E20 | Macro Paper Warehouse

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.

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.

Disaggregated Economic Accounts

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

This paper develops and implements a system of disaggregated economic accounts that breaks down national accounting positions into bilateral flows between small groups of consumers, producers, the government, and the rest of the world. Standard national accounts document aggregate income and production plus input-output trade between producer industries; they contain no comprehensive data on which consumers buy from which producers or which producers pay income to which consumers. The paper fills this gap by measuring, for Denmark, all 36 positions in the UN System of National Accounts (SNA) — consumer spending, labor compensation, profit income, intermediates trade, government transfers and taxes, and foreign trade — as bilateral cell-to-cell flows, satisfying all national accounting identities at the level of individual cells and at the aggregate level. The data reveal systematic stylized facts about domestic spending shares, gravity of spending, urban bias, and assortative matching between consumer and producer characteristics. Combining the disaggregated accounts with a general equilibrium model with nominal wage rigidities, the paper shows that fiscal transfer multipliers vary substantially across consumer cells — from below 1 to above 2 — depending on the spending intensity of recipient cells on the slack (unemployed) portion of the economy. Applying the framework to a hypothetical U.S. tariff shock on Denmark (calibrated to July 2025 effective tariff levels on China), the paper demonstrates that the cells generating the highest multipliers are not those directly exposed to the shock or even those made slack, but those whose spending intensity on slack cells is high. The disaggregated accounts allow the government to select more effective fiscal policies: choosing transfers targeting high-spending-intensity cells saves approximately 0.4–0.7% of Danish GDP relative to programs targeting low-intensity cells, for the same GDP stimulus.

Measurement framework (Section II): The paper assigns every Danish adult to one of approximately 2,744 consumer cells, defined by the interaction of 98 municipalities (regions) and 28 industries (industry of main employment). Every production establishment is assigned to one of approximately 2,646 producer cells by region and industry. Median consumer cell contains 658 adults; median producer cell contains 47 establishments. The circular flow includes: (i) consumer spending on domestic and foreign producers; (ii) labor compensation paid by producer cells to consumer cells; (iii) profit income (dividends, mixed income, owner-occupied housing surplus) from producers to consumers; (iv) intermediates trade between domestic producers; (v) foreign trade; (vi) government taxes, transfers, and spending. A “bottom-up” approach uses microdata — geocoded transaction records from Danske Bank (largest Danish bank) and administrative government registers — to directly measure bilateral flows; a “top-down” approach distributes aggregate flows using assignment algorithms. Year: 2018. Data available at disaggregatedaccounts.com.

Stylized facts (Section IV):

1. Domestic spending shares (§IV.B): The share of a consumer cell’s spending going to domestic rather than foreign producers ranges from 75% to almost 100% (average 92%). Rural (small-population) cells, older cells, and less college-educated cells have higher domestic spending shares. Population size, average age, and college share jointly explain about half of the cross-cell variation in domestic shares; the patterns hold within industry and within region. The majority of foreign spending goes to travel-related and specialized retail categories (hotels, airlines, food away from home, clothing).

2. Gravity (§IV.C): Consumer spending declines with distance (log-log gradient = −1.33, column 1 of Table II). On average, roughly 50% of spending stays in the home region and an additional 10% goes to regions within 25 km. The distance gradient is steeper for groceries and fuel (local, in-person purchases) and shallower for telecommunications, insurance, and hotels. Rural, older, and less college-educated consumers spend more locally (stronger distance gradient, consistent with higher domestic shares).

3. Urban bias (§IV.E): Consumer spending flows disproportionately toward large cities. The 15 largest regions receive 34% of national consumer spending while accounting for only 27% of consumers. Urban bias is absent for everyday purchases (groceries) and strong for irregular or remote purchases (telecommunications, specialized retail). Rural consumers also visit urban regions in person, so urban bias is present in card payments too.

4. Assortative spending (§IV.D): Consumers tend to spend on producer cells employing workers with similar characteristics. Age of consumers and average age of workers in receiving cells are positively correlated (β = 0.178); college share similarly (β = 0.120); domestic spending share similarly (β = 0.203). The slopes are well below 1 (consumers purchase from many cells), but mild assortative spending reinforces first-order domestic spending patterns through higher-order connections.

5. Triangular flows (§IV.F): A distinctive cross-regional pattern: consumer spending and intermediates trade flow on net from rural to urban regions (urban regions run a net internal trade surplus); rural regions run a net external surplus (rural manufacturers export; e.g., Novo Nordisk in Kalundborg, Vestas in Nakskov); urban regions import relatively more from abroad. This triangular flow arises from urban consumption amenities and urban business service concentration.

Spending intensity (§IV.G): The paper constructs a reduced-form measure capturing, for each consumer cell i, how much its spending contributes to the income of a target group of cells — accounting for all higher-order connections (the infinite sum over indirect spending chains). The domestic spending intensity of cell i is defined recursively as the sum over all domestic producer cells j of (spending share αji × domestic spending intensity of producer cell j). Values range from roughly 0.4 to 0.9. The measure is strictly greater than the direct domestic spending share because the recursive formula incorporates second- and higher-order domestic connections. Domestic spending intensity is higher for rural, older, and less college-educated cells (consistent with the stylized facts). A spending intensity on slack cells can be constructed in the same way by replacing the target group with cells experiencing demand-driven unemployment.

General equilibrium model (Sections V–VI): The model is a static small open economy with many consumer and producer cells. Consumer utility is Cobb-Douglas over goods from all producer cells and foreign goods. Each producer cell’s production function is Cobb-Douglas with decreasing returns to scale (equivalent to a fixed factor). The key friction is downward nominal wage rigidity: Wi ≥ (1−δ)W̄i. When demand for a cell’s labor falls sufficiently (more than fraction δ), the wage rigidity binds and some workers in that cell become slack (unemployed demand-determined). A fiscal transfer to consumer cell i raises its income, which stimulates spending, which flows through the disaggregated network to raise labor demand across cells. The multiplier is higher when recipient spending flows disproportionately to slack cells, generating additional employment. The model is calibrated using the measured disaggregated accounts: spending shares αji, profit shares κij, labor shares λij, intermediates shares ωjj′, and tax rates are all taken directly from the disaggregated data. Baseline elasticity of substitution = 1 (Cobb-Douglas); robustness checks use short-run elasticities (< 1) and long-run elasticities (> 1), with no material change in conclusions.

Analytical result (Proposition 1): In an economy-wide recession (all cells slack), the vector of transfer multipliers is µ = ϕ′ · (I − M)⁻¹ · M · D((1 − τ̄ᵢ)⁻¹), where M is a transformed Leontief-style spending matrix incorporating the disaggregated accounts and τ̄ᵢ are fiscal externalities. The key insight is that the multiplier of cell i’s transfer is closely linked to its spending intensity on all other domestic cells, with all higher-order connections captured by the (I − M)⁻¹ M term. A cell’s multiplier is high when: (i) it spends domestically rather than on imports; (ii) it spends on producers that in turn employ domestic workers in slack cells; and (iii) these higher-order effects amplify through the circular flow.

Economy-wide recession: quantitative multipliers (Table III):

Transfer policy	Multiplier	Cost to raise GDP by 5% (bn DKK)
Uniform (all adults)	1.04	96.08
Top 10% domestic spending intensity	1.21	81.99
2018 child tax credit	1.02	97.85
2022 inflation relief to elderly	1.13	88.11
2023 housing rent inflation support	1.03	96.45
Construction worker support	1.23	81.16
Consulting/IT worker support	0.95	105.22

High-multiplier policies (construction workers, 2022 elderly relief) target rural, older, less college-educated cells with high domestic spending intensity. Low-multiplier policies (consulting/IT workers, 2023 housing relief, 2018 child tax credit) target urban, young, or college-educated cells with lower domestic intensity. The gap between the best and worst policies amounts to savings of roughly 15 bn DKK (≈ 2.4 bn USD), or 0.4–0.7% of Danish GDP, for the same aggregate GDP impact.

U.S. tariff shock application (Section VII): The paper analyzes a hypothetical U.S. tariff increase to 41.4% (the July 2025 effective U.S. tariff on China) on Danish exports, motivated by Greenland tensions. The shock reduces export revenue by 41.4% for each producer cell, with direct exposure varying by region: Billund (Lego headquarters), Kalundborg (pharmaceuticals), and a Copenhagen manufacturing hinterland face the largest direct declines — up to 8% of total regional sales. The shock propagates through the disaggregated network; cells whose income falls by more than 4% become slack. Key findings:

Regional slackness follows direct exposure but is also shaped by proximity to other exposed regions (urban bias propagates the shock to cities) and isolation (Billund has high direct exposure but low slackness relative to exposure because it is geographically isolated from other high-exposure cells)
Transfer multipliers for this heterogeneous recession (Proposition 2) depend on spending intensity on slack cells, not on direct exposure or own slackness
Table IV (R² for multiplier): slack cell indicator alone explains R² = 0.015; direct spending share on slack raises R² to 0.366; spending intensity on slack cells raises R² to 0.769 (column 3); adding both spending share and spending intensity on slack reaches R² = 0.840 (column 4)
Billund, despite high exposure, has low multiplier because its spending (often local to a low-exposure vicinity) does not create labor demand for slack cells elsewhere
Some of the highest-multiplier regions are themselves non-slack but are surrounded by many slack cells, so their spending effectively employs slack workers

Dynamic model (Section VIII): The paper extends to a dynamic OLG (Blanchard-Yaari) model with heterogeneous marginal propensities to consume (MPCs) calibrated from a 2009 Danish fiscal policy. Key result: static and year-4 dynamic multipliers are closely correlated (slope ≈ 0.898). Long-run cumulative multipliers exactly equal static multipliers (formally proved in Appendix V.F): in the long run, all transfers are fully spent. MPCs and domestic spending intensity are complementary determinants of dynamic multipliers — targeting high-MPC cells amplifies short-horizon (year 0–2) multipliers, while targeting high-spending-intensity cells shapes both short- and long-run multipliers. The paper’s main mechanism (spending intensity on slack cells) is robust at all horizons.

Robustness (Section IX): (i) Counterfactual accounts with reversed stylized patterns (e.g., rural cells spending like urban cells) lead to substantially different multipliers — the specific measured patterns drive the results. (ii) Imposing standard simplifying assumptions (consumer spending flows only to local producers; spending flows across regions in proportion to intermediate trade) misses most of the multiplier variation. (iii) The mechanism is similarly important in less open economies. (iv) Low short-run and high long-run substitution elasticities (from the trade literature) produce similar multiplier rankings across cells.

Scope conditions: The implementation is a proof of concept for Denmark, using existing micro data from a single large bank and government registers; full coverage of all banks and complete data on within-firm flows would strengthen measurement. Capital-related transactions (saving, investment, financial assets) are aggregated into a single capital accumulation cell — disaggregating these would require different data. The model is intentionally static (with a dynamic extension), abstracting from price adjustment dynamics beyond the NK wage rigidity. The analysis is a partial equilibrium in the sense that monetary policy response is not modeled; the fixed exchange rate assumption is realistic for Denmark (pegged to the Euro) but may not transfer to economies with flexible rates. The proof of concept suggests that national statistical agencies could benefit substantially from measuring disaggregated flows through refined surveys.

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 missing from standard national accounts that this paper’s system provides?

Standard national accounts measure aggregate consumer spending, income, and output, plus intermediates trade among producer industries (input-output tables); what they do not measure is which specific consumer groups buy from which specific producer groups, or which specific producer groups pay labor and profit income to which specific consumer groups. This means that propagation of a shock through the circular flow — e.g., a tariff shock that reduces exports by rural manufacturers, which reduces income for rural workers, who then reduce spending on urban services, which reduces urban workers’ income — cannot be traced without simplifying assumptions (like “spending flows only to local producers”) that the disaggregated data shows to be empirically inaccurate. The paper provides a proof of concept demonstrating that measuring these bilateral consumer-to-producer and producer-to-consumer flows, while satisfying all national accounting identities, is feasible with existing micro data and yields policy-relevant variation in fiscal multipliers.

Q2. Why do rural, older, and less college-educated consumer cells have higher fiscal multipliers during an economy-wide recession?

These groups have higher domestic spending intensity — a higher fraction of their spending reaches domestic consumers rather than leaking abroad — because they spend less on international tourism, less on imported goods accessed through online retail or urban services, and more on local goods purchased in person. The gravity patterns (stronger distance gradient) and direct domestic spending shares document this directly: rural consumers allocate ~92–100% of spending to domestic producers versus ~75–80% for urban young college-educated consumers. When all cells are slack, a transfer to a high-domestic-intensity cell circulates more within the country, generating more rounds of domestic income and employment before leaking to imports. The mild assortative spending pattern further reinforces the first-order effect: spending by rural older consumers flows toward producer cells employing workers with similar characteristics, who also spend domestically, so higher-order connections amplify rather than dilute the domestic spending effect.

Q3. Why does targeting directly exposed or slack cells not guarantee a high transfer multiplier after the U.S. tariff shock?

A transfer raises GDP by increasing spending, which creates labor demand for other consumer cells; a transfer to a slack cell only generates a high multiplier if that cell’s spending flows toward other slack cells (directly or through indirect chains) — not if it flows toward non-slack cells or abroad. The tariff shock creates isolated pockets of slackness in rural manufacturing regions (e.g., Billund for Lego) that are geographically far from other slack regions; Billund consumers spend locally (gravity) and their locality is not itself a center of other slack cells. In contrast, regions near Copenhagen with moderate direct exposure may have high multipliers if they are close to many other slack manufacturing cells — their spending generates employment across the slack network. The R² decomposition confirms this: knowing a cell is slack explains only 1.5% of multiplier variation (R² = 0.015), while knowing its spending intensity on slack cells explains 76.9% (R² = 0.769).

Q4. How does the paper ensure that the disaggregated flows satisfy national accounting identities?

The system is designed so that every cell’s total inflows equal total outflows (a cell-level balance sheet constraint), and the sum of all cell-level flows equals the corresponding national aggregate from the SNA — both conditions are imposed by construction, not just approximated. For most positions, a bottom-up approach uses observed bilateral microdata (e.g., card payments from Danske Bank directly measure consumer spending by consumer cell i at producer cell j); for positions without direct microdata, a top-down algorithm distributes an aggregate total across cells using assignment rules grounded in the microdata. This dual approach ensures national comprehensiveness (the sum of disaggregated flows equals aggregate national accounts) and individual consistency (cell-level identities hold), unlike existing regional accounts or social accounting matrices that satisfy only one of these constraints.

Q5. What is the relationship between spending intensity and the standard fiscal multiplier formula?

The cell-level multiplier (dGDP/dTi) in Proposition 1 equals approximately the cell’s spending intensity on domestic cells, corrected for fiscal externalities and price effects of the fixed factor. The formal difference is that the model multiplier involves the matrix (I − M)⁻¹M where M incorporates both spending and production shares (through which price changes for the fixed factor enter), while the reduced-form spending intensity uses only the spending matrix. Despite this difference, the two measures are highly correlated empirically: the regression of cell-level multipliers on domestic spending intensity has a slope of approximately 1.66 for static multipliers. The spending intensity can thus be calculated directly from the disaggregated accounts without solving the full general equilibrium model, making it a practical statistic for policy guidance.

Q6. How does the dynamic model reconcile the fact that rural, older, and less college-educated cells have high spending intensities but typically lower MPCs?

MPCs and spending intensities are complementary but distinct determinants of dynamic multipliers at short horizons: high-MPC cells spend the transfer quickly (year 0–1), generating a large immediate impact, while high-spending-intensity cells ensure that spending, whenever it occurs, circulates domestically and reaches slack labor markets. At long horizons (year 4+) the two effects converge because all cells eventually spend their full transfer (long-run MPC = 1) and the multiplier converges to the static model’s value, which depends only on spending intensity. The practical implication is that policies targeting rural/older/less-educated cells (high intensity, lower MPC) may have lower immediate multipliers than policies targeting high-MPC urban consumers, but converge to higher long-run multipliers. The year-4 cumulative multipliers from the dynamic model closely resemble the static model, suggesting a 3–5 year business cycle horizon is well captured by the static analysis.

Q7. What does the triangular flow pattern imply for understanding regional inequality and fiscal redistribution?

The triangular flow — rural regions receive net income from foreign exports; rural consumers spend net inflows toward urban regions; urban consumers spend net toward abroad — means that rural regions’ incomes depend on export competitiveness while urban regions’ incomes depend on domestic consumption demand; fiscal transfers to rural consumers thus have high domestic multipliers because their spending boosts urban income (via the rural-to-urban spending flow), which then circulates domestically before leaking abroad. This pattern is also consistent with the political economy finding that high-multiplier cells (rural, older, less educated) are more likely to vote for right-wing populists and feel politically disenfranchised — they are the “left behind” groups that economic research associates with exposure to globalization and automation, but whose spending patterns happen to generate large domestic multipliers during recessions.

Key concepts

disaggregated economic accounts : a system that breaks down all national accounting positions — consumer spending, labor and profit income, intermediates trade, government transactions, foreign trade — into bilateral flows between consistently defined region-by-industry consumer cells and producer cells, satisfying national accounting identities both at the cell level and in aggregate; the paper’s proof of concept is implemented for Denmark using 2,744 consumer cells and 2,646 producer cells in 2018.

spending intensity : a cell-level, reduced-form statistic capturing how much a consumer cell’s spending contributes to the income of a target group of cells (e.g., all domestic cells or all slack cells), accounting for all indirect higher-order connections through the circular flow; formally defined as a recursive sum that incorporates the full disaggregated network structure; ranges from 0.4 to 0.9 for domestic spending intensity and is systematically higher for rural, older, and less college-educated cells.

slack cell : in the paper’s NK model, a consumer cell for which demand-driven unemployment occurs because the nominal wage rigidity binds — labor supply exceeds demand when the cell’s income declines by more than a threshold δ due to a negative demand shock; fiscal transfers with high multipliers are those whose spending reaches slack cells (directly or through higher-order network connections).

triangular flows : the cross-regional spending pattern documented for Denmark in which net consumption spending flows from rural regions to urban regions (urban bias), net foreign export revenue flows to rural regions (rural manufacturing), and net foreign import spending flows from urban regions; implies that rural-to-urban spending flows act as an important transmission channel for fiscal stimulus targeted at rural consumers.

bottom-up vs top-down disaggregation : the two methodological approaches for constructing bilateral cell-to-cell flows; the bottom-up approach uses individual-level microdata (e.g., bank transaction records) to directly observe cell-to-cell payment flows; the top-down approach allocates an aggregate national accounting position across cells using assignment algorithms informed by microdata; both approaches are designed so that the resulting disaggregated flows sum to the corresponding SNA aggregate.

Monopsony Makes Firms Not Only Small but Also Unproductive: Why East Germany Has Not Converged

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

Layer 1 — Summary

When employers face a trade-off between growing large and paying low wages — that is, when they have monopsony power — some productive employers will decide to acquire fewer customers, forgo sales, and remain small; these decisions have adverse consequences for aggregate labor productivity beyond the standard monopsony result that firms are too small. The paper documents that East German plants (compared to West German ones) face a steeper size-wage curve, invest less into marketing, and remain smaller, with the share of employment at plants with more than 249 employees standing at roughly 25% in East Germany versus 39% in West Germany in 2014 (and 31% versus 55% in manufacturing specifically). The steeper size-wage curve in East Germany is traceable to the historically determined underrepresentation of collective bargaining and union membership in small East German plants — a legacy of communist-era labor organization that caused union membership to collapse after reunification. The authors combine this evidence with a heterogeneous-plant model in which plants have product market power and choose how many customers to acquire subject to an upward-sloping size-wage schedule; two channels reduce aggregate productivity: a love-of-variety loss (fewer active plants means consumers bundle from a smaller variety of suppliers) and a compositional reallocation loss (labor is shifted from more productive to less productive plants, an effect exacerbated by product market power). When the model is calibrated to West Germany and the steeper East German size-wage trade-off is imposed, it predicts 10 percentage points lower aggregate labor productivity in East Germany — and for manufacturing, where East-West differences in plant size and the size-wage trade-off are particularly pronounced, the model predicts 18 percentage points lower productivity; in both cases the compression of the plant size distribution accounts for the largest share of the predicted productivity loss. The paper thus offers an explanation for why, more than thirty years after reunification, labor productivity and wages remain roughly 25% lower in the East German private sector despite uniform legal institutions across the two regions.

Layer 2 — Q&A

Q1: What is the core mechanism by which monopsony power reduces aggregate productivity, and how does it differ from the standard “firms are too small” result?

In the standard monopsony account, firms face an upward-sloping labor supply curve and choose to employ fewer workers than the competitive optimum, so individual firms are below efficient scale. The paper identifies an additional, investment-distortion channel: plants must also decide how large a customer base to acquire, and doing so requires marketing expenditure as well as the labor to service additional customers — labor whose cost rises with plant size along the size-wage schedule. A steeper size-wage curve therefore makes customer acquisition more expensive at the margin, and some productive plants optimally choose to acquire fewer customers, forgo sales, and remain small. The new aggregate productivity loss stems from this distorted investment margin: plants that could generate high value added at large scale instead operate at sub-optimal customer networks, suppressing aggregate output through both a love-of-variety effect (fewer active large plants means consumers access a smaller product variety) and a misallocation effect (the compressed size distribution shifts employment toward less productive plants).

Q2: What empirical patterns do the authors document to link the East-West productivity gap to missing large plants and steeper size-wage curves?

The authors document three nested empirical facts using the German Structure of Earnings Survey (SES) pooled across 2006, 2010, and 2014, supplemented by administrative wage panel data (AWFP) and national accounts (VGR). First, East German labor productivity in the private non-primary sector is about 25% below West Germany’s and has not converged since roughly 1995. Second, the share of employment at large plants (>249 employees) is substantially smaller in the East, and this gap is present both cross-sectionally across survey years and conditionally: East German plants enter smaller and remain smaller over their life-cycles, so plant age does not explain the difference. Third, industries where missing large plants are most pronounced in East Germany relative to West Germany are also the industries with the largest East-West productivity and wage gaps — the employment-weighted correlation between the large-plant share gap and the productivity gap is 0.53 across industries. The steeper size-wage curve itself is documented using within-industry comparisons: on average the plant size elasticity of wages is one-fifth larger in East Germany, and those industries with a steeper East-West size-wage differential are also the industries with the most missing large plants and the lowest average wages in the East.

Q3: Why is the steeper size-wage curve specific to East Germany, and why does it persist decades after reunification?

In communist East Germany, trade unions did not have the role of representing worker interests; consequently, after reunification, union membership fell dramatically. The key institutional consequence is that collective bargaining coverage in East Germany is underrepresented specifically in small plants. Workers at small plants in East Germany are more likely to have individually rather than collectively bargained wages than their West German counterparts, whereas workers at large plants in both regions are more similarly covered. Because collective bargaining flattens the size-wage curve (larger plants pay a smaller premium over small plants’ wages when both are covered by the same bargaining agreement), its absence in small East German plants produces a steeper gradient of wages with plant size in the East. This is a persistent structural feature rather than a transitional one: government policies and their enforcement are essentially uniform across regions, so the asymmetric bargaining coverage, which originates in communist-era institutional history, has not been erased by market forces or policy since 1990.

Q4: How is the model structured, and what are the three decision stages for plants?

The model is a static, long-run heterogeneous-plant framework that yields closed-form solutions. Within a period, plants face a three-stage decision problem. First, they decide whether to enter the market. Second, after entry, they choose how many customers to acquire, trading off additional sales revenue against marketing costs and the labor cost of servicing a larger customer base — a cost that rises with the number of customers because the upward-sloping size-wage curve means each additional worker hired requires a higher wage for all infra-marginal workers. Third, taking into account their product market power (each plant is a monopolistic competitor with its own customers), plants set prices to each customer and thereby determine how many workers they need. The size-wage schedule enters the second stage directly, so a steeper schedule reduces optimal customer acquisition across all plants, with the distortion being largest for the most productive plants (which would otherwise grow the largest).

Q5: Through what two channels does the steeper size-wage trade-off reduce aggregate labor productivity in the model?

The first channel is a love-of-variety effect in the product market: because more productive plants acquire fewer customers and operate at smaller scale under a steeper size-wage schedule, the average consumer bundles goods from a smaller number of distinct plants, and aggregate efficiency falls through the standard CES love-of-variety mechanism. The second channel is a misallocation effect in the labor market: the steeper size-wage schedule compresses the employment distribution across plants, reallocating labor from more productive to less productive plants relative to the benchmark with a flatter schedule. The paper shows that this second channel is exacerbated by product market power, because plants with stronger pricing power respond more aggressively to the changed labor cost trade-off. In the model’s decomposition, the compression of the plant size distribution (the misallocation channel) accounts for the largest part of the predicted 10 percentage point productivity shortfall.

Q6: What quantitative predictions does the model make, and how does it perform in untargeted moments?

The model is calibrated to two moments for West Germany: average plant size and the share of large plants (>249 employees). When the steeper East German size-wage trade-off is imposed without re-calibrating other parameters, the model predicts 10 percentage points lower aggregate labor productivity in East Germany — accounting for at least 10 of the roughly 25 percentage point observed gap. For the manufacturing sector alone, where East-West differences in plant size, the size-wage trade-off, and aggregate productivity are particularly pronounced, the calibrated model predicts 18 percentage points lower productivity. As an untargeted validation, the model also replicates the plant size distribution in East Germany, matching both the smaller average plant size and the relatively small number of large plants. These untargeted predictions provide additional support for the mechanism.

Q7: What alternative explanations for East Germany’s non-convergence does the paper rule out or place in context?

The paper addresses several confounds. In Appendix A, the authors show that East-West aggregate labor productivity differences are driven by differences in aggregate total factor productivity, not by labor quality differences, capital intensity differences, or capital quality differences — confirming within-country the finding that TFP explains a large fraction of productivity dispersion. The TFP differences are shown to be unlikely the result of greater labor market flexibility in West Germany or differences in industry composition. Appendix B shows that the East-West plant size distribution gap is not driven by differences in urbanization (West Germany has more metropolitan areas). The paper also addresses plant age: East German plants enter smaller and remain smaller at every age and across entry cohorts, ruling out the hypothesis that the size gap is purely a transitional legacy of the restructuring that destroyed many large East German plants at reunification.

Q8: How does this paper relate to the Heise and Porzio (2021) finding that plant productivity differences, not worker quality differences, drive the East-West wage gap?

Heise and Porzio (2021) use matched employer-employee data to document that plant productivity differences (as opposed to worker quality differences) account for most of the East-West wage differential, and they explain why low worker mobility does not remove these differences. The present paper complements this by providing an explanation for why plant productivity is lower in East Germany in the first place and why firm-level convergence does not occur: the steeper size-wage curve induced by the legacy of missing collective bargaining coverage in small East German plants distorts the investment and customer acquisition decisions of productive plants, keeping them small and unproductive. The two papers are thus complementary: Heise and Porzio take the plant productivity gap as given; Bachmann et al. endogenize it through the size-wage mechanism.

Key Concepts

Size-wage curve: The empirical relationship between plant size (measured by employment) and wages paid to workers, conditional on worker characteristics. A steeper size-wage curve means that the wage premium for working at a large plant relative to a small plant is larger. In this paper’s model, plants internalize that expanding their customer base and workforce requires paying higher wages to all workers (not just the marginal hire), making growth more costly when the size-wage curve is steeper.

Monopsony power (monopsonistic competition): The market structure in which an individual employer faces an upward-sloping labor supply curve — i.e., it must raise wages to attract additional workers. The paper uses “monopsonistic competition” to describe a setting with many such employers, each with some wage-setting power, in contrast to oligopsony. The paper focuses on allocative effects of this power, not on normative efficiency questions.

Customer capital / customer acquisition: Plants must incur marketing expenses to build a customer base; each customer relationship generates a stream of sales but requires labor to service. The size of the customer network is a long-run investment decision. Under monopsonistic labor markets, the cost of expanding the customer base includes not only marketing expenses but also the higher wages that a larger workforce requires, making customer acquisition a margin that is distorted by labor market power.

Love-of-variety effect: A welfare loss that arises in models with monopolistic competition and CES preferences when the number of active product varieties declines. In this paper it applies to the product market: when plants remain small and acquire fewer customers, the effective number of distinct varieties consumed falls, reducing aggregate efficiency even holding plant-level productivity fixed.

Misallocation / compressed size distribution: A situation in which factors of production are not allocated to their highest-value uses. Here, the steeper size-wage curve induces productive plants to remain small, so labor that would otherwise be employed at high-productivity large plants is instead employed at lower-productivity small plants. The resulting compression of the plant size distribution — fewer very large plants, more mass in the middle — is both the key empirical fact and the primary quantitative driver of the predicted aggregate productivity shortfall.

Collective bargaining coverage: The fraction of workers whose wages are set by collective agreements between employers (or employer associations) and trade unions, rather than by individual negotiation. The paper establishes that collective bargaining flattens the size-wage curve by compressing wages across plants of different sizes. The historically low collective bargaining coverage among small East German plants — a legacy of communist-era labor relations — is the institutional root cause of the steeper East German size-wage schedule.

Summary based on IZA Discussion Paper 15293. AI-assisted, human review pending.

Rent Guarantee Insurance

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

Abramson and Van Nieuwerburgh study Rent Guarantee Insurance (RGI), a product in which an insurer pays the landlord on behalf of a tenant who defaults on rent due to a negative income or health expenditure shock, in exchange for a monthly premium proportional to rent. The central question is whether RGI can be designed to be both welfare-improving and financially viable, given the frictions of moral hazard and adverse selection.

The authors develop a dynamic overlapping-generations equilibrium model of the rental market that features endogenous rent default, security deposits, evictions, and homelessness. Households face idiosyncratic persistent and transitory income risk, idiosyncratic medical expenditure risk, and aggregate (cyclical) income risk. Rental contracts are non-contingent, households face borrowing constraints, and housing is indivisible with a minimum quality floor. Landlords set deposits to break even in expectation given observed tenant characteristics. An insurance agency can offer RGI and must also break even in the long run. The model is calibrated to the United States at monthly frequency. Income dynamics are estimated from CPS data (1994–2023) and incorporate transitions among employment, unemployment, out-of-labor-force, and retirement states along with transfer income (unemployment insurance, disability, food stamps) and a progressive tax system. Key moments targeted by Simulated Method of Moments include a delinquency rate of 12.15% (model: 12.69%), average security deposit of $984 (model: $992, from approximately 500,000 Craigslist listings across the 100 largest MSAs), homelessness rate of 1.43% (model: 1.42%), and home-ownership rate of 63.6% (model: 63.2%).

The model’s pre-RGI analysis establishes that persistent income shocks — not transitory shocks or medical shocks — are the primary driver of rent defaults. Default risk remains elevated for 3–6 months following a persistent shock, implying that short-duration RGI coverage is insufficient to prevent eviction; coverage must span multiple months.

The paper’s main policy experiments introduce RGI under different access rules and provider types. Unrestricted RGI (available to all renters) generates large welfare gains through improved risk-sharing and lower security deposits — because insured tenants pose less default risk, landlords lower deposit requirements — but is not financially viable for either a public or private insurer due to moral hazard and adverse selection. Even a public insurer that internalizes the fiscal savings from reduced homelessness cannot break even under unrestricted access.

Restricting access changes the viability calculus sharply. A publicly provided RGI targeted to households at the bottom of the wealth distribution can achieve financial viability: these households are precisely those most prone to homelessness, so the reduction in homelessness expenses — which the public insurer internalizes — offsets the insurance deficit. This restricted public RGI generates substantial welfare gains for the most vulnerable households.

A privately provided RGI must instead target higher-wealth renters to break even, because these households have low default risk (limiting claim payouts) while remaining sufficiently risk averse to pay the premium. The intersection of financial viability and take-up is small, yielding a limited target audience. The private program has minimal impact on housing insecurity, and the most vulnerable households derive little benefit. This pattern matches observed private RGI markets, where providers restrict access to renters in good financial condition.

An RGI mandate — requiring all renters to purchase coverage — mitigates adverse selection by improving the pool of insured tenants, dramatically increasing financial viability and allowing the insurer to reduce the premium substantially while still breaking even. Mandated RGI is highly effective at preventing housing insecurity and generates welfare gains concentrated among the most financially vulnerable households.

Scope conditions: results are calibrated to U.S. income, medical, and housing market parameters as of 2019. The insurer’s borrowing cost matters: the public insurer faces lower, counter-cyclical municipal bond spreads, whereas private insurers face higher, pro-cyclical corporate spreads, which constrains the generosity of private contracts in recessions.

Q: What is Rent Guarantee Insurance and how does it work mechanically in the model? A: RGI is a contract under which a tenant pays a flat monthly premium equal to a fraction kappa of rent. When the insured tenant defaults, the insurer pays the landlord directly and deducts one period from the tenant’s stock of “insurance credit.” The tenant remains housed. Once insurance credit is exhausted, the insurer no longer covers defaults. The insurer sets the premium and the maximum coverage duration to break even in the long run.

Q: Why do most rent defaults arise from persistent rather than transitory shocks? A: The model shows that the renter population is disproportionately exposed to persistent unemployment and labor-force-exit spells, and that negative persistent income shocks are harder to smooth through savings than transitory ones. Default risk remains elevated for 3–6 months after a persistent shock but dissipates quickly after a transitory shock. This implies that RGI coverage periods of only a few months would fail to prevent eviction for the majority of defaulting tenants.

Q: How does RGI affect security deposits in equilibrium? A: Because landlords observe the tenant’s insurance status at lease signing and deposits are set to make landlords break even in expectation, insured tenants pose lower default risk and thus face lower upfront deposit requirements. This deposit reduction is a key welfare channel of RGI, as large deposits tie up a disproportionate share of poor households’ wealth and price the most vulnerable out of housing entirely.

Q: Why is unrestricted RGI financially non-viable even for the public insurer? A: Unrestricted access induces both adverse selection — riskier households self-select into coverage — and moral hazard — insured households alter their default and savings behavior. These effects cause the insurer to run a persistent deficit. Even a public insurer that internalizes the fiscal cost savings from reduced homelessness cannot recoup enough to break even, implying that an unrestricted program would require an ongoing subsidy.

Q: How does publicly provided restricted RGI achieve financial viability? A: By targeting households at the bottom of the wealth distribution — precisely those most prone to homelessness — the public RGI program produces large reductions in homelessness. Because the public insurer internalizes the fiscal expenses associated with shelters, health services, and policing that accompany homelessness, these savings are passed through to the insurer and are sufficient to offset the insurance deficit. No such mechanism is available to a private insurer.

Q: Why must private RGI target higher-wealth renters, and what are the consequences? A: Private insurers must break even using only premium revenue, without access to homelessness cost savings. Higher-wealth renters have lower default probabilities, which limits claim payouts, while remaining sufficiently risk averse to demand coverage and pay the premium. The viable target audience is small given these competing requirements. As a result, private RGI covers few households, has minimal effect on housing insecurity, and provides essentially no benefit to the most vulnerable renters. This pattern is consistent with observed private RGI markets.

Q: What are the two differences between public and private insurers in the model? A: First, the public insurer internalizes the fiscal costs of homelessness (shelters, health services, policing), raising its net benefit from offering coverage. Second, the public insurer borrows at municipal bond spreads — which are lower than corporate spreads and counter-cyclical — whereas the private insurer faces higher, pro-cyclical corporate spreads. Counter-cyclical borrowing costs allow the public insurer to extend more generous coverage precisely when aggregate conditions deteriorate and claims rise.

Q: How does an RGI mandate improve financial viability? A: Mandatory enrollment forces all renters, including low-risk ones, into the insurance pool, which counteracts adverse selection. The expanded and higher-quality pool dramatically reduces per-insured expected claim costs, allowing the insurer to lower the premium substantially while still breaking even. The low-premium mandated policy is then both affordable and effective at preventing housing insecurity, with welfare gains concentrated among the most financially vulnerable renters.

Q: What novel data does the paper use for calibration of security deposits? A: The authors construct a dataset of approximately 500,000 Craigslist rental listings scraped across the 100 largest U.S. metropolitan statistical areas between November 2022 and March 2024 to measure the cross-sectional distribution of security deposits. The average deposit in this dataset is $984, which the model matches closely at $992. The data also reveal that the deposit-to-rent ratio is decreasing in house quality, reflecting the higher default risk of low-income renters in lower-quality units.

Q: What is the paper’s definition of homelessness and what rate does the model match? A: Homelessness is defined broadly to include sheltered homeless, unsheltered homeless (0.6% of households), and doubled-up families (0.83% of households), for a total of 1.43% of U.S. households. The model matches this rate closely at 1.42%.

Q: What is the paper’s key implication for the design of housing policy? A: The central implication is that financial viability and impact on housing insecurity are in tension for private insurers, and cannot both be achieved simultaneously. Only a publicly provided program that internalizes homelessness fiscal costs and faces counter-cyclical borrowing spreads can target the most vulnerable renters, break even, and materially reduce housing insecurity. Private RGI, while viable for a narrow segment, cannot substitute for public provision as a tool against homelessness.

Q: How does RGI relate conceptually to rental assistance programs? A: The paper distinguishes RGI from rental assistance on a structural basis: insurance contracts require tenants to pay premiums, making them potentially self-financing for private providers, whereas rental assistance is a net transfer that can never be self-financing. This conceptual distinction motivates studying whether RGI can be designed to eliminate the need for ongoing fiscal transfers, though the analysis ultimately shows that a public subsidy or mandate is required to serve the most vulnerable renters.

Rent Guarantee Insurance (RGI): A contract under which an insured tenant pays a monthly premium equal to a flat percentage of rent; when the tenant defaults, the insurer pays the landlord directly, preserving tenancy, for a limited number of periods governed by the tenant’s stock of insurance credit.

Insurance Credit: An endowment of periods of RGI coverage that households receive upon entry into the model; each time the insurer pays on behalf of a defaulting tenant, one unit of credit is consumed, and no further coverage is available once credit is exhausted.

Housing Insecurity: In the paper’s framework, the set of outcomes — rent delinquency, eviction, and homelessness — arising from the combination of non-contingent rental contracts, borrowing constraints, and idiosyncratic or aggregate income and medical shocks.

Security Deposit: An upfront payment from tenant to landlord, set by the competitive landlord to break even in expectation given the tenant’s characteristics and insurance status; a key channel through which RGI affects welfare by reducing the upfront cost barrier to obtaining housing.

Moral Hazard (in RGI context): The change in a tenant’s default, savings, and housing choices induced by the presence of insurance coverage, which increases expected claim costs for the insurer relative to a world where behavior is held fixed.

Adverse Selection (in RGI context): The tendency of renters with higher default risk to self-select into RGI when access is unrestricted, worsening the insurer’s risk pool and driving up expected payouts relative to premiums.

Homelessness Externality: The fiscal costs borne by government — for shelters, health services, and policing — that accompany homelessness; the public insurer internalizes these costs, creating a net benefit from RGI that private insurers cannot capture.

Counter-cyclical Borrowing Spread: The feature of public (municipal bond) financing whereby borrowing costs fall during recessions, allowing the public insurer to expand coverage when claims are highest; contrasted with private insurers’ pro-cyclical corporate bond spreads that tighten precisely when aggregate conditions worsen.

The Macroeconomics of Irreversibility

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

Overview

Research question. How does partial capital irreversibility — arising from a wedge between the purchase price and the resale (discounted) price of capital — shape the persistence and amplitude of aggregate capital fluctuations? And what is the quantitative magnitude of the capital price wedge that is needed to simultaneously reconcile micro-level investment behavior with macroeconomic propagation?

Methodology. Baley and Blanco build a continuous-time investment model for a continuum of firms facing (i) idiosyncratic productivity shocks (geometric Brownian motion), (ii) fixed capital adjustment costs proportional to productivity, and (iii) a capital price wedge ω, under which firms buy capital at price p and sell at p(1−ω). The key state variable is the log capital-productivity ratio k̂. The optimal policy takes the form of an inaction region with two distinct reset points — one for upsizing (k̂*₋) and one for downsizing (k̂*₊) — instead of the single reset point that arises without the wedge.

Their central innovation is the Cumulative Impulse Response (CIR): the cumulative deviation of average capital-productivity ratios following a small, permanent, unanticipated aggregate productivity shock. They show the CIR can be expressed analytically through three sufficient statistics derived entirely from the steady-state cross-sectional distribution of k̂ and capital age a: (i) Var[k̂], (ii) Cov[k̂, a], and (iii) an “irreversibility term” reflecting how idiosyncratic shocks change the anticipated direction of the next adjustment. Because idiosyncratic and aggregate shocks enter the law of motion symmetrically, steady-state moments encode the aggregate propagation.

To handle the path dependence introduced by the dual reset points, they condition all behavior on the previous reset (upsizing or downsizing) and characterize transitions across reset points via a Markov chain. They then derive explicit mappings from observable microdata — size and direction of investment adjustments, duration of inaction spells, and cross-spell transition probabilities — back to the unobservable capital-productivity distributions and sufficient statistics. These mappings require no revenue or productivity data; investment actions alone suffice.

They extend the baseline model to a generalized hazard framework (stochastic, asymmetric fixed costs), enabling the model to match the full empirical investment-rate distribution, and apply everything to annual establishment-level manufacturing data from Chile (Encuesta Nacional Industrial Anual, 1980–2011), restricting to plants observed for at least ten years with more than ten workers.

Main findings.

Price wedge estimate. A capital price wedge of ω = 0.12 (12%) is selected as the preferred value because it maximizes joint consistency between the model’s predicted CIR decomposition and the data, while also matching the distribution of investment rates. At ω = 0 the model generates a CIR of 0.92 and a negative covariance term, inconsistent with the data. At ω = 0.18 the aggregate CIR level (2.39) is close to data (2.33) but the decomposition diverges. At ω = 0.12, the CIR is 1.93 and the decomposition into sufficient statistics closely mirrors the data structure.
Irreversibility doubles persistence. In the analytically tractable case of zero drift and only a price wedge (no fixed costs), the CIR equals exactly twice the ratio Var[k̂]/σ², compared to the single fixed-cost case. This means irreversibility doubles the persistence of aggregate capital fluctuations for a given cross-sectional dispersion. More generally, under the calibrated model, a 1% decrease in aggregate productivity generates a nearly 2% cumulative deviation of average capital-productivity ratios from steady state. Without irreversibility, the CIR collapses to approximately 1.
Decomposition of the CIR. At ω = 0.12, the variance term Var[k̂]/σ² accounts for 72% of the CIR; the covariance term ν·Cov[k̂,a]/σ² accounts for 10%; and the irreversibility term accounts for 18%. The positive covariance (Cov[k̂,a] = 0.152 > 0) reflects that firms subject to downward rigidity accumulate older capital stocks above the economy’s average, amplifying persistence. This positive covariance arises because the price wedge’s downward-rigidity force dominates the drift’s negative effect.
Micro-level evidence. In the Chilean data, the inaction rate is 40%. More than 96% of adjustments are positive (upsizing), fewer than 4% are negative. The probability of upsizing after a previous upsize is P⁻⁻ = 0.958; the probability of downsizing after a downsize is P⁺⁺ = 0.124. A logistic regression yields an odds ratio of 3.3, meaning a firm is more than three times as likely to purchase capital following a prior purchase than following a prior sale. The average duration of inaction conditional on a prior purchase is E⁻[τ] = 1.72 years; conditional on a prior sale it is E⁺[τ] = 1.98 years. These patterns are qualitatively consistent with the serial correlation in adjustment sign predicted by the model.
Comparison with existing wedge estimates. The calibrated ω = 0.12 lies between micro-level studies based on liquidating firms (Ramey and Shapiro, 2001: ω ≈ 0.72; Kermani and Ma, 2023: ω ≈ 0.65) and structural models calibrated to static moments of investment distributions (Cooper and Haltiwanger, 2006; Khan and Thomas, 2013: ω = 0.025–0.07). The lower value relative to liquidation studies is attributed to selection effects (liquidating firms face fire-sale dynamics) and firm-internal capital reallocation that mitigates irreversibility for continuing firms.

Scope conditions. The analysis is a partial equilibrium characterization of transitional dynamics, maintaining constant interest rates and steady-state investment policies throughout the transition (a general equilibrium extension delivering constant prices as an equilibrium outcome is provided in Appendix D). Results apply to small, permanent, unanticipated aggregate productivity shocks; nonlinearities for shocks below 5% are found to be tiny. The empirical application is specific to Chilean manufacturing establishments, 1980–2011.

Q&A

Q1. What is the economic mechanism by which capital irreversibility generates persistence in aggregate capital fluctuations?

Irreversibility creates two distinct reset points rather than one. When a negative aggregate productivity shock hits, it shifts more firms into the downsizing region. Downsizing firms, because they have been selling capital sequentially, maintain capital-productivity ratios persistently above the economy’s average and continue to do so for multiple periods. This increases the share of firms in a persistent “downsizing phase,” which prolongs the aggregate deviation from steady state. Two channels compound: first, the population tilts toward more downsizing firms; second, their mean deviations become larger and converge more slowly. Both channels increase the CIR. Crucially, without irreversibility, firms become identical after their first adjustment and there is no additional persistence beyond what fixed costs alone generate.

Q2. How are the three sufficient statistics derived, and what does each capture?

The CIR is characterized as a steady-state cross-sectional average of a recursive function m(k̂). Integrating over firms first and then time, and splitting each firm’s horizon at its first adjustment, yields three steady-state terms (Proposition 4). The first statistic, Var[k̂]/σ², measures how far firms allow their capital-productivity ratio to drift from the frictionless optimum — the “insensitivity of incomplete spells” to idiosyncratic productivity shocks. The second statistic, ν·Cov[k̂,a]/σ², is a bias-correction term that removes drift effects from the variance, ensuring only Brownian-shock sensitivity is captured. The third statistic, unique to the irreversibility case, measures how much idiosyncratic shocks alter the anticipated direction of the next adjustment — the “insensitivity of complete spells” — and equals the difference in expected cumulative deviations between departing and ending points of an inaction spell, scaled by duration.

Q3. Why is the CIR exactly twice as large under pure irreversibility (no fixed costs) as under pure fixed costs, for a given level of dispersion?

Proposition 5, case (ii) shows that with zero drift and only a price wedge, the CIR = 2 × Var[k̂]/σ², because the first and third sufficient statistics are identical and the covariance term is zero. In contrast, with only fixed costs (case (i)), the CIR = Var[k̂]/σ². The doubling arises because the price wedge generates history-dependence through the dual reset: after a firm adjusts, whether it upsized or downsized predicts its future adjustment direction. This “anticipated terminal condition” effect (captured by the third statistic) adds an equal contribution to the CIR as the pure inaction effect (the first statistic), doubling total persistence for the same cross-sectional dispersion.

Q4. How does the empirical strategy recover the capital price wedge?

The price wedge cannot be identified from the investment rate distribution alone: for any price wedge ω, the generalized hazard framework can find an adjustment hazard function Λ(k̂) such that the product Λ(k̂)·g(k̂) matches the observed investment density h(Δk̂). Instead, the authors use the CIR’s sufficient statistics — specifically the covariance term and the irreversibility term — as additional discriminating moments. At ω = 0, the model produces a negative covariance (inconsistent with the positive Cov[k̂,a] = 0.152 in the data) and no irreversibility term. At ω = 0.12, all three sufficient statistics simultaneously align with their data counterparts in relative importance (72%, 10%, 18%), selecting this wedge as preferred. The CIR level at ω = 0.12 is 1.93, somewhat below the data value of approximately 2.54–2.60, but the preferred criterion is mechanistic consistency, not just level matching.

Q5. What is the role of the Markov chain across reset points in handling path dependence?

Because optimal investment features serial correlation in the sign of adjustment (P⁻⁻ = 0.958 and P⁺⁺ = 0.124 in the data), firms’ future behavior depends on their most recent reset point. To maintain tractability, the authors condition all densities, durations, and expectations on the previous reset (upsizing g⁻(k̂) or downsizing g⁺(k̂)). The transition matrix P encoding probabilities P⁻⁻, P⁻⁺, P⁺⁻, P⁺⁺ determines the steady-state shares of upsizing and downsizing firms (as the eigenvector of P) and the renewal weights r⁻ and r⁺ that rescale conditional densities to account for observational bias (firms with longer inaction spells contribute more to the cross-section). This Markov structure is sufficient because one adjustment erases all heterogeneity except the direction of adjustment.

Q6. What do the microdata mappings recover, and how are the reset points identified?

Stage I mappings (Propositions 6–9) recover: drift ν = E[Δk̂]/E[τ]; volatility σ² from cross-spell moment E[(k̂τ’ + ντ’)² − (k̂*)²]/E[τ]; conditional means E±[k̂] as midpoints of inaction spells weighted by relative adjustment size; Var[k̂] from differences in cubed stopped values; Cov[k̂,a] from variance, average age, and the dynamic covariance E[(k̂τ’ − E[k̂])²τ’]/E[τ]; and the irreversibility term from differences in expected deviations at departing vs. ending reset points. Stage II (Proposition 10) recovers the two reset points k̂*₋ and k̂*₊ from optimality conditions that equalize the investment price to the expected discounted marginal product of capital during inaction plus the expected value of undepreciated capital, conditioning on the prior reset. The inner inaction region width k̂*₊ − k̂*₋ = 0.813 in the Chilean data, of which 45% is attributed to the exogenous price wedge and 55% to the endogenous response to the wedge.

Q7. How does the sign of Cov[k̂,a] depend on the price wedge vs. the drift?

With zero price wedge and negative drift ν < 0 (depreciation exceeding productivity growth), firms with older capital have capital-productivity ratios below average, yielding Cov[k̂,a] < 0. The drift makes old capital-productivity ratios negative. Introducing a price wedge creates downward rigidity: unproductive firms delay selling, so old firms accumulate capital-productivity ratios above average, pushing Cov[k̂,a] toward positive values. The covariance turns positive once ω > 0.08 (in the illustrative parametrization in Figure V). In the Chilean calibration at ω = 0.12, Cov[k̂,a] = 0.152 > 0, confirming that the price wedge’s effect dominates the drift’s negative effect. A positive covariance amplifies the CIR (through the second sufficient statistic with ν > 0).

Q8. What is the generalized hazard extension and why is it needed?

The baseline model with a single fixed cost θ generates an investment distribution concentrated at two mass points (purchases and sales of fixed size), which does not match the empirical distribution’s coexistence of large and small investment rates and its convex shape. The generalized hazard model replaces the deterministic fixed cost with a stochastic, state-dependent adjustment cost, parameterized by a hazard function Λ(k̂) giving the probability of adjusting per unit time at any capital-productivity ratio in the outer inaction region. This function is recovered non-parametrically from the data by fitting a Gamma distribution to the investment density and inverting the Kolmogorov Forward Equation. The generalized hazard model nests the baseline model, random fixed cost models (Thomas 2002, Khan and Thomas 2008), and asymmetric adjustment models, while preserving the sufficient statistics characterization.

Q9. How does the model handle the “problem with reinjection” that arises from path dependence after the first adjustment?

Without irreversibility, a firm’s initial state k̂₀ does not affect behavior after the first adjustment, because there is a unique reset point; subsequent behavior is independent of the aggregate shock magnitude. With irreversibility, firms only partially absorb the aggregate shock at the first adjustment, since the initial state affects the probability of subsequently upsizing or downsizing. In principle, one must track firms through infinitely many adjustments. The paper’s resolution (Proposition 2) is to note that the first adjustment erases all heterogeneity except the direction (upsizing vs. downsizing), allowing subsequent behavior to be summarized by just two numbers m(k̂*₋) and m(k̂*₊), combined with the transition probabilities P⁻(k̂₀) and P⁺(k̂₀). This yields a recursive formulation for m(k̂) governed by an HJB equation with two boundary conditions at the reset points, making the problem tractable.

Q10. What is the role of the stationarity condition in pinning down the CIR?

The HJB for m(k̂) has infinitely many solutions (m(k̂) + a for any constant a). The stationarity condition, requiring that the cross-sectional average of m(k̂) in steady state is zero (no fluctuations without shocks), pins down the unique solution. Economically, it says that average cumulative deviations from complete upsizing spells and complete downsizing spells must exactly balance the deviations from incomplete inaction spells. For upsizing firms, deviations are negative (they hold too little capital relative to average); for downsizing firms, deviations are positive (they hold too much capital). The stationarity condition imposes a linear relationship between m(k̂*₋) and m(k̂*₊) that together with the HJB uniquely determines the solution.

Q11. How are the results extended to assess nonlinearities and robustness?

Appendix G studies nonlinearities numerically in the generalized hazard model for different signs and magnitudes of the aggregate productivity shock. The authors find tiny nonlinearities and asymmetries for productivity shocks below ε = 5%, validating the first-order approximation used throughout. Appendix E.7 provides comparative statics on the output-capital elasticity α. The model is estimated with an inaction threshold of ι = 0.01 (investment rates below 1% in absolute value are treated as inaction), consistent with Cooper and Haltiwanger (2006). The investment distribution is truncated at the 2nd and 98th percentiles to remove outliers.

Q12. What broader applicability do the authors claim for the CIR sufficient statistics framework?

The authors argue the framework applies wherever path-dependent lumpy adjustments occur, including: inventory management (with two types of ordering decisions), durable goods consumption, and labor markets with sticky wages. The key requirement is the existence of a finite number of reset points and sufficient microdata to discipline the transition probabilities across them. Future extensions noted in the paper include: analysis of other aggregate shocks (profitability, capital prices, interest rates); corporate tax reform; monetary policy interacting with investment frictions; time-varying and endogenous price wedges in secondary markets; and higher-order cross-sectional moment responses (variance, skewness of capital-productivity ratios) by choosing different functions f(k̂) for the generalized CIR.

Key Concepts

Capital price wedge (ω). The fractional discount between the purchase price of capital p and its resale price p(1−ω). In the model this creates two distinct reset points for investment (one for buying at price p, one for selling at the discounted price) and represents the core source of irreversibility. It reflects asset specificity, adverse selection, intermediary fees, and obsolescence. The preferred calibrated value for Chilean manufacturing is ω = 0.12.

Cumulative Impulse Response (CIR). The integral over all future dates of the impulse response function of the average capital-productivity ratio following a small, permanent, unanticipated aggregate productivity shock. It summarizes both the impact and persistence of aggregate capital fluctuations in a single scalar. Without investment frictions, the CIR is zero (firms adjust instantaneously); the calibrated CIR at ω = 0.12 is 1.93, meaning a 1% aggregate shock generates a 1.93% cumulative deviation.

Dual reset points (k̂₋ and k̂₊).** The two levels to which firms reset their capital-productivity ratio upon adjustment: k̂*₋ after a capital purchase (upsizing) and k̂*₊ after a capital sale (downsizing). With a price wedge, k̂*₊ > k̂*₋, creating an “inner inaction region” [k̂*₋, k̂*₊] with path-dependent behavior. The inner inaction region width is 0.813 in the Chilean data.

Sufficient statistics for the CIR. Three steady-state cross-sectional moments that together fully characterize the CIR up to first order: (i) Var[k̂]/σ², the scaled cross-sectional variance of capital-productivity ratios (captures insensitivity of incomplete spells to idiosyncratic shocks); (ii) ν·Cov[k̂,a]/σ², the scaled covariance of capital-productivity ratios with capital age (a drift-bias correction); (iii) the “irreversibility term” measuring how idiosyncratic shocks change the anticipated direction of the next adjustment (unique to the irreversibility case, zero without a price wedge).

Serial correlation in adjustment sign. The property, implied by the dual-reset structure, that a firm is more likely to purchase capital following a prior purchase and more likely to sell following a prior sale. In the Chilean data, P⁻⁻ = 0.958 (probability of upsizing after a prior upsize) vs. P⁺⁺ = 0.124 (probability of downsizing after a prior downside), and a logistic regression yields an odds ratio of 3.3.

Generalized hazard function Λ(k̂). A state-dependent adjustment probability per unit time, allowing for stochastic and asymmetric fixed costs, that generates the full empirical investment rate distribution. It replaces the single deterministic fixed cost of the baseline model. The hazard function is recovered non-parametrically from microdata by fitting a Gamma distribution to the investment density and inverting the Kolmogorov Forward Equation, conditional on the price wedge.

Renewal weights (r⁻, r⁺). Weights used to construct the unconditional density of capital-productivity ratios from the two conditional densities (conditional on prior purchase g⁻(k̂) and prior sale g⁺(k̂)). They rescale adjustment shares by relative average duration, correcting for the observational bias that firms with longer inaction spells are over-represented in the cross-section: r± = (N±/N) × (E±[τ]/E[τ]).

Endogenous irreversibility. The component of the inner inaction region width (k̂*₊ − k̂*₋) that arises not from the exogenous price wedge directly but from firms’ endogenous responses to the wedge — specifically, the differences in expected marginal products and user costs across the two types of inaction spells. At ω = 0.12, 45% of the inner inaction region is attributed to the exogenous wedge and 55% to endogenous amplification.