The Macroeconomics of Irreversibility
What this paper finds — and why it matters
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.