E30 | Macro Paper Warehouse

Are Inflationary Shocks Regressive? A Feasible Set Approach

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

Layer 1 — Overview

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

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

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

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

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

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

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

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

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

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

In depth

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Key Concepts

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

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

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

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

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

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

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

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.

In depth

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.