Firm dynamics and random search over the business cycle

Layer 1 — Overview

Research Question

How do aggregate economic fluctuations reallocate workers across the firm productivity distribution over the business cycle? In particular, to what extent do recessions impede workers’ movement up the job ladder toward more productive firms?

Model and Methodology

The paper develops a tractable random search model combining three features that had not previously been integrated in a single quantitative framework: (i) firm dynamics driven by idiosyncratic productivity shocks, with endogenous entry and exit; (ii) on-the-job search, generating a job ladder in which workers gradually move toward more productive firms; and (iii) aggregate productivity shocks. Multi-worker firms post employment contracts, choose hiring rates, and decide whether to continue or exit. The key tractability result — called “size-independence” (Result 1) — shows that, under a constant-returns hiring cost technology, firms’ optimal policies (contract value, hiring rate, exit decision) are all independent of firm size, so the relevant state space reduces from the full joint distribution of firm productivity and size to the employment-weighted distribution of firm productivity alone. A further result (“rank-monotonic equilibrium,” Result 2) guarantees, under a sufficient convexity condition on hiring costs (hc’’(h)/c’(h) ≥ 1), that the optimal employment contract is increasing in firm productivity, so the job ladder maps one-for-one onto the firm productivity ladder. The optimal wage contract then admits a closed-form solution.

The model is calibrated to British data for 1997–2018. Worker-level transition rates (unemployment-to-employment, employment-to-unemployment, and job-to-job) are drawn from the British Household Panel Survey (BHPS). Firm-level data on labor productivity (value added per worker) and employment costs per worker come from the Annual Respondents Database (ARD) and Annual Business Survey (ABS), merged with the Business Structure Database (BSD). The numerical solution adapts ideas from Krusell and Smith (1998), approximating the employment-weighted productivity distribution by a small set of moments and parameterizing value functions as polynomials in the aggregate state; standard linearization methods are inapplicable because endogenous firm entry and exit introduces a discontinuity in value functions.

Main Findings

Model validation via the OP decomposition. The paper’s central validation exercise uses the Olley-Pakes (OP) decomposition of a labor productivity index constructed from firm-level data. The aggregate employment-weighted labor productivity index is decomposed into (a) the unweighted average firm productivity and (b) an interaction term (the “OP term”), which captures the covariance between employment shares and productivity — i.e., how well workers are allocated to productive firms. In the British firm-level data, approximately 20 percent of the variance of the aggregate labor productivity index is accounted for by this interaction (OP) term, with the remaining ~80 percent attributable to the unweighted average of firm productivity. The baseline model, with this moment untargeted, successfully replicates this 80/20 split. By contrast, the leading benchmark model of Moscarini and Postel-Vinay (2016) (MPV2016), calibrated to the same British data, attributes nearly all of the variance of labor productivity to the OP/worker reallocation term, grossly overstating the importance of job-ladder dynamics.

Structural decomposition of labor productivity. Using the calibrated baseline model to decompose the variance of aggregate labor productivity over the post-war British business cycle (“GDP shocks” going back to 1955), the baseline model attributes approximately 30 percent to the direct effect of the aggregate productivity shock, approximately 50 percent to changes in the distribution of active firms (the “firm ladder” or firm selection component), and approximately 20 percent to the worker reallocation component (the OP interaction term). This result is robust to an alternative calibration with a lower curvature of the hiring cost function (c1 = 1).

Persistence and mechanisms. The impact of recessions on the job ladder is persistent: while the aggregate productivity shock is typically close to its pre-recession value four years after a typical recession onset, the overall allocation of workers to firms remains clearly worse relative to the pre-recession level at that same horizon. The Great Recession, viewed through the lens of the model, is a large but not unusually large recession.

Firm selection with multiple aggregate shocks. An unexpected finding concerns the direction of firm selection. With a single aggregate productivity shock, the model generates a standard “cleansing” mechanism: negative shocks raise the firm exit threshold, so surviving firms are on average more productive. However, when additional shocks to the exogenous separation rate (δ) and hiring cost scale (c0) are included — as required to match the volatility of labor market flows — firm selection instead amplifies the decline in labor productivity. The mechanism is a general equilibrium one: a higher separation rate lowers the optimal wage contract (since greater separation risk is passed on to workers), which in turn lowers the entry-exit threshold. Less productive firms become viable because their employees face higher unemployment risk and therefore accept lower wages; moreover, a larger pool of unemployed workers makes it easier for low-productivity firms to recruit.

Wage flexibility tension. The model implies a pass-through elasticity of wages to productivity shocks of approximately 0.7, well above the 0.05–0.2 range typically found empirically.

Scope Conditions

All calibration and quantitative results pertain to Britain for the period 1997–2018 (firm-level data) and 1955–2018 (GDP-based aggregate shocks). The model abstracts from decreasing returns to scale in production and from nominal rigidities. The tractability results rely on specific assumptions about the hiring cost function; the rank-monotonicity condition requires sufficient convexity (hc’’(h)/c’(h) ≥ 1).

Layer 2 — Q&A

Q1: What is the central tractability result and why does it matter for computational feasibility?

A: Result 1 (“size-independence”) shows that, because both the production technology and the hiring cost function are constant returns to scale, the firm’s present discounted value of profits is linear in employment. As a result, per-worker profits are independent of firm size, and optimal firm policies — the hiring rate, the contract value offered to workers, and the continuation/exit decision — all depend only on the firm’s current productivity, not on its size. This collapses the state space from the full joint distribution of firm productivity and employment size to the employment-weighted measure of firm productivity Lt(p), a uni-dimensional object. Without this result, the model would require tracking the entire joint firm distribution, making it computationally intractable.

Q2: What is a rank-monotonic equilibrium (RME) and what conditions guarantee it?

A: An RME is a recursive equilibrium in which the optimal contract offered by a firm is weakly increasing in that firm’s current productivity realization, for all aggregate states. Result 2 provides sufficient conditions: (i) the Markov process for firm-specific productivity satisfies first-order stochastic dominance (more productive firms today are more likely to be more productive tomorrow), (ii) the distribution of offered contracts is everywhere differentiable (ruling out mass points), and (iii) the hiring cost function satisfies hc’’(h)/c’(h) ≥ 1 — a sufficient convexity condition. The economic interpretation of the convexity condition is that firms must find retention (offering higher wages) sufficiently costly relative to new hiring that more productive firms optimally choose to use the wage margin to limit quits. The baseline calibration yields c1 ≈ 5.9 (so costs are highly convex in the hiring rate), though results are also reported for the minimum permissible c1 = 1.

Q3: What does the optimal employment contract look like in a rank-monotonic equilibrium, and what does it reveal about rent extraction?

A: In an RME, the optimal contract V(p,ω,L) is a weighted average of the value of unemployment U(ω,L) and the firm-workers’ joint surplus S(p,ω,L), where the weights are determined endogenously by the employment-weighted measure of firm productivity L. Specifically, the contract integrates the surplus of all firms with productivity below p, weighted by the share of employed workers at those firms, and divided by the mass of job seekers willing to accept the contract. As the employed workers’ relative search intensity s approaches zero, the contract converges to the value of unemployment — workers receive no rents. The endogenous bargaining weight evolves with the aggregate state over the business cycle, unlike standard Nash bargaining models with a fixed exogenous weight.

Q4: What firm-level moments are used to calibrate the steady-state model, and what is the logic behind the parameter-moment mapping?

A: Eight moments are targeted. From the BHPS worker data: the average UE rate (0.058) pins down the scale of hiring costs c0; the average EU rate (0.003) pins down the exogenous separation rate δ; and the average EE (job-to-job) rate (0.016) pins down the relative search intensity s. From the firm-level ARD/BSD data: average firm size (12.1 employees) pins down the entry probability µ; the share of job destruction from firm exits (0.526) disciplines the flow value of unemployment b; the autocorrelation of firm employment ln(n) (0.949 annually) disciplines the persistence of idiosyncratic productivity ρp; the interquartile range of firm-level labor productivity (1.129 log points) disciplines the volatility of idiosyncratic shocks σp; and the regression coefficient of firm employment growth on lagged labor productivity (0.136) disciplines the curvature of hiring costs c1. The baseline calibration fits all eight moments closely.

Q5: How does the calibrated model match non-targeted moments, and what does this establish?

A: The model generates several realistic features not targeted in calibration. It produces a realistic Pareto tail for the employment-size distribution (Pareto tail exponent of 1.033 in the model vs. 1.066 in the data), which arises from the combination of size-independent growth rates and firm entry and exit — conditions identified in the literature as generating power law distributions. The model also matches the dispersion of employment costs per worker across firms (capturing about 70 percent of the interquartile range of ECi,t), the slope of a regression of employment costs on labor productivity (model: 0.685 vs. data: 0.704), and the slope of a regression of employment growth on employment costs (model: 0.162 vs. data: 0.131). These non-targeted matches provide independent validation of the model’s wage-determination mechanism.

Q6: Why is a single aggregate productivity shock insufficient to match labor market fluctuations, and what additional shocks are needed?

A: With a single aggregate productivity shock calibrated to match the autocorrelation and standard deviation of log GDP, the model generates labor market fluctuations that are roughly an order of magnitude smaller than in the data. For example, the standard deviation of the EU transition rate is 4.1×10⁻⁴ in the single-shock model versus 2.3×10⁻³ in the data. Adding a discount rate shock (ω,r) partially helps but still leaves the job-finding rate (UE) more than 50 percent too smooth. Adding a separation rate shock (ω,δ) substantially increases EU and UE volatility but generates insufficient EE (job-to-job) volatility. The combination (ω,δ,c0) — adding a shock to the scale of hiring costs c0 — brings the standard deviations of EU and UE close to the data (2.0×10⁻³ and 4.0×10⁻⁴ vs. data 2.3×10⁻³ and 2.7×10⁻⁴), though the model still generates slightly under half the observed volatility in EE rates. This combination is the baseline for the quantitative analysis.

Q7: What is the OP decomposition, how is it computed from the firm-level data, and what does it measure in the model?

A: The aggregate labor productivity index LPt is constructed from firm-level data as the employment-share-weighted average of log value added per worker across firms. The OP decomposition writes this as LPt = LPt_bar + OPt, where LPt_bar is the unweighted (simple) average of firm-level productivity and OPt is the covariance between employment shares and labor productivity (the “interaction term”). In the data, OPt increases when workers are disproportionately employed at above-average-productivity firms. In the model, LPt_bar maps onto the average (log) productivity of active firms — the support of the job ladder — while OPt maps onto the difference between the employment-weighted and the unweighted averages of firm productivity, directly measuring how high up the ladder workers are located relative to the set of active firms. Around 20 percent of the variance of LPt in the British data is accounted for by OPt, and the model replicates this.

Q8: How does the Great Recession appear in the OP decomposition, and does the model fit the decomposition during this episode?

A: During the Great Recession (2008q2–2009q3 in the UK), around 20 percent of the overall fall in the labor productivity index is accounted for by the fall in the OP interaction term, with the remaining 80 percent coming from the fall in the unweighted average firm productivity. The model, even though it does not target this decomposition in calibration, successfully matches both the average firm productivity component and the interaction (OP) component during the Great Recession. This matching holds both in the baseline calibration (c1 ≈ 5.9) and in the alternative calibration with c1 = 1. The model also matches the analogous decomposition for employment costs per worker (ECt), an additional non-targeted validation.

Q9: Why does firm selection amplify rather than cleanse in the baseline multi-shock calibration?

A: In the single-shock (productivity ω only) model, a negative productivity shock lowers surplus at all firms, raising the exit threshold pE and thus selecting out low-productivity firms — the standard “cleansing” mechanism. In the multi-shock baseline, the additional separation rate shock (δ) generates a less intuitive mechanism. A higher δ lowers the optimal wage contract (since increased separation risk is passed on to workers: ∂V/∂δ ≤ 0), which reduces the value of continued employment. This lowers the joint firm-worker surplus threshold for exit, making it viable for low-productivity firms to remain active. Moreover, the larger pool of unemployed workers (generated by the δ shock) depresses the outside option of workers and makes it easier for low-productivity firms to recruit. As a result, the entry-exit threshold pE,t falls — the set of active firms becomes less productive on average — producing a negative firm selection contribution to labor productivity and a positive (amplifying rather than cleansing) contribution to the variance of LPt.

Q10: What is the structural variance decomposition of labor productivity in the baseline model?

A: Simulating the baseline model over the post-war British business cycle (1955–2020, GDP shocks), the variance of aggregate labor productivity LPt decomposes into three structural terms: approximately 30 percent (0.296) from the direct effect of the aggregate productivity shock ln(ωt); approximately 50 percent (0.541) from changes in the average productivity of active firms E[KP bar_t(ln p)] — the “firm ladder” or firm selection component; and approximately 20 percent (0.163) from the worker reallocation component OPt = E[LP bar_t(ln p)] − E[KP bar_t(ln p)]. This decomposition implies that roughly 70 percent of fluctuations in labor productivity are driven by worker reallocation broadly defined (the firm ladder plus the interaction term), with the firm selection component being the largest single driver. The result is robust to the alternative c1 = 1 calibration (30/49/22 percent split).

Q11: How does the baseline model compare to MPV2016 in the variance decomposition?

A: In the multi-shock calibration (ω,δ,c0), the MPV2016 model calibrated to the same British data attributes approximately 97.7 percent (0.977) of the variance of LPt to the worker reallocation (OP) term, with essentially none attributed to a firm selection term (since there is no firm entry and exit in MPV2016). This is nearly five times the 20 percent share attributed to worker reallocation in the data and in the baseline model. In the single-shock (ω) calibration, both models attribute a more modest share to worker reallocation (7.2 percent for the baseline model, 0.1 percent for MPV2016 with c1=5), and the difference narrows considerably. The contrast thus stems from the interaction of firm dynamics with multiple aggregate shocks: allowing for endogenous firm entry and exit is critical to prevent the model from overstating the role of the job ladder.

Q12: How persistent is the impact of recessions on the job ladder, based on the model simulations?

A: The paper simulates the structural decomposition of labor productivity starting from each of seven post-war British recessions (defined by two consecutive quarters of negative GDP growth). On average across these recessions, the aggregate productivity shock ln(ωt) is close to its pre-recession level by four years after the recession onset. However, the overall employment-weighted average productivity E[LP bar_t(ln p)] — reflecting workers’ position on the job ladder — remains clearly below its pre-recession value at the four-year horizon, indicating persistent misallocation. The OP interaction term accounts for approximately 20 percent of the total drop in the employment-weighted productivity measure three years after a typical recession onset. Through the model’s lens, the Great Recession is a large recession but not an outlier relative to the historical distribution.

Q13: What does the counterfactual with countercyclical unemployment benefits reveal about the tradeoff between firm selection and worker reallocation?

A: When the flow value of unemployment is made countercyclical (falling in recessions, rising in expansions — mimicking US unemployment insurance extension programs), the model generates a sign reversal in the firm selection (“firm ladder”) component. With countercyclical b, the unemployment value rises in recessions, which raises the minimum wage firms must offer and raises the exit threshold pE,t: fewer low-productivity firms survive, improving the composition of active firms. However, countercyclical benefits also amplify the slowdown in job-to-job reallocation: the higher value of unemployment reduces workers’ willingness to accept job offers, and all firms cut recruitment since optimal wage contracts must rise. The OP interaction term therefore falls more sharply than in the baseline model. The counterfactual with ϵb,ω ∈ {−100, −50} finds that the positive “firm ladder” effect dominates on net, so the overall allocation of workers to firms improves relative to the baseline after a typical recession under countercyclical unemployment benefits.

Q14: What is the numerical solution method, and why are standard linearization approaches inapplicable?

A: The model is solved in two steps. First, aggregate shocks are shut down and the steady-state rank-monotonic equilibrium is solved numerically by discretizing the firm productivity process (401 grid points via Tauchen’s method) and iterating on the value function and the employment-weighted productivity measure until convergence. Second, aggregate shocks are reintroduced using a simulation-based approach adapted from Krusell and Smith (1998): the employment-weighted distribution of productivity is summarized by Nm = 2 moments (plus the unemployment rate), and the value functions are parameterized as polynomials in the aggregate state, with coefficients updated by regression until convergence. Standard linearization methods (Reiter 2009) are inapplicable because the endogenous entry-exit decision creates a kink (discontinuity) in value functions at the productivity threshold pE, making first-order approximations around the steady state inaccurate. Accuracy tests based on den Haan (2010) show that the polynomial approximation generates errors of at most 0.065 percent for value functions and at most 1 percentage point for the unemployment rate across simulation paths.

Key Concepts

1. Rank-Monotonic Equilibrium (RME) A recursive equilibrium in which the optimal state-contingent employment contract V(p,ω,L) offered by a firm is weakly increasing in the firm’s current productivity realization p, for all aggregate states (ω,L). This property implies that the job ladder maps one-for-one onto the firm productivity ladder: workers always prefer to work at more productive firms. The paper shows this property holds under a sufficient convexity condition on hiring costs (hc’’(h)/c’(h) ≥ 1) and first-order stochastic dominance of the productivity process.

2. Size-Independence The property that a firm’s optimal policies — the hiring rate h(p), the employment contract V(p), and the entry/exit decision χ(p) — are all independent of the firm’s current employment size n. This follows from constant returns to scale in production and hiring, which implies that firm profits are linear in employment. Size-independence reduces the model’s relevant state space to the employment-weighted distribution of firm productivity, enabling tractability.

3. Employment-Weighted Distribution of Firm Productivity (L_t(p)) The measure recording, for each productivity level p, the total employment at firms with productivity at most p. This is the sufficient statistic for the state of the job ladder at any point in time: combined with the aggregate shock ω, it determines all equilibrium policy functions and value functions. In the model, it replaces the full joint distribution of firm productivity and employment size that would otherwise be required.

4. OP Decomposition (Olley-Pakes Decomposition) The decomposition of the aggregate employment-weighted labor productivity index LPt into: (a) the unweighted average firm productivity LPt-bar, which summarizes the productivity of active firms (the support of the job ladder); and (b) an interaction term OPt, the covariance between employment shares and firm-level productivity, which measures how well workers are allocated across the productivity distribution (i.e., how high up the ladder workers sit given the set of active firms). In the model, (a) maps to E[KP bar_t(ln p)] and (b) maps to OPt = E[LP bar_t(ln p)] − E[KP bar_t(ln p)].

5. Contract Posting The wage-setting protocol in which each firm commits upon entry to a full state-contingent employment contract — a schedule mapping each future realization of aggregate and idiosyncratic productivity to a wage and continuation decision — and is bound by an equal treatment constraint to offer the same contract to all employees. Workers cannot renegotiate based on outside offers. This protocol produces a well-defined closed-form for the optimal contract in an RME and differs from alternating-offer bargaining (Nash bargaining) in that the bargaining weights are endogenous rather than fixed.

6. Firm-Workers’ Joint Surplus (S_t(p)) The total present discounted value accruing to the firm-worker pair: firm profits per worker plus the contract value promised to workers. Because utility is transferable (risk neutrality) and the firm fully commits to its contract, this surplus depends only on the firm’s current productivity and the aggregate state — not on the promised contract value V. The surplus S_t(p) is the key object determining firm entry/exit (the firm continues if and only if S_t(p) ≥ U_t) and optimal hiring (the marginal return to an additional hire equals S_t(p) − V(p)).

7. Cleansing vs. Anti-Cleansing Firm Selection In models with endogenous firm entry and exit, a negative aggregate shock can either raise or lower the productivity threshold for firm survival. “Cleansing” refers to the standard mechanism where a negative productivity shock raises the exit threshold, selecting out low-productivity firms and improving the average quality of survivors. “Anti-cleansing” (as in the baseline multi-shock calibration) occurs when separation rate or hiring cost shocks lower the optimal wage contract and reduce the exit threshold, allowing less productive firms to survive and worsening average firm productivity.