This result is robust to differences in distance to policy meetings, with size differences across the full term structure persisting for periods far from meetings; near meetings, differences narrow but the average market reaction to leaks never falls below that to attributable statements.

Q7. Do the market effects of leaks differ by topic?

Yes. Leaks about policy rates primarily move the short and medium end of the yield curve. Leaks about UMP tools affect the long end of the curve and equities. Leaks about general economic conditions (growth, inflation, euro exchange rate) do not produce statistically significant market reactions, consistent with the interpretation that economic condition leaks require more interpretation before their implications for the policy path become apparent.

Q8. Do leaks move market expectations in the direction of the subsequent policy outcome?

No. The average pre-meeting leak does not reduce the absolute distance of market rates to post-announcement levels. This result holds across maturities from 3M to 10Y and is robust to separating leaks inside and outside the ECB’s quiet period. Attributable statements, by contrast, systematically reduce this distance (Table 7). The failure of leaks to align expectations with outcomes is interpreted as evidence that leaks predominantly reflect minority views within the Governing Council rather than information held by the decisive voter.

Q9. Do leaks counteract or reinforce prevailing trends in market expectations?

Leaks counteract prevailing trends. The regression of market reactions to leaks and placebo events on the seven-day pre-event trend reveals a significantly negative interaction between the trend and the leak dummy at the short end of the yield curve. This result is driven specifically by leaks about policy rates.

Q10. Do post-announcement leaks dampen the transmission of monetary policy surprises to longer-term rates?

Yes. In the augmented Tillmann (2021) framework, the interaction of the high-frequency 2Y monetary policy surprise with a dummy for post-announcement leaks is negative and significant for 2Y, 5Y, and 10Y OIS rates. In contrast, the interaction with a dummy for post-announcement attributable statements is positive and significant across maturities, indicating that attributed communication reinforces the official policy signal. These two results jointly show that leaks weaken official policy announcements while attributed communication strengthens them.

Q11. Does more intense pre-leak attributable communication reduce the market impact of subsequent leaks?

Yes. Using an intensity measure that weights each attributable statement by the inverse of its distance in hours to the subsequent leak (covering a window from 36 hours to 30 minutes before the leak), the paper finds a significant negative relationship between pre-leak communication intensity and the absolute market reaction to the leak, controlling for year, month, weekday, and hour fixed effects. This holds across most maturities.

Q12. Does the market impact evidence support the “plant” hypothesis?

No. If leaks were institutional plants intended to prepare markets for new policy, one would expect the ECB Executive Board — which controls official communication — to subsequently reinforce the signal from leaks. Instead, attributable statements by ECB-affiliated Governing Council members are systematically negatively correlated with the market direction of the preceding leak across the yield curve, with significant coefficients at medium maturities. NCB Governor statements show weaker and more ambiguous effects, potentially because their statements generate smaller average market movements rather than reflecting a lack of willingness to counteract leaks.

Q13. Why do markets react to leaks even though leaks are generally uninformative about policy outcomes?

The paper offers three candidate explanations: (1) automated trading algorithms that do not distinguish between attributed and anonymous communication; (2) leaks serve as a coordination device in the spirit of Morris and Shin (2002), amplifying even noisy signals; (3) media-reporting models such as Nimark (2014) and Chahrour et al. (2021) predict that “man-bites-dog” news — unusual events such as revelations of committee disagreement — shift beliefs beyond their true information content. Leaks are unusual both in frequency (far less common than attributed statements) and in content (they reveal disagreement that rarely surfaces in official communication).

Q14. What are the implications for the measurement of monetary policy shocks from high-frequency identification?

The paper notes that Eurosystem leaks frequently occur shortly before or after official policy announcements. Pre-announcement leaks can shift market expectations before the start of standard event windows, reducing the measured surprise component of official announcements. Post-meeting leaks dampen the end-of-day effects of announcements. In both cases, standard high-frequency surprise instruments extracted from official announcements alone may miss the full extent of new information available to market participants, suggesting that accounting for leaks could improve the relevance of high-frequency instruments used in monetary policy identification.

Q15. What are the implications for the design of central bank quiet periods?

The ECB’s quiet period ends with the policy announcement, whereas the Federal Reserve’s extends to the day after the meeting. Based on the finding that post-announcement leaks dampen policy announcement effects while post-announcement attributed statements reinforce them, the paper suggests that permitting attributed communication shortly after policy decisions may help mitigate the market impact of post-announcement leaks.

Key Concepts

Monetary policy leak (“sources story”): In this paper, a leak is defined as a disclosure of confidential information emanating from an insider within the Eurosystem (ECB or NCB staff or policy-makers) that is transmitted to financial media with an expectation of anonymity for the source. The paper excludes whistle-blower cases and focuses on leaks where anonymity keeps attention on the content rather than the identity of the source. Leaks are distinct from “plants” (formally authorized institutional disclosures intended to advance the institution’s goals) and from “pleaks” (the middle ground).

Plant: An authorized or semi-authorized anonymous disclosure of confidential information made for the purpose of advancing the public institution’s own goals and interests, as distinct from a leak that originates from an individual insider’s personal agenda. The paper tests and rejects the plant hypothesis for most Eurosystem leaks on the basis that ECB Executive Board members’ attributed statements systematically counteract the market impact of leaks.

Single voice principle: The ECB’s communication norm requiring that Governing Council members discuss and resolve disagreements internally while publicly representing the official policy stance. This principle creates a setting where individual members with minority views may resort to anonymous communication as a way to express dissent “off-protocol.”

Quiet period (purdah): The ECB’s rule requiring policy-makers to refrain from public statements on policy-related topics in the seven days before each Governing Council monetary policy meeting. Leaks cluster during this period despite the restriction, supporting the non-random interpretation of leaks.

Attributable (named) statement: A public statement clearly attributed to a specific, named member of the ECB Governing Council, reported as a breaking-news headline. Attributable statements serve both as a comparison benchmark for measuring the market impact of leaks and as a mitigation instrument when they counteract leak-induced market moves.

Pre-leak communication intensity (lambda): The paper’s measure of the intensity of attributable communication in the 36-hour window before a given leak, defined as the sum of inverse time distances (in hours) from each attributable statement to the leak. A higher value means more recent and/or more numerous attributed statements precede the leak.

High-frequency event study window: The paper uses an asymmetric window starting 5 minutes before and ending 30 minutes after a leak’s timestamp. Market reactions are measured as the change in the median OIS quote during the 10 minutes after the window versus the 10 minutes before, matching methodology used for both leaks and attributable statements to ensure comparability across communication types.

Post-announcement leak dummy: An indicator taking the value of one if at least one leak occurs between the end of the official ECB monetary policy announcement window (15:50 CET) and end of trading hours on the announcement day. Used in the augmented Tillmann (2021) regression to measure whether leaks dampen the transmission of monetary policy surprises to longer-term rates.

Central bank reputation with noise

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. How does noise in the mapping from central bank actions to realized inflation affect the existence and character of reputational equilibria in monetary policy? Specifically, can a central bank that faces uncertainty about whether it is perceived as “hawkish” or “dovish” sustain a pure strategy separating equilibrium, and how should each type behave as a function of its current reputation?

Model and Methodology. Amador and Phelan build on the monopolistic-competition, cash-in-advance framework of Chari, Christiano, and Eichenbaum (1998) and extend it to allow for (i) two central bank types — hawkish (type 1, high penalty γ₁ for inflationary actions) and dovish (type 2, lower penalty γ₂ < γ₁) — whose identity is private information; (ii) type switching governed by a Markov process, with probability δ that a hawkish bank is replaced by a dovish one and probability ε that a dovish bank is replaced by a hawkish one; and (iii) noise between the central bank’s chosen action μᵢ and realized money growth μₐ, which is drawn from a density f(μₐ|μᵢ) with full support. The equilibrium concept is pure symmetric Markov perfect equilibrium, in which all strategies are functions only of the public Bayesian posterior ρ that the current central bank is hawkish. The paper proceeds analytically to characterize no-pooling results and then computationally to demonstrate existence of separating equilibria.

Main Findings.

No pooling equilibria exist (analytical). Propositions 2 and 3 establish that no pure symmetric Markov equilibrium can have both types choosing the same positive action for any reputation ρ, as long as γ₁ ≠ γ₂ and Assumption 1 (pricing distortion sufficiently severe) holds. The intuition: if both types pool, realized inflation is uninformative, reputation does not change, and there are no dynamic incentives — but different static incentives (γ₁ ≠ γ₂) then imply different optimal actions, a contradiction.
Without sufficient noise, separating equilibria also fail to exist. In the no-noise limit, Bayesian updating forces the dovish bank’s reputation to jump to its maximum after one period of mimicking the hawkish action, making mimicry cheap when the discount factor β is high or the type-persistence probability ε is low. This makes the incentive-compatibility constraint for the dovish bank very difficult to satisfy, potentially precluding existence of a separating equilibrium.
With sufficient noise, pure strategy separating equilibria exist and have appealing properties (computational). The benchmark parameterization sets α = 1, σ = 5, β = 0.99, h(μ) = 0.5μ², ε = δ = 0.02, and the noise distribution such that the hawkish type’s unconstrained target would deliver mean inflation of 2% and the dovish type’s 3%. Under these parameters:
- In the full-information (known-type) world: price P = 1.313 for the hawkish type and P = 1.338 for the dovish type, with E[log(c) − αc] = −1.0297 and −1.0320 respectively, versus the efficient benchmark of −1.
- In the reputational equilibrium, both types choose lower inflationary actions than they would absent reputation considerations — because reputation is valuable (higher ρ lowers household prices and thus improves welfare for both types).
- Both types’ optimal actions are U-shaped in reputation ρ: they are most restrained — choosing the lowest inflationary actions — when ρ is middling (interior), because Bayesian updating is most sensitive (and thus the reputation cost of inflating is greatest) at interior beliefs, while it is difficult to move extreme beliefs.
- Average equilibrium inflation is 2.1%, which lies below the weighted average of unconstrained type targets (2.5% given equal switching probabilities), demonstrating that reputation concerns compress inflation outcomes.
Ergodic distribution of reputation remains interior. Starting from ρ = 0.5, expected reputation conditional on being hawkish stays below 0.63 and conditional on being dovish stays above 0.38, reflecting that noise and type switching prevent reputation from collapsing to its extremes.
Welfare implications. The hawkish type is made worse off by ongoing household uncertainty (relative to the reference game in which type is immediately revealed), while the dovish type is made better off. Households are better off under continuing uncertainty than under immediate revelation, unless reputation is near its maximum — because uncertainty suppresses inflationary temptations for both types.

Scope Conditions. Results apply within a monopolistic-competition, cash-in-advance economy with discrete time, infinite horizon, and Markov strategies. The no-pooling result requires Assumption 1 (the pricing distortion is sufficiently severe that the central bank has a positive incentive to inflate from μ = 0). The no-noise existence failure is an informal argument holding fixed discount and type-switching parameters. Computational results are specific to the benchmark parameterization but are verified to be robust to variation in β, σ, γ₁, γ₂, ε, and δ.

In depth

Q1. What is the fundamental time-inconsistency problem in the underlying Chari et al. (1998) economy, and how does the paper extend it?

A1: In the Chari et al. (1998) monopolistic-competition cash-in-advance economy, households exploit market power when setting prices, and the cash-in-advance constraint depresses consumption efficiency; this creates an ex-post temptation for the central bank to inflate and partially offset these distortions, even though in equilibrium such inflation is anticipated and only worsens inefficiencies. Equilibrium consumption equals (1/α) × ((σ−1)/σ) × (β/(1+μ)), compounding a monopoly distortion (σ−1)/σ < 1 and a cash-in-advance distortion β/(1+μ) < 1 below the efficient level 1/α. Amador and Phelan add household uncertainty about the central bank’s type — captured by the Bayesian posterior ρ that the bank is hawkish — allowing reputation to be endogenously determined and to feed back into equilibrium pricing.

Q2. Why does reputation matter only through differences in inflation costs γᵢ and not through differences in effective discount factors alone?

A2: Proposition 1 establishes that if γ₁ = γ₂ (equal inflation penalties), then even if the two types have different effective discount factors β₁ = β(1−δ) ≠ β₂ = β(1−ε), there exists a pooling Markov equilibrium in which both types choose the same action μ* and reputation plays no role. When both types have identical static incentives, they will always choose the same action given that reputation doesn’t affect payoffs in such an equilibrium. Hence the relevant dimension of heterogeneity for reputation to matter is the inflation cost parameter γᵢ, not patience.

Q3. What is the formal argument that no pooling equilibrium can exist when γ₁ ≠ γ₂?

A3: Propositions 2 and 3 provide the formal argument. If both types pool at any reputation ρ with a common positive action μ, Bayesian updating implies that ρ⁺ is independent of the money growth realization μₐ. The first-order condition for type i then reduces to the static condition (∂E[log(c) − αc|μ]/∂μ) = γᵢh’(μ), which cannot hold simultaneously for types 1 and 2 since γ₁ ≠ γ₂ and h’(μ) > 0 for μ > 0. This logic rules out pooling at the stationary reputation ρ* = ε/(δ+ε) in Proposition 2 and at any reputation where μ > 0 in Proposition 3.

Q4. Why does noise facilitate the existence of separating equilibria?

A4: Without noise, if types separate, observing the hawkish action reveals the bank is hawkish with certainty, pushing reputation to its maximum (1−δ) in a single period. This makes mimicry extremely cheap for the dovish type when β₂ is large or ε is small: the incentive compatibility condition requires that the dovish type’s static gain from choosing its own action exceeds the value gain from jumping to the best possible reputation, which is a very stringent requirement. With noise, mimicry generates only a probabilistic shift in beliefs rather than a discrete jump to the extreme, so the dovish type must maintain the hawkish action repeatedly to achieve a reputational gain — making mimicry costly enough that the incentive compatibility condition can be satisfied.

Q5. What is the “reference game” and what analytical purpose does it serve?

A5: The reference game is a variant in which the central bank’s type is fixed and is revealed to households immediately after they set prices at date t = 0. From t = 1 onward, the game reduces to the full-information, single-type game of Section 4. This allows the authors to isolate the “direct” effect of reputation — the fact that expected type affects equilibrium prices today — from the “indirect” or strategic effect of the central bank actively managing its reputation. In the numerical example, the reference-game prices form the upper dashed line in Figure 1, while the actual game’s prices form the lower solid line, with the gap between them attributable to the central bank’s incentive to restrain inflation in order to protect reputation.

Q6. What are the equilibrium price and welfare levels in the benchmark numerical example, and how do they compare to efficient and full-information benchmarks?

A6: The efficient benchmark delivers log(c) − αc = −1 with consumption c* = 1/α = 1. Under full information with only the hawkish type present, P = 1.313 and E[log(c) − αc] = −1.0297; under only the dovish type, P = 1.338 and E[log(c) − αc] = −1.0320. In the reputational equilibrium, prices lie below the full-information mixed benchmark for any given ρ (the solid line in Figure 1 lies below the dashed reference-game line), reflecting that the central banks’ desire to maintain reputation leads both types to restrain inflation beyond what the direct price effect alone would induce.

Q7. How does the U-shape of optimal central bank actions in reputation arise, and what does it imply for policy?

A7: The U-shape arises because Bayesian updating is most powerful at interior beliefs: for extreme reputations (near ε or 1−δ), any given realization of money growth moves the posterior relatively little, so the reputational cost of inflating is small. For interior (middling) reputations, the same action shifts the posterior substantially, making reputation more sensitive to inflation choices and thus increasing the marginal cost of inflating. Both types therefore choose their minimum inflationary actions at middling reputations. The policy implication is that a hawkish central bank with a very low reputation (following a run of high realized inflation outcomes) should not dramatically tighten, because further contraction does relatively little for its reputation until nature delivers enough favorable realizations to move it to a more interior range.

Q8. What happens to the ergodic distribution of reputation and inflation, and what does this imply about the persistence of reputational dynamics?

A8: Starting from ρ = 0.5, expected reputation remains in the interior: above 0.38 for the dovish type and below 0.63 for the hawkish type. The ergodic distribution of ρ (Figure 5) concentrates at interior values rather than the poles, showing that noise and type switching prevent reputation from stabilizing at extremes. The ergodic inflation distribution (Figure 6) has an average of 2.1%, compared to 2% under an all-hawkish world and 3% under an all-dovish world. Because ε = δ (types are equally likely in the long run), the unconstrained-type-weighted average would be 2.5%, so reputational incentives reduce equilibrium average inflation by approximately 0.4 percentage points.

Q9. Who gains and who loses from ongoing type uncertainty relative to immediate revelation?

A9: The hawkish type’s value function (Figure 3a) lies below the reference-game dashed line for intermediate reputations, indicating that the hawkish type is made worse off by uncertainty — it must bear the cost of restraining inflation beyond what is statically optimal in order to signal its type, but the households partially “blame” it for high realized inflation regardless. The dovish type (Figure 3b) is made better off under continuing uncertainty because its reputation benefits from households’ inability to perfectly distinguish types. Households (Figure 3c) are better off under uncertainty unless reputation is very high, because uncertainty suppresses inflation temptations for both types and keeps prices lower.

Q10. What happens to equilibrium behavior under robustness checks on key parameters?

A10: When the discount factor β or the elasticity of substitution σ decreases, both types inflate more and prices rise. When the hawkish type’s penalty γ₁ decreases (becomes less hawkish), both types inflate more and prices rise. When the dovish type’s penalty γ₂ decreases (becomes more dovish), the dovish type inflates more and, somewhat counterintuitively, the hawkish type inflates less, leaving prices roughly unchanged but slightly higher. When switching probabilities ε or δ increase, prices rise and both types inflate more, analogously to a decrease in β. Across all robustness exercises, the dovish type never inflates less than the hawkish type — consistent with Proposition 1’s implication that the inflation-cost difference γ₁ − γ₂ is the fundamental driver of separation.

Key Concepts

Hawkish type (type 1): A central bank that receives a relatively large negative payoff γ₁h(μᵢ) for taking inflationary actions, where γ₁ > γ₂. In the paper’s own sense, this type is not behavioral — it optimizes fully and can choose any action — but has a strong intrinsic cost to inflation, making it prefer lower money growth rates ceteris paribus.

Dovish type (type 2): A central bank with a lower penalty parameter γ₂ < γ₁ for inflationary actions. Like the hawkish type, it is fully strategic and optimizing, differing only in the magnitude of its intrinsic inflation cost.

Reputation (ρ): The Bayesian posterior probability that households assign to the current central bank being the hawkish type. It is the single payoff-relevant state variable in the Markov equilibrium, evolving through Bayes’ rule applied to realized money growth and type-switching probabilities.

Pure symmetric Markov perfect equilibrium: An equilibrium in which all households set the same price and consume the same amount (symmetry), and all strategies — prices P(ρ), central bank actions μ₁(ρ) and μ₂(ρ), and household consumption c(μₐ, ρ) — depend on history only through the current reputation ρ (Markov). The paper focuses exclusively on pure (non-mixed) strategy equilibria.

Pooling equilibrium: An equilibrium in which both types choose the same action μ₁(ρ) = μ₂(ρ) at some reputation ρ. The paper proves analytically that no pooling equilibrium can exist when γ₁ ≠ γ₂ and the pricing distortion is sufficiently severe (Assumption 1).

Separating equilibrium: An equilibrium in which μ₁(ρ) ≠ μ₂(ρ) for all ρ, so that realized money growth outcomes are informative about type and reputation evolves non-trivially. The paper argues that sufficient noise is necessary for such equilibria to exist.

Effective discount factor (βᵢ): The discount factor net of type-switching: β₁ = β(1−δ) for the hawkish type (which survives as hawkish with probability 1−δ) and β₂ = β(1−ε) for the dovish type. Central banks care only about payoffs while they are active, so effective discounting captures both time preference and expected tenure.

Noise (disconnection between actions and outcomes): The stochastic wedge between the central bank’s chosen action μᵢ and realized money growth μₐ, governed by a density f(μₐ|μᵢ) with full support. In the paper’s framework, noise is not merely a nuisance but a structural feature that makes reputational equilibria possible by preventing single-period complete revelation of type.

Changing Opportunity: Sociological Mechanisms Underlying Growing Class Gaps

Mon, 01 Jan 0001 00:00:00 +0000

This paper documents sharp divergent trends in intergenerational economic mobility by race and class in the United States across the 1978 to 1992 birth cohorts, and investigates the causal mechanisms driving those changes. The core empirical facts are two: between 1978 and 1992 birth cohorts, the earnings gap between white children from high-income versus low-income families grew by approximately 28–30% (the “white class gap”), while the earnings gap between white and Black children from low-income families shrank by approximately 27–30% (the “white-Black race gap”). These twin trends — growing class gaps and shrinking race gaps — appear consistently across earnings, employment rates, educational attainment, SAT/ACT scores, incarceration, marriage, and mortality, and they hold in nearly every region of the country.

The data are drawn from de-identified federal income tax returns linked to decennial census records and the Numident database, covering 57 million children born between 1978 and 1992, with information on parental and child incomes, employment, marital status, mortality, and residential location, supplemented by ACS educational attainment and linked SAT/ACT records covering 24.8 million students. Children’s outcomes are measured primarily as household income percentile ranks at age 27.

In dollar terms, the white class gap (mean income difference between children raised at the 25th vs. 75th parental income percentile) grew from $17,720 to $20,950 in real 2023 dollars, while the white-Black race gap for low-income families fell from $20,810 to $14,910. The intergenerational rank-rank slope for white children increased from 0.23 to 0.29. The racial gap in intergenerational persistence of poverty — the probability of a child born to the bottom income quintile remaining there — shrank from 14.7 percentage points to 4.1 percentage points (a 72% reduction), driven roughly equally by improvement in Black children’s chances of escaping poverty and deterioration in low-income white children’s chances. The white class gap in early-adulthood mortality more than doubled, while the white-Black race gap in mortality fell by 77%.

The paper systematically rules out three alternative explanations. Observable family characteristics (parental education, wealth, occupation, and marital status) explain only 7% of the growing white class gap and none of the shrinking white-Black race gap. Neighborhood-level common shocks, tested by including childhood county or Census tract-by-cohort fixed effects, similarly explain only 7% of the class gap and none of the race gap. The divergent trends persist even among children raised in the same Census tract, pointing to forces that operate differentially across race and class groups within the same neighborhood.

The paper’s central finding is that changes in children’s outcomes across cohorts are strongly and positively correlated (r = 0.91 across subgroups) with changes in parental employment rates within the child’s social community, defined as families sharing the same race, class, and childhood county. Low-income white communities experienced sharp relative declines in parental employment rates; low-income Black communities experienced relative improvements. These community-level parental employment changes account for nearly all of the divergent trends.

To establish causation, the paper exploits variation in the age at which children move to counties with changing parental employment rates. Children who moved at younger ages (before age 8) to counties where parental employment was increasing experienced larger improvements in earnings than those who moved at older ages (after age 13), consistent with a causal exposure effect with greater impact for longer durations of exposure. Sibling comparisons — comparing outcomes of younger versus older siblings who moved together — confirm that the age gradient reflects causal exposure rather than family-level selection.

The social interaction mechanism is supported by two sources of variation: children’s outcomes are more strongly related to parental employment rates of their own birth cohort than adjacent cohorts (cohort specificity unlikely to be explained by resources), and outcomes are primarily driven by the employment rates of same-race, same-class community members, with cross-racial influence appearing only in counties where cross-racial interaction is greater (counties with small Black population shares or higher interracial marriage rates). The unified explanation the paper proposes is that children’s outcomes mimic those of the adults in their social communities, following Borjas (1992).

Q: What are the precise magnitudes of the growing white class gap and shrinking white-Black race gap in income percentile ranks? A: The white class gap — the difference in mean household income ranks between white children raised at the 25th versus 75th parental income percentiles — increased from 11.1 to 14.1 percentile ranks between the 1978 and 1992 birth cohorts, a 28% increase. The white-Black race gap for children from low-income families fell from 14.9 to 10.9 percentile ranks, a 27% decrease. The intergenerational rank-rank slope for white children increased from 0.23 to 0.29 (a 28% rise in persistence).

Q: How did the trends in poverty persistence versus upward mobility differ? A: The convergence in white-Black outcomes was driven almost entirely by changes in poverty persistence rather than upward mobility. The racial gap in the probability of remaining in the bottom income quintile shrank from 14.7 percentage points to 4.1 percentage points (a 72% reduction), with roughly half from Black children being less likely to remain at the bottom and half from white children being more likely to remain. By contrast, the white-Black gap in the probability of rising from the bottom quintile to the top quintile fell by only 1.9 percentage points (17%).

Q: How widespread geographically were the divergent trends? A: Outcomes declined for low-income white families in nearly every county, but the largest declines occurred in historically high-mobility areas such as the Great Plains and the coasts. For low-income Black families, outcomes improved in most areas, with the largest gains in historically low-mobility regions including the Southeast and the industrial Midwest. The correlation between county-level changes for low-income white versus low-income Black children is a positive 0.58, meaning the areas where Black families improved most tended to be areas where white families declined least, not most.

Q: Do the trends persist when using non-rank, inflation-adjusted dollar outcomes? A: Yes. The white class gap in mean household income grew from $17,720 to $20,950 in real 2023 dollars, and the white-Black race gap for low-income families narrowed from $20,810 to $14,910. The paper also reports similar patterns for individual earnings (as opposed to household income), ruling out changes in household composition as a driver.

Q: What do the pre-labor-market outcomes show? A: The divergent trends emerge before children enter the labor market. The white class gap in educational attainment grew by 20%, driven by growing gaps in four-year college completion. The white-Black race gap in educational attainment disappeared by the 1992 cohort, driven by narrowing gaps in high school graduation. The white class gap in the share of students taking the SAT/ACT increased by 12.1 percentage points between the 1980 and 1991 birth cohorts, while the white-Black race gap in SAT/ACT-taking decreased by 20.3 percentage points. The white class gap in mean SAT/ACT scores grew by 62% between the 1980 and 1997 birth cohorts among test-takers.

Q: How large is the mortality dimension of these trends? A: The white class gap in early-adulthood mortality (ages 24–27) more than doubled between the 1978 and 1992 birth cohorts, while the white-Black race gap in early-adulthood mortality decreased by 77%. These non-monetary outcomes are invariant to inflation and income measurement choices, confirming the robustness of the broader trends.

Q: How much do family-level characteristics explain? A: Controlling jointly for parental education, wealth, occupation, and marital status reduces the estimated growth in the white class gap by only 7% (from 3.37 to 3.13 percentile ranks). The same controls do not explain the shrinking white-Black race gap — the estimated reduction in the race gap actually becomes slightly larger (4.56 rather than 4.16 percentiles) after controlling for family characteristics, indicating that observable family factors work against the observed convergence.

Q: How much do neighborhood-level common shocks explain? A: Including childhood county fixed effects interacted with birth cohort explains only 7% of the growing white class gap and none of the shrinking white-Black race gap. Including Census tract fixed effects yields essentially identical results. The divergent trends persist among children growing up in the same Census tract, ruling out explanations based on differential exposure to neighborhood-level economic shocks.

Q: What is the community-level parental employment correlation, and what does it explain? A: Changes in children’s earnings, SAT/ACT scores, and educational attainment across cohorts are strongly positively correlated with changes in parental employment rates within the child’s community (same race, same class, same county), controlling for the employment status of the child’s own parents. The correlation between changes in children’s outcomes and changes in community parental employment rates across all race and class subgroups is 0.91. This single community-level factor — as proxied by parental employment rates — accounts for nearly all of the divergent trends by race and class.

Q: What is the quasi-experimental design for estimating causal effects, and what does it assume? A: The paper compares outcomes of children who moved to counties with increasing parental employment rates at younger versus older ages, across earlier versus later birth cohorts. The identification assumption is “constant selection by age”: any selection of families into moving to a given county in years when parental employment is higher may differ across cohorts, but those selection differences must not themselves vary systematically with the age at which children move. The paper treats this as a “constant selection by age” assumption standard in the neighborhood effects literature.

Q: What do the causal exposure results show? A: Children who moved before age 8 to communities where parental employment was increasing show systematically higher earnings in later birth cohorts, while children who made the same move after age 13 show little difference in earnings across cohorts. This pattern — larger effects at younger ages — is consistent with a causal exposure effect of growing up in an improving community, with effects proportional to the duration of exposure.

Q: How do sibling comparisons validate the identification assumption? A: When siblings move together to a community with increasing parental employment rates, the younger sibling — who receives more years of exposure to the higher-employment environment — earns significantly more than the older sibling. The earnings difference is proportional to the age gap between siblings. This rules out explanations based on fixed unobserved family characteristics and supports the constant-selection-by-age assumption.

Q: What evidence distinguishes social interaction mechanisms from economic resource mechanisms? A: Two sources of variation are used. First, children’s outcomes are much more strongly related to the parental employment rates of peers in their own birth cohort than peers in adjacent cohorts — a cohort-specificity that is implausible for economic resource channels (school budgets, local tax bases) which would not vary sharply across adjacent cohorts. Second, outcomes of low-income white children are driven primarily by the employment rates of low-income white parents, not by low-income Black or high-income white parents’ employment rates, and vice versa for low-income Black children — consistent with interaction patterns being stratified by race and class.

Q: What role does cross-racial interaction play? A: In counties where Black children constitute a small share of the population (making cross-racial interaction more likely), Black children’s outcomes are also related to low-income white parental employment rates. Similarly, in counties with higher interracial marriage rates (a proxy for cross-racial interaction), Black children’s outcomes are related to white parental employment rates even after controlling for racial composition. This cross-sectional variation supports the interpretation that the influence channel is social interaction rather than parallel economic shocks.

Q: How do the findings for Hispanic, Asian, and AIAN children compare? A: Changes in economic mobility for Hispanic, Asian, and AIAN children between 1978 and 1992 birth cohorts were much more modest than for white and Black children. For children from low-income families, mean household income ranks were essentially unchanged for Asian children and rose by only about 0.5 percentiles for Hispanic and AIAN children. However, the same community-level parental employment rate mechanism explains the (smaller) changes for these groups as well; the correlation between changes in children’s outcomes and changes in community parental employment rates is 0.91 across all subgroups.

Q: What is the paper’s unified theoretical account of all the divergent trends? A: The paper concludes that a parsimonious theory — that children’s outcomes mimic those of the parents in their social communities, following Borjas (1992) — explains the divergent trends by race and class. Because social interaction is stratified by race and class even within neighborhoods, changes in parental outcomes in the parent generation propagate differentially to white versus Black and high-income versus low-income children, producing growing class gaps and shrinking race gaps through the same underlying mechanism.

Q: What does the paper imply about the malleability of economic mobility disparities? A: Because the causal exposure effects of community environments on children’s outcomes can be detected within a 14-year span (1978 to 1992 birth cohorts), the paper implies that differences in economic mobility by race and class may be malleable in policy-relevant timeframes. This is despite the fact that long-standing disparities partly trace back to historical factors such as slavery, Jim Crow laws, redlining, and the Great Migration.

White class gap: The difference in mean household income ranks in adulthood for white children born to families at the 25th versus 75th percentiles of the national parental income distribution; increased from 11.1 to 14.1 percentile ranks (28%) between the 1978 and 1992 birth cohorts.

White-Black race gap: The difference in mean household income ranks in adulthood for white versus Black children born to families at the 25th percentile of the national parental income distribution; decreased from 14.9 to 10.9 percentile ranks (27%) between the 1978 and 1992 birth cohorts.

Social community: In this paper’s usage, other families who share the same race, class category, and childhood county as a given child; the unit within which community-level parental employment rates are measured and found to be predictive of children’s outcomes.

Causal exposure effect: The effect on a child’s adult outcomes of an additional year spent growing up in a community with higher parental employment rates, estimated quasi-experimentally by comparing children who moved to counties with changing parental employment rates at younger versus older ages; larger effects at younger ages imply a causal, duration-sensitive exposure channel.

Constant selection by age: The identification assumption underlying the quasi-experimental design; requires that any systematic differences in the types of families who move to a county when parental employment is high versus low do not themselves vary with the age at which children move to that county.

Intergenerational rank-rank slope: The OLS slope coefficient from regressing child income percentile rank on parental income percentile rank; for white children, increased from 0.23 in the 1978 birth cohort to 0.29 in the 1992 birth cohort, indicating greater persistence of economic status.

Cohort-specificity of community effects: The empirical pattern that children’s outcomes are more strongly related to the parental employment rates of peers in their own birth cohort than those of adjacent cohorts, used in the paper as evidence favoring social interaction over economic resource channels as the mediating mechanism.

Choice and Opportunity Costs

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

This paper develops a unified choice-theoretic framework in which agents evaluate alternatives not in isolation but relative to their opportunity costs — the alternatives they forgo. The central departure from classical theory is the relaxation of additive separability between benefits and costs. In the standard additive model, accounting for opportunity costs is behaviourally equivalent to simple utility maximisation: a decision maker who correctly perceives the feasible set and maximises an additively separable utility will make identical choices whether or not opportunity costs are explicitly considered (the paper calls this the irrelevance of opportunity costs under additivity, formally establishing it as a general result). Once additive separability is relaxed, however, opportunity costs become non-trivial and generate a genuinely distinct theory of choice.

The primitive of the model is a net preference — an asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers obtaining x while forgoing y over obtaining w while forgoing z. Because the opportunity cost of a chosen alternative depends on what else the agent would choose, and vice versa, choice emerges from an intrapersonal equilibrium rather than from direct maximisation.

The paper defines and axiomatically characterises two nested models. The Recursive Opportunity Model (ROM) adopts a behavioural definition of opportunity costs: the cost of the chosen alternative x in menu A is c(A \ x), the alternative that would actually be chosen were x unavailable; the cost of every unchosen alternative is x itself. This recursive structure is completely characterised by a single observable condition — Weak Path Independence (WPI): if x is chosen when added to a menu A, then x must also be chosen in a pairwise comparison against c(A). WPI is shown to imply Always Chosen (AC) — that a Condorcet winner is always selected — but it permits pairwise cycles of choice (failures of No Binary Cycles). Rationality within the ROM requires additionally that the net preference be a strict order satisfying Congruence, an acyclicity condition on the gross preference induced by the net preference. Even then, the utility function being maximised need not coincide with the gross preference naturally implied by the underlying psychological net preference, raising a welfare identification problem.

The Opportunity Model (OM) generalises the ROM by allowing the opportunity cost of the chosen alternative to be any unchosen alternative rather than the recursively determined one. This relaxation permits both pairwise cycles and menu effects (Condorcet violations). The OM is completely characterised by Never Chosen (NC): an alternative that loses every pairwise comparison within a menu (a Condorcet loser) cannot be chosen. Imposing a strict order and Congruence on the net preference of an OM rules out only pairwise cycles, leaving menu effects intact. Full rationality within the OM is restored only with the additional assumption that opportunity costs are non-decreasing in the induced gross preference as the feasible set expands (the Increasing Opportunity Model).

Extensions characterise multivalued versions of both models (M-ROM and M-OM) via adapted axioms on choice correspondences, and show that several known behavioural models in the literature — including list-rationalizable choice and game-tree rationalizable choice — satisfy WPI and thus are instances of ROM. Applications demonstrate that OMs can represent the attraction effect and the multiple decoy effect, providing a preference-maximisation account without appealing to bounded cognition, and that ROMs can represent intransitive pairwise choices via smooth parametric net preferences, avoiding the discontinuities of lexicographic semiorder models.

Q: What is the paper’s foundational definition of opportunity cost, and how does it differ from the standard textbook definition? A: The paper defines the opportunity cost of the chosen alternative x in menu A as the alternative that would actually be chosen from A \ {x} — that is, c(A \ {x}). The opportunity cost of any unchosen alternative y is the actual choice x. The standard textbook definition — “the next-best feasible alternative” — presupposes context-independent, additively separable preferences, precisely the assumption the paper relaxes. The behavioural definition is grounded directly in the agent’s own choice function, making it consistent with non-separable evaluations.

Q: Under what conditions do opportunity costs become irrelevant, and why? A: If preferences admit an additively separable utility representation u, then for any finite menu A and any two alternatives x and y, u(x) ≥ u(y) if and only if u(x) − max_{a ∈ A{x}} u(a) ≥ u(y) − max_{a ∈ A{y}} u(a). Net utility maximisation and gross utility maximisation rank alternatives identically. Opportunity costs become non-trivial only when additive separability is relaxed — at that point, the agent’s comparative evaluation of (alternative, cost) pairs can produce choices that no gross utility function rationalises.

Q: What is the Recursive Opportunity Model (ROM) and what single axiom characterises it? A: A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen alternative x, the chosen alternative evaluated at its opportunity cost is preferred to x evaluated at c(A). This is equivalent to the choice function satisfying Weak Path Independence (WPI): if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). WPI is necessary and sufficient for a ROM (Theorem 1). It is not sufficient for full rationality, as it permits pairwise cycles while ruling out menu effects.

Q: What kinds of irrationality can a ROM exhibit, and what kinds does it preclude? A: The paper establishes (Corollary 1) that WPI implies Always Chosen — a ROM always selects the Condorcet winner when one exists. Therefore, the only admissible form of irrational behaviour in a ROM is pairwise cycles (failures of No Binary Cycles). Condorcet violations (menu effects) are precluded. A ROM becomes fully rational if and only if it additionally satisfies No Binary Cycles.

Q: What additional condition on the net preference guarantees that a ROM is rational? A: Theorem 2 establishes that a choice function is rational if and only if it is a ROM generated by a net preference that is a strict order (complete, asymmetric, transitive) satisfying Congruence. Congruence requires that the induced binary relation P≻ on alternatives — defined by xP≻y whenever there exists z such that (x, z) ≻ (y, z) or (z, y) ≻ (z, x) — is acyclic. For a (u, v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent.

Q: Can rational behaviour generated by a ROM be welfare-analysed using revealed preference in the standard sense? A: No — and this is a key warning in the paper. Even when a ROM with a strict order and Congruence produces fully rational behaviour, the utility function being maximised need not coincide with the gross preference P≻ naturally induced by the underlying net preference. The paper provides an explicit example (Remark 1, equation 10) in which the choice-rationalising order P is xPyPz while the induced preference is xP≻zP≻y. The utility “revealed” by choice may diverge from the psychological primitive driving that choice, undermining the normative authority of standard revealed preference welfare analysis.

Q: What is the Opportunity Model (OM) and how does it extend the ROM? A: The OM relaxes the recursive assumption by allowing the opportunity cost of the chosen alternative to be any unchosen element of the menu rather than specifically c(A \ c(A)). This breaks the recursive structure while preserving the intrapersonal equilibrium character (the choice still affects the net value of alternatives). The OM is completely characterised by Never Chosen (NC): no Condorcet loser can be chosen (Theorem 3). Unlike the ROM, an OM may fail to select the Condorcet winner, permitting both pairwise cycles and Condorcet violations.

Q: What is the Increasing Opportunity Model and when does it restore full rationality? A: An IOM is an OM in which the opportunity function o is monotone in the sense that if A ⊃ B and o(A) ≠ o(B), then o(A) is ranked higher than o(B) in the induced gross preference P≻. Intuitively, opportunity costs do not decrease as the feasible set expands. Theorem 5 establishes that a choice function is rational if and only if it is an IOM generated by a net preference that is a strict order satisfying Congruence. Full rationality within the OM thus requires both the internal consistency of the net preference (strict order, Congruence) and this monotonicity of opportunity costs.

Q: How does the paper explain the attraction effect using the OM? A: In the canonical formulation, c({x,y}) = x, c({y,d}) = y, c({x,d}) = x, and c({x,y,d}) = y, where d is a decoy. This pattern is incompatible with gross preference maximisation. The paper represents it as an OM with opportunity function o({x,y,d}) = d and a strict net preference order yd ≻ xy ≻ yx ≻ xd ≻ dx ≻ dy. The psychological interpretation is that the introduction of the decoy shifts the comparator for y from x to d; y looks more favourably comparable to d than x does, so the equilibrium where y is chosen is selected. No bounded cognition or imperfect attention is assumed.

Q: How does the framework account for multiple decoys? A: With decoys dx and dy specific to x and y respectively, the observed pattern c({x,y}) = x and c({x,y,dy}) = y and c({x,y,dx,dy}) = y can be represented as an OM with a transitive net preference satisfying xdx ≻ ydy ≻ xy ≻ yx ≻ dyy ≻ dxx and opportunity function o({x,y,dx,dy}) = dx, o({x,y,dy}) = dy. The paper notes this net preference can be extended to a strict order while preserving the choice pattern. This accommodates a phenomenon that poses a challenge to standard theoretical choice literature (per Masatlioglu, Nakajima and Ozbay [25]).

Q: How does the ROM explain intransitive choices more smoothly than lexicographic semiorder models? A: The paper shows that the Tversky (1969) cyclical pattern c({x,y}) = x, c({y,z}) = y, c({x,z}) = z with x=(115,7), y=(117,3), z=(120,0) can be generated by net preferences that admit smooth parametric representations. Specifically, for any two alternatives w=(a,b) and z=(c,d), the paper proposes (w,z) ≻ (z,w) iff (max{a−c, b−d})² > k(min{a−c, b−d})², where k is a relative sensitivity parameter. For k=1/2 this yields the required cycle. Lexicographic models require sharp discontinuities in preference and systematic avoidance of trade-offs, which are often viewed as implausible within the standard economic paradigm; the smooth parametric form avoids these features.

Q: What is the relationship between ROMs and previously studied choice models in the literature? A: Several known models satisfy WPI and are therefore, by Theorem 1, instances of ROMs: specifically, Rationalizability by Game Trees (Xu and Zhou) and List-Rationalizable Choice (Yildiz) are shown to satisfy WPI. The two-stage choice model of Bajraj and Ulku satisfies NC but not WPI, making it an OM but not a ROM. The net preference being maximised in each case can in principle be recovered using the explicit construction in the proof of Theorem 1.

Q: How does the ROM relate to Koszegi-Rabin personal equilibrium? A: Both models involve preferences that depend on a variable determined endogenously by choice, requiring an intrapersonal equilibrium concept in which the agent’s conjectures about their own behaviour must be internally consistent. The key difference is that in Koszegi-Rabin the psychological primitive is a set of reference-dependent preferences ≻r on alternatives in X (where r is the reference point), and equilibrium requires c(A) ≻{c(A)} y for all y ∈ A \ c(A). In the ROM, the primitive is a preference on pairs of distinct alternatives, and the opportunity cost differs for each alternative being compared (the chosen alternative has one opportunity cost, each unchosen alternative has a different one, namely c(A) itself).

Net preference: An asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers to be in a situation where they choose x while forgoing y over a situation where they choose w while forgoing z. The primitive is defined on X = {(x, y) ∈ X × X : x ≠ y}, without imposing additive separability.

Recursive Opportunity Model (ROM): A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen x, the pair (c(A), c(A \ c(A))) ≻ (x, c(A)). The opportunity cost of the chosen alternative is defined recursively as c(A \ c(A)); choice results from intrapersonal equilibrium rather than simple maximisation.

Opportunity Model (OM): A generalisation of the ROM in which the opportunity cost of the chosen alternative can be any unchosen alternative in the menu (not necessarily the recursively determined one). Characterised by Never Chosen: no Condorcet loser can be chosen. Permits both pairwise cycles and Condorcet violations.

Weak Path Independence (WPI): The axiom characterising ROMs: if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). Equivalently, if an alternative is chosen upon being added to a menu, it must also win in a pairwise comparison with what was previously chosen from the original menu.

Congruence: A consistency condition on net preferences requiring that the induced binary relation P≻ — defined by xP≻y whenever there exists z such that (x,z) ≻ (y,z) or (z,y) ≻ (z,x) — is acyclic. For a (u,v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent. Together with a strict net preference order, Congruence in a ROM is equivalent to rational choice.

Intrapersonal equilibrium: The concept underlying both models: an agent is in equilibrium when selecting x from A if they correctly anticipate their own contingent behaviour across hypothetical scenarios (i.e., they use the actual choice function c to evaluate what they would choose from A \ {x}), and the chosen alternative is net-preference-maximal given those consistent conjectures.

Never Chosen (NC): The axiom characterising OMs: an alternative that is a Condorcet loser — losing in every pairwise comparison within a menu — cannot be chosen from that menu. NC is weaker than WPI (which implies both Always Chosen and Never Chosen) and is the precise behavioural content of the OM.

Climate change and the macroeconomics of bank capital regulation

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper asks two related questions about the intersection of climate policy and bank capital regulation. First, can differentiated bank capital requirements — imposing higher equity charges on loans to fossil energy firms — serve as a quantitatively meaningful climate policy instrument, in particular relative to carbon taxes? Second, how should optimal bank capital requirements respond to a carbon-tax-induced clean energy transition?

Methodology

The authors build a quantitative multi-sector DSGE model with two layers of default: corporate default at the firm level and bank failure at the bank level. Three intermediate goods sectors are modeled — non-energy, fossil energy, and clean energy — linked via a nested CES final-good production structure. Banks collect deposits from households (who value deposits for liquidity services) and issue defaultable loans to all three sectors. Deposit insurance, combined with limited liability for bank owners, generates an inefficiently high bank risk-taking motive, creating a role for capital regulation. The Ramsey-optimal capital requirement balances the social benefit of liquid deposit provision to households against the social cost of bank failure.

The model is calibrated to quarterly data, targeting a 0.7% annualized bank failure rate, a 2% annualized corporate default rate, a 30% loan recovery rate, a deposit spread of -100 basis points, and a baseline Ramsey-optimal equity requirement of 8% (consistent with Basel III). Sectoral parameters follow Bartocci, Notarpietro, and Pisani (2022) and Fried, Novan, and Peterman (2022): the energy-to-non-energy elasticity of substitution is 0.2, the clean-to-fossil energy elasticity is 3, and full abatement occurs at carbon taxes exceeding 125 $/tonne of carbon (ToC). The clean transition experiment imposes a linear carbon tax path from zero to 10 $/ToC over 40 quarters, announced as an unanticipated but fully credible shock.

Main Findings

Finding 1 — Fossil-penalizing capital requirements are quantitatively negligible as climate policy. Raising the capital requirement on fossil loans from the baseline 8% to 12% (a 150% risk-weight, consistent with current BB- treatment) reduces the fossil capital share within the energy sector by only 0.06 percentage points (from 80.00% to 79.94%) and cuts aggregate emissions by only 0.08%. A 1 $/ToC carbon tax, by contrast, achieves a 5.23% emission reduction while modestly reducing the fossil capital share to 79.80%. The difference arises because capital requirements affect only the size and financing cost of fossil firms, leaving abatement incentives unchanged; the loan-rate effect on fossil firms is small (loan rate rises from 124 bps to 128 bps), consistent with Kashyap, Stein, and Hanson (2010).

Finding 2 — Sustainability-linked capital requirements remain insufficient. Conditioning the fossil capital requirement on firms’ abatement effort (κ_f = 0.12 − η_t) induces an optimal abatement effort of 2.69% and an effective fossil requirement of approximately 9.5%. The implied emission reduction remains far below even a modest carbon tax: the authors state the induced emission reduction falls short by a factor of almost 100 relative to full abatement.

Finding 3 — Ramsey-optimal capital requirements decline monotonically along the transition (in the baseline real model). When a carbon tax gradually rises from zero to 10 $/ToC over 40 quarters, aggregate loan demand contracts permanently because clean, fossil, and non-energy goods are imperfect substitutes and the shock is recessionary for GDP. Banks reduce balance sheets, deposit supply falls, the deposit spread widens by approximately 8 basis points in the long run, and corporate default rates across all sectors rise by almost 0.1 percentage points from the baseline of 2.05% (in steady state). To counteract the deposit scarcity and associated firm risk-taking, the Ramsey-optimal capital requirement declines symmetrically and monotonically to a lower long-run level. Bank capital regulation cannot affect impact default rates because leverage decisions are made before the transition is announced.

Finding 4 — Nominal rigidities produce a temporary tightening before the long-run relaxation. When debt is denominated in nominal terms and Rotemberg price adjustment costs are added, the clean transition is inflationary in the short run (consistent with Ciccarelli and Marotta 2021). Inflation makes deposit financing more attractive, inducing firms to temporarily increase nominal loan issuance; real deposits rise briefly, the deposit spread narrows by around 2 basis points, and the optimal capital requirement tightens over the initial phase of the transition before converging to the same lenient long-run level as the baseline. The short-run tightening is followed by a permanent relaxation.

Finding 5 — Differentiated sector-specific capital requirements are only warranted when banks are not diversified across sectors. In the baseline, perfectly diversified banks face a symmetric aggregate loan demand contraction, so uniform adjustment suffices. When sector-specific banks are introduced (an extreme case meant to bound concentration effects), fossil banks experience a strong reduction in deposit supply while clean banks experience the opposite. The optimal response is temporarily tighter capital requirements for clean banks and relaxed requirements for fossil banks. In the long run, both converge to an aggregate risk-weight of approximately 99.85% relative to the baseline (a small but symmetric relaxation), very close to the diversified baseline.

Scope Conditions

All results are derived within a model calibrated to match broad financial-market and macroeconomic regularities rather than a specific country. Physical risk from climate change is abstracted away throughout. The carbon tax is set exogenously (not derived from a climate policy optimum). Firms cannot switch technologies, providing a conservative lower bound on the sectoral reallocation. Results are robust to halving the deposit demand elasticity parameter (γ_D = 0.6 versus 1.5 in the baseline) and to raising the energy/non-energy substitution elasticity to 3 from 0.2.

In depth

Q1. What is the core trade-off that determines the optimal level of bank capital requirements in this model?

A: The optimal capital requirement balances two welfare-relevant effects of bank leverage. Tighter requirements reduce bank failure rates, limiting the resource losses (proportional to deposits under DIA management) and the inefficient risk-taking that deposit insurance induces. At the same time, tighter requirements force banks to reduce deposit-financed lending, shrinking the supply of liquid deposits that households value directly in utility. The Ramsey planner chooses the capital requirement that equates the marginal welfare benefit of lower bank failure against the marginal welfare cost of reduced deposit provision. In the baseline calibration this optimum is at 8%.

Q2. Why does raising capital requirements on fossil loans have such a small effect on carbon emissions?

A: Capital requirements affect the deposit-financing wedge for fossil loans — the share of loans that can be funded via cheap, deposit-financed sources — but they do not enter firms’ first-order condition for abatement. Firms respond by modestly reducing leverage and investment (the loan rate for fossil energy firms rises from 124 bps to 128 bps), but the emission intensity of fossil production is unchanged. In equilibrium, the fossil capital share within the energy sector declines by only 0.06 percentage points (from 80.00% to 79.94%), reducing total emissions by 0.08%. A 1 $/ToC carbon tax produces a 5.23% emission reduction, many times larger, because carbon taxes directly alter the return to abatement and the profitability of fossil relative to clean production.

Q3. How does the sustainability-linked capital requirement work and why is it still insufficient?

A: Under sustainability-linked capital requirements, the fossil loan charge is set as κ_f = κ̃ − η_t, so firms that abate more face lower capital requirements on their loans and thus lower financing costs. This creates a direct financial incentive for abatement that the simple penalizing factor lacks. With κ̃ = 0.12, the equilibrium abatement effort is 2.69% and the effective fossil requirement falls to approximately 9.5%. Despite this improvement relative to the plain fossil factor, the climate impact remains far smaller than even a modest carbon tax: the induced emission reduction falls short by a factor of almost 100 relative to full abatement. The fundamental limitation is that the feedback from abatement to financing cost is attenuated by deposit-financing wedge mechanics, making the instrument too weak to substitute for direct carbon pricing.

Q4. What are the impact, short-run, and long-run effects of the clean transition on default rates and bank failure?

A: On impact, the unexpected compliance cost increase raises fossil firms’ default threshold, causing a sharp but short-lived uptick in fossil firm default rates (from 2.05% to approximately 2.08% in the baseline transition) and a brief increase in bank failure. Clean firm defaults fall slightly on impact due to higher clean energy prices. In the short run, clean firms increase risk-taking (higher leverage) because the relative attractiveness of debt financing improves as deposit spreads widen; fossil firms deleverage. In the long run, aggregate corporate default rates rise by almost 0.1 percentage points from the baseline of 2.05% (equivalently 2.7% in the Appendix B long-run analysis), driven by the widening of the deposit spread (approximately 8 bps), which raises the deposit financing wedge for all firms. Bank failure rates are always tied to binding capital requirements and revert quickly to their steady-state level.

Q5. Why can bank capital regulation not mitigate the impact default spike when the transition is announced?

A: At the moment of announcement, leverage decisions for the current period have already been made. The bank capital requirement binds on new lending decisions but cannot alter the existing capital structure of banks or firms. Therefore the regulator faces a “bygone” on impact: changing the capital requirement in the announcement period does not affect current corporate default rates or bank failure rates. The regulator’s tool only becomes effective for lending decisions going forward, implying that the transition-induced impact default surge cannot be smoothed by macroprudential policy.

Q6. Why do Ramsey-optimal capital requirements decline along the transition rather than tighten to address higher default risk?

A: The key channel is that aggregate loan demand contracts permanently as imperfect substitutability across sectors makes the carbon tax recessionary. Banks shrink their balance sheets, reducing deposit supply. The resulting deposit scarcity makes deposits more valuable to households (widening the spread), which also makes deposit financing cheaper for banks, partially offsetting the loan demand decline but at the cost of higher corporate leverage. The welfare loss from reduced liquidity provision and higher firm default rates dominates, so the planner relaxes capital requirements to stimulate deposit supply. The dominant effect is the large, permanent decline in credit demand, which makes it welfare-improving to allow banks to operate at lower capital ratios to rebuild deposit provision.

Q7. What is the role of the deposit financing wedge in transmitting carbon tax shocks to the entire corporate sector?

A: The deposit financing wedge (Ξ_t) reflects the benefit for banks of funding loans through deposits rather than equity, combining the liquidity premium households pay on deposits and the deposit insurance put (expected repayment is only 1 − F(μ_{t+1}) per unit of deposits issued). When aggregate loan demand falls due to carbon taxes, deposits become scarcer relative to their steady-state level, making the wedge larger. Through the loan pricing condition, all sectors — not just fossil — face more attractive deposit-financed debt, causing clean and non-energy firms to also increase their leverage and default risk along the transition. This is the mechanism through which a sector-specific shock has symmetric aggregate effects that shape optimal bank regulation.

Q8. How do nominal rigidities change the optimal path of capital requirements along the clean transition?

A: With Rotemberg price adjustment costs and nominally denominated debt, the clean transition is inflationary in the short run (consistent with empirical evidence in Ciccarelli and Marotta 2021). Inflation lowers the real value of outstanding nominal loan obligations, incentivizing firms across all sectors to temporarily increase nominal borrowing. Banks accommodate this demand by increasing deposit issuance, which briefly narrows the deposit spread by around 2 basis points. With deposit supply temporarily elevated, the regulator’s trade-off tilts toward reducing bank failure rather than stimulating deposit provision, so optimal capital requirements tighten during the inflationary phase before reverting to the lenient long-run path of the baseline model. The long-run level is unchanged.

Q9. Under what conditions are sector-specific capital requirements welfare-improving?

A: Sector-specific requirements are only welfare-improving when banks are not perfectly diversified across sectors, so that the transition has heterogeneous effects on sector-specific deposit supply and bank failure rates. In the baseline with perfectly diversified banks, the loan demand decline affects all banks uniformly, so a symmetric uniform adjustment is optimal. When sector-specific banks are introduced as an extreme case of carbon concentration, fossil banks experience a sharp reduction in deposit provision while clean banks see deposits temporarily increase. The planner responds by temporarily relaxing requirements for fossil banks and tightening them for clean banks. In the long run, both converge to approximately the same aggregate relaxation as the diversified baseline (aggregate risk-weight of 99.85%).

Q10. How does the carbon tax shock experiment relate to the perfect-foresight transition analysis?

A: In the carbon tax shock experiment, the tax level follows an AR(1) process with persistence ρ_τ = 0.9, starting from a long-run level of 10 $/ToC, with a one-standard-deviation shock implying an additional 10 $/ToC on impact. Fossil firm default rates spike from 2% to approximately 2.8% on impact and revert relatively quickly. Emissions decline by slightly more than 10% on impact and revert as the shock dissipates. The macroeconomic dynamics — GDP, investment, loan demand, and bank failure rate responses — closely resemble the impact and short-run effects of the perfect-foresight transition. Optimal capital requirements decline temporarily in both cases, confirming that the transition-path results are not an artifact of the specific perfect-foresight assumption.

Q11. What is the “forced safety effect” and how does it interact with the model’s capital requirement trade-off?

A: The “forced safety effect” (following Bahaj and Malherbe 2020) refers to the positive effect of tighter capital requirements on loan supply that operates through reducing bank failure probability. When banks are less likely to fail (lower F(μ_{t+1})), the expected bank productivity conditional on not failing — (1 − G(μ_{t+1})) — rises toward one, reducing the discount applied to future loan payoffs in the bank’s stochastic discount factor. This improves the profitability of lending and expands loan supply. In the model, this effect partially offsets the direct loan-supply reduction from higher equity requirements but does not dominate, so the overall effect of tighter requirements on deposit supply is still negative, preserving the core trade-off.

Q12. What robustness checks are performed and do they materially change the main results?

A: The authors consider three main robustness checks. First, reducing the deposit demand elasticity parameter from γ_D = 1.5 to γ_D = 0.6 (recalibrating ω_D = 0.012 to preserve the -100 bp deposit spread target) has almost no effect on the optimal path of capital requirements. Second, raising the energy/non-energy substitution elasticity from ε̃ = 0.2 to ε̃ = 3 (and adjusting the energy weight to maintain a 10% energy share) produces much stronger fossil investment declines and smaller clean investment responses, but aggregate loan demand and bank deposits contract only slightly less, so the relaxation in capital requirements is slightly smaller than in the baseline. Third, recalibrating to a 2% annualized bank failure rate (versus the baseline 0.7%) does not materially change results. The conclusion that capital requirements should decline along the transition is robust across all specifications.

Key Concepts

Deposit financing wedge (Ξ_t): The gain for banks from funding loans via deposits rather than equity. It comprises two components: (i) the liquidity premium — households value deposits for their liquidity services, so the deposit rate lies below the risk-free rate; and (ii) the deposit insurance put — the expected repayment obligation per unit of deposits is only 1 − F(μ_{t+1}), not one, since the DIA covers depositors in the event of bank failure. A larger wedge makes deposit-financed lending more profitable, expanding loan supply. In this paper the wedge is the central transmission mechanism through which capital requirements and aggregate loan demand interact.

Bank failure threshold (μ_t): The realization of the bank-specific idiosyncratic risk shock below which a bank cannot service depositors and transfers all assets and liabilities to the deposit insurance agency. It depends on the ratio of deposit repayment obligations to the aggregate realized loan portfolio return. In the model the threshold increases when aggregate loan payoffs fall (as in a carbon tax shock), temporarily raising bank failure rates.

Ramsey-optimal capital requirement: The sequence of sector-specific (or uniform) capital ratios chosen by a benevolent government planner to maximize household welfare, treating the capital requirement as the sole policy instrument. In this model the Ramsey problem is solved nonlinearly along the perfect-foresight transition path. The planner internalizes that tighter requirements simultaneously reduce bank failure probability and shrink deposit supply; the optimum trades off these two objectives.

Sustainability-linked capital requirement: A capital requirement on fossil loans that explicitly depends on the abatement effort undertaken by fossil firms (κ_f = κ̃ − η_t), creating a direct financing-cost incentive for emission reduction. This contrasts with a plain fossil penalizing factor, which affects only the financing cost of fossil capital without altering abatement incentives. The paper shows that even sustainability-linked requirements are quantitatively negligible as climate policy relative to carbon taxes.

Carbon compliance cost per unit of fossil production (ξ_t): A summary statistic combining the direct carbon tax payment and the abatement cost at the optimal abatement effort. It measures the total policy-induced wedge that reduces the profitability of fossil capital and raises fossil firms’ break-even default threshold. In the transition experiment, compliance costs rise from zero to approximately 4% of fossil production value as the tax increases from 0 to 10 $/ToC.

Asset stranding channel: The mechanism through which an unanticipated tightening of carbon policy raises fossil firms’ default probability on impact (by increasing compliance costs above the level priced into existing loan contracts) and subsequently reduces their loan demand permanently. The paper contrasts its treatment of this channel — where stranding affects bank regulation through aggregate deposit supply effects — against models (such as Carattini, Melkadze, and Heutel 2023) where stranding causes an inefficient credit crunch via a financial accelerator.

Deposit spread (s^D_t): Defined as the annualized difference between the deposit rate and the risk-free rate, expressed in basis points. Because households value deposits for liquidity services, the deposit rate lies permanently below the risk-free rate (spread is negative). In the baseline calibration the target is -100 bps. The spread widens (becomes less negative) when deposits become scarcer, which is the case along the carbon tax transition as bank balance sheets contract.

Coarse Bayesian Updating

Mon, 01 Jan 0001 00:00:00 +0000

This paper introduces and axiomatically characterizes Coarse Bayesian updating, a generalization of Bayes’ rule designed to accommodate the wide empirical evidence that individuals systematically deviate from standard Bayesian belief revision. The research question is: what is the minimal, tractable, axiomatically grounded generalization of Bayes’ rule that can accommodate heterogeneous non-Bayesian behaviors — including under-reaction, over-reaction, asymmetric updating, limited perception, and motivated reasoning — while remaining portable to standard economic settings?

The paper takes as primitive a finite state space Omega = {1, …, N} and an updating rule mu: S -> Delta assigning posterior beliefs to signals, where signals represent likelihood profiles from stochastic information structures. No data are used; the methodology is axiomatic decision theory combined with analysis of the model’s implications in static, dynamic, and decision-theoretic settings.

A Coarse Bayesian agent is characterized by (i) a partition of the probability simplex Delta into convex cells, and (ii) a representative distribution for each cell, one of which is the prior. Upon observing a signal, the agent determines which cell contains the Bayesian posterior and adopts the representative of that cell as his posterior belief. The agent need not point-identify the Bayesian posterior; he merely approximates it by identifying which cell it belongs to.

The central characterization result (Theorem 1) establishes that an updating rule has a Coarse Bayesian representation if and only if it satisfies three axioms: Homogeneity (beliefs depend only on likelihood ratios of the signal, not its scale), Cognizance (if two signals induce the same belief, then a garbled signal indicating one of them was generated also induces that belief), and Confirmation (if a signal is perfect evidence of some feasible belief, the agent adopts that belief). The representation — partition, representative points, and prior — is unique.

Proposition 1 shows that, under mild regularity conditions, strengthening any of the three axioms to an if-and-only-if form forces the agent to be perfectly Bayesian. This identifies the Coarse Bayesian framework as a qualitatively small but substantively rich departure from Bayes’ rule. The converse statements identify three necessary non-Bayesian behaviors exhibited by any proper Coarse Bayesian: (i) treating some signals as equivalent when a Bayesian would not; (ii) collapsing to a default belief when uncertain between two signals the agent would otherwise distinguish; (iii) false extrapolation — arriving at a belief via signals that are not perfect evidence of it.

In dynamic settings, Pooled Coarse Bayesian rules (which apply the full signal history at each period) are invariant to signal ordering and pooling and converge whenever Bayesian beliefs do, though to the representative point of the cell containing the true state rather than the true state itself. Sequential Signal Distortion rules are invariant to signal ordering but not pooling, and beliefs converge almost surely — but not necessarily to the true state (Example 1 illustrates convergence to the wrong state in a two-state setting). Sequential Coarse Bayesian rules need not satisfy either form of path-independence and need not converge at all.

In the decision-theoretic application (Section 4), a Coarse Bayesian’s value of information is posterior-separable and generally violates the Blackwell (1951) information ordering — more informative experiments need not be valued more highly. Two Coarse Bayesians are shown to be identical (same cells and representative points) if and only if they benefit from the same Blackwell improvements, providing a behavioral identification result. Agents with finer partitions are more sophisticated (higher ex-ante value of information), while agents with larger distortions from Bayesian posteriors are more biased (larger worst-case losses relative to a Bayesian). Neither greater sophistication nor lower bias implies being better off at all menus or signal realizations.

Q: What are the three axioms that characterize Coarse Bayesian updating, and what property of Bayes’ rule does each capture? A: Homogeneity requires that beliefs depend only on likelihood ratios of the signal — if two signals are proportional (s ~ t), they induce the same posterior. Cognizance requires that if two signals induce the same belief, then a garbled signal indicating that one of them was generated also induces that belief (mu_{s+t} = mu_s when mu_s = mu_t). Confirmation requires that if a signal is perfect evidence of some feasible belief — i.e., the Bayesian posterior at that signal equals a candidate belief — then the agent adopts that belief. Each axiom is satisfied by standard Bayesian updating.

Q: In what sense is Coarse Bayesian updating a “small” departure from Bayes’ rule? A: Proposition 1 establishes that strengthening any one of the three axioms to an if-and-only-if form forces the agent to be perfectly Bayesian. The converses are: (i) different likelihood ratios lead to different posteriors; (ii) if a garbled signal does not change beliefs, then the two signals must induce the same belief individually; (iii) if a signal induces the same posterior as another, then it must be perfect evidence of that posterior. Any Coarse Bayesian satisfying any one of these is in fact perfectly Bayesian, meaning the three axioms together come very close to fully characterizing Bayesian rationality.

Q: What non-Bayesian behaviors does the model generate as special cases? A: The framework generates under-reaction (representative points of cells close to the prior boundary), over-reaction (representative points at the far boundary), asymmetric updating (favoring one state, making upward revision easier than downward), limited perception (the agent retains the prior unless the Bayesian posterior is sufficiently far from the prior), extreme-belief aversion (the agent applies Bayes’ rule except when posteriors are near degenerate distributions), and reactions to unexpected news (non-Bayesian behavior only when signals have low prior probability). In each case the Coarse Bayesian Representation provides an axiomatic foundation via Axioms 1–3.

Q: What are the three necessary non-Bayesian behaviors exhibited by any proper (non-Bayesian) Coarse Bayesian? A: These follow from the negations of properties (i)-(iii) in Proposition 1. First, there exist signals s and t that are not proportional yet induce the same posterior — the agent treats informationally distinct signals as equivalent. Second, there exist signals s and t such that mu_s ≠ mu_t but mu_{s+t} = mu_s — signals the agent distinguishes individually collapse to a default when the agent is uncertain which one was generated. Third, there exist signals s and t with mu_s = mu_t where t is not perfect evidence of mu_s — a form of false extrapolation. Together, these three biases account for all non-Bayesian behavior the model generates.

Q: How does the model accommodate globally uniform biases like always-under-reaction, and how common does it predict such behavior to be? A: Global under-reaction requires representative points of cells to sit on their cell boundaries (as close to the prior as possible given the partition). This is a non-generic, hairline case — representative points generically lie in the interior of their cells, so a typical Coarse Bayesian under-reacts to some signals and over-reacts to others depending on which cell the Bayesian posterior falls into. The model additionally predicts local stability: if an agent over-reacts to signal s, nearby signals typically produce the same response; if an agent is Bayesian at s, nearby signals are almost surely also Bayesian.

Q: What does the model imply about dynamic updating under sequential signal-by-signal processing versus pooled processing? A: Pooled Coarse Bayesian rules apply the full signal history at each period, are invariant to both signal ordering and signal pooling, and converge almost surely whenever Bayesian beliefs converge — but to the representative point of the cell containing the true state, not necessarily the true state itself. Sequential Signal Distortion rules are invariant to signal ordering but not signal pooling, and also yield almost-sure convergence though potentially to the wrong state (Example 1 shows this for a two-state setting). Sequential Coarse Bayesian rules need not be invariant to either form of path-dependence and need not converge at all.

Q: How does the paper provide a behavioral identification of the model’s parameters? A: Theorem 1 establishes that the partition, representative points, and prior are uniquely determined by the agent’s updating rule alone — they are identifiable from observable updating behavior without additional assumptions. In the decision-theoretic setting of Section 4, a stronger result holds: two Coarse Bayesians are identical (same cells and same representative points) if and only if they benefit from the same Blackwell improvements across all menus (decision problems). This means the model’s parameters can be uniquely identified from menu-contingent rankings of Blackwell-comparable experiments.

Q: Does the Coarse Bayesian framework respect the Blackwell information ordering, and what characterizes when Blackwell improvements are beneficial? A: Unlike Bayesians, Coarse Bayesians typically violate the Blackwell ordering — they need not assign higher ex-ante value to more informative experiments. The paper characterizes the menus (decision problems) for which a given Coarse Bayesian benefits from Blackwell improvements, and shows this characterization runs deep: the complete set of such menus fully identifies the agent’s representation.

Q: How do the sophistication and bias orderings relate to welfare? A: An agent is more sophisticated if he employs a finer partition; more-sophisticated agents have a higher ex-ante value of information. An agent is more biased if his updating rule exhibits larger distortions from Bayesian posteriors; greater bias is characterized by greater worst-case losses relative to a Bayesian. Crucially, neither greater sophistication nor lower bias implies the agent is better off at all menus or signal realizations — welfare improvements require the agent to be perfectly Bayesian on a strictly larger set of signal realizations, giving rise to a third ordering that jointly refines the other two.

Q: How does the model relate to Wilson (2014) and Ortoleva (2012)? A: Wilson (2014) studies optimal updating for a boundedly rational agent with K memory states over binary decisions: each memory state is associated with a convex set of posteriors and a representative, so the optimal protocol is a dynamic Coarse Bayesian updating procedure. However, Wilson’s parameters are endogenous (determined by signal structure, stakes, and the bound K), whereas Coarse Bayesian updating does not require optimality or a bound on the number of cells — the model can accommodate behavior (e.g., Bayesian updating except at “extreme” signals) that Wilson’s model cannot. Ortoleva’s (2012) Hypothesis Testing model applies Bayes’ rule when the prior probability of a signal exceeds a threshold epsilon and otherwise uses a maximum-likelihood criterion; Coarse Bayesian updating can accommodate similar behavior, and the paper shows that Coarse Bayesian rules can be expressed as Maximum-Likelihood rules when there are only two states, but neither class subsumes the other in general — Maximum-Likelihood rules may violate Confirmation.

Q: What are the main limitations of the Coarse Bayesian framework? A: The paper identifies four. First, only likelihood ratios of the realized signal matter — sensitivity to framing and extraneous environmental features are ruled out. Second, beliefs must be probability distributions, so phenomena like the conjunction fallacy (where subjects assign higher probability to a conjunction than a component event) are outside the model’s scope. Third, the model exhibits discontinuities when signal perturbations move the Bayesian posterior across a cell boundary — a feature shared with Wilson (2014), Ortoleva (2012), and related models. Fourth, cells must be convex (driven by Cognizance); dropping Cognizance allows non-convex cells but removes the normative foundation that agents correctly forecast their own updating behavior.

Coarse Bayesian Representation: A pair consisting of a partition P of the probability simplex Delta into convex cells and a profile of representative distributions (one per cell, including the prior), such that the agent’s posterior after observing signal s equals the representative of the cell containing the Bayesian posterior B(mu_e|s).

Homogeneity: The axiom that if two signals are proportional (s ~ t, meaning s = lambda*t for some lambda > 0), they induce the same posterior belief — updating depends only on likelihood ratios, not signal scale.

Cognizance: The axiom that if signals s and t induce the same posterior, then the garbled signal s+t (indicating that either s or t was generated) also induces that belief — the agent correctly forecasts his own updating behavior.

Confirmation: The axiom that if a signal constitutes perfect evidence of some feasible belief (i.e., the Bayesian posterior equals a candidate belief), the agent adopts that belief — candidate beliefs are adopted when the signal confirms them exactly.

Signal Distortion Representation: An equivalent representation of Coarse Bayesian behavior as a function d: S -> S that distorts signals before Bayesian updating is applied (mu_s = B(mu_e|d(s))), satisfying properties analogous to the three axioms; equivalent to the partition representation in static settings but distinct in dynamic settings.

Blackwell Information Ordering: The partial order on experiments under which sigma is more informative than sigma’ if sigma can be obtained from sigma’ by a garbling; Bayesians always weakly prefer more informative experiments in this ordering, but Coarse Bayesians typically do not.

Sophistication Ordering: The partial order under which one Coarse Bayesian is more sophisticated than another if he employs a finer partition; more-sophisticated agents exhibit greater responsiveness to information as measured by ex-ante value of information.

Bias Ordering: The partial order under which one Coarse Bayesian is more biased than another if his updating rule exhibits larger distortions away from Bayesian posteriors; greater bias is characterized by larger worst-case losses relative to a Bayesian benchmark.

Collusion with Optimal Information Disclosure

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how a third-party intermediary (an “algorithm”) that observes market demand or costs superior to competing firms should optimally disclose that information to maximize the firms’ collusive profit in a repeated Bertrand competition setting. The motivation is the rise of algorithmic pricing intermediaries such as RealPage in apartment rentals, A2i Systems in retail gasoline, and Rainmaker in hotel rooms, as well as offline cartel facilitators like AC-Treuhand.

The model extends the canonical Rotemberg–Saloner (1986) repeated Bertrand framework with stochastic demand. The key technical assumption is that firm profit is affine in the unknown state s, so expected profit depends only on the expected state. This holds for binary states, linear demand with unknown intercept (D(p,s) = s − p), and linear demand with unknown per-unit cost. The algorithm observes s and commits to a known disclosure policy mapping s to a public signal. The solution concept is pure-strategy subgame-perfect equilibrium, and the paper solves for the disclosure policy and equilibrium that jointly maximize collusive profit.

The main result (Theorem 1) is that the unique optimal disclosure policy is upper censorship: there is a cutoff ŝ such that demand states s < ŝ are disclosed and result in the corresponding monopoly price p^m(s), while demand states s ≥ ŝ are pooled — only the event {s ≥ ŝ} is disclosed — and result in the monopoly price for the mean concealed state, p^m(s*), where s* = E[s | s ≥ ŝ]. The reduction to a static information design problem (Lemma 1) is the key technical step: optimal collusive profit equals V*, the greatest fixed point of V = max_{G ∈ MPC(F)} E_G[min{π^m(s), δV/((1−δ)(n−1))}]. The “capped monopoly profit” min{π^m(s), π^max} is convex-then-concave in s, and classical results from the static information design literature (Kolotilin 2018; Dworczak and Martini 2019) then imply upper censorship is uniquely optimal.

Two features of the optimal equilibrium are notable. First, prices are rigid (constant at p^m(s*)) whenever s ≥ ŝ — the opposite of Rotemberg–Saloner’s “price wars during booms.” The logic is that pooling high demand states with a lower average state is more profitable than cutting prices, because pooling reduces the current-period deviation gain without sacrificing as much on-path profit. Second, for demand states s ∈ (ŝ, s*), the equilibrium price p^m(s*) exceeds the monopoly price p^m(s) — supra-monopoly pricing occurs for a range of intermediate states. Monopoly pricing is attainable at each such state in isolation, but recommending the higher price p^m(s*) is necessary to make the pooling incentive-compatible at states s > s*.

Comparing to full disclosure, Proposition 1 shows that optimal disclosure leads to strictly higher prices at every demand state, and hence unambiguously lower consumer surplus. Proposition 3 shows that improving the algorithm’s accuracy (a mean-preserving spread of F) reduces expected consumer surplus whenever consumer surplus under monopoly pricing is concave in s — a natural condition. This result is more pessimistic than prior work (Sugaya–Wolitzky 2018; Miklos-Thal–Tucker 2019), which found ambiguous effects because those papers assumed full disclosure.

Comparative statics (Proposition 2): fewer firms or a higher discount factor δ increases collusive profit V* and makes prices more flexible (raises ŝ). Collusion is impossible if and only if δ < (n−1)/n, the same threshold as under full disclosure.

Extensions maintain the core results. With Markov (persistent) demand (Section 4 / Theorem 2), upper censorship remains optimal but the cutoff ŝ(s) depends on last-period demand s: under positive serial correlation, ŝ(s) is decreasing in s, so the algorithm discloses less information following high demand. With differentiated products under a symmetric linear demand system (Section 5 / Theorem 3), the optimal policy censors an intermediate interval [ŝ_L, ŝ_H] and discloses both the lowest and highest demand states, because at high states the absence of an upper bound on equilibrium profit makes disclosure with price-cutting optimal.

Q: What is the core research question and why is it policy-relevant? A: The paper asks how an informed intermediary should optimally disclose demand or cost information to competing firms to maximize their collusive profit. It is directly motivated by antitrust cases against RealPage (sued by the US DOJ in August 2024), A2i Systems/Kalibrate, and Rainmaker, all of which gather market data from competing firms and recommend prices. The theory also applies to offline facilitators like AC-Treuhand, prosecuted by the European Commission for disclosing competitively sensitive information.

Q: What is the affinity assumption and why does it matter? A: The paper assumes that firm profit π(p, s) is affine (linearly increasing) in the demand or cost state s for each price p. This implies that expected profit for any distribution over states equals profit evaluated at the expected state: E[π(p,s)] = π(p, E[s]). As a consequence, any disclosure policy is equivalent, from a profit standpoint, to choosing a distribution G of the firms’ posterior mean beliefs over s, and G must be a mean-preserving contraction of the prior F (by Blackwell 1953). The assumption is satisfied for binary states, linear demand with unknown intercept, and linear demand with unknown cost.

Q: What is the key reduction result (Lemma 1) and what does it achieve? A: Lemma 1 reduces the problem of finding an optimal repeated-game equilibrium to a static information design problem. Optimal collusive profit equals V*, the greatest fixed point of V = max_{G ∈ MPC(F)} E_G[min{π^m(s), δV/((1−δ)(n−1))}], and this is attained by a symmetric, stationary, grim-trigger equilibrium. The reduction works because, under Bertrand competition, static deviation gains are proportional to on-path payoffs, creating a one-to-one correspondence that allows the repeated-game constraint to be folded into a single-period objective.

Q: Why is upper censorship the uniquely optimal disclosure policy? A: The static information design problem has a “capped monopoly profit” objective: min{π^m(s), π^max}, where π^max = δV*/((1−δ)(n−1)) is the maximum per-period profit that satisfies incentive constraints. Because π^m(s) is convex (as the maximum of affine functions) and the cap π^max is constant, the overall objective is convex for s below the cap and constant (then concave) above it — i.e., convex-then-concave in s. Classical results for linear information design (Kolotilin 2018; Dworczak and Martini 2019) imply that the unique optimal policy for a convex-then-concave objective is upper censorship.

Q: What is the supra-monopoly pricing result and why does it arise? A: For demand states s ∈ (ŝ, s*), the equilibrium price is p^m(s*) > p^m(s), meaning firms charge above the monopoly price for the current state. This arises because the pooling policy must recommend a single price for all states s ≥ ŝ, and the recommended price is p^m(s*) where s* = E[s | s ≥ ŝ]. At intermediate states s ∈ (ŝ, s*), this price exceeds the local monopoly price. The algorithm accepts lower profit at these states because it is necessary to maintain the pooled recommendation at higher states where monopoly pricing would otherwise require a price cut.

Q: How does optimal disclosure compare to full disclosure in terms of consumer surplus? A: Proposition 1 shows that collusive prices under optimal disclosure are strictly higher at every demand state compared to full disclosure (Rotemberg–Saloner). In Rotemberg–Saloner, high demand states trigger price cuts (“price wars during booms”) to deter deviation; under optimal disclosure, high states are pooled and prices are instead rigid at p^m(s*). Because prices are higher at all states, consumer surplus is unambiguously lower under optimal disclosure.

Q: What does Proposition 3 say about the effect of algorithmic accuracy on consumer surplus? A: Proposition 3 states that if consumer surplus under monopoly pricing, CS(s), is concave in s, then a mean-preserving spread of F (i.e., improved algorithmic accuracy) reduces expected consumer surplus. This result is more pessimistic than prior work by Sugaya–Wolitzky (2018) and Miklos-Thal–Tucker (2019), which found ambiguous effects. The difference is that those papers assumed full disclosure, so better accuracy tightened incentive constraints and sometimes forced price cuts. Under optimal selective disclosure, a more accurate algorithm always raises average prices because the algorithm withholds information that would have forced price cuts.

Q: What are the comparative statics with respect to the number of firms and the discount factor? A: Proposition 2 establishes that a decrease in the number of firms n or an increase in the discount factor δ increases collusive profit V* and makes collusive prices more flexible (raises ŝ). The intuition for fewer firms making prices more flexible is that with fewer firms, incentive constraints bind for a narrower range of demand states, so less pooling is needed. Collusion is impossible if and only if δ < (n−1)/n, the same threshold as under full disclosure.

Q: How does the model generate empirically testable predictions distinct from other collusion models? A: The model predicts: (1) the equilibrium price distribution has support on an interval [p^m(s_bar), p^m(ŝ)] plus a single mass point at the higher price p^m(s*); (2) prices are pro-cyclical overall but rigidly fixed at p^m(s*) for all but the lowest demand states; (3) the gap p^m(s) − p(s) is non-monotone — zero at low states, negative (supra-monopoly) at intermediate states, and positive at high states; (4) prices are more flexible when firms are more patient or fewer. The rigid high price combined with a flexible interval of lower prices is described as a distinctive collusive marker not present in other models.

Q: How does the model relate to the empirical literature testing Green–Porter versus Rotemberg–Saloner? A: Rotemberg–Saloner predicts counter-cyclical prices (price wars during booms), while Green–Porter predicts pro-cyclical prices. Empirical tests (e.g., Porter 1983, Ellison 1994) have typically found pro-cyclical prices, favoring Green–Porter. The present model generates pro-cyclical prices through a different mechanism — perfect monitoring plus selectively disclosed demand information — showing that pro-cyclical prices are consistent with perfect monitoring when the information intermediary optimally pools high demand states. The paper suggests that distinguishing the theories requires estimating the gap between price and monopoly price over the cycle: under Green–Porter, collusion succeeds better in high demand states; under this model, collusion succeeds better in low demand states.

Q: What narrative evidence from the RealPage case corroborates the model’s predictions? A: The US DOJ complaint against RealPage states that “in down markets… [RealPage] instills pricing discipline in landlords, curbing normal fully independent competitive reactions by substituting them with interdependent decision-making,” and that RealPage advertised that its AI helps clients “avoid the race to the bottom in down markets.” This is consistent with the model’s prediction of flexible monopoly prices at low demand states and a rigid, supra-monopolistic price in normal times. The Kumatori Contractors Cooperative case (studied by Kawai, Nakabayashi, and Ortner 2024) corroborates the censorship result: that organization took drastic steps to limit bidders’ information about costs on the largest projects — exactly the states where deviation is most tempting.

Q: How do results change with persistent (Markov) demand? A: Theorem 2 shows that upper censorship remains uniquely optimal with Markov demand, but the cutoff ŝ(s) now depends on last-period demand s. Under positive serial correlation, ŝ(s) is decreasing in s: the algorithm discloses less information after high demand because firms are more optimistic and thus more tempted to deviate. Under negative serial correlation, ŝ(s) is increasing. The optimal collusive price is no longer always equal to the monopoly price for the disclosed mean demand, and the expected price conditional on last-period demand can be countercyclical (similar to Rotemberg–Saloner), even though the current-period price is always monotone in current demand.

Q: How does the optimal disclosure policy change with differentiated products? A: With a symmetric linear demand system (Section 5, Theorem 3), the optimal policy censors an intermediate interval [ŝ_L, ŝ_H] and discloses both the lowest and the highest demand states. At high demand states s > ŝ_H, the algorithm discloses the state and recommends a price below monopoly (to satisfy incentive constraints), because with differentiated goods there is no upper bound on equilibrium profit and profit is convex in s at high states, making disclosure with price-cutting optimal. Mathematically, the capped monopoly profit is piecewise-convex rather than convex-then-concave, so the optimal policy is intermediate-interval censorship rather than upper censorship. The Appendix A version extends to general demand systems and capacity constraints with the same qualitative logic.

Q: What are the main limitations and directions for future work acknowledged by the authors? A: The paper identifies three main limitations. First, if profit is not affine in s (i.e., expected profit depends on more than the mean state), the information design problem becomes non-linear and upper censorship is typically suboptimal, though it remains approximately optimal when the problem is close to linear. Second, the model assumes the algorithm’s objective is to maximize industry profit; if the intermediary is a profit-maximizing seller of software (as in Harrington 2022), the objective may instead be to maximize the profit differential between adopters and non-adopters. Third, the model assumes all firms use the algorithm; allowing partial adoption would require modeling firms’ incentives to subscribe. The paper notes that incorporating these considerations “could be an interesting direction for future research.”

Upper Censorship (disclosure policy): A disclosure policy in which demand states below a cutoff ŝ are revealed to firms (along with the corresponding monopoly price recommendation), while states above ŝ are pooled — only the event {s ≥ ŝ} is disclosed — with a single monopoly price recommendation p^m(s*) for the mean concealed state s* = E[s | s ≥ ŝ]. This is the uniquely optimal disclosure policy in the baseline model.

Capped Monopoly Profit: The per-period profit objective in the reduced static information design problem: min{π^m(s), π^max}, where π^max = δV*/((1−δ)(n−1)) is the maximum industry profit attainable in a single period without violating incentive constraints. This function is convex-then-concave in s, which drives the optimality of upper censorship.

Supra-Monopoly Pricing: Equilibrium prices that exceed the monopoly price for the realized demand state. In the model, this occurs for states s ∈ (ŝ, s*), where the algorithm’s pooled recommendation p^m(s*) is above the local monopoly price p^m(s). It arises because the pooled recommendation must be incentive-compatible at the highest concealed states.

Price Rigidity: The feature of the optimal equilibrium in which the collusive price is constant at p^m(s*) for all demand states s ≥ ŝ. The algorithm achieves this by withholding information about high demand states, preventing the “price wars during booms” predicted by Rotemberg–Saloner (1986) under full disclosure.

Algorithmic Accuracy: In the paper’s terms, the informativeness of the algorithm’s signal about s, formalized as the precision of the distribution F. Improving accuracy corresponds to a mean-preserving spread of F (Blackwell 1953). A more accurate algorithm always increases collusive profit; under the concavity condition on consumer surplus, it also reduces expected consumer surplus.

Mean-Preserving Contraction (MPC(F)): The set of distributions G of firms’ posterior mean beliefs over s that are consistent with Bayesian updating of the prior F. By Blackwell (1953), a disclosure policy is feasible if and only if it induces a distribution G ∈ MPC(F). This is the feasibility constraint in the static information design problem.

Affinity in the state: The assumption that π(p, s) is affine (linearly increasing) in s for each price p. This implies E[π(p,s)] = π(p, E[s]), so expected profit is determined entirely by the expected state, enabling the reduction of the disclosure problem to choosing a distribution of posterior means.

Comment on "Artificial Intelligence and Technological Unemployment" by Wang and Wong

Mon, 01 Jan 0001 00:00:00 +0000

This comment, written by J. Carter Braxton (University of Wisconsin), discusses the paper “Artificial Intelligence and Technological Unemployment” by Wang and Wong (2025), which develops and quantifies an equilibrium labor search model to evaluate the employment effects of spreading AI. Wang and Wong’s central finding is that improvements in AI quality will increase productivity by a factor of three while reducing employment by 23%, with approximately half of the employment decline occurring within the next five years. Braxton’s comment serves two purposes: first, to clarify the model’s structural channels through which AI affects employment; and second, to bring empirical evidence from the spread of computers in the 1980s–2000s to bear on the relative magnitude of those channels.

Braxton identifies two competing forces within Wang and Wong’s framework. The job destruction channel arises from endogenous separations: as AI quality improves, firms increasingly replace matched workers with AI, raising outflows from employment. The job creation channel arises from the free-entry condition: rising AI quality increases firm profits on all matches, inducing firms to post more vacancies, which raises workers’ job-finding rates and employment inflows. Whether aggregate employment rises or falls depends on which channel dominates — a quantitative question the authors resolve through calibration, finding the job destruction channel dominant. Braxton notes that three modeling choices (learning-by-using, the requirement that firms must be matched with a worker to adopt AI, and disembodied technological change) each push against the job-destruction result, making the authors’ findings more striking.

Braxton then evaluates the relative strength of these channels using the historical spread of personal computers. Drawing on Bick, Blandin, and Deming (2024), he notes that workplace AI adoption in 2024 follows nearly the same time trend and income-distribution profile as computer adoption in 1984, making computers a plausible historical analog. Using the CPS Computer Supplement (1984–2003), Braxton measures the change in computer usage by occupation and regresses it against the change in employment-to-unemployment (EU) transition rates by occupation. The estimated coefficient is 0.0146 (robust SE 0.0064), indicating that occupations with higher computer adoption rates saw higher flows into unemployment — confirming that a job destruction channel was active during the computer era. However, regressing the change in log occupation-level employment (1980–2000 Census) on the change in computer usage yields a coefficient of 0.7761 (robust SE 0.2658), with a positive slope indicating that occupations more exposed to computers saw higher employment growth. For the computer episode, therefore, the job creation channel dominated the job destruction channel — the opposite of Wang and Wong’s AI projection.

Braxton also cites his own prior work showing that even when job creation and destruction balance in aggregate, workers displaced by technological change face lasting earnings losses and elevated permanent income risk, raising the question of how to optimally insure these workers.

The comment concludes by identifying avenues for future research: introducing occupational heterogeneity (with some occupations more exposed to AI than others) and worker heterogeneity (skills that are complements versus substitutes to AI). The central open question is whether AI is qualitatively different from prior episodes of technological change, and if so, why.

Q1: What are the two central channels through which AI quality affects employment in Wang and Wong’s model, and how do they operate? The job destruction channel operates through endogenous separations: as AI quality (At) improves, firms that are matched with workers are more likely to replace them with AI at rate ρ, adding the term ρµAt Ht It to outflows from employment in the law of motion for employment. The job creation channel operates through the free-entry condition: higher AI quality raises firm profits on all existing matches (because technological change is disembodied, benefiting matches formed today with future AI gains), inducing firms to post more vacancies, which via free entry reduces the firm’s matching probability but raises the worker’s job-finding rate αt and thereby increases employment inflows. The net employment effect depends on which channel quantitatively dominates.

Q2: What is Wang and Wong’s quantitative finding about the aggregate employment and productivity effects of AI? Using a calibrated equilibrium labor search model, Wang and Wong find that the spread of AI will increase productivity by a factor of three while reducing employment by 23%. Approximately half of the employment decline is projected to occur within the next five years. A version of the model holding job-finding rates fixed yields a similar result, indicating that through the lens of their model the job creation channel is quantitatively small and the job destruction channel dominates.

Q3: What three modeling choices push against Wang and Wong’s job-destruction result, and why does Braxton view this as making the finding more striking? First, AI improves through “learning by using” — it learns from all output being produced — which creates an incentive for employment to remain elevated to accelerate AI learning, dampening job destruction. Second, firms can only adopt AI if currently matched with a worker, which creates an incentive for vacancy posting and pushes in favor of job creation. Third, AI improvements are disembodied (raising productivity in all matches, including those formed before the improvement), which increases the value of forming new matches today and strengthens job creation. Because each of these assumptions pushes against the job destruction result, Braxton argues that finding job destruction dominant despite these model features makes the result more striking.

Q4: How does Braxton use the historical spread of computers to assess the job destruction and job creation channels? Braxton measures occupation-level computer adoption as the change in the share of CPS Computer Supplement respondents who reported using a computer at work between 1984 and 2003 (denoted ΔCPUo,84–03), using occupation codes from Autor and Dorn (2013). He then regresses the occupation-level change in EU transition rates (ΔEUo,84–03, from monthly CPS micro data) on ΔCPUo,84–03 to measure the job destruction channel, and separately regresses the change in log occupation-level employment (Δlog Eo,80–00, from the 1980 and 2000 Census IPUMS) on ΔCPUo,84–03 to assess the net employment effect. A positive coefficient on the employment regression indicates job creation dominates; a negative coefficient indicates job destruction dominates.

Q5: What do the regression results show about the job destruction and job creation channels during the computer era? The job destruction regression yields a coefficient of β = 0.0146 (robust SE = 0.0064, R² = 0.0178), indicating that occupations with higher computer adoption rates did see higher employment-to-unemployment transition rates — the job destruction channel was present. However, the employment-level regression yields a coefficient of β = 0.7761 (robust SE = 0.2658, R² = 0.0348), with a positive slope indicating that occupations more exposed to computers experienced higher employment growth between 1980 and 2000. Thus, for the computer episode, the job creation channel dominated the job destruction channel — the opposite of what Wang and Wong project for AI.

Q6: What is the basis for treating the computer episode as a relevant analog to the spread of AI? Braxton cites Bick, Blandin, and Deming (2024), who show that AI adoption in the workplace in 2024 is following nearly the same aggregate time trend as the spread of personal computers in the early 1980s. Moreover, the distribution of AI usage across the income distribution in 2024 is nearly identical to computer usage across the income distribution in 1984: for both technologies, workplace usage peaks between the 80th and 90th percentiles of the income distribution before declining modestly at the top. Bick et al. (2024) also show the similarities hold by education level and age.

Q7: Even if job creation and destruction balance in aggregate, what does prior work suggest about the distributional consequences for workers? Braxton and Taska (2023) show that workers in occupations more exposed to technological change (measured by changes in computer and software task requirements) suffered larger earnings losses following displacement. Braxton, Herkenhoff, Rothbaum, and Schmidt (2024, forthcoming AER) show that workers in occupations more exposed to technological change experienced larger increases in permanent income risk between the 1980s and 2010s. These findings imply that even if AI does not reduce aggregate employment, workers who are displaced will face deteriorating labor market prospects, raising the question of how to optimally provide insurance.

Q8: What policy implication does Braxton draw from the distributional consequences of technological change? Braxton and Taska (2025, forthcoming Review of Economic Dynamics) show that technological change expands the motive for governments to provide retraining subsidies. Braxton argues that if AI represents an acceleration of technological change, even larger retraining subsidies — and potentially other forms of insurance — may be needed for displaced workers.

Q9: What are the main avenues for future research identified in the comment? Braxton identifies two principal directions. First, introducing occupational heterogeneity into the Wang-Wong framework, so that some occupations are more exposed to AI displacement than others, would allow the model to generate richer distributional implications. Second, allowing worker heterogeneity in skills — distinguishing skill dimensions that are complements to AI from those that are substitutes — would permit the model to capture differential effects across the workforce. The overarching research question is whether AI is qualitatively different from prior technological change episodes, and if so, to identify the precise mechanisms that make it different.

Job destruction channel: In the Wang-Wong model, the increase in endogenous separations driven by firms replacing matched workers with AI as AI quality improves. Formally, this is the term ρµAt Ht It in the law of motion for employment, representing separations that occur when a firm adopts AI and the worker and firm cannot renegotiate a mutually acceptable wage.

Job creation channel: The increase in vacancy posting and worker job-finding rates induced by rising AI quality. Because higher AI quality raises firm profits on all matches (via disembodied technological change), the free-entry condition implies firms post more vacancies, lowering the firm’s matching probability but raising the worker’s job-finding rate αt, increasing employment inflows.

Free-entry condition: The equilibrium condition equating the cost of posting a vacancy (κt) to the expected benefit (the probability of matching ft times the firm’s match surplus Πt). This condition pins down the job-finding rate for workers: when firms find it more profitable to post vacancies, αt rises.

Disembodied technological change: The modeling assumption that AI quality improvements raise productivity in all existing matches, not just those formed after the improvement. This means future AI gains benefit matches formed today, increasing the incentive to create new matches and pushing in favor of the job creation channel.

Learning by using: The mechanism in Wang-Wong whereby AI quality (At) improves as a function of current aggregate employment (Ht) and the learning rate µ. Because AI learns from all output being produced, maintaining higher employment accelerates AI improvement, creating a motive that partially offsets the job destruction channel.

Employment-to-unemployment (EU) transition rate: The rate at which employed workers flow into unemployment in a given occupation, used by Braxton as the empirical measure of the job destruction channel during the computer episode. Measured from monthly CPS micro data.

Capitalization effect: The tendency for firms to post more vacancies today in anticipation of future productivity improvements, because the cost of posting is paid upfront while the benefits of a future-better-AI accrue to the match going forward. Referenced by Braxton as relevant to understanding the job creation channel in Wang-Wong’s framework (citing Pissarides (2000), Chapter 3).

Comment on: Is it AI or data that drives market power?

Mon, 01 Jan 0001 00:00:00 +0000

This paper is a published comment by Miao Ben Zhang (USC Marshall School of Business) on Mihet, Rishabh, and Gomes (2025), “Is It AI or Data That Drives Market Power?” Zhang identifies three contributions of the commented paper and benchmarks each against the existing literature, offering targeted suggestions for strengthening the analysis.

The first contribution Zhang discusses is the commented paper’s distinction between raw data, AI capability, and processed data. Raw data is modeled as a by-product of production linearly related to firm size; processed data is modeled as the abundance of signals improving the precision of firms’ next-period productivity predictions. The commented paper’s key modeling innovation is a formula linking raw data (n_{i,t}), firm-level AI capability (z_i), and processed data (n_{i,t}-tilde): processed data equals a weighted sum of an information entropy effect — e^(-z_i) * (-n_{i,t} * ln(n_{i,t})) — and an AI capability effect — (1 - e^(-z_i)) * n_{i,t} * e^(n_{i,t}). Zhang notes this formula implies that the marginal value of raw data can turn negative for firms with low AI capability, consistent with information-theoretic constraints from the rational inattention literature (Sims, 2003). Zhang requests more empirical support for this equation, specifically asking whether low-AI firms exhibit lower TFP than high-AI firms at similar data-intensity levels, and encouraging discussion of existing measures of data-processing ability such as human capital in data engineering and ML pipeline automation.

The second contribution is the commented paper’s modeling of a secondary market for trading processed data among firms. Zhang notes that facilitating processed data markets — for example via APIs or structured knowledge sharing — can, per the commented paper’s simulation and empirical analysis, democratize innovation and reduce market concentration, enabling even low-AI firms to compete. Zhang flags that the paper is silent on firm acquisition as an alternative channel for accessing processed data, arguing this omission is significant given that processed data, unlike ideas or technologies, is less portable and cannot be obtained simply by poaching skilled employees.

The third contribution is the commented paper’s empirical strategy. The commented paper constructs firm-level proxies for AI intensity and data intensity, then exploits two exogenous technological shocks — the advent of AWS cloud computing and transformer-based architectures — to identify causal effects of improvements in compute and processed data accessibility. The evidence shows that compute improvements disproportionately benefit data-rich firms, while processed data access disproportionately benefits low-AI firms. The central empirical message is that access to raw data tends to foster market concentration, whereas access to processed data tends to reduce market concentration. Zhang raises a measurement concern: the commented paper relies on firm-level Herfindahl-Hirschman Index (HHI) calculations based on time-varying, text-based industry definitions (Hoberg and Phillips, 2016). Zhang argues a positive effect on this HHI could reflect either genuine firm growth relative to competitors or reclassification of the firm into different, possibly more concentrated, sectors — making the HHI measure alone insufficient to support claims about product market concentration. Zhang recommends complementing this with industry-level concentration measures anchored to fixed baseline industry codes (FIC codes from Hoberg and Phillips, 2016), constructed at the FIC-year level, following the approach of Gutierrez and Philippon (2017) on industries’ growth and median Q.

No quantitative magnitudes from regressions or calibrations are reported in the comment itself, as this is a discussion piece rather than an original empirical paper. All claims above are drawn directly from the text.

Q: What are the three contributions of Mihet, Rishabh, and Gomes (2025) that Zhang identifies? A: First, the paper explicitly models the distinct roles of raw data, AI capability, and processed data, linking the information entropy literature to firm production. Second, it models a secondary market for trading processed data among firms, relevant for policy on data sharing platforms. Third, it empirically tests the model’s predictions using firm-level proxies and two exogenous technological shocks.

Q: What is the core formula linking raw data, AI capability, and processed data in the commented paper? A: Processed data (n_{i,t}-tilde) equals e^(-z_i) * (-n_{i,t} * ln(n_{i,t})) plus (1 - e^(-z_i)) * n_{i,t} * e^(n_{i,t}), where z_i is firm-level AI capability and n_{i,t} is raw data. The first term captures the information entropy effect (which can reduce or negate the value of raw data for low-AI firms) and the second captures the AI capability effect (where AI turns raw data into abundant useful signals).

Q: Why can the marginal value of raw data turn negative, according to the framework? A: Information-theoretic constraints — long studied through concepts like Shannon entropy and Sims’s rational inattention — imply that unprocessed raw data may harm rather than help firms that lack adequate processing capabilities. Zhang situates this in the broader macro-finance literature on information choice (Sims, 2003; Veldkamp, 2011).

Q: What empirical suggestion does Zhang make regarding the raw data versus AI capability distinction? A: Zhang asks whether, in the commented paper’s sample of publicly-traded firms with measures of data intensity and AI intensity, low-AI firms exhibit lower TFP (following Imrohoroglu and Tuzel, 2012) than high-AI firms when controlling for similar levels of data intensity. Zhang also encourages discussion of anecdotal evidence for negative information entropy effects and of existing measures of data processing ability such as human capital in data engineering, annotation, cleaning, or ML pipeline automation (Abis and Veldkamp, 2024).

Q: What is the policy relevance of the secondary market for processed data? A: The commented paper’s simulation and empirical analysis shows that facilitating processed data markets (e.g., via APIs or structured knowledge sharing) can democratize innovation and reduce market concentration, enabling even low-AI firms to compete. This aligns with recent literature on secondary markets for structured data and foundation model outputs (Gans, 2018, 2024; Conti et al., 2023, 2024; Athey, 2019). Platforms may have incentives to restrict processed data access, potentially reinforcing incumbent power (Carballa Smichowski et al., 2023).

Q: What channel does Zhang argue the commented paper neglects in its analysis of market concentration? A: Zhang argues the paper is silent on firm acquisition as an alternative means by which firms access processed data, noting that processed data is less portable than ideas or technologies — it cannot be obtained simply by poaching a skilled employee. Zhang contends this acquisition channel appears central to the paper’s focus on market concentration and encourages the authors to include a discussion of it.

Q: What is the central empirical finding of the commented paper regarding raw versus processed data and market concentration? A: Access to raw data tends to foster market concentration, while access to processed data tends to reduce market concentration. The evidence shows that compute improvements (proxied by the AWS shock) disproportionately benefit data-rich firms, while processed data accessibility (proxied by the transformer architecture shock) disproportionately benefits low-AI firms, consistent with theoretical predictions.

Q: What is Zhang’s specific concern about the HHI measure used in the commented paper? A: The commented paper constructs firm-level HHI using time-varying, text-based industry definitions (Hoberg and Phillips, 2016). Zhang argues a positive effect on this HHI is ambiguous: it could reflect genuine firm growth relative to competitors or reclassification of the firm into different, possibly more concentrated, sectors. Zhang concludes that the HHI measure alone is not strong enough to support claims about product market concentration.

Q: What robustness check does Zhang recommend for the empirical analysis? A: Zhang recommends constructing industry-level concentration measures at the FIC-year level using fixed baseline FIC codes from Hoberg and Phillips (2016), available at the Hoberg-Phillips Data Library. The authors could then analyze how industries with high versus low average or median AI intensity and data intensity respond to the two technological shocks in terms of concentration. Zhang cites Gutierrez and Philippon (2017) as an example of this approach and notes it would help distinguish within-industry dynamics from shifts in firm business focus, aligning with best practices from De Loecker, Eeckhout, and Unger (2020) on persistent market power.

Raw data: A by-product of firms’ production, modeled as linearly related to firm size; represents unprocessed observations that have not yet been transformed into useful signals. Distinguished from processed data, which is what actually improves productivity predictions.

Processed data: Modeled as the abundance of signals that improves the precision of firms’ predictions of their next-period productivity (following Farboodi and Veldkamp, 2022). Unlike ideas or technologies, processed data is less portable and cannot easily be transferred by poaching skilled employees.

AI capability (z_i): Firm-level ability to transform raw data into processed data. Firms with low AI capability may receive negative marginal value from additional raw data due to information entropy effects; firms with high AI capability extract large gains from the same raw data.

Information entropy effect: The component of the raw-to-processed-data transformation — e^(-z_i) * (-n_{i,t} * ln(n_{i,t})) — that captures the information-theoretic cost of possessing raw data without adequate processing capability. At low AI capability, this effect can reduce or negate the precision of signals.

Secondary market for processed data: A market in which firms trade processed data, modeled in the commented paper as a platform or API-based exchange. The commented paper’s analysis shows this market can democratize innovation and reduce market concentration by enabling low-AI firms to access processed data they cannot produce internally.

Firm-level HHI (text-based): Herfindahl-Hirschman Index calculated using time-varying, text-based industry definitions (Hoberg and Phillips, 2016). Zhang identifies a measurement ambiguity: a positive effect on this measure could reflect genuine competitive gains or reclassification into more concentrated sectors.

Competing under Information Heterogeneity: Evidence from Auto Insurance

Mon, 01 Jan 0001 00:00:00 +0000

This paper studies imperfect competition in selection markets where competing firms have heterogeneous information about consumers — a layer of asymmetry distinct from the classic buyer-seller information gap. The central questions are: how do inter-firm information asymmetries shape equilibrium pricing, consumer sorting, and market efficiency; and whether a centralized bureau that aggregates and equalizes firms’ risk information can promote competition and improve welfare.

The empirical setting is the Italian mandatory motor vehicle liability insurance market (Responsabilità Civile Auto). The authors use the IPER dataset from IVASS, a nationally representative panel of matched insurer-insuree contracts covering 124,428 liability insurance contracts for new customers in the province of Rome from 2013 to 2021. The panel tracks consumers across insurer switches, enabling construction of individual-specific risk estimates from ex-post claim records using Poisson regressions for claim frequency and log-normal regressions for claim severity. The analysis focuses on the top 10 largest firms plus a composite fringe firm.

The paper’s empirical strategy proceeds in three stages. First, individual risk types are estimated from multi-year claim panels. Second, demand parameters — price sensitivity and firm-level unobserved product attributes — are recovered using a novel fixed-point algorithm (extending Berry et al. 1995) that infers the full offered-price distribution from observed transaction prices alone, without parametric restrictions on price distributions across firms. Third, supply-side parameters — pricing coefficients, signal variances, and cost parameters — are identified by exploiting the monotone mapping between offered prices and private signals, borrowing from the nonparametric auction literature.

The model features firms that each draw a private Gaussian signal about a consumer’s true risk type theta, with firm-specific signal standard deviation sigma_j. Lower sigma_j means higher information precision. Firms set prices as a linear function of their posterior risk rating: p_j = alpha_j + beta_j * E(theta | theta_j, D=j). Firms simultaneously choose pricing coefficients to maximize expected profits.

Key empirical findings: (1) Firms differ substantially in how sensitively their premiums respond to realized consumer risk — a reduced-form measure of information precision — with Figure 2 showing wide cross-firm variation in premium-to-risk coefficients. (2) Structural estimation confirms substantial heterogeneity in signal standard deviations sigma_j across all 11 firms. Firms with less accurate risk-rating algorithms (higher sigma_j) tend to have more efficient cost structures (lower claim-processing cost parameter k_j), generating distinct comparative advantages. (3) Baseline pricing coefficients alpha_j and risk-sensitivity coefficients beta_j vary dramatically across firms. (4) Senior drivers are less price sensitive; urban drivers are more price sensitive. Lower-risk consumers show stronger preferences for Firms 3 and 5, while higher-risk consumers disproportionately choose Firm 8.

Counterfactual simulations assess three information policies relative to the baseline. Under a centralized risk bureau — which collects each firm’s signal, aggregates them weighted by precision, and distributes the combined signal equally — average premiums fall by 21.6% and consumer surplus rises by 15.7%. The efficiency benchmark (firms observe true risk perfectly) yields a 25.7% premium reduction and a 16.9% consumer surplus gain, so the bureau recovers almost all the efficiency gap. The privacy benchmark (all firms restricted to the coarsest signal in the market) raises surplus for high-risk consumers by 6.9% but harms low-risk consumers.

The bureau’s price reduction operates through two channels: it eliminates the market power that accrues to firms with superior private information, and it aligns firms’ risk evaluations, enabling sharper undercutting. The bureau also reduces average costs by 12 euros per contract by enabling more efficient insurer-insuree matching — cost-efficient claim processors can better target the consumer types they have a comparative advantage in serving.

The analysis is confined to new customers in Rome’s provincial market to avoid complications from dynamic pricing and consumer-firm learning. The model abstracts away from optional contract clauses (treated as observable characteristics) and does not model the specific mechanisms generating information heterogeneity.

Q: What is the paper’s core research question? A: The paper asks how information asymmetries between competing firms (not just between buyers and sellers) shape equilibrium pricing strategies, consumer sorting, and market efficiency in a selection market, and whether a centralized bureau that equalizes firms’ access to aggregated risk information can improve competition and welfare. This extends the classic Akerlof-Rothschild-Stiglitz framework by introducing a second layer of asymmetry — across sellers themselves.

Q: Why is the Italian auto insurance market well suited for this study? A: Italy mandates liability insurance for all drivers and prohibits rejections, so the analysis focuses entirely on how consumers sort across insurers rather than on participation margins. The IPER dataset from IVASS is a nationally representative panel tracking policyholders even across insurer switches, providing both premium and ex-post claim records needed to construct individual risk types. The market has roughly 50 competing firms using demonstrably heterogeneous pricing algorithms, documented through a survey of major insurers and reduced-form regressions.

Q: How do the authors measure firm-level information precision in the reduced-form analysis? A: They estimate individual-specific risk types from a panel of claim records using Poisson regressions (claim frequency) and log-normal regressions (claim severity), then regress each firm’s premiums on those estimated risk measures. Firms whose premiums respond more sensitively to realized risk are inferred to have higher information precision. Figure 2 shows that these premium-to-risk coefficients vary significantly across firms — for example, Firm 7’s premiums are considerably more sensitive to risk than Firm 8’s — providing reduced-form evidence of heterogeneous information precision before any structural estimation.

Q: What is the structural model’s signal structure? A: Each firm j draws a private signal theta_j ~ N(theta, sigma_j^2) about a consumer’s true risk type theta, where sigma_j is the firm-specific signal standard deviation. A smaller sigma_j means higher precision. Signals are independent across firms conditional on theta, analogous to common-value auctions where firms receive noisy estimates of a shared unknown value (expected claim payouts). The parameter sigma_j is the key structural object the paper identifies and estimates.

Q: What is novel about the demand estimation strategy? A: Standard demand estimation assumes the same price is offered to all consumers or that the full price menu is observed. Here, only transaction prices are observed — the prices of unchosen insurers are not in the data. The authors apply the Wu and Xin (2024) fixed-point algorithm, which jointly estimates consumers’ sorting probabilities, offered price distributions, and demand parameters by adding an outer loop over sorting propensities to the Berry (1994) contraction mapping. No parametric restrictions are imposed on the offered price distributions, and they are allowed to vary fully across firms.

Q: How are firms’ signal variances identified separately from pricing coefficients? A: There is a one-to-one mapping between a firm’s offered price and its signal (prices increase monotonically in the signal, analogous to bids in auctions). After recovering the offered price distribution from the demand step, the authors observe price dispersion at a fixed risk level. By focusing on average prices conditional on each risk level, signal noise averages out, identifying the pricing coefficients beta_j. The residual price dispersion at fixed risk then identifies signal variance sigma_j^2.

Q: What does structural estimation reveal about the relationship between information precision and cost efficiency? A: Firms with higher signal standard deviations (less precise risk evaluation) tend to have lower claim-processing cost parameters k_j — they are more efficient at handling claims. This creates distinct comparative advantages: some firms excel at risk identification but face higher processing costs, while others process claims cheaply but evaluate risk less precisely. This heterogeneity means information-equalizing policies have differentiated firm-level impacts.

Q: What are the quantitative effects of the centralized risk bureau on premiums and consumer surplus? A: The bureau reduces average premiums by 21.6% relative to baseline and increases consumer surplus by 15.7%. The efficiency benchmark — where firms observe consumers’ true risk perfectly — produces a 25.7% premium reduction and a 16.9% consumer surplus gain. The bureau therefore closes nearly all of the gap to the first-best allocation in surplus terms (15.7% vs. 16.9%).

Q: Through what mechanisms does the bureau reduce prices? A: Two distinct channels are identified. First, equalizing information precision eliminates the informational market power held by firms with superior signals, compelling them to compete more aggressively on price. Second, when all firms share the same risk evaluation of a consumer, they can undercut each other more precisely, which intensifies price competition further. Both channels operate simultaneously under the bureau.

Q: How does the bureau affect consumer surplus distribution across risk types? A: The bureau primarily benefits low-risk consumers because improved information allows firms to price discriminate more accurately on risk type, lowering prices for those who are low risk. High-risk consumers see smaller benefits and may face relatively higher premiums. This contrasts with the privacy benchmark, where restricting all firms to the coarsest signal in the market raises high-risk consumers’ surplus by 6.9% — because it becomes harder for firms to distinguish them from low-risk consumers.

Q: What is the cost efficiency effect of the bureau? A: Under the centralized risk bureau, average costs per contract fall by 12 euros. This reflects more efficient insurer-insuree matching: when firms have equal and better information, those with cost advantages in claims processing can better identify and attract the consumer types they are relatively best equipped to serve. The authors note that given the scale of the Italian auto insurance market (approximately 31 million contracts annually), this per-contract saving implies a substantial aggregate impact.

Q: What happens to firm profits under the bureau, and is the impact uniform? A: Average profits decline overall due to lower prices. However, the impact is heterogeneous across firms. Firms that rely most heavily on superior information precision — often smaller, more specialized firms — experience greater profit losses, since the bureau most directly erodes their competitive advantage.

Q: How does the privacy benchmark differ from the bureau scenario? A: The privacy benchmark simulates a regulation that restricts all firms to using only basic consumer information, setting signal variance to the highest level observed in the market. Unlike the bureau (which improves and equalizes information), this benchmark degrades information uniformly. It produces opposite distributional effects: high-risk consumers gain 6.9% in surplus as cross-subsidization from low-risk to high-risk consumers increases, while low-risk consumers are worse off.

Q: Why does the paper focus on new customers only? A: Focusing on new customers avoids complications from dynamic pricing, where insurers update premiums based on accumulated claim history with a specific consumer, and from consumer-firm learning dynamics. This follows standard practice in the empirical asymmetric information literature, as cited in Chiappori and Salanie (2000) and Crawford et al. (2018).

Q: How does this paper relate to and extend prior work on selection markets? A: Prior empirical work on imperfect competition in selection markets — including Einav et al. (2010), Crawford et al. (2018), and related studies — assumes that competing firms have symmetric information about consumers. This paper is described as introducing the first tractable empirical framework for analyzing selection markets where firms have heterogeneous information. It also incorporates multidimensional cost heterogeneity on the supply side, adding to work by Salanié (2017) and Nelson (2025).

Q: What do the reduced-form regressions reveal about pricing heterogeneity across insurers? A: Firm-level regressions of premiums on observable risk factors show R-squared values ranging from 0.39 to 0.59. Estimated coefficients on key risk factors vary dramatically: being one year older reduces premiums by 0.25 to 1.68 euros depending on the firm; a higher bonus-malus class increases premiums by 12 to 32 euros; one additional accident in the previous five years raises premiums by 74 to 181 euros. These ranges reflect genuine differences in actuarial algorithms, not just sampling variation.

Q: What is the bonus-malus system and why does its saturation matter for the paper’s setting? A: Italy’s bonus-malus (BM) system assigns drivers to one of 18 risk classes based on accident history. Because approximately 80% of policyholders are in the best class (BM class 1), the public BM system provides limited granularity for risk evaluation. This saturation creates strong incentives for firms to develop proprietary risk-rating algorithms, which is the institutional basis for the substantial information heterogeneity that the paper documents and models.

Information Precision (sigma_j): In the paper’s model, the firm-specific parameter measuring the dispersion of a firm’s private signal about a consumer’s true risk type. Firm j draws signal theta_j ~ N(theta, sigma_j^2); 1/sigma_j is information precision. A smaller sigma_j means the firm more accurately identifies consumer risk. This is not merely a theoretical construct — the paper identifies and estimates sigma_j structurally for each of the 11 firms.

Heterogeneous Information: The condition where competing firms hold signals of different precision about the same consumer’s unobserved risk type, introducing asymmetry not just between buyers and sellers (as in Akerlof 1970) but among sellers themselves. This is the paper’s central departure from prior literature on selection markets, which assumed symmetric information among firms.

Centralized Risk Bureau: A policy institution that collects each firm’s analyzed risk signal, aggregates them weighted by each firm’s information precision (producing a combined signal more precise than any individual firm’s signal), and makes the aggregated information equally accessible to all firms. The bureau is the paper’s primary policy counterfactual, and it is modeled as equalizing both the level and heterogeneity of information precision across competitors.

Offered vs. Accepted Price Distribution: A distinction central to the paper’s identification strategy. The accepted price distribution is what is observed in transaction data — prices conditional on the consumer having chosen that firm. The offered price distribution is the full set of prices the firm would charge across all consumers, including those who did not select it. The paper recovers the offered distribution from the accepted distribution using a fixed-point algorithm, without imposing parametric restrictions.

Selection Loop: The paper’s methodological extension of the Berry (1994) BLP contraction mapping for mean utilities. An outer loop iterates over consumers’ sorting propensities to jointly recover offered price distributions, sorting probabilities, and demand parameters when only transaction prices are observed. This technique handles the endogeneity of which prices are accepted.

Risk Rating: The firm’s posterior assessment of a consumer’s expected cost, computed as the posterior mean E(theta | theta_j, D=j) — the expected true risk type conditional on the firm’s private signal and the consumer selecting that firm. Firms set prices as a linear function of their risk rating: p_j = alpha_j + beta_j * E(theta | theta_j, D=j).

Comparative Advantage (information vs. cost): The paper’s finding that firms with lower information precision (higher sigma_j) tend to have more efficient cost structures (lower k_j), and vice versa. This cross-sectional negative correlation between information advantage and cost advantage means that policy interventions that equalize information precision shift the basis of competition from information asymmetry to cost specialization.

Competition and the Phillips curve

Mon, 01 Jan 0001 00:00:00 +0000

Fujiwara and Matsuyama ask whether the well-documented flattening of the New Keynesian Phillips curve (NKPC) and the concurrent rise in market concentration and markup rates are causally linked or merely coincidental. Under the canonical New Keynesian model with CES demand, competition is irrelevant to the Phillips curve regardless of whether entry is endogenous — concentration neither changes its slope nor affects inflation directly. This paper overturns that irrelevance result by extending the canonical model in two directions: (1) incorporating endogenous firm entry and exit following Bilbiie, Ghironi, and Melitz (2008) and Bilbiie, Fujiwara, and Ghironi (2014), and (2) replacing CES with the Homothetic Single Aggregator (HSA) demand system (Matsuyama and Ushchev 2017, 2020b), a flexible, tractable class of homothetic demand systems that nests CES and Translog as special cases.

The paper’s theoretical results depend on two of Marshall’s laws of demand. The Second law states that the price elasticity of demand rises with the firm’s own price; the Third law states that the rate of increase in that elasticity falls with price. Together these conditions imply that the markup rate and pass-through rate are endogenous to the competitive environment.

The main findings, delivered under both Rotemberg (1982) and Calvo (1983) pricing, are that higher entry costs — leading to market concentration — cause Phillips curve flattening through two distinct, complementary channels:

Structural (steady-state) effect. Under Rotemberg pricing, the slope of the NKPC is proportional to the price elasticity zeta(z); market concentration reduces z, hence reduces zeta(z) under the Second law, directly flattening the curve. Under Calvo pricing, the slope is proportional to the pass-through rate rho(z); the Third law implies that concentration reduces rho(z), again flattening the curve. The Calvo–Rotemberg equivalence, which holds under CES to first order (Roberts 1995), breaks down under HSA: each pricing mechanism highlights a different channel.
Observational (omitted variable bias) effect. Endogenous entry generates an endogenous cost-push shock through strategic complementarity in price setting. Because the number of firms N_t is omitted from a naive regression of inflation on real marginal cost, and because N_t is positively correlated with the marginal cost under the Second law, the omitted variable bias is negative — the estimated slope is biased downward. This bias is amplified with greater concentration under the Third law (Rotemberg case) and under both the Second and Third laws (Calvo case).

Quantitatively, the paper simulates under three parametric HSA families — CES, Translog, and Co-PaTh (Constant Pass-Through). De Loecker, Eeckhout, and Unger (2020) document that aggregate markups rose from 21% above marginal cost to 61% — a rise of approximately 40 percentage points. The authors’ simulations imply this increase corresponds to an entry cost roughly 3.5 times higher under Translog and roughly 2.5 times higher under Co-PaTh with pass-through rate rho = 0.5. Under these parameterizations, the accompanying market concentration can halve the slope of the NKPC. Impulse responses confirm that the responses of inflation to both technology shocks and monetary policy shocks become smaller as market concentration deepens.

Scope conditions: results require departure from CES (the Second and/or Third law must hold); endogenous entry is necessary for the dynamic cost-push channel; the structural flattening requires only the Second law under Rotemberg but additionally the Third law under Calvo; the omitted variable bias requires the Second law under Rotemberg and both laws under Calvo. The model is closed-economy, with symmetric monopolistic competition and Rotemberg or Calvo price adjustment.

Q1: What is the irrelevance result the paper overturns, and why does CES produce it? Under CES, the market share function takes the form s(z) = gamma * z^(1-theta), yielding a constant price elasticity zeta = theta and a pass-through rate rho = 1, regardless of the number of firms or entry costs. As a result, concentration neither alters the slope of the NKPC nor generates any endogenous cost-push shock; competition is simply irrelevant to inflation dynamics. This irrelevance holds even with endogenous entry under CES.

Q2: What is the Homothetic Single Aggregator (HSA) and why is it used? HSA is a class of homothetic demand systems, originally proposed by Matsuyama and Ushchev (2017), in which the market share of each intermediate input variety depends solely on its own price normalized by a single price aggregator A_t. This single aggregator serves as a sufficient statistic summarizing all competitive pressure effects on pricing behavior, including the markup rate and pass-through rate. HSA nests CES and Translog as special cases, is analytically tractable (equilibrium existence and uniqueness are straightforward to ensure with endogenous entry), and is flexible enough to accommodate both the Second and Third laws of demand.

Q3: What are Marshall’s Second and Third laws as defined in the paper? The Second law states that the price elasticity of demand zeta(z) is increasing in the normalized price z (equivalently, increasing in the single price aggregator A_t, which rises with fewer firms). The Third law, as defined by Matsuyama and Ushchev (2023b), states that the rate of increase in the price elasticity is decreasing in z. Together they ensure that both markup rates and pass-through rates respond systematically to changes in competitive pressure.

Q4: How does market concentration structurally flatten the NKPC under Rotemberg pricing? Under Rotemberg pricing, the slope of the NKPC equals (zeta(z) - 1) / chi, where chi is the Rotemberg price adjustment cost parameter. Higher entry costs reduce the equilibrium number of firms, which reduces competitive pressure and lowers z. Under the Second law, lower z reduces zeta(z), directly shrinking the slope coefficient. This is the steady-state effect of concentration: the structural slope of the curve declines because the price elasticity falls.

Q5: How does market concentration structurally flatten the NKPC under Calvo pricing? Under Calvo pricing, the slope of the NKPC is positively related to the pass-through rate rho(z) rather than the price elasticity. The Third law implies that lower z (more concentration) reduces rho(z). Market concentration therefore causes structural flattening through the pass-through channel under Calvo. This is why the Calvo–Rotemberg equivalence — which holds to first order under CES — breaks down under HSA: Rotemberg highlights the Second law / price elasticity channel and Calvo highlights the Third law / pass-through channel.

Q6: What is the endogenous cost-push shock and how does it arise? When the number of operating firms N_t changes endogenously, it alters the single price aggregator A_t and therefore the competitive environment facing each firm. Under the Second law, firms exhibit strategic complementarity in price setting: a firm reduces its markup when other firms lower their prices (A_t falls with more entry). Consequently, movements in N_t directly enter the NKPC as an additional term — (1/chi) * (1 - rho(z)) / rho(z) * N_hat_t — acting as an endogenous cost-push shock. This channel is absent under CES because rho = 1 makes the coefficient zero.

Q7: How does the endogenous cost-push shock create a negative omitted variable bias? A naive regression of inflation on real marginal cost omits the N_hat_t term. Under the Second law, N_t is positively correlated with the marginal cost (more entry drives markups down, consistent with marginal cost movements), so the omitted variable N_hat_t is positively correlated with the included regressor. Because the true coefficient on N_hat_t in the NKPC is negative, omitting it biases the estimated slope on marginal cost downward (negative omitted variable bias). The estimated relationship between inflation and marginal cost is therefore weaker than the true structural relationship.

Q8: How is the omitted variable bias amplified by concentration? Under the Third law (Rotemberg case) and under both the Second and Third laws (Calvo case), greater market concentration amplifies the magnitude of this negative bias. The intuition is that higher concentration makes the pass-through rate rho(z) smaller, which increases the coefficient on N_hat_t in the NKPC and thereby raises the magnitude of the bias when N_hat_t is omitted. Greater concentration thus generates both more structural flattening and more observational flattening simultaneously.

Q9: What are the quantitative magnitudes of Phillips curve flattening in the simulations? De Loecker, Eeckhout, and Unger (2020) document that aggregate markups rose from 21% above marginal cost to 61% — approximately 40 percentage points. The paper’s simulations imply this corresponds to an entry cost increase of roughly 3.5 times under Translog and roughly 2.5 times under Co-PaTh with rho = 0.5. According to Figure 2, the accompanying market concentration can halve the slope of the NKPC. The slope declines more steeply for demand systems with smaller pass-through rates (rho further from 1).

Q10: How do impulse responses change with market concentration? As entry costs rise (deeper concentration), the responses of the inflation rate to both technology shocks and monetary policy shocks become smaller in magnitude. Under the Second law, a positive technology shock increases the number of firms through a wealth effect, but strategic complementarity in price setting reduces markups, muting the inflation response relative to CES. The dynamic effect of endogenous entry thus weakens the transmission of real economic shocks to inflation — a supply side effect of monetary policy that parallels Baqaee, Farhi, and Sangani (2021) but operates through firm entry rather than the misallocation channel.

Q11: What is the cyclicality of the markup rate under HSA, and why is it ambiguous? Under CES with flexible prices, the markup is constant. Under CES with sticky prices, the markup is procyclical (marginal cost falls with a positive technology shock but the price is rigid in the short run). Under the Second law with flexible prices, a positive technology shock increases firm entry, which reduces markups, making the markup countercyclical. In a sticky price equilibrium under the Second and Third laws, the cyclicality is therefore ambiguous: it depends on the tension between nominal rigidities (pushing toward procyclicality) and the pass-through rate (pushing toward countercyclicality).

Q12: Why do the three price indices in the model differ, and which is used for the NKPC? The model features three aggregate price measures: the final goods price (CPI) P_t, which captures productivity effects of entry; the single price aggregator A_t, which captures competitive effects of entry and is the reference price for firms; and the average price index (PPI) p_t, which is not affected by entry effects and is the measured price index. Because entry effects shift P_t and A_t in ways that are not directly observed, the paper evaluates NKPC responsiveness in terms of p_t (PPI inflation), the measurable index.

Q13: How does this paper relate to Wang and Werning (2022) and Baqaee, Farhi, and Sangani (2021)? Wang and Werning (2022) use a dynamic oligopoly model with exogenous entry and CES/Kimball demand, showing that higher concentration amplifies real effects of monetary policy and generates inflation persistence and endogenous cost-push shocks. Baqaee, Farhi, and Sangani (2021) use monopolistic competition with exogenous entry and Kimball demand under Calvo pricing, showing flattening through real rigidities and a misallocation channel (supply side effects of monetary policy). This paper uses monopolistic competition with endogenous entry and HSA under both Rotemberg and Calvo pricing; it produces supply side effects through firm entry rather than misallocation, and uses HSA rather than Kimball because HSA more readily guarantees equilibrium uniqueness with endogenous entry.

Q14: What parametric families of HSA are used in simulations and what are their properties? Three families are used: CES (constant price elasticity theta, pass-through rho = 1, benchmark); Translog (satisfies the Second law, variable markups and pass-through); and Co-PaTh or Constant Pass-Through (proposed by Matsuyama and Ushchev 2020a, constant pass-through rate rho in (0,1) under flexible prices, containing CES as a limit as rho approaches 1). For Calvo pricing, a fourth family — PEM (Power Elasticity of Markup, proposed by Matsuyama and Ushchev 2023b) — is used; PEM satisfies the Third law in its strong form and contains Co-PaTh as a limit case. Translog is noted to behave similarly to Co-PaTh with rho = 0.5.

Q15: What are the policy implications for central banks? Rising market concentration, by flattening the NKPC both structurally and observationally, reduces the effectiveness of monetary policy in achieving price stability through real economic activity — consistent with the concerns expressed by Federal Reserve officials (Clarida, Daly, Williams) quoted in the paper. The results suggest that empirical estimates of the NKPC slope that omit endogenous entry dynamics will be systematically biased downward, potentially leading central banks to underestimate the true structural responsiveness of inflation to demand conditions. Competition policy and barriers to entry thus have macroeconomic consequences beyond standard allocative efficiency considerations.

Homothetic Single Aggregator (HSA): A class of homothetic demand systems in which the market share of each input variety depends solely on its own price normalized by a single price aggregator A_t, which serves as a sufficient statistic for all competitive pressure effects on firm pricing behavior including the markup rate and pass-through rate. Nests CES and Translog as special cases.

Marshall’s Second Law of Demand (as used in the paper): The condition that the price elasticity of demand zeta(z) is strictly increasing in the firm’s normalized price z. Under this condition, markup rates and pass-through rates vary endogenously with competitive pressure, and strategic complementarity in price setting arises.

Marshall’s Third Law of Demand (as used in the paper): The condition, defined by Matsuyama and Ushchev (2023b), that the rate of increase in the price elasticity is decreasing in z. This law determines how the pass-through rate responds to concentration changes and is the relevant condition for structural flattening under Calvo pricing.

Pass-through rate rho(z): The fraction of a cost change that a monopolistically competitive firm passes through to its price under flexible pricing, defined as rho(z) = [1 - dln(zeta/(zeta-1))/dln(z)]^(-1). Under CES, rho = 1 (complete pass-through); under the Second law, rho < 1 (incomplete pass-through); it declines with concentration under the Third law.

Endogenous cost-push shock: The direct effect of changes in the endogenous number of firms N_t on inflation in the NKPC, arising from strategic complementarity in price setting under HSA. This term is absent under CES (where the coefficient is zero) and generates an omitted variable bias in naive regressions of inflation on marginal cost.

Steady-state (structural) flattening: The reduction in the true structural slope of the NKPC caused by market concentration operating through lower price elasticity (Rotemberg channel) or lower pass-through rate (Calvo channel). This is the first of the paper’s two reasons for observed Phillips curve flattening.

Observational (omitted variable bias) flattening: The downward bias in empirically estimated NKPC slopes arising because naive regressions omit the endogenous cost-push shock term. The bias is negative and is amplified by greater market concentration under the Third law and/or Second law depending on the pricing mechanism.

Competition in a Spatially-Differentiated Product Market with Negotiated Prices

Mon, 01 Jan 0001 00:00:00 +0000

Research Question

How does individually negotiated pricing — where buyers make discrete choices among differentiated products and negotiate transaction-specific prices — affect market power and merger effects in oligopoly markets, and how do these effects differ from the uniform-pricing benchmark?

Data and Setting

The paper estimates the model using 13,788 transactions between the four main UK brick manufacturers and national house-building firms over 2003–2006. For each transaction (defined as a unique buyer-variety-destination-year combination), the data record the chosen product, negotiated price, production and delivery locations, volume, transport costs, and brick characteristics. The market is highly concentrated: four manufacturers held an 85% share of brick sales, with a two-firm concentration ratio of 0.60 and an HHI of 2,113. Spatial differentiation is a central feature — transport costs vary substantially by project location, and prices for the same brick product vary across the different projects of the same buyer depending on local competitive conditions.

Model

The paper develops an empirical model that adapts the Berry, Levinsohn, and Pakes (1995) differentiated-products framework to individually negotiated pricing. In the model, each buyer negotiates simultaneously and bilaterally with the sellers of the first-best and runner-up products (defined by surplus — value minus cost). The equilibrium first-best markup equals the minimum of (i) the unconstrained Nash bargaining solution, bj(wj(1) − w0), and (ii) the first-best seller’s surplus advantage over the runner-up, (wj(1) − wj(2)). Runner-up and lower-ranked sellers earn zero markups in equilibrium. This outcome is shown to be consistent with a range of non-cooperative bargaining models (Binmore 1985, Bolton and Whinston 1993, Manea 2018) and lies in the core of the associated coalition game. The TIOLI posted-price model is nested as the special case where seller bargaining skill equals one. A tractable likelihood for the joint probability of observed product choice and negotiated price is derived under the assumption that idiosyncratic taste terms follow a Generalized Extreme Value (GEV) distribution.

Main Findings

The estimated mean seller bargaining skill is b̄ = 0.41 (s.e. 0.03), and a likelihood ratio test rejects the TIOLI restriction with a chi-squared statistic of 847 (p < 0.001), confirming that buyer bargaining power is economically and statistically significant. The model-implied price-cost margins (Lerner index) are low on average — mean of 0.08 — but vary widely across transactions (coefficient of variation of 0.78). Project location matters: sellers extract higher margins from buyers that are relatively close, taking advantage of their transport-cost proximity. Multi-product ownership also affects markups, but its relevance varies by project.

Switching from negotiated to uniform pricing raises average markups by 34% at the observed market structure. However, effects are heterogeneous: approximately 15% of transactions see markup decreases. Buyers who benefit from uniform pricing are those with relatively little runner-up competition — precisely the buyers who face weak bargaining positions under negotiated pricing, and for whom the seller’s ability to use that position is constrained under a uniform rule.

Under negotiated pricing, a merger affects a transaction’s markup only if it brings the first-best and runner-up products for that transaction under joint ownership. A demerger to single-product manufacturers reduces total manufacturer surplus by 25%. The merger of the two largest firms increases total manufacturer surplus by 19%, but with highly unequal transaction-level effects. Comparing the same mergers across pricing regimes, negotiated pricing abates average markup-increasing merger effects but worsens them for a minority of transactions — those where the merger creates a first-best/runner-up pairing.

Scope Conditions

The model applies to complete-information settings where prices are negotiated transaction-by-transaction, buyers single-source for each discrete purchase occasion, and sellers have multiple spatially differentiated products. It is most directly applicable to business-to-business markets where individual transaction values are large enough to justify project-level negotiation.

Q: What is the fundamental difference between negotiated pricing in this paper and the standard Nash-in-Nash (NiN) bargaining framework? A: In standard NiN (Horn and Wolinsky 1988), a buyer negotiates one price per product and trades positive quantities of all products with negotiated prices, so all negotiated prices are observed in transaction data. In this paper, buyers make discrete single-sourcing choices — each project uses exactly one product — so only the chosen product’s price appears in data; the runner-up product and its counterfactual price are unobserved. Additionally, under NiN, prices are set at the buyer level and apply uniformly to all the buyer’s needs, whereas here prices are negotiated separately for each project, generating intra-buyer cross-project price variation.

Q: What is the equilibrium markup formula, and what determines whether the Nash bargaining solution or the TIOLI constraint binds? A: The equilibrium first-best markup is ρ*j(1) = min[bj(1)(wj(1) − w0), (wj(1) − wj(2))], the minimum of the unconstrained Nash bargaining solution and the first-best seller’s surplus advantage over the runner-up. The TIOLI constraint (surplus advantage) binds when the seller’s bargaining skill is sufficiently high that the unconstrained NBS would exceed the surplus advantage — that is, when bj(1)(wj(1) − w0) > (wj(1) − wj(2)). Runner-up and all lower-ranked sellers earn zero markups in equilibrium because competition from the first-best drives their outside-option constraint to bind.

Q: Why do third-best and lower-ranked sellers have no effect on equilibrium outcomes? A: Because the most attractive offer any seller below the runner-up could make is a zero markup, and the runner-up already offers a zero markup due to competition from the first-best. Since the runner-up at zero markup already offers the buyer at least as much utility as any third-best product, the third-best cannot improve the buyer’s position. Proposition 1 (part iii) shows that the equilibrium markup and choice are invariant to N for N in {2, …, N̄}.

Q: How does the paper address the econometric challenge that the runner-up product and its price are unobserved? A: The paper derives a tractable closed-form likelihood for the joint probability of the observed product choice and the observed negotiated price, integrating out the unobserved idiosyncratic taste terms along with their implications for the identity and surplus of the unobserved runner-up product. The GEV distributional assumption on taste terms is crucial: it ensures that (1) choice probabilities have a closed form, (2) the surplus advantage can be expressed in terms of observed surpluses and GEV terms, and (3) the probability that the NBS is constrained has a closed form. This reduces the full problem to a lower-dimensional numerical integral over the normally distributed random effects.

Q: What empirical evidence motivates the negotiated pricing model over simpler alternatives? A: Four data patterns motivate the model. First, prices vary across projects even after controlling for product identity and buyer identity — intra-buyer cross-project variation that is inconsistent with standard NiN where prices are set at the buyer level. Second, prices are lower, other things equal, when there is greater local competition from manufacturers not chosen for a project — inconsistent with standard NiN where excluded products play no competitive role. Third, buyers have many projects and make a discrete single-sourcing choice for each. Fourth, sellers are multi-product firms with products differentiated spatially and in other dimensions.

Q: What do the price regressions reveal about price determinants? A: Adding year effects to a simple regression explains only a small share of price variation (R² rises from 0.000 to 0.118 for the full sample). Adding variety-year effects raises R² to 0.775 and adding buyer-variety-year effects to 0.918, but still leaves substantial unexplained variation. Panel B regressions show that prices decrease with quantity, increase with input prices (gas price coefficient 27.2, wage coefficient 8.3), decrease with buyer-to-seller size ratio (coefficient −2.51), and decrease with greater local competition (a distance advantage indicator raises price by about 0.48–2.20 and N(DST) count reduces price by about 1.49–1.53 depending on specification).

Q: What do the parameter estimates imply about spatial differentiation and buyer preferences? A: Transport costs have a strongly negative effect on value (coefficient on distance is −1.27, s.e. 0.04), and the interaction of distance with fuel costs is also negative and significant. The nesting parameter σJ is estimated at 0.47, indicating substantial within-group taste correlation across products from the same firm. Product characteristics matter: red and wire-cut bricks are preferred, and there are significant interactions between weather conditions and technical brick characteristics (frost positively interacts with strength; rainfall negatively interacts with absorption), indicating that buyers value bricks whose technical performance is suited to their project’s climate.

Q: How is the mean seller bargaining skill estimated, and how is the TIOLI model rejected? A: The mean seller bargaining skill b̄ is estimated at 0.41 (s.e. 0.03), substantially below one. The TIOLI restriction corresponds to b̄ = 1 (all markup determined by surplus advantage). A likelihood ratio test rejects this restriction with a chi-squared statistic of 847 (p < 0.001), providing strong statistical evidence that buyer bargaining power — not just competitive pressure — constrains markups below the TIOLI level.

Q: What are the main findings regarding the distribution of price-cost margins? A: Price-cost margins (Lerner index form) are low on average, with a mean of 0.08, but vary widely across transactions, with a coefficient of variation of 0.78. Sellers set higher margins to buyers located relatively close to them (lower transport costs make the seller more attractive to the buyer, strengthening the seller’s position). Multi-product manufacturer portfolios also affect markups, but the relevance of multi-product ownership varies across projects depending on whether different products from the same firm compete as first-best and runner-up for a given project.

Q: What does the uniform pricing counterfactual show, and how does it differ from the Hotelling benchmark? A: Switching from individually negotiated to uniform pricing raises average markups by 34% at the observed market structure. However, effects are heterogeneous: approximately 15% of transactions see markup decreases. Buyers who benefit from the switch are those in transactions with relatively weak runner-up competition — who had weak bargaining positions under negotiated pricing — and who gain because uniform pricing prevents sellers from exploiting that weakness. This contrasts with the result from the simple Hotelling linear city model (Thisse and Vives 1988), where switching to uniform pricing raises all markups.

Q: How does the demerger counterfactual quantify multi-product effects? A: Decomposing the observed market to single-product manufacturers reduces total manufacturer surplus by 25%. This large reduction reflects the role of multi-product ownership in determining who the runner-up is for each transaction: when a manufacturer owns multiple products, it can avoid internal competition between its own first-best and runner-up products, preserving its surplus advantage. The impact is highly unequal across individual transactions, however, because the relevance of multi-product effects depends on whether any of a manufacturer’s other products would have been the runner-up for a given project.

Q: What does the merger of the two largest firms imply for markups and surplus? A: The merger of the two largest firms (by market share) increases total manufacturer surplus in the industry by 19%. Markup increases are very unequal across transactions: the merger affects only those transactions for which the merging firms jointly become the first-best and runner-up, which is the mechanism highlighted in the 2010 US Merger Guidelines for negotiated pricing markets. The heterogeneity of effects means that aggregate market-level concentration measures (such as HHI changes) can be poor proxies for merger effects in these markets.

Q: How does the pricing regime interact with merger effects? A: Comparing the same mergers under negotiated versus uniform pricing, negotiated pricing abates the average markup-increasing effects of mergers. However, for a minority of transactions — specifically those where the merger creates a first-best/runner-up pairing that did not exist pre-merger — negotiated pricing makes the merger’s markup effect worse than it would be under uniform pricing. This implies that the direction of the pricing-regime effect on merger harm is not uniform across buyers, and that transaction-level analysis is required for accurate antitrust assessment.

Q: How does the paper relate to the Competition Commission’s 2007 assessment of the Wienerberger/Baggeridge merger? A: The CC (2007) found the market highly concentrated (HHI 2,113, implied HHI increase of 390 from the merger, both exceeding guideline thresholds) but approved the merger, judging profitability to be at or below average for comparable industries and competition to be more intense than the concentration level alone would suggest. This paper’s model provides formal underpinning for that assessment: with negotiated pricing and buyer bargaining power, markups are constrained by the runner-up competitive threat at the transaction level, not by market-wide concentration, and the low mean Lerner index of 0.08 is consistent with the CC’s profitability finding.

Q: What external validity evidence supports the model’s cost specification? A: The paper compares the marginal costs implied by the estimated model to plant-month level production cost data that were not used in estimation. A good match between the two provides external validation of the cost specification and supports the model’s structural interpretation of the markup decomposition.

First-best and runner-up products: Defined at the project level in terms of surplus (value minus cost). The first-best product j(i,1) is the inside good yielding the highest surplus for project i; the runner-up j(i,2) is the highest-surplus inside good not sold by the first-best seller. These two products — and only these two — determine the equilibrium markup and buyer choice; third-best and lower-ranked products are irrelevant.

Surplus advantage: The difference wj(i,1) − wj(i,2) ≥ 0 between the first-best product’s surplus and the runner-up’s surplus for a given project. This is the competitive constraint on the first-best seller’s markup under TIOLI pricing and the binding ceiling on the negotiated markup whenever the unconstrained Nash bargaining solution would exceed it.

Negotiated pricing: A pricing arrangement in which buyers negotiate prices specific to the individual purchase occasion (here, each construction project), as opposed to uniform pricing where the pre-transport price is the same for all buyers. Prices are determined bilaterally between buyer and competing sellers, with the buyer’s outside option — buying the runner-up at its anticipated negotiated price — serving as the competitive constraint.

Outside option principle (Binmore et al. 1989): The principle that a rival offer (outside option) has no effect on a bilateral Nash bargaining problem unless it would leave the receiving party better off than the Nash bargaining solution — i.e., it constrains rather than shifts the disagreement point. In the paper’s model, the runner-up seller’s zero-markup offer serves as the first-best seller’s constraining outside option when seller bargaining skill is high.

GEV (Generalized Extreme Value) taste distribution: The distributional assumption on project-product idiosyncratic match terms that makes the joint likelihood of observed product choice and negotiated price tractable. The GEV structure yields closed-form choice probabilities (nested logit) and allows the surplus advantage — which depends on unobserved runner-up surplus — to be expressed analytically, enabling joint estimation from transaction-level data.

Price-cost margin (Lerner index): Markup (price minus cost) divided by price, used here at the transaction level. The estimated mean Lerner index is 0.08 with a coefficient of variation of 0.78, reflecting wide dispersion driven by spatial variation in local competition and first-best surplus advantage across transactions.

Nash-in-Nash (NiN) vs. single-sourcing bargaining: NiN (Horn and Wolinsky 1988) applies when a buyer trades positive quantities of all products with negotiated prices (multi-sourcing); the paper’s model applies when a buyer makes a discrete single-sourcing choice per occasion, so only the chosen product’s price is observed. The distinction generates different data observability and different competitive mechanisms — in NiN, excluded products play no role; in this paper, the runner-up’s potential zero-markup offer disciplines the first-best seller’s markup.

Competitive Advertising and Pricing

Mon, 01 Jan 0001 00:00:00 +0000

Hwang, Kim, and Boleslavsky study how firms in an oligopoly simultaneously choose prices and advertising strategies, where advertising is modeled as the choice of how much product information to disclose to consumers. The paper extends the canonical Perloff-Salop (1985) random-utility discrete-choice framework — in which n firms engage in Bertrand competition for a consumer whose value for each product is independently drawn from a common distribution F — by endogenizing the information environment: each firm may choose any mean-preserving contraction (MPC) of F as its advertising strategy, with no structural restriction on feasible content. This full flexibility, drawn from the information design literature, allows each firm to choose the consumer’s effective value distribution, ranging from full information (choosing F itself) to complete concealment (a degenerate distribution at the mean). The model is silent on advertising costs, which are assumed to be zero throughout.

The central result is that intense competition forces firms to provide precise product information. Formally, the full information equilibrium — in which every firm chooses F — exists in the advertising game (the subgame in which prices are fixed symmetrically) if and only if F^(n-1) is convex over its support. Because F^(n-1) represents the distribution of the consumer’s best outside option, convexity means the consumer likely faces an attractive alternative, incentivizing each firm to maximize the chance of offering the highest possible value. Crucially, this convexity condition is guaranteed to hold when n is sufficiently large, regardless of the shape of F, because the power function x^(n-1) becomes more convex as n rises. This establishes that under sufficiently intense competition, full information disclosure is the unique symmetric equilibrium.

The general equilibrium advertising strategy G* — which governs cases where full information is not an equilibrium — satisfies two necessary and sufficient conditions: (i) (G*)^(n-1) is convex over the support of G*, and (ii) for almost all values in the support, G* either coincides with F (where the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (where the firm is locally risk-neutral and has no incentive to alter its distribution). The paper proves existence and uniqueness of G* for any F satisfying the stated regularity conditions (density positive, continuously differentiable, bounded, with finitely many peaks). When F has log-concave density, a unique symmetric pure-price equilibrium (p*, G*) exists in the full game.

The paper demonstrates that strategic advertising has ambiguous implications for prices and consumer welfare. Strategic advertising necessarily reduces social surplus through information loss, since consumers select suboptimal products with positive probability when G* differs from F. However, it compresses the support of the value distribution relative to F, which — by a new result (Proposition 3) — tends to lower the equilibrium price. Offsetting this, strategic advertising also redistributes marginal consumers in ways that may raise or lower the price. In the duopoly case with power distributions F(v) = v^alpha on [0,1], strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) (approximately 0.7071), and raises consumer surplus if and only if alpha > 0.7928.

The paper examines three extensions: (1) a binding consumer outside option, (2) multi-unit (k-out-of-n) demand, and (3) asymmetric firms with two types. In all three cases, full information cannot be a strict equilibrium for any finite n under the relevant structural condition, yet the equilibrium distribution G* converges pointwise to F as n tends to infinity, preserving the paper’s core asymptotic insight.

Q: What is the main research question? A: The paper asks how much product information firms will voluntarily disclose when they compete both on price and advertising content in an oligopoly. Unlike the monopoly literature, the oligopoly context creates strategic interdependencies — each firm’s optimal disclosure depends on rivals’ disclosure choices — that the paper characterizes fully.

Q: How is advertising modeled, and why use mean-preserving contractions? A: Each firm’s advertising strategy is modeled as a choice of any mean-preserving contraction (MPC) of the true value distribution F. An MPC preserves the expected value but reduces dispersion, capturing the idea that a firm can selectively conceal information (moving toward a degenerate distribution) but cannot fabricate value dispersion beyond what F allows. Because consumers are risk-neutral and buy based on expected values net of prices, this MPC formulation captures full flexibility in information design without loss of generality.

Q: What is the precise necessary and sufficient condition for the full information equilibrium in the advertising game? A: The full information equilibrium — in which every firm chooses F — exists if and only if F^(n-1) is convex over its support [v, v̄]. The “only if” direction follows from Lemma 1: in any equilibrium, (G*)^(n-1) must be convex, so if F^(n-1) is not convex, F is not an equilibrium. The “if” direction follows because a convex F^(n-1) makes each firm locally risk-loving, so no MPC of F yields a higher payoff than F itself.

Q: Why does sufficiently intense competition force full information disclosure? A: For any distribution F with positive, continuously differentiable, bounded density f with bounded derivative f’, the second derivative of F^(n-1) satisfies F(v)^(n-1)’’ >= (n-1)F(v)^(n-3)[(n-2)epsilon^2 - M], where epsilon = min f(v)^2 > 0 and M = max |f’(v)| < infinity. This expression is strictly positive for n sufficiently large, so F^(n-1) is convex and the full information equilibrium exists. Economically, with many competitors each firm wins the consumer only when it offers the highest possible value, so providing full information is optimal.

Q: What are the two necessary and sufficient properties characterizing the general equilibrium advertising strategy G?* A: First (Lemma 1), (G*)^(n-1) must be convex over the support of G* — this prevents any firm from profitably concentrating mass to reduce dispersion. Second (Lemma 2), for almost all values in the support, either G* = F locally (the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (the firm is locally risk-neutral and indifferent over distributions with the same local mean). Theorem 1 proves these two conditions are both necessary and sufficient, and that G* is unique for any F satisfying the stated regularity conditions.

Q: What structure does G take when F^(n-1) has strictly quasi-concave density?* A: By Corollary 2(1), there exists a cutoff v* in [v, v̄] such that G*(v) = F(v) for v <= v* (full information below the cutoff) and (G*)^(n-1) is linear above v*. As n increases, v* rises, meaning the region of full disclosure expands, and G* increases in convex order — so consumers receive strictly more information. One immediate implication is that consumer surplus strictly increases in n: consumers benefit both from more options and from more accurate information about each product.

Q: What happens when F^(n-1) is concave? A: By Corollary 3, when F^(n-1) is concave, (G*)^(n-1) is linear over the entire support, with lower bound v. In the illustrative Example 1 (truncated exponential with n=2), this yields G* = U[0, 2*mu_F] — a uniform distribution on an interval whose upper bound is twice the mean of F.

Q: Does strategic advertising raise or lower equilibrium prices, and consumer surplus? A: Both effects are ambiguous and depend on the shape of F. Strategic advertising compresses the support of the value distribution (since G* is an MPC of F), which by Proposition 3(1) tends to lower equilibrium prices. But it also reshapes the distribution of marginal consumers, which may raise or lower prices. In the power distribution example (n=2, F(v) = v^alpha on [0,1]), strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) ≈ 0.7071, and raises consumer surplus if and only if alpha > 0.7928. Thus even with deadweight loss from information suppression, consumers can be better off under strategic advertising than under forced full disclosure.

Q: What does Proposition 3 contribute about equilibrium prices in the Perloff-Salop model? A: Proposition 3 delivers two results about how the distribution of marginal consumers (integral (F^(n-1))’ dF) determines equilibrium prices. First, the measure of marginal consumers decreases if F is proportionally stretched over a larger support, confirming that longer support raises equilibrium prices. Second — presented as novel — among all distributions with support in [v, v̄], the power distribution F(v) = ((v-v)/(v̄-v))^(2/n) minimizes the measure of marginal consumers, corresponding to the maximum equilibrium price. The key property is that marginal consumers are uniformly distributed under this power distribution, and any deviation from uniformity allows a “flattening” adjustment that increases the measure of marginal consumers and lowers the price.

Q: Under what condition does the full game (price plus advertising) have a unique symmetric pure-price equilibrium? A: Theorem 2 states that log-concavity of the density f is sufficient for existence and uniqueness of a symmetric pure-price equilibrium (p*, G*) as characterized in Theorems 1 and 2. Log-concavity ensures that the equilibrium distribution G* has a convex-linear structure (as in Corollary 2), which preserves log-concavity of each firm’s profit function even under compound deviations (simultaneous changes to both price and advertising strategy), making the first-order conditions sufficient for global optimality.

Q: Can strategic advertising create or destroy pure-price equilibria relative to the Perloff-Salop benchmark? A: Yes, both directions are possible. When F^(n-1) is convex (so G* = F), equilibrium existence in the Perloff-Salop (PS) model is necessary but not sufficient for existence in the full model, because compound deviations (changing both price and advertising) may be profitable even when pure price deviations are not. Conversely, when G* differs from F, the changed distribution of marginal consumers can sustain an equilibrium in the full model even when none exists in PS. Appendix E of the paper provides a specific example of the latter phenomenon.

Q: What happens with a binding consumer outside option? A: Proposition 4 shows that a full information equilibrium never exists in the advertising game when the consumer has a binding outside option (p* in (v, v̄)). The firm’s value function acquires a discrete jump at p* due to the indicator 1_{v >= p*}, making it optimal to pool mass around p* rather than disclose fully. Nevertheless, Proposition 5 proves that G* converges pointwise to F as n tends to infinity, because the jump of size F(p*)^(n-1) vanishes exponentially fast as n grows.

Q: Does the full information result survive multi-unit demand? A: No. Proposition 6 shows that with k > 1 units demanded (out of n products), the full information equilibrium never exists for any finite n or F. The reason is that phi’(v; F) — the firm’s marginal value of offering value v — is zero at v̄ when k > 1, so the firm can profitably pool values near the top of the support. However, Proposition 7 shows that G* converges pointwise to F as n tends to infinity (with k fixed), preserving the asymptotic full information result.

Q: What happens with asymmetric firms differing in their value distribution supports? A: Proposition 8 shows a sharp dichotomy. If both firm types share the same upper bound of their value supports (v̄_1 = v̄_2), the full information equilibrium exists whenever both F_1^(n1-1) and F_2^(n2-1) are convex. If the supports have different upper bounds (v̄_1 < v̄_2), the full information equilibrium never exists regardless of n_1 and n_2, because type-2 firms face a downward kink in their winning probability at v̄_1 and always have an incentive to pool mass there. The authors conjecture that G*_1 and G*_2 still converge to F_1 and F_2 asymptotically but do not prove this due to technical complexity.

Q: How does this paper relate to Ivanov (2013)? A: Ivanov (2013) also uses the Perloff-Salop framework and shows that full information is an equilibrium when n is sufficiently large, but restricts advertising to rotation-ordered strategies (in the sense of Johnson and Myatt, 2006). The present paper imposes no structural restriction and strengthens Ivanov’s result by: (a) providing a necessary and sufficient condition for the full information equilibrium (not just a sufficient condition for large n); (b) fully characterizing G* when full information is not an equilibrium; and (c) demonstrating robustness across multiple model variants.

Q: What policy implication does the ambiguity result carry? A: The paper warns against assuming that mandating full information disclosure is unambiguously consumer-beneficial. While strategic advertising creates deadweight loss through information suppression, it can simultaneously compress support and alter the marginal consumer distribution in ways that lower equilibrium prices significantly. The power distribution example (alpha > 0.7928) shows consumers can be strictly better off under strategic advertising than under forced full disclosure. This ambiguity is a cautionary tale for disclosure regulation.

Mean-Preserving Contraction (MPC): A distribution G_i is an MPC of F if it has the same mean as F but less dispersion (in the sense of second-order stochastic dominance). In the paper, each firm’s feasible advertising strategies are exactly the set MPC(F) — this captures all informationally feasible disclosures without structural restriction on content.

Advertising Game: A restricted subgame of the full market game in which firms choose their advertising strategies G_i taking the symmetric price as given. An equilibrium in the advertising game is a necessary condition for equilibrium in the full game. The advertising game’s equilibrium uniquely pins down G* independently of the price level (under the baseline model without binding outside option).

Full Information Equilibrium: An equilibrium of the advertising game in which every firm chooses the true underlying distribution F as its advertising strategy. This corresponds to complete, unobstructed product disclosure. The paper’s central result is that this equilibrium exists if and only if F^(n-1) is convex over its support.

Convexity of F^(n-1): The key distributional condition governing advertising equilibria. F^(n-1) is the distribution of the consumer’s best alternative among (n-1) rivals’ products. Convexity of F^(n-1) means its density is increasing, signaling a likely attractive outside option, which makes each firm risk-loving and induces full disclosure. This convexity is guaranteed for n sufficiently large.

Locally Linear (G)^(n-1):* A region of the equilibrium distribution where (G*)^(n-1) has constant slope, making the firm locally risk-neutral. Over such a region, the firm is indifferent among all distributions with the same local mean, and the equilibrium G* need not coincide with F — it is only required to be an MPC of F on that interval. This alternating structure (coinciding with F on strictly convex regions; linear elsewhere) fully characterizes G*.

Marginal Consumers: In the Perloff-Salop pricing formula, the equilibrium price p* = (1/n) / integral [(G*(v)^(n-1))’ dG*(v)]. The integrand (G*(v)^(n-1))’ * g*(v) is the density of consumers who are indifferent between a given firm’s product and their best alternative at value v. A larger measure of marginal consumers implies lower equilibrium prices through greater competitive pressure.

Compound Deviation: In the full game, a deviation by a firm that changes both its price p_i and its advertising strategy G_i simultaneously, rather than varying only one dimension. The possibility of compound deviations is what distinguishes equilibrium existence conditions in the full model from those in the standard Perloff-Salop model, even when G* = F.

Consistent Evidence on Duration Dependence of Price Changes

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. This paper asks two related questions. First, can one develop a robust, distribution-free estimator for the discrete-time mixed proportional hazard (MPH) model of duration with unobserved heterogeneity? Second, what does that estimator reveal about the shape of the hazard of price changes, the role of heterogeneity in shaping aggregate price dynamics, and the distinction between regular price changes and sales?

Methodology. The authors develop a linear generalized method of moments (GMM) estimator for the discrete-time MPH model, building on identification results in Honoré (1993). The model specifies that the probability a price spell ends at duration t, conditional on surviving to t, equals the product of a product-specific frailty parameter θ (unobserved, fixed over time) and a common baseline hazard bt. The estimator exploits repeated price spells per product via moment conditions that are linear in bt, making estimation and inference straightforward. It accommodates right- and left-censored data, competing risks, and spell-specific observable characteristics, without requiring any parametric assumption on the frailty distribution. The estimator is consistent as the number of products grows, even with a short time dimension. A Hansen-Sargan J-test of overidentifying restrictions and a test of the monotone-average-type prediction are also developed.

The estimator is applied to two datasets: (1) IRI weekly store data (2001–2011), covering 30 product categories and more than 21 million products, yielding 684,919,778 pairs of durations; and (2) Online Micro Price data from Cavallo (2018), comprising approximately 250,000 products at daily frequency.

Main Findings with Quantitative Magnitudes.

Baseline hazard and heterogeneity. In the pooled IRI data, the Kaplan-Meier hazard is steeply declining throughout the entire range from 2 to 60 weeks. In contrast, the estimated baseline hazard is roughly constant until week 4 and then declines only modestly, with a noticeable spike at week 52. The ratio of the Kaplan-Meier hazard to the baseline hazard — the average type, E[θ|t] — drops by approximately 60 percent within the first 20 weeks, and continues to decline, reaching roughly 0.3 of its initial value after one year. This decomposition reveals substantial unobserved heterogeneity that accounts for a large fraction of the observed decline in the Kaplan-Meier hazard.

Implications for structural models. The finding of a decreasing baseline hazard is inconsistent with canonical state-dependent pricing models (Golosov and Lucas, 2007), which predict an increasing hazard, conditional on a given firm’s type. The decreasing baseline hazard is instead broadly consistent with time-dependent pricing models, though not with a constant-hazard (Calvo, 1983) specification.

Monetary policy impulse response. In a calibrated time-dependent pricing model with strategic complementarity (α = 0, 0.5, 0.95), the aggregate price level dynamics in the estimated heterogeneous-firm MPH economy are close to those of a homogeneous-firm economy that uses the Kaplan-Meier hazard as the common price-change hazard. The homogeneous-firm approximation is substantially closer to the MPH economy than a Taylor (1979, 1980) staggered-contract economy with the same Kaplan-Meier hazard, particularly when strategic complementarity is strong (α = 0.95). The Calvo economy provides a poor approximation due to its exponential (constant-speed) price convergence structure.

Regular versus temporary price changes. Using the competing-risks extension with spell-specific observables — classifying spells by whether they start and end with a price increase (+) or decrease (−) — the authors separately estimate four baseline hazards. The baseline hazard for consecutive price increases (b++t) is relatively flat, especially for the first 6 weeks, then flat until week 45, with a spike near one year, consistent with price-plan models. The baseline hazard for reversals (particularly b−+t, price decreases followed by price increases, associated with sales) is steeply declining. The J-test statistics are substantially lower for price trends (J++ = 3,920; J−− = 3,401) than for reversals (J+− = 8,737; J−+ = 7,910), and markedly lower than the pooled-model J = 10,498, indicating that the MPH structure fits regular price changes considerably better than sales.

Scope Conditions. Results are conditional on weekly store-level price data for mostly packaged consumer goods (30 IRI product categories). The analysis focuses on price spells of at least 2 weeks to avoid spurious duration-one spells from mid-week price changes. The maximum duration examined is 60 weeks. The comparison of estimation methods relies on the IRI data only; the Online Micro Price data confirm weekly decision-making through a spike in the daily hazard every 7 days. Comparisons with maximum likelihood estimates show that GMM recovers more heterogeneity (average type declines to 0.37 at 6 months by GMM versus 0.48 by continuous-time MLE), and that time aggregation explains most of the discrepancy between the two methods.

In depth

Q1. What is the mixed proportional hazard (MPH) model as used in this paper, and what does the estimator identify?

A1. The MPH model specifies that the hazard that a price spell ends at duration t, conditional on surviving to t, equals θ·bt, where θ is a product-specific frailty parameter drawn from an unknown distribution G and bt is a baseline hazard common to all products. The estimator, which is linear in bt, identifies the baseline hazard up to a multiplicative constant using moment conditions derived from repeated spell data, without restricting the shape of the frailty distribution. Identification relies on comparing the joint survival probabilities of two consecutive spells for the same product and exploits the symmetry implied by the MPH structure across spells.

Q2. How does the Kaplan-Meier hazard relate to the baseline hazard, and what does this relationship imply about heterogeneity?

A2. The paper proves that the Kaplan-Meier hazard Ht equals bt times E[θ|t], the mean frailty among spells surviving to duration t. Because higher-type products (those with a higher propensity to change prices) exit the pool of surviving spells earlier, E[θ|t] is strictly decreasing in t — a form of dynamic selection. The ratio Ht/bt, normalized to 1 at the start, falls to approximately 0.4 by week 20 in the pooled IRI data and to approximately 0.3 after one year, documenting that a large share of the decline in the Kaplan-Meier hazard reflects heterogeneity rather than structural negative duration dependence.

Q3. What does the estimated baseline hazard imply about structural models of price setting?

A3. A decreasing baseline hazard is inconsistent with the canonical state-dependent model of Golosov and Lucas (2007), in which a firm’s hazard of price change is increasing in the time since the last change, because larger deviations from the desired price accumulate with duration. The decreasing baseline hazard is instead consistent with time-dependent pricing models and with price-plan models where within-plan switches are costless. The mild spike at week 52 in the baseline hazard is consistent with Taylor-type annual pricing rules.

Q4. What is the approximate aggregation result for monetary policy, and how quantitatively accurate is it?

A4. In the time-dependent pricing model without strategic complementarity (α = 0), the impulse response of the aggregate price level to a monetary shock in a heterogeneous-firm economy is exactly the same as in a homogeneous-firm economy whose single firm uses the Kaplan-Meier survival function. This extends Carvalho and Schwartzman (2015) to an approximation in the case with strategic complementarity (α = 0.5 and α = 0.95). Numerically, the path of aggregate prices in the estimated MPH economy is close to that in the homogeneous-firm Kaplan-Meier economy, and substantially closer to it than to the Taylor-contract economy — the difference is most pronounced at horizons beyond about half a year when α = 0.95, where the Taylor economy shows notably slower initial convergence and faster later convergence relative to the MPH and homogeneous economies.

Q5. How do the paper’s results differ from those obtained using maximum likelihood estimation of the continuous-time MPH model?

A5. The GMM estimator recovers substantially more heterogeneity than maximum likelihood (MLE) applied to the continuous-time model with continuous records (assumed gamma frailty). The average type falls from 1 to 0.37 at six months under GMM, versus only 0.48 under MLE. The authors investigate two sources of this discrepancy: the assumed frailty distribution family (gamma) and time aggregation. They conclude that time aggregation is quantitatively more important in the IRI weekly data — that is, the continuous-time MLE approach fails to properly account for the discrete nature of the data-generating process, leading it to understate heterogeneity and recover a steeper baseline hazard.

Q6. How does the paper distinguish regular price changes from sales without directly observing a sales flag?

A6. The competing-risks extension classifies each spell by whether it starts with a price increase or decrease (observable characteristic χ ∈ {+, −}) and by whether it ends with a price increase or decrease (competing risk ρ ∈ {+, −}). Price trends — spells where the direction is the same at both the start and end (++ or −−) — are interpreted as regular price changes; price reversals (especially −+, i.e., price decrease followed by increase) are associated with sales. This approach is consistent with the statistical model used for estimation, avoids the bias from simply dropping suspected sales spells before estimation, and allows the MPH structure to hold only for the risks of interest even if it fails for others.

Q7. How well does the MPH model fit regular price changes versus sales?

A7. The J-test of overidentifying restrictions yields test statistics of J++ = 3,920 for consecutive price increases and J−− = 3,401 for consecutive price decreases, compared with J = 10,498 for the pooled model and J+− = 8,737 and J−+ = 7,910 for the reversal hazards. All rejections are at conventional significance levels (critical value 1,749 at 5%), but the rejection is substantially milder for price trends than for price reversals. For individual product categories, the model cannot be rejected for 8 categories (out of 30) for b++ and 21 categories for b−−, suggesting the MPH structure is a much better description of regular price changes than of sales.

Q8. What role do one-week price spells play in the data, and why are they excluded?

A8. In the IRI data, prices are measured as the ratio of weekly revenue to quantity, so a price change occurring mid-week generates a spurious price spell of duration one week. If all spells including one-week spells are retained, the autocorrelation of spell durations is only 0.029 in levels and even negative (−0.042) in logs, which is inconsistent with a mixture model. Once one-week spells are excluded, the autocorrelation rises to 0.235 in levels and 0.233 in logs, and is stable when two-week spells are also excluded (0.248 and 0.256). The paper therefore sets the lower duration bound at T̲ = 2 weeks.

Q9. What does the daily Online Micro Price data add relative to the weekly IRI data?

A9. The daily data reveal a sharp spike in the price-change hazard every seven days, suggesting that even when prices are observed daily, the decision to change prices is made at the weekly frequency. This justifies the use of a discrete-time model with a one-week period. The estimates from daily and weekly aggregations of the same data are broadly similar, though weekly data recovers somewhat less heterogeneity than daily data. Aggregating IRI weekly data to monthly frequency understates heterogeneity even more, confirming that frequency matters for measuring heterogeneity.

Q10. What are the computational advantages of the GMM estimator relative to maximum likelihood?

A10. Because the moment conditions are linear in the baseline hazard bt, the GMM estimator is obtained in closed form, making estimation fast and inference straightforward. On the pooled IRI sample, GMM estimation (including standard errors) required 70 minutes on a machine with 60 GB memory, whereas the maximum likelihood estimator required 15 hours on a machine with 256 GB memory and failed entirely on the 60 GB machine. The GMM approach also avoids the need to specify the frailty distribution family and guarantees a global solution (proved by the identification result), whereas the likelihood function is non-linear in bt and may have multiple local maxima.

Q11. What is the shape of the b++ baseline hazard for regular price increases, and what models does it support?

A11. The baseline hazard for spells starting and ending with a price increase (b++) is decreasing during the first 6 weeks — dropping by almost 50% — and then flat until approximately week 45, with a pronounced spike at around one year. This shape is consistent with price-plan models (Eichenbaum, Jaimovich, and Rebelo, 2011) with Calvo-type switching between plans, where within-plan changes are costless and the hazard of between-plan switching is approximately constant. The annual spike is consistent with Taylor-type pricing. Approximately 76.8% of complete spells starting after a price increase last at most 6 weeks.

Key Concepts

Baseline hazard (bt). The component of the MPH hazard that is common to all products and may vary arbitrarily with elapsed duration t. It represents structural duration dependence — the tendency for a given product to be more or less likely to change price as a function of how long its current spell has lasted — net of heterogeneity. It is identified only up to a multiplicative constant.

Frailty parameter (θ) / frailty distribution (G). The product-specific scaling factor in the MPH model, fixed over all spells for a given product, that captures permanent unobserved differences in price-change frequency across products. The paper treats G as a nuisance parameter and does not require a parametric assumption on its shape. A higher θ means the product has a higher baseline propensity to change its price.

Average type (E[θ|t]). The mean frailty parameter among spells that have survived to at least duration t. Because high-type products change price earlier and exit the pool of surviving spells first, the average type is provably strictly decreasing in t under the MPH model. It is measured as the ratio of the Kaplan-Meier hazard to the baseline hazard, and its rate of decline measures the importance of dynamic selection.

Kaplan-Meier hazard (Ht). The probability that a randomly drawn spell ends at duration t, conditional on having lasted at least t periods. It mixes together structural duration dependence (captured by bt) and dynamic selection (captured by changes in the average type). It can be estimated without imposing the MPH structure, requiring only stationarity of the duration process.

Competing risks. The framework in which a price spell can end for multiple distinct reasons — here, ending with a price increase or a price decrease — each with its own hazard function. The paper’s GMM approach allows the MPH structure to hold for only a subset of risks and observables, without imposing any structure on the remaining risks.

Price trends vs. price reversals. A classification of spells based on the direction of the surrounding price changes. Price trends are spells where the direction of the price change at the start and end of the spell is the same (++ or −−), interpreted as regular price changes. Price reversals are spells where the direction switches (e.g., −+, a price decrease followed by a price increase), associated with sales and other temporary price changes.

Strategic complementarity in pricing (α). The degree to which a firm’s target price responds to the average price set by other firms. Parameterized by α ∈ [0, 1), where α = 0 yields the exact aggregation result (only the Kaplan-Meier hazard matters) and higher α increases aggregate price stickiness by making firms reluctant to deviate from the average price when few others are adjusting.

Dynamic selection. The mechanism by which the composition of the pool of surviving price spells shifts toward lower-type (more price-sticky) products as duration increases, because higher-type products change price sooner and exit the pool. This is the source of the gap between the steeply declining Kaplan-Meier hazard and the more modestly declining baseline hazard.

Consumer durables and monetary policy according to HANK

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

Consumer durables account for a disproportionately large share of household expenditure fluctuations despite their small share of total private consumption. Two stylized facts motivate the paper: (1) durable expenditure is far more interest-rate sensitive than nondurable expenditure following monetary policy shocks, and (2) durable and nondurable expenditures comove positively and persistently—both reaching trough in the same quarter. Standard two-sector New Keynesian models struggle to generate this positive conditional comovement because asymmetric sectoral price rigidity induces large relative-price movements that push the two sectors in opposite directions. This paper asks what model features are necessary and sufficient to reproduce both the sectoral comovement pattern and the hump-shaped aggregate dynamics observed in the data, and how the answer changes across households sorted by liquid asset holdings.

Data and Methodology

Empirical identification. The authors employ a local projection instrumental variables (LP-IV) strategy using Romer-Romer monetary policy shocks updated by Wieland and Yang (2020), over the sample 1969:Q1–2007:Q3. Impulse response functions (IRFs) are normalized to a cumulative 100 basis-point increase in the Federal Funds Rate over five years. Household-level evidence is drawn from the Consumer Expenditure Survey (CEX) and the Survey of Consumer Finances (SCF); households are classified as liquidity-constrained if liquid assets are below $1,000.

Model. The authors develop a two-sector Heterogeneous Agent New Keynesian (HANK) model in which households maximize utility over nondurable consumption and a durable stock (Cobb-Douglas aggregation), face convex adjustment costs on durable purchases, and update expectations infrequently in the Mankiw-Reis sense (probability of not updating: Xi = 0.918 per period). The general equilibrium version features asymmetric Rotemberg price stickiness (Calvo probability 0.671 for nondurables, 0.797 for durables), nominal wage stickiness (Calvo 0.802), and a Taylor rule with inflation coefficient 1.105, output coefficient 1.440, and smoothing 0.988.

Main Findings and Quantitative Magnitudes

Sectoral magnitude gap. At trough (approximately 8 quarters after the shock), the durable expenditure response to monetary tightening is an order of magnitude larger than the nondurable response—a fact the calibrated HANK model is designed to match.
Positive comovement. Both durable and nondurable expenditures contract and reach trough in the same quarter, contradicting TANK models (Monacelli 2009) in which savers shift portfolios toward durables and generate negative comovement for that group.
Relative-price dynamics. The relative price of durables rises following monetary tightening (nondurables deflate more), but the rise is modest and cannot overturn the positive comovement result.
Role of the direct interest-rate effect. Across liquid-asset groups, the direct effect accounts for 73–87% of the cumulated durable expenditure response and 37–91% of the cumulated nondurable expenditure response. This direct channel—operating through intertemporal substitution—is quantitatively first-order for durables in a way it is not in standard single-sector HANK models where income effects dominate.
Role of sticky information. A full-information HANK variant produces a counterfactually high durable elasticity (35.24 times the baseline) and no hump-shaped dynamics. Infrequent information updating (Xi = 0.918) is essential to match the hump-shaped propagation of both sectoral and aggregate expenditures.
Income effects and fiscal policy. For a fiscal subsidy specifically targeting durable purchases, intertemporal substitution incentives generate a large shift toward durables and, without income effects, a counterfactual crowding-out of nondurable spending. Income effects are essential to protect nondurable spending, and the aggregate consumption effect of such a policy is at best modest—consistent with Mian and Sufi’s (2012) evidence on cash-for-clunkers.

Scope Conditions

All empirical results are conditional on the LP-IV sample 1969:Q1–2007:Q3 and Romer-Romer shocks as instrumented by Wieland-Yang. The household-level comovement result is established for both liquidity-constrained (liquid assets below $1,000) and unconstrained savers using CEX/SCF data. Model quantitative results are specific to the calibration targeting moments from Fagereng et al. (2021) marginal propensities and BEA depreciation data (delta = 0.054).

In depth

Q1. What is the core empirical puzzle the paper addresses, and why do standard models fail?

Standard two-sector New Keynesian models predict that asymmetric sectoral price stickiness generates large relative-price movements between durables and nondurables following a monetary shock. These relative-price shifts tend to produce negative conditional comovement—when durables contract, nondurables expand—contradicting the data. The authors document that both categories exhibit positive and persistent comovement, both reaching their trough at approximately 8 quarters, which standard models cannot replicate.

Q2. What are the key empirical facts established via LP-IV?

Using Romer-Romer shocks over 1969:Q1–2007:Q3, normalized to a cumulative 100bp Federal Funds Rate increase, the authors find: (1) aggregate expenditure follows a hump-shaped contraction with trough at roughly 8 quarters; (2) the durable expenditure response is an order of magnitude larger than the nondurable response at trough; (3) both categories reach their trough in the same quarter; and (4) the relative price of durables rises modestly after monetary tightening (nondurables deflate more), but not enough to reverse comovement.

Q3. How is the partial equilibrium model calibrated, and which moments does it target?

Key calibrated parameters include CRRA sigma = 2.640, Cobb-Douglas weight on nondurables theta = 0.607 (implying durable expenditure share 0.193), adjustment cost alpha = 8.299, information stickiness Xi = 0.918, depreciation rate delta = 0.054, steady-state real rate r = 0.03/4, discount factor beta = 0.915 (matching a 30% share of liquidity-constrained households with liquid assets-to-income ratio of 0.26), and borrowing wedge kappa = 0.05. Moments matched include quarterly MPC on nondurables (22.94%), quarterly MPX on durables (24.15%), interest-rate elasticity of durable expenditure (3.35, within the empirical range of 1.1–5.0), price elasticity of durable demand (29.59), and durable stock skewness relative to nondurable consumption (0.695, consistent with Bertola et al. 2005).

Q4. How does the paper decompose monetary policy transmission?

The paper decomposes transmission into three channels: (1) the direct effect of real interest rate changes, which operates through intertemporal substitution and accounts for the quantitatively largest share of the durable response; (2) the relative-price effect, which is modest and redistributive but cannot overturn positive comovement; and (3) pure income effects, which are key for persistence of the nondurable response but not for the sign of comovement.

Q5. What do counterfactual models reveal about the role of each model ingredient?

A sticky-information RANK produces positive comovement but the dynamics are front-loaded and less inertial than in the data. A sticky-information TANK delivers results similar to RANK—income effects do not qualitatively change the story. A full-information HANK produces a counterfactually high durable interest-rate elasticity (35.24 times the baseline) and no hump-shaped dynamics, demonstrating that sticky information is the ingredient generating realistic propagation, not heterogeneity per se.

Q6. What does the household-level evidence from CEX and SCF show about comovement across the wealth distribution?

Classifying households as liquidity-constrained if liquid assets are below $1,000, the LP-IV estimates show positive comovement between durables and nondurables for both constrained and unconstrained savers. This contradicts TANK models (Monacelli 2009), in which savers shift portfolios toward durables following a monetary shock, generating negative comovement for the saver group. After controlling for income and relative prices, the direct interest-rate effect operates uniformly across financial status groups.

Q7. How does the direct effect vary across liquid asset groups quantitatively?

Decomposing across four liquid asset groups (below $1k, $1k–$10k, $10k–$20k, above $20k), the direct effect accounts for 73–87% of the cumulated durable expenditure response and 37–91% of the cumulated nondurable expenditure response. Income effects are more important for nondurable spending prolongation among liquidity-constrained households, but the direct channel dominates durable expenditure for all groups.

Q8. How does the general equilibrium two-sector HANK model differ from the partial equilibrium setup?

The GE model adds asymmetric sectoral price stickiness (Calvo probabilities 0.671 for nondurables and 0.797 for durables), nominal wage stickiness (Calvo 0.802), a Taylor rule (inflation coefficient 1.105, output coefficient 1.440, smoothing 0.988), and fiscal lump-sum taxes responding to debt (coefficient 0.191). These features generate the relative-price dynamics observed in the data while preserving the positive comovement result.

Q9. What does the fiscal policy application reveal about the role of income effects?

A fiscal subsidy targeting durable purchases generates a much larger shift in the relative price of durables than monetary policy does. Without income effects, intertemporal substitution dominates and nondurable spending falls—a counterfactual result inconsistent with the data. With income effects present, nondurable spending is protected. The aggregate consumption effect of such a durable-targeted fiscal policy is at best modest, consistent with Mian and Sufi’s (2012) evidence from the cash-for-clunkers program.

Q10. What is the broader implication for the literature on HANK versus RANK transmission?

In standard single-sector HANK models, income effects (the indirect channel) typically dominate monetary transmission. The presence of consumer durables restores a quantitatively important role for the direct interest-rate channel, which operates through intertemporal substitution in durable purchases. This rebalances the direct-versus-indirect decomposition relative to the conventional HANK wisdom and shows that the durable goods sector is essential to understanding the full transmission mechanism.

Key Concepts

Sectoral comovement (conditional on monetary policy shocks) The empirical regularity that durable and nondurable expenditures both contract following monetary tightening and reach their respective troughs in the same quarter. In this paper, comovement is defined conditional on identified monetary policy shocks (LP-IV with Romer-Romer instruments), not unconditionally. Standard two-sector NK models predict negative conditional comovement due to relative-price effects; replicating positive comovement is the central discipline imposed on the model.

Direct effect (of real interest rate changes) The component of monetary transmission that operates through the intertemporal substitution incentive induced by changes in the real interest rate, holding income and relative prices fixed. Distinct from the income effect (indirect channel) and the relative-price effect. In this paper’s decomposition, the direct effect accounts for 73–87% of the cumulated durable expenditure response across liquid-asset groups.

Sticky information (Mankiw-Reis) Households update their information sets infrequently, with probability (1 - Xi) per period; Xi = 0.918 means only about 8.2% of households update each quarter. This mechanism is essential in the model for generating the hump-shaped, inertial impulse response dynamics observed in the data. Without it (full-information HANK), the durable elasticity is counterfactually large (35.24 times baseline) and dynamics are front-loaded.

MPX (Marginal Propensity to Expend on durables) Analogous to the MPC for nondurables, the MPX measures the additional durable expenditure flow induced by an income windfall. Calibrated to 24.15% quarterly, matching estimates from Fagereng et al. (2021). Distinct from the MPC because durable purchases represent investment in a stock, not immediate consumption flow.

Liquidity-constrained households Households with liquid assets below $1,000, identified in the CEX and SCF. In the model, the 30% share of such households is targeted by the discount factor (beta = 0.915) and the borrowing wedge (kappa = 0.05). The paper’s key finding is that positive comovement holds for both constrained and unconstrained households, contradicting TANK predictions.

HANK (Heterogeneous Agent New Keynesian model) A New Keynesian general equilibrium model in which households are heterogeneous in their liquid asset holdings (and thus face binding borrowing constraints), so that the distribution of assets matters for aggregate dynamics. Distinguished from RANK (Representative Agent NK) and TANK (Two-Agent NK, which approximates heterogeneity with one unconstrained and one hand-to-mouth agent). In this paper, HANK is extended to a two-sector setting with durables and nondurables.

Convex adjustment costs on durable purchases A cost of adjusting the durable stock that is convex in the size of the adjustment (calibrated parameter alpha = 8.299). This smooths the durable expenditure response and prevents counterfactually sharp jumps in durable purchases following interest rate changes, contributing to realistic propagation dynamics alongside sticky information.

Contextually Private Mechanisms

Mon, 01 Jan 0001 00:00:00 +0000

Haupt and Hitzig introduce a framework for comparing the privacy properties of different mechanism protocols. The core research question is: when a designer commits to implementing a social choice rule, how much superfluous private information must they inevitably learn about agents, and how should they design the elicitation protocol to minimize that exposure?

The setting is a finite-player extensive-form game in which a designer elicits agents’ private types through a dynamic protocol to compute a social choice function. The authors explicitly exclude cryptographic tools and trusted mediators, working under the minimal assumption that the designer learns information if and only if an agent discloses it. This assumption is motivated by the historical prevalence of live dynamic auction formats — ascending formats at Sotheby’s, descending formats at Aalsmeer, oral ascending formats used by the U.S. Forest Service for timber, multi-round clock auctions for radio-spectrum allocation — and by settings where mediating technology is unavailable or costly.

The central object is the contextual privacy violation. A protocol produces a contextual privacy violation for agent i at type profile θ if the designer can distinguish θ_i from some alternative type θ’_i while holding other agents’ types fixed, yet the social choice rule assigns the same outcome at both profiles. Violations are defined at the level of individual agent–state pairs, not aggregated ex ante. A protocol is fully contextually private if it produces no violations; it is maximally contextually private if its set of violations is inclusion-minimal among all protocols that implement the same rule.

The main characterization result (Theorem 1) connects privacy to pivotality: a social choice function admits a fully contextually private protocol if and only if, on every product subset of the type space where agents are collectively pivotal, at least one agent is individually pivotal. The contrapositive is what drives the paper’s impossibility results: whenever a rule contains a region where no single agent’s report changes the outcome but a group’s joint report does, any implementing protocol must produce contextual privacy violations.

Using this characterization, the authors establish that the first-price auction rule (Proposition 2) and serial dictatorship (Proposition 3) admit fully contextually private protocols. Conversely, k-item Vickrey auction rules (Proposition 4) and any stable school-choice rule (Proposition 5) do not admit fully contextually private protocols, because these rules contain type-space regions where agents are only collectively — not individually — pivotal.

For k-item Vickrey auctions, the authors study maximally contextually private protocols. They establish (Proposition 6) that, for a class of social choice rules on totally ordered type spaces that contains k-item Vickrey auctions, it is without loss to consider only protocols consisting of threshold queries that are monotonically increasing or decreasing after an initial guess. This reduction identifies two key design dimensions: the initial query posed to each agent, and the order in which agents are queried.

The main constructive result (Theorem 2) proves that an ascending-join protocol is maximally contextually private for the k-item Vickrey auction. Proposition 7 formalizes the sense in which this protocol protects privacy by delaying queries to certain bidders — it repeatedly asks agents whether they can rule out a particular outcome, and postpones questioning agents whose privacy it is protecting.

The authors also show (Proposition 19) that the ascending-join protocol is minimally relatively informative among protocols that are maximally contextually private. Extensions cover group contextual privacy (Proposition 11) and individual contextual privacy (Proposition 8), showing that individual contextual privacy violations equal the union of contextual privacy violations and nonbossiness violations.

Q: What is a contextual privacy violation, precisely? A: A protocol produces a contextual privacy violation for agent i at type profile θ if the designer can distinguish θ_i from some alternative type θ’_i — holding all other agents’ types fixed — yet the social choice rule assigns the same outcome at both profiles. The violation is defined at the level of individual agent–state pairs. A single additional superfluous distinction at the same (i, θ) pair does not register as a second violation; the framework records whether any unnecessary disclosure occurs for that agent at that state, not the degree of overexposure.

Q: How does contextual privacy differ from relative informativeness? A: Relative informativeness compares two protocols by whether one distinguishes every pair of type profiles the other does, treating all disclosures as equally undesirable. Contextual privacy conditions the notion of a “violation” on the social choice rule: a distinction between θ_i and θ’_i counts as a violation only when the rule assigns the same outcome at both profiles. Relative informativeness thus penalizes the designer for learning information that is necessary to implement the rule, whereas contextual privacy imposes no penalty for learning pivotal information.

Q: What is the pivotality characterization (Theorem 1)? A: A social choice function admits a fully contextually private protocol if and only if, on every product subset of the type space where agents are collectively pivotal, at least one agent is individually pivotal. The necessity direction shows that if a collectively pivotal set exists where no agent is individually pivotal, any implementing iterative partition must contain an earliest node that distinguishes two type profiles leading to the same outcome. The sufficiency direction constructs a contextually private protocol inductively by always querying an individually pivotal agent, ensuring every distinction implies a different outcome.

Q: Which social choice rules admit fully contextually private protocols? A: The first-price auction rule (Proposition 2) and serial dictatorship (Proposition 3) admit fully contextually private protocols. The authors use Theorem 1 to show this: in both rules, any collectively pivotal region contains an individually pivotal agent. By contrast, k-item Vickrey auction rules (Proposition 4), any stable school-choice rule (Proposition 5), efficient allocations in housing assignment, and generalized median voting rules (Section B) do not admit fully contextually private protocols.

Q: Why do k-item Vickrey auctions fail full contextual privacy? A: Proposition 4 shows that k-item Vickrey auctions for k ≥ 1 do not admit fully contextually private protocols. The argument uses the necessary conditions from Theorem 1 (Corollaries 1 and 2): the Vickrey payment rule creates type-space regions where multiple agents together determine the price but no single agent is individually pivotal over the price, so any protocol implementing the Vickrey rule must produce violations for at least some agents at some type profiles.

Q: What is the ascending-join protocol and what does Theorem 2 establish? A: The ascending-join protocol is a specific dynamic elicitation protocol for k-item Vickrey auctions that repeatedly asks agents whether they can rule out a particular outcome, structured as threshold queries ascending from an initial guess. Theorem 2 proves that the ascending-join protocol is maximally contextually private for the k-item Vickrey auction. Proposition 7 formalizes the protection mechanism: the protocol delays queries to the bidders whose privacy it is protecting, querying them only when their responses become necessary for determining the outcome.

Q: What does Proposition 6 establish about the structure of maximally contextually private protocols? A: For a class of social choice rules on totally ordered type spaces that contains k-item Vickrey auctions, Proposition 6 shows it is without loss of generality to consider only protocols consisting of threshold queries that are monotonically increasing or decreasing in the threshold after an initial guess. This result serves as a theoretical reduction (enabling proofs that certain protocols are maximally private) and as a practical design principle (identifying the initial query and the ordering of agents as the two key design dimensions).

Q: How does contextual privacy relate to obviously dominant strategies? A: The paper treats privacy properties and incentive properties as largely orthogonal questions, to be analyzed separately. For the ascending-join protocol specifically, the authors verify obvious dominance — the most demanding incentive notion they consider — which requires that at every history, the worst-case payoff from the equilibrium action exceeds the best-case payoff from any deviation. This analysis proceeds after the contextual privacy properties of the protocol are established.

Q: What is group contextual privacy and why do the authors focus on individual-level violations instead? A: Group contextual privacy requires that whenever the designer learns any property of the joint type profile, that property must affect the outcome. The authors show (Proposition 11) that a protocol is fully group contextually private if and only if every query rules out at least one outcome. They argue this standard is extremely demanding and produces a very coarse partial order: improving in the group privacy order requires restructuring the entire protocol tree rather than making agent- or state-specific improvements. They also note that normative accounts of privacy, including Nissenbaum’s contextual integrity theory, center on individual rather than group information.

Q: How does individual contextual privacy relate to nonbossiness? A: Individual contextual privacy (Proposition 8) requires that if two type profiles differing only in agent i’s type are distinguished, they must lead to different allocations for agent i — presuming a private allocation domain. The paper shows that the set of individual contextual privacy violations equals the union of contextual privacy violations and nonbossiness violations: individual contextual privacy is violated precisely when either (a) agent i’s superfluous type information is revealed, or (b) agent i is “bossy” — able to change others’ outcomes without changing their own.

Q: What is the relationship between the ascending-join protocol and minimal relative informativeness? A: Proposition 19 shows that the ascending-join protocol is not only maximally contextually private but also minimally relatively informative among protocols that are maximally contextually private. That is, among all maximally contextually private protocols, the ascending-join protocol reveals the smallest total amount of information about the type profile in the relative informativeness order. This establishes relative informativeness as a useful refinement for selecting among contextually privacy-equivalent protocols.

Q: What motivates the exclusion of cryptographic tools and trusted mediators from the framework? A: The authors work under the minimal assumption that the designer learns information if and only if an agent directly discloses it — no commitment to forget, anonymize, or cryptographically conceal. They motivate this on two grounds: first, many real-world auction formats are live and dynamic with no mediating technology; second, advanced cryptography is often costly in time, money, or computation, and studying the no-mediator benchmark can explain the historical prevalence of dynamic protocols and inform auction design in environments where cryptography may become unavailable (for example, due to quantum computing). The authors cite a Danish sugar-beet auction as a case where designers themselves questioned whether full multiparty computation was necessary.

Contextual privacy violation: A protocol produces a contextual privacy violation for agent i at type profile θ if the designer can distinguish θ_i from some alternative type θ’_i — holding other agents’ types fixed — yet the social choice rule assigns the same outcome at both profiles. The violation is assigned at the level of individual agent–state pairs.

Maximally contextually private protocol: A protocol whose set of contextual privacy violations is inclusion-minimal among all protocols that implement the same social choice rule — equivalently, a protocol that lies on the Pareto frontier of implementation and contextual privacy, such that no other implementing protocol weakly reduces every violation and strictly reduces at least one.

Iterative partition: A directed rooted tree whose nodes are subsets of the type space, where each non-leaf node is split into children by partitioning on a single agent’s type. Any protocol is equivalent (in terms of what the designer learns) to a partitional protocol induced by an iterative partition (Proposition 1).

Individual pivotality: On a product set of type profiles, agent i is individually pivotal if there exist two subsets of agent i’s types such that every type profile from one subset leads to a different outcome than every type profile from the other subset, holding others’ types fixed.

Collective pivotality: Agents are collectively pivotal on a product set if there exist two type profiles in that set with different outcomes. Collective pivotality without any agent being individually pivotal is precisely the condition that forces contextual privacy violations (Theorem 1).

Ascending-join protocol: A specific dynamic protocol for k-item Vickrey auctions that poses threshold queries in ascending order after an initial guess, repeatedly asking agents whether they can rule out a particular outcome. It is maximally contextually private (Theorem 2) and minimally relatively informative among maximally contextually private protocols (Proposition 19), and it achieves privacy protection by delaying queries to agents whose privacy it protects (Proposition 7).

Relative informativeness: A partial order on protocols defined by: protocol P is less relatively informative than P’ if every pair of type profiles P distinguishes is also distinguished by P’. Unlike contextual privacy, relative informativeness treats all disclosures as equally undesirable and does not condition on the social choice rule. The paper positions it as a useful refinement for selecting among contextually privacy-equivalent protocols.

Costly Multidimensional Screening

Mon, 01 Jan 0001 00:00:00 +0000

This paper studies when a principal can improve upon simple one-dimensional mechanisms by also deploying costly nonprice screening instruments — actions that are socially wasteful yet potentially informative about the agent’s private type.

The model features a principal and an agent with quasilinear, additively separable preferences across two components: (i) a productive component, where allocations lie in a one-dimensional compact space X and generate genuine surplus, and (ii) a costly component, where any allocation y in an arbitrary measurable space Y satisfies sB(y, θB) ≤ 0 — it destroys or at best does not create social surplus. The agent’s private type is multidimensional, θ = (θA, θB), drawn from a commonly known distribution. Both components allow for nonlinear valuations and, on the principal’s side, interdependent preferences.

The central result (Theorem 1) establishes that if the agent’s preferences between the productive and costly components are positively correlated — meaning that a higher θA implies a stochastically higher θB — then there exists an optimal mechanism that involves no costly screening. Moreover, if instruments are strictly costly, every optimal mechanism involves no costly screening almost everywhere. Positive correlation is defined in terms of stochastic dominance: θB | θA is stochastically nondecreasing in θA. A sufficient but not necessary condition is affiliation in the sense of Milgrom and Weber (1982).

The intuition centers on two observations. First, under positive correlation, costly instruments can only help relax upward incentive constraints (deterring lower types from mimicking higher types). Second, under the surplus condition — a single-crossing condition on the surplus function sA(x, θA) requiring that if x generates more surplus than x’ at some type, it continues to do so at all higher types — the principal can safely ignore upward incentive constraints at the optimum. The Downward Sufficiency Theorem (Theorem 2) formalizes the second observation: in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program (with only downward IC constraints) that also satisfies all upward IC constraints. Because monetary transfers fully substitute for costly instruments in relaxing downward constraints without destroying surplus, the costly instruments add no value under positive correlation.

The proof proceeds via a monotone path decomposition of the multidimensional type space, exploiting a measurable monotone coupling (Lemma 1) to write θ = (θA, h(θA; ε)) where ε is independent of θA and h is nondecreasing. This reduces the problem to a family of one-dimensional paths, on each of which the Reconstruction Lemma (Lemma 2) shows that any costly mechanism can be weakly improved upon by one with no costly screening that satisfies all downward IC constraints.

A partial converse (Proposition 1) shows that under negative correlation — when some dimension of θB is stochastically nonincreasing in θA — there exist utility functions satisfying the surplus condition for which any mechanism screening only the productive component is strictly dominated.

The paper derives three applications. In monopoly pricing with costly signals (waiting in line, climbing stairs, collecting coupons), profit-maximizing mechanisms require no costly signals when higher-willingness-to-pay consumers also face weakly lower signal costs (Proposition 2). In monopsonistic labor market screening, the firm need not make offers contingent on costly credentials when higher-ability workers find credentialing easier — in contrast to the competitive Spence (1973) model where all screening must occur through costly effort because wages are pinned down by expected output (Proposition 3). In multiproduct pricing, the paper reinterprets bundle components as costly instruments for screening grand-bundle values, recovering Haghpanah and Hartline’s (2021) pure bundling optimality result and extending it to nested bundling (Proposition 4), under conditions that the incremental value of adding items to nested bundles is strictly increasing in type while the value of any non-nested bundle is nonincreasing relative to some nested superset.

Q: What is the paper’s central research question? A: The paper asks whether a principal can improve upon simple one-dimensional mechanisms by also deploying costly nonprice screening instruments when the agent has multidimensional private information. The goal is to characterize conditions under which augmenting a standard price menu with surplus-destroying actions — such as waiting in line, climbing stairs, or obtaining credentials — is or is not beneficial for the principal.

Q: What does “positively correlated preferences” mean precisely in this model? A: Positive correlation means that θB is stochastically nondecreasing in θA: for any θA < θ̂A, the conditional distribution of θB given θA first-order stochastically dominates that given θ̂A — i.e., θB | θA ≤_st θB | θ̂A. Observing a high θA conveys good news about θB in the stochastic dominance sense. A sufficient but not necessary condition is affiliation in the sense of Milgrom and Weber (1982). The condition is asymmetric and does not require full independence or monotone dependence in a deterministic sense.

Q: What is the surplus condition and why does it matter? A: The surplus condition is a single-crossing condition on the productive surplus function: for any x < x̂ and θA < θ̂A, if sA(x̂, θA) > sA(x, θA) then sA(x̂, θ̂A) > sA(x, θ̂A). It says that if a higher allocation generates more total surplus at some type, it continues to do so at all higher types. This condition ensures the existence of a monotone efficient allocation rule, and it is the key enabling condition for the Downward Sufficiency Theorem. It is automatically satisfied when the principal has no interdependent preferences and the agent satisfies increasing differences, and also when sA is strictly increasing in x or has nonnegative cross partial derivative.

Q: What is the Downward Sufficiency Theorem and why is it the key technical result? A: Theorem 2 states that in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program — which ignores all upward IC constraints — that also satisfies all upward IC constraints. This means the principal can solve the easier downward-IC-only problem and the solution is fully incentive compatible. The result is novel and uncovers a general property of one-dimensional screening problems beyond the standard monotone allocation rule setting. It is key because, combined with the observation that costly instruments under positive correlation can only relax upward constraints, it implies there is no benefit to using costly screening.

Q: How does the proof handle the case of multidimensional types? A: The proof uses a monotone path decomposition. By Lemma 1 (measurable monotone coupling), under positive correlation there exists a random variable ε independent of θA and a nondecreasing measurable function h such that θ =^d (θA, h(θA; ε)). This writes the joint type distribution as a family of monotone paths indexed by ε. On each path ε = e, the types are ordered by θA alone, reducing the problem to a one-dimensional screening problem. The Reconstruction Lemma (Lemma 2) then shows that on each such path, any mechanism involving costly screening can be replaced by one without costly screening that weakly improves principal payoff and satisfies all downward IC constraints.

Q: What does the partial converse (Proposition 1) establish? A: Proposition 1 shows that when some dimension i of the costly component satisfies that θi is stochastically nonincreasing in θA (negative correlation), and the type distribution has a density with |X| > 1 and |Y| > 1, then there exist utility functions satisfying the surplus condition for which any mechanism screening only the productive component is strictly dominated by one involving costly screening. This is not a full converse — it establishes existence of cases where costly screening is strictly beneficial, not that it is always beneficial under negative correlation.

Q: How does the insurance example illustrate the two correlation cases? A: In Example 1 (negative correlation), a low-risk type (θA = 0) values insurance at 2, a high-risk type (θA = 1) values it at 3; costs are 0 and 5/2 respectively; and the high-risk type also has higher disutility for the costly action. Without costly screening, the optimal mechanism sells full insurance at price 2 to both types for a profit of 3/4. With costly screening (e.g., requiring the agent to climb stairs to get full insurance), only the low-risk type purchases, yielding profit of 1 > 3/4. In Example 2 (positive correlation), the high-risk type has lower disutility for the costly action; any mechanism using the costly instrument is strictly dominated by simply selling full insurance at price 2 to both types.

Q: How does the labor market application differ from Spence (1973)? A: In Spence (1973), wages are competitive and pinned down by expected output, leaving no room to screen workers via monetary payments, so all screening must occur through costly credentials. In Yang’s model, the monopsonistic firm sets wages and all types face the same outside option, so monetary transfers can screen types. Proposition 3 says that when θB is stochastically nondecreasing in θA — higher-ability workers find credentials easier — no credential is needed in the optimal mechanism. The paper thus shows that costly screening is a feature of competitive, not monopsonistic, labor markets, under positive correlation of preferences.

Q: What is the bundling application and what new results does it yield? A: The paper reinterprets the multiproduct pricing problem by treating the grand bundle as the productive component and sub-bundles as costly instruments (since selling a sub-bundle instead of the grand bundle destroys social surplus relative to selling the grand bundle). Proposition 4 (nested bundling) establishes that a nested menu B of bundles is optimal among deterministic mechanisms if: (i) the incremental value of adding items to move from bundle b to b’ ⊃ b in B is strictly increasing in θ, and (ii) for any bundle b not in B, there exists a nested superset b’ ∈ B such that the value of b relative to b’ is nonincreasing in θ. This extends and complements Haghpanah and Hartline (2021), which is recovered as the special case of pure bundling (Proposition 5).

Q: What are the key scope conditions that delimit when Theorem 1 applies? A: Theorem 1 requires: (i) additive separability of preferences across productive and costly components; (ii) the surplus condition on sA (single-crossing of total surplus in the productive component); (iii) the positive correlation condition (stochastic monotonicity of θB in θA); and (iv) the costly instruments satisfy sB(y, θB) ≤ 0 for all y, θB. The productive allocation space X must be compact and one-dimensional; Y can be any measurable space. The agent’s type space can be multidimensional. The result holds for both private values and interdependent valuations on the principal’s side.

Q: Under what conditions does costly screening arise in practice, according to the model? A: The model predicts that if costly screening instruments are observed in practice, the consumers or agents with higher willingness to pay (or ability) for the productive good must tend to face higher costs for the screening action. For instance, higher-willingness-to-pay consumers who find waiting in line more costly (positively correlated preferences) would not be subjected to waiting as a screening device. If a firm uses waiting in line, it must be because higher-willingness-to-pay consumers find waiting less costly — consistent with negative correlation.

Costly Instruments: Allocations in the space Y such that the ex post social surplus sB(y, θB) = uB(y, θB) + vB(y, θB) ≤ 0 for all y and all θB. These include actions like waiting in line, collecting coupons, or obtaining credentials that destroy social surplus but may convey private information useful for screening.

Productive Component: The one-dimensional allocation dimension X in which both principal and agent derive non-negative surplus, representing the intrinsically valuable output of the mechanism (e.g., insurance coverage, job placement, bundle of goods).

Positive Correlation (Stochastic Monotonicity): The condition that θB is stochastically nondecreasing in θA: for any θA < θ̂A, the conditional distribution of θB given θA first-order stochastically dominates that given θ̂A. Equivalently, observing a higher θA conveys good news about θB. A sufficient condition is affiliation (Milgrom-Weber), but positive correlation is strictly weaker.

Surplus Condition: A single-crossing condition on the total surplus function sA(x, θA) for the productive component: for any x < x̂ and θA < θ̂A, if x̂ generates strictly more surplus than x at type θA, it continues to do so at θ̂A. This ensures a monotone efficient allocation rule exists and is the enabling condition for the Downward Sufficiency Theorem.

Downward Sufficiency Theorem (Theorem 2): The result that in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program (which ignores upward IC constraints) that also satisfies all upward IC constraints. This implies the principal need only enforce downward incentive constraints at the optimum.

Monotone Path Decomposition: A proof technique that writes the multidimensional type distribution as θ =^d (θA, h(θA; ε)) where ε ⊥ θA and h is nondecreasing in θA. Borrowed from dynamic mechanism design (Eso-Szentes, Pavan-Segal-Toikka), it reduces multidimensional IC problems to families of one-dimensional paths indexed by the independent residual ε.

Nested Bundling: A menu B of product bundles that can be totally ordered by set inclusion (b1 ⊂ b2 ⊂ … ⊂ bK). The paper shows that nested bundling is optimal under conditions that the incremental value of nesting is strictly increasing in type for bundles within B, and nonincreasing relative to any nested superset for bundles outside B.

Counterfactual Analysis for Structural Dynamic Discrete Choice Models

Mon, 01 Jan 0001 00:00:00 +0000

Research Question. Discrete choice data identify only differences in agents’ utilities, not utility levels. In dynamic discrete choice (DDC) models this means many policy-relevant counterfactuals — those requiring knowledge of utility in levels — are not point-identified. Kalouptsidi, Kitamura, Lima, and Souza-Rodrigues ask: how much can researchers learn about counterfactual outcomes under mild, verifiable restrictions, without imposing the strong normalizations that are standard in applied work but often hard to justify and potentially sign-reversing in their effects?

Setting and Methodology. The paper works within a canonical infinite-horizon DDC framework where an agent chooses among a finite action set each period, with additively separable per-period payoffs and i.i.d. unobservables. The econometrician observes conditional choice probabilities (CCPs) and state transition functions from panel data, but the payoff vector is underidentified by X free parameters (one per state), which is the source of non-identification of many counterfactuals. The authors characterize the sharp identified set for counterfactual CCPs, for low-dimensional outcomes such as average welfare, and develop both identification theory and a feasible inference procedure.

Main Identification Results. The sharp identified set for counterfactual CCPs is a smooth, connected manifold whose dimension equals the rank of a specific matrix (CJ*QJ) that the econometrician can compute directly from the data. This rank is at most X minus the number of linearly independent equality restrictions imposed. Two classes of commonly used restrictions reduce the dimension further without requiring full point identification: (i) local counterfactuals — experiments affecting only a subset of the state-action space — reduce the dimension to at most the number of eigenvalues of the relevant transformation matrix that differ from one; (ii) parametric payoffs with ηγ free parameters reduce the dimension to at most ηγ. Combining both achieves the tightest bound. Point identification is the special case where the rank equals zero.

For scalar low-dimensional outcomes (e.g., average welfare), the identified set is a compact interval whose endpoints are obtained by solving constrained optimization programs implementable in standard nonlinear solvers (e.g., Knitro), feasible even when the state space is large.

Quantitative Illustration. In the firm entry/exit Monte Carlo with state space X = 4 and a counterfactual entry subsidy removal: under Restriction 1 alone (outside option = 0, non-negative costs, known variable profits), the identified set for the change in the long-run probability of being active is [-0.1235, 0.0000], correctly signed and containing the true value of -0.0638. Adding shape restrictions (Restrictions 1–2) tightens the upper bound to -0.0341; adding the scrap-value exclusion restriction (Restrictions 1–3) tightens it to -0.0421. Analogous patterns hold for consumer surplus (true: -0.0875; bounds narrowing from [-0.1735, 0.0000] to [-0.1735, -0.0573]) and firm value (true: 0.9513; bounds from [0.0000, 1.8229] to [0.6388, 1.8229]). Critically, the authors show that setting scrap values to zero — the standard identifying assumption — is rejected by the data under Restrictions 1 and 2, because that payoff vector does not lie in the identified set.

Empirical Application. Revisiting Das, Roberts, and Tybout (2007) on Colombian exporters, the paper re-examines the horserace among export revenue, fixed cost, and entry cost subsidies. The DRT ranking (revenue subsidies dominate, entry cost subsidies rank last) survives under weaker restrictions than originally imposed, but hinges on the assumption that scrap values do not vary across states. Without that restriction, entry cost subsidies can potentially outperform the other types, reversing the original conclusion.

Inference. The paper develops a subsampling-based inference procedure that is asymptotically uniformly valid (bootstrap fails here due to non-regularity of the set boundary). The confidence set is constructed by inverting a quadratic-form distance test statistic. The critical practical recommendation is subsample size hN = N^{2/3}. The procedure remains feasible in binary choice models with state spaces up to X = 240 (dimension of the optimization problem: 720), where standard moment-inequality approaches are computationally infeasible.

Q: Why are many counterfactuals not point-identified in DDC models, even after the model is estimated? A: Choice data identify only differences in value functions across actions, not utility levels. The identifying matrix M has rank AX, leaving X free payoff parameters undetermined. Counterfactuals that depend on utility levels — such as the welfare impact of an entry subsidy when scrap values are unknown — therefore cannot be recovered uniquely from the data, even with a fully estimated model.

Q: What is the key object the paper characterizes, and what does it look like geometrically? A: The paper characterizes the sharp identified set for the counterfactual CCP vector p̃. Proposition 1 establishes that this set is a smooth, connected manifold with boundary, whose interior dimension equals rank(CJ*QJ). Connectedness is important because it means the set has no gaps and boundary tracing is sufficient to characterize it.

Q: How does the dimension of the identified set depend on the type of model restrictions imposed? A: Equality restrictions (d of them) reduce the maximum possible dimension from X to X–d. Local counterfactuals (affecting L state-action pairs) reduce the dimension further to at most the number of eigenvalues of the payoff transformation H(L) that differ from one, which is at most L. Parametric payoffs with ηγ free parameters cap the dimension at ηγ. Combining local counterfactuals with parametric payoffs gives the tightest bound: at most the number of eigenvalues of a related matrix D that differ from one, which is at most min(L, ηγ).

Q: Under what conditions does the identified set for counterfactual behavior collapse to a point? A: When rank(CJ*QJ) = 0, every payoff vector in the identified set PI maps to the same counterfactual CCP — that is, p̃ is point-identified even though the structural payoff π may not be. This can occur through a combination of equality restrictions and specific structure of the counterfactual experiment, without requiring full identification of all model parameters.

Q: What properties does the identified set for a scalar low-dimensional outcome have, and how is it computed? A: Under continuity of the outcome function φ and boundedness of the payoff identified set, the identified set for a scalar outcome θ is a compact interval [θL, θU]. The endpoints are computed as the minimum and maximum of a constrained optimization program over the joint space of counterfactual CCPs and payoff vectors, subject to the model’s Bellman equations, model restrictions, and equality constraints linking observed to counterfactual behavior. These programs can be solved with standard nonlinear solvers.

Q: What do the Monte Carlo results show about the informativeness of the bounds? A: In the firm entry/exit example with X = 4, the identified sets under only mild restrictions (non-negative costs, known variable profits, zero outside option) are already informative and correctly signed. For the change in the probability of being active (true value: -0.0638), the set under Restriction 1 alone is [-0.1235, 0.0000], establishing that the probability does not increase. Adding shape restrictions and exclusion restrictions progressively tightens the interval. All intervals contain the true parameter value, confirming sharpness.

Q: What does the paper show about the assumption of zero scrap values, which is standard in the entry cost literature? A: The paper shows that setting scrap values to zero can be rejected by the data: in the firm entry/exit example, the payoff vector with s = 0 does not belong to the identified set PI under Restrictions 1 and 2. This is empirically important because Kalouptsidi, Scott, and Souza-Rodrigues (2021) had previously shown that mistakenly setting scrap values to zero not only biases estimated entry costs downward but can also reverse the sign of a subsidy’s predicted effect.

Q: What is the main finding of the empirical application to export subsidies? A: Revisiting Das, Roberts, and Tybout (2007), the paper finds that the DRT ranking — export revenue subsidies dominate, entry cost subsidies rank last — can be confirmed under restrictions weaker than those DRT originally imposed. However, the ranking is not robust to allowing scrap values to vary across states: under that generalization, entry cost subsidies can potentially outperform the other subsidy types, reversing the original policy conclusion.

Q: Why does the bootstrap fail for inference in this setting, and why does subsampling work? A: The test statistic ĴN(θ0) involves the minimum of a quadratic form over a non-regular (kinked), random, and possibly nonconvex set. Bootstrap critical values are not asymptotically uniformly valid in this non-regular setting. Subsampling with subsample size hN → ∞, hN/N → 0 (the paper recommends hN = N^{2/3}) delivers asymptotically uniformly valid critical values under weak conditions, because it does not require regularity of the constraint set boundary.

Q: How does the inference approach handle the high dimensionality of DDC settings? A: The paper develops a computational algorithm specifically tailored to the structure of DDC models, exploiting the linear Bellman equation constraints to reduce the effective dimensionality of the optimization problem. In a binary choice model with X = 90, the joint optimization is over a 270-dimensional space; with X = 240 (as in Blundell, Gowrisankaran, and Langer, 2020), the dimension is 720. Standard moment-inequality inference methods (Kaido, Molinari, Stoye, 2019; Bugni, Canay, Shi, 2017) are computationally infeasible at these scales; the authors’ algorithm remains tractable.

Q: How does the paper relate to Norets and Tang (2014), the closest alternative approach? A: Norets and Tang (2014) partially identify structural parameters and high-dimensional counterfactual CCPs by relaxing the assumed distribution of idiosyncratic shocks, focusing on binary choice models and using a pointwise-valid Bayesian approach. The present paper instead targets low-dimensional policy outcomes (nonlinear functions of payoffs and counterfactual CCPs), accommodates multinomial choice, provides asymptotically uniformly valid frequentist inference via subsampling, and restricts the source of underidentification to the payoff function rather than the error distribution. The two contributions are non-nested and complementary.

Q: What is the practical workflow the paper enables for applied researchers? A: A researcher can (i) select any combination of model restrictions (equality or inequality, parametric or shape), (ii) specify any counterfactual experiment via an affine payoff transformation (H, g), and (iii) define any low-dimensional outcome of interest φ, then directly compute the identified set and a valid confidence interval by solving two constrained optimization programs — without deriving new analytical identification results for each specification. The rank condition for checking the dimension of the identified set is computable from the data.

Dynamic Discrete Choice (DDC) Model. A discrete-time infinite-horizon model where agents choose among a finite action set each period, with per-period utilities additively separable into an observed payoff function π and an i.i.d. unobservable shock, and agents maximize expected discounted lifetime utility. The model is parameterized by payoffs π, transition function F, discount factor β, and shock distribution G.

Conditional Choice Probability (CCP). The probability that an agent selects a given action in a given state, integrating out the unobservable shocks. CCPs and state transitions are directly identifiable from panel data and serve as the sufficient statistics for the identified set, in place of the unidentified payoff vector.

Sharp Identified Set for Counterfactual CCPs. The set PĨ(p, F) of all counterfactual CCP vectors p̃ that are consistent with the observed data (p, F) and the imposed model restrictions, given the specified counterfactual transformation. Characterized as a smooth connected manifold with dimension equal to rank(CJ*QJ).

Local Counterfactual. A counterfactual experiment in which the payoff transformation H modifies only a subset L of the state-action pairs, leaving the rest unchanged. Local counterfactuals reduce the dimension of the identified set relative to global experiments, because only the payoffs in the affected subset matter for the unidentified component of the counterfactual response.

Partial Identification / Identified Set for Outcomes. Rather than seeking a unique estimate of a counterfactual outcome θ, partial identification recovers the set ΘI of all values of θ consistent with the data and restrictions. For scalar outcomes this is a compact interval [θL, θU] whose endpoints solve constrained optimization problems over payoff and counterfactual CCP spaces.

Subsampling Inference. A procedure for constructing asymptotically uniformly valid confidence sets by repeatedly computing the test statistic on subsamples of size hN < N, approximating the sampling distribution of ĴN(θ0) without requiring regularity (smoothness) of the boundary of the constraint set — a requirement that fails here due to the kinked, nonconvex nature of the identified set.

Rank Condition for Dimension. The dimension of the identified set for counterfactual CCPs is determined by the rank of the matrix CJ*QJ, which depends on the counterfactual transformation H, the model restrictions, and the observed data. The econometrician can compute this rank from observables to assess, before imposing any strong assumptions, how many dimensions of freedom remain in the identified set.

Customer accumulation, returns to scale, and secular trends

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how rising returns to scale in production contributed to three concurrent U.S. secular trends since 1980: declining business dynamism, rising markups, and growing firm expenditures on customer acquisition. The author constructs a firm dynamics model in the Hopenhayn (1992) tradition with endogenous entry and exit, heterogeneous markups, and customer accumulation grounded in directed search in the product market. Firms compete for customers through both prices and selling activities; larger firms gain a competitive edge when returns to scale rise because their marginal costs fall more than those of smaller firms—even though the technological shift is uniform across firms. This demand-based channel triggers winners-and-losers dynamics and the rise of superstar firms.

The empirical foundation rests on Compustat data for U.S. publicly traded firms (1977–2014) and Business Dynamics Statistics (BDS) for aggregate and sector-level dynamism measures. Production-function estimation using Ackerberg, Caves, and Frazer (2015) augmented with sales-share controls documents that aggregate returns to scale rose from approximately 1.0 in 1980 to approximately 1.05 by 2014—a within-sector increase, not a reallocation effect. Over the same period, the cost-weighted markup rose by 42%, the firm entry rate fell by 33%, the excess reallocation rate fell by 29%, and selling costs relative to production costs rose by 60%–90% depending on the measure used.

The model is calibrated to 1980 steady-state moments (firm life-cycle patterns, markups, entry and reallocation rates). A 5% increase in returns to scale—matching the empirical estimate—accounts for: a +15 percentage point rise in the average cost-weighted markup (vs. +42% in the data); a 33% decline in the entry rate (exactly matching the data); a 21% decline in the reallocation rate (vs. 29% in the data); and a 23% increase in selling costs relative to production costs (vs. 60%–90% in the data). The model also generates a 53% rise in the share of firms aged 11 years or older (vs. 50% in the data) and a 58% decline in the employment share of firms aged 5 years or younger (vs. 56% in the data), closely tracking the aging of the U.S. firm population. Firm-level responsiveness to productivity shocks declines by 0.08 in the model, versus about 0.01 in Compustat and 0.09 in Decker et al. (2020).

Sector-level panel regressions with sector fixed effects confirm the model’s directional predictions: within-sector increases in returns to scale are associated with lower entry rates (coefficient −2.89, significant at 1%), lower reallocation rates (−1.16, significant at 1%), higher markups (+3.15, significant at 1%), and higher selling costs relative to production costs (+1.85 for the advertising-based measure; +8.52 for adjusted SG&A).

A key scope condition is that the model yields a constrained-efficient allocation: directed search and full internalization of returns to scale imply decentralized equilibrium efficiency, making the paper a laboratory for assessing how far efficient firm responses to technological change can explain the secular trends without invoking market failures. The model fits the post-2000 transition dynamics better than the 1980s–1990s period, and explains a substantial but incomplete share of the trends, suggesting complementary—possibly inefficient—forces also contributed.

Q: What is the core mechanism through which rising returns to scale generate winners-and-losers dynamics?

A: The marginal cost of production under increasing returns to scale (alpha > 1) is MC(z,n) = l(n,z)^(1−alpha) × (1/alpha) × (W/e^z), which depends on firm size l(n,z). A uniform rise in alpha rotates the marginal cost schedule clockwise by firm size: larger firms see a proportionally larger cost reduction than smaller firms, even though the technological change is identical across all firms. Because firms compete for the same pool of customers, this asymmetric cost advantage allows large firms to offer lower prices while sustaining higher margins, attracting customers away from small firms. The result is a demand-based channel that generates winners-and-losers dynamics and increases market concentration.

Q: How does the model capture customer accumulation, and why is it central to the paper’s argument?

A: The model introduces directed search in the product market, where firms post advertisements and customers—including those already matched with a firm—choose which submarket to enter by trading off offered utility against matching probability. A constant-returns-to-scale matching function governs match creation; in submarket with tightness theta, customers match with probability m(theta) = theta(1+theta)^(−1) and firms attract customers with probability q(theta) = (1+theta)^(−1). The customer accumulation motive creates an investment-harvest trade-off: firms can either post high promised utility (low prices) to grow their customer base or extract surplus through high prices. Rising returns to scale amplify large firms’ ability to resolve this trade-off favorably, linking the technological change directly to markup dynamics, entry incentives, and selling expenditures.

Q: What is the directed search framework’s role in ensuring equilibrium uniqueness and efficiency?

A: The author introduces firm-side commitment contracts—specifying price, separation probability, and continuation utility contingent on productivity realizations—combined with directed search. Because search is directed on both sides and firms fully internalize returns to scale, the decentralized equilibrium is constrained-efficient. This delivers uniquely determined heterogeneous prices in equilibrium (solving the indeterminacy problem common in customer-market models) and establishes the paper’s efficient-mechanism benchmark: it tests how far profit-maximizing firm responses to technological change—without any market failure—can account for the secular trends.

Q: How are prices structured in the model, and what life-cycle pattern do they generate?

A: Each firm charges two distinct prices in each period: one to incumbent customers (the same for all incumbents, since they are identical conditional on being attached to the same firm) and one to newly acquired customers (which varies based on the promised utility in the submarket searched). Firms that are expanding their customer base offer greater promised utility and therefore charge lower prices to attract customers; firms harvesting their existing base charge higher prices. Because firms enter small and grow, this dynamic generates a price life cycle: young firms invest via low prices and mature firms harvest through higher prices, which the model reproduces as a rising markup pattern over the firm life cycle—an untargeted moment the model fits well.

Q: What does the calibration target and what untargeted moments does the model reproduce?

A: The model is calibrated to 1980 using: the number of employees of entrant firms (pinning entry customer base n_e), employees of age-5 firms (pinning convex cost chi_1), share of firms aged 11+ years (pinning chi_2), average firm size (operating cost f), entry rate (entry cost kappa), excess reallocation rate (exit shock delta), and average cost-weighted markup (linear cost c). Untargeted moments reproduced include: a sales-weighted markup of 0.28 (vs. 0.25 in De Loecker et al. 2020), endogenous customer turnover of approximately 9% (vs. 15% in Gourio and Rudanko 2014), and an elasticity of customer base shrinkage to price of 0.08 (within the 0.01–0.16 range from Paciello et al. 2019). The model also matches markup and selling-cost life-cycle patterns that are typically overlooked.

Q: How large is the quantitative contribution of the 5% rise in returns to scale to each secular trend?

A: Comparing the 1980 steady state (alpha = 1) to the 2014 steady state (alpha = 1.05): the average cost-weighted markup rises by 15% in the model versus 42% in the data; the entry rate declines by 33% in the model, exactly matching the data; the reallocation rate declines by 21% in the model versus 29% in the data; and selling costs relative to production costs rise by 23% in the model versus 60%–90% in the data. The model thus explains a substantial share of each trend while leaving a residual requiring additional mechanisms.

Q: How does the model explain the aging of U.S. firms, and how well does it match the data?

A: The winners-and-losers mechanism shifts activity toward larger, older firms, which mechanically ages the firm population. The model generates a 53% increase in the share of firms aged 11 years or older (vs. 50% in the data) and a 58% decline in the employment share of firms aged 5 years or younger (vs. 56% in the data). This aging arises because rising returns to scale increase the cost of customer acquisition, acting as a barrier to entry that disproportionately hurts new, small firms while allowing large incumbents to remain viable at lower productivity thresholds.

Q: What is the channel through which rising returns to scale reduce business dynamism specifically?

A: The unequal reduction in marginal costs intensifies competition for customers and raises customer acquisition costs. This operates through two simultaneous effects on the exit threshold: (i) lower marginal costs allow large firms to remain viable at lower productivity levels despite higher customer acquisition costs; and (ii) heightened competition forces smaller firms to require higher productivity to survive in a market that has become increasingly costly to operate in. Higher customer acquisition costs therefore function as an endogenous barrier to entry, reducing the entry rate and the reallocation of resources across firms.

Q: Does the model attribute the secular trends entirely to efficient firm behavior, and what does it conclude about residual explanations?

A: No. The model is explicitly designed as a constrained-efficient benchmark, and the paper finds that while rising returns to scale account for a substantial share of the trends—particularly in magnitude—the transition dynamics show a less accurate fit before the 2000s. The author concludes that complementary mechanisms, likely involving inefficiencies (such as market power from horizontal product differentiation or barriers to entry beyond those captured by the model), played a significant role in the earlier evolution of these trends and in the portion of the trends not explained by the efficient channel.

Q: What evidence supports the rising returns to scale finding, and what are its limitations?

A: Production-function estimation using the Ackerberg-Caves-Frazer method with sales-share controls on Compustat data shows returns to scale rising from approximately 1.0 in 1980 to approximately 1.05 by 2014, driven primarily by within-sector increases rather than reallocation toward high-returns sectors. A translog production function finds limited evidence of heterogeneous increases across firm sizes within Compustat. However, Compustat predominantly covers large publicly traded firms; smaller firms outside the sample may have experienced minimal or no increase in returns to scale. If technology adoption involves fixed costs, the aggregate impact could be larger than estimated, meaning the quantitative exercises likely represent a conservative lower bound.

Q: How does the paper relate to and extend the directed search literature in product markets?

A: The paper builds on Gourio and Rudanko (2014) and Roldan-Blanco and Gilbukh (2020), where customers are locked in once matched, by introducing labor-search tools from Schaal (2017) to allow: (i) incumbent customer switching between firms at rates of 10%–25% annually (Gourio and Rudanko 2014), and (ii) a non-zero price sensitivity of incumbent customers (Paciello et al. 2019). It also allows firms to invest in demand through selling expenditures, which prior directed search models in product markets typically abstracted from, making it possible to study how technological changes affect customer reallocation and firms’ cost structures jointly.

Customer capital: The stock of customers a firm has accumulated through prior selling and pricing decisions; treated as a state variable that firms invest in (by offering low prices and spending on advertisements) or harvest from (by charging high markups), with a customer turnover rate estimated at 10%–25% annually in the literature.

Directed search in the product market: A market structure in which both firms and customers choose which submarket (indexed by the promised utility level) to enter, trading off match probability against terms; delivers constrained-efficient equilibrium and uniquely determined heterogeneous prices.

Investment-harvest trade-off: The firm’s dynamic choice between offering high promised utility (low prices, low current markups) to grow the customer base versus extracting surplus through high prices from an existing customer base; shaped by the firm’s current size, productivity, and the cost structure implied by returns to scale.

Returns to scale (alpha): The curvature of the production function y = e^z × l^alpha; equals 1.0 under constant returns and approximately 1.05 by 2014 in the empirical estimates; the paper’s central technological change parameter, whose rise disproportionately reduces marginal costs for larger firms.

Winners-and-losers dynamics: The reallocation of customers and market share from small to large firms triggered by the asymmetric cost advantage large firms obtain when returns to scale rise; the demand-based channel through which superstar firms emerge.

Cost-weighted markup: The average markup aggregated using each firm’s costs as weights, as opposed to sales-weighted markup; the primary measure of market power used in the paper, rising by 42% in the data between 1980 and 2014.

Constrained-efficient allocation: An equilibrium outcome in which, given the frictions present (search-and-matching in the product market), no social planner operating under the same constraints could improve welfare; the paper uses this as a benchmark to assess how far efficient firm responses explain secular trends without invoking market failures.

Selling costs relative to production costs: The ratio of customer acquisition expenditures (advertising or adjusted SG&A) to cost of goods sold; rose by 60%–90% in the data between 1980 and 2014 and by 23% in the model’s steady-state comparison.

Customer Acquisition, Business Dynamism and Aggregate Growth

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks whether firm-level customer acquisition — distinct from productivity differences — is a quantitatively important driver of aggregate economic growth, and whether ignoring it distorts predictions about growth policy efficacy. The authors build a novel endogenous growth model in which innovating firms must first accumulate customers to sell their products, with two channels of customer acquisition operating simultaneously: costly sales-and-marketing expenditure and below-static-markup pricing (sales-driven accumulation). The model is estimated using indirect inference against a combination of aggregate data (U.S. real GDP per worker growth of 1.43% annually, 1979–2019), Business Dynamics Statistics (BDS) life-cycle profiles, and firm-level data from Compustat matched to Capital IQ’s sales-and-marketing expense records covering 1997–2019.

The benchmark model yields four closed-form propositions. First, a “firm-level market size effect”: higher customer retention raises a firm’s future profit base, strengthening incentives to conduct R&D. Second, an endogenous feedback loop: more productive firms invest more in customer acquisition, which expands their customer base and further strengthens R&D incentives. Third, customer base accumulation raises aggregate growth, but only indirectly — by boosting firm-level innovation rates — since aggregate productivity is a customer-weighted average of firm productivity levels. Fourth, the sensitivity of innovation to R&D subsidies increases with customer base growth, because firms with faster-growing customer bases discount future profits less steeply.

In the quantitatively estimated full model — which relaxes the benchmark’s perfect-scaling restrictions and endogenizes firm entry and exit — the authors conduct two decomposition exercises. In a counterfactual scenario where expected customer retention is reduced to make average customer base growth zero among continuing businesses, firm-level innovation rates fall by approximately 40% relative to the full model. Of this 40% decline, only about 6 percentage points are attributable to the direct firm-level market size effect alone; the vast majority is driven by the endogenous feedback loop between innovation and customer acquisition. In a second decomposition focused on aggregate growth, the firm-level market size effect and a reallocation effect — whereby the feedback loop concentrates customers among high-productivity firms — together account for 44% of aggregate growth in the full model.

On policy, the authors compare R&D subsidies and operational subsidies in the full model against an otherwise identical model that ignores customer accumulation. R&D subsidies are approximately twice as effective at boosting aggregate growth in the full model as in the model without customer accumulation. Conversely, operational subsidies produce a stronger decline in aggregate growth in the full model than in the benchmark-without-customer-accumulation, because aggregate growth in the full model is a customer-weighted average of firms’ productivity growth rates, making the joint distribution of productivity and customer bases the relevant object of study.

Firm-level data support three empirical predictions. Marketing expenditure, R&D intensity, and markups co-move in model-consistent directions both contemporaneously and over the life cycle. The estimated relative weight of marketing versus pricing as channels of customer accumulation is γ = 0.745, indicating marketing is the dominant channel. A model-consistent proxy for the severity of customer-base frictions, estimated in the cross-section of industries, shows that stronger frictions correlate with lower R&D investment, as predicted. The customer-base depreciation rate is estimated at ζ = 0.375, R&D cost scaling at σx = 1.264, and marketing cost scaling at σa = 1.405.

Q: What is the firm-level market size effect and why does it arise? A: When a firm retains more customers, successful innovations apply to a larger market, raising the profitability of each unit reduction in production costs. This increases the marginal benefit of R&D investment. In the benchmark model, Proposition 2(a) shows formally that firm-level innovation increases with customer base growth: ∂x/∂(1−ζ) > 0, where ζ is the customer separation rate.

Q: What is the endogenous feedback loop between innovation and customer accumulation? A: More productive firms have lower production costs and can therefore afford greater investment in marketing and can set lower markups, both of which attract more customers. A larger customer base raises firm value and strengthens R&D incentives further (Proposition 2(b)). This bidirectional feedback means that productivity growth and customer accumulation are jointly determined in equilibrium, not independent processes.

Q: How large is the quantitative effect of customer accumulation on firm-level innovation? A: In the counterfactual where expected customer retention is reduced so that average customer base growth among continuing firms is zero, firm-level innovation rates are approximately 40% lower than in the full model. Of this, only about 6% (of the total drop) is attributable to the direct market size effect in isolation; the feedback loop accounts for the remaining roughly 34 percentage points.

Q: How much of aggregate growth do customer-acquisition channels explain? A: The firm-level market size effect and a customer reallocation effect together account for 44% of aggregate growth in the full model. The firm-level market size effect alone reduces aggregate growth by about one-fifth (20%) in the relevant counterfactual. The reallocation effect — by which productive firms accumulate disproportionate market share — contributes the remainder of the 44%.

Q: What is the reallocation channel for aggregate growth? A: Because highly productive firms can invest more in customer acquisition, the feedback loop endogenously concentrates customers (market shares) among high-productivity firms. Since aggregate productivity in the model is a customer-weighted average of firm productivity levels (equation 16), this reallocation raises aggregate productivity growth beyond what the firm-level R&D incentive effect alone would produce.

Q: How does customer accumulation change the efficacy of R&D subsidies? A: R&D subsidies are approximately twice as effective at raising aggregate growth in the full model (with customer accumulation) as in an otherwise identical model that ignores customer accumulation. The mechanism is Proposition 4(b): faster customer base growth makes firms weight future profits more heavily, increasing their sensitivity to any change in R&D costs, including that brought about by a government subsidy.

Q: What happens to aggregate growth under operational subsidies in the two models? A: Operational subsidies lead to a stronger decline in aggregate growth in the full model than in the model without customer accumulation. The reason is that aggregate growth in the full model depends on the joint distribution of firm productivity and customer bases; operational subsidies alter this distribution in ways that reduce the customer-weighted average of productivity growth rates, an effect absent when customer accumulation is ignored.

Q: How are the two customer-acquisition channels (marketing and pricing) measured empirically? A: Marketing is measured using sales-and-marketing expenses from Capital IQ, available for 48% of the Compustat sample (34% report directly; an additional 14% report advertising or marketing sub-components). Markups are measured following De Loecker et al. (2020) as the inverse share of variable costs in sales multiplied by the cost-output elasticity, with variation across firms identified from balance sheet data under the assumption that cost-output elasticities are constant within industry-year cells.

Q: What is the estimated relative strength of marketing versus pricing in customer accumulation? A: The relative weight on marketing is γ = 0.745, estimated by targeting the coefficient βµ = 0.04 (standard error 0.01) from a reduced-form regression of firm-level sales growth on changes in markups (equation 29). This implies that marketing is the dominant channel, consistent with evidence in Afrouzi et al. (2021) and Fitzgerald et al. (forthcoming).

Q: What is the estimated customer-base depreciation rate and how is it disciplined? A: The depreciation rate ζ is estimated at 0.375, targeted to match average firm-level employment growth from the BDS. This falls toward the lower end of existing estimates, which range from about 0.3 to 0.7 across studies.

Q: How do R&D costs scale with firm size in the estimated model? A: The R&D cost scaling parameter is σx = 1.264, estimated by targeting the reduced-form coefficient of −0.01 from a regression of log R&D intensity on log sales with industry-time fixed effects (equation 28). This is close to the estimate in Akcigit and Kerr (2018).

Q: How do marketing costs scale with firm size? A: The marketing cost scaling parameter is σa = 1.405, estimated by targeting a reduced-form coefficient of −0.01 from a regression of log sales-and-marketing intensity on log sales with industry-time fixed effects (equation 30).

Q: What empirical co-movement evidence supports the model’s predictions? A: In the cross-section of firms, marketing expenditure, R&D intensity, and markups all co-move in model-predicted directions, for both static (contemporaneous) relationships and dynamic (life-cycle) patterns. Additionally, a model-consistent industry-level proxy for the severity of customer-base frictions shows that stronger frictions are associated with lower R&D investment, as the model predicts.

Q: How does endogenous firm exit work in the full model and why does it differ from standard models? A: Firms pay a stochastic per-period operational cost and exit when that cost exceeds a threshold κ*_j = v(q_j, b_j)/W. Unlike standard growth models where exit depends only on productivity, here the exit threshold depends on both productivity and accumulated customers, so customer loss can trigger exit even for relatively productive firms.

Q: What data sources are used and what are their key limitations? A: The three primary firm-level sources are the Census Bureau’s BDS (broad coverage, employment-focused), Compustat (rich financial data but limited to publicly traded firms and lacking direct customer-acquisition measures), and Capital IQ (sales-and-marketing expenses available from 1997, matched to 91% of the Compustat sample). To address Compustat’s non-representativeness, employment-based weights aligning Compustat and BDS firm-size distributions are applied when computing model moments against Compustat targets.

Firm-level market size effect: The mechanism by which higher customer retention raises a firm’s future profit base — because lower production costs from successful innovation apply to a larger market — thereby strengthening incentives to conduct R&D. This is the primary channel linking customer accumulation to innovation.

Customer base (b_j): The mass of household members consuming a firm’s product variety, which varies endogenously across firms. It enters demand directly (equation 4) and serves as a state variable in the firm’s value function alongside productivity.

Endogenous feedback loop: The bidirectional reinforcement between productivity growth and customer accumulation. More productive firms invest more in customers; a larger customer base raises the value of innovation; higher innovation raises productivity further.

Reallocation effect: The concentration of customers (market shares) toward high-productivity firms that arises endogenously from the feedback loop, contributing to aggregate growth because aggregate productivity is a customer-weighted average of firm-level productivity.

Customer-base depreciation rate (ζ): The exogenous rate at which a firm loses its existing customers each period, estimated at 0.375 in the paper’s calibration. It governs the baseline speed of customer attrition and is the key parameter for the firm-level market size effect.

Sales-and-marketing expenses: Expenditures on sales force, brand development, customer service, advertising, and customer data acquisition — measured from Capital IQ — that directly drive marketing-based customer accumulation (the dominant channel with estimated weight γ = 0.745).

Perfect scaling (Assumption 1): The benchmark restriction that R&D and marketing costs, and the sales-driven customer accumulation benefit, all scale one-for-one with a composite of firm productivity and customer base. This assumption enables closed-form solutions and is relaxed in the full model using estimated scaling parameters.

De Gustibus and Disputes about Reference Dependence

Mon, 01 Jan 0001 00:00:00 +0000

This paper examines whether heterogeneity in individual gain-loss attitudes — the degree to which people weigh losses more or less severely than equivalent gains — contaminates prior tests of expectations-based reference dependence (EBRD). The central question is: do prior experiments that appear to yield mixed or null evidence against EBRD actually reflect a failure of the expectations-based reference point, or instead reflect a methodological flaw — the implicit assumption that all individuals are uniformly loss averse?

All prior tests of EBRD models (e.g., Kőszegi and Rabin 2006, 2007) have proceeded under what the authors call “universal loss aversion,” the assumption that every individual weighs losses more heavily than commensurate gains (λ > 1). The authors argue that this assumption — a form of the classic De Gustibus conjecture — is empirically incorrect and theoretically distorting: within EBRD designs, loss-averse and gain-seeking subjects are predicted to respond in opposite directions to expectations manipulations, so aggregating across them suppresses or reverses treatment effects.

The authors run two pre-registered laboratory experiments totaling 1,524 subjects. The labor supply experiment (N = 500, UC San Diego) uses a two-stage design. Stage 1 elicits each subject’s gain-loss attitude parameter λ_i from their effort responses to fixed versus uncertain piece rates in a real-effort transcription task, exploiting the prediction that loss-averse workers reduce effort under wage uncertainty while gain-seeking workers increase it. Stage 2 manipulates expectations by varying the probability of a high outside payment (p = 0.05 in Condition Low vs. p = 0.45 in Condition High), holding the piece-rate probability constant at 50%; under EBRD, this shifts the reference point and should change effort in a direction governed by λ_i.

The exchange experiment (N = 1,024, University of Bonn, with a pre-registered 2018 replication of N = 417) uses Stage 1 preference statements over randomly endowed objects to estimate λ_i, and Stage 2 manipulates expectations via a 0% vs. 50% probability of forced exchange. Under EBRD, loss-averse subjects should become more willing to exchange in the High condition; gain-seeking subjects should become less willing.

Both experiments document substantial heterogeneity in gain-loss attitudes. In the labor supply study, approximately 70.6% of subjects exhibit loss aversion (λ̂ > 1) and 29.4% exhibit gain-seeking (λ̂ < 1), with an average structural estimate of λ̂ = 1.65 and median 1.66. In the exchange study, 76% are loss averse and 24% are gain-seeking, with mean λ̂ = 1.49 and median 1.34. Lottery-based elicitation in the labor supply experiment yields 28% gain-seeking, consistent with prior literature estimates of roughly 22% gain-seeking from Chapman et al. (2018).

Crucially, Stage 1 gain-loss attitudes are strongly predictive of Stage 2 treatment effects in both experiments. In the labor supply study, the aggregate treatment effect of approximately 26% greater effort in Condition High — reproducing Abeler et al. (2011) — masks strongly heterogeneous responses: higher λ̂ predicts larger positive treatment effects (raw correlation ρ = 0.18, p < 0.01), and controlling for heterogeneous gain-loss attitudes raises R² by more than a factor of 10. In the exchange study, the aggregate treatment effect is precisely zero (coefficient = 0.00, clustered s.e. = 0.03), a result that prior literature would interpret as contradicting EBRD; but once gain-loss heterogeneity is accounted for, treatment effects are strongly positive for loss-averse subjects and negative for gain-seeking subjects, again raising R² by more than a factor of 10.

Gain-seeking subjects exhibit negative treatment effects in the exchange study, consistent with EBRD predictions, but in the labor supply study the average treatment effect for gain-seeking subjects remains slightly positive, representing a partial deviation from the model’s quantitative predictions. The authors interpret this as evidence that expectations-based reference points are an important but likely incomplete determinant of behavior, with attention-based, status-quo-based, or anchoring-based reference points potentially playing supplementary roles.

Q: What is the central methodological problem with prior tests of expectations-based reference dependence?

A: All prior tests assumed universal loss aversion — that every individual has λ > 1, i.e., weighs losses more severely than equivalent gains. The authors show this is both empirically wrong (roughly 24–29% of subjects are gain-seeking across both studies) and theoretically distorting: within EBRD designs, gain-seeking individuals are predicted to respond in the opposite direction from loss-averse individuals, so averaging across heterogeneous types can suppress, zero out, or even reverse the true treatment effect. This makes standard aggregate tests of EBRD unreliable.

Q: How do the authors measure gain-loss attitudes in the labor supply experiment?

A: In Stage 1, subjects make 30 effort decisions across fixed piece rates and uncertain piece rates with the same mean. Under the Kőszegi-Rabin CPE model, a loss-averse individual reduces effort when the wage is uncertain (because outcomes can fall below the reference point), while a gain-seeking individual increases effort under uncertainty. The authors estimate individual-level parameters by regressing log(e_i + 10) on log(w) and Δw/w in a random-coefficients framework; the coefficient l̂_i on Δw/w is the reduced-form measure of gain-loss attitudes, with λ̂_i = 1 + 4·(l̂_i/ĝ_i) as the structural estimate. The correlation between the two measures is ρ = 0.85 (p < 0.01).

Q: How do the authors measure gain-loss attitudes in the exchange experiment?

A: In Stage 1, subjects are randomly endowed with one of two objects and provide three unincentivized preference statements (relative liking, relative wanting, and hypothetical choice) before any possibility of exchange is introduced. Under CPE, an individual endowed with object X will prefer X to the extent that (1 + λ_i) − 2(Y/X) > 0, so subjects with higher λ_i should more strongly favor their endowment. A principal components analysis reduces the three statements to one factor (capturing ~70% of variation), and residuals from regressing that factor on object assignment constitute the reduced-form measure l̂_i. The structural estimate λ̂_i is obtained via a mixed logit using a log-normal distribution for λ_i; the reduced form and structural measures are correlated at r = 0.95 (p < 0.01).

Q: What does the distribution of gain-loss attitudes look like across the two experiments?

A: In the labor supply experiment (N = 453 estimable subjects), 70.6% are loss averse and 29.4% are gain-seeking, with mean λ̂ = 1.65 and median λ̂ = 1.66. In the exchange experiment (N = 1,024), 76% are loss averse and 24% are gain-seeking, with mean λ̂ = 1.49 and median λ̂ = 1.34. A separate lottery-based elicitation in the labor supply study finds 28% gain-seeking subjects. These proportions are consistent with the weighted average of 22% gain-seeking found by Chapman et al. (2018) across seven prior lottery-choice studies.

Q: What is the aggregate treatment effect in the labor supply experiment, and what does it look like once heterogeneity is accounted for?

A: Without accounting for gain-loss heterogeneity, Condition High is associated with roughly a 26% increase in effort relative to Condition Low (individual-clustered s.e. = 0.03, p < 0.01), reproducing the Abeler et al. (2011) result and consistent with EBRD under universal loss aversion. However, R² = 0.03. Once interactions of Condition High with l̂_i and λ̂_i are included, R² rises to 0.40 and 0.39 respectively — more than a tenfold increase. Higher λ̂_i predicts larger positive treatment effects (raw correlation ρ = 0.18, p < 0.01), and the interaction of Condition High with λ̂_i is highly significant (F(1,452) = 49.14, p < 0.01).

Q: What is the aggregate treatment effect in the exchange experiment, and what does it look like once heterogeneity is accounted for?

A: Without heterogeneity, the treatment effect of Condition High on the probability of exchanging is precisely 0.00 (clustered s.e. = 0.03), which prior literature would read as a failure of EBRD. Once heterogeneity is introduced via interactions with l̂_i and λ̂_i, the pattern changes markedly: loss-averse subjects show positive treatment effects (greater willingness to exchange in High), while gain-seeking subjects show negative treatment effects (less willingness to exchange in High), consistent with Predictions 4–6. R² again rises by more than a factor of 10. In Condition Low, 38% of subjects exchange, reflecting a significant endowment effect (F(1,1022) = 25.66, p < 0.01).

Q: Why does the aggregate treatment effect in the exchange experiment equal zero?

A: The authors show in Appendix B.4 that the relationship between λ_i and exchange probability treatment effects can be concave — negative effects for gain-seeking subjects can be of greater absolute magnitude than positive effects for loss-averse subjects. With roughly 24% gain-seeking and 76% loss-averse subjects, aggregation can yield a near-zero average even when heterogeneous effects are substantial and directionally consistent with EBRD. This aggregation problem, not a failure of the expectations-based reference point mechanism, explains the null aggregate result.

Q: Do gain-loss attitudes measured in one domain predict behavior in another domain?

A: The lottery-based measure of gain-loss attitudes (from Multiple Price Lists administered after the real-effort task in the labor supply experiment) has mean λ̂ = 1.48 and median 1.42, with 28% gain-seeking subjects — proportions similar to the labor supply estimates. However, the correlation between the lottery-based and labor-supply-based structural estimates of λ̂ is only Pearson’s r = 0.091 (p = 0.03) and Spearman’s ρ = 0.084 (p = 0.075). Furthermore, the lottery measure has no predictive power for Stage 2 treatment effects. This suggests that while the prevalence of gain-seeking is similar across domains, gain-loss attitudes at the individual level are more domain-specific than prior work has appreciated.

Q: How do the authors address the “generated regressor problem” when using estimated λ̂_i as a regressor?

A: Since λ̂_i is itself estimated from Stage 1 data, using it directly as a regressor in Stage 2 regressions treats imprecise preference estimates as ideal data, which can distort inference (the Murphy-Topel problem). The authors address this by bootstrapping the entire pipeline — re-estimating gain-loss attitudes from Stage 1 in each of 500 bootstrap iterations and re-running the Stage 2 regressions — then reporting the average bootstrap coefficient and its standard deviation. The bootstrapped conclusions are qualitatively identical to the original regression results in both experiments.

Q: What limitations do the authors acknowledge in the EBRD model’s fit?

A: Even after accounting for heterogeneity, the EBRD model does not provide a complete quantitative account of behavior. In the labor supply experiment, gain-seeking subjects exhibit slightly positive average treatment effects (not negative as predicted), and loss-averse subjects’ empirical treatment effects fall short of theoretical predictions, despite a significant correlation between predicted and empirical treatment effects (ρ = 0.25, p < 0.01). The authors attribute these deviations to potential measurement error (which would attenuate estimated relationships), and to the possibility that reference points have multiple determinants — including status quo-based, attention-based, and anchoring-based factors — beyond expectations alone.

Q: What are the broader implications for other applications of gain-loss attitudes?

A: The paper’s findings have implications for any application that relies on universal loss aversion as a maintained assumption, including Rabin’s (2000) calibration argument for risk aversion at small and large stakes, insurance demand for small losses (Slovic et al., 1977), and preferences for bunched resolution of uncertainty (Kőszegi and Rabin, 2009). Admitting heterogeneity in gain-loss attitudes will require more nuanced predictions in each of these settings. The paper provides a methodology — measuring individual-level gain-loss attitudes within the experimental context of interest — for investigating and controlling for such heterogeneity.

Q: What design features prevent confounds between Stage 1 measurement and Stage 2 treatment in the exchange experiment?

A: Stage 1 uses a different pair of objects (USB stick and pens) than Stage 2 (picnic mat and thermos), or vice versa — each subject encounters each pair exactly once, with counterbalancing at the session level. Stage 1 preference statements are unincentivized and made before any possibility of exchange is introduced, so they do not contaminate the Stage 2 expectations manipulation. The random reassignment of objects at the end of Stage 1 generates exogenous variation in endowments, preventing mechanical confounds. The authors also verify that interpreting Stage 1 variation as reflecting heterogeneity in object valuations (rather than gain-loss attitudes) would predict zero heterogeneous treatment effects in Stage 2 — a prediction rejected by the data.

Expectations-Based Reference Dependence (EBRD): The formulation, due to Kőszegi and Rabin (2006, 2007), in which an individual’s reference point is the entire distribution of outcomes they rationally expected, rather than a fixed status quo. Behavior is governed by a Choice-Acclimating Personal Equilibrium (CPE) in which the chosen action is optimal given that the expectation of that action serves as the reference.

Gain-Loss Attitudes (λ_i): The individual-specific parameter governing how outcomes above versus below the reference point affect utility. Under piecewise-linear gain-loss utility, an outcome that falls short of the reference by z reduces utility by η·λ_i·z, while an outcome above it raises utility by η·z. Loss aversion is λ_i > 1; gain-seeking is λ_i < 1; loss neutrality is λ_i = 1. In this paper, λ_i is treated as heterogeneous across individuals rather than assumed uniform.

Universal Loss Aversion: The implicit homogeneity assumption maintained in all prior tests of EBRD — that every individual has λ > 1. The authors characterize this as a form of the De Gustibus Non Est Disputandum conjecture applied to gain-loss attitudes, and document that it fails empirically in both experimental settings.

Choice-Acclimating Personal Equilibrium (CPE): The rational expectations equilibrium concept from Kőszegi and Rabin (2006, 2007) used throughout the paper to derive comparative statics. A choice is a CPE if its expected utility given its own expectation as the reference exceeds the expected utility of any alternative given that alternative’s expectation as the reference.

Reduced-Form Gain-Loss Measure (l̂_i): In the labor supply context, the individual-level OLS coefficient on Δw/w in a log-effort regression — capturing how strongly a subject reduces (or increases) effort under wage uncertainty relative to a fixed wage of equal mean. A positive l̂_i identifies loss aversion; negative identifies gain-seeking. In the exchange context, the analogous measure is the residual from regressing the first principal component of Stage 1 preference statements on object assignment.

Aggregation Problem: The paper’s central methodological contribution — when gain-loss attitudes are heterogeneous and the EBRD treatment effect is non-linear in λ_i, the average treatment effect across a heterogeneous population need not equal the treatment effect at the average λ. In the exchange experiment, the aggregate treatment effect is precisely zero even though loss-averse and gain-seeking subjects each respond in the theoretically predicted (opposite) direction, because the concave relationship between λ_i and the exchange probability treatment effect causes negative gain-seeking effects to dominate in the aggregate.

Debiasing and T-Tests for Synthetic Control Inference on Average Causal Effects

Mon, 01 Jan 0001 00:00:00 +0000

Chernozhukov, Wüthrich, and Zhu propose a debiased synthetic control (SC) estimator and an accompanying self-normalized t-test for making inferences on the average treatment effect on the treated (ATT) in aggregate panel data settings with one treated unit. The inferential target is the time-averaged treatment effect τ = (1/T1) Σ_{t=T0+1}^{T} (Y0t(1) − Y0t(0)), a one-number summary of the overall causal impact that admits standard-form confidence intervals, in contrast to per-period effects (which cannot be consistently estimated with one treated unit) and sharp null hypotheses (which do not inform effect magnitude).

The method addresses two structural challenges in SC inference. First, the canonical SC estimator τ_SC is biased because the weights are estimated from high-dimensional pre-treatment data, and the bias can be substantial under misspecification. Second, even if true weights were known, constructing standard errors requires estimating the long-run variance (LRV), for which classical estimators such as Newey-West are unreliable in the small samples typical of SC applications.

The debiasing procedure is a K-fold cross-fitting scheme applied to the pre-treatment period. The pre-treatment sample is split into K consecutive blocks. For each fold k, SC weights w_(k) are estimated on the leave-one-block-out pre-treatment data H_{(-k)}, and a component estimator τ_k is formed as the difference between the post-treatment SC residual (using w_(k)) and the in-block pre-treatment SC residual. The latter serves as an estimator of the bias, which under the model assumptions is stable across the pre- and post-treatment periods. The final estimator τ_hat is the average of τ_k across folds. A self-normalized t-statistic T_K = sqrt(K)(τ_hat − τ)/σ_τ is constructed using the cross-fold variance; its asymptotic distribution is t_{K-1}, so no LRV estimation is required and (1−α) confidence intervals take the textbook form τ_hat ± t_{K-1}(1−α/2) × σ_τ/sqrt(K).

The t-test is proven valid with both stationary and non-stationary data. With stationary data (Theorem 2), it is valid under arbitrary misspecification. With non-stationary data, validity holds either when all units share a common nonstationarity (Theorem 3, also misspecification-robust) or when units deviate from a common nonstationarity under restrictions on the magnitude and heterogeneity of deviations but SC is correctly specified (Theorem 4). The latter covers heterogeneous deterministic time trends and certain cointegration structures. Researchers therefore need not pre-test for unit roots and select inference procedures accordingly.

A formal efficiency result (Section 3.3) shows that the asymptotic variance of the debiased SC estimator is no larger than that of difference-in-differences (DID), because SC minimizes prediction error and w* dominates the equal-weight DID vector. The relative asymptotic efficiency (RAE) of the t-test versus DID rises with K: K=3 yields RAE of 63.56%; K=5 yields 82.08%; K=10 yields 92.25%.

Simulations calibrated to Andersson’s (2019) Swedish carbon tax application — T0=30, T1=16, N=14, Gaussian AR(1) errors — show that the t-test at K=3 achieves coverage close to the nominal 90% level across correct-specification and misspecification DGPs, while Newey-West standard errors produce substantial undercoverage (coverage = 0.72–0.84) at moderate to high AR(1) coefficients. The method performs comparably to or better than subsampling (Li, 2020) and synthetic DID (Arkhangelsky et al., 2021), and avoids bandwidth selection.

In the empirical application, the debiased SC t-test (K=3) applied to annual CO2 emissions from transport across Sweden (treated, 1990) and 14 OECD control countries over 1960–2005 yields a negative and statistically significant ATT, with a 90% confidence interval lying entirely below zero, implying approximately an 11% average reduction in per capita CO2 emissions from transport attributable to the Swedish carbon tax over 1990–2005. The pre-treatment AR(1) coefficient of SC residuals is approximately 0.31, supporting K=3 as appropriate. These findings corroborate and extend Andersson’s (2019) permutation-based results by providing a confidence interval for the magnitude of the average effect. The method is implemented in the R package scinference.

Q: What is the primary inferential target and why is it preferred over per-period effects or sharp nulls? A: The target is the ATT τ = (1/T1) Σ_{t=T0+1}^{T} (Y0t(1)−Y0t(0)), the time-averaged treatment effect on the treated unit over the post-treatment period. Per-period effects cannot be consistently estimated when there is only one treated unit, yielding wide and uninformative confidence intervals. Sharp nulls (e.g., of no effect whatsoever) are useful starting points but do not inform policy decisions about effect magnitude. The ATT provides an interpretable one-number summary and admits standard-form confidence intervals.

Q: What are the two main inferential challenges that the paper addresses? A: First, the canonical SC estimator τ_SC is biased due to estimation error in the high-dimensional weights, even under correct specification, and the bias can be substantial under misspecification. Second, even with known true weights, standard error estimation requires the long-run variance (LRV), for which classical estimators such as Newey-West (1987) and Andrews (1991) are not sufficiently accurate in the small samples typical of SC applications.

Q: How does the K-fold cross-fitting procedure debias the SC estimator? A: The pre-treatment period is divided into K consecutive blocks H1,…,HK. For each fold k, SC weights w_(k) are estimated using leave-one-block-out pre-treatment data H_{(-k)}. The component estimator τ_k subtracts the in-block pre-treatment SC residual (an estimator of the bias in period Hk) from the post-treatment SC residual (using w_(k)). Because the bias is assumed stable across pre- and post-treatment periods, this subtraction removes it. The final estimator τ_hat averages τ_k across k=1,…,K.

Q: How does the self-normalized t-statistic avoid LRV estimation? A: The statistic T_K = sqrt(K)(τ_hat − τ)/σ_τ uses σ_τ = sqrt(1 + Kr/T1) × sqrt[(1/(K−1)) Σ_k (τ_k − τ_hat)^2], which is the cross-fold standard deviation of the component estimators scaled by a factor reflecting the ratio of pre- to post-treatment block lengths. Under the asymptotic theory, T_K converges to a t_{K-1} distribution, which is pivotal and requires no bandwidth or kernel choice. The cross-fold structure acts as a self-normalizer analogous to the fixed-b approach in the LRV literature.

Q: What does the paper prove about validity with non-stationary data? A: Theorem 3 establishes that when all units share a common nonstationarity (Assumption 4: Yt(0) = Vt(0)+θt and Xt = Zt+1_N·θt where {Vt(0),Zt} is stationary and θt is unrestricted), T_K → t_{K-1} under arbitrary misspecification. Theorem 4 establishes validity when units deviate from common nonstationarity (Assumption 5) under restrictions on the magnitude and heterogeneity of deviations, but requires SC to be correctly specified. These results jointly imply that researchers need not pre-test for unit roots before applying the t-test.

Q: How does the paper formally show that debiased SC is more efficient than DID? A: The pseudo-true SC weights w* minimize mean squared prediction error over W_SC, so the residual variance σ^2_* = E(Yt(0)−Xt’w*)^2 ≤ E(Yt(0)−Xt’w_DID)^2 = σ^2_DID, where w_DID = (1/N,…,1/N)’ is the equal-weight DID vector. This inequality holds regardless of whether SC is correctly specified or not, so the efficiency gain over DID is unconditional. The t-test is also valid when the parallel trends assumption underlying DID is violated, making it more robust.

Q: What is the trade-off in choosing K, and what does the paper recommend? A: A larger K produces shorter confidence intervals (higher RAE: 63.56% at K=3 versus 92.25% at K=10) but may reduce coverage accuracy in finite samples because the t_{K-1} approximation improves with K while each block becomes smaller. The paper recommends K=3 as a starting point for typical SC applications where T0 is small, based on simulation evidence showing excellent 90% coverage at K=3. When T0 is moderate or large, K can be increased without loss of coverage accuracy.

Q: What do the simulations show about the performance of Newey-West standard errors versus the t-test? A: In simulations calibrated to the Swedish carbon tax application (T0=30, T1=16, N=14, AR(1) errors), the t-test at K=3 achieves coverage close to the nominal 90% level across both correct-specification and misspecification DGPs. Newey-West standard errors produce coverage of only 0.72–0.84 when the AR(1) coefficient of the error process is moderate to high. DID achieves nominal coverage when parallel trends hold but is biased and has poor coverage under violations of parallel trends.

Q: How does the method compare with Li (2020) subsampling and synthetic DID (Arkhangelsky et al., 2021)? A: Compared with Li (2020), the t-test allows N to grow with (T0,T1) rather than treating N as fixed, directly corrects for SC estimation bias via cross-fitting, avoids the need to pre-process data for stationarity, and does not require a subsampling bandwidth choice. Compared with SDID (Arkhangelsky et al., 2021), the t-test is simpler, does not require homoskedasticity across units as SDID’s placebo variance estimator does, and is developed under a linear prediction model rather than a factor model. Simulations show the t-test performs comparably to or better than both alternatives in the application-calibrated DGP.

Q: What are the empirical findings for the Swedish carbon tax application? A: Using annual CO2 emissions from transport for Sweden and 14 OECD control countries over 1960–2005, with T0=30 (1960–1989) and T1=16 (1990–2005), the debiased SC t-test at K=3 yields a negative and statistically significant ATT. The 90% confidence interval lies entirely below zero. The estimated average effect is approximately an 11% reduction in per capita CO2 emissions from transport attributable to the carbon tax over 1990–2005. The pre-treatment SC residuals show an estimated AR(1) coefficient of approximately 0.31, confirming moderate persistence and supporting the use of K=3.

Q: When does the paper recommend against using the t-test? A: The paper advises against the t-test when T1 is very small (T1 < 8–10), as asymptotic approximations may be inaccurate; when there are structural breaks shortly after T0 (making the ATT ill-defined); and when SC fit is poor because the treated unit is very different from controls. The method requires T0, T1, N → ∞ for asymptotic validity, and T1 ≥ 10–15 is suggested for reliable finite-sample performance.

Q: How does the paper cover higher-order improvements in finite samples? A: Appendix D formally establishes that the coverage error of the confidence interval I_K(1−α) is O(1/T) rather than O(1/sqrt(T)), analogous to the fixed-b approach in the LRV literature. This provides a formal justification for the excellent finite-sample coverage observed in the simulations and distinguishes the t-test from Gaussian approximations whose coverage error is of larger order.

K-fold cross-fitting debiasing: A procedure that splits the pre-treatment period into K consecutive blocks, estimates SC weights on the leave-one-block-out pre-treatment data for each fold, and subtracts the in-block pre-treatment prediction error as an estimator of the bias. Under the model, the bias is assumed stable across pre- and post-treatment periods, so this subtraction removes it from the final estimator.

Self-normalized t-statistic: A scale-free test statistic T_K = sqrt(K)(τ_hat − τ)/σ_τ whose denominator is the cross-fold standard deviation of the K component estimators, scaled to account for the ratio of pre-treatment block length to post-treatment period length. The statistic converges to a t_{K-1} distribution without requiring any LRV estimation.

Average treatment effect on the treated (ATT): The target parameter τ = (1/T1) Σ_{t=T0+1}^{T} (Y0t(1)−Y0t(0)), representing the time-averaged causal effect of the treatment on the treated unit over the post-treatment period. It provides an interpretable one-number summary that admits standard-form confidence intervals, in contrast to per-period effects (not consistently estimable with one unit) and sharp null hypotheses (informative about presence but not magnitude of effect).

Common nonstationarity: The condition (Assumption 4) that all units share the same nonstationary component θt — formally, Yt(0) = Vt(0)+θt and Xt = Zt+1_N·θt with {Vt(0),Zt} stationary and θt unrestricted. Under this condition, the t-test is valid under arbitrary misspecification of SC weights, without requiring the researcher to specify or pre-test the type of nonstationarity.

Relative asymptotic efficiency (RAE): The ratio of the asymptotic expected confidence interval length of the debiased SC t-test to a benchmark (taken as K→∞), quantifying the cost in interval length from using a finite K. At K=3, RAE = 63.56%; at K=5, RAE = 82.08%; at K=10, RAE = 92.25%.

Long-run variance (LRV): The quantity that governs the asymptotic variance of time-averaged quantities in settings with serially correlated data. The paper argues that classical LRV estimators (Newey-West, Andrews) are insufficiently accurate in the small samples typical of SC applications, motivating the self-normalization approach that avoids LRV estimation entirely.

Pseudo-true SC weights: The population minimizer w* = argmin_{w ∈ W_SC} E(Yt(0)−Xt’w)^2, defined as the best linear predictor of the treated unit’s counterfactual outcome within the SC simplex constraint. These weights exist and satisfy the efficiency bound even under model misspecification, providing the foundation for the efficiency comparison with DID.

Decision Theory for Treatment Choice Problems with Partial Identification

Mon, 01 Jan 0001 00:00:00 +0000

This paper applies classical statistical decision theory (Wald 1950) to treatment choice problems where the data only partially identify payoff-relevant parameters. The policy maker chooses an action a in [0,1] — interpreted as the share of the population assigned to a new policy — to maximize welfare that is linear in the action. The data are Gaussian, and the key departure from prior literature is that the mean function mapping parameters to data need not be injective, so even infinite data may not reveal the optimal action.

The paper evaluates decision rules under three classical criteria: admissibility, maximin welfare, and minimax regret (MMR).

Admissibility result (Theorem 1): Under nontrivial partial identification, every decision rule — however exotic — is welfare-admissible. No rule is dominated. This is a sharp reversal from point-identified settings, where admissibility meaningfully restricts the rule class: in the scalar point-identified case (n=1, m(theta)=theta), Karlin and Rubin’s (1956) result implies that any non-threshold rule is dominated. The proof exploits completeness of the Gaussian statistical model: if a dominating rule d’ existed, it would have to agree almost everywhere with d, yielding a contradiction. Theorem 5 generalizes this result beyond Gaussian likelihoods, tying it to bounded completeness of the statistical model.

Maximin welfare result (Theorem 2): The maximin criterion selects the no-data rule d(y) = 0 — preserve the status quo regardless of data — whenever the status quo welfare is the infimum over states with non-positive welfare contrast. In the running example, maximin welfare equals zero and is achieved by never assigning the new policy. This echoes critiques from Savage (1951) and Manski (2004) about ultra-pessimism.

Minimax regret result (Theorem 3): In point-identified problems, the MMR rule is essentially unique and nonrandomized (Canner 1970; Stoye 2009a; Tetenov 2012). Under partial identification, when the identified set is large enough — formally, when I(0) is large enough and there exists mu in the identified set with I(mu) > I(0) — there are infinitely many MMR optimal rules, and any symmetric, weakly increasing MMR rule depending only on the sufficient statistic (w*)^T Y must randomize for some data realizations. Moreover, if I(mu) is differentiable at zero, no linear threshold rule is MMR optimal.

Least randomizing MMR rule (Theorem 4): Because policy randomization is difficult to implement in practice, the authors uniquely characterize the MMR optimal rule that randomizes least frequently. Among all symmetric, weakly increasing, unimodal MMR optimal rules depending on (w*)^T Y, the rule d*_linear has the smallest randomization region — every other distinct such rule has a strictly wider randomization region. This rule can be profiled-regret dominant over the Stoye (2012a)/Yata (2023) MMR rule (Proposition 2), and the uniformly randomizing rule is inadmissible under profiled regret (Proposition 3). Under some conditions, d*_linear can also be obtained as the MMR rule within a class that penalizes randomized assignments equally (Proposition 4).

Three applications ground the theory. First, in Ishihara and Kitagawa’s (2021) evidence aggregation framework — extrapolating treatment effects from n source countries to a target country — the least randomizing rule randomizes only when estimated bounds on the target treatment effect straddle zero, linking decision rules directly to identified-set estimators. Second, in LATE extrapolation (Mogstad et al. 2018), all decision rules are admissible and IV-based threshold rules are not dominated. Third, in the omitted-variable-bias setting of Diegert et al. (2022), the decision-theoretic breakdown point — the largest confounding magnitude under which the seemingly better policy should be adopted without hedging — tolerates strictly more confounding than Diegert et al.’s breakdown point, where the threshold is k = sqrt(pi/2) * sigma.

Q: What is the central research question? A: The paper asks how classical statistical decision theory — admissibility, maximin welfare, minimax regret — applies when the data only partially identify the payoff-relevant parameters governing a binary treatment choice. Prior literature had developed these criteria for point-identified settings; this paper characterizes how partial identification fundamentally changes the answers.

Q: What is the formal framework? A: The policy maker chooses a in [0,1] (population share assigned to the new policy) with welfare W(a,theta) = a*W(1,theta) + (1-a)*W(0,theta), linear in a. The data are Y ~ N(m(theta), Sigma) with known m and Sigma. Partial identification arises when m is not injective, so distinct parameter values theta and theta’ with opposite-sign welfare contrasts U(theta) = W(1,theta) - W(0,theta) can produce the same data distribution.

Q: Why does admissibility lose all refinement power under partial identification? A: Theorem 1 shows that every decision rule is admissible when there is nontrivial partial identification. The mechanism is Gaussian completeness: if a dominating rule d’ existed, then for every data distribution in the model, d and d’ would have equal expected values, which by completeness implies d = d’ almost everywhere — a contradiction. This relies on the fact that nontrivial partial identification ensures that each data distribution is compatible with both positive and negative welfare contrasts, preventing the construction of a uniformly dominating rule.

Q: What is the contrast with point-identified settings? A: In the scalar point-identified case (n=1, m(theta)=theta, W(1,theta)=theta, W(0,theta)=0), Karlin and Rubin’s (1956) theorem implies any non-threshold rule is dominated; admissibility restricts attention to threshold rules. Partial identification completely eliminates this refinement: even randomized or otherwise arbitrary rules are admissible.

Q: What does the maximin welfare criterion recommend? A: Theorem 2 shows that when the status quo welfare equals the infimum of welfare over states with non-positive welfare contrast, the maximin optimal rule is d(y) = 0 for all y — preserve the status quo regardless of the data. In the running evidence-aggregation example, maximin welfare equals zero and is achieved by never assigning the new policy. The criterion ignores all data because the worst case is always achieved at states where the new policy performs no better than the status quo.

Q: What is the minimax regret criterion and why is it preferred? A: Expected regret at state theta is R(d,theta) = U(theta)*{1{U(theta)>=0} - E[d(Y)]} — the expected welfare loss relative to the oracle who knows theta. A rule is MMR optimal if it minimizes worst-case expected regret. Unlike maximin welfare, MMR uses data and balances risks across states. In point-identified settings it yields essentially unique, nonrandomized rules.

Q: How does partial identification change the MMR solution set? A: Theorem 3 shows that when the identified set is large enough — I(0) is sufficiently large and there exists mu with I(mu) > I(0) — there are infinitely many MMR optimal rules, and every symmetric, weakly increasing MMR rule depending on the sufficient statistic (w*)^T Y must randomize for some data realizations. If I(mu) is differentiable at zero, no linear threshold rule is MMR optimal. Different MMR rules can recommend different policies for the same data, creating a nontrivial multiplicity problem.

Q: How is the least randomizing MMR rule characterized? A: Theorem 4 shows that among all symmetric, weakly increasing, unimodal MMR optimal rules that depend on data only through (w*)^T Y, the rule d*_linear has the smallest randomization region: every other distinct rule in this class has a strictly wider randomization region, V(d*_linear) ⊆ V(F∘w*) with strict inclusion when F ≠ d*_linear. This characterization is essentially unique and provides a pragmatic refinement of the MMR solution set.

Q: What is profiled regret and why is it used? A: Profiled regret reports worst-case expected regret at each fixed value of the point-identified parameters, rather than worst-case over all parameters jointly. Proposition 2 shows that the least randomizing rule d*_linear can profiled-regret dominate the Stoye (2012a)/Yata (2023) MMR rule in the running example. Proposition 3 shows that the uniformly randomizing rule is profiled-regret inadmissible when profiling over point-identified parameters. This concept provides an additional selection criterion within the MMR solution set.

Q: Can the least randomizing rule be derived from an explicit welfare penalty? A: Proposition 4 shows that, under some conditions, d*_linear is minimax regret optimal within the class of rules that penalize all randomized assignments equally. This connects the least randomizing criterion to a modified welfare function that treats randomization itself as costly, providing an interpretation for the refinement beyond mere pragmatics.

Q: What does the evidence aggregation application show? A: In the Ishihara-Kitagawa (2021) framework — extrapolating effects from n source countries to a target country using Lipschitz smoothness — the least randomizing rule randomizes only (though not always) when the estimated bounds on the target treatment effect contain both positive and negative values. When bounds are entirely positive or entirely negative, the rule recommends a deterministic action. This shows how identified-set estimators directly enter decision-theoretically optimal rules.

Q: What does the LATE extrapolation application show? A: In the Mogstad et al. (2018) setting with a binary instrument and no covariates, where the payoff-relevant parameter is a policy-relevant treatment effect corresponding to expanding the complier subpopulation, Theorem 1 applies: all decision rules are admissible. In particular, the IV threshold rule — implement the policy for large IV estimates — is not dominated, providing decision-theoretic grounding for a common empirical practice.

Q: What does the omitted variable bias application show? A: In the Diegert et al. (2022) setting where the identified set for the long regression coefficient given the medium regression coefficient is [beta_med - k, beta_med + k], the least randomizing MMR rule is d*_linear(beta_hat_med) when k > sqrt(pi/2) * sigma. The decision-theoretic breakdown point — the largest k under which the seemingly better policy should be adopted without randomization — is strictly larger than Diegert et al.’s sensitivity breakdown point, meaning the decision-theoretic approach tolerates more confounding before recommending hedging.

Q: How does Theorem 5 generalize Theorem 1 beyond Gaussian likelihoods? A: Theorem 5 extends the admissibility result by connecting it to bounded completeness of the statistical model rather than Gaussian-specific completeness. This shows that the collapse of admissibility’s refinement power is not an artifact of normality but a general consequence of partial identification combined with a sufficiently rich statistical model.

Q: What is the paper’s broader implication for empirical practice? A: The results show that under partial identification, two of the three classical decision-theoretic criteria (admissibility and maximin welfare) provide no useful guidance — the former because everything passes, the latter because it ignores data entirely. MMR remains the operative criterion but yields infinitely many rules, all requiring some randomization. The least randomizing refinement provides a unique, practically implementable rule that connects to estimated identified sets and tolerates more ambiguity than purely statistical sensitivity analyses.

Partial identification: A setting where even infinite data cannot uniquely determine payoff-relevant parameters, because the mean function m mapping parameters to data distributions is not injective. Distinct parameter values with opposite-sign welfare contrasts may be observationally equivalent.

Welfare contrast U(theta): The difference W(1,theta) - W(0,theta) between the welfare under the new policy and under the status quo at parameter theta. The oracle optimal action is 1{U(theta) >= 0}.

Admissibility (welfare): A rule d is admissible if no rule d’ weakly dominates it in expected welfare at every theta with strict improvement at some theta. Under partial identification with Gaussian likelihood, every rule is admissible — admissibility has no refinement power.

Maximin welfare optimality: A rule is maximin optimal if it attains the highest worst-case expected welfare. Under partial identification, this criterion selects the no-data rule (always preserve status quo) whenever the status quo welfare equals the infimum over states with non-positive welfare contrast.

Minimax regret (MMR) optimality: A rule minimizes the worst-case expected welfare loss relative to the oracle action. Under severe enough partial identification, MMR optimal rules are non-unique and all require randomizing policy recommendations for some data realizations.

Least randomizing MMR rule (d*_linear): The unique MMR optimal rule with the smallest randomization region among all symmetric, weakly increasing, unimodal MMR rules depending on the sufficient statistic. Characterized in Theorem 4; randomizes only when estimated identified set bounds straddle zero in the running example.

Profiled regret: The worst-case expected regret at each fixed value of the point-identified parameters, treating them as a parameter of interest and profiling out the partially identified parameters. Provides a finer ranking within the MMR solution set and renders the uniformly randomizing rule inadmissible.

Demand Analysis under Latent Choice Constraints

Mon, 01 Jan 0001 00:00:00 +0000

Agarwal and Somaini study demand estimation in markets where consumers face latent choice constraints — situations where a consumer’s effective choice set is determined not only by her preferences but also by supply-side rationing or information frictions that restrict which options are actually available to her. Standard discrete choice methods assume consumers pick freely from the full product set, but this assumption fails in school and college admissions, entry-level labor markets, healthcare with selective admissions, and consumer markets with incomplete consideration sets. The paper provides a unified non-parametric identification framework for this class of models, proves necessity of the identifying instruments, proposes a computationally tractable estimator, and applies the framework to the California kidney dialysis market.

The model combines a general random utility specification — accommodating multi-dimensional unobserved heterogeneity and product-level unobservables correlated with observed characteristics as in Berry (1994) and BLP (1995) — with a reduced-form acceptance policy function that governs which products accept which consumers. The consumer’s latent choice set is the set of products that accept her, and she picks her most preferred option within that set. Crucially, the acceptance decision may be arbitrarily correlated with consumer preferences, ruling out the independence assumptions common in the consideration-set literature.

Identification rests on two sets of instruments. The first is a preference shifter, a consumer-product observable that affects utility but is excluded from the acceptance policy — distance to facility in the application. The second is a choice-set shifter, an observable that affects the acceptance decision but is excluded from consumer utility — short-term deviation of a facility’s caseload from its estimated target in the application. The main result (Theorem 1) establishes non-parametric point identification of the joint distribution of indirect utilities and acceptance decisions given both instruments. Proposition 1 establishes that the model is not identified when the choice-set shifter is absent — even when the preference shifter has full support — making both instruments necessary rather than merely sufficient.

The application uses USRDS data on 41,913 new dialysis patients treated at 552 California facilities between 2015 and 2018. Most facilities are owned by Fresenius or DaVita. The choice-set shifter is the facility’s caseload deviation from target when a patient enters the market; facility and quarter fixed effects are included so that only short-term caseload variation drives identification. A reduced-form regression shows that higher caseload deviation significantly reduces the inflow of new patients to a facility, consistent with supply-side rationing. Patients also choose more distant facilities when nearby facilities have above-normal caseloads, providing further reduced-form evidence that rationing shapes allocations.

A Gibbs sampler with data augmentation — drawing alternately from the distribution of latent choice sets conditional on utilities and from utility parameters conditional on choice sets — circumvents the curse of dimensionality that makes direct likelihood maximization over all possible choice sets infeasible.

Estimation results show that the probability a patient is accepted at her first-choice facility is only 73.0%, with variation across facilities. Standard discrete choice models that ignore rationing misestimate facility quality, systematically assigning high desirability to low-caseload facilities in a manner that conflates easy access with genuine patient preference. A naive correction that includes the caseload measure in the utility function mischaracterizes the diversion pattern: rationed patients are marginal for the facility but strictly prefer it, so they divert differently from patients who voluntarily switch because of quality changes. Fresenius and DaVita facilities are estimated to be more selective than independent facilities, consistent with chain networks enabling coordinated patient-flow management across locations.

Q: What is the core empirical problem the paper addresses? A: Standard demand estimation inverts market shares to recover preference parameters under the assumption that consumers choose freely from the full product set. When choice sets are constrained by supply-side rationing or information frictions, the largest market share product need not be the one most preferred — it may simply be the one that accepts the most consumers. This makes the standard inversion inapplicable, and ignoring constraints yields biased preference estimates.

Q: What does the paper’s model consist of? A: The model has two components: (1) a random utility model for consumer preferences with rich observed and unobserved heterogeneity, allowing product-level unobservables correlated with observed characteristics; and (2) a reduced-form acceptance policy function sigma_jt taking values in {0,1} that determines whether product j accepts consumer i. The consumer’s latent choice set is the set of products that accept her; she picks her most preferred option within it. Utilities and acceptance decisions may be arbitrarily correlated.

Q: What examples of latent choice constraints are covered by the framework? A: The reduced form encompasses: selective admissions in healthcare (facility accepts patient if profitability exceeds a caseload-dependent threshold); two-sided matching markets where a pairwise stable allocation is described by cutoff scores (school admissions, entry-level labor markets); consideration set models where brand awareness advertising or inattention determines which products a consumer sees; fixed-sample consumer search; and product stock-outs. Each of these implies an acceptance policy function of the form specified in the paper’s reduced-form model.

Q: What are the two identifying instruments and the intuition behind each? A: The preference shifter yij is a consumer-product observable that affects the consumer’s indirect utility for product j but is excluded from that product’s acceptance decision. In the application this is distance: dialysis requires multiple weekly visits, so distance affects patient utility, but a facility’s decision to accept a patient does not depend on how far the patient lives. The choice-set shifter zij is an observable that affects the acceptance decision but is excluded from consumer preferences. In the application this is the deviation of facility caseload from its estimated target: short-term caseload swings affect whether a facility can take a new patient but, conditional on facility fixed effects, do not reflect facility quality as perceived by patients.

Q: What does Theorem 1 establish and under what conditions? A: Theorem 1 establishes non-parametric point identification of (i) the function gj mapping the preference shifter to its utility contribution, and (ii) the joint distribution of indirect utilities and acceptance indicators, for every consumer attribute vector and every value in the interior of the joint support of the instruments. Conditions required include: monotonicity of the acceptance policy in the choice-set shifter (higher z makes acceptance weakly less likely, with sigma=1 as z approaches negative infinity and sigma=0 as z approaches positive infinity); conditional independence of unobservables from the instruments given observed consumer attributes; and at least two products available.

Q: What does Proposition 1 establish about necessity of the choice-set shifter? A: Proposition 1 shows that if the choice-set shifter z has singleton support (no variation), then even when the preference shifter g has full support on R^|J|, the distribution of preferences is not identified wherever a choice set strictly smaller than the full product set has positive probability. The non-identification result applies on any open set where a constrained choice set has positive probability — it is not a knife-edge case. This makes the choice-set shifter a necessary condition for identification, not merely a convenient one.

Q: How does the paper handle endogeneity of product characteristics? A: Corollary 2 extends the baseline identification result to allow product-level unobservables that may be correlated with observed product characteristics, as in Berry (1994) and BLP (1995). Identification in this case requires an additional instrument that shifts product characteristics but is excluded from both preferences and choice sets — analogous to BLP supply-side instruments — alongside the two shifters already required. This extends Berry and Haile (2010) to settings with constrained choice sets.

Q: What is the Gibbs sampler estimator and why is it needed? A: With J products per market, the number of possible choice sets is 2^J, making direct likelihood computation infeasible for even moderate J. The Gibbs sampler uses data augmentation to alternate between: (a) drawing latent choice sets conditional on current utility parameters and observed choices; and (b) drawing utility parameters conditional on the augmented choice sets. Each conditional draw reduces to a standard problem, avoiding the curse of dimensionality. The Bernstein-von Mises theorem implies that the posterior mean of the sampling chain is asymptotically equivalent to the maximum likelihood estimator.

Q: What is the reduced-form evidence for supply-side rationing in dialysis? A: The regression of log(1 + new patient inflows to facility j in quarter q) on facility fixed effects, quarter fixed effects, and the caseload deviation z_jq yields a statistically significant negative coefficient on caseload deviation: above-target caseloads reduce new patient admissions even after controlling for facility-level and time-level averages. Additionally, patients whose nearest facilities have above-normal caseloads travel to more distant facilities, providing complementary evidence that rationing displaces patients geographically.

Q: What is the estimated probability of acceptance at a first-choice facility? A: The structural estimates imply that a patient is accepted at her first-choice facility with probability only 73.0%, with variation across facilities. The implied 27.0% rejection rate is economically substantial, meaning a large share of observed allocations do not reflect unconstrained patient preference.

Q: How do estimates from the constrained model differ from a standard discrete choice model? A: The standard model, which ignores selective admissions, assigns higher utility to facilities with lower caseloads — a bias that conflates easy access with genuine patient preference. The constrained model separately identifies the facility’s acceptance propensity from the patient’s underlying preference, yielding different facility quality rankings. The largest facilities are not necessarily the most desirable once selective admissions are accounted for.

Q: Why is the naive correction — including caseload in the utility function — insufficient? A: The naive correction treats caseload as a quality attribute, implying that a patient turned away because of high caseload and a patient who voluntarily avoids a high-caseload facility are pulled from the same margin. In the constrained model, a rationed patient is marginal for the facility but strictly prefers it, so she diverts to a different set of alternatives than a patient who voluntarily switches. Not capturing this distinction produces quantitatively different diversion ratios.

Q: What do the estimates say about chain versus independent facilities? A: Fresenius and DaVita facilities are estimated to be more selective in their admissions than independent facilities. The paper interprets this as consistent with large chains having better ability to coordinate patient flows across their network of facilities, potentially directing turned-away patients to other chain locations.

Q: What is the scope of the identification results? A: Identification is established within each market, for consumer attribute vectors in the interior of support, and for utility-acceptance pairs in the interior of the joint support of the instruments. The results are non-parametric in that they do not restrict the functional form of preferences or acceptance policies beyond monotonicity and support conditions, and they allow unobservables affecting choice sets to be arbitrarily correlated with preference unobservables. The empirical application implements a parametric version for tractability.

Latent choice constraint: A restriction on a consumer’s effective choice set arising from supply-side rationing or information frictions, such that the consumer can only choose among the products that accept her rather than freely among all products in the market. Distinct from price-based market clearing.

Acceptance policy function: A reduced-form function mapping consumer attributes, consumer unobservables, and the choice-set shifter to a binary accept/reject decision by product j. Indexed by product and market, allowing arbitrary variation in selectivity across products and time. The consumer’s latent choice set is defined as the set of products whose acceptance policy equals 1.

Choice-set shifter: A consumer-product observable that shifts the acceptance probability — making product j more or less likely to accept consumer i — while being excluded from consumer indirect utility. In the application: short-term deviation of facility caseload from its estimated target. Necessary (not merely sufficient) for non-parametric identification of the model.

Preference shifter: A consumer-product observable that shifts consumer utility for product j and is separable from consumer-specific unobservables, but is excluded from that product’s acceptance policy function. In the application: distance from patient’s residence to the facility. Also necessary for identification.

Curse of dimensionality in constrained choice: The computational problem that the number of possible latent choice sets grows as 2^J with the number of products J, making direct likelihood integration over choice sets infeasible for even moderate J. Resolved in this paper by a Gibbs sampler with data augmentation that conditions alternately on latent choice sets or utility parameters.

Diversion ratio under selective admissions: The share of patients lost by a facility who are captured by each alternative facility. In a model with selective admissions, rationed patients (marginal for the facility) divert differently from patients who voluntarily switch (marginal for the consumer), because rationed patients strictly prefer the rejecting facility. The naive correction conflates these two margins, yielding quantitatively different and biased diversion ratio estimates.

Non-parametric necessity of instruments: The property that both the preference shifter and the choice-set shifter are individually necessary conditions for point identification of the joint distribution of preferences and acceptance decisions, not merely convenient sufficient conditions. Absence of either instrument leaves the model non-identified on any open set where a constrained choice set has positive probability.

Designing Disability Insurance Reforms: Tightening Eligibility Rules or Reducing Benefits?

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a sufficient statistics framework for the welfare analysis of disability insurance (DI) policy reforms and applies it to two reform episodes in Austria. The framework derives social optimality conditions for the two main DI policy instruments — eligibility rules and benefit levels — expressed in terms of estimable reduced-form objects (fiscal multipliers and insurance losses). The fiscal multiplier of a DI policy instrument is defined as the ratio of total fiscal cost savings to the mechanical (counterfactual-behavior-held-fixed) fiscal cost savings; it measures how much the program shrinks per dollar mechanically removed, and values above 1 indicate behavioral crowd-out of DI enrollment. The paper then evaluates two Austrian reforms: (1) a 2013 increase in the Rehabilitation Stricter Assessment (RSA) age threshold from 57 to 58 (and separately to 59), which tightened eligibility for DI applicants aged 57 by requiring them to demonstrate inability to be retrained for alternative work; and (2) a 2003 reform that reduced DI benefit generosity for workers aged 30–60 as a side effect of a pension reform. Using difference-in-differences with cohorts just above and below the relevant thresholds, the paper finds that the RSA reform generated a fiscal multiplier of 2.50 (RSA to 58) and 2.05 (RSA to 59), while the benefit reduction generated a multiplier of only 1.41 (ages 57–60) and 1.36 (ages 30–56). The large gap implies that for a given mechanical cost saving, tighter eligibility rules generate 1.8 times more total fiscal savings than benefit cuts. The paper further provides empirical evidence that the insurance losses associated with stricter eligibility rules are, in all likelihood, smaller than those from benefit reductions, strengthening the dominance of eligibility tightening over benefit cuts as a DI reform instrument.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the sufficient statistics framework and what does it deliver?

The paper derives social optimality conditions for two DI policy instruments — tighter eligibility rules and lower benefits — in terms of two sufficient statistics: the fiscal multiplier (total fiscal savings / mechanical fiscal savings) and the insurance loss (marginal utility of consumption of the affected recipients). An eligibility reform that tightens the threshold θ from θ* to θ* + dθ is welfare-improving if and only if the fiscal multiplier exceeds the social value of one dollar in the hands of the marginally excluded applicant; a benefit cut from b to b − db is welfare-improving if the multiplier exceeds the social value of one dollar held by the average current DI recipient. Because the fiscal multiplier is estimable from reduced-form variation and the insurance loss gives the welfare benchmark, the framework converts the welfare question into: “Is the multiplier large enough relative to the insurance value?”

Q2. How is the fiscal multiplier decomposed, and why does this decomposition matter?

The fiscal multiplier equals 1 + B/M where B is the behavioral fiscal effect (savings from deterred applications and enrollment) and M is the mechanical fiscal effect (savings from unchanged behavior on the affected population); the multiplier exceeds 1 whenever the behavioral response amplifies the direct savings. The decomposition matters because the behavioral effect operates through marginal applicants (who apply only under lenient rules) while the mechanical effect operates through always-applicants (who apply regardless). These groups have different characteristics: in the Austrian data, marginal applicants are more likely to be employed at age 56 (73% vs. 60% for always-applicants) and more likely to be blue-collar workers with musculoskeletal impairments, while always-applicants are more likely to be on sick leave — a proxy for genuine disability. Confusing the two groups would misidentify who bears the insurance loss.

Q3. How is the mechanical fiscal effect identified when marginal and always-applicants cannot be directly observed?

The paper exploits previously-rejected DI applicants (those who filed applications between ages 50–56 and were rejected) as a proxy group for always-applicants: these individuals qualify as always-applicants by revealed preference (they applied under strict rules), and the paper shows that their DI benefit receipt and net fiscal expenditures after a simulated age-57 application are statistically indistinguishable from those of all-age-57 applicants in the whole population. The mechanical fiscal effect per always-applicant in the whole population is estimated as M = 5,585 Euro × 0.070 = 391 Euro per capita (the product of the mechanical effect among pre-57 re-applicants and the share of always-applicants in the population, πAA = 0.070). The behavioral fiscal effect is then computed as the residual: B = total fiscal savings – M = 976 − 391 = 585 Euro, yielding the multiplier of 2.50.

Q4. What are the reduced-form effects of the RSA reform on DI enrollment and employment?

A one-year increase in the RSA from age 57 to 58 reduces DI inflow by approximately 21 percentage points at age 57 (relative to a control-group mean), increases employment by roughly 15 percentage points, increases other benefit receipt (unemployment insurance and social assistance) by about 16 percentage points, and generates net fiscal cost savings of approximately 976 Euro per person per year in the two years after the reform. The pattern of employment and benefit substitution shows that the behavioral response is substantial: a large share of those deterred from DI enrollment at 57 transition to employment or other social benefits rather than remaining without any income support, which is why the fiscal multiplier of 2.50 substantially exceeds 1.

Q5. What are the effects of the 2003 DI benefit generosity reduction?

A 1-percentage-point reduction in DI benefit generosity for the ages 57–60 cohort produces a behavioral fiscal effect of 18.69 Euro per year (through reduced DI inflow and application), a mechanical fiscal effect of 45.16 Euro per year (1% of the pre-reform mean benefit expenditure of 4,516 Euro among those aged 57–60), and a total fiscal effect of 63.85 Euro — yielding a multiplier of 1.41. For the younger cohort (ages 30–56), the multiplier is 1.36, with a behavioral effect of 1.18 Euro and mechanical effect of 3.24 Euro per year. The lower multipliers for benefit cuts relative to eligibility tightening reflect the fact that benefit reductions affect all current recipients uniformly (generating large mechanical savings) rather than targeting a group with a strong behavioral response at the margin.

Q6. How does the paper compare the insurance losses of the two DI instruments?

The paper derives a sufficient condition under which the insurance loss from tighter eligibility rules is smaller than the insurance loss from benefit cuts: it requires that the per-dollar income loss borne by the marginally excluded applicant (upper-bounded by their DI benefit minus available social welfare benefits) is weakly smaller than the per-dollar income loss of current recipients (lower-bounded by the benefit reduction itself), evaluated within each income quintile. Implementing this condition empirically using the Austrian income data, the paper finds that the income losses borne by marginally excluded applicants fall short of those borne by current recipients at all income quintile comparisons — meaning tighter eligibility rules both generate higher fiscal multipliers and impose smaller insurance losses than benefit cuts, making eligibility tightening the dominant instrument when the goal is to reduce DI program costs.

Q7. What welfare conclusion follows from combining the fiscal multipliers with the insurance benchmark?

Combining the multiplier estimates with hand-to-mouth CRRA assumptions as an upper-bound calculation, the RSA increase to 58 is welfare-improving if the coefficient of relative risk aversion of affected DI recipients is below 2.8; the RSA increase to 59 is welfare-improving if risk aversion is below 2.2. The corresponding critical risk aversion level for the benefit cut (ages 57–60) is 1.1 — below the range typically estimated in the literature for low-income individuals — suggesting the benefit cut was likely welfare-reducing while the eligibility reform was likely welfare-improving. For a given dollar of mechanical budget reduction, stricter eligibility rules generate 1.8 times the total fiscal savings (= 2.50 / 1.41) relative to benefit cuts.

Q8. What is the complier analysis and what does it reveal about who is affected by each instrument?

Using the complier analysis method adapted for difference-in-differences settings, the paper estimates that the RSA-58 reform affects three types of individuals in the age-57 population: marginal applicants (πMA = 0.014, who apply only under lenient rules), always-applicants (πAA = 0.070, who apply regardless), and never-applicants (πNA = 0.916, who never apply). Marginal applicants differ from always-applicants in that they are more likely to be employed at age 56 (73% vs. 60%) and less likely to be on sick leave; they are more likely to apply with musculoskeletal impairments than with mental impairments — consistent with these workers facing the largest relaxation in disability eligibility when reaching the RSA and being on the borderline of eligibility under strict rules.

Key concepts

fiscal multiplier of a DI instrument : total fiscal cost savings divided by mechanical fiscal cost savings from that instrument; equals 1 + B/M where B is the behavioral savings (from deterred applications) and M is the mechanical savings (from unchanged behavior); values above 1 indicate behavioral crowd-out and are the policy-relevant benchmark against which insurance losses must be compared.

mechanical fiscal effect (M) : the fiscal cost savings that would accrue if DI application and enrollment behavior were held fixed at pre-reform levels; for eligibility tightening, this equals the DI benefits that would have been paid to always-applicants who are now rejected; identified using the subpopulation of previously-rejected DI applicants as a proxy for always-applicants.

behavioral fiscal effect (B) : the additional fiscal savings generated by deterred applications and enrollment that result from the reform; equals total fiscal savings minus the mechanical fiscal effect; operates through marginal applicants who adjust their application behavior in response to stricter rules or lower benefits.

always-applicants : individuals who apply for DI regardless of whether eligibility rules are strict or lenient; they bear the mechanical cost of eligibility tightening (being rejected under stricter rules); in the Austrian data, their population share at age 57 is estimated at 7.0%.

rehabilitation stricter assessment (RSA) age : the Austrian policy threshold above which DI applicants are evaluated under more lenient standards that do not require demonstration that the applicant can be retrained for alternative work; increasing the RSA age from 57 to 58 subjects the age-57 cohort to stricter evaluation criteria.

insurance loss : the welfare cost to DI recipients or excluded applicants from the income reduction caused by a DI reform; the right-hand side of the social optimality condition; the paper bounds it using income losses by income quintile rather than requiring utility function assumptions.

Destabilizing Capital Flows amid Global Inflation

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

Bengui and Coulibaly ask whether the pattern of capital flows observed during the 2021–2023 global monetary tightening cycle — whereby capital flowed from low-inflation to high-inflation countries — was a stabilizing or destabilizing force for the global economy’s adjustment to cost-push shocks. Among the G7 and a broader sample of 26 jurisdictions, those with higher average CPI inflation (October 2021–March 2023) and larger cumulative interest rate hikes ran more negative current account balances over the same period, with the slope of the cross-sectional relationship between cumulative hikes and the current account equal to −1.29 (significant at 1%) and the slope between average inflation and the current account equal to −0.99 (significant at 1%), and over 75% of the top two quartile hikers running deficits while over 75% of the bottom two quartiles ran surpluses.

Model and Methodology

The authors build a standard continuous-time two-country general equilibrium model with nominal rigidities (Calvo price-setting), internationally traded bonds, and cost-push shocks modeled as wage markup shocks that create an output-inflation trade-off. The baseline model features no home bias (equal weights on domestic and foreign goods) and two tradable goods. Extensions introduce (i) consumption home bias (parameter α ∈ [0, 1/2]) and (ii) non-tradable goods. Policy is analyzed under two regimes: (a) free capital mobility (no taxes on financial transactions) with optimal cooperative monetary policy, and (b) a managed capital flow regime in which a planner jointly optimizes both monetary policy and a tax wedge on the international bond (τ^D_t). A second-order approximation of household utility yields a loss function penalizing world and cross-country output gaps, PPI inflation differentials, and the demand imbalance term θ_t. The quantitative section replaces optimal monetary policy with standard Taylor rules (φ_π = 1.5, φ_y = 0.25) and calibrates a Home cost-push shock to generate a peak CPI inflation rate of about 7%, with an annual autocorrelation of 0.65.

Main Findings

The paper’s central theoretical result (Proposition 2, “Topsy-Turvy Capital Flows”) is that, under the Marshall-Lerner condition (trade elasticity η > 1), a free capital mobility regime channels capital into the country with the most acute inflationary pressures — the very country whose central bank is most aggressively tightening — while the constrained-efficient managed regime would channel capital in the opposite direction. The mechanism operates through the supply side: capital inflows raise domestic households’ wealth, reducing their labor supply and thereby raising real wages and firms’ marginal costs. In the presence of non-tradable goods, an additional channel operates through the real exchange rate — capital inflows appreciate the domestic real exchange rate and inflate tradable-sector firms’ marginal costs independently of labor supply. Both channels worsen the central bank’s output-inflation trade-off.

In the quantitative exercise (Taylor rule setting, home bias α = 0.25, trade elasticity χ = 3), following the calibrated inflationary cost-push shock in Home:

Under free capital mobility: Home inflation rises to 8% on impact; Home output gap reaches −8.4%; Foreign output gap reaches +2.4%; Home runs a trade deficit of 2.5% of GDP on impact; Home’s initial policy rate hike is nearly 10% while Foreign’s is less than 1%.
Under the managed capital flow regime (capital flows reversed to outflows from Home): Home inflation on impact falls to nearly 6% (a reduction of approximately 2 percentage points); Home output gap is −6.8% (improvement of about 1.5 percentage points); Foreign output gap is 0.8% (improvement of about 1.5 percentage points); Home runs a trade surplus of 0.6% of GDP; Home’s initial hike falls to approximately 8% (roughly 2 percentage points lower) while Foreign’s rises to approximately 2.5% (roughly 1.5 percentage points higher).
The managed regime delivers average welfare gains of 0.78% of current consumption (0.03% of permanent consumption). Welfare gains are increasing in the trade elasticity η: at η = 10 (consistent with Yi 2003’s bilateral trade flow estimates), gains reach approximately 0.08% of permanent consumption or 1.9% of current consumption.

Scope Conditions

The topsy-turvy result (free mobility channels capital in the wrong direction) holds conditional on the Marshall-Lerner condition (η > 1 in the baseline; equivalently, the trade elasticity χ > 1). With consumption home bias, the condition weakens to: the trade elasticity exceeds the degree of home bias (χ > 1 − 2α, which is weaker than Marshall-Lerner). When home bias is strong relative to the trade elasticity, a purchasing power effect may dominate the wealth effect, and free capital mobility may instead deliver too little capital flow toward the depressed country — the opposite inefficiency. The welfare analysis throughout assumes symmetric initial net foreign asset positions. The key insight is specific to environments in which monetary policy faces an output-inflation trade-off from cost-push shocks; it is directionally opposite to the aggregate demand externality prescription that arises in demand-shortage environments (e.g., currency unions with productivity shocks), where optimal policy instead calls for capital to flow toward the more depressed country.

In depth

Q1. What is the empirical motivation for the paper, and how is the stylized fact documented?

A1: During October 2021–March 2023, jurisdictions with higher average CPI inflation and larger cumulative policy rate hikes ran more negative current account balances. The cross-sectional slope between average inflation and the current account-to-GDP ratio is −0.99 (R² = 0.22, significant at 1%), while the slope between cumulative hikes and the current account is −1.29 (R² = 0.27, significant at 1%). Among the top two quartiles of cumulative hikers, over 75% of jurisdictions ran current account deficits, while among the bottom two quartiles over 75% ran surpluses. Data come from the BIS (inflation and policy rates) and the OECD Main Economic Indicators (quarterly current accounts), covering 26 jurisdictions excluding Argentina, Russia, and Turkey.

Q2. What is the core externality the paper identifies, and why do atomistic agents fail to internalize it?

A2: When a household in the high-inflation country borrows from abroad for consumption smoothing (as the domestic central bank tightens), it raises domestic consumption and thereby reduces labor supply through a wealth effect, pushing up real wages and firms’ marginal costs. The central bank must then tighten further to achieve the same inflation stabilization, or accept a worse inflation outcome. Because this effect operates through economy-wide wages and prices (general equilibrium), atomistic households do not internalize it when making individual borrowing decisions. The paper shows formally that a marginal increase in Home borrowing dθ_t raises welfare losses by an amount proportional to the product of the Phillips curve slope κ, the co-state variable φ^D_t (equal to the cross-country output gap differential y^D_t under optimal monetary policy), and the direct effect on cross-country marginal cost differences (1/2). When output is more depressed in Home (y^D_t < 0), additional borrowing by Home tightens the constraint and lowers welfare.

Q3. What does the optimal capital flow management targeting rule say, and what is its economic interpretation?

A3: Proposition 1 states that under jointly optimal monetary and capital flow management, the demand imbalance (relative consumption) should satisfy θ_t = 2y^D_t. This means the planner generates a demand imbalance in favor of the less depressed country, reallocating spending away from the country with the most acute inflationary pressure. This is counterintuitive from a pure output stabilization view: policy deliberately shifts demand away from the country with the most depressed output. The logic is that reducing the domestic wealth of the high-inflation country lowers real wages, reduces firms’ marginal costs, and thereby relaxes the output-inflation trade-off for that country’s central bank.

Q4. What is the “topsy-turvy” capital flows result (Proposition 2), and under what condition does it hold?

A4: Under free capital mobility, standard neoclassical consumption-smoothing motives lead capital to flow into the country with the most depressed output (the high-inflation country): the trade deficit equals [(η−1)/η]·y^D_t. Under managed capital flows, the optimal regime instead mandates a trade surplus for the most depressed country: the trade balance equals −(1/η)·y^D_t. Comparing signs, the direction of capital flows is literally reversed — hence “topsy-turvy.” The result holds whenever Assumption 1 (η > 1, the Marshall-Lerner condition in the baseline model) is satisfied, which the authors argue has compelling empirical support (trade elasticities estimated at 7–17 in the literature).

Q5. How does the presence of home bias in consumption affect the externality and the topsy-turvy result?

A5: With home bias (α < 1/2), capital inflows also appreciate the terms of trade, which lowers the relative price of imports in terms of domestic goods and reduces marginal costs for domestic tradable firms — a “purchasing power effect” that partially offsets the wealth effect. The optimal capital flow targeting rule becomes θ_t = [1 − (1−2α)/(2(1−α)η)]·2y^D_t. Under the condition that the trade elasticity exceeds the degree of home bias (χ > 1 − 2α, strictly weaker than Marshall-Lerner), the wealth effect dominates the purchasing power effect and the topsy-turvy result is preserved. Below a knife-edge curve in the (α, η) parameter space, the purchasing power effect dominates and free capital mobility results in too little rather than too much capital flowing toward the high-inflation country.

Q6. Does the externality always imply excessive capital flow volatility?

A6: No — this is a novel contribution relative to the prior literature. In the limiting case of a unit intratemporal elasticity (η → 1, the Cole-Obstfeld case), trade is balanced at all times under free capital mobility. Under managed capital flows, however, capital should flow from the most depressed to the least depressed country. This means the externality can result in too little rather than too much capital flow. The standard normative literature (e.g., Bianchi 2011) has focused on excessive capital flow volatility; the supply-side channel identified here shows that market failures can sometimes lead to insufficient external imbalances.

Q7. How does the paper’s mechanism differ from aggregate demand externalities as in Farhi and Werning (2016)?

A7: Farhi and Werning (2016) study demand-shortage environments (fixed exchange rates or zero lower bound) where constraints on monetary policy mean output is demand-constrained. Their prescription is to channel capital toward the most depressed country to stimulate demand for undersupplied goods. In Bengui and Coulibaly, monetary policy is unconstrained but faces an output-inflation trade-off from cost-push shocks. Here, the depressed output reflects the central bank’s deliberate demand contraction to fight inflation, not an inability to stimulate. The optimal response is therefore to shift spending away from the high-inflation (most depressed) country to reduce supply pressure — the opposite direction. Formally, in the demand-shortage case with unit elasticity and home bias, the optimal trade balance targeting rule is nxt = [(1−2α)/(4(1−α))]·ỹ^D_t (trade deficit for most depressed country), while in the supply pressure case it is nxt = −[α/(1−α)]·y^D_t (trade surplus for most depressed country).

Q8. What does the non-tradable goods extension add to the baseline mechanism?

A8: The baseline model (two tradable goods, no home bias) transmits the externality only through the wealth effect on labor supply: capital inflows raise consumption, reduce labor supply, and raise real wages and marginal costs. In the non-tradable goods extension, a second channel operates through the real exchange rate. Capital inflows raise demand for non-tradable goods, appreciating the domestic real exchange rate and inflating the price of the consumption basket relative to domestically produced tradable goods. This raises marginal costs for tradable-sector firms independently of any labor supply response, and is therefore unaffected by whether preferences exhibit a wealth effect on labor supply. The paper shows that the optimal policy problem in this extension is isomorphic to the baseline: the loss decomposition (equation 42) yields two additive terms proportional to the share of tradable goods (wealth effect on labor supply) and the share of non-tradable goods (wealth effect on demand for non-tradables), respectively.

Q9. What does the quantitative exercise show about cross-country policy rate dispersion?

A9: Under free capital mobility with Taylor rules, the initial policy rate hike in Home following the calibrated shock is nearly 10%, while in Foreign it is less than 1% — a cross-country dispersion of roughly 9 percentage points. Under managed capital flows, Home’s initial hike falls to approximately 8% and Foreign’s rises to approximately 2.5% — a dispersion of roughly 5.5 percentage points. The authors interpret this as evidence that free capital mobility leads high-inflation countries to tighten excessively and low-inflation countries to tighten too little, generating an inefficiently large cross-country dispersion in monetary policy.

Q10. How does the welfare gain from managed capital flows vary with the trade elasticity?

A10: Welfare gains are increasing in the elasticity of substitution between domestic and foreign goods (η). At the baseline calibration of η = 2 (trade elasticity χ = 3, near the lower bound of empirical estimates), the gain is 0.78% of current consumption (0.03% of permanent consumption). At η = 10 (consistent with Yi 2003’s estimate needed to match bilateral trade flows), the gain rises to approximately 1.9% of current consumption (0.08% of permanent consumption). The welfare gain is defined as the percentage increase in permanent consumption required by a household under free capital mobility to be as well off as under managed capital flows.

Q11. What is the role of Lemma 1 (irrelevance of capital flow regime for world variables)?

A11: Lemma 1 shows that under optimal cooperative monetary policy, the paths of world output gap and world inflation are independent of the capital flow regime (i.e., independent of the path of θ_t). This follows because the “world” block of the model can be solved independently of the “difference” block and the demand imbalance. As a result, the entire normative analysis of capital flows reduces to the behavior of cross-country difference variables (y^D_t, π^D_t, and θ_t), greatly simplifying the analysis. It also implies that switching capital flow regimes does not affect the global total of output or inflation, only its distribution across countries.

Q12. What extensions do the authors suggest would enrich the analysis without invalidating the main insight?

A12: Three extensions are noted. First, additional monetary policy constraints — discretionary (non-commitment) policy, non-cooperative policy setting, or a currency union — would introduce extra stabilization constraints and generate additional terms in the capital flow management targeting rule but would not overturn the supply-side channel. Second, alternative goods pricing specifications (local currency pricing, deviations from the law of one price) would make additional variables like cross-country consumer price differentials relevant measures of policy tightness, again adding terms to the rule. Third, the insight is argued to apply more generally in heterogeneous-agent or multi-sector closed-economy models with nominal rigidities whenever private financial decisions affect the economy’s supply side through general equilibrium price effects.

Key Concepts

Cost-push shock (wage markup shock): In the paper’s model, a cost-push shock is a positive deviation of the wage markup (µ^w_t) from its steady-state value. It shifts the New Keynesian Phillips curve, creating an output-inflation trade-off: the central bank must accept either higher inflation or a larger negative output gap. It is not a demand shock; its policy implications are directionally opposite to demand shortage shocks.

Demand imbalance (θ_t): The log ratio of Home to Foreign consumption, defined as c_t − c^*_t = θ_t in the linearized model. Under free capital mobility and symmetric initial wealth, θ_t = 0 (consumption shares are equalized). Under managed capital flows, θ_t is the instrument of capital flow policy: setting θ_t > 0 shifts spending toward Home; θ_t < 0 shifts it toward Foreign. The loss function penalizes deviations of θ_t from zero as an independent inefficiency (cross-country consumption misallocation).

Topsy-turvy capital flows: The paper’s central finding that, following a cost-push shock, the direction of capital flows prescribed by constrained-efficient policy is opposite to the direction that free capital mobility generates. Under free mobility, capital flows into the high-inflation country (trade deficit there); under managed flows, capital should flow out of the high-inflation country (trade surplus there). The term is used to describe the directional reversal, not merely excessive magnitude.

Macroeconomic externality (supply-side): The failure of atomistic agents to internalize the general equilibrium effect of their borrowing decisions on domestic firms’ marginal costs (via real wages or the real exchange rate). This is the paper’s label for the source of inefficiency. It is classified as a supply-side externality to distinguish it from aggregate demand externalities (Farhi and Werning 2016), where the operative mechanism runs through demand for specific goods rather than through factor costs.

Trade elasticity (χ): In the baseline model, χ = η (elasticity of substitution between domestic and foreign tradable goods). With home bias, χ = 2(1−α)η. The trade elasticity plays the key role in determining whether the topsy-turvy result holds: the result requires χ > 1 (Marshall-Lerner in baseline) or, with home bias, χ > 1 − 2α (weaker condition). At χ = 1 (Cole-Obstfeld case), trade is balanced under free mobility, and managed flows call for capital to move from the most to the least depressed country — implying insufficient rather than excessive capital flows under free mobility.

Purchasing power effect: In the model with home bias, a capital inflow appreciates the terms of trade (the relative price of exports over imports), which raises the purchasing power of domestic firms and lowers their marginal costs. This effect partially offsets the wealth-effect-driven rise in marginal costs. Its strength is proportional to the degree of home bias (1−2α) relative to the trade elasticity 2(1−α)η. Under the paper’s weaker-than-Marshall-Lerner condition, the wealth effect dominates the purchasing power effect.

Managed capital flow regime: A policy regime in which the government imposes taxes on international financial transactions (τ_t for Home, τ^_t for Foreign) to control the demand imbalance θ_t, subject to the targeting rule θ_t = 2y^D_t (or its home-bias-adjusted counterpart). This regime accounts for the macroeconomic externality and delivers a constrained-efficient allocation given the presence of nominal rigidities. The tax wedge τ^D_t = (τ_t − τ^_t)/2 represents the gap in returns on the international bond faced by Home versus Foreign households.

World and difference formulation: Following Engel (2011) and Groll and Monacelli (2020), the model is decomposed into “world” variables (averages: y^W_t, π^W_t) and “difference” variables (cross-country gaps: y^D_t, π^D_t). The targeting rules and Phillips curves separate additively into world and difference blocks, and Lemma 1 establishes that the capital flow regime affects only the difference block. This decomposition is the analytical device that isolates the role of capital flows.

Devaluations, Deposit Dollarization, and Household Heterogeneity

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

Ferrante and Gornemann study the aggregate and redistributive effects of currency devaluations in emerging market economies, focusing on a feature that prior open-economy HANK models had not jointly incorporated: households hold dollar-denominated deposits that are disproportionately concentrated among wealthier agents, and these deposits sit on the liability side of leveraged, agency-constrained banks. The paper asks how this combination of deposit dollarization and household wealth heterogeneity shapes the macroeconomic and distributional consequences of a currency depreciation, and what it implies for the optimal degree of exchange-rate smoothing by the central bank.

Data and Empirical Motivation

The model is calibrated to match cross-sectional micro-data from the 2013 Uruguayan Household Financial Survey, which records the currency denomination of household assets and liabilities. As documented by Drenik et al. [2018] and confirmed by the authors for Uruguay, the top quintile of the wealth distribution holds close to 70% of liquid savings in dollars, while households with zero or negative net wealth have essentially no direct foreign-currency exposure. The baseline calibration targets a deposit dollarization rate of 40% of aggregate bank deposits, in line with the cross-country average reported for Latin America. The spread between bank lending and deposit rates is calibrated at 8% annualized for household loans (consistent with Uruguayan bank data over the prior 15 years) and 2% for capital returns, implying a bank leverage ratio of approximately 6.

Model

The framework is a small open economy New Keynesian model with two non-standard elements layered on a Bewley-Huggett-Aiyagari incomplete-markets household sector. First, households face idiosyncratic labor productivity risk and a borrowing constraint, generating a non-degenerate wealth distribution in which, at the calibrated steady state, approximately 8% of households are constrained borrowers, 22% are unconstrained borrowers, 27% hold zero liquid wealth and behave hand-to-mouth (HtM), 52% are net savers, and 1% are capitalists. Second, financial intermediaries face a Gertler-Karadi [2011] agency problem that generates an endogenous, time-varying spread between lending and deposit rates. Households can save in local- or foreign-currency bank deposits and in foreign bonds, but can only borrow through domestic banks. The currency composition of household portfolios, which is a linear function of household wealth in the baseline, maps through market clearing into the banks’ currency mismatch, so that a wealthier-household preference for dollar deposits directly determines the bank’s foreign-currency liability share.

Main Findings with Quantitative Magnitudes

The paper’s central experiment is a 100 basis-point annualized increase in the foreign interest rate with persistence 0.85, which induces a currency depreciation.

Aggregate amplification: Combining a HANK household sector with leverage-constrained banks exposed to currency mismatch causes aggregate consumption to drop approximately twice as much as in a representative-agent New Keynesian (RANK) model with constrained banks, and output to decline more than 1% — roughly 30% larger than the 0.75% decline in the RANK model with financial frictions. In contrast, absent banking frictions, a bank-less HANK model would generate an output expansion because the standard expenditure switching channel dominates.
Channels: The paper decomposes the consumption decline into (a) a labor income channel — lower hours and wages caused by the financial accelerator contraction account for approximately two-thirds of the aggregate consumption decline — and (b) a borrowing rate channel — the endogenous rise in household lending spreads accounts for approximately one-third. In a counterfactual model in which the spread on household loans is held fixed, the decline in consumption and output is approximately 50% smaller than in the baseline, confirming that the borrowing rate channel and its general-equilibrium feedback onto wages and asset prices are responsible for more than half of the baseline output decline.
Distributional effects: Within the baseline model, unconstrained borrowers see their consumption fall on average by more than 3.5% on impact; constrained borrowers’ consumption falls by more than 5% in the second period as interest payments jump. Zero-wealth HtM agents cut consumption roughly one-for-one with the more-than-2% decline in real labor income. Wealthier savers and capitalists are partially insulated through their dollar holdings, which gain real value during the depreciation.
Portfolio composition and deposit dollarization: When the deposit dollarization rate is raised from the baseline 40% to 80% (to match high-dollarization countries such as Uruguay at the extreme), investment declines approximately 12% (versus 6% in the baseline) and aggregate consumption falls approximately 1.7% (versus 1% in the baseline), with the output decline more than twice as large as in the baseline. Wealthier households’ consumption path is actually higher in the high-dollarization calibration because of larger windfall gains on their dollar portfolios, while poorer households bear the amplified downturn through stronger labor income and borrowing rate channels. This produces a novel distributional result: stronger currency hedging by richer households deepens the aggregate recession and worsens outcomes for poorer agents.
Monetary policy: In the baseline 40% dollarization calibration, reacting to exchange rate changes by raising domestic interest rates is welfare-detrimental for most households: the gain from partially stabilizing banks’ balance sheets is more than offset by the contractionary effect of higher rates on aggregate demand and spreads. A modest response (κ_e ≈ 0.04 in the ex-ante welfare experiment) is preferred, conditional on aggregate dynamics. When dollarization is 80%, a small degree of exchange rate leaning (κ_e = 0.5) can improve welfare for most agents, as the benefit from protecting banks’ balance sheets becomes larger relative to the cost of tighter monetary conditions.

In depth

Q1. What three stylized facts about liability dollarization motivate the model, and how does the model’s structure capture each?

A1: The three facts are: (i) banks and firms borrow in foreign currency; (ii) foreign-currency bank debt is matched by dollar-denominated deposits from domestic households; (iii) those deposits are held predominantly by wealthier households. The model captures (i) and (ii) by having the bank hold a currency mismatch on its balance sheet — local-currency loans on the asset side, foreign-currency deposits on the liability side. Fact (iii) is captured by assuming a linear portfolio rule in which household dollar deposit share is an increasing function of wealth, calibrated to the slope observed in Uruguayan micro-data, with borrowers restricted to local-currency debt.

Q2. Why does a bank-less HANK open-economy model produce an output expansion rather than a contraction following a foreign interest rate shock in the calibration used?

A2: Without banking frictions, the expenditure switching channel dominates. A rise in the foreign interest rate depreciates the real exchange rate by roughly 1%, making domestic goods cheaper and raising exports by approximately 2%. In the bank-less HANK, this export boost causes hours and real labor income to increase, and high-MPC households (HtM and constrained borrowers) raise consumption. There is no financial accelerator operating through the bank’s balance sheet to offset this stimulus, so output expands rather than contracts.

Q3. Through what exact mechanism does bank currency mismatch transform an exchange rate depreciation into a financial accelerator event?

A3: A weaker domestic currency raises the real cost of repaying foreign-currency deposits (R_Dt jumps on impact), directly eroding bank net worth (N_t). As net worth falls and leverage rises, the bank’s incentive constraint tightens, requiring spreads on both capital loans and household loans to increase jointly (per equation 21, the ratio of spreads moves one-for-one with the ratio of diversion parameters). Lower asset prices further reduce the return on capital, feeding back into net worth in the standard Gertler-Karadi financial accelerator loop. In the RANK with banks benchmark, investment declines approximately 6% compared to only 1% in the frictionless RANK.

Q4. What is the borrowing rate channel, and how is it distinct from the balance-sheet exposure channel studied in De Ferra et al. [2020]?

A4: The borrowing rate channel operates through the endogenous widening of bank lending spreads following a net worth erosion: when banks’ leverage constraint binds more tightly, both the spread on firm capital and the spread on household loans rise simultaneously (equation 21). This forces even households who borrow only in local currency — and thus have no direct exchange-rate exposure on their liabilities — to face sharply higher borrowing costs, causing their consumption to fall steeply. De Ferra et al. [2020] study a different channel in which households borrow in foreign currency and suffer a direct balance-sheet loss from depreciation; the borrowing rate channel in this paper is distinct because it operates through financial intermediary frictions rather than through direct currency exposure of household debt.

Q5. How much of the aggregate consumption decline is attributable to the borrowing rate channel versus the labor income channel, and how do the authors establish these shares?

A5: The decomposition exercise (Figure 6) simulates each household’s response to a single price path at a time while holding all other prices at steady state. The labor income channel — the decline in real wages and hours caused by the contraction in output — accounts for approximately two-thirds of the aggregate consumption decline. The borrowing rate channel accounts for approximately one-third. Separately, a counterfactual model in which the household loan spread is held fixed produces consumption and output declines roughly 50% smaller than the baseline, showing that the borrowing rate channel and its second-round effects on wages and asset prices together account for more than half of the output decline in general equilibrium.

Q6. How does the distribution of dollar deposits across the wealth distribution affect the severity of the downturn, and what is the novel redistribution result?

A6: Through market clearing for local-currency deposits (equation 44), a larger household demand for dollar deposits directly raises the bank’s foreign-currency liability share (x^D_bt), magnifying the bank’s currency mismatch. Raising the deposit dollarization rate from 40% to 80% causes bank net worth to decline twice as much as in the baseline, investment to fall roughly 12% versus 6%, and aggregate consumption to fall roughly 1.7% versus 1%, with output declining more than twice as much. The novel distributional result is that wealthier savers and capitalists are actually better off in the high-dollarization scenario because their windfall dollar gains are larger, while poorer households suffer a more severe recession through the labor income and borrowing rate channels. Hence, stronger currency hedging by the rich deepens the aggregate recession and worsens distributional outcomes for the poor.

Q7. What happens when borrowers are assumed to hold foreign-currency debt rather than local-currency debt, as in De Ferra et al. [2020]?

A7: In this alternative calibration, borrowers face a direct balance-sheet loss from depreciation, causing constrained borrowers’ consumption to drop more steeply on impact. However, since household loans represent only approximately 5% of annual GDP in the baseline, the boost to bank net worth from having dollar-denominated loan assets is modest compared to the reduction in the dollar deposit liability. As a result, the path for investment is very similar to the baseline, while on impact consumption drops about 20% more and output declines about 10% more than in the baseline model.

Q8. What welfare implications arise from removing dollar deposits entirely from savers’ portfolios?

A8: In a calibration where households hold only local-currency assets (with banks’ currency mismatch maintained through external dollar borrowing), savers lose their windfall dollar gains during depreciation. The consumption of savers drops about 25% more than in the baseline on impact, and capitalists experience even larger changes. Because of general equilibrium feedback through wages and prices, poorer households also cut consumption more, causing aggregate consumption to fall approximately 20% more than in the baseline and output to decline approximately 5% more on impact.

Q9. Under what dollarization conditions does exchange rate stabilization through monetary tightening improve welfare, and why?

A9: Under the baseline 40% dollarization, raising domestic interest rates in response to depreciation is welfare-detrimental for most households because higher rates depress asset prices, tighten the bank’s leverage constraint, worsen the borrowing rate channel and the labor income channel for low-net-worth agents, more than offsetting the benefit from partially stabilizing the bank’s balance sheet. Only a very modest response (κ_e ≈ 0.04) is preferred. When deposit dollarization is 80%, the benefit from protecting the bank’s balance sheet is proportionally larger; a moderate reaction (κ_e = 0.5) can improve welfare for most households, though further tightening (κ_e = 5) causes bank net worth to fall more than 20% and leads to a deeper recession, reversing the gains.

Q10. How does the quarterly average MPC in the model compare to external estimates, and why is the MPC distribution central to the paper’s mechanism?

A10: The quarterly average MPC in steady state is approximately 27%, which implies an annual MPC of approximately 71%, consistent with Hong [2020b]’s estimates for Peru. The MPC distribution is central because the amplification mechanisms — both the borrowing rate channel and the labor income channel — work by hitting high-MPC agents (HtM households and constrained borrowers) hardest. Without a sufficiently high mass of high-MPC agents, changes in spreads and labor income would have muted aggregate consumption effects. The presence of approximately 27% of households with zero liquid wealth at the borrowing spread is itself endogenously generated by the bank’s agency problem, which creates a wedge between saving and borrowing rates.

Q11. How does the HANK model without banks compare to the RANK model without banks in transmitting the foreign interest rate shock?

A11: Both HANK-without-banks and RANK-without-banks generate output expansions through the expenditure switching channel. However, in the bank-less HANK, aggregate consumption declines only half as much as in the frictionless RANK because high-MPC households amplify the positive real income effect from rising labor income. Some household groups (HtM agents and constrained borrowers) actually increase consumption on impact due to higher real labor income, the Fisher channel reducing the real value of domestic-currency debt, and portfolio gains for savers holding dollar assets.

Q12. What role does the monetary policy Taylor rule play during the baseline devaluation, and how does it interact with the financial accelerator?

A12: The standard Taylor rule (coefficient 1.5 on domestic inflation) causes the central bank to raise rates in response to the CPI inflation spike accompanying the depreciation. Higher domestic rates compress the real exchange rate depreciation and reduce the boost to exports, but also directly increase banks’ funding costs, contributing to the financial accelerator by compressing the return on capital. This interaction means that the baseline monetary policy passively amplifies the banking-sector contraction relative to a model with no monetary response.

Key Concepts

Deposit dollarization: The share of domestic bank deposits denominated in foreign currency, held by domestic households. In the paper’s calibration this is set at 40% of aggregate bank deposits (baseline) or 80% (high-dollarization alternative), reflecting the empirical range across Latin American countries. It determines the bank’s foreign-currency liability share and thus the severity of currency mismatch.

Currency mismatch (banks): The gap between the currency denomination of a bank’s assets (local-currency loans to households and firms) and its liabilities (foreign-currency deposits from households). In the model, when the domestic currency depreciates the real cost of dollar deposits rises, directly eroding bank net worth without any offsetting appreciation of loan assets.

Borrowing rate channel: The mechanism by which a decline in bank net worth, caused by currency mismatch losses, tightens the bank’s incentive constraint and forces up the spread on household loans. This raises borrowing costs for households who have no direct foreign-currency exposure on their balance sheets, causing high-MPC borrowers to cut consumption sharply and thereby depressing aggregate demand and wages. This channel is distinct from the direct balance-sheet channel studied in De Ferra et al. [2020].

Labor income channel (in an open economy with banking frictions): The mechanism by which the financial accelerator — reduced credit supply and lower capital demand following bank net worth erosion — depresses output, hours, and wages, causing a decline in real labor income that hits high-MPC workers regardless of their asset-portfolio currency composition. Accounts for approximately two-thirds of the aggregate consumption decline in the baseline experiment.

Hand-to-mouth (HtM) agents: In this paper’s setting, HtM behavior is not a permanent household state but arises endogenously for households who hold zero liquid wealth because the bank’s endogenous lending spread makes both saving and borrowing suboptimal for them in a given period. Their consumption moves approximately one-for-one with current labor income, making them a key amplifier of real income fluctuations.

Financial accelerator (with currency mismatch): The Gertler-Karadi [2011] mechanism as augmented by exchange-rate exposure: a currency depreciation erodes bank net worth through the dollar deposit liability, tightening the leverage constraint, raising spreads on capital and household loans simultaneously, lowering the price of capital, further reducing net worth, and feeding back to reduce credit supply. The currency mismatch channel and the asset-price channel interact to amplify the initial shock.

Portfolio dollarization rule: The assumption that each household’s share of savings held in foreign-currency deposits is a linear function of net wealth (x_i = λ_bar + λ·b_i, with λ > 0 and x_i = 0 for borrowers). This rule is calibrated to match the wealth-gradient of dollar holdings in the 2013 Uruguayan Household Financial Survey, and through market clearing it pins down the aggregate bank deposit dollarization rate and the distributional exposure of households to exchange rate shocks.

Exchange rate stabilization trade-off: The central bank’s choice of how much to raise domestic interest rates in response to a depreciation (parameterized by κ_e in the augmented Taylor rule). A higher κ_e reduces the bank’s currency mismatch loss but simultaneously depresses asset prices and raises borrowing costs, potentially worsening the financial accelerator. The paper shows the net welfare effect depends critically on the level of deposit dollarization: at 40% dollarization aggressive leaning is harmful for most agents; at 80% dollarization a moderate response (κ_e = 0.5) can be welfare improving.

Digital Distractions with Peer Influence

Mon, 01 Jan 0001 00:00:00 +0000

This paper estimates the causal effects of mobile app usage on college students’ academic performance, physical health, and labor market outcomes, while separately identifying behavioral (endogenous) and contextual (exogenous) peer effects in app usage — the first study to do so within a unified empirical framework. The analysis draws on administrative data for three freshman cohorts (2018–2020) at a mid-tier Chinese university, linked to individual-level mobile phone usage records from a major telecommunications carrier covering 6,430 students over four years (excluding COVID semester). High-frequency GPS data, hourly app usage records for the 2020 cohort, and two waves of university surveys supplement the main dataset.

The identification strategy addresses three challenges: endogeneity of own app usage, endogeneity of peer group formation, and the reflection problem in peer effects. For own usage, two instrumental variables are used: (1) a shift-share instrument interacting the September 2020 launch of the blockbuster game Yuanshen with students’ pre-college app usage intensity; and (2) China’s October 2019 minors’ game restriction policy (prohibiting under-18s from playing online games 10 p.m.–8 a.m. and capping weekday gaming at 90 minutes/day) interacted with the evolving number of underage pre-college friends. For peer effects, the university’s random dormitory assignment within gender-class units provides exogenous peer variation; behavioral peer effects are further isolated using the minors’ restriction policy interacted with roommates’ pre-college underage friend networks, an instrument that affects roommates but not the focal student. Contextual peer effects are recovered by subtracting the estimated behavioral component from reduced-form estimates.

The main findings are as follows. First, app usage is contagious: a one standard deviation (s.d.) increase in roommates’ in-college total app usage raises a student’s own usage by 5.8% (IV). Behavioral peer effects dominate: contextual peer effects are small and statistically insignificant. Second, own app usage severely harms academic performance: a one s.d. increase in total app usage reduces GPA for required courses by 36.2% of a within-cohort-major s.d. (IV), and a one s.d. increase in game app usage alone reduces GPA by 56.6% of a within-cohort-major s.d. The direct disruption effect of roommates’ app usage reduces GPA by a further 20.6% of a within-cohort-major s.d.; combining the indirect channel (behavioral contagion), the total roommate effect reaches 22.7% of a within-cohort-major s.d., more than 60% of the own-usage effect. Third, the effect on physical education scores is roughly four times larger than on required-course GPA: a one s.d. increase in own app usage reduces PE scores by 2.74 points, while roommates’ app usage has no direct effect on PE. Fourth, a one s.d. increase in own in-college app usage reduces initial wages upon graduation by 2.3% (12.1% of within-cohort-major wage s.d.); a one s.d. increase in roommates’ usage reduces wages by 0.9% directly, with a total effect (including the contagion channel) of approximately 1.0% (5.3% of within-cohort-major s.d.). Controlling for cumulative GPA reduces the gaming-to-wage coefficient by roughly one-third, indicating that academic performance is an important but partial mediator.

A back-of-the-envelope policy simulation extending the minors’ gaming cap (3 hours/week) to college students — binding for 34.3% of student-month observations — projects an average wage increase of 0.9% at graduation, approximately half the wage premium from one additional year of work experience in developing countries.

Mechanism evidence from GPS data shows that Yuanshen’s launch caused students to arrive at study halls 18.2 minutes later and leave 23.4 minutes earlier per day. High-frequency sleep data show that a one s.d. increase in nighttime app usage reduces sleep duration by approximately 30 minutes and raises the probability of sleeping late by 34 percentage points. Survey evidence indicates that heavy app users recognize the addictive nature of gaming, pointing to self-control problems rather than lack of awareness.

The scope conditions are: single mid-tier Chinese university; 2018–2020 cohorts; outcomes through initial job placement only; peer group restricted to dormitory roommates; findings rely on IV exclusion restrictions conditional on student and time fixed effects.

Q: What is the core research question? A: The paper asks how individual and peer mobile app usage affect college students’ academic performance, physical health, and early labor market outcomes, and it separately identifies the behavioral (endogenous) versus contextual (exogenous) components of peer influence in app usage. This is claimed as the first study to disentangle these two types of peer effects within a unified empirical framework.

Q: What data does the paper use? A: Administrative records for 7,479 undergraduates across three freshman cohorts (2018–2020) at a medium-sized mid-tier Chinese university are linked to monthly mobile app usage records from a telecommunications provider covering 75% of the provincial population; 6,430 students are matched. The dataset also includes GPS location data at 5-minute intervals, hourly app usage for the 2020 cohort (used to infer sleep), and two waves of voluntary annual surveys with 1,798 respondents (24% response rate). Labor market outcomes — employment status, wages, post-graduate admissions — are available for the 2018 and 2019 cohorts.

Q: How does the paper address the endogeneity of own app usage? A: Two sets of instruments are used. The first interacts the September 2020 launch of Yuanshen (the most popular game in China, with over 13 million Chinese users by 2021, the majority under age 25) with students’ pre-college app usage, forming a shift-share instrument under the assumption that the game launch is orthogonal to unobserved GPA determinants conditional on student fixed effects. The second interacts China’s October 2019 minors’ game restriction policy with the evolving count of a student’s underage pre-college friends; event studies confirm no pre-trends and a sharp, transitory drop in app usage post-policy that dissipates as friends age out of the restricted group.

Q: How does the paper solve the reflection problem and separate behavioral from contextual peer effects? A: Three-step procedure: (1) random dormitory assignment within gender-class units yields reduced-form peer effect estimates using roommates’ pre-college app usage as the exogenous peer shifter; (2) behavioral peer effects are isolated via an IV using the minors’ restriction policy interacted with roommates’ (not the focal student’s) underage pre-college friend networks — an instrument that shifts roommates’ app usage but is orthogonal to the focal student’s outcomes; (3) contextual peer effects are recovered as the residual from subtracting the estimated behavioral effect from the reduced-form estimate.

Q: How large and significant are the behavioral versus contextual peer effects in app usage? A: A one s.d. increase in roommates’ in-college total app usage raises own usage by 5.8% (IV estimate, significant). For game apps alone the behavioral spillover is 10.7%, and for games plus video it is 6.5%. Contextual peer effects (identified from roommates’ pre-college characteristics) are much smaller and statistically insignificant, indicating that peer influence operates primarily through the direct imitation of peers’ actions rather than their background traits.

Q: What is the effect of own app usage on GPA? A: The IV estimate shows a one s.d. increase in total in-college app usage reduces GPA for required courses by 0.716 points, equivalent to 36.2% of a within-cohort-major GPA s.d. (significant at 1%). For game apps alone, a one s.d. increase reduces GPA by 1.119 points, or 56.6% of a within-cohort-major s.d. OLS estimates are biased toward zero, likely because negative health shocks reduce both GPA and app usage simultaneously.

Q: How large is the total peer effect of roommates’ app usage on a student’s GPA? A: Roommates’ app usage directly lowers GPA by 0.408 points (20.6% of within-cohort-major s.d.) through disruption of the dormitory study environment or crowding out of group study. The behavioral contagion channel (5.8% increase in own usage per s.d. of roommates’ usage) adds an additional 0.042 points, bringing the total effect to approximately 0.450 points, or 22.7% of a within-cohort-major s.d. — over 60% of the own-usage effect.

Q: What is the effect on physical education (PE) scores, and why do roommates’ app usage not matter there? A: A one s.d. increase in own total app usage reduces PE scores by 2.74 points (IV), approximately four times the magnitude of the effect on required-course GPA, consistent with health literature on excessive screen time. Roommates’ app usage has no statistically significant direct effect on PE, which the authors attribute to the irrelevance of dormitory noise and study disruptions for outdoor physical activity.

Q: What are the effects of app usage on wages at graduation? A: Doubling total app usage during college reduces initial wages by approximately 2% (IV). A one s.d. increase in own usage reduces wages by 2.3%, or 12.1% of a within-cohort-major wage s.d. A one s.d. increase in roommates’ usage directly reduces wages by 0.9% (4.8% of within-cohort-major s.d.); including the behavioral contagion channel, the total roommate effect is approximately 1.0% (5.3% of within-cohort-major s.d.). Controlling for cumulative GPA reduces the game-usage-to-wage coefficient by about one-third, implying GPA is a partial but not complete mediator.

Q: What does the policy simulation of the gaming cap say? A: Extending the minors’ game restriction (3 hours/week cap) to college students would bind for 34.3% of student-month observations, reducing average monthly gaming from 12.1 hours to 8 hours (a one-third decrease). Incorporating the behavioral peer multiplier for gaming (0.078), average gaming further converges to approximately 7.65 hours in steady state. The implied wage gain at graduation is 0.9%, approximately half the wage premium from one additional year of work experience in developing countries (Lagakos et al., 2019 estimate).

Q: What does the GPS evidence show about time allocation? A: Following Yuanshen’s launch, the average student arrives at the study hall 18.2 minutes later and returns to the dormitory 23.4 minutes earlier per day. The minors’ restriction reverses this: students with the average number of minor friends arrive at study halls 17.4 minutes earlier and return to the dorm 19.8 minutes later. Both game shocks also shift tardiness and absence rates for major-required courses in the expected directions, and the effects intensify over time with Yuanshen’s growing popularity.

Q: What do the sleep data show? A: A one s.d. increase in nighttime app usage (9 p.m.–3 a.m.) is associated with roughly 30 minutes less sleep (7% of the mean), a 34 percentage point higher probability of sleeping late, and a 4.5 percentage point higher probability of waking up late. Daytime app usage (8 a.m.–9 p.m.) is also associated with 7.2 fewer minutes of sleep (1.8% of mean) and a 3.7 percentage point higher probability of late wake-up. These results are descriptive (from the 2020 cohort hourly data) rather than IV-based.

Q: What does the survey evidence show about mechanisms and self-awareness? A: Heavier app users report worse physical health and higher stress, are less likely to have obtained professional certifications by graduation, submit fewer job applications, and express lower satisfaction with job offers. Notably, heavier users are more likely to acknowledge the addictive nature of apps and games, suggesting a self-control problem rather than informational deficiency. They also report better relationships with roommates and greater likelihood of following roommates’ advice on post-graduation choices, a potential direct channel for peer labor market effects.

Q: How representative is the sample, and what are the key scope conditions? A: The university is a mid-tier institution in southern China with students predominantly from the 30th–80th CEE score percentile among provincial college-admitted applicants; it is less female (42% vs. 53% nationally) and more rural (40% vs. 27% nationally). Survey respondents oversample less advantaged backgrounds and are re-weighted. Findings pertain to dormitory roommates as the peer group; all labor market outcomes are initial wages upon graduation; the sample covers 2018–2021 with COVID semester excluded. The peer effects estimates rest on random dormitory assignment, which the authors verify by showing no within-dorm correlation in pre-college characteristics.

Behavioral (endogenous) peer effects: The mechanism by which a peer’s actual behavior — here, contemporaneous app usage — directly influences a focal individual’s own behavior. In this paper, identified via IV using the minors’ game restriction policy interacted with roommates’ underage pre-college friend networks, which shifts roommates’ usage but not the focal student’s characteristics.

Contextual (exogenous) peer effects: The influence of peers’ pre-determined background characteristics (e.g., pre-college app usage, reflecting motivation, study habits, attitudes toward academics) on a focal individual’s outcomes, independent of peers’ actual in-college behavior. Recovered as the residual after subtracting estimated behavioral peer effects from reduced-form estimates; found to be small and insignificant in this setting.

Shift-share instrument (Yuanshen): A quasi-experimental instrument constructed by interacting the mid-sample launch date of the blockbuster game Yuanshen (September 2020) with students’ pre-college app usage intensity, under the assumption that pre-college usage predicts differential susceptibility to the shock while the launch itself is orthogonal to the university’s academic environment.

Minors’ game restriction policy: China’s October 2019 policy prohibiting individuals under 18 from playing online games between 10 p.m. and 8 a.m. and capping weekday gaming at 90 minutes per day (tightened to 3 hours/week in September 2021). Used both as an instrument for own app usage (via underage pre-college friends) and as an instrument for roommates’ usage (via roommates’ underage friends) to isolate behavioral peer effects.

Reflection problem: The identification challenge first articulated by Manski (1993) arising because an individual’s behavior both affects and is affected by peers simultaneously, making it impossible to separately identify the direction of influence from observational data without exogenous variation in peer behavior.

Source text origin: The paper’s own data provenance category distinguishing whether summaries are based on full working paper text (pdf or oa-html) versus abstract only — a distinction the paper itself does not use but that is relevant to the review pipeline running this analysis.

Within-cohort-major GPA standard deviation: The unit used to scale all GPA effect sizes, defined as the standard deviation of GPA within students of the same graduation cohort and declared major. This normalization accounts for systematic differences in grading across fields and years, making effect magnitudes comparable across specifications.

Disincentive effects of unemployment insurance benefits

Mon, 01 Jan 0001 00:00:00 +0000

This paper isolates the disincentive effects of pandemic unemployment insurance (UI) benefits on employment recovery, separating them from the simultaneously operating stimulative (demand) effects that previous studies conflate. The authors study the largest UI expansion in U.S. history — the CARES Act of March 2020 — which introduced three simultaneous provisions: a $600 weekly income supplement (FPUC) through end of July 2020, a 13-week extension of maximum benefit duration (PEUC), and expanded eligibility to workers previously ineligible for UI (PUA), together raising the median replacement rate to 145% and more than doubling the number of UI recipients.

The empirical strategy uses high-frequency establishment-level data from Homebase (HB), a scheduling and payroll provider covering approximately 140,000 small U.S. businesses — predominantly restaurants and retailers — matched to Yelp price-tier data and Safegraph foot-traffic and spending data. The final estimation sample is 4,595 businesses within 1,195 local-industry cells, observed at weekly frequency from January 2019 to December 2020.

The identification rests on comparing employment recovery of low-wage versus high-wage businesses within the same narrow local labor market (four-digit zip code), industry (two-digit NAICS), and price tier. Because neighboring businesses largely share the local demand stimulus from UI, differencing within local-industry cells removes common demand effects. The key variation is the expiration of the $600 supplement, which differentially compresses the replacement-rate gap between low- and high-wage businesses depending on local average wages — labor markets where the gap falls more sharply are the treated group.

The main empirical finding is that a 100 percentage point decline in the replacement rate gap is associated with a 5.7 percentage point rise in low-wage business employment recovery relative to high-wage business employment recovery at 12 weeks after the $600 expiration. For the average labor market, the expiration of the $600 supplement decreased the replacement rate gap by 46 percentage points, implying a 2.6 percentage point closing of the low-versus-high-wage employment gap within 12 weeks. Importantly, hours per employee and hourly wages grew faster in low-wage businesses over the same period, consistent with a labor supply rather than a demand mechanism. When the comparison is conducted at the U.S. state level rather than within local-industry cells — as in Finamor and Scott (2021) — the effect disappears and reverses sign, illustrating how local demand effects obscure disincentive effects at broader geographic aggregations.

To quantify the aggregate employment impact, the authors build and calibrate a McCall-style labor search model with heterogeneous firm wages, a UI-eligible and non-UI unemployed pool, and equilibrium reservation wages. The model is extended to include a probability (calibrated at 16.5%) that workers lose UI eligibility upon refusing a job offer, which reconciles the model with the empirical estimates; without this feature the baseline model substantially overstates the differential employment effect of the $600 expiration.

The full model-implied aggregate employment loss from all CARES Act UI provisions combined is 3.4 percentage points on average between April and December 2020, representing approximately 20% of the average employment shortfall in the Leisure and Hospitality sector over that period. When each provision is implemented in isolation, the effects are modest ($600 supplement: 0.2 pp; extended duration: 0.2 pp; expanded eligibility: 1.0 pp), but their interaction generates the large combined effect. Expanded eligibility is identified as the most disruptive provision, particularly for low-wage businesses, because it depletes the pool of non-UI unemployed who are the primary source of hires for these firms. The unemployment duration elasticities implied by the model are modest and in line with the low-to-middle range of pre-pandemic estimates.

The paper’s scope is restricted to the disincentive channel and deliberately excludes the stimulative effects of UI; it studies small, in-person service sector businesses and the April–December 2020 recovery period only.

Q: What is the core identification challenge this paper addresses? A: Prior empirical studies find only modest net effects of pandemic UI on employment, but it is unclear whether this reflects small disincentive effects or the near-cancellation of two opposing forces — UI suppressing labor supply while simultaneously stimulating local consumer demand. Identifying the disincentive effect alone requires a design that neutralizes the demand channel. The authors accomplish this by comparing low-wage and high-wage businesses within the same narrow local market, industry, and price tier, so that common local demand shifts from UI are differenced out.

Q: What data does the empirical analysis use, and how is the sample constructed? A: The primary data source is Homebase, covering approximately 140,000 small U.S. businesses with daily employment, hourly wages, and hours worked. The estimation sample is restricted to 4,595 businesses present throughout 2019, matched to Yelp price-tier classification and Safegraph weekly foot traffic and credit-card spending. Businesses are grouped into 1,195 local-industry cells defined by four-digit zip code, two-digit NAICS industry, and Yelp price tier (inexpensive vs. expensive). Within each cell, businesses are classified as low-wage or high-wage, with high-wage businesses paying on average $1.80 per hour more — about 8% above the average hourly wage of $10.90.

Q: How is the replacement rate defined in the empirical framework? A: The business-specific replacement rate is the ratio of average UI receipts (state benefit plus the pandemic supplement, converted to hourly units) to the pre-pandemic average hourly wage of that business. Because the supplement is uniform across workers, businesses with lower pre-pandemic wages face higher replacement rates; the replacement rate gap between low- and high-wage businesses within a local market is therefore a function of both state benefit levels and the local wage dispersion.

Q: What does the event-study analysis around the $600 expiration show? A: The event study exploits cross-labor-market variation in how much the replacement rate gap between low- and high-wage businesses declined when the $600 FPUC supplement expired at end of July 2020. Labor markets with a larger decline in the gap see faster relative recovery in low-wage business employment after expiration. A 100 percentage point decline in the replacement rate gap is associated with a 5.7 percentage point rise in the low-versus-high-wage employment recovery gap at 12 weeks post-expiration. For the average labor market, the $600 expiration reduced the replacement rate gap by 46 percentage points, implying a 2.6 percentage point narrowing of the employment recovery gap.

Q: Why does the estimated effect disappear when broader geographic aggregations are used? A: When businesses are compared within U.S. state borders rather than within local-industry cells, the estimated coefficient on the replacement rate gap turns positive and statistically insignificant. This occurs because at the state level, low-wage areas benefit disproportionately from the purchasing power increase that generous UI provides to local unemployed workers, so demand effects swamp and reverse the supply-side disincentive. This finding explains why Finamor and Scott (2021), using Homebase data with state fixed effects, find no negative association between replacement rates and labor market re-entry.

Q: What evidence supports a labor supply rather than demand interpretation of the differential recovery? A: During the period of the $600 supplement, hours per employee and hourly wages grew faster in low-wage businesses than in high-wage businesses, even as low-wage businesses lagged in employment levels. If the differential recovery reflected demand deficiencies at low-wage businesses, hours per employee and wages should have grown faster at high-wage businesses instead. The observed pattern is consistent with labor supply shortfalls at low-wage firms.

Q: What is the structure of the quantitative labor search model? A: The model features a unit measure of workers and a fixed measure of firms, each posting a constant idiosyncratic wage drawn from an exogenous distribution. Unemployed workers receive job offers at a rate determined by labor market tightness and accept offers above their reservation wage. Reservation wages are equilibrium objects because UI benefits depend on the worker’s previous wage. The unemployed are split into UI-eligible and non-UI pools; the non-UI pool accepts jobs from lower in the wage distribution and is the primary supply source for low-wage firms. The model is calibrated to pre-pandemic U.S. service sector averages, with a pre-pandemic UI replacement rate of 0.51, a UI recipiency probability of 14%, and a non-UI replacement rate of 0.15.

Q: Why does the baseline model overstate the empirical effect, and how is this reconciled? A: The baseline model dramatically overstates the differential employment impact of the $600 expiration because the CARES Act’s expanded eligibility (modeled as a rise in the recipiency probability from 14% to 70%) nearly empties the non-UI unemployed pool, which is the dominant labor supply source for low-wage firms. In the data, the share of unemployed receiving UI nearly tripled for in-person leisure and hospitality workers, but not to the degree that the model’s implied employment collapse would require. The model is reconciled by introducing a 16.5% probability that a worker loses UI eligibility upon refusing a suitable job offer — consistent with UI law — which reduces the effective outside option and raises acceptance rates for low-wage firms.

Q: What are the aggregate employment losses implied by the model? A: When all three CARES Act provisions are implemented jointly, the model estimates that the disincentive effects held back aggregate employment recovery by 3.4 percentage points on average between April and December 2020 — approximately 20% of the average employment shortfall in the Leisure and Hospitality sector. Implemented in isolation, each provision generates only modest losses: the $600 supplement alone accounts for 0.2 percentage points, extended duration for 0.2 percentage points, and expanded eligibility for 1.0 percentage points. The large combined effect arises from the interaction of all three provisions, not from any single one.

Q: What are the conditional (interaction) effects of each provision when the other two are in place? A: Conditional on the other two provisions being active, the income supplement holds back employment recovery by 1.6 percentage points, the extended duration by 1.5 percentage points, and expanded eligibility by 2.9 percentage points. This interaction effect is the central quantitative finding: individually modest provisions combine to produce effects far exceeding their sum when implemented simultaneously.

Q: What are the implied unemployment duration elasticities, and how do they compare to the literature? A: The $600 supplement alone raises average unemployment duration by 8% against a 343% rise in the replacement rate, implying an elasticity of 0.02. Extended duration alone raises unemployment duration by 6% against a 150% increase in potential benefit duration, implying an elasticity of 0.03. Expanded eligibility alone raises unemployment duration by 19%, implying an elasticity of 0.04. When each provision is activated on top of the other two, the implied elasticities rise substantially: 0.24 for the $600 supplement, 0.43 for extended duration, and 0.28 for expanded eligibility. These are in the low-to-middle range of pre-pandemic estimates (Katz and Meyer, 1990: 0.3–0.5; Johnston and Mas, 2018: 0.4–0.8; Rothstein, 2011: 0.06; Farber and Valletta, 2015: 0.15).

Q: What is the role of expanded eligibility specifically? A: Expanded eligibility is identified as the most disruptive CARES Act provision, accounting for 1.0 percentage points of employment loss alone and 2.9 percentage points conditional on the other provisions. Mechanically, expanded eligibility converts non-UI unemployed workers into UI-eligible workers, draining the pool of workers willing to accept low-wage job offers. Because low-wage firms depend disproportionately on the non-UI pool for hiring, this provision disproportionately depresses their employment. Using CPS data, the authors document that the share of unemployed workers receiving UI in the in-person leisure and hospitality sector nearly tripled in 2020 relative to the pre-pandemic period.

Q: What are the scope conditions and limitations of the analysis? A: The empirical analysis is restricted to small, in-person service sector businesses (restaurants and retailers) in the Homebase sample, which may not be representative of the broader labor market. The quantitative model is explicitly focused on disincentive effects only and does not capture the stimulative or demand effects of UI. The model also abstracts from re-opening restrictions and other pandemic-specific confounders. The analysis covers April to December 2020; the 2021 pandemic UI extensions are not studied. The job-refusal probability (chi = 16.5%) is a reduced-form calibration target rather than a structurally identified parameter.

Replacement rate gap: The difference in business-specific UI replacement rates between low-wage and high-wage businesses within the same local labor market; defined as UI benefits (state benefit plus supplement) divided by the business’s pre-pandemic average hourly wage. Larger gaps indicate greater relative disincentive for workers to accept jobs at low-wage firms.

Disincentive effect: The negative impact of higher UI replacement rates on workers’ willingness to accept job offers and thus on business employment recovery, isolated from the simultaneous stimulative demand effect of UI spending.

Non-UI unemployed pool: Workers who are ineligible for or have exhausted UI benefits and therefore receive only social benefits at a lower replacement rate (calibrated at 0.15 in the model). This group has a lower reservation wage and constitutes the primary labor supply source for low-wage firms.

Local-industry cell: The paper’s unit of comparison — businesses sharing the same four-digit zip code (covering on average four neighboring zip codes), two-digit NAICS industry, and Yelp price tier. Within-cell differencing is the mechanism that removes common local demand effects.

Benefit recipiency probability: The probability that a newly separated worker enters the UI-eligible unemployed pool, combining UI eligibility and takeup. Pre-pandemic this is calibrated at 14%; under the CARES Act it rises to 70%, targeting the observed near-tripling of UI recipients in the CPS data.

Job-refusal eligibility loss: A probability (calibrated at 16.5%) that a UI-eligible worker who rejects a job offer loses UI status and transitions to the non-UI pool. Motivated by UI law prohibiting refusal of suitable work; reduces the effective outside option and reconciles the model’s predicted employment gap with the empirical estimate.

Equilibrium residual wage dispersion: The wage dispersion observed in equilibrium conditional on worker observables. The model generates realistic dispersion by calibrating the non-UI replacement rate to match the lower half of the wage distribution and the firm wage offer variance to match the upper half; the presence of the non-UI state substantially increases residual dispersion relative to standard search models.

Distorted prices and targeted taxes in the New Keynesian Network model

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how governments should optimally adjust sector-specific taxes in response to sectoral shocks when monetary policy cannot be tailored to individual sectors. The authors work within a variant of Rubbo’s (2023) New Keynesian Network (NKN) model, augmented to include time-varying sectoral sales taxes and production subsidies. The model features N sectors connected through input-output linkages, with Calvo-type price rigidity that is heterogeneous across sectors, and encompasses both sectoral productivity (supply) shocks and demand shocks.

The central finding, stated as Proposition 1, is that the first-best tax policy requires exactly 2N instruments—one sales tax and one production subsidy per sector—not just instruments in the shocked sector. The mechanism turns on a twofold distortion created by sticky prices. Because only a fraction of firms adjust prices at any time, relative prices are distorted both within sectors (price dispersion among firms) and across sectors (misalignment of relative prices). The production subsidy offsets the effect of shocks on marginal costs, incentivizing price-adjusting firms to leave seller prices unchanged and thereby eliminating within-sector dispersion. The sales tax—which applies to both household purchases and intermediate goods trade—steers demand across sectors so that market prices move as if fully flexible, closing sectoral output gaps even as seller prices remain constant. The optimal sales tax moves exactly one-for-one with the vector of natural prices. Crucially, budget neutrality holds to first order: the sales tax revenues fund the production subsidies.

The strength of each instrument’s response depends on network proximity rather than price rigidity. For supply shocks, adjustment propagates downstream (governed by the Leontief inverse), so sectors that intensively use inputs from the shocked sector require larger responses. For demand shocks, adjustment propagates upstream first and then back downstream, so upstream suppliers to the shocked sector face the largest responses.

Because the first-best policy requires observing sectoral shocks directly, the authors propose a simple 2N rule (Proposition 2) that responds only to observable sectoral seller-price inflation, with rule strength parameter ϕ_i per sector. As ϕ_i → ∞ the simple rule converges to the first-best. Crucially, the rule can be implemented by observing inflation only in the shocked sector and adjusting taxes and subsidies in other sectors proportionally to their input-output distance from that sector.

The quantitative assessment calibrates the model to the U.S. economy using BEA 2017 input-output accounts with N = 373 sectors at the 6-digit classification. Sectoral price flexibility is drawn from Antonova (2025), ranging from 0.052 to 0.989 with a median of 0.277 (implying a median price duration of roughly 4.3 months). Shocks follow AR(1) processes with persistence ρ = 0.97. Supply shocks hit 10 energy-related sectors (roughly 10% of total sales); demand shocks hit 22 service-related sectors (roughly 7% of total sales). The key quantitative finding is that the simple 2N policy—both subsidy and tax together—delivers substantially greater welfare improvement than a subsidy-only policy (N instruments), particularly for supply shocks. When the subsidy is not accompanied by the corresponding sales tax, welfare gains are much smaller.

The paper extends to an open economy with import-price shocks that act simultaneously as supply and demand shocks. Applied to the 2022 Ukraine war energy crisis: a 24% world-energy-price increase (IMF Global Energy Price index, 2022M1–2022M4) is used, with high-dependence Europe (energy import share γ_EU = 0.63, substitution elasticity η_EU = 1) contrasted against low-dependence U.S. (γ_US = 0.17, η_US = 4). In Europe, adverse supply effects dominate so the domestic energy sector contracts; in the U.S., demand substitution effects dominate so domestic energy expands. Simple 2N rules correlate 0.89 with the optimal policy across sectors for Europe and 0.94 for the U.S. A notable normative implication: the optimal policy raises sales taxes on energy to discourage consumption, in contrast to the actual European policy of subsidizing energy consumption during the 2022 crisis.

Q: Why can monetary policy not achieve the first-best allocation in the NKN model?

A: Monetary policy sets a single nominal interest rate that applies uniformly across all sectors, but sectoral shocks generate heterogeneous natural rates. Even if monetary policy stabilizes aggregate output, it cannot simultaneously close all sectoral output gaps and eliminate within-sector price dispersion. Rubbo (2023) shows that optimal monetary policy improves welfare but leaves a significant welfare loss remaining.

Q: What is the core tradeoff in each sector that motivates the 2N result?

A: With Calvo-type staggered pricing, adjusting a sector’s relative price to close its output gap creates price dispersion within the sector because not all firms adjust simultaneously; but holding seller prices constant to avoid dispersion leaves output gaps open due to the absence of relative price adjustment. Two instruments—production subsidy and sales tax—are required to address both sides of this distortion simultaneously, in keeping with the Tinbergen principle.

Q: How exactly do the production subsidy and sales tax each work under the optimal policy?

A: The production subsidy is paid to producers and affects the optimal seller price for a given marginal cost, incentivizing firms that can adjust prices to leave them unchanged. The sales tax is levied on buyers (households and downstream firms) and, because it is applied to both household consumption and intermediate goods trade, it steers demand across sectors to replicate the efficient allocation of expenditure. Under the optimal policy, seller prices are fully stabilized (ps_t = 0) while buyer (market) prices move as pt = τs_t = pn_t, mimicking flexible-price outcomes.

Q: What determines which sectors receive larger optimal tax and subsidy responses?

A: For supply (productivity) shocks, responses are governed by the matrix L̄ = XL, where L is the Leontief inverse measuring downstream proximity; sectors that are more intensive downstream users of the shocked sector require larger responses. For demand shocks, the relevant matrix measures upstream proximity, so sectors that supply inputs to the shocked sector face stronger responses. Critically, the level of the policy response is independent of sector-specific price rigidity; only the network structure matters.

Q: Is the optimal 2N policy budget-neutral, and why only approximately?

A: Budget neutrality holds to first order around the zero-profit steady state. The production subsidy applies to costs while the sales tax applies to sales; at the steady state these coincide, so the subsidy is exactly funded by the tax revenue. The approximation breaks down away from the zero-profit steady state because costs and sales diverge.

Q: What is the simple 2N rule and how does it relate to the first-best?

A: The simple rule sets sp_t = Iϕ · πs_t and τs_t = sp_t, where Iϕ = diag{ϕ_i} is a diagonal matrix of response coefficients for each sector’s seller-price inflation. As ϕ_i → ∞ for all i, the allocation converges to first-best; larger ϕ_i produces a stronger commitment to stabilize sectoral inflation, resulting in muted inflation rather than large tax and subsidy levels. In practice, the rule can be implemented by observing inflation only in the shocked sector and scaling responses in other sectors by their input-output distance from that sector.

Q: What does the three-sector example (Energy, Manufacturing, Services) illustrate about supply vs. demand shocks?

A: Under an adverse energy productivity shock, the optimal policy subsidizes Energy and Manufacturing (proportional to energy use in manufacturing) but not Services, since Services are not energy-intensive and thus not closely connected downstream. Under a positive manufacturing demand shock, the optimal policy subsidizes both Manufacturing and upstream Energy equally, reflecting that demand shocks propagate upstream first.

Q: What does the calibrated quantitative exercise show about the welfare gains from using both instruments versus one?

A: For both supply and demand shock scenarios, the simple 2N policy (subsidy plus tax) delivers substantially greater welfare improvement than using only monetary policy. When the subsidy is not accompanied by the corresponding sales tax, welfare gains are much smaller, confirming that both instruments together—not subsidies alone—are essential. This is identified as a key quantitative finding of the paper.

Q: How robust are results to decreasing returns to scale in production?

A: Under decreasing returns to scale, the optimal policy response is highly similar to the baseline: correlations between the two are 0.98 for supply shocks and 0.99 for demand shocks across sectors. The simple 2N rule continues to deliver significant welfare improvements. One difference is that demand shocks generate relatively higher welfare losses under decreasing returns, while productivity shocks lead to lower losses.

Q: How does the open-economy extension change the analysis for import-price shocks?

A: Import-price shocks enter the model as both supply shocks (raising input costs) and demand shocks (shifting expenditures toward domestic substitutes), so they require a policy response that accounts for both propagation channels simultaneously. The optimal open-economy policy is formally isomorphic to the closed-economy counterpart but with redefined upstream and downstream matrices and shock vectors. The relative importance of the supply versus demand channel depends on the economy’s import dependence and substitution elasticity.

Q: How does the 2022 energy crisis illustrate the difference between the optimal policy and actual European policy?

A: Using a 24% world-energy-price increase (IMF Global Energy Price index, 2022M1–2022M4), the model implies that with high European energy dependence (γ_EU = 0.63, η_EU = 1), adverse supply effects dominate and the optimal policy raises sales taxes on energy to discourage consumption and subsidizes domestic energy users proportional to downstream proximity. Actual European policy partly subsidized energy consumption, which the model identifies as welfare-reducing relative to the optimal response. For the low-dependence U.S. (γ_US = 0.17, η_US = 4), demand substitution toward domestic energy dominates, requiring additional subsidies to domestic energy producers.

Q: How does this paper relate to the Diamond-Mirrlees result on intermediate good taxation?

A: Diamond-Mirrlees (1971) recommends against taxing intermediate goods in an otherwise efficient economy to avoid introducing additional distortions. This paper considers an economy already subject to pricing frictions (Calvo staggered pricing), and shows that taxing intermediate goods through the sales tax—which applies to intermediate goods trade—is part of the optimal policy precisely because it corrects the pre-existing distortions. The paper thus does not contradict Diamond-Mirrlees but operates in a different setting where frictions are already present.

New Keynesian Network (NKN) model: A multi-sector general equilibrium framework with N sectors connected through input-output linkages, Calvo-type staggered price setting that is heterogeneous across sectors, and monopolistically competitive firms; provides the canonical system of sectoral IS curves and Phillips curves used in this paper.

2N policy: The paper’s central result that the first-best tax policy requires exactly two instruments per sector—one production subsidy and one sales tax—for a total of 2N instruments; characterized in Proposition 1 and named for this instrument count.

Production subsidy (sp_t,i): A sector-specific transfer paid to producers that affects the optimal seller price for a given marginal cost; under the optimal policy it offsets the effect of shocks on marginal costs, incentivizing price-adjusting firms to leave seller prices unchanged and thereby eliminating within-sector price dispersion.

Sales tax (τs_t,i): A sector-specific tax levied on buyers—both households and downstream firms purchasing intermediate goods—such that the buyer (market) price equals (1 + τs_t,i) times the seller price; under the optimal policy it replicates the efficient allocation of expenditure across sectors even when seller prices are fully stabilized.

Downstream proximity (Leontief inverse L̄ = XL): A measure of the total direct and indirect use of a sector’s output by other sectors, governing the propagation and optimal policy response to supply (productivity) shocks; the ij-th element of L̄ captures how strongly a shock in sector j affects policy in sector i through downstream input-output linkages.

Upstream proximity: A measure of how closely a sector supplies inputs to another sector, governing the propagation of demand shocks; demand shocks propagate first upstream (to input suppliers) before feeding back downstream.

Budget neutrality: The property that the optimal 2N policy is self-financing to first order—sales tax revenues exactly fund the production subsidies around the zero-profit steady state—so the fiscal intervention does not require net government expenditure.

Simple 2N rule: A practically implementable approximation to the first-best policy that sets subsidies and taxes proportional to observed sectoral seller-price inflation with response coefficients ϕ_i; converges to the first-best as ϕ_i → ∞ and can be implemented using only the inflation rate of the shocked sector plus network-distance weights from the input-output table.

Diversification, Market Entry, and the Global Internet Backbone

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates how buyer demand for supplier diversification shapes entry incentives and market structure, using the global undersea fiber-optic cable industry as the empirical setting. The research question has two parts: first, how much of observed cable entry and surplus generation is attributable to buyers’ diversification motives rather than standard price competition; and second, whether market forces produce too much or too little diversification relative to the social optimum.

The empirical setting spans 2005–2021 and covers the worldwide network of undersea cables that carries more than 98% of all international internet traffic. Cables fail frequently — hundreds of faults per year — and industry professionals confirm that “no customer would buy capacity on a single cable.” The median monthly price for a 10Gbps lease fell from $55,500 in 2005 to $2,200 in 2021, and the number of active cables roughly doubled over the sample period.

The authors use proprietary data from TeleGeography covering cable characteristics (construction costs, capacity, landing points, entry dates), quarterly bandwidth prices at the city-pair level, annual used bandwidth at the country-pair level, and 168 documented cable faults. Markets are defined as country-pairs in calendar quarters.

The theoretical model begins with a representative buyer who splits bandwidth purchases equally across n symmetric cable operators to minimize expected disruption costs. Because disruption shocks are i.i.d. across cables, adding suppliers reduces the variance of realized bandwidth delivery, lowering the required over-provisioning buffer. This generates a “market expansion” channel: entry increases aggregate demand holding prices fixed, not just through price competition. The aggregate demand equation takes log-linear form with cable count indicators alongside price and demand shifters.

The structural model adds a dynamic oligopoly game where firms make entry and exit decisions as a non-stationary Markov Perfect Equilibrium, with Cournot competition in each period. The three-step estimation procedure recovers: (1) price elasticities and diversification parameters from an IV demand regression using electricity generation cost shares as instruments; (2) marginal costs from firms’ first-order conditions; (3) entry and fixed costs from a nested pseudo-likelihood (NPL) estimator, supplemented by construction cost data to separately identify entry costs given the near-absence of observed exits.

Key demand results: the IV price elasticity is −1.36. The market expansion effect is large and exhibits decreasing marginal returns — entry of a second cable expands demand by as much as a 28.3% price decrease; a third cable is equivalent to a 19.3% price decrease; an eighth cable is equivalent to a 7.5% price decrease. The demand model achieves R² = 95%.

The first counterfactual removes the diversification channel entirely (entry raises competition only). Without diversification, cable investment falls by 12%. The net present value of total surplus per market over the sample period averages $1.11 billion under the observed equilibrium; supplier diversification accounts for 11% of total surplus and 27% of consumer surplus.

The second counterfactual quantifies two opposing distortions relative to the social optimum. Business-stealing creates excessive entry (entrants reduce incumbents’ output), while diversity effects create insufficient entry (marginal entrants generate surplus through diversification they cannot fully capture). At end-of-sample (2021-Q4), diversity distortions in terms of number of entrants range from 54% to 125% of the business-stealing distortion. Business-stealing tends to dominate for most markets, producing moderately excessive entry. Relative to the market outcome, total surplus under the social planner’s solution is on average 10% higher: 53% of this welfare gap is attributable to diversity effects and 47% to business-stealing effects. These findings hold across market heterogeneity in entry costs, market size, and demand growth.

The paper concludes that profit-maximizing suppliers fail to fully internalize diversification-related social benefits, and that targeted entry subsidies would pass cost-benefit tests in settings where diversity distortions dominate.

Q: What is the core mechanism by which supplier diversification expands demand? A: When buyers split purchases across n cable operators whose disruption shocks are i.i.d., adding a supplier reduces the variance of realized delivered bandwidth. The buyer therefore needs to hold a smaller over-provisioning buffer to achieve the same expected level of used bandwidth B. This lowers the effective cost of a given quantity of used bandwidth, shifting the aggregate demand curve outward. As the number of suppliers grows to infinity, the expected disruption cost converges to zero.

Q: How large is the market-expansion effect of diversification empirically? A: The effect is large but exhibits decreasing marginal returns. Entry of a second cable expands demand by as much as a 28.3% price reduction holding prices fixed; the third cable is equivalent to a 19.3% price reduction; and the eighth cable is equivalent to a 7.5% price reduction. All cable-count coefficients are positive and statistically significant in the IV demand model.

Q: How is price endogeneity addressed in the demand estimation? A: Bandwidth prices are instrumented using the marginal cost of electricity generation — specifically, country-level electricity generation shares (coal, gas, oil) interacted with quarterly commodity price series for coal, gas, and oil (Brent crude, Australian coal price, EU natural gas price). The first-stage results indicate electricity costs are strong predictors of bandwidth prices. Accounting for endogeneity raises the price elasticity from an OLS level to −1.36 in absolute value, consistent with the expected direction of OLS bias.

Q: What share of cable investment and surplus is attributable to diversification motives? A: In the counterfactual where the diversification channel is eliminated — entry raises competition and lowers prices but provides no diversification benefit — cable investment falls by 12%. Under the observed equilibrium, the net present value of total surplus per market over 2005–2021 averages $1.11 billion; supplier diversification accounts for 11% of this total surplus and 27% of consumer surplus.

Q: How are the two distortions — business-stealing and diversity — defined and separated? A: Business-stealing distortion arises because entrants reduce incumbents’ outputs and revenues, so private entry benefits exceed social benefits, leading to excessive entry. Diversity distortion arises because entrants create surplus for buyers through diversification but cannot fully capture it without perfect price discrimination (following Spence (1976) and Mankiw and Whinston (1986)), leading to insufficient entry. The authors disentangle these by comparing: (i) the social planner’s solution (eliminates both distortions), and (ii) a coordinated entry solution maximizing producer surplus (eliminates only business-stealing). The residual gap between the two identifies the diversity distortion.

Q: What is the net direction and magnitude of distortion in equilibrium market structure? A: At 2021-Q4, for most markets, business-stealing dominates, leading to moderately excessive entry. Diversity distortions in number of entrants range from 54% to 125% of the business-stealing distortion across markets. Relative to the market outcome, the social planner’s solution yields average total surplus that is 10% higher. Of that welfare gap, 53% is attributable to diversity effects and 47% to business-stealing effects.

Q: How do market characteristics affect which distortion dominates? A: The paper analyzes cross-market heterogeneity and identifies market features — including the size of entry costs, market size, and the rate of demand growth over time — as determinants of whether insufficient diversification or excessive entry is the binding distortion. Markets with higher entry costs or slower demand growth are more likely to exhibit insufficient diversification.

Q: How are entry costs identified given the near-absence of cable exits in the data? A: Because exit events are rare in a nascent industry — only a handful of exits observed, mostly after 2020 — entry and fixed costs cannot be separated by exit decisions alone. The authors address this by using cable-level construction cost data from TeleGeography to estimate entry costs outside the dynamic model. With entry costs in hand, firms’ optimal entry decisions identify fixed costs. Scrap values are normalized to zero, consistent with industry reports that retired cables are typically abandoned on the seabed.

Q: What role does the non-stationarity of the market environment play in the model? A: The data covers the industry’s earliest growth phase, with demand growing by roughly three orders of magnitude (used bandwidth from 5 Tbps in 2005 to 2,886 Tbps in 2021) and prices falling by a factor of roughly 25. The authors use a non-stationary Markov Perfect Equilibrium concept in which strategies and transition functions are indexed by time, aligning with the treatment of high-tech commodities in Igami (2017).

Q: What are the policy implications of the findings? A: Because profit-maximizing suppliers do not fully internalize the diversification-related social benefits of entry, entry rates can be sub-optimal from a welfare perspective when diversity distortions dominate. The authors suggest targeted entry subsidies would pass cost-benefit tests in such cases. For antitrust analysis, regulators who ignore the demand-expansion effect of incremental suppliers may incorrectly judge a market as sufficiently competitive. In merger review, authorities must account for firms’ private incentives to provide diversification to reach accurate welfare conclusions.

Q: How does the paper verify that diversification demand is not a spurious empirical artifact? A: Several checks support the causal interpretation. The estimated demand parameters are consistent with the predictions of the consumer-level utility maximization problem derived analytically: decreasing marginal returns to diversification and a positive relationship between the number of suppliers and demand. The demand model achieves R² = 95%, suggesting limited unobserved confounders. Additionally, 78% of cable faults involve only a single cable, confirming that disruptions are geographically isolated and that cross-cable diversification provides genuine insurance value.

Q: What are the main data limitations acknowledged by the authors? A: The authors cannot observe cable-level revenue or market shares, nor contracts between buyers and sellers; only aggregate country-pair used bandwidth is observed. Price coverage is not comprehensive — TeleGeography collects prices on a voluntary basis from dozens of providers. The cable faults dataset (168 faults) represents only a subset of total faults, as collection focuses on publicly disclosed events. The demand model also does not explicitly account for substitution patterns across firms due to lack of firm-level market share data, though the high R² partly mitigates this concern.

Diversification (in this paper’s sense): Buyers’ practice of splitting bandwidth purchases across multiple cable operators to reduce exposure to idiosyncratic disruption risk. Diversification across n cables with i.i.d. disruption shocks reduces the variance of realized delivered bandwidth and lowers the required over-provisioning buffer, making the effective cost of a given usage level B a decreasing function of n.

Market Expansion Effect: The channel through which entry of additional cable suppliers raises aggregate demand holding prices fixed. This occurs because each additional supplier reduces disruption risk, allowing buyers to demand more used bandwidth for the same price. It is distinct from the conventional competition channel (entry lowering prices).

Diversity Distortion: The tendency toward insufficient entry arising because marginal entrants generate consumer surplus through diversification benefits but cannot fully capture this surplus absent price discrimination. Follows Spence (1976) and Mankiw and Whinston (1986).

Business-Stealing Distortion: The tendency toward excessive entry arising because entrants reduce incumbents’ output and revenues, creating a gap between private and social returns to entry.

Non-Stationary Markov Perfect Equilibrium: The equilibrium concept used for the dynamic entry game, in which strategies and equilibrium selection rules are indexed by calendar time to accommodate substantial secular trends in demand and costs — as opposed to a stationary MPE which assumes a stable long-run distribution.

Used Bandwidth vs. Purchased Bandwidth: Used bandwidth B is the amount the buyer is committed to delivering (to downstream customers or for internal use). Purchased bandwidth Q is what the buyer actually contracts for across all cables; Q > B because the buyer holds an over-provisioning buffer against disruption risk. The ratio B/Q is a decreasing function of the disruption cost parameter gamma and an increasing function of the number of suppliers n.

Nested Pseudo-Likelihood (NPL) Algorithm: The baseline estimator for the dynamic game, following Aguirregabiria and Mira (2007). It iterates on the best-response mapping to impose equilibrium restrictions. The authors supplement NPL with two-step estimators (1-PML, 1-MD) and the spectral algorithm of Aguirregabiria and Marcoux (2021), which solves for the root of a nonlinear system using a quasi-Newton method and is robust to fixed-point instability.

Do Credit Conditions Move House Prices?

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. To what extent did an expansion and contraction of credit drive the 2000s housing boom and bust? The existing literature offers sharply divergent answers — ranging from credit explaining virtually none of the boom (Kaplan, Mitman, and Violante 2020) to credit explaining the majority of it (Favilukis, Ludvigson, and Van Nieuwerburgh 2017, who find credit alone explains 60% of the rise in price-to-rent ratios). Greenwald and Guren argue that the source of these divergent findings is a single structural assumption: the degree to which credit-insensitive agents (landlords and unconstrained savers) can absorb credit-driven demand for housing, which in turn depends on the degree of segmentation between the owner-occupied and rental housing markets.

Key Mechanism. The paper organizes the literature around a “tenure supply” curve, defined in price-rent ratio versus homeownership rate space. A perfectly inelastic (vertical) supply curve — corresponding to perfect segmentation, in which housing cannot move between the owner-occupied and rental sectors — implies that credit expansion bids up house prices with no change in the homeownership rate. A perfectly elastic (horizontal) supply curve — corresponding to a frictionless rental market with deep-pocketed landlords who price at the present value of rents — implies that credit expansion raises the homeownership rate but not the price-rent ratio, because landlord reservation prices are unaffected by credit. Intermediate degrees of segmentation produce intermediate outcomes: credit raises both the price-rent ratio and the homeownership rate, with the relative magnitudes determined by the slope of the tenure supply curve.

Empirical Strategy. To measure where reality falls on this spectrum, the authors estimate the relative elasticity of the price-rent ratio to an identified credit supply shock, compared to the elasticity of the homeownership rate to the same shock. This ratio is a sufficient statistic for the slope of the tenure supply curve. They use three distinct identification strategies from prior literature — (1) Loutskina and Strahan (2015), instrumenting for local credit supply using differential city-level exposure to changes in the conforming loan limit (CLL); (2) Di Maggio and Kermani (2017), exploiting the 2004 OCC preemption of state anti-predatory-lending laws for national banks; and (3) Mian and Sufi (2019), using differential city-level exposure to the 2003 private label securitization (PLS) expansion through bank funding composition. Regressions are estimated on annual CBSA-level panels using local projection IV (LP-IV) or event-study reduced-form methods. Key data include the CoreLogic repeat-sales house price index, the CBRE Torto-Wheaton same-store rent index (a repeat-rent index for multi-unit apartment buildings, constructed from newly-leased units), and Census Housing Vacancy Survey homeownership rates.

Main Empirical Findings. All three instruments consistently find that credit supply shocks generate a significant increase in house prices and the price-rent ratio but a much smaller, rarely statistically significant, effect on the homeownership rate. Under the LS LP-IV, the price-rent ratio peaks at an increase of 0.471, while the homeownership rate response reaches only 0.037 at the 2-year horizon and peaks at 0.101 after 5 years. The ratio of price-rent to homeownership responses ranges from 3 to infinity across the three instruments and horizons. These estimates imply a substantial degree of segmentation — the no-segmentation model falls far outside the 95% confidence intervals at all horizons.

Structural Model and Calibration. The authors construct a general equilibrium model featuring a representative borrower, landlord, and saver, with long-term fixed-rate mortgages subject to loan-to-value (LTV) and payment-to-income (PTI) limits following Greenwald (2018). The key modeling innovation is within-type heterogeneity in the benefit of owning versus renting, captured by logistic distributions for both borrowers and landlords. The dispersion parameter of the landlord distribution (σω,L) governs the slope of the tenure supply curve and is calibrated to minimize weighted distance to the LS empirical impulse responses. The resulting benchmark calibration yields σω,L = 2.877, with the benchmark model’s price-rent-to-homeownership ratio between 6.98 and 9.31 depending on the horizon — consistent with the empirical estimates.

Quantitative Results on the 2000s Boom. The paper then uses the calibrated model to simulate a credit standard relaxation (LTV limits relaxed from 85% to 99%, PTI limits from 36% to 65%) from 1998 Q1 through 2007 Q1, with a reversion at the start of the bust. This credit relaxation alone explains 34% of the peak rise in price-rent ratios observed in the boom, with a lower bound of 26% accounting for parameter uncertainty. In contrast, the no-segmentation model explains -1%, while the full segmentation model explains 38%. Adding a 2 percentage point permanent decline in mortgage spreads alongside the credit standard relaxation allows the benchmark model to explain 72% of the observed rise in price-rent ratios and 80% of the rise in loan-to-income ratios, compared to only 4% in the no-segmentation model. In a “full boom” scenario where additional demand and supply shocks are added to match the entire boom in price-rent ratios and homeownership, removing the credit relaxation reduces the rise in price-rent ratios by 55% in the benchmark economy — larger than the 34% explained in isolation due to nonlinear interactions — compared to only 5% in the no-segmentation economy.

Scope Conditions and Extensions. These results apply to the benchmark calibration in which landlords do not use credit and saver housing demand is fixed. When landlords are allowed to use credit (LTV limit of 65% relaxed to 85% during the boom), the role of credit is strengthened: the recalibrated model explains 80% of the rise in price-rent ratios from combined credit and rate changes, suggesting the benchmark is a lower bound. When savers are allowed to frictionlessly trade housing with borrowers, credit explains 54% of the rise in price-rent ratios even after recalibration — a roughly 25% reduction relative to the benchmark 72%, representing what the authors characterize as an extreme lower bound given that saver housing markets are in practice substantially segmented due to indivisibility, quality, and location differences.

Policy Implications. The findings imply that macroprudential policies tightening LTV and PTI ratios can be effective at restraining house price growth, but only in the presence of the significant rental market segmentation found in the benchmark economy. In the no-segmentation economy, removing the credit relaxation from the full boom reduces price-rent ratio growth by only 5%.

In depth

Q1. What is the core theoretical insight that reconciles the divergent findings in the prior literature on credit and house prices?

The key difference is the degree to which credit-insensitive agents — specifically landlords and unconstrained savers — can absorb credit-driven demand for housing. Models with perfectly segmented rental markets (no rental sector or fixed homeownership rate) feature borrowers competing only with each other for a fixed stock, so credit expansion bids up prices. Models with frictionless rental markets feature deep-pocketed landlords who supply housing at a price equal to the present value of rents, which is unaffected by credit; credit expansion then raises the homeownership rate rather than prices. Intermediate degrees of frictions produce intermediate outcomes. This mechanism had not been recognized as the source of the literature’s divergence before this paper.

Q2. What is the “tenure supply curve” and why is its slope the key empirical object?

The tenure supply curve describes the menu of price-rent ratios at which landlords are willing to supply varying amounts of owner-occupied housing (given total housing stock), traced out in price-rent ratio versus homeownership rate space. Its slope determines how the equilibrium responds to a credit-induced demand shift: a steep (inelastic) supply curve translates credit expansion primarily into price-rent ratio increases; a flat (elastic) supply curve translates it primarily into homeownership rate increases. Identifying this slope empirically is therefore sufficient to discipline any macro-housing model’s predictions about the role of credit in price dynamics, for arbitrary underlying shocks.

Q3. How do the authors identify the slope of the tenure supply curve empirically?

They estimate the slope as the ratio of the causal elasticity of the price-rent ratio to that of the homeownership rate, with respect to an identified credit supply shock. Three instruments are used: (1) the Loutskina-Strahan shift-share instrument based on differential exposure to changes in the conforming loan limit, estimated by LP-IV on an unbalanced panel of 62 CBSAs from 1992 to 2016; (2) the Di Maggio-Kermani event study based on the 2004 OCC preemption of state anti-predatory-lending laws, covering 262 CBSAs for house prices and 82 CBSAs for homeownership from 2001 to 2010; and (3) the Mian-Sufi event study based on differential exposure to the 2003 PLS expansion via non-core deposit share, covering 245 CBSAs using ACS and FHFA data. In practice, they estimate the inverse slope (ratio of homeownership to price-rent response) because the first stage is far stronger using price-rent ratios as the endogenous variable.

Q4. What are the empirical results on the relative price-rent and homeownership responses?

Across all three instruments, credit supply shocks significantly raise the price-rent ratio but have a much smaller, rarely statistically significant effect on the homeownership rate. Under the LS LP-IV, the price-rent ratio peaks at 0.471 after 2 years, while the homeownership rate reaches only 0.037 at 2 years and peaks at 0.101 at 5 years. The naive point-estimate ratios range from 2.93 to 12.83 at horizons 2 through 5, with the 4-year estimate negative (implying an infinite slope). The directly estimated inverse slope coefficients are small (0.05 to 0.24) and never statistically different from zero. The DK instrument yields slopes of 6.72 in 2005, 3.67 in 2006, and 3.40 in 2007. The MS instrument yields a slope of approximately 4.49 in both 2006 and 2007. The lower bound of the 95% confidence intervals corresponds to slopes of at least 1.8 to 8.4.

Q5. What is the key modeling contribution on the structural side?

The key innovation is the introduction of within-type heterogeneity in ownership preferences for both borrowers and landlords, modeled as logistic distributions. This heterogeneity allows the model to generate a fractional and time-varying homeownership rate — a feature absent from most prior macro-housing models — and maps directly into the slopes of the demand and tenure supply curves. The dispersion in landlord ownership costs (σω,L) governs the supply curve slope and is calibrated to match the empirical impulse responses. Without this heterogeneity, the model would produce corner solutions with all housing owned by one type.

Q6. How is the landlord dispersion parameter σω,L calibrated, and what is the estimated value?

The calibration minimizes a weighted sum of squared deviations between model and data impulse responses for the price-rent ratio and homeownership rate, using the LS LP-IV estimates. Deviations are weighted by the inverse of empirical standard errors. Because model impulse responses jump on impact while empirical responses are hump-shaped (due to search frictions), the calibration uses only horizons 2 through 5 years. The minimum-distance estimate yields σω,L = 2.877, alongside a mortgage spread shock persistence of 0.965 and a shock size of -0.041 (corresponding to an annualized CLL subsidy of approximately 17 basis points, within the 10-24bp range found in prior literature). The benchmark model’s implied price-rent-to-homeownership response ratio ranges from 6.98 to 9.31, consistent with the empirical estimates.

Q7. What lower bound does the paper derive for σω,L, and how does the no-segmentation model compare?

A credible set for σω,L is derived by targeting the upper and lower bounds of the 95% confidence interval for the estimated inverse slope. The lower bound for σω,L (targeting the top of the confidence interval) is 0.810; the lower bound targets the bottom of the confidence interval but is best matched by the full segmentation case (σω,L → ∞). The no-segmentation economy (σω,L = 0) produces inverse ratios between 4 and 32 times the empirical upper bound, placing it far outside the credible set.

Q8. What is the model’s quantitative finding on the role of credit standard relaxation in isolation?

A credit standard relaxation (LTV from 85% to 99%, PTI from 36% to 65%) implemented from 1998 Q1 to 2007 Q1 and then reverted explains 34% of the peak rise in price-rent ratios in the benchmark model, with a lower bound of 26% conditional on parameter uncertainty. In the full segmentation model, the same relaxation explains 38%, while in the no-segmentation model it explains -1%. Credit standard relaxation also explains 51% of the rise in loan-to-income ratios in the benchmark, compared to 31% in the no-segmentation model.

Q9. What does adding a decline in mortgage rates contribute?

Adding a permanent 2 percentage point decline in mortgage spreads alongside the credit standard relaxation increases the benchmark model’s explained share of the price-rent ratio boom from 34% to 72%, and the loan-to-income ratio share from 51% to 80%. The no-segmentation model explains only 4% of the price-rent ratio boom and 38% of the loan-to-income ratio boom under the same combined experiment.

Q10. How does the “full boom” counterfactual estimate the marginal contribution of credit?

The full boom experiment adds exogenous demand shocks (shifts to µω,B) and supply shocks (shifts to µω,L) on top of the credit relaxation and rate decline, calibrated to exactly reproduce the observed peak increase in both the price-rent ratio and the homeownership rate during the boom. Removing the credit relaxation from this full boom scenario reduces the rise in price-rent ratios by 55% and the rise in loan-to-income ratios by 74% in the benchmark economy. This exceeds the 34% figure from the credit-alone experiment due to strong nonlinear interactions: without the credit relaxation, binding PTI limits constrain households’ ability to finance properties even when ownership preferences rise, dampening both price and credit growth. In the no-segmentation economy, removing the credit relaxation reduces price-rent ratio growth by only 5%.

Q11. What are the implications of allowing landlords to use credit?

When landlords face an LTV limit of 65% relaxed to 85% during the boom, the credit expansion also shifts the tenure supply curve upward (as in Panel (d) of the supply-demand framework), leading to a larger price-rent ratio response and a smaller homeownership rate response than in the baseline. Without recalibration, this model explains 81% of the price-rent ratio rise. After recalibration of σω,L (which is required because landlord credit changes the mapping from empirical moments to structural parameters), the model explains 80% of the price-rent ratio rise. This implies the benchmark results are a lower bound on the role of credit in driving house prices.

Q12. What are the implications of allowing savers to frictionlessly trade housing with borrowers?

When savers are allowed to frictionlessly adjust their housing demand (purchasing housing from or selling to borrowers as credit conditions change), the price-rent ratio response is dampened because savers absorb excess borrower demand. After recalibrating σω,L, the combined credit-and-rate experiment explains 54% of the price-rent ratio boom — roughly 25% less than the benchmark 72%. The authors regard this as an extreme lower bound because in practice saver and borrower housing markets are substantially segmented due to indivisibility, location, and quality differences.

Q13. What are the implications for macroprudential policy?

Macroprudential policies that tighten LTV and PTI limits are effective at slowing house price growth in the benchmark economy, where rental market frictions are substantial. In the full boom counterfactual, tightening credit standards reduces the rise in price-rent ratios by 55%. However, in the no-segmentation economy, the same tightening reduces price-rent ratio growth by only 5%, because landlords readily absorb credit-driven demand and pin prices to the present value of rents. The effectiveness of macroprudential policies is therefore deeply dependent on the degree of rental market segmentation.

Q14. Why do the authors prefer the CBRE Torto-Wheaton rent index over typical rent measures?

The TW index uses a repeat-rent methodology on newly-leased multi-unit apartments, which better captures current market conditions than median rent measures, which are biased by composition changes and are sticky due to long-term lease contracts. Since the price-rent ratio is meant to capture the rent a unit could command if leased instead of sold, newly-leased apartment rents are more appropriate for constructing this ratio. The TW index is available for 53 CBSAs from 1989 and 62 CBSAs from 1994.

Q15. Why do the authors estimate the inverse slope rather than the slope directly?

The first stage for the homeownership rate response is very weak — the estimated coefficients are small and imprecise, so using the homeownership rate as an endogenous variable would suffer severe weak instrument problems. Instead, the authors use the price-rent ratio as the endogenous variable (with a much stronger first stage) and the homeownership rate as the outcome, obtaining the inverse slope (homeownership response per unit price-rent ratio response). The upper bounds of the 95% confidence intervals for the inverse slope range from 0.12 to 0.56 across horizons, corresponding to lower bounds on the slope of 1.8 to 8.4.

Key Concepts

Tenure Supply Curve. The menu of price-rent ratios at which landlords are willing to supply varying quantities of owner-occupied housing (i.e., sell rental units to potential homeowners) at a given total housing stock. Defined in price-rent ratio versus homeownership rate space. Distinct from the absolute supply of housing via the construction sector; shifts in the construction margin affect absolute quantities and prices but not necessarily the price-rent ratio or the ownership share. The slope of this curve — not the level — is the central empirical and structural object of the paper.

Market Segmentation (in the paper’s sense). The degree to which credit-insensitive agents (landlords, unconstrained savers) cannot absorb credit-driven demand from constrained borrowers. Perfect segmentation means owner-occupied and rental housing are entirely non-fungible, so all credit-driven demand falls on a fixed supply of owned units. Zero segmentation means landlords (or savers) can frictionlessly convert between owned and rented housing at a price tied to present discounted rents. In this paper, segmentation is measured continuously by the slope of the tenure supply curve.

Sufficient Statistic (for segmentation). The ratio of the causal elasticity of the price-rent ratio to the causal elasticity of the homeownership rate, both with respect to the same identified credit supply shock. This ratio identifies the slope of the tenure supply curve and is sufficient to calibrate a structural model to recover the role of credit in driving house prices for arbitrary combinations of shocks, even when those shocks differ from the identifying variation.

Ownership Benefit Heterogeneity. An additional idiosyncratic utility flow (positive or negative) that borrowers or landlords receive from owning versus renting a given unit, modeled as a logistic distribution. This within-type heterogeneity generates a fractional and time-varying homeownership rate in the model and maps directly into the slope of the demand and tenure supply curves. The dispersion parameter σω,L for landlords governs the slope of the tenure supply curve; higher dispersion implies a steeper (more segmented) supply curve and larger price-rent ratio responses to credit shocks.

Marginal Collateral Value (CB,t). The shadow value to borrowers of the additional credit that can be collateralized by an additional dollar of housing value, equal to µB,t × FLTV × θLTV in the model. A relaxation of credit standards (raising θLTV or θPTI) or a decline in credit costs raises CB,t, increasing borrower reservation prices and shifting the housing demand curve outward. This is the channel through which credit conditions enter house price dynamics.

Local Projection IV (LP-IV). A generalization of Jordà (2005) local projections to instrumental variables settings, as in Ramey (2016) and Ramey and Zubairy (2018), extended to a panel context with CBSA and time fixed effects. Used to estimate impulse responses of price-rent ratios, house prices, and homeownership rates to credit supply shocks at horizons 0 through 5 years, instrumenting for endogenous credit growth using the conforming loan limit shift-share instrument.

Conforming Loan Limit (CLL) Instrument. A shift-share instrument for local credit supply constructed by interacting the share of mortgage originations in the prior year falling within 5% of the current year’s CLL with the percentage change in the national CLL. Cities where a larger fraction of loans cluster near the CLL threshold experience a larger credit supply shock when the CLL increases, because more loans shift from unsubsidized to GSE-subsidized rates. The instrument is constructed using the change in the national CLL only to avoid endogeneity from high-cost area adjustments.

Do Financial Concerns Make Workers Less Productive?

Mon, 01 Jan 0001 00:00:00 +0000

Do Financial Concerns Make Workers Less Productive?

Research Question

The paper tests whether financial concerns distract workers sufficiently to meaningfully reduce their productivity, and whether receiving cash — by alleviating those concerns — can raise output even when total compensation is held fixed.

Setting and Sample

The experiment involves 408 low-income male agricultural casual laborers in rural Odisha, India, recruited from 47 villages across five worksites in four districts. The study takes place during the lean agricultural season (March–June 2017 and 2018), when formal employment is scarce (workers found paid wage work on only 1.9 days per week on average). During this period, 86% of workers reported being “worried” or “very worried” about their finances, 68–71% carried outstanding loans, and 64–66% said they would have difficulty coming up with Rs. 1,000 (roughly four days of wages) in an emergency. Workers bring these burdens to the job: on a given day, approximately one in two workers reported thinking about financial worries while working.

Experimental Design

Workers were employed for twelve days in a piece-rate manufacturing task — stitching sal tree leaves into disposable plates for restaurants. The payment-timing manipulation is the core of the identification strategy. Control workers received all accrued earnings as a lump sum on the final day (day 12). Treatment workers received their earnings in two installments: an interim payment of earnings to date on day 8 or 9 (randomly staggered across waves), with the balance paid on day 12. Total compensation was held constant across groups; only the timing of receipt differed. On day 5 (the “announcement day”), each worker learned his payment schedule individually. The design thus separates the announcement period (days 5 through the interim payment day, when workers know their schedule but have not yet received cash) from the post-pay period (days after the interim payment until the contract end). This enables the authors to test whether productivity effects arise from information about impending cash, or only once cash is physically in hand.

First Stage: Effects on Financial Strain

Within three days of receiving the interim payment, treated workers increased loan repayments by Rs. 271, a 287% increase relative to the control group mean (p < 0.001), and were 40 percentage points (222%) more likely to repay any loan (p < 0.001). The majority of repayments occurred on the same evening as the cash disbursement — a 746% single-day increase in loan payments. Household expenditures on food, clothing, and essentials rose by 40% (Rs. 150) over three days (p < 0.001). Treatment workers also reported feeling more focused on the work task (11.5 percentage points more likely, p = 0.032) and were less likely to report thinking about financial worries while making plates (13.7 percentage points, p = 0.044).

Main Productivity Results

In the post-pay period, treated workers increased output by 0.109 SD (6.9%) relative to the control group (p = 0.020). No treatment effect emerged during the announcement period (0.014 SD, p = 0.685); the post-pay and announcement-period effects are statistically distinguishable (p = 0.008). Because work hours are fixed and daily attendance is 98.3% with no treatment effect on attendance, these gains reflect improvements in how quickly workers produce plates per hour of work.

Effects are concentrated among workers with below-median baseline wealth (fewer assets, less liquidity): for this subgroup, the interim payment increases output by 0.204 SD (13.0%, p = 0.003). For workers with above-median wealth, the effect is close to zero and statistically insignificant (p = 0.819).

Attentiveness Results

Beyond total output, the authors measure attentiveness through three markers embedded in the finished plates: the number of “double holes” (paired stitching holes indicating a removed mistaken stitch), the number of leaves used, and the number of stitches used. These measures are collected unbeknownst to workers and combined into an “attentiveness index.” After receiving the interim payment, treated workers’ attentiveness index increased by 0.077 SD across all workers (p = 0.092); among poorer workers, attentiveness increased by 0.17 SD (p = 0.041). This improvement occurred simultaneously with higher output speed — workers were producing plates faster while also making fewer mistakes, suggesting improved cognitive engagement rather than mere effort intensification.

Piece-Rate Comparison

In separate supplementary rounds with 150 experienced workers, the authors varied piece rates (Rs. 2, 3, or 4) while holding overall earnings constant. Each one-rupee increase in the piece rate raised output by 0.020 SD (p = 0.042). Critically, piece-rate increases produced no detectable change in the attentiveness index (point estimate negative, statistically insignificant), and the piece-rate effect on output differs significantly from the attentiveness effect (p = 0.001). This indicates that consciou effort and automatic attentiveness can move independently: higher incentives increase pace but do not reduce attentional lapses, whereas financial relief increases both pace and attentiveness.

Alternative Explanations Ruled Out

The authors systematically address gift exchange/fairness, trust, nutrition, and sleep. Fairness and gift-exchange stories are inconsistent with: (i) no detectable announcement-period effect; (ii) no decline in control-worker effort when treatment workers are paid before them; (iii) the pattern of effects being concentrated among poorer workers; and (iv) attentiveness being affected when it is not a sanctioned quality dimension for payment. Nutritional channels are inconsistent with overnight effect onset (nutritional stock changes are too slow biologically), no treatment effect on breakfast consumption patterns, and productivity effects persisting through the end of each workday. Sleep channels are inconsistent with no treatment effect on hours or quality of sleep.

Scope Conditions and Implications

The effect operates through the actual arrival of cash, not its anticipation, consistent with a model in which automatic cognitive inputs — unlike consciously chosen effort — respond to current financial strain rather than expected future income. Effects are concentrated among more financially constrained workers within an already-poor sample. The authors do not identify the specific psychological mechanism (worry, anxiety, affect, or rumination) but interpret results as evidence that financial strain, at least partly through psychological channels, reduces earnings exactly when money is most needed.

In depth

Q1. Why does the experiment focus on payment timing rather than an outright transfer of additional money?

Varying only payment timing — not total pay — holds constant both the piece-rate incentive and total wealth across treatment and control. An outright cash transfer would raise total lifetime income, potentially reducing effort through a neoclassical income effect (more lifetime wealth lowers the marginal utility of current consumption). By holding total compensation fixed and only shifting when it arrives, the design isolates the effect of financial strain per se, separable from any wealth or incentive effect.

Q2. Why is there no treatment effect during the announcement period, and why does this matter?

Between day 5 (when workers learn their payment schedule) and the interim payment date, treated workers know cash is coming but have not yet received it. Output in this window shows no treatment effect (0.014 SD, p = 0.685), and the announcement effect is significantly smaller than the post-pay effect (p = 0.008). This matters because it rules out mechanisms that should operate on information alone — including gift exchange, trust updating, or effort responses to higher discounted expected income — and is consistent with a model in which financial strain falls only when cash is physically received (e.g., moneylenders do not relent until the loan is actually repaid).

Q3. What is the attentiveness index and how was it constructed?

The attentiveness index averages three plate-level markers: (i) number of “double holes” — pairs of stitching holes indicating a mistaken stitch was removed; (ii) number of leaves used; and (iii) number of stitches used. Each component was normalized using the control group’s post-pay mean and standard deviation, then averaged and reverse-coded so that higher values denote better attentiveness (fewer mistakes, fewer leaves, fewer stitches). Workers were unaware these dimensions were being measured. The index thus captures the number of unforced steps a worker took to complete a plate — a behavioral trace of cognitive lapses.

Q4. How do the piece-rate rounds demonstrate that effort and attentiveness are separable?

In supplementary rounds (150 workers, 2019), piece rates were experimentally varied among Rs. 2, 3, and 4 per plate with the base wage adjusted to hold total earnings constant, so financial strain was unchanged. A one-rupee increase in the piece rate raises output by 0.020 SD (p = 0.042), consistent with a standard effort response. The same increase produces no discernible change in the attentiveness index (point estimate: negative but not significant), and the output and attentiveness effects are significantly different from each other (p = 0.001). This shows that workers can speed up via conscious effort without reducing attentional lapses, whereas the cash infusion raises both pace and attentiveness simultaneously — a pattern inconsistent with pure motivation as the mechanism.

Q5. What does the staggered timing within the treatment group (Wave A vs. Wave B) contribute to identification?

Treatment workers were randomized to receive their interim payment on day 8 (Wave A) or day 9 (Wave B). On day 9, Wave B workers have not yet been paid while Wave A workers have. If fairness concerns drove control workers to reduce effort upon seeing colleagues paid first, control workers on day 9 — having observed Wave A payments the evening before — should work less hard relative to Wave B treatment workers (who have also not yet been paid). The authors find no such pattern: the triple interaction (Cash × Payment Day × Wave B) is close to zero and insignificant, ruling out effort reductions from seeing peers paid earlier.

Q6. What are the magnitudes and timing of the spending response to the cash infusion?

Within three days of the interim payment, treatment workers spent Rs. 900 in total — roughly two-thirds of the average interim payment of over Rs. 1,400. On the day of the payment itself, loan repayments rose by Rs. 169 (746% increase), and household expenditures rose by Rs. 70 (68% increase). Over three days, loan repayments increased by Rs. 271 (287%), the probability of repaying any loan rose by 40 percentage points (222%), and total household spending rose by 65% (Rs. 371). These patterns indicate that the two main sources of financial stress cited by workers — outstanding debt and inability to meet household essentials — were directly addressed, suggesting a meaningful reduction in financial strain.

Q7. Why are the productivity effects concentrated among poorer workers, and what are the two interpretations?

Workers with below-median baseline wealth (fewer assets, lower liquidity) show a 0.204 SD (13.0%) productivity gain, while workers above the median wealth threshold show essentially no effect. The authors offer two interpretations. First, poorer workers may start from a higher level of financial strain, giving the intervention more scope to reduce it. Second, since all workers in the sample are objectively poor and report similar baseline financial worries and loan levels, the more likely explanation is that the interim payment is larger relative to the wealth and income buffer of poorer workers, making the same nominal cash infusion more meaningful for them. Both richer and poorer workers in the sample use the interim payment to repay loans and cover household needs.

Q8. How do the authors rule out nutritional channels?

Two tests address nutrition. First, workers were not at subsistence — 94% reported missing no meals the prior week — and increased food spending cannot change the nutritional stock overnight (the medical literature indicates nutritional-stock effects on cognition operate over longer time horizons). Second, and more precisely, all food consumed at the worksite during the workday was provided by the researchers, so differential pre-worksite breakfast consumption is the only plausible same-day biological channel. The authors find no treatment effect on breakfast consumption (whether workers had breakfast, how much, or what they ate). Further, if blood sugar or satiety drove effects, they should attenuate over the workday as all workers are given the same afternoon meal; instead, treatment effects persist and if anything increase through the final hours of the workday.

Q9. What does the self-report evidence on focus and worry show, and why is it treated as suggestive rather than primary?

Two days after the interim payment, workers were asked an open-ended question about what they were thinking about while working. Treatment workers were 11.5 percentage points (15.5%) more likely to report feeling focused on the task (p = 0.032) and 13.7 percentage points (32.7%) less likely to report thinking about financial worries (p = 0.044). A supplementary test showed treated workers were 10 percentage points (31%) more likely to generate explanations for a low-income person’s negative affect that were unrelated to financial concerns (p < 0.05), suggesting a broadening of cognitive scope. These measures are treated as suggestive because they were collected only at a single point and are self-reported; the primary evidence rests on objective production data because it is more objective and collected at fine hourly resolution throughout the post-pay period.

Q10. What does the paper say about optimal payment frequency as a policy implication?

The authors are cautious in drawing a direct policy inference about paying workers more frequently. While the positive productivity effect of early payment points toward more frequent paydays reducing financial strain, this must be weighed against workers’ self-control problems in consumption. In settings where workers face lumpy expenditure needs (e.g., monthly rent), more frequent payments could cause under-saving and worsen strain at the time of lumpy bills. The authors suggest payment frequency or size that matches expenditure needs, or more generally financial products that allow workers to time income receipts to coincide with expenses, as potentially more robust solutions — noting that such products appear largely absent in these markets.

Key Concepts

Financial strain (as used in the paper): A psychological burden arising from pressing present needs for resources — defined in the authors’ model as increasing in both the current marginal utility of consumption (i.e., how valuable an additional rupee would be today) and the level of outstanding debt (including lender harassment pressure). Strain is present-oriented: it responds to current cash-on-hand and debt levels, not to expected future income, which is why anticipating a payment does not fully relieve it.

Automatic input (a): In the authors’ behavioral model, one of two inputs into production. Unlike “effortful” input (e), which the worker consciously controls (speed of hands, consciously directed attention), the automatic input captures cognitive functions that are beyond the worker’s full control — background attentional processes that can be degraded by financial strain even when a worker is motivated and exerting high effort. The key behavioral assumption is that a falls when financial strain is high, independently of chosen effort.

Attentiveness index: A composite measure constructed from three unincentivized physical markers embedded in completed leaf plates: (i) number of double holes (pairs indicating a stitch was removed to correct a mistake); (ii) number of leaves used; (iii) number of stitches used. The index is normalized to the control group’s post-pay distribution and reverse-coded so higher values denote better attentiveness. Workers were unaware these dimensions were measured. The index captures attentional lapses — unforced errors that increase the number of steps and time needed to complete each plate.

Announcement period: The days between when workers are individually informed of their payment schedule (day 5) and when the interim payment is actually disbursed (day 8 or 9). This window serves as a within-experiment control: if effects arose from information about impending cash (e.g., through discounting, gift exchange, or trust), they should appear here. The consistent absence of treatment effects during this period is a key identification result.

Post-pay period: The days from the interim payment until the contract end (day 12). The main productivity and attentiveness treatment effects are estimated in this window, comparing treatment workers (who have received cash) to control workers (who have not yet been paid).

Lean season: The months outside the peak agricultural planting and harvesting periods (roughly six to eight months per year in the study area) during which agricultural workers seek intermittent casual employment in manufacturing, construction, and other sectors. Employment rates are low (1.9 paid days per week on average), income is low and variable, and financial strain is correspondingly high. The experiment is intentionally conducted during this period to maximize baseline levels of financial concern.

Piece-rate elasticity of effort: The responsiveness of output to changes in the marginal return per unit produced (the piece rate), holding financial strain constant. In the supplementary rounds, a one-rupee increase in the piece rate raises output by 0.020 SD. The authors interpret this as the upper bound on how much pure motivational effort can move output in this task, and use it to benchmark the cash infusion effects, which are roughly five times larger per unit of treatment variation and additionally move attentiveness (which piece-rate changes do not).

Do The Effects of Nudges Persist? Theory and Evidence from 38 Natural Field Experiments

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks why the Home Energy Report (HER) — a widely deployed social-comparison nudge that shows households how their electricity consumption compares to their neighbors — produces behavioral changes that persist long after the nudge is discontinued, while analogous nudges in other domains (charitable giving, financial savings, voter turnout, tax compliance) fade almost entirely within a year or two. The authors formalize a research design to decompose the HER’s long-run effectiveness into two channels: technology adoption (a change in the stock of energy-efficient capital in the home) and habit formation (a change in the stock of habits or skills in the resident).

The identifying strategy exploits the administrative rule that when the initial resident in an HER experiment moves out, HER mailings stop immediately — but electricity consumption in the home continues to be observed as new residents occupy it. Under three assumptions — (1) treatment assignment did not influence the initial resident’s decision to move; (2) treatment assignment did not influence the type of resident who moved in; and (3) energy-efficient technology adopted in response to the HER remained in the home after the move — the post-move HER effect identifies the fraction of the long-run treatment effect attributable to technology adoption (ATK), and the remainder identifies the fraction attributable to habit formation (ATH).

Data come from 38 natural field experiments administered by Opower between 2008 and 2013 across 21 U.S. residential energy providers, comprising 61,310,166 electricity bills for 1,810,096 homes. The mover sample, restricted to homes where the initial resident deactivated service at or after the receipt of their fourth HER, contains 5,890,855 bills for 139,908 homes. Treatment and control homes enter the mover sample at statistically indistinguishable rates and have similar baseline electricity consumption.

The main findings: the HER reduced electricity consumption by 2.1 percent in the long run (the pre-move ATE). After the initial resident moved and the HER was discontinued, 1.1 percent of the reduction persisted in the home — attributable to technology. The habit channel accounts for the remaining 1.0 percent reduction. Normalizing by the ATE, 51.4 percent (s.e. = 13.1) of the long-run effectiveness is attributable to technology adoption and 48.6 percent to habit formation. The persistence of the post-move effect is robust across alternative specifications, different HER-receipt cutoffs, balanced panels, and exclusion of low-consumption move-period homes. A falsification test using rental homes — where tenants do not typically own appliances and the technology channel is therefore shut down — yields a null post-move effect, consistent with the balanced-habits assumption.

The authors use these results to explain a broader empirical pattern: one year after discontinuation, social comparison nudges targeting compliance, charitable giving, savings, and voter turnout retain on average only 4 percent of their initial effect, while nudges targeting energy and water conservation retain 65 percent. The paper argues this divergence reflects the relative abundance of enabling technologies in conservation contexts versus their absence in compliance or voting contexts. The findings also have cost-benefit implications: ignoring HER-induced technology adoption overstates net benefits by as much as 65 percent, depending on assumed technology cost per kWh saved (ranging from $0.03 per kWh saved per Gillingham et al. 2018 to $0.12 per kWh saved per Billingsley et al. 2014).

Scope conditions: results are specific to electricity-consumption nudges in the U.S. residential sector; the technology channel identification requires that adopted equipment stays in the home after a move; the decomposition rests on a linear production function for outcomes in habits and technology.

Q: What is the Home Energy Report and how was it administered in these experiments? A: The HER is a mailed social-comparison report that contrasts a household’s electricity consumption with that of similar neighbors. In each of the 38 waves, homes were observed for a 12-month baseline, then randomly assigned to treatment (receiving HERs) or control. HERs were mailed monthly, bimonthly, or quarterly; generation ceased when the initial resident deactivated electricity service.

Q: What is the paper’s central identification strategy? A: The authors exploit a discontinuity created when the initial treated resident moves out: HER mailings stop, but the home’s electricity consumption continues to be measured as new residents move in. Under three assumptions about non-interference of treatment with moving decisions, balanced habits of subsequent residents, and stability of adopted technology, the post-move HER effect point-identifies the technology-adoption component (ATK) of the long-run average treatment effect (ATE). The habit-formation component (ATH) is then inferred as ATE minus ATK.

Q: What are the three identifying assumptions and how are they tested? A: Assumption 1 (no effect of treatment on moving rates) and Assumption 2 (balanced habits of subsequent residents) are tested with the data; treatment and control homes enter the mover sample at statistically indistinguishable rates and have similar baseline consumption, supporting Assumption 1. The rental-home falsification test supports Assumption 2: rental homes show a null post-move effect, consistent with renters having balanced habits because the technology channel is inactive in rentals. Assumption 3 (stable technology after a move) is untestable from the data; the authors note that violation of this assumption would imply the post-move effect is a lower bound on ATK, making the technology-adoption estimate conservative.

Q: What are the main quantitative estimates of the decomposition? A: The pre-move (long-run) ATE is -2.1 percent of baseline electricity consumption. The post-move effect (ATK) is -1.1 percent, and the habit-formation component (ATH) is -1.0 percent. Normalizing by the ATE, 51.4 percent (s.e. = 13.1) is attributed to technology adoption and 48.6 percent to habits.

Q: How large is the HER effect in absolute terms during the comparison period? A: During the comparison period, the HER reduced average daily electricity consumption by approximately -1.8 to -2.3 percent in the first year and -1.5 to -2.0 percent in the second year, with 95 percent confidence intervals excluding zero. In levels, these correspond to roughly -0.6 to -0.9 kWh per day — equivalent to using 2 to 4 sixty-watt incandescent bulbs for 5 fewer hours per day.

Q: How persistent is the HER effect during the move period? A: In the first year of the move period the HER continues to produce reductions of -1.7 and -1.4 percent; more than a year after the initial resident’s departure the estimated effect is -1.2 percent. All move-period estimates are statistically significant at conventional levels.

Q: How does the paper explain variation in persistence across social-comparison nudge contexts? A: One year after discontinuation, nudges targeting compliance, charitable giving, savings, and voter turnout retain on average only 4 percent of their initial effect, while nudges targeting energy or water conservation retain 65 percent on average. The paper argues the divergence reflects the relative availability of enabling technologies: households can adopt long-lived, input-efficient technologies (appliances, fixtures) to reduce energy and water use, but analogous technologies to facilitate compliance, donations, or voting are largely unavailable or absent.

Q: How does this paper’s finding about technology adoption compare to Allcott and Rogers (2014)? A: Allcott and Rogers (2014) used participation in utility-sponsored energy-efficiency programs as a proxy for technology adoption and found it explained no more than 2 percent of the HER’s long-run effectiveness. The authors reject this conclusion: their decomposition attributes 51.4 percent to technology, which is estimated precisely enough to statistically reject the 2 percent figure from Allcott and Rogers (2014). They attribute the discrepancy to the imperfect proxy used by Allcott and Rogers and low statistical power in analogous analyses.

Q: What are the cost-benefit implications of accounting for HER-induced technology adoption? A: Assuming monthly HERs for one year, a household electricity price of $0.10/kWh, and benefits accruing over two years, the baseline net benefit (ignoring technology costs) is $32.38 per household (electricity savings of $44.38 minus $12 administration cost). Using a technology cost of $0.03/kWh saved (Gillingham et al. 2018), net benefits fall to $27.14. Using $0.12/kWh saved (Billingsley et al. 2014), net benefits drop to $11.43 — a reduction of up to 65 percent from the baseline estimate. The HER still passes cost-benefit analysis but prior evaluations that ignore technology costs overstate net benefits substantially.

Q: How robust are the decomposition results to alternative sample definitions and specifications? A: The qualitative findings are stable across: alternative sets of control variables (Table A1); mover samples defined by receiving as few as 1 or as many as 5 HERs before moving (Table A2, with pre-move effects of -2.08 and post-move effects of -0.93 to -1.04 across cutoffs); balanced panels requiring fixed observation windows in each period (Table A3); and exclusion of homes showing unusually low consumption in the move period (Table A4, post-move effects of -1.19 to -1.48).

Q: What policy implications does the paper draw for nudge design? A: Policymakers seeking persistent nudge effects should target behaviors that can be augmented by readily available technologies, or pair social-comparison nudges with opportunities to adopt new technologies. In voting contexts, combining social-comparison nudges with opt-in mail-in or online ballot defaults could produce more persistent effects. In savings and charitable giving, pairing social comparisons with automatic contribution-rate defaults (as in Madrian and Shea 2001; Thaler and Benartzi 2004) is predicted to produce longer-lived effects than the nudge alone.

Q: What methodological contribution does the paper offer beyond the HER application? A: The mover-based decomposition is a generalizable research design for separating human capital (habits, skills) from physical capital (technology, infrastructure) as channels of policy effectiveness. The authors suggest it can be applied using other natural separation events — such as student graduation or employee departure — to assess the extent to which nudges build human capital in both recipients and the organizations in which they are embedded.

Technology adoption channel (ATK): The component of the HER’s long-run average treatment effect attributable to increases in the stock of energy-efficient technologies in the home — identified empirically as the post-move HER effect that persists after the treated resident departs and the HER is discontinued.

Habit formation channel (ATH): The component of the HER’s long-run treatment effect attributable to changes in the habits or skills of the resident — inferred as the residual after netting the technology component (ATK) from the total long-run effect (ATE).

Post-move effect: The estimated difference in electricity consumption between treatment and control homes after the initial resident has moved out, the HER has been discontinued, and a new resident has taken occupancy; under the paper’s identifying assumptions this equals ATK.

Balanced-habits assumption: The identifying assumption that treatment assignment did not influence the characteristics or habits of residents who subsequently moved into homes in the experimental sample, so that the habits of incoming residents are comparable across treated and control homes.

Stable-technology assumption: The identifying assumption that energy-efficient technologies adopted in response to the HER remain in the home after the initial resident moves; relaxing this assumption implies the post-move effect is a lower bound on ATK.

Home Energy Report (HER): A mailed social-comparison report that contrasts a recipient household’s electricity consumption with that of similar neighboring households; the treatment studied across all 38 experiments in this paper.

Enabling technologies: Long-lived, input-efficient capital goods (appliances, lighting, insulation) that reduce the marginal cost of conservation and thereby lock in behavioral changes induced by a nudge; their relative abundance in energy and water conservation contexts — versus their absence in voting, giving, or compliance contexts — is the paper’s proposed explanation for cross-context variation in nudge persistence.

Does Deposit Insurance Promote Deposit Stability? Evidence from the Postal Savings System during the 1920s

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research question. Does deposit insurance promote financial depth by arresting the outflow of deposits from the banking system during periods of bank distress? The paper tests and quantifies the deposit-stabilizing effect of state-level deposit insurance schemes operating in the United States during the 1920s.

Setting and identification. Between 1908 and 1929, eight primarily Midwestern states adopted some form of deposit insurance. The paper exploits the discontinuity in deposit insurance coverage at state borders to identify the causal effect of insurance on depositor behavior. The identification strategy compares outcomes in contiguous city pairs straddling deposit-insurance (DI) and non-deposit-insurance (NDI) state borders — a quasi-experimental design that controls for observed and unobserved confounders by using narrow geographic areas where the only relevant policy difference is the presence or absence of deposit insurance.

Proxy for “mattress money.” The paper uses postal savings deposits as a proxy for money withdrawn from the banking system. The U.S. Postal Savings System (established 1911) was backed by the full faith and credit of the federal government, with a maximum individual account limit of $2,500, and was widely viewed as a far safer alternative to commercial bank deposits. The authors validate this proxy by demonstrating, via Johansen cointegration tests, that the nationwide ratio of postal savings balances to total bank deposits is cointegrated (rank 1) with the currency-deposit ratio — a well-established indicator of banking distress.

Data. The empirical analysis covers 1921–1929. The main postal savings dataset is drawn from Annual Reports of the Postmaster General. Bank suspension data are drawn from FDIC manuscript lists compiled in the 1930s by FDIC economist Clark Warburton, providing location, charter type, and suspension/reopening dates. The sample includes 74 city pairs across 14 states (7 DI: North Dakota, South Dakota, Nebraska, Kansas, Oklahoma, Texas, Mississippi; 7 NDI: Minnesota, Iowa, Missouri, Arkansas, Louisiana, Tennessee, Alabama), with an average distance between paired cities of approximately 18 miles.

Main findings — postal savings regressions (Table 4). Using OLS with city-pair and year fixed effects and standard errors clustered at the NDI city level, the paper finds that following a bank suspension within a 10-mile radius, postal savings deposits in NDI cities grew 16 percent more than deposits in the corresponding DI city. The effect is positive and statistically significant at the 20-mile radius but smaller — approximately 9 percent — and is statistically indistinguishable from zero at the 30-mile radius. The localized decay with distance is consistent with a geographically contained flight-to-safety response. Critically, when the same specification is estimated for periods after deposit insurance was discontinued, the effect at all radii is statistically nil, providing a falsification test ruling out omitted unobserved factors as the driver.

Persistence of effects (Table 5). Arellano-Bond GMM dynamic panel regressions confirm that the disintermediation effects are persistent. The lagged dependent variable enters with a negative and statistically significant coefficient (approximately −0.20 for the 10-mile regression), indicating mean reversion, but the bank suspension coefficients remain robust. Implied long-run effects for the 10-mile and 20-mile equations are approximately 0.151 and 0.100, respectively, suggesting sustained rather than transitory deposit diversion away from the banking system in the absence of deposit insurance.

Banking capacity (Table 6). Because the postal savings deposit limit constrained the intake of funds — particularly severely during distress episodes, as documented through narrative evidence from the 1915 Congressional Record — the postal savings regressions underestimate the true effect of deposit insurance. The paper therefore estimates an alternative specification at the county level, comparing deposits at state-chartered banks in paired DI and NDI border counties. The results indicate that deposit insurance is associated with approximately a 56 percent increase in county-level deposits at state-chartered banks (coefficient 0.574, significant at 5 percent, robust to inclusion or exclusion of year fixed effects). By contrast, the analogous coefficient for national banks — which were prohibited by the OCC from participating in state deposit insurance schemes — is positive but statistically insignificant, providing a placebo test consistent with the interpretation that deposit insurance, not unobserved county characteristics, drove the banking capacity difference.

Scope conditions. All effects are estimated for state-chartered bank deposits in predominantly agricultural, Midwestern border counties during 1921–1929, a period characterized by an average annual bank suspension rate of 2.22 percent (versus 0.3 percent during 1911–1920). The paper acknowledges that state deposit insurance schemes of this era generated moral hazard (as established by prior literature), and frames the contribution as quantifying the stability-enhancing component rather than the net welfare effect.

Policy implication. The 56 percent banking capacity differential implies that deposit runoffs in the absence of insurance are substantially higher than the 3–10 percent runoff rates assumed in the Basel III Liquidity Coverage Ratio (LCR) framework, and more consistent with the 25–50 percent runoffs observed in non-systemic institutions in Denmark following an exogenous reduction in deposit insurance limits (Iyer et al., 2016).

In depth

Q1. Why is the Postal Savings System a valid proxy for “mattress money,” and what evidence supports this?

The postal savings system was backed by the full faith and credit of the United States, making it categorically safer than commercial bank deposits, and was explicitly designed to attract savings hidden in mattresses. The authors validate the proxy empirically by showing that the nationwide ratio of postal savings balances to total bank deposits is cointegrated (Johansen test, rank 1) with the currency-deposit ratio — a series that rises during banking distress as depositors convert bank funds to currency. Contemporary narrative accounts from the 1915 Congressional Record further confirm that postal savings offices experienced sharp deposit inflows during local banking distress, with deposit intake frequently constrained by the $2,500 individual account cap.

Q2. What is the identification strategy, and why does it address endogeneity concerns?

The strategy exploits the discontinuity in deposit insurance at state borders by comparing relative postal savings deposit growth in contiguous city pairs — one city in a DI state, one in an adjacent NDI state — conditioning on bank suspensions within 10, 20, or 30 miles. The authors argue that deposit insurance legislation was a statewide political decision driven largely by partisan composition (Democrats favored it, Republicans opposed it), making it implausible that interests concentrated at border cities systematically determined which states adopted it. Six of the seven NDI control states introduced deposit insurance legislation but failed to pass it, underscoring that the policy variation was not determined by border-specific characteristics. A falsification test using the same city pairs after deposit insurance was discontinued shows zero effects, ruling out time-invariant unobserved heterogeneity as the driver.

Q3. What are the main quantitative results from the city-pair postal savings regressions?

Following a bank suspension within 10 miles, postal savings deposits in NDI cities grew 16 percent more than in DI cities (coefficient 0.162, significant at 5 percent). At the 20-mile radius the differential is approximately 9 percent (coefficient 0.0933, significant at 5 percent). At the 30-mile radius the coefficient is 0.0997 and statistically indistinguishable from zero. These results are estimated with OLS using city-pair and year fixed effects and standard errors clustered at the NDI city level, based on 524 observations for the 10- and 20-mile specifications and 66 observations for the post-discontinuation falsification regressions.

Q4. How does the paper establish that distance matters for the flight-to-safety effect?

The monotonic decline in the estimated coefficient from 0.162 (10 miles) to 0.093 (20 miles) to a statistically insignificant 0.100 (30 miles) indicates that the diversion of deposits into postal savings was geographically localized. This pattern is consistent with depositors responding primarily to nearby bank failures rather than to distant ones, and it supports the interpretation that the effect is driven by local banking distress rather than by state-level or regional macroeconomic shocks that would affect all pairs symmetrically.

Q5. Are the disintermediation effects of bank suspensions temporary or persistent?

The Arellano-Bond GMM dynamic panel regressions (Table 5) show that the effects are persistent. The lagged dependent variable coefficient is approximately −0.205 (10-mile) and −0.188 to −0.201 (20-mile), indicating partial mean reversion but not full reversal. Year-1, Year-2, and implied long-run dynamic effects are all statistically significant and of similar magnitude (approximately 0.145–0.152 for the 10-mile equation and 0.096–0.100 for the 20-mile equation), indicating that once depositors shift funds to postal savings in response to bank suspensions, a substantial portion of the effect persists in subsequent years. This is consistent with prior literature showing that deposits leave the banking system quickly but return slowly.

Q6. Why are the postal savings coefficient estimates considered a lower bound on the true effect of deposit insurance?

Two institutional features constrained the postal savings system from fully capturing flight-to-safety deposits. First, individual accounts were capped at $2,500, and narrative evidence shows that this limit was severely binding during distress — depositors attempted to place far more than the ceiling allowed. Second, the re-depositing rate of postal savings funds back into local banks was not 100 percent: during 1921–1923 only 32–47 percent of postal savings deposits were re-deposited in banks, compared to 72–82 percent in calmer years. Because the postal savings system could not absorb unlimited deposits and did not fully recycle absorbed funds into local banking, its level understates the true flight of deposits from the banking system in NDI states.

Q7. How does the county-level banking capacity test address the censoring problem?

The paper estimates log-ratio regressions comparing county-level deposits at state-chartered banks in DI versus NDI border counties, using a “DI Active” indicator that switches on when deposit insurance is in effect in a given state-year and switches off when schemes are discontinued. Because different states discontinued their insurance at different times, there is sufficient within-county variation to identify the DI coefficient even with year fixed effects. The estimated coefficient of 0.574 (without year FE) and 0.557 (with year FE) translates to approximately a 56 percent higher deposit level in state-chartered bank counties with deposit insurance, with virtually identical estimates across specifications.

Q8. What is the placebo test for national banks, and what does it show?

National banks were prohibited by the Office of the Comptroller of the Currency from participating in state deposit insurance schemes. If deposit insurance — rather than unobserved county characteristics — is responsible for the 56 percent banking capacity premium, then county deposits at national banks in DI states should show no corresponding premium. The Table 6 results confirm this: the DI Active coefficient for national bank deposits is positive (0.165 to 0.267) but statistically insignificant, providing a falsification result consistent with the causal interpretation for state-chartered banks.

Q9. How does the paper situate deposit insurance’s stabilizing benefits relative to its moral hazard costs?

The paper explicitly frames its contribution as quantifying the stability-enhancing component of deposit insurance separately from the moral hazard component. It cites extensive prior literature (Calomiris 1992, 1993; Wheelock 1992, 1993; Wheelock and Wilson 1994) establishing that the 1910s–1920s state schemes generated moral hazard: insured banks reduced capital-to-asset ratios, relaxed lending standards, and increased risk exposure. The paper does not contest those findings but argues that the two effects are analytically separable and that the stabilization benefit had significant quantitative magnitude — a benefit that should be accounted for when assessing the net welfare effects of deposit insurance design.

Q10. What are the implications for the Basel III Liquidity Coverage Ratio framework?

The Basel III LCR formula assumes that during distress 3 percent of “stable deposits” and 10 percent of “less stable deposits” run off. The paper’s finding that deposit insurance is associated with a 56 percent increase in banking capacity implies that in the absence of insurance, deposit runoffs are far higher than these Basel assumptions — substantially larger than 10 percent and more consistent with the 25–50 percent runoffs observed for non-systemic banks in Denmark following an insurance limit reduction (Iyer et al. 2016). The authors argue their results suggest that empirical grounding for the LCR runoff assumptions remains insufficient, consistent with critiques by Allen (2014) and Diamond and Kashyap (2016).

Key Concepts

Postal Savings System (as “mattress money” proxy). The U.S. Postal Savings System (1911–) accepted deposits up to $2,500 per individual, backed by the full faith and credit of the United States. In this paper, postal savings deposits are used as a quantitative proxy for money withdrawn from the banking system during distress — “money under the mattress” — validated by cointegration with the currency-deposit ratio.

Policy discontinuity / border-pair design. The identification strategy exploits the fact that deposit insurance was adopted at the state level, creating a sharp policy discontinuity at state borders. Contiguous city pairs straddling DI and NDI state borders are treated as quasi-experimental units, with the within-pair difference in postal savings deposit growth serving as the outcome, controlling for time-invariant city-level heterogeneity and common time effects.

Relative Postal Savings Deposit Growth (RPS). The dependent variable defined as the log-ratio of postal savings deposits in the NDI city to postal savings deposits in the DI city within a pair, and then first-differenced over time. This construction controls for city-pair-level time-invariant characteristics and isolates the differential response to bank suspensions.

Bank suspension. In this paper’s context, a bank suspension is any closure of a bank (state-chartered or national) at a specific geographic location, as recorded in FDIC manuscript lists compiled by Clark Warburton during the 1930s. The variable used in regressions is the change in the number of suspensions within R miles (R = 10, 20, 30) of the paired postal savings offices.

Financial depth / local banking capacity. The paper uses county-level deposits at state-chartered banks as a measure of local banking market size. Deposit insurance is hypothesized to increase financial depth by preventing the diversion of funds out of the banking system during distress, and the 56 percent estimated premium is the paper’s primary measure of the insurance’s capacity-enhancing effect.

DI Active indicator. A time-varying binary variable equal to 1 when deposit insurance was legally in effect in a given state at a given time, and 0 otherwise (including after repeal). Because different states repealed their schemes at different times (Oklahoma 1923, Texas 1927, South Dakota 1927, North Dakota 1929, Kansas 1929, Nebraska 1930, Mississippi 1930), this variable provides within-county variation that identifies the banking capacity coefficient after controlling for county and year fixed effects.

Moral hazard vs. stability-enhancing components. The paper distinguishes analytically between the moral hazard effect of deposit insurance (insured banks undertake riskier projects, reduce capital buffers, relax lending standards) and the stability-enhancing effect (depositors retain funds in the banking system, preventing runs). The paper’s contribution is to quantify the latter component in isolation, using a setting where the two effects can be separated by focusing on depositor — rather than banker — behavior.

Double Robustness of Local Projections and Some Unpleasant VARithmetic

Mon, 01 Jan 0001 00:00:00 +0000

This paper provides formal theoretical results on the relative robustness of local projection (LP) and vector autoregression (VAR) confidence intervals for impulse response inference when the data generating process (DGP) is locally misspecified. The research question is whether the widely held belief that LP estimators are more robust to misspecification than VARs is theoretically justified, and if so, precisely under what conditions and with what consequences for VAR inference.

The analytical framework models the DGP as a stationary structural VARMA(1, ∞) that is local to an SVAR(1), of the form y_t = Ay_{t-1} + H[I + T^{-ζ}α(L)]ε_t, where the MA component T^{-ζ}α(L)ε_t represents misspecification that vanishes at rate T^{-ζ} as sample size T grows. The key rate parameter is ζ ∈ (1/4, 1/2), which corresponds to misspecification large enough to be detected with probability approaching 1 by conventional Hausman-type specification tests, yet small enough that the bias-variance trade-off between LP and VAR remains non-trivial asymptotically. The framework encompasses under-specification of lag length, omitted variables, temporal aggregation, measurement error, and failure of shock invertibility — essentially all sources of dynamic misspecification relevant to linearized DSGE models.

The main finding on LP is a “double robustness” result: the conventional LP confidence interval achieves correct asymptotic coverage for all ζ > 1/4, even when misspecification is large enough to be detected with certainty. The mechanism is that the omitted-variable bias in the LP regression is of order T^{-2ζ} = o(T^{-1/2}) when ζ > 1/4, because both the direct effect of omitted lags on the outcome and the covariance of the residualized regressor with omitted lags are each of order T^{-ζ}, so their product is negligible relative to the T^{-1/2} standard deviation. This is formally analogous to double robustness in partially linear regression and debiased machine learning: LP is consistent if either the outcome-equation controls or the first-stage controls are correctly specified.

In stark contrast, the VAR estimator carries asymptotic bias of order T^{-ζ}, which is non-negligible relative to its T^{-1/2} standard deviation for ζ ≤ 1/2. This causes the conventional VAR confidence interval to severely undercover: for ζ ∈ (1/4, 1/2) the coverage converges to zero, and for ζ = 1/2 it converges to a level strictly below the nominal level.

The “no free lunch” result formalizes the trade-off. Setting ζ = 1/2 and bounding the noise-to-signal ratio at M²/T, the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1). This worst-case bias is small if and only if the VAR asymptotic variance is close to that of LP. When the VAR standard error is less than half that of LP — which is typical in applied practice — worst-case coverage falls below 48% even for M = 1. Moreover, the least favorable misspecification takes the form of exponentially decaying MA coefficients peaking at horizon h, a pattern consistent with standard economic theories of adjustment costs, learning, or overshooting, and is difficult to rule out on prior grounds. The Hausman test also provides weak protection: when M = 1, the odds of the test failing to reject are nearly 3-to-1 at the 10% significance level.

Simulations using the Smets and Wouters (2007) model with T = 240 observations confirm these results. With lag length selected by AIC (median selected p = 2), VAR confidence intervals materially undercover at all but very short horizons while LP achieves close to nominal coverage throughout. Increasing lag length to p = 4 or p = 8 ameliorates VAR undercoverage at short horizons but at the cost of making VAR confidence intervals essentially as wide as LP intervals, with substantial undercoverage persisting at longer horizons. For p = 4 the total misspecification measure is M ≈ 3.23; for p = 8, M ≈ 1.89.

Scope conditions: results are pointwise asymptotic in fixed model parameters and horizon; they abstract from order-T^{-1} small-sample biases from persistence or the nonlinearity of the impulse response transformation. The LP robustness result requires controlling for lags that are strong predictors of the outcome or impulse variables; omitting lags with small-to-moderate predictive power does not threaten coverage.

Q: What is the precise sense in which LP confidence intervals are “doubly robust”?

A: LP is doubly robust in the sense of partially linear regression: its bias from misspecified MA dynamics is the product of two errors, the estimation error in the outcome-equation lag controls γ̂ − γ_0 and the estimation error in the first-stage lag controls ν̂ − ν_0. In the local-to-SVAR model each error is of order T^{-ζ}, so their product is of order T^{-2ζ} = o(T^{-1/2}) whenever ζ > 1/4, making the omitted-variable bias negligible relative to the T^{-1/2} standard deviation. This means the asymptotic distribution of the LP estimator is completely invariant to the misspecification parameters α(L) and ζ.

Q: How large does misspecification need to be before LP coverage is threatened?

A: The LP double robustness result holds for all ζ > 1/4 regardless of the magnitude parameter M of the MA misspecification. Misspecification with ζ ∈ (1/4, 1/2) can be detected with probability approaching 1 asymptotically by standard specification tests — in particular, the Hausman test is consistent for this range — yet LP coverage remains exactly correct. There is no threshold M below which LP fails; robustness is structural, not contingent on misspecification being small.

Q: Under what conditions does the VAR estimator have zero asymptotic bias?

A: The VAR asymptotic bias is zero if and only if the lagged shocks ε_{j*,t-ℓ} for ℓ = 1, …, h lie in the span of the lagged data used for estimation. Two sufficient conditions from Corollary 3.2 are: (i) the true model is SVAR(p_0) and the estimation lag length p satisfies h ≤ p − p_0, so the extra lags absorb the residual MA structure; or (ii) the shock of interest is directly observed and ordered first, and h ≤ p. In these cases the VAR estimator is asymptotically equivalent to LP, with equal variance.

Q: What is the “no free lunch” result for VARs?

A: For ζ = 1/2 and noise-to-signal ratio bounded by M²/T, the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1) (Proposition 4.1). This quantity is small if and only if aVar(δ̂_h) ≈ aVar(β̂_h), meaning the VAR has little efficiency advantage over LP. Put differently, the only way to guarantee robust VAR coverage is to include enough lags that the VAR confidence interval becomes as wide as the LP interval. There is no procedure that simultaneously offers narrower intervals than LP and reliable coverage.

Q: How severe is the worst-case undercoverage of conventional VAR confidence intervals?

A: From Corollary 4.3, even for M = 1 (a noise-to-signal ratio of just 1/T), worst-case VAR coverage falls below 48% whenever the VAR asymptotic standard deviation is less than half that of LP — a configuration typical in applied practice. For larger M the undercoverage is worse: the formula 1 − r(M√(aVar(β̂_h)/aVar(δ̂_h) − 1); z_{1-α/2}) can approach zero. Furthermore, the worst-case probability that VAR fails to cover AND the Hausman test fails to reject misspecification simultaneously exceeds 46% when the VAR standard deviation is less than half that of LP (Corollary 4.4).

Q: Can the researcher detect the problematic misspecification using a Hausman test before it causes undercoverage?

A: Only weakly. When M = 1, the Hausman test fails to reject misspecification with probability approximately 74% (odds of nearly 3-to-1) at the 10% significance level, since r(1; z_{0.95}) = 26%. At the 5% level the odds of non-rejection are nearly 5-to-1, since r(1; z_{0.975}) = 17%. The least favorable misspecification also cannot be ruled out on economic-theory grounds: the least favorable MA polynomial has exponentially decaying coefficients peaking at horizon h, consistent with adjustment costs, learning, or overshooting.

Q: Does using a bias-aware critical value (Armstrong-Kolesár approach) resolve the VAR undercoverage problem?

A: The bias-aware VAR confidence interval CI_B(δ̂_h; M) achieves correct asymptotic coverage by inflating the critical value based on the known bound M on misspecification. However, the bias-aware VAR interval tends to be wider than the LP interval. Specifically, M must be quite small — apparently below 1 — for the bias-aware VAR to dominate LP in width regardless of DGP and horizon. For M ≥ 2 (noise-to-signal ratio above 4/T), bias-aware VAR is dominated by LP in interval width. The practical conclusion is that the simpler LP interval is preferable in most empirically relevant settings.

Q: What does the minimax model-averaging result say about optimal weighting of LP and VAR?

A: From Corollary 4.2, the minimax optimal weight on LP when estimating a convex combination of LP and VAR estimators is M²/(1 + M²). For M = 1 (equal noise-to-signal threshold), the optimal weight is 50% on each. For M = 2, the LP estimator receives 80% weight. In the Smets and Wouters simulations, M ≈ 3.23 for p = 4 lags, corresponding to an optimal LP weight of approximately 91%, and M ≈ 1.89 for p = 8 lags, giving an optimal LP weight of approximately 78%.

Q: What do the Smets and Wouters simulations show about AIC-selected VARs?

A: In 5,000 simulated samples of T = 240 observations from the Smets and Wouters (2007) model, the AIC selects a median lag length of p = 2. At all but very short horizons, VAR confidence intervals materially undercover while LP confidence intervals throughout achieve close to nominal coverage. A bootstrap correction for VARs somewhat improves coverage but leaves large distortions. Increasing lag length to p = 4 or p = 8 moves coverage closer to nominal at short horizons (h ≤ p) but makes VAR confidence intervals essentially as wide as LP, and substantial VAR undercoverage persists at longer horizons.

Q: Is the no-free-lunch result specific to univariate impulse responses?

A: No. Proposition 4.2 extends the result to simultaneous inference on multiple impulse responses. For any k × 1 linear combination R of the impulse response vector, the worst-case squared bias is M² λ_max(R[aVar(β̂) − aVar(δ̂)]R’), where λ_max denotes the largest eigenvalue. Because VAR impulse response estimates are often highly correlated across horizons, undercoverage can be particularly severe in the multivariate (joint confidence ellipsoid) case. The no-free-lunch principle holds: the VAR ellipsoid offers non-negligible worst-case bias as long as it offers any efficiency gain relative to LP for any linear combination of horizon-specific impulse responses.

Q: What is the practical recommendation for lag selection in LP and VAR?

A: The paper offers three practical guidelines. First, LP researchers should control for those lags of the data that are strong predictors of the outcome or impulse variables, using conventional information criteria (such as AIC) applied to a VAR in all variables to select the number of lags for LP control — omitting lags with small-to-moderate predictive power does not threaten coverage. Second, VAR researchers should increase the lag length until the VAR confidence interval is no longer substantially narrower than the corresponding LP interval. Third, conventional specification tests do not suffice to guard against VAR coverage distortions.

Local Projection (LP) Estimator: The LP estimator for the impulse response at horizon h is the OLS coefficient on the shock variable y_{j*,t} in a direct regression of y_{i*,t+h} on y_{j*,t}, the variables ordered before it, and lagged data. It is a “direct” estimator in that it does not iterate a one-step VAR forward.

Double Robustness: A property of LP whereby its asymptotic bias from MA misspecification equals the product of two estimation errors — in the outcome-equation lag controls and in the first-stage residualization controls — each of order T^{-ζ}, making their product of order T^{-2ζ} = o(T^{-1/2}) for ζ > 1/4. This is the LP analogue of the double robustness of partially linear regression estimators in debiased machine learning.

Local-to-SVAR Misspecification: A DGP of the form y_t = Ay_{t-1} + H[I + T^{-ζ}α(L)]ε_t in which the MA term T^{-ζ}α(L)ε_t represents misspecification that vanishes at rate T^{-ζ}. The rate parameter ζ governs the magnitude; ζ ∈ (1/4, 1/2) is the empirically relevant range where bias is detectable by specification tests yet the bias-variance trade-off between LP and VAR remains non-trivial.

No Free Lunch (for VARs): The result that the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1), implying that the VAR confidence interval has reliable (robust) coverage if and only if the VAR asymptotic variance is close to that of LP — i.e., there is no way to simultaneously have shorter confidence intervals than LP and guaranteed coverage robustness.

Noise-to-Signal Ratio: The quantity T^{-1}||α(L)||² = trace{Var(T^{-1/2}α(L)ε_t) Var(ε_t)^{-1}}, which measures the total magnitude of the MA misspecification relative to the variance of the shocks. The paper bounds this at M²/T and uses M as the sufficient statistic for worst-case bias and coverage.

Bias-Aware Critical Value: An inflated critical value cv_{1-α}(b) solving r(b; cv_{1-α}(b)) = α, used to construct a VAR confidence interval CI_B(δ̂_h; M) that achieves correct asymptotic coverage by accounting for the worst-case bias M√(aVar(β̂_h)/aVar(δ̂_h) − 1). The paper shows this approach typically produces intervals at least as wide as LP for M ≥ 2.

Asymptotic Bias of VAR (aBias): The scaled bias term T^{ζ}E[δ̂_h − θ_{h,T}] converging to aBias(δ̂_h) = trace{S^{-1}Ψ_h H Σ_{ℓ=1}^∞ α_ℓ D H’(A’)^{ℓ-1}} − e’{i*,n} Σ{ℓ=1}^h A^{h-ℓ} H α_ℓ e_{j*,m}. This term is structurally absent from the LP asymptotics due to the double robustness mechanism.

Dynamic Concern for Misspecification

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper asks how an agent who fears that none of their probabilistic models is the correct description of the data-generating process (DGP) should update that fear as evidence accumulates, and what long-run behavior such an agent exhibits. The central contribution is making the concern for misspecification endogenous: the better the agent’s structured models explain past observations, the less concerned the agent becomes.

Decision Criterion

The agent posits a finite-dimensional parametric set of structured models Θ, holds a prior µ over Θ, and evaluates each action according to an average robust control criterion. This criterion takes a weighted average (over models) of robust control assessments, where each assessment penalizes expected utility for probability distributions that deviate from the structured model in terms of relative entropy, scaled by a misspecification concern parameter λ > 0. A standard subjective expected utility maximizer is the limiting case as λ → 0 (no concern), and a maxmin agent is approached as λ → ∞.

Endogenous Misspecification Concern

The concern parameter λ is updated each period as a function of the likelihood ratio test (LRT) statistic of the structured models against unstructured alternatives, scaled by a time-normalizing sequence βₜ: λ(hₜ) = LRT(hₜ, Θ) / (2βₜ). The sequence βₜ determines how demanding the agent is in evaluating model fit.

Taxonomy of Agent Types

Three types emerge based on the speed of βₜ:

Statistician type (βₜ = ct, linear): applies a time scaling that keeps the LRT asymptotically informative about the degree of misspecification. This is the unique type satisfying both safety (long-run average payoff at least ε-close to the maxmin guarantee, almost surely) and consistency under almost correct specification (no ε-regret when misspecification is small).
Lenient type (t = o(βₜ)): attributes unexplained evidence to sampling variability; corresponds to the Law of Large Numbers intuition.
Demanding type (βₜ = o(t)): overly penalizes small discrepancies, analogous to the Law of Small Numbers fallacy (Tversky and Kahneman, 1971).

Standard SEU maximization fails safety; robust control with an invariant λ (Hansen and Sargent, 2001; 2022) fails consistency under almost correct specification.

Long-Run Convergence Results (Theorem 1)

For a misspecified agent (no θ ∈ Θ with qθ_{a*} = p*_{a*}), the nature of the limit action a* depends on the agent type:

Lenient type: a* is a Berk-Nash equilibrium — an SEU best reply to beliefs supported on the models with minimum relative entropy from the true DGP.
Demanding type: a* is a maxmin equilibrium — a worst-case best reply to all models absolutely continuous with respect to the true DGP.
Statistician type: if behavior converges, a* is a c-robust equilibrium — a robust control best reply to beliefs on the relative entropy minimizers, with the concern for misspecification endogenously set at minθ R(p*{a*} || qθ{a*}) / c.

For a correctly specified agent (Proposition 2), every limit action is a self-confirming equilibrium, regardless of the agent type.

Cycles and Limit Frequency (Section 4, Theorem 2)

The statistician type’s behavior need not converge. In natural settings, the agent cycles between actions: playing a “safe” action whose consequences are well-explained by Θ reduces concern for misspecification, eventually leading to a riskier action whose poorly-explained consequences raise concern again, inducing a return to the safe action. The paper proves that every limit frequency (empirical distribution over actions) is a mixed c-robust equilibrium — a generalization that allows mixing while tying the concern for misspecification to the frequency-weighted average relative entropy of each action.

Empirical Applications

Monetary policy cycles (Sargent 1999, 2008): In a central bank model where the true DGP includes increased inflation variability under aggressive policy (a feature absent from the bank’s structured models), no pure c-robust equilibrium exists for small c. The model predicts persistent cycles between conservative and aggressive policy. The frequency of the conservative policy is increasing in the strength of the exploitable inflation-unemployment trade-off (θ₁π + θ₁a).
Labor supply under complex tax schedules (Rees-Jones and Taubinsky, 2020): Agents with a “schmeduling” heuristic (linearizing the tax schedule) are misspecified. Berk-Nash equilibrium predicts these agents exert excess effort, with the bias increasing in the complexity (convexity) of the tax code. The c-robust equilibrium attenuates this bias: conditional on the equilibrium, minθ R(p*_a || qθ_a) > 0, so agents maintain positive concern for misspecification and pull back from the biased recommendation. The paper rationalizes the empirical finding that approximately 40% of agents hold the schmeduling belief but only about 20% fewer agents act on it — consistent with endogenous concern reducing the behavioral impact of the biased model.

Axiomatization (Section 5)

The paper axiomatizes the static average robust control criterion (Theorem 3) using: a Variational Axiom (from Maccheroni, Marinacci, and Rustichini, 2006a), a Structured Savage axiom (Sure-Thing Principle for bets on the model identity), an Intramodel Sure-Thing Principle (STP for bets conditional on the model), and Uniform Misspecification Concern (the agent is equally concerned about misspecification regardless of which model is identified as best-fitting). Three additional dynamic axioms characterize preference evolution: Constant Preference Invariance (utility index stable over time), Dynamic Consistency over Models (Bayesian updating over structured models), and Q-Likelihood (misspecification concern increases in the LRT). A novel Asymptotic Frequentism axiom characterizes the statistician type: preferences must become arbitrarily similar (in a precise quantitative sense) after sufficiently long histories with the same outcome frequency.

In depth

Q1. What is the average robust control criterion and how does it generalize prior decision criteria?

A: An agent evaluates action a by averaging over structured models θ a robust control assessment: for each θ, minimize expected utility over probability distributions within relative entropy distance (penalized by 1/λ) of qθ_a, then integrate over θ with prior µ. This nests SEU (λ → 0, perfect trust in models), standard robust control of Hansen and Sargent (2001) (µ is Dirac, single benchmark model), and maxmin expected utility of Gilboa and Schmeidler (λ → ∞). The key extension is allowing µ to be nondegenerate, so the agent is simultaneously uncertain about the best-fitting model and about whether any model is exact.

Q2. What is the role of the likelihood ratio test statistic in driving misspecification concern?

A: The LRT statistic compares the maximum likelihood of the structured models against the best unstructured alternative. It diverges almost surely when the agent is misspecified, regardless of how close the structured models are to the true DGP. The concern parameter λ(hₜ) = LRT(hₜ, Θ) / (2βₜ) uses a time-scaling sequence βₜ to keep this statistic interpretable. Without scaling, a misspecified agent’s concern would always explode to infinity.

Q3. Why does linear time scaling (βₜ = ct) uniquely characterize the statistician type as rational?

A: Proposition 1 establishes two properties: (1) ε-safety — every βₜ = ct-optimal policy achieves average payoff at least ε below the maxmin guarantee, almost surely; (2) ε-consistency under almost correct specification — for DGPs sufficiently close to Θ, the agent avoids long-run regret. Part 2 of Proposition 1 shows that no βₜ with βₜ = o(t) or t = o(βₜ) satisfies both properties simultaneously. SEU fails safety; invariant-λ robust control fails consistency.

Q4. What is a c-robust equilibrium and how does it differ from a Berk-Nash equilibrium?

A: A Berk-Nash equilibrium (Esponda and Pouzo, 2016) requires the action to be an SEU best reply to beliefs supported on the relative entropy minimizers of the true DGP. A c-robust equilibrium requires the same support condition but with the best reply taken under the average robust control criterion, where the concern for misspecification λ equals minθ R(p*{a*} || qθ{a*}) / c — that is, the minimum relative entropy scaled by 1/c. The endogenous λ is positive whenever the agent is misspecified, so the agent does not fully trust even the best-fitting model.

Q5. How does the paper explain that misspecified lenient types converge to Berk-Nash while demanding types converge to maxmin?

A: For the lenient type (t = o(βₜ)), the time scaling makes the concern for misspecification converge to 0 (the LRT grows slower than βₜ relative to t), so the agent effectively behaves as an SEU maximizer with beliefs on the KL-minimizing models — the Berk-Nash condition. For the demanding type (βₜ = o(t)), the LRT diverges relative to βₜ, so λ → ∞ and the agent’s preferences converge to worst-case evaluation over all models absolutely continuous with the true DGP — the maxmin condition. These are Theorem 1, parts 1 and 2.

Q6. Why does the statistician type exhibit cycles rather than convergence?

A: Section 4 and Corollary 1 show in the monetary policy application that no pure c-robust equilibrium exists for small c. Intuitively, the conservative policy (a=0) is a best reply to a high misspecification concern, but it produces outcomes well-explained by Θ, which drives concern down. The aggressive policy (a=1) is a best reply to a low concern, but it generates increased inflation variability not captured in Θ, which drives concern up sharply. There is no fixed point that is self-sustaining, so the agent cycles. Theorem 2 shows that the empirical frequency of actions still converges to a mixed c-robust equilibrium.

Q7. What are the quantitative comparative statics for the monetary policy cycles?

A: Corollary 1 establishes that there exists a threshold c̄ > 0 such that for all c ≤ c̄: (1) no pure c-robust equilibrium exists; (2) a mixed c-robust equilibrium exists; and (3) in the maximal and minimal equilibria, the frequency of the conservative policy α*(0) is increasing in θ₁π + θ₁a — a larger exploitable trade-off between inflation and unemployment implies more time spent on the aggressive policy.

Q8. How does the model rationalize the Rees-Jones and Taubinsky (2020) labor supply finding?

A: Rees-Jones and Taubinsky (2020) find that approximately 40% of agents have incentive-compatible beliefs consistent with the schmeduling heuristic (linearizing a convex tax schedule), but approximately 20% fewer agents act according to that heuristic. In a Berk-Nash equilibrium, the schmeduling agent exerts excess effort relative to the optimum; the more convex the tax code, the larger the excess. In a c-robust equilibrium, the agent retains a positive misspecification concern proportional to the deviation between the convex tax schedule and the linear approximation. Higher effort levels are more exposed to uncertainty in the marginal rate (the misspecified term θ+ε multiplies a higher average income z), so the concern for misspecification provides a natural force that reduces effort below the Berk-Nash prediction. The paper notes this finding is also consistent with an alternative interpretation in Rees-Jones and Taubinsky where all agents hold schmeduling beliefs but under-respond behaviorally.

Q9. What is the mixed c-robust equilibrium and why does it always exist?

A: A mixed c-robust equilibrium is a mixed action α* ∈ Δ(A) such that beliefs ν are supported on the relative entropy minimizers Θ(α*) — computed as the parameter minimizing the α*-weighted average relative entropy across actions — and every action in the support of α* is a best reply under the average robust control criterion with λ = minθ Σ_a α*(a) R(p*_a || qθ_a) / c. Proposition 3 proves existence by mapping this fixed-point condition to a Nash equilibrium in an auxiliary game between the agent and two adversarial Nature players, then invoking Reny (1999) on that game. A pure c-robust equilibrium need not exist, but mixing over actions allows the concern for misspecification to be calibrated to the frequency of poorly-explained actions.

Q10. How does Theorem 2 formally connect cycles to mixed c-robust equilibria?

A: Theorem 2 states that if βₜ = ct for all t and α* is a βₜ-limit frequency (i.e., the empirical action distribution converges to α* with positive probability under some optimal policy), then α* is a mixed c-robust equilibrium. The intuition is that when α* places weight on both a well-explained action and a poorly-explained action, the time-averaged relative entropy stabilizes at a fixed level, producing a stable endogenous concern for misspecification that makes the agent asymptotically indifferent between the actions in the support — sharply reducing the incentive to break the cycle.

Q11. What does the axiomatization contribute beyond the learning results?

A: The axiomatization (Section 5, Theorem 3) provides behavioral foundations observable from choices, without assuming the internal LRT mechanism. Two primary axioms pin down the average robust control criterion within the variational class: Structured Savage (Sure-Thing Principle for bets over model identity) and Uniform Misspecification Concern (equal concern for misspecification regardless of which model is revealed as best-fitting). Dynamic Consistency over Models pins down Bayesian updating. Q-Likelihood axiomatizes that the concern for misspecification is ordinally increasing in the LRT. The novel Asymptotic Frequentism axiom (Axiom 9) pins down the quantitative speed of adjustment: long histories with the same empirical frequency must induce asymptotically similar preferences, and Proposition 5 shows this implies λ_{hₜ} / (LRT(hₜ, Q) / (2tₙ)) converges to a finite limit — exactly the statistician type’s linear scaling.

Q12. What is the correlation between behavioral biases that the model predicts?

A: The paper derives three novel empirical predictions about the cross-sectional and time-series correlation of uncertainty attitudes: (1) long-run uncertainty aversion positively correlates with initial misspecification and with belief in the Law of Small Numbers; (2) these correlations are causal — repeated model failures and overly demanding evaluation induce a shift toward cautious behavior; (3) even holding misspecification and probability reasoning fixed, limit uncertainty attitudes are stochastic, depending on whether the limit action’s outcomes are well-explained by the structured models.

Q13. How does Example 2 (Correlation Neglect) show that endogenous concern can amplify rather than attenuate biases?

A: In a double auction, a buyer who mistakenly treats their own valuation and the ask price as independent (Correlation Neglect, Esponda, 2008) bids below the optimum in Berk-Nash equilibrium. In a c-robust equilibrium, the positive correlation between valuations and prices produces a strictly positive minθ R(p*{a*} || qθ{a*}), so the agent maintains misspecification concern. Since lower bids are accepted with lower probability (and thus are less sensitive to model misspecification), the endogenous concern drives the agent to bid even lower — amplifying the bias rather than attenuating it. This example illustrates that the direction of the correction depends on the geometry of how the misspecification interacts with the payoff structure.

Key Concepts

Average Robust Control Criterion: The decision criterion proposed in the paper. An agent evaluates action a by taking the expectation over structured models θ (with prior µ) of min_{p_a ∈ Δ(Y)} [E_{p_a}[u(a,y)] + (1/λ) R(p_a || qθ_a)]. This is a weighted average of robust control assessments, each penalizing distributions that deviate from a structured model in relative entropy. The parameter λ > 0 governs the intensity of misspecification concern, with SEU as the limit at λ → 0 and maxmin at λ → ∞.

Endogenous Misspecification Concern: Unlike prior robust control models where λ is fixed or set externally, here λ(hₜ) = LRT(hₜ, Θ) / (2βₜ) is a function of how well the structured models explain the observed history hₜ via the likelihood ratio test statistic. The better the models explain past data, the smaller λ becomes and the less the agent hedges.

Statistician Type: An agent who scales the likelihood ratio test statistic with a linear time sequence βₜ = ct for some c > 0. This is the unique agent type satisfying both ε-safety (guaranteed long-run average payoff above the maxmin guarantee minus ε) and ε-consistency under almost correct specification (no long-run regret when misspecification is small). The statistician type’s linear scaling is the only one for which the LRT statistic retains asymptotic informativeness about the degree of misspecification.

c-Robust Equilibrium: A fixed-point concept for the long-run behavior of the statistician type. Action a* is a c-robust equilibrium if it is an average robust control best reply to beliefs supported on Θ(a*) = argmin_θ R(p*{a*} || qθ{a*}), with misspecification concern λ = minθ R(p*{a*} || qθ{a*}) / c. This generalizes Berk-Nash equilibrium by incorporating an endogenous hedging motive proportional to the minimum relative entropy between the true DGP and the best structured model.

Mixed c-Robust Equilibrium: A generalization of c-robust equilibrium to mixed actions α* ∈ Δ(A) for environments where no pure equilibrium exists. The beliefs are supported on the models minimizing the α*-weighted average relative entropy, and the misspecification concern is tied to that average entropy. Every βₜ-limit frequency is a mixed c-robust equilibrium (Theorem 2). This concept characterizes the long-run time-average behavior when the statistician type cycles.

Law of Small Numbers (LSN) Type / Demanding Type: An agent for whom βₜ = o(t), meaning the time scaling grows sub-linearly. This agent is excessively sensitive to early model failures (analogously to the Law of Small Numbers fallacy of Tversky and Kahneman, 1971, where short-run frequencies are treated as the long-run norm). The long-run behavior of such a type converges to maxmin behavior rather than robust control.

Asymptotic Frequentism (Axiom 9): A novel axiom requiring that conditional preferences after sufficiently long histories with the same empirical outcome frequency must be arbitrarily similar (in a quantitative sense defined by measuring rods x, y, E) to a limiting preference. This axiom axiomatically pins down the statistician type’s linear time scaling: it implies that the ratio λ_{hₜ} / (LRT(hₜ, Q) / (2t)) converges to a finite limit c, exactly characterizing βₜ = ct.

Berk-Nash Equilibrium: The equilibrium concept (Esponda and Pouzo, 2016) that describes the long-run behavior of lenient (SEU) agents learning under misspecification. An action a* is a Berk-Nash equilibrium if it is an SEU best reply to beliefs supported on Θ(a*) — the KL-minimizing models — without any additional hedging against misspecification. The current paper shows that lenient types converge to Berk-Nash equilibria, while statistician types converge to c-robust equilibria that differ by incorporating a positive misspecification concern.

Education and the Margins of Cyclical Adjustment in the Labor Market

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research question. This paper asks how the cyclical sensitivity of wages varies with workers’ educational attainment, what mechanisms drive the differences, and what the welfare consequences are of ignoring this heterogeneity. The starting point is a well-known asymmetry: less-educated workers have much higher and more volatile job separation rates, yet the standard macroeconomic literature has treated wages as roughly acyclical for a representative worker. Doniger asks whether this employment-centric picture is incomplete—and finds that it is, in a direction opposite to what the employment pattern would suggest.

Data and methodology. The paper uses two primary data sources: the National Longitudinal Survey of Youth 1979 (NLSY), which provides detailed job histories enabling identification of current and completed employer tenure, and the Current Population Survey (CPS) from 1995 to 2020, used both for employment flow statistics and, via biennial Job Tenure Supplements, for replication of the main wage findings. The sample is restricted throughout to males with 0–30 years of potential experience, following the conventions of the user-cost-of-labor (UCL) literature (Kudlyak, 2014; Basu and House, 2016). Workers are grouped into three educational categories: less than high school, high school or some college, and bachelor’s degree or more.

A key methodological contribution is a new, more parsimonious estimator for the cyclical sensitivity of the UCL. Rather than the multi-step indicator-variable approach of Kudlyak (2014), the paper recovers the UCL sensitivity from interaction terms between a flexible function of tenure and the cyclical position at the time of hiring, estimated within an augmented Mincer regression. This estimator admits higher-frequency identification, enables transparent inference via the delta method, and facilitates nonparametric impulse response estimation via the Jorda (2005) local projection method. Cyclical position is measured primarily as the deviation of the unemployment rate from an HP-filtered trend (lambda = 100,000), with robustness checks using the Hamilton (2018) filter and GDP-based detrending.

Main findings — employment. Monthly separation rates from the CPS (1995–2020) show that workers with less than a high school degree separate at a rate of 9.4 percent per month, more than twice the 3.4 percent rate for workers with a bachelor’s degree or more, regardless of cyclical position. The volatility of the separation rate (measured by the time-series standard deviation) is also larger for the least educated (1.7) than for the most educated (0.6). All sub-components of separation-to unemployment, to inactivity, and job-to-job transitions-exhibit the same ordering. In response to a 100 basis point monetary policy contraction (Romer and Romer, 2004 shocks), employment of workers with less than a high school education falls significantly, while employment of college graduates or more is statistically unaffected.

Main findings — wages. Using the NLSY, the cyclical sensitivity of the UCL to a 1 percentage point deviation of the unemployment rate from trend is estimated at approximately −15.5 percent for workers with a bachelor’s degree or more, −4.9 percent for high school or some college workers, and −1.4 percent (statistically indistinguishable from zero) for workers without a high school degree. In contrast, average hourly earnings (AHE) show much smaller and more compressed differences across education groups (−1.4, −1.1, and −1.0 percent respectively). The pattern of increasing procyclicality with education holds for new hires’ wages (NHW) as well but is considerably less stark than for the UCL. Replication in the CPS confirms the ordering: UCL sensitivities are −7.0 percent for college graduates, −2.9 percent for high school or some college, and effectively zero for those without a high school degree.

Mechanism. Counterfactual decompositions show that differences in the cyclical sensitivity of the wage-tenure profile—not just differences in job duration (separation rates)-account for the vast majority of the divergence across education groups. When separation rates are held constant across groups, the UCL sensitivity of the college-educated falls from -15.5 to −13.0 percent; when wage-tenure profile sensitivities are held constant, it falls to −6.3 percent, and the ordering across groups largely disappears. This finding is consistent with implicit contracting theory (Thomas and Worrall, 1988): longer expected employment durations for the more educated make it optimal to defer a greater share of the wage response to shocks over time, rendering near-term rigidities functionally less binding and producing more persistent effects of hiring-period conditions on subsequent wages.

Robustness. After controlling for cyclical sorting in match quality using the Hagedorn and Manovskii (2013) proxies (cumulated market tightness during tenure and leading up to the present job), the UCL sensitivity for college graduates falls modestly to −12.4 percent, confirming that match-quality composition effects account for only a minority of the documented pattern. The monetary policy shock analysis (Romer-Romer shocks identified from Greenbook forecast errors) yields a 35 percent decrease in the UCL for the most educated at the two-year horizon following a 100 basis point contraction, with no discernible effect for the least educated.

Welfare consequences. Using a stylized New Keynesian model extended to two labor varieties with heterogeneous wage flexibility, the paper shows that ignoring the documented heterogeneity leads to underestimating the welfare costs of business cycle fluctuations by more than 15 percent under the baseline calibration (unit Frisch elasticity and unit elasticity of intertemporal substitution). Conditional on this model, the welfare loss due to fluctuations for the least educated is more than 15 times larger than for the most educated. The paper explicitly notes this is a conservative lower bound, because the model assumes pooled household consumption, and admitting idiosyncratic consumption risk would disproportionately burden less-educated workers who bear adjustment on the extensive (employment) rather than intensive (wage) margin.

In depth

Q1. What is the user cost of labor (UCL), and why does the paper use it rather than average hourly earnings or new hires’ wages?

The UCL, formalized by Kudlyak (2014), is the present discounted value of wage payments an employer expects to make to a worker over the duration of the employment relationship, net of the continuation value of retaining that worker. It equals the new hire’s wage plus the expected wage wedge—the discounted stream of future wage differences between workers hired in the current period versus workers hired one period later. Unlike average hourly earnings or new hires’ wages, the UCL captures the persistent effects of macroeconomic conditions at the time of hiring on all future remitted wages, making it the appropriate allocative wage concept from a macroeconomic standpoint. The paper documents that AHE understates the cyclicality of wages for all groups but especially for the most educated, because AHE omits the highly cyclically sensitive expected wage wedge that characterizes college-educated employment relationships.

Q2. How does the paper’s new estimator for the cyclical sensitivity of the UCL differ from the existing method, and what does this enable?

The existing Kudlyak (2014)/Basu and House (2016) method recovers the UCL by estimating a very large set of date-of-hire x current-date indicator interactions, constructing a time series of the UCL, and then analyzing that series—a multi-step procedure that loses covariances across steps and makes cross-sectional disaggregation or high-frequency identification impractical. The new method instead estimates the UCL sensitivity directly from coefficients on the interaction between a flexible tenure function and the cyclical position at hiring, estimated within a single augmented Mincer regression. The UCL semi-elasticity is recovered analytically from these coefficients via a formula that sums discounted weighted differences in the tenure-interaction coefficients across the tenure horizon. This single-step approach allows transparent inference via the delta method, enables fully interacted specifications for heterogeneous subgroups, permits the hiring-date frequency (e.g., weekly in NLSY) to differ from the wage observation frequency (annual or biannual), and permits estimation from repeated cross-sections—all of which were infeasible in the prior approach.

Q3. What are the quantitative magnitudes of the education gradient in UCL cyclicality, and how do they compare across wage measures?

Using the NLSY with unemployment deviations from HP-filtered trend as the cyclical indicator: the UCL sensitivity is −15.5 percent (se 3.86) for workers with a bachelor’s degree or more, −4.9 percent (se 1.52) for high school or some college, and −1.4 percent (se 2.48, statistically insignificant) for those without a high school degree. By contrast, new hires’ wages show sensitivities of −3.4, −1.8, and −1.2 percent respectively, and average hourly earnings show −1.4, −1.1, and −1.0 percent. The gradient is largest and most statistically significant for the UCL, indicating that the bulk of the education gap in cyclical wage sensitivity operates through the persistent effect of hiring-period conditions on subsequent wages rather than through the contemporaneous wage alone.

Q4. What mechanism accounts for the UCL gradient — differential job durations or differential sensitivity of the wage-tenure profile?

The paper decomposes the UCL into the new hire’s wage and the expected wage wedge, and performs counterfactual exercises holding either separation rates or wage-tenure profile sensitivities constant across education groups (Table 3). Holding separation rates constant while allowing wage-tenure profiles to differ reduces the college-educated UCL sensitivity only modestly, from -15.5 to −13.0 percent; holding wage-tenure profile sensitivities constant while allowing separation rates to differ reduces the college-educated sensitivity to −6.3 percent and compresses the education gradient substantially. Thus, differential sensitivity of the wage-tenure profile—the degree to which wages continue to respond to hiring-period conditions over the course of the job-is the primary driver of the UCL gradient, with differential separation rates playing a secondary but non-trivial role. This finding confirms the prediction of Thomas and Worrall (1988) that lower separation rates support greater use of deferred payment and intertemporal risk sharing in optimal wage contracts.

Q5. How does the paper rule out cyclical sorting in match quality as the explanation for the UCL gradient?

Workers hired during recessions may be of systematically lower match quality, producing persistently lower wages not because wages are more cyclically sensitive for the same quality match but because recession hires are worse matches. Using the Hagedorn and Manovskii (2013) proxies for match quality - cumulated market tightness during the worker’s tenure on the present job (mjob) and on all prior jobs leading to it (mctj) - the paper augments the wage regression with full interactions between these proxies and the tenure-cyclicality terms. After controlling for match quality, the UCL sensitivity for college graduates falls from -15.5 to −12.4 percent (se 5.56); the point estimate remains large, statistically significant, and well above the estimates for lower-education groups. Figure 4 shows that match-quality adjustment primarily affects the first two years of the wage-tenure profile, after which the bias from cyclical sorting fades, confirming that scarring in remuneration for college graduates hired in recessions persists beyond what sorting can explain.

Q6. What do monetary policy shocks reveal about the education gradient in wage sensitivity?

Monetary policy shocks (identified from Greenbook forecast errors as in Romer and Romer, 2004) subject all labor markets to the same aggregate demand shock simultaneously, providing a cleaner test of differential responsiveness than cyclical regressions that may conflate demand composition and supply factors. Using Jorda (2005) local projections, a 100 basis point monetary policy contraction is associated with a 35 percent decrease in the UCL for workers with a bachelor’s degree or more at the two-year horizon, with statistically insignificant effects on the UCL of workers without a high school degree. The employment results are symmetric: less-educated workers’ employment falls significantly after a monetary contraction, while college-educated workers’ employment is unaffected. This cross-validation using monetary policy shocks supports the main thesis that more-educated workers absorb aggregate demand variation through the wage margin, while less-educated workers absorb it through the employment margin.

Q7. How does acyclical wages for the least educated affect interpretation of the existing macro literature on wage rigidity?

The aggregate finding of Kudlyak (2014) and Basu and House (2016)-that the UCL is more procyclical than new hires’ wages or average hourly earnings, casting doubt on wage rigidity as an amplification mechanism—holds only for educated workers. The paper finds that the UCL for workers without a high school degree is statistically acyclical by all three wage measures. This result restores a potential role for nominal wage rigidity in generating amplification and persistence of shocks for less-educated labor markets, including in the Diamond-Mortensen-Pisarides class of search models criticized by Kudlyak (2014) and in New Keynesian models criticized by Basu and House (2016). The paper therefore reconciles the literature on wage rigidity with the empirical finding of cyclical employment volatility concentrated among the less educated.

Q8. What is the welfare calculation, and what are its key results and limitations?

The welfare exercise uses a parsimonious New Keynesian model with two labor varieties (capturing more- and less-educated workers) and price and wage rigidities. The model is extended to admit heterogeneous wage flexibility, and the welfare costs of fluctuations are evaluated following the second-order approximation method of Gali et al. (2007). Under the baseline calibration (unit Frisch elasticity, unit elasticity of intertemporal substitution), the heterogeneous-worker economy incurs welfare costs of fluctuations that exceed those of the output-gap-equivalent representative agent economy by more than 15 percent. The welfare loss of the least-educated workers is more than 15 times that of the most educated. The paper explicitly characterizes this as a conservative lower bound: the model assumes pooled household consumption (within varieties), which implies equal consumption sensitivity across education groups, whereas in reality less-educated workers face income loss on the extensive margin without the wage smoothing available to the more educated. Relaxing this assumption, as in Krusell et al. (2009), could yield welfare losses an order of magnitude larger.

Q9. What does the CPS replication add, and what are its limitations relative to the NLSY baseline?

The CPS replication (Table 7) confirms the main ordering: UCL sensitivities are −7.0, −2.9, and approximately 0 percent for college graduates, high school or some college, and less than high school respectively. This rules out the concern that the NLSY findings are artifacts of the single aging cohort that characterizes the NLSY 1979. However, the CPS must be treated as a repeated cross-section because the tenure data are only available biennially and individual-level panel linkage across tenure supplement waves is infeasible. As a result, the CPS estimates cannot include individual fixed effects and must rely more heavily on observable controls (industry, occupation) to absorb cyclical variation in workforce composition. The CPS also precludes the match-quality controls of Hagedorn and Manovskii (2013). Despite these limitations, the main qualitative and directional findings replicate.

Q10. What policy implications does the paper draw for monetary policy?

The paper argues that because less-educated workers bear adjustment to aggregate demand shocks disproportionately through the employment margin while their wages are acyclical, welfare assessments that focus on the aggregate output gap underweight the costs borne by less-educated workers. The paper suggests that re-optimizing the monetary policy rule to account for documented heterogeneity would entail placing greater weight on the unemployment rate of the least-educated when measuring the output gap. More broadly, the K-shaped nature of labor market adjustment across education groups — wage scarring for the educated versus employment volatility for the less educated - implies that policies targeting either margin in isolation will miss welfare costs concentrated in the other group.

Key Concepts

User Cost of Labor (UCL). The allocative wage from the employer’s perspective, defined as the present discounted value of expected future wage payments to a worker hired at date t, net of the continuation value of retaining that worker in the next period. Formally, UCL_t = w_{t,t} + E_t[sum beta^j(1-s)^j (w_{t+j,t} - w_{t+j,t+1})], decomposing into the new hire’s wage and the expected wage wedge. In this paper’s usage, the UCL is the appropriate measure of the cyclical impact of shocks on labor costs because it captures persistent effects of hiring-period conditions on the entire subsequent wage sequence, not just the contemporaneous wage.

Expected Wage Wedge (EWW). The component of the UCL beyond the new hire’s wage: the discounted stream of differences between wages a worker hired at date t will receive in future periods and the wages a worker hired one period later would receive in those same future periods. The EWW is non-zero whenever wages are history-dependent - i.e., whenever current macroeconomic conditions at the time of hiring affect future remitted wages. The paper finds that the EWW is larger, more negative, and more persistent for more-educated workers conditional on being hired during a cyclical downturn.

Self-enforcing implicit wage contract. A labor contract in which the sequence of remitted wages is not pinned down period-by-period by spot-market forces but instead reflects an intertemporal risk-sharing arrangement between employer and worker that is sustained by the mutual benefit of the ongoing employment relationship. In this paper’s framework (drawing on Thomas and Worrall, 1988), lower separation rates make longer planning horizons feasible, which in turn expands the scope for deferring wage adjustments across time - effectively allowing more-educated workers and their employers to smooth the effects of cyclical shocks over longer horizons than is possible for less-educated workers with shorter expected job durations.

Cyclical sorting / match quality bias. The compositional concern that workers hired during recessions may be of systematically different (in this context, lower) match quality than those hired during booms, so that the persistent wage depression observed for recession hires could reflect poor match quality rather than cyclically sensitive wages for equivalent-quality matches. The paper uses the Hagedorn and Manovskii (2013) proxies - cumulated labor market tightness during the current job and prior employment history - to control for cyclical variation in match quality and assess the residual sensitivity of the UCL for average-quality matches.

Extensive versus intensive margin of labor market adjustment. The distinction between adjustment through changes in the number of workers employed (extensive margin: hiring and separation) versus adjustment through changes in wages or hours conditional on employment (intensive margin). A central finding of the paper is that less-educated workers bear cyclical adjustment disproportionately on the extensive margin (more volatile separation rates, employment losses following monetary contractions) while their wages are acyclical, whereas more-educated workers exhibit the reverse: stable employment but highly cyclically sensitive wages, especially as measured by the UCL.

Wage scarring. The persistent negative effect of hiring-period macroeconomic conditions on wages throughout the subsequent employment spell, beyond what is explained by contemporaneous market conditions. In this paper’s context, wage scarring is concentrated among more-educated workers: being hired when the unemployment rate is one percentage point above trend is associated with wages that remain depressed for several years, with the depression being larger and more persistent for college-educated workers than for those with less education. This is demonstrated via the expected wage wedge profiles in Figure 3 and is confirmed to survive controls for match-quality sorting.

Output-gap-equivalent representative agent economy. A conceptual benchmark constructed in the paper’s welfare analysis: a single-worker-type New Keynesian economy whose wage and labor supply elasticities are set equal to the output-elasticity-weighted averages of the two labor variety types in the heterogeneous economy. The paper shows that the heterogeneous-worker economy and this representative-agent benchmark produce identical aggregate output gap and price level paths (under Cobb-Douglas production, earnings elasticities are identical across varieties), but welfare diverges because period utility is more volatile for the variety with more rigid wages. The 15 percent excess welfare cost of the heterogeneous economy relative to this benchmark is the paper’s headline welfare result.

Energy Transitions in Regulated Markets

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how rate-of-return (RoR) regulation in U.S. electricity markets affects the speed and efficiency of energy transitions, specifically the transition from coal to combined-cycle natural gas (CCNG) generation driven by fracking-induced cost declines. The authors build and estimate a structural model of regulated utility behavior in which utilities optimize investment, retirement, and hourly operations decisions against an incentive structure set by state Public Utility Commissions (PUCs).

The regulatory environment combines two instruments: (1) an allowable rate of return that is decreasing in consumer electricity rates (incentive regulation), parameterized as s = (r/r₀)^{-γ}, where higher γ penalizes high-cost outcomes more severely; and (2) a “used-and-useful” standard in which a coal plant’s contribution to the rate base depends on its capacity utilization via a logit function. These two instruments create a tension: utilities want to lower costs to earn a higher RoR, but also want to run existing coal plants—even when uneconomical—to prove they are “used and useful” and thus maximize their rate base and profits.

The authors estimate the model using publicly available EIA and EPA CEMS data spanning 2006–2017, covering 39 unique regulated utilities in the Eastern Interconnection across more than 4 million utility-hour observations (459 utility-years). Structural parameters are recovered via a nested fixed-point indirect inference approach that matches simulated regression coefficients to actual data; investment and retirement costs are estimated with a GMM nested fixed-point approach.

Key reduced-form findings confirm the model’s two core mechanisms. First, a 10% increase in total variable costs is associated with a 2.5% decrease in variable profits per MW of capacity (with utility fixed effects), consistent with incentive regulation. Second, regulated utilities reduce coal generation by only a statistically insignificant 4.2 percentage points when coal fuel costs exceed import prices, compared to 16.1 percentage points for restructured utilities—consistent with regulated utilities running coal out-of-dispatch order to preserve used-and-useful status.

In counterfactual simulations that impose 2018–20 natural gas prices ($2.01/MMBtu versus the 2006 price of $7.24/MMBtu) on utilities with their 2006 capital stocks, regulated utilities retire only 53% of coal capacity over 30 years and increase CCNG capacity by 296%, whereas a cost minimizer would retire most coal capacity while increasing CCNG by only 58%. The Averch-Johnson over-investment effect dominates: regulated utilities over-invest in CCNG while simultaneously over-using legacy coal.

Carbon taxes on regulated utilities reduce short-run coal generation only 48% as much as when imposed on a cost minimizer (because the used-and-useful incentive partially offsets the carbon price signal), but in the long run result in 68% lower coal capacity and 77% lower coal generation relative to baseline by year 30—larger effects than for the cost minimizer. Eliminating the coal usage incentive (μ₂ = 0) produces 82% lower coal capacity and 92% lower coal generation over 30 years but requires utility variable profits to fall by over $300 million, threatening reliability without compensating transfers.

Scope conditions: Results apply to regulated (non-restructured) utilities in the Eastern Interconnection, 2006–2017. The model estimates the coal-to-CCNG transition only; it explicitly does not model the ongoing transition to renewables and storage due to insufficient data variation.

In depth

Q1. What is the central research question?

The paper asks whether and how rate-of-return regulation in U.S. electricity markets slows energy transitions, and what alternative regulatory structures or carbon tax policies could accelerate the transition away from coal. It addresses this both theoretically—through a structural model of regulated utility behavior—and empirically, through estimation and counterfactual simulation using data on 39 regulated utilities over 2006–2017.

Q2. What are the two key regulatory instruments in the model, and what distortions do they create?

The first instrument is incentive regulation: the allowable rate of return declines as consumer electricity rates rise (s = (r/r₀)^{-γ}), so utilities have an incentive to lower costs. The second is the used-and-useful standard: a coal plant’s contribution to the rate base depends on its capacity utilization via a logit function, creating an incentive to run coal plants even when their fuel costs exceed import prices. Together, these instruments generate a tension between cost-reduction incentives and legacy-capacity-preservation incentives, causing the regulated utility to both over-invest in new CCNG capacity (Averch-Johnson effect) and over-use existing coal capacity relative to the cost-minimizing benchmark.

Q3. What does the reduced-form evidence show about uneconomical coal usage?

In a triple-difference specification, regulated utilities reduce coal generation by only 4.2 percentage points (statistically insignificant) when coal fuel costs exceed import prices, compared to a 16.1 percentage point reduction for restructured utilities. CCNG generation responds similarly under both regulatory regimes (21.1 vs. 19.7 percentage points), confirming that the distortion is specific to legacy coal under RoR regulation and not a general feature of high-cost generation. The six states with the largest responsiveness of coal usage to low market prices are all restructured states; out-of-dispatch-order coal generation also correlates strongly with utility ownership share across states.

Q4. What do the structural parameter estimates reveal about the rate base?

Each MW of CCNG capacity increases the rate base by $229,000. When fully utilized, each MW of coal capacity contributes 1.144 times as much as CCNG. When coal is not fully used, unused coal capacity contributes only 40% as much to the rate base as CCNG. NGT capacity contributes 79% more to the rate base than CCNG per MW. Operations cost estimates include O&M costs of $12.89/MWh for coal, $8.82/MWh for CCNG, and $44.63/MWh for NGT; a 100 MW coal ramp in one hour costs $4,770 versus $3,860 for CCNG.

Q5. What happens in the 30-year long-run counterfactual under the baseline regulated utility?

Facing a sudden drop to 2018–20 natural gas prices ($2.01/MMBtu vs. $7.24/MMBtu in 2006), regulated utilities retire 53% of coal capacity and increase CCNG capacity by 296% over 30 years. The Averch-Johnson over-investment effect dominates: utilities invest heavily in CCNG while retaining and using legacy coal far longer than a cost minimizer would. The social planner effectively eliminates coal generation immediately (99% reduction in the first period) and retires almost all coal capacity over the horizon.

Q6. How does a cost minimizer behave relative to the regulated utility in the same long-run counterfactual?

A cost minimizer immediately reduces coal generation by 50% in the first period and retires most coal capacity over 30 years while increasing CCNG capacity by only 58%—versus the regulated utility’s 296% CCNG increase. Thirty years after the shock, the cost minimizer has retired 71% more coal capacity than the regulated utility. The cost minimizer’s much smaller CCNG expansion reflects that it does not face Averch-Johnson incentives to over-invest in rate-base capital.

Q7. What is the short-run vs. long-run impact of carbon taxes on regulated utilities compared to cost minimizers?

In the short run, carbon taxes on regulated utilities reduce coal generation only 48% as much as when imposed on a cost minimizer (34% vs. ~100% in immediate generation drop), because the used-and-useful incentive counteracts the carbon price signal. In the long run (30-year horizon), however, carbon taxes on regulated utilities result in 68% lower coal capacity and 77% lower coal generation relative to baseline—larger percentage reductions than for a cost minimizer—because the regulatory structure amplifies the retirement incentive over time once carbon costs erode the economic rationale for keeping coal in the rate base.

Q8. What is the short-run operations counterfactual finding for carbon taxes in the sample period?

Using each utility-year in the analysis sample, imposing carbon taxes on regulated utilities reduces carbon costs by only about $500 million relative to baseline—41% of the $1.3 billion carbon cost savings from imposing the same carbon taxes on a cost minimizer. Despite this limited carbon reduction, electricity rates nearly triple from $77.58/MWh to $224.18/MWh under the regulated utility with carbon taxes, as the utility passes through most carbon costs to consumers; regulated utility variable profits also fall by over $500 million.

Q9. What happens when the coal usage incentive is eliminated (μ₂ = 0)?

Setting the coal usage incentive parameter μ₂ = 0 (eliminating the logit slope on capacity utilization) causes coal capacity to fall 82% and coal generation to fall 92% relative to baseline over 30 years—a slightly larger generation decline than for the cost minimizer. However, this comes at the cost of more than twice the CCNG capacity due to the Averch-Johnson effect, and requires utility variable profits to fall by over $300 million, raising reliability concerns unless accompanied by compensating transfers.

Q10. How does the paper’s mechanism relate to observed differences in coal exit rates between regulated and restructured states?

Between 2006 and 2018, 26.0% of coal capacity exited in restructured states versus only 17.2% in regulated states—a gap the authors attribute primarily to the used-and-useful incentive structure in RoR regulation. The structural model quantifies how this regulatory feature specifically distorts coal usage and retirement decisions; it is not explained by demand or cost differences across states, as confirmed by the triple-difference evidence showing the gap is specific to coal (not CCNG) and to regulated (not restructured) utilities.

Q11. Why does the paper argue that alternative regulatory adjustments are insufficient to replicate cost-minimizing transitions?

Changing regulatory parameters—such as increasing the coal usage incentive or adjusting the electricity rate penalty—does not come close to replicating the speed of the energy transition under a cost minimizer in the long-run simulations. Regulatory adjustments that do approach cost-minimizing outcomes (such as eliminating μ₂) require large reductions in utility variable profits sufficient to risk reliability, consistent with why the 2022 Inflation Reduction Act relied on substantial investment transfers rather than carbon taxes as its primary clean energy instrument.

Q12. What is the paper’s identification strategy?

Identification exploits the sharp, exogenous decline in natural gas fuel prices from fracking, which had heterogeneous implications across utilities depending on their initial capital mixes (coal-heavy vs. CCNG-heavy). By comparing investment, retirement, and operations decisions across utilities and over time—particularly between utilities that had CCNG exposure before the price decline and those that did not—the authors recover the structural regulatory and cost parameters. The IV specification for reduced-form evidence uses the current natural gas price interacted with the utility’s initial CCNG generation share as an instrument for fuel and import costs.

Q13. What are the paper’s explicit limitations?

The paper estimates the coal-to-CCNG transition only and cannot speak to the transition to renewables and storage, because there is insufficient variation in the data to identify how regulators would treat CCNG as a legacy technology subject to used-and-useful standards, or how renewables and storage would contribute to the rate base. The authors note that over-investment in CCNG capacity may create future stranded asset problems for ratepayers and that usage incentives for CCNG are likely to further hinder the transition to renewables—but these are conjectures rather than estimated findings.

Rate-of-return (RoR) regulation: A regulatory structure in which the PUC sets electricity rates so that utility revenues cover total variable costs plus an allowable return on the utility’s rate base (capital stock), with the allowable return parameterized as s = (r/r₀)^{-γ}, declining as consumer electricity rates rise.

Used-and-useful standard: A prudence criterion under which a capital asset’s contribution to the rate base depends on its capacity utilization, modeled as a logit function of the generation-to-capacity ratio; fully used coal capacity contributes 1.144 times as much as CCNG per MW, while unused coal contributes only 40% as much.

Rate base: The capital stock on which the PUC grants the utility its allowable rate of return; adjusted by prudence and used-and-useful assessments and described in the paper as “at best an arduous task” to quantify precisely.

Averch-Johnson (AJ) over-investment effect: The tendency of regulated utilities to over-invest in capital because profits are proportional to the rate base; in this paper’s setting, this causes regulated utilities to increase CCNG capacity by 296% over 30 years following the natural gas price shock, compared to 58% for a cost minimizer.

Incentive regulation: A modification of cost-plus RoR regulation in which the allowable rate of return declines as electricity rates rise; it provides efficiency incentives for cost reduction but does not achieve first-best outcomes and is insufficient to overcome the used-and-useful distortion for legacy coal.

Out-of-dispatch-order generation: Running a generation unit when its fuel costs exceed the market import price; regulated utilities engage in this behavior with coal plants to maintain used-and-useful status and rate base contribution, whereas restructured utilities do not face this incentive.

Nested fixed-point indirect inference: The estimation approach used to recover structural regulatory and operations parameters by minimizing the distance between regression coefficients from actual data and those from model-simulated data via a non-linear parameter search.

Enlightenment Ideals and Belief in Progress in the Run-up to the Industrial Revolution

Mon, 01 Jan 0001 00:00:00 +0000

This paper tests Joel Mokyr’s claim that Britain’s industrialization was preceded and enabled by a cultural shift — specifically, that Enlightenment ideals produced a “progress-oriented” view of science that diffused to artisans and craftsmen. The central research question is whether and when the language of science became more progress-oriented in the build-up to the Industrial Revolution, and whether this shift was concentrated in volumes directly linked to industrial production.

The authors assemble 173,031 unique volumes printed in England and written in English between 1500 and 1900, drawn from the Hathitrust Digital Library. Because copyright law prohibits downloading full text, they use HDL’s Extracted-Features “bag of words” dataset. After removing duplicates and Latin-language volumes from an initial set of 420,081, they apply Latent Dirichlet Allocation (LDA) with cross-validated perplexity minimization to identify an optimal T=60 topics. Topic-pair co-occurrence analysis identifies three categories — science, religion, and political economy — each anchored by three defining topics. Volume-level category weights are derived by multiplying each topic’s weight by its category coefficient. The resulting classification yields 50,090 science volumes, 102,565 political economy volumes, and 14,124 religion volumes.

Progressive sentiment is measured using a seven-word dictionary (progress, improvement, stride, betterment, advance, rise, amelioration) assembled from thesaurus synonyms for “progress,” manually vetted by all four authors, and restricted to words attested in the Oxford English Dictionary before 1643 (Newton’s birth year). Sentiment for each volume equals the count of progress-dictionary words divided by total word count. An analogous optimism-sentiment placebo dictionary is constructed separately.

Industrial relevance is scored using the digitized indexes of all five volumes of Appleby’s Illustrated Handbook of Machinery (1877–1903); the top industrial root words are crane (weight 51), electr (42), weight (37), rope (27), and cost (27). Each volume receives an industry score equal to the weighted occurrence of industrial root words normalized by volume length.

Three main findings emerge. First, the language of science and religion showed little overlap beginning in the 17th century — that is, the secularization of science predates the onset of industrialization. Science volumes shifted from approximately 40 percent religious content around 1700 to only about 10 percent by 1850, with scientific content rising correspondingly from roughly 40 percent to over 60 percent. This trend was stable from 1650 through 1900.

Second, while scientific volumes became more progress-oriented during the Enlightenment, this progressive shift was concentrated in volumes at the nexus of science and political economy. Volumes of “pure” science were largely neutral with respect to progress sentiment, and those at the science-religion nexus had on average negative progress sentiment. The marginal effect of scientific content on progress sentiment was greatest for volumes mixing science and political economy, and most of the increase in predicted sentiment at that nexus occurred during the 18th century, remaining stable thereafter. A placebo test using optimism sentiment finds the opposite pattern: volumes at the science-political economy nexus were among the least optimistic, while the most optimistic language appeared at the religion-political economy nexus. This rules out the interpretation that the measured shift reflects a general increase in positive affect rather than specifically progress-oriented language.

Third, volumes employing industrial terminology that also sat at the science-political economy nexus were distinctively progressive beginning in the mid-18th century. At the 90th percentile of industry score, predicted progress sentiment at the science-political economy nexus was positive throughout the sample; at zero industry score, it was negative until the mid-18th century. Volumes at the religion-political economy nexus showed modestly positive and time-stable progress sentiment regardless of industry score.

The paper concludes that it was the pragmatic, applied volumes — those bridging science and political economy, written for artisans and a broader literate public rather than for the human-capital elite alone — that embodied the cultural values Mokyr identifies as central to Britain’s industrialization.

Q: What gap in the existing literature does this paper address?

A: Prior work on the cultural deep roots of economic growth rarely tracks how culture changes over time, relying instead on cross-sectional variation or qualitative case studies. Quantitative evidence that the language of science itself became more progress-oriented — and that this change reached beyond elite thinkers to artisans and craftsmen — had not been marshaled before. The paper provides inaugural quantitative support by analyzing 173,031 volumes spanning four centuries.

Q: Why does the paper restrict the progress-sentiment dictionary to words attested before 1643?

A: Words that entered English only after 1643 (Newton’s birth year) could not have appeared in volumes from the early Enlightenment, so including them would bias sentiment scores toward the later part of the sample. The restriction ensures the dictionary is applicable and unbiased across the full 1500–1900 period. The final retained words are: progress, improvement, stride, betterment, advance, rise, amelioration.

Q: How does LDA classify volumes, and how is T=60 selected?

A: LDA treats each volume as a bag of words and derives a Dirichlet distribution such that observed documents are generated by repeated topic sampling. The number of topics T is selected by minimizing perplexity on held-out data via 4-fold cross-validation, rotating training and test sets across folds; this procedure yields T=60 as optimal. Each volume is then represented as a mixture over those 60 topics.

Q: What are the three categories and their anchor topics?

A: Political Economy is anchored by topics on law/public opinion, governance/parliament, and trade/price/labour. Religion is anchored by topics on church/Christian doctrine, God/faith/sin, and virtue/fame/religion. Science is anchored by topics on engineering/steam/electricity, chemistry/acid/heat, and geometry/equations/trigonometry. These three sets of topics were selected for high corpus-wide importance and mutual independence.

Q: What does the finding on science-religion separation imply for timing?

A: The separation of scientific and religious language was already visible by 1600 and firmly established by the mid-17th century, well before the Industrial Revolution conventionally dated to the mid-18th century. This supports Mokyr’s argument that the secularization of science was an Enlightenment-era precursor to industrialization rather than a product of it. The trend remained stable from 1650 through 1900.

Q: How does the progressive sentiment differ between pure science and the science-political economy nexus?

A: Volumes of pure science were largely neutral with respect to progress-oriented language and in some periods showed slightly negative predicted progress sentiment. The science-religion nexus showed consistently negative progress sentiment. By contrast, volumes at the science-political economy nexus showed the highest level of progressive sentiment beginning in the mid-18th century, and most of this growth in predicted sentiment occurred during the 18th century, after which it remained stable.

Q: What does the placebo optimism test show?

A: The optimism sentiment scores are nearly the mirror opposite of the progress scores: the most optimistic language appears at the religion-political economy nexus, while volumes at the science-political economy nexus are among the least optimistic. This dissociation rules out the interpretation that the measured progress-sentiment rise reflects a general shift toward positive language rather than a specific cultural embrace of science as a tool for improving human welfare.

Q: How is the industrial score constructed and what are the most heavily weighted terms?

A: The authors digitized the detailed indexes of all five volumes of Appleby’s Illustrated Handbook of Machinery (1877–1903), restricted to words attested before 1643, and weighted each industrial root word by its index frequency. Each corpus volume’s industry score equals the sum of (word count × index weight) across all industrial words, normalized by volume length, yielding a score between 0 and 1. The top-weighted terms are crane (51), electr (42), weight (37), rope (27), and cost (27).

Q: What is the key result linking industrial scores to progressive sentiment?

A: At the science-political economy nexus, volumes with industry scores at the 90th percentile had persistently positive predicted progress sentiment throughout the sample, while volumes at that nexus with zero industry score had negative predicted sentiment until the mid-18th century. The shift to positive sentiment for high-industry volumes at this nexus occurred in the mid-18th century — roughly coinciding with the onset of Britain’s industrialization — and those volumes remained the most progress-oriented in the corpus thereafter.

Q: What is the paper’s interpretation of the science-political economy nexus finding in relation to Mokyr?

A: The authors interpret volumes at the science-political economy nexus as pragmatic, applied works aimed at a broader literate audience including artisans and craftsmen, not exclusively the human-capital elite. These are precisely the volumes Mokyr’s “Industrial Enlightenment” thesis predicts would carry progress-oriented cultural values into the mechanical and artisanal pursuits that drove industrialization. The finding that pure-science volumes were not especially progressive, while applied volumes bridging science and political economy were, is consistent with Mokyr’s argument that it was the diffusion of Enlightenment ideals to skilled practitioners — not just to elite scientists — that mattered.

Q: What qualitative examples support the quantitative findings?

A: Martin Clare’s The Motion of Fluids (1735) explicitly addresses “the Unlearned” and states in its preface that the work is meant to be “of singular Use and Benefit to Mankind” — a direct expression of the progress-oriented language the algorithm detects. George Stephenson’s 1831 railway report argues that rail infrastructure would allow Ireland to “reciprocate with England and with other nations, the products of industry,” exemplifying how progress-oriented language pervaded industrial writing by the early 19th century. These examples confirm that the high progress-sentiment scores for industrial volumes at the science-political economy nexus reflect genuine rhetorical content, not measurement artifacts.

Q: What are the paper’s limitations regarding early sample periods?

A: The corpus is thin in earlier eras, particularly around 1550, so results from the earliest decades must be interpreted with caution. The HDL data derive from digitized scans with OCR output of very old books, introducing errors such as the “long-S” misread (e.g., “juftice” for “justice”) that require manual correction. Additionally, the bag-of-words model discards word order, which may obscure some semantic distinctions.

Q: What future research directions do the authors identify?

A: The authors propose applying the same textual analysis techniques to test whether English-language volumes began reflecting greater freedom of expression in the run-up to Britain’s economic takeoff, connecting to the literature on European political fragmentation and the marketplace of ideas. They also suggest applying the approach to corpora in other languages — Dutch (following McCloskey’s argument about bourgeois values) and Spanish (to examine whether the Counter-Reformation and Spain’s economic lag are reflected in cultural attitudes toward progress and science).

LDA (Latent Dirichlet Allocation): An unsupervised generative statistical model that treats each document as a bag of words and extracts latent topics as multinomial distributions over vocabulary; used here to reduce 173,031 volumes to mixtures of 60 topics without imposing prior scholarly interpretations.

Progressive Sentiment Score: The fraction of words in a volume belonging to a seven-word dictionary of progress synonyms (progress, improvement, stride, betterment, advance, rise, amelioration), normalized by total word count; measures the cultural orientation toward the betterment of humankind as embedded in text.

Industrial Score: A volume-level measure equal to the weighted count of industrial root words — derived from the indexes of Appleby’s Illustrated Handbook of Machinery (1877–1903) — normalized by volume length; captures the degree to which a volume’s vocabulary overlaps with industrial production terminology.

Science-Political Economy Nexus: The region of the topic simplex where volumes carry substantial weight in both the science and political economy categories but low weight in religion; the paper finds this is where progress-oriented language was most concentrated from the mid-18th century onward, interpreted as applied science aimed at artisans and a broader literate public.

Industrial Enlightenment: Joel Mokyr’s (2009) concept describing the diffusion of Enlightenment ideals about the practical utility of science into the mechanical and artisanal pursuits that drove Britain’s industrialization; the paper provides quantitative support for this thesis by showing that industrial volumes at the science-political economy nexus were distinctively progress-oriented.

Culture of Growth: Mokyr’s (2016) broader argument that a pan-European network of elite intellectuals fostered a progress-oriented view of science — the idea that scientific understanding could improve the human condition — and that this cultural norm, in combination with Britain’s stock of skilled craftsmen, made industrialization possible.

Bag of Words: A representation of text that records only word frequencies within a document, discarding word order; used here both because HDL copyright restrictions prevent full-text download and because it is the input format required by LDA.

Environmental Consequences of Hydrocarbon Infrastructure Policy

Mon, 01 Jan 0001 00:00:00 +0000

Covert and Kellogg study policies that aim to “keep carbon in the ground” by blocking fossil fuel infrastructure investment, with the Dakota Access Pipeline (DAPL) as their empirical application. DAPL moves more than 500,000 barrels per day of oil from the Bakken Shale of North Dakota to the U.S. Gulf Coast and was completed in June 2017 amid substantial opposition. The central research question is whether blocking pipeline construction actually keeps oil in the ground or merely shifts transport to alternative modes — specifically crude-by-rail — and what the net environmental and economic consequences are.

The paper develops a two-period model of crude oil production and transportation mode choice. In the model, oil shippers decide in period 1 whether to commit to pipeline capacity under ship-or-pay contracts, then in period 2 allocate flows between the committed pipeline and the more flexible but costlier railroad alternative. Pipeline construction is an irreversible sunk cost with zero ongoing marginal cost; rail involves no sunk cost but substantial ongoing marginal costs including quadratic adjustment costs that capture capital investment in rail cars and loading/unloading facilities. Equilibrium pipeline capacity is determined by a shippers’ indifference condition: expected per-barrel returns from pipeline access equal the FERC-regulated tariff.

The empirical model is estimated using monthly Bakken oil production and transportation data, price differentials across three coastal destinations (Gulf, East, West), and drilling productivity data. Crude-by-rail marginal costs are estimated via 2SLS, yielding static marginal cost intercepts of $9.49/bbl to the East Coast, $12.64/bbl to the Gulf Coast, and $8.69/bbl to the West Coast, plus a dynamic adjustment cost of $1.28/bbl per mbbl/d of flow change. The upstream supply model follows Anderson, Kellogg, and Salant (2018), with old-well production following exponential decline (estimated decay parameter β = 0.955) and new-well drilling responding to current and lagged prices with a total long-run elasticity of 1.32. Shippers’ beliefs about future oil prices are calibrated to an AR(1) process fit to historical price volatility (persistence φ₁ = 0.9925, volatility σ_G = 0.098). Model validation confirms a predicted expected return to pipeline commitment of $6.17/bbl against DAPL’s actual tariff of $5.50–$6.25/bbl.

The main counterfactual asks what would have happened had DAPL’s construction been enjoined. In expectation, blocking DAPL reduces pipeline flows by 306 mbbl/d. Expected crude-by-rail flows increase by 248 mbbl/d, offsetting 81% of the pipeline reduction. Bakken oil production falls by only 58 mbbl/d, a 4% reduction. The modal shift from pipeline to rail worsens local environmental outcomes: per-barrel local pollution damages from rail transport substantially exceed those from pipelines, dominated by locomotive NOx emissions in populated areas. Foreclosing DAPL increases net local pollution damages by $444,000 per day (the decrease in pipeline-related harm of $144,000/day is more than offset by the increase from rail of $588,000/day). The total cost of blocking DAPL is $45/tonne of CO2 abated — $28/tonne from lost producer surplus and $17/tonne from increased local pollution damages — a figure comparable to the contemporaneous U.S. government social cost of carbon estimate of $42/tonne.

An upstream production tax achieving the same CO2 reduction costs only $1.01–$2.68/tonne CO2 abated, an order of magnitude less, because it does not induce the distortionary modal shift to rail. Two caveats apply: if 57% of Bakken production reductions leak to other basins, the cost of blocking DAPL rises from $45/tonne to $104/tonne; and if reductions represent production delays rather than permanent reductions, effective abatement is further diminished. The analysis is scoped to Bakken crude oil and land transportation alternatives. The finding that blocking infrastructure increases local pollution is atypical of CO2 abatement policies, which usually generate local pollution co-benefits.

Q: What is the core economic mechanism by which blocking a pipeline can keep oil in the ground? A: When a pipeline is foreclosed, crude oil can still move by railroad, but rail transport involves substantial ongoing marginal costs. These costs create a wedge between upstream (Bakken) and downstream (Gulf Coast) prices that depresses upstream supply. Only when downstream prices are high enough to cover both rail marginal cost and this wedge will rail fully substitute for the pipeline; at lower prices, some production is uneconomical and stays in the ground. In the model, this price-depressing wedge is the mechanism that reduces production — but it operates only partially, since rail can substitute for much of the pipeline’s flow.

Q: How much of the blocked pipeline flow substitutes to rail versus stays in the ground? A: In expectation, blocking DAPL reduces pipeline flows by 306 mbbl/d. Expected crude-by-rail flows increase by 248 mbbl/d, offsetting 81% of the pipeline reduction. Bakken oil production falls by only 58 mbbl/d, or approximately 4%. In a specific simulated month (December 2019), 348 mbbl/d (67%) of the 520 mbbl/d of foregone pipeline flows would still move by rail.

Q: How are crude-by-rail costs estimated, and what is the role of adjustment costs? A: The authors estimate a 2SLS model of rail flows on price differentials, allowing for quadratic adjustment costs to capture investments and disinvestments in rail cars and loading facilities. Static marginal costs are $9.49/bbl (East Coast), $12.64/bbl (Gulf Coast), and $8.69/bbl (West Coast). The adjustment cost parameter γ is estimated at $1.28/bbl per mbbl/d, meaning a 10 mbbl/d monthly increase in rail flows raises marginal shipping cost by $12.76/bbl — a substantial share of total rail costs. Adjustment costs are necessary to reconcile the model with the sluggish observed response of rail flows to price differentials.

Q: What is the structure of the upstream oil supply model and what are its key parameter estimates? A: The model distinguishes “old” production from pre-existing wells, which follows exponential decline with estimated decay parameter β = 0.955, and “new” production from newly drilled wells, which is price-responsive with a total long-run elasticity of 1.32 — comparable to the 1.1–1.2 estimated by Newell and Prest (2019) across major U.S. shale plays. This structure implies that total production is highly inelastic in the short run (dominated by old wells) but responds to persistent price shocks over the long run through changes in drilling rates.

Q: How do the local pollution damages of rail compare to those of pipeline transport? A: At a social cost of carbon of $100/tonne, local air pollution damages from rail transport to the Gulf Coast are $1.66/bbl (plus $0.73/bbl in spill/accident costs), versus only $0.35/bbl local pollution (plus $0.11/bbl spills) for pipelines. Locomotive NOx emissions are the dominant factor, both because locomotives have high NOx emission factors and because these emissions often occur in densely populated areas. CO2 damages at $100/tonne SCC are roughly similar across modes ($0.79–0.83/bbl), so local pollution is the key differentiator.

Q: What is the net welfare impact of foreclosing DAPL, and how is it decomposed? A: Foreclosing DAPL reduces producer surplus by $716,000/day, increases net local pollution damages by $444,000/day (the $588,000/day increase from rail more than offsets the $144,000/day decrease from pipeline), and reduces CO2 emissions by 25.2 mtonnes/day from the 58 mbbl/d production reduction. The cost per tonne of CO2 abated is $28/tonne from lost producer surplus and $17/tonne from increased local pollution damages, totaling $45/tonne — broadly comparable to the U.S. government’s contemporaneous SCC estimate of $42/tonne. This means the policy’s abatement cost is approximately equal to the social value of each tonne abated, leaving little or no net social gain even before accounting for leakage.

Q: How does the model validate against observed data and institutional parameters? A: The model predicts an expected return to committed DAPL pipeline shipment of $6.17/bbl, which closely matches the actual DAPL tariff for committed shippers of $5.50–$6.25/bbl. The authors also validate simulated crude-by-rail flows against actual flows across destinations. The close match on the tariff is particularly meaningful because it tests the model’s equilibrium condition for pipeline capacity investment rather than a within-sample fit.

Q: How does an upstream production tax compare to blocking DAPL as a policy instrument? A: A production tax normalized to achieve the same CO2 reduction requires only $3.68/bbl if imposed after shippers have committed to DAPL (holding capacity fixed), or $3.24/bbl if announced before commitments are made (reducing pipeline capacity to 443 mbbl/d). The production tax reduces combined producer surplus and government revenue by only $96,000–$109,000/day versus $716,000/day under the DAPL ban, and reduces local pollution damages by $82,000/day rather than increasing them. The resulting cost per tonne CO2 abated is $1.01–$2.68 — an order of magnitude smaller than the $44.63/tonne for blocking DAPL.

Q: What is the production leakage caveat and how large is its effect? A: If blocking DAPL causes Bakken production to fall, production from other U.S. or global oil basins may increase, partially or fully offsetting the CO2 reduction. Following Prest (2022) and Prest et al. (2023), the authors note that if 57% of the Bakken production reduction leaks to other basins, the cost of blocking DAPL rises from $45/tonne to $104/tonne. Leakage would increase the cost per tonne for the upstream tax as well, but the relative advantage of the tax over the pipeline ban is unaffected by this caveat.

Q: What is the production delay caveat? A: Even absent leakage, the paper cautions that production reductions from either policy may represent production delays rather than permanent reductions — oil not extracted today may be extracted later as prices rise or technology improves. To the extent that reductions are temporary, the effective carbon abatement is smaller than the authors compute, and the cost per tonne of CO2 abated is correspondingly higher. The paper does not quantify this effect but flags it as a material caveat.

Q: What institutional features drive pipeline capacity investment and risk allocation? A: Pipelines are irreversible investments subject to ex-post holdup, so construction financing requires firm ship-or-pay commitments from shippers before construction and before future prices are known, meaning oil price risk is borne primarily by shippers rather than the pipeline owner. Pipeline tariffs are regulated by FERC on a cost-of-service basis. In the DAPL case, shippers executed binding ten-year ship-or-pay contracts in June 2014, and shippers’ beliefs about future oil prices at that date — calibrated to historical price volatility using an AR(1) process with estimated persistence φ₁ = 0.9925 and volatility σ_G = 0.098 — determine equilibrium capacity investment.

Q: How does the paper’s finding relate to the typical co-benefit structure of climate policies? A: Most CO2 abatement policies generate local pollution co-benefits (reduced NOx, SOx, particulates), so the abatement cost is partially offset by local pollution gains. Blocking DAPL reverses this: the pipeline-to-rail modal shift increases local pollution damages, making local pollution a cost rather than a co-benefit of the policy. The authors note this is atypical but not unprecedented — urban densification and post-combustion emissions controls in fossil fuel boilers also present CO2–local pollution trade-offs.

Infrastructure foreclosure policy: A “keep it in the ground” strategy that blocks construction of specialized fossil fuel transportation infrastructure (pipelines) with the aim of inhibiting production of the fuels that would have been transported, without requiring direct acquisition or buyout of mineral rights.
Ship-or-pay agreement: A firm, up-front capacity commitment in which a pipeline shipper agrees to pay for reserved pipeline capacity whether or not they ultimately use it, made before construction and before future prices are realized; the institutional mechanism by which oil price risk is transferred from pipeline owners to shippers.
Crude-by-rail adjustment costs: Quadratic costs modeled as linear in the period-to-period change in rail volumes to a given destination, capturing capital investments and disinvestments in rail cars, loading facilities, and unloading terminals needed to expand or contract crude-by-rail capacity; estimated at $1.28/bbl per mbbl/d of monthly flow change.
Production leakage: The partial or full offset of production reductions in one oil basin (Bakken) by production increases in other U.S. or global basins in response to the same price signals; at 57% leakage, the cost of blocking DAPL rises from $45/tonne to $104/tonne of CO2 abated.
Old-well vs. new-well production dynamics: The distinction between production from pre-existing wells (which follows an exponential decline path insensitive to current prices, β = 0.955) and production from newly drilled wells (which responds to current and lagged upstream prices with long-run elasticity 1.32); this structure makes total short-run supply highly inelastic while allowing substantial long-run price responsiveness through drilling adjustments.
Local pollution damages from NOx: The dominant component of environmental harm from crude-by-rail transport, arising from locomotive NOx emissions that are both large in magnitude and concentrated in densely populated areas along rail corridors; at $100/tonne SCC, monetized local pollution damages from rail exceed CO2 damages for all three coastal destinations, whereas for pipelines CO2 damages exceed local pollution costs.
Cost per tonne of CO2 abated: The authors’ metric for comparing infrastructure foreclosure to alternative policies; computed as the sum of lost producer surplus and net change in local pollution damages divided by the quantity of CO2 emissions avoided from reduced oil production and consumption; equals $45/tonne for blocking DAPL versus $1.01–$2.68/tonne for an equivalent upstream production tax.

Evaluating macroeconomic outcomes under asymmetries: Expectations matter

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper investigates whether and how assumptions about household and firm expectations alter the macroeconomic implications of asymmetries commonly embedded in macroeconomic models. Specifically, it asks: when a model features a nonlinearity — such as an asymmetric monetary policy rule or a nonlinear Phillips curve — do the longer-run average outcomes and the distributional properties of inflation and unemployment depend on whether agents have rational expectations (RE, accounting for the possibility of future shocks) versus perfect foresight (PF, not anticipating future shocks)?

Methodology

The paper works within a standard three-equation New Keynesian model comprising an IS curve (linking the unemployment gap to the policy rate and the natural rate of interest via Okun’s law with coefficient c ≈ 2), a forward-looking Phillips curve, and a monetary policy rule. The model is parameterized at a quarterly frequency with β = 0.99, κ = 0.01, φπ = 1.5, φu = −0.25, shock persistence ρ_rn = 0.9, and shock standard deviation σ_rn = 0.0025 (calibrated to match a 1-percentage-point standard deviation of the unemployment gap under the symmetric baseline rule).

The key methodological distinction is the specification of the expectations operator. Under RE, agents use the true stochastic transition matrix for the natural rate (approximated via the Rouwenhorst method with 105 grid points). Under PF, agents instead use a transition matrix that always places probability one on the steady-state value of the natural rate next period — i.e., they do not anticipate future shocks. The model is solved globally with a discrete state space projection (parameterized expectations) method, applied identically to RE and PF cases. The authors first derive analytical results in a simplified three-state environment and then present numerical results from 3,000 simulations of 1,000 periods each.

Two types of asymmetry serve as case studies: (i) an asymmetric monetary policy rule — the “Shortfalls rule” — under which the central bank does not tighten in response to a tight labor market (negative unemployment gap), in the spirit of the FOMC’s 2020 framework update; and (ii) a nonlinear (kinked) Phillips curve that steepens by a factor of three when the labor market is tight (unemployment gap < 0), consistent with empirical evidence in Smith, Timmermann, and Wright (2025).

Main Findings

The core finding is that the sign and magnitude of longer-run average outcomes under asymmetric macroeconomic environments can differ substantially — and can even reverse — depending on whether agents have rational expectations or perfect foresight.

For the Shortfalls rule, under PF the model implies a longer-run tradeoff: average unemployment gap is −0.32 percentage points and average inflation gap is +0.25 annualized percentage points relative to the symmetric Deviations rule. PF thus suggests policymakers can lower average unemployment at modest inflationary cost. Under RE, however, this apparent tradeoff disappears entirely: the average unemployment gap is essentially zero (−0.05 percentage points) while average inflation is elevated by approximately 1.02 annualized percentage points. The gap in average inflation outcomes between RE and PF thus exceeds one percentage point, and the labor market benefit implied by PF is absent under RE.

For the nonlinear Phillips curve (under a symmetric deviations rule with φu = 0), the results again diverge across expectations assumptions, and the direction of the effects reverses. Under PF, the kinked Phillips curve implies average inflation of +0.41 annualized percentage points and a near-zero unemployment gap (+0.30 percentage points). Under RE, the average inflation gap is essentially zero while the average unemployment gap rises to +0.63 percentage points — the opposite directional pattern from PF.

The mechanism driving the RE–PF divergence is the interaction between forward-looking price-setters and an inflation-stabilizing central bank. Under RE, anticipated future episodes in which the asymmetry may bind (e.g., the Shortfalls rule providing accommodation, or the Phillips curve steepening) cause firms to set higher prices today. The central bank responds to the resulting pickup in inflation expectations with tighter policy, generating a persistent contractionary offset. This channel is absent under PF because agents expect no future shocks.

Robustness

The main conclusions are robust across three extensions: (i) Bounded rationality (following Gabaix 2020, with m_br = 0.97): outcomes move toward the PF case, confirming that what matters is the degree to which agents internalize the probability of future shocks; (ii) Cost-push shocks instead of natural rate shocks: the RE–PF divergence under a Shortfalls rule is broadly similar in direction and magnitude to the baseline; (iii) Alternative shock specifications: the qualitative conclusions are maintained.

Crucially, under the symmetric Deviations rule the RE and PF solutions are identical in all cases, confirming that the divergence is specific to models with macroeconomic asymmetries, not an artifact of the solution method.

In depth

Q1. What is the central methodological claim about perfect foresight solutions in asymmetric models?

The paper argues that in macroeconomic models with asymmetries or nonlinearities, perfect foresight solutions — in which agents do not account for the possibility that future shocks may occur — can yield longer-run average outcomes and distributions that differ from their rational expectations counterparts in magnitude and potentially in sign. The paper is explicit that this is not a critique of PF methods per se, as PF is often necessary for estimating larger models; rather, the point is that researchers should check the robustness of conclusions about longer-run averages using simplified models solvable under both approaches.

Q2. How is the difference between RE and PF operationalized in the model?

The sole technical distinction lies in the specification of the conditional expectations operator Et. Under RE, this operator uses the true stochastic Markov transition matrix for the natural rate (P^RE), which assigns positive probability to all feasible future states. Under PF, agents use a degenerate transition matrix (P^PF) that assigns probability one to the mean value of the natural rate next period regardless of the current state — effectively, agents expect no future innovations. The same global solution method (discrete state space projection with 105 Rouwenhorst grid points) is applied to both, so differences in equilibrium outcomes are entirely attributable to the expectation specification.

Q3. What are the analytical results for the Shortfalls rule in the simplified three-state model?

In the simplified environment with the natural rate taking three equiprobable values (low, steady-state, high) and no persistence, the analytical solution shows that under PF the average unemployment gap is −Δ/(1 + φπκ) < 0 and the average inflation gap is Δκ/(1 + φπκ) > 0, where Δ parameterizes the degree of additional accommodation in the high-demand state. Under RE, the average unemployment gap is exactly zero and the average inflation gap is Δ/(φπ − 1) > 0. The inflation gap under RE exceeds that under PF by Δ(1 + κ)/[(φπ − 1)(1 + φπκ)] > 0, and the unemployment gap under RE exceeds that under PF by Δ/(1 + φπκ) > 0. Thus, PF spuriously implies an exploitable long-run tradeoff that vanishes under RE.

Q4. What are the analytical results for the nonlinear Phillips curve in the simplified model, and how do the directions of the effects compare to the Shortfalls rule case?

Under PF with a nonlinear (kinked) Phillips curve, the average inflation gap is positive (= Δpc > 0) while the average unemployment gap is zero. Under RE, the signs reverse: the average unemployment gap is positive (= Δpc/κ > 0) and the average inflation gap is zero. The difference is ūRE − ūPF = Δpc/κ > 0 and π̄RE − π̄PF = −Δpc < 0. This sign reversal relative to the Shortfalls rule case illustrates that the directional error introduced by PF is not uniform but depends on the specific asymmetry — the key feature is always the absence, under PF, of the forward-looking price-setting channel interacting with monetary policy.

Q5. What is the quantitative magnitude of the RE–PF divergence in the numerical model for the Shortfalls rule?

In the fully parameterized numerical model (Table 2), under a Shortfalls rule the average inflation gap is 1.02 annualized percentage points under RE versus 0.25 annualized percentage points under PF — a difference of roughly 0.77 percentage points. The average unemployment gap is −0.05 percentage points under RE versus −0.32 percentage points under PF — a difference of 0.27 percentage points. The paper also notes that model-implied averages for inflation and nominal interest rates “under perfect foresight can easily differ by at least one percentage point from their rational expectations counterparts.”

Q6. How do the simulated distributions differ between RE and PF under a Shortfalls rule?

Under PF, the simulated distributions of unemployment and inflation gaps exhibit a pronounced kink near the steady-state value (zero gap), reflecting the asymmetric treatment of expansions and contractions. Under RE, the distributions are substantially more symmetric, shifted to the right for inflation (mean of 1.0 versus 0.25 under PF). Standard deviations of the unemployment and inflation gaps are somewhat larger under PF (1.42 and 1.10, respectively) than under RE (1.33 and 1.03), because under RE the contractionary force from inflation expectations moderates the amplitude of fluctuations. These distributional differences have direct implications for how policymakers interpret the risks associated with state-contingent policies.

Q7. What is the role of the forward-looking pricing–central bank interaction in generating RE–PF differences?

The key mechanism is as follows: under RE, the possibility that the asymmetry may bind in the future (e.g., a positive demand shock triggering more accommodation under the Shortfalls rule, or a tight labor market steepening the Phillips curve) causes forward-looking firms to raise prices today in anticipation of future inflation. This increase in current inflation leads the central bank — whose mandate includes inflation stabilization — to raise policy rates, generating a contractionary offset even when the economy is not currently in the high-demand state. Under PF, agents do not form these anticipatory expectations, so this channel is entirely absent, and the asymmetry affects outcomes only when it directly binds.

Q8. Does the RE–PF divergence arise under a symmetric Deviations rule?

No. The paper shows analytically and numerically that when the monetary policy rule is symmetric (the Deviations rule, responding equally to deviations above and below target), the RE and PF solutions are identical. Unemployment and inflation gaps are both zero on average under either expectations assumption, and the policy rate gap is essentially zero (0.01 annualized percentage points) in both cases. This equivalence result confirms that the RE–PF divergence is not an artifact of the solution method or parameterization but is specifically generated by the interaction between an asymmetry and agents’ forward-looking behavior.

Q9. What do the bounded rationality results imply about the mechanism?

The extension following Gabaix (2020), with a myopia parameter m_br = 0.97, produces results that lie between the full-RE and PF cases: the adoption of the Shortfalls rule yields average unemployment of −0.26 percentage points (intermediate between RE’s −0.05 and PF’s −0.32) and average inflation of 0.62 annualized percentage points (between RE’s 1.02 and PF’s 0.25). This gradient confirms that the key driver is the extent to which agents internalize the probability of future shocks: the more forward-looking agents are, the more strongly the anticipatory pricing channel operates and the less favorable (and more inflationary) the apparent policy tradeoff becomes.

Q10. What are the results for the nonlinear Phillips curve in the numerical model?

Under the numerically calibrated nonlinear Phillips curve model (Panel B.3 of Table 3, with the slope increasing by a factor of three when the unemployment gap is negative), the average unemployment gap under RE is 0.63 percentage points versus 0.30 under PF, and the average inflation gap under RE is essentially zero (0.01 annualized percentage points) versus 0.41 under PF. The authors note that “the average outcomes for both unemployment and inflation can differ by roughly 0.3 to 0.4 percentage points between rational expectations and perfect foresight” in this case.

Q11. What is the paper’s advice for researchers who must use perfect foresight methods?

The paper explicitly states that PF methods remain valuable, especially for estimating or simulating larger models with heterogeneity at the micro level where RE solutions are computationally prohibitive. The authors recommend that researchers relying on PF to solve larger models “check the robustness of their conclusions on longer-run averages and the distribution of outcomes using simplified models which can be solved under both perfect foresight and rational expectations.” To support this, the authors provide multiple versions of code for solving simple macroeconomic models under various asymmetries and expectations assumptions.

Q12. How does the paper position its contribution relative to prior work on RE vs. PF in asymmetric models?

The paper acknowledges that Adam and Billi (2007) and Nakov (2008) previously documented that, at the zero lower bound, households’ anticipation of future ZLB episodes leads to lower average inflation — an RE–PF difference in the spirit of this paper’s findings. However, the paper’s contribution is to show that the sign and quantitative implications of a given asymmetry can change depending on the expectations assumption, and to systematically characterize this sensitivity across multiple types of asymmetry (asymmetric policy rules and nonlinear Phillips curves). The paper also categorizes the existing literature by expectations assumptions in Table A.1, showing that many papers examining macroeconomic asymmetries use only one approach.

Key Concepts

Shortfalls Rule: A monetary policy rule, motivated by the FOMC’s 2020 Statement on Longer-Run Goals and Monetary Policy Strategy, under which the central bank responds only to shortfalls of employment from its maximum level — i.e., it does not tighten policy in response to a tight labor market (negative unemployment gap) during an expansion. Formally, it = φπ πt + φu ut when ut ≥ 0 (labor market slack), and it = φπ πt only when ut < 0 (labor market tight). Contrasts with the symmetric Deviations rule that responds to deviations of employment in both directions.

Deviations Rule: A symmetric monetary policy rule in which the central bank responds to the unemployment gap regardless of its sign — tightening in expansions and easing in contractions. Serves as the baseline against which the Shortfalls rule is compared, and as the case in which RE and PF solutions are identical.

Perfect Foresight (PF) Equilibrium: An equilibrium in which agents solve their optimization problems assuming that no future shocks will occur — they expect all endogenous variables to converge to their longer-run (steady-state) values next period, regardless of the current state. In the paper’s notation, the PF transition matrix P^PF assigns probability one to the mean state next period. In linear models, PF and RE yield identical outcomes; in models with asymmetries, they diverge.

Rational Expectations (RE) Equilibrium: An equilibrium in which households and firms correctly account for the full stochastic distribution of future shocks in forming their expectations. Agents use the true Markov transition matrix P^RE for the natural rate process. This allows forward-looking pricing behavior to incorporate the possibility that the economy may enter states in which asymmetries bind in the future.

Nonlinear (Kinked) Phillips Curve: A Phillips curve in which the slope coefficient κ̃t is state-contingent, increasing when the unemployment gap is negative (labor market is tight). In the paper’s numerical implementation, the slope triples (κ̃ = 3κ) when ut < 0, consistent with empirical evidence in Smith, Timmermann, and Wright (2025) on structural breaks in the Phillips curve. The nonlinearity generates an asymmetric inflationary response: a given level of unemployment produces more inflation when the labor market is tight than when it is slack.

Stochastic Steady State: The equilibrium to which the economy converges in the absence of additional shocks, taking into account the stochastic nature of the environment (i.e., accounting for the possibility of future shocks). Used as the initial condition for computing impulse response functions under RE. Contrasts with the deterministic steady state (zero gaps), which serves as the initial condition under PF.

Parameterized Expectations (Global Solution) Method: The numerical solution algorithm used in the paper to solve for equilibrium policy functions for unemployment and inflation gaps over the state space. Implemented identically for RE and PF cases, differing only in the transition matrix used. Applied with 105 Rouwenhorst grid points for the natural rate. The paper shows this method is orders of magnitude faster than the more common shooting algorithm (0.04 seconds vs. 10.8 seconds) while yielding identical policy functions.

Bounded Rationality (Gabaix 2020): An extension of the baseline model in which agents discount the influence of future expectations by a myopia parameter m_br ∈ (0, 1), applied to both the IS curve and the Phillips curve. The parameter m_br = 0.97 (following McKay, Nakamura, and Steinsson 2017) limits the degree to which distant future states affect current decisions. Produces outcomes intermediate between full RE and PF, confirming that the key dimension of variation is the extent to which agents internalize the probability of future shocks.

Expectation-driven term structure of equity and bond yields

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. What drives the joint historical dynamics of the term structure of equity yields and nominal bond yields — and can a single unified equilibrium model explain the procyclical equity yield slope, the switch in bond-stock correlation from positive to negative after the late 1990s, the maturity-declining predictability of dividend strip returns, and standard aggregate stock market puzzles?

Key Departure from Prior Literature. Existing equilibrium models (habit formation, long-run risk, disaster risk) rely on time-varying risk premia to explain asset prices. Recent survey evidence challenges this: De La O and Myers (2021) show that most aggregate stock price movements are driven by cash-flow growth expectations rather than return expectations, and Van Binsbergen et al. (2013) show that equity yields are driven mainly by dividend growth expectations. This paper constructs an equilibrium model in which equity (bond) yield variation is attributable to subjective dividend growth (GDP growth) expectations, with a constant subjective risk premium implied by CRRA utility.

Model Architecture. The representative agent has CRRA utility with risk-aversion coefficient γ = 4 and subjective discount factor β = 1.0065 (calibrated to the average 10-year equity yield). The agent departs from rational expectations by having the “belief in the law of small numbers” (Tversky and Kahneman 1971): she perceives small samples to represent their population as well as large samples, leading to subjective learning gains that differ from the rational Kalman gain. The subjective belief updating rule is a modified Kalman filter in which the likelihood is exaggerated by factor (1+θ), producing a subjective learning gain ν that exceeds the Kalman gain K when overreaction applies and falls below it when underreaction applies.

The model has three blocks of fundamentals, each decomposed into a stable and a transitory component. (1) Real GDP growth is decomposed into PCE growth (stable, with a random-walk trend state µ_g) and a volatile gap component (stationary state x_g, persistence ρ_g = 0.941). (2) Inflation is decomposed into core inflation (stable, with trend state µ_π) and a volatile gap (persistence ρ_π = 0.932). (3) Real aggregate dividend is decomposed into a long-duration dividend component dl (levered on log real GDP with leverage λ = 3) and the share of long-duration dividend ds (stationary with persistence ρ_d = 0.94). This cross-sectional decomposition uses firm-level long-term earnings growth (LTG) forecasts from IBES as a model-free equity duration measure.

Estimation. State-space parameters are estimated by maximum likelihood with the Kalman filter on data from NYSE/NASDAQ/AMEX firms (CRSP/Compustat), quarterly, from 1987Q4 to 2019Q4. Subjective learning gains are estimated by minimizing RMSE between model-implied expectations and consensus forecasts: 1-year real GDP growth and inflation from the Survey of Professional Forecasters (SPF, 1981Q3–2019Q4), and 1-year aggregate dividend growth extended from De La O and Myers (2021) to 2019Q4. Equity yield data are from Giglio et al. (2021); bond yields are end-of-quarter zero-coupon nominal yields from Gürkaynak et al. (2007).

Main Findings.

Equity Term Structure Dynamics. The model’s subjective dividend growth expectations drive equity yields. The 1-year model-implied equity yield correlates 0.68 with data; the 10-year correlates 0.79; the 10Y–1Y slope correlates 0.59 with data. Consistent with “belief in the law of small numbers,” the agent overreacts to dividend news (estimated learning gains νl_d = 0.166 and νs_d = 0.458, both below their Kalman gains, which under the level-to-growth translation implies overreaction to dividend growth news, confirmed by negative CG(2015) regression slope coefficients of −0.69 at 1Y and −0.97 at 5Y).
Procyclical Equity Yield Slope. During recessions, the average equity yield slope (10Y–1Y) in the model is −3.77%; during expansions it is +3.96%, matching the data (−5.50% in recessions, +3.93% in expansions). The sign reversal is driven primarily by the dividend-specific component of the decomposition: in recessions, short-run dividend growth expectations fall much more sharply than long-run expectations.
Bond Pricing. The model’s 1-year and 10-year nominal bond yields achieve correlations of 0.92 and 0.95 with their data counterparts, inheriting the explanatory power of Zhao (2020) for the bond market. The agent underreacts to GDP growth and inflation news (estimated learning gains well below Kalman gains, confirmed by positive CG(2015) slope coefficients of +2.08 at 1Y for GDP growth and +1.01 at 1Y for inflation).
Bond-Stock Correlation Switch. In data, 10Y bond vs. dividend strip return correlation (5Y strip) goes from +0.46 before 2000 to −0.49 after 2000. The model produces +0.14 before and −0.56 after (for the 5Y strip). Decomposing the change in bond-stock return covariance: the “inflation real effect” (correlation between expected inflation and real growth) accounts for approximately 27–31% of total changes (for 5Y to 10Y strips); the “real growth correlation” channel — stronger co-movement between real GDP and real dividend growth expectations after 2000 — accounts for approximately 89–95% of total changes. The paper identifies this real bond hedging channel as the dominant and previously unexamined driver.
Dividend Strip Return Predictability. The price-dividend ratio predicts annual market excess returns with R² of 10.3% (data) vs. 9.0% (model). Strip return predictability is downward-sloping by maturity: in data, the R² is 20.2% for 5-year strips and 14.5% for 10-year strips; the model generates 14.2% and 10.4% respectively. This is decomposed into three sources: bond return predictability (small contribution), dividend forecast error predictability (dominant for short maturities), and forecast revision predictability (negative contribution that offsets). The downward slope occurs because current news has smaller impact on long-term dividend expectations.
Aggregate Market Puzzles. The model-implied log dividend-price ratio correlates 0.86 with data, with AR(1) coefficient 0.96 (data: 0.95). Model-implied average market return is 9% (data: 8%); annualized return volatility 12% (data: 16%). The model replicates the switch of the bond-stock aggregate return correlation from +0.13 before 2000 to −0.46 after 2000 (data: +0.39 to −0.64).

Scope Conditions. Results apply to U.S. equity and bond markets over 1987Q4–2019Q4 (with bond learning using data back to 1959Q1). The model assumes a representative agent with CRRA utility and constant subjective risk premium. It is silent on the term structure of expected returns in the statistical sense (which requires identification of latent states under the physical measure). The aggregate market results require a reduced-form specification for stochastic equity duration H_t linked to the value-weighted LTG average.

In depth

Q1. What is the core psychological mechanism generating subjective beliefs, and how does it differ from the diagnostic expectations approach?

The agent has the “belief in the law of small numbers” (Tversky and Kahneman 1971): she treats small samples as equally representative of their population as large samples. Formally, this is embedded by exaggerating the likelihood in the Bayesian update: p(x_t|I_t) ∝ p(y_t|x_t)^{1+θ} × p(x_t|I_{t-1}), where θ captures the magnitude of cognitive bias. The resulting subjective learning gain ν = (1+θ)P̃ / [(1+θ)P̃ + σ²_ε] can exceed the Kalman gain K when θ is large (overreaction) or fall below it when θ is small (underreaction). This differs from diagnostic expectations (Bordalo et al. 2019, 2020a,b), which are based on the representativeness heuristic; the paper notes the two notions of news are highly correlated in simulation (Table IA.2) and that both can imply overreaction.

Q2. Why does the model generate overreaction to dividend growth news even though the dividend-level learning gains are smaller than the Kalman gains?

The model separates dividend learning into level and growth. Section 2.2 derives that underreaction to dividend level news (νl_d < Kl_d, νs_d < Ks_d, estimated values 0.166 and 0.458 against Kalman gains 0.19 and 0.49 respectively) translates into overreaction to dividend growth news. This is confirmed by the CG(2015) rationality test: regressing forecast errors on lagged forecast revisions yields slope coefficients of −0.69 (1Y) and −0.97 (5Y) for real dividend growth, both statistically significant (t-statistics −3.63 and −3.22). In contrast, the same test yields positive slope coefficients for GDP growth (2.08 at 1Y) and inflation (1.01 at 1Y), confirming underreaction for these series.

Q3. How well does the model match subjective dividend growth expectations in the survey data?

The model-implied 1-year subjective dividend growth forecast is estimated by minimizing RMSE against the consensus dividend growth forecast series (extended from De La O and Myers 2021 to 2019Q4, with a replication correlation of 0.92 over the overlapping sample). The unconditional correlation between model-implied and data 1-year forecasts is 0.80. Although only 1-year forecasts are used in estimation, the model also achieves a correlation of 0.80 for 2-year forecasts, providing an out-of-sample validation.

Q4. What explains the higher volatility of short-term equity yields relative to long-term equity yields?

Short-term subjective dividend growth expectations are more volatile because the agent’s short-run expectation mean-reverts toward the less volatile long-run (levered) GDP growth expectation. In the model’s two-component dividend structure, the transitory dividend-share component xd has persistence ρ_d = 0.94 and its effect on equity yields decays as maturity increases (via the factor (1−ρ^n_d)/n). Similarly, the effect of the transitory GDP growth state x_g decays with maturity. Long-term equity yields are thus anchored by the slower-moving trend components µ_g and µ_d. In the data from Giglio et al. (2021), 1-year yields have a standard deviation of 8.89% annualized vs. 2.70% for 10-year yields; the model generates 8.22% and 1.89% respectively.

Q5. What is the quantitative importance of the “real growth correlation” channel vs. the “inflation real effect” channel in explaining the bond-stock correlation switch?

For the switch in bond-stock return correlation (using the 10-year nominal bond and various maturity dividend strips), the decomposition in Table 4 shows that the “real growth correlation” channel accounts for 89.1% (5Y strip), 92.1% (7Y strip), and 94.8% (10Y strip) of total bond-stock covariance changes, while the “inflation real effect” (correlation between expected inflation and expected real growth) accounts for 27.3%, 29.3%, and 31.1% respectively. The “volatility of shocks to expected inflation and real growth” makes a negative contribution (−16.4%, −21.4%, −25.9%), mostly attributable to more volatile beliefs during the 2008 global financial crisis. The real growth correlation channel reflects that after 2000, real bonds provide a better hedge to aggregate real dividend risks because real GDP growth expectations and real dividend growth expectations became more positively correlated.

Q6. Does the same real growth correlation story hold for the “Fed model” (bond-stock yield correlation)?

Yes, but with a quantitatively different balance. For yield correlations (Table 5), the “real growth correlation” channel accounts for 72.4%–80.1% of bond-stock yield covariance changes (5Y to 10Y strip), while the “inflation real effect” now accounts for 41.2%–43.9%. The inflation real effect is proportionally larger for yield levels because persistent expected inflation correlates strongly with the level of expected real GDP growth — even though inflation expectations do not move fast enough at high frequency to explain return correlation, they co-move strongly with expected growth at low frequency.

Q7. How does the model generate a downward-sloping term structure of return predictability?

The strip excess return is decomposed into three components (Equation 44): maturity-matched bond excess return (Bond), dividend forecast error within the holding period (FE), and forecast revision regarding dividend growth after the holding period (FR). For short maturities, bond predictability contributes little (R² ≈ 6.7% for 5Y strip), while FE predictability (R² ≈ 31.5%) and FR predictability (R² ≈ 35.6%) dominate. As maturity increases, the current news has smaller impact on long-term dividend expectations, reducing the predictability of FE (R² ≈ 26.6% for 10Y) and FR (R² ≈ 26.5% for 10Y). Taken together, total model-implied strip R² declines from 14.2% (5Y) to 10.4% (10Y), matching the data pattern (20.2% to 14.5%). The paper identifies forecast revision predictability as a new channel not previously documented.

Q8. Why do forecast errors and forecast revisions have opposite signs in the predictability regressions?

Bad news (high equity yields, i.e., low current stock prices) triggers excessively pessimistic subjective dividend growth expectations because the agent overreacts to dividend news. These overly pessimistic forecasts tend to be disappointed in the future — actual dividend realizations exceed the forecast — producing positive subsequent forecast errors (FE is positively predicted by high yields, with R² ≈ 31.5% for 5Y strips). However, as dividend levels mean-revert, higher subsequent realizations cause the agent to revise down the forecast for dividend growth thereafter, leading to negative forecast revisions (FR is negatively predicted by high yields, with R² ≈ 35.6% for 5Y strips, opposite sign from FE). The net effect on return predictability is thus a combination of positive (FE) and negative (FR) contributions.

Q9. How does the model handle the aggregate market dividend-price ratio and its persistence?

The aggregate stock price is modeled as the sum of dividend strip prices up to a stochastic horizon H_t, which is parameterized as a linear function of the value-weighted average of LTG forecasts: H_t = a + b·LTG_t. Parameters a and b are estimated by minimizing RMSE between model-implied and data log dividend-price ratio. The model-implied ratio achieves a correlation of 0.86 with data, an AR(1) coefficient of 0.96 (data: 0.95), and an annualized volatility of 26% (data: 30%). The time-variation is driven entirely by strip yield variations and exogenous LTG movements.

Q10. Is the overreaction to dividend news and underreaction to GDP/inflation news consistent in a single framework?

Yes. The model’s subjective learning framework (based on “belief in the law of small numbers”) generates both over- and underreaction depending on the estimated subjective learning gain relative to the Kalman gain. For GDP growth and inflation, the learning gains (ν*_g = 0.012, νgap_g = 0.065; ν*_π = 0.049, νgap_π = 0.228) are below their Kalman gains (0.29 and 0.67 for GDP components; 0.67 and 0.48 for inflation components), producing underreaction. The paper hypothesizes this is related to the Fed’s dual mandate: agents rationally assign lower weight to GDP and inflation shocks expecting the Fed will stabilize them. For dividend growth, a level-to-growth translation converts level underreaction into growth overreaction.

Q11. What are the robustness checks, and what do they show?

The paper checks three alternative equity duration measures: those from Dechow et al. (2004), Weber (2018), and Gonçalves (2021b), as well as the book-to-market ratio following Lettau and Wachter (2007). Table IA.1 shows that replacing LTG with these measures still produces model-implied equity yields that replicate key data moments with high time-series correlations. Changing the cross-sectional breakpoint for long-duration dividends from the median LTG to the 40th or 60th percentile leaves results similar. The paper also presents an Internet Appendix extension in which the agent has ambiguity about real GDP and dividend growth (model misspecification fear), yielding equity yields and returns even closer to data.

Q12. What is the paper’s contribution to the bond market relative to Zhao (2020)?

The bond pricing block closely follows Zhao (2020), inheriting its explanatory power for bond market stylized facts. The model’s 1-year and 10-year nominal bond yields achieve correlations of 0.92 and 0.95 with data, respectively. The new contribution is the joint model covering both equity and bond markets simultaneously, enabling the decomposition of bond-stock covariance and the identification of the real growth correlation as the dominant driver of the bond-stock correlation switch — a channel not addressed by Zhao (2020), which focused on bond market puzzles alone.

Key Concepts

Equity Yield (Dividend Strip Yield). Defined as ey^(n)_t = (1/n)(d$_t − p^(n)_t), where p^(n)_t is the log price of the n-period dividend strip (a claim to the nominal dividend n periods ahead) and d$_t is the log nominal aggregate dividend. It decomposes into the bond yield, a subjective dividend growth component, and a (constant) risk premium component.

Belief in the Law of Small Numbers. A cognitive bias (Tversky and Kahneman 1971) in which the agent perceives small samples to represent their population as well as large samples. Modeled by exaggerating the likelihood in Bayesian updating: p(x_t|I_t) ∝ p(y_t|x_t)^{1+θ} × p(x_t|I_{t-1}). This generates a subjective learning gain ν that can exceed the Kalman gain (overreaction) or fall below it (underreaction) depending on θ and the signal-to-noise ratio.

Subjective Learning Gain. The coefficient ν in the subjective Kalman filter update ẽ_t x_t = ρẽ_{t-1}x_{t-1} + ν(y_t − ρẽ_{t-1}x_{t-1}). It equals (1+θ)P̃ / [(1+θ)P̃ + σ²_ε], where P̃ is the subjective predictive variance. When ν > K (the rational Kalman gain), the agent overreacts to news; when ν < K, the agent underreacts.

Long-Duration Dividend Component. The portion of aggregate real dividend (dl_t) attributable to “long-duration” firms — those with above-median analyst LTG forecasts in CRSP/Compustat/IBES data. Levered on log real GDP with leverage parameter λ = 3, it carries aggregate risk. The complementary short-duration dividend share ds_t is stationary and carries no aggregate risk. The decomposition allows the model to exploit cross-sectional cash-flow duration information when learning about future aggregate dividend growth.

Real Growth Correlation Channel. A bond-stock covariance component defined as Cov(RGDP^(N), RDIV^(n)), where RGDP^(N) is the real GDP growth expectation component of 10-year nominal bond returns and RDIV^(n) is the real dividend growth expectation component of n-period strip returns. This channel captures whether real bonds hedge aggregate real dividend risks. The paper shows this channel accounts for approximately 89–95% of the post-2000 bond-stock covariance change for dividend strips.

Inflation Real Effect. The covariance component Cov(INFL^(N)_B, RGDP^(n) + RDIV^(n)), defined as the correlation between shocks to expected inflation (embedded in nominal bond returns) and shocks to expected real growth (in strip returns). In the paper’s framework this is distinct from the standard inflation risk premium story, as it concerns the correlation between subjective beliefs rather than realized covariances under the physical measure.

Forecast Error (FE) and Forecast Revision (FR) Predictability. Two of three components of realized strip excess return (Equation 44). FE = ∆d${t+1:t+h} − ẽ_t∆d${t+1:t+h} is the realized dividend growth forecast error within the holding period; FR = (ẽ_{t+h} − ẽ_t)∆d$_{t+h+1:t+n} is the forecast revision for dividend growth beyond the holding period. Because the agent overreacts to dividend news, bad news triggers overly pessimistic forecasts (positive subsequent FE) and, as dividends mean-revert, downward forecast revisions (negative FR). These two have opposite signs in predictive regressions, generating the downward-sloping term structure of return predictability.

Fed Model. The empirical positive correlation between equity yields (real) and nominal bond yield levels. The paper shows that this yield-level correlation switched from strongly positive (≈ 0.85 before 2000) to significantly negative (≈ −0.60 to −0.62 after 2000) for 5Y–10Y dividend strips, and that the same real growth correlation and inflation real effect decomposition applies, albeit with the inflation real effect proportionally larger (≈ 40%) for yield levels than for returns (≈ 30%) because persistent inflation expectations co-move with the level of expected real GDP growth.

Explicit consumption functions with borrowing constraints: A continuous-time approach

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research question. The paper asks whether an explicit, global, closed-form solution exists for the consumption function in the standard income fluctuation problem with a borrowing constraint and constant income, a problem that has resisted closed-form solution since at least Schechtman (1976). All prior continuous-time work (Park 2006, Holm 2018, Fischer 2024) produced only implicit expressions; Achdou et al. (2022) produced explicit expressions valid only locally, near zero assets or as assets diverge to infinity, and only for r > 0.

Model. A single agent with CRRA utility (coefficient of relative risk aversion γ > 0) maximizes discounted utility over an infinite horizon, subject to the flow budget constraint da/dt = ra + y − c, with a borrowing constraint a(t) ≥ 0. The agent receives a constant, deterministic income stream y ≥ 0 and discounts at rate ρ, with the impatience condition ρ > r maintained throughout. The paper takes a continuous-time formulation arrived at by letting the discrete period length Δ → 0, nesting Helpman (1981)’s discrete-time analysis as a special case.

Key analytical device. A one-to-one mapping exists between initial assets a and the time T it takes for the consumer to fully run down her assets. This map, denoted T = h(a; y), is well-defined, strictly increasing, and concave in a (established in Proposition 1 via the Hadamard-Lévy theorem). Expressing the optimal consumption function as c*(a; y) = y · exp(ρh(a;y)/γ) evaluated at t = 0 reduces the problem to explicitly inverting the transcendental equation relating a to T.

Main result (r = 0). For the case of a zero net real interest rate, the transcendental equation can be solved explicitly using the second branch W₋₁(·) of the Lambert W function. The closed-form consumption function is (Theorem 2 and Corollary 2.1):

c*(a; y) = y · exp(ρ h(a;y) / γ), where h(a; y) = −(a/y + γ/ρ) − (γ/ρ) W₋₁(f(a;y)), and f(a;y) = −exp(−b(a + γy/ρ)/y), b := ρ/γ.

This is a global solution (valid for all a ≥ 0), in contrast to the local solutions in prior work. The paper notes that for the illustrative parameter values r = 0.01, γ = 0.5, ρ = 0.08, y = 3 (broadly consistent with average U.S. real interest rates in 2025), there is a visually sizable gap between the constrained and unconstrained consumption functions except as a → ∞, where the two converge (in line with the asymptotic linearity result of Benhabib et al. 2015).

Main result (r > 0). For positive interest rates, the Lambert W function cannot invert a sum of exponentials with different exponents (an open mathematical problem). The paper instead derives a global closed-form approximation valid for r ∼ 0, by expanding e^(−rT) ≈ 1 − rT to first order and applying the same Lambert W inversion. The approximating consumption function has the same structural form but with modified coefficients b_r, c_r, d_r that collapse to their r = 0 counterparts as r → 0 (Proposition 2). Numerical comparison against the implicit-expression solution of Park (2006) confirms the approximation is close for small r.

Characterization of the MPC and supermodularity (Section 3). Leveraging the explicit expression, the paper derives the full Jacobian vector and Hessian matrix of c*(a; y) in closed form (Propositions 3 and 4). Key findings, all proved formally and holding under the impatience condition ρ > r:

Consumption is increasing in both assets and permanent income (both entries of the Jacobian are strictly positive — Corollary 2.2). The second result (∂c*/∂y > 0 for all a) is new for the borrowing-constrained setting; Achdou et al. (2022) provided only suggestive evidence for the limiting case a ∼ 0.
Consumption is strictly concave in both assets and permanent income (both diagonal entries of the Hessian are strictly negative — Corollary 2.3). Concavity in assets was known (Carroll and Kimball 1996); concavity in permanent income under borrowing constraints is new.
The consumption function is supermodular: the cross-derivative ∂²c*/∂a∂y is strictly positive (Corollary 2.3). This means assets and permanent income are complements in generating consumption. Equivalently, the MPC out of permanent income is strictly increasing in the level of initial assets — a counter-intuitive result, since high MPCs are usually associated with poor (low-asset) agents. An identical result was obtained by Commault (2025) for a life-cycle model without borrowing constraints; the current paper confirms it holds in the presence of a borrowing constraint. By symmetry of the Hessian, the MPC out of assets is also strictly increasing in permanent income.

Intuition for supermodularity. When assets are low, an increase in permanent income produces little additional consumption because the risk of hitting the borrowing constraint is high. When assets are higher, the agent has buffer savings, faces a lower constraint-risk, and can smooth the higher future income stream into current consumption.

Scope conditions. Results are derived under CRRA utility, constant (deterministic) income, no stochastic variation, and the impatience condition ρ > r. The exact closed form applies to r = 0; the approximation is characterized as valid for r ∼ 0 and is not a local expansion in assets.

In depth

Q1. What is the longstanding gap in the literature that this paper addresses?

A: Since Zeldes (1989) noted that no closed-form solution exists for the consumption function with stochastic income and CRRA utility, researchers settled for numerical solutions or local analytical approximations. In the constant-income/borrowing-constraint version studied here, Park (2006), Holm (2018), and Fischer (2024) derived only implicit continuous-time expressions. Achdou et al. (2022) gave explicit local solutions valid near a ∼ 0 or a → ∞ under r > 0. No prior work produced an explicit, global closed-form for any case.

Q2. Why does moving to continuous time enable progress that discrete time did not?

A: In discrete time, the consumption function is piecewise linear (Helpman 1981), with kinks at the sequence of asset thresholds µ(T) for T = 0, Δ, 2Δ, …. As Δ → 0, the piecewise-linear function converges to a smooth function whose governing ODE can be solved analytically. This convergence to smoothness, illustrated in Figure 1, is what enables the application of the Lambert W function to invert the resulting transcendental equation.

Q3. What is the role of the Lambert W function, specifically its second branch W₋₁?

A: The optimal asset-depletion time T satisfies the transcendental equation e^(bT) = yT + c (for r = 0), which cannot be solved with elementary functions. Via the change of variables z := −bT − bc/y, the equation reduces to ze^z = α, whose solution is z = W(α). The argument α lies in (−1/e, 0) for a ∈ (0, +∞), and it is precisely on this interval that the Lambert W function is double-valued; the relevant branch is W₋₁ (the second, lower branch), which is well-defined and strictly less than −1 on (−1/e, 0). It is the properties of W₋₁ on this domain — specifically that 1 + W₋₁(α) < 0 — that drive the sign conclusions for the Hessian.

Q4. Why does the Lambert W approach fail for r > 0, and what is the approximation strategy?

A: For r > 0, Equation (8) contains two exponentials with different exponents — e^((ρ−r)T/γ) and e^(−rT) — and their sum cannot be inverted by the Lambert W function, which handles only a linear-plus-single-exponential structure. Inverting a sum of exponentials with different exponents is stated in the paper to be an open problem. The approximation strategy exploits the fact that for r ∼ 0, e^(−rT) ≈ 1 − rT + o(r), reducing the equation to a single-exponential transcendental form (Equation 15) with modified coefficients b_r, d_r, c_r, all of which converge to their r = 0 analogues as r → 0.

Q5. What does Proposition 1 establish, and why is it necessary before stating the main theorem?

A: Proposition 1 establishes that the mapping µ(T) from depletion time T to initial assets a is smooth (infinitely differentiable), bijective (one-to-one and onto) on ℝ₊, and strictly convex. The Hadamard-Lévy theorem then guarantees that its inverse h(a;y) = µ⁻¹(a) exists, is unique, is strictly increasing, and is strictly concave in a. This is a necessary prerequisite for Theorem 2 because h(a;y) is the central object in the closed-form consumption function; without establishing its existence and uniqueness, Theorem 2 would have no well-defined object.

Q6. What does the Jacobian characterization (Proposition 3 and Corollary 2.2) contribute?

A: Proposition 3 gives explicit formulas for ∂c*/∂a = (ρ/γ) · w/(1+w) and ∂c*/∂y in terms of w = W₋₁(f(a;y)). Corollary 2.2 proves both are strictly positive using the property w < −1 on (−1/e, 0), which ensures w/(1+w) > 0 and that the bracketed term in the expression for ∂c*/∂y is strictly positive. The contribution is that the positivity of ∂c*/∂y for all a was previously unproven in a borrowing-constrained setting with constant income.

Q7. What is the structure of the Hessian matrix and what signs do its entries take?

A: All four entries of Hc are proportional to w/(1+w)³. Since w < −1, we have 1 + w < 0, so (1+w)³ < 0, making w/(1+w)³ > 0. The diagonal elements ∂²c*/∂a² = −(ρ²/γ²y) · w/(1+w)³ and ∂²c*/∂y² = −(ρ²a²/γ²y³) · w/(1+w)³ are both strictly negative (concavity). The off-diagonal elements ∂²c*/∂a∂y = (aρ²/γ²y²) · w/(1+w)³ are strictly positive (supermodularity/complementarity).

Q8. What is the precise counter-intuitive implication of supermodularity for MPC heterogeneity?

A: Supermodularity (∂²c*/∂a∂y > 0) means the MPC out of permanent income — conventionally associated with low-wealth households — is in fact increasing in the level of initial assets. This contradicts the conventional narrative that high MPCs are a hallmark of poor agents. The paper’s intuition is that low-asset agents face high risk of hitting the constraint, suppressing their consumption response to income news, while high-asset agents can freely smooth the increased income stream. The same supermodularity implies, by the symmetry of the Hessian, that the MPC out of assets is also increasing in permanent income.

Q9. How does this result relate to Commault (2025)?

A: Commault (2025) proved, in a life-cycle model with a permanent/transitory stochastic income process but without borrowing constraints, that the MPC out of permanent income is increasing in assets. The current paper obtains the same qualitative finding in the opposite environment — constant income with a borrowing constraint. The paper treats these as complementary, noting that the result thus appears robust to these different modeling choices.

Q10. What does concavity in permanent income (∂²c*/∂y² < 0) add that was not previously known?

A: Carroll and Kimball (1996) established concavity of the consumption function in assets for a broad utility class. Concavity in permanent income — that the marginal consumption response to a windfall increase in y is diminishing — had been proved by Commault (2025) only in the absence of borrowing constraints. The current paper provides the first formal proof of this property in a setting with a borrowing constraint (albeit for constant, deterministic income and CRRA utility in continuous time).

Q11. What is the potential use of these closed-form results for numerical methods?

A: The paper notes in the conclusion that the closed-form solutions for r = 0 and the approximation for r ∼ 0 can serve as benchmarks for assessing the reliability of continuous-time numerical methods when computing objects such as the MPC out of assets. Because the exact solution is known analytically, numerical implementations can be compared against it to detect discretization errors or convergence failures.

Q12. What parameter values are used to illustrate the consumption function, and what do they imply?

A: The paper uses r = 0.01, γ = 0.5, ρ = 0.08, y = 3, where r = 0.01 is described as roughly in line with the average real interest rate in the U.S. in 2025. With these values, Figure 1 shows a visually sizable gap between the constrained and unconstrained consumption functions at low to moderate asset levels, with the two converging as a → ∞ as guaranteed by asymptotic linearity (Benhabib et al. 2015).

Key Concepts

Income fluctuation problem (with borrowing constraint): The standard infinite-horizon single-agent savings problem in which the agent faces a non-negativity constraint on assets (a(t) ≥ 0), so that the agent cannot borrow. In the paper’s formulation: maximize ∫ e^(−ρt)u(c(t))dt subject to da/dt = ra + y − c and a(t) ≥ 0, with constant income y and CRRA utility. The borrowing constraint creates the concavity of the consumption function and was the source of intractability in prior closed-form attempts.

Lambert W function (second branch W₋₁): A special transcendental function defined as the solution to we^w = x. It is double-valued on (−1/e, 0); the second branch W₋₁ takes values strictly less than −1 on this interval. In this paper, the transcendental equation linking initial assets to asset-depletion time is reduced to the form ze^z = α, enabling explicit inversion via W₋₁. The property that 1 + W₋₁(α) < 0 on (−1/e, 0) is the algebraic engine driving all sign results in the Hessian.

Asset-depletion time T = h(a; y): The time it takes for the optimal consumer to fully run down her initial assets before settling into perpetual income consumption of y. The paper establishes a bijective mapping from initial assets a to depletion time T (Proposition 1); the closed-form solution is obtained by explicitly inverting this mapping. In the paper’s formulation, h(a; y) = µ⁻¹(a) where µ(T) is derived from the ODE governing the consumption path.

Supermodularity of the consumption function: The property that the cross-derivative ∂²c*/∂a∂y is strictly positive, meaning assets a and permanent income y act as complements in generating consumption. This is an equilibrium property of the consumption function (not an assumption on the utility function), and the paper identifies it as new to the income fluctuation literature. It implies the MPC out of permanent income is increasing in a, and the MPC out of assets is increasing in y.

MPC out of permanent income (∂c/∂y):* The marginal increase in current consumption per unit increase in the constant income stream y, holding initial assets constant. This object is less studied than the MPC out of a transient asset windfall. In the paper’s setting, it is shown to be strictly positive for all a (Corollary 2.2) and, counter-intuitively, strictly increasing in a (supermodularity).

Global vs. local closed-form solution: A global solution holds for all values of the state variable (here, all a ≥ 0), while a local solution is valid only in the neighborhood of a particular value (e.g., a ∼ 0 or a → ∞). Achdou et al. (2022) produced local closed-form expressions; the current paper’s Theorem 2 (r = 0) is the first global explicit closed-form for this class of problems.

Piecewise-linear consumption function (discrete time): In Helpman (1981)’s discrete-time formulation with period length Δ = 1, the optimal consumption function is piecewise linear in assets, with slope changes at the asset thresholds µ(T) for integer T. As Δ → 0, this becomes a smooth function, enabling the passage to the continuous-time closed form derived in the paper.

Failing Banks

Mon, 01 Jan 0001 00:00:00 +0000

Correia, Luck, and Verner ask a foundational question in banking: why do banks fail? Specifically, they seek to adjudicate between two theoretical views — the solvency view (failures caused by deteriorating asset quality and insolvency) and the bank runs view (failures caused by depositor coordination failure that can bring down otherwise solvent banks) — using the longest micro-level panel of U.S. commercial bank balance sheets assembled to date.

The authors construct a panel covering approximately 37,000 distinct banks across two samples: a historical sample of all national banks from 1863 to 1941 (sourced from OCC Annual Reports, digitized via OCR) and a modern sample of all commercial banks from 1959 to 2024 (from FFIEC Call Reports merged with the FDIC failure list). More than 5,000 banks fail across the full sample, with 2,887 failures before 1935 and 2,233 after 1959. The sample spans institutional regimes before and after the Federal Reserve (founded 1913) and the FDIC (founded 1933/1934).

Three sets of findings emerge. First, failing banks are characterized by deteriorating fundamentals well before failure: rising non-performing loans and declining solvency (equity-to-assets falls by 8 percentage points in the five years before failure in the modern sample), increasing reliance on expensive noncore funding (rising by 18% of assets in the decade before modern-era failures), and a boom-bust pattern in real assets (expanding by 34% from ten years to three years before failure before contracting). These patterns are consistent across the pre-FDIC and modern eras.

Second, bank failures are highly predictable from publicly available accounting data. Using simple regression models with insolvency risk, noncore funding reliance, and asset growth as predictors, the area under the ROC curve (AUC) for predicting failure within one year reaches 86% in the historical sample and 90–95% in the modern sample. Pseudo-out-of-sample performance is nearly as strong as in-sample performance. A bank in the top 5th percentile of both insolvency risk and noncore funding vulnerability faces a three-year failure probability of 27% in both the historical and modern samples, compared to unconditional rates of 2.5% (historical) and 1% (modern) — a 10- to 25-fold increase.

Third, while large deposit outflows consistent with bank runs were common in pre-FDIC failures — deposits declined on average by 14% immediately before failure in 1880–1934, and by 21% in the period before the banking holiday — failures with runs are as predictable as failures without runs, and they occur in banks with similarly weak fundamentals. Recovery rates on failed banks’ assets averaged only 52% of book value in pre-FDIC failures. Using a framework comparing recovery rates to leverage, the majority of pre-FDIC failed banks appear to have been fundamentally insolvent. Even under the extreme assumption of zero value destruction from failure, runs on banks that were not fundamentally insolvent account for fewer than 8% of pre-FDIC failures; under an assumption of 20% value destruction from failure, this share rises to 22%.

OCC bank examiners classified fewer than 2% of pre-FDIC failures as caused by runs or liquidity issues; most were attributed to losses, fraud, or external shocks. The aggregate failure rate is also largely predictable: regressing the actual bank failure rate on predicted aggregate failure risk yields an R-squared of 40%.

Scope conditions: the historical sample covers only national banks (market share ranging from ~80% in the 1870s to ~45% in the 1930s); the modern sample excludes de novo banks (younger than three years); deposit outflow data for the historical period begin in 1880; and FDIC failure transaction data for the modern period begin in 1993.

Q: What are the two main theoretical views the paper evaluates, and how does the paper distinguish between them? A: The solvency view holds that bank failures are caused by deteriorating asset quality and insolvency, with the runnable nature of liabilities playing no essential causal role. The bank runs view holds that the runnable nature of demandable deposits is central, with depositor coordination failure capable of bringing down otherwise solvent banks (Diamond and Dybvig, 1983) or weak-but-solvent banks (Goldstein and Pauzner, 2005). The paper distinguishes between them using three empirical tests: predictability of failures from fundamentals, deposit outflows before failure, and asset recovery rates in failure.

Q: How predictable are bank failures, and what does predictability imply for the bank runs view? A: In the historical pre-FDIC sample (1863–1934), the in-sample AUC for predicting failure within one year is 86%; in the modern sample (1959–2024) it is 90–95%. Pseudo-out-of-sample AUC is nearly as strong as in-sample AUC. High predictability is consistent with the solvency view and fundamental-based panic run models, but is inconsistent with non-fundamental self-fulfilling runs (Diamond and Dybvig, 1983), which should strike randomly. Predictability also cuts against the assumption of rational, forward-looking depositors in fundamental-run models, since attentive depositors would act on observable signals and accelerate failure, reducing predictability.

Q: What is the boom-bust pattern in failing banks’ assets? A: In the decade before failure, failing banks’ real total assets expand by 34% from ten years to three years before failure, then contract over the final two years. The boom-and-bust pattern is present in both the historical and modern samples but is more pronounced in the modern period. The boom is driven primarily by loan growth (particularly real estate lending and C&I lending in the modern sample) rather than by growth in liquid assets, consistent with the view that rapid credit expansion produces future credit losses.

Q: How does noncore funding behave in failing banks, and why does it matter? A: In failing banks in the modern sample, noncore funding (time deposits plus wholesale funding) rises by 18% of assets over the decade before failure, while demand deposits decline as a share of assets. In the historical sample, noncore (wholesale) funding also rises gradually. Noncore funding is a signal of failure for multiple reasons: it is more expensive than core deposits, eroding profitability; it can finance risky asset growth; it reflects realized losses being funded at the margin; and it increases funding fragility, making banks more vulnerable to shocks.

Q: How strong is the joint signal from insolvency and noncore funding? A: A bank in the top 5th percentile of both insolvency risk and noncore funding vulnerability faces a three-year failure probability of 27% in the historical sample and 27% in the modern sample. The unconditional three-year failure probability is 2.5% in the historical sample and 1% in the modern sample. This amounts to a 10- to 20-fold increase in failure probability, illustrating that the combination of solvency and funding weakness is a powerful joint predictor.

Q: Were deposit outflows common before the FDIC, and did they decline after its introduction? A: In the 1880–1934 historical sample, deposits in failing banks declined on average by 14% between the last call report and failure, with 25% of pre-FDIC failures preceded by outflows exceeding 20%; during the period before the banking holiday the average deposit decline was 21%. In contrast, in the modern sample (1993–2024), average pre-failure deposit outflows were only 2.5%, and outflows exceeding 20% occurred in only 3% of failures, consistent with deposit insurance insulating most depositors.

Q: Are failures with large deposit outflows (runs) less connected to weak fundamentals than other failures? A: No. The paper finds that failures with large deposit outflows are as predictable as failures without large deposit outflows. The relationship between insolvency risk or noncore funding and three-year failure probability is similar for failures with and without large deposit outflows. This implies that runs did not disproportionately strike banks with otherwise strong fundamentals.

Q: What do asset recovery rates reveal about the insolvency status of pre-FDIC failed banks? A: Recovery rates on pre-FDIC failed banks averaged 52% of book value of assets. Under the extreme assumption that receivership destroys zero bank value, runs on non-fundamentally-insolvent (weak but solvent) banks account for fewer than 8% of pre-FDIC failures. Under the equally extreme assumption that failure destroys 20% of bank value, this share rises to 22%. The majority of pre-FDIC failed banks therefore appear to have been fundamentally insolvent.

Q: What did contemporary OCC bank examiners attribute as the causes of bank failures? A: OCC bank examiners classified most pre-FDIC failures as caused by losses, fraud, or external economic shocks. Runs and liquidity issues together account for fewer than 2% of OCC-classified failures, notwithstanding the common occurrence of large deposit outflows before many of these failures. This examiner evidence supports the solvency view.

Q: Can bank-level fundamentals predict systemic banking crises and aggregate failure waves? A: Yes. The authors aggregate out-of-sample predicted failure probabilities to construct a predicted aggregate bank failure rate. The R-squared from regressing the actual aggregate bank failure rate on this predicted rate is 40%, indicating that spikes in bank failures during systemic crises are substantially accounted for by the prior deterioration of bank-level fundamentals.

Q: Why is predictability higher in the modern sample than in the historical sample? A: The authors identify several reasons. Accounting data quality is higher in the modern sample. Historical national banks operated as unit branches with less geographic diversification, making idiosyncratic shocks more important and harder to predict. Modern-era failures are preceded by larger lending booms that produce more predictable downstream losses. Additionally, in the modern context bank failures are largely supervisory decisions, and frictions in the supervisory process may delay closure and thereby increase predictability.

Q: What role do the authors assign to depositor inattention? A: The high predictability of failures combined with the finding that many failing banks had high predicted failure probabilities before actually failing suggests that depositors were often slow to react to observable signals of bank weakness. The authors note this points to behavioral frictions such as neglect of downside risk (Gennaioli et al., 2012) and sleepy or inattentive depositors (Hanson et al., 2015; Jiang et al., 2023), rather than the rational, forward-looking depositor assumption embedded in standard bank run models.

Q: What is the paper’s overall interpretive conclusion about the relative importance of solvency versus runs? A: The primary cause of bank failures is almost always and everywhere a deterioration of bank solvency. Runs were more common in the historical pre-FDIC data as a mechanism triggering failure, but they typically closed banks that were already fundamentally insolvent. Non-fundamental, self-fulfilling runs on otherwise healthy banks appear to be an uncommon cause of bank failures. Under the solvency view, even when runs occur, they are the trigger and final mechanism rather than the root cause.

Insolvency risk: A bank’s proximity to default, proxied in the historical sample by surplus profits relative to equity (capturing profitability and capitalization) and in the modern sample by net income to assets. High insolvency risk reflects declining profitability and eroding capital buffers.

Noncore funding: Expensive, risk-sensitive funding sources outside core demand deposits, including time deposits, wholesale funding (bills payable, rediscounts), and non-deposit wholesale borrowings. Banks relying heavily on noncore funding face higher funding costs, reduced profitability, and greater fragility to funding shocks.

Fundamental run: A run triggered when bank fundamentals are so weak (theta at or below the lower threshold in the Goldstein-Pauzner framework) that all depositors have an incentive to withdraw regardless of others’ actions — the bank is effectively insolvent and failure is inevitable.

Panic-based run: A run triggered when bank fundamentals are moderately weak (below the threshold equilibrium in Goldstein-Pauzner) but the bank would have been able to pay all creditors absent the run; the run itself destroys value and causes failure.

Non-fundamental (self-fulfilling) run: A run on an otherwise solvent bank driven purely by depositor coordination failure, as in Diamond and Dybvig (1983); failure arises from one of two equilibria and is not predicted by fundamentals.

Recovery rate: Funds ultimately collected by the receiver throughout receivership proceedings divided by the book value of assets at suspension; used as a proxy for the degree of fundamental insolvency at failure. Pre-FDIC recovery rates averaged 52% of book value.

Area Under the ROC Curve (AUC): A measure of binary classification performance used to quantify the predictability of bank failures; an uninformative predictor has AUC of 0.5, while AUC of 1.0 indicates perfect classification. In this paper, AUC ranges from 86% (historical, one-year horizon) to 95% (modern).

Boom-bust pattern: The systematic tendency of failing banks to experience rapid loan-driven asset growth in the years preceding failure followed by asset contraction in the final two years before failure — present in both the historical and modern samples, more pronounced in the latter, with real assets expanding by 34% from ten to three years before failure.

Financial Frictions: Micro versus Macro Volatility

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. How do consumer credit spreads — the gap between household borrowing rates and deposit rates — affect aggregate business cycle dynamics and the distribution of consumption across the wealth distribution? And what is the welfare trade-off between macroeconomic stabilization and household-level consumption volatility when bank capital requirements are tightened?

Data and Empirical Approach. The empirical analysis draws on Danish administrative register data for 2003–2018, combining approximately 15.5 million household-year observations. Income tax return data, which capture housing wealth, portfolio wealth, bank deposits, and bank and mortgage debt, are merged with bank-level reporting of interest rates submitted to Danmarks Nationalbank (MFI data). Household-specific credit spreads are constructed as the difference between the loan rate at a household’s primary loan bank and the deposit rate at its primary deposit bank in a given year. Consumption is imputed from household balance sheets following the method of Crawley and Kuchler (2023). The empirical specifications include household and time fixed effects, and quantile regressions are run across bins of the net wealth distribution.

Model. The authors develop a Heterogeneous Agent New Keynesian (HANK) model with explicit banking intermediation. Banks, subject to an agency friction following Gertler and Karadi (2011) — in which bankers can divert a fraction λ = 0.381 of assets — combine household deposits with net worth to invest in corporate equity and consumer loans. This leverage constraint generates an endogenous, countercyclical spread between borrowing and saving rates. Households face idiosyncratic income risk and a kink in their budget constraint at zero net worth due to the spread. The supply side features New Keynesian sticky prices (Rotemberg quadratic adjustment costs) and a Taylor rule. Aggregate shocks include monetary policy surprises, total factor productivity (TFP), and capital quality shocks (affecting bank net worth). The model is solved by first-order perturbation using the method of Bayer and Luetticke (2020) and calibrated to Danish macro and micro moments for 2003–2018.

Main Empirical Findings.

The average consumer credit spread in Denmark is strongly countercyclical, with a cross-correlation with HP-filtered output of −0.44 in the data (−0.31 in the model).
Higher credit spreads increase the transition rate into the zero net wealth state for households with moderately positive wealth at the beginning of the year, and reduce the outflow rate for households already at zero net wealth.
Pooled OLS (with household and time fixed effects) finds that a higher spread is negatively associated with consumption (coefficient −0.266), and the interaction between spread and log income is positive (coefficient 1.366), indicating that higher spreads raise income sensitivity of consumption. For below-median wealth households, the income–consumption link is stronger and the negative spread effect on consumption is larger.
The consumption-income elasticity derived from quantile regression estimates has a standard deviation of 2.4 percent and a cross-correlation with output of −0.53 when spread variation is incorporated; holding spreads constant roughly halves the volatility (to 1.3 percent) and reduces the countercyclicality (cross-correlation −0.31).

Model Aggregate Findings.

Consumer credit is procyclical (cross-correlation with output 0.56 in data, 0.67 in model) and more than twice as volatile as output (standard deviation ratio 2.11 in data, 1.51 in model).
Capital quality shocks and monetary policy shocks are amplified at the aggregate level through a financial accelerator working through endogenous spread movements. TFP shocks generate little spread amplification because households’ labor supply responses partially insulate banks’ net worth.
A 1 percentage point contractionary monetary policy shock leads to a sharp, persistent decline in aggregate output and investment, and is amplified relative to a constant-spread HANK benchmark.

Distributional Findings.

In response to a contractionary monetary policy shock, consumption of households at the 10th percentile of the consumption distribution (who are indebted) falls sharply in the short run, while consumption of the 90th percentile (wealthy households) rises in the short run due to higher returns on savings. The responses converge across the distribution in the medium run as spreads normalize.
When the consumer credit spread is held constant, consumption paths move in parallel across the wealth distribution, demonstrating that endogenous spread movements are the key driver of distributional effects for monetary policy and capital quality shocks.
The MPC is countercyclical in the model, with a cross-correlation with output of −0.60 (unconditional), compared with −0.53 for the empirically-estimated consumption-income elasticity. The consumption-income elasticity and MPC are correlated at 90 percent in the model at the annual rate.

Macroprudential Regulation.

A tightening of bank capital requirements reducing leverage by 10 percent (diversion parameter λ rising from 0.381 to 0.445) reduces output volatility by 5.5 percent and investment volatility by 10.1 percent, and does so at apparently no long-run aggregate cost in the HANK setting (precautionary savings stimulate output and consumption in the stationary equilibrium).
However, the regulation increases the annual consumer credit spread by 40 basis points, raises household consumption volatility across the wealth distribution (from about 8 percent to 10 percent for the poorest households under idiosyncratic shocks alone), and generates welfare losses across all deciles equivalent to 0.24–4.28 percent of consumption (with aggregate welfare loss of 0.79 percent).
When aggregate shocks are included, the lower cyclical sensitivity of spreads partially mitigates welfare losses for the poorest 80 percent of the population, but the overall welfare effect remains negative with an aggregate loss equivalent to 0.58 percent of consumption. The paper thus documents a trade-off between macro volatility (stabilized) and micro volatility (increased).
Results are robust to the extension of the model to three assets (including illiquid assets), which provides a better fit to micro data without materially changing the welfare conclusions.

In depth

Q1. What is the specific Danish dataset used, and how is consumption constructed?

A: The dataset covers 2003–2018 from Statistics Denmark administrative registers, combining income tax return data (which report end-of-year balances on all bank accounts, housing wealth, portfolio wealth, bank deposits, bank loans, and mortgage debt) with bank-level MFI interest rate reporting submitted to Danmarks Nationalbank. The total sample is approximately 15.5 million household-year observations (about 1.76–1.97 million households per year). Consumption is imputed as after-tax labor income plus after-tax financial income minus the change in end-of-year net worth, following Crawley and Kuchler (2023). Households with self-employment, housing transactions in the current or prior year, negative imputed consumption, or in the bottom and top 1 percent of wealth or income distributions are excluded.

Q2. How are household-specific credit spreads constructed from the administrative data?

A: Each household’s primary loan bank is defined as the bank where it holds the largest loan balance at end of calendar year, and the primary deposit bank as the one holding the largest deposit balance. The household-specific spread is the difference between the loan rate applied by the primary loan bank and the deposit rate applied by the primary deposit bank, both measured as averages over the calendar year. If a household has no loans, the loan rate of the primary deposit bank is used. This construction yields a household-level interest rate spread that moves countercyclically at the aggregate level (cross-correlation with HP-filtered output of −0.44).

Q3. What do the empirical results say about the relationship between spreads and the probability of a household reaching zero net wealth?

A: Equation (2) is estimated as a linear probability model for the transition to zero net wealth (defined as net assets within plus or minus two weeks of 2007 median weekly income). Higher spreads significantly increase the transition rate into zero net wealth for households with moderately positive net wealth at the beginning of the year (those in the third to sixth net wealth bins), and reduce the outflow rate from zero net wealth for households already in that state. Higher spreads also appear to increase debt repayments for indebted households (third to fifth bins), making it more difficult for them to accumulate wealth. Households at the extremes of the wealth distribution (very poor or very wealthy) show essentially no sensitivity of transition rates to spread movements.

Q4. What do the consumption regressions in Table 1 find, and what is the key identification caveat?

A: The pooled regression (column 1) finds a positive income–consumption coefficient of 0.372, a negative spread coefficient of −0.266, and a positive income–spread interaction of 1.366, all statistically significant with standard errors clustered at the household level (15,610,327 observations, R² = 0.591). When interacted with below-median wealth (column 2), the income coefficient is larger (0.397 versus 0.335 for above-median), the spread effect is more negative for below-median wealth (−0.362 versus −0.101 for above-median), and the income–spread interaction is stronger for below-median wealth (1.640 versus 0.875). The authors explicitly note that these results should not be given a causal interpretation, as income and consumption are likely jointly determined. Institutional features of the Danish mortgage market (covered bonds, competitive market, rates independent of borrower credit situation) minimize confounding from mortgage rate correlation with consumer credit spreads.

Q5. How do the quantile regression results and the derived consumption-income elasticity demonstrate countercyclical MPC?

A: Quantile regressions across five-percent bins of the net wealth distribution show that income coefficients decline with wealth (from nearly 0.5 for the poorest to about 0.35 for the wealthiest households), spread coefficients are negative for households with negative, zero, and moderately positive wealth and positive for significantly wealthy households, and the income–spread interaction term is positive for all but the richest households (largest near zero net wealth). The consumption-income elasticity is computed as β₀,ⱼ + β₂,ⱼ × spread at the household level, then averaged cross-sectionally. When only wealth distribution shifts are allowed, the elasticity’s standard deviation is 1.3 percent and its cross-correlation with HP-filtered output is −0.31. When spread variation is also incorporated, standard deviation rises to 2.4 percent and the cross-correlation becomes −0.53. This measure is highly correlated (90 percent) with the model MPC, supporting the inference that the MPC is countercyclical.

Q6. What is the structure of the banking sector in the HANK model, and how does the agency friction generate a countercyclical spread?

A: A continuum of banks combines household deposits with net worth to invest in corporate equity and consumer loans. Bankers can divert a fraction λ = 0.381 of assets, and if they do so, depositors can recover only the remaining fraction (1 − λ). This threat of diversion constrains the supply of deposits, resulting in banks needing to earn excess returns — Et(RK,t+1 − RS,t+1) > 0 — on their assets relative to the deposit rate. The leverage ratio is bounded above by ϱt/λ, where ϱt is a value multiplier that depends on current and expected future excess returns. When an adverse shock (capital quality shock or monetary tightening) reduces banking sector net worth, the leverage constraint tightens, banks reduce asset supply, and the spread between the return on capital (and hence the consumer loan rate, which is proportional to RK at markup ωB = 0.0075) and the deposit rate rises. This generates the observed countercyclical credit spread.

Q7. In the model, how do aggregate shocks affect the distribution of consumption, and why is the monetary policy shock particularly distributional?

A: A one-percent capital quality shock reduces both wages and bank net worth, causing spreads to rise. In the baseline economy, rising borrowing rates lead to a large reduction in consumption for indebted households (10th percentile) while the constant spread model shows near-parallel movements across the distribution. A one-percentage-point monetary policy shock reduces equity returns, depressing bank net worth and (with a lag) raising spreads. Indebted households face both lower labor income and higher borrowing costs, producing a sharp consumption decline at the 10th percentile; wealthy households gain from higher returns on savings, so their consumption rises in the short run. Responses converge as spreads return to normal over the medium run. This matches empirical evidence from Holm, Paul, and Tischbirek (2021) for Norway. For TFP shocks, banks’ net worth is less affected because households’ higher labor supply partially offsets the productivity decline, so spreads move little and distributional effects are smaller (driven mainly by wage effects across the distribution).

Q8. How does the financial accelerator in the HANK model compare to the RANK version?

A: In response to capital quality shocks and monetary policy shocks, the HANK model with banking frictions generates amplification relative to a constant-spread HANK benchmark, confirming the presence of a financial accelerator. However, relative to the RANK model, the incomplete markets model implies slightly less amplification of aggregate investment and consumption. This is because, in the HANK model, households facing higher credit spreads increase their labor supply (precautionary motive), which partially stabilizes aggregate income and moderates the financial accelerator. The finding that heterogeneous agent aspects are less important at the aggregate level is consistent with Berger, Bocola, and Dovis (2020). For TFP shocks, the financial accelerator through spreads is largely absent in both HANK and RANK, as spread changes are minor.

Q9. What are the long-run aggregate effects of tightening bank capital requirements (reducing leverage by 10 percent) in the HANK versus RANK model?

A: In the RANK model, higher capital requirements increase the annual spread between the return on capital and the deposit rate by 25 basis points, reduce the aggregate capital stock by 2.4 percent, output by 0.5 percent, and aggregate consumption by 0.8 percent. In the HANK model, the spread increases by 40 basis points annually, but the mechanism differs: much of the spread change is absorbed by a reduction in the deposit rate (from 3.81 percent to 3.54 percent annually) rather than an increase in the capital return. Households respond to the lower deposit rate and higher credit costs by increasing precautionary savings and labor supply, so aggregate output and consumption actually rise slightly in the HANK stationary equilibrium. The capital requirements thus appear costless at the aggregate level in the HANK model — but this masks welfare costs that operate through the idiosyncratic risk channel.

Q10. What are the quantitative welfare costs of macroprudential regulation, and how do they vary across the wealth distribution and between idiosyncratic and aggregate shocks?

A: Welfare is measured as the fraction of lifetime consumption households are willing to give up to stay in the unregulated baseline. In the face of idiosyncratic shocks only, welfare losses range from 0.24 to 0.43 percent of consumption for the first seven wealth deciles, and reach 4.28 percent for the richest decile (primarily because of the reduction in the return on their savings), with an average welfare loss of 0.79 percent. When aggregate shocks are added, the losses are substantially reduced for the poorest 80 percent (due to lower cyclical sensitivity of spreads), but remain large for the wealthiest decile (4.23 percent) and in aggregate (0.58 percent). These results are robust to the three-asset model extension, where the poorest households are approximately welfare-neutral under the regulation when aggregate shocks are included (0.00 percent), but aggregate welfare losses remain at 0.75 percent.

Q11. How does the three-asset model extension (with illiquid assets) affect the key results?

A: In the three-asset extension, households can hold illiquid capital (calibrated with an adjustment probability of φk = 0.0025 per quarter, targeting the Danish ratio of bank deposits to output of 34 percent), creating wealthy hand-to-mouth households who have illiquid assets but no liquid assets. The consumption impulse responses across the wealth distribution remain very similar to the two-asset baseline: endogenous spread movements generate heterogeneous consumption dynamics in response to capital quality and monetary shocks, while constant-spread models produce near-parallel responses. The three-asset model provides a better fit to the micro data (consumption-spread-income relationship across the wealth distribution), but the welfare conclusions from macroprudential regulation are essentially unchanged: welfare losses across the distribution in the stationary equilibrium, partially mitigated when aggregate shocks are added, with losses concentrated in the richest decile.

Q12. What robustness checks are reported for the empirical consumption regressions?

A: Three robustness exercises are reported. First, capitalizing car purchases using their official tax value (rather than treating car purchases as current expenditure) yields coefficients similar to the baseline (Table 10). Second, excluding households who purchase a car in the current or prior year (reducing the sample to 13.24 million observations) also leaves results unchanged. Third, first-differenced specifications (equation 42, with and without household fixed effects) produce results similar to the levels specification; the main exception is the spread effect for above-median wealth households when household fixed effects are omitted from the differenced specification (Table 11). The income–spread interaction is consistently positive and significant across all robustness checks.

Q13. What evidence does the paper provide that the model’s MPC is countercyclical and that credit spreads are the primary driver?

A: Figure 7 shows impulse response functions of the average MPC to each of the three aggregate shocks. In all three cases, the MPC rises in recessions (countercyclical). The key mechanism is that adverse shocks cause spreads to rise, increasing the mass of households at the kink in the budget constraint (zero liquid assets), where MPCs are highest. When the consumer credit spread is held constant, the MPC remains countercyclical but close to constant, indicating that spread movements account for most of the cyclical variation in MPC. Eliminating the spread altogether implies an acyclical MPC (Table 12, Appendix D). The unconditional cross-correlation of the model MPC with output is −0.60, compared with −0.53 for the empirically estimated consumption-income elasticity in the Danish data.

Key Concepts

Consumer credit spread (borrowing-saving spread): In the paper, this is the difference between the gross real interest rate on consumer loans (RL,t) charged by banks and the gross real return on deposits (RS,t) received by savers. It is not an abstract measure of credit conditions but a household-specific, bank-derived rate gap that moves countercyclically due to banking agency frictions and creates a kink in households’ budget constraints at zero net worth. Distinct from mortgage spreads (which in Denmark are market-determined and independent of borrower credit conditions).

Kink in the budget constraint: The household budget constraint has a kink at zero net assets because borrowers face RL,t > RS,t; households at exactly zero liquid assets (type IV in the paper’s taxonomy) face a discrete jump in the cost of additional borrowing. This kink creates a mass point in the wealth distribution at zero net wealth, and households at this kink have higher MPCs than unconstrained savers or borrowers. The size of the mass point increases when the spread rises.

Financial accelerator (in the HANK-with-banking context): The amplification mechanism in which shocks that reduce banking sector net worth tighten banks’ leverage constraints, raise credit spreads, reduce asset supply to both the corporate sector and households, and further depress investment and consumption — which in turn reduces bank net worth further. In this paper, the accelerator operates through the consumer credit spread channel in addition to the standard corporate lending channel, and is present for capital quality and monetary policy shocks but not materially for TFP shocks.

Countercyclical MPC: The MPC — defined as the response of consumption to a small transitory income shock — rises during recessions and falls during expansions in this model. The mechanism is that recessions are associated with higher consumer credit spreads, which expand the mass of households at or near the zero net wealth kink (high MPC), and contract the mass of unconstrained savers (low MPC). This is a distinct source of MPC cyclicality from the wealth distribution channel alone.

Agency friction (diversion problem): Banks can divert a fraction λ of their assets; if they do so, depositors can recover only the fraction (1 − λ) and the bank is liquidated. This threat limits depositors’ willingness to supply funds, resulting in an incentive-compatibility constraint on bank leverage: assets cannot exceed ϱt/λ (where ϱt is the bank’s franchise value multiplier). When ϱt declines (because expected excess returns fall), the constraint binds more tightly and the spread between the return on assets and the deposit rate must be positive to sustain bank participation.

Macro versus micro volatility trade-off: The paper uses this phrase to describe the finding that tighter bank capital requirements (restricting leverage) reduce the cyclical volatility of aggregate output and investment (macro volatility falls) while simultaneously increasing the volatility of individual household consumption streams due to higher credit spreads and lower deposit returns (micro volatility rises). Welfare costs from increased micro volatility outweigh the aggregate stabilization benefits.

Consumption-income elasticity (d log c / d log y): A time-varying cross-sectional average measure derived from quantile regression parameter estimates, equal to β₀,ⱼ + β₂,ⱼ × RSi,t for household i in wealth bin j. It is used in the paper as an empirical proxy for the MPC (not a direct estimate), and is shown to be highly correlated with the model MPC (cross-correlation of 90 percent at the annual rate). Its cyclicality is stronger when spread variation is incorporated (standard deviation 2.4 percent, cross-correlation with output −0.53) than when spreads are held fixed (standard deviation 1.3 percent, cross-correlation −0.31).

Financial shocks and leverage of financial institutions: When do they matter?

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates the role of leverage of financial institutions in amplifying the transmission of financial shocks to the macroeconomy, with particular attention to whether that amplification differs across economic regimes. The authors develop a new endogenous regime-switching structural vector autoregression (RS-SVAR) model with time-varying transition probabilities, in which the probability of switching regime depends on the contemporaneous state of the economy (endogenous switching). The model extends the Sims and Zha (2006) and Sims, Waggoner, and Zha (2008) Markov-switching SVAR framework by: (1) incorporating a time-varying transition matrix in which the probability of staying in a regime is a logistic function of lagged endogenous variables; and (2) introducing new identification techniques for RS-SVARs, including non-recursive zero restrictions, sign restrictions, and narrative sign restrictions, which can in some cases uniquely identify structural shocks rather than merely set-identify them.

The leverage measure is market-based — book assets divided by market equity — constructed from CRSP/Compustat institution-level data covering publicly listed depository institutions, bank holding companies, and nonbank financial institutions. The sample runs monthly from December 1988 to December 2019. The five-variable VAR includes industrial production growth, core CPI inflation, the 2-year Treasury rate, market leverage of financial institutions, and the Chicago Fed’s National Financial Conditions Index (NFCI). The authors estimate three model variants that substitute in turn the leverage of: (i) all depository institutions, (ii) Global Systemically Important Banks (GSIBs), and (iii) securities brokers and dealers.

The model identifies two coefficient regimes — a “financial constraint” regime and “normal times” — using the criterion that the first regime has higher smoothed probability during September 2008 to August 2009. The financial constraint regime covers the end of the Savings and Loan crisis, the 1990/91 recession, the Russian debt default, the Global Financial Crisis (GFC), and the European sovereign debt crisis.

The core finding is that real effects of financial shocks are amplified in the financial constraint regime but not in normal times. In the financial constraint regime, the output response to a financial shock is significantly negative, large, and protracted; GSIB leverage initially rises sharply (as falling asset prices erode equity) and then declines as institutions deleverage. In normal times, the output growth response is negative but non-persistent, and market leverage remains insignificant over the entire horizon.

The counterfactual experiment holding GSIB market leverage constant as of October 2008 is the sharpest quantitative result: if GSIB leverage had not risen further at the onset of the GFC, the decline in industrial production growth would have been approximately 20 percentage points smaller, with a faster subsequent recovery in output growth and inflation and higher short-term interest rates. The counterfactual probability of staying in the financial constraint regime would have fallen as low as 0.1 for some draws, compared to the actual probability remaining elevated. By contrast, for a system using depository institution leverage, the lower-bound counterfactual probability of staying in the constraint regime does not fall below 0.90, indicating substantially weaker heterogeneity effects for the broader depository sector.

Securities brokers and dealers show leverage that rises more on impact than other institutions and then declines immediately, consistent with their willingness to expand balance sheets going into the crisis amplifying losses and forcing a sharp post-crisis contraction.

A separate counterfactual holding the NFCI constant (rather than leverage) shows that the probability of staying in the constraint regime does not decline, confirming that market leverage and the financial conditions index provide distinct characterizations of the financial system and have different implications for shock propagation and regime persistence. Results are robust to substituting the GZ corporate spread for the NFCI and to imposing narrative restrictions for shock identification.

Q: What is the central research question? A: The paper asks whether and how the leverage of financial institutions amplifies the transmission of financial shocks to the real economy, and whether this amplification differs between a financial constraint regime and normal times. A secondary question concerns heterogeneity: do GSIBs, depository institutions broadly, and nonbank securities dealers transmit shocks differently?

Q: What is novel about the econometric framework? A: The RS-SVAR model allows the probability of remaining in a given coefficient regime to vary over time as a logistic function of lagged endogenous variables, so regime switching is endogenous to the state of the economy rather than governed by a fixed transition matrix. The paper also introduces sign restrictions, zero restrictions, and narrative sign restrictions into the RS-SVAR class, enabling identification of both structural shocks and regimes within a single framework; in roughly 20 percent of posterior draws these sign restrictions uniquely identify the financial shock.

Q: Why does the paper use market leverage rather than book leverage? A: Market leverage (book assets divided by market equity) is argued to be more timely than book leverage because book equity incorporates losses with a delay, giving institutions time to adjust book leverage to avoid regulatory limits. Market capitalization reflects market participants’ assessment of an institution’s creditworthiness, and low market-to-book ratios signal that institutions are more leveraged than their books indicate. Market leverage is therefore a more informative early-warning indicator of financial fragility and the need for rapid deleveraging.

Q: How are the two regimes identified? A: For each estimated regime, the authors count the number of months between September 2008 and August 2009 (inclusive) for which the smoothed probability of being in that regime exceeds 0.70; the regime with the higher count is labeled “financial constraint” and ordered first. Shock identification uses sign restrictions: in the financial constraint regime, a positive financial shock must have a contemporaneously negative effect on output, inflation, and the short-term interest rate, but positive effects on the financial conditions index and leverage; in normal times, only the financial conditions index is required to respond positively on impact.

Q: What regimes does the model assign historically? A: The smoothed probability of the financial constraint regime is elevated during the end of the Savings and Loan crisis, the 1990/91 recession, the Russian debt default, the GFC and associated recession (where the probability reaches 1.0 at end-2008 and beginning-2009 before declining sharply to approximately 0.6 percent in 2009/2010), and the European sovereign debt crisis.

Q: What do the impulse responses show in the financial constraint regime? A: In the financial constraint regime, the output response to a positive financial shock (tightening) is significantly negative, large, and protracted. GSIB leverage initially rises due to a sharp decline in asset prices eroding market equity, then falls as GSIBs deleverage in response. The authors interpret this pattern as evidence that deleveraging produces procyclical financial amplification effects with adverse real consequences.

Q: What do the impulse responses show in normal times? A: In normal times, the output growth response is large and negative but non-persistent, in contrast to the financial constraint regime. Market leverage remains statistically insignificant across the entire horizon in normal times, indicating that the leverage amplification channel is inactive outside of financial constraint episodes.

Q: What does the GSIB leverage counterfactual show quantitatively? A: Holding GSIB market leverage constant as of October 2008 implies a decline in industrial production growth that is approximately 20 percentage points smaller than actually occurred, along with a faster recovery in output growth and inflation and higher short-term interest rates. The counterfactual probability of staying in the financial constraint regime declines to as low as 0.1 for some posterior draws, compared to remaining elevated in the actual data.

Q: How do depository institutions compare to GSIBs in the counterfactual? A: For the model using broad depository institution leverage, the lower-bound counterfactual probability of staying in the financial constraint regime does not fall below 0.90, compared to as low as 0.1 for the GSIB specification. This implies that GSIB deleveraging has substantially more detrimental macroeconomic effects and a much larger effect on regime persistence than the broader depository sector.

Q: What is distinctive about securities brokers and dealers? A: Broker-dealer market leverage rises more on impact than leverage of other financial institutions following a financial shock, and then immediately declines due to rapid deleveraging. The authors interpret this as reflecting that dealers’ willingness to expand balance sheets ahead of the crisis amplified growth and losses, followed by a sharp post-crisis contraction — a pattern consistent with the procyclical leverage mechanism described in Adrian and Shin (2014).

Q: How do the authors distinguish the role of market leverage from the financial conditions index? A: A counterfactual holding the NFCI constant (rather than leverage) as of October 2008 shows that the probability of staying in the financial constraint regime does not decline, unlike the leverage counterfactual. This demonstrates that market leverage and the NFCI provide distinct characterizations of financial conditions and have different implications for the propagation of shocks and the persistence of the constraint regime.

Q: How robust are the results? A: Substituting the GZ corporate bond spread for the NFCI yields very similar results, specifically that the probability of staying in the constraint regime declines much more in the counterfactual than in the actual data, suggesting the findings are not driven by the choice of financial conditions proxy. Imposing narrative restrictions for shock identification (exploiting the known high-stress period around Lehman’s failure in September 2008) yields results that are “rather robust” relative to the baseline sign-restriction identification.

Q: What are the policy implications? A: The results confirm the leverage ratio as a useful financial stability indicator, with particular emphasis on market leverage as providing timely information for monitoring. The heterogeneity findings suggest that regulatory attention to GSIB leverage is especially warranted, since GSIB deleveraging can have substantially more detrimental macroeconomic effects and a much larger influence on the persistence of financial constraint regimes than deleveraging by the broader depository sector. The leverage ratio is characterized as complementary to the risk-weighted capital ratio as a regulatory tool.

Market leverage: Measured as book assets divided by market equity (not book equity), constructed from CRSP/Compustat institution-level data at monthly frequency. The paper argues market leverage is more timely than book leverage because market equity immediately reflects losses, preventing institutions from masking fragility through delayed book adjustments.

Financial constraint regime: One of two identified coefficient regimes in the RS-SVAR, characterized by a significantly negative, large, and protracted output response to financial shocks and by active leverage amplification. Identified empirically as the regime with the highest smoothed probability during September 2008 to August 2009.

Endogenous regime switching: A modeling approach in which the probability of transitioning between regimes depends on lagged values of the endogenous variables themselves (via a logistic function), rather than being governed by a fixed constant transition matrix. This allows regime dynamics to respond to the state of the economy.

Time-varying transition probabilities: The diagonal elements of the coefficient-regime transition matrix follow a logistic transformation of a linear function of lagged endogenous variables, so the probability of remaining in any given regime changes each period as a function of current financial and macroeconomic conditions.

Procyclical financial amplification: The mechanism by which financial institution deleveraging in response to falling asset prices further tightens financial conditions and reduces real output, generating a feedback loop. The paper provides empirical evidence for this channel operating specifically in financial constraint regimes.

Heterogeneity of financial institutions: The finding that GSIBs, broad depository institutions, and securities brokers and dealers differ substantially in how their leverage affects the transmission of financial shocks. GSIB deleveraging is shown to have much more detrimental macroeconomic effects and a much larger influence on the probability of remaining in the financial constraint regime than depository institution deleveraging more broadly.

Narrative sign restrictions in RS-SVARs: An identification technique extended from Antolin-Diaz and Rubio-Ramirez (2018) to the regime-switching context, which uses known historical episodes (here, the Lehman failure in September 2008) to impose restrictions on which regime the economy was in or on the sign of structural shocks at particular dates, thereby aiding identification of both shocks and regimes.

Firm dynamics and random search over the business cycle

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

How do aggregate economic fluctuations reallocate workers across the firm productivity distribution over the business cycle? In particular, to what extent do recessions impede workers’ movement up the job ladder toward more productive firms?

Model and Methodology

The paper develops a tractable random search model combining three features that had not previously been integrated in a single quantitative framework: (i) firm dynamics driven by idiosyncratic productivity shocks, with endogenous entry and exit; (ii) on-the-job search, generating a job ladder in which workers gradually move toward more productive firms; and (iii) aggregate productivity shocks. Multi-worker firms post employment contracts, choose hiring rates, and decide whether to continue or exit. The key tractability result — called “size-independence” (Result 1) — shows that, under a constant-returns hiring cost technology, firms’ optimal policies (contract value, hiring rate, exit decision) are all independent of firm size, so the relevant state space reduces from the full joint distribution of firm productivity and size to the employment-weighted distribution of firm productivity alone. A further result (“rank-monotonic equilibrium,” Result 2) guarantees, under a sufficient convexity condition on hiring costs (hc’’(h)/c’(h) ≥ 1), that the optimal employment contract is increasing in firm productivity, so the job ladder maps one-for-one onto the firm productivity ladder. The optimal wage contract then admits a closed-form solution.

The model is calibrated to British data for 1997–2018. Worker-level transition rates (unemployment-to-employment, employment-to-unemployment, and job-to-job) are drawn from the British Household Panel Survey (BHPS). Firm-level data on labor productivity (value added per worker) and employment costs per worker come from the Annual Respondents Database (ARD) and Annual Business Survey (ABS), merged with the Business Structure Database (BSD). The numerical solution adapts ideas from Krusell and Smith (1998), approximating the employment-weighted productivity distribution by a small set of moments and parameterizing value functions as polynomials in the aggregate state; standard linearization methods are inapplicable because endogenous firm entry and exit introduces a discontinuity in value functions.

Main Findings

Model validation via the OP decomposition. The paper’s central validation exercise uses the Olley-Pakes (OP) decomposition of a labor productivity index constructed from firm-level data. The aggregate employment-weighted labor productivity index is decomposed into (a) the unweighted average firm productivity and (b) an interaction term (the “OP term”), which captures the covariance between employment shares and productivity — i.e., how well workers are allocated to productive firms. In the British firm-level data, approximately 20 percent of the variance of the aggregate labor productivity index is accounted for by this interaction (OP) term, with the remaining ~80 percent attributable to the unweighted average of firm productivity. The baseline model, with this moment untargeted, successfully replicates this 80/20 split. By contrast, the leading benchmark model of Moscarini and Postel-Vinay (2016) (MPV2016), calibrated to the same British data, attributes nearly all of the variance of labor productivity to the OP/worker reallocation term, grossly overstating the importance of job-ladder dynamics.

Structural decomposition of labor productivity. Using the calibrated baseline model to decompose the variance of aggregate labor productivity over the post-war British business cycle (“GDP shocks” going back to 1955), the baseline model attributes approximately 30 percent to the direct effect of the aggregate productivity shock, approximately 50 percent to changes in the distribution of active firms (the “firm ladder” or firm selection component), and approximately 20 percent to the worker reallocation component (the OP interaction term). This result is robust to an alternative calibration with a lower curvature of the hiring cost function (c1 = 1).

Persistence and mechanisms. The impact of recessions on the job ladder is persistent: while the aggregate productivity shock is typically close to its pre-recession value four years after a typical recession onset, the overall allocation of workers to firms remains clearly worse relative to the pre-recession level at that same horizon. The Great Recession, viewed through the lens of the model, is a large but not unusually large recession.

Firm selection with multiple aggregate shocks. An unexpected finding concerns the direction of firm selection. With a single aggregate productivity shock, the model generates a standard “cleansing” mechanism: negative shocks raise the firm exit threshold, so surviving firms are on average more productive. However, when additional shocks to the exogenous separation rate (δ) and hiring cost scale (c0) are included — as required to match the volatility of labor market flows — firm selection instead amplifies the decline in labor productivity. The mechanism is a general equilibrium one: a higher separation rate lowers the optimal wage contract (since greater separation risk is passed on to workers), which in turn lowers the entry-exit threshold. Less productive firms become viable because their employees face higher unemployment risk and therefore accept lower wages; moreover, a larger pool of unemployed workers makes it easier for low-productivity firms to recruit.

Wage flexibility tension. The model implies a pass-through elasticity of wages to productivity shocks of approximately 0.7, well above the 0.05–0.2 range typically found empirically.

Scope Conditions

All calibration and quantitative results pertain to Britain for the period 1997–2018 (firm-level data) and 1955–2018 (GDP-based aggregate shocks). The model abstracts from decreasing returns to scale in production and from nominal rigidities. The tractability results rely on specific assumptions about the hiring cost function; the rank-monotonicity condition requires sufficient convexity (hc’’(h)/c’(h) ≥ 1).

In depth

Q1. What is the central tractability result and why does it matter for computational feasibility?

A: Result 1 (“size-independence”) shows that, because both the production technology and the hiring cost function are constant returns to scale, the firm’s present discounted value of profits is linear in employment. As a result, per-worker profits are independent of firm size, and optimal firm policies — the hiring rate, the contract value offered to workers, and the continuation/exit decision — all depend only on the firm’s current productivity, not on its size. This collapses the state space from the full joint distribution of firm productivity and employment size to the employment-weighted measure of firm productivity Lt(p), a uni-dimensional object. Without this result, the model would require tracking the entire joint firm distribution, making it computationally intractable.

Q2. What is a rank-monotonic equilibrium (RME) and what conditions guarantee it?

A: An RME is a recursive equilibrium in which the optimal contract offered by a firm is weakly increasing in that firm’s current productivity realization, for all aggregate states. Result 2 provides sufficient conditions: (i) the Markov process for firm-specific productivity satisfies first-order stochastic dominance (more productive firms today are more likely to be more productive tomorrow), (ii) the distribution of offered contracts is everywhere differentiable (ruling out mass points), and (iii) the hiring cost function satisfies hc’’(h)/c’(h) ≥ 1 — a sufficient convexity condition. The economic interpretation of the convexity condition is that firms must find retention (offering higher wages) sufficiently costly relative to new hiring that more productive firms optimally choose to use the wage margin to limit quits. The baseline calibration yields c1 ≈ 5.9 (so costs are highly convex in the hiring rate), though results are also reported for the minimum permissible c1 = 1.

Q3. What does the optimal employment contract look like in a rank-monotonic equilibrium, and what does it reveal about rent extraction?

A: In an RME, the optimal contract V(p,ω,L) is a weighted average of the value of unemployment U(ω,L) and the firm-workers’ joint surplus S(p,ω,L), where the weights are determined endogenously by the employment-weighted measure of firm productivity L. Specifically, the contract integrates the surplus of all firms with productivity below p, weighted by the share of employed workers at those firms, and divided by the mass of job seekers willing to accept the contract. As the employed workers’ relative search intensity s approaches zero, the contract converges to the value of unemployment — workers receive no rents. The endogenous bargaining weight evolves with the aggregate state over the business cycle, unlike standard Nash bargaining models with a fixed exogenous weight.

Q4. What firm-level moments are used to calibrate the steady-state model, and what is the logic behind the parameter-moment mapping?

A: Eight moments are targeted. From the BHPS worker data: the average UE rate (0.058) pins down the scale of hiring costs c0; the average EU rate (0.003) pins down the exogenous separation rate δ; and the average EE (job-to-job) rate (0.016) pins down the relative search intensity s. From the firm-level ARD/BSD data: average firm size (12.1 employees) pins down the entry probability µ; the share of job destruction from firm exits (0.526) disciplines the flow value of unemployment b; the autocorrelation of firm employment ln(n) (0.949 annually) disciplines the persistence of idiosyncratic productivity ρp; the interquartile range of firm-level labor productivity (1.129 log points) disciplines the volatility of idiosyncratic shocks σp; and the regression coefficient of firm employment growth on lagged labor productivity (0.136) disciplines the curvature of hiring costs c1. The baseline calibration fits all eight moments closely.

Q5. How does the calibrated model match non-targeted moments, and what does this establish?

A: The model generates several realistic features not targeted in calibration. It produces a realistic Pareto tail for the employment-size distribution (Pareto tail exponent of 1.033 in the model vs. 1.066 in the data), which arises from the combination of size-independent growth rates and firm entry and exit — conditions identified in the literature as generating power law distributions. The model also matches the dispersion of employment costs per worker across firms (capturing about 70 percent of the interquartile range of ECi,t), the slope of a regression of employment costs on labor productivity (model: 0.685 vs. data: 0.704), and the slope of a regression of employment growth on employment costs (model: 0.162 vs. data: 0.131). These non-targeted matches provide independent validation of the model’s wage-determination mechanism.

Q6. Why is a single aggregate productivity shock insufficient to match labor market fluctuations, and what additional shocks are needed?

A: With a single aggregate productivity shock calibrated to match the autocorrelation and standard deviation of log GDP, the model generates labor market fluctuations that are roughly an order of magnitude smaller than in the data. For example, the standard deviation of the EU transition rate is 4.1×10⁻⁴ in the single-shock model versus 2.3×10⁻³ in the data. Adding a discount rate shock (ω,r) partially helps but still leaves the job-finding rate (UE) more than 50 percent too smooth. Adding a separation rate shock (ω,δ) substantially increases EU and UE volatility but generates insufficient EE (job-to-job) volatility. The combination (ω,δ,c0) — adding a shock to the scale of hiring costs c0 — brings the standard deviations of EU and UE close to the data (2.0×10⁻³ and 4.0×10⁻⁴ vs. data 2.3×10⁻³ and 2.7×10⁻⁴), though the model still generates slightly under half the observed volatility in EE rates. This combination is the baseline for the quantitative analysis.

Q7. What is the OP decomposition, how is it computed from the firm-level data, and what does it measure in the model?

A: The aggregate labor productivity index LPt is constructed from firm-level data as the employment-share-weighted average of log value added per worker across firms. The OP decomposition writes this as LPt = LPt_bar + OPt, where LPt_bar is the unweighted (simple) average of firm-level productivity and OPt is the covariance between employment shares and labor productivity (the “interaction term”). In the data, OPt increases when workers are disproportionately employed at above-average-productivity firms. In the model, LPt_bar maps onto the average (log) productivity of active firms — the support of the job ladder — while OPt maps onto the difference between the employment-weighted and the unweighted averages of firm productivity, directly measuring how high up the ladder workers are located relative to the set of active firms. Around 20 percent of the variance of LPt in the British data is accounted for by OPt, and the model replicates this.

Q8. How does the Great Recession appear in the OP decomposition, and does the model fit the decomposition during this episode?

A: During the Great Recession (2008q2–2009q3 in the UK), around 20 percent of the overall fall in the labor productivity index is accounted for by the fall in the OP interaction term, with the remaining 80 percent coming from the fall in the unweighted average firm productivity. The model, even though it does not target this decomposition in calibration, successfully matches both the average firm productivity component and the interaction (OP) component during the Great Recession. This matching holds both in the baseline calibration (c1 ≈ 5.9) and in the alternative calibration with c1 = 1. The model also matches the analogous decomposition for employment costs per worker (ECt), an additional non-targeted validation.

Q9. Why does firm selection amplify rather than cleanse in the baseline multi-shock calibration?

A: In the single-shock (productivity ω only) model, a negative productivity shock lowers surplus at all firms, raising the exit threshold pE and thus selecting out low-productivity firms — the standard “cleansing” mechanism. In the multi-shock baseline, the additional separation rate shock (δ) generates a less intuitive mechanism. A higher δ lowers the optimal wage contract (since increased separation risk is passed on to workers: ∂V/∂δ ≤ 0), which reduces the value of continued employment. This lowers the joint firm-worker surplus threshold for exit, making it viable for low-productivity firms to remain active. Moreover, the larger pool of unemployed workers (generated by the δ shock) depresses the outside option of workers and makes it easier for low-productivity firms to recruit. As a result, the entry-exit threshold pE,t falls — the set of active firms becomes less productive on average — producing a negative firm selection contribution to labor productivity and a positive (amplifying rather than cleansing) contribution to the variance of LPt.

Q10. What is the structural variance decomposition of labor productivity in the baseline model?

A: Simulating the baseline model over the post-war British business cycle (1955–2020, GDP shocks), the variance of aggregate labor productivity LPt decomposes into three structural terms: approximately 30 percent (0.296) from the direct effect of the aggregate productivity shock ln(ωt); approximately 50 percent (0.541) from changes in the average productivity of active firms E[KP bar_t(ln p)] — the “firm ladder” or firm selection component; and approximately 20 percent (0.163) from the worker reallocation component OPt = E[LP bar_t(ln p)] − E[KP bar_t(ln p)]. This decomposition implies that roughly 70 percent of fluctuations in labor productivity are driven by worker reallocation broadly defined (the firm ladder plus the interaction term), with the firm selection component being the largest single driver. The result is robust to the alternative c1 = 1 calibration (30/49/22 percent split).

Q11. How does the baseline model compare to MPV2016 in the variance decomposition?

A: In the multi-shock calibration (ω,δ,c0), the MPV2016 model calibrated to the same British data attributes approximately 97.7 percent (0.977) of the variance of LPt to the worker reallocation (OP) term, with essentially none attributed to a firm selection term (since there is no firm entry and exit in MPV2016). This is nearly five times the 20 percent share attributed to worker reallocation in the data and in the baseline model. In the single-shock (ω) calibration, both models attribute a more modest share to worker reallocation (7.2 percent for the baseline model, 0.1 percent for MPV2016 with c1=5), and the difference narrows considerably. The contrast thus stems from the interaction of firm dynamics with multiple aggregate shocks: allowing for endogenous firm entry and exit is critical to prevent the model from overstating the role of the job ladder.

Q12. How persistent is the impact of recessions on the job ladder, based on the model simulations?

A: The paper simulates the structural decomposition of labor productivity starting from each of seven post-war British recessions (defined by two consecutive quarters of negative GDP growth). On average across these recessions, the aggregate productivity shock ln(ωt) is close to its pre-recession level by four years after the recession onset. However, the overall employment-weighted average productivity E[LP bar_t(ln p)] — reflecting workers’ position on the job ladder — remains clearly below its pre-recession value at the four-year horizon, indicating persistent misallocation. The OP interaction term accounts for approximately 20 percent of the total drop in the employment-weighted productivity measure three years after a typical recession onset. Through the model’s lens, the Great Recession is a large recession but not an outlier relative to the historical distribution.

Q13. What does the counterfactual with countercyclical unemployment benefits reveal about the tradeoff between firm selection and worker reallocation?

A: When the flow value of unemployment is made countercyclical (falling in recessions, rising in expansions — mimicking US unemployment insurance extension programs), the model generates a sign reversal in the firm selection (“firm ladder”) component. With countercyclical b, the unemployment value rises in recessions, which raises the minimum wage firms must offer and raises the exit threshold pE,t: fewer low-productivity firms survive, improving the composition of active firms. However, countercyclical benefits also amplify the slowdown in job-to-job reallocation: the higher value of unemployment reduces workers’ willingness to accept job offers, and all firms cut recruitment since optimal wage contracts must rise. The OP interaction term therefore falls more sharply than in the baseline model. The counterfactual with ϵb,ω ∈ {−100, −50} finds that the positive “firm ladder” effect dominates on net, so the overall allocation of workers to firms improves relative to the baseline after a typical recession under countercyclical unemployment benefits.

Q14. What is the numerical solution method, and why are standard linearization approaches inapplicable?

A: The model is solved in two steps. First, aggregate shocks are shut down and the steady-state rank-monotonic equilibrium is solved numerically by discretizing the firm productivity process (401 grid points via Tauchen’s method) and iterating on the value function and the employment-weighted productivity measure until convergence. Second, aggregate shocks are reintroduced using a simulation-based approach adapted from Krusell and Smith (1998): the employment-weighted distribution of productivity is summarized by Nm = 2 moments (plus the unemployment rate), and the value functions are parameterized as polynomials in the aggregate state, with coefficients updated by regression until convergence. Standard linearization methods (Reiter 2009) are inapplicable because the endogenous entry-exit decision creates a kink (discontinuity) in value functions at the productivity threshold pE, making first-order approximations around the steady state inaccurate. Accuracy tests based on den Haan (2010) show that the polynomial approximation generates errors of at most 0.065 percent for value functions and at most 1 percentage point for the unemployment rate across simulation paths.

Key Concepts

1. Rank-Monotonic Equilibrium (RME) A recursive equilibrium in which the optimal state-contingent employment contract V(p,ω,L) offered by a firm is weakly increasing in the firm’s current productivity realization p, for all aggregate states (ω,L). This property implies that the job ladder maps one-for-one onto the firm productivity ladder: workers always prefer to work at more productive firms. The paper shows this property holds under a sufficient convexity condition on hiring costs (hc’’(h)/c’(h) ≥ 1) and first-order stochastic dominance of the productivity process.

2. Size-Independence The property that a firm’s optimal policies — the hiring rate h(p), the employment contract V(p), and the entry/exit decision χ(p) — are all independent of the firm’s current employment size n. This follows from constant returns to scale in production and hiring, which implies that firm profits are linear in employment. Size-independence reduces the model’s relevant state space to the employment-weighted distribution of firm productivity, enabling tractability.

3. Employment-Weighted Distribution of Firm Productivity (L_t(p)) The measure recording, for each productivity level p, the total employment at firms with productivity at most p. This is the sufficient statistic for the state of the job ladder at any point in time: combined with the aggregate shock ω, it determines all equilibrium policy functions and value functions. In the model, it replaces the full joint distribution of firm productivity and employment size that would otherwise be required.

4. OP Decomposition (Olley-Pakes Decomposition) The decomposition of the aggregate employment-weighted labor productivity index LPt into: (a) the unweighted average firm productivity LPt-bar, which summarizes the productivity of active firms (the support of the job ladder); and (b) an interaction term OPt, the covariance between employment shares and firm-level productivity, which measures how well workers are allocated across the productivity distribution (i.e., how high up the ladder workers sit given the set of active firms). In the model, (a) maps to E[KP bar_t(ln p)] and (b) maps to OPt = E[LP bar_t(ln p)] − E[KP bar_t(ln p)].

5. Contract Posting The wage-setting protocol in which each firm commits upon entry to a full state-contingent employment contract — a schedule mapping each future realization of aggregate and idiosyncratic productivity to a wage and continuation decision — and is bound by an equal treatment constraint to offer the same contract to all employees. Workers cannot renegotiate based on outside offers. This protocol produces a well-defined closed-form for the optimal contract in an RME and differs from alternating-offer bargaining (Nash bargaining) in that the bargaining weights are endogenous rather than fixed.

6. Firm-Workers’ Joint Surplus (S_t(p)) The total present discounted value accruing to the firm-worker pair: firm profits per worker plus the contract value promised to workers. Because utility is transferable (risk neutrality) and the firm fully commits to its contract, this surplus depends only on the firm’s current productivity and the aggregate state — not on the promised contract value V. The surplus S_t(p) is the key object determining firm entry/exit (the firm continues if and only if S_t(p) ≥ U_t) and optimal hiring (the marginal return to an additional hire equals S_t(p) − V(p)).

7. Cleansing vs. Anti-Cleansing Firm Selection In models with endogenous firm entry and exit, a negative aggregate shock can either raise or lower the productivity threshold for firm survival. “Cleansing” refers to the standard mechanism where a negative productivity shock raises the exit threshold, selecting out low-productivity firms and improving the average quality of survivors. “Anti-cleansing” (as in the baseline multi-shock calibration) occurs when separation rate or hiring cost shocks lower the optimal wage contract and reduce the exit threshold, allowing less productive firms to survive and worsening average firm productivity.

FraNK: Fragmentation in the NK Model

Mon, 01 Jan 0001 00:00:00 +0000

Moro and Nispi Landi develop FraNK, a multi-country New Keynesian model designed to study geoeconomic fragmentation — defined, following Aiyar et al. (2023), as a policy-driven reversal of economic integration guided by strategic considerations. The model extends Gali and Monacelli (2005) along three dimensions: it is multi-country rather than small-open-economy; it assumes incomplete international financial markets, relaxing perfect risk sharing; and it incorporates commodities as intermediate inputs in production, capturing both domestic and imported commodity sourcing. A fragmentation shock is modeled as a simultaneous increase in three tax rates imposed on rival countries: a tax on imports of final goods, a tax on imports of commodities, and a tax on the purchase of foreign bonds (capital controls).

The paper proceeds in two stages. First, under a symmetric two-bloc calibration, closed-form analytical results establish the distinct macroeconomic channels of each tax. The good import tax operates through both demand (households reduce consumption of foreign goods) and supply (firms face higher real marginal costs), with the demand channel dominating: output falls unambiguously and PPI inflation decreases, though CPI inflation rises on impact due to the direct pass-through of import prices. The commodity import tax operates exclusively through supply — raising intermediate input costs — so both output and PPI inflation move in the same direction: output falls and PPI inflation rises. The bond tax is neutral under symmetric calibration: because each country’s net foreign asset position is unchanged (each country reduces its holdings of rival-bloc bonds by exactly as much as it reduces its own issuance), output and inflation are unaffected.

Second, the model is calibrated to four asymmetric regions: the United States (US), US-allied countries including the European Union (WE), the China-Russia-aligned bloc (CR), and a neutral rest of the world (NE). Bloc assignment follows Den Besten et al. (2023), using a political alignment index combining sanctions data, military imports, Belt and Road Initiative participation, and UNGA voting on Russia’s invasion of Ukraine. The US and WE impose all three taxes on CR, and vice versa; NE neither imposes nor receives taxes.

Five main findings emerge from the asymmetric simulation. First, fragmentation predominantly affects CR and WE: both experience substantial declines in consumption and production across all three tax scenarios, with CR most affected when goods or asset taxes are applied. Second, the US is largely insulated: its lower trade and financial exposure to the rival bloc relative to WE limits the pass-through of fragmentation. Third, spillovers to neutral NE are nearly negligible: the expenditure-switching channel (which raises demand for untaxed NE goods) and the global income channel (which reduces demand for all goods as the world becomes poorer) roughly cancel each other out. Fourth, fragmentation is not necessarily inflationary: whether PPI inflation rises or falls depends on the relative weight of commodities in production and the mix of taxes applied — a goods tax lowers PPI inflation, while a commodity tax raises it. Fifth, the bilateral exchange rates most affected are those of the CR bloc, which appreciate under goods and asset taxes and depreciate under commodity taxes.

Sensitivity analyses confirm robustness across higher elasticity of substitution between domestic and foreign goods (eta raised from 1.5 to 5), lower elasticity of substitution between labor and commodities (xi lowered from 0.4 to 0.1), tighter financial market integration (bond transaction costs multiplied by 5), and permanent shocks (persistence rho raised to 1). Under permanent shocks, the goods-tax effect on PPI inflation approaches zero — consistent with the closed-form result — while commodity-tax effects on production become larger and more persistent.

Q: What is the core research question of FraNK? A: The paper asks how geoeconomic fragmentation — modeled as policy-driven increases in taxes on rival countries’ goods, commodities, and bonds — affects output, inflation, exchange rates, and capital flows at both the global and country level. It also asks whether different sources of fragmentation (real versus financial) have distinct macroeconomic implications, and whether neutral countries experience meaningful spillovers.

Q: How does the model depart from the Gali-Monacelli (2005) benchmark? A: Three departures are made. The model is multi-country (N countries) rather than a single small open economy facing the rest of the world. Financial markets are incomplete, so international risk sharing is imperfect — a realistic assumption in a fragmented world. And intermediate-good production uses a CES bundle of labor and a commodity bundle that includes both domestic and imported commodities, which is essential for capturing commodity market disruptions such as those following Russia’s invasion of Ukraine.

Q: What are the three tax instruments and what does each represent? A: The goods import tax (tau_ijt) is a tariff on final goods imports, representing trade barriers. The commodity import tax (tau_O_ijt) is a tariff on imported commodity inputs, representing sanctions or restrictions on energy and raw material trade. The bond tax (theta_ijt) is a capital control discouraging purchases of bonds issued by rival countries, representing financial fragmentation or sanctions on financial assets.

Q: What does the closed-form symmetric-calibration result establish about output? A: Under the symmetric calibration, both the goods import tax and the commodity import tax reduce output unambiguously (Proposition 3.3). The bond tax is neutral for output under symmetry because each country’s net foreign asset position is unchanged — any reduction in holdings of rival-bloc bonds is exactly matched by a reduction in own-bond issuance, leaving net positions and aggregate demand unaffected (Proposition 3.4).

Q: Why does the goods import tax reduce PPI inflation while the commodity import tax raises it? A: The goods import tax operates through two opposing channels: a demand channel (households substitute away from foreign goods, reducing aggregate demand) and a supply channel (import taxes raise firms’ real marginal costs). The closed-form solution establishes that the demand channel dominates, so PPI inflation falls. The commodity import tax operates only through the supply channel — raising the cost of intermediate inputs directly — so PPI inflation rises unambiguously. CPI inflation rises on impact under the goods tax because import prices are directly included in the CPI even as PPI falls.

Q: Under what condition does simultaneous fragmentation (goods and commodity taxes together) produce PPI inflation? A: When both taxes are imposed simultaneously, the net effect on PPI inflation is ambiguous. The paper shows analytically that PPI inflation rises if and only if omega * gamma_O_tilde > gamma_tilde * (phi/sigma), where omega is the commodity weight in production, gamma_O_tilde captures commodity import weights, and gamma_tilde captures goods import weights. That is, fragmentation tends to be stagflationary the larger the weight of commodities in the production function, consistent with the empirical finding in Caldara et al. (2024) of stagflationary effects from elevated geopolitical risk.

Q: Why is the US more shielded from fragmentation than its WE allies? A: The US has relatively lower trade and financial exposure to the CR bloc compared to WE. Because the trade and financial weights calibrated from UN Comtrade, IMF CPIS, BIS LBS, and IMF CDIS data place WE in closer economic relationships with CR countries, a tax on CR imports or assets falls more heavily on WE than on the US. This asymmetry is a direct consequence of the calibration: no structural or strategic advantage of the US is assumed beyond its actual pattern of trade and financial linkages.

Q: What happens to the CR bloc’s exchange rate under each tax scenario? A: Under the goods import tax, the CR exchange rate appreciates: CR’s own tax reduces demand for US/WE goods, increasing domestic demand relative to the rest of the world, and the reduced demand for CR bonds from abroad raises CR interest rates, further attracting capital. Under the commodity import tax, the CR exchange rate depreciates: lower commodity demand reduces CR commodity prices and production, shifting labor toward goods, increasing goods supply, and lowering the CR price level relative to trading partners. Under the bond tax, the CR exchange rate also appreciates, as reduced CR demand for US/WE bonds is interpreted by markets as a shift in capital flows favoring CR assets.

Q: What explains the near-zero spillovers to neutral countries? A: Two forces operate on NE in opposite directions. The expenditure-switching channel raises demand for NE goods and commodities, as taxing countries divert purchases away from taxed rival goods toward untaxed NE products — a positive demand shock for NE. The global income channel reduces demand for all goods, including NE’s, as the taxing and taxed regions become poorer and reduce imports from everywhere. In the calibration these two forces approximately cancel, leaving NE macroeconomic variables nearly unchanged.

Q: How is the commodity sector modeled, and why does this matter for the commodity tax result? A: Each country has a representative commodity firm using a linear production function (Y_iOt = A_iO * H_iOt), where A_iO is interpretable as a per-capita endowment of natural resources. Intermediate-good firms use a CES bundle of labor and commodities (domestic and imported) with elasticity xi=0.4 between the two. When the commodity import tax is imposed, firms face higher commodity input costs, raising real marginal costs and PPI inflation while depressing production. The asymmetry between commodity exporters (CR, NE) and importers (WE) under this tax is the main source of differential regional effects.

Q: How are financial openness differences across country pairs captured, and what effect do they have? A: Bond transaction costs psi_ijF differ across pairs: psi_12F = psi_21F = 0.01 for the US-WE pair (reflecting high financial integration), while all other pairs have psi_ijF = 1 — one hundred times higher — reflecting limited cross-bloc financial integration. The sensitivity analysis multiplies all psi_ijF by 5 (less open financial markets) and finds that bond position volatility falls but qualitative results are unchanged, confirming that the financial openness calibration does not drive the main results.

Q: What are the main caveats acknowledged by the authors? A: The model omits capital accumulation, so investment dynamics are absent. Cross-country production networks (global value chains) are not modeled, which the authors acknowledge limits the richness of the production structure relative to Baqaee-Farhi (2024) style models. Domestic financial markets are assumed frictionless. The model has no role for dollar dominance in the global economy, which may matter for exchange rate and capital flow dynamics in reality. These are flagged as directions for future research.

Q: What is the key result for permanent (rho=1) versus temporary (rho=0.9) fragmentation shocks? A: Under permanent shocks, output reductions become permanent rather than transitory. For the goods import tax, the effect on PPI inflation approaches zero in the permanent case, consistent with the closed-form prediction that the demand channel effect on PPI vanishes when the tax persists indefinitely (households no longer have an intertemporal substitution motive). The commodity tax permanent shock induces a larger and more persistent fall (rise) in production for commodity importers (exporters). Bond tax permanent shock has larger magnitude effects but is otherwise qualitatively similar to the temporary case.

Q: How does FraNK relate to the existing DSGE literature on sanctions and trade wars? A: The paper positions FraNK as providing a unified framework covering all three forms of fragmentation (goods, commodity, and financial) simultaneously, with nominal rigidities allowing for inflation analysis, closed-form analytical results for transparency, and a multi-country setup rather than small-open-economy. Ghironi et al. (2024) study sanctions in a three-country model but without nominal rigidities. Itskhoki and Mukhin (2022) analyze sanctions on Russia but in a small-open-economy. Attinasi et al. (2023) and Conteduca et al. (2024b) use richer production networks (Baqaee-Farhi) but are static and exclude financial fragmentation. FraNK trades production network richness for dynamics, nominal rigidities, financial fragmentation, and analytical tractability.

Geoeconomic fragmentation: A policy-driven reversal of economic integration, often guided by strategic or geopolitical considerations, operationalized in FraNK as simultaneous increases in taxes on rival countries’ goods imports, commodity imports, and bond purchases.

Fragmentation shock: A simultaneous increase in three tax rates — goods import tax (tau), commodity import tax (tau_O), and bond tax (theta) — applied by each bloc against the other, representing the policy instruments through which integration is reversed.

Demand channel (goods tax): The mechanism by which a goods import tax reduces aggregate demand, as households substitute away from now-more-expensive foreign goods, reducing output and — because this channel dominates the supply channel — lowering PPI inflation.

Supply channel (commodity tax): The mechanism by which a commodity import tax raises intermediate input costs for firms, increasing real marginal costs and PPI inflation while reducing output — a purely cost-push effect with no offsetting demand-side force.

Bond tax neutrality: Under symmetric calibration, capital controls on rival-bloc bonds are macroeconomically neutral because each country’s net foreign asset position is unchanged: the reduction in holdings of rival bonds is exactly matched by a reduction in own-bond issuance, leaving the IS curve and Phillips curve unaffected.

Expenditure-switching channel: The force by which fragmentation between two blocs diverts import demand toward untaxed third-country (neutral) goods, generating a positive demand spillover for NE countries that roughly offsets the global income channel.

Global income channel: The negative spillover to neutral countries arising from the reduction in world income caused by fragmentation between the taxing blocs, which reduces demand for all goods including those of neutral producers, approximately canceling the expenditure-switching channel.

From Doubt to Devotion: Trials and Learning-Based Pricing

Mon, 01 Jan 0001 00:00:00 +0000

This paper studies a dynamic mechanism design problem in which an informed seller sells an experience good to a skeptical buyer who learns about the product through consumption. The central question is: how does a seller leverage proprietary data about product-buyer match quality together with the buyer’s ability to learn, and what are the welfare implications in equilibrium?

The model features a seller who privately observes a binary match quality (theta in {H, L}) between their service and the buyer. The buyer does not observe match quality and has an initially unknown private value v for the good, drawn from a Myerson-regular distribution F with support [v_low, v_high] and normalized mean E[v] = 1. If the match is high, the buyer receives instantaneous utility rewards according to a Poisson process with flow rate lambda*I, where I in [0,1] is the seller-controlled access level. Upon receiving the first reward, the buyer perfectly learns both match quality theta and their own value v. The seller commits to a dynamic mechanism over time horizon T = [0, T] specifying access and prices conditional on reported histories. Both parties are risk-neutral and there is no discounting in the baseline.

Two benchmark cases show the first-best is attainable absent both key features simultaneously. If trade is static (prices set only at time 0) or if the seller is uninformed about theta, the seller achieves first-best revenue of lambdamu_0T by selling the entire service upfront. Proposition 1 establishes both cases; this implies that consumer data on theta is not required for maximizing social welfare, and it is weakly dominant for a seller to never collect consumer data in static environments.

The central result is that the combination of dynamic pricing and seller private information breaks the first-best. A high-type seller can deviate by offering a “Myersonian free trial”: provide full access up to time tM (defined as argmax_t {(1 - exp(-lambdat))(T - t)}), then offer the remaining service at post-trial price lambdavM(T - tM), where vM is the Myerson monopoly price. The buyer accepts the trial regardless of beliefs (participation is weakly dominant) and purchases the post-trial service if and only if v >= vM. This deviation yields payoff pi_F = (1 - exp(-lambdatM))(1 - F(vM))lambdavM*(T - tM). Proposition 2 states that the first-best cannot be implemented in any equilibrium if and only if pi_F > lambdamu_0T. Corollary 1 shows this condition holds for sufficiently large T, since pi_F grows proportionally with T while the first-best also grows with T but the ratio converges to a constant less than 1 only for some parameter configurations and exceeds 1 for others.

Theorem 1 (the main mechanism design result) characterizes the boundary of the IC-IR feasible payoff set: any mechanism on this boundary is outcome-uniquely implemented by a trial mechanism, defined by a triple (v0, t0, p0) — a trial length, a post-trial value threshold, and a trial price. During [0, t0] uninformed buyers receive full access; after t0 only buyers who received a reward with v >= v0 continue at a premium. Trial length t0 is weakly increasing in the weight placed on the low-type seller and in the prior mu_0; post-trial threshold v0 is weakly decreasing in the same objects (Proposition 3).

Equilibrium payoffs (Proposition 5) are precisely the IC-IR feasible pairs satisfying pi_H >= pi_F, implemented by pooling trial mechanisms in which both seller types propose identical mechanisms and the buyer updates beliefs only through private consumption signals. Under the D1 refinement (Proposition 6), only mechanisms with trial length tM and post-trial threshold vM survive. These have the shortest trial and highest post-trial price of all equilibrium mechanisms, minimize social surplus, and may leave both seller types strictly worse off than in a world without private information — directly contrasting the static informed principal result of Koessler and Skreta (2016) where data always helps the seller.

When the seller can control service quality q in addition to access I (Section 6), the relevant equilibrium mechanisms become dynamic tiered pricing rather than binary trials: a low-quality, high-ad-load free tier provides learning opportunities while reducing information rents; convinced buyers upgrade to a premium ad-free tier. Counterintuitively, enriching the seller’s screening technology can reduce both revenue and social efficiency in equilibrium because additional instruments create additional signaling opportunities that distort outcomes further.

Q: What is the core tension that prevents the first-best from being an equilibrium?

A: When the seller is privately informed and pricing is dynamic, the high-type seller anticipates a greater likelihood of the buyer receiving a utility shock than the buyer’s own prior implies. This belief gap makes it profitable for the high-type seller to deviate from a proposed first-best mechanism by offering a free trial that “proves” high match quality and then extracting rent from convinced buyers. Because this deviation is profitable — yielding pi_F > lambdamu_0T under some parameters — the first-best pooling contract unravels. The interaction of both ingredients (dynamic pricing and informed seller) is necessary: either ingredient alone is insufficient to break the first-best (Proposition 1).

Q: What exactly is the Myersonian free trial and why does the buyer always accept it?

A: The Myersonian free trial provides full service access up to time tM = argmax_t {(1 - exp(-lambdat))(T - t)} at (approximately) zero price, then offers the remaining service at price lambdavM(T - tM) where vM is the Myerson monopoly price. The buyer accepts the trial regardless of their prior belief about match quality because the trial itself is free and provides non-negative payoff. After the trial, the buyer purchases the post-trial service if and only if they received a reward with v >= vM; otherwise they exit. The deviation payoff is pi_F = (1 - exp(-lambdatM))(1 - F(vM))lambdavM*(T - tM).

Q: Under what parametric conditions can the first-best not be supported in equilibrium?

A: By Proposition 2, the first-best cannot be implemented if and only if pi_F > lambdamu_0T. Corollary 1 states that for sufficiently large T this always fails, since as T grows, pi_F grows proportionally (the post-trial term (T - tM) dominates) while tM converges to a finite value. More precisely, for large T, pi_F / (lambdamu_0T) converges to (1 - exp(-lambda*tM)) * (1 - F(vM)) * vM / mu_0, which exceeds 1 under appropriate parameter configurations. Conversely, when mu_0 is high or the service horizon is short, the first-best may remain implementable.

Q: What is a trial mechanism and how does Theorem 1 characterize it?

A: A trial mechanism is defined by a triple (v0, t0, p0): uninformed buyers receive full access on [0, t0] and no access thereafter; a buyer who reports a reward of value v >= v0 at time t receives full service for the remainder [t, T] at a price increment of lambdav0(T - t0); the trial itself is priced at p0. Theorem 1 states that any payoff pair on the boundary of the IC-IR feasible set is outcome-uniquely attained by such a trial mechanism with appropriately determined (v0, t0, p0). The proof uses a relaxed problem retaining only two key constraint families: local incentive constraints on value reporting (IC-V) and a global intertemporal constraint preventing buyers from hiding the arrival of rewards forever (IC-U).

Q: How does the trial length respond to changes in prior belief mu_0 and distributional spread?

A: Proposition 3 states that t0 is weakly increasing in mu_0: as market belief becomes more optimistic, both seller types extract higher revenue from the trial, so the mechanism designer extends the trial. Proposition 4 adds that for a uniform distribution on [1-delta, 1+delta], trial length t0 is weakly increasing in delta (greater spread). The post-trial threshold v0 is weakly decreasing in mu_0, meaning that a more optimistic prior leads to a less exclusive post-trial cutoff.

Q: What are the equilibrium payoffs and how does the high-type seller’s free-trial option constrain them?

A: Proposition 5 states that (pi_L, pi_H) is an equilibrium payoff if and only if it lies in the IC-IR feasible set and pi_H >= pi_F. The lower bound pi_H >= pi_F reflects the high-type seller’s outside option: they can always deviate to the Myersonian free trial. Corollary 4 then shows that all “reasonable” equilibrium payoffs (those with pi_H >= pi_L, surviving a mild off-path refinement) are implemented by trial mechanisms with complete pooling — both seller types propose the same mechanism and the buyer updates beliefs only through private consumption signals, not the mechanism’s structure.

Q: What does the D1 refinement select and why do it lead to worse outcomes?

A: Proposition 6 shows that the only equilibrium trial mechanisms surviving the D1 criterion have trial length tM and post-trial threshold vM — the Myersonian free trial parameters. These have the shortest trial and highest post-trial price among all equilibrium mechanisms, resulting in the minimum social surplus. The intuition is that the high-type seller signals credibly by proposing mechanisms that generate high revenue from post-trial price discrimination (which the low type cannot profit from), pushing toward maximum learning-based discrimination. All D1-surviving payoffs are Pareto dominated by the point H (the unconstrained IC-IR optimum) for any prior mu_0, and Pareto dominated by point B when mu_0 is small.

Q: Can having consumer preference data hurt the seller, and under what conditions?

A: Yes. The distortion from signaling incentives can be so large that both seller types earn strictly less in the D1-surviving equilibrium than they would if neither possessed private information (where the first-best is attained). This result holds when the condition of Proposition 2 is satisfied — i.e., when pi_F > lambdamu_0T. This contrasts sharply with the static result of Koessler and Skreta (2016), in which the ex-ante profit-maximizing mechanism is always supportable in equilibrium and data always (weakly) helps sellers.

Q: How do trial mechanisms differ from the prior literature on signaling through introductory prices?

A: The earlier literature (Milgrom and Roberts 1986; Bagwell 1987; Bagwell and Riordan 1991; Judd and Riordan 1994) uses two-period models with no seller commitment, so all pricing behavior is necessarily trial-like by model restriction. The present model instead allows the seller full flexibility to design any dynamic mechanism — including selling everything ex-ante, which would prevent buyers from gaining information rent. Trials emerge endogenously as the equilibrium outcome rather than being imposed by the model structure, and the paper provides new economic content on what determines trial length and price thresholds.

Q: What happens when the seller controls service quality in addition to access?

A: Section 6 extends the baseline by allowing the seller to choose (I, q) from a subset of [0,1]^2, where I governs the Poisson arrival rate and q scales the reward value (utility from a reward is v*q). Theorem 2 shows that the relevant equilibrium mechanisms now take the form of dynamic tiered pricing: a low-quality tier (interpreted as high ad load) provides learning opportunities while reducing information rents; once convinced, buyers upgrade to a premium high-quality tier. Enriching the screening technology in this way can reduce both revenue and social efficiency in equilibrium, because additional instruments create additional signaling opportunities that distort outcomes further from the revenue-maximizing benchmark.

Q: What are the two sources of welfare loss relative to the first-best in D1-surviving equilibria?

A: The welfare analysis in Appendix F identifies two sources. First, exclusion inefficiency: buyers with values v in [v_low, vM) who would generate positive surplus are excluded from post-trial service. Second, service truncation inefficiency: service access is cut off after trial length tM for buyers who were never convinced (theta = L type realizations and high-type buyers with v < vM), reducing total surplus below the first-best of mu_0 * lambda * T. Both losses are minimized (welfare is maximized) among trial mechanisms by longer trials and lower post-trial cutoffs, precisely the opposite of what D1 selects.

Q: Does the model extend to continuous seller types or multiple buyer types?

A: Appendix K outlines an extension to continuous seller types theta drawn from a distribution G on [theta_low, theta_high], where rewards arrive at rate lambdaItheta. The main economic forces persist: higher seller types anticipate faster buyer learning and have stronger incentives to offer trials. The main results generalize: equilibrium mechanisms are trial mechanisms, and under D1, pooling equilibria with maximum post-trial discrimination are selected. Appendix G similarly notes that the multiple-buyer-type extension preserves complete pooling and the D1 selection result.

Q: What is the role of the “global intertemporal constraint” (IC-U) in the proof of Theorem 1?

A: The canonical approach to dynamic mechanism design (Eso and Szentes 2007; Pavan, Segal, and Toikka 2014) relaxes the problem to only local incentive constraints on the initial report. This fails here because the informed seller causes buyer and seller to disagree on the evolution of buyer beliefs, making the timing of trade matter and requiring tracking of incentive constraints at every point in time. The paper identifies two key binding constraints in the relaxed problem: (IC-V) the buyer does not misreport their reward value, and (IC-U) the buyer does not remain silent about the arrival of a reward forever. Retaining only these two constraint families yields a tractable bang-bang solution for the optimal access policy, which is then verified to satisfy all original IC-IR constraints.

Q: What are the implications for platform design and data collection strategy?

A: The results imply that the value of consumer data depends critically on market dynamics. In static markets, collecting data about consumer match quality is weakly beneficial for sellers (Proposition 1, first point). In dynamic markets with buyer learning and sufficiently long service horizons, the same data can strictly reduce seller revenue by enabling a deviation that unravels first-best pricing. This suggests platforms in dynamic digital markets should weigh whether possessing and acting on proprietary match data improves or worsens their equilibrium position, and that regulatory attention to consumer data collection in dynamic markets may have welfare-ambiguous effects.

Trial mechanism: A dynamic mechanism parameterized by (v0, t0, p0) in which the seller provides full service access during [0, t0] for uninformed buyers, offers continued service after t0 only to buyers who received a reward with value v >= v0, and charges a post-trial price of p0 + lambdav0(T - t0) for those who qualify. In the paper’s usage, this is the unique outcome-implementing mechanism on the boundary of the IC-IR feasible payoff set.

Myersonian free trial: The limiting trial mechanism as the trial price epsilon approaches zero, with trial length tM = argmax_t {(1 - exp(-lambdat))(T - t)} and post-trial threshold vM equal to the Myerson monopoly price. It yields payoff pi_F = (1 - exp(-lambdatM))(1 - F(vM))lambdavM*(T - tM) to the high-type seller, and constitutes the binding outside option constraining equilibrium payoffs.

Belief gap: The divergence between the seller’s and buyer’s beliefs about the rate at which the buyer will receive Poisson rewards. Because the high-type seller knows theta = H, they anticipate a higher probability of reward arrival than the buyer’s prior implies. This gap makes the buyer’s belief process non-martingale from the seller’s perspective, breaking the standard dynamic mechanism design approach and creating profitable deviation incentives.

IC-IR feasible payoff set: The set of seller payoff pairs (pi_L, pi_H) achievable by mechanisms satisfying both incentive compatibility (for seller type reports and buyer learning reports) and individual rationality (non-negative ex-ante payoffs for all parties). Theorem 1 establishes that the boundary of this set is uniquely implemented by trial mechanisms.

Dynamic tiered pricing: The equilibrium mechanism form that emerges when the seller controls both access I and service quality q. It features a low-quality tier (high ad load) providing learning opportunities at reduced information rent, and a premium tier offering full quality to buyers convinced of high match quality. This generalizes trial mechanisms to settings with richer screening technology.

Global intertemporal constraint (IC-U): The constraint requiring that, upon receiving a Poisson reward, the buyer finds it suboptimal to remain silent about its arrival forever. Together with the local value-reporting incentive constraint (IC-V), these two constraints constitute the binding restrictions in the paper’s relaxed mechanism design problem, replacing the full continuum of incentive constraints that would otherwise be intractable.

D1 criterion: A standard equilibrium refinement from signaling games applied here to the space of mechanism proposals. Among all pooling equilibrium trial mechanisms, D1 selects only those with parameters (tM, vM) — the shortest trial length and highest post-trial threshold — because the high-type seller has a strictly larger set of buyer responses for which deviation to a high-discrimination mechanism is profitable. These surviving mechanisms Pareto dominate no other equilibrium mechanism and minimize social surplus.

Gendered Spheres of Learning and Household Decision-Making over Fertility

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates whether information asymmetries within households about maternal health risk can explain persistent spousal disagreement over fertility in a high-fertility, high-maternal-mortality setting. The authors develop a theoretical model and conduct a randomized field experiment among approximately 500 couples in peri-urban Lusaka, Zambia, where the lifetime risk of maternal death is 1 in 59 women and the maternal mortality ratio is 398 deaths per 100,000 live births.

The central mechanism is a communication barrier that arises from conflicting fertility preferences between spouses. When husbands have higher desired fertility than wives (4.43 vs. 4.19 children on average in the study sample), wives who are better informed about maternal health risk lack the incentive to credibly transmit that information to their husbands. Strategic communication concerns — not a generically lower propensity of men to learn from women — drive this asymmetry. The model predicts a pooling equilibrium in which no informative communication flows from wives to husbands when preference divergence is sufficiently large.

The experiment randomized whether the maternal mortality information curriculum was delivered to the husband or the wife in each couple, with both spouses in all arms also receiving a family planning curriculum. This design isolates the incremental effect of the maternal mortality information and permits identification of direct versus spillover effects within the household.

Consistent with the model, treated husbands significantly update their beliefs about maternal health risk factors, and their wives also update — information flows from husbands to wives. By contrast, treated wives update their own beliefs, but their husbands do not update at all. The test that spillover effects are symmetric is rejected (p-value = 0.097 for risk factors index; p-value < 0.001 for direct vs. indirect effects on men). The communication asymmetry is most pronounced among husbands who, at baseline, want a child as soon as possible — precisely the households with the greatest preference conflict.

Both treatment arms reduce fertility. Households in which the husband is treated experience a 43% reduction in the probability of having a child or being pregnant in the year following the intervention. The fertility reduction is strongest when the wife faces higher ex ante risk based on her birth history, consistent with the model’s prediction that treatment effects are concentrated among households with high maternal health costs.

The transfers evidence is the key differentiator between the two arms. When the wife is treated, fertility declines but is accompanied by a significant reduction in transfers from husband to wife, consistent with the wife updating her own beliefs without being able to convey them to her husband, who then reduces compensation. When the husband is treated, fertility declines without the same reduction in transfers — and treated husbands report higher communication with their spouse about family planning and higher relationship satisfaction. This combination is consistent with the husband treatment resolving the information gap directly, enabling efficient contracting, whereas the wife treatment leaves the information asymmetry in place.

The study is conducted in informal settlements of Lusaka, a prime-age urban sample in which the average woman is 28 years old with 2.6 children at baseline. Scope conditions: results apply to a setting with very high maternal mortality, large baseline spousal fertility gaps, and strong traditional beliefs (55.5% of men cite marital infidelity as a leading cause of maternal complications). Generalizability to lower-risk or lower-preference-gap settings is explicitly circumscribed by the model’s comparative statics.

Q: What is the baseline gender gap in knowledge of maternal health risk? A: Men are less likely than women to identify high parity (72.0% vs. 77.7%) and advanced maternal age (74.3% vs. 84.6%) as risk factors. In seven hypothetical scenarios rating complication likelihood on a 0–10 scale, men report lower scores than women in six out of seven cases. Despite Zambia’s 1-in-59 lifetime maternal mortality risk, only 27.6% of men (vs. 53.4% of women) report having attempted to discuss maternal health risk with their spouse.

Q: What drives the gender gap in knowledge? A: The authors argue the gap stems from “gendered spheres of direct and indirect knowledge accumulation of maternal labor and delivery outcomes.” Women are embedded in social networks where maternal mortality episodes are more salient: 11.0% of women report knowing a close friend who died giving birth, vs. 6.8% of men knowing a close friend whose wife died. The gap widens with social distance to the victim, suggesting women’s networks give them systematically more exposure to maternal mortality events.

Q: How does the model explain the failure of within-household communication? A: The model places husband and wife preferences as minimizing the distance between realized fertility and their respective net fertility optima (ideal fertility minus weighted maternal health cost). When the husband’s ideal fertility is high enough, he makes transfers to induce the wife to bear more children than her private optimum. Given these incentives, a wife who is informed about high health costs has an interest in exaggerating the cost to extract larger transfers. Because the husband anticipates this, no informative communication occurs in equilibrium — the only equilibrium is a pooling equilibrium where the wife’s message is uninformative regardless of her true cost realization.

Q: What is the specific asymmetry in belief updating observed in the experiment? A: Among treated husbands, both husbands and their wives update beliefs about maternal risk factors — information flows from husband to wife. Among treated wives, only the wife updates; her husband does not. The Wald test rejects equal direct and indirect effects on men at p < 0.001 and rejects symmetric spillovers at p = 0.097 for the risk factors index. There is no symmetric restriction binding for women’s updating across arms.

Q: How large is the fertility effect and which arm drives it? A: Households in which the husband is treated experience a 43% reduction in the probability of having a child or being pregnant in the year following the intervention. This effect is described as of the same order of magnitude as other household-level interventions shown to reduce pregnancy (citing Ashraf, Field, and Lee 2014). The fertility reduction is strongest among households where the woman faces higher ex ante risk based on birth history, consistent with the model’s Prediction 5 that effects are concentrated where theta_j is high.

Q: How do transfers differ between the wife-treated and husband-treated arms? A: When the wife is treated, the fertility decline is accompanied by a significant reduction in transfers from husband to wife. When the husband is treated, the fertility decline is not accompanied by a similar reduction in transfers. The authors interpret this pattern as: wife treatment leaves the husband uninformed, so he reduces transfers when he observes her reducing fertility without understanding why; husband treatment resolves the information gap, allowing efficient renegotiation without penalizing the wife.

Q: Which husbands fail to update beliefs even when their wife is treated? A: Husbands who at baseline want a child “as soon as possible” do not update their beliefs in response to their wife’s treatment status. These men also reduce transfers to their wife more than other groups when she is treated. In the model, these are precisely the households with the highest conflict of interest (high alpha_H), where the pooling equilibrium prediction is sharpest.

Q: What is the role of traditional beliefs about maternal mortality? A: 55.5% of men and 42.0% of women report (without prompting) marital infidelity as a leading cause of maternal labor and delivery complications — greater weight than assigned to lack of healthcare and poor health status combined. This stigma directly reduces women’s willingness to raise concerns about birth complications with their spouse, reinforcing the communication barrier the model formalizes.

Q: What are the welfare implications of targeting men vs. women with information? A: The fertility reduction from husband treatment is not inferior to that from wife treatment, but husband treatment also produces improvements in marital surplus — treated husbands report higher communication with spouse about family planning, higher relationship satisfaction, and greater closeness — whereas wife treatment reduces transfers to the wife, indicating she bears a financial cost. The authors argue male-targeted information can reduce unmet need for family planning while enhancing rather than exacerbating household conflict.

Q: Does this paper provide field experimental evidence on strategic communication models? A: The authors claim this is the first field experimental evidence directly testing models of strategic communication (Crawford and Sobel 1982; Mailath 1987; Crawford 1998, 2019), wherein persistent preference differences and conflict of interest impede communication and beliefs updating. Prior tests of these models were conducted in the lab; this paper provides the first real-world behavioral test with consequential decisions (fertility) in a high-stakes setting.

Q: What is the unmet need for family planning in the study sample? A: Overall, 32% of women in the sample report not using modern contraceptives at baseline. Of the 33% of women who want no more children, 27% are not using any modern contraceptive (8% of the overall sample). Of the 52% of women who wish to delay giving birth by at least one year, 23% are not using any modern contraceptive (12% of the overall sample).

Q: How does the model characterize the husband’s partial internalization of maternal health costs? A: The husband’s utility function includes the maternal health cost theta_j scaled by delta (0 ≤ delta ≤ 1), capturing how much weight he places on his wife’s risk. When delta is sufficiently high and the husband’s ideal fertility (alpha_H) is sufficiently low, or when his disutility of transfers (gamma) is sufficiently low, informative communication can occur after the husband is treated. When delta is low, the husband discounts his wife’s risk and communication barriers are more severe regardless of treatment.

Maternal health cost (theta): A random variable representing the welfare cost borne by the wife from childbearing, including mortality risk and morbidity. In Zambia, distributed with a higher mean than the worldwide distribution. Enters the wife’s utility directly and the husband’s utility only scaled by delta, his degree of internalization of her cost.

Gendered spheres of learning: The paper’s term for the systematic differential in experiential exposure to maternal mortality outcomes between men and women, arising from gender-segregated social networks. Women witness maternal mortality events more directly through closer social ties, while men’s networks provide systematically less exposure.

Communication barrier (pooling equilibrium): The equilibrium outcome in the model where no informative signal is transmitted from an informed wife to her uninformed husband about the true realization of maternal health cost. Arises because the wife’s incentives to misreport are independent of the true cost realization, making any message uninformative when preference conflict is sufficiently large.

Intra-household information spillover: The transmission of information learned by one spouse to the other as a consequence of the treated spouse’s belief update. The paper documents asymmetric spillovers: information flows from treated husbands to their wives, but not from treated wives to their husbands.

Husband’s demand for children (alpha_H): The husband’s ideal fertility level, which governs the degree of preference conflict within the household. Baseline husband desire for a child as soon as possible serves as the empirical proxy for high alpha_H and is the key moderator of spillover and transfer effects.

Degree of internalization (delta): The parameter in the husband’s utility function (0 ≤ delta ≤ 1) capturing how much weight he places on his wife’s maternal health cost. When delta is high and gamma (disutility of transfers) is low, communication can occur in equilibrium after the husband is treated.

Unmet need for family planning: Women who wish to space or limit births but are not using modern contraception. In the study sample, 32% of women report not using modern contraceptives at baseline, with substantial shares among both those wanting no more children and those wishing to delay.

Genetic Prediction and Adverse Selection

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how much adverse selection would arise in critical illness insurance (CII) markets if consumers can observe polygenic indexes (PGIs) — genetic risk scores derived from millions of genetic variants — while insurers are legally barred from using genetic information. The authors develop an econometric method that measures selection under current PGI technology, then extends identification to expected future PGI accuracy using heritability bounds, even though future PGIs are not yet observable in data.

The primary dataset is the UK Biobank (UKB), comprising approximately 446,570 genotyped individuals of European-like ancestry linked to NHS electronic health records. The authors study seven single-disease CII contracts (Alzheimer’s disease, breast cancer, coronary artery disease, colorectal cancer, prostate cancer, schizophrenia, and type 2 diabetes) and multiple-disease bundled contracts paying a lump sum upon onset. The econometric model assumes a probit disease probability, Gaussian PGI structure, and identification relies on published heritability estimates to pin down future PGI predictive power. The key selection metric is the implicit tax proposed by Hendren (2013): the percentage markup a marginal consumer must pay above her actuarially fair price due to adverse selection. The authors use the minimum implicit tax up to the 80th percentile of risk (t80) as their summary statistic, with market unraveling benchmarked at t80 between 43% and 83% from prior literature.

The paper reports three main findings, all scoped to a population of 35-year-olds in the standard insurer risk class (those whose predicted risk falls within 0.75–1.25 times the population mean).

First, under current PGI technology with full consumer adoption, selection is noticeable but heterogeneous across diseases. t80 ranges from 17.9% for coronary artery disease to 117.9% for Alzheimer’s disease. Coronary artery disease and colorectal cancer fall in the middle of the no-unraveling range; breast cancer, schizophrenia, and type 2 diabetes fall between the no-unraveling and unraveling ranges; Alzheimer’s disease and prostate cancer (t80 = 59.8%) reach or exceed the unraveling range. The current prostate cancer PGI explains 9.9% of liability variance, adding 8.3 percentage points over the 22.9% explained by non-genetic covariates.

Second, under expected future PGI accuracy — bounded below by SNP heritability and above by twin heritability — selection becomes potentially crippling. Under the lower bound (Scenario 3L), t80 ranges from 57.5% for breast cancer to above 1,000% for Alzheimer’s. Under the upper bound (Scenario 3U), t80 exceeds 100% for all seven single-disease contracts and exceeds 1,000% for three of them. For prostate cancer, the reference case, t80 reaches 86.8% under Scenario 3L and 426.9% under Scenario 3U — far above Hendren’s unraveling benchmarks. For multiple-disease male contracts, t80 = 30.8% under current technology, rising to 54.4% (Scenario 3L) and 243.9% (Scenario 3U).

Third, variation in selection across contracts is driven primarily by: the predictive power of the future PGI, the incremental predictive power over non-genetic covariates, and disease prevalence. Alzheimer’s and schizophrenia — high heritability, low prevalence — display the highest implicit taxes; breast and colorectal cancer — lower SNP heritability, lower incremental R2 — display the lowest.

These findings are corroborated by a calibrated Akerlof-Einav-Finkelstein equilibrium model using HRS data: current PGI availability reduces equilibrium market quantity from 30% to 21.4%; future PGI availability drives equilibrium quantity to zero in a full adverse selection death spiral. Partial take-up robustness checks show that even at 50% consumer adoption, selection remains problematically high under future PGI accuracy for most contracts. The analysis is restricted to individuals of European-like ancestry due to data availability constraints.

Q: What is the core market failure the paper analyzes? A: The paper analyzes adverse selection arising from an asymmetric information gap: consumers can observe PGI-based disease risk predictions from consumer genetic tests (e.g., 23andMe), while insurers in many jurisdictions are legally prohibited from requesting or using genetic information. This creates a situation where high-risk consumers have private information allowing them to sort into insurance, driving up average claims costs and potentially unraveling the market.

Q: What is a polygenic index (PGI) and why does it differ from classical genetic testing? A: A PGI is a weighted sum of millions of genetic variants (typically over one million) each with individually tiny effects, constructed using effect-size estimates from genome-wide association studies (GWASs). This contrasts with traditional genetic testing focused on rare single-gene mutations (e.g., BRCA for breast cancer or PKD for kidney disease), which are rare, explain small shares of population-level disease variance, and can largely be inferred from family history. PGIs target common polygenic diseases and are the primary driver of the adverse selection concern because they aggregate diffuse genetic signals into a meaningful risk prediction.

Q: What are the current PGI R2 values for the seven diseases studied? A: Estimated on the liability scale in the UKB, current PGI R2 values are: Alzheimer’s disease 7.1%, breast cancer 6.7%, coronary artery disease 2.5%, colorectal cancer 2.2%, prostate cancer 9.9%, schizophrenia 4.9%, and type 2 diabetes 7.4%. These represent the share of liability variance explained by each disease’s current PGI in the study sample.

Q: How does the paper identify the degree of selection under future PGI technology that does not yet exist in the data? A: The identification strategy combines three elements: the normality of PGI distributions, the relationship between current and future PGIs (the current PGI is modeled as a noisy version of the future PGI with an independent Gaussian error), and published heritability estimates that bound the future PGI’s predictive power. Theorem 1 establishes that under five stated assumptions — including a probit disease model and known future R2 from heritability studies — the full joint distribution of loss, current PGI, future PGI, and non-genetic covariates is identified from observed data.

Q: What heritability bounds are used for the future PGI scenarios, and why two bounds? A: Scenario 3L sets future PGI R2 equal to each disease’s SNP heritability (estimated from common genetic variants), which the authors treat as a conservative lower bound because future PGIs will also incorporate rarer variants with better effect-size precision. Scenario 3U sets future PGI R2 equal to twin heritability, treating it as an upper bound since the theoretical maximum predictive power of a PGI is the trait’s narrow-sense heritability. For prostate cancer, these bounds are 18.0% (SNP) and 57.0% (twin); for Alzheimer’s, SNP heritability is 33.1% and twin heritability is 58%.

Q: What is the implicit tax and how is it used as a benchmark? A: The implicit tax t(r) for a consumer with private risk r equals the percentage by which her insurance cost exceeds her own actuarially fair price when she must pool with all consumers of equal or higher risk. It measures how much the marginal buyer overpays due to adverse selection. The authors follow Hendren (2013) in reporting t80, the minimum implicit tax up to the 80th percentile. Hendren’s benchmarks: t80 between 7–35% for markets that did not unravel; t80 between 43–83% for markets that had unraveled.

Q: What are the single-disease contract results under current PGI technology (Scenario 2)? A: With full consumer adoption of current PGI technology, t80 ranges from 17.9% for coronary artery disease to 117.9% for Alzheimer’s disease. Coronary artery disease (17.9%) and colorectal cancer (26.5%) fall in the middle of Hendren’s no-unraveling range. Breast cancer (36.9%), schizophrenia (42.1%), and type 2 diabetes (37.0%) fall between the no-unraveling and unraveling ranges. Alzheimer’s disease (117.9%) and prostate cancer (59.8%) reach or exceed the unraveling range.

Q: What are the single-disease contract results under future PGI technology? A: Under the lower bound (Scenario 3L, R2 = SNP heritability), t80 ranges from 57.5% for breast cancer to above 1,000% for Alzheimer’s disease. Under the upper bound (Scenario 3U, R2 = twin heritability), t80 exceeds 100% for all seven contracts and exceeds 1,000% for three (Alzheimer’s, schizophrenia, and at least one other). These figures substantially exceed Hendren’s unraveled-market benchmarks for virtually all contracts.

Q: What drives cross-disease variation in the implicit tax? A: The authors identify three main drivers: the expected accuracy of future PGI (higher heritability → higher implicit tax), the incremental predictive power of the future PGI over non-genetic covariates observable by insurers (more incremental information → more adverse selection), and disease prevalence (lower prevalence concentrates risk heterogeneity, amplifying selection). Alzheimer’s disease and schizophrenia — high heritability and low prevalence — have the highest implicit taxes. Breast and colorectal cancers — lower SNP heritability and lower incremental R2 — have the lowest.

Q: What do the multiple-disease bundled contract results show? A: For the male multiple-disease contract under Scenario 2 (current PGI), t80 = 30.8%, comparable to Hendren’s no-unraveling range. Under Scenario 3L, t80 = 54.4%; under Scenario 3U, t80 = 243.9%, both in or above the unraveling range. The female contract yields qualitatively similar results. Implicit taxes in bundled contracts are generally lower than in single-disease contracts, suggesting some diversification of genetic risk across diseases.

Q: What does the calibrated equilibrium model find? A: Using an Akerlof (1970) / Einav-Finkelstein-Cullen (2010) supply-and-demand model calibrated to match a 30% market participation rate and a 50% loss ratio in the UK CII market, and using HRS data on individual risk aversion, the model finds that current PGI availability reduces equilibrium quantity from 30% to 21.4%. Future PGI availability (both Scenario 3L and 3U) drives equilibrium quantity to zero — a complete adverse selection death spiral with no trade.

Q: How robust are results to partial consumer adoption of genetic testing? A: At 10% consumer take-up, selection is low regardless of PGI accuracy. At 50% take-up, selection remains problematically high for all single-disease contracts under future PGI accuracy (Scenarios 3L and 3U). For multiple-disease contracts at 50% take-up, t80 falls just below Hendren’s unraveling threshold under Scenario 3L but enters the unraveling range under Scenario 3U. This suggests market problems would materialize once predictive power exceeds the SNP heritability bound and take-up exceeds roughly 50%.

Q: What role do risk preferences play, and do they confound the results? A: The authors test whether risk tolerance correlates with disease risk in the UKB using a self-reported general risk tolerance measure. They find extremely low correlations between risk tolerance and each disease. This is consistent with low correlation between relative risk aversion and disease risk in the HRS calibration, and supports the finding that correlation between risk and risk preferences is unlikely to meaningfully affect the main results.

Q: What is the paper’s assessment of preventive treatment as a mitigating factor? A: The authors acknowledge that genetic testing could enable personalized preventive medicine, which would reduce actual disease incidence among high-risk individuals. However, they argue this is unlikely to substantially affect their main findings because the most commonly covered diseases under CII are cancers, for which preventive behaviors have bounded effectiveness.

Q: What are the paper’s policy implications? A: The paper situates the genetic information problem within the standard regulatory framework for selection markets, distinguishing laissez-faire (allow genetic underwriting — efficient but potentially unfair to high-risk consumers), government provision (unattractive for non-essential CII), and managed competition (community rating combined with subsidies and risk adjustment). The authors argue that a full ban on genetic underwriting — the current policy in many countries — may become untenable as PGI accuracy improves, because it generates potentially crippling adverse selection. Some level of community rating may remain desirable for redistribution, but needs to be paired with subsidies or risk adjustment to prevent market collapse.

Q: What are the main data and scope limitations? A: The analysis is restricted to individuals of European-like ancestry because most large GWASs were conducted in European ancestry samples and PGIs perform poorly across ancestries. The UKB sample was aged 40–69 at recruitment and the analysis adjusts for age-dependent covariates; the HRS replication uses approximately 20,000 individuals. The equilibrium model ignores moral hazard and uses a parsimonious binary loss framework. The paper does not specify a timeline for when PGI accuracy will reach heritability bounds.

Polygenic Index (PGI): A weighted sum of an individual’s genetic variants across the genome (typically over one million variants), constructed using effect-size estimates from a genome-wide association study (GWAS) conducted in an independent sample. It is a noisy proxy for the individual’s true additive genetic factor for a disease, and its predictive power is bounded above by the trait’s narrow-sense heritability.

Implicit Tax: A measure of adverse selection defined by Hendren (2013) as the percentage by which a consumer with private risk r must overpay relative to her own actuarially fair price if she is pooled with all consumers of equal or higher risk. The minimum implicit tax up to the 80th percentile of risk (t80) serves as the paper’s primary summary statistic; t80 above roughly 43% is associated with market unraveling in prior literature.

SNP Heritability: The share of variance in a disease’s liability attributable to the set of common genetic variants (SNPs) used in heritability estimation. Used in this paper as a conservative lower bound on the predictive power of future PGIs, because future PGIs will additionally capture rarer variants.

Twin Heritability: An estimate of a trait’s narrow-sense (additive) heritability computed by comparing resemblance of monozygotic twins (sharing 100% of their genomes) to dizygotic twins (sharing ~50% on average). Used as an upper bound on future PGI predictive power, since heritability is the theoretical maximum R2 for a PGI.

Standard Risk Class: The set of consumers whose predicted disease risk (based on non-genetic covariates observable to insurers) falls between 0.75 and 1.25 times the population-wide average risk, following standard insurance underwriting practice. Insurers charge the same premium to all consumers in this class; any variation in risk within the class due to private genetic information constitutes the source of adverse selection analyzed in this paper.

Private Risk Function: The probability rho(g, w) of contracting the disease conditional on both the consumer’s observed PGI g and non-genetic factors w. Contrasted with the non-genetic private risk function pi(w), which conditions only on non-genetic covariates. The dispersion of the private risk distribution across consumers in the same risk class determines the degree of adverse selection.

Adverse Selection Death Spiral: The Akerlof (1970) mechanism in which high-risk consumers disproportionately purchase insurance, causing insurers to raise premiums, which deters low-risk consumers, which further raises the average risk of purchasers, ultimately driving equilibrium quantity to zero. The paper’s calibrated equilibrium model finds this outcome under future PGI accuracy for the HRS CAD contract.

Growth Experiences and Trust in Government

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates whether individuals who have experienced stronger GDP growth over their lifetimes are more likely to trust their national government. The authors — Besley, Dann, and Dray — assemble a newly harmonized global dataset comprising approximately 3.3 million respondents across 166 countries since 1990, drawn from 11 major opinion surveys (Afrobarometer, Americasbarometer, Arabarometer, Asiabarometer, European Social Survey, Gallup World Poll, Integrated Values Survey, Latinobarometer, Life in Transition Survey, South Asia Barometer, and World Justice Project). They supplement this with longer-run U.S. evidence from the American National Election Studies (ANES) going back to 1958, covering respondents born as early as the 1880s, and longitudinal Swiss evidence from the Swiss Household Panel (SHP) which allows individual fixed-effects estimation.

The core methodological contribution is the exploitation of country-cohort variation in lifetime GDP growth experiences. Following Malmendier and Nagel (2011), the authors construct a weighted average of past growth realizations across an individual’s lifetime, with weights decaying linearly over time (lambda = 1), so that more recent growth receives greater weight. The baseline specification includes country fixed effects, cohort-by-subcontinent fixed effects, survey-by-survey-year fixed effects, controls for log GDP per capita at year of birth, and individual characteristics (sex, marital status, education, religious denomination). More demanding specifications add country-by-survey-year and country-by-age fixed effects. For Switzerland, individual fixed effects are included, fully absorbing time-invariant personal characteristics.

The main finding is that a one standard deviation increase in lifetime GDP growth experience — corresponding to approximately 2 percentage points of additional growth — is associated with a 2.1 percentage point increase in the probability of trusting the national government, significant at the 1 percent level. This corresponds to roughly 0.042 standard deviations of the trust outcome and approximately 5 percent of the global mean trust in government. The effect is quantitatively meaningful: it approximates between one-quarter and one-half of the difference in average trust between older and younger cohorts in India and Italy, respectively. For the U.S. ANES sample, a one standard deviation increase in growth experience (about 0.2 percentage points) increases trust in the federal government by 2.4 percentage points, explaining more than two-thirds of the average trust gap between Baby Boomers (born 1946–1964) and Millennials (born 1981–1996).

Several scope conditions and heterogeneity findings sharpen the interpretation. First, the growth-trust link is specific to government institutions: there is no statistically significant effect of growth experience on interpersonal trust or trust in religious organizations, indicating the channel runs through perceptions of state performance rather than generalized social capital. Second, a recency heuristic operates: the linearly decaying weighting function (lambda = 1) outperforms both an unweighted lifetime average (lambda = 0) and a formative-years weighting. Growth experienced during formative years (ages 18–25) or before birth has no detectable effect on trust in government; the pre-birth result serves as a placebo test. Third, the positive growth-trust relationship is stronger in democracies than in autocracies, which the authors interpret as democracies producing citizens more responsive to government performance signals. Fourth, a “trust paradox” emerges: unconditionally, average trust in government is lower in democracies than in autocracies, and longer democratic experience is associated with lower trust, which the authors attribute to democratic institutions generating greater citizen skepticism about government performance. Fifth, core results are robust to controlling for other lifetime politico-economic experiences including inflation, banking and currency crises, epidemics, political unrest, executive turnover, stock market returns, and income inequality. The Swiss evidence further shows that private income growth experience does not drive the result — only aggregate macroeconomic growth does.

Q: What is the paper’s core quantitative finding on the growth-trust relationship? A: Using the global harmonized dataset of 3.3 million respondents across 166 countries, a one standard deviation increase in lifetime GDP growth experience (corresponding to approximately 2 percentage points of additional growth) is associated with a 2.1 percentage point increase in the probability of trusting the national government, significant at the 1 percent level. Using only the Gallup World Poll subsample (roughly half the observations), the estimated effect is somewhat larger at 3.6 percentage points per standard deviation increase. These estimates remain statistically significant under more demanding specifications with country-by-survey-year and country-by-age fixed effects, though the magnitudes decrease as these interacted fixed effects absorb variation in recent growth experiences.

Q: How do the authors measure individual lifetime growth experience? A: The growth experience variable is a weighted average of all past annual GDP per capita growth rates since an individual’s birth, with weights that decay linearly over time (lambda = 1 in the Malmendier-Nagel framework). Under this parameterization, the measure simplifies to how much recent economic performance (in the year prior to the survey) exceeds the long-run mean over the respondent’s lifetime, scaled by the respondent’s midpoint of life. This implies younger individuals are more sensitive to recent growth outcomes because their shorter life histories give recent events relatively greater weight. The authors validate this lambda = 1 choice via a grid search over alternative weighting structures using minimum residual sum of squares as the criterion.

Q: How is reverse causality addressed? A: The empirical strategy identifies the relationship using past, cumulative growth experiences measured prior to the survey, so current trust in government cannot cause past growth. Survey-year fixed effects absorb all aggregate time trends simultaneously affecting trust and growth. The authors also conduct a placebo test showing that GDP growth occurring before an individual’s birth has a precisely estimated null effect on their trust in government, which would not be the case if unobserved societal trends were jointly driving both growth histories and political perceptions.

Q: Does growth experience affect interpersonal trust or trust in non-state institutions? A: No. The estimated coefficient on lifetime growth experience is statistically insignificant at conventional levels when interpersonal trust replaces trust in government as the dependent variable, with narrow confidence intervals indicating a precisely estimated null. Similarly, growth experience has no systematic effect on trust in religious organizations such as churches or mosques. The authors interpret these null results as evidence against the alternative explanation that broad modernizing social changes are jointly driving both growth experiences and political trust.

Q: What do the U.S. ANES results add? A: The ANES data, which extends back to 1958 and captures cohorts born as early as the 1880s, provide a within-country test controlling for state fixed effects, generation dummies, and rich individual characteristics including partisan affiliation and partisan strength. A one standard deviation increase in U.S. growth experience (approximately 0.2 percentage points) raises trust in the federal government by 2.4 percentage points, significant at the 1 percent level. This estimate is quantitatively large enough to explain more than two-thirds of the average trust gap between Baby Boomers and Millennials. Results are robust to adding state-by-survey-year fixed effects and birth-state-by-generation fixed effects, and hold for a broader “trust in government index” covering beliefs about waste, corruption, and responsiveness of the federal government.

Q: What do the Swiss Household Panel results contribute? A: The SHP allows individual fixed-effects estimation, exploiting within-person changes in growth experience and trust over time from 1999 onward, which absorbs all time-invariant individual characteristics that could confound the global and U.S. cross-cohort results. The growth experience coefficient remains positive and significant, with a one standard deviation increase yielding a 1.9 percentage point increase in trust in the Swiss federal government (significant at the 1 percent level). The Swiss data also uniquely allow the authors to test whether personal income growth experience drives the result; they find no significant effect of private income growth experience on trust in government, only aggregate macroeconomic growth matters.

Q: Does the recency heuristic hold — does growth in formative years matter? A: No. The authors find no detectable effect of growth experienced specifically during formative years (ages 18–25) on trust in government. Additionally, in a grid-search exercise assessing model fit across different lambda values, the linearly decaying weighting scheme (lambda = 1, giving more weight to recent growth) outperforms both equal-weighted lifetime averages (lambda = 0) and weighting schemes that emphasize earlier life experiences (lambda less than 0). The pre-birth placebo result (null effect) and the absence of a formative-years effect together indicate that the operative mechanism is about evaluating current government performance based on recent macroeconomic experience, not the imprinting of long-lasting political dispositions during youth.

Q: What is the “trust paradox” and how is it documented? A: The trust paradox refers to the empirical finding that average trust in government is lower in democracies than in autocracies at the cross-country level, and that longer experience with democratic institutions within countries is associated with lower levels of trust in government in the micro data. This is counterintuitive given the standard view that good institutions should foster confidence in government. The authors suggest the paradox likely reflects democracies cultivating greater citizen skepticism and more critical judgment of government performance, rather than indicating that democratic governance actually performs worse. Importantly, the positive effect of growth experience on trust remains present in democracies, and the growth-trust relationship is actually stronger in democratic regimes, consistent with citizens in democracies being more responsive to government performance signals.

Q: How is the growth-trust finding related to corruption perceptions and living standards? A: Using the Gallup World Poll, the authors find that stronger lifetime growth experience is associated with lower perceived corruption in government, greater satisfaction with personal living standards, and higher likelihood of feeling one lives comfortably on one’s present income. These results are consistent with citizens attributing economic success to government competence and integrity, and with growth translating into perceptions of improved personal circumstances through both direct income effects and indirect public goods provision.

Q: Are the results robust to controlling for other lifetime politico-economic experiences? A: Yes. When the authors include lifetime experience measures for political unrest, executive turnover, epidemic exposure, banking crises, currency crises, and inflation (both levels and volatility) simultaneously in equation (3), the growth experience coefficient remains consistently positive, stable, and significant across all specifications. Among the other experience variables, only lifetime unrest and epidemic exposure are independently negative and statistically significant at conventional levels. F-tests reject the null hypothesis that the crisis and growth experience coefficients are equal in magnitude. The U.S. results are also robust to adding lifetime experiences with S&P 500 returns, unemployment, and top-income-share inequality measures.

Q: What are the policy implications of the findings? A: The authors note that sustained economic growth may itself be a mechanism for building political trust, with positive downstream effects for policy compliance — a connection they document has been relevant during the COVID-19 pandemic (where higher-trust societies showed lower mobility during lockdowns and higher vaccine acceptance). The growth-trust channel could have implications for increasing compliance across a range of policy domains including climate action and tax morale. Governments that deliver sustained economic growth can expect citizens to update their trust upward, particularly in democracies where citizens are more performance-responsive, while governments that preside over stagnation or contraction face predictable erosion of political legitimacy across cohorts.

Growth experience: A weighted average of all past annual GDP per capita growth realizations since an individual’s birth, with weights that decay linearly over time following Malmendier and Nagel (2011), so that more recent growth receives greater weight. Under the paper’s preferred parameterization (lambda = 1), the measure equals how much last year’s GDP per capita exceeds the respondent’s lifetime mean, scaled by the respondent’s midpoint of life.

Trust in government: A binary dummy variable equal to one if a survey respondent expresses “a great deal” or “quite a lot” of trust or confidence in the national government, constructed from harmonized responses across 11 major opinion surveys. The paper treats this as reflecting respondents’ perceptions of government performance rather than a deep interpersonal trust relationship.

Trust paradox: The empirical regularity documented in the paper whereby average trust in government is unconditionally lower in democracies than in autocracies at the cross-country level, and whereby longer democratic experience within countries is associated with lower individual trust in government. The authors attribute this to democratic institutions generating more critical citizen judgment of government performance.

Recency heuristic: The finding that more recent growth experiences carry greater weight in forming trust in government, as captured by the linear decay weighting scheme (lambda = 1) outperforming equal-weighted or early-life-weighted alternatives. Growth before birth and growth during formative years (ages 18–25) have no detectable effect, while recent macroeconomic performance is the operative signal.

Cohort-level variation: The within-country differences in lifetime growth experiences across birth cohorts that form the paper’s primary identification strategy. Because different cohorts in the same country have lived through different sequences of growth episodes, differences in trust across cohorts within a country can be attributed to differential growth exposure rather than time-invariant country characteristics.

Formative years effect: The hypothesis, tested and rejected in the paper, that economic experiences during ages 18–25 have a lasting imprint on political attitudes analogous to formative-years effects found in other political behavior literatures. The paper finds no statistically significant association between growth experienced during these years and trust in government.

Source text origin: In the pipeline context relevant to this paper’s acquisition, this refers to whether a summary was generated from full working paper text (“pdf” or “oa-html”) versus abstract only (which is hard-blocked). The working paper was obtained from LSE Research Online (eprint 129614), classified as published version under CC BY 4.0.

Heterogeneity and the Macro-Economic Effects of Changes in Loan-to-Value Limits

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

De Veirman and de Jong develop a new approach to estimating the macroeconomic effects of changes in regulatory loan-to-value (LTV) limits on mortgage loans. The central questions are: (1) how do changes in an LTV cap translate into changes in the average LTV and, through that channel, into house prices and real output; and (2) how do heterogeneity in the cross-sectional LTV distribution, non-linearity, and asymmetry shape those effects?

Motivation and Gap

Prior empirical literature on macroprudential LTV policy typically pools across countries using coded indicator variables, which imposes the restriction that all LTV policy actions have the same effect regardless of the size of the change or the position of the limit relative to the distribution. Standard TANK models with homogeneous borrowers imply either full symmetry or threshold asymmetry precisely at the point where the constraint ceases to bind. The authors are the first to relate borrower heterogeneity to non-linearity and asymmetry in LTV policy effects.

Data and Setting

The empirical application focuses on the Netherlands, which introduced an LTV cap of 106 percent on August 1, 2011, subsequently reduced in annual one-percentage-point steps to 100 percent by January 2018. Cross-sectional LTV distributions are constructed from the De Nederlandsche Bank Loan Level Data (LLD), covering 77-81 percent of outstanding Dutch mortgage debt in 2012Q4-2014Q4, restricted to borrowers aged 35 or younger as a proxy for first-time buyers. A survey-based average LTV series spanning 1979-2015 was fielded in January 2016 across the CentERpanel and LISS panel (7,943 respondents combined; 2,238 usable observations after cleaning), measuring LTV at the time of first home purchase. This survey-based annual LTV series, together with the log relative house price, log real GDP, and the real mortgage rate, forms a four-variable Vector Error Correction Model (VECM) estimated over 1981-2015, with a single cointegrating vector identified by Johansen maximum likelihood.

Methodology

The authors’ core innovation is to translate changes in the LTV cap into changes in the cross-sectional average LTV by applying each successive cap level to the underlying distribution: observations above the cap are moved to the cap value (with adjustments for exceptions in the ex post variant). These implied annual changes in the average LTV serve as a succession of impulses fed into the VECM. Two variants are implemented: an ex ante approach using only the pre-cap 2010M8-2011M7 distribution, and an ex post approach that uses the most recent empirical distribution prior to each cap change. The Cholesky identification ordering is [LTV, house prices, GDP, mortgage rate].

Main Findings with Quantitative Magnitudes

Non-trivial macroeconomic effects of Dutch LTV policy: Under the ex post approach (the preferred estimate), the imposition of the cap at 106 percent in 2011 and its gradual reduction to 100 percent by 2018 imply, twenty years after the first shock, that relative house prices are 4.84 percent lower and real GDP is 1.15 percent lower than they would have been in the absence of the cap sequence. The bulk of these responses materializes within ten years, at 4.18 percent and 1.05 percent respectively.
Non-linearity: For a given underlying distribution, changes in the cap have progressively larger effects as the cap tightens. In the ex ante approach, the fraction of households constrained by the cap rises from approximately 20 percent at a limit of 105 percent to approximately 40 percent at a limit of 100 percent. A 10 percentage point tightening from 110 to 100 percent implies a long-run relative house price response of 6.12 percent, while a tightening from 100 to 90 percent implies a response of 14.27 percent — a pronounced non-linearity traceable to the substantial mass of observations in the 90-110 range of the Dutch distribution.
Heterogeneity matters substantially: In mean-preserving comparisons using Pearson-family approximations to the pre-cap Dutch distribution, the macroeconomic effects of the actual Dutch LTV policy sequence are 2.58 times larger in the high standard deviation case (standard deviation 25 percent above the Dutch baseline of 17.09) than in the low standard deviation case (standard deviation 25 percent below). Specifically, twenty-year house price responses are 12.34 percent (high SD) versus 4.79 percent (low SD), and GDP responses are 2.93 percent versus 1.14 percent.
Asymmetry is conditional on the position of the cap relative to the distribution: For the Dutch distribution, symmetry is a good approximation for LTV limits at around 80 percent or lower, where the cap is binding for the bulk of households. Asymmetry is pronounced for higher levels. At an initial cap of 100 percent, the absolute effect of a ten-percentage-point tightening is 2.33 times that of a ten-percentage-point loosening. At 80 percent, the asymmetry ratio is only 1.17. Tightenings have smaller effects when they start from a point where few households are constrained; conversely, loosenings can have larger effects when starting from a point where many are constrained.
Homogeneity assumption understates effects above the mean LTV: Under the homogeneous-borrower benchmark (all borrowers at the Dutch mean of 93.72 percent), asymmetry is infinite at cap levels of 100 and 95 percent but zero at other levels — a feature that causes effects to be entirely absent for caps above the mean. In the heterogeneous Dutch setting, an increase in the LTV limit from 95 to 105 percent raises house prices by 10.72 percent in the long run; the homogeneous case implies no effect at all.

Scope Conditions and Caveats

The paper does not address welfare or financial stability effects. The VECM impulse responses do not establish economic causality. Anticipation effects — if households front-loaded high-LTV purchases before the cap — would cause the procedure to overstate the effect. The LTI robustness check (which smooths the loan-to-income ratio due to noisy survey responses) yields twenty-year responses of 3.32 percent (house prices) and 0.74 percent (GDP), somewhat lower than the baseline, indicating that not controlling for LTI tends to overstate the LTV-macroeconomy connection. The approach requires a usable pre-cap or recent-prior LTV distribution; it is not directly portable to settings where a loosening is studied and no recent pre-cap distribution is available.

In depth

Q1. What is the fundamental identification challenge this paper faces, and how does the proposed approach address it?

A: The standard challenge is that LTV caps are changed infrequently and have no long time series suitable for regression, so panel studies typically pool countries and use coded dummy variables that impose size-independence of effects. The authors bypass this by using the cross-sectional LTV distribution itself: they measure how each cap level would truncate the underlying distribution and track the implied change in the cross-sectional mean LTV, which is then fed as a shock into a time-series VECM. This approach does not require the cap to have been in place previously, imposes no cross-country coefficient restrictions, and explicitly accounts for the size of the policy change.

Q2. What are the ex ante and ex post approaches to translating cap changes into average LTV changes, and how do their cumulative estimates differ?

A: The ex ante approach applies all successive cap levels to the single pre-cap distribution of 2010M8-2011M7 (after correcting for the June 2011 sales-tax reduction from 6 to 2 percent), without allowing for exceptions. The ex post approach uses the most recent empirical distribution prior to each cap change and accounts for the observed share of borrowers above the cap as exceptions. The ex ante approach yields a cumulative decline in the average LTV of 3.08 percentage points over 2011-2018; the ex post approach yields 1.96 percentage points, roughly one percentage point less. The difference is largely concentrated in 2011-2012 and stems from the ex ante approach not accounting for exceptions to the cap.

Q3. How does the paper correct for the coincident 2011 sales-tax reduction, and why does this matter?

A: In June 2011, the Dutch sales tax on housing purchases fell from 6 to 2 percent, approximately coinciding with the August 2011 imposition of the LTV cap. Without correction, the observed drop in high LTVs in the 106-cap period would conflate the two policy changes. The authors apply a tiered correction: LTVs at or below 100 percent are left unchanged (the data show no notable change in that range); LTVs between 100 and 110 percent are reduced proportionally to the share of total closing costs attributable to the tax; LTVs at or above 110 percent are reduced by the full magnitude of the tax decline. This yields the “tax-adjusted pre-cap distribution” with a mean of 93.72 percent, down from 94.46 percent in the unadjusted data.

Q4. Why does the fraction of constrained households matter so much, and how does it drive non-linearity?

A: The key mechanism is that the average LTV changes when and only when the cap binds for a given borrower. The larger the share of borrowers whose LTV (in the counterfactual uncapped distribution) would exceed the cap, the larger the share of individual LTVs that move in lockstep with any change in the cap, and therefore the larger the aggregate average LTV response and, through the VECM, the house price and GDP response. As the Dutch cap tightened from 105 to 100 percent, the constrained fraction rose from roughly 20 percent to roughly 40 percent, and the annual implied decline in the average LTV grew from 22 basis points to 42 basis points — illustrating monotonically increasing non-linearity within the ex ante approach.

Q5. How does the survey design address the risk of selection bias relative to alternative data sources such as the American Housing Survey?

A: The survey, fielded in January 2016 across both the CentERpanel and LISS panel, asks retrospectively about respondents’ first home purchase, irrespective of whether they still reside there. This avoids the selection bias in the American Housing Survey, where the first-time-buyer flag captures only those still living in the first home — disproportionately selecting homes that are traded less frequently. A single-wave design also avoids the methodological discontinuities that arise from combining multiple survey waves. The resulting series covers 2,238 observations over 1979-2015 (average 60.49 per year).

Q6. What does the VECM cointegration evidence suggest about the long-run relationship between LTV, house prices, GDP, and the real mortgage rate?

A: Augmented Dickey-Fuller tests do not reject a unit root in any of the four series in levels, while all four are stationary in first differences (with the borderline case of log relative house price inflation when an intercept is included). Both the Johansen L-Max and Trace tests reject no cointegration at the 1 percent level, and neither test indicates more than one cointegrating vector. The authors therefore estimate a single-cointegrating-vector VECM with one lag (selected by the Schwarz Information Criterion) over 1981-2015. The long-run relation is normalized so that the coefficient on the log relative house price is one.

Q7. What do the impulse responses in the baseline VECM specification imply for the long-run macro effects of Dutch LTV policy?

A: Under the preferred ex post approach, twenty years after the first shock in 2011 the VECM implies that relative house prices are 4.84 percent lower and real GDP is 1.15 percent lower than the no-cap counterfactual. The bulk of the response materializes within ten years, with house prices 4.18 percent lower and GDP 1.05 percent lower at the ten-year horizon. The twenty-year real mortgage rate response is positive but negligibly small. When the ex ante approach is used instead, responses are larger owing to the larger cumulative LTV impulse.

Q8. How does the paper conduct the mean-preserving heterogeneity exercise, and what are the key quantitative results?

A: The authors generate Pearson-family distributions that match the first four moments of the Dutch pre-cap distribution (mean 93.72, standard deviation 17.09, skewness -1.16, kurtosis 5.97 under the convention that a normal has kurtosis 3), truncated to support (0, 200]. Two alternative distributions are constructed with standard deviations 25 percent below (12.97) and 25 percent above (21.61) the Pearson proxy, holding mean, skewness, and kurtosis constant. The same VECM and Cholesky ordering are applied. Twenty-year house price responses are 12.34 percent (high SD), 8.46 percent (Pearson proxy), and 4.79 percent (low SD). Twenty-year GDP responses are 2.93, 2.01, and 1.14 percent respectively. The ratio of high-to-low-SD responses is 2.58 for both variables.

Q9. How does asymmetry vary across different initial levels of the LTV cap for the Dutch distribution, and what is the intuition?

A: At a starting cap of 100 percent, a ten-percentage-point tightening produces a long-run house price response 2.33 times larger (in absolute value) than a ten-percentage-point easing from the same starting point. At 80 percent the asymmetry ratio falls to 1.17, meaning the effects of tightening and easing are nearly symmetric. The intuition is that at 80 percent the cap is binding for the bulk of the distribution, so both tightenings and easings move a similarly large fraction of borrowers and have large, roughly comparable effects. At 100 percent, far fewer borrowers are currently constrained, so an easing from 100 to 110 moves almost no one whereas a tightening from 100 to 90 moves substantially more.

Q10. What does the comparison of the heterogeneous-borrower and homogeneous-borrower cases reveal about the implications for TANK and HANK models?

A: Under the homogeneous benchmark — all borrowers at the mean Dutch LTV of 93.72 percent — changes in the cap produce infinite asymmetry at cap levels of 100 and 95 percent (tightening has a full effect, easing has zero effect) but zero asymmetry and zero effect for any cap level above 95 percent. For example, an increase in the cap from 95 to 105 percent has no effect in the homogeneous case but raises house prices by 10.72 percent in the heterogeneous case. In sum, homogeneous-borrower models — including TANK frameworks and linearized models with always-binding constraints such as Iacoviello (2005) — overstate asymmetry in a narrow range around the mean LTV and simultaneously understate the effects of cap changes above the mean LTV. The results are more consistent with heterogeneous-agent frameworks, though the authors note they are not aware of any existing HANK paper that investigates asymmetry and non-linearity specifically in response to changes in the borrowing limit.

Q11. What do the robustness checks show about sensitivity of results to LTV measurement choices?

A: The results are robust to all alternative Cholesky orderings, to using the real mortgage rate computed as the nominal rate minus current (rather than two-year moving average) inflation, to using the computed LTV without cross-checking, and to using the directly reported LTV after cross-checking. The most notable alternative is the directly reported LTV without cross-checking, which yields a twenty-year house price response of 3.81 percent and a GDP response of 0.72 percent (ex post approach), somewhat lower than the baseline of 4.84 and 1.15 percent but in the same direction. A further robustness check using an LTV series that extrapolates 2011-2015 values from the Loan Level Data yields larger estimates (cumulative twenty-year house price response of 6.65 percent and GDP response of 1.40 percent), reflecting the LLD series’ more moderate drop in 2014.

Q12. What is the policy implication regarding the importance of distributional information for gauging LTV policy effects?

A: The results imply that knowing the mean of the LTV distribution is not sufficient for estimating the effects of cap changes: the variance — and specifically the fraction of borrowers constrained by the cap — is critical. This is analogous in spirit to the finding of Krueger, Mitman, and Perri (2016) that matching the tails of the wealth distribution, and not just the mean, is essential for determining the aggregate consumption effects of shocks. Existing empirical literature that focuses on the first moment of the LTV distribution will therefore systematically mismeasure the macro effects of LTV limits, and the direction of the bias depends on where the cap stands relative to the distribution.

Key Concepts

Loan-to-value (LTV) cap / limit: The regulatory maximum on the ratio of total mortgage loan amount to the purchase price of the property (excluding buyer-incurred closing costs such as sales taxes and notary fees). In the Netherlands, this was set at 106 percent from August 2011 and reduced annually by one percentage point to 100 percent by January 2018. The paper explicitly distinguishes the cap (the regulatory threshold) from the average LTV (the cross-sectional mean of the distribution, which the cap may or may not bind for all borrowers).

Underlying (or pre-cap) LTV distribution: The cross-sectional distribution of LTV ratios that would prevail in the absence of any LTV cap — approximated in the paper by the empirical distribution in the twelve months before the cap was introduced (2010M8-2011M7, adjusted for the June 2011 sales-tax cut). The shape, mean, and variance of this distribution determine the fraction of borrowers who are constrained by any given cap level and therefore govern the magnitude and symmetry of policy effects.

Mean-preserving change in heterogeneity: A change in the standard deviation of the LTV distribution that holds the mean (and, in the paper’s stylized scenarios, also the skewness and kurtosis) constant. The paper uses this construct to isolate the effect of dispersion per se on the macroeconomic consequences of cap changes, showing that a 25 percent increase in the standard deviation relative to the Dutch baseline more than doubles the macro effects relative to a 25 percent decrease.

Ex ante approach: The method of translating cap changes into average LTV changes that uses only the pre-cap distribution, applying successive cap levels to that single distribution. It does not require an LTV cap to have been in place and is therefore applicable for prospective analysis. It does not account for exceptions to the cap.

Ex post approach: The method that uses the most recent empirical LTV distribution preceding each cap change as the proxy for the counterfactual uncapped distribution, and that explicitly accounts for the observed share of borrowers above the cap (treated as exceptions). Preferred by the authors when feasible because it incorporates information about how the underlying distribution has evolved for reasons unrelated to the current cap change.

Asymmetry ratio: The ratio of the absolute value of the long-run house price (or GDP) response to a ten-percentage-point tightening in the cap to the absolute value of the response to a ten-percentage-point easing from the same initial cap level. A ratio exceeding one indicates that tightenings have larger effects than easings of equal magnitude from the same starting point. In the paper, this ratio is shown to depend critically on where the initial cap sits relative to the underlying distribution.

Non-linearity in LTV effects: The property that changes in the cap from a lower starting point have larger macroeconomic effects than changes from a higher starting point, for a given underlying distribution. This arises because the fraction of constrained borrowers increases as the cap is tightened, so a further tightening moves a larger share of individual LTVs. In the paper, this is documented through the increasing year-on-year effects in Table 1 and the large difference between the house price response to a tightening from 110 to 100 percent (6.12 percent) versus from 100 to 90 percent (14.27 percent).

Pearson system (as used in this paper): A parametric family of distributions in which every combination of the first four moments (mean, variance, skewness, kurtosis) corresponds to a unique distribution. The authors use it to construct smooth approximations to the empirical Dutch distribution with the same mean, skewness, and kurtosis but varying standard deviations, enabling a controlled comparison of heterogeneity scenarios.

Heterogeneous innovations and growth under imperfect technology spillovers

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. Jo and Kim ask two related questions: (1) How do firms use different types of innovation when learning others’ technology takes time? (2) How does this process alter the aggregate implications of firm innovation, particularly in the context of increasing competition?

Model. The paper develops a discrete-time infinite-horizon endogenous growth model with multi-product firms pursuing two types of innovation — “own-innovation” (improving existing product quality) and “creative destruction” (entering new product markets by displacing incumbents) — subject to a novel friction called “imperfect technology spillovers.” The friction takes the specific form of lagged learning: creative destruction builds on the one-period-lagged technology of the target market’s incumbent, while only the incumbent can observe the current frontier technology level. This one-period lag creates a technology gap (Δ = q_t / q_{t−1}) between the incumbent’s frontier and the level available to rivals. Four possible technology gap values arise in equilibrium: Δ₁ = 1 (no gap), Δ₂ = λ (one successful own-innovation), Δ₃ = η (one successful creative destruction), and Δ₄ = η/λ. The step sizes satisfy λ² > η > λ, meaning a single creative destruction improves quality more than a single own-innovation, but two consecutive own-innovations dominate a single creative destruction.

Key Mechanisms. The learning friction generates two novel mechanisms. First, the “market-protection effect”: incumbents with a technology advantage (Δ > 1) intensify own-innovation to widen the gap and protect their product lines when competitive pressure rises. Formally, own-innovation probability is highest for Δ₂ products and declines monotonically (z₂ > z₃ > z₄ > z₁), and ∂z₂/∂x > ∂z₃/∂x > 0 while ∂z₁/∂x < 0, conditional on value coefficients. Second, the “technological barrier effect”: higher overall own-innovation and creative destruction intensity widens the average technology gap across products, reducing rivals’ conditional probability of successfully taking over a product market. This is distinct from the standard Schumpeterian effect (lower expected future profits) and from the escape-competition effect in step-by-step models (which apply only to neck-and-neck, single-product firms).

Data and Empirical Strategy. The empirical analysis combines the USPTO PatentsView database, the Longitudinal Business Database (LBD), the Longitudinal Firm Trade Transactions Database (LFTTD), the Census of Manufactures (CMF), Compustat, and NBER-CES data, covering the universe of U.S. patenting firms from 1976 to 2016, with main analyses from 1982 to 2007. Own-innovation is proxied by the self-citation ratio of patents (the ratio of self-citations to total backward citations); creative destruction by new products added and low-self-citation patents. Exogenous competitive pressure comes from China’s WTO accession in 2001, instrumented by the industry-level NTR tariff gap (the gap between non-NTR and NTR rates in 1999) following Pierce and Schott (2016).

Empirical Findings. Pre-shock (1982–1999): patents with lower self-citation ratios (closer to creative destruction) have significantly longer backward citation gaps (coefficient −2.29 to −2.59, p < 0.01 across specifications), confirming that learning others’ technology takes more time. Creative-destruction-type patents also have higher market value (Kogan et al. stock return measure) and scientific value (forward citations), with self-citation ratio negatively associated with both (e.g., coefficient on self-citation for market value: −0.289 without firm FE; −0.110 with firm FE, p < 0.01). Conditional on patenting, higher self-citation ratios are negatively associated with employment growth (coefficient −0.256, p < 0.05), number of industries added (−0.158, p < 0.05), and products added (−0.274, p < 0.01).

Post-shock (DID): foreign competition had no statistically significant effect on overall patent counts, but firms with above-average innovation intensity in industries with high NTR gaps significantly increased their self-citation ratio — indicating a shift toward own-innovation. The triple-interaction coefficient is 0.795 (p < 0.01) with baseline controls. For a firm with average lagged innovation intensity (0.18) in an industry with an average NTR gap (0.291), this corresponds to a 4.2 percentage point increase in the seven-year growth rate of the self-citation ratio, representing a 15.0% increase relative to the average growth rate of 28.2 percentage points. Consistent with the technological barrier effect, firm entry rates are lower in industries with higher TFPR-skewness-based technological barriers (coefficient −0.012 to −0.016, p < 0.05).

Quantitative Analysis. Calibrated to the U.S. manufacturing sector in 1992, the model matches six target moments including average number of products (2.3), products added (0.3), firm entry rate (7.6%), average productivity growth (1.9%), high-growth-firm employment growth (22.5%), and import penetration (15.3%). Creative destruction contributes approximately 1.88 times more to growth per unit than own-innovation (step size ratio 0.075/0.04). The aggregate R&D-to-sales ratio (untargeted) is 4.6% in the model vs. 4.1% in data.

A counterfactual increasing outside entrants by 83% (matching the rise in import penetration from 15.3% to 25.1% between 1992 and 2007) generates a 1.51% increase in aggregate creative destruction arrival rate x, but firm-level creative destruction probability falls 1.33% and startup creative destruction also falls 1.33%. The aggregate R&D-to-sales ratio falls 1.6% and creative destruction R&D intensity falls 1.2%. Average domestic productivity growth declines 11.0%, with growth from creative destruction falling 13.0% and growth from domestic startups falling 1.7%. The total mass of domestic firms falls 6.4%.

In economies with creative destruction costs 80 times higher than the U.S. baseline, the same competitive pressure shock raises rather than lowers total R&D (by 1.0%), but domestic growth still falls 9.7%, because the marginal decline in creative destruction impedes the growth contribution and firm entry even when aggregate innovation spending rises.

In depth

Q1. What is the key friction that distinguishes this model from the existing multi-product firm literature (e.g., Klette and Kortum 2004; Akcigit and Kerr 2018)?

A: The key friction is “imperfect technology spillovers,” modeled as lagged learning: creative destruction can only build on the one-period-lagged technology of the target product (q_{j,t−1}), while the product’s current owner observes the frontier technology (q_{j,t}). In models without this friction — such as Akcigit and Kerr (2018) — rivals can instantly learn and copy frontier technology, so firms have no technological advantage and cannot protect their markets. In the current model, own-innovation by the incumbent widens the gap between q_{j,t} and q_{j,t−1}, creating a barrier that a rival must overcome even after successful creative destruction. This makes own-innovation an endogenous function of the technology gap, a feature absent from existing multi-product firm frameworks.

Q2. Why does the model predict that own-innovation increases with the technology gap up to a point, then decreases?

A: From Corollary 1, the ordering z₂ > z₃ > z₄ > z₁ reflects competing forces. Products with gap Δ₂ = λ gain the most from additional own-innovation in terms of reducing the probability of losing the product line (equation 2), so own-innovation is highest there. Products with Δ₃ = η or Δ₄ = η/λ already have substantial technological advantages from prior creative destruction, so the marginal value of own-innovation in reducing market loss probability is lower. Products with Δ₁ = 1 have no advantage at all: if a rival succeeds in creative destruction, the incumbent loses the product regardless of own-innovation (equation 1), so z₁ is lowest. Beyond a certain gap level, the incumbent is sufficiently protected that additional own-innovation has diminishing returns in deterrence.

Q3. What is the market-protection effect formally, and for which products is it strongest?

A: The market-protection effect (Corollary 2) is the positive response of a firm’s own-innovation to an increase in the aggregate creative destruction arrival rate x, conditional on the value coefficients A₁ and A₂ being fixed. It is strongest for products with Δ₂ = λ (∂z₂/∂x is the largest and positive), positive but weaker for Δ₃ = η (∂z₃/∂x > 0), of ambiguous sign for Δ₄ = η/λ, and negative for Δ₁ = 1 (∂z₁/∂x < 0). The asymmetry reflects the asymmetric payoff to own-innovation across gap levels: for Δ₂ products, successful own-innovation can turn a losing situation into a winning one because it shifts the technology gap from Δ₁ to Δ₂ from the rival’s perspective, effectively defeating the rival’s creative destruction attempt. This mechanism provides a micro-foundation for why frontier firms (like Google or NVIDIA) keep innovating intensely despite their technological leads, a pattern the standard step-by-step model cannot explain.

Q4. What is the technological barrier effect and how does it differ from the Schumpeterian effect?

A: The technological barrier effect refers to the reduction in rivals’ incentive for creative destruction caused by an increase in the average technology gap across product lines. When incumbents do more own-innovation or when outside firms do more creative destruction, the distribution of technology gaps shifts rightward (density at Δ₁ falls; density at Δ₂, Δ₃, Δ₄ rises). This raises the average technology barrier rivals must overcome to successfully take over a product market, reducing the conditional takeover probability x^{takeover} and the expected value of creative destruction B. In the U.S. counterfactual, the technological barrier effect accounts for 17.0% of the total change in the aggregate creative destruction rate x and 15.0% of the change in startup creative destruction x_e. In contrast, the Schumpeterian effect refers to the reduction in expected future profits from owning a product due to increased displacement risk (through the value coefficient A₂), a mechanism present in standard quality-ladder models. Both operate simultaneously but the technological barrier effect is a novel feature of this framework.

Q5. How is own-innovation vs. creative destruction measured empirically, and what validates this measure?

A: The self-citation ratio (the share of a patent’s backward citations that cite the same assignee’s earlier patents) is used as the primary measure: a higher ratio indicates greater reliance on the firm’s own prior knowledge, hence a higher probability that the innovation improves an existing product line (own-innovation). This is validated empirically in three ways. First, patents with lower self-citation ratios have significantly larger backward citation gaps (coefficient −2.29 to −2.59 across fixed-effect specifications on 728,721 observations), consistent with creative destruction requiring more time to learn others’ technology. Second, lower self-citation patents have higher market value and scientific value (forward citations), consistent with η > λ (creative destruction contributes more per event to quality). Third, firm-level regressions show that lower self-citation ratios are associated with higher employment growth, more products added, and more industries entered, consistent with creative destruction contributing more to firm expansion.

Q6. How does the DID identification strategy work, and what are the main results?

A: The identification exploits the removal of trade policy uncertainty (TPU) after China’s WTO accession in 2001. The treatment variable is the industry-level NTR gap (the gap between non-NTR and NTR tariff rates in 1999): industries with larger gaps experienced a larger reduction in uncertainty and thus a greater increase in Chinese import competition. The DID compares patenting firms across periods (1992–1999 vs. 2000–2007) and across high- vs. low-NTR-gap industries, with a triple interaction for firm-level innovation intensity (lagged five-year average patents per employee, normalized within two-digit NAICS). The main finding (Table 4): the NTR gap × Post interaction has no significant effect on overall patent counts (coefficient 0.238 without controls, standard error 0.237), but the triple interaction (NTR gap × Post × innovation intensity) has a positive and significant effect on the growth rate of the self-citation ratio (0.732 without controls, p < 0.05; 0.795 with baseline controls, p < 0.01). This implies that innovation-intensive firms in high-competition industries shifted their composition toward own-innovation, while overall patenting was unchanged — consistent with an offsetting rise in own-innovation and fall in creative destruction.

Q7. What are the aggregate growth effects of increasing competitive pressure in the calibrated model?

A: Using an 83% increase in outside entrants (matching the 1992–2007 rise in import penetration from 15.3% to 25.1%), average domestic productivity growth falls 11.0%. Decomposing: growth from domestic own-innovation falls 11.4%, growth from domestic creative destruction falls 13.0%, and growth from domestic startups falls 1.7% (Table 9). The aggregate R&D-to-sales ratio falls 1.6% and the creative destruction R&D intensity falls 1.2%, indicating that the decline in creative destruction R&D outweighs the rise in own-innovation R&D. The total mass of domestic firms falls 6.4% and the average number of products per firm falls 5.5%.

Q8. How do results differ in economies with high creative destruction costs vs. the U.S.?

A: When creative destruction costs (χ̃) are set 80 times higher than the U.S. baseline, the initial equilibrium has much lower creative destruction: R&D-to-sales ratio is 1.39% (vs. 4.58% in U.S.), creative destruction R&D intensity is 8.6% (vs. 63.9%), average number of products is 1.0 (vs. 2.3), and average domestic productivity growth is 1.4% (vs. 1.9%). Under the same competition shock, total R&D actually rises by 1.0% in this high-CD-cost economy (because own-innovation increases more than creative destruction falls, given the already low baseline of creative destruction), in contrast to the −1.6% in the U.S. However, domestic growth still falls 9.7% even in this economy, driven by reductions in creative destruction by incumbents and startups combined with a decline in the mass of domestic incumbents. This result holds even with a fixed firm mass (Table E5), confirming the mechanism is not solely due to entry/exit dynamics.

Q9. What is the technological barrier effect’s quantitative contribution to the decline in creative destruction?

A: In the U.S. counterfactual (Table 8 and associated decomposition), 17.0% of the total change in the aggregate creative destruction arrival rate x and 15.0% of the total change in startup creative destruction x_e are attributable specifically to the technological barrier effect — that is, to the shift in the technology gap distribution µ(Δℓ) holding all else equal. The conditional takeover probability x^{takeover} declines from 73.2% to 73.0%. The density at Δ₁ (the easiest gap to overcome) falls 0.4%, while densities at Δ₃ and Δ₄ rise 1.1% and 1.4% respectively, driven by increased creative destruction by outside firms and intensified own-innovation by incumbents.

Q10. What are the policy implications the paper draws from its framework?

A: The paper argues that policies evaluating innovation should account for composition, not just aggregate R&D levels or patent counts. Increased overall innovation driven by defensive own-innovation contributes less to economic growth than creative destruction and restricts firm entry — so it is less beneficial than it appears. In low-creativity economies (e.g., European economies with high regulatory barriers to creative destruction), increased foreign competition may raise aggregate R&D while still lowering domestic growth, misleading policymakers who track only total innovation spending. The model also suggests that the mixed empirical findings in the competition-innovation literature (Aghion et al. 2005; Bloom et al. 2016; Autor et al. 2020) can be reconciled by accounting for compositional shifts: the net effect of competition on total innovation is ambiguous because it raises own-innovation for technologically advantaged firms while reducing creative destruction for all firms.

Key Concepts

Imperfect Technology Spillovers: The novel friction introduced in this paper, modeled as lagged learning: firms attempting creative destruction can only access the one-period-lagged technology of the target product market (q_{j,t−1}), while the incumbent product owner observes and can improve from the current frontier (q_{j,t}). This asymmetry creates a persistent technological advantage for incumbents and enables strategic defensive innovation.

Own-Innovation: R&D investment by a firm to improve the quality of its existing product lines. Successful own-innovation raises product quality by a step size λ > 1. Own-innovation does not require learning others’ technology and, in the model, constitutes the incumbents’ defensive margin against creative destruction. At the aggregate level, it contributes more to total growth than creative destruction because it succeeds more frequently, but per successful event it contributes less (λ < η).

Creative Destruction: R&D investment enabling a firm to enter a new product market by displacing the incumbent. Successful creative destruction improves the lagged quality of the target product by a step size η > λ, where λ² > η > λ. It requires learning the incumbent’s one-period-lagged technology, takes longer to develop (evidenced empirically by longer backward citation gaps), and contributes more to firm growth and product expansion per event than own-innovation.

Technology Gap (Δ): The ratio of a product’s current-period technology to its previous-period technology (Δ_{j,t} = q_{j,t}/q_{j,t−1}). This gap summarizes the technological advantage the incumbent holds in a product market under imperfect spillovers. Four values are possible in equilibrium: Δ₁ = 1, Δ₂ = λ, Δ₃ = η, Δ₄ = η/λ. The gap determines both the incumbent’s own-innovation incentive and the rival’s probability of successfully completing a product takeover conditional on creative destruction.

Market-Protection Effect: The mechanism by which incumbents with a technological advantage (Δ > 1) increase own-innovation in response to heightened competitive pressure (an increase in the aggregate creative destruction arrival rate x). This effect is maximized for products with Δ₂ = λ and positive but diminishing for Δ₃. It is absent for Δ₁ = 1 products (where own-innovation cannot prevent displacement) and is formally distinct from the escape-competition effect in step-by-step innovation models, which applies only to neck-and-neck single-product firms.

Technological Barrier Effect: The reduction in rivals’ incentive for creative destruction caused by an increase in the average technology gap across the economy’s product lines. When incumbents intensify own-innovation and/or when outside creative destruction increases, the distribution of technology gaps shifts toward higher Δ values, reducing the conditional probability that a rival successfully takes over any given product market. This feedback mechanism endogenously suppresses creative destruction and firm entry beyond what the Schumpeterian effect alone would predict.

Self-Citation Ratio: The share of a patent’s backward citations that cite patents previously owned by the same firm. Used in the paper as a continuous proxy for the likelihood that a patent represents own-innovation vs. creative destruction: a ratio of 1 (100% self-citations) implies 100% probability of own-innovation; a ratio of 0 implies 100% probability of creative destruction. This measure follows Akcigit and Kerr (2018) and is validated in the paper against learning time, quality, and firm growth outcomes.

NTR Gap (Trade Policy Uncertainty Shock): The industry-level difference between non-NTR (column 2) and NTR (column 1) U.S. tariff rates in 1999, used as an instrument for the exogenous increase in Chinese competitive pressure following China’s WTO accession and the U.S. granting of Permanent Normal Trade Relations (PNTR) in 2002. Industries with larger NTR gaps experienced a greater reduction in trade policy uncertainty and thus a larger increase in competitive pressure from foreign firms.

How Do Rising U.S. Interest Rates Affect Emerging and Developing Economies? It Depends

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper examines how the effects of rising U.S. interest rates on emerging market and developing economies (EMDEs) depend on the underlying source of the interest rate increase. Specifically, it asks: what mix of inflation, reaction, and real shocks has driven changes in U.S. interest rates in recent years; how do these different shock types affect EMDE financial markets, capital flows, borrowing costs, and fiscal outcomes; and how do they affect the likelihood of EMDE financial crises?

Motivation and Context

Written in late 2022 against the backdrop of the Federal Reserve’s most aggressive tightening cycle since the 1990s, the paper argues that the standard practice of treating all interest rate increases as equivalent is misleading. Whether rising U.S. rates reflect strengthening growth, rising inflation expectations, or a perceived hawkish shift in the Fed’s reaction function carries very different implications for EMDEs already burdened by post-COVID debt at record highs and scarring from the pandemic.

Methodology

Three distinct empirical approaches are used, chosen to match the data frequency and parsimony requirements of each research question.

A sign-restricted Bayesian VAR model with stochastic volatility is estimated on monthly U.S. data (January 1982 - September 2022) using four variables: 2-year Treasury yield, 10-year Treasury yield, S&P 500 index, and 5-year breakeven inflation expectations. Sign restrictions identify three shocks: (i) real shocks raise both yields, equity prices, and inflation expectations; (ii) inflation shocks raise yields and inflation expectations but lower equity prices; (iii) reaction shocks raise yields but lower both equity prices and inflation expectations.
Panel local projection models (Jorda 2005) are estimated at quarterly frequency for 17-38 EMDEs over 1997Q2-2019Q4, excluding the 2008Q4-2009Q4 global financial crisis and the COVID-19 pandemic. The models link the VAR-identified quarterly shock series (normalized to represent a 25-basis-point move in the 2-year yield) to EMDE financial, real, and fiscal variables, including local-currency bond yields, EMBI+ sovereign spreads, capital flows, real GDP components, CPI inflation, the real effective exchange rate, primary fiscal balance, government revenues, expenditures, gross debt, and debt composition.
A panel logit model with random effects is estimated on annual data for 139 EMDEs over 1985-2018, linking the three shock types to the probability of banking, currency, and sovereign debt crises (as defined by Laeven and Valencia 2020).

Key Findings

Shock decomposition: Real shocks account for the largest share of variance in 2-year U.S. yields over the full sample (39 percent at a 10-month horizon); inflation shocks explain 14 percent and reaction shocks 13 percent. However, since the start of 2022, reaction and inflation shocks together account for approximately three-quarters of the cumulative increase in yields, with real shocks playing a negligible role.

Financial market and macroeconomic spillovers: Conditional on a 25-basis-point shock, reaction shocks produce significantly adverse EMDE outcomes: widening sovereign spreads (EMBI+), declining capital flows, real exchange rate depreciation, and unlike inflation shocks, statistically significant declines in private consumption and fixed investment. Inflation shocks raise domestic EMDE CPI significantly. By contrast, real shocks are associated with declining sovereign spreads, rising capital flows, real exchange rate appreciation, and higher real exports, with other real GDP components unaffected.

Fiscal outcomes: In response to inflation and especially reaction shocks, EMDE governments improve their primary balances almost exclusively through expenditure cuts, consistent with tighter credit availability constraining fiscal space. Real shocks also improve primary balances, but through both revenue gains and expenditure reductions. Government debt declines in response to all three shock types, though the decline is statistically significant only for real shocks.

Debt composition: Reaction shocks shift debt composition toward shorter maturities and foreign-currency instruments (the latter reflecting exchange rate depreciation mechanically raising the local-currency value of foreign-currency debt). Real shocks shift composition toward longer maturities and higher external creditor participation, consistent with improved market access.

Heterogeneity by credit rating: Investment-grade and noninvestment-grade EMDEs show broadly similar responses to reaction shocks, with the exception of statistically larger yield responses for noninvestment-grade economies. The paper notes this finding contrasts with several prior studies that find stronger fundamentals buffer spillovers.

Crisis probabilities: A 25-basis-point increase in 2-year U.S. yields driven by a reaction shock almost doubles the baseline probability of financial crisis in the average EMDE, from 3.5 percent to 6.6 percent. Extrapolating the nonlinear logit relationship to the 114-basis-point reaction-shock-driven increase in 2-year yields that occurred from January through September 2022 implies the probability of financial crisis in the average EMDE rising approximately 36 percentage points, to nearly 40 percent. The paper cautions that no comparable yield episode occurred in the 1985-2018 estimation sample, so this extrapolation carries substantial uncertainty. Inflation shocks are associated with only small, statistically insignificant changes in crisis probability; real shocks reduce the probability of sovereign debt crisis while raising currency crisis probability by less than reaction shocks do.

Historical episode analysis: The 2013 taper tantrum was dominated by reaction shocks, causing 10-year yields to rise by approximately 100 basis points; sovereign spreads widened by 60 basis points in the May-June 2013 window and capital flows dropped sharply. The 2022 tightening episode was driven by reaction and inflation shocks (reaction shocks adding 114 basis points to 2-year yields through September 2022), with five-year breakeven inflation expectations breaching 3 percent for the first time in the two-decade history of the series. The 2004-2006 build-up to the global financial crisis involved a mix of all three shock types with real shocks prominent, and EMDE financial conditions remained broadly benign.

In depth

Q1. How are the three shock types identified, and what makes this identification strategy credible?

The identification uses sign restrictions imposed on a Bayesian VAR with stochastic volatility. A real shock is identified as one that simultaneously raises 2-year yields, 10-year yields, S&P 500 equity prices, and inflation expectations. An inflation shock raises all yields and inflation expectations but lowers equity prices the equity decline signals that higher rates are not accompanied by stronger growth prospects. A reaction shock raises all yields but lowers both equity prices and inflation expectations the fall in inflation expectations distinguishes it from an inflation shock and signals that markets perceive the Fed is tightening beyond what current inflation warrants. Covering both short- and long-maturity yields in the sign restrictions ensures the identified shocks capture both conventional and unconventional (e.g., quantitative easing tapering) policy moves.

At a 10-month horizon, real shocks explain 39 percent of the forecast error variance in 2-year U.S. Treasury yields, making them the dominant driver over the full sample (January 1982 - September 2022). Inflation shocks account for 14 percent and reaction shocks for 13 percent. Together the three identified shocks explain roughly two-thirds of total yield variation; the remaining one-third reflects residual or unclassified movements.

Q3. How did the composition of shocks driving 2-year yields change from 2021 into 2022?

Starting in September 2021, as inflation mounted and the Fed pivoted toward aggressive tightening, reaction and inflation shocks became the dominant drivers of 2-year yield increases. By September 2022, reaction and inflation shocks together accounted for approximately three-quarters of the cumulative increase in yields from the beginning of 2022, with reaction shocks alone contributing 114 basis points to the 2-year yield.

Q4. What are the financial market effects of a 25-basis-point reaction shock on EMDEs?

Reaction shocks produce significant adverse effects on EMDE financial markets within one quarter: 10-year local-currency government bond yields rise significantly, EMBI+ sovereign spreads widen significantly, capital flows decline significantly, and the real effective exchange rate depreciates significantly. Short-term (3-month) yields and equity prices also deteriorate, but these movements are not statistically significant at conventional levels.

Q5. How do financial market effects of inflation shocks compare to reaction shocks?

Inflation shocks generate adverse directional effects similar to reaction shocks rising 10-year yields, declining capital flows, real exchange rate depreciation, and falling equity prices but with the notable difference that, except for equity prices, these effects are generally not statistically significant. The paper thus finds that reaction shocks are more potent drivers of EMDE financial market tightening than inflation shocks.

Q6. How do real shocks affect EMDE financial conditions?

Real shocks produce outcomes broadly opposite to those from inflation and reaction shocks. They are associated with significant declines in EMBI+ sovereign spreads, significant increases in capital flows, significant real effective exchange rate appreciation, and significant increases in equity prices. Ten-year government bond yields do rise consistent with global bond market integration but this occurs alongside improving risk sentiment, not financial stress.

Q7. What are the macroeconomic (real activity) effects of the three shock types?

Reaction shocks produce a statistically significant decline in real GDP components, particularly in private consumption expenditure and gross fixed capital formation (fixed investment), within one quarter. Real shocks lead to higher real exports consistent with beneficial demand spillovers from stronger U.S. activity while leaving other GDP components unchanged. Inflation shocks induce a large and statistically significant increase in domestic EMDE CPI inflation, while real shocks reduce it; neither produces significant real GDP effects beyond the export channel.

Q8. How do EMDE fiscal balances respond differently to the three shock types?

Both inflation and especially reaction shocks are followed by an improvement in the EMDE primary balance (smaller deficit or larger surplus), achieved almost exclusively through declines in government expenditure. The paper attributes this to tighter credit availability and higher borrowing costs constraining fiscal space. Real shocks also improve primary balances, but the mechanism differs: both revenue increases and expenditure decreases contribute to the improvement. Declines in gross government debt occur in response to all three shocks but are statistically significant only for real shocks.

Q9. How does the composition of government debt shift in response to the different shocks?

Following inflation and reaction shocks, debt held by external creditors declines significantly as a share of total government debt, consistent with reduced access to global credit markets. Short-term debt eventually rises following both shock types. Foreign-currency debt rises considerably following reaction shocks likely reflecting the mechanical effect of currency depreciation boosting the local-currency value of pre-existing foreign-currency obligations. Conversely, following real shocks, external creditor participation rises significantly (improved market access), foreign-currency debt shares remain broadly stable, and short-term debt declines significantly (consistent with maturity extension by fiscal authorities seeking to minimize rollover risk under favourable conditions).

Q10. Do investment-grade and noninvestment-grade EMDEs respond differently to reaction shocks?

The paper finds little evidence of important differences between investment-grade and noninvestment-grade EMDEs in their responses to reaction shocks across most variables. Noninvestment-grade economies do show statistically larger increases in 10-year bond yields, and larger increases in EMBI+ spreads and 3-month yields than investment-grade economies though the latter two differences are not statistically distinguishable. For fiscal, GDP, and capital flow outcomes, the two groups respond similarly. The paper notes this finding is inconsistent with several prior studies but consistent with others, concluding the role of fundamentals remains unresolved.

Q11. How does the probability of financial crisis in EMDEs respond to the three shock types?

In the baseline (explanatory variables at sample means), the average EMDE faces a 3.5 percent probability of experiencing any type of financial crisis in a given year, with currency and banking crises the most common and sovereign debt crisis the least. Reaction shocks drive by far the largest increase: a 25-basis-point increase in 2-year yields from a reaction shock almost doubles the crisis probability to 6.6 percent. Inflation shocks produce small and statistically insignificant effects. Real shocks reduce the probability of sovereign debt crisis (consistent with their benign effects on financial markets) while raising currency crisis probability by less than reaction shocks.

Q12. What does the nonlinear logit relationship imply for the 2022 tightening cycle specifically?

Because the logit function is nonlinear, a doubling of the shock size leads to a more-than-proportional increase in crisis probability. Applying the estimated model to the 114-basis-point reaction-shock contribution to 2-year yields from January to September 2022, the model implies that the probability of financial crisis in the average EMDE increased by approximately 36 percentage points, to nearly 40 percent. The paper emphasizes this estimate carries wide uncertainty because no comparable yield increase occurred during the 1985-2018 estimation period, placing this extrapolation well outside the sample’s support.

Q13. What crisis dynamics were already materializing in 2022 consistent with the model predictions?

By the time of writing (late 2022), seven EMDEs had experienced currency depreciations of at least 30 percent against the U.S. dollar meeting the Laeven and Valencia (2020) threshold for a currency crisis and 21 EMDEs had reached agreements with the IMF for additional financing. The paper notes these developments had occurred despite standard macroeconomic factors (interest rate differentials and flight-to-safety flows) not fully explaining the magnitude of depreciations.

Q14. What robustness tests were conducted, and did they alter the main conclusions?

The VAR decomposition was re-estimated using weekly rather than monthly data. The three-shock model was simplified to two shocks (real versus monetary, combining inflation and reaction). The VAR was extended to include real GDP and PCE inflation with contemporaneous exclusion restrictions to insulate shock identification from current macroeconomic conditions. Inflation expectations were replaced with the Haubrich, Pennacchi, and Ritchken (2012) model-based measure throughout, rather than only pre-2003. For the crisis probability models, panel probit with random effects and panel logit with fixed effects were estimated alongside the baseline panel logit with random effects. In all cases, the results were not materially different: inflation and reaction shocks remained more adverse than real shocks for EMDE financial and fiscal variables, and only reaction shocks produced statistically significant increases in overall crisis probability. One noteworthy robustness finding: when combining inflation and reaction into a single monetary shock, the relative importance of the inflation component appears somewhat larger than when the two are separated.

Q15. What are this paper’s main contributions relative to existing literature?

The paper makes three stated contributions. First, it is the first to decompose the evolution of U.S. interest rates over the COVID-19 pandemic recession, subsequent recovery, and 2021-22 inflation surge into the separate contributions of real, inflation, and reaction shocks. Second, it extends prior work on EMDE spillovers (e.g., Arteta et al. 2015; Hoek, Kamin, and Yoldas 2021, 2022) by showing how different shock types affect government budget balances, revenues, expenditures, and debt composition, and by expanding the EMDE country sample. Third, it is the first to examine how real, inflation, and reaction shocks differentially affect the probability of banking, currency, and sovereign debt crises in EMDEs.

Key Concepts

Reaction shock: In this paper’s framework, a change in U.S. interest rates caused by a perceived shift in the Federal Reserve’s reaction function toward a more hawkish policy stance. Identified as a shock that raises both 2-year and 10-year Treasury yields while simultaneously lowering equity prices and lowering inflation expectations. The fall in inflation expectations distinguishes this shock from an inflation shock and signals that markets believe the Fed is tightening beyond what current inflation alone would warrant.

Inflation shock: A change in U.S. interest rates caused by rising expectations of U.S. inflation. Identified as a shock that raises both yields and inflation expectations but lowers equity prices. The equity decline signals that higher rates reflect inflationary pressure rather than improved growth prospects.

Real shock: A change in U.S. interest rates driven by improved prospects for U.S. real economic activity. Identified as a shock that simultaneously raises both yields, equity prices, and inflation expectations. The equity increase distinguishes this shock from the other two and signals that higher rates are accompanied by strengthening U.S. growth.

Sign-restricted Bayesian VAR with stochastic volatility: The paper’s primary model for decomposing U.S. yield movements. Sign restrictions on four variables (2-year yield, 10-year yield, S&P 500, 5-year inflation expectations) identify the three shock types without requiring timing restrictions. Stochastic volatility is incorporated to handle the heteroskedastic financial data and the COVID-19 period’s unusual size and nature; the model covers February 1982 to September 2022 at monthly frequency.

Panel local projection (Jorda 2005): The empirical framework linking the VAR-identified shock series to EMDE outcomes at quarterly frequency. Direct estimation of impulse responses at each horizon h avoids the misspecification accumulated in iterated VAR forecasts and permits straightforward incorporation of state-dependent (investment-grade vs. noninvestment-grade) heterogeneity via a dummy-variable interaction specification.

Capital flows (as used in this paper): Defined specifically as increases in net portfolio and other investment liabilities of EMDEs, excluding foreign direct investment liabilities. This definition isolates the more volatile, financially driven flows rather than the longer-horizon FDI component.

Financial crisis typology (Laeven and Valencia 2020): The crisis classification underlying the logit analysis. Sovereign debt crises are defined as a government default or restructuring of debt owed to private creditors. Banking crises require significant distress in the banking system combined with significant policy intervention measures. Currency crises are defined as a sharp nominal depreciation of at least 30 percent against the U.S. dollar. The paper uses these definitions from Laeven and Valencia (2020), extended through 2018 in Kose et al. (2021).

Primary budget balance improvement via expenditure compression: In the paper’s framework, the fiscal adjustment mechanism triggered specifically by inflation and reaction shocks: EMDE governments improve their primary balance (reduce deficits or increase surpluses) almost exclusively by cutting expenditures, rather than raising revenues, as a response to the credit tightening and higher borrowing costs associated with adverse U.S. interest rate shocks.

Identification and Estimation of Dynamic Random Coefficient Models

Mon, 01 Jan 0001 00:00:00 +0000

This paper studies linear panel data models where regression coefficients are individual-specific (random coefficients) and regressors may be predetermined — that is, sequentially exogenous rather than strictly exogenous, as occurs when a lagged dependent variable appears on the right-hand side. The canonical example is the AR(1) model Yit = gamma_i + beta_i * Yi,t-1 + epsilon_it, where both the intercept and the autoregressive coefficient vary across individuals. The setting is short panels (small T), which rules out learning about individual-level coefficient values.

The paper’s central finding, building on Chamberlain (1993, 2022), is that the mean of the coefficient distribution is not point-identified in this dynamic setting. Chamberlain established this for discrete regressors; the paper’s Proposition 1 extends the non-identification result to continuous regressors under stronger assumptions. The paper then characterizes finite lower and upper bounds for the mean, variance, and CDF of the random coefficient distribution. The identification strategy recasts the problem as an infinite-dimensional linear program and exploits the dual representation of that program (following Galichon and Henry (2009) and Schennach (2014)) to derive tractable closed-form bounds for the mean and optimization-based bounds for the variance and CDF.

For the mean parameter, the bounds take a closed-form expression involving the individual OLS estimator, the pooled OLS estimator, and cross-sectional moments of the data. The bounds remain finite even when the data are unbounded, provided certain moments of the data are finite. Tighter (refined) bounds are available when instrumental variables are brought in as additional unconditional moment restrictions. A numerical illustration shows how the outer identified set for E(beta_i) with a true value of 0.5 shrinks as T increases: at T=3 the outer set is approximately [0.216, 0.617]; at T=5 it narrows to approximately [0.306, 0.613]; the corresponding sharp identified sets (available for T=3 through T=5) range from [0.401, 0.593] at T=3 to [0.473, 0.532] at T=5.

The paper proposes computationally tractable inference procedures matched to each parameter. For mean parameters, the closed-form bounds permit a delta-method asymptotic approach augmented with Stoye’s (2020) smooth approximation to handle cases where the sample analog of the bound width can be negative (due to overidentification or mild misspecification). The resulting confidence intervals are valid and robust to overidentification. For the variance and CDF of the coefficient distribution, the paper uses the Andrews and Shi (2017) procedure for inference on a continuum of moment inequalities, which remains computationally feasible.

The empirical application estimates a generalization of Guvenen’s (2007, 2009) lifecycle earnings models using the Panel Study of Income Dynamics (PSID). Where Guvenen compared a restricted income profile (RIP, homogeneous persistence rho) against a heterogeneous income profile (HIP, heterogeneous time trend beta_i), this paper allows persistence rho itself to vary across households (rho_i). The key empirical findings are: (1) under both the RIP and HIP specifications, the estimated average earnings persistence E(rho_i) is significantly below 1; (2) the two specifications produce similar mean-persistence estimates once heterogeneity in rho_i is permitted, suggesting that misspecifying HIP as RIP or vice versa may not cause serious model misspecification when earnings persistence is allowed to vary; (3) the identified sets for the variance of rho_i provide evidence of genuine heterogeneity in earnings persistence across households, implying that households face different levels of earnings risk, which in turn contributes to heterogeneity in their consumption and savings behavior.

Q: Why is the mean of the random coefficient not point-identified in a short dynamic panel? A: Chamberlain (1993, 2022) first established this non-identification for discrete regressors. The paper’s Proposition 1 extends the result to continuous regressors under stronger assumptions. The fundamental obstacle is Lemma 1: E(beta_i) is point-identified if and only if there exists an unbiased estimator of beta_i in the individual time series, and no such estimator exists in short panels where T is small relative to the number of individual parameters.

Q: How does the paper characterize the identified set for the mean parameter? A: The identification problem is recast as an infinite-dimensional linear program. Using the dual representation (Galichon and Henry, 2009; Schennach, 2014), Theorem 1 yields a closed-form interval [L, U] = [BR - (1/2)sqrt(ERDR), BR + (1/2)sqrt(ERDR)], where BR is a weighted average of the individual OLS estimator and the pooled OLS estimator, ER is a non-negative term capturing cross-sectional variation in design matrices, and DR is a non-negative term related to residual variation. The bounds are finite whenever the relevant moments of the data are finite, even with unbounded data.

Q: How are the bounds tightened using instruments? A: Proposition 2 introduces refined bounds [LS, US] by incorporating additional unconditional moment restrictions from instruments Sit. The refined bounds use a larger set of restrictions and are weakly tighter than the baseline bounds. The empirical application employs up to 59 regressors with homogeneous coefficients (handled by Proposition 3), and instruments from lagged earnings levels and differences, substantially increasing the number of moment conditions.

Q: How are the variance and CDF of the coefficient distribution identified? A: Theorem 2 provides a general duality result for any parameter theta of the coefficient distribution. The lower bound is the maximum of E[min_{b} {m(Wi,b) + sum_k lambda_k phi_k(Wi,b)}] over Lagrange multipliers lambda, and the upper bound is the minimum of the corresponding maximum. Proposition 5 and Proposition 6 specialize this to the second moment (variance) of beta_i, with the upper bound requiring an eigenvalue assumption (Assumption 9) that the smallest eigenvalue of the individual design matrix R’R is bounded away from zero. Proposition 7 derives lower and upper bounds for the CDF P(e’Bi <= c) using a two-step optimization that separates the support into two regions.

Q: What guarantees computational tractability of the optimization problems? A: Proposition 4 establishes that GL(lambda, w) is globally concave in lambda for every w, and GU(lambda, w) is globally convex in lambda for every w. This means the optimization problems for the lower and upper bounds are concave maximization and convex minimization problems respectively, which can be solved with standard convex optimization methods.

Q: How does the inference procedure for mean parameters handle overidentification and misspecification? A: In finite samples, the sample analog of the bound-width term D_hat_S can be negative, which would make the estimated bounds degenerate. The paper adopts Stoye’s (2020) approach using the smooth approximation s(x,y) = sqrt((xy + sqrt((xy)^2 + r^2))/2). The (1-alpha)-level confidence interval combines a standard bound-based interval with an interval for a pseudo-true parameter mu*_e, ensuring validity under both correct specification and mild overidentification or misspecification.

Q: How does this paper’s approach to inference on the variance and CDF differ from that for the mean? A: For the mean, closed-form bounds permit a straightforward delta-method asymptotic argument and explicit confidence intervals. For the variance and CDF, the paper uses the Andrews and Shi (2017) procedure for inference on a continuum of moment inequalities, constructing a test statistic TAS(theta) = sup_{lambda} max{sqrt(N)(mu_hat_GL - theta)/sigma_hat_GL, sqrt(N)(theta - mu_hat_GU)/sigma_hat_GU}^2, 0, with the confidence set being the set of theta values not rejected. This procedure is computationally more demanding but remains feasible.

Q: What are the main empirical findings from the PSID application? A: In both the RIP and HIP specifications extended to allow heterogeneous persistence rho_i, the estimated average earnings persistence E(rho_i) is significantly below 1. Both specifications produce similar mean-persistence estimates once rho_i heterogeneity is permitted, suggesting that the HIP vs. RIP misspecification debate may be less consequential when persistence itself varies across households. The identified sets for the variance of rho_i provide evidence of genuine unobserved heterogeneity in earnings persistence.

Q: What is the economic significance of heterogeneous earnings persistence? A: Heterogeneity in earnings persistence rho_i means households face different levels of earnings risk: a household with high rho_i experiences earnings shocks that are more persistent, reducing its ability to smooth consumption over time and strengthening its motive for precautionary savings. The paper argues this heterogeneity contributes directly to heterogeneity in consumption and savings behavior, making rho_i a first-order parameter in lifecycle consumption models such as those of Hall and Mishkin (1982), Blundell, Pistaferri, and Preston (2008), and Arellano, Blundell, and Bonhomme (2017).

Q: How does the paper situate itself relative to Guvenen (2007, 2009)? A: Guvenen showed that allowing for heterogeneity in the time trend of earnings (HIP: heterogeneous income profile) yields estimated persistence significantly below 1, whereas imposing no such heterogeneity (RIP: restricted income profile) yields persistence near 1. This paper generalizes both models by additionally allowing persistence itself to vary across households (rho_i). The finding that both HIP and RIP deliver similar E(rho_i) estimates significantly below 1 suggests that Guvenen’s contrast may be partly an artifact of restricting persistence to be homogeneous.

Q: What is the scope of the identification results? A: The results apply to short panels (small T, large N), accommodate discrete, continuous, and unbounded data, and require the idiosyncratic error epsilon_it to be mean-independent of the full history of strictly exogenous regressors and of the current history of predetermined regressors. The bounds for the mean are finite under finite moment conditions on the data. The bounds for the variance additionally require the eigenvalue assumption (Assumption 9). The paper notes that the results extend to probit and logit models with individual-specific coefficients, panel VAR models, and systems of panel data regressions, though these extensions are not developed in detail.

Dynamic random coefficient model: A linear panel data model in which both the intercept and slope coefficients are individual-specific (gamma_i, beta_i), the regressor is predetermined (sequentially exogenous rather than strictly exogenous), and T is small — so individual coefficient values cannot be estimated from the time series alone.

Partial identification: The property that a parameter of interest (such as E(beta_i)) cannot be consistently estimated from the data (it is not point-identified), but finite lower and upper bounds on its value can be characterized. The paper shows this is the generic situation for dynamic random coefficient models in short panels.

Dual representation of infinite-dimensional linear programs: The technique, following Galichon and Henry (2009) and Schennach (2014), of converting an infinite-dimensional linear programming problem (which arises when data or coefficients are continuous) into an equivalent dual problem that yields tractable closed-form or convex-optimization-based bounds.

Refined bounds (instrument-augmented bounds): Tighter identified sets for the mean parameter obtained by incorporating additional unconditional moment restrictions from instruments Sit, beyond the baseline moment conditions. These correspond to Proposition 2 and make the identification interval weakly narrower.

Sequential exogeneity (predetermined regressor): The assumption E(epsilon_it | gamma_i, beta_i, Zi1,…,ZiT, Xi1,…,Xit) = 0, which allows the regressor Xit (e.g., Yi,t-1) to be correlated with future errors but not current or past errors. This is weaker than strict exogeneity and is what makes the model dynamic and identification challenging.

Heterogeneous income profile (HIP) vs. restricted income profile (RIP): In Guvenen’s framework, HIP allows the time trend of earnings to vary across individuals (heterogeneous beta_i), while RIP does not. The paper extends both by also allowing the AR(1) persistence parameter rho to vary across individuals (rho_i), yielding an empirically more general earnings process.

Earnings persistence (rho_i): The individual-specific autoregressive coefficient in the lifecycle earnings process. High rho_i means earnings shocks last longer, increasing earnings risk, reducing the household’s ability to smooth consumption, and strengthening precautionary savings motives. The paper finds evidence that rho_i varies meaningfully across U.S. households in the PSID.

Identification of Time-Inconsistent Models: The Case of Insecticide-Treated Nets

Mon, 01 Jan 0001 00:00:00 +0000

This paper addresses two related problems: the formal identification of time-inconsistent preferences in dynamic discrete choice models with unobserved heterogeneous types, and the structural estimation of those preferences using data from a health intervention in rural Orissa, India. The identification challenge is fundamental — even the standard exponential discount factor delta is generically not identified in dynamic choice models (Rust 1994; Magnac and Thesmar 2002), and this non-identification extends a fortiori to the hyperbolic (beta, delta) parameterization. The paper’s first contribution is constructing identification conditions that overcome these results through two exclusion restrictions: a variable z that affects utility only through the perceived value of future states (played in the application by elicited beliefs about state evolution), and a variable r that acts as an imperfect signal of agent type but is uninformative about choices conditional on type.

The general model accommodates a finite but unknown number of agent types — time-consistent (beta=1), time-inconsistent naive (beta<1, unaware of future present-bias), and time-inconsistent sophisticated (beta<1, aware of future present-bias) — as well as sub-types within each class. The paper proceeds in four identification steps when types are unobserved: identifying the total number of types (via the rank of an observable matrix), recovering type-specific choice probabilities, assigning type identities, and recovering preference parameters. For time-consistent and sophisticated agents, both beta and delta are point-identified. For naive agents, the parameters are set-identified in general, with point identification available under a monotonicity condition (Assumption 14) or by imposing a common exponential discount factor across types (Assumption 15).

The empirical application studies demand for insecticide-treated nets (ITNs) and their periodic retreatment — a health-protective technology with low up-front cost but substantial future benefits — among households in malarious areas of rural Orissa. A key design feature is that households were offered either a standard ITN contract (with the option to purchase retreatment later) or a commitment contract bundling two consecutive retreatments, allowing the commitment product choice to serve as a noisy type signal r. Elicited beliefs about future state variables serve as the excluded z variable.

The main empirical findings are: approximately 21% of the population is time-consistent, 49% are naive time-inconsistent, and 30% are sophisticated time-inconsistent — so time-inconsistent agents account for approximately 79% of the sample. The preferred estimates of the hyperbolic parameter beta are 0.16 for naive agents and 0.08 for sophisticated agents, indicating substantial present-bias in both groups. These estimates of the population type distribution and type-specific beta parameters are described as new to the literature.

A counterfactual exercise quantifies the welfare cost of present-bias: the median undiscounted additional expected total cost of malaria during the study period attributable to under-investment in ITNs exceeds the price of a treated net by a factor of approximately six. However, because time-inconsistent households heavily discount future malaria costs, the discounted total costs of malaria are low for many inconsistent agents relative to the ITN price, explaining low demand from the agents’ own subjective perspective. The paper also finds that commitment products are not disproportionately chosen by sophisticated agents — take-up of the commitment contract is actually higher among naive households — contradicting the deterministic mapping from commitment product purchase to sophistication that is commonly assumed in the literature. Finally, differences in per-period utilities across agent types exist but are not substantively important in explaining differential outcomes in the sample.

Q: What is the core identification problem the paper addresses, and why is it hard? A: Even the standard exponential discount factor delta is generically not identified in dynamic discrete choice models (Rust 1994; Magnac and Thesmar 2002). This non-identification extends a fortiori to both beta and delta in the hyperbolic (beta, delta) model. When agents are also heterogeneous in unobserved type, the additional problem of identifying the population distribution of types — itself a key policy parameter — must be solved jointly with preference identification.

Q: What two exclusion restrictions provide the key identifying variation? A: The first restriction is a variable z that affects utility only via the perceived value of future states but not per-period utility (Assumption 3); in the application this is played by elicited subjective beliefs about future state evolution. The second is a variable r that predicts agent type but, conditional on type and observables, provides no additional information about choices (Assumption 16); in the application r includes elicited time-preference indicators and the choice of the commitment versus standard ITN contract.

Q: Why does the paper require at least three periods? A: Three periods are the minimum required to capture the notions of time-inconsistency studied here: with only two periods, no time-inconsistency problem would arise. Three periods allow the researcher to separately observe how an agent plans in period 1, how the agent actually behaves in period 2 (potentially deviating from the period-1 plan), and how the agent behaves in the terminal period 3 where the problem reduces to a static discrete choice.

Q: What is point-identified versus set-identified across agent types? A: For time-consistent agents, all per-period utilities and the (single) discount factor delta are point-identified. For sophisticated agents, both beta and delta are separately point-identified under the rank conditions in Assumptions 10-11. For naive agents, the parameters are in general only set-identified (Lemma 4 provides sharp bounds); point identification holds under either a monotonicity condition (Assumption 14) or the assumption that naive and sophisticated agents share the same exponential discount factor (Assumption 15).

Q: How does the paper identify the total number of types in the population? A: The number of types equals the rank of a directly identified matrix P formed from the joint distribution of actions and states in adjacent time periods (Proposition 1). The rank provides a lower bound in general and equals the true number of types when the state space is sufficiently rich and type-specific choice probabilities vary sufficiently across the state space (Assumptions 17 and 19).

Q: How does the paper distinguish naive from sophisticated agents among the identified type-specific choice probabilities? A: A key diagnostic is the function delta_hat_tau(x2,z2), which compares an agent’s period-1 view of the future against what would be expected given period 2-3 choices. For time-consistent and sophisticated agents, this function is constant across the state space (x2,z2); for naive agents it varies across the state space (Lemma 7, Proposition 2). This variation arises because naive agents incorrectly anticipate their future behavior in period 1, generating a wedge between planned and actual continuation values that shifts with the state.

Q: What fraction of the sample is time-inconsistent, and what are the estimated beta parameters? A: Approximately 79% of the sample is time-inconsistent: 49% are naive and 30% are sophisticated. The preferred estimates of the hyperbolic (present-bias) parameter beta are 0.16 for naive agents and 0.08 for sophisticated agents. Both estimates indicate substantial present-bias. The paper states that these estimates of the population type distribution and the type-specific beta values are new to the literature.

Q: What is the welfare cost of present-bias in terms of malaria risk? A: Present-bias leads to lower ITN purchases and fewer retreatments, which increases the likelihood of contracting malaria. The median undiscounted additional expected total cost of malaria during the study period attributable to under-investment in ITNs exceeds the price of a treated net by a factor of approximately six. However, because inconsistent agents heavily discount future health costs, the discounted total costs of malaria are low relative to the ITN price for many such agents, which explains low demand from the agents’ own subjective perspective despite large social costs.

Q: What does the paper find about commitment products and agent sophistication? A: The commitment contract — bundling two consecutive retreatments — was designed to appeal to sophisticated present-biased agents who anticipate their future self-control problems. Contrary to the deterministic mapping from commitment product purchase to agent sophistication commonly assumed in the literature, take-up of the commitment contract is actually higher among naive households than sophisticated ones. The paper argues this is possible because the model allows commitment product choice to only imperfectly predict type, enabling a richer analysis than prior work that rules out type heterogeneity by assumption.

Q: Are differences in per-period utilities across types an important alternative explanation for observed behavior? A: Per-period utilities do vary across agent types, but the paper finds they are not substantively important in explaining differential outcomes in the sample. This finding supports the interpretation that time-inconsistent preferences — rather than heterogeneity in static preferences over states — are the primary driver of the behavioral differences observed across agent types in this context.

Q: What is the role of elicited beliefs in the identification strategy? A: Elicited beliefs about the future evolution of state variables serve as the excluded variable z that shifts the forward-looking component of the value function while leaving per-period utility unchanged. The use of expectational data, as advocated by Manski (2004), provides a natural and interpretable source of identifying variation for the discount parameters. The paper argues that this plausible exclusion restriction contributes to the encouraging Monte Carlo simulation results relative to other work in the identification literature.

Q: What happens to identification under partial sophistication? A: When agents are partially sophisticated — aware of some but not all of their future present-bias, so that beta_tilde in [beta, 1] rather than exactly equal to beta or 1 — the three time-preference parameters (delta, beta, beta_tilde) are not point-identified in general (Proposition 4 provides a set identification result). Point identification requires that the exponential discount factor delta be identified separately. The paper shows that partial and complete sophistication can be distinguished from time-consistency by whether the function delta_hat varies across the state space, and partially sophisticated types can be distinguished from fully sophisticated types under an additional variability condition (Assumption 23, Proposition 3).

Hyperbolic (beta-delta) discounting: A model of time-inconsistent preferences in which future utility at time s discounted from time t carries the factor beta*delta^(s-t), where beta<1 introduces an additional present-bias relative to pure exponential discounting. The parameter beta governs the wedge between the discount rate applied to immediate versus purely future tradeoffs; delta governs the intertemporal rate of substitution between any two future periods.

Sophisticated vs. naive agents: Both types are time-inconsistent (beta<1) and both are aware of their current present-bias. Sophisticated agents (tau_S) also correctly anticipate the extent of their future present-bias (beta_tilde = beta), while naive agents (tau_N) incorrectly believe their future self will behave as if beta_tilde = 1. This difference in beliefs about future behavior drives distinct choice dynamics across the three periods, providing the key observable variation used to distinguish the two types.

Exclusion restriction (z variable): A state variable that enters the transition probabilities and thus the value of future states but does not enter the current per-period utility function (Assumption 3). Variation in z shifts the forward-looking component of the Bellman equation while holding current utility fixed, providing the identifying variation needed to separately recover discount parameters from per-period utility parameters.

Type indicator / type proxy (r): An observed variable that is informative about an agent’s time-preference type but, conditional on type and other observables, provides no additional information about choices (Assumption 16). In the application, r includes elicited time-preference indicators and whether the agent chose the commitment versus standard ITN contract. Critically, the mapping from r to type is imperfect, so r does not directly reveal type for each individual.

Conditional choice probability (CCP) inversion: Following Hotz and Miller (1993), the type-specific conditional choice probabilities P_tau(a_t|x_t, z_t) — directly identified from data given type — can be inverted to recover per-period utility differences and combinations of discount parameters without solving the full dynamic programming problem. This approach underpins the constructive identification arguments throughout the paper.

Commitment contract: A product design in which two consecutive ITN retreatments are bundled at purchase, intended to mitigate the time-inconsistency problem by removing the future self-control decision about retreatment. The commitment contract is theoretically predicted to be preferred by sophisticated present-biased agents; the paper finds this prediction fails empirically, with naive households showing higher take-up.

Present-bias welfare cost: The undiscounted additional expected total cost of malaria attributable to under-investment in ITNs driven by present-bias. The paper estimates this cost exceeds the price of a treated net by a factor of approximately six at the median, capturing the gap between the social planner’s valuation of ITN adoption and the discounted valuation of time-inconsistent agents.

Ideological Alignment and Evidence-Based Policy Adoption

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates how the ideological alignment between knowledge-disseminating institutions and policymakers affects the adoption of evidence-based policies. The core research question is whether, and through which mechanisms, the ideology of the messenger — rather than the content of the message — determines whether local policymakers act on rigorous research evidence.

The authors conduct a country-wide randomized controlled trial (RCT) across 5,678 touristic Spanish municipalities. The policy recommendation derives from Hinnosaar et al. (2021), an RCT demonstrating that minor improvements to municipalities’ Wikipedia pages (adding photographs, local festival information, touristic landmark details) increased overnight tourist stays by 9%. This policy was chosen because it is ideologically neutral, low cost, within local policymakers’ remit, and its implementation is directly traceable via Wikipedia edit histories.

Municipalities were randomized into five treatment arms and a control group (approximately 950 municipalities each), stratified by ruling party ideology, population, and touristic accommodation count. Three arms received the same policy brief endorsed by: (1) an ideologically aligned think tank (FAES for right-wing municipalities, Fundación Alternativas for left-wing), (2) the ideologically opposite think tank, or (3) an ideologically nonsalient researcher from the London School of Economics. Two further arms received links to newspaper articles covering the same research from either an ideologically aligned outlet (El Mundo for right, Eldiario.es for left) or an ideologically opposite outlet. The control group received no information. The experiment ran from May to December 2022, with multiple reminder emails sent across the period.

The main outcome is a binary indicator for whether a municipality’s Wikipedia page was changed in line with the recommended guidelines during the study period, coded blind to treatment status by two independent coders.

Key findings: Pooled across all treatment arms, information provision increased the probability of policy adoption by approximately 0.98 percentage points (a 38% relative increase over the control group baseline), but this effect is only marginally above conventional significance thresholds (p-value = 0.13). The aggregate effect masks sharp heterogeneity by ideological alignment. When the informing institution’s ideology aligns with the policymaker’s, policy adoption increases by 1.68 percentage points (think tank) and 1.67 percentage points (newspaper) relative to the control group — equivalent to a 66% and 65% relative increase, respectively, both statistically significant at the 5% level. By contrast, information from an ideologically opposite institution produces a coefficient that is negligible and statistically indistinguishable from zero, indicating that misaligned information is no more effective than receiving no information at all. The ideologically nonsalient LSE researcher arm produced an intermediate effect (0.94 percentage points, 37% relative increase), but the p-value (0.27) exceeds conventional thresholds, and the effect is not statistically distinguishable from either the aligned or the control condition. Policy briefs and newspaper articles are equally effective when ideologically aligned (difference of 0.1 percentage points, p-value = 0.82).

To decompose mechanisms, the authors propose a three-stage framework: (1) selective exposure to information, (2) belief updating, and (3) policy implementation. Email click-through rates (access to the full policy brief or article once the informing institution is revealed) do not differ significantly across treatment arms, ruling out selective exposure as the operative mechanism. A post-intervention online survey experiment with 1,600 policymakers from 1,196 municipalities shows that those receiving information from an aligned or nonsalient institution updated their beliefs about policy effectiveness significantly more than those receiving information from an opposite institution, implicating belief updating as one operative channel. However, comparing the survey experiment (where nonsalient and aligned treatments produce similar belief updating) with the main experiment (where the aligned arm adopts at nearly twice the rate of the nonsalient arm, though not statistically distinguishable) suggests that ideological alignment also affects the third stage — policy implementation — beyond mere belief updating.

The estimated monetary cost of ideological misalignment is 2,192 euros per municipality per year, calculated using the impact of Wikipedia changes on touristic revenues from Hinnosaar et al. (2021).

Scope conditions: The context is Spanish local government, a policy that is explicitly non-ideological, low-cost, and easily implemented. Generalizability to ideologically charged or costly policies is not established. Left-wing municipalities show larger responses to aligned information, though this heterogeneity is not statistically significant at conventional levels.

Q: What is the baseline rate of policy adoption in the control group, and what does the aligned-institution treatment achieve in absolute terms?

A: The paper reports that ideologically aligned institutions increase the share of municipalities implementing recommended Wikipedia changes by 1.68 percentage points (think tank) and 1.67 percentage points (newspaper) relative to the control group. Working backward from the stated 66% and 65% relative increases, this implies a control group baseline of approximately 2.5 percentage points. The aligned effects are statistically significant at the 5% level.

Q: Does information from an ideologically opposite institution have any effect on policy adoption?

A: No. The coefficient for opposite-ideology treatment arms is negligible in magnitude, closely resembling the near-zero coefficients from the placebo analysis conducted for the same months in 2019 (pre-intervention). The authors conclude that receiving information from an ideologically opposite institution is statistically indistinguishable from receiving no information at all. This null result is consistent across heterogeneity analyses by mayor ideology, municipality population, Wikipedia page length, and party type.

Q: How does the ideologically nonsalient (LSE researcher) treatment compare to aligned and opposite arms?

A: The nonsalient arm increases policy adoption by 0.94 percentage points (a 37% relative increase), approximately half the effect of the aligned arm (1.68 percentage points). However, the p-value is 0.27, and the effect is not statistically different from either the aligned arm (p-value = 0.34) or the control group at conventional confidence levels. The result should therefore be interpreted with caution.

Q: Are policy briefs or newspaper articles more effective in promoting policy adoption?

A: Neither format is significantly more effective than the other. Conditional on ideological alignment, the difference between policy brief and newspaper article effects is 0.1 percentage points with a p-value of 0.82. Both are equally effective when ideologically aligned with the receiving policymaker, a finding the authors describe as a novel contribution to the policy communication literature.

Q: Does ideological alignment affect whether policymakers choose to access the full information (selective exposure)?

A: No. Click-through rates on the links to policy briefs or newspaper articles — measured after policymakers have seen the informing institution’s identity — do not differ significantly across treatment arms. The observed average click-through rate is 6.42%. This null result is consistent with the hypothesis that policymakers do not strategically filter information acquisition based on the messenger’s ideology, at least for non-ideological policies.

Q: What does the survey experiment reveal about belief updating?

A: In the post-intervention survey experiment with 1,600 policymakers, participants first reported beliefs about a purportedly beneficial (but actually harmful) policy, then were randomly assigned to receive information about its negative effects from an aligned, opposite, or nonsalient think tank. Those receiving information from an aligned or nonsalient institution updated their beliefs significantly more than those receiving information from an ideologically opposite institution. This implicates belief updating — not just selective exposure — as a channel through which ideological alignment affects policy adoption.

Q: Why do the authors conclude that ideological alignment also affects the third stage (policy implementation) beyond belief updating?

A: In the survey experiment, aligned and nonsalient institutions produce statistically similar belief updating. Yet in the main field experiment, the aligned arm adopts policy at nearly twice the rate of the nonsalient arm (1.68 vs. 0.94 percentage points), although this difference is not statistically significant. The authors interpret this gap as suggestive evidence that ideological alignment affects policy implementation through channels beyond belief updating — such as career concerns, party cues, or the political economy of implementation — though they acknowledge the evidence is indirect and the treatment difference is not statistically distinguishable.

Q: What is the estimated economic cost of ideological misalignment?

A: The authors estimate a cost of 2,192 euros per municipality per year attributable to ideological misalignment between the informing institution and the receiving policymaker. This calculation uses the estimated impact of Wikipedia changes on touristic revenues from Hinnosaar et al. (2021) and reflects not the cost of not implementing the policy, but the marginal cost of using an ideologically opposite rather than aligned institution to disseminate the research evidence.

Q: How did outside researchers’ predictions compare to actual results?

A: Researchers surveyed on the Social Science Prediction Platform correctly anticipated the rank ordering of treatment effectiveness (aligned > nonsalient > opposite > control) but substantially overestimated adoption rates in every arm. They predicted relative increases of 144%, 103%, and 48% for aligned, nonsalient, and opposite conditions respectively, compared to actual relative increases of roughly 65%, 37%, and ~0%. Email opening rates were the most accurately predicted (49% predicted vs. 38% actual). The results highlight the difficulty of translating evidence into policy even for simple, low-cost interventions.

Q: What are the main threats to validity and how are they addressed?

A: Three main threats are considered. First, differential email opening rates across treatment arms: addressed by showing the informing institution was revealed only after email opening, and confirmed by finding no significant differences in opening rates across groups. Second, spillovers between municipalities: the endline survey shows only 5 of 236 control-group respondents reported receiving any information from external sources; spillover distance analyses in Table D.II find no significant effect on control municipalities’ adoption rates. Third, contamination bias in multi-arm RCTs with strata fixed effects: addressed by replicating main results using the Goldsmith-Pinkham et al. (2022) method, yielding nearly identical estimates.

Q: What heterogeneity is observed across left- and right-wing municipalities?

A: The positive effect of receiving information from an ideologically aligned institution appears larger for left-wing municipalities, with coefficients approximately three times larger than for right-wing municipalities, but this difference is not statistically significant at conventional confidence levels. The authors caution that the strength of ideological alignment may differ systematically between the partner think tanks on the left and right, making direct comparisons between left- and right-wing effects difficult to interpret cleanly.

Q: How does the paper relate to prior work on evidence-based policymaking?

A: The closest prior work is Hjort et al. (2021) and Mehmood et al. (2024), which examine the impact of scientific evidence access on actual policy adoption, and DellaVigna and Kim (2022), which identifies ideology as a factor in the diffusion of innovative policies across governments. The present paper’s main contribution is being the first to isolate the causal effect of ideological alignment on policy adoption using a large-scale field experiment with real, authoritative ideological institutions — rather than surveys or hypothetical scenarios — while using a non-ideological policy recommendation to avoid confounding messenger ideology with policy ideology.

Ideological alignment: In this paper’s usage, the congruence between the political ideology of the institution disseminating research evidence (think tank or newspaper) and the political ideology of the local government receiving that information. Alignment is operationalized by matching right-wing municipalities with right-leaning institutions (FAES, El Mundo) and left-wing municipalities with left-leaning institutions (Fundación Alternativas, Eldiario.es).

Evidence-based policy adoption: The actual implementation by local policymakers of a policy recommendation derived from published peer-reviewed research — measured here as whether a municipality’s Wikipedia page was edited in line with specific recommended guidelines during the study period, not merely expressed intention or stated support.

Knowledge brokers: Institutions, such as think tanks, that serve as intermediaries between academic researchers and policymakers, translating and disseminating research findings in accessible formats (policy briefs) to bridge the gap between evidence and policy.

Nonsalient ideology: A condition in which the informing institution carries no salient or recognizable partisan affiliation, operationalized here by a foreign research university professor (LSE) whose institutional identity does not carry a clear left-right signal in the Spanish political context.

Three-stage policy adoption framework: The authors’ conceptual structure positing that ideology can interfere at three sequential stages: (1) selective exposure — whether policymakers choose to access information once the messenger’s ideology is revealed; (2) belief updating — whether policymakers revise their assessment of a policy’s effectiveness upon receiving evidence; and (3) policy implementation — whether policymakers act on updated beliefs to adopt the policy.

Selective exposure: The tendency of individuals to avoid information from sources whose ideology conflicts with their own prior beliefs; in this paper, operationalized as differential click-through rates on links to policy briefs or news articles after the informing institution’s identity is revealed.

Motivated reasoning: A documented tendency, also observed in policymakers, to reject or discount evidence that contradicts ideologically held prior beliefs — the mechanism proposed to explain why opposite-ideology information fails to update beliefs as effectively as aligned-ideology information.

Inference Based on Time-Varying SVARs Identified with Sign Restrictions

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. The paper asks how to conduct valid Bayesian inference in time-varying structural vector autoregressions (SVARs) identified with sign restrictions, a setting in which existing algorithms are shown to be theoretically flawed. As an empirical illustration, the authors use the new framework to examine three questions about the 2022–2023 Federal Reserve tightening cycle: (i) how did the Fed respond to the state of the economy; (ii) how would more dovish or hawkish stances have fared; and (iii) was the Fed behind the curve in 2021, and at what cost?

Methodology. The paper defines a class of rotation-invariant time-varying SVARs, building on Bognanni (2018). A model belongs to this class when its prior over sequences of structural parameters is invariant to orthogonal transformations of those sequences—i.e., it assigns equal prior density to all observationally equivalent structural parameter sequences (Proposition 1 establishes that observational equivalence corresponds exactly to orthogonal rotation of the sequence). The authors prove an if-and-only-if characterization (Proposition 2): a prior belongs to this class if and only if the induced prior over sequences of orthogonal matrices is uniform and independent of the time-varying reduced-form parameters.

A specific member of this class, the Random Correlations SVAR (RC-SVAR), is constructed by combining a prior over time-varying reduced-form parameters based on Archakov and Hansen’s (2021) parametrization of correlation matrices with a uniform prior over sequences of orthogonal matrices. The RC-SVAR is preferred over alternatives (Primiceri 2005’s decomposition, which is order-dependent; Bognanni’s 2018 discounted Wishart model, whose marginal likelihood significantly underperforms) because, for the type of empirical applications considered, it generally implies a higher log-predictive score than most orderings of the Primiceri (2005) model.

The authors introduce three algorithms. Algorithm 1 (simple acceptance sampling) is theoretically correct but computationally infeasible when sign restrictions span many periods because the probability of satisfying all restrictions simultaneously converges to zero as sample length T grows. Algorithm 2, the current approach in the literature (Baumeister and Peersman 2013; Bognanni 2018; Debortoli, Galí and Gambetti 2020), draws orthogonal matrices period-by-period from the sign-restriction-truncated uniform distribution; the authors show this does not draw from the correct target posterior because the resulting prior over orthogonal matrices is not independent of the reduced-form parameters and therefore the prior does not satisfy the rotation-invariance condition. Algorithm 3, the paper’s contribution, uses a Gibbs sampler that incorporates the Particle Gibbs with Ancestor Sampling (PGAS) method of Lindsten, Jordan and Schon (2014) to draw sequentially from the correct target posterior conditional on sign restrictions over an arbitrary number of periods.

An important additional contribution is the allowance for time-varying sign restrictions—restrictions that are imposed only in selected periods—enabling researchers to tailor identification to institutional knowledge about when particular restrictions are economically appropriate.

Data and Empirical Application. The RC-SVAR is estimated at a quarterly frequency with five variables: output growth (log difference of real GDP), core inflation (log difference of core PCE price index), the federal funds rate, money growth (log difference of M2), and the Moody’s Baa corporate bond yield relative to the 10-year Treasury yield (credit spread). The sample runs from 1959:Q1 to 2023:Q2, with a constant and two lags (n=5, p=2, m=11). Four independent MCMC chains of 20,000 draws are used, keeping every tenth draw after discarding the first 2,500; 1,800 particles approximate the reduced-form posterior and 3,600 particles approximate the posterior of the orthogonal matrices.

Main Findings. Decomposing the unexpected change in the federal funds rate from 2022:Q2 to 2023:Q2 into contributions from the predictable component, the systematic monetary policy response to non-monetary-policy shocks, and pure monetary policy shocks, the authors find that the lion’s share of the unpredictable rate increase was a systematic response to non-monetary policy shocks. Monetary policy shocks contributed about 100 basis points of the unexpected change in the federal funds rate by 2023:Q2 (out of roughly 4.99 percentage points of cumulative actual funds rate).

In the Dovish Fed counterfactual—where the response of the federal funds rate to contemporaneous inflation is halved for the first quarter of 2022—the economy would have marginally overheated, with inflation running persistently above 5 percent. In the Hawkish Fed counterfactual—where the response to inflation is doubled—inflation would have quickly declined at a small output cost: focusing on posterior medians, real GDP in 2023:Q2 would have been about 0.7 percent lower than in the data, though the lower envelope of the 68 percent probability bands indicates the output cost could have been as large as 3.1 percent.

Regarding the “behind the curve” question, the model finds evidence that the Fed was accommodative in 2021 (expansionary monetary policy shocks in that period), consistent with Summers (2021b). However, monetary policy shocks contributed only about 0.6 percentage points to annualized core inflation during 2021:Q2–2021:Q4 on a cumulative basis; the larger and dominant source of the unexpected inflation surge was non-monetary policy shocks. A comparison of the RC-SVAR with a constant-parameter SVAR identified only by Restriction 1 (Uhlig 2005) shows substantively different conclusions: the constant-parameter model attributes the unexpected increase in the federal funds rate to shocks that affect money growth and credit spreads, without a clear connection to the real economy, whereas the RC-SVAR links the rate increases to shocks that made the economy run hotter.

In depth

Q1. What is the fundamental theoretical flaw in existing algorithms for time-varying SVARs identified with sign restrictions, and why does it matter?

Existing algorithms (e.g., Baumeister and Peersman 2013; Bognanni 2018; Debortoli, Galí and Gambetti 2020) draw orthogonal matrices period-by-period from the uniform distribution restricted to those matrices satisfying the sign restrictions at each t. This construction implicitly defines a marginal density for the orthogonal matrices conditional on the reduced-form parameters that is not uniform: it is proportional to the reciprocal of the volume of the sign-restriction-satisfying subset of the orthogonal group, which depends on the reduced-form parameters. Consequently, the prior over structural parameters implied by these algorithms does not assign equal density to observationally equivalent sequences of structural parameters, violating Proposition 2’s necessary and sufficient condition. The resulting posteriors are therefore not correctly targeted to the desired posterior, meaning inference is distorted in a way that cannot be corrected by importance reweighting without prohibitive computation.

Q2. What does Proposition 1 establish, and how does it generalize the constant-parameter case?

Proposition 1 proves that two sequences of time-varying structural parameters are observationally equivalent if and only if there exists a sequence of orthogonal matrices such that one sequence is obtained from the other by post-multiplying each period’s structural parameters by the corresponding orthogonal matrix. This directly mirrors the constant-parameter result in Rubio-Ramírez, Waggoner and Zha (2010) and Uhlig (2005), where a single orthogonal matrix produces observational equivalence. The extension to sequences is non-trivial because the law of motion couples parameter draws across time, but the likelihood’s separability across periods preserves the period-by-period orthogonal rotation structure.

Q3. What is Proposition 2, and what is its practical implication for constructing valid priors?

Proposition 2 states that the prior over time-varying structural parameters satisfies the rotation-invariance condition (Equation 3) if and only if the induced prior over the time-varying orthogonal reduced-form parameters does not depend on the sequence of orthogonal matrices—equivalently, the prior over (Qt) is uniform over the product of orthogonal groups and is independent of the reduced-form parameters (Bt, Σt). The practical implication is constructive: any prior over time-varying reduced-form parameters (Bt, Σt), combined with an independent uniform prior over sequences of orthogonal matrices, automatically produces a rotation-invariant SVAR. This means that widely-used priors for reduced-form time-varying VARs (Primiceri 2005, Bognanni 2018, the new RC prior) can all be adapted for structural analysis without modification, as long as the orthogonal matrices are drawn uniformly and independently of the reduced-form parameters.

Q4. Why do models with heteroskedastic structural shocks (identification via heteroskedasticity) not belong to the class of rotation-invariant SVARs?

In models identified through heteroskedasticity, the time-varying structural parameters take the form (A Ψt^{-1/2}, F Ψt^{-1/2}), where Ψt is a time-varying diagonal matrix. For any permissible sequence, post-multiplying by a non-diagonal orthogonal matrix at one period produces a sequence where the ratio of structural parameters across consecutive periods is not diagonal, which violates the permissibility constraint of those models. Thus, the class of rotation-invariant SVARs and models identified through heteroskedasticity are mutually exclusive when the heteroskedastic specification has constant impulse responses up to scale—a restriction that the authors note has been criticized as a potential weakness of the heteroskedasticity-based approach.

Q5. Why is the Random Correlations SVAR (RC-SVAR) chosen as the baseline, and how does it compare to alternatives?

The RC-SVAR uses the Archakov and Hansen (2021) parametrization of correlation matrices to define a prior over time-varying reduced-form parameters that is order-invariant (unlike Primiceri 2005, which produces n! different elements depending on variable ordering) and avoids the highly restrictive structure of Bognanni’s (2018) discounted Wishart model, which significantly underperforms in marginal likelihood. For the empirical applications considered, Arias, Rubio-Ramírez and Shin (2023) show the RC-SVAR generally achieves a higher log-predictive score than most orderings of the Primiceri (2005) model, motivating its use as the baseline. The theoretical results apply to any member of the rotation-invariant class, so the algorithm is not specific to the RC-SVAR.

Q6. Why are time-varying sign restrictions important, and how are they implemented in the monetary policy application?

Time-varying sign restrictions allow researchers to impose identification restrictions only in periods where those restrictions are economically appropriate, adhering to the principle “If you know it, impose it; if you do not know it, do not impose it” (Uhlig 2017). In the monetary policy application, Restriction 2 (which constrains the contemporaneous elasticities in the policy rule to plausible ranges, following Arias, Caldara and Rubio-Ramírez 2019) is not imposed during three exceptional periods: 1979:Q4–1982:Q4 (non-borrowed reserves targeting under Volcker), 2009:Q1–2015:Q3 (quantitative easing following the Great Recession), and 2020:Q2–2021:Q4 (QE and effective zero lower bound during COVID-19). Restriction 1 (sign restrictions on impulse responses to a monetary policy shock, following Uhlig 2005) is imposed throughout the entire sample.

Q7. What do the estimated contemporaneous elasticities reveal about how monetary policy has changed over time?

The model estimates show substantial time variation. The contemporaneous elasticity of the federal funds rate to output growth exhibits three peaks: during Arthur Burns’s chairmanship in 1974 (capturing the sharp rate cut during the 1974–1975 recession), during Volcker’s chairmanship in 1983–1984 (when annualized real GDP growth averaged 6.8 percent), and during Greenspan’s tenure in 2001 (when the federal funds rate fell from 6.4 percent in December 2000 to 1.8 percent by end-2001). Outside these peaks, the elasticity averaged about 0.1, implying a 0.1 percentage point rise in the annualized federal funds rate per 1 percentage point increase in annualized GDP growth. The elasticity to inflation averaged about 0.3 percentage points per 1 percentage point rise in annualized core inflation, with a range from above 0.5 in the early 1970s and early Volcker years down to about 0.15 during Yellen’s tenure. The elasticity to the credit spread moved from about −1.4 at the beginning of Burns’s tenure to −2.2 at the end of Nixon’s presidency, then declined through the mid-1970s to the Great Recession, and stood at about −1 by mid-2023.

Q8. What is the exact decomposition of the 2022–2023 tightening cycle into predictable, systematic non-monetary, and monetary policy shock components?

Table 1 from the paper shows the federal funds rate decomposition. In 2022:Q2, the predictable component was 0.27 percentage points, the unpredictable component due to systematic response to non-monetary shocks was 0.24 pp, and the unpredictable component due to monetary policy shocks was 0.26 pp, summing to 0.77 pp. By 2023:Q2, these were 1.70 pp (predictable), 2.25 pp (systematic/non-monetary), and 1.04 pp (MP shocks), totaling 4.99 pp. Thus, at the tightening cycle’s end in 2023:Q2, the systematic response to non-monetary shocks accounted for about two-thirds of the unpredictable component (2.25 / (2.25 + 1.04) ≈ 68 percent), consistent with the broader literature finding that most variation in policy instruments is driven by the systematic component of policy.

Q9. How do the Hawkish and Dovish Fed counterfactuals work, and what do they imply?

The Hawkish (Dovish) counterfactual replaces the estimated contemporaneous response to inflation in the policy rule with one that is twice (half) as large as the estimated response for the first quarter of 2022, then simulates history forward from 2022:Q2 under the modified rule. Under the Dovish Fed, the economy would have marginally overheated with output rising above CBO potential GDP estimates, and inflation would have run persistently above 5 percent. Under the Hawkish Fed, posterior medians show inflation quickly declining at a cost of about 0.7 percent of real GDP in 2023:Q2 relative to the data; the lower envelope of the 68 percent probability bands shows the output cost could have been as large as 3.1 percent. A parallel set of counterfactuals, designed to be robust to the Lucas critique by working through one-time monetary policy shocks rather than changes to the reaction function, yields broadly similar results.

Q10. What does the comparison with Romer and Romer (2023a) reveal about the model’s monetary policy shock series?

Romer and Romer (2023a) identify a contractionary monetary policy shock in July 2022 (2022:Q3) using a narrative approach. The RC-SVAR’s estimated monetary policy shock series is broadly consistent with this finding: the model detects a contractionary shock in 2022:Q3 and, like Romer and Romer, also finds some evidence of a contractionary shock in 2022:Q2 (though they characterized it as “signs but not definitive evidence”). Beyond the Romer-Romer estimation window, the RC-SVAR additionally finds evidence of an expansionary monetary policy shock in 2023:Q1, when the Fed decelerated the pace of rate increases from 50 to 25 basis points.

Q11. How does the RC-SVAR’s inference on the 2022–2023 tightening cycle differ from that of a constant-parameter SVAR identified only with Restriction 1?

Two salient differences emerge. First, through the lens of the constant-parameter SVAR, monetary policy shocks contribute insignificantly to unexpected output growth between 2022:Q2 and 2023:Q2; in fact, the posterior median output response to a contractionary monetary policy shock is positive in that model (consistent with Uhlig 2005’s finding), implying that the positive monetary policy shocks needed to explain the rate increase would propel rather than reduce output. In the RC-SVAR, the posterior median output response to a contractionary shock is negative, so contractionary monetary policy shocks worked to decelerate output against a backdrop of non-monetary shocks that made the economy run hotter. Second, in the constant-parameter SVAR, non-monetary policy shocks that drive the unexpected increase in the federal funds rate do not propagate through output or inflation, whereas in the RC-SVAR they do—yielding a much more coherent macroeconomic narrative for the tightening cycle.

Q12. What does the model find about whether the Fed was behind the curve in 2021, and what were the consequences?

The model’s 2021:Q1 forecasts predicted the federal funds rate would reach about 0.6 percent by end-2021, consistent with a view that rate normalization was already warranted. The actual federal funds rate remained at its effective lower bound through 2021:Q4, and the shock decomposition shows that the cumulative unexpected change in the funds rate during 2021:Q2–2021:Q4 was driven by expansionary monetary policy shocks—supporting the view that monetary policy was accommodative and the FOMC fell behind the curve. However, monetary policy shocks contributed only about 0.6 percentage points (annualized) to the unexpected increase in core inflation during this period; the dominant and larger source of the inflation surge was non-monetary policy shocks. The model therefore finds that the delay in tightening was not the primary driver of the 2021 inflation surge.

Q13. Do time-varying sign restrictions materially affect inference, as demonstrated in Section 6.8?

Yes. Comparing the baseline identification scheme (Restrictions 1 and 2, with Restriction 2 not imposed during exceptional periods) against an alternative scheme that imposes both restrictions throughout the entire sample reveals differences in the estimated monetary policy shocks, particularly in 2021:Q4. Under the alternative scheme, there was an expansionary monetary policy shock in 2021:Q4, while the baseline finds the shock was nearly centered around zero. Additionally, for 2021:Q2, the alternative scheme implies the contemporaneous output response to an expansionary monetary policy shock is more likely to have been positive, whereas the baseline scheme yields a different posterior distribution for this response. These differences illustrate that imposing or omitting restrictions in specific periods affects inference about structural shocks and impulse responses at economically important junctures.

Key Concepts

Rotation-Invariant Time-Varying SVAR: A class of time-varying SVAR models whose prior over sequences of structural parameters satisfies: for every permissible sequence of structural parameters and every sequence of orthogonal matrices, the orthogonally-rotated sequence is also permissible and receives the same prior density. This ensures the prior does not break the observational equivalence among structural parameter sequences related by orthogonal rotation, so that identification comes solely from the imposed restrictions.

Observational Equivalence in Time-Varying SVARs: Two sequences of time-varying structural parameters are observationally equivalent if and only if there exists a sequence of orthogonal matrices such that one sequence equals the other sequence post-multiplied period-by-period by the corresponding orthogonal matrix. This definition extends Rothenberg’s (1971) concept to the time-varying setting and directly implies the rotation-invariance restriction.

Random Correlations SVAR (RC-SVAR): A specific member of the rotation-invariant class constructed by using the Archakov and Hansen (2021) parametrization of correlation matrices to define the prior over time-varying reduced-form parameters, combined with a uniform prior over sequences of orthogonal matrices. The prior is order-invariant and, for the empirical applications considered, generally achieves higher log-predictive scores than the workhorse Primiceri (2005) model.

Time-Varying Sign Restrictions: Sign restrictions imposed only on selected time periods rather than uniformly across the sample, implemented by allowing the restriction function St() to differ across t (including the possibility that no restriction is imposed at some t). This allows researchers to tailor identification to periods in which the theoretical or institutional knowledge motivating the restriction is deemed applicable—e.g., imposing policy-rule contemporaneous restrictions only when the federal funds rate is the primary policy instrument.

Particle Gibbs with Ancestor Sampling (PGAS): The sequential Monte Carlo method (from Lindsten, Jordan and Schon 2014) used in the paper’s Algorithm 3 to draw the sequence of structural parameters At from its conditional posterior given the sign restrictions. PGAS conditions on the previous Gibbs draw of the structural parameter sequence to ensure an invariant distribution, which is the key property that makes the Gibbs sampler valid for drawing from the correct target posterior.

Systematic Component of Monetary Policy: In the paper’s structural monetary policy equation, the linear combination of contemporaneous endogenous variables (output growth, inflation, money growth, credit spread) that enters the federal funds rate equation, weighted by the contemporaneous elasticities ψ. It represents the portion of interest rate variation that is a predictable, rule-based response to economic conditions, as distinguished from the monetary policy shock (the residual).

Contemporaneous Elasticity: The coefficient ψi,t in the monetary policy equation measuring the response of the federal funds rate to a one-unit contemporaneous change in variable i at time t, defined directly in terms of the structural parameter matrix At. The paper’s time-varying framework allows these elasticities to evolve over the sample, revealing historically distinct episodes of how aggressively the Fed responded to output growth, inflation, money growth, and credit spreads.

Input Sourcing under Climate Risk: Evidence from U.S. Manufacturing Firms

Mon, 01 Jan 0001 00:00:00 +0000

Blaum, Esposito, and Heise study how supply chain risk — specifically, the risk of unexpected shipping delays caused by ocean weather conditions — affects U.S. manufacturing firms’ import sourcing decisions. The paper asks three related questions: Do weather-induced shipping delays harm firm performance? Do firms adapt their sourcing strategies ex ante in response to shipping time risk? And what are the aggregate welfare costs of heightened supply chain risk from climate change, geopolitical tensions, and port congestion?

The empirical foundation is the U.S. Census Bureau’s Longitudinal Firm Trade Transactions Database (LFTTD), covering the universe of U.S. import transactions from 1992 to 2016, merged with the Longitudinal Business Database and Annual Survey of Manufacturers for firm-level outcomes. For ocean shipments, the authors reconstruct vessel routes using vessel names, foreign port stops, and U.S. ports of entry, then map those routes to hourly wave height and direction data from NOAA’s WaveWatch III model at 0.5-degree resolution across more than 40,000 distinct maritime routes (period: 2011–2016 for weather data).

The identification strategy proceeds in two steps. First, observed shipping times are regressed on a rich set of fixed effects — supplier, product, route-month, vessel, buyer, relationship status — plus controls for shipping charges and weight, to strip out anticipated determinants of delivery time. Second, the residuals are projected onto realized wave height and direction along the vessel’s route to isolate the weather-induced, unexpected component of shipping time variation. The identifying assumption is that realized wave conditions along the entire multi-week ocean crossing are not predictable by importers at the time orders are placed, beyond seasonal patterns absorbed by route-month fixed effects. This assumption is supported by the literature on weather forecasting, which finds accuracy degrades sharply beyond seven days.

The paper’s first empirical result concerns the consequences of weather-induced delays. Defining an extreme delay as a weather-induced shipping time above the 95th percentile for a given product-route, the authors estimate that a one standard deviation increase in the share of input costs that are weather-delayed (2.66 percentage points) reduces firm sales by 6.5%, profits by 3.5%, and employment by 1.0% within the same year. These effects are estimated from panel regressions for 2011–2016, with importer, product, and year fixed effects. The magnitudes indicate that firms are typically unable to fully hedge supply chain disruptions through insurance or financial instruments.

The paper’s second empirical result concerns ex ante adaptation. Risk exposure is measured as the standard deviation of weather-induced shipping times over three-year rolling windows for each supplier-route-product combination, then aggregated to the importer-product-year level using pre-determined import shares as weights (Bartik shift-share). Moving from the 25th to the 75th percentile of this shipping risk distribution increases the number of routes used by 7.7% and the number of foreign suppliers by 4.9%, while reducing total import value by 5.1%, route concentration (HHI) by 4.6%, and supplier concentration (HHI) by 3.2%. The risk effect on imports is estimated conditional on average shipping time, indicating that uncertainty exerts an additional, independent negative effect on import demand beyond the level of delays.

To rationalize these findings, the authors build a quantitative general equilibrium model of importing with firm heterogeneity. Firms source domestic and foreign inputs; foreign input quality is reduced when delivery is late, and firms face uncertainty about shipping times when placing orders. Risk-neutral firms nonetheless face a concavity in expected revenues from monopolistic competition, so higher variance in input quality reduces expected profits. Firms can diversify by adding foreign suppliers (at a per-supplier fixed cost), and a key theoretical result is that a mean-preserving spread in supplier quality variance increases the optimal number of suppliers but, because the extensive-margin elasticity is less than one, total import value necessarily falls.

The calibrated model is used to evaluate three counterfactual scenarios. Ocean wave height volatility increased by 0.34% per year on average between 2011 and 2023; projecting this trend forward 50 years generates a climate change scenario. The Houthi attacks in the Red Sea caused rerouting that raised both the mean and variance of navigation time. Post-Covid port congestion (2021–2022) increased the variance of port waiting times. Across all three scenarios, U.S. real income falls by 0.4% to 1.33%, driven by firms substituting toward more expensive domestic inputs as they reduce exposure to risky foreign sourcing.

The sample scope is U.S. manufacturing importers using ocean shipping during 2011–2016 for the main empirical results (weather data period), with an extended robustness sample of 1992–2016 using residualized shipping time volatility. The study covers 43,080 origin-destination port pairs, 401,700 unique vessels, and approximately 35.8 million seaborne transactions.

Q: What is the paper’s core research question? A: The paper asks how supply chain risk — specifically, the risk of unexpected delays in ocean shipping caused by weather conditions — affects U.S. manufacturing firms’ import sourcing decisions and aggregate welfare. It examines both the disruption effects of realized delays and the ex ante adaptation of sourcing strategies to risk exposure, then quantifies aggregate costs through a calibrated general equilibrium model.

Q: What data sources underpin the empirical analysis? A: The primary dataset is the LFTTD, which covers the universe of U.S. import transactions from 1992 to 2016, recording importer and exporter identities, HS-10 product codes, values, quantities, shipping dates, vessel names, and port pairs. This is merged with the Longitudinal Business Database for employment and industry, and with Census of Manufactures and Annual Survey of Manufacturers for sales, material costs, and payroll. Weather data come from NOAA’s WaveWatch III model at hourly, 0.5-degree resolution for 2011–2016. Ocean routes are constructed using Eurostat’s SeaRoute program, covering over 40,000 distinct routes across approximately 10,500 route segments.

Q: How do the authors isolate the unexpected component of shipping time variation? A: They use a two-step residualization. In step one, observed log shipping times are regressed on supplier, product, route-month, vessel, buyer, and relationship-status fixed effects, plus controls for log shipping charges and log weight; the residuals capture variation not explained by anticipated factors. In step two, these residuals are projected onto realized average wave height and relative wave direction along the vessel’s route to extract the weather-induced component. The identifying assumption is that importers cannot forecast realized wave conditions beyond seasonal patterns when placing orders that initiate multi-week ocean crossings, consistent with evidence that weather forecasts lose accuracy beyond seven days and that ocean wave height is particularly hard to predict.

Q: What are the estimated effects of weather-induced shipping delays on firm performance? A: A one standard deviation increase in the share of input costs that are weather-delayed (2.66 percentage points) reduces firm sales by 6.5%, profits by 3.5%, and employment by 1.0% within the same year. Using a broader measure of residualized shipping time delays (not restricted to the weather-induced component) produces similar results: a one standard deviation increase reduces sales by 6%, profits by 3.2%, and employment by 0.9%. These effects are estimated from panel regressions for 2011–2016 with importer, product, and year fixed effects.

Q: How do firms adjust their sourcing strategies in response to higher shipping time risk? A: Moving from the 25th to the 75th percentile of the shipping risk distribution (a 61 log-point increase) raises the number of routes used by 7.7% and the number of foreign suppliers by 4.9%, while reducing route HHI by 4.6%, supplier HHI by 3.2%, and total import value by 5.1%. The margin of route diversification is larger than supplier diversification, consistent with shipping risk being determined primarily at the route level. Higher risk also increases the likelihood of switching to air freight by 1.0% over the same interquartile range.

Q: Does the risk effect on imports operate independently of the level of shipping times? A: Yes. The regressions of total import demand on risk exposure control for average shipping time, and the coefficient on risk remains negative and significant after this control. This indicates that the variance of shipping times has an independent negative effect on import demand beyond the first-moment effect of longer average delays.

Q: What is the theoretical mechanism through which shipping time risk reduces import demand? A: In the model, firms are risk-neutral but face monopolistically competitive output markets, which introduces curvature in the revenue function. Higher variance in input quality (stemming from unpredictable shipping times) reduces expected revenues even for risk-neutral firms. Firms can diversify by adding foreign suppliers at a per-supplier fixed cost, which reduces variance in average input quality. However, the elasticity of the optimal number of suppliers with respect to quality variance is less than one, so total import expenditure necessarily falls as variance rises — diversification is incomplete and firms substitute toward domestic inputs.

Q: What does Proposition 1 state about the extensive margin response to risk? A: Proposition 1 establishes that, under the condition that shipping time risk is small relative to expected revenues, a mean-preserving spread in the variance of supplier quality increases the optimal number of foreign suppliers. However, the elasticity of the optimal number of suppliers with respect to quality variance is strictly less than one, which implies that total import value necessarily falls whenever quality variance increases, regardless of the extensive margin diversification response.

Q: How is the calibration structured and what moments does it target? A: The model features firm heterogeneity in both productivity and shipping time risk (variance of delivery times). The calibration targets three sets of moments: the estimated effect of shipping time risk on the extensive margin of importing (number of suppliers), the negative association between firm sales and average shipping times (which disciplines the timeliness elasticity parameter tau), and the joint distribution of firm size and risk observed in the data — specifically, the empirical finding that larger importers are matched with safer (lower-risk) foreign suppliers, with a correlation of -0.12. The calibrated model replicates the key moments of shipping time risk and import demand.

Q: What are the three counterfactual scenarios and their aggregate welfare costs? A: (1) Climate change: ocean wave height volatility increased by 0.34% per year on average between 2011 and 2023; projecting this trend forward 50 years and passing the resulting increase in shipping time variance through the model. (2) Red Sea/Houthi attacks: re-routing around the Suez Canal raises both the mean and variance of navigation time. (3) Post-Covid port congestion: greater variability in port waiting times during 2021–2022. Across all three scenarios, U.S. real income falls by 0.4% to 1.33%, driven by firms substituting from cheaper foreign inputs toward more expensive domestic production to reduce risk exposure.

Q: What is the role of the shift-share (Bartik) instrument in the risk exposure measure? A: The exposure measure aggregates supplier-route-product level risk (standard deviation of weather-induced shipping times over three-year rolling windows) to the importer-product-year level using pre-determined import shares from the prior three years as weights. Using lagged shares rather than contemporaneous shares ensures that the weights are not endogenous to current sourcing decisions. This construction is standard in the Bartik shift-share literature and helps isolate variation in risk that is plausibly exogenous to the firm’s current sourcing choices.

Q: How do the authors handle the endogeneity concern that firms may select into riskier routes? A: The weather-induced component of shipping time variation is by construction driven by realized ocean conditions that are unpredictable at the time orders are placed. The residualization removes all fixed-effect variation associated with route, season, vessel, supplier, and buyer characteristics. Additionally, the shift-share construction uses pre-determined weights, so risk exposure does not mechanically reflect current sourcing decisions. The authors also show robustness using the longer 1992–2016 sample with residualized (rather than weather-specific) shipping time volatility, obtaining qualitatively and quantitatively similar results.

Q: What does the paper contribute relative to the literature on shipping times and trade? A: Prior work by Evans and Harrigan (2005) and Hummels and Schaur (2010, 2013) focused on the level of shipping times (the first moment) as a trade cost. This paper is the first to systematically study the variance of shipping times (the second moment) as an independent determinant of import demand and sourcing structure, both empirically and theoretically. The authors show that uncertainty around delivery times has negative effects on trade that are separate from the effects of longer average delays.

Q: What are the robustness checks reported for the main empirical results? A: For the effects of risk on sourcing behavior, the authors show that using residualized shipping time volatility over the longer 1992–2016 sample (rather than the weather-induced measure over 2011–2016) produces similar results: moving from the 25th to the 75th percentile increases routes by 6.6%, suppliers by 3.7%, decreases route HHI by 3.9%, and supplier HHI by 2.5%, while reducing total imports by 10.5%. For the effects of delays on firm performance, applying the same specification with residualized (not weather-induced) delay shares yields coefficients on sales, profits, and employment that are very close to the baseline estimates.

Q: What are the welfare implications for firms that cannot hedge through financial markets? A: The large negative effects of weather-induced delays on sales, profits, and employment — and the finding that firms respond by ex ante restructuring their supply chains rather than relying on insurance — indicate that financial hedging instruments are largely unavailable or insufficient for managing input delivery risk. This motivates the model’s assumption that firms must manage risk through sourcing diversification, which is costly because of per-supplier fixed costs and because it ultimately requires substituting toward more expensive domestic inputs.

Weather-induced unexpected shipping time: The component of shipping time variation explained by realized ocean wave height and direction along the vessel’s route, after removing all variation attributable to anticipated factors (route, season, vessel, supplier, buyer characteristics, shipping charges, weight). Interpreted as unexpected because multi-week ocean crossings begin before accurate weather forecasts are available.

Shipping time risk: Measured as the standard deviation of weather-induced residualized shipping times over three-year rolling windows for each foreign supplier-route-product combination. This captures the second moment (variance) of delivery time uncertainty, distinct from the first moment (average shipping time level).

Shift-share risk exposure: An importer-product-year level risk measure constructed as a weighted average of supplier-route-product level risk, using pre-determined import shares from the prior three years as weights. This Bartik-style construction ensures exposure weights are not endogenous to current sourcing decisions.

Timeliness elasticity (tau): A structural parameter in the model governing how rapidly input quality degrades when delivery is later than expected. Specifically, when a shipment arrives di days late, quality is reduced by the factor exp(-tau*(di - E[di])). Calibrated to match the observed negative association between firm sales and average shipping times in the data.

Extensive margin diversification: The response of firms to higher shipping time risk by increasing the number of foreign suppliers and shipping routes used for a given product, rather than increasing the volume sourced from existing suppliers. In the model and data, this margin is the primary channel through which firms hedge delivery risk.

Mean-preserving spread condition: The theoretical condition (Proposition 1) under which higher variance in supplier quality increases the optimal number of foreign suppliers. The condition requires that shipping time risk be small relative to expected revenues, so that the diversification benefit of adding suppliers (reducing variance in average quality) dominates the revenue-reducing effect of higher variance.

Per-supplier fixed cost: A fixed cost in the model that must be paid for each foreign supplier relationship maintained. This cost limits the extent of diversification, ensuring that firms cannot fully eliminate shipping time risk by adding arbitrarily many suppliers, and that higher risk raises (rather than eliminates) per-unit sourcing costs.

Insuring Peace: Index-Based Livestock Insurance, Droughts, and Conflict

Mon, 01 Jan 0001 00:00:00 +0000

This paper provides quasi-experimental evidence that Index-Based Livestock Insurance (IBLI) — a remote-sensing-triggered, automated payout scheme for pastoralists — substantially reduces drought-induced conflict in Kenya over the 2001–2020 period.

The research question is whether a market-based financial instrument can mitigate the causal chain running from drought shocks to violent conflict between nomadic pastoralists and sedentary farmers and other land users. The authors motivate the study by documenting that droughts force pastoralists out of their traditional grazing grounds and into mixed-land-use areas (farms, ranches, urban settlements, nature reserves), where miscoordination with other land users escalates into violence. A case study of the Samburu-Laikipia-Isiolo-Meru region in central Kenya — drawing on georeferenced survey data from Lengoiboni et al. (2010) and ACLED conflict events — validates this spatial mechanism: during droughts, roughly 60–90% of non-pastoral land users report encounters with pastoralists, and conflicts accumulate precisely where drought migration routes cross into non-pastoral land.

The empirical design combines two sources of variation: (1) plausibly exogenous changes in rainfall deficits at the 0.1 × 0.1-degree grid-cell level (roughly 10 × 10 km), derived from NASA GPM satellite data; and (2) the staggered, five-wave rollout of IBLI across 146 insurance districts in Kenya from 2010 onward, which the authors argue was driven primarily by technical challenges rather than pre-existing conflict or drought patterns. The unit of observation is 94,300 cell-periods. Because conflicts due to pastoralist drought migration occur in the neighborhood of affected areas rather than within them, both drought and IBLI coverage are measured as inverse-distance-weighted averages over surrounding cells. The estimating equation is a linear probability model with cell and period fixed effects, interacting neighborhood rainfall deficit with neighborhood IBLI coverage; the coefficient on this interaction term (delta3) is the parameter of interest.

The main finding is that a one-standard-deviation increase in neighborhood IBLI coverage reduces the semi-elasticity of neighborhood rainfall deficit on conflict probability by approximately 23%. In absolute terms, a one-percentage-point increase in the rainfall deficit raises the probability of conflict by 6.92 percentage points at average IBLI coverage; with one additional standard deviation of neighborhood IBLI, that same deficit raises conflict probability by only 5.34 percentage points — a reduction of 1.58 percentage points against a baseline conflict probability of roughly 2.5%.

Scope conditions: the effect is estimated for Kenya specifically, over a pastoralist-heavy population of approximately 8.8 million out of 53 million Kenyans, during 2001–2020. The conflict-mitigating effect is approximately four times larger in mixed-land-use areas (nine times when rollout-cluster-times-period fixed effects are included), consistent with the theoretical expectation that IBLI matters most where pastoralists are most likely to encounter other land users during drought migration.

Two mechanisms are identified. First, IBLI reduces migratory pressure: when pastoral homelands have IBLI coverage, the distance between the ethnic homeland centroid and conflict events involving that group decreases, indicating reduced drought migration. Second, IBLI smooths incomes — corroborated with Afrobarometer geo-coded data — raising the opportunity cost of fighting. An instrumental-variable specification finds that actual IBLI payouts in the neighborhood reduce conflict probability by approximately 150% relative to the baseline risk.

A cost-effectiveness analysis finds that even using conservative World Health Organization or World Bank estimates of the value of statistical life, IBLI delivers fatality savings of between 10 and 22 cents per dollar spent on government subsidies for the program, making it a cost-effective complement to political and institutional conflict-mitigation approaches.

Q: What is the core causal mechanism linking droughts to conflict that IBLI interrupts?

A: Droughts deplete forage in pastoralists’ traditional grazing grounds, forcing them to migrate into mixed-land-use areas — farms, ranches, urban settlements, and nature reserves — where encounters with other land users are more likely to escalate into violence. Without insurance, pastoralists hold excess livestock as precautionary savings, amplifying the extent of necessary migration during dry periods. IBLI payouts allow pastoralists to purchase forage locally, reducing migration distance and intensity, and also smooth income, raising the opportunity cost of engaging in violence.

Q: How does IBLI work technically, and why does it overcome problems of traditional livestock insurance?

A: IBLI uses satellite remote sensing to calculate whether a district-specific drought threshold has been crossed; if so, automated payments are triggered immediately without requiring direct loss assessment or field inspections. This design eliminates moral hazard and adverse selection problems inherent in traditional indemnity insurance, reduces monitoring costs, and enables fast delivery via mobile payment platforms such as MPESA even to remote households. The Kenyan government rebranded the program as the Kenyan Livestock Insurance Program (KLIP) in 2015 and fully subsidizes coverage for up to five tropical livestock units per household.

Q: What is the magnitude of the main conflict-mitigation result?

A: A one-standard-deviation increase in neighborhood IBLI coverage reduces the semi-elasticity of the neighborhood rainfall deficit on conflict probability by approximately 23% (delta3/delta1 = -0.0158/0.0692). In absolute terms, this translates to a reduction from a 6.92 percentage-point increase in conflict probability per one-percentage-point rainfall deficit to a 5.34 percentage-point increase — a decline of 1.58 percentage points against a mean conflict probability of roughly 2.5%.

Q: Why do the authors use a neighborhood rather than cell-level treatment measure?

A: Drought-induced pastoralist conflicts occur primarily not in the pastoral home areas themselves but in neighboring regions where drought migration routes cross into non-pastoral land. The case study documents this pattern directly: ACLED conflict events accumulate where migration routes from Namelok, Lodungokwe, and Ngaremara communities intersect urban or agricultural areas, not within the pastoral zones. The neighborhood approach, using inverse-distance-weighted averages, captures both the probability of migration from surrounding cells and the declining probability of migration with distance.

Q: What is the main identification concern and how do the authors address it?

A: The main concern is that the timing of the IBLI rollout is endogenously determined — areas with a higher latent drought-conflict elasticity might receive coverage earlier or later, biasing the interaction coefficient. The authors show that the pre-treatment drought-conflict elasticity has no systematic correlation with either IBLI eligibility or the timing of coverage receipt. Placebo tests interacting the neighborhood rainfall deficit with pre-treatment eligibility or eventual coverage indicators yield positive, statistically insignificant coefficients, suggesting any bias would run in the direction of underestimating the mitigation effect. A permutation test randomly reassigning IBLI coverage across the six rollout clusters finds the actual point estimate is in the bottom 2.2% of the simulated distribution, indicating it is unlikely to arise from cluster-level confounders.

Q: How do the authors rule out that other programs — cash transfers or development aid — explain the result?

A: The authors control for cell-level and neighborhood-level coverage of Kenya’s Hunger Safety Net Programme (HSNP), which provides unconditional cash transfers to vulnerable households and covers most IBLI-eligible areas, as well as for World Bank agricultural aid projects. Across these specifications, the estimated conflict mitigation ranges from -19.16% to -42.24%, with the baseline estimate of -22.79% remaining robust, indicating neither HSNP nor development aid is a plausible alternative explanation.

Q: What is the alternative identification strategy using within-rollout-cluster variation?

A: The authors exploit pre-determined (1984 government land-use map) variation in mixed-land-use status across cells within the same IBLI rollout cluster-period, including rollout-cluster-times-period fixed effects that absorb any omitted variable related to the potentially endogenous rollout steps. The conflict-mitigating effect of IBLI is approximately four times larger in mixed-land-use cells, and approximately nine times larger in the most restrictive specification with rollout-cluster-times-period fixed effects, consistent with the prediction that IBLI matters most where pastoralists encounter other land users.

Q: How do the authors establish the migratory pressure mechanism?

A: Following Eberle et al. (2023), the authors match conflict actors to ethnic homelands using Murdock (1967) boundaries and test whether IBLI coverage in a homeland reduces the distance between the homeland centroid and conflict events involving that group. They find that it does, indicating that IBLI coverage reduces the spatial range of pastoralist drought migration and thus the probability of conflict-generating encounters with other land users.

Q: How do the authors establish the income-smoothing mechanism?

A: Using geo-coded Afrobarometer survey data, the authors show that IBLI coverage is associated with higher reported incomes among pastoralist households, consistent with Jensen et al. (2017). Higher incomes raise the opportunity cost of fighting (following Grossman, 1991), contributing to the overall conflict-mitigating effect alongside reduced migratory pressure.

Q: What does the instrumental variable specification find?

A: The authors instrument inverse-distance-weighted IBLI payouts in the neighborhood with the interaction of neighborhood rainfall deficit and neighborhood IBLI coverage. The first stage confirms that rainfall deficits trigger payouts conditional on coverage. The second stage finds that the occurrence of payouts in the neighborhood reduces the probability of conflict by approximately 150% relative to the baseline risk, corroborating the reduced-form results.

Q: How do the authors assess cost-effectiveness?

A: The authors predict plausible drought-induced conflict fatalities in Kenya over the pre-treatment period and calculate yearly lives saved from the main estimates, then compare the monetary value of saved lives to government subsidy expenditures on IBLI. Using conservative VSL estimates from the WHO and World Bank, IBLI delivers between 10 and 22 cents of pure fatality savings per dollar of public subsidy expenditure.

Q: How robust are the results to alternative drought and conflict measures?

A: Results are qualitatively similar using an Aridity Index or Dry Matter Productivity (DMP) as drought proxies instead of rainfall deficit. The estimated interaction effect maintains a t-statistic above two for spatial decay functions ranging from distance^-0.5 to distance^-1.5 and for Conley standard error cutoffs from 200 km up to 400 km. Results also hold when restricting to conflict events not involving the government, or to battles, riots, and violence against civilians only, and when excluding the pre-IBLI period (2000–2009) entirely.

Q: What are the policy implications regarding scalability?

A: Pastoralism covers 43% of the African landmass across 36 countries, supporting approximately 268 million people (FAO, 2018). The World Bank and private equity were planning to invest close to 900 million dollars in East African pastoralist programs over 2023–2027. The authors argue that IBLI’s cost structure — high fixed costs of technology and setup but low marginal costs of expansion — gives it a scalability advantage over cash transfer programs or public works schemes that require sustained state capacity. Market-based IBLI complements rather than substitutes for political and institutional reforms.

Index-Based Livestock Insurance (IBLI): A financial instrument that uses satellite remote sensing to automatically trigger preemptive cash payouts to pastoralists when a pre-determined district-specific drought threshold is crossed, bypassing direct loss assessment and thereby eliminating moral hazard and adverse selection problems inherent in traditional indemnity insurance.

Drought-conflict semi-elasticity: The percentage-point change in the probability of conflict associated with a one-percentage-point increase in the rainfall deficit; the paper’s main outcome quantity, estimated at 6.92 percentage points at mean IBLI coverage, reduced by 23% for a one-standard-deviation increase in neighborhood IBLI coverage.

Neighborhood approach: An empirical strategy that measures both drought severity and IBLI coverage as inverse-distance-weighted averages over all surrounding grid cells, reflecting the authors’ finding that pastoralist drought-migration generates conflicts not in the pastoral home area but in neighboring mixed-land-use zones where migration routes intersect other land users.

Migratory pressure: The mechanism by which drought forces pastoralists — who hold excess livestock as precautionary savings in the absence of insurance — to migrate farther from traditional grazing grounds into mixed-land-use areas, increasing the probability of encounters and violent miscoordination with farmers, urban dwellers, and protected-area managers.

Mixed land use: Areas, designated using a 1984 Kenyan government land-use map, where pastoral grazing zones are proximate to farms, ranches, urban settlements, or nature reserves; the paper identifies these as the locations with the highest expected treatment intensity, where IBLI coverage reduces drought-induced conflict approximately four to nine times more than elsewhere.

Tropical Livestock Unit (TLU): The standard unit of account for IBLI contracts in Kenya; one TLU corresponds to one head of cattle or ten goats or sheep; the Kenyan government fully subsidizes IBLI for up to five TLUs per household.

Rollout-cluster-times-period fixed effects: A restrictive set of fixed effects included in the alternative identification strategy that absorbs all omitted variables varying at the level of the six IBLI spatial rollout clusters over time, allowing the authors to identify the conflict-mitigating effect purely from within-cluster variation in mixed-land-use exposure.

Jackknife Standard Errors for Clustered Regression

Mon, 01 Jan 0001 00:00:00 +0000

Hansen (2025) makes a theoretical case for replacing the conventional cluster-robust variance estimator (CRVE) and heteroskedasticity-consistent (HC) standard errors with a specific jackknife variance estimator, V5, in linear regression with heteroskedastic and/or cluster-dependent observations.

The paper identifies two fundamental problems with conventional CRVE1 and CRVE2 estimators. First, these estimators can be fully downward biased: Theorem 2 establishes that the infimum of E[v̂1²]/v² and E[v̂2²]/v² over all admissible regressor and covariance matrix configurations equals zero, meaning expected variance can be arbitrarily close to zero relative to the true variance. This pathology arises from extreme regressor leverage — specifically when one cluster dominates the sample — and holds even under homoskedasticity and clusterwise invertibility. Second, Theorem 5 shows that confidence intervals constructed from CRVE1 and CRVE2 standard errors have worst-case coverage probability equal to zero for any finite critical value c, making them unable to achieve any target coverage level uniformly over regression designs.

Crucially, Hansen shows that even the conventional jackknife estimators V3 and V4, which are already in use (e.g., via Stata’s vce(jackknife) option), share these pathologies when clusterwise noninvertibility is present. Clusterwise noninvertibility occurs when deleting a single cluster renders the regressor matrix singular — as in regressions with cluster-level fixed effects, a single treated cluster, or sparse dummy variables. Stata’s existing fix of simply dropping noninvertible clusters is shown to be insufficient: under clusterwise noninvertibility, the infimum of E[v̂3²]/v² and E[v̂4²]/v² over the broader model class equals zero (Theorem 2, equations 19–20), and the corresponding confidence intervals also achieve worst-case coverage of zero.

The proposed estimator V5 resolves these problems through three modifications to the conventional jackknife: (1) it uses a generalized (Moore-Penrose) inverse rather than dropping noninvertible clusters, ensuring all clusters are included; (2) it centers at the full-sample estimator β̂ rather than the mean of delete-one estimates; and (3) it omits the (G−1)/G degrees-of-freedom correction. Theorem 1 proves that E[V̂5] ≥ V in the positive semidefinite sense for all sample sizes, regressor matrices, and covariance structures — the estimator is never downward biased. Theorem 3 then shows that jackknife-based confidence intervals C̃5(c) have coverage probability bounded below by the Cauchy distribution for any c ≥ 1. With the conventional critical value c = 1.96, this guarantees finite-sample coverage of at least 70% and test size of at most 30%, regardless of regression design or error variance structure.

To improve upon the conservative Cauchy bound in practice, the paper proposes a Satterthwaite adjusted t approximation for the jackknife t-ratio. The adjustment derives degrees of freedom K and a scale factor a from the eigenvalue structure of a design-dependent matrix D. Theorem 7 shows that a → 1 and K → ∞ as n → ∞ under mild regularity conditions (no single cluster dominates). Simulation evidence across six regression designs — varying regressor distributions (Normal, LogNormal with cluster dependence, sparse Dummy) and error structures (clustered normal, heteroskedastic) — with G ∈ {6, 12, 40, 100} clusters confirms that the Satterthwaite jackknife interval achieves coverage rates uniformly above 93% at the nominal 95% level even with G = 6, while CRVE1 intervals fall as low as 57% coverage in the LogNormal/heteroskedastic design. The empirical application extends Meng, Qian, and Yared (2015) on Chinese TV access and redistribution preferences, finding that the jackknife standard error for the TV access coefficient exceeds the CRVE1 standard error and the Satterthwaite interval is wider, affecting conclusions about statistical significance.

The theory holds under Assumptions 1–4: correctly specified linear regression with zero conditional mean errors, full rank X, finite second moments, arbitrary cluster sizes and within-cluster covariance structure, and (for Theorem 3) normal errors. Results hold for fixed k and G, arbitrary n, and allow clusterwise noninvertibility subject to Assumption 3 (inference targets the well-identified regressors).

Q: What is the central claim of the paper? A: Conventional CRVE and HC variance estimators should be replaced by the jackknife estimator V5 in all linear regression contexts with heteroskedastic or clustered errors. V5 is never downward biased (its expectation weakly exceeds the true variance matrix), whereas CRVE1 and CRVE2 can be arbitrarily downward biased. The Satterthwaite-adjusted V5 confidence interval has excellent finite-sample coverage.

Q: What is the worst-case bias of CRVE1? A: The infimum of E[v̂1²]/v² over all admissible regressor matrices and covariance matrices equals zero (Theorem 2, equation 15). This means that for some data-generating process, the expected CRVE1 variance estimate is arbitrarily close to zero relative to the true variance — full downward bias. Importantly, this pathology holds even under homoskedasticity (Σ = Iₙ) and clusterwise invertibility; it is driven entirely by extreme regressor leverage.

Q: Why is CRVE2 also fully downward biased, and how does its failure differ from CRVE1’s? A: Theorem 2 (equation 16) shows that the infimum of E[v̂2²]/v² over F* also equals zero. The difference is that the proof for CRVE2 requires non-i.i.d. errors, meaning CRVE2’s failure requires manipulation of the covariance matrices in addition to extreme leverage, whereas CRVE1 can fail under i.i.d. errors from leverage alone.

Q: What is clusterwise noninvertibility and why does it matter? A: Clusterwise noninvertibility occurs when deleting a single cluster renders the regressor design matrix X’X − Xg’Xg singular. This happens in regressions with cluster-level fixed effects, with a cluster-level treatment indicator when only one cluster is treated, or with sparse dummy variables. The paper shows that the conventional jackknife estimators V3 and V4 become fully downward biased (infimum of expectation ratio equals zero) under clusterwise noninvertibility, even though Stata’s existing fix of dropping noninvertible clusters was explicitly designed to handle this case.

Q: What is the key innovation in V5 that makes it robust to clusterwise noninvertibility? A: V5 uses the Moore-Penrose generalized inverse in the delete-one-cluster estimator β̂₋g, ensuring all G clusters are included in the sum rather than discarding noninvertible clusters. It also centers at the full-sample β̂ rather than the mean β̄ of delete-one estimates, and omits the (G−1)/G degrees-of-freedom correction. The paper shows these three differences together imply V̂5 ≻ V̂4 ≻ V̂3 in the positive semidefinite ordering.

Q: What does Theorem 1 establish about V5? A: Theorem 1 proves E[V̂5] ≥ V in the positive semidefinite sense for all sample sizes, all regressor matrices, all covariance matrices, and under clusterwise noninvertibility. This conservative property holds without any assumption on cluster sizes, regressor leverage, within-cluster correlation, or heteroskedasticity beyond Assumption 1 (correct specification and finite second moments). The infimum of E[v̂5²]/v² equals 1 (equation 21), meaning the inequality is sharp.

Q: What does the Cauchy distribution bound say, and how useful is it in practice? A: Theorem 3 shows that for any c ≥ 1, the jackknife confidence interval C̃5(c) has coverage probability at least P[|ζ| ≤ c] where ζ is Cauchy. With c = 1.96, this guarantees coverage of at least 70% and test size of at most 30% uniformly over all regression designs and error structures (under normality). The bound is not tight in typical applications — actual coverage is much higher — but it provides the first generally applicable uniform guarantee for clustered/heteroskedastic regression. The Cauchy critical value at 5% is 12.7, far too large for practical use, so the bound is more useful as a theoretical guarantee than as a practical inference tool.

Q: What does Theorem 5 establish about confidence intervals from CRVE1–CRVE4? A: Under normality, the worst-case coverage probability of confidence intervals constructed from any of the four estimators v̂1 through v̂4 equals zero for any finite critical value c (equations 26–29). For v̂1 and v̂2, this holds over the clusterwise-invertible model class F*; for v̂3 and v̂4 it holds over the broader class F allowing noninvertibility. Zero worst-case coverage cannot be fixed by enlarging c, since the result holds for all finite c. This is not an impossibility result in the Bahadur-Savage sense; it is a statement that specific commonly-used intervals fail, while V5-based intervals succeed.

Q: What is the Satterthwaite approximation and how is it implemented? A: The Satterthwaite adjustment replaces the jackknife t-ratio’s exact finite-sample distribution — a ratio of a normal to the square root of a weighted sum of chi-squares — with a scaled t distribution with K degrees of freedom, where K and a scale factor a are matched by moment conditions on the eigenvalues of a design matrix D. The confidence interval is θ̂ ± v̂5 · t^{1−α/2}_K / a, and the p-value uses a Student t or F distribution with the same K and scale. These quantities can be computed without explicit eigendecomposition using trace formulas (equations 38–39), which are preferred computationally when G > k.

Q: What do the simulations show about coverage rates? A: Across six designs (three regressor types × two error types) and G ∈ {6, 12, 40, 100}, CRVE1 falls as low as 57% coverage in the LogNormal regressor/heteroskedastic error design with G = 6. CRVE2 has somewhat better but still substantially undercovering intervals. The conventional jackknife interval undercovers (as low as 85%) in leveraged/heteroskedastic designs. The Satterthwaite jackknife interval achieves coverage uniformly exceeding 93% across all designs, though it can be excessively conservative (100%) in some cases. All simulation estimates have standard errors less than 0.003 (20,000 replications).

Q: Does the Satterthwaite adjustment vanish in large balanced samples? A: Yes. Theorem 7 shows that if the design matrix is uniformly non-singular and no single cluster dominates (maxg ||Xg||² = o(n)), then a → 1 and K → ∞ as n → ∞. Consequently, the Satterthwaite interval converges to the standard normal interval in well-balanced large samples.

Q: How does V5 relate to the classical HC3 estimator? A: Under independent sampling (no clustering, ng = 1), V5 reduces to the HC3 estimator of Andrews (1991) and Davidson and MacKinnon (1993), which uses the Moore-Penrose inverse. The conventional jackknife V3/V4 reduce to the HC3 of MacKinnon and White (1985). The paper’s results thus provide a formal theoretical basis for the longstanding recommendation (by Efron-Stein 1981, MacKinnon-White 1985, Andrews 1991, and others) to use HC3/jackknife standard errors.

Q: What is the practical recommendation for empirical researchers? A: Replace all CRVE1/CRVE2/HC standard errors with V5, computed via the Moore-Penrose generalized inverse including all clusters. Report V5-based standard errors (which are never downward biased) alongside Satterthwaite-adjusted confidence intervals and p-values using equations (30)–(31). The adjustment parameters a and K differ per coefficient and must be computed separately for each. The paper advises against reporting a/v̂5 as an “adjusted standard error” since that quantity loses the never-downward-biased property.

Q: What is the empirical application and what does it find? A: The paper extends Meng, Qian, and Yared (2015), which studies the effect of TV access on demand for redistribution in China using provincial household survey data (30 provinces, multiple years), and Canay, Santos, and Shaikh (2021), who found CRVE1 standard errors may be unreliable in that setting. Applying V5, the jackknife standard error for the TV access coefficient exceeds the CRVE1 standard error, the Satterthwaite interval is wider than the conventional interval, and conclusions about statistical significance are affected.

Q: What are the scope conditions and limitations? A: The bias results (Theorems 1–2) require only correct specification (zero conditional mean) and finite second moments. The Cauchy bound (Theorem 3) additionally requires normal errors; whether a similar bound holds without normality or in G → ∞ asymptotics is left open. The Satterthwaite adjustment applies only to inference on real-valued (scalar) parameters and does not extend to joint hypothesis tests. Assumption 3 limits inference to “well-identified” regressors (those whose leave-cluster-out coefficients are uniquely defined after partialling out controls).

V5 (jackknife variance estimator): The paper’s proposed estimator, defined in equation (10) as the sum over all G clusters of outer products of (β̂₋g − β̂), where β̂₋g uses the Moore-Penrose generalized inverse. Unlike conventional jackknife estimators, V5 includes all clusters (no dropping), centers at the full-sample β̂, and omits the (G−1)/G correction. Its key property is E[V̂5] ≥ V for all regression designs.

Never-downward-biased (conservative) estimator: A variance estimator whose expectation is weakly greater than the true variance in the positive semidefinite sense, for all admissible regressor matrices and covariance structures. V5 has this property; CRVE1, CRVE2, and conventional jackknife estimators do not.

Full downward bias: The worst-case property that the infimum of E[v̂²]/v² equals zero over the model class — meaning the expected variance estimate can be arbitrarily close to zero relative to the true variance. CRVE1 is fully downward biased under clusterwise invertibility alone; CRVE2 requires non-i.i.d. errors; conventional jackknife estimators become fully downward biased under clusterwise noninvertibility.

Clusterwise noninvertibility: The condition where deleting a single cluster g renders the matrix X’X − Xg’Xg singular, so the standard delete-one-cluster estimator β̂₋g is undefined. This occurs in regressions with cluster-level fixed effects, a single treated cluster, or sparse dummy variables. V5 handles this via the Moore-Penrose generalized inverse; Stata’s existing fix of dropping such clusters is shown to be non-robust.

Cauchy distribution bound: Theorem 3’s result that the jackknife confidence interval C̃5(c) has coverage probability at least P[|ζ| ≤ c] for all c ≥ 1, uniformly over all regression designs and error variances (under normality). With c = 1.96, this gives a guaranteed coverage floor of 70%. This is the first generally applicable uniform coverage guarantee for clustered/heteroskedastic regression.

Satterthwaite adjusted t approximation: A data-dependent distributional approximation for the jackknife t-ratio that approximates the denominator’s weighted chi-square distribution by a scaled chi-square with K degrees of freedom, where K and scale factor a are computed from trace formulas involving the design matrix. The resulting confidence interval θ̂ ± v̂5 · t^{1−α/2}_K / a converges to the standard normal interval in well-balanced large samples.

Regressor leverage: The degree to which variation in a coefficient of interest is concentrated in a small number of clusters. High leverage (when one cluster dominates the regressor of interest) is the mechanism by which CRVE1/CRVE2 achieve worst-case downward bias even under homoskedasticity.

Labor Market Shocks and Monetary Policy

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research question. The paper asks two related questions: (1) How much, and through which channels, do employer-to-employer (EE) worker transitions affect macroeconomic outcomes — particularly inflation? (2) What is the optimal monetary policy within a class of Taylor rules when EE flows are taken explicitly into account?

Motivation. Standard monetary policy frameworks condition on the unemployment rate as the primary labor market slack measure and underemphasize the “quality” dimension of employment. The paper documents a striking empirical pattern: the 2016–2019 recovery and the 2021–2022 recovery from COVID-19 featured nearly identical declines in the unemployment rate, yet exhibited dramatically different EE rate dynamics and inflation outcomes. During 2016–2019, the EE rate remained flat despite a roughly 25 percent decline in the unemployment rate from trend. During 2021–2022, the EE rate rose by around 8 percent above trend over a comparable unemployment decline. Correspondingly, unit labor cost (ULC) growth reached approximately 6 percent during the COVID-19 recovery when unemployment fell below 4 percent, compared with only about 2 percent ULC growth in the 2016–2019 period at similar unemployment levels.

Methodology. The authors develop a Heterogeneous Agent New Keynesian (HANK) model with a frictional labor market featuring on-the-job search (OJS). Workers are heterogeneous in wealth (mutual fund shares), human capital, match-specific productivity, and endogenous piece-rate wages. Human capital stochastically appreciates when employed and depreciates when unemployed, capturing scarring effects and job-stayer wage growth. Wage determination follows a Bertrand competition protocol based on flow output: workers switch to higher-productivity matches and extract the full surplus from the new firm, while outside offers from lower-productivity firms can still trigger rebargaining with the incumbent firm and raise the piece rate without a job switch. Three vertically integrated sectors — labor services, intermediate goods, and final goods — are linked so that the real price of labor services pl is the real marginal cost for intermediate firms and the sole driver of inflation in the New Keynesian Phillips curve (absent aggregate productivity shocks). The economy is subject to AR(1) shocks to the discount rate β (demand), aggregate labor productivity z (supply), and OJS efficiency ν (the relative search efficiency of employed workers). The model is solved using the Sequence-Space Jacobian (SSJ) method, extended to handle discretized worker distributions as direct inputs to equilibrium conditions.

The model is calibrated to U.S. pre-Great Recession data (2004–2006), targeting the fraction of hand-to-mouth individuals (16 percent of SIPP sample), unemployment rate (5.1 percent), EU separation rate (3.8 percent quarterly), EE rate (2 percent quarterly from LEHD), earnings drop upon job loss (35 percent), wage growth of job switchers (9 percent), and the labor share (0.67). Shock processes are estimated by minimizing deviations from empirical correlations and standard deviations of output, unemployment, EE rate, and inflation over 1995:Q3–2008:Q4.

Main findings — positive analysis. Shocks to OJS efficiency account for 43.1 percent of fluctuations in inflation in the variance decomposition, and 78.7 percent of fluctuations in the EE rate. The mechanism: a higher OJS efficiency lowers the expected match value EJ for labor services firms through three channels — (i) a compositional shift toward employed job seekers who extract the entire match surplus, (ii) shorter expected match duration as workers face higher poaching probabilities, and (iii) more frequent wage rebargaining where outside offers bid up wages without accompanying productivity gains. To maintain the free-entry condition, the real price of labor services pl must rise, increasing the real marginal cost and inflation. This direct labor market effect explains 139 percent of the total increase in pl; general equilibrium effects through reduced tightness θ — which raises expected match values by making vacancies easier to fill and workers less likely to be poached — offset −42 percent; the remainder (3 percent) comes from real rate changes driven by the monetary policy reaction.

In two historical simulations, muted OJS efficiency during 2016–2019 generated approximately 0.23 percentage points lower annualized inflation at the peak relative to a counterfactual economy with the same unemployment path but an endogenously rising EE rate. Conversely, elevated OJS efficiency during 2021–2022 generated approximately 0.56 percentage points higher annualized inflation compared to the flat-EE-rate counterfactual. The paper notes that strong worker mobility accounts for roughly 10 percent of the approximately 6 percentage point total rise in annual inflation during the COVID-19 recovery episode.

An important cross-model comparison shows that the Representative Agent New Keynesian (RANK) version of the model overestimates the decline in demand, output, and labor market tightness upon a positive OJS shock, and underestimates the rise in real rate, marginal cost, and inflation. Household heterogeneity is therefore quantitatively important: hand-to-mouth households’ demand responds directly to labor income increases from job switches, mitigating the demand decline and amplifying inflation.

Main findings — normative analysis. The optimal monetary policy within an augmented Taylor rule — adding an EE gap term ΦEE(EEt − EE*) alongside the standard inflation and unemployment gap terms — prescribes Φ*_u = −3.18 and Φ*_EE = 2.22 (with Φπ fixed at 1.5). This yields a 78.7 percent reduction in the central bank loss relative to the baseline Taylor rule. A policy that ignores EE dynamics and optimizes only the unemployment gap coefficient (finding Φu = −2.71, ΦEE = 0) produces a 12 percent larger central bank loss than the full optimal policy. In terms of welfare, the optimal policy delivers 0.16 percent additional lifetime consumption equivalent in the aggregate. Workers at the bottom of the match quality distribution gain the most (0.24 percent), as do the unemployed (0.20 percent), while those at the top of the wealth distribution gain the least due to larger share price fluctuations under the more aggressive policy.

Scope conditions. Results are derived conditional on a dual-mandate central bank objective (variance of inflation and output gaps), within a class of Taylor-type rules (not fully optimal Ramsey policy), under first-order approximation around a non-stochastic steady state. The historical simulations abstract from supply shocks active in the normative exercises and assume the economy starts from steady state in 2016.

In depth

Q1. What is the OJS efficiency shock, and how does it differ from a standard demand or supply shock?

An OJS efficiency shock is modeled as a time-varying shift in νt, the relative job search efficiency of employed workers compared with unemployed workers. Unlike demand shocks (discount rate β innovations) and productivity shocks (aggregate z innovations), which move inflation and unemployment in opposite directions under standard New Keynesian logic (divine coincidence), OJS efficiency shocks move inflation and unemployment in the same direction: a positive OJS shock raises inflation while also raising unemployment (because the higher real rate induced by the central bank’s reaction reduces demand and employment). This makes OJS shocks behave like cost-push shocks and introduces a genuine policy trade-off for a dual-mandate central bank.

Q2. What are the three mechanisms through which higher OJS efficiency raises the real price of labor services, and what is the quantitative contribution of each?

The decomposition (Figure 8) shows that the direct effect of ν on EJ — encompassing the composition channel (more employed job seekers who extract the full surplus), the match-duration channel (shorter expected match lives), and the wage rebargaining channel (outside offers raise wages without productivity gains) — explains 139 percent of the total increase in pl. The general equilibrium reduction in labor market tightness θ, which raises EJ and partially offsets the cost increase, explains −42 percent in total: −18 percent through increased supply of labor services L (productivity-enhancing job switches improve the match distribution) and −24 percent through reduced output Y (lower aggregate demand). Real rate effects account for the remaining 3 percent net (8 percent from the inflation channel and −5 percent from the unemployment channel). Labor market effects in total therefore explain 97 percent of the marginal cost increase.

Q3. Does the positive relationship between EE rates and inflation require wage increases upon job switches?

No. The paper demonstrates (Section 2.4.2, Figure 3) that even when the piece rate for workers hired from unemployment is set to α = 0.95 (so that outside offers have negligible wage effects), a positive OJS efficiency shock still generates a decline in output and a rise in inflation in both the RANK and TANK models. Quantitatively, the inflation response is similar across the baseline and near-zero composition-channel specifications, confirming that the shorter expected match duration is the primary driver of the increase in the real price of labor services. The match duration channel operates independently of wage increases: firms anticipate shorter matches and require a higher flow price to break even on vacancy costs.

Q4. How does household heterogeneity change the quantitative effects of OJS shocks relative to the RANK benchmark?

Under a constant real rate, in the RANK model a higher OJS efficiency increases the real price of labor services and inflation but has no effect on aggregate demand or output (because higher labor income for the PIH household is exactly offset by lower firm profits). In the TANK model, hand-to-mouth households consume their entire labor income, so the rise in labor income from job switches directly boosts their demand, raising output and tightness and further amplifying inflation. Under an endogenous real rate, the RANK model overestimates the decline in demand and output, and underestimates the rise in real rate and inflation, compared with the TANK model. The TANK model requires a substantially larger equilibrium real rate increase to contain inflation because HtM households’ demand is less elastic to the real rate than PIH households'.

The six AR(1) parameters governing β, z, and ν (three persistence parameters ρj and three standard deviations σj) are estimated by minimizing the sum of squared deviations between model-generated and empirical moments: the autocorrelation of output; correlations of the unemployment rate, EE rate, and inflation with output; and standard deviations of output, unemployment rate, EE rate, and inflation. Data cover 1995:Q3–2008:Q4. Estimated values are ρβ = 0.909, ρz = 0.332, ρν = 0.936 and σβ = 0.001, σz = 0.002, σν = 0.003. The variance decomposition (Table 4) assigns 43.1 percent of inflation variance to OJS efficiency shocks ν, 52.0 percent to demand shocks β, and 4.9 percent to productivity shocks z.

Q6. How is the “missing inflation” during 2016–2019 quantified, and what is the counterfactual?

The exercise simulates two economies both replicating the same unemployment path — a 15 percent decline in unemployment relative to its 5.2 percent steady state, spread linearly over 16 quarters, followed by mean reversion. The first economy uses only positive demand shocks, which generate an endogenously rising EE rate consistent with the historical unemployment-EE correlation. The second economy additionally introduces negative OJS efficiency shocks to keep the EE rate unchanged, as observed in the data during 2016–2019. Annualized inflation in the second economy is 0.23 percentage points lower at the peak (16 quarters after the shock), implying that had the EE rate risen normally, inflation would have been around 2 percent in 2019 rather than the observed 1.8 percent.

Q7. How is the inflationary role of elevated EE transitions during 2021–2022 quantified?

Using the same unemployment path as the 2016–2019 exercise, the COVID-19 recovery economy combines positive demand shocks with positive OJS efficiency shocks to replicate the observed 0.16 percentage point (8 percent above trend) increase in the EE rate. Comparing this economy to the flat-EE-rate economy from the prior exercise, the elevated EE rate generates 0.56 percentage points higher annualized inflation. Because annual inflation rose approximately 6 percentage points in the data during this episode, the model attributes roughly 10 percent of the total inflation increase to strong worker mobility.

Q8. What are the optimal Taylor rule coefficients when EE dynamics are included, and what is the welfare cost of ignoring them?

The optimal policy over the augmented Taylor rule it = i* + Φπ(πt − π*) + Φu(ut − u*) + ΦEE(EEt − EE*), with Φπ fixed at 1.5 and a dual-mandate loss function W = var(πt − π*) + 0.25·var(Yt − Y*), prescribes Φ*_u = −3.18 and Φ*_EE = 2.22. This reduces the central bank loss by 78.7 percent relative to the baseline rule (Φu = −0.25, ΦEE = 0). If the EE gap term is excluded and only the unemployment gap coefficient is re-optimized (finding Φu = −2.71), the central bank loss is 12 percent higher than under the full optimal policy.

Q9. How does the optimal policy affect macroeconomic volatility, and who gains most from it?

Table 5 shows that the optimal policy substantially reduces volatility of inflation (standard deviation falls from 0.0013 to 0.0011), output (0.0059 to 0.0020), consumption (0.0059 to 0.0020), unemployment (0.0047 to 0.0013), labor market tightness (0.0600 to 0.0175), and the real marginal cost pl (0.0203 to 0.0081), at the cost of higher real rate volatility (0.0019 to 0.0033) and share price volatility (0.1975 to 0.3051). In terms of welfare (Table 6), the unemployed gain 0.20 percent in lifetime consumption equivalents (versus 0.15 percent for the employed), workers at the bottom quintile of match quality gain 0.24 percent (versus 0.16 percent at the top), and wealth-poor individuals in the bottom share quintile gain 0.23 percent (versus 0.11 percent at the top, whose gains are eroded by larger share price fluctuations).

Q10. How does the model extend the SSJ computational method, and why is this extension necessary?

The standard SSJ method of Auclert, Bardoczy, Rognlie, and Straub (2021) handles settings where only scalar aggregates enter equilibrium conditions in sequence space. In this model, the discretized distributions of employed workers µE(h, x) and unemployed workers µU(h) at the job search stage enter directly into the expected match value EJ (because human capital and current match productivity determine output and wage levels upon new contacts), and the distribution λE(h, x, α) at the production stage enters into labor services firm profits ΓS. The authors treat worker distributions as histograms and compute Jacobians for each mass point, combining the SSJ method with Reiter (2009)-style projection. This substantially increases computation time but remains feasible, extending the SSJ method to multi-stage models with search frictions where endogenous distributions are state variables.

Q11. What are the three sources of wage growth in the HANK model, and what is their relevance for inflation dynamics?

First, human capital h stochastically appreciates during employment (at rate πE = 0.018 per quarter, calibrated to annual job-stayer wage growth of approximately 2 percent), raising wages through a higher piece-rate base. Second, job switches to higher-productivity matches yield wage increases as the worker extracts the full surplus from the new firm (the new piece rate equals x/x’, the ratio of old to new match productivity). Third, outside offers with productivity x’ satisfying αx < x’ < x — not good enough to trigger a switch but better than the current bargaining threat — cause the incumbent firm to raise the piece rate to x’/x via rebargaining, increasing wages without a job change. The second and third channels are the ones directly affected by OJS efficiency shocks and are inflationary: they raise labor costs beyond productivity gains.

Q12. Why do OJS shocks have a shorter match duration channel even without wage increases?

When OJS efficiency ν rises, each employed worker faces a higher probability νtf(θt) of contacting another firm each period. Even if wages do not change upon contact (as in the α = 0.95 robustness exercise), a labor services firm posting a vacancy expects that any match it forms will be shorter-lived: the worker is more likely to be poached in the future. This shortens the expected present discounted value of the match for the firm, reducing EJ. To satisfy the free-entry condition (expected profit = vacancy cost κ), the price of labor services pl must rise, increasing the real marginal cost and inflation. Figure 3 confirms a nearly identical inflationary response under α = 0.95 as under the baseline, isolating this match-duration mechanism.

Key Concepts

OJS efficiency shock (νt shock). A time-varying shift in the relative job search efficiency of employed workers compared with unemployed workers. Modeled as an AR(1) process for νt (estimated persistence ρν = 0.936). An increase in νt raises the probability that employed workers contact outside firms each period, boosting the EE rate. In the model, this acts as a cost-push shock: it raises inflation and unemployment simultaneously, breaking divine coincidence and creating a policy trade-off for a dual-mandate central bank.

Expected match value (EJt). The ex-ante expected value to a labor services firm of a filled vacancy, conditional on contacting a worker, defined as a weighted average of match values J across the pool of job seekers (unemployed and employed). The free-entry condition Vt = κ/q(θt) = EJt pins down the real price of labor services pl: when EJt declines (due to shorter match durations or compositional shifts toward high-surplus-extracting workers), pl must rise to maintain zero expected profit for vacancy posters.

Composition channel. The mechanism by which a rise in OJS efficiency shifts the composition of the job-seeker pool toward employed workers, who (under Bertrand competition) extract the entire flow surplus of a new match and receive wage equal to plF(h,x). Since firms receive zero rent from poached workers, an increase in the fraction of employed in the applicant pool lowers EJt and requires a compensatory increase in pl.

Match duration channel. When OJS efficiency ν rises, each existing match faces a higher probability of dissolution because the worker is more likely to be poached. The reduced expected match duration lowers the present discounted value of a match for the firm (even holding wages fixed), reducing EJt and raising pl. Demonstrated as the primary driver of inflation in the α = 0.95 robustness exercise where wage increases upon job switches are near zero.

Piece-rate α (endogenous). The share of match output F(h,x) that the worker receives as wage, determined through Bertrand competition on flow output following Postel-Vinay and Robin (2002). A worker hired from unemployment starts at α = x̄/x’ (where x̄ is the lowest match productivity). Job switches to higher-x’ firms reset α = x/x’. Rebargaining upon a credible outside offer from a firm with αx < x̃ < x raises α to x̃/x. The piece rate endogenizes wage dynamics for switchers, stayers, and job losers, allowing the model to discipline these moments in the data.

Divine coincidence (and its breakdown under OJS shocks). In standard New Keynesian models, demand and productivity shocks move inflation and unemployment gaps in opposite directions, so stabilizing inflation also stabilizes the output gap. OJS efficiency shocks break this property: they generate simultaneous increases in inflation and unemployment, introducing a genuine trade-off between the two mandates and making EE-augmented Taylor rules welfare-improving relative to rules that respond only to unemployment.

Sequence-Space Jacobian (SSJ) method with distributed worker states. An extension of the Auclert, Bardoczy, Rognlie, and Straub (2021) computational method to settings where discretized distributions of workers (µE(h,x) and µU(h)) enter directly into equilibrium conditions — specifically into the free-entry condition via EJt and into firm profits. The authors treat distributions as histograms and compute Jacobians for each mass point, combining SSJ with Reiter (2009)-style projection to efficiently solve for transitional dynamics under aggregate uncertainty.

Latent Heterogeneity in the Marginal Propensity to Consume

Mon, 01 Jan 0001 00:00:00 +0000

Lewis, Melcangi, and Pilossoph estimate the unconditional distribution of the marginal propensity to consume (MPC) using the 2008 Economic Stimulus Act (ESA) rebate payments, deploying Gaussian mixture linear regression (GMLR) — a clustering regression approach — rather than the standard practice of interacting the rebate with observable household characteristics. The key methodological departure is that households are assigned to groups not by any presupposed observable, but by how well estimated group-specific MPCs describe each household’s actual consumption response; this allows recovery of the full unconditional MPC distribution, including heterogeneity driven by latent (unobservable) factors.

Data come from the 2008 Consumer Expenditure Survey (CEX), which contains household-level expenditure data and supplemental questions on ESA payments. Identification exploits the quasi-random timing of rebate receipt, determined by the last two digits of recipients’ Social Security Numbers, following the design of Parker, Souleles, Johnson, and McClelland (2013). The specification is updated following Borusyak et al. (2024) to avoid “forbidden comparisons” in staggered treatment settings. The number of groups G is selected by BIC, which selects G = 3 for total expenditures, confirmed by K-fold cross-validation.

The main finding is substantial MPC heterogeneity. For total expenditures, the three estimated group-level MPCs are 0.04, 0.23, and 1.33, with population shares of 30%, 48%, and 23% respectively. The implied aggregate (share-weighted average) MPC is 0.42, compared to 0.24 in the homogeneous Parker et al. (2013) specification estimated on the same data. Splitting by consumption category: for nondurables, two groups have MPCs of 0.09 and 0.18, with roughly equal population shares, and the lower bound of 0.09 is statistically distinguishable from zero — evidence against strict adherence to the Permanent Income Hypothesis even among the lowest-MPC group. For durables, the MPC distribution is dichotomous: about 29% of households have a durable MPC statistically indistinguishable from zero, while 21% have an MPC of 0.67. The cross-good correlation between household-level nondurable and durable predicted MPCs is only 0.13, ruling out strong substitution but indicating weak complementarity.

Turning to observable determinants, the paper finds that many household characteristics are individually correlated with estimated MPCs — including homeownership, mortgage status, income, and the average propensity to consume (APC) — despite the fact that the same dataset and similar identification strategies previously yielded insignificant relationships. Homeowners have significantly higher MPCs than renters; households with a mortgage have even higher MPCs than outright homeowners. In salary income, households in the top tercile spend 0.17 more per rebate dollar than the baseline group; households in the top tercile of non-salary income spend 0.19 more. However, in joint regressions, only two characteristics remain robustly and positively correlated with MPCs: total income (both salary and non-salary components) and the APC. The APC relationship is particularly notable: a one-percentage-point higher prior spending rate is associated with 0.19 additional cents spent per rebate dollar in the full multivariate specification.

The paper identifies three groups in the joint income-APC space: “poor savers” (low income, low APC, lowest MPCs), an intermediate group (high income or high APC but not both), and “rich spenders” (high income and high APC, highest MPCs). The “rich spender” group has received little prior attention in consumption-savings models.

Critically, observable characteristics jointly explain at most 8% of MPC variation (adjusted R-squared from a measurement-error correction). With 92% of MPC heterogeneity unexplained by standard observables, the authors conclude that a substantial share of variation reflects latent household traits — plausibly heterogeneity in discount rates or intertemporal elasticities of substitution. This finding also limits the practical scope for government targeting of fiscal transfers: because observable characteristics predict little MPC variation, any targeting strategy can exploit only a small fraction of the overall distribution.

Scope conditions: results apply to household expenditure responses (marginal propensities to spend, not to consume in the strict sense) within one quarter of rebate receipt. The income-MPC positive correlation is confined to households within the income range eligible for the 2008 ESA (phased out above $150,000 for joint filers). The sample excludes the top and bottom 1.5% of consumption changes as outliers.

Q: What is the core methodological innovation of this paper? A: The paper applies Gaussian mixture linear regression (GMLR) to the 2008 tax rebate setting, jointly estimating group-level MPCs and household group membership probabilities without imposing any prior restriction on which observable characteristics drive heterogeneity. Because groups are determined by how well group-specific MPCs explain consumption patterns rather than by presupposed observables, the method recovers the full unconditional distribution of MPCs, including latent heterogeneity. This contrasts with sample-splitting approaches that can only recover co-variation with chosen characteristics.

Q: What are the three group-level MPCs for total expenditures, and what shares of the population do they represent? A: The three estimated MPCs are 0.04 (30% of households), 0.23 (48%), and 1.33 (23%), all with precisely estimated group shares (standard errors of 0.01). The largest MPC of 1.33 is statistically significant at the 1% level. The lowest MPC of 0.04 is not statistically different from zero even under the more favorable conditional standard errors that treat group assignment as known.

Q: How does the average MPC implied by the GMLR distribution compare to the homogeneous specification? A: The share-weighted average MPC from the three-group GMLR is 0.42, compared to 0.24 from the homogeneous (G=1) specification on the same data and identification strategy. This gap arises partly because the homogeneous estimate averages across households with very heterogeneous responses, and partly because the distribution has a right-skewed tail with a meaningful mass at MPC above 1.

Q: What are the MPC distributions for nondurable and durable goods separately? A: For nondurables, BIC selects two groups with MPCs of 0.09 and 0.18 and roughly equal population shares (48% and 52%); crucially, the lower bound of 0.09 is statistically distinguishable from zero at the 5% level, providing evidence that no household strictly follows the Permanent Income Hypothesis for nondurables. For durables, BIC selects three groups: MPCs of 0.03 (not distinguishable from zero, 29% of households), 0.15 (50%), and 0.67 (21%), reflecting the discrete, lumpy nature of durable goods purchases.

Q: How correlated are nondurable and durable MPCs at the household level? A: The correlation between household-level posterior predicted MPCs for nondurables and durables is 0.13, statistically significant at the 1% level. This rules out substitution between goods categories, but the positive complementarity is quantitatively small. The authors interpret this as possibly reflecting a small share of “spender” types who adjust multiple consumption categories in response to transitory income shocks.

Q: Which observable characteristics are individually correlated with MPCs? A: Homeowners have significantly higher MPCs than renters; households with a mortgage display even greater MPCs than outright homeowners. Both salary and non-salary income are positively correlated: households in the top tercile of salary income have MPCs about 0.13 higher than the omitted group, and top-tercile non-salary income households have MPCs about 0.015 higher (though the latter is individually less precisely estimated). The average propensity to consume (APC) is significantly positively correlated with the MPC, with a coefficient of 0.075 in univariate regression and 0.166 in the full joint specification.

Q: Which observable characteristics remain significant in the joint (multivariate) regression? A: When all household characteristics are included jointly, only income (both salary and non-salary components) and the APC remain robustly and positively correlated with MPCs. Top-tercile salary income is associated with 0.112 higher MPCs and top-tercile non-salary income with 0.049 higher MPCs, while the APC coefficient rises to 0.166 (from 0.075 univariate). Homeownership, age, education, and most demographic controls become statistically insignificant in the joint specification.

Q: What fraction of MPC variation is explained by observable characteristics? A: The adjusted R-squared from the full multivariate regression of predicted MPCs on all observable characteristics is approximately 6%. After a measurement-error correction proposed in Supplement A.6 to account for noise in estimated posterior MPCs, the corrected R-squared rises to 8%. Either way, the vast majority — over 90% — of MPC heterogeneity is unexplained by standard observables, implicating latent household traits such as heterogeneous discount rates or intertemporal elasticities of substitution.

Q: How does the extent of MPC heterogeneity recovered by GMLR compare to sample-splitting on observables? A: Table 4 shows that splitting by age terciles yields MPC estimates ranging from 0.13 to 0.34; splitting by total income yields a range of 0.18 to 0.45; splitting by the APC yields 0.06 to 0.21. All of these ranges are far narrower than the GMLR-recovered range of 0.04 to 1.33. The authors argue that sample-splitting on individual observables, which are noisy and correlated with only a portion of MPC heterogeneity, systematically understates the true extent of heterogeneity.

Q: What is the “rich spender” finding and why is it theoretically notable? A: Households with both high total income and a high prior average propensity to consume have the largest MPCs. This “rich spender” group is poorly accommodated by standard consumption-savings models: the canonical one-asset incomplete markets model typically predicts a negative MPC-APC correlation conditional on income, and the two-asset Kaplan-Violante (2014) model can generate wealthy hand-to-mouth households with high income and high MPCs, but not necessarily high APCs. Preference heterogeneity — e.g., heterogeneous intertemporal elasticities of substitution as in Aguiar, Boar, and Bils (2019) — can rationalize the positive income-APC-MPC nexus.

Q: What explains the positive income-MPC correlation, and how does the paper relate it to the prior literature? A: The paper notes that this positive correlation is consistent with Kueng (2018), who finds higher spending propensities among high-income recipients of Alaska Permanent Fund payments, and rationalizes it via near-rationality or mental accounting: when a rebate is small relative to income, the perceived cost of deviating from consumption smoothing is low. The authors also note that low-income households still exhibit large absolute MPCs, suggesting sizable deviations from consumption smoothing at the bottom of the income distribution, even if relatively lower than for high-income households.

Q: What are the policy implications for targeting fiscal transfers? A: The paper finds that the 2008 ESA increased spending for all households in partial equilibrium (minimum group MPC of 0.04, nondurable lower bound 0.09, all statistically positive or near-positive). Among observable characteristics, targeting relatively higher-income households (including retirees and entrepreneurs via non-salary income) would maximize aggregate consumption effects. However, since observables explain only 8% of MPC variation, any targeting strategy can exploit only a small fraction of the overall heterogeneity; the government faces fundamental limits on feasible targeting. This also implies a tension between stimulus and distributional/insurance motives for transfer programs.

Q: How does the paper confirm that recovered heterogeneity is not spurious? A: The authors generate 250 Monte Carlo samples from the estimated homogeneous model, impose G=3, and re-run the GMLR and observable regressions; they find significant relationships with observable characteristics in virtually none of these samples. Additionally, applying the BIC to homogeneous Monte Carlo samples, the BIC selects G=1 in all 250 samples, confirming that the selected G=3 in actual data reflects genuine heterogeneity rather than overfitting.

Q: How does GMLR compare to quantile regression for recovering the MPC distribution? A: Quantile regression (as used by Misra and Surico (2014) on the same data) recovers relationships at percentiles of the overall conditional distribution of consumption changes, so the ranking of households is driven by all sources of variation in consumption, not just the rebate response. If factors unrelated to the rebate dominate the conditional distribution, MPC heterogeneity will be underestimated in the presence of noise. The authors illustrate this formally in Supplement B and note that Misra and Surico (2014) find a substantial share of MPCs at or below zero for nondurables, in contrast to the GMLR lower bound of 0.09 that is statistically positive.

Q: What do the longer-run (lagged) MPC estimates show? A: The specification includes up to two lags of rebate indicators, allowing measurement of spending responses in subsequent quarters after rebate receipt. The paper reports these results (Section 4.4) but the text provided does not fully detail them; the heterogeneous structure is maintained across horizons.

Gaussian Mixture Linear Regression (GMLR): A probabilistic clustering regression approach that jointly estimates group-specific regression coefficients (here, MPCs) and population group shares by maximizing an expected log-likelihood via the EM algorithm. Households receive continuous posterior weights (gamma_{jg}) reflecting uncertainty about their group membership rather than binary hard assignment, with identification from a Gaussianity assumption on within-group errors.

Unconditional MPC Distribution: The full marginal distribution of MPCs across all households in the population, capturing heterogeneity from both observable and latent (unobservable) sources. Contrasted in the paper with the conditional distributions recovered by sample-splitting on observables, which by construction can only reflect co-variation with the chosen splitting variable.

Posterior Predicted MPC: For each household, the expectation of the group-specific MPC weighted by the household’s posterior group membership probabilities (lambda-tilde_{0,j} = sum_g gamma_{jg} lambda_{0g}). This object is the optimal (MSE-minimizing) individual-level MPC prediction and is the relevant input for targeted fiscal policy design.

Latent Heterogeneity: MPC variation that cannot be attributed to any observable household characteristic and is instead driven by unobserved traits — plausibly heterogeneous discount rates, intertemporal elasticities of substitution, or other preference parameters. Operationalized as the share of MPC variance unexplained by observable regressors (approximately 92% in this paper).

Rich Spenders: A group identified jointly in the APC-income space: households with both high total income and a high average propensity to consume, displaying the largest marginal propensities to consume out of the rebate. This group is not well-accommodated by standard one-asset or two-asset incomplete markets models under homogeneous preferences.

Average Propensity to Consume (APC): Defined empirically as average lagged consumption expenditures divided by total income, intended to capture persistent preference heterogeneity — a “spender type” — by measuring how much of income a household habitually spends before receiving the rebate. A one-percentage-point higher APC is associated with 0.19 additional cents spent per rebate dollar in the full multivariate specification.

Forbidden Comparisons: A bias identified by Borusyak et al. (2024) in event-study designs with staggered treatment, arising when newly treated units are compared to previously treated units rather than true controls. The paper addresses this by regressing consumption changes on rebate receipt indicators (iota_{jl}) directly rather than on rebate amounts, and including lagged rebate indicators to account for persistent effects.

Lender concentration of external debts and sudden stops

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper studies how the lender structure of external debt — specifically, the degree to which a borrowing country’s external debt is concentrated among a small number of large lenders — affects open economies’ credit conditions, borrowing behavior, and the severity of sudden stops.

Core Mechanism

The paper argues that the pecuniary externality arising from collateral foreclosure can be internalized not only by borrowers (as in the standard Bianchi 2011 framework) but also by lenders. When a large lender holds a substantial share of total loans, it has an incentive to foreclose only partially on seized collateral. Selling foreclosed collateral injects asset supply and depresses the collateral price; a sufficiently large lender internalizes this price impact and therefore restrains foreclosure. Atomistic lenders, by contrast, take the collateral price as given and sell all seized collateral (foreclosure rate = 1). Consequently, concentrating external debt in fewer, larger lenders supports a higher collateral price during financial downturns. This higher collateral price raises borrowing capacity, weakens borrowers’ precautionary saving motive, and causes them to overborrow relative to the social optimum.

Empirical Evidence

Using FFIEC 009a data — quarterly exposure of individual U.S. banks to the external debts of other countries, covering 2003Q1–2022Q2 — the paper documents two new empirical facts. First, lender concentration of emerging countries’ external debt has been considerably higher than that of advanced countries since the Global Financial Crisis. The average difference in the mean top-3 lender concentration (LTop3) between emerging and advanced economies is 0.11 (= 0.93 − 0.82), with a t-statistic of 13.87. Second, higher lender concentration alleviates sudden stop events in terms of both current account reversal and the decline in asset price proxies. In a difference-in-differences specification interacting sudden stop indicators with lagged lender concentration, the coefficient on the interaction term is negative and statistically significant across all concentration measures. A one-standard-deviation increase in LTop3 (7.2 percentage points) results in a 2.6 percentage point reduction in current account-to-GDP reversal during sudden stops, constituting 7.5% of the overall sudden stop increase. Lender concentration also mitigates real effective exchange rate depreciation during sudden stops, consistent with the mechanism operating through the collateral price channel. Results hold when controlling for rollover risk motives.

Model

The model extends a standard small open economy DSGE framework (Bianchi 2011) by introducing one large lender who holds share eta of total loans and internalizes the pecuniary externality of collateral foreclosure, alongside atomistic lenders who hold share (1 − eta) and take the collateral price as given. When tradable endowment falls short of debt obligations (foreclosure state), lenders optimally choose their foreclosure rate: atomistic lenders set foreclosure rate = 1 (sell all seized collateral), while the large lender sets foreclosure rate < 1 (partial foreclosure to maintain the collateral price). Higher lender concentration (larger eta) leads to lower aggregate foreclosure, less collateral sold, a higher nontradable goods price, a higher borrowing capacity, more tradable consumption, and a weaker precautionary saving motive — generating overborrowing relative to the social planner’s allocation.

Two channels through which concentration affects overborrowing are identified: (1) a debt capacity channel, whereby concentration raises the nontradable price in foreclosure states and thereby increases borrowing capacity; and (2) an amplification channel, whereby concentration steepens the decline in nontradable price per unit fall in tradable consumption, amplifying the pecuniary externality that the social planner internalizes.

Quantitative Results (Calibrated to Argentina)

In the competitive equilibrium, agents encounter foreclosure with probability 2%, and the large lender sells two-thirds of seized collateral. The social planner’s allocation eliminates foreclosure entirely. The social planner’s allocation can be implemented via a state-dependent debt tax; the implied consumption-equivalent welfare gain is 0.78%. The pecuniary externality internalized by lenders is estimated to equal two-thirds of the externality internalized by borrowers. Overborrowing is increasing in lender concentration.

Optimal Lender Structure

When lender countries optimally choose their lender structure, they select further concentration relative to the baseline in order to gain higher foreclosure repayment. Under optimal lender structure, domestic agents consume and borrow more and encounter sudden stops with higher probability, but completely avoid foreclosure events. Borrower welfare improves by 0.1% in consumption-equivalent terms relative to the baseline competitive equilibrium. The paper concludes that managing lender structure benefits both sides of the international credit market, and notes that policies targeting creditor coordination — such as collective action clauses — may be insufficient to fully correct the efficiency implications of lender structure.

Key Implication

Because lender concentration alleviates crisis severity, emerging economies (which are documented to have substantially more concentrated lender structures than advanced economies) face a reduced precautionary saving motive and therefore tend to overborrow more than advanced economies, compounding their vulnerability to sudden stops.

In depth

Q1. What is the paper’s central departure from the Bianchi (2011) sudden stops framework?

The standard Bianchi (2011) model features atomistic lenders who take the collateral price as given, so the pecuniary externality of collateral fire-sales is internalized only by the borrower’s social planner. This paper introduces a large lender who holds a non-trivial share eta of total loans and therefore internalizes the price impact of selling foreclosed collateral. This creates a second source of pecuniary externality internalization — on the lender side — that is absent from the canonical framework.

Q2. Why do atomistic lenders sell all seized collateral, while the large lender does not?

Atomistic lenders take the collateral price as given and therefore face no downside from selling their entire share of seized collateral — they cannot individually affect the price. The large lender, holding share eta of total loans, recognizes that selling a large quantity of collateral depresses the nontradable goods price, which reduces the value of any remaining collateral claims. It therefore optimally sets foreclosure rate < 1, retaining some seized collateral to support the equilibrium price.

Q3. What are the two channels through which lender concentration amplifies overborrowing, and how do they differ?

The debt capacity channel operates in foreclosure states: higher concentration reduces foreclosure, raises the nontradable price, and increases the collateral value that backs borrowing. This directly expands the borrowing capacity available to agents and weakens their precautionary saving motive. The amplification channel operates through the slope of the nontradable price response: greater concentration steepens the decline in the nontradable price per unit fall in tradable consumption, which amplifies the pecuniary externality that the social planner internalizes. The two channels reinforce each other in driving overborrowing.

Q4. What empirical dataset is used, and what does it measure?

The paper uses FFIEC 009a data, which records the quarterly exposure of individual U.S. banks to the external debts of other countries, covering 2003Q1–2022Q2. From these data, the paper constructs lender concentration measures — including LTop3, the combined share of the top three lenders — at the borrowing-country level for each quarter.

Q5. What is the quantitative magnitude of the lender concentration gap between emerging and advanced economies?

The average difference in mean top-3 lender concentration (LTop3) between emerging countries and advanced countries is 0.11 (= 0.93 − 0.82), and this difference is highly statistically significant, with a t-statistic of 13.87. This gap emerged and persisted notably since the Global Financial Crisis.

Q6. How does lender concentration affect sudden stop severity in the empirical specification, and how large is the effect?

The paper estimates a difference-in-differences specification in which current account reversal (and other sudden stop outcome variables) is regressed on a sudden stop indicator, lagged lender concentration, and their interaction, with country and time fixed effects. The coefficient on the interaction term is negative and statistically significant across all concentration measures. A one-standard-deviation increase in LTop3 (7.2 percentage points) reduces current account-to-GDP reversal by 2.6 percentage points, which corresponds to 7.5% of the overall increase in the current account during a sudden stop episode.

Q7. Does higher lender concentration also mitigate exchange rate and asset price pressures during sudden stops?

Yes. Lender concentration is also found to mitigate real effective exchange rate depreciation during sudden stops, which is consistent with the model’s proposed mechanism: higher concentration supports the collateral (nontradable goods) price, which in turn limits the depreciation of the real exchange rate. The paper reports results on asset price proxy declines as well.

Q8. What is the welfare cost of overborrowing under the baseline calibration to Argentina?

The social planner’s allocation, implemented by a state-dependent debt tax, delivers a consumption-equivalent welfare gain of 0.78% relative to the competitive equilibrium. This measures the efficiency cost of overborrowing under the calibrated model in which the large lender sells two-thirds of seized collateral and competitive equilibrium agents encounter foreclosure with probability 2%.

Q9. How large is the lender-side pecuniary externality relative to the borrower-side externality?

Under the baseline calibration, the pecuniary externality internalized by lenders is estimated to be two-thirds of the externality internalized by borrowers. This is described as a “plausible parameterization,” meaning that lender-side internalization of the externality is quantitatively substantial relative to the classic borrower-side effect.

Q10. What does the optimal lender structure exercise find, and what does it imply for welfare?

When lender countries are allowed to optimally choose lender structure, they select a more concentrated structure than the baseline in order to maximize foreclosure repayment. Under this optimal structure, domestic (borrowing-country) agents consume and borrow more, face sudden stops with higher probability, but completely avoid foreclosure events. Borrower welfare improves by 0.1% in consumption-equivalent terms relative to the baseline competitive equilibrium. This implies that concentrating lender structure can be mutually beneficial for both sides of the international credit market.

Q11. Why might collective action clauses be insufficient to correct the efficiency implications of lender structure?

Collective action clauses are policies designed to improve creditor coordination in sovereign debt restructuring. The paper argues that the efficiency distortions arising from lender structure go beyond pure coordination failures: because a concentrated lender structure generates welfare-relevant pecuniary externalities through the collateral price channel — affecting overborrowing and crisis severity — addressing creditor coordination alone is insufficient to fully resolve these inefficiencies.

Key Concepts

Lender concentration (LTop3): The combined loan share held by the top three lenders in a borrowing country’s external debt. Measured using FFIEC 009a data. Used as the primary empirical proxy for the degree to which external debt is concentrated in a few large creditors rather than dispersed among many atomistic lenders.

Pecuniary externality (lender-side): The price impact that a large lender imposes on the collateral market when selling foreclosed assets. Unlike in the standard Bianchi (2011) framework where only borrowers (via the social planner) internalize this externality, a sufficiently large lender also internalizes it by restraining collateral sales to support the collateral price.

Foreclosure rate (zeta): The fraction of seized collateral that a lender sells after foreclosure. Atomistic lenders set zeta = 1 (sell everything); the large lender sets zeta < 1 (partial foreclosure) to prevent collateral price depression. The aggregate foreclosure rate is a weighted average across lender types.

Overborrowing: Borrowing in excess of the social planner’s optimal level, arising because competitive equilibrium agents do not internalize the pecuniary externality of their borrowing on the collateral price. In this model, overborrowing is increasing in lender concentration because a more concentrated lender structure supports a higher collateral price, reducing precautionary saving.

Sudden stop: An abrupt reversal of capital inflows to an emerging economy, typically associated with a sharp current account reversal, real exchange rate depreciation, and a decline in asset prices. In the model, sudden stops are associated with foreclosure states in which tradable endowment falls short of debt obligations.

Debt capacity channel: The mechanism by which higher lender concentration raises the nontradable goods price in foreclosure states, thereby increasing the collateral value and expanding agents’ borrowing capacity, which weakens the precautionary saving motive.

Amplification channel: The mechanism by which higher lender concentration steepens the slope of the nontradable price response to a fall in tradable consumption, amplifying the magnitude of the pecuniary externality that the social planner internalizes and thus increasing the social planner’s incentive to restrict borrowing.

Life-cycle worker flows and cross-country differences in aggregate employment

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research question. The paper asks: what are the sources of cross-country differences in aggregate employment across European economies, and which types of worker flows — between employment (E), unemployment (U), and nonparticipation (N) — drive those differences? The authors pay particular attention to heterogeneity by gender and age, motivated by the observation that cross-country employment dispersion is concentrated among women, youth, and older workers, and that a large portion of the dispersion is traceable to differences in labor force participation rather than unemployment rates alone.

Data. The empirical analysis draws on microdata from the EU Statistics on Income and Living Conditions (EU-SILC), an annual survey covering 32 European countries for 2004–2019. Germany is covered using the German Socio-Economic Panel (GSOEP, 2003–2018) because GSOEP longitudinal coverage begins earlier. The combined sample contains 7,064,306 individual-year observations for 2,221,672 individuals. Labor force status is recorded monthly via a retrospective calendar; transition probabilities are estimated at the quarterly frequency after correcting for measurement error (a “de-NUN-ification” procedure following Elsby et al. [2015]) and time-aggregation bias (Shimer [2012]).

Methodology — empirical. Six quarterly transition probabilities among E, U, and N are estimated by gender and single year of age (16–65). The life-cycle profile of each probability is extracted nonparametrically by regressing age-time cells on age and time dummies, removing business-cycle variation. To decompose cross-country employment differences into contributions of the six transition rates while handling the path-dependence of the decomposition (6! = 720 possible orderings), the authors apply the Shapley-Owen decomposition, which assigns to each transition rate its average marginal contribution across all orderings. An initial first-pass decomposition allocates the aggregate employment gap between any two countries into three parts: demographics, initial conditions (distribution across E, U, N at age 16), and transition probabilities. Transition probabilities account for 93–105% of the cross-country variance in aggregate employment, while demographics and initial conditions together explain less than 10%.

Methodology — structural model. The authors build a life-cycle Diamond-Mortensen-Pissarides (DMP) model with three labor market states, calibrated separately by gender and country for France, Germany, Italy, Spain, and the U.K. — the five largest economies in the sample. A key feature is that all primitives (technology, search and matching) are age-independent; life-cycle variation in worker flows arises endogenously from the finite retirement horizon and from two search margins: (i) an intensive margin — variable search intensity s in [0,1] chosen optimally each period — and (ii) an extensive margin — the endogenous labor force participation decision modeled as a discrete choice with i.i.d. extreme-value utility shocks. The model also incorporates permanent match quality (an experience good revealed stochastically with probability alpha per period following Jovanovic [1979]), transitory match-quality shocks (persistent AR(1) process), exogenous job-destruction shocks (per-period probability delta), a two-tier UI system, a two-tier EPL system capturing temporary vs. permanent contracts, and proportional value-added and social-security taxes.

Main empirical findings.

For male workers, employment-to-unemployment (EU) transitions account for approximately half of the cross-country variance in aggregate male employment across all 32 countries, rising to about three-quarters when looking at the five largest economies, and exceeding 85% for prime-age males (ages 25–54). Transitions in the reverse direction (UE) explain less than 30% of the variance across all 32 countries and play almost no role among the five largest economies. The labor force participation margin (combining NE and EN transitions) explains a non-negligible 25–30% of the aggregate male employment gap.
For female workers, at least half of the cross-country variance in employment is explained by participation-related flows, primarily transitions from nonparticipation to employment (NE). In the full 32-country sample, NE alone explains 65% of the variance in female employment rates across all ages (16–65). Its role is somewhat smaller in the five largest economies, where EN transitions also play a larger role. Crucially, the sum of NE and EN variance contributions for women is at least as large as the sum of UE and EU contributions, underlining the indispensability of a three-state model.

Main quantitative (model-based) findings. The model decomposes cross-country employment differences into technology (the distribution of permanent match quality, job-separation risk delta, and information frictions alpha), search parameters (vacancy costs, non-work utility, search-cost parameters), and policies (UI generosity, firing costs, taxes). The total employment variance across the five economies and two gender groups is 0.36 percentage points squared. Technology differences over-explain this variance (contribution of 0.65), while policies play almost no role (contribution of -0.04) and search frictions have a negative variance contribution (-0.25). The negative sign of search and policy contributions reflects the negative cross-country correlation between these factors and technology: countries with high employment rates (e.g., France) tend to have more generous UI and higher taxes, which the model attributes to compensating technology advantages. For individual countries: France is about 4.4 percentage points above the cross-country benchmark, driven by technology and partly offset by the highest replacement ratios and labor tax rates in the sample (67% and 56%, respectively). Spain is about 7 percentage points below the benchmark, driven by the lowest measured labor productivity (78% of Germany’s level) and the highest employment outflow rates (~4–5% per quarter vs. ~2% in France).

The channels through which technology affects employment are predominantly the employment inflows, not outflows. The exogenous job-separation risk delta affects aggregate employment mostly through its impact on expected duration of future employment spells, which reduces search incentives and job-finding rates from both unemployment and nonparticipation, and lowers labor force attachment. Similarly, mean permanent match quality (mu_x) and labor taxes (tau_ss) operate mainly through the inflow margin. Technology effects are amplified by search effort margins, particularly for women and youth: women face higher non-work utility (interpreted as labor-market frictions or opportunity costs), implying a lower employment surplus and therefore a higher surplus elasticity; for young workers, the long remaining horizon amplifies the effect of technology variations on discounted lifetime earnings, generating relatively higher search-effort responses.

Scope conditions. The analysis is confined to European countries. The structural decomposition covers only the five largest European economies. The authors acknowledge that parameters labeled as “job-separation risk” may also capture employment protection and temporary contracts not explicitly modeled, or non-monetary quit motives, so the attribution to “technology” should be interpreted with that caveat in mind. The model operates in a complete-markets, no-savings environment without on-the-job search.

In depth

Q1. What fraction of cross-country employment variance is explained by transition probabilities vs. demographics and initial conditions?

A: In the full 32-country sample, transition probabilities account for 94.7% of the cross-country variance in aggregate male employment and 99.9% for female employment. In the five largest economies, the corresponding figures are 93.5% (men) and 104.9% (women) — the slight excess above 100% reflects the negative contribution of initial conditions for women. Demographics and initial conditions together explain less than 10% of the variance, with somewhat larger demographic effects in Baltic and Eastern European countries, plausibly due to emigration-driven changes in age composition.

Q2. For male workers, which specific transition probability dominates the cross-country employment variance, and how does this vary by age and across country groupings?

A: EU (employment-to-unemployment) transitions account for approximately 51% of the cross-country variance in aggregate male employment (ages 16–65) across all 32 countries, rising to 77% in the five largest economies, and to 89% for prime-age males (ages 25–54) in the same group. By contrast, UE (job-finding from unemployment) explains at most 29% across all 32 countries and virtually nothing in the five largest economies. For prime-age men, EU remains dominant throughout; toward the end of the working life, EN (employment-to-nonparticipation) transitions become the main driver as workers move into retirement.

Q3. For female workers, what is the primary driver of cross-country employment variance, and does the pattern differ from men?

A: For women, transitions from nonparticipation to employment (NE) explain 65% of the cross-country variance in female employment across all ages in the 32-country sample. This dominance is more concentrated at ages 20–30, when participation entry is particularly heterogeneous across countries, likely reflecting fertility and child-rearing patterns. The sum of NE and EN contributions for women equals or exceeds the combined UE and EU contributions in both country groupings, demonstrating a fundamentally different demographic structure of employment differences for women relative to men.

Q4. How does the model generate life-cycle variation in transition rates despite having age-independent primitives?

A: The model produces age-varying transition rates through two mechanisms operating on age-independent fundamentals. First, variable search intensity declines as workers age because the remaining time to retirement shortens, reducing the expected lifetime returns to job search — the “horizon effect” (Cheron et al. [2011, 2013]). This mechanism explains virtually all of the life-cycle variation in the NE job-finding rate and an overwhelmingly large share of the variation in the UE rate, as shown by counterfactual exercises that fix search intensity at its life-cycle average. Second, information frictions about permanent match quality generate declining separation rates over the working life: young workers disproportionately hold matches with unrevealed quality and thus face higher reallocation risk upon quality revelation; as workers age, their employment share shifts toward matches with revealed quality, which have lower separation rates due to sorting.

Q5. What does the structural decomposition (Table 7) reveal about the role of technology vs. policies in explaining cross-country employment differences?

A: The variance decomposition in Table 7 shows that technology parameters (permanent match-quality distribution, job-separation risk delta, and match-quality revelation probability alpha) account for a variance contribution of 0.65 (against total employment variance of 0.36), over-explaining the cross-country dispersion. Labor market policies (UI benefits, firing costs, taxes) have a near-zero variance contribution of -0.04. Search parameters contribute -0.25. The result that policies explain little does not mean they have no level effect: in simple comparative statics, the model predicts that more generous UI and higher labor taxes lower employment. However, in the cross-country calibration, countries with higher employment rates tend to have more interventionist policies, so the cross-country correlation between policies and technology masks individual policy effects at the variance level.

Q6. How do technology effects propagate to employment differences through worker flows, and why is the inflow channel dominant?

A: Table 8 decomposes employment elasticities with respect to delta (job-separation risk), mu_x (mean log permanent match quality), and tau_ss (social security tax rate) into contributions from (i) the NE job-finding rate, (ii) the share of nonemployed in the labor force (labor force attachment, u-tilde), (iii) the differential between UE and NE rates, and (iv) the employment outflow rate (pEO). At the aggregate level, the separation risk delta has an employment elasticity of -0.28, of which the outflow contribution (dpEO = -0.08) is smaller in absolute magnitude than the sum of inflow contributions (dpNE = -0.06, du-tilde = -0.07, dpDelta = -0.06). Mean match quality mu_x has an employment elasticity of 0.53, primarily mediated through inflows. The mechanism is that changes in delta or mu_x alter expected lifetime earnings, which in turn change search incentives and participation decisions, generating correlated movements in job-finding rates and labor force attachment that amplify the employment impact beyond what a simple outflow change would imply.

Q7. Why do women and youth show larger search-effort responses to technology variations?

A: For women, the calibrated non-work utility yo is higher in all five countries than for men (interpreting this as extra costs and wedges on the returns to working), which implies a smaller employment surplus. A smaller surplus generates a higher elasticity of surplus with respect to parameter changes, and since search intensity and participation decisions depend on expected surplus, women exhibit larger employment elasticities to technology variations. The aggregate employment elasticity of delta is -0.39 for women vs. -0.19 for men; for mu_x, it is 0.78 for women vs. 0.33 for men. For youth (ages 20–29), the long remaining horizon amplifies the effect of technology changes on discounted expected lifetime earnings, which in turn amplifies participation incentives: the labor force attachment channel (du-tilde) contributes -0.13 for youth compared to -0.07 at the aggregate, while dE = -0.31 for youth vs. -0.28 aggregate for delta.

Q8. What is the quantitative role of individual technology sub-components (match quality, job-separation risk, information frictions)?

A: Panel B of Table 7 breaks down technology into three sub-components. Match quality (mean mu_x and variance sigma^2_x) and job-separation risk (delta) are the key drivers; the match-quality revelation probability (alpha, “match revelation”) plays almost no independent role (variance contribution approximately 0.00). For France, the primary positive technology contributor is mean match quality (consistent with France’s labor productivity slightly above the German benchmark). For Germany and the U.K., the low job-separation risk is the primary positive contributor. For Spain, the high job-separation risk — calibrated to match Spain’s employment outflow rate of around 4–5% per quarter versus 2% in France — is the main negative contributor, reflecting the widespread prevalence of temporary contracts.

Q9. What role do labor market policies play at the country-specific level, even though they explain little cross-country variance?

A: Panel C of Table 7 shows that employment protection legislation plays almost no role for any country. Labor taxes are quantitatively important: they explain the relatively high employment rate in the U.K. (the country with the lowest social security contribution rate, about 20%), contributing positively. In France, where labor taxes exceed 50% of the average wage, the policy contribution is strongly negative, roughly offsetting the large positive technology contribution. UI benefits lower aggregate employment — Italy, with calibrated UI benefits lower than France’s, has a smaller employment gap vis-a-vis the benchmark partly because of this. The finding that policies explain little variance while having large individual-country effects is explained by the negative cross-country correlation: countries with generous policies also tend to have favorable technology, so policy and technology contributions partially offset each other in the variance decomposition.

Q10. How does the model fit untargeted moments, particularly the empirical Shapley-Owen variance decomposition?

A: The model is calibrated to aggregate transition rates by gender, and to moments describing labor productivity, vacancy rates, and policy targets. Despite having age-independent primitives, the calibrated model captures the empirical life-cycle profiles of transition rates as untargeted moments: declining NE and UE rates with age, rising EN rates near retirement, and the hump-shaped patterns. More stringently, the model replicates the empirical Shapley-Owen variance decomposition: it correctly predicts that EU separations account for most of the employment variance for men, and that NE inflows are relatively more important for women and youth. A notable limitation is that the model overshoots the UN (unemployment-to-nonparticipation) transition rate for a significant share of data points — but the authors note that flows between U and N play almost no role in cross-country employment variance.

Q11. What is the “horizon effect” and how does it operate in this model?

A: The horizon effect, coined by Cheron et al. [2011, 2013] in a two-state (E/U) DMP model, refers to the phenomenon that as workers approach retirement, the expected returns to job search fall because the remaining period of employment is shorter. This reduces search intensity from both unemployment and nonparticipation, lowering job-finding rates, and in the present model also affects the match-acceptance probability: workers near retirement find it optimal to remain in unemployment to collect UI benefits rather than accept a job offer, further reducing the UE rate. The current paper generalizes this effect to a three-state setting by incorporating the labor force participation margin alongside search intensity, generating plausible declining job-finding rates and increasing EN rates at older ages from age-independent parameters.

Q12. How does the paper handle the gender dimension in the model calibration?

A: The model assumes that men and women share the same production and matching technology parameters within a country (A, cv, delta, alpha, mu_x, sigma^2_x, sigma^2_z), but allows the search-cost and non-work-utility parameters (ceu, cnu, cu, kappa_u, kappa_n, yo) to differ by gender. The gender-specific search parameters are identified from the gender-specific transition rates: for example, kappa_u (marginal search cost in unemployment) for women is inferred from the female UE transition rate, relative to the normalization for men. The non-work utility yo is consistently higher for women in all five countries, rationalizing lower female employment through a lower employment surplus. This generates a higher surplus elasticity for women, which in turn explains why women’s employment is more responsive to technology variations across countries.

Key Concepts

Shapley-Owen Decomposition. A method from cooperative game theory (Shapley [1953], Owen [1977]) used here to decompose cross-country differences in employment into contributions of individual worker-flow transition rates (or structural parameters). It computes the marginal contribution of each component averaged over all 6! = 720 orderings of the six transition rates, yielding a unique, symmetric, exact decomposition that sums to the total employment gap. Unlike sequential decompositions, it is path-independent.

Extensive Margin of Search Effort. The binary labor force participation decision: whether a nonemployed worker enters the unemployment state (and thus accesses the superior search technology at a flow cost) or remains in nonparticipation. In the paper’s model, this is captured as a discrete choice between states U and N, governed by i.i.d. extreme-value utility shocks, yielding a closed-form logit participation probability.

Intensive Margin of Search Effort. The continuous choice of search intensity s in [0,1] by nonemployed workers (both unemployed and nonparticipants), which scales the probability of meeting a vacancy per period. The optimal intensity equates the marginal cost of search (convex in s) to the marginal benefit (the expected surplus from meeting a firm times the contact rate). Search intensity declines with age because the remaining working life shortens, reducing the discounted value of a job.

Permanent Match Quality (x). A time-invariant, match-specific productivity component drawn from a log-normal distribution upon meeting a firm, but initially unobserved by both worker and firm (an experience good). With per-period probability alpha, the quality is revealed; prior to revelation, the parties form expectations over the distribution. Revelation triggers reallocation of bad matches, generating a negative relation between job tenure and separation probability (following Jovanovic [1979]).

Horizon Effect. The mechanism by which workers reduce search effort as they approach retirement because the expected present value of future employment spells shortens. In this paper the concept, coined by Cheron et al. [2011, 2013] in a two-state DMP setting, is extended to include the labor force participation margin: near-retirement workers not only search less intensively but also become more likely to choose nonparticipation (or to remain unemployed to collect benefits rather than accept a job), generating the observed life-cycle decline in job-finding rates from age-independent parameters.

Technology Parameters (theta). In the paper’s structural decomposition, “technology” refers specifically to the vector (mu_x, sigma^2_x, alpha, delta) — the mean and variance of log permanent match quality, the match-quality revelation probability, and the exogenous job-destruction probability. These are contrasted with search-cost parameters (phi) and policy parameters (psi). The label “technology” is acknowledged to potentially also capture employment protection and quit motives not explicitly modeled.

Life-Cycle DMP Model. A finite-horizon version of the Diamond-Mortensen-Pissarides search-and-matching framework in which workers live for J periods, all primitives are age-independent, and life-cycle variation in worker flows arises endogenously from the interaction of the finite horizon with search intensity, labor force participation, and match-learning mechanisms. The model distinguishes three labor market states (E, U, N) and uses Nash bargaining to split the employment surplus.

Linking Social and Personal Preferences: Theory and Experiment

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks whether an individual’s attitude toward risk in the personal domain (choices affecting only oneself) can be linked to that same individual’s attitude toward risk in the social domain (choices affecting both oneself and others). The authors provide a theoretical answer in the form of necessary and sufficient conditions, and then test those conditions experimentally.

The formal model posits a decision maker (DM) with a preference relation over lotteries on a set of social states, where a distinguished subset of states are personal (consequences for the DM alone). The authors assume preferences satisfy Completeness, Transitivity, Continuity, and State Monotonicity — the last being equivalent to respect for First-Order Stochastic Dominance (FOSD), a condition weaker than the Expected Utility Independence Axiom and satisfied by virtually all extant decision theories including Weighted Expected Utility, Rank-Dependent Utility, and Prospect Theory. The key theoretical result (Theorem 1) establishes that the full preference relation over all social lotteries can be uniquely deduced from the partial observations of (i) riskless social choices and (ii) risky personal choices if and only if the DM finds every social state indifferent to some personal state. When this condition fails, there exist social lotteries whose ranking cannot be recovered from the partial data.

For two empirically relevant preference types, this condition generates directly testable predictions: for selfish subjects (who allocate nothing to others in deterministic social choices), risky personal preferences must coincide with risky social preferences; for impartial subjects (who treat self and other symmetrically in deterministic social choices), riskless social preferences must coincide with risky social preferences.

The experiment was conducted at the University of Bergen and NHH Norwegian School of Economics with 276 undergraduate subjects. Each subject faced 50 budget-line choice problems in each of three domains: Personal Risk (equiprobable binary lotteries over own payoffs only), Social Choice (deterministic splits between self and an anonymous other), and Social Risk (equiprobable binary lotteries over symmetric payout pairs for self and other). The graphical interface of Choi et al. (2007b) was used throughout. One randomly selected decision per domain was paid out; each token was worth 1.2 NOK (approximately 0.2 USD), with average earnings of approximately 270 NOK.

Within-domain consistency, measured by the Critical Cost Efficiency Index (CCEI), is high: mean CCEIs are 0.959, 0.952, and 0.902 in the Personal Risk, Social Choice, and Social Risk domains respectively. At the CCEI > 0.90 threshold, 89.9%, 85.9%, and 69.9% of subjects pass in the three domains. Using a 0.95 share-to-self threshold, 103 subjects (37.3%) are classified as selfish; using revealed-preference criteria at the 5% significance level, 33 subjects (12.0%) are classified as impartial.

Testing is done via an individual-level nonparametric permutation test that draws 10,000 random data sets per subject and compares simulated CCEI distributions to actual cross-domain CCEIs, with Bonferroni correction. At the 1% significance level, the null that Personal Risk and Social Risk preferences coincide is rejected for only 5.9%–9.3% of selfish subjects (varying by classification threshold), compared with 14.7%–16.3% rejection rates for non-selfish subjects. For impartial subjects at the 1% level, the null that Social Choice and Social Risk preferences coincide is rejected for 0.0%–11.1%, compared with 19.8%–26.8% for non-impartial subjects. The theory’s predictions are thus supported for a large majority of both selfish and impartial subjects.

A theoretical extension (Theorem 2) shows that if one additionally observes comparisons between social states and personal lotteries, unique deduction of the full preference relation requires that preferences in both personal and social domains satisfy Expected Utility (Independence Axiom) and that every social state is indifferent to some personal lottery — a strictly stronger set of conditions.

Q: What is the central theoretical question and why does it matter? A: The paper asks whether preferences over risky social choices (lotteries over outcomes for self and others) can be deduced from observing only riskless social choices and risky personal choices. This matters because people frequently observe or predict the risky social choices of leaders and representatives, but may have access only to those leaders’ personal risk-taking behavior and their expressed social preferences under certainty.

Q: What is the main theoretical result (Theorem 1)? A: Under Completeness, Transitivity, Continuity, and State Monotonicity, the unique extension of the partial preference relation (over social states and personal lotteries) to the full domain of social lotteries exists if and only if every social state is indifferent to some personal state. When this condition is not met, multiple distinct preference relations can extend the partial observations, making deduction impossible.

Q: What is State Monotonicity and how does it relate to standard axioms? A: State Monotonicity requires that if each social state in one lottery dominates the corresponding state in another lottery, then the first lottery is weakly preferred. The paper shows this is equivalent to respect for First-Order Stochastic Dominance (FOSD) given the other axioms, and is strictly weaker than the von Neumann–Morgenstern Independence Axiom. It is satisfied by Weighted Expected Utility, Rank-Dependent Utility, and Prospect Theory, making it a broadly applicable assumption.

Q: What are the testable predictions for selfish subjects? A: Proposition 2 establishes that if a subject’s Social Choice preferences are selfish — meaning any bundle (x, y) is indifferent to (0, y), so the subject is indifferent between keeping x for self and giving it to other — then preferences in the Personal Risk domain must coincide with preferences in the Social Risk domain. In the experiment, selfish subjects are those allocating more than 95% of tokens to themselves in the Social Choice domain (103 of 276 subjects, or 37.3%).

Q: What are the testable predictions for impartial subjects? A: Proposition 3 establishes that if a subject’s Social Choice preferences are symmetric — meaning (x, y) is indifferent to (y, x) for all pairs — then preferences in the Social Choice domain must coincide with preferences in the Social Risk domain, implying risk neutrality toward social lotteries. The intuition is that such a subject treats self and other identically, so risky splits are evaluated by expected value alone. In the experiment, 33 subjects (12.0%) are classified as impartial by the revealed-preference criterion at the 5% significance level.

Q: How does the experiment measure within-domain rationality? A: Choices within each domain are evaluated using the Critical Cost Efficiency Index (CCEI, following Afriat 1967), which measures how much a budget constraint must be relaxed to remove all GARP violations. Mean CCEIs are 0.959 (Personal Risk), 0.952 (Social Choice), and 0.902 (Social Risk). At the CCEI > 0.90 threshold, 248 subjects (89.9%), 237 (85.9%), and 193 (69.9%) pass in the three domains respectively, compared to a simulated mean CCEI of only 0.585 for subjects randomizing uniformly.

Q: How does the cross-domain test work and why is it nonparametric? A: The test uses individual-level permutation inference: under the null that preferences in domains I and J are identical, any 50-element subset drawn from the pooled 100 choices should satisfy GARP as well as the actual domain-specific choices. For each subject, 10,000 such random draws are generated, their CCEI scores are computed, and the distribution is compared to the actual cross-domain CCEI with Bonferroni correction. The test makes no functional form assumptions about utility and accommodates the observed within-domain errors without parametric error modeling.

Q: What are the rejection rates for the selfish-subject prediction? A: At the 1% significance level, the null that Personal Risk and Social Risk preferences coincide is rejected for only 5.9%–9.3% of selfish subjects (range across four classification thresholds from 0.99 to 0.90 share-to-self), compared to 14.7%–16.3% for non-selfish subjects. At the 5% level, rejection rates rise to 20.4%–25.6% for selfish and 22.4%–31.8% for non-selfish subjects.

Q: What are the rejection rates for the impartial-subject prediction? A: At the 1% significance level, the null that Social Choice and Social Risk preferences coincide is rejected for 0.0%–11.1% of impartial subjects (range depending on threshold and classification method), compared to 19.8%–26.8% for non-impartial subjects. At the 5% and 10% levels, rejection rates for impartial subjects range from 0.0% to 22.2%.

Q: Does the theory predict how risk aversion should map across domains for non-selfish, non-impartial subjects? A: The theory does not directly produce testable cross-domain predictions for subjects who are neither selfish nor impartial without additional parametric assumptions, because the specific personal-state equivalent of each social state depends on the form of preferences. The paper restricts its nonparametric tests to the two polar cases where the equivalence mapping is determinate from social choice behavior alone.

Q: What is the extended result (Theorem 2) and what stronger conditions does it require? A: When one additionally observes comparisons between social states and personal lotteries (not just within each domain separately), unique deduction of the full preference relation is possible if and only if preferences in both the personal and social domains are consistent with an Expected Utility representation and every social state is indifferent to some personal lottery. This requires the Independence Axiom — a strictly stronger condition than State Monotonicity — highlighting that the main Theorem 1 result exploits the weaker observational structure.

Q: What is the distribution of social preferences in the sample? A: Of 276 subjects, 103 (37.3%) are classified as selfish at the 0.95 share-to-self threshold. Only 6 subjects (2.2%) kept fewer than 0.45 of tokens on average, making purely altruistic subjects rare. In the Personal Risk domain, 41 subjects (14.9%) allocated more than 95% to the cheaper account (consistent with risk neutrality), while 9 (3.3%) allocated fewer than 55% (consistent with infinite risk aversion). In the Social Risk domain, 30 subjects (10.9%) are consistent with utilitarianism in money and 9 (3.3%) with Rawlsianism in money.

Q: How does the Social Risk domain compare to the Personal Risk and Social Choice domains in terms of rationality scores? A: The Social Risk domain shows lower consistency than the other two: mean CCEI is 0.902 versus 0.959 and 0.952, and only 69.9% of subjects exceed the 0.90 threshold versus 89.9% and 85.9%. The CCEI distribution is shifted left for Social Risk, suggesting the novel combined dimension of social and risky choice introduces more decision complexity or error.

Q: What is the relationship to the prior experimental literature on social and risk preferences? A: The Personal Risk domain replicates the symmetric risk experiment of Choi et al. (2007a), and the Social Choice domain replicates the linear two-person dictator experiment of Fisman et al. (2007). The Social Risk domain is new to this paper. The theoretical framework connects to Saito (2013) on social preferences under risk, and to the preference extension literature of Grant et al. (1992) and Nishimura et al. (2017).

State Monotonicity: The axiom requiring that if each social state in one lottery weakly dominates the corresponding social state in another lottery, the first lottery is weakly preferred. The paper proves this is equivalent to respect for First-Order Stochastic Dominance given Completeness, Transitivity, and Continuity, and distinguishes it from the stronger Independence Axiom by noting that Independence compares lotteries over lotteries while State Monotonicity only compares lotteries over states.

Selfish preferences (in the paper’s sense): Preferences in the Social Choice domain such that (x, y) is indifferent to (0, y) for all bundles — the subject is indifferent between receiving x themselves versus giving x to the other person. Operationally measured as allocating more than a threshold share (e.g., 95%) of tokens to self across Social Choice decisions.

Impartial preferences (in the paper’s sense): Preferences in the Social Choice domain such that (x, y) is indifferent to (y, x) for all bundles — the subject treats self and other symmetrically. Operationally identified by the revealed preference criterion that choices in the Social Choice domain satisfy GARP and are consistent with symmetric treatment.

Unique extension (deducibility): The property that there exists exactly one complete preference relation over all social lotteries that is consistent with the axioms and agrees with the observed partial relation over social states and personal lotteries. Theorem 1 identifies the necessary and sufficient condition for unique extension under State Monotonicity.

Personal state indifference condition: The condition that for every social state omega in Omega minus P, there exists some personal state in P to which the DM is indifferent. This is the necessary and sufficient condition in Theorem 1 for deducibility of the full preference relation. Interpreted as: for every proposed social allocation, there exists a “bribe” — a personal allocation with nothing for others — that the DM finds equally desirable.

Critical Cost Efficiency Index (CCEI): A measure of how much budget constraints must be scaled down to eliminate all GARP violations in a dataset of choices from budget lines (following Afriat 1967). A CCEI of 1 indicates perfect rationality; the paper uses 0.90 as a practical threshold. Mean values are 0.959, 0.952, and 0.902 in the Personal Risk, Social Choice, and Social Risk domains respectively.

Nonparametric permutation test: The individual-level test used to assess consistency across choice domains. Under the null that preferences are identical in domains I and J, any random 50-element draw from the pooled 100 choices should achieve CCEI scores no worse than the actual domain scores. The test draws 10,000 permuted datasets per subject and uses the Bonferroni correction for multiple comparisons, making no assumptions about the functional form of utility.

Lives Versus Livelihoods: The Impact of the Great Recession on Mortality and Welfare

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. Does the Great Recession reduce or increase mortality, and what are the welfare implications of incorporating recession-induced mortality changes into standard macroeconomic welfare frameworks?

Setting and Identification. The authors exploit spatial variation in the severity of the 2007–2009 Great Recession across 741 U.S. Commuting Zones (CZs), following the empirical design of Yagan (2019). The primary shock variable is the percentage-point change in the CZ unemployment rate between 2007 and 2009. The key identifying assumption is that no concurrent shocks to mortality coincide with the timing and geographic pattern of the Great Recession shock. Pre-trend evidence supports this: CZs subsequently harder hit experienced a slight relative increase in mortality before 2007, which is the opposite sign from the main effect, supporting the validity of the design.

Data. Mortality data come from CDC restricted-use death certificate microdata (2003–2016) covering the universe of U.S. deaths, combined with SEER population denominators. A 20 percent random sample of Medicare enrollees aged 65–99 provides an individual-level panel that directly addresses concerns about endogenous migration. The main outcome is the log age-adjusted CZ mortality rate; economic indicators come from BLS, BEA, and FHFA; air pollution data from the EPA AQS monitor network (PM2.5); morbidity from the BRFSS; nursing home characteristics from federal certification inspections.

Main Mortality Finding. A one-percentage-point increase in the local unemployment rate between 2007 and 2009 is associated with a 0.50 percent decline (SE = 0.15) in the annual age-adjusted mortality rate in 2007–2009, and a 0.58 percent decline (SE = 0.34) in 2010–2016; the two periods are statistically indistinguishable (p = 0.78). Because the national average unemployment rate rose by 4.6 percentage points, the Great Recession on average reduced the annual age-adjusted mortality rate by approximately 2.3 percent, with effects persisting for at least 10 years. The authors note this is equivalent to approximately two years of secular mortality improvement at the pre-recession trend pace of 1.1 percent per year. For a 55-year-old, the estimates imply that 1 in 25 gained an extra year of life from a shock of this magnitude.

Heterogeneity by Cause of Death. Mortality declines appear across most major causes. Cardiovascular disease (34 percent of 2006 deaths) declines by 0.65 percent per percentage-point unemployment increase (SE = 0.21) and accounts for approximately 48 percent of the total estimated mortality reduction. Motor vehicle mortality falls by 1.7 percent (SE = 0.56) and liver disease by 1.1 percent (SE = 0.43). Suicides show a statistically significant 1.7 percent decline (SE = 0.5) in the 2010–2016 period. The notable exception is cancer (the second-largest cause of death), for which the estimated effect is a precise null of 0.02 percent (SE = 0.11). The null cancer result is interpreted as a specification check: if mortality declines were spurious (e.g., driven by population mismeasurement), cancer mortality should also decline.

Heterogeneity by Demographics. Recession-induced mortality declines are similar in percentage terms across gender and race/ethnicity, and statistically equi-proportional across age groups (p-value for equality across 25–64 versus 65+: 0.76). Because mortality is heavily concentrated in the elderly, those aged 65 and over account for approximately 74.3 percent of averted deaths, roughly proportional to their 72.5 percent share of 2006 mortality. The most striking heterogeneity is by education: the entire mortality decline is concentrated among the approximately 52 percent of the population with a high school degree or less. The estimated 2007-2016 effect is −1.3 percent per percentage-point unemployment increase (SE = 0.56) for those with high school or less, compared to +0.34 percent (SE = 0.68) for those with more than high school (statistically distinguishable at p < 0.01).

Mechanisms. The authors distinguish internal effects (own reduced employment or consumption improving health) from external effects (externalities from reduced aggregate economic activity, holding own employment/consumption fixed). Evidence strongly favors external effects as the primary driver. Three-quarters of averted deaths accrue to the elderly, who experienced no direct income effects from the labor market shock. Moreover, the timing pattern—an immediate mortality drop that does not grow over time—is inconsistent with health-behavior channels (e.g., smoking cessation, improved diet) that would build up gradually. Direct tests find no statistically significant impact on self-reported health behaviors (smoking, drinking, exercise) and no impact on healthcare use among Medicare enrollees.

Among external channels, neither reduced spread of infectious disease nor improved nursing home staffing receives empirical support. Reduced air pollution (PM2.5) is identified as a quantitatively important channel. A one-percentage-point increase in CZ unemployment is associated with a 0.16 µg/m³ decline in PM2.5 (SE = 0.04), a 1.3 percent decline relative to the 2006 national average of 12 µg/m³. A mediation analysis (controlling for the PM2.5 shock) attenuates the estimated mortality effect by 37 percent, from −0.52 percent to −0.33 percent per percentage-point unemployment increase. Back-of-the-envelope calculations combining the PM2.5 decline with external estimates of PM2.5-mortality elasticities suggest pollution can explain 17 to 35 percent of total recession-induced mortality declines.

Lag Structure. Exploiting variation in the speed of post-recession labor market recovery (measured by 2010–2016 EPOP ratio changes) conditional on the initial shock, the authors find that mortality reductions persist in areas that have fully recovered economically by 2016, suggesting lagged mortality effects of the initial economic downturn beyond what contemporaneous economic conditions alone explain.

Welfare Analysis. The authors extend the Krebs (2007) consumption-based welfare cost-of-recessions model to incorporate endogenous mortality. For a 45-year-old with γ = 2 and a value of a statistical life-year (VSLY) of $250k (five times annual consumption), accounting for endogenous mortality reduces the willingness to pay to avoid all future recessions from 2.00 percent of average annual consumption to 0.91 percent—a reduction of approximately 55 percent. Starting around age 55, recessions become welfare-improving on net. For the Great Recession specifically, at age 55 endogenous mortality reduces the welfare cost by approximately 25 percent (from 2.39 to 1.80 percent of average annual consumption). Because mortality declines are concentrated among those with high school or less, accounting for endogenous mortality also substantially mitigates—and at older ages reverses—the finding that the Great Recession was more costly for the less educated.

Scope Conditions and Caveats. (i) The design captures only differential local effects, not nationwide impacts (e.g., stock market collapse, nationwide malaise). (ii) Mortality impacts may not generalize to milder recessions, though the relationship appears approximately linear in shock size. (iii) The analysis excludes morbidity, though limited evidence suggests morbidity is also pro-cyclical and roughly equi-proportional across ages. (iv) The welfare analysis begins at age 35 and does not account for longer-run mortality costs of recession entry for younger cohorts.

In depth

Q1. What is the baseline empirical specification, and why does the design exploit cross-sectional variation rather than time-series panel regressions?

The estimating equation regresses the log age-adjusted CZ mortality rate on an interaction of the CZ-level Great Recession shock (2007–2009 unemployment change) with year indicators, plus CZ and year fixed effects, weighted by 2006 CZ population. The authors prefer this to the standard two-way fixed effects panel approach (area and year FE with contemporaneous unemployment rate) for three reasons: (1) it directly identifies the full dynamic lag structure of the shock rather than imposing contemporaneity; (2) exploiting a single spatially differentiated shock reduces risk of confounding from other concurrent area-level shocks; (3) the panel can be linked to individual-level Medicare data, allowing explicit control for endogenous migration, which the existing literature cannot do.

Q2. How does the paper address the concern that mortality rate declines might simply reflect unmeasured population outflows from hard-hit areas rather than genuine reductions in deaths?

The authors offer two main responses. First, cancer mortality shows a precise null effect despite being the second-leading cause of death; if unmeasured population losses were driving the results, cancer deaths should decline proportionally. Second, using the Medicare individual-level panel, they fix each enrollee’s location at their 2003 CZ and find a statistically significant mortality decline of 0.35 percent per percentage-point unemployment increase in the reduced-form (2007–2009 period). A control function approach that instruments current-year location with 2003 location yields an estimate of −0.37 percent (SE = 0.17), similar to the baseline −0.50 percent from the aggregate specification, confirming that migration bias is not the primary driver.

Q3. How long do the mortality reductions from the Great Recession persist, and does the paper identify whether these are contemporaneous or lagged effects?

The 2007–2009 period estimate is −0.50 percent per percentage-point unemployment increase and the 2010–2016 period estimate is −0.58 percent, and these are statistically indistinguishable (p = 0.78). To identify whether persistence reflects ongoing economic effects or true lagged mortality effects, the authors compare CZs with above- vs. below-median 2010–2016 EPOP recovery (conditional on initial shock decile). Both groups show similar 2010–2016 mortality declines despite the above-median recovery CZs having returned to pre-recession employment levels by 2016. This finding is consistent with lagged mortality effects of the initial economic downturn that persist independently of current economic conditions.

Q4. Are mortality reductions concentrated among individuals already near death (“harvesting”), or do they represent meaningful longevity gains?

The authors use a Medicare auxiliary model to predict counterfactual remaining life expectancy for each enrollee based on age, demographics, and chronic conditions. The marginal life saved has only about 6 percent lower counterfactual remaining life expectancy than a typical decedent of the same age, and this difference is statistically insignificant. Because effects persist over 10 years (not just days or weeks), short-run mortality displacement (harvesting) is not the operative concern. The 6 percent difference is also small enough that the authors do not adjust their welfare analysis for it.

Q5. What is the educational gradient in mortality impacts, and is it explained by age composition or other confounders?

Mortality declines are entirely concentrated among those with a high school degree or less: the 2007–2016 estimate is −1.3 percent per percentage-point unemployment increase (SE = 0.56) for this group versus +0.34 percent (SE = 0.68) for those with more than high school, distinguishable at p < 0.01. This gradient holds within age groups (confirmed in Appendix analysis), and further disaggregation shows no mortality declines for those with some college or college-or-more separately. In Medicare data, the elderly mortality effect is concentrated among the approximately 12 percent enrolled in Medicaid (a proxy for low income), reinforcing the socioeconomic concentration.

Q6. What evidence rules out improved health behaviors (increased exercise, reduced smoking, reduced alcohol) as the main mechanism?

Two types of evidence argue against this channel. First, three-quarters of averted deaths are among the elderly, who experienced no direct income or employment effects from the local labor market shock and would not plausibly change their health behaviors in response to someone else losing employment. Second, the mortality decline is immediate in 2007 and flat through 2016 rather than growing over time; smoking cessation, for example, takes 10–15 years to accumulate mortality effects. Direct tests of behavioral outcomes from BRFSS find no statistically significant impact on smoking, drinking, exercise, or flu vaccination rates, individually or pooled. The pooled average treatment effect on six morbidity measures is statistically significant and negative (suggesting morbidity improvements), but behavioral covariates show no movement.

Q7. What is the evidence for and against improved nursing home care as a mechanism?

Prior literature (Stevens et al. 2015; Konetzka et al. 2018; Antwi and Bowblis 2018) documents that recessions increase nursing home staffing and reduce nursing home deaths in earlier decades. However, the authors find no evidence for this channel in the Great Recession context. Estimated mortality impacts are virtually identical (approximately 0.5 percent per percentage-point unemployment increase) for the 7 percent of the elderly in nursing home care and the 93 percent not in nursing home care. Direct measures of nursing home staffing (direct-care staff hours per resident-day, highly skilled nurses ratio) show no statistically significant change in harder-hit areas: the point estimate for direct-care hours is −0.11 percent (SE = 0.22) in 2007–2009. Nursing home occupancy rates and resident characteristics also show no significant changes.

Q8. How is the quantitative importance of the air pollution channel estimated, and what are the two complementary approaches used?

Approach 1 (back-of-the-envelope): The authors combine their estimate that a one-percentage-point unemployment increase reduces PM2.5 by 0.16 µg/m³ with external estimates from Deryugina et al. (2019) of PM2.5’s effect on elderly daily mortality, rescaled to annual exposure. This calculation implies pollution explains 17–35 percent of total recession-induced mortality declines, depending on which Deryugina et al. mortality estimates are used. Approach 2 (mediation analysis): Adding the county-level PM2.5 shock as an additional control in the mortality regression attenuates the Great Recession mortality coefficient from −0.52 percent to −0.33 percent per percentage-point unemployment increase—a 37 percent attenuation. Both approaches are suggestive rather than definitive, as the mediation analysis requires the strong assumption that the recession shock and PM2.5 shock are conditionally independent of other unmeasured mediators.

Q9. What are the specific calibration parameters in the welfare model and how does the paper set the mortality decline parameter?

The authors extend Krebs (2007)’s income process calibration (pH = 0.03, pL = 0.05, dH = 0.09, dL = 0.21, g = 0.02, σ = 0.01, πH = 0.5) and use 2007 SSA life tables for age-specific mortality rates in normal times. The recession mortality parameter is set to dm = −0.015 for all ages, derived from a 3.1 percentage-point unemployment increase in a typical recession multiplied by the estimated 0.5 percent mortality decline per percentage-point. VSLY values are parameterized at two, five, or eight times annual consumption ($100k, $250k, or $400k at $50k annual consumption). Risk aversion γ takes values 1.5, 2, and 2.5. For the Great Recession-specific exercise, dmA = −0.023 (4.6 × 0.5 percent), dmHS = −0.037, and dmC = 0.0006.

Q10. How does accounting for endogenous mortality change the distributional welfare analysis of the Great Recession by education group?

Under exogenous mortality, the welfare cost of the Great Recession at age 35 is 2.89 percent of average annual consumption for those with high school or less versus 1.23 percent for those with more than high school—the less educated bear roughly twice the burden. Under endogenous mortality, the mortality declines are concentrated entirely among the less educated (dmHS = −0.037 vs. dmC ≈ 0), so accounting for mortality disproportionately offsets welfare losses for that group. By around age 65, the welfare costs of the Great Recession converge across education groups, and after age 65, the less educated bear lower welfare costs than the more educated, reversing the exogenous-mortality ranking. This result depends on the same education differential in mortality impacts that drives the main empirical finding.

Q11. What robustness checks demonstrate that the baseline mortality estimates are not driven by geographic or functional-form choices?

The baseline CZ-level estimate of −0.50 percent (SE = 0.15) is replicated almost exactly at the state level (−0.62, SE = 0.25) and county level (−0.49, SE = 0.10). A Poisson regression yields −0.45 percent (SE = 0.14). Dropping the top/bottom decile of CZs by shock size yields −0.46 percent (SE = 0.16). Adding Census-division-by-year fixed effects attenuates the estimate slightly to −0.38 percent (SE = 0.14) but retains statistical significance. Dropping CZs with high fracking activity and dropping the ten most populous CZs both produce estimates similar to baseline. Quartile regressions show monotone mortality reductions across quartiles of the unemployment shock, consistent with approximate linearity.

Q12. What does the expert survey reveal about prior beliefs, and how does the paper’s finding compare?

In a spring 2023 survey of over 300 experts, 50 percent predicted the Great Recession would increase mortality and only 27 percent predicted a decrease. Of those predicting a decrease, 93 percent gave a magnitude larger (in absolute value) than the paper’s negative point estimate of 0.50 percent per percentage-point unemployment increase, and 82 percent gave a prediction larger than the upper bound of the 95 percent confidence interval. This illustrates that the paper’s finding—mortality is meaningfully pro-cyclical during the Great Recession—was highly surprising to the empirical and policy economics community.

Key Concepts

Pro-cyclical mortality: The phenomenon whereby mortality rates fall during economic downturns and rise during expansions. The paper documents this for the Great Recession using a spatial identification strategy, in contrast to the time-series correlation that had weakened in the two decades before the Great Recession. The term “pro-cyclical” means mortality moves in the same direction as the business cycle (up in booms, down in recessions), implying recessions are associated with fewer deaths.

Internal vs. external effects (of recessions on mortality): The paper distinguishes internal effects—whereby an individual’s own reduced employment or consumption affects her own mortality—from external effects, which are changes in mortality from reduced aggregate economic activity that hold constant one’s own employment and consumption. This distinction has direct welfare implications: external effects (e.g., less pollution from lower industrial output) are genuine welfare improvements for people who did not lose income, while internal effects of behavioral change are mitigated by the envelope theorem if behavior is privately optimal.

Commuting Zone (CZ) shock: The paper’s primary treatment variable, defined as the percentage-point change in the CZ unemployment rate between 2007 and 2009. CZs are aggregations of counties (741 total) designed to approximate local labor markets. The median CZ experienced a 4.6-percentage-point increase, with substantial variation ranging from roughly 2.9 points (bottom quartile) to 6.7 points (top quartile).

Value of a Statistical Life-Year (VSLY): The dollar value placed on one additional year of life in expectation, used in the welfare calibration. In the paper’s framework it equals VSLY = bcγ − c/(γ−1), where b is a preference parameter governing the marginal utility of life-years. Results are reported for VSLYs of $100k, $250k, and $400k corresponding to two, five, and eight times average annual consumption of $50k, following Hall and Jones (2007).

Endogenous mortality in welfare analysis: The paper’s central theoretical contribution is augmenting the Krebs (2007) welfare cost-of-recessions framework to allow mortality to vary with the aggregate state of the economy. When mortality is endogenously lower in recessions, the willingness to pay to eliminate recession risk falls—and at high enough VSLY or old enough ages, recessions become welfare-improving because the mortality benefit outweighs the consumption cost.

Mortality displacement (harvesting): The possibility that short-run mortality declines merely reflect the premature death of already-frail individuals being slightly delayed, without meaningful longevity gains. The paper argues this is not the operative concern given 10-year persistence and uses auxiliary Medicare models to show marginal lives saved have only 6 percent shorter counterfactual life expectancy than average decedents of the same age.

PM2.5 mediation analysis: An empirical approach in which the county-level change in fine particulate matter (PM2.5, in µg/m³) between 2006 and 2010 is added as a covariate in the mortality regression. Under the assumption that the recession shock and the PM2.5 shock are conditionally independent of other unmeasured mediators, the attenuation in the recession-mortality coefficient when controlling for PM2.5 identifies the share of the mortality effect operating through the pollution channel. A 37 percent attenuation is found in the 2007–2009 period.

Local Projection-Based Inference under General Conditions

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a uniform asymptotic theory for local projection (LP) regression under general conditions, addressing a gap in the literature where existing results required restrictive assumptions about lag order, data persistence, and shock processes. The research question is: how can one conduct valid statistical inference on impulse responses from LP regressions when the true lag order is unknown (possibly infinite), data exhibit arbitrary persistence including unit roots and near-unit roots, horizons are allowed to grow with sample size, and shocks follow general conditionally heteroskedastic martingale difference sequences (MDS)?

The paper works within a VAR(infinity) data-generating process framework, where the vector autoregression may have an unknown and potentially infinite number of lags. The LP regression truncates this at a chosen model order p, with the truncation bias controlled by tail decay conditions on the VAR coefficients. The theoretical framework accommodates a class of VARMA models as a specific illustration, showing that Assumptions 1 and 2 hold for VARMA(q+1, r) processes when the model lag order p diverges at least as fast as log n.

The main theoretical result (Theorem 1) establishes uniform asymptotic normality of the LP estimator, simultaneously over: the coefficient parameter space A, model lag orders p in [p_low, p_high], horizons h in [1, h_bar], and configurations of the linear combination vector gamma (covering both individual and cumulated impulse responses). The convergence rate is pi_1(h; gamma)^{-1/2} n^{1/2}, which depends on persistence level and horizon. For an AR(1) process, the individual response rate is (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} and the cumulative response rate is h^{-3/2} n^{1/2}, which is slower.

The paper makes two principal contributions. First, LP is shown to be semiparametrically efficient when the controlled lag order diverges. Under classical assumptions (homoskedastic MDS shocks, stationarity, fixed horizon), the LP estimator achieves the same asymptotic distribution as the VAR-implied iterative estimator, and reaches the semiparametric efficiency bound of Chamberlain (1987) under the conditional moment restriction model. Under Gaussianity, LP is asymptotically Cramer-Rao efficient. This extends Plagborg-Moller and Wolf (2021) from distributional equivalence of estimands to equivalence of asymptotic distributions. The commonly held view that LP is inefficient relative to VAR-implied methods holds only under finite small-order VAR models; with a diverging lag order, the efficiency gain from the parsimonious VAR structure vanishes. The alternative LP estimator of Lusompa (2022), shown to be more efficient than standard LP under a known AR(1) model, is likewise shown (Proposition 2) to be asymptotically equivalent to standard LP when a sufficiently large lag order is used (p_u/sqrt(n) -> 0 and sqrt(n)(1-|rho|)^{p_u} -> 0).

Second, two new standard errors are proposed, neither involving HAR-type correction or bandwidth selection. SE_1 is a White-style heteroskedasticity-robust standard error applied after partialling out controls; it is uniformly consistent under a zero fourth cumulant condition on shocks (e.g., zero excess kurtosis with conditional homoskedasticity), but not for general MDS shocks. SE_2, the paper’s main methodological contribution, constructs the variance estimator using martingale-transformed scores: the LP residual Delta_t is projected onto forward residuals (Delta_{t+1}, …, Delta_{t+h-1}) to partial out serial dependence, recovering the true MDS error xi_{1t}(h; gamma) asymptotically. SE_2 is uniformly consistent for general MDS shocks (Proposition 4) and, under a finite-order VAR DGP, requires only p = p_true lags (rather than p >= p_true + 1 required by SE_1 and HAR-type methods).

Simulations using univariate ARMA(1,1) models with rho in {0, 0.5, 0.95, 1} and theta in {-0.5, 0, 0.5}, and bivariate VAR(1) models, confirm that SE_2-based 95% confidence intervals maintain coverage close to the nominal level across all cases including unit roots, while SE_1 shows degraded coverage under conditional heteroskedasticity (GARCH). Both outperform MOPM for cumulated responses at longer horizons.

Scope conditions: the framework accommodates data with unit roots and near-unit roots but not explosive roots or integration of order greater than one (for which differencing is prescribed before applying the LP). The growing-horizon rate condition p^2 h^2 / n -> 0 becomes binding as h grows, requiring h and p to grow at comparable rates or p more slowly. The results are for the VAR framework and do not directly apply to structural (SVAR) identification without additional assumptions.

Q: What is the central inferential problem that motivates this paper?

A: Applied macroeconomists estimating impulse responses via LP regressions face a trilemma: the true lag order is unknown and may be infinite, data may be highly persistent or integrated, and shocks may be conditionally heteroskedastic. Existing uniform validity results (chiefly Montiel Olea and Plagborg-Møller 2021) assume a finite and known model order and require mean-independent shocks, leaving inference potentially invalid when these conditions fail. The paper constructs a theory and inference procedures that remain valid simultaneously over all these dimensions.

Q: What is the VAR(infinity) data-generating process assumed, and what are the key restrictions on it?

A: The DGP is yt = sum_{j=1}^{infinity} a_j y_{t-j} + u_t, where u_t is serially uncorrelated. Assumption 1 bounds the impulse responses uniformly over the parameter space (ruling out explosive roots and integration of order greater than one). Assumption 2 imposes that the tail coefficients a_j decay fast enough that the truncation bias is asymptotically negligible: the rate condition requires sqrt(n) * p * sum_{j=1}^{infinity} j |a_{p+j}| -> 0, implying p must diverge for infinite-order processes. For VARMA models, p need only diverge as slowly as log n.

Q: What does Theorem 1 establish, and what is the convergence rate?

A: Theorem 1 establishes uniform asymptotic normality of the LP estimator, with the supremum taken jointly over the coefficient space A, lag orders p in [p_low, p_high], horizons h in [1, h_bar], and the linear combination vector gamma. The convergence rate is pi_1(h; gamma)^{-1/2} n^{1/2}, where pi_1(h; gamma) = sum_{i=1}^{h} |phi_{1i}|^2 captures persistence and horizon effects. For an AR(1) process, the individual response rate is (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} and the cumulative response rate is the slower h^{-3/2} n^{1/2}.

Q: In what sense is LP semiparametrically efficient, and under what assumptions?

A: Under classical assumptions — homoskedastic MDS shocks, stationarity, and fixed horizon — when the controlled lag order p diverges at the appropriate rate, the LP estimator reaches the semiparametric efficiency bound of Chamberlain (1987) under the conditional moment restriction model E(yt - sum a_j y_{t-j} | ys, s <= t-1) = 0. It achieves the same asymptotic distribution as the VAR-implied estimator, which itself has the same distribution as the LP estimator under these conditions (established by extending Lutkepohl 1990). Under Gaussianity, LP is asymptotically Cramer-Rao efficient.

Q: Why does the efficiency advantage of VAR-implied methods over LP vanish with a large lag order?

A: Under a finite, small-order VAR model, imposing the functional relationship between all impulse responses and a small set of VAR slope parameters — analogous to dimension reduction in a factor model — yields an efficiency gain for the iterative VAR-implied estimator. However, as the model lag order grows, the number of parameters to estimate grows correspondingly, eroding the dimension-reduction benefit. With a diverging lag order, the extraction of common parameters through a parsimonious model no longer tightens the asymptotic variance of the VAR-implied estimator relative to the direct LP estimator.

Q: How does SE_2 avoid the need for HAR (heteroskedasticity and autocorrelation robust) bandwidth selection?

A: The LP regression error Delta_t(h; gamma) is serially correlated for h >= 2 (it contains MA terms of order h-1), which would normally require HAR correction. SE_2 avoids this by constructing the variance estimator from the martingale-transformed score: the LP residual Delta_t is regressed on the forward residuals (Delta_{t+1}, …, Delta_{t+h-1}) and the fitted residual hat{xi}{1t} is used in place of Delta_t. Asymptotically, hat{xi}{1t} recovers the true LP(infinity) error xi_{1t}(h; gamma) = sum_{i=1}^{h} phi’{1i} u{t+i}, which is a MDS with respect to {u_t, u_{t-1}, …}. Since MDS sums have a martingale structure, their variance can be estimated as a simple sum of squares without bandwidth selection.

Q: Under what condition is SE_1 uniformly consistent, and when does it fail?

A: SE_1 is the standard White heteroskedasticity-robust variance estimator applied to the partialled-out score. It is uniformly consistent under the zero fourth cumulant condition on shocks — that is, when u_t has zero excess kurtosis and is conditionally homoskedastic. This condition fails for general MDS shocks (e.g., GARCH-type shocks), because the cross-moment Cov((tau’w_0)^2, (tau’w_k)^2) does not vanish in general. Simulation results confirm that SE_1-based confidence intervals show degraded coverage under GARCH shocks, while SE_2 maintains coverage.

Q: What is the relationship between this paper and Montiel Olea and Plagborg-Møller (2021)?

A: Montiel Olea and Plagborg-Møller (2021) (MOPM) established uniform validity of LP inference under a finite-order, known VAR model and required mean-independent (not merely MDS) shocks. The current paper extends MOPM in five dimensions: it allows an unknown and potentially infinite true lag order; allows the controlled lag order to diverge; develops new asymptotic theory for general MDS shocks; proposes SE_2 whose consistency does not require mean-independent shocks; and unifies inference for both individual and cumulated impulse responses. The lag-augmented LP regression of MOPM (setting p = p_true + 1) is a special case of the framework here.

Q: What does the paper show about the alternative LP estimator of Lusompa (2022)?

A: Lusompa (2022) showed that, under a known AR(1) model with the true lag order, an alternative LP estimator that exploits the serial dependence structure of the LP error is asymptotically more efficient than standard LP across horizons. Proposition 2 of the current paper shows this efficiency gain does not survive when a sufficiently large lag order is used for the preliminary VAR used to compute the transformation. Specifically, when p_u/sqrt(n) -> 0 and sqrt(n)(1-|rho|)^{p_u} -> 0, the alternative and standard LP estimators are asymptotically equivalent: sqrt(n)[tilde{beta}_1(h) - beta_1(h)] - sqrt(n)[hat{beta}_1(h) - beta_1(h)] = o_p(1). The discrepancy arises from estimation errors in the preliminary residuals entering the asymptotic distribution.

Q: What are the rate conditions on the lag order p and horizon h, and how do they compare to VAR-implied methods?

A: Under a fixed horizon, the condition p^2/n -> 0 suffices for LP, which is weaker than the p^3/n -> 0 typically required for VAR-implied methods (the stricter condition arises because VAR-implied methods must estimate all p slope matrices jointly, while LP treats all but the first as nuisance). Under growing horizons (h -> infinity), the rate condition is p^2 h^2/n -> 0, and the analysis shows p = O(h) is sometimes optimal — p and h should grow at the same rate or p more slowly. By contrast, VAR-implied methods require p = o(n^{1/3}/h^{2/3}) under growing horizons.

Q: What is the lag order flexibility advantage of SE_2 under a finite-order VAR DGP?

A: When the true DGP is a finite-order VAR(p_true), SE_2 achieves consistent inference using exactly p = p_true lags — the exact order. In contrast, SE_1 and HAR-type standard errors require p >= p_true + 1 (at least one extra lag) because at p = p_true the LP residuals Delta_t(h; gamma) contain MA terms of order h-1 that create serial dependence. SE_2’s martingale transformation handles this serial dependence directly, without requiring the extra lag to purge it.

Q: What scope conditions limit the paper’s framework?

A: The framework rules out explosive roots (violating the uniform impulse response bound in Assumption 1) and integration of order two or higher (violating Assumption 1(iii)). For I(2) variables, the prescribed solution is to take differences before applying the LP, and then use the cumulated response (gamma = gamma_CIR) to recover original level responses. The growing-horizon results require the tension condition h_bar * p^2 / n -> 0 (for gamma with ||gamma||_1 = O(1)), implying a binding tradeoff between the range of allowed horizons and the range of allowed lag orders. Results do not directly extend to structural identification without additional assumptions.

Local Projection (LP) regression: A direct regression of the outcome h periods ahead on current and lagged endogenous variables, as in Jorda (2005). The LP estimator of the horizon-h impulse response is the OLS coefficient on the current endogenous variable in this regression, with p-1 lags included as controls. It estimates impulse responses directly for each horizon without imposing the recursive structure of a VAR model.

Uniform asymptotic validity: A distributional approximation (here, standard normal) that holds simultaneously over a parameter space A, a range of model lag orders [p_low, p_high], a range of horizons [1, h_bar], and specifications of the linear combination vector gamma — not merely pointwise for fixed parameter values. Uniformity is the operative concept ensuring finite-sample reliability across empirically relevant configurations.

Semiparametric efficiency: In the paper’s usage, the LP estimator achieves the efficiency bound of Chamberlain (1987) for the semiparametric conditional moment restriction model E(yt - sum a_j y_{t-j} | ys, s <= t-1) = 0 when the controlled lag order diverges. Under Gaussianity, this coincides with Cramer-Rao efficiency. The key result is that the efficiency loss of LP relative to VAR-implied methods — well-documented under finite small-order VAR — is asymptotically negligible once the lag order diverges.

Martingale difference sequence (MDS) shocks: The shock process u_t satisfying E(u_t | u_s, s <= t-1) = 0 almost surely — a condition weaker than mean independence (E(u_t | u_s, s <= t-1) = 0 for all functions of past shocks). MDS shocks include GARCH and stochastic volatility processes. The paper’s SE_2 is designed to be consistent for general MDS shocks, while SE_1 and MOPM require the stronger mean-independence condition.

SE_2 (martingale-transformed standard error): The paper’s proposed standard error, constructed by first regressing LP residuals Delta_t on their forward values (Delta_{t+1}, …, Delta_{t+h-1}) to partial out serial dependence, then using the residual hat{xi}{1t} in the variance estimator as a simple sum of squares. SE_2 is uniformly consistent for general MDS shocks and requires no bandwidth selection, because the residual hat{xi}{1t} asymptotically recovers the MDS LP(infinity) error xi_{1t}(h; gamma).

VAR(infinity) model: A vector autoregression yt = sum_{j=1}^{infinity} a_j y_{t-j} + u_t with potentially infinitely many lags. The paper’s framework treats the true lag order as unknown and possibly infinite, requiring the controlled lag order p in the LP regression to diverge (at a rate constrained by Assumption 2) so that truncation bias becomes asymptotically negligible. VARMA processes are a special case shown to satisfy the paper’s assumptions.

Cumulated impulse response: The linear combination beta_1(h; gamma_CIR) = sum_{j=1}^{h} beta_1(j), corresponding to gamma = (1, …, 1)’. Cumulated responses exhibit slower convergence rates than individual responses — h^{-3/2} n^{1/2} versus (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} for an AR(1) — and are especially relevant when the response variable is in differences and the researcher seeks level responses of the original variable.

Manipulation-Robust Prediction

Mon, 01 Jan 0001 00:00:00 +0000

This paper addresses the problem of algorithmic manipulation: when consequential decisions are encoded in machine learning algorithms, individuals strategically alter their behavior to achieve desired outcomes, undermining the predictive validity of the algorithm. The authors develop a “strategy-robust” approach to training decision rules that explicitly models the incentives and costs of manipulation, producing rules that remain stable even when fully transparent. They then deploy and evaluate this approach in a large field experiment in Kenya — the first real-world implementation and evaluation of such a strategy-robust empirical decision rule.

The theoretical framework considers a policymaker who observes training data with features x_i and optimal decisions y_i, and wishes to estimate a decision rule to apply to new instances where behavior may be manipulated. While the standard approach (OLS or LASSO) selects a rule optimal for the training distribution, the strategy-robust approach models how individuals will adjust behavior in response to the incentive structure implied by any given rule. Under linear decision rules and quadratic manipulation costs, each individual shifts behavior by C_i^{-1} * beta away from their “bliss level,” where C_i captures individual- and behavior-specific manipulation costs. The strategy-robust estimator finds the rule that minimizes prediction error in the counterfactual world where people manipulate — a “Stackelberg” solution that commits the policymaker to a rule while anticipating equilibrium behavioral responses. Unlike LASSO, which penalizes all features equally without regard to their manipulability, the strategy-robust approach attenuates the weight on features that are both easily manipulated and subject to manipulation noise.

The empirical setting is a smartphone app (“Smart Sensing”) deployed to 1,557 participants in Nairobi, Kenya, in collaboration with the Busara Center. The app passively collected over 1,000 behavioral indicators (calls, texts, app usage, mobility, etc.) and delivered weekly financial “challenges” that rewarded participants based on decision rules randomly assigned to them. Average weekly payouts were calibrated to approximate typical digital credit loan amounts in Kenya at the time (approximately $4.80). The experiment has two phases: a training phase using control (beta = 0) and simple single-behavior incentive rules to estimate manipulation cost parameters via GMM, and an implementation phase using complex multi-feature decision rules to compare strategy-robust versus LASSO classifiers.

The main findings are as follows. First, participants demonstrably manipulate behavior: a joint F-test that incentive diagonals all equal zero is rejected with p < 0.001. The number of texts sent was 49 times more responsive to incentives than the number of people called during the workday. Outgoing communications are cheaper to manipulate than incoming, and simple behaviors (e.g., average talk time) more manipulable than complex ones (e.g., standard deviation of talk time). Individuals who self-report higher tech skills find manipulation 9% easier on average, and the 90th percentile of gaming ability finds manipulation twice as easy as the 10th percentile.

Second, in the implementation phase, strategy-robust decision rules outperform LASSO when the decision rule is made transparent to participants. Across all pooled outcomes, strategy-robust rules reduce RMSE by 11% (p = 0.024) relative to LASSO under transparency. For the single income-prediction outcome alone, the improvement is 5% ($0.19 RMSE reduction) but not statistically significant (p = 0.507).

Third, the framework enables estimation of the “cost of transparency.” Making naive LASSO rules transparent lowers performance by 23%. Switching to strategy-robust rules under full transparency reduces that performance decline to 9.2% — a 60% reduction in the cost of transparency. The model predicts this cost to be 9.8%, close to the implemented value of 11.3%.

The scope of the findings is bounded by the linear model with quadratic manipulation costs, a particular population of Kenyan smartphone users, and financial incentive magnitudes comparable to small digital credit loans. The mechanism relies on experimentally estimating manipulation cost parameters, though the authors also show that expert elicitation provides a correlated but noisier substitute (correlation 0.30 with experimental estimates).

Q: What is the core market failure the paper addresses, and why do standard fixes fail?

A: Standard machine learning training assumes the relationship between observed features and outcomes is stable, but implementing a consequential decision rule creates incentives for individuals to manipulate the features on which the rule is based (Goodhart’s Law; Lucas critique). The two common industry responses — restricting to “stable” predictors and keeping rules secret — are inadequate: restricting predictors amounts to a dogmatic prior that manipulation costs are either infinite or zero, while secrecy is increasingly at odds with demands for algorithmic transparency and fails anyway when sophisticated actors reverse-engineer the rule. Periodic retraining treats manipulation as generic covariate shift, can produce non-converging oscillations, and requires observing mistakes before learning from them.

Q: How does the strategy-robust estimator differ from OLS and LASSO?

A: OLS maximizes fit within the unincentivized training sample but ignores that implementing beta will shift behavior; LASSO adds a regularization penalty but still assumes behavior remains fixed at bliss levels and so penalizes all features equally regardless of manipulability. The strategy-robust estimator replaces each individual’s observed behavior x_i with their anticipated counterfactual behavior x_tilde_i(beta) = x_i + C_i^{-1} * beta, and finds the beta that minimizes prediction error in this manipulated distribution — a Stackelberg equilibrium. It attenuates features that are easily manipulated or subject to high manipulation noise, shifting weight toward harder-to-manipulate features even when the latter are less predictive in the training data.

Q: What are the three ways the strategy-robust estimator differs from standard estimators?

A: First, it anticipates level shifts in behavior: behaviors respond to beta, so observed training behaviors are replaced by counterfactual manipulated behaviors. Second, it accounts for signaling and noise: when manipulation ability correlates with the outcome of interest, manipulation can be informative about type (as in Spence 1973), but unobserved heterogeneity in gaming ability that is unrelated to outcomes introduces noise that attenuates coefficients on manipulable behaviors. Third, it achieves subgame perfection by anticipating how behaviors would respond to off-path deviations in beta, rather than assuming behaviors are fixed when beta deviates — yielding a Stackelberg rather than a one-step best-response solution.

Q: How were manipulation cost parameters estimated in the Kenya experiment?

A: In the training phase, each participant was randomly assigned to simple single-behavior incentive rules (e.g., “earn 12 Ksh. per incoming call this week, up to 250 Ksh.”) or control rules (beta = 0). This random variation in per-behavior incentives identifies how sensitive each behavior vector is to incentives, enabling GMM estimation of individual and behavior-specific cost parameters C and the heterogeneity scaling parameter omega. Off-diagonal elements of C were regularized to zero due to noisy estimation; diagonal elements used LASSO penalization with lambda = 1.0 set by cross-validation. Observable heterogeneity was allowed to vary with self-reported tech skills, which explained the most variation in preliminary analysis.

Q: What patterns were found in manipulation costs across behaviors?

A: Outgoing communications are cheaper to manipulate than incoming communications. Text messages, being relatively cheap to send, are more manipulable than calls. Simple behaviors such as average call duration are more manipulable than complex behaviors such as the standard deviation of talk time. Cross-behavior elasticities exist but are mostly noisy: 94.5% of off-diagonal incentive effects are not statistically significant (p < 0.05), 3.6% are significantly positive, and 1.8% are significantly negative.

Q: How large is heterogeneity in gaming ability, and what predicts it?

A: Individuals who self-report advanced or higher tech skills find it on average 9% easier to manipulate behaviors. Including unobserved heterogeneity, the 90th percentile of gaming ability finds manipulation twice as easy as the 10th percentile. Much of the heterogeneity arises from unobservables not captured by observables in the model.

Q: What happened when the naive LASSO rule was made transparent versus when the strategy-robust rule was made transparent?

A: Under the transparent treatment, participants received the full coefficients of the decision rule plus access to an interactive earnings calculator. Making naive LASSO rules transparent lowered performance by 23% relative to the opaque naive rule (RMSE $3.780 versus $4.641 in pooled outcomes). Switching to strategy-robust rules under full transparency reduced the performance decline to 9.2% — corresponding to a 60% reduction in the cost of transparency. The model predicted this cost to be 9.8%, which is close to the implemented value of 11.3%.

Q: What does the reduced-form evidence on behavior change under complex decision rules show?

A: Under the opaque treatment, participant behavior responses to complex decision rules were largely statistically insignificant and often in the wrong direction — 38.5% of estimated behavioral effects are in the same direction as the incentivized behavior. Under the transparent treatment, 75.4% of point-estimated effects are in the same direction as the incentive, confirming that transparency is a prerequisite for meaningful manipulation in this setting.

Q: How does the paper compare strategy-robust estimation to iterative retraining?

A: Simulation results show that iterative retraining of a naive LASSO model approaches the performance of the strategy-robust method after approximately 4 iterations. However, simulated performance of iterative retraining then begins to deteriorate; for the intelligence outcome, performance eventually falls below baseline performance before any retraining began. This illustrates that myopic best responses can produce non-convergent or suboptimal dynamics, while the strategy-robust approach finds the equilibrium rule directly.

Q: How does the paper compare strategy-robust estimation to the “intuitive” approach of simply excluding highly manipulable features?

A: The intuitive approach of excluding features above a manipulability threshold reduces predicted manipulability but also discards useful predictors. In some cases, the exclusions leave LASSO with no behaviors predictive enough to include, reducing performance. The strategy-robust approach can extract signal even from manipulable behaviors by adjusting their weights to account for manipulation noise, and outperforms the intuitive exclusion approach in the simulations reported in the Supplemental Appendix.

Q: Can manipulation costs be estimated without an experiment?

A: The authors briefly explore expert elicitation as a nonexperimental alternative: 171 individuals were surveyed to predict how Kenyans would manipulate phone behaviors when incentivized. Experts generally predicted lower costs (more manipulability) than observed experimentally, but the correlation between expert predictions and experimental estimates is 0.30. Using expert-elicited costs to train the strategy-robust model improved simulated performance substantially for one focal outcome and had an inconsequential negative effect for the other. Costs can also potentially be estimated from market prices and first principles when a structural model of underlying manipulations is available.

Q: What is the paper’s interpretation of its results through the lens of the Lucas critique?

A: The paper frames its contribution as a machine learning interpretation of Lucas (1976): just as implementing an economic policy changes the behavioral relationships on which the policy was calibrated, implementing a predictive decision rule beta changes the distribution of the very features the rule is based on. The key insight is that this counterfactual world has predictable structure — including a feature in the model tends to induce manipulation in that feature of a magnitude directly related to beta — so counterfactual fit can be estimated and rules can be optimized to perform well in the equilibrium they induce.

Q: What are the policy implications for algorithmic transparency?

A: The framework allows a policymaker to quantify and reduce the performance cost of transparency. The estimated equilibrium cost of transparency is roughly 10% when using strategy-robust rules, substantially less than the approximately 23% cost of making naive rules transparent. This means that strategy-robust rules can be disclosed — satisfying demands for a “right to explanation” under regulations such as GDPR — while losing far less performance than opaque naive rules would lose if disclosed.

Strategy-robust decision rule: A decision rule trained to anticipate that individuals will manipulate the features on which it is based, by replacing observed training behaviors with anticipated counterfactual manipulated behaviors in the loss function. It yields a Stackelberg equilibrium in which the policymaker commits to a rule while correctly forecasting the equilibrium behavioral response.

Manipulation costs (C_i): Individual- and behavior-specific quadratic costs that determine how far an individual shifts behavior from their bliss level in response to the incentive implied by a decision rule’s coefficient vector beta. Higher costs imply less behavioral response; costs are parameterized to allow separable heterogeneity by person and by behavior.

Bliss level (x_i): An individual’s unincentivized behavior — the behavior they would exhibit absent any decision rule (i.e., when beta = 0). Estimated from control periods in the experiment.

Gaming ability (gamma_i): Individual-level scaling factor for manipulation costs; a higher value means lower costs and easier manipulation. Modeled as a function of observable characteristics (e.g., self-reported tech skills) and unobservable heterogeneity.

Counterfactual fit: Predictive fit evaluated in the counterfactual state of the world where the decision rule is implemented and agents manipulate their features in response. The strategy-robust approach maximizes counterfactual fit, sacrificing within-sample fit (as measured on unmanipulated training data) to improve performance in deployment.

Cost of transparency: The reduction in predictive performance of a decision rule when its coefficients are disclosed to the individuals being evaluated. In the experiment, disclosure reduces performance of naive LASSO rules by 23% and strategy-robust rules by 9.2%, implying strategy-robust rules reduce the cost of transparency by 60%.

Stackelberg equilibrium: The solution concept in which the policymaker (leader) commits to a decision rule, correctly anticipating the best-response behavior of individuals (followers), rather than taking behavior as fixed or updating myopically. The strategy-robust estimator implements this equilibrium concept.

Performative prediction: The broader phenomenon, drawing on Perdomo et al. (2020), whereby a decision rule changes the distribution of the data it is applied to. The paper’s strategy-robust approach is an empirically estimable solution within this framework.

Marginal Propensity to Consume and Personal Characteristics: Evidence from Bank Transaction Data and Survey

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. This paper asks whether heterogeneity in the marginal propensity to consume (MPC) stems from temporary circumstances (e.g., transient wealth shocks that tighten liquidity) or persistent personal characteristics (e.g., high time discount rates or strong risk aversion that permanently shape saving behavior). Because liquidity constraints are endogenous — they can reflect either bad luck or impatient preferences — disentangling these two sources requires independently measured individual characteristics, which are not available in standard transaction datasets.

Data and Setting. The study combines two data sources drawn from Mizuho Bank, one of Japan’s three largest banks (approximately 24 million individual accounts). First, weekly bank account transaction data for January 2019 to November 2022 covering all outflows (ATM withdrawals, credit card debits, utility payments, interbank transfers) for the approximately 5,282 survey respondents. Second, a bespoke survey conducted in November–December 2022 among 400,000 randomly selected salary-receiving account holders (response rate 1.32%, yielding 5,282 usable observations). The survey elicits the Arrow–Pratt measure of absolute risk aversion, quantitative time discount rates for one-week, one-year, and ten-year horizons, self-reported liquidity constraints, homeownership, education, age, and gender, among other variables.

Three Income Shocks. MPC is estimated against three distinct income events: (1) the Japanese government’s Special Cash Payments (SCP) — a 100,000 JPY (approximately 800 USD) per-person lump-sum transfer during COVID-19, likely transitory, unexpected, and nearly randomly timed across municipalities due to administrative bottlenecks; (2) regular salary receipts (recurring, expected in both timing and amount); and (3) semi-annual bonus payments (received twice yearly, with timing known in advance but amount largely unknown — intermediate between SCP and salary in terms of expectedness).

Estimation Strategy. A two-way fixed effects regression with event-study leads and lags (windows of five weeks before and after each income event) is used to estimate consumption responses. Individual and week fixed effects absorb time-invariant heterogeneity and aggregate shocks (including COVID-19 emergency declarations). Standard errors are clustered at the individual level. For heterogeneity analysis, the income shock variable is interacted with individual characteristics from the survey (treated as proxies for persistent characteristics) and with time-varying log wealth and a liquidity constraint dummy (wealth below one-twelfth of annual income, proxying temporary circumstances).

Main Findings — Average MPC. Across all three income types, the on-impact MPC (week of receipt) is approximately 0.2: specifically γ₀ = 0.23 for the SCP (significant at 5%), 0.20 for salary, and 0.22 for bonus. When estimated jointly in a single regression, coefficients are γ_SCP = 0.21, γ_salary = 0.19, and γ_bonus = 0.21. This uniformity holds despite the sharply different properties of these shocks (transitory-unexpected vs. regular-expected vs. semi-known).

Main Findings — Heterogeneity. Significant heterogeneity in MPC is found primarily in the bonus subsample, where statistical power is greatest. The following cross-term coefficients are significant at the 5% level in the multivariate specification: (a) liquidity constraint dummy — positive and significant, indicating that individuals temporarily below one month’s income in deposits spend a larger fraction of their bonus, with a one standard deviation increase raising MPC by 0.094 (9.4 percentage points); (b) time discount rate (quantitative measure) — positive and significant, with a one standard deviation increase in impatience raising MPC by 0.084; (c) risk aversion (quantitative Arrow–Pratt measure) — positive and significant, conditional on controlling for wealth and liquidity, with a one standard deviation increase raising MPC by 0.031; (d) education — negative and significant irrespective of wealth/liquidity controls, with a one standard deviation increase in education reducing MPC by 0.041.

These magnitude estimates are sizable relative to the baseline MPC of approximately 0.2. For SCP and salary shocks, cross-term coefficients are uniformly insignificant at the 5% level, which the author attributes partly to smaller sample sizes and shorter observation windows for the SCP subsample.

Scope Conditions. The sample consists of Mizuho Bank account holders who receive salary payments directly into their Mizuho account, overrepresenting metropolitan areas and salaried workers relative to the national census. Wealth at Mizuho captures only deposits at that institution and excludes securities accounts, postal savings, and intra-household transfers. Age and gender do not yield significant cross-term coefficients in any specification; the self-reported survey measure of liquidity constraints (ability to cover one month’s income by drawing on savings, assets, or borrowing) is also insignificant, in contrast to the transaction-based liquidity constraint dummy.

In depth

Q1. Why is separating temporary circumstances from persistent characteristics important for MPC estimation?

Liquidity constraints — the standard proximate predictor of high MPC — are endogenous. An individual may be liquidity-constrained because of a temporary adverse income shock (bad luck) or because of persistently high impatience (high time discount rate) that leads to chronically low saving. If policy evaluation treats all constrained households symmetrically, it conflates these two very different channels. The paper follows Jappelli and Pistaferri (2020), Gelman (2021), and Aguiar, Bils, and Boar (2021) in arguing that both channels matter and that their relative contributions need empirical separation.

Q2. Why are Japanese bonuses particularly well-suited to identifying MPC heterogeneity?

Bonuses are paid semi-annually to most regular employees in Japan (accounting for roughly 15–30% of annual income), with timing known in advance but amount largely unknown until receipt. This intermediate nature — partially anticipated in timing but uncertain in magnitude — provides meaningful variation in consumption responses across individuals while maintaining a clean event-study design. The bonus subsample (3,722 individuals who received a bonus at least once) is also large enough to detect cross-term effects that are statistically insignificant in the SCP subsample (2,446 individuals) and in the salary analysis, likely due to greater statistical power.

Q3. How is the Arrow–Pratt measure of risk aversion constructed from the survey?

Respondents are asked whether they would purchase a lottery ticket at prize value Z = 100,000 JPY and price p = 10,000 JPY for varying winning probabilities α. The threshold α at which a respondent switches from accepting to rejecting identifies their risk attitude. The absolute risk aversion σ = −U’’/U’ is then calculated as (αZ² − 2αZp + p²) / (2(αZ − p)). This yields σ ranging from −4.5 (when α = 0.01, i.e., risk-loving) to 0.891 (when α = 1, i.e., refusing to buy even at a 90% win probability). Risk neutrality corresponds to σ = 0 (at α = 0.1).

Q4. How are time discount rates measured, and what is the range?

Respondents are asked the minimum amount X they would require to wait one week, one year, or ten years to receive a payment instead of receiving 100,000 JPY one week from now (using a one-week anchor to address hyperbolic discounting). The discount rate is calculated as r = X/100,000. The range is 0.01 (X = 100 JPY) to 100 (X = 10,000,000 JPY, i.e., would not wait even for 1,100,000 JPY in ten years). The unweighted average across one-week, one-year, and ten-year horizons is used as the composite discount rate in the multivariate specifications.

Q5. What is the transaction-based liquidity constraint dummy, and how does it differ from the survey-based measure?

The transaction-based dummy equals one if end-of-month deposits at Mizuho Bank (the previous month) are below one-twelfth of the individual’s annual income — i.e., if the individual holds less than one month’s equivalent income in liquid deposits. This is a time-varying measure. The survey-based measure asks respondents to self-report whether they could cover one month’s income by drawing on savings, selling assets, or borrowing. The transaction-based measure is significant at the 5% level in the bonus and salary heterogeneity regressions, while the survey-based measure is insignificant, indicating that the precise definition and data source of the liquidity constraint measure matters materially for detecting its effect on MPC.

Q6. What are the estimated on-impact MPC values for each income shock, and how stable are they across robustness checks?

The point estimates from the event-study regression (γ₀) are: 0.23 for SCP in the baseline sample (SCP recipients in 2020, N = 2,446 individuals), 0.20 for salary (all 5,282 survey respondents), and 0.22 for bonus (3,722 bonus recipients). In a robustness specification restricting to only year-2020 data for the SCP, γ₀ = 0.235; using cash withdrawals from ATMs as a proxy for consumption instead of total outflows, γ₀ = 0.162 for SCP. In a joint regression including all three income types simultaneously, γ_SCP = 0.21, γ_salary = 0.19, and γ_bonus = 0.21. The SCP MPC for the smaller second-wave subsample (200 individuals, 2021–22) is 0.104 and insignificant, consistent with insufficient statistical power rather than a structural difference.

Q7. Why is the similarity in MPC across the three shock types potentially surprising, and what does the paper say about it?

Standard theory predicts divergent MPCs: transitory unexpected windfalls (SCP) should have a higher MPC than permanent salary changes under the permanent income hypothesis, while Ricardian equivalence might reduce the MPC to fiscal transfers like the SCP if households anticipate future tax increases. The paper finds the MPCs are approximately equal (around 0.2 across all three types), and if anything the SCP MPC is slightly higher than the salary MPC. The paper acknowledges this uniformity without offering a structural explanation, using it primarily as a robustness check on the baseline estimate rather than a substantive puzzle to resolve.

Q8. Which personal characteristics are significantly associated with higher MPC, and in which income shock samples?

In the multivariate heterogeneity regression, significant cross-term coefficients at the 5% level are found exclusively in the bonus subsample (columns 5–6 of Table 6): the quantitative risk aversion measure (positive, coefficient 0.042–0.049), the quantitative discount rate (positive, coefficient 0.004), and education (negative, coefficient −0.034 to −0.037). The liquidity constraint dummy (transaction-based) is also positive and significant for bonuses. In the univariate robustness regressions (Table 7), the own-house dummy is negative and significant at 5% for bonuses (controlled and uncontrolled); discount rates for one-week and ten-year horizons are positive and significant at 5% for bonuses; risk aversion A (direct self-report) is negative and significant at 5% for SCPs in the uncontrolled specification.

Q9. Do age and gender matter for MPC heterogeneity?

No. In all specifications across all three income shock types, the cross-term coefficients on age and the male dummy are uniformly insignificant at the 5% level. The lack of significance for age and gender is noted as a notable result, since both are commonly used demographic proxies in heterogeneous agent models that assume they reflect economically meaningful differences in consumption behavior.

Q10. How does the paper quantify the economic magnitude of each significant heterogeneity factor?

Table 8 reports the product of each cross-term coefficient and the standard deviation of the corresponding variable. For the bonus subsample: a one standard deviation increase in the liquidity constraint dummy raises MPC by 0.094 (9.4 percentage points); a one standard deviation increase in the discount rate raises MPC by 0.084; a one standard deviation increase in risk aversion raises MPC by 0.031; and a one standard deviation increase in education reduces MPC by 0.041. All four magnitudes are described as sizable relative to the baseline MPC of approximately 0.2 (20%).

Q11. Why does the paper focus on bonuses for the heterogeneity analysis rather than the SCP?

The SCP events provide cleaner identification of transitory, exogenous income shocks (near-random timing due to municipal administrative bottlenecks, as documented by Kubota, Onishi, and Toyama 2021), but the subsample of SCP recipients is smaller (2,446 in 2020, 200 in the second wave), reducing statistical power for detecting heterogeneity in cross-term coefficients. The salary sample is large (5,282 individuals) but salaries are expected, recurring, and may partially update permanent income, complicating interpretation of cross-term estimates. Bonuses offer a balance: a relatively large subsample (3,722) and a partially unexpected income component, making them the most informative sample for heterogeneity analysis.

Q12. What are the main caveats and limitations the paper identifies?

Four caveats are noted. First, the personal characteristics from the survey — including time discount rates and risk aversion — are treated as exogenous, but they may themselves be endogenous to economic circumstances or short-term conditions at the time of the survey. Second, only Mizuho Bank deposits are observed; financial assets at other institutions (securities, postal savings) are missing, meaning the liquidity constraint measure understates true wealth for some respondents. Third, the sample is tilted toward metropolitan salaried workers and toward wealthier individuals compared to the full Mizuho customer base (median log wealth of 7.4 vs. 5.9 in Kubota et al. 2021). Fourth, the multiple-testing problem is acknowledged: with many cross-term tests conducted, some rejections of the null at the 5% level may be spurious.

Key Concepts

Marginal Propensity to Consume (MPC, on-impact). In this paper, MPC is operationalized as the coefficient γ₀ from the two-way fixed effects event-study regression — specifically, the fraction of an income shock spent during the same week the shock is received, estimated from total bank account outflows. This is a weekly, within-account measure, not a lifetime or annual consumption response.

Arrow–Pratt Absolute Risk Aversion (σ). A quantitative measure of risk preferences computed from the paper’s survey by eliciting the probability threshold α at which a respondent is indifferent between buying and not buying a lottery with prize Z = 100,000 JPY and price p = 10,000 JPY. Calculated as σ = (αZ² − 2αZp + p²) / (2(αZ − p)). Ranges from −4.5 to 0.891 in the sample, with σ = 0 indicating risk neutrality.

Time Discount Rate (r). Measured by asking respondents the minimum additional amount X (beyond 100,000 JPY) they would require to delay receipt by one week, one year, or ten years, with r = X/100,000. The paper uses the unweighted average of three horizon-specific rates as a composite measure. Ranges from 0.01 to 100 in the sample. Used as a proxy for impatience or myopia — a persistent personal characteristic.

Liquidity Constraint Dummy (transaction-based). A time-varying binary indicator that equals one if individual i’s end-of-month Mizuho Bank deposit balance in month t−1 is below one-twelfth of annual income at t−1 — i.e., less than one month’s equivalent income in liquid deposits. Distinguished in the paper from a survey-based self-report of liquidity constraints, which is found to be insignificant.

Special Cash Payment (SCP). The Japanese government’s COVID-19 pandemic transfer program, providing 100,000 JPY (approximately 800 USD) per person in 2020 (universal) and 100,000 JPY per child in 2021–22 (restricted to households with children under 18 and income below 9.6 million JPY annually). Used in this paper as a transitory, salient, and largely unexpected income shock because municipal administrative bottlenecks made the exact timing unpredictable and nearly random across households.

Two-Way Fixed Effects Event-Study Regression. The paper’s primary estimator, which includes individual fixed effects (controlling for time-invariant person-level heterogeneity) and week fixed effects (absorbing aggregate shocks such as COVID-19 emergency declarations and seasonal patterns). Event-study leads and lags (k = −5 to +5 weeks around each income receipt) allow pre-trend testing and tracing of the dynamic consumption response. Normalized to γ_{−1} = 0.

MPC Heterogeneity Cross-Term. A regression augmentation (equation 3 in the paper) in which the contemporaneous income shock X⁰_{it} is interacted with individual characteristic Z_{it}. The coefficient δ on this cross-term identifies how the MPC varies with Z — the marginal effect of characteristic Z on the MPC. Persistent characteristics (e.g., risk aversion, discount rate, education from the survey) and temporary circumstances (e.g., log wealth, liquidity constraint dummy from transaction data) are included as separate Z variables.

Market Segmentation through Information

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks what market outcomes an information designer — modeled as an internet platform that knows consumers’ preferences — can achieve by choosing what information to disclose to competing oligopolistic firms who then make personalized price offers. The model features n firms each producing a single differentiated product at zero cost, a continuum of consumers with unit demand and multidimensional valuations (one per product), and a designer who commits to a mapping from consumer types to joint distributions over messages sent to firms before they play a simultaneous pricing game. The designer’s objective spans the full range from maximizing producer surplus to maximizing consumer surplus.

The paper establishes two main results. First, under a necessary and sufficient condition called Aggregate Incentive Compatibility (AIC), the designer can implement full surplus extraction by firms — the producer-optimal outcome — in which every consumer buys her most preferred product at a price exactly equal to her valuation for it, capturing 100% of available surplus for producers. The AIC condition requires, for each firm i and each candidate deviation price p_hat_i, that the infra-marginal losses firm i would bear on its natural customers (those in Ei who value i most) from lowering price to p_hat_i must be weakly greater than the maximum business-stealing profit available from consumers who prefer other products but have valuation for i above p_hat_i. The condition is easier to satisfy when consumer preferences are more polarized, i.e., when consumers have stronger relative preferences for their most-preferred product. When firms offer homogeneous products the condition fails everywhere and no information structure can generate any producer surplus — Bertrand competition drives all profits to zero under any signal structure.

Second, the paper characterizes the consumer-optimal information structure, which achieves the maximum possible consumer surplus across all equilibria induced by any information structure. The upper bound on consumer surplus is CS* = (total surplus) minus sum_i Pi*_i, where Pi*_i is the profit firm i can guarantee itself by ignoring the designer’s signal and setting the best uniform price assuming all rivals price at zero. This bound is tight: the designer can implement it by publicly partitioning consumers into groups by most-preferred product, inducing rival firms to price at marginal cost (zero) for consumers who prefer another firm’s product, and then applying the Bergemann-Brooks-Morris (2015) extremal segmentation within each firm’s natural customer set to preserve each firm’s guarantee profit while achieving efficiency.

The illustrative two-firm example shows the quantitative stakes concretely. With no information disclosure, firms charge 4/5 and total producer surplus is about 76% of total surplus S*, consumer surplus is just under 10% of S*, and some consumers are excluded. With full disclosure, producer surplus rises to about 81% of S* and consumer surplus to 19%. The producer-optimal information structure (Case 3) achieves 100% of S* as producer surplus by pooling consumers who prefer different products into the same message submarket, giving each firm an incentive to price for its highest-valuing customers and ignore the others. The consumer-optimal information structure (Case 4) brings producer surplus down to about 57% of S* — its guaranteed lower bound — and delivers roughly 43% of S* to consumers, an outcome unattainable by full disclosure alone.

Both producer-optimal and consumer-optimal outcomes are efficient: all consumers buy their most-preferred product in both cases. The paper further characterizes the full efficient frontier between consumer- and producer-optimal outcomes, showing that mixing the consumer-optimal and full-information structures (or consumer-optimal, full-information, and producer-optimal structures when the latter is implementable) spans every point on the frontier.

The model assumes firms will price-discriminate if they can, that the designer has full knowledge of consumer types, and that the game is played once. The core results extend to continuous type distributions as shown in Online Appendix B.2. The analysis is restricted to a monopoly platform; competition among platforms is left for future work.

Q: What is the central research question and why does the two-benchmark comparison used by antitrust authorities miss important possibilities?

A: The paper asks what market outcomes — combinations of consumer and producer surplus — an information designer (a platform) can achieve by choosing among all possible information structures, not just the two benchmarks of no-information and full-information. Antitrust analysis that compares only those two cases misses a vast middle ground: an intermediary can package information in ways that, for instance, implement perfect collusion (extracting all surplus as producer surplus) while appearing to use privacy-protective technologies, or can intensify competition well beyond the full-information benchmark to benefit consumers.

Q: What is the producer-optimal information structure and when does it exist?

A: A producer-optimal information structure is one that induces an equilibrium in which every consumer buys her most-preferred product at a price exactly equal to her valuation — full surplus extraction. It exists if and only if, for every firm i and every candidate deviation price p_hat_i, the Aggregate Incentive Compatibility (AIC) condition holds: the aggregate infra-marginal losses firm i would suffer on its natural customers Ei from lowering price to p_hat_i must be at least as large as the maximum business-stealing profit from consumers outside Ei who have valuation for i weakly above p_hat_i. This is a condition on the distribution of consumer valuations, not on the information structure per se.

Q: What is the economic mechanism behind the producer-optimal structure — how does pooling consumers implement full surplus extraction?

A: The designer assigns consumers who prefer product A to the same message submarket as consumers who prefer another product but have a lower valuation for A. Firm A is then price-recommended its highest-valuing customers’ willingness to pay. The presence of the “outside” consumers in the same message makes it unprofitable for firm A to deviate downward to capture them, because the infra-marginal loss on the natural customers exceeds the additional revenue. Simultaneously, the rival firm cannot identify and undercut for A’s natural customers because the messages do not allow it to distinguish them. The result is that each firm plays a niche strategy, setting price equal to the valuation of its highest-type natural customers and excluding the others from its offer.

Q: When does polarization of consumer preferences help achieve the producer-optimal outcome?

A: Proposition 1 states that if a producer-optimal information structure exists under distribution f, it also exists under any distribution f_tilde that is more polarized than f — where more polarized means the mass of consumers who prefer i and have valuation above any threshold for i increases, and the mass of consumers who prefer j but have valuation above that threshold for i decreases. Intuitively, polarization slackens the Firm IC constraints because it reduces the business-stealing temptation: fewer consumers with high cross-product valuations are available for firm i to capture by undercutting. Concrete continuous-distribution examples include: uniform over the unit square (producer-optimal always exists), Hotelling anti-correlated values (exists everywhere), and truncated normal with mean 1/2 — producer-optimal is feasible for all standard deviations sigma > 0.15.

Q: Why does the producer-optimal outcome fail entirely when products are homogeneous?

A: Proposition 2 states that when all consumer types have equal valuations across products (the support of f lies on the diagonal of V^n), then for any information structure and any induced equilibrium, every consumer buys at price zero and all firms earn zero profit. The logic extends the standard Bertrand undercutting argument: with homogeneous products, any positive price a firm charges is undercut by a rival who can always profitably steal demand, and this applies to any posterior distribution induced by any signal realization. Even private signals cannot prevent this outcome because no signal realization can give a firm a non-contestable position.

Q: How is the consumer-optimal information structure constructed, and what is its key economic logic?

A: Theorem 2 shows the consumer-optimal structure has three layers. First, consumers are partitioned into n groups by most-preferred product (Ei). Second, firms j not equal to i are induced — by publicly revealing which group a consumer belongs to — to set price zero for consumers outside their group, because competing for those consumers is hopeless when their preferred firm is identified. Third, within each Ei, consumers are further partitioned into submarkets using the Bergemann-Brooks-Morris (2015) extremal segmentation applied to residual valuations (theta_i minus the maximum of competing valuations), ensuring firm i earns exactly its guarantee profit Pi*_i. By holding each firm down to its guarantee profit, the residual goes to consumers, maximizing CS.

Q: What is the guarantee profit Pi*_i and how does it bound consumer surplus?

A: Pi*i is the maximum profit firm i can achieve by ignoring all designer signals and setting a single uniform price to all consumers, against the worst-case scenario in which all other firms price at zero. Formally, Pi*i = max{pi} sum{theta in Ei: theta_i - pi >= max_{j not equal i} theta_j} pi * f(theta). Since firm i can always achieve Pi*_i regardless of the information structure (by simply ignoring signals), no information structure can push firm i’s profit below Pi*_i. The sum of these guarantee profits across all firms provides a lower bound on total producer surplus — and therefore an upper bound on consumer surplus — achievable by any information structure.

Q: In the two-firm numerical example, what is the quantitative comparison across the four cases?

A: Total available surplus S* = 0.84. Under no information (Case 1): producer surplus approximately 76% of S*, consumer surplus just under 10% of S*, and consumers of types (3/5, 2/5) and (2/5, 3/5) do not trade. Under full disclosure (Case 2): producer surplus approximately 81% of S*, consumer surplus 19% of S*, efficient. Under the producer-optimal structure (Case 3): producer surplus = 100% of S* (all surplus extracted), consumer surplus = 0%, efficient. Under the consumer-optimal structure (Case 4): producer surplus approximately 57% of S*, consumer surplus approximately 43% of S*, efficient. All cases except Case 1 are efficient; the no-information case excludes some consumers from trading.

Q: Is the full-information disclosure structure consumer-optimal?

A: Not in general. Proposition 3 states that full information is consumer-optimal if and only if all consumers in Ei have identical residual valuations (theta_i minus their second-best alternative) — a condition that generically fails. When residual valuations within Ei are heterogeneous, the designer can do strictly better for consumers by applying the extremal segmentation within each Ei rather than revealing full information, which would allow firms to price-discriminate on individual residual valuations and extract more surplus.

Q: Can the designer trace out the entire efficient frontier between consumer- and producer-optimal outcomes?

A: Yes, under two conditions. First, by mixing the consumer-optimal structure (point A) with the full-information structure (point B) using fractions lambda and 1-lambda respectively, the designer can implement any point on the efficient frontier between A and B. Second, when the producer-optimal outcome (point C) is also implementable, mixing the full-information structure with the producer-optimal structure by applying them to fractions lambda and 1-lambda of the consumer population respectively spans every point between B and C. The key insight is that the AIC condition, if it holds for f, also holds for any rescaled sub-distribution of f (it is scale-invariant), so the producer-optimal sub-problem remains feasible.

Q: What are the regulatory implications of the analysis?

A: The paper identifies a fundamental tension: banning information use sacrifices efficiency (some consumers excluded, wrong products purchased), but unrestricted use permits platforms to implement perfect collusion through information design. Critically, the paper shows that privacy-enhancing technologies that pool consumers into cohorts — like Google’s Privacy Sandbox — are equally consistent with the producer-optimal (collusive) and consumer-optimal (competitive) structures; the two differ only in the principle by which consumers are grouped. The paper suggests regulators could mandate that consumers in the same cohort share the same most-preferred product and that information be disclosed symmetrically across firms — the defining features of the consumer-optimal structure. This would block the producer-optimal grouping (which mixes consumers with different most-preferred products) while preserving efficiency.

Q: How does this paper relate to and extend Bergemann, Brooks, and Morris (2015)?

A: Bergemann, Brooks, and Morris (2015) characterize achievable consumer and producer surplus outcomes when a designer discloses information to a single monopolist who can price-discriminate. The present paper extends this to oligopoly, where competition between firms creates both additional constraints (firms may undercut each other) and additional instruments (the designer can play firms against each other). The consumer-optimal construction directly applies the BBM (2015) extremal segmentation within each firm’s natural customer set Ei, but the outer layer — using public revelation of group membership to induce rival firms to price at zero — is new and arises specifically from the oligopoly setting.

Information designer: An entity (modeled as a platform) that observes the full joint distribution of consumer valuations over all products and commits, before firms price, to a mapping from consumer types to joint distributions over messages sent to competing firms; the designer can be interpreted as an internet intermediary choosing how to package and share consumer data.

Aggregate Incentive Compatibility (AIC): The necessary and sufficient condition on the distribution of consumer valuations for the existence of a producer-optimal information structure; for each firm i and each candidate deviation price p_hat_i, the aggregate infra-marginal losses firm i would incur on its natural customers by lowering price to p_hat_i must weakly exceed the maximum revenue firm i could gain by attracting consumers who prefer rival products but have valuation for i above p_hat_i.

Producer-optimal information structure: An information structure that induces an equilibrium in which every consumer buys her most-preferred product at a price exactly equal to her full valuation for it, extracting 100% of available surplus as producer surplus — the outcome equivalent to the firms’ fully collusive joint surplus maximum.

Consumer-optimal information structure: An information structure that achieves the maximum consumer surplus attainable across all equilibria induced by any information structure, holding each firm to its guarantee profit Pi*_i (the best uniform-price profit the firm can secure by ignoring all signals) and allocating all residual surplus to consumers while maintaining allocative efficiency.

Guarantee profit (Pi*i): The maximum profit firm i can secure unilaterally by ignoring the designer’s signal and setting an optimal uniform price, computed against the worst case in which all rival firms price at zero; it equals max{pi} times the sum of f(theta) over all types in Ei for which theta_i minus pi exceeds all rival valuations.

Polarization of preferences: A stochastic dominance condition under which, relative to a baseline distribution, the mass of consumers who prefer product i and have high valuations for it increases while the mass of consumers who prefer rival products but have high valuations for i decreases; higher polarization weakens the Firm IC constraints and makes the producer-optimal outcome easier to implement (Proposition 1).

Separation and Consistency: Two structural properties any producer-optimal information structure must satisfy: Separation requires that the messages firm i sends to different consumers in Ei who have distinct valuations for i are disjoint in support; Consistency requires that every message firm i can send to any consumer type is contained in the union of messages firm i sends to consumers in Ei, preventing firm i from ever inferring that a consumer prefers a rival’s product.

Markov-Perfect Equilibria in Differential Games—With an Application to Climate Policy

Mon, 01 Jan 0001 00:00:00 +0000

This paper by Jaakkola and Wagener addresses a long-standing open problem in the theory of differential games: how to make Markov-perfect equilibria (MPE) well-defined when best-response policy functions are generically discontinuous in the state variable. The paper’s primary contribution is methodological — it introduces discontinuous Markovian strategies into differential games and proves that, under this extension, (i) payoffs can always be computed and (ii) unique best responses exist for almost all strategy profiles of opponents. The authors then apply this framework to derive the entire set of symmetric MPE in a canonical non-cooperative climate mitigation model (van der Ploeg and de Zeeuw, 1992), finding welfare results that are quantitatively large and policy-relevant.

The technical difficulty the paper resolves is that discontinuous policy functions can cause the ordinary differential equation governing state dynamics to lack classical solutions, making payoffs undefined. Prior literature responded either by restricting strategies to continuous functions — which rules out many natural best responses and imposes an unjustified constraint on the strategy space — or by allowing discontinuities only in “admissible” profiles, which makes each player’s strategy set depend on opponents’ choices and thus violates the basic structure of non-cooperative game theory. The authors’ solution is to adopt Filippov solutions (differential inclusions that convexify dynamics at discontinuities), so that a well-defined state trajectory and payoff exist for every strategy profile, not just admissible ones.

The paper’s three main theorems cover existence (Theorem 1), characterization (Theorem 2), and symmetric equilibrium conditions (Theorem 3). Theorem 1 establishes that, given any fixed set of potential jump points, the best-response correspondence maps almost all opponent strategy profiles to a unique Markovian best response — “almost all” in the sense of prevalence on infinite-dimensional function spaces. Theorem 2 provides necessary and sufficient conditions for a strategy to be a best response: it must satisfy the maximum principle where the value function is differentiable, value discontinuities may only occur at jump points of opponents’ strategies where the player cannot unilaterally push the state back to the low-stock side, and the value at any such interface must exceed the static optimum. Theorem 3 translates these into conditions for symmetric Nash equilibrium.

Applied to the van der Ploeg–de Zeeuw climate model — N symmetric countries choosing emissions a_i, with carbon stock x evolving as x-dot = sum(a_i) - deltax, and flow utility u(x, a_i) = a_i - (1/2)a_i^2 - dx — the paper characterizes the complete set of symmetric MPE. The unique continuous globally defined equilibrium (the linear MPE, previously established by Rowat 2007) is shown to be weakly Pareto-dominated by every other MPE with a continuous value function. The best equilibria feature discontinuous strategies that act like stock-conditioned trigger strategies: when the carbon stock falls below a target steady state x, players respond with a discrete upward jump in emissions to rapidly return the economy to x*; when carbon rises above x*, players increase emissions only gradually, creating a threat of drifting to a higher-pollution steady state that disciplines deviations. In a calibrated example with N=10, delta=0.02, rho=0.02, and damage parameter d=0.5, the linear equilibrium steady state is approximately 2.5 times the first-best level, while the best continuous-value MPE steady state is approximately 1.2 times the first-best level. Choosing the best equilibrium rather than the linear equilibrium closes between 50 and 100 percent of the welfare gap to the first-best outcome, depending on initial conditions. The paper also identifies particularly bad equilibria involving value-function discontinuities — coordination failures in which no single country can unilaterally stop the carbon stock from rising past a threshold — that can yield welfare outcomes worse than the linear equilibrium at high carbon levels.

The scope of the methodological results covers differential games with a single state variable and strategies that are real-analytic except at finitely many points. Extension to multiple state variables is left for future work. The climate application is restricted to the symmetric linear-quadratic van der Ploeg–de Zeeuw framework, chosen to facilitate comparison with prior literature.

Q: What is the fundamental technical problem with MPE in differential games that this paper resolves?

A: In differential games with Markovian strategies, best-response policy functions are generically discontinuous in the state variable. Discontinuous right-hand sides in the state dynamics ODE can prevent existence or uniqueness of classical solutions, making payoffs undefined for some strategy profiles. Prior literature either restricted attention to continuous strategies (causing non-existence of best responses to many profiles) or defined “admissible” strategy sets that depend on opponents’ choices (violating non-cooperative game theory structure). This paper resolves both problems for the single-state-variable case.

Q: How does the paper make payoffs well-defined under discontinuous strategies?

A: The paper adopts Filippov solutions — differential inclusions that replace the dynamics at a discontinuity point with a convex hull of the left and right limits. At a “push-push” discontinuity (where dynamics push the state toward the jump point from both sides), the Filippov solution remains at the jump point and flow payoffs are a weighted average of left and right actions. This ensures a well-defined trajectory and payoff for every strategy profile, not just “admissible” ones, restoring the standard non-cooperative game-theoretic structure.

Q: What does Theorem 1 establish, and what does “almost all” mean in this context?

A: Theorem 1 establishes that, for any fixed collection of jump points, each player has a unique Markovian best response to almost every profile of opponents’ strategies. “Almost all” is in the sense of prevalence on infinite-dimensional function spaces (following Hunt, Sauer, and Yorke 1992): the set of profiles for which a unique best response fails to exist is shy (measure-zero analog in infinite dimensions) and nowhere dense. This resolves the long-standing open problem of making MPE well-founded in differential games.

Q: What are the necessary and sufficient conditions for a best response given by Theorem 2?

A: A strategy phi_i is the best response to opponents’ profile if and only if: (i) at all points where the value function is differentiable, the strategy satisfies the maximum principle; (ii) the value function is decreasing in the state (monotonicity); (iii) value discontinuities may occur only at opponents’ jump points where player i cannot unilaterally move the state back to the low-stock region; (iv) at any such interface, the value must be at least as large as the static optimum u(x, a_i)/rho; and (v) the value is differentiable at push-push steady states. These conditions extend the standard maximum principle with local requirements that restrict which discontinuities are possible.

Q: What is the van der Ploeg–de Zeeuw model and why is it used here?

A: The van der Ploeg–de Zeeuw (1992) model has N symmetric countries choosing emissions a_i, with carbon stock evolving as x-dot = sum(a_i) - delta*x, and flow utility u(x, a_i) = a_i - (1/2)a_i^2 - dx. It is linear-quadratic, so a linear MPE exists and is analytically tractable, and prior literature (Dockner and Long 1993; Rowat 2007; Dockner and Wagener 2014) has studied it extensively. The paper uses it as a benchmark to demonstrate that the new methods yield novel and economically important results for even well-understood models.

Q: What is the linear equilibrium and why does it produce poor welfare outcomes?

A: The linear equilibrium phi_L(x) = alpha + beta*x, with beta negative, is the unique continuous globally defined MPE (Rowat 2007). In it, emissions decrease with the carbon stock because each player anticipates that opponents will also reduce emissions when carbon is high. This strategic substitutability creates adverse dynamic free-riding: players try to exploit the fact that high carbon stock will cause opponents to cut back, so each has an incentive to emit more when carbon is low. In the calibrated example, the linear equilibrium steady state is approximately 2.5 times the first-best level.

Q: What do the best equilibria look like, and why do they achieve high welfare?

A: The best equilibria feature a target steady state x* near the first-best level and a discontinuous upward jump in emissions when carbon falls slightly below x*. This threat rapidly returns any carbon reduction back to x*, eliminating the strategic incentive to free-ride on others’ reductions. When carbon rises above x*, emissions increase only slightly, causing the economy to drift slowly toward a higher-pollution steady state — the threat of this bad outcome disciplines overshooting. This mechanism is analogous to a trigger strategy but is conditioned on the stock level rather than on past actions, making it compatible with Markovian strategies.

Q: How large are the welfare gains from the best equilibrium relative to the linear equilibrium?

A: In the calibrated example with N=10, delta=0.02, rho=0.02, and d=0.5, the best continuous-value MPE steady state is approximately 1.2 times the first-best level, compared to 2.5 times for the linear equilibrium. Choosing the best equilibrium closes between 50 and 100 percent of the welfare gap between the linear equilibrium and the first-best outcome, depending on initial conditions. The paper characterizes this as a quantitatively large, first-order welfare improvement.

Q: What are “coordination failure” equilibria and when do they arise?

A: Coordination failure equilibria feature discontinuities not only in the strategy (emission rate) but also in the value function itself. They arise when no single country can unilaterally prevent the carbon stock from rising past a threshold — formally, when N * a_max < delta * x at the discontinuity point. In such cases, if opponents are emitting heavily, no individual country can stop atmospheric carbon from rising even if it emits nothing, making heavy emission a best response. All players following this logic simultaneously produce a self-fulfilling collapse to high emissions. At high carbon levels these equilibria can yield welfare outcomes worse than the linear equilibrium.

Q: What is the paper’s main policy implication for climate negotiations?

A: The paper argues that international climate negotiations should be understood as a coordination problem over which of many MPE is played, rather than as bargaining over a limited cooperative surplus in a dynamic prisoners’ dilemma. Since the best equilibria are self-enforcing (they are Nash equilibria, not cooperative solutions), they do not require external enforcement. The paper suggests effective agreements may involve threshold-based commitments — sharp decarbonisation if a carbon target is met, but acceptance of a substantially higher stabilisation target (e.g., 2.5 degrees C rather than 2 degrees C) if the first target is missed — to create the discontinuous strategic incentives that support good equilibria.

Q: How does the paper handle the previously identified “local MPE” that could not be extended to the entire state space?

A: Prior work (Dockner and Long 1993; Rubio and Casino 2002; Dockner and Wagener 2014) constructed nonlinear equilibria that were only locally defined, and the validity of such equilibria was questioned (Rowat 2007; Bernhard 2024) because they were undefined on the full state space. The present paper’s framework allows discontinuous strategies, so these locally defined equilibria can be extended into globally defined, discontinuous MPE. Most previously discovered equilibria are shown to be nested within the larger set of all symmetric MPE identified here.

Q: What mathematical tools are used to prove the main results?

A: The proofs rely on the theory of viscosity solutions to Hamilton-Jacobi-Bellman equations (Bardi and Capuzzo-Dolcetta 2008), building on and extending results of Barles, Briani, and Chasseigne (2013, 2014) on optimal control with discontinuous dynamics. A key departure from Barles et al. is that the paper cannot assume controllability of the dynamics near discontinuities without imposing undue restrictions on opponents’ strategies. The application of these results to a fixed-point condition of the best-response correspondence to construct MPE conditions is described as entirely novel.

Q: What are the scope conditions and limitations of the methodological results?

A: The main results (Theorems 1–3) apply to differential games with a single state variable and strategies that are real-analytic except at finitely many points with one-sided derivatives everywhere. The climate application is further restricted to the symmetric linear-quadratic van der Ploeg–de Zeeuw framework. Extension to multiple state variables is acknowledged as future work. The welfare calibration results are specific to the parameter values N=10, delta=0.02, rho=0.02, d=0.5.

Markov-perfect equilibrium (MPE): A Nash equilibrium in Markovian strategies, where each player’s strategy conditions only on the current state variable and not on the history of play. The paper makes this concept well-founded in differential games by allowing discontinuous strategies, ensuring payoffs can be computed for all strategy profiles and unique best responses exist almost everywhere.

Filippov solution: A solution concept for ordinary differential equations with discontinuous right-hand sides, which replaces the dynamics at a discontinuity point with a convex hull of the left and right limits. Used in this paper to define well-specified state trajectories and payoffs even when players’ strategies have jumps, eliminating the need to restrict strategy sets to “admissible” profiles.

Discontinuous Markovian strategy: A policy function phi: X -> A that maps the state to an action and is real-analytic except at finitely many points, with well-defined one-sided derivatives everywhere. The key innovation of the paper — allowing such strategies makes differential games well-behaved as standard non-cooperative games while capturing the generically discontinuous nature of optimal policy functions.

Push-push steady state: A steady state at a discontinuity point of a strategy where the dynamics push the state toward that point from both sides. Under Filippov solutions the state remains at such a point, with flow payoffs being a weighted average of left and right actions. Theorem 2 requires the value function to be differentiable at these points in equilibrium.

Coordination failure equilibrium: An MPE featuring discontinuities in both the strategy and the value function, arising when no single player can unilaterally move the state across a threshold. At high carbon levels, if opponents emit heavily, individual emission cuts are ineffective; heavy emission becomes a best response for all, sustaining a self-fulfilling high-emission outcome. These equilibria can yield welfare outcomes worse than the linear equilibrium.

Linear equilibrium: The unique continuous globally defined symmetric MPE in the van der Ploeg–de Zeeuw model, characterized by emissions decreasing linearly in the carbon stock. It involves adverse strategic substitutability — each player reduces emissions in response to high carbon because opponents do likewise — and is weakly Pareto-dominated by every MPE with a continuous value function.

Skiba point: A state at which the optimal policy is discontinuous because the value function has distinct left and right derivatives, corresponding to the boundary between two basins of attraction with different long-run outcomes. In this paper, the steady state of a best equilibrium is a Skiba-type point: below it, emissions jump up to return rapidly to the target; above it, emissions increase only gradually.

Markups Across Space and Time

Mon, 01 Jan 0001 00:00:00 +0000

Anderson, Rebelo, and Wong study the behavior of markups in the retail sector across regions and over time, using a combination of firm-level Compustat data and product-level scanner data from two large retailers — one operating over 100 stores across U.S. states (quarterly data from 2006 Q1 to 2009 Q3, covering roughly 3.6 million SKU-store pairs across 79 product categories) and one operating hundreds of stores across Canadian provinces (quarterly data from 2016 Q1 to 2018 Q4, covering 15.6 million item-store pairs across 41 product groups). Markups are measured using gross margins — sales minus cost of goods sold as a fraction of sales — computed at the product level using the replacement cost for every item. This measurement approach is appropriate for retail because cost of goods sold accounts for over 80 percent of total retail firm costs, making it a reliable proxy for marginal cost. The replacement cost data, available at the store level, is the cost used by managers in actual pricing decisions, distinguishing these datasets from typical scanner data that contain only average costs.

The paper documents five main facts. First, markups are remarkably stable over time and display a mild procyclical pattern. At the aggregate level, gross margins are roughly acyclical or mildly procyclical while sales and cost of goods sold are highly procyclical. The elasticity of gross margins with respect to real GDP is statistically insignificant at both the aggregate and firm level. The conditional response of gross margins to high-frequency monetary policy shocks and oil price shocks is also statistically insignificant, while net operating profit margins fall significantly in response to both shocks. Operating profit margins are 3.4 times more volatile than gross margins at a quarterly frequency, and sales and costs are roughly 2.6 times more volatile.

Second, there is large regional dispersion in gross margins. A variance decomposition shows that the regional variance of gross margins (0.103) is substantially larger than the time-series variance (0.013), with a near-zero covariance between the two components. Third, regions with higher incomes and more expensive houses have higher markups — gross margins are positively correlated with log household income and log median house value in both the U.S. and Canadian data.

Fourth, these higher regional markups do not result from less intense competition or regional differences in marginal costs. Gross margins are uncorrelated with the Herfindahl index (a measure of competition) and with a rural dummy (a proxy for higher transportation costs). The cyclicality of markups is acyclical or mildly procyclical regardless of whether the underlying product costs are themselves acyclical, procyclical, or countercyclical.

Fifth, and most distinctively, regional variation in markups arises from differences in assortment composition across regions rather than from deviations from uniform pricing. A decomposition of regional gross margin variance confirms that the dominant component is the term capturing differences in product assortment across markets; the term capturing differences in gross margins for the same item — which would be nonzero under geographic price discrimination — accounts for very little of the regional variation. When the same item is available in different regions, the retailer charges a uniform price, consistent with Della Vigna and Gentzkow (2019).

To rationalize these five facts, the authors propose a model with non-homothetic, quadratic preferences (following Melitz and Ottaviano 2008). In the model, higher-productivity regions choose higher-quality goods, which have less elastic demand and therefore higher markups. The markup is procyclical with respect to productivity shocks (A) but acyclical with respect to labor supply shocks (N), so a mixture of both types of shocks produces mildly procyclical markups. The model generates uniform pricing across regions for the homogeneous good, with regional markup differences arising through quality and assortment selection rather than price discrimination.

Q: How do the authors measure markups, and why is this approach appropriate for retail? A: Markups are measured as gross margins — (sales minus cost of goods sold) divided by sales — computed at the product level using the replacement cost for every item. This is appropriate for retail because cost of goods sold is the predominant variable cost, accounting for over 80 percent of total retail firm costs. The replacement cost is the marginal cost concept used by managers in pricing decisions and is available at the store level rather than as a national average.

Q: What is the cyclical behavior of gross margins at the aggregate retail level? A: Gross margins are roughly acyclical or mildly procyclical. Sales and cost of goods sold are highly procyclical, suggesting that the business cycle primarily affects quantities sold rather than markups. Operating profit margins are 3.4 times more volatile than gross margins at a quarterly frequency, while sales and costs are roughly 2.6 times more volatile.

Q: What is the conditional response of gross margins to monetary policy and oil price shocks? A: The response of gross margins to both high-frequency monetary policy shocks (identified from Federal Funds futures data) and oil price shocks (identified via the Ramey-Vine 2010 VAR approach) is statistically insignificant. In contrast, net operating profit margins fall in a statistically significant manner in response to both types of shocks, indicating that fixed cost absorption rather than markup adjustment drives profit volatility.

Q: How large is the regional dispersion in gross margins relative to their time-series variation? A: The variance decomposition shows that the regional variance of gross margins is 0.103, compared to a time-series variance of only 0.013, with a covariance term close to zero. The vast majority of gross margin variation is therefore cross-sectional rather than time-series.

Q: What variables explain the regional variation in gross margins? A: In the U.S. data, gross margins are positively correlated with log household income and log median house value. Gross margins are uncorrelated with the Herfindahl index (a competition measure) and with the rural county dummy (a transportation cost proxy). Canadian data confirms the positive correlation between gross margins and both log household income and log median house value.

Q: What is the mechanism through which higher-income regions have higher markups? A: Regional markup differences are driven by assortment composition differences, not price discrimination. When the same item is sold in multiple regions, it sells at a uniform price. Higher-income regions carry different (higher-quality, higher-margin) products. The correlation between unique items sold and regional household income is 0.42 for the Canadian retailer and 0.17 for the U.S. retailer.

Q: How is the variance of regional gross margins decomposed into assortment versus pricing components? A: The variance decomposition separates total regional gross margin variance into: (1) a term for differences in gross margins for the same item across regions (would be nonzero with geographic price discrimination), (2) a term for differences in assortment composition holding gross margins fixed, and (3) an interaction term plus covariance terms. The dominant term is the assortment composition component; the same-item price difference term accounts for very little of the regional variation.

Q: Does the acyclicality of gross margins hold for products with procyclical costs? A: Yes. The authors divide products into those with acyclical, procyclical, and countercyclical costs and show (Table 7) that gross margins are acyclical or mildly procyclical for all three groups in both the U.S. and Canadian data. This implies that retailer pricing behavior contributes to price inertia even for products whose wholesale costs move with the cycle.

Q: What fraction of gross margin changes are active versus passive? A: In the U.S. data, 91 percent of margin changes are active (resulting from price changes, regardless of whether replacement cost has changed); 9 percent are passive (replacement cost changes with no price change). In the Canadian data, 93 percent of changes are active. Both the probability of active margin changes and the size of margin changes are acyclical with respect to unemployment and local house prices.

Q: How does the Hall approach compare to gross-margin-based markup estimates? A: When the Hall approach is implemented using output elasticities (deflating sales by a product-level price deflator to obtain quantity), the resulting markup estimates are very close to those from gross margins — the ratio is 1.014 for the U.S. firm and 0.991 for the Canadian firm. However, when revenue elasticities are used instead of output elasticities (the common practice in the literature due to data limitations), the implied markup is 14 percent lower for the U.S. firm and 13 percent lower for the Canadian firm, confirming the bias documented by Bond et al. (2020).

Q: What are the key features of the theoretical model and what facts does it explain? A: The model uses non-homothetic quadratic preferences (Melitz-Ottaviano form) in which demand elasticity falls as consumption quality rises. Higher-productivity regions optimally consume higher-quality varieties, which face less elastic demand and hence carry higher markups. The markup is procyclical in productivity (A) with an elasticity less than one (incomplete cost passthrough) and acyclical in labor supply (N), so a mixture of shocks generates mild procyclicality. Uniform pricing across regions for the homogeneous good holds by construction, and regional markup differences arise through quality-assortment selection.

Q: Which existing macroeconomic models are consistent with the time-series evidence, and which are not? A: The evidence is inconsistent with models featuring countercyclical markups (Rotemberg-Woodford 1992 imperfect competition, Ravn-Schmitt-Grohe-Uribe deep habits, Jaimovich-Floetotto entry-exit, and standard New Keynesian models with sticky prices and procyclical marginal costs). The time-series evidence is consistent with models featuring sticky retail prices and acyclical marginal costs (Nakamura-Steinsson 2010, Coibion-Gorodnichenko-Hong 2015) and models with price and wage rigidities at the manufacturing level (Erceg-Henderson-Levin 2000, Christiano-Eichenbaum-Evans 2005). Mildly procyclical search models (Alessandria 2009) are also consistent when procyclicality is mild.

Q: Which existing trade and regional models are consistent or inconsistent with the regional evidence? A: The spatial price discrimination models of Greenhut-Greenhut (1975) and Thisse-Vives (1988), which predict higher markups in less competitive regions, are inconsistent with the data. The Bertoletti-Etro (2017) non-homothetic model predicts that regional markup variation is driven by deviations from uniform pricing, which is also inconsistent. The Fajgelbaum-Grossman-Helpman (2011) model predicts countercyclical markups when costs are procyclical, contradicting the time-series results. Most existing macroeconomic models rely on homothetic preferences, predicting markups independent of regional income, inconsistent with the regional facts.

Q: What are the scope conditions on the measurement approach? A: Gross margins are valid proxies for markups only in the retail sector, where cost of goods sold is the dominant variable cost (over 80 percent of total costs). In manufacturing, where labor and other costs represent a larger fraction of total variable costs, gross margins would not be a reliable markup measure. The product-level scanner data cover the 2006-2009 period for the U.S. and 2016-2018 for Canada; the U.S. sample includes a recession while the Canadian sample covers a moderate expansion.

Gross margin as markup proxy: The ratio of (sales minus cost of goods sold) to sales, computed at the product level using the replacement cost for each item at each store and time period. Used as a proxy for the price-cost markup because cost of goods sold is the dominant variable cost in retail (over 80 percent of total costs), and the replacement cost is the marginal cost concept managers use in pricing decisions.

Replacement cost: The cost at which the retailer would replenish a unit of inventory at current prices, available at the store level in the scanner datasets. Distinct from average historical cost and used here as a direct proxy for marginal cost, eliminating one of the main sources of markup mismeasurement in prior empirical work.

Assortment composition: The set of products stocked and the expenditure weights of those products within a region. The paper’s central mechanism for regional markup variation — higher-income regions carry different (higher-quality, higher-margin) goods rather than charging different prices for the same goods.

Uniform pricing: The practice of charging identical prices for the same item across different geographic regions. Confirmed empirically in both the U.S. and Canadian scanner datasets, and embedded structurally in the theoretical model for the homogeneous good.

Active versus passive margin changes: A decomposition of gross margin changes into active changes (arising from retailer price decisions, irrespective of cost changes) and passive changes (arising when replacement cost changes but the retailer holds price fixed). Ninety-one percent of U.S. margin changes and 93 percent of Canadian changes are active.

Non-homothetic quadratic preferences: The utility specification (following Melitz and Ottaviano 2008) in which the absolute value of the own-price demand elasticity falls as quality consumption rises. This property implies that higher-quality goods carry higher markups and that richer regions, which demand higher quality, have higher average markups — the key mechanism linking income to markups in the model.

Hall approach to markup estimation: A production-function-based method in which the markup equals the output elasticity with respect to a variable input divided by that input’s cost share in revenue. The paper shows this yields estimates close to gross-margin estimates when implemented with true output quantities, but produces markups roughly 13-14 percent lower when revenue is substituted for output (a common approximation), confirming the Bond et al. 2020 bias.

Marriage, Fertility, and Cultural Integration in Italy

Mon, 01 Jan 0001 00:00:00 +0000

Bisin and Tura study the cultural integration of immigrants in Italy by estimating a structural model of marital matching embedded with intra-household decisions — fertility, socialization of children, and divorce — along cultural-ethnic lines. The central research question is how to decompose the demand for integration (from immigrants) and the supply of cultural acceptance (from natives) in explaining the pace and heterogeneity of cultural convergence.

The empirical analysis exploits administrative individual-level data from ISTAT’s ADELE Laboratory covering the universe of marriages formed in Italy from 1995 to 2012 and the universe of births and separations over the same period. After matching marriage, birth, and separation records, the final sample comprises more than 4 million marriages, representing 92.6% of all marriages celebrated in Italy over the period. Seven cultural-ethnic groups are studied: Italian (majority), Europe-EU15, Other Europe, North Africa–Middle East, Sub-Saharan Africa, East Asia, and Latin America. The model is a transferable-utility (TU) frictionless marriage market in which the joint marital surplus depends on a systematic component — itself the outcome of a collective household decision problem — and an idiosyncratic component capturing unobserved individual heterogeneity (following Choo and Siow, 2006). Parameters are estimated via method of moments, with identification drawing on cross-sectional variation across ethnic-group pairings and across Italy’s 20 administrative regions. Cultural socialization is proxied by language transmission (whether Italian is spoken at home with children).

The data confirm strong positive assortative mating along cultural-ethnic lines, with particularly high homogamy rates for Sub-Saharan African and East Asian minorities. Homogamous minority households show notably lower rates of Italian-language use at home — for East Asian parents, 20% in a homogamous marriage versus 92% in a heterogamous marriage. Heterogamous marriages have higher separation rates (7.5% for mixed families with at least one Italian spouse versus 6.4% for homogamous Italian couples) and lower fertility.

The estimated cultural intolerance parameters — measuring the psychological value a parent places on socializing a child to his/her own ethnic identity relative to a child acquiring a different identity — are strictly positive, asymmetric across directions, and highly heterogeneous across groups. North Africa–Middle East immigrants exhibit the highest minority intolerance (estimated at 97.85), more than six times that of Europe-EU15 immigrants (6.69). Latin America (93.13), Sub-Saharan Africa (87.08), and East Asia (81.22) also show high intolerance. On the native side, Italian intolerance is highest toward Sub-Saharan African immigrants (78.23) and lowest toward Europe-EU15 immigrants.

Long-run simulations over successive generations show that all minorities eventually converge to the Italian majority along the language dimension, but at heterogeneous rates. Seventy-five percent of second-generation immigrants speak Italian at home with their children (one-generation integration rate). Europe-EU15 and Other Europe minorities converge almost completely within a single generation. Latin America shows the slowest path, with only 70% integration after four generations. East Asia and Sub-Saharan Africa also integrate more slowly, driven respectively by high fertility rates and strong selection into homogamous marriages.

A counterintuitive counterfactual result is central to the paper: if Italian cultural intolerance were reduced to zero (full acceptance), cultural integration of minorities would slow by 15 percentage points over a generation (from 93% to 78% by the third generation). The mechanism is that greater native acceptance enables immigrants to sustain their own language even within heterogamous (mixed) marriages, increasing demand for such marriages and raising minority fertility, thereby preserving cultural distinctiveness.

Finally, doubling immigration inflows while holding population shares constant reduces third-generation integration from 93% to 86% (a 7-percentage-point reduction). Effects are concentrated among Sub-Saharan African (20-percentage-point reduction) and East Asian (6-percentage-point reduction) minorities, with little impact on European and North African minorities. When inflows are reweighted toward Sub-Saharan African and East Asian groups, integration losses for those minorities range from 20 to 60 percentage points by the third generation.

Q: What is the paper’s core methodological contribution? A: The paper embeds a collective household decision problem — covering fertility, socialization, and divorce — within a transferable-utility frictionless marriage matching framework. This allows marital utility to emerge endogenously from intra-household decisions rather than being specified exogenously. The key innovation is that socialization incentives and technologies differ systematically between homogamous and heterogamous marriages, and these differences feed back into marital matching and long-run cultural dynamics.

Q: What does “cultural intolerance” mean in this model, and how is it identified? A: Cultural intolerance is the psychological value a parent obtains from socializing a child to his/her own ethnic identity, relative to having a child adopt a different cultural-ethnic identity. It is the main parameter driving socialization effort and resistance to cultural integration. Identification relies on two sources of cross-sectional variation: differences in matching patterns, fertility, separation, and socialization rates across cultural-ethnic group pairings, and exogenous variation in the ethnic composition of the regional population across Italy’s 20 administrative regions.

Q: How heterogeneous are the estimated cultural intolerance parameters across minority groups? A: The parameters are highly heterogeneous. North Africa–Middle East immigrants have the highest estimated minority intolerance (97.85), more than six times the EU15 estimate (6.69). Latin America (93.13), Sub-Saharan Africa (87.08), and East Asia (81.22) are also substantially higher than EU15. The matrix is asymmetric: Italian intolerance toward Sub-Saharan Africans (78.23) is higher than toward North Africans (67.88), even though those two groups show comparable minority intolerance levels.

Q: What are the three mechanisms beyond intolerance parameters that explain heterogeneous integration dynamics? A: First, selection into homogamous marriages: Sub-Saharan Africa’s particularly strong selection into homogamy gives those households access to superior coordinated socialization technology, sustaining cultural heterogeneity despite similar intolerance levels to other groups. Second, fertility rates: East Asian minorities have particularly high estimated fertility, which amplifies the transmission of their cultural identity across generations. Third, socialization effectiveness in heterogamous marriages: Latin American immigrants are uniquely able to socialize children to their own language even when married to native Italians, making their integration the slowest despite being in many mixed marriages.

Q: What is the counterintuitive result about Italian cultural intolerance and integration speed? A: Lowering Italian cultural intolerance to zero would reduce minority integration by 15 percentage points over one generation, with third-generation integration falling from 93% to 78%. The intuition is that higher native acceptance enables immigrants to maintain their own language more effectively within heterogamous marriages, which in turn increases immigrant demand for intermarriage with natives and raises minority fertility — both of which slow cultural convergence rather than accelerating it.

Q: How do divorce dynamics differ between homogamous and heterogamous households? A: Heterogamous households exhibit higher separation rates than culturally homogeneous unions: 7.5% for mixed families with at least one Italian spouse versus 6.4% for homogamous Italian couples. In the model, divorce by heterogamous households can be a strategic choice by mothers with high cultural intolerance, since custody grants single mothers greater unilateral control over socialization. Divorce probabilities are decreasing in the number of children for both family types. Interestingly, heterogamous households invest more in socialization when divorced than when married, because the high-intolerance parent can act without spousal opposition.

Q: How well does the model fit the data? A: The raw correlation between predicted and observed gains to marriage is 0.84. The correlation between predicted and observed foreign-language socialization rates is 0.83, for both homogamous and heterogamous families. The dataset covers 92.5% of all marriages in Italy from 1995 to 2012, representing over 4 million marriages matched with birth and separation records at a 98.5% one-to-one match rate.

Q: What happens to cultural integration when immigration inflows are doubled with an overweighting of North Africa–Middle East, Sub-Saharan Africa, and East Asian immigrants? A: North Africa–Middle East immigrants reduce third-generation convergence by only 4 percentage points. By contrast, East Asian and Sub-Saharan African minorities produce integration losses ranging from 20 to 60 percentage points by the third generation. This wide range reflects how the interaction between high fertility, strong homogamy selection, and effective socialization in heterogamous marriages amplifies cultural persistence when these groups constitute a larger share of inflows.

Q: What is the one-generation cultural integration rate, and which groups diverge most from it? A: Seventy-five percent of second-generation immigrants speak Italian at home with their children, constituting the one-generation baseline integration rate. Europe-EU15 and Other Europe minorities converge almost completely within one generation, as does North Africa–Middle East. Latin America diverges most sharply downward, with only 70% integration even after four generations, and shows a partial retreat from integration in the first generation. Sub-Saharan Africa and East Asia also fall below the 75% one-generation benchmark.

Q: How does the paper relate to the debate on native labor market effects of immigration? A: The paper notes that sizeable negative labor market effects of immigration on natives are far from well-documented in the empirical literature, with results ranging from negative wage effects (Borjas) to positive or heterogeneous effects (Card, Ottaviano-Peri, Dustmann et al.). The authors therefore focus on the cultural externalities channel, which they argue better explains voter opposition to immigration, and study cultural integration structurally rather than examining wage outcomes.

Cultural intolerance: The psychological value a parent obtains from socializing a child to his/her own ethnic identity, relative to having a child adopt a different cultural-ethnic identity. It is specific to the household type (homogamous vs. heterogamous) and is the primary parameter measuring the strength of a group’s resistance to cultural integration.

Cultural socialization / language transmission: The costly investments parents make to transmit their own cultural-ethnic traits to children. In the empirical model, socialization is proxied by whether a parent speaks his/her own non-Italian language at home with children. Socialization technologies are more efficient in homogamous (same-ethnicity) marriages than heterogamous ones.

Homogamous vs. heterogamous marriage: A homogamous marriage is one in which both spouses share the same cultural-ethnic identity; a heterogamous marriage is one in which spouses differ. The distinction is load-bearing throughout the model: homogamous households have coordinated socialization incentives and superior technology, higher fertility, and lower separation rates.

Transferable utility (TU) matching: A marriage market framework in which utility is transferable between spouses, so that the equilibrium allocation maximizes aggregate marital surplus and equilibrium transfers are determined by outside options. The model is frictionless, meaning matching is driven purely by preferences over the characteristics of potential spouses.

Cultural integration (language dimension): In the paper’s long-run simulations, cultural integration is defined as the share of second- (or later-) generation immigrants who speak Italian at home with their own children. It is the empirical outcome used to track convergence to the majoritarian culture across generations.

Assortative mating along cultural-ethnic lines: The tendency for individuals to match with spouses of the same cultural-ethnic group. The paper finds positive assortative mating for all groups, with particularly strong homogamy for Sub-Saharan African and East Asian minorities, and explains it as the equilibrium outcome of the TU matching model given cultural intolerance preferences.

Socialization technology asymmetry: The model’s assumption that homogamous married parents hold a more efficient socialization technology than heterogamous parents, but that divorced heterogamous households invest more in socialization than married heterogamous ones, because the high-intolerance parent can act unilaterally without spousal opposition.

Merger Effects and Antitrust Enforcement: Evidence from US Consumer Packaged Goods

Mon, 01 Jan 0001 00:00:00 +0000

This paper by Bhattacharya, Illanes, and Stillerman makes two contributions to the debate over US antitrust enforcement stringency. First, it documents the price, quantity, and assortment effects of a comprehensive set of consummated mergers in US consumer packaged goods (CPG). Second, it develops and estimates a model of agency enforcement decisions to quantify antitrust stringency and simulate counterfactual outcomes under stricter regimes.

Data and scope. The analysis covers 129 product markets across 47 transactions in US CPG from 2006 to 2017, using the NielsenIQ Retail Scanner Dataset (covering 35,000–50,000 stores and 2.6–4.5 million UPCs). The sample is restricted to all deals valued at $280 million or more where both the acquirer and target sold products in at least one overlapping product market-DMA. Geographic markets are NielsenIQ designated market areas (DMAs). The sample is defined to avoid selection bias from studying only mergers that attracted press attention or were litigation targets.

Identification strategy. The empirical approach is a before-after event study within geography and product. For each merger, a brand-specific linear time trend is estimated from the 36 months prior to the merger announcement, controlling for UPC-DMA fixed effects, month-of-year fixed effects, input cost indices, and log median household income. Post-merger outcomes (24 months after completion) are measured as deviations from the extrapolated pre-merger trend. The identifying assumption is that secular demand and cost trends are gradual and well-captured by a linear trend. Pre-trend placebo tests show no significant departures from trend in the pre-period, and randomized-date placebos confirm that the linear trend is a better predictor of post-period outcomes under random merger dates than under actual merger dates, supporting the interpretation that observed post-period departures reflect merger effects.

Price effects. The average price effect of consummated CPG mergers is small: across specifications, estimates range from -0.6% to 1.0%, with a baseline mean of 0.3%. However, heterogeneity is substantial. The standard deviation of merger-level price effects is 4.0–7.5 percentage points. In the baseline specification, the first quartile of price effects is -2.1% and the third quartile is 3.7%. Merging and non-merging party price changes are positively correlated (correlation = 0.49), consistent with strategic complementarity. Thirty-six percent of mergers lead both groups to lower prices; 36% lead both groups to raise prices.

Quantity and assortment effects. Total quantities fall on average by 0.4–1.0% across specifications, with 60% of mergers producing quantity reductions. Merging parties exhibit a larger average quantity decline of 6.4%. Mergers also lead to a 2.7% average reduction in the number of stores served by merging parties, a 2.2% reduction in the number of brands sold in a DMA by merging parties, and a 3.2% reduction for non-merging parties. Brands with less than 5% of the merged entity’s sales are 6 percentage points more likely to be dropped post-merger.

Enforcement model. To interpret these outcomes relative to enforcement, the authors develop a model in which the agency receives a noisy signal of a merger’s price effect and challenges the merger if the posterior mean exceeds a threshold that is decreasing in deal size. They estimate the model by maximum likelihood using data on enforcement actions (6 mergers receiving remedies, 4 withdrawn under antitrust pressure) and realized price changes. The estimated sales-weighted average threshold is 4.8–6.3%: agencies act as if they challenge CPG mergers only when they expect a price increase exceeding this level. The posterior standard deviation of the agency’s assessment is 2.5–3.2 pp (aggregate prices) to 4.1–4.8 pp (merging-party prices).

Counterfactual stringency. Tightening the threshold from approximately 6.1% to 2.5% would roughly quadruple the challenge probability (from 0.075 to 0.30), reduce aggregate price changes of consummated mergers by approximately 1.4 pp, and lower the share of allowed anti-competitive mergers from roughly 50% to 35%. Critically, type I errors (blocking pro-competitive mergers) remain negligible at thresholds down to approximately 3%; at 0% threshold only 10% of blocked mergers would be type I errors. The primary cost of tighter enforcement is a significantly larger agency workload, not an increase in blocked pro-competitive mergers.

Scope conditions. Results pertain specifically to large CPG mergers (deal size ≥ $280 million) sold through US retail outlets, 2006–2017. Findings on structural presumptions show DHHI and merging share have predictive value for price changes, but structural metrics alone explain less than 10% of the variance in price effects (adjusted R-squared never exceeds 10% even with third-order interactions).

Q: What is the average price effect of consummated CPG mergers and how should it be interpreted? A: Across specifications, the average price effect is between -0.6% and 1.0%, with a baseline mean of 0.3%. This small average does not imply that enforcement is strict: Carlton (2009) shows that with perfect foresight, the largest observed price change — not the average — would indicate stringency. Because agencies face uncertainty, the distribution of realized price changes reflects both inframarginal approved mergers and the noise in agency forecasts.

Q: How large is the heterogeneity in merger price effects? A: The standard deviation of merger-level price effects is 4.0–7.5 percentage points across specifications. In the baseline, the first quartile of price effects is -2.1% and the third quartile is 3.7% for all parties combined. Merging parties specifically show a first quartile of -3.2% and third quartile of 3.7%, meaning a full quarter of mergers raise merging-party prices by more than 3.7%.

Q: How do merging and non-merging party prices co-move? A: Price changes for merging and non-merging parties are positively correlated (correlation = 0.49, s.e. = 0.08), consistent with strategic complementarity in pricing. Thirty-six percent of mergers lead both groups to lower prices, 36% lead both to raise prices, 13% cause merging parties to lower while non-merging parties raise, and 15% cause the reverse. The timing evidence shows merging-party prices begin changing upon merger completion, with rivals following suit.

Q: What happens to quantities following mergers? A: Total quantities fall on average between 0.4% and 1.0% across specifications, with 60% of mergers producing quantity reductions. Merging parties bear the bulk of quantity adjustment, with an average quantity decline of 6.4% and a standard deviation and interquartile range both around 30 pp. Non-merging party quantity changes are much less variable. The correlation between merging and non-merging party quantity changes is 0.36 (s.e. 0.08), which is positive — at odds with theoretical predictions from demand systems with the “type aggregation property” (Nocke and Schutz, 2018, 2024), where mergers should produce negatively correlated quantity changes.

Q: What non-price competitive responses do mergers trigger? A: Merging parties reduce the number of stores they serve by 2.7% on average, though in 38% of mergers store networks expand. Both merging and non-merging parties reduce product portfolios: merging parties drop the number of brands in a DMA by 2.2% on average and non-merging parties by 3.2%. Brands most likely to be dropped are those with less than 5% of the merged entity’s sales (6 pp more likely to be dropped), brands in small DMAs, and brands with small DMA shares.

Q: Do the Merger Guidelines’ structural presumptions (HHI, DHHI, merging share) predict price effects? A: DHHI and merging share have statistically significant but quantitatively modest predictive power. A 100-point increase in average DHHI is associated with a 0.2 pp increase in merging-party price changes and 0.3 pp for non-merging parties. Price effects are significantly larger when merging share exceeds 30%. However, structural metrics alone explain very little variance: adjusted R-squared never exceeds 10% even with third-order interactions of HHI, DHHI, merging share, private label share, and market size. Within-merger, DHHI is positively correlated with local price changes, and markets with DHHI above 200 exhibit significantly higher price effects than those below.

Q: How do the authors model antitrust enforcement and identify its stringency? A: The agency observes a noisy signal of a merger’s price effect, forms a posterior distribution combining a normally distributed prior (mean X’beta, standard deviation sigma_p*) with a normally distributed signal error (standard deviation sigma_epsilon), and challenges the merger if the posterior mean exceeds a threshold that is decreasing in deal size. The model is estimated by maximum likelihood: for approved mergers, the realized price change is observed; for withdrawn/remedied mergers, the posterior mean must have exceeded the threshold. Six mergers (from four deals) received remedies for horizontal market power concerns and four mergers (from two deals) were withdrawn under antitrust pressure, forming the challenged set.

Q: What is the estimated enforcement threshold and how does it vary across mergers? A: The sales-weighted average threshold is 4.8–6.3% using aggregate price changes and 6.6–7.8% using merging-party price changes. The threshold is lower for larger mergers: a 10% increase in merging-party sales is associated with an approximately 0.06 pp decrease in the threshold. The first quartile of thresholds across mergers is 4.5–5.6% and the third quartile is 5.6–6.9%, reflecting that the agencies apply stricter standards to larger deals.

Q: How accurate are the agencies’ forecasts of merger price effects? A: Using only the prior (structural characteristics), the agency’s accuracy in classifying mergers as anti-competitive versus pro-competitive is 56% (s.e. 3 pp). Adding the signal increases accuracy to 83% (s.e. 9 pp). The correlation between the prior mean and the true price change is 0.29 (s.e. 0.08); the correlation between the posterior mean and the true price change is 0.85 (s.e. 0.15). The posterior standard deviation is 2.5–3.2 pp for aggregate price changes and 4.1–4.8 pp for merging-party price changes.

Q: What would happen under stricter antitrust enforcement? A: Tightening the average threshold from 6.1% to 2.5% would raise the challenge probability from approximately 0.075 to 0.30 — roughly quadrupling it — and would reduce aggregate price changes of consummated mergers by approximately 1.4 pp (from roughly 0.2% to -1.2%). Moving to a 0% threshold would result in challenges to 57% of mergers, with 60–70% of consummated mergers then causing price decreases.

Q: How large are type I and type II errors at the current and counterfactual thresholds? A: At the current threshold (~6.1%), approximately 50% of allowed mergers are type II errors (anti-competitive mergers that should have been challenged). Type I errors (pro-competitive mergers wrongly blocked) are negligible at the current threshold and only become non-trivial starting around a 3% threshold. At a 2.5% threshold, the type II error share falls to 35%; at a 0% threshold, to 16%, while type I errors reach 10% of blocked mergers. The primary trade-off of stricter enforcement is therefore a larger agency workload, not an increase in blocking pro-competitive mergers.

Q: What identification strategy is used and how is it validated? A: The strategy is a within-product, within-geography before-after comparison using a brand-specific linear pre-merger trend as the counterfactual. Validation proceeds through three checks: (1) coefficient plots from an extended event study show no significant pre-trends after controlling for the linear trend; (2) a plot of brand trends against estimated price effects shows little explanatory power (statistically significant negative correlation but small magnitude, not consistent with results being driven by trend extrapolation); (3) placebo tests randomizing merger dates within the same markets yield a distribution centered at zero, narrower than the true distribution, and a significantly higher mean squared prediction error in the post-period, confirming that the linear trend is a better predictor under randomly assigned merger dates than under true dates.

Q: Why do the authors not use alternative control group approaches? A: Non-merging firms in the same market are rejected as controls because they may strategically respond to the merger. Synthetic controls using similar-industry untreated markets are rejected because deals often treat multiple similar markets (ruling out natural donors) and estimates prove sensitive to individual donors. Geographic controls (markets where merging parties have small shares) are rejected because they omit all 39 national mergers, untreated markets are not randomly selected, and regional pricing by non-merging parties could propagate effects into untreated regions, biasing estimates toward zero.

Merger retrospective. In this paper’s usage, an ex-post empirical study of the price, quantity, and assortment effects of a consummated merger, using pre-merger trends as the counterfactual, as opposed to forward-looking merger simulation.

Enforcement stringency. The marginal price increase at which the antitrust agency would expect to challenge a merger. Measured here as the sales-weighted average posterior-mean threshold: the value above which the agency acts as if it would propose a remedy, estimated at 4.8–6.3% for US CPG mergers.

Type I error (antitrust). The mistake of challenging (blocking) a merger that would have reduced prices (a pro-competitive merger). In the model, this occurs when an adverse signal causes the agency to block a merger whose true price effect is below the threshold.

Type II error (antitrust). The mistake of allowing a merger that increases prices (an anti-competitive merger). In the model, this occurs when a favorable signal causes the agency to approve a merger whose true price effect is above the threshold. Estimated at approximately 50% of allowed mergers at the current enforcement threshold.

Structural presumptions. The HHI-based rules in the 2010 and 2023 Merger Guidelines that create a presumption of competitive harm when DHHI exceeds specified thresholds (e.g., DHHI > 200 and post-merger HHI > 2,500 for the “red zone”). The paper finds DHHI and merging share have statistically significant but low explanatory power (adjusted R-squared below 10%) for actual price changes.

Prior and signal (in the enforcement model). The agency’s prior is a normal distribution over the merger’s true price effect, parameterized by structural characteristics (HHI, DHHI). The signal is a noisy draw centered on the true price effect, capturing information gathered through due diligence (e.g., evidence of efficiencies). The posterior mean — combining prior and signal — determines whether the agency challenges the merger.

Product market-deal pair (merger). The unit of observation in the empirical analysis: a specific NielsenIQ product module (e.g., soluble coffee) within a specific acquisition transaction (e.g., a food conglomerate merger). The sample contains 129 such pairs across 47 deals.

Micro MPCs and Macro Counterfactuals: The Case of the 2008 Rebates

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research question. Do the high marginal propensities to consume (MPCs) estimated in the leading household studies of the 2008 U.S. tax rebates—particularly Parker et al. (2013), which found MPCs of 50–90 percent within three months—imply plausible macroeconomic counterfactuals? And if not, what combination of micro-level bias corrections and general equilibrium forces reconciles the micro evidence with aggregate data?

Setting. The 2008 Economic Stimulus Act distributed approximately $100 billion in tax rebates, totaling eleven percent of January 2008 monthly disposable income. Among the 85 percent of households receiving a check, the average amount was $1,000. Rebates were distributed primarily from April through July 2008, with nearly half delivered in May alone. The timing of receipt was determined by the last two digits of Social Security numbers, providing quasi-random variation exploited by the household-level literature.

Methodology. The paper proceeds in two halves. In the first, the authors construct macro counterfactuals by calibrating a standard medium-scale two-good, two-agent New Keynesian (TANK) model with the micro MPCs from the literature and simulating what aggregate consumption would have been absent the rebate. The model contains life-cycle permanent income households and hand-to-mouth households whose dynamic spending propensities are calibrated directly to match the household-level estimates. General equilibrium effects—including Keynesian income multipliers, real interest rate movements, and changes in the relative price of durable goods—are incorporated. Counterfactual consumption paths are constructed by subtracting model-simulated deviations from steady state from actual NIPA consumption data.

In the second half, the authors revisit both the micro estimates and the macro model. On the micro side, they identify three upward biases in standard two-way fixed effects (TWFE) estimates applied to CEX data: (1) omitted variable bias from excluding the lagged rebate indicator; (2) “forbidden comparisons” bias arising from comparing cohorts with heterogeneous treatment effects, following Borusyak et al. (2022) and Sun and Abraham (2020); and (3) a rebate reporting bias in which households are systematically more likely to report receiving the rebate in the month that coincides with large expenditure increases, causing spurious positive correlation between reported receipt and contemporaneous spending. On the macro side, the baseline model is modified to incorporate an upward-sloping supply curve for durable goods (calibrated to a supply elasticity of 5, midway between House and Shapiro (2008) and Goolsbee (1998)), replacing the baseline assumption of frictionless conversion between nondurable and durable intermediates.

Main findings with quantitative magnitudes.

Implausibility of baseline counterfactuals. When calibrated to Parker et al.’s (2013) micro MPC of 0.9, the baseline model implies that real PCE absent the rebate would have collapsed by 6.0 percent from April through July 2008—a decline exceeded historically only by the Covid-19 lockdowns. Even the more modest micro MPC of 0.5 implies a 2.7 percent three-month PCE decline, comparable only to the 1980 Volcker disinflation with credit controls. For motor vehicle expenditures, the counterfactual drops range from 38 percent (micro MPC = 0.3) to 67 percent (micro MPC = 0.9)—larger than any historical experience, including the 30 percent Covid decline. Contemporaneous professional forecasters (Federal Reserve Greenbooks, Survey of Professional Forecasters, Goldman Sachs) predicted at most small consumption declines in summer 2008. Even the authors’ own pessimistic forecast model—incorporating actual oil price paths and a Lehman Brothers bankruptcy dummy—implies that the cumulative difference between actual and forecast consumption attributable to the rebate was at most $20 billion out of $100 billion in rebates, for an implied GE-MPC of at most 0.2.

Bias correction in micro MPC estimates. Applying all three bias corrections to CEX data (the preferred specification with lagged rebate indicator, cohort-level treatment effects, and lagged expenditure controls), the estimated three-month MPC falls from 0.50 to 0.28 in the full sample and from 0.82 to 0.34 in the rebate-recipients-only sample, with both rounding to approximately 0.3. The Borusyak-Jaravel-Spiess (BJS) imputation method yields an MPC of 0.20 in the full sample and 0.37 in the rebate-only sample, consistent with the OLS corrections.

Composition of spending. In the preferred corrected specification, essentially all of the total expenditure MPC of 0.3 is accounted for by motor vehicle spending: the MPC on motor vehicles is 0.30 in the full sample and 0.26 in the rebate-only sample, while the MPC on all other expenditures is −0.02 (full sample) and 0.08 (rebate-only sample).

General equilibrium dampening via inelastic durable supply. In the model with a calibrated durable supply elasticity of 5, rebate-induced demand for motor vehicles raises the relative vehicle price by approximately 1.1 percent in July 2008. This price increase crowds out durable expenditure by optimizing households through intertemporal substitution. At the preferred micro MPC of 0.3, the general equilibrium MPC (GE-MPC) for total PCE is only 0.07, well below the 0.3 micro estimate. At a micro MPC of 0.5, the GE-MPC is 0.22. The combination of the bias-corrected micro MPC and dampening general equilibrium forces implies a general equilibrium consumption multiplier below 0.2 for the 2008 rebates.

Importance of durable goods composition for HANK models. A model that abstracts from durable goods and calibrates the full expenditure micro MPC to nondurable spending predicts a GE-MPC of 0.36 when the micro MPC is 0.30—five times larger than the 0.07 implied by the model with durable goods. This contrast illustrates that the distribution of spending across nondurable and durable goods is a key determinant of the aggregate fiscal multiplier, in addition to heterogeneity in wealth and income emphasized by the existing HANK literature.

In depth

Q1. What is the central empirical puzzle the paper addresses?

A. The leading household studies of the 2008 rebates estimate very high three-month MPCs (50–90 percent). When these estimates are plugged into a standard New Keynesian model to construct counterfactual consumption paths absent the rebate, the model implies that PCE would have collapsed by 2.7–6.0 percent from April through July 2008 and then sharply recovered just as Lehman Brothers failed in September. No contemporaneous forecaster or narrative evidence suggests such extreme, short-lived macroeconomic stress was present. The Lehman collapse itself caused only a 1.1 percent three-month PCE decline—smaller than all three counterfactual declines implied by micro MPCs of 0.3, 0.5, or 0.9.

Q2. What are the features of the TANK model used to construct the counterfactuals?

A. The model is a two-good (nondurable and durable), two-agent (optimizing life-cycle and hand-to-mouth) New Keynesian model calibrated at monthly frequency, building on Ramey (2021) and Galí et al. (2007). Intermediate goods can, in the baseline, be frictionlessly converted into either nondurable or durable goods (implying a fixed relative price of one). Durable goods (interpreted as motor vehicles) enter household utility, with optimizing households facing a Calvo-type adjustment friction motivated by Evans and Ramey (1992) calculation costs. The fraction of hand-to-mouth consumers and their dynamic propensities to spend are calibrated directly to match the micro MPC estimates from the household literature. The model incorporates a Calvo-style price-adjustment structure for nondurables, sticky wages set by unions, capital with adjustment costs and variable utilization, and an inertial monetary policy rule.

Q3. How does the model translate micro MPCs into macro counterfactuals, and why does it amplify rather than dampen the micro estimates in the baseline?

A. The model’s GE-MPC equals the micro MPC’s direct demand effect plus Keynesian income multiplier effects. Because the rebate is highly transitory, there is little movement in the real interest rate (the Phillips curve is flat and monetary policy is inertial), so the dominant general equilibrium force is the income multiplier. This amplifies, rather than dampens, the micro MPCs. As a result, the GE counterfactuals exhibit even sharper V-shapes than the pure micro counterfactuals.

Q4. What narrative and forecast evidence do the authors use to argue the baseline counterfactuals are implausible?

A. Contemporary forecasts from the Federal Reserve Greenbooks, the Survey of Professional Forecasters, and Goldman Sachs all predicted at most small consumption declines in summer 2008—Goldman Sachs forecast only −0.125 percent (not annualized) per quarter in Q2–Q3 2008. The authors also construct their own “pessimistic” time-series forecast that incorporates actual oil price paths (which rose from $98 to $140 per barrel by July 2008) and a Lehman Brothers bankruptcy dummy; even this forecast lies above all three model counterfactuals in summer 2008 and displays no V-shape. Furthermore, the cumulative difference between actual PCE and the pessimistic forecast over April–October 2008 totals only $20 billion—implying a GE-MPC of at most 0.2 even if the entire gap were attributed to the rebate.

Q5. What is the first bias in standard TWFE estimates of the MPC, and how large is its effect?

A. The first bias is omitted variable bias from excluding the lagged rebate indicator. In a first-differenced panel regression, lagged treatment enters the error term. Because current treatment reduces the probability of past treatment, current and lagged treatment are negatively correlated, and omitting the lag inflates the OLS estimate of the contemporaneous effect. Including a lagged rebate indicator reduces the contemporaneous spending response by $40 in the full CEX sample (from $470 to $434) and by approximately $237 in the rebate-only sample (from $764 to $527).

Q6. What is the “forbidden comparisons” bias and how is it corrected?

A. When treatment effects are heterogeneous across cohorts (e.g., the June rebate cohort has a larger MPC than the September cohort), standard homogeneous TWFE estimates use later-treated cohorts as control groups for earlier-treated cohorts even after accounting for average mean-reversion. Because the mean-reversion of the earlier (larger-effect) cohort is larger than that of the later cohort, this comparison is contaminated, inflating the estimate. The authors correct for this by allowing cohort-specific treatment effects, following Sun and Abraham (2020). This reduces the contemporaneous effect by a further $90 in the full sample; in the rebate-only sample the correction raises the estimate slightly (by $70) because later treatment effects are larger in that sample.

Q7. What is the rebate reporting bias and what mechanism underlies it?

A. The rebate reporting bias arises because households in the CEX are systematically more likely to report receiving the rebate in the interview month that coincides with high expenditure. Although the true timing of rebate checks is determined by Social Security number last-digits (and is thus random), the reported timing may reflect recall issues: households more readily remember and report receiving the rebate when it was accompanied by a large purchase. The empirical signature is a statistically significant negative effect of future rebate receipt on current expenditure (−$863 in the full sample, −$575 in the rebate-only sample at the 10% level), indicating that rebate reporters had unusually low spending in the period prior to reporting receipt. Controlling for lagged expenditure and income decile fixed effects corrects for this bias, reducing the three-month MPC in the full sample from 0.37 to 0.28.

Q8. What are the authors’ preferred bias-corrected MPC estimates, and how do they compare across specifications and estimators?

A. After correcting for all three biases (preferred specification, column 4 of Table 3), the implied three-month MPC is 0.28 in the full sample and 0.34 in the rebate-only sample, both approximately 0.3. The Borusyak-Jaravel-Spiess imputation method, which imposes weaker assumptions and overcomes the first two biases by construction, yields an MPC of 0.20 (full sample) and 0.37 (rebate-only sample), with an average consistent with the OLS-corrected estimates. Both methods point to an MPC around 0.3, substantially below the 0.5–0.9 range from the baseline Parker et al. (2013) approach.

Q9. How is almost all of the total expenditure MPC concentrated in motor vehicles?

A. After bias correction, the MPC on motor vehicles is 0.30 in the full sample and 0.26 in the rebate-only sample. The MPC on all other PCE is −0.02 (full sample) and 0.08 (rebate-only sample), neither statistically significant. This concentration in durables is consistent with Adams et al. (2009) and Aaronson et al. (2012), and is corroborated by CEX vehicle-expenditure data showing a car-purchase response concentrated in the three months surrounding receipt of the rebate.

Q10. How does introducing an upward-sloping supply curve for durable goods change the model’s general equilibrium predictions?

A. In the modified model, durable goods producers face a production externality (or fixed factor) that makes the short-run supply of motor vehicles upward-sloping, with supply elasticity calibrated to 5. When rebate recipients increase demand for motor vehicles, the relative price of motor vehicles rises by approximately 1.1 percent in July 2008 (consistent with the observed 1.5 percent spike in the BLS new vehicle price index relative to core CPI around the rebate distribution). This price increase induces optimizing households to intertemporally substitute away from durable goods. Because durable demand is highly price-elastic (long-run elasticity of −1 to −15 depending on the study), even a modest relative price increase generates substantial crowding out of durable expenditure by non-recipients.

Q11. What are the GE-MPC estimates in the modified model with less elastic durable supply, and how do they decompose?

A. At the preferred micro MPC of 0.3, the GE-MPC for total PCE is 0.07—general equilibrium forces dampen the micro effect. At micro MPC of 0.5, GE-MPC is 0.22 (modest dampening). At micro MPC of 0.9, the GE-MPC rises to 1.42 (amplification). Decomposing by good type at micro MPC of 0.3: the GE-MPC on motor vehicles is 0.09 and the GE-MPC on nondurables is −0.03. The dampening is concentrated almost entirely in durable expenditure.

Q12. How sensitive are the GE-MPC results to the calibration of durable demand elasticity?

A. The baseline calibration uses a long-run vehicle demand elasticity of −15, based on household-level evidence from Bachmann et al. (2021). When the authors instead use the lower-bound estimate of −6.4 from Baker et al. (2019), the GE-MPC at micro MPC of 0.3 rises from 0.07 to 0.12. Even at this lower demand elasticity there is substantial crowding out in general equilibrium, so the qualitative conclusion is robust.

Q13. Why does a nondurables-only model with the same overall MPC substantially overstate the fiscal multiplier?

A. When abstracting from durable goods and calibrating a nondurable MPC of 0.30 (to match the overall expenditure MPC), the model predicts a GE-MPC of 0.36—five times larger than the 0.07 from the two-good model. This occurs because nondurable demand is far less price-elastic than durable demand, and the nearly-flat Phillips curve makes nondurable supply very elastic, so there is no relative-price-driven crowding out channel. The comparison illustrates that the distribution of spending across nondurable and durable goods is a quantitatively important determinant of the fiscal multiplier, independent of the level of the MPC.

Q14. What evidence is provided that the control group in the household regressions is itself affected by the rebate in general equilibrium?

A. Figure 9 in the paper plots motor vehicle spending per household by rebate-receipt status using CEX data. When rebate recipients begin reporting receipt in June 2008, motor vehicle expenditure in the rebate group rises while simultaneously falling in the never-rebate group. This pattern is consistent with the model’s prediction that the rebate-induced rise in relative motor vehicle prices crowds out purchases by non-recipient households. This general equilibrium spillover means the difference-in-differences micro MPC estimate remains valid as a micro estimate (the symmetric crowding out does not affect the treated-versus-control difference), but the aggregate GE-MPC is less than the micro MPC.

Q15. How do the authors verify that their preferred corrected specification recovers true MPCs?

A. In Appendix C.6 the authors simulate household-level data from the modified Section 5 model and apply both the original Parker et al. (2013) specification (Equation 1) and their preferred corrected specification (Equation 5). The Parker et al. specification produces upward-biased MPC estimates in the simulated data, consistent with Kaplan and Violante’s (2014) theoretical argument. The preferred corrected specification recovers the true MPCs from the model, validating the correction methodology.

Key Concepts

GE-MPC (General Equilibrium Marginal Propensity to Consume). The paper’s term for the aggregate increase in total consumer spending per dollar of tax rebate, incorporating both the direct micro-level demand effect of the rebate on hand-to-mouth households’ consumption and the induced macroeconomic income effects from Keynesian multipliers and relative price changes. Distinct from the micro MPC, which captures only the household-level spending response before any general equilibrium feedbacks.

Micro MPC. The causal effect of receiving a temporary lump-sum transfer on a household’s own consumer expenditure, expressed as a fraction of the transfer amount, estimated from household panel data via difference-in-differences event studies. In the paper’s usage, this is a partial equilibrium concept that excludes any impact of the policy on prices, wages, or other households’ incomes.

Forbidden comparisons bias. A form of bias in two-way fixed effects event study estimates that arises when treatment effects are heterogeneous across cohorts and later-treated units are used as control groups for earlier-treated units whose outcomes are still reverting after treatment. Named and formalized in Borusyak and Jaravel (2017) and Borusyak et al. (2022); in this paper it manifests because cohorts receiving rebates in June have systematically larger spending responses than those receiving in September, so using September recipients as a “clean” control for June reversal yields contaminated estimates.

Rebate reporting bias. A bias specific to the CEX survey data in which the timing of a household’s self-reported rebate receipt is correlated with unusually high contemporaneous expenditure (and correspondingly low prior-period expenditure), likely due to recall effects. Because the true rebate timing is random but the reported timing is not, this correlation inflates the difference-in-differences estimate of the spending effect.

Two-good, two-agent New Keynesian (TANK) model. A medium-scale New Keynesian model containing two types of households (optimizing life-cycle consumers and hand-to-mouth consumers who exhaust current income) and two goods (nondurables and durable goods interpreted as motor vehicles). The model is used in this paper as a framework to translate micro MPC estimates into aggregate general equilibrium counterfactuals, calibrated at monthly frequency.

Durable supply elasticity. The elasticity of real durable goods production with respect to the relative price of durable goods, calibrated in the paper to 5. In the baseline model, this elasticity is infinite (the relative price is fixed at one because intermediates convert frictionlessly). With a finite supply elasticity of 5, rebate-induced durable demand causes the relative vehicle price to rise, generating crowding out of optimizing households’ durable expenditure.

Calvo durable adjustment friction. An adjustment friction imposed on optimizing households’ durable goods purchases, motivated by Evans and Ramey’s (1992) calculation cost model. Only a fraction 1−θd of households reoptimize their durable stock each period (with probability drawn randomly), producing a Calvo-type reduced form. This friction limits both the extensive and intensive margins of durable adjustment and prevents unrealistically large intertemporal substitution of durable purchases in response to price changes.

Macro counterfactual. In this paper’s usage, the simulated path of aggregate consumption that would have occurred in the absence of the 2008 tax rebate, constructed by subtracting the model-implied impulse response to the rebate from the actual observed NIPA consumption series. Plausibility of the counterfactual is assessed by comparison to contemporaneous forecasts and to historical episodes of large consumption declines.

Motivating banks to lend? Credit spillover effects of the Main Street Lending Program

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. Minoiu, Zarutskie, and Zlate ask whether participation in the Main Street Lending Program (MSLP)—a Federal Reserve emergency facility launched in mid-2020 to channel credit to small and mid-sized firms during the COVID-19 pandemic—caused banks to lend more outside the program. The authors focus on credit spillover effects: did MSLP-participating banks ease standards and expand volumes on their general commercial and industrial (C&I) loan books, beyond the direct loans originated under the program itself?

Institutional Context. The MSLP opened for lender registration on June 15, 2020 and began accepting loan submissions on July 6, 2020, expiring December 31, 2020. Of $600 billion in available SPV capacity, only $16.05 billion was actually deployed, making overall take-up approximately 2.7% of capacity. Despite this, the program required participating banks to retain 5% of each loan’s credit risk while offloading 95% to the SPV, and charged borrowers LIBOR plus 300 bps. Registration rate among all Call Report banks was 11.7% (614 out of 5,242 banks), with participation rising steeply with bank size: from 6.5% of banks in the below-$1-billion asset group to 63.8% of banks with assets above $50 billion.

Data and Methodology. The analysis draws on multiple data sources: (a) supervisory Y-14Q H1 loan-level data covering C&I loans above $1 million commitments, reported by 32 bank holding companies (BHCs) that account for roughly three-quarters of total U.S. C&I loans; (b) Y-14Q A9 loan portfolio segment data for small business C&I loans (below $1 million commitments) from 22 BHCs; (c) quarterly Senior Loan Officer Opinion Survey (SLOOS) microdata for April, July, and October 2020, providing bank-level assessments of lending standard changes, loan terms, demand shifts, and stated reasons for tightening; (d) Dealscan syndicated loan originations for 262 banks (51 MSLP participants); and (e) bank balance sheet data from Call Reports, including the Ellul-Yerramilli risk management index (RMI) for 16 BHCs. The core empirical design is a difference-in-differences (DiD) comparing MSLP-participating vs. non-participating banks before (2020:Q1–Q2) and after (2020:Q3) program implementation. To address nonrandom selection, the authors instrument MSLP participation with three variables: (i) a dummy for banks that cited registration as “too burdensome” in the September 2020 supplementary SLOOS; (ii) a dummy for banks with prior experience pledging loan collateral at the Fed’s discount window; and (iii) a dummy for banks with prior experience pledging securities collateral at the discount window. Firm×quarter fixed effects absorb time-varying credit demand at the borrower level (Khwaja-Mian design), and bank×borrower fixed effects further control for relationship-specific lending patterns.

Main Findings — Extensive Margin (Large Business Loans). In the Y-14Q H1 data, MSLP banks were 30–32% more likely to renew existing loans than non-MSLP banks in 2020:Q3, with the probability of renewal 1.6–1.7 percentage points higher (against a sample average renewal rate of 5.3%). New loan originations were 22–27% more likely at MSLP banks, or 1.1–1.4 percentage points higher (against a sample average origination rate of 5.1%). 2SLS estimates are similar in magnitude to OLS, indicating selection bias is modest.

Main Findings — Extensive Margin (Small Business Loans and Survey Data). In the A9 small business segment data, MSLP lenders had 17.3% more small business loan accounts outstanding in 2020:Q3 than non-MSLP banks. In SLOOS microdata, MSLP banks were approximately 15 percentage points less likely to report tightening C&I lending standards in 2020:Q3 (conditional on demand controls), compared to an actual tightening rate of 37.5%. This effect is larger for small (more financially constrained) firms (16–17 percentage points) than for large firms (13–14 percentage points).

Main Findings — Intensive Margin. On loan terms, MSLP banks charged spreads that were approximately 9 basis points lower on renewed/originated C&I loans in the Y-14Q data, and 13.5 basis points lower in the Dealscan syndicated loan sample, compared to non-MSLP banks in 2020:Q3. 2SLS estimates are somewhat larger (19–30 bps). In the Dealscan sample, MSLP banks also extended syndicated loans that were 11.2% larger (about $2.4 million more given a $22 million average loan size). Survey data confirm MSLP banks were less likely to tighten most individual loan terms.

Aggregate Magnitude. The authors estimate that, in the absence of the MSLP, total loan renewals and originations at Y-14Q reporting banks in 2020:Q3 would have been approximately 10% lower. Scaling to the broader banking sector, the estimated credit spillover effect is approximately $44.8 billion in C&I lending—nearly three times the $16.05 billion in direct MSLP loan purchases.

Mechanism. Survey and objective evidence both point to reduced risk aversion as the primary channel, rather than immediate balance sheet constraint relief. MSLP banks were significantly less likely to cite “reduced tolerance for risk” as a reason for tightening lending standards after the program’s introduction, while showing no differential propensity to cite capital or liquidity deterioration. Banks with higher risk management index scores (more risk-averse institutions) exhibited larger spillover effects on two of three lending margins. Indicators of immediate balance sheet tightness (excess capital cushions, cost of capital, core deposit reliance) do not predict larger spillovers, with a partial exception for lower excess capital and higher loan loss reserves — suggesting future rather than current balance sheet constraints may have played some role.

Scope Conditions and Robustness. The backstop mechanism is explicitly tied to the program’s credibility period: the spillover effects are smaller in 2020:Q4, consistent with the Treasury’s November 19, 2020 announcement that the program would not be extended, which diminished its backstop role. Placebo regressions using 2018 and 2019 data find no differential lending behavior between MSLP and non-MSLP banks before the program, supporting parallel trends. Results are robust to controls for PPP participation, credit line drawdown exposure, loan loss provisioning, and bank-level loan portfolio cyclicality.

In depth

Q1. What precisely is the “spillover effect” that the paper measures, and how does it differ from the direct effect of the MSLP?

A: The direct effect is the $16.05 billion in MSLP loans purchased by the SPV — credit extended specifically through the program. The spillover effect refers to changes in banks’ general C&I lending behavior outside the program: renewals and originations of non-MSLP loans, changes in lending standards and terms for all business borrowers, and changes in small business loan volumes. The sample in the Y-14Q regression explicitly excludes MSLP loans themselves, so the estimates reflect only the indirect, broader credit effects.

Q2. What instruments does the paper use for MSLP participation, and why are they plausibly exogenous?

A: Three IVs are employed: (1) a dummy for banks that cited program registration as “too burdensome” as a very important reason for not joining (from the September 2020 supplementary SLOOS); (2) a dummy for banks that pledged loan collateral at the Fed’s discount window in December 2019; and (3) a dummy for banks that pledged securities collateral at the discount window in the same period. The exclusion restriction argument is that (1) reflects banks’ administrative capacity and prior Fed engagement rather than underlying balance sheet strength or lending appetite, and that (2) and (3) reflect familiarity with Fed collateral processes in ways that made a loan-based program easier to understand and join — without independently affecting lending standards or volumes in 2020:Q3.

Q3. How large are the spillover effects on the extensive margin of large corporate lending?

A: In the Y-14Q H1 data across 32 BHCs, MSLP banks renewed loans 1.6–1.7 percentage points more frequently and originated new loans 1.1–1.4 percentage points more frequently in 2020:Q3, relative to non-MSLP banks. Against sample averages of 5.3% renewal rate and 5.1% origination rate, these translate to MSLP banks being 30–32% more likely to renew and 22–27% more likely to originate loans. The 2SLS estimates are broadly similar in magnitude, suggesting that self-selection bias in OLS is limited.

Q4. What are the estimated aggregate dollar spillovers from the MSLP?

A: The paper calculates that, in the absence of the program, total loan renewals and originations at Y-14Q H1 MSLP banks in 2020:Q3 would have been lower by approximately $33.6 billion (derived from 44,274 bank-borrower pairs × 1.38 existing loans per pair × 3.06 percentage points of extra loan activity × $17.98 million average loan size). Scaling to all Y-14Q banks (MSLP and non-MSLP alike), the shortfall would represent roughly a 10% reduction in total 2020:Q3 loan renewals and originations. Extrapolating to the full banking sector (since Y-14Q banks cover about 75% of total C&I lending), and assuming similar spillover magnitudes for banks outside the sample, total MSLP spillovers amount to roughly $44.8 billion — approximately three times the $16.05 billion in direct MSLP loan purchases.

Q5. What is the estimated effect on C&I lending standards using survey data?

A: Using SLOOS microdata, the paper estimates that MSLP banks were approximately 15 percentage points less likely to tighten C&I lending standards in 2020:Q3 compared to non-MSLP banks, after controlling for demand conditions. The actual tightening rate in 2020:Q3 was 37.5%, meaning the counterfactual tightening rate absent the program would have been approximately 5 percentage points higher. In a further hypothetical where all SLOOS sample banks had participated, the counterfactual tightening rate would have been nearly 10 percentage points higher than actual.

Q6. Are spillover effects larger for small or large borrowers, and what does this imply?

A: The SLOOS-based estimates show that MSLP banks were 16–17 percentage points less likely to tighten lending standards for small firms (annual sales below $50 million), compared to 13–14 percentage points less likely for large and middle-market firms — a statistically significant difference. The authors interpret this as consistent with the MSLP reducing risk aversion broadly, with the largest effect on borrowers facing greater credit constraints where uncertainty about creditworthiness was highest.

Q7. What evidence supports the risk aversion (psychological backstop) mechanism over the balance sheet constraint mechanism?

A: From SLOOS data, MSLP banks were significantly less likely (at the 1% level) to cite “reduced tolerance for risk” as a reason for tightening lending standards after the program’s introduction, while showing no differential likelihood of citing deteriorating capital or liquidity positions as reasons. Furthermore, splitting banks by the risk management index (RMI), the spillover effects are stronger for high-RMI (more risk-averse) banks on two of three lending outcomes. Conversely, proxies for immediate balance sheet constraints — excess capital cushions, core deposit ratios, equity issuance, and cost of capital — do not yield consistently stronger spillover effects for more constrained banks. The only partial exception is lower excess capital and higher loan loss reserves, which are associated with more loan renewals, suggesting future rather than current balance sheet constraints may have contributed.

Q8. What is the risk management index (RMI), and how is it used here?

A: The RMI is an index developed by Ellul and Yerramilli (2013) that captures the strength of a bank’s internal risk management function, constructed from variables including whether the bank has a chief risk officer (CRO), the CRO’s executive status and relative compensation, risk committee member experience, and meeting frequency. Available for 61 BHCs over 2011–2013, it is matched to 16 BHCs in the Y-14Q H1 sample and used as a pre-COVID proxy for institutional risk aversion. Banks above the median RMI show larger MSLP spillover effects on loan renewals and tightening standards, consistent with the interpretation that the MSLP reduced effective risk aversion more for banks that had higher baseline risk-consciousness.

Q9. How do the authors address the concern that PPP participation — not MSLP participation — might drive the results?

A: First, they test directly that MSLP participation does not predict outstanding PPP/federally-guaranteed loan balances (in Q2 or Q3 2020) in the A9 loan segment data, finding no correlation. Second, they add an interaction of PPP loan balances (divided by total assets) × Post to the baseline regression in Table A10 and find that while PPP lending is positively associated with loan renewals and originations, the MSLP bank × Post coefficient remains statistically significant and similar in magnitude to the baseline, ruling out PPP participation as the driver of the baseline results.

Q10. What explains the low take-up of the MSLP despite its large designed capacity?

A: Survey responses from the September 2020 supplementary SLOOS indicate several demand- and supply-side constraints: banks reported they could generally meet credit demand outside the program; borrower leverage limits (capped at 4–6× EBITDA depending on facility) were seen as too restrictive; the LIBOR plus 300 bps interest rate was high relative to historical pricing for eligible firms; and registration and loss-sharing arrangements were viewed as burdensome and uncertain. The paper interprets these findings as consistent with banks treating the MSLP primarily as a backstop — a facility they would activate only if economic conditions deteriorated significantly — rather than a primary lending channel.

Q11. How does the paper address the threat that MSLP participation reflects bank-level cyclicality in loan portfolios?

A: Table 10 controls for bank-specific C&I loan portfolio cyclicality, measured as the correlation between each bank’s C&I loan growth and aggregate banking-sector C&I loan growth estimated over 1985:Q1–2021:Q2 using two functional forms. The MSLP bank × Post coefficient estimates remain very similar to the baseline after including these controls, ruling out the concern that MSLP participants were simply banks with naturally more procyclical or countercyclical lending patterns.

Q12. What happens to the estimated spillover effects in 2020:Q4, and what does this reveal?

A: The paper shows (Table A6) that extending the sample to include 2020:Q4 yields somewhat smaller estimated spillover effects than in the baseline 2020:Q3 period. The authors attribute this to the November 19, 2020 announcement by Treasury Secretary Mnuchin that the MSLP would not be extended beyond year-end, which effectively ended the program’s backstop role and — consistent with the psychological backstop mechanism — reduced banks’ confidence in the program’s future availability and thus the spillover motivation.

Q13. Does the paper find spillover effects on intensive margin loan terms, and how large are they?

A: On loan spreads, MSLP banks charged approximately 9 basis points lower spreads on floating-rate C&I loans renewed or originated in 2020:Q3 in the Y-14Q data (2SLS: 19 bps), and approximately 13.5 bps lower spreads in the Dealscan syndicated loan sample (2SLS: 30 bps). The 9 bps OLS estimate implies the average spread across all LIBOR-indexed C&I loans in 2020:Q3 would have been approximately 4 bps higher absent the program (i.e., 0.43 × 9 bps), relative to an actual average spread of 235 bps — an effect the authors characterize as economically small. On loan size, the Dealscan evidence indicates MSLP banks extended syndicated loans that were 11.2% larger (2SLS: 25% larger).

Key Concepts

Credit Spillover Effects: As used in this paper, spillover effects refer to the impact of MSLP participation on participating banks’ lending behavior outside and beyond the program itself — specifically, changes in loan renewal rates, new loan origination rates, lending standards, and loan terms for non-MSLP C&I loans. This is distinct from the direct effect (i.e., loans originated through the MSLP proper).

Psychological Backstop: The paper’s term for the mechanism by which the MSLP reduced participating banks’ effective risk aversion without necessarily easing their immediate balance sheet constraints. By committing to provide lending support if conditions deteriorated, the program built banks’ confidence to lend ex ante, functioning as “insurance” against bad outcomes rather than a direct funding facility. The mechanism is distinguished from balance sheet easing by the fact that constrained and unconstrained banks exhibited similar spillover effects.

Extensive Margin of Lending: The binary dimension of lending activity — specifically, whether a bank renews an existing loan or originates a new loan within a bank-borrower pair. In this paper, measured as the share of existing loan commitments within each bank-borrower pair that are renewed or newly originated each quarter. Contrasted with the intensive margin.

Intensive Margin of Lending: The quantitative dimension of existing lending relationships — specifically, the average loan size and average spread on loans renewed or originated in a given period, conditional on a loan being extended.

Senior Loan Officer Opinion Survey (SLOOS): A quarterly Federal Reserve survey of senior lending officers at large U.S. banks covering self-reported changes in C&I lending standards, terms (including spreads, maximum loan size, maturity, covenants, collateral requirements), demand conditions, and — in supplementary editions — reasons for changing standards. Used in this paper both as an outcome variable (tightening standards) and as a control variable (changes in loan demand) and as a source of IV variation (burden of MSLP registration).

Risk Management Index (RMI): An index developed by Ellul and Yerramilli (2013) measuring the strength of a bank’s internal risk management function, combining information on the presence and compensation of a chief risk officer, risk committee composition, and meeting frequency. Used in this paper as a pre-pandemic proxy for institutional risk aversion to test whether the MSLP disproportionately reduced risk aversion in banks with stronger risk controls.

Difference-in-Differences with Granular Fixed Effects: The primary identification strategy, comparing changes in lending outcomes between MSLP-participating and non-participating banks before (2020:Q1–Q2) and after (2020:Q3) program implementation. The paper uses firm×quarter fixed effects following Khwaja and Mian (2008) to absorb borrower-level credit demand, and bank×borrower fixed effects following Chodorow-Reich (2013) to absorb relationship-specific supply factors — isolating the bank credit supply effect attributable to MSLP participation.

Originate-and-Distribute Feature (of MSLP): The MSLP’s design in which banks originate MSLP loans but sell 95% of the credit exposure to the SPV, retaining only 5%. This feature was intended to free up balance sheet capacity for further lending. The paper tests whether this channel (easing current balance sheet constraints) explains the observed spillovers, finding limited support relative to the risk aversion reduction channel.

Narratives about the Macroeconomy

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper investigates two related empirical questions in the context of the historic surge in US inflation in late 2021 and 2022: (1) What narratives—causal stories—do people invoke to explain why inflation increased? (2) How do those narratives shape economic expectations? A companion theoretical component asks how narrative heterogeneity affects aggregate macroeconomic outcomes.

Data and Methodology

The authors recruit more than 10,000 US households across five descriptive survey waves (November 2021, December 2021, January 2022, March 2022, May 2022) via Lucid, plus a separate expert survey of 111 academic economists with JEL-E publications in top journals, recruited simultaneously with the November 2021 household wave. Household samples are broadly representative of the US population in terms of gender, age, region, and income. The expert sample is highly credentialed: on average 18.6 years post-PhD, 2.7 top-five publications, and 5,534 Google Scholar citations.

Narratives are elicited through open-ended questions asking respondents to explain in their own words why inflation increased. Each text response is coded by two independent, blinded research assistants as a Directed Acyclic Graph (DAG) — a network of causal nodes representing factors (demand-side: government spending, monetary policy, pent-up demand, demand shift; supply-side: supply chain disruptions, labor shortage, energy crisis; miscellaneous: pandemic, government mismanagement, price gouging, Russia-Ukraine war) connected by directed causal edges. Inter-rater reliability is high: if one coder identifies a factor, the other does so 88% of the time; for specific causal connections between factors, agreement is 77%.

Three experiments study the causal effect of narratives on expectations: (1) A pent-up demand vs. energy crisis narrative provision experiment (April 2022, n=2,397 baseline, n=1,329 follow-up); (2) A monetary policy vs. energy crisis narrative provision experiment (June 2022, n=1,069 baseline, n=736 follow-up); (3) A 2×2 belief-updating experiment crossing narrative type (government spending vs. energy crisis) with information type (low vs. high government spending forecast) (April 2022, n=997).

Main Findings with Quantitative Magnitudes

Households’ narratives are substantially coarser than experts’: expert DAGs contain on average 4.3 factors and 3.6 causal links, while household DAGs contain only 3.5 factors and 2.8 links (both differences p < 0.01). Households focus predominantly on supply-side explanations: 57% invoke at least one supply-side factor vs. only 32% invoking any demand-side factor. The most common household narrative factors are supply chain disruptions (30%), labor shortage (27%), and general supply-side factors (22%); the leading demand-side factor is government spending, appearing in only 17% of household narratives, while loose monetary policy appears in just 5%. By contrast, 90% of experts invoke at least one supply-side factor and 84% at least one demand-side factor, with government spending mentioned by 50% of experts and monetary policy by 38%.

Among households who invoke at least one supply or demand narrative, only 34% mention both supply and demand factors; among the corresponding subsample of experts, 77% mention both. Government mismanagement—a politicized judgment of policy failure—appears in 32% of household narratives but only 1% of expert narratives. Price gouging appears in 8% of household narratives and 0% among experts.

Partisan polarization is large: Democrat-leaning respondents are 26 pp more likely to attribute inflation to the pandemic as a root cause (p < 0.01); Republican-leaning respondents are 38 pp more likely to blame government mismanagement (p < 0.01), and 19 pp more likely to mention high government spending (p < 0.01) and 14 pp more likely to mention high energy prices (p < 0.01).

Narratives are correlated with inflation expectations in OLS regressions controlling for demographics and survey wave fixed effects (n=2,951): households invoking government mismanagement predict 1.155 pp higher 1-year-ahead inflation (p < 0.01) and 0.805 pp higher 5-year-ahead inflation (p < 0.01). Energy crisis narratives predict 0.661 pp higher 1-year-ahead inflation (p < 0.01). Pent-up demand narratives predict 0.640 pp lower 5-year-ahead inflation (p < 0.05). Narrative variables explain approximately 10% of the out-of-sample variation in 1-year-ahead inflation expectations via LASSO, comparable to or exceeding the explanatory power of demographics and inflation experiences found in prior work.

In Experiment 1 (pent-up demand vs. energy crisis), providing the pent-up demand narrative reduces 12-month inflation expectations by 0.71 pp relative to the energy crisis treatment (p < 0.01, in the main survey), corresponding to 24% of a standard deviation. This effect persists in the follow-up survey one day later (−0.63 pp, p < 0.01).

In Experiment 2 (monetary policy vs. energy crisis), the monetary policy narrative reduces 12-month inflation expectations by 0.40 pp at the time of the main survey (p < 0.01) and by 0.62 pp in the follow-up (p < 0.01).

In Experiment 3 (information updating), respondents exposed to the government spending narrative increase 12-month inflation expectations by 1.79 pp in response to a high-spending forecast (p < 0.01), while those exposed to the energy crisis narrative show no significant reaction (0.34 pp, p = 0.205). In IV regressions instrumenting government spending expectations with the high/low forecast treatment, a 1 pp increase in perceived government spending growth raises inflation expectations by 0.378 pp among those holding the government spending narrative (p < 0.01) versus only 0.051 pp among those holding the energy narrative (p = 0.184; difference p < 0.01).

The New Keynesian DSGE model shows that a modest shift in perceived importance of monetary policy relative to productivity (raising ω_ν from 0.1 to 0.2, holding ω_g fixed) raises equilibrium consumption by 27 basis points and reduces equilibrium inflation by 27 basis points in the calibrated model with φ = 1.5; with a less reactive central bank (φ = 1.25), the same shift raises consumption by 30 basis points and reduces inflation by 62 basis points.

Scope Conditions

All empirical results are drawn from the US context during the 2021–2022 inflation surge. The authors note that the extent of partisan polarization in US narratives may not generalize to less politically polarized countries. The test-retest correlation of narrative factors across a three-day interval is 0.63 (p < 0.01), indicating significant but not perfect stability. The experiment results may partly reflect that narratives were especially malleable because the inflation surge was a relatively recent and salient phenomenon at the time of data collection.

In depth

Q1. How do the authors define and operationalize “narratives”?

A: The paper defines economic narratives as causal accounts for why an economic event occurred — agents’ assessments of cause-effect relationships across events. Each text response is coded as a Directed Acyclic Graph (DAG) where nodes are economic factors and directed edges represent perceived causal links. DAGs can represent both simple mono-causal accounts and complex multi-factor chains. The authors use a predefined coding scheme of 16+ factor categories spanning demand-side, supply-side, and miscellaneous nodes, with inflation as the terminal node.

Q2. What is the inter-rater reliability of the DAG coding, and what does it imply for the quality of the narrative data?

A: Two independent, blinded coders annotate each response. If one coder assigns a given factor, the other does so 88% of the time; for specific causal connections between factors, agreement is 77%. Approximately 95% of assigned factors and 89% of assigned connections make it to the final coded version. At the coarser level of “any demand-side factor,” agreement rises to 94%; for “any supply-side factor,” to 93%. Test-retest reliability across a three-day interval averages a correlation of 0.63 across all narrative factors (p < 0.01), comparable in magnitude to the measured persistence of economic preferences in prior work.

Q3. How do expert and household narratives differ in their structural complexity?

A: Expert DAGs contain on average 4.3 factors and 3.6 causal links, compared to 3.5 factors and 2.8 links for households (both p < 0.01). These differences persist even after controlling for response time and word count, indicating genuine differences in economic understanding rather than effort. Among agents who invoke at least one supply or demand factor, 77% of experts mention both, compared to only 34% of households.

Q4. What are the most prevalent factors in household narratives versus expert narratives, and why does this matter?

A: Supply chain disruptions (30%), labor shortage (27%), and general supply-side factors (22%) top household narratives, while monetary policy appears in only 5% of household DAGs. Expert narratives are more balanced: 90% cite supply-side factors and 84% cite demand-side factors, with government spending mentioned by 50% and monetary policy by 38%. This matters because factors with different persistence imply different trajectories for future inflation; households’ supply-side emphasis, combined with low awareness of monetary policy, shapes their inflation expectations in systematically different ways than experts.

Q5. What is the structure of household narrative clusters, and how fragmented are they?

A: Agglomerative hierarchical clustering using the Jaccard distance between DAG edge lists reveals 15 optimal clusters (Silhouette criterion), of which eight have at least 30 members. Four supply-side clusters account for 55% of households: pandemic-related supply chain disruptions (20%), general supply-side causes (18%), energy crisis often attributed to government mismanagement (11%), and labor shortages attributed to the pandemic or government spending (7%). The only clear demand-side cluster—combining government spending and loose monetary policy—captures just 8%. Simple mono-causal clusters attributing inflation to the pandemic alone (15%), government mismanagement alone (11%), and price gouging alone (4%) are collectively prominent, underscoring how fragmented and often single-factor household reasoning is.

Q6. How do partisan affiliations correlate with narrative content?

A: Republicans are 38 pp more likely than Democrats to attribute inflation to government mismanagement (p < 0.01), 19 pp more likely to mention high government spending (p < 0.01), and 14 pp more likely to mention high energy prices (p < 0.01). Democrats are 26 pp more likely to cite the pandemic as a root cause of inflation (p < 0.01) and more frequently cite pandemic-related supply chain issues and corporate greed. Government mismanagement appears in 32% of all household narratives (and is often portrayed as a root cause of spending, monetary policy, and energy prices) but in only 1% of expert narratives.

Q7. How did the composition of household narratives shift over time (November 2021 to May 2022)?

A: The energy crisis narrative rose sharply from 12% in January 2022 to 28% in March 2022, coinciding with Russia’s invasion of Ukraine in late February 2022. The Russia-Ukraine war narrative went from virtually zero before February 2022 to 28% in March 2022. By contrast, pandemic references, which climbed from 44% in November 2021 to 55% in January 2022, fell back to 47% in March 2022 and 39% in May 2022. Labor shortage references fell sharply from 32% in January 2022 to 15% in May 2022. These abrupt shifts suggest household narratives respond to major news events and, by extension, could drive rapid revisions in inflation expectations around such events.

Q8. What is the correlational evidence that narratives predict inflation expectations, and how large is the explanatory power?

A: OLS regressions on pooled data from November 2021–January 2022 (n=2,951), controlling for survey wave fixed effects and sociodemographics, show: government mismanagement narratives predict 1.155 pp higher 1-year inflation expectations (p < 0.01) and 0.805 pp higher 5-year expectations (p < 0.01); energy crisis narratives predict 0.661 pp higher 1-year expectations (p < 0.01); monetary policy narratives predict 1.005 pp higher 1-year expectations (p < 0.01); pent-up demand narratives predict 0.640 pp lower 5-year expectations (p < 0.05). LASSO out-of-sample prediction using DAG factor dummies and connection dummies explains approximately 10% of variation in 1-year-ahead inflation expectations — comparable to the 10% within-sample R² found by D’Acunto et al. (2021) for grocery price exposure, and substantially above the 2–7% found by Giglio et al. (2021) for investor characteristics explaining stock return expectations.

Q9. What does Experiment 1 (pent-up demand vs. energy crisis) show about the causal effect of narratives?

A: Providing the pent-up demand narrative (relative to the energy crisis narrative) increases the fraction of respondents invoking pent-up demand by 37.8 pp in the follow-up survey (baseline: 2.8%, p < 0.01) and reduces the fraction invoking the energy crisis by 7.9 pp (p < 0.01), establishing successful first-stage uptake. In the main survey (n=2,397), the pent-up demand treatment reduces 12-month inflation expectations by 0.71 pp relative to the energy treatment (p < 0.01), equivalent to 24% of a standard deviation; the effect persists at −0.63 pp in the follow-up one day later (p < 0.01). The energy crisis treatment has no significant effect on expectations relative to a pure control (−0.02 pp, p = 0.911), suggesting that energy crisis implications were already salient at the time.

Q10. What does Experiment 2 (monetary policy vs. energy crisis) add, given it was conducted after significant Fed tightening?

A: The experiment was run in June 2022, when 61% of respondents were already aware the Fed had raised rates. The monetary policy narrative increases the fraction invoking monetary policy by 39 pp and reduces the energy fraction by 50 pp relative to the energy group (both p < 0.01). The monetary policy narrative reduces 12-month inflation expectations by 0.40 pp in the main survey (p < 0.01) and 0.62 pp in the follow-up (p < 0.01). The mechanism is that attributing past inflation to loose monetary policy — which has since been tightened — leads respondents to infer lower future inflation, consistent with the narrative about persistence of the underlying cause.

Q11. What does Experiment 3 demonstrate about how narratives filter the interpretation of new information?

A: In the 2×2 design, all respondents first receive either a government spending narrative or an energy crisis narrative, then either a low (−4%) or high (+6%) government spending forecast from the Survey of Professional Forecasters. Among those with the government spending narrative, the high-spending forecast raises 12-month inflation expectations by 1.79 pp (p < 0.01); among those with the energy crisis narrative, the high-spending forecast raises inflation expectations by a non-significant 0.34 pp (p = 0.205). The IV estimate shows that a 1 pp increase in expected government spending growth raises inflation expectations by 0.378 pp for those holding the spending narrative (p < 0.01) vs. 0.051 pp for those holding the energy narrative (p = 0.184); this difference is highly significant (p < 0.01). Importantly, the first-stage effect on expected government spending growth is similar across narrative groups (4.7 pp vs. 6.8 pp, difference not significant), ruling out differential interpretation of the forecast itself as the mechanism.

Q12. How do the authors formalize narratives in the DSGE model, and what is the key mapping result?

A: Narratives are formalized as subjective causal models (SCMs): linear mappings from N observable factors to inflation, π_t = ψ_1(i)z_{1,t} + … + ψ_N(i)z_{N,t}, combined with perceived AR(1) processes for each factor. The “subjective inflation narrative” of agent i is summarized by perceived contribution shares ω_z(i). The paper’s Proposition 2 gives closed-form expressions for equilibrium inflation and consumption as functions of these perceived shares, without imposing that they be correct or identical across agents. The key result is that subjective causal models always affect equilibrium outcomes so long as the perceived persistence parameters differ across factors — the mechanism being that different narratives produce different inflation expectations, which feed back into consumption and pricing decisions.

Q13. What are the quantitative implications of narrative shifts in the calibrated DSGE model?

A: The baseline calibration uses standard New Keynesian parameters (β=0.99, γ=1, ς=5, Calvo price duration=4 quarters, φ=1.5, ρ_a=0.9, ρ_g=0.8, ρ_ν=0.5) with a scenario of a 10% productivity decline, 10% government spending increase, and policy rate 2 pp below the Taylor rule. Under rational expectations, π_t=3.68% and c_t=−11.79%. Raising the perceived importance of monetary policy in household and firm inflation narratives from ω_ν=0.1 to ω_ν=0.2 (lowering ω_a by the same amount, holding ω_g fixed) increases equilibrium consumption by 27 basis points and reduces equilibrium inflation by 27 basis points. With a less reactive central bank (φ=1.25), the same narrative shift raises consumption by 30 basis points and reduces inflation by 62 basis points. The paper notes that these effects are approximately linear in the narrative shift, meaning the directional implication holds across a wide range of narrative configurations.

Q14. How does narrative heterogeneity across households affect aggregate outcomes in the model?

A: When households hold heterogeneous narratives, aggregate outcomes depend on the joint distribution of perceived factor importance (ω_z(i)) and perceived factor persistence (ρ_z(i)) across agents, rather than on average values alone. Specifically, the model shows that if households who assign higher importance to a given factor also perceive that factor as more persistent, the aggregate effect on expectations and consumption is amplified beyond what the average narrative predicts. Additionally, narrative heterogeneity generates consumption heterogeneity even when the efficient allocation requires all households to consume the same amount, representing a welfare-relevant distortion absent under rational expectations.

Q15. What is the practical implication for central bank communication?

A: Under full-information rational expectations, central bank narrative communication about the drivers of inflation is irrelevant because agents already hold the correct model. Once subjective causal models can deviate from the truth, central bank narrative provision shifts aggregate equilibrium outcomes (inflation and consumption) in a benchmark New Keynesian model. The paper argues that central banks need to measure the distribution of household narratives to know whether their communication shifts agents toward or away from the rational expectations equilibrium — moving agents in the direction of the correct narrative produces better aggregate outcomes from the central bank’s perspective, conditional on inflation being above target and output below first-best.

Key Concepts

Economic Narrative (as used in this paper): An agent’s causal account for why a given economic event occurred — specifically, an assessment of cause-effect relationships that explains the drivers of an economic outcome. Distinguished from more general notions of “story” in that causality is the core; the paper does not count descriptions of correlation or simple statements of fact as narratives.

Directed Acyclic Graph (DAG) representation of narratives: Each narrative is coded as a network of factor nodes connected by directed edges indicating perceived causation. Acyclicity rules out feedback loops in a respondent’s causal account. Factors with nonzero ψ(i) are included; the direction of edges indicates causal flow. This representation allows quantitative comparison across respondents via adjacency matrices or Jaccard distances between edge lists.

Subjective Causal Model (SCM) of inflation: The paper’s formal theoretical counterpart to a narrative: a linear mapping π_t = Σ_n ψ_n(i) z_{n,t} in which individual i assigns perceived marginal effect ψ_n(i) to each factor z_n, combined with a perceived AR(1) law of motion for each factor. The SCM does not need to be correct or shared across agents. The rational expectations equilibrium is the special case where all agents’ SCMs match the true data-generating process.

Perceived contribution share (ω_z): The ratio ψ_z(i)·z_t / π_t — agent i’s perceived percentage contribution of factor z to current inflation. This is the sufficient statistic for the effect of household narratives on inflation expectations and, through the NK model, on equilibrium aggregate outcomes. The aggregate distribution of ω_z(i) and perceived persistence ρ_z(i) determines the consumption Euler equation at the aggregate level.

Government mismanagement (as a narrative factor): A coding category that captures explicit reference to policy failure or low-quality decision-making by policymakers in a politicized sense — distinct from the economic factors of government spending or monetary policy. It represents households’ attribution of inflation to the incompetence or malfeasance of officials, rather than to any specific economic mechanism. This factor appears in 32% of household narratives but only 1% of expert narratives.

Narrative cluster: A group of respondents whose DAGs are mutually similar (measured by Jaccard distance between edge lists) and whose typical DAG differs from other clusters. Identified via agglomerative hierarchical clustering. The paper identifies eight substantively meaningful clusters, ranging from supply-chain-focused to mono-causal pandemic or mismanagement narratives, with no single cluster capturing more than 20% of households.

Test-retest reliability of narratives: The correlation between the same respondent’s narrative elicited on two occasions three days apart. The paper estimates an average correlation of 0.63 across all narrative factors (p < 0.01), interpreted as indicating significant stability in households’ causal beliefs rather than survey noise. Comparable in magnitude to test-retest correlations of economic preferences in other studies.

Normal Approximation in Large Network Models

Mon, 01 Jan 0001 00:00:00 +0000

This paper proves a central limit theorem (CLT) for network formation models with strategic interactions and homophilous agents, addressing a foundational inferential gap in the econometrics of large networks. The setting is one where the econometrician observes a single large network — the asymptotic framework sends network size n to infinity — which is the empirically relevant case for most network datasets. The network moments of interest are averages of node-level statistics (1/n) Σ ψ_i, where ψ_i can capture degree, clustering coefficients, or subnetwork counts (triangles, k-stars) that have been used for structural inference in network formation games.

The model is a pairwise-stability network formation game augmented onto a latent-space/geometric-graph structure. Each node i has an i.i.d. type (X_i, Z_i), where X_i is a continuously distributed position vector capturing homophilous attributes. Two nodes i and j form a link if a joint-surplus function V(·) exceeds zero, where V depends on the scaled distance r_n^{-1}‖X_i − X_j‖ between positions, a vector of strategic interaction statistics S_{ij} (functions of neighboring links), node attributes Z_i, Z_j, and an i.i.d. utility shock ζ_{ij}. Homophily enters as a monotonicity requirement: V is decreasing in the distance component, so dissimilar nodes are less likely to link. Sparsity is ensured by setting r_n = (κ/n)^{1/d}, which keeps expected degree asymptotically bounded.

Strategic interactions enter through S_{ij}, which depends on links involving neighbors of i or j (local externalities), generating chains of cross-sectional dependence that are the central obstacle to the CLT. The paper identifies two distinct sources of dependence: (1) link interdependencies from best-response chains, where the realization of one link influences neighboring links; and (2) global coordination in equilibrium selection, where agents may condition on a common signal.

The main technical contribution is adapting “stabilization” conditions from the literature on geometric graphs (Penrose and Yukich 2003, 2008) to the strategic setting. Exponential stabilization (Assumption 5) requires that the radius of stabilization R_i — the smallest neighborhood of i such that ψ_i depends only on nodes within that neighborhood — has a distribution with exponential tails. This bounds the effective dependence neighborhood and provides the weak dependence structure needed for the CLT.

To verify stabilization from primitive conditions, the paper employs branching process theory. The key construct is the “strategic neighborhood” C_i^+, the component of i in the network of non-robust links D (pairs where strategic interactions can change the link outcome). The paper bounds |C_i^+| by a subcritical Galton-Watson branching process: if the mean offspring is below 1 (subcriticality, Assumption 7, stated as ‖h*‖_m < 1), the process is non-explosive and its size has exponential tails, yielding the required stabilization. The subcriticality condition directly restricts the strength of strategic interactions and is the network analog of the condition ‖β‖ < 1 in linear autoregressive models. A second condition (Assumption 8, decentralized selection) requires that equilibrium selection operates independently across disjoint strategic neighborhoods, ruling out global coordination; this holds under myopic best-response dynamics.

For inference, the paper proposes a network HAC variance estimator hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where k(·) is a kernel, d_{ij} is the path distance in A, and b_n is a bandwidth, and a network bootstrap that resamples nodes with replacement. Both are shown to be consistent (Theorem 3). Simulation results with n up to 500, varying strategic interaction strength θ_2 from 0 to 0.5, show that the network HAC estimator achieves nominal 5% rejection rates and 95% coverage for n ≥ 500, while the bootstrap slightly over-rejects in small samples and performance degrades as θ_2 increases.

The scope conditions are explicit: the CLT applies to sparse networks (expected degree bounded), undirected networks with local externalities, models admitting a pairwise-stability equilibrium, and equilibrium selection satisfying decentralization. Extensions to directed or denser networks are left for future work.

Q: What is the primary research question and why does it require new theory? A: The paper asks when sample averages of network statistics — degree, clustering, subnetwork counts — satisfy a CLT in strategic network formation models observed as a single large network. Standard CLT proofs require weakly dependent observations, but strategic interactions generate chains of link dependence of a priori unbounded length, and multiple equilibria allow global coordination, both of which can destroy asymptotic normality. Prior work (Leung 2019b; Menzel 2024) established laws of large numbers but not CLTs, which require stronger conditions.

Q: What is the stabilization condition and why is it the right formulation of weak dependence? A: Exponential stabilization (Assumption 5) requires that the radius of stabilization R_i — the smallest K such that ψ_i depends only on the K-neighborhood of i in the network — has a distribution with exponential tails: lim sup_{w→∞} w^{-η} max{log τ_{b,ε}(w), log τ_p(w)} < 0 for some η ∈ (0,1]. This implies that each node’s statistic depends effectively only on a bounded fraction of the network, making {ψ_i} weakly dependent. The condition is a modification of stabilization conditions from the geometric graph literature (Penrose and Yukich 2003, 2008) adapted to allow strategic interactions.

Q: How does the paper connect the abstract stabilization condition to primitive model conditions? A: The paper defines the strategic neighborhood C_i^+ as the union of one-step network neighborhoods of nodes in i’s component in the non-robust link network D (where D_{ij} = 1 iff the link A_{ij} can be switched by strategic interactions). The size |C_i^+| controls the radius of stabilization. By mapping exploration of C_i via breadth-first search onto a Galton-Watson branching process, subcriticality (mean offspring < 1, i.e., ‖h*‖_m < 1) implies that |C_i^+| has exponential tails, which yields exponential stabilization with η = 1 (Theorem 2).

Q: What is the subcriticality condition and what does it restrict? A: Subcriticality (Assumption 7) requires that the mean interaction-strength measure satisfies ‖h*‖_m < 1, where h* bounds the probability that a given link is non-robust as a function of node attributes. This restricts how strongly the existence of one link influences the probability of neighboring links. The authors explicitly analogize this to the condition ‖β‖ < 1 in linear autoregressive models: both bound the magnitude of “autoregressive” dependence below one to prevent explosive propagation of dependence.

Q: What is the decentralized selection condition and what does it rule out? A: Assumption 8 (decentralized selection) requires that the equilibrium selection mechanism operates independently across disjoint strategic neighborhoods: A_{H_l} = λ_{|H_l|}(r^{-1}T_{H_l}, ζ_{H_l}) for each disjoint strategic neighborhood H_l. This rules out global coordination where agents condition on a common signal (such as the type of a particular node) to jointly select an equilibrium. The condition is satisfied by myopic best-response dynamics and is described as the single-network analog of requiring equilibrium selection to be independent across networks under many-network asymptotics.

Q: What is the structure of the CLT proof? A: The proof has two steps. Step 1 proves a CLT for the Poissonized model where the number of nodes N_n ~ Poisson(n), leveraging results from Penrose and Yukich (2008) for geometric graphs extended to the strategic setting. Step 2 is a de-Poissonization argument that transfers the Poissonized CLT back to the fixed-n model. The abstract CLT (Theorem 1) requires Assumptions 5 and 6, and Theorem 2 establishes that Assumptions 1–8 imply Assumption 5 with η = 1.

Q: How does the network HAC estimator work and what are its consistency conditions? A: The estimator is hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where d_{ij} is the path distance between i and j in the observed network A, k(·) is a kernel function, b_n is a bandwidth, and hat_ψ_i = ψ_i(N_n) − (1/n) Σ_j ψ_j(N_n) is the demeaned statistic. Consistency (hat_Σ_n →^p Σ_n) is established under appropriate conditions on the bandwidth b_n (Theorem 3). The bandwidth plays the same role as in time-series HAC estimation, controlling the window over which covariances are summed.

Q: What do the simulations show about finite-sample performance? A: Using a DGP with X_i ~ U([0,1]^2), ζ_{ij} ~ N(0,1), and θ_2 varying from 0 to 0.5 to control strategic interaction strength, the network HAC estimator achieves nominal 5% rejection rates and 95% coverage at n ≥ 500 across all settings. The bootstrap slightly over-rejects in small samples. Performance of all procedures degrades as θ_2 increases (stronger strategic interactions), consistent with the theoretical condition that subcriticality must hold. These results support practical use of the inference procedures based on Theorem 1.

Q: How does this paper relate to prior work on CLTs for network data? A: Kojevnikov et al. (2021) prove a CLT for node-level data conditional on the network, but this does not apply to network formation because the network is the outcome, not a conditioning variable. Leung (2019b) and Menzel (2024) prove laws of large numbers for strategic network formation but not CLTs. Kuersteiner (2019) takes a different approach using a conditional mixingale assumption. The paper’s abstract CLT extends Penrose and Yukich (2008) by modifying the stabilization condition to accommodate strategic interactions; the primitive conditions are new and use branching process tools that build on Leung (2019b).

Q: What network moments can the CLT be applied to? A: The CLT applies to any average of node statistics ψ_i that depends only on the K-neighborhood of i in the network (Assumption 4 with finite K). Explicit examples include average degree (ψ_i = Σ_j A_{ij}), average clustering coefficient, and counts of connected subnetworks such as triangles and k-stars. Subnetwork counts have been used as the basis for structural identification and estimation of network formation games (Sheng 2020), making the CLT directly applicable to inference in those models.

Q: What are the scope limitations and directions for future work? A: The CLT applies to sparse undirected networks with local externalities (Assumption 2), homophily in positions (Assumption 1), and equilibrium selection satisfying decentralization (Assumption 8). It does not cover directed networks, denser networks where expected degree grows with n, or models with global link externalities. The authors identify extending results to directed and denser networks and developing more powerful inference procedures exploiting network structure as priorities for future work.

Stabilization (exponential): The condition that the radius of stabilization R_i — the smallest neighborhood of i beyond which ψ_i does not depend on further nodes — has a distribution with exponential tails (lim sup_{w→∞} w^{-η} log τ(w) < 0 for η ∈ (0,1]). This is the paper’s operative formulation of weak dependence for network statistics and is adapted from geometric graph theory to the strategic setting.

Strategic neighborhood (C_i^+): The union of one-step neighborhoods of nodes in i’s component in the non-robust link network D. A link (i,j) is non-robust (D_{ij} = 1) if strategic interactions can change its realization — i.e., the surplus V can be positive under some interaction configurations and non-positive under others. The size of C_i^+ governs the radius of stabilization and hence the degree of cross-sectional dependence.

Subcriticality (‖h*‖_m < 1): The condition that the mean-field interaction strength measure satisfies ‖h*‖_m < 1, where h* bounds the conditional probability that a link is non-robust. Subcriticality ensures that breadth-first search of the strategic neighborhood is dominated by a subcritical Galton-Watson process (mean offspring < 1), preventing explosive growth of the dependence neighborhood. The paper explicitly frames this as the network analog of ‖β‖ < 1 in autoregressive models.

Decentralized selection (Assumption 8): The requirement that the equilibrium selection mechanism assigns outcomes independently across disjoint strategic neighborhoods: A_{H_l} = λ_{|H_l|}(r^{-1}T_{H_l}, ζ_{H_l}) for each disjoint H_l. This rules out global coordination — agents conditioning on a common signal to select among equilibria — while permitting local coordination within strategic neighborhoods. Satisfied by myopic best-response dynamics.

Pairwise stability: The solution concept underlying the model. A network A satisfies pairwise stability under transferable utility if A_{ij} = 1{V_{ij} > 0}, meaning a link forms exactly when the joint surplus is positive. This is the equilibrium condition from which the strategic interaction statistics S_{ij} and non-robustness indicators D_{ij} are derived.

Network HAC estimator: The variance estimator hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where d_{ij} is the path distance in the observed network, k(·) is a kernel, and b_n is a bandwidth. It is the network analog of heteroskedasticity- and autocorrelation-consistent (HAC) estimators in time series, using path distance in place of temporal lag distance.

Homophily (in this paper’s sense): The property that the joint-surplus function V is decreasing in the first argument r_n^{-1}‖X_i − X_j‖ (scaled positional distance), so nodes that are more dissimilar in position are strictly less likely to form links. Combined with the sparsity scaling r_n = (κ/n)^{1/d}, this ensures that links decay with distance in social space and that the network remains sparse as n grows.

Open Rule Legislative Bargaining

Mon, 01 Jan 0001 00:00:00 +0000

This paper revisits the open rule legislative bargaining model of Baron and Ferejohn (1989) — the dominant workhorse model in political economy for analyzing how legislatures divide a surplus — and provides a more complete characterization of its stationary equilibria. The core research question is whether the equilibrium typically cited in the literature as the “open rule equilibrium” is actually the unique equilibrium, or whether it rests on implicit and unstated assumptions that, once relaxed, reveal a much richer equilibrium set.

The model features n=3 negotiators dividing a surplus normalized to one, operating under simple majority rule (2 of 3 votes required). The common discount factor is Delta in (0,1). In each period, a proposer is selected uniformly at random; under the open rule, an amender is then selected uniformly at random from the two non-proposers and may either accept or counter-propose. Sincere voting determines the outcome. The authors analyze stationary subgame perfect equilibria (SSPE), in which strategies depend only on current role, not history.

The existing literature implicitly adopted what the authors call the “standard assumption”: when given the opportunity to amend, the amender proposes the same allocation she would propose as a proposer in a closed rule game. Under this assumption, the unique SSPE has the proposer receiving share 1-Delta and each of the other two negotiators receiving Delta/2 (in the Pareto-efficient equilibrium). The literature treated this as the definitive open rule solution.

The paper’s first main result is that this standard-assumption equilibrium is indeed a valid SSPE, but it is not the only one. The key mechanism generating multiplicity is the treatment of off-path behavior: what the amender does when the proposer deviates to a non-equilibrium proposal. With n=3, a deviating proposer can exploit the structure so that the amender becomes a “free” coalition member — the proposer does not need to buy the amender’s vote separately, because the amender is already included in the majority once she counter-proposes. This expands the set of credible threats and supports a continuum of additional Pareto-undominated SSPEs.

The paper’s second main result characterizes the broader equilibrium set: all Pareto-undominated SSPEs belong to a class in which the proposer offers (1-Delta) to herself and equal shares to both other negotiators. In the non-standard equilibria, the amender always amends, generating equilibrium delay — agreements are not reached immediately, and payoffs are discounted by Delta^(t-1) for each period of delay.

The third main result is that among all Pareto-undominated SSPEs, the unique Pareto-efficient one is the standard-assumption equilibrium (no delay). All other equilibria involve delay and are therefore Pareto-inferior in expectation.

The institutional design implication reverses a widely held view: the open rule was thought to promote more egalitarian allocations relative to the closed rule. The authors show this is not the case for Pareto-efficient equilibria. The Pareto-efficient open rule equilibrium is actually a special case of the closed rule equilibrium — the proposer captures 1-Delta and offers Delta to the coalition. More broadly, open rule bargaining tends to generate longer equilibrium delays and less egalitarian surplus allocations than previously predicted by Baron and Ferejohn. Scope conditions: the formal analysis is restricted to n=3 negotiators; generalization to larger legislatures is noted as an open direction.

Q: What is the “standard assumption” and why does the existing literature rely on it?

A: The standard assumption holds that when an amender gets the opportunity to counter-propose, she proposes the same allocation she would choose if she were the proposer in a closed rule game. The existing open rule literature — including Baron and Ferejohn (1989), Jackson and Morelli (2004), Baron (2012), van Weelden (2013), and Austen-Smith and Banks (1999) — accepted this assumption implicitly, treating the resulting equilibrium as the unique open rule equilibrium. The assumption sidesteps the question of off-path behavior: what happens when the proposer deviates to a non-equilibrium proposal that the amender would want to amend. Because deviations are resolved within the same bargaining session under the open rule, off-path specifications are consequential.

Q: What is the unique SSPE under the standard assumption, and what are its payoff implications?

A: Under the standard assumption with n=3 and discount factor Delta, the unique SSPE has the proposer receiving a share of 1-Delta of the surplus and each of the other two negotiators receiving Delta/2. There is no delay: the proposal passes immediately in the period it is made. This equilibrium is Pareto-efficient relative to all other stationary equilibria identified in the paper.

Q: What is the mechanism by which the equilibrium set is larger than the standard assumption predicts?

A: With n=3, when a proposer deviates to a non-equilibrium proposal, the amender — who responds by counter-proposing — automatically becomes part of the passing coalition without the proposer needing to separately compensate her. This makes the amender a “free” coalition member in the deviation subgame, which changes the cost structure of deviations and expands the range of proposals the proposer can credibly make. Consequently, a wider set of strategies by the amender can be sustained as equilibrium responses, yielding a continuum of additional Pareto-undominated SSPEs beyond the standard-assumption equilibrium.

Q: What do the non-standard equilibria look like in terms of proposals, delay, and payoffs?

A: In the non-standard Pareto-undominated SSPEs, the proposer offers (1-Delta) to herself and equal shares (Delta/2 each) to the other two negotiators — note the proposer’s own share is the same as in the standard equilibrium, but the off-path behavior differs — and the amender always chooses to amend rather than accept. The amendment triggers a vote in which the amendment fails (or the process repeats), pushing resolution to the next period. This generates equilibrium delay: agreements take multiple periods to reach, and all payoffs are discounted by Delta^(t-1) per period of delay, making these equilibria Pareto-inferior to the no-delay equilibrium.

Q: Which equilibrium is Pareto-efficient among all Pareto-undominated SSPEs, and why?

A: The unique Pareto-efficient SSPE is the standard-assumption equilibrium, because it is the only one that involves no delay. All other Pareto-undominated SSPEs involve at least one period of delay, which destroys surplus through discounting (payoffs shrink by a factor of Delta per period). Since delay is costly for all negotiators and generates no compensating redistribution, any equilibrium with delay is Pareto-dominated by the no-delay equilibrium.

Q: What are the implications for the classic efficiency comparison between open and closed rules?

A: The closed rule always generates an efficient outcome (no delay in SSPE). The open rule can also generate an efficient outcome — under the standard-assumption equilibrium — but uniquely admits a continuum of inefficient equilibria involving delay. Therefore the open rule is weakly dominated by the closed rule from an efficiency standpoint: at best it matches the closed rule (one efficient equilibrium), and at worst it generates costly delay. This reverses the common inference that open rule unambiguously improves outcomes.

Q: What are the implications for the classic fairness comparison between open and closed rules?

A: The open rule was commonly believed to promote more egalitarian surplus divisions relative to the closed rule, which allows the proposer to extract a large share. The paper shows this view is misleading. In the Pareto-efficient open rule equilibrium, the proposer still captures 1-Delta — the same as under the closed rule — and the result is no more egalitarian. In the delay equilibria, the proposer does offer equal shares to both other negotiators, but this comes at the cost of inefficiency (delay). There is no Pareto-undominated open rule equilibrium that is both efficient and more egalitarian than the closed rule.

Q: What is the class of “Pareto-undominated stationary strategies” and why does the paper focus on it?

A: A stationary strategy profile is Pareto-undominated if no other stationary strategy profile gives every negotiator at least as high an expected payoff with at least one strictly better off. The paper focuses on this class to provide a tractable but principled selection criterion within the large set of SSPEs: it eliminates equilibria that are dominated from every player’s perspective, retaining only those that could plausibly arise if players coordinate on mutually beneficial outcomes. The characterization of this class reveals that equilibrium multiplicity is already substantial even after imposing this selection.

Q: What is the scope of the formal results, and what is left open?

A: The formal analysis is restricted to n=3 negotiators with simple majority rule (2 of 3 votes). The authors acknowledge that generalization to larger n is an important open question. The three-legislator case is the simplest non-trivial instance of the majority-rule bargaining problem, and the authors use it to isolate the mechanism cleanly. The model assumes sincere voting, a common discount factor Delta in (0,1), and stationary strategies.

Q: How does this paper relate to Baron and Ferejohn (1989)?

A: Baron and Ferejohn (1989) originated both the closed rule and open rule bargaining frameworks and derived the standard-assumption equilibrium for the open rule. Subsequent literature (Eraslan 2002, Cho and Duggan 2003, 2009, Banks and Duggan 2000) extended various aspects of the B&F framework. The present paper takes the B&F open rule model as given but demonstrates that B&F’s open rule analysis was incomplete: it did not systematically address off-path behavior, and as a result the equilibrium it identified is not unique. The paper’s main contribution is to show that the B&F open rule predictions — more egalitarian allocations and prompt agreement — do not hold generally across the full equilibrium set.

Open Rule: A bargaining protocol in which, after an initial proposal is made, a nominated amender may make a counter-proposal before a vote is taken; contrasted with the closed rule, under which the initial proposal is voted on without amendment.

Closed Rule: A bargaining protocol in which a vote is taken directly on the first proposal, with no opportunity for amendment.

Standard Assumption: The implicit assumption, used by Baron and Ferejohn (1989) and subsequent literature, that when the amender counter-proposes under the open rule, she proposes the same allocation she would choose as a proposer in a closed rule game; the paper shows this assumption is consequential for equilibrium uniqueness.

Stationary Subgame Perfect Equilibrium (SSPE): An equilibrium concept in which each player’s strategy depends only on her current role (proposer, amender, or voter) and not on the history of play; the paper characterizes SSPEs of the open rule model.

Pareto-Undominated Stationary Strategy Profile: A stationary strategy profile for which no other stationary strategy profile gives every negotiator weakly higher expected payoff with at least one strictly higher; used as a selection criterion to prune the large equilibrium set.

Equilibrium Delay: The phenomenon in which agreement is not reached in the current period because the amender always counter-proposes and the counter-proposal also fails, pushing resolution to a future period and discounting payoffs; all non-standard-assumption Pareto-undominated SSPEs involve delay.

Off-Path Behavior: The specification of what strategies players use following a deviation from equilibrium play; the paper shows that different specifications of off-path behavior by the amender support different equilibria, and that the existing literature was not systematic about this.

Optimal Decision Rules When Payoffs are Partially Identified

Mon, 01 Jan 0001 00:00:00 +0000

This paper derives asymptotically optimal statistical decision rules for discrete choice problems when the payoffs associated with some choices are only partially identified. The research question is: how should a decision maker who can bound but not point-identify a payoff-relevant parameter θ use data to make optimal policy choices?

The framework separates two parameter types. The reduced-form parameter µ is point-identified and can be estimated from data. The structural parameter θ — such as the average treatment effect (ATE) in a target population — is set-identified, meaning only that θ ∈ Θ0(µ) can be established, where the identified set is indexed by µ. The decision maker confronts both ambiguity (arising from partial identification of θ given µ) and statistical uncertainty (µ must be estimated).

The authors propose a hybrid optimality criterion that applies minimax reasoning to the partially-identified parameter θ — choosing actions that minimize maximum risk over Θ0(µ) — while applying average (integrated) risk minimization over µ, reflecting the asymmetric nature of the two identification problems. This asymmetric treatment follows the generalized Bayes-minimax principle of Hurwicz (1951).

The optimal decision rule is implemented by computing, for each action, the maximum risk (or regret) over θ ∈ Θ0(µ) conditional on µ, then averaging this maximum risk across either (i) a bootstrap distribution for an efficient estimator µ̂, (ii) a posterior distribution for µ in parametric models, or (iii) a quasi-posterior based on a limited-information criterion in semiparametric models. The optimal action is whichever choice has the smallest average maximum risk.

A central theoretical result (Theorems 1 and 4) establishes formal asymptotic optimality for both parametric and semiparametric settings: Bayes and quasi-Bayes decisions with any prior whose density is positive, bounded, and continuous are asymptotically equivalent and optimal. Critically, the optimality of these rules is asymptotically independent of the choice of prior for µ. The authors also establish a necessity result (Theorems 2 and 5): any decision rule not asymptotically equivalent to the Bayes or bootstrap rule is strictly sub-optimal.

A key finding is that “plug-in” rules — which substitute an efficient point estimate µ̂ directly into the oracle decision rule — can be sub-optimal. This failure occurs generically under partial identification because the maximum risk function R(d,µ) is typically only directionally differentiable (not fully differentiable) in µ, owing to max and min operators in intersection bounds, linear program value functions, or other bound constructions. When full differentiability holds, Corollary 1 confirms plug-in rules are optimal; otherwise they are not. The empirical illustration demonstrates the practical consequence: for German male youths deciding whether to adopt a job-training program based on 14 RCT studies from Card, Kluve, and Weber (2017), the optimal rule recommends treatment (average quasi-posterior robust welfare contrast b̄n > 0) while the plug-in rule recommends against treatment (plug-in value b(µ̂) < 0). The lower bound maximum of µ̂k − C‖x0 − xk‖ is −0.3190 for the leading US study and −0.3298 for the second-best Brazilian study; because these two values are close relative to the average standard error of 0.034 across studies, the lower bound distribution is right-skewed (behaving like the maximum of two Gaussians), pushing b̄n positive even though b(µ̂) is negative.

The paper extends optimality theory to semiparametric models via a least favorable parametric submodel, introduces the concept of σ-optimality for cases where the average maximum risk criterion is infinite (relevant when the dimension K of µ exceeds 1), and provides detailed implementation guides for treatment assignment under intersection bounds, IV-like estimands, and non-separable panel data, as well as for optimal pricing decisions where revealed-preference demand theory bounds counterfactual demand responses via linear programming.

Scope conditions: optimality results apply to discrete action spaces, require efficient estimation of µ, require the identified set Θ0(µ) to be known as a set-valued mapping, and assume no “first-order ties” (the oracle decision is unique at µ0). The asymptotic framework is local, mimicking the finite-sample problem where µ is not known with certainty.

Q: What is the core decision problem this paper addresses?

A: A decision maker must choose from a finite set of actions D = {0, 1, …, D}. Payoffs depend on a structural parameter θ that is only set-identified — the data can establish θ ∈ Θ0(µ) but not pin down θ exactly. The reduced-form parameter µ is point-identified and estimated from data. The decision maker faces both ambiguity (which θ in Θ0(µ) is true?) and sampling uncertainty (what is µ?). The paper asks how to construct decision rules that are optimal in large samples under this dual uncertainty.

Q: What is the proposed optimality criterion, and why is it asymmetric across parameters?

A: The criterion applies minimax reasoning to the partially-identified θ — the maximum risk over Θ0(µ) given µ is the relevant loss — and integrates this maximum risk over µ using Lebesgue measure on local perturbations h = √n(µ − µ0) of a fixed µ0. The asymmetry reflects the fact that θ is not updated by the data (the prior for θ is not identified), while µ can be learned efficiently from the data. Full minimax over both (θ, µ) is rarely tractable even for simple binary treatment problems; the asymmetric approach yields tractable optimal rules for a broad empirically relevant class of settings.

Q: What are the Bayes, bootstrap, and quasi-Bayes implementations of the optimal rule?

A: In all three cases, the decision maker computes R̄n(d) — the average maximum risk for action d — and chooses the action that minimizes it. The Bayes rule averages R(d, µ) over the posterior πn(µ|Xn) for µ using Bayes’ theorem with a prior π on M. The bootstrap rule averages R(d, µ̂*) over bootstrap redraws µ̂* of the efficient estimator µ̂. The quasi-Bayes rule (for semiparametric models) uses a limited-information quasi-posterior N(µ̂, (nÎ)−1) combining a Gaussian quasi-likelihood with a prior for µ. All three implementations are asymptotically equivalent and optimal under the regularity conditions of Theorems 1 and 4.

Q: What do Theorems 1 and 2 (and their semiparametric analogues Theorems 4 and 5) establish?

A: Theorem 1 establishes sufficiency: Bayes decisions with any prior in the class Π are asymptotically equivalent to each other and are optimal; any rule asymptotically equivalent to such a Bayes decision is also optimal. Theorem 2 establishes necessity: any rule in the admissible class D that is not asymptotically equivalent to the Bayes rule has strictly higher average excess risk at any µ0 where asymptotic equivalence fails. Together, these theorems fully characterize the class of asymptotically optimal rules and show that the Bayes/bootstrap class is not merely sufficient but also necessary for optimality.

Q: When are plug-in rules sub-optimal, and when are they optimal?

A: Plug-in rules substitute an efficient point estimate µ̂ directly into the oracle decision δo(µ̂). If R(d, µ) is fully differentiable at µ0 for all oracle-optimal actions d, then the directional derivative is linear and plug-in and Bayes rules are asymptotically equivalent; Corollary 1 confirms plug-in rules are then optimal. However, under partial identification, max and min operators in bound constructions — intersection bounds, linear program value functions, revealed-preference bounds — generically induce only directional (non-linear) differentiability of R(d, µ). In these cases asymptotic equivalence can fail, and Theorem 2 implies plug-in rules are sub-optimal. Manski (2021, 2023) documents poor finite-sample performance of plug-in rules numerically; the authors’ necessity result provides a general theoretical explanation under the asymptotic average risk criterion.

Q: How does the treatment assignment empirical illustration demonstrate the difference between optimal and plug-in rules?

A: Using data from Ishihara and Kitagawa (2021) with K = 14 RCT studies from Card, Kluve, and Weber (2017) and Lipschitz constant C = 0.25, the decision is whether to adopt a job-training program for German male youths or female youths in 2010 (GDP growth 3.48%, unemployment 9.45%). For male youths, the largest lower bound value µ̂k − C‖x0 − xk‖ is −0.3190 (US study) and the second-largest is −0.3298 (Brazilian study), separated by only 0.0108 against an average standard error of 0.034 across studies, so the lower bound distribution is right-skewed (maximum of two near-tied Gaussians). This right-skew pushes the quasi-posterior mean b̄n positive, yielding a treatment recommendation, while the plug-in value b(µ̂) is negative, yielding a non-treatment recommendation — a concrete reversal of the policy decision. For female youths, the minima and maxima are better separated, the distribution is near-Gaussian, and b̄n ≈ b(µ̂), so both rules agree on treatment.

Q: What are intersection bounds and why do they generate directional differentiability?

A: Intersection bounds arise when the ATE is bounded in K separate observational studies by lower bounds bL,k(µk) and upper bounds bU,k(µk). The combined identified set uses bL(µ) = max_{1≤k≤K} bL,k(µk) and bU(µ) = min_{1≤k≤K} bU,k(µk). Even if each component bound is smooth in µk, the max and min operators make bL and bU only directionally differentiable (not fully differentiable) in µ. The directional derivative is positively homogeneous of degree one but non-linear, which is the property that drives the wedge between Bayes and plug-in rules.

Q: How does the paper extend to semiparametric models, and what technical tool does it use?

A: In semiparametric models, the data distribution depends on both µ ∈ R^K and an infinite-dimensional nuisance parameter η. Integrating over local perturbations of η as well as µ raises measure-theoretic problems in infinite-dimensional spaces. The authors instead restrict attention to local perturbations of µ0 within a least favorable parametric submodel, which is the direction that makes the problem hardest. The quasi-posterior N(µ̂, (nÎ)−1) is then used as the averaging distribution, combining a Gaussian quasi-likelihood with a prior for µ. Theorem 4 establishes optimality and Theorem 5 establishes necessity under these semiparametric conditions, mirroring the parametric Theorems 1 and 2.

Q: What is σ-optimality and why is it needed?

A: When the dimension K of µ exceeds 1, the integrated average excess risk criterion R({δn}; µ0) — which integrates over Lebesgue measure on R^K — may be infinite for all decision sequences in D, making the criterion uninformative. σ-optimality approximates the improper Lebesgue prior on h by a sequence of proper priors indexed by σ, and requires that the decision rule minimize the resulting criterion for all σ. Theorem 3 shows that the limiting behavior of σ-optimal rules coincides with that of the Bayes rule δ*n(·; π), preserving the practical implementation.

Q: How is the optimal pricing application structured and what role do revealed-preference bounds play?

A: A monopolist observes repeated cross-sections of individual demands across B budget sets and must choose a price vector from D = O ∪ C, where O contains observed prices and C contains counterfactual prices. For observed prices, average demand is identified; for counterfactual prices, only bounds are available. Following Kitamura and Stoye (2019), the space of goods is partitioned into GARP-compatible regions, and sharp bounds on counterfactual demand are computed by solving linear programs over the mass allocated to each region subject to GARP consistency constraints. The reduced-form parameter µ collects empirical choice probabilities across observed budget-region cells, estimated consistently by sample frequencies. The optimal pricing decision averages the linear-program bound solutions across quasi-posterior draws of µ.

Q: How does this approach relate to minimax and conditional Γ-minimax approaches?

A: Full minimax over (θ, µ) requires strong distributional assumptions and tractable finite-sample distributions; the authors note that no minimax treatment rule exists even for binary treatment with binary outcomes and estimated bounds. Conditional Γ-minimax (DasGupta and Studden, 1989; Giacomini, Kitagawa, and Read, 2021) fixes a prior for µ and takes minimax over the set of priors for θ conditional on µ; this is closely related to the authors’ approach but can be conservative when the marginal prior for µ varies. The authors’ framework fixes the marginal prior for µ and takes minimax over θ ∈ Θ0(µ) conditional on µ, which is shown to arise as the equilibrium of a two-player zero-sum game where adversarial nature chooses a prior for θ ∈ Θ0(µ) conditional on µ and the available data for µ.

Q: What is the technical contribution regarding directionally differentiable functions?

A: Hirano and Porter (2009) derived asymptotic optimality for treatment rules under fully differentiable welfare contrasts. This paper extends that theory to settings with directional (but not full) differentiability — a generic feature whenever bounds involve max/min operators or linear program values. The key technical building block is the asymptotic distribution of the quasi-posterior mean of directionally differentiable functions (Propositions 2 and 3 in Appendix C). While Kitagawa, Montiel Olea, Payne, and Velez (2020) characterized the asymptotic behavior of the posterior distribution of such functions, this paper instead characterizes the frequentist distribution of the posterior mean — a distinct and novel contribution to the literature on asymptotics for non-smooth functions (Dümbgen, 1993; Fang and Santos, 2019).

Q: What are the key scope conditions and limitations of the optimality results?

A: The action space D must be finite and discrete (continuous pricing must be approximated by a grid of whole-currency units, as noted in the introduction). The identified set mapping Θ0(·) must be known. Efficient estimation of µ is required, along with a consistent estimator of its asymptotic variance for quasi-Bayes implementation. The optimality criterion assumes “no first-order ties” — the oracle decision must be unique at µ0. The framework is asymptotic (local perturbations around a fixed µ0), and the theory is designed for settings where deriving exact finite-sample optimal rules is intractable. The results do not cover the case where θ affects the data distribution (only payoffs are partially identified, not identification of µ itself).

Partially-identified parameter (θ): A structural parameter — such as the ATE in a target population — about which the data can establish only set membership θ ∈ Θ0(µ), not a point value. The identified set Θ0(µ) is indexed by the point-identified reduced-form parameter µ.

Oracle decision (δo(µ)): The infeasible first-best decision that minimizes maximum risk over the identified set Θ0(µ) for a known value of µ. It serves as the benchmark against which practical rules are evaluated; any data-dependent rule can only do weakly worse.

Maximum risk (R(d, µ)): The supremum of risk r(d, θ, µ) = Eθ[l(d, Y, θ, µ)] over all θ ∈ Θ0(µ) conditional on µ. Under the regret criterion for binary treatment, R(0, µ) = (bU(µ))+ and R(1, µ) = −(bL(µ))−.

Robust welfare contrast (b(µ)): In the treatment assignment application, b(µ) = (bU(µ))+ + (bL(µ))−, whose sign determines the oracle decision: treat if b(µ) ≥ 0. The optimal rule replaces b(µ) with its quasi-posterior mean b̄n.

Directional differentiability: A function f : M → R^k is directionally differentiable at µ0 if limits of (f(µ0 + tn hn) − f(µ0))/tn exist for all sequences tn ↓ 0 and hn → h, yielding a directional derivative ḟµ0[·] that is positively homogeneous but not necessarily linear. Max/min operators and linear program value functions are generically only directionally differentiable, not fully differentiable. This property is what causes plug-in rules to fail.

Quasi-posterior: In semiparametric models, a posterior-like distribution for µ formed by combining a limited-information Gaussian quasi-likelihood N(µ̂, (nÎ)−1) with a prior π, yielding πn(µ|Xn) ∝ exp(−½(µ − µ̂)T(nÎ)(µ − µ̂))π(µ). Used in place of a full Bayesian posterior when the exact likelihood of the data-generating process is unavailable.

σ-optimality: An optimality concept that replaces the improper Lebesgue prior on local perturbations h ∈ R^K with a sequence of proper priors indexed by σ, used when the average excess risk criterion is infinite for K > 1. Theorem 3 establishes that the σ-optimal decision rule converges to the Bayes rule as σ → ∞.

Plug-in rule (δplug_n): A decision rule formed by substituting an efficient point estimate µ̂ directly into the oracle decision: δplug_n = δo(µ̂). Optimal when R(d, µ) is fully differentiable (Corollary 1), but generically sub-optimal under partial identification because directional differentiability of R(d, µ) breaks the asymptotic equivalence between the plug-in and Bayes rules.

Optimal Public Transportation Networks: Evidence from the World's Largest Bus Rapid Transit System in Jakarta

Mon, 01 Jan 0001 00:00:00 +0000

This paper studies how commuter preferences over wait times, travel times, and transfers should shape the design of urban bus networks, using the world’s largest Bus Rapid Transit (BRT) system — TransJakarta in Jakarta, Indonesia — as the empirical laboratory. The setting provides unusually rich identification: between January 2016 and February 2020, TransJakarta launched 93 new BRT and non-BRT feeder routes in a staggered, city-wide expansion, during which the operating bus fleet more than doubled from roughly 700 to over 1,600 vehicles. The authors combine over 500 million smart-card tap records, GPS tracking of every bus at 5–10 second intervals, and anonymized smartphone location data covering 35 million weekday trips from 2.3 million devices.

The paper proceeds in three steps. First, the authors classify new route launches into three event types and estimate their causal impact on ridership via difference-in-differences. Event 1: a new direct connection between an origin-destination pair already served by transfer only, with no travel-time improvement — raises BRT ridership by 0.16 log points. Event 2: a new direct connection that also reduces travel time (by 0.29 log points on average) — raises ridership by 0.27 log points. Event 3: additional buses on an already-directly-connected pair, which increases the bus arrival rate by 0.32 log points and reduces wait times — raises ridership by 0.09 log points, implying a ridership elasticity with respect to wait times of approximately −0.29 for BRT. For non-BRT routes the implied wait-time elasticity is −1.05, raising the possibility of multiple equilibria in service levels. Crucially, none of the three event types produce detectable increases in aggregate trip volumes measured by smartphone data, implying the ridership gains reflect modal substitution toward the bus rather than trip generation.

Second, the authors estimate a structural demand model. At its core is a route-choice model in which bus arrivals follow independent Poisson processes, so wait times are exponentially distributed and idiosyncratic. This formulation avoids the red-bus/blue-bus aggregation problem endemic to logit models. Commuters are also allowed to be partially inattentive to routes whose travel time exceeds the fastest available option by more than an estimated threshold. Structural parameters are recovered by classical minimum distance, matching seven reduced-form moments. Key findings: wait time is valued 2.4 times more than time on the bus for BRT routes, and 4.2 times more for non-BRT routes. There is no additional transfer penalty beyond the wait time and travel time costs of the second leg. Commuters pay significantly less attention to options with travel time more than roughly 34–44 percent above the fastest option in their choice set.

Third, the authors use the estimated preference parameters to characterize optimal bus networks. Because the optimization problem is high-dimensional (418 grid cells, 1,536 possible edges, yielding on the order of 10^500 configurations) and exhibits neither global convexity nor simple complementarity, they reformulate the social planner’s problem as a discrete choice over networks with additive logit shocks — effectively sampling from a multinomial logit distribution via simulated annealing. The result: optimal networks cover approximately 66 percent of grid cells versus 42 percent under the actual TransJakarta network, and would give 91 percent of Jakarta residents bus access versus 73 percent currently. Bus frequency in the city center is somewhat lower in the optimal network. Despite commuters’ high sensitivity to wait times, the current network concentrates too many buses in the city center where wait times are already short, rather than extending reach to underserved areas. Comparative statics show that doubling the wait-time cost parameter produces much more concentrated optimal networks (23 percent of origin-destination pairs connected, 41 percent fewer than baseline), while increasing the transfer penalty by the equivalent of 15 minutes of wait time raises the direct-connection share of served pairs from 12 to 16 percent.

Q: What are the three event types and why are they analytically distinct?

A: Event 1 is the launch of the first direct route between an origin-destination pair already connected by transfer, where the direct route is not faster than the existing transfer option; it isolates the effect of directness absent a travel-time change. Event 2 is the same but with a faster direct route (average reduction of 0.29 log points in travel time), combining directness and speed improvements. Event 3 is the launch of a new route that overlaps an existing direct route, increasing bus frequency and cutting wait times (arrival rate up 0.32 log points) without substantially changing travel time or directness. The three events together provide variation across the key dimensions — directness, speed, and frequency — needed to separately identify commuter preference parameters.

Q: What are the main ridership effects and how large are they in levels?

A: For BRT routes, Event 1 raises ridership by 0.16 log points (approximately 19 additional riders per week for a treated origin-destination pair with a baseline of 111 weekly riders), Event 2 by 0.27 log points (approximately 24 additional riders per week), and Event 3 by 0.09 log points (approximately 20 additional riders per week). For non-BRT routes, proportional effects are larger but level effects are similar: Event 1 yields roughly 34 additional weekly riders, Event 2 roughly 21, and Event 3 roughly 15. Event-study graphs show clear, discrete jumps in ridership at route launch with no pre-trends, and some gradual adjustment in the months following.

Q: What does the paper find about aggregate trip generation versus modal substitution?

A: Using smartphone location data to measure all trips regardless of mode, the authors find no statistically significant increase in aggregate trip volumes for any of the three event types. For BRT Event 1, the estimated aggregate-trip coefficient is −0.008 with a standard error of 0.051, allowing rejection at the 95 percent level of any positive impact above roughly 0.091 log points — small relative to the precise 0.11 log-point bus ridership effect in the same sample. The authors interpret this as evidence that the ridership gains over the 10-month post-event window reflect substitution from private modes (motorcycles, cars, taxis) toward TransJakarta rather than trip generation, and they use this null result to justify holding destination choices fixed in the structural model.

Q: How does the model avoid the red-bus/blue-bus aggregation problem?

A: The paper’s route-choice model assumes bus arrivals follow independent Poisson processes, so wait times are exponentially distributed. A key proposition (Proposition 1) proves that splitting one route into two identical routes with half the buses each produces exactly the same choice probabilities and expected utility as the original single route — because the sum of two independent Poisson processes is itself Poisson with the summed rate. Standard logit models fail this invariance because splitting a route creates two options with independent error draws, artificially inflating expected utility. The invariance property is essential for the optimal network design exercise, where the planner freely reallocates buses across routes.

Q: What are the estimated preference parameters and what do they imply about commuter behavior?

A: The paper estimates that wait time is valued 2.4 times more than time on the bus for BRT routes and 4.2 times more for non-BRT routes. There is no additional transfer disutility beyond the wait time and travel time costs implied by the extra leg. Commuters become substantially inattentive to routes with travel time more than approximately 34 percent above the fastest available option (BRT threshold) or 44 percent (non-BRT). The high relative cost of waiting versus riding reflects both the discomfort of waiting at exposed non-BRT stops and the fact that TransJakarta runs without a published schedule, so commuters cannot minimize wait time by timing arrivals.

Q: What explains the non-BRT wait-time elasticity exceeding −1?

A: For non-BRT routes, Event 3 raises ridership by 0.450 log points while raising the bus arrival rate by 0.425 log points, yielding an implied elasticity of ridership with respect to wait times of −1.05. Because the baseline arrival rate for non-BRT treated pairs is 2–4 times lower than for BRT pairs, the absolute reduction in wait time per additional bus is much larger. An elasticity exceeding −1 in absolute value implies that adding buses on some non-BRT routes could increase ridership enough to maintain or even raise average ridership per bus — the extreme form of the Mohring effect — suggesting the possibility of a high-ridership/low-wait-time equilibrium distinct from the current low-ridership/high-wait-time one.

Q: How is the optimal network characterized and what algorithm is used?

A: The social planner chooses a network to maximize utilitarian welfare (average expected utility across all commuters) from the estimated demand model, plus a network-level logit shock capturing cost and other factors outside the model. This transforms the combinatorially explosive optimization into sampling from a multinomial logit distribution over networks, which the authors approximate using simulated annealing. They run the algorithm multiple times to obtain a sample of networks drawn asymptotically from the planner’s distribution, then estimate optimal network characteristics and comparative statics from sample analogs. The theoretical framework is general and, the authors note, applicable to other high-dimensional spatial planning problems where welfare differences can be computed for pairs of counterfactuals.

Q: How does the optimal network differ from the current TransJakarta network?

A: The typical optimal network covers approximately 66 percent of 2km grid cells versus 42 percent for the actual network, and 91 percent of Jakarta residents would have bus access versus 73 percent currently. The optimal network reduces bus frequency in the city center relative to the current network, accepting longer wait times there in order to extend reach to peripheral areas. The paper finds no tension between distributional and efficiency concerns in this setting — expanding coverage improves both aggregate welfare and access for underserved areas.

Q: What do the comparative statics reveal about the sensitivity of optimal network design to preference parameters?

A: Doubling the wait-time cost parameter leads to substantially more concentrated optimal networks: only 23 percent of origin-destination pairs are connected, 41 percent fewer than in the baseline optimal network. This is because higher wait-time costs make it more valuable to concentrate buses on fewer routes to achieve short headways. Increasing the transfer penalty by the equivalent of 15 minutes of wait time raises the share of connected location pairs with a direct (non-transfer) connection from 12 to 16 percent. These comparative statics link micro-level preference parameters to macro-level network topology, clarifying which parameters most influence design choices.

Q: How does the paper validate the destination imputation from tap-in-only smart card data?

A: For the subset of BRT stations where tap-out is enforced (36 percent of stations), the authors estimate bivariate regressions of imputed daily ridership shares against actual observed ridership shares, obtaining R-squared of 0.85. They also show robustness by varying the grid cell size from 500 meters to 2 kilometers, finding no systematic decline in treatment effect magnitudes, which rules out large displacement effects within the network as an explanation for the results.

Q: Does the response to network improvements vary by local poverty rates?

A: The authors interact all six event types with an indicator for above-median poverty rate at the origin grid cell (from SMERU 2014 data), controlling for population. They find no clear pattern of heterogeneity by income level — richer and poorer areas respond similarly to service improvements. The paper notes this absence of heterogeneity as relevant context for interpreting optimal network design: the case for extending reach is not offset by a differential preference for frequency among poorer commuters.

Mohring Effect: The externality arising from ridership responsiveness to wait times — more riders justify more buses, which reduce wait times for all riders, further increasing ridership. The paper estimates a BRT wait-time elasticity of −0.29, confirming the effect operates in Jakarta; for non-BRT the elasticity of −1.05 suggests the possibility of multiple equilibria in service levels.

Negative Exponential Distribution Model (Daganzo 1979): The route-choice model used in the paper, in which bus arrivals on each route follow independent Poisson processes and wait times are exponentially distributed. The model is invariant to aggregation of identical routes (avoids the red-bus/blue-bus problem) and yields tractable closed-form expressions for choice probabilities and expected utility.

Partial Inattention: The model feature whereby commuters assign near-zero effective arrival rates to bus options whose travel time exceeds the fastest available option by more than an estimated threshold (34–44 percent depending on route type). Captures the empirical finding that commuters in a large, complex network do not appear to consider all available options.

Event Types (1, 2, 3): The paper’s taxonomy of service improvements induced by new route launches. Event 1 isolates the value of directness (new direct route, no speed gain). Event 2 combines directness and speed (new direct route that is also faster). Event 3 isolates the value of frequency (additional buses on an already-direct route, reducing wait time without changing travel time).

Optimal Network Characterization via Social Planner’s Logit: The paper’s approach to the combinatorially intractable network optimization problem. The planner is modeled as making a logit discrete choice over all possible networks, with welfare from the demand model plus a network-level idiosyncratic shock. Sampling via simulated annealing yields estimates of optimal network characteristics and comparative statics without requiring identification of a single globally optimal network.

Network Concentration vs. Extensiveness Tradeoff: The core design tension the paper formalizes — for a fixed bus fleet, concentrating buses on fewer routes reduces wait times on served routes but leaves more areas without coverage, while spreading buses across more routes extends reach at the cost of longer headways. The estimated preference parameters (high wait-time sensitivity) make this tradeoff non-trivial; nonetheless, the paper finds the current network is too concentrated relative to the optimum.

Optimal Resilience in Multitier Supply Chains

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

Grossman, Helpman, and Sabal ask what market failures arise in vertical supply chains with multiple production tiers, limited (non-anonymous) supply networks, arms-length transactions, and recurrent risks of disruption at every node. They then ask what government policies would be required to implement the socially efficient (first-best) allocation as a decentralized equilibrium, and — in a second-best environment where subsidies to firm-to-firm transactions are politically infeasible — how optimal policies to promote resilience and network formation differ.

Model and Methodology

The paper develops a general-equilibrium model of a closed economy with an arbitrary number S+1 of vertical production tiers (tier 0 through tier S). A finite measure of “lead” firms in tier S produce differentiated consumer goods under monopolistic competition using labor and a CES bundle of intermediate inputs from tier S-1 suppliers. Firms in each intermediate tier combine labor and inputs from the tier above using a Cobb-Douglas production function. Tier 0 firms produce from labor alone.

Every firm faces an independent, non-zero probability of a catastrophic disruption (complete inability to produce). Firms may invest labor up front to moderate this risk — endogenous “resilience” — or may invest to forge relationships with a larger fraction of potential suppliers in the next upstream tier — endogenous “network thickness.” Each formed relationship costs k units of labor.

After disruption shocks are realized, surviving firms negotiate quantities and payments bilaterally. Bargaining is sequential (beginning with lead firms negotiating with tier S-1, then tier S-1 with tier S-2, and so on to tier 0), and within each round is governed by Nash-in-Nash equilibrium (Horn and Wolinsky, 1988): each firm takes as given the outcomes of its negotiations with all other partners. The Nash surplus is split with exogenous bargaining weight β_s for the downstream buyer in the s-to-s−1 negotiation.

The paper solves the planner’s direct-control problem and then characterizes the three sets of policy instruments needed to decentralize the first best: subsidies to input transactions between adjacent tiers, subsidies to investments in resilience (agility), and subsidies to network formation (redundancy). It then solves the second-best problem in which transaction subsidies are constrained to zero.

Main Findings

Transaction subsidies. In the competitive bargaining equilibrium, each pair of firms undervalues input transactions because the upstream firm anticipates paying a marked-up price when it bargains with its own suppliers. This cascading distortion means the private marginal cost of producing a tier-s good exceeds the social marginal cost. The optimal first-best transaction subsidy on sales by tier s firms (τ*s) equals [γ_s + (1−γ_s)μ{s−1}]^{−1}, where γ_s is the labor share in tier s production and μ_{s−1} is the endogenous markup factor from bargaining at the s-to-s−1 interface. This subsidy depends only on production function parameters and bargaining weights at the immediately adjacent tier. No subsidy is needed at tier 0 (the most upstream tier), and no subsidy is applied to final-good sales. Under Assumption 1 — inputs become weakly less substitutable as goods proceed downstream — the optimal purchase subsidies rise monotonically as one moves downstream along the supply chain.

Resilience subsidies (first best). Two offsetting forces govern the optimal subsidy to resilience investments θ*s at intermediate tiers: (i) firms capture only the fraction (1−β{s+1}) of the joint surplus that their resilience creates for downstream customers, creating underinvestment; (ii) optimal transaction subsidies inflate private profitability, creating a countervailing overinvestment incentive. The net optimal first-best subsidy for intermediate-tier firms is θ*s = (1−β{s+1}) / τ*_s. This formula depends only on technological and bargaining parameters of tier s and the tier immediately adjacent; it does not depend on conditions elsewhere in the chain. When production parameters and bargaining weights are uniform across tiers, the first-best resilience subsidy is the same at every interior tier. If goods become strictly less substitutable downstream, the first-best subsidy for resilience declines monotonically as one moves downstream, and may turn into an optimal tax for middle tiers where the transaction subsidy is large enough to over-incentivize resilience investment. The first-best resilience subsidy always applies at both extreme ends of the chain.

Network formation subsidies (first best). Despite firms’ private incentive to manipulate their number of upstream suppliers to improve bargaining position, the net strategic effect of network formation in general equilibrium exactly cancels the off-equilibrium spillovers to non-partners. As a result, the optimal first-best policy toward network formation at every tier is identical to the optimal policy toward resilience investment.

Second-best policies. When transaction subsidies are unavailable, uncorrected markups downstream from tier s depress demand for tier-s output, reducing profitability and incentives to invest in resilience below the first-best level. Second-best optimal subsidies for resilience and network formation therefore reflect production function parameters and bargaining weights throughout the entire downstream supply chain, not just at the immediately adjacent tier. Specifically, when buyer bargaining weights are non-increasing along the chain (β_{s+1} ≤ β_s for all s), the second-best subsidy to resilience falls monotonically as one moves downstream. This is the opposite pattern from what might be inferred from the first-best analysis when transaction subsidies are available: with non-increasing bargaining weights, second-best subsidies are larger for upstream producers than for downstream producers.

Scope Conditions

Results are derived for a closed economy. Welfare is measured by the CES utility of the representative consumer over differentiated final goods. The sequential bargaining structure assumes contracts are written after disruption shocks are realized. Assumption 1 (σ_1 ≥ σ_2 ≥ … ≥ σ_S > ε, where σ_s is the elasticity of substitution between inputs at tier s and ε is the demand elasticity for final goods) is maintained for sharper monotonicity results on the structure of optimal subsidies across tiers.

In depth

Q1. What is the precise structure of the supply chain in the model, and why does the bargaining take place sequentially rather than simultaneously across all tiers?

A: The economy has S+1 tiers. Tier 0 firms use only labor; tier s firms (s = 1,…,S−1) use labor and a CES bundle of tier s−1 inputs with elasticity of substitution σ_s > 1; tier S firms produce final differentiated goods using labor and tier S−1 inputs under Cobb-Douglas technology. Sequential bargaining is imposed because the vast number of simultaneous negotiations across all tiers makes a grand coalition impractical. The timing is that lead firms (tier S) first negotiate input quantities and payments with their tier S−1 suppliers; those suppliers, now contractually obligated to their downstream customers, then negotiate with tier S−2, and so on up the chain until tier 1 firms contract with tier 0 suppliers.

Q2. How is the markup factor defined, and what parameters determine it?

A: The markup factor μ_s is the ratio of the payment per unit made by tier s+1 firms to the production cost of tier s firms. It equals μ_s = (1−β_{s+1}) · [σ_{s+1}/(σ_{s+1}−1)] + β_{s+1}, where β_{s+1} is the exogenous bargaining weight of the downstream (tier s+1) buyer. When the downstream firm has all bargaining power (β_{s+1} = 1), the markup equals unity (competitive outcome). When the upstream firm has all bargaining power (β_{s+1} = 0), the markup equals the standard monopoly markup σ_{s+1}/(σ_{s+1}−1). For intermediate bargaining weights, the markup is a weighted average. The markup enters the optimal transaction subsidy formula by inflating the private marginal cost of producing tier-s inputs above the social marginal cost.

Q3. Why are no subsidies needed for the most upstream (tier 0) transactions or for final-good sales?

A: For tier 0 transactions: when tier 0 and tier 1 firms bargain, the negotiations occur last sequentially and so do not affect any prior agreements. There are no downstream cascading markup effects — tier 0 firms produce from labor alone, so their private marginal cost equals their social marginal cost. The joint surplus maximization by the pair thus aligns with the planner’s objective, yielding τ*_0 = 1 (no intervention needed). For final-good sales: final producers do mark up above marginal cost under monopolistic competition, but all varieties are symmetric, so the markup affects all goods equally and does not distort relative consumption choices. Hence τ*_S = 1.

Q4. What are the two offsetting forces that determine the optimal first-best subsidy to resilience investments at an intermediate tier?

A: First, a firm in tier s captures only the fraction (1−β_{s+1}) of the joint surplus that its survival creates for its downstream customers (the rest is appropriated through bargaining by those customers), leading to underinvestment relative to the social optimum. Second, the optimal transaction subsidy τ*_s < 1 raises the private profitability of firms in tier s above its social value, because public finances bear part of the cost of their input purchases. This inflated private profitability encourages resilience investment beyond what the planner desires. The net optimal policy is θ*s = (1−β{s+1}) / τ*_s, which may be a subsidy (θ*_s < 1) or a tax (θ*_s > 1) depending on which force dominates.

Q5. Why does the first-best subsidy for resilience at an intermediate tier depend only on local parameters (at tier s and its immediate neighbors), even though resilience investments generate spillovers to firms throughout the network?

A: When optimal transaction subsidies are in place at all tiers, a firm’s value becomes independent of the joint surplus in sales that occur between firms in tiers other than its own. That is, the positive spillovers to all firms farther upstream and downstream in a firm’s own network are exactly offset by the negative spillovers to firms in rival networks (including rival firms in the same tier). What remains after this general-equilibrium cancellation is only the benefit to the firm’s immediate downstream customers and the wedge created by the transaction subsidy. This result implies that the formula θ*s = (1−β{s+1}) / τ*_s does not involve conditions at tiers other than s and s−1.

Q6. Why does the optimal policy for network formation (supplier link investment) equal the optimal policy for resilience investment, despite the fact that network formation also strategically improves a firm’s bargaining position?

A: Firms in intermediate tiers do have a private incentive to form additional supplier links specifically to improve their bargaining position vis-à-vis their upstream suppliers (by improving their outside options) and vis-à-vis their downstream customers (by the same mechanism). However, the authors show by comparing the firm’s first-order condition for link formation with the planner’s first-order condition that this strategic motivation exactly balances the offsetting general-equilibrium effects from rival firms doing the same. After this cancellation, the residual wedge between private and social incentives for network formation is identical to that for resilience investment. Hence #_s = θ_s for all tiers.

Q7. How do second-best policies differ from first-best policies in terms of both the magnitude of subsidies and the information required to set them?

A: In the first best, the subsidy for resilience at tier s depends only on the bargaining weight β_{s+1} and the markup factor μ_{s−1} — parameters relevant to tier s and its immediate neighbors. In the second best, when transaction subsidies are unavailable, the optimal resilience subsidy at tier s is θ†s = J^{−1} · [1 − (cumulative distortion of all downstream tiers)] · (1−β{s+1}), where J captures aggregate labor-market effects of all markups throughout the chain. This formula requires knowledge of production function parameters (labor shares γ_j, markups μ_j, elasticities σ_j) for every tier j downstream from s. The second-best subsidy may be larger or smaller than the first-best subsidy; it is more likely to exceed the first-best subsidy for upstream tiers, where the cumulative downstream distortions (uncorrected markups contracting demand) produce a larger shortfall in private profitability and hence a larger underinvestment in resilience.

Q8. Under what condition do second-best subsidies fall monotonically as one moves downstream, and how does this compare to the first-best pattern?

A: The ratio of second-best subsidies at adjacent tiers (θ†_{s−1} / θ†s) equals [(1−β_s) / (1−β{s+1})] · [τ*s]^{−1}, where τ*s is the first-best transaction subsidy. If buyer bargaining weights are non-increasing along the chain — β{s+1} ≤ β_s for all s — then (1−β_s) ≤ (1−β{s+1}) and, combined with τ*s ≤ 1, the second-best subsidy is larger upstream than downstream (θ†{s−1} ≥ θ†_s). This contrasts with the first-best policy: when parameters are uniform across tiers, first-best resilience subsidies are the same at every interior tier, while second-best subsidies are strictly larger upstream than downstream.

Q9. What role does Assumption 1 (elasticities of substitution non-increasing as goods move downstream) play in the results?

A: Assumption 1 (σ_1 ≥ σ_2 ≥ … ≥ σ_S > ε) ensures that the operating profit function ~v_s(η) is concave in a firm’s network size η, which in turn ensures interior solutions to the network formation problem. It also delivers sharper monotonicity results: under this assumption, if other production parameters and bargaining weights are similar across tiers, the optimal purchase subsidies rise monotonically downstream, and the optimal first-best resilience subsidies decline monotonically downstream (potentially turning into taxes at some interior tiers). The assumption reflects the realistic view that inputs become more differentiated and specialized as they approach the final consumer good.

Q10. What are the limitations the authors identify regarding their model, and what extensions do they suggest?

A: Three main limitations are identified. First, the model assumes bargaining occurs after disruption shocks are realized, ruling out contingent contracts. Pre-disruption bargaining with contingent payments could mitigate double-marginalization inefficiencies and help internalize resilience externalities, though complex network-wide contingent contracts would likely be needed for full efficiency even in the second-best environment. Second, the model assumes symmetric firms within each tier, so downstream firms cannot sort on upstream firms’ observable resilience levels; if observable differences existed, downstream firms could seek out more reliable partners, partially internalizing the resilience externality. Third, the model covers only a closed economy with idiosyncratic (uncorrelated) shocks. Extensions to global supply chains, correlated (geographic) shocks, cross-country differences in wages and technologies, and optimal cooperative versus unilateral policy are identified as important directions for future research.

Key Concepts

Resilience (agility): In the paper’s usage, a firm’s endogenous investment in reducing the probability of a catastrophic disruption to its own operations. A firm in tier s hires r_s units of labor up front, which raises its survival probability φ_s(r_s), with φ’_s > 0 and φ’’_s < 0. Resilience is a relationship-specific investment in the sense that its payoff is realized only conditional on the firm surviving and then trading with its downstream customers.

Network thickness (redundancy): The fraction η_s of firms in the next upstream tier with whom a firm in tier s forms a supply relationship prior to the disruption shock. Forming k units of labor per link creates a thicker network that hedges against supplier disruption, increases input variety (and thus CES productivity), and improves bargaining positions vis-à-vis both upstream suppliers and downstream customers. Distinct from resilience: resilience reduces the firm’s own probability of disruption; network thickness provides substitutability across suppliers should some fail.

Markup factor (μ_s): The ratio of the per-unit payment made by tier s+1 firms to the production cost of tier s firms, as determined by Nash bargaining. Specifically, μ_s = (1−β_{s+1}) · [σ_{s+1}/(σ_{s+1}−1)] + β_{s+1}. The markup distorts private marginal costs above social marginal costs, causing underinvestment in transactions between firms and, transitively, in resilience and network formation.

Nash-in-Nash equilibrium: The bargaining solution concept used in the paper (following Horn and Wolinsky, 1988). Each pair of firms negotiates as if all other bilateral negotiations involving either party proceed at their equilibrium outcomes, both on and off the equilibrium path. This is the appropriate equilibrium concept when grand coalitions across all firms and all tiers are impractical.

Sequential bargaining: The specific timing structure in which negotiations proceed from the most downstream tier (lead firms bargaining with tier S−1 suppliers) sequentially upstream until tier 1 firms bargain with tier 0 suppliers. Each tier of firms, at the time they bargain with their own suppliers, are already contractually obligated to deliver specified quantities to their downstream customers. This obligation anchors the downstream firm’s outside option in any given bilateral negotiation.

First-best transaction subsidy (τ_s):* The fraction of the cost of a tier-s input that, under the optimal policy, the downstream (tier s+1) buyer must pay. Equals [γ_s + (1−γ_s) · μ_{s−1}]^{−1} < 1 for all intermediate tiers, i.e., it is always a subsidy. Designed to align private marginal cost in the bilateral negotiation with the social marginal cost by offsetting the distortion introduced by anticipated markups on the upstream firm’s own inputs.

Second-best subsidy: The optimal policy toward resilience and network formation when subsidizing firm-to-firm transactions is infeasible (constrained to τ_s = 1 for all s). Unlike first-best subsidies — which depend only on local tier parameters — second-best subsidies depend on production function parameters and bargaining weights throughout the entire downstream supply chain due to the uncorrected cumulative markup distortions.

Optimal Tests Following Sequential Experiments

Mon, 01 Jan 0001 00:00:00 +0000

This paper addresses a practical gap in the inference literature for sequential and adaptive experiments: while the design of such experiments has been studied extensively, there is little theory characterizing which tests are optimal once the experiment concludes. Adusumilli asks what the best hypothesis test looks like after a sequential experiment — a costly sampling design, a group sequential trial, or a bandit experiment — and whether the complexity of the adaptive protocol can be reduced to a manageable set of sufficient statistics for inference purposes.

The methodological core is the derivation of two Asymptotic Representation Theorems (ARTs). The first ART applies to stopping-time experiments, where the sampling rule is fixed in advance but the stopping time is fully adaptive (updated after every observation). The second ART allows the sampling rule itself to be adaptive, but requires that both the sampling and stopping decisions are updated only a finite number of times after observing batches of data. Both ARTs establish that the asymptotic power function of any test in the original sequential experiment can be matched by a test in a limit experiment in which a Gaussian process is observed for each treatment and inference is made on the drifts of those processes.

The key sufficiency result is a dimension reduction: regardless of the number of batches or the complexity of the adaptive protocol, any candidate test’s asymptotic power can be reproduced by a test that depends only on a fixed, finite set of statistics. For stopping-time experiments, the sufficient statistics are the stopped value of the score process (parametric) or the efficient influence function process (non-parametric), together with the stopping time. For batched experiments with adaptive sampling, the sufficient statistics are the final allocation proportions for each treatment (q_1, q_0) and the final values of the influence function processes (x_1, x_0) — a fixed dimension of 2d+2 regardless of the number of batches. This stands in contrast to the earlier ART of Hirano and Porter (2023), whose state variables grow linearly with the number of batches.

The paper then characterizes optimal tests within the limit experiment under several criteria. Under no restriction, the Neyman-Pearson lemma yields the uniformly most powerful (UMP) test for a point alternative. For testing linear combinations of the parameter vector, a further dimension reduction applies and a UMP test exists in the limit experiment, depending only on a scalar projection of the sufficient statistic. Under unbiasedness, any valid test must satisfy an orthogonality condition on the stopped process. Under an alpha-spending constraint — where the overall size alpha is pre-allocated across stages — optimal stage-specific thresholds are derived. Under a weighted average power criterion, the optimal test takes the form of a likelihood ratio statistic integrated against the weight function.

Three application classes are treated with explicit optimal procedures. For horizontal boundary designs (stopping when a test statistic crosses a fixed threshold, including the SPRT and the Neyman-allocation design from Adusumilli 2022), the most powerful asymptotically unbiased test rejects when the stopping time falls below a specific quantile of its null distribution. Monte Carlo simulations show the test achieves nominal 5% size even for small n, while the standard two-sample test has actual size near 9% in the same setting. For group sequential trials (including O’Brien-Fleming designs with T=2 stages), the paper derives stage-specific critical values satisfying the alpha-spending constraint, with numerical simulations confirming the asymptotic approximation is close to nominal for small n, though accuracy degrades for larger values of the null mean. For bandit experiments run with a batched Thompson-sampling algorithm (K=2 treatments, J=10 batches), the paper constructs the power envelope and shows it is asymmetric: distinguishing (a, 0) from (0, 0) is easier than distinguishing (-a, 0) from (0, 0) for a > 0, because Thompson sampling directs more observations to the arm with higher estimated mean, reducing informativeness from the other arm. Simulations confirm the asymptotic approximation is accurate for as few as n=20 observations per batch (200 total).

The framework covers both parametric and non-parametric models. The non-parametric setting replaces the score process with the efficient influence function process, and the asymptotic power bound translates directly. Results also apply to conditional power given the stopping time.

Q: What is the core methodological contribution of the paper? A: The paper derives two Asymptotic Representation Theorems (ARTs) showing that the asymptotic power function of any test following a sequential experiment can be matched by a test in a Gaussian-diffusion limit experiment. The first ART covers stopping-time experiments with fully adaptive stopping rules; the second covers batched experiments with adaptive sampling rules. These ARTs reduce the infinite-dimensional adaptive experiment to a tractable limit object.

Q: What are the sufficient statistics for inference, and why does this matter? A: For stopping-time experiments, the sufficient statistics are the stopped value of the score (parametric) or efficient influence function (non-parametric) process, together with the stopping time. For batched experiments with adaptive sampling over K treatments, the sufficient statistics are the final allocation fractions (q_1, q_0) and the final influence function process values (x_1, x_0), a fixed dimension of 2d+2. This matters because it establishes that all the adaptive complexity of the protocol can be discarded: a test that uses only these statistics is asymptotically as powerful as any test that uses the full sample path.

Q: How does this paper extend or differ from Hirano and Porter (2023)? A: Hirano and Porter (2023) derive an ART for batched sequential experiments whose state variables grow linearly with the number of batches, making the limit experiment increasingly complex. Adusumilli shows that only a fixed number of sufficient statistics (2d+2) are needed to match unconditional asymptotic power, irrespective of the number of batches. The paper also extends to non-parametric models, derives optimal conditional tests given stopping times, and covers fully adaptive stopping-time experiments via a different route (Le Cam 1979) that does not require the batching restriction.

Q: What is the result for testing linear combinations of the parameter? A: When the null hypothesis is H0: a^T h = 0 in the limit experiment, a further dimension reduction applies: the UMP test depends only on a scalar projection x-tilde(tau) = sigma^{-1} a^T I^{-1/2} x(tau) and the stopping time tau. Because under the null this projection is a standard Brownian motion evaluated at the stopping time, the test is pivotal and uniformly most powerful for the composite hypothesis, regardless of the nuisance components of h.

Q: What is the unbiasedness condition in the limit experiment? A: A test phi is unbiased if its power exceeds its size under all alternatives. In the Gaussian limit experiment, Proposition 2 shows that any unbiased test of H0: h=0 vs H1: h≠0 must satisfy the moment condition E_0[x(tau) phi(tau, x(tau))] = 0, which is obtained by differentiating the power function at h=0 and applying the unbiasedness constraint. This condition restricts which tests can be considered, and the optimal unbiased test is characterized within this class.

Q: What is the alpha-spending criterion and what does the paper show about it? A: Alpha-spending (introduced by Gordon Lan and DeMets, 1983) pre-allocates the total size alpha across T stages via a spending vector (alpha_1, …, alpha_T) with sum equal to alpha, and requires that the conditional rejection probability at stage t not exceed alpha_t. Theorem 2 shows that for discrete stopping times, the asymptotic conditional power beta_n(h|t) converges to beta(h|t) in the limit experiment on subsequences, enabling the derivation of optimal stage-specific thresholds satisfying the spending constraint.

Q: What is the key finding for horizontal boundary designs with a fixed sampling rule? A: For experiments that stop when the influence function process first crosses a fixed threshold gamma — including the SPRT and the Neyman-allocation costly-sampling design of Adusumilli (2022) — Lemma 1 establishes that the UMP asymptotically unbiased test of H0: mu_1 = mu_0 is the test that rejects when the stopping time tau-hat falls below the alpha-quantile of its null distribution. Monte Carlo evidence shows this test achieves nominal 5% size even for small n, while a naive two-sample test ignoring the adaptive stopping rule has actual size near 9%.

Q: What does the power envelope look like for Thompson-sampling bandit experiments, and why is it asymmetric? A: For Thompson-sampling bandit experiments with K=2 arms and J=10 batches, the power envelope for testing H0: (mu_1, mu_0) = (0, 0) is asymmetric: it is easier to distinguish the alternative (a, 0) from the null than to distinguish (-a, 0) for the same a > 0. The mechanism is that Thompson sampling allocates more observations to the arm with the higher estimated mean, so a positive treatment effect leads to more data for treatment arm 1 and less for arm 0, making the joint test more informative in one direction than the other.

Q: How accurate are the asymptotic approximations in finite samples? A: For horizontal boundary designs, Monte Carlo simulations show size is close to nominal 5% even for small n. For group sequential trials with an O’Brien-Fleming design (T=2 stages), the approximation is close to nominal for small n but degrades for larger values of the null mean mu-bar. For Thompson-sampling bandit experiments with K=2 arms and J=10 batches, the approximation is accurate for as few as n=20 observations per batch (200 total observations).

Q: How does the paper handle non-parametric models? A: In non-parametric settings, the sufficient statistic is the efficient influence function process x_n(t) = (sigma^{-1}/sqrt(n)) sum_{i=1}^{floor(nt)} psi(Y_i), where psi is the efficient influence function for the functional of interest and sigma^2 = E[psi^2]. Proposition 3 establishes that the asymptotic power of any test is bounded above by the power envelope in the Gaussian limit experiment indexed by this process. The non-parametric and linear-combination parametric cases share the same limit structure.

Q: What are the open questions identified by the author? A: Two main limitations are noted. First, the ART for adaptive sampling rules is established only for batched experiments; whether it extends to fully adaptive (non-batched) sampling rules without loss of power is conjectured but not formally verified. Second, for fully adaptive experiments, the alpha-spending characterization is not yet available, and the author suggests exploring invariance restrictions or conditional inference as alternative optimality criteria.

Asymptotic Representation Theorem (ART): A result showing that the asymptotic power function of any test in the original sequential experiment can be matched by that of a test in a Gaussian-diffusion limit experiment; used to transfer optimality results from the limit to the original problem.

Limit experiment (Gaussian diffusion): The limiting statistical model in which one observes a Gaussian process x(t) = I^{1/2} h t + W(t) for each treatment, with unknown drift vector h; inference on h in this experiment characterizes optimal tests in the original sequential experiment.

Sufficient statistics (for sequential inference): The finite set of statistics that, in the limit experiment, capture all power-relevant information from the adaptive experiment: for stopping-time experiments, the stopped score/influence function process value and the stopping time; for batched adaptive experiments, the final allocation fractions (q_a) and final influence function values (x_a) for each treatment arm.

Alpha-spending constraint: A strengthened size requirement in group sequential trials that pre-allocates the total Type I error alpha across stages via a spending vector (alpha_1, …, alpha_T); requires that conditional rejection probability at each stage t not exceed alpha_t, and sum alpha_t = alpha.

Efficient influence function process: In a non-parametric model, the partial-sum process x_n(t) = (sigma^{-1}/sqrt(n)) sum_{i=1}^{floor(nt)} psi(Y_i), where psi is the efficient influence function for the target functional; this process is the non-parametric analogue of the score process and serves as the sufficient statistic for non-parametric sequential inference.

Stopping-time experiment: A sequential experiment in which the sampling rule (how to allocate observations across treatments) is fixed before the experiment begins but the stopping rule (when to terminate) is fully adaptive and updated after every observation.

Power envelope: The supremum of the asymptotic power function over all tests of a given size; computed in the limit experiment via the Neyman-Pearson lemma and the Girsanov theorem, and serves as an upper bound on the power of any feasible test in the original sequential experiment.

Organizational Change and Reference-Dependent Preferences

Mon, 01 Jan 0001 00:00:00 +0000

Schmidt and von Wangenheim develop a dynamic model of organizational change in which workers have reference-dependent preferences — specifically loss aversion and social comparisons — to explain several empirically observed patterns that standard models cannot easily account for: organizational inertia in normal times, sudden productivity jumps during crises, persistent total factor productivity (TFP) differences across firms in the same industry, and effort and wage compression within firms.

The motivating empirical puzzle is the early-1980s collapse of the Great Lakes iron ore and steel industry, which had been geographically shielded from foreign competition for over 100 years. When Brazilian competitors undercut prices, the industry responded by roughly doubling labor productivity within a few years — not through new technology or capital investment, but through organizational improvements and more efficient use of existing capital (Schmitz 2007). The broader puzzle is Syverson’s (2004) finding that at the four-digit industry level, the 90th-percentile firm has TFP 1.9 times that of the 10th-percentile firm, a gap that cannot be explained by observable input differences.

The model features a principal (firm owner) bargaining with loss-averse workers (represented by a union) over organizational change — represented as a worker effort level x that adapts the firm to the state of technology θ. Workers’ reference point is a convex combination of the status quo contract and their rational expectations of the agreed contract, with weight α on the status quo. Loss aversion parameter λ > 0 means that losses relative to the reference point are weighted more heavily than gains.

The core static result (Proposition 1) is that loss aversion drives a wedge of 1 + αλ between the workers’ marginal cost and the firm’s marginal benefit of organizational change. Below a threshold θ defined by ∂v(x₀,θ)/∂x = 1 + αλ, there is complete inertia: the firm does not change the effort level at all. Above θ, the firm adjusts effort, but to x(θ) < x^ME(θ), undershooting the materially efficient level. Higher λ or higher α both widen the inertia range and reduce the amount of implemented change (Proposition 2).

A crisis — modeled as a cost shock that makes the status quo contract generate negative profits, threatening firm closure — changes workers’ outside option from their current utility U₀ to the unemployment utility of zero. Workers are now willing to accept either wage cuts or effort increases to keep their jobs. Crucially, because both concessions are perceived as losses of equal size by workers, the firm prefers to increase effort rather than cut wages, since increasing effort is more productive when x < x^ME. The model thus provides a microfoundation for downward nominal wage rigidity: in a recession, workers make concessions through harder work rather than wage cuts.

In the infinite-horizon dynamic model, workers accumulate a quasi-rent over time equal to αλ(x_{t-1} − x₀), which represents compensation paid for past effort increases. This quasi-rent is what the firm expropriates during a crisis, allowing a discontinuous jump in effort toward the materially efficient level. Firms founded at different times or hitting different idiosyncratic shocks will therefore have different effort histories and different productivity levels, generating persistent TFP differences even among firms with identical technologies. When forward-looking players anticipate the possibility of crisis, inertia in normal times actually widens further (x̃(θ) ≤ x(θ)), because firms rationally delay effort adaptation knowing it will be cheaper to implement change during a crisis.

The expectations-management extension (Section 4) introduces a moral-hazard problem with a manager who chooses the probability of successful change. Because a higher probability of change raises the workers’ expectation-based reference point and reduces their perceived adaptation cost, the firm’s optimization problem becomes convex when the cost of effort for management is sufficiently low relative to (1−α)λΔx. This delivers a bang-bang result: the principal induces either full implementation (p = 1) or no change (p = 0), never an interior probability. This formalizes the management-consulting advice that commitment and urgency are essential to organizational change.

The social-comparisons extension (Section 5) shows that when workers compare their wages and effort to colleagues, the firm optimally compresses effort differences across workers — inducing the less productive worker to work more than efficiency requires and the more productive worker to work less. If productivity differences between workers are sufficiently small, the firm sets identical effort levels. Wage compression follows from effort compression. To avoid the cost of social comparisons entirely, it may be optimal for the firm to split into separate legal entities whose workers no longer form a common reference group — a new explanation for organizational unbundling.

Q: What is the core mechanism by which loss aversion generates organizational inertia in normal times? A: Workers have a reference point that is a convex combination (weight α on status quo, weight 1−α on rational expectations) of their current contract and the expected new contract. Because workers perceive an effort increase above their reference effort as a loss, the firm must pay a wage premium of αλ per unit of additional effort on top of the material effort cost of 1. This raises the effective marginal cost of implementing change from 1 to 1 + αλ, so the firm only implements change when the marginal revenue of effort strictly exceeds 1 + αλ. Below the threshold technology level θ (defined by ∂v(x₀,θ)/∂x = 1 + αλ), there is complete inertia and the firm keeps x* = x₀.

Q: How does a crisis break the inertia? A: A crisis is a cost shock large enough to make the firm’s profits negative under the status quo contract, so the firm would close unless workers make concessions. Workers’ outside option shifts from their accumulated utility U₀ to the unemployment utility of zero. Because wage cuts and effort increases are both perceived as losses of equal magnitude, the firm prefers to demand effort increases (which raise revenue) over wage cuts (which do not). At the margin, when workers are at zero utility, the loss-aversion terms cancel from the marginal rate of substitution, and the firm can push effort up to the materially efficient level x^ME — a discontinuous jump.

Q: Why do wages not fall during a recession in this model? A: Workers perceive both wage cuts and effort increases as losses of equal per-unit utility cost. Since increasing effort by one unit and cutting wages by one unit impose the same utility cost on workers but effort increases raise firm revenue while wage cuts do not, it is always more efficient for the firm to extract concessions through higher effort rather than lower wages. The firm therefore first drives effort to x^ME before cutting wages, and cuts wages only if the zero-utility constraint still is not binding at x^ME. This provides a microfoundation for Bewley’s (1999) observation that wages do not fall during recessions.

Q: Where does the quasi-rent exploited during a crisis come from? A: Every time the firm implements an effort increase in normal times it must compensate workers with a permanent wage increase to cover both the permanent higher effort cost (x_{t}−x_{t-1}) and the one-time behavioral adaptation cost αλ(x_{t}−x_{t-1}). Because the compensation for the adaptation cost must be spread over all future periods as a permanent payment, workers accumulate a quasi-rent that by period t equals αλ(x_{t-1}−x₀) above their initial utility U₀ = w₀−x₀. This is the rent the firm expropriates in a crisis to fund the discontinuous effort increase.

Q: How does the dynamic model generate persistent TFP differences across firms in the same industry? A: Firms founded at different times start with different initial status-quo effort levels relative to the current technology θ. Because each firm’s path of organizational adaptation is history-dependent — inertia regions, timing of crises, and accumulated quasi-rents all depend on when the firm was founded and what idiosyncratic shocks it experienced — firms that start later (or hit crises earlier) can remain more productive than older firms for extended periods. The numerical example with v(x,θ) = θ ln(x), α = 0.5, λ = 1, δ implied parameters shows that a firm founded when θ = 7 at the materially efficient point can maintain a substantial productivity advantage over a firm founded when θ = 4 that has accumulated inertia, even though both firms have access to the same technology.

Q: Does rational anticipation of a future crisis increase or decrease inertia in normal times? A: It strictly increases inertia. When players assign probability µ > 0 to a crisis each period, forward-looking workers demand higher compensation for effort increases in normal times — specifically, the per-period compensation for behavioral adaptation cost rises from (1−δ)αλ to γ = (1−δ(1−µ))αλ, which is increasing in µ. Simultaneously, the firm anticipates that effort adaptation will be cheaper to achieve in a crisis and therefore delays effort increases. The result is that the inertia threshold shifts from x(θ) to x̃(θ) ≤ x(θ), a strictly wider inertia region (Proposition 6).

Q: What is the expectations-management result and what drives it? A: When a manager chooses the probability of successful change p at cost c(p) = (c/2)p², the wage the firm must pay workers is concave in p (equation 22): w = x₀ + p(1+λ)Δx − p²(1−α)λΔx + U₀. The concavity arises because a higher p raises the expectation-based component of the reference point, lowering workers’ perceived adaptation cost. When c < (1−α)λΔx, this makes the principal’s profit function convex in p, so the optimum is at a corner: the principal induces either p = 1 (full implementation) or p = 0 (no change). Even when an interior solution obtains, a decrease in α (more weight on expectations) increases p. This formalizes the practitioner prescription that organizational change requires convincing everyone that change is certain and unavoidable.

Q: What is the effort and wage compression result under social comparisons? A: When each worker compares his situation to his colleague’s, with weight β on the peer’s wage and effort in forming the reference point, the firm must pay both workers a social-comparison premium of λβ(x₂−x₁) per unit of effort difference (Lemma 5). The firm therefore optimally compresses effort differences: it induces the less productive worker to exert effort above his efficient level and the more productive worker below his efficient level, at first-order conditions ∂v₁/∂x = 1 − 2λβ and ∂v₂/∂x = 1 + 2λβ respectively. If the productivity difference is small enough (specifically if ∂v₂(x*,θ)/∂x < 1 + 2λβ at the equal-effort point), the firm sets x₁* = x₂* = x*, eliminating wage inequality entirely (Proposition 8).

Q: Why might it be optimal for a firm to split into separate entities? A: Social comparisons impose costs on the firm by requiring higher wages for both workers (each receives a premium of λβ(x₂−x₁) regardless of their relative rank) and by distorting effort levels away from their efficient values. If workers employed by legally separate firms no longer treat each other as part of their reference group — because β falls to zero across firm boundaries — the firm can eliminate these comparison costs by spinning off activities into independent entities. This provides an efficiency rationale for organizational unbundling that does not rely on asset specificity or transaction costs, addressing what the authors call the “Williamson puzzle.”

Q: What are the implications for older workers and for social insurance policy? A: Older workers have two compounding reasons to be more resistant to organizational change: shorter remaining time horizons reduce the present value of permanent wage compensation for adaptation costs, and Gächter, Johnson, and Herrmann (2022) report that loss aversion λ increases with age, income, and wealth. Both factors raise the cost of implementing change with older workers. For social insurance, generous unemployment benefits or policies preventing layoffs (such as short-time work schemes) reduce workers’ concession costs in a crisis, weakening the mechanism by which crises trigger change. The model suggests this may contribute to slower technology adoption in countries with stronger labor market protections.

Q: What empirical facts from the existing literature does the model account for? A: The model accounts for: (1) Syverson’s (2004) finding of a 90th/10th percentile TFP ratio of 1.9 in four-digit US industries; (2) the iron ore and steel case study (Schmitz 2007) in which labor productivity doubled within a few years of a competitive shock with no new technology; (3) Bloom et al.’s (2014) correlation between more intense competition and higher TFP; (4) Holmes and Schmitz’s (2010) survey finding that competitive shocks raise industry productivity mainly through survival and improvement of existing firms; (5) Bewley’s (1999) downward nominal wage rigidity; and (6) Hjort, Li, and Sarsons (2022) on multinational firms using headquarters wages as reference points for wages in low-wage locations.

Loss aversion (λ): The parameter measuring the degree to which workers weight losses relative to their reference point more heavily than gains. A meta-analysis (Brown et al. 2023) across 607 empirical estimates finds an average loss aversion parameter of 1 + λ = 1.955. In this paper, λ > 0 means workers perceive a wage cut and an effort increase as losses, raising the effective marginal cost of organizational change by a factor of 1 + αλ.

Reference point (w^r, x^r): The benchmark wage and effort level against which workers evaluate outcomes. Defined as a convex combination of the status quo contract (w₀, x₀) with weight α and the rational expectation of the agreed contract (w^e, x^e) with weight 1−α. Losses occur when the realized wage falls below w^r or the realized effort exceeds x^r.

Organizational inertia: The firm’s failure to implement materially efficient organizational change even when doing so would increase total surplus. In the model, inertia arises because the effective marginal cost of effort to the firm is 1 + αλ rather than 1, so the firm only implements change above a threshold technology level θ. The range of inertia widens with higher λ, higher α, and higher initial effort x₀.

Quasi-rent: The utility accumulated by workers above their initial utility U₀ = w₀−x₀ as compensation for past effort increases. By period t it equals αλ(x_{t-1}−x₀). This quasi-rent is the source of concessions the firm can extract in a crisis: workers accept higher effort (or lower wages) in exchange for keeping their jobs rather than losing this accumulated utility through unemployment.

Behaviorally efficient effort x(θ): The effort level that maximizes joint surplus taking behavioral adaptation costs into account, defined by ∂v(x,θ)/∂x = 1 + (1−δ)αλ in the dynamic model. This is strictly below the materially efficient effort x^ME(θ) (defined by ∂v/∂x = 1) and strictly above the firm’s privately optimal effort in normal times.

Effort compression: The result under social comparisons that the principal optimally reduces the effort difference between workers relative to the efficient allocation — inducing the less productive worker to work more and the more productive worker to work less than efficiency requires. Driven by social-comparison costs λβ(x₂−x₁) that both workers receive as premiums regardless of relative rank.

Expectations management: The strategic use of commitment to high probability of change in order to shift workers’ expectation-based reference point and reduce the perceived adaptation cost. When α is small (rational expectations dominate the reference point), making change more certain lowers the wage cost of implementation, creating a complementarity between commitment and cost reduction that produces the bang-bang result: implement with certainty or not at all.

Macroeconomics | Macro Paper Warehouse

A Cognitive Theory of Reasoning and Choice

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A Monetary-Fiscal Theory of Sudden Inflations

Overview

In depth

Q1. What is the fundamental analogy that drives the paper’s theory, and what economic literature does it build on?

Q2. What is the key information friction, and how does it generate the switch between “normal times” and crisis?

Q3. How does the bond payoff structure explain the absence of sudden deflations?

Q4. How does the first numerical example (lower surplus expectations) work quantitatively?

Q5. How does the second numerical example (interest rate increase) work quantitatively, and what does it imply for monetary policy?

Q6. What are the two regions of the deterministic model and how do they differ in fiscal and price-level dynamics?

Q7. What are the two interest rate regions in the deterministic model, and what is their implication for monetary policy effectiveness?

Q8. How does the paper relate to and extend prior FTPL literature?

Q9. What is the welfare role of consumer ignorance in this framework?

Q10. What are the scope conditions and limitations of the model?

Key Concepts

A Theory of Supply Function Choice and Aggregate Supply

Research Question

Methodology

Main Findings

Scope Conditions

In depth

Q1. What is wrong with assuming price-setting or quantity-setting as a primitive restriction on firm behavior?

Q2. What is the formal equivalence between supply function choice and conditioning on realized demand?

Q3. How is the optimal inverse supply elasticity α₁ derived, and what is the 2SLS interpretation?

Q4. How do uncertainty and demand elasticity shape the firm’s optimal supply function in partial equilibrium?

Q5. How does the model generate an aggregate supply and demand representation from supply function choices?

Q6. What is the equilibrium fixed-point equation and why can there be multiple equilibria?

Q7. What determines the slope of aggregate supply in the special case of balanced strategic interactions (ηγ = 1)?

Q8. What is the role of total uncertainty versus the composition of uncertainty?

Q9. How does market power (demand elasticity) affect the slope of aggregate supply, and why does this differ from the New Keynesian prediction?

Q10. How does the model rationalize the steepening of aggregate supply in the 1970s and 2020s?

Q11. How does the cross-country evidence help distinguish the model from alternatives based on the level of inflation?

Q12. How can supply functions be integrated into larger linearized macroeconomic models?

Q13. What are the implications of supply function choice for monetary policy discretion?

A traffic-jam theory of growth

A Welfare Analysis of Policies Impacting Climate Change

Additionality and Asymmetric Information in Environmental Markets: Evidence from Conservation Auctions

Aggregate demand externality and self-fulfilling default cycles

Layer 1 — Overview

In depth

Q1. What is the core empirical motivation for the model, and what does the regime-switching analysis establish?

Q2. How does the Dixit-Stiglitz CES structure generate a demand externality that links aggregate output to individual firm default decisions?

Q3. What is the two-dimensional dynamical system that summarizes the equilibrium, and what do the two loci look like in the phase diagram?

Q4. Under what conditions do multiple steady states exist, and what does each look like?

Q5. What are the local dynamics around each steady state, and how does parameter κ determine whether the bad steady state is a sink or a source?

Q6. What does the global dynamics analysis reveal that local analysis misses?

Q7. How does the paper simulate the clustered default pattern and what is the role of the “ugly” equilibrium?

Q8. Can purely sentiment-driven cycles generate the clustered default pattern?

Q9. How does the collateral constraint on working capital create dynamic strategic complementarity?

Q10. What is the policy remedy for the bad equilibrium, and how does it work?

Q11. How does this paper differ from Cui and Kaas (2021), the most closely related predecessor?

Q12. What is the markup prediction of the model, and is it consistent with empirical evidence?

Key Concepts

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An endogenous gridpoint method for distributional dynamics

Anatomy of the Phillips Curve: Micro Evidence and Macro Implications

Layer 1 — Overview

In depth

Q1. What is the primitive formulation of the NKPC, and how does it differ from the conventional formulation?

Q2. What structural parameters govern the slope of the cost-based NKPC, and what is the formula?

Q3. How does the dynamic pass-through regression differ from the static (long-run) pass-through regressions used in prior literature?

Q4. What are the baseline estimates of the structural parameters, and how robust are they?

Q5. What is the short-run elasticity of a firm’s own price to a permanent marginal cost shock, and how do nominal rigidities and strategic complementarities each contribute?

Q6. How is marginal cost measured in the data, and why is the inclusion of intermediate input costs important?

Q7. What instruments are used for marginal cost and competitors’ prices, and what are the identifying assumptions?

Q8. What evidence supports the validity of the TFPQ instrument against capacity utilization concerns?

Q9. How does the model track aggregate Belgian manufacturing PPI inflation, and what does this imply for cost-push shocks?

Q10. How do the authors estimate the output elasticity of marginal cost σ_y, and what do they find?

Q11. How does the oil shock validation exercise confirm the cost-based NKPC slope estimate?

Q12. What do the estimates imply about why the conventional NKPC appears flat in normal times?

Q13. Under what conditions does the cost-based Phillips curve decompose cleanly into a product of the two elasticities?

Q14. How do macroeconomic complementarities from aggregate decreasing returns to scale affect the NKPC slope?

Key Concepts

Are Inflationary Shocks Regressive? A Feasible Set Approach

Layer 1 — Overview

In depth

Q1. What is the “feasible set approach” and how does it differ from prior work on inflation incidence?

Q2. Why is the asset price channel — rather than consumption prices — the dominant channel in both shocks?