Capital Reallocation and Aggregate Productivity

Capital Reallocation and Aggregate Productivity Russell W. Cooper and Immo Schott This Version: June 4, 206 First Version: December 203 Abstract This paper studies the effects of cyclical capital reallocation on aggregate productivity. Frictions in the reallocation process are a source of factor misallocation and lead to variations in measured aggregate productivity over the business cycle. The effects are quantitatively important in the presence of fluctuations in the cross-sectional dispersion of plant-level productivity shocks. The cyclicality of the productivity losses depends on the joint distribution of capital and plant-level productivity. Even without aggregate productivity shocks, the model has quantitative properties that resemble those of a standard stochastic growth model: (i) persistent variation in the Solow residual, (ii) positive comovement of output, investment and consumption and (iii) consumption smoothing. The estimated model with dispersion shocks alone accounts for nearly 85% of the time series variation in the observed Solow residual. Contrary to a model with productivity shocks, the model driven by dispersion shocks can mimic the dynamics of reallocation and the cross sectional dispersion in average capital productivity. Instead of relying on approximative solution techniques we show analytically that a higher-order moment is needed to solve the model accurately. Motivation With heterogenous plants the assignment of capital, labor and other inputs across production sites impacts directly on aggregate productivity. Frictions in the reallocation process thus lead to the misallocation of factors of production (relative to a frictionless benchmark) and effects aggregate productivity. This point Thanks to Dean Corbae for lengthy discussions on a related project. We are grateful to Nick Bloom, Michael Elsby, Matthias Kehrig, Thorsten Drautzburg, and Sophie Osotimehin for comments and suggestions on the project and to seminar participations at the European University Institute, the European Central Bank, the CEA, and the Scuola Superiore Sant Anna in Pisa for comments and questions. The first author thanks the NSF under grant #089682 for financial support. This version adds an endogenous adjustment decision to our December 203 NBER Working Paper. Department of Economics, the Pennsylvania State University and NBER, russellcoop@gmail.com Department of Economics, Université de Montréal and CIREQ, Immo.Schott@umontreal.ca

MOTIVATION lies at the heart of the analysis of productivity both within and across countries in Maksimovic and Phillips (200), Hsieh and Klenow (2009), Bartelsman, Haltiwanger, and Scarpetta (203) Restuccia and Rogerson (2008) and others. In this paper we consider the cyclical dimension of reallocation in the presence of capital reallocation costs. The model is estimated to match both reallocation and aggregate moments. In important empirical contributions, Eisfeldt and Rampini (2006), Kehrig (20), Kehrig and Vincent (203), and Osotimehin (206) show that capital reallocation is procyclical and that the cross-sectional productivity dispersion behaves countercyclically. 2 This not only underlines the significance of heterogeneity in the production sector but also suggests that frictions in the reallocation of capital may produce cyclical effects on output and aggregate productivity over the business cycle. The cyclical reallocation process generates an important distinction between the cyclical behavior of aggregate total factor productivity (TFP) and the Solow residual (SR), calculated from an aggregate production technology. 3 In the standard Real Business Cycle model, these are the same. But in a model with heterogenous producers and costly reallocation, the SR reflects both TFP and the assignment of factors of production to heterogeneous production sites. cyclical. In fact, this latter reallocation effect is itself The primary objective of this paper is to integrate these findings about cyclical reallocation along with the distinction between aggregate TFP and the SR with more standard properties of aggregate fluctuations. Using a dynamic general equilibrium model we ask: What are the driving processes and propagation mechanisms that generate the observed moments in economic aggregates as well as procyclical reallocation and countercyclical dispersion of productivity? We consider two shocks: (i) aggregate total factor productivity and (ii) shocks to the dispersion of idiosyncratic shocks. The focus on aggregate total factor productivity is traditional, as in the vast literature starting from the contributions of Kydland and Prescott (982) and King, Plosser, and Rebelo (988). While successful in matching some aggregate moments, those exercises study homogenous production units and thus ignore the significance of factor reallocation for aggregate productivity. Moreover, those models are understood to lack endogenous propagation, making the serial correlation of exogenous productivity key to matching the data. With heterogenous plants, we argue that a model economy driven only by shocks to total factor productivity fails to match the joint dynamics of reallocation and the cross-sectional dispersion of productivity. In contrast to Midrigan and Xu (204) there are no borrowing frictions. They argue that these frictions do not create large losses from misallocation between firms, but potentially large losses by deterring entry. In Cui (204) capital reallocation is procyclical because partial irreversibility interacts with financial constraints. 2 Eisfeldt and Rampini (2006) use dispersion in firm level Tobin s Q, dispersion in firm level investment rates, dispersion in total factor productivity growth rates, and dispersion in capacity utilization. Kehrig (20) constructs dispersion measures based on TFPR estimates. Kehrig and Vincent (203) find that the dispersion of the cross-sectional distribution of capital productivity is countercyclical as well. Osotimehin (206) finds that the efficiency of resource allocation across firms within the same sector is procyclical and quantitatively more important than entry and exit. 3 Thanks for Susanto Basu for urging us to make these terms clear. 2

MOTIVATION Matching these other moments requires the presence of a different shock and, as we shall see, an internal source of propagation through the endogenous evolution of higher order moments. Our second shock, directly to the dispersion of idiosyncratic productivity and hereafter termed a dispersion shock, is motivated by the evidence cited earlier from Eisfeldt and Rampini (2006), Kehrig (20), Kehrig and Vincent (203), Osotimehin (206) and others which point to the quantitative significance of cyclical factor reallocation and its contribution to measures of aggregate productivity. From Olley and Pakes (996) and related contributions, the combination of heterogeneous plants and adjustment frictions means that aggregate output depends on the allocation of capital across plants. In our analysis, this assignment of capital is captured by the covariance between capital and plant-level productivity. This covariance appears in the state vector of the planner s problem and plays a central role in generating the cyclical properties of reallocation and the dispersion in the average productivity of capital. Moreover, this covariance is a slow-moving object and thus creates endogenous propagation. The fact that the covariance matters as a moment for determining the optimal allocation is indicative of the significance of reallocation effects. If the covariance was not needed for characterizing optimal allocations, then reallocation could not have a cyclical effect on aggregate output. 4 reflects the cyclical gains to capital reallocation. Thus the covariance Relative to the literature emerging from Krusell and Smith (998), almost all papers find that approximating a joint distribution with its first moment is sufficient and higher order moments are not necessary. 5 Our results indicate that in the presence of reallocation shocks, these higher order moments do matter. Besides the issue of approximation, not properly taking cross-sectional heterogeneity into account will lead to a mis-measurement of TFP. So, for example, it is possible for the Solow residual, i.e. measured aggregate TFP, to fall due to the misallocation of aggregate resources rather than from an actual fall in TFP. Thus there is a potentially powerful interaction between the traditionally measured aggregate TFP and these two exogenous aggregate shocks. This is a central point in Foster, Haltiwanger, and Krizan (200) and related studies that isolate the contribution of reallocation to aggregate productivity. Our analysis is distinguished from the existing literature by our joint focus on these shocks and assessing their quantitative implications for a rich set of facts. Other studies either ignore dispersion shocks or do not include facts about reallocation in their analysis. Neither Eisfeldt and Rampini (2006) nor Kehrig (20) include shocks to the distribution of plant-level productivity in their models. These shocks are prominent in, for example, Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (202) and Bachmann and Bayer (203), but their implications are not included in the set of moments under consideration. 6 4 As discussed below, even if the covariance is constant, reallocation may be important for average productivity. 5 An exception is Bachmann and Bayer (203). 6 Specifically, Bloom et al. (202) estimate, using SMM, the parameters governing their uncertainty process to match the distribution of plant-level TFP shocks and the coefficients of a GARCH representation of the growth in the Solow residual. Importantly, the aggregate TFP process is set based upon calibrations that match the dynamics of the Solow residual. For our analysis, the process of the Solow residual is the outcome of the interaction of fundamental shocks and the reallocation process and thus is not treated as an input into the quantitative analysis. Moreover, higher order moments do not appear 3

2 FRICTIONLESS ECONOMY This emphasis on dispersion shocks is not misplaced. Matching aggregate moments along with the cyclical patterns of both reallocation and the cross sectional distribution of productivity requires the presence of the dispersion shock and a role for cross-sectional heterogeneity. This is captured by the covariance between plant-level productivity and capital in the state vector. If the only shocks in the economy are to aggregate TFP, then the productivity loss from costly reallocation has no cyclical element, and the model is unable to match the reallocation facts. We estimate the aggregate TFP and dispersion shocks along with the parameters of adjustment costs to match the reallocation and aggregate business cycle moments. With the estimated model, the dispersion shocks capture about 85% of the observed variability in the Solow residual. Adding an aggregate TFP shock to this estimated model does not improve the fit. 2 Frictionless Economy To fix basic ideas and notation, consider an economy with heterogeneity in plant-level productivity and no frictions in the accumulation of capital nor in its reallocation. The planner maximizes for all (A, K). The constraints are V (A, K) = max K,k(ε) u(c) + βe A AV (A, K ) () c + K = y + ( δ)k, (2) ε k(ε)f(ε)dε = K, (3) y = A εk(ε) α f(ε)d(ε). (4) ε The objective function is the lifetime utility of the representative household. The state vector has two elements: A is aggregate TFP and K is the aggregate stock of capital. There is a distribution of plant specific productivity shocks, f(ε) which is (provisionally) fixed and hence omitted from the state vector. At the beginning of the period, A, as well as the idiosyncratic productivity shocks, ε, realize. There are two controls in (). The first is the choice of aggregate capital for the next period. The second is the assignment function, k(ε), which allocates the given stock of capital across the production sites, indexed by their current productivity. While aggregate capital K requires one period time-to-build, the reallocation of existing capital takes place instantaneously and is given by k(ε). The resource constraint for the accumulation of aggregate capital is given in (2). The constraint for explicitly in their state space. Bachmann and Bayer (203) also calibrate treating the Solow residual as aggregate TFP and focus on business cycle, not reallocation moments. 4

2. Optimal Choices 2 FRICTIONLESS ECONOMY the allocation of capital across production sites in given in (3). From (4), total output, y, is the sum of the output across production sites. The production function at any site is y(k, A, ε) = Aεk α (5) where k is the capital used at the site with productivity ε. 7 Both idiosyncratic and aggregate productivity shocks ε and A can be persistent. We assume α < as in Lucas (978). 8 In this frictionless environment, a plants optimal capital stock is entirely determined by ε. The assumption of diminishing returns to scale, α <, implies that the allocation of capital across production sites is non-trivial. There are gains to allocating capital to high productivity sites but there are also gains, due to α <, from spreading capital across production sites. 2. Optimal Choices Within a period, the condition for the optimal allocation of capital across production sites is given by Substituting into (4) yields k(ε) = K y = AK α ( ε α ε ε ε α f(ε)dε. 9 (6) ε α f(ε)dε ) α. (7) This is a standard aggregate production function, AK α, augmented by a term that captures a love of variety effect from the optimal allocation of capital across plants. With a given distribution f( ) the idiosyncratic shocks magnify average aggregate productivity as the planner can reallocate inputs to the more productive sites. The condition for intertemporal optimality is u (c) = βev K (A, K ) so that the marginal cost and expected marginal gains of additional capital are equated. Using (), this condition becomes 7 Labor and other inputs are not made explicit. One interpretation is that these inputs have no adjustment costs and are optimally chosen each period, given the state. In this case, the marginal product of labor (and other inputs) will be equal across production sites. This does not imply equality of the marginal products of capital. Adding labor adjustment, perhaps interactive with capital adjustment, would be a natural extension of our model. Presumably, adding labor frictions would enhance our results. Bloom et al. (202) include labor adjustment costs while Bachmann and Bayer (203) assume flexible labor. 8 As in Cooper and Haltiwanger (2006), estimates of α are routinely below unity. This is interpreted as reflecting both diminishing returns to scale in production and market power due to product differentiation. For simplicity, our model ignores product differentiation and treats the curvature as reflecting diminishing returns. The analysis in Kehrig (20) includes product differentiation at the level of intermediate goods. 9 The first order condition implies αaεk(ε) α = η for all ε, where η is the multiplier on (3). It implies k(ε) = η Using (3), η = AαK α ( ε ε α f(ε)dε ) α. Putting these two conditions together yields (6). α αaε. 5

2.2 Aggregate Output and Productivity 2 FRICTIONLESS ECONOMY ( ) ] α u (c) = βeu (c ) [( δ) + A αk α ε α f(ε)dε. (8) ε The left side is the marginal cost of accumulating an additional unit of capital. The right side is the discounted marginal gain of capital accumulation. Part of this gain comes from having an extra unit of capital to allocate across production sites in the following period. The productivity from these production sites depend ons two factors, the expected future values of aggregate productivity, A, and the cross sectional distribution of idiosyncratic shocks, f(ε). The choice of k for each plant within a period is independent of the choice between consumption and saving. The planner optimally allocates capital to maximize the level of output and then allocates output between consumption and capital accumulation. Clearly, once we allow for limits to reallocation, the capital accumulation decision will depend upon the future allocation of capital across production sites. 2.2 Aggregate Output and Productivity For this economy, there is a fundamental link between productivity and the assignment of capital to plants. 0 Let k(ε) = ξ(ε)k, so that ξ(ε) is the fraction of the capital stock going to a plant with productivity ε. Then (4) becomes: Define a measure of productivity à as y = AK α ε εξ(ε) α f(ε)dε (9) à A εξ(ε) α f(ε)dε. (0) ε As is well understood from the Olley and Pakes (996) analysis of productivity, the level of aggregate output will depend on the covariance between the plant-level productivity and the factor allocation. Let µ = ε ε ξ(ε)α f(ε)d(ε), and φ = cov(ε, ξ(ε) α ), where ε is the mean of ε. Total output depends on these two moments: y = ÃKα () where à is given by à A(µ + φ). (2) This connection between aggregate productivity and the cross sectional allocation of capital will be fundamental to our analysis. As we shall see, the presence of adjustment frictions implies that these moments are the source of endogenous movements in aggregate productivity. 0 This builds on Olley and Pakes (996). 6

3 COSTLY REALLOCATION Researchers interested in measuring TFP from the aggregate data will typically uncover à rather than A. This is the mis-measurement referred to earlier. As the discussion progresses, we will refer to à as the Solow residual, as distinct from aggregate TFP. From (0) there are three factors which influence Ã. The first one is A. The influence of A, aggregate TFP, on the Solow residual à is direct and has been central to many studies of aggregate fluctuations. Second, fluctuations in f(ε) influence à because variations in the cross sectional distribution of the idiosyncratic shocks lead to different marginal productivities of plants and thus changes in the Solow residual. Without any costs of reallocation, a mean-preserving spread in the distribution of idiosyncratic shocks, for example, creates opportunities to assign more capital to higher productivity sites and thus output as well as productivity will increase. Finally, there is the allocation of factors, ξ. If factors are optimally allocated, then the distribution of capital over plants does not have an independent effect on Ã. However, the presence of frictions may imply that, in a static sense, capital is not efficiently allocated. In that case, even with f(ε) fixed, the reallocation process will lead to variations in Ã. This is the topic of the next section. 3 Costly Reallocation The allocation of capital over sites has significant effects on measured total factor productivity in the presence of idiosyncratic productivity shocks. In a frictionless economy with fixed f(ε) there are no cyclical effects of reallocation on productivity. However, there is ample evidence in the literature for both non-convex and convex adjustment costs associated with changes in plant-level capital. Introducing these adjustment costs will enrich the analysis of productivity and reallocation. There are two distinct frictions to study, corresponding to the two dimensions of capital adjustment. The first, our focus here, is costly reallocation in which the friction is associated with the allocation of capital across the production sites. The second is costly accumulation in which the adjustment cost refers to the cost of accumulating rather than allocating capital. Given the emphasis on reallocation, we study a tractable yet rich model of reallocation costs. 3. The Planner s Problem For the dynamic program of the planner in the presence of adjustment costs, the state vector includes aggregate productivity A, the aggregate capital stock K, and Γ, the joint distribution over beginning-ofperiod capital and productivity shocks across plants. Γ is needed in the state vector because the presence of adjustment costs implies that a plant s capital stock may not reflect the current draw of ε. Following the discussion above, variations in f(ε), the distribution of idiosyncratic shocks, influence measured aggregate productivity. To study this effect further, we introduce shocks to the variance of 7

3. The Planner s Problem 3 COSTLY REALLOCATION idiosyncratic productivity shocks, parameterized by λ. Such changes can be interpreted as variations in uncertainty. Specifically, consider a mean-preserving spread (MPS) in the distribution of ε. In a frictionless economy such a spread would incentivize the planner to carry out more reallocation of capital between plants because capital can be employed in highly productive sites. Let s = (A, λ; Γ, K) denote the vector of aggregate state variables. Note the assumed timing: changes in the distribution of idiosyncratic shocks are known in the period they occur, not in advance. 2 As a consequence production and reallocation depend on the current realizations of A and λ. Each period, the planner has the opportunity to reallocate capital across all plants. However, in order to learn about the plant specific state, (ε, k), before reallocation and production take place, the planner must pay a fixed adjustment cost, denoted F, scaled by the aggregate capital stock, as in (4). 3 The adjustment cost is independently and identically distributed across time and plants. The distribution of F is denoted G. The adjustment status of a plant is given by j = a, n, where a stands for adjusting, while n stands for non-adjusting, depending on whether or not the planner decides to pay the plant s fixed cost of adjusting. Denote the fraction of adjusting plants as π. This specification of adjustment costs has a couple of key advantages. First, gains to adjustment will be procyclical: i.e. the gains to reallocation will increase in A and the costs of adjustment are independent of the current value of productivity. In this way, the countercyclical adjustment costs of Eisfeldt and Rampini (2006) emerge endogenously in our model. Second, the cost of adjustment is scaled by the aggregate capital stock at the plant, K α. This will not matter for the analysis of reallocation but will add tractability since, as in (), output net of adjustment costs will be proportional to the capital stock. Given the state, the planner makes an investment decision K, determines π, and chooses how much capital to reallocate between adjusting plants, (k j, ε j ) a. Let k j (k, ε, s) for j = a, n denote the capital allocation to a plant that enters the period with capital k and productivity shock ε in group j after reallocation. The capital of a plant in group j = a is adjusted and is optimally set by the planner to the level k a (k, ε, s). The capital of a plant in group j = n is not adjusted so that k n (k, ε, s) = k. The choice problem of the planner is: V (A, λ; Γ, K) = max π, ka(k,ε,s),k u(c) + βe [A,Γ,λ, A,Γ,λ]V (A, λ ; Γ, K ) (3) subject to the resource constraint (2), amended to include adjustment costs and reflecting the fact that A number of recent papers such as Bloom (2009) and Gilchrist, Sim, and Zakrajšek (204) find that time-varying uncertainty can have effects on aggregate output, while Bachmann and Bayer (203) contest the importance of these shocks. 2 Other models, such as Bloom et al. (202), include future values of λ in the current state as a way to generate a reduction in activity in the face of greater uncertainty about the future. This is not a focus of our analysis. 3 Importantly, this cost is independent of A. This is part of the mechanism that creates cyclical reallocation with TFP shocks alone. Else, variations in A would have no impact on reallocation moments. Our chosen specification thus provides an opportunity for TFP shocks to match some of the reallocation moments. Even with this choice, we find that variations in TFP alone are not enough to match the reallocation moments. We return to alternative specifications in section 5.2. 8

3. The Planner s Problem 3 COSTLY REALLOCATION some but not all plants adjust their capital stocks: with y = (k,ε) a F (π) c + K = y + ( δ)k K α F dg(f ), (4) Aε k a (k, ε, s) α dγ(k, ε) + 0 (k,ε) n Aε k n (k, ε, s) α dγ(k, ε). (5) Here output is simply (4) split into adjusting and non-adjusting plants, where j = a, n indexes the state of plant adjustment. The adjustment cost in (4) is linked to the fraction of plants π chosen for adjustment through the CDF of adjustment costs. Specifically, given π, the planner selects plants starting with the lowest adjustment costs until the desired fraction π of plants are adjusted. Through this process, the maximal cost incurred is denoted F (π) and given implicitly by π = G(F ). Once the maximal adjustment cost is determined, the total amount paid is the integral over the distribution of adjustment costs up to F (π), as in the last term of (5). Given π, the amount of capital over all plants must sum to total capital K: π k a (k, ε, s)dγ(k, ε) + ( π) k n (k, ε, s)dγ(k, ε) = K. (6) (k,ε) a (k,ε) n As the capital is plant specific, it is necessary to specify transition equations at the plant level. Let i = K K denote the gross investment rate so that K K = ( δ + i)k is the aggregate capital accumulation equation. To distinguish reallocation from aggregate capital accumulation, assume that the capital at all plants, regardless of their reallocation status, have the same capital accumulation. The transition for the capital this period (after reallocation) and the initial plant-specific capital next period is given by k j(k, ε, s) = ( δ + i) k j (k, ε, s), (7) for j = a, n. Due to the reallocation frictions, k a (k, ε, s) is not given by (6). The capital reallocation decision is now forward looking due to the recognition that adjustment may be prohibitively expensive in the future. The quantitative analysis will focus on reallocation of capital, defined as the fraction of total capital that is moved between adjusting plants within a period. Following a new realization of idiosyncratic productivity shocks, the planner will reallocate capital from less productive to more productive sites. Aggregate output is thus increasing in the amount of capital reallocation. As k a (k, ε, s) denotes the post-reallocation capital stock of a plant with initial capital k, the plant-level reallocation level would be r(k, ε, s) = k a (k, ε, s) k. Aggregating over all the plants who adjust, the aggregate reallocation level is 9

3.2 Joint Distribution of Capital and Productivity 3 COSTLY REALLOCATION R(s) 0.5 r(k, ε, s)dγ(k, ε). (8) (k,ε) a The multiplication by 0.5 avoids double counting flows between adjusting plants. Throughout the analysis, R is a level of reallocation not a reallocation rate. 3.2 Joint Distribution of Capital and Productivity In the presence of reallocation frictions, the state space of the problem includes the cross sectional distribution, Γ. Consequently, when making investment and reallocation decisions the planner needs to forecast Γ. It is computationally not feasible to follow the joint distribution of capital and productivity shocks over plants. However, our setup allows us to represent the joint distribution by two of its moments. Unlike the literature following Krusell and Smith (998) this is an exact solution, not an approximation. The appropriate set of moments, as developed above for an economy without reallocation frictions, comes from the representation of aggregate output as y = ÃKα with à = A(µ + φ).4 For the economy with costly adjustment and an endogenous reallocation rate of π, aggregate output, taken from (5), becomes F (π) y = AK α [π(µ a + φ a ) + ( π)(µ n + φ n )] K α F dg(f ), (9) where µ j εe( k j (k, ε, s) α ) and φ j Cov(ε, k j (k, ε, s) α ), for j = a, n. Instead of Γ we retain µ n and φ n in the state vector of (3). This allows us to write the Solow residual as: 0 à = A[π(µ a + φ a ) + ( π)(µ n + φ n )]. (20) The two moments, (µ n, φ n ), contain all the necessary information about the joint distribution of capital and productivity among non-adjusting plants. 5 over plants, which maps into values of µ a and φ a. Each period the planner chooses an allocation of capital Note that by keeping (µ n, φ n ) in the state space, we are not approximating the joint distribution over capital and productivity since the two moments can account for all the variation of the joint distribution. That is, the covariance appears in (9) precisely because output depends on the assignment of capital to 4 As noted earlier, this decomposition of productivity taken from Olley and Pakes (996) highlights the interaction between the distribution of productivity and factors of production across firms. Gourio and Miao (200) use a version of this argument, see their equation (46), to study the effects of dividend taxes on productivity. Khan and Thomas (2008) study individual choice problems and aggregation in the frictionless model with plant specific shocks. Basu and Fernald (997) also discuss the role of reallocation for productivity in an aggregate model. 5 The information about capital in plants in F A, captured in µ a and φ a is not needed in the state vector as capital is freely adjusted within the period. 0

3.3 Laws of Motion 3 COSTLY REALLOCATION plants, based on the realization of ε. This feature of our choice of moments allows us to compare it with common approximation techniques in the spirit of Krusell and Smith (998) in Section 5.3. The covariance term φ is crucial for understanding the impact of reallocation on measures of aggregate productivity. If the covariance is indispensable in the state vector of the planner, then the model is not isomorphic to the stochastic growth model. That is, if the covariance is part of the state vector, then the existence of heterogeneous plants along with capital adjustment costs matters for aggregate variables like investment over the business cycle. This returns us to a main point raised in section : the interaction between the presence of higher order moments and reallocation. As the quantitative analysis develops, the link from the source of variation to the movements in these higher order moments will be stressed. 3.3 Laws of Motion Though the model is rich due to plant heterogeneity and non-convex adjustment costs, we can characterize the evolution of the endogenous state variables of the cross sectional distribution. Each period, for a given aggregate state, the choice of k a maps into the two moments (µ a, φ a ). This choice determines output at the adjusting plants. The joint distribution over capital and ε for all plants at the end of the current period is summarized by µ and φ, with and µ = ( π)µ n + πµ a. (2) φ = ( π)φ n + πφ a. (22) These are convex combinations of the moments from the adjusting and non-adjusting plants, weighted by their respective weights π and π. Next period s state variables µ n and φ n are given by µ n = µ. (23) and φ n = ρ ε φ. (24) Here ρ ε is the serial correlation of the idiosyncratic shock. For this analysis, a plant is assumed to stay with its current productivity with probability ρ ε and to draw a new value of productivity with probability ρ ε as in Elsby and Michaels (203). 6 Together, (2) - (24) define the law of motion of the joint distribution Γ, allowing us to follow the 6 This uses the following steps: Cov(ε, k j (k, ε, s) α ) = Cov(ρ ε ε + ( ρ ε )η, k j (k, ε, s) α ) = ρ ε φ where η is the new draw from the distribution of idiosyncratic shocks. See (27).

4 QUANTITATIVE RESULTS: CAPITAL REALLOCATION AND CROSS SECTIONAL DISPERSION evolution of this component of the aggregate state. The planner faces a trade-off regarding the reallocation of capital across sites. The planner can increase contemporaneous output by reallocating capital from lowto high-productivity sites. This will increase the covariance between productivity and capital, φ a, while at the same time decreasing µ a because α <. However, the evolution of ε will create a mismatch between k n (k, ε, s) = k and the realization of ε for non-adjusting plants tomorrow. The planner therefore has to trade off the higher instantaneous output from reallocation with the higher costs of adjusting mis-matched plants tomorrow. Suppose A and λ are time-invariant. In this environment a stationary distribution Γ and a value π exist. In the stationary distribution µ n = µ a = µ. Furthermore, the economy converges to a stationary value φ n = φ π ρ ε a ( π )ρ ε, so that φ [ = φ π a ( π )ρ ε ]. Total output in (9) becomes y = K α ( µ + φ ). (25) 4 Quantitative Results: Capital Reallocation and Cross Sectional Dispersion The point of the quantitative analysis is to understand the role of productivity and dispersion shocks, along with frictions in reallocation, in matching key moments. This section focuses on the two moments stressed in the introduction: the cyclical patterns of reallocation and the dispersion of the cross sectional distribution of productivities. It also highlights the interplay between the Solow residual and movements in the cross sectional distribution of capital and plant-level productivity. In keeping with the distinction noted earlier between reallocation and accumulation, the quantitative analysis presented here is of an economy with a fixed capital stock, thus highlighting reallocation. The insights from the reallocation process are key to understanding a model with accumulation to jointly match all the aggregate moments discussed in the next section. 4. Parameterization For this analysis, the parameters are taken from other studies. In section 5. a subset of the parameters are estimated. We solve the model at the annual frequency, using these baseline parameters. Following the estimates in Cooper and Haltiwanger (2006), we set α = 0.6. 7 We assume log-utility and a depreciation rate δ = 0.. Assuming an annual interest rate of 4% implies an annual discount factor β = 0.965. Aggregate 7 This curvature is 0.44 in Bachmann and Bayer (203) and 0.4 in Bloom et al. (202). 2

4 QUANTITATIVE RESULTS: CAPITAL REALLOCATION AND CROSS SECTIONAL 4.2 Effects of Productivity and Dispersion Shocks DISPERSION productivity takes the form of an AR() in logs ln a t = ρ a ln a t + ν a,t, ν a N(0, σ a ), (26) where ρ a = 0.9 and σ a = 0.007. As noted earlier, the distinction between the aggregate TFP and the SR is important in our analysis. We return to this later when the model is estimated. Idiosyncratic productivity shocks are log-normally distributed and evolve according to a law of motion with time-varying variance. Each period, there is a probability ρ ε of drawing a new value of ε. With the counter-probability, the site produces with last period s ε. The stochastic process is given by: { } ε t, with probability ρ ε ε t = (27) ln N (0, λ t σ ε ), with probability ρ ε The parameters of the idiosyncratic shock process are ρ ε = 0.9 and σ ε = 0.2. The parameter λ governs the mean-preserving spread of the normal distribution from which idiosyncratic productivity ε is drawn. It has a mean of and variance σ λ. The stochastic process for λ is given by: ln λ t = ρ λ ln λ t + ν λ,t, ν λ,t N(0, σ λ ). (28) We set ρ λ = 0.95 and σ λ = 0.04. To parameterize the adjustment costs, we assume that G(F ) is uniform between zero and an upper limit denoted B as in Thomas (2002). For the analysis in this section, we set B = 0.4 to match an average reallocation rate of 40% from the post-990 Compustat data. This parameter is estimated below. The computational strategy is discussed in detail in Appendix A. 4.2 Effects of Productivity and Dispersion Shocks We study the effects of shocks to A and λ on capital reallocation to better understand the workings of the model. Table shows measures of capital reallocation and productivity. The column labeled E t (σ arpk ) measures the time series average of the cross sectional standard deviation of the average revenue product of capital. The column labeled σ(ã/a) reports the standard deviation of the SR relative to TFP. Recall from (20) that Ã/A is precisely the part of measured aggregate productivity which is stemming from the allocation of factors: Ã/A = π(µ a + φ a ) + ( π)(µ n + φ n ). This is a key moment as it measures the extent to which the cross-sectional distribution f(ε) and the allocation affect aggregate productivity; i.e. this measures the cyclicality of productivity which does not come from A alone. The columns C(R, Ã) and C(σ arpk, Ã) are the correlations between the SR and, respectively, capital reallocation and the standard deviation of the average revenue product of capital. These two columns provide a link back to the facts, noted in the introduction, about the cyclical behavior of reallocation and 3

4 QUANTITATIVE RESULTS: CAPITAL REALLOCATION AND CROSS SECTIONAL 4.2 Effects of Productivity and Dispersion Shocks DISPERSION dispersion in productivity. As in the data analysis that follows, these are correlations with the SR not aggregate TFP. Case E t (σ arpk ) σ(ã/a) C(R, Ã) C(σ arpk, Ã) stochastic A 0.3602 0.0006 0.650-0.4733 stochastic λ, ρ λ = 0.90 0.366 0.0 0.2390-0.4274 stochastic λ, ρ λ = 0 0.363 0.06 0.5933-0.928 Table : Simulation of Capital Reallocation Model: Productivity Implications Based on Simulation with T=2,000. The columns show the mean of σ (the cross-sectional standard deviation of average products of capital), the standard deviation of Ã/A, the correlation of log HP-filtered reallocation and the Solow residual, and the correlation between σ and the log HP-filtered Solow residual. As a benchmark, consider the economy without any adjustment costs from Section 2. Without frictions, the average product of capital is equalized across plants, E t (σ arpk ), is zero. Although capital is reallocated each period, the total amount is time-invariant and hence plays no role in the cyclicality of aggregate productivity. There are three experiments for the economy with adjustment costs. The first allows a shock to aggregate TFP, holding the distribution of the idiosyncratic shocks fixed. The second and third assume the distribution of shocks is stochastic, holding aggregate TFP constant. To better understand the mechanism at work, the second and third cases differ by the assumed serial correlation of the shock to λ. For all the treatments, the introduction of adjustment costs creates a non-degenerate distribution of the average product of capital, as indicated in the first column. This reflects the assumed distribution of adjustment costs, with an upper bound on adjustment costs set at B = 0.4. This distortion is a level effect. Our interest is in the cyclical patterns of this misallocation of capital. The row labeled stochastic A allows for randomness in aggregate productivity. In this case, there is almost no variation in the Solow residual independent of TFP: i.e. σ(ã/a) is nearly zero. Still, there is some response of the reallocation process to variations in A because the adjustment costs do not depend directly on productivity. Thus, reallocation is procyclical as the gains to reallocation rise with A. Further, C(σ arpk, Ã) is countercyclical since the dispersion in productivity is reduced when reallocation increases. Figure illustrates the response to both an iid shock and a serially correlated shock to A. As the figure makes clear, the response of reallocation and the adjustment rate are procyclical, while the dispersion in average capital productivity is countercyclical. However, the magnitudes are very small. This portends the estimation results reported below. Essentially, the aggregate TFP shock alone is unable to generate the observed patterns of reallocation and dispersion. It is precisely for this reason that the dispersion shock is so important for matching the reallocation patterns in the data. The question is whether the dispersion shock can also match the business cycle moments. 4

5 AGGREGATE IMPLICATIONS The rows labeled stochastic λ study the effects of time-variation in f(ε) with A fixed. Variations in λ do not lead to direct fluctuations in output. Instead, output variations come from the reallocation choices of the planner, as indicated by the positive correlation between reallocation and the Solow residual. This variation produces procyclical reallocation. The reallocation choices also appear as variations in the Solow residual relative to TFP, as measured as σ(ã/a). It is precisely in these simulations that the distinction between TFP and the variations in the SR due to reallocation is clear. There are two cases explored when λ is stochastic: one with serially correlated shocks and another with iid shocks. The persistence of the shock matters for the cyclical properties of reallocation and the standard deviation of capital productivity. This is seen by the difference in the impulse response functions in Figure 2. In response to a positive shock in λ, (i.e. an increase in the dispersion of idiosyncratic productivity shocks), reallocation increases as does the Solow residual. Importantly, the increase in reallocation decreases the cross sectional dispersion in the average productivity of capital. This is clear from Figure 2. Thus the initial response to the shock in λ is a negative co-movement between dispersion and the Solow residual. The pattern of response clearly depends on the persistence of the shock. When ρ λ = 0.90, the Solow residual remains above trend through the entire transition. The dispersion of the average productivity of capital remains below its steady state value in the transition as the reallocation effect dominates the spread in the distribution of idiosyncratic productivities. With iid shocks, the initial response of reallocation and dispersion is very similar. However, once the dispersion shock has ended, reallocation falls, leading to a positive response of the dispersion of average products of capital. 5 Aggregate Implications This section returns to the themes of the introduction: the cyclical properties of reallocation and business cycles. The ultimate goal is to estimate parameters of the model to match moments. This analysis is conducted in economies with capital accumulation. As noted in the introduction, our analysis adds to existing studies in two dimensions. First, both productivity shocks and shocks to the distribution of plant-level productivity are present and their relative importance is estimated. Second, we enlarge the traditional set of macroeconomic moments to include procyclical reallocation and the countercyclical standard deviation of productivity. We estimate the two models, one with TFP shocks and one with dispersion shocks, to match three sets of moments: aggregate business cycle moments, moments pertaining to reallocation, and then all of these moments together. We find evidence that the dispersion shock model alone is able to generate patterns of reallocation as well as aggregate variations consistent with the data. Importantly, the model with dispersion shocks is able to generate a procyclical and persistent Solow residual. While the model with aggregate TFP shocks alone can generate the business cycle moments, these shocks simply cannot generate cyclical movements in reallocation and dispersion that match the data. In fact, mixing the two 5

5. Matching Business Cycles 5 AGGREGATE IMPLICATIONS.04 Exogenous shock A.04 Mis measured TFP.06 π.03.03.04.02.02.02.0.0 0 20 30 0 20 30 0.98 0 20 30 µ.008 φ.06 Total Reallocation.006.04 0.9999.004.02.002 0.9999 0 20 30 0 20 30 0.98 0 20 30.04 Output.04 Consumption.005 XS std of MPK.03.03.02.02 0.995.0.0 0.99 0 20 30 0 20 30 0.985 0 20 30 Figure : Impulse Response of a positive shock to A. Note: The green dash-dotted line represents responses after a one-time iid shock, while the blue solid line represents responses after a persistent shock. The panels show (clockwise) the exogenous shock, the mis-measured part of TFP, the fraction of adjusters, capital reallocation, the cross-sectional standard deviation of the average product of capital, consumption, output, and µ, the center panel shows φ. shocks does not improve the fit of the model. This contrasts with the results reported in Bloom et al. (202) where the aggregate TFP process is assumed to follow the standard estimates of a model without dispersion shocks, thus equating movements in TFP with the SR. 5. Matching Business Cycles This sub-section compares the aggregate properties of our model with those of the standard RBC model. A standard criticism of the RBC model is technological regress: i.e. apparent reductions in total factor productivity. As emphasized in Bloom et al. (202) as well, model economies which induce variations in the 6

5. Matching Business Cycles 5 AGGREGATE IMPLICATIONS.06 Exogenous shock λ.05 Mis measured TFP.3 π.05.04.03.02.0.0.005.2. 0.9 0 20 30 0 20 30 0.8 0 20 30.005 0.995 0.99 µ 0 20 30.5..05 φ 0 20 30.3.2. 0.9 0.8 0.7 Total Reallocation 0 20 30.05 Output.05 Consumption.04 XS std of MPK.0.005.0.005.02 0.98 0.96 0 20 30 0.995 0 20 30 0.94 0 20 30 Figure 2: Impulse Response of a positive shock to λ. Note: The green dash-dotted line represents responses after a one-time iid shock, while the blue solid line represents responses after a persistent shock. The panels show (clockwise) the exogenous shock, the mis-measured part of TFP, the fraction of adjusters, capital reallocation, the cross-sectional standard deviation of the average product of capital, consumption, output, and µ, the center panel shows φ. Solow residual have the ability to explain technological regress and can potentially match other correlation patterns. The parameters we estimate are given by Θ = (B, ρ λ, σ λ, ρ A, σ A ). The first of these is the upper bound on the uniform distribution of capital adjustment costs. The other four are the serial correlation and standard deviation of the innovation to the dispersion shock and aggregate TFP shock, respectively. Other parameters, such as (α, β, δ, σ ε, ρ ε ) are taken from other sources. 8 Given the theme of the paper, the moments mix those reflecting the reallocation process as well as the traditional business cycle moments. Unless stated otherwise, the data are annual, in logs and are HP 8 Specifically, α = 0.60, σ ε = 0.2, ρ ε = 0.9, β = 0.965, δ = 0.0 throughout. We experimented with estimating (ρ ε, σ ε ) and ended up very close to the calibrated values. 7

5. Matching Business Cycles 5 AGGREGATE IMPLICATIONS filtered. The reallocation moments include:. C(R, y): Correlation between output and reallocation from Eisfeldt and Rampini (2006). 9 2. C(σ, y): Correlation between output and the standard deviation of average revenue product of capital from Kehrig and Vincent (203). 20 3. µ(adj): Mean fraction of adjusters (reallocation), from Compustat where adjustment is defined as non-zero sales of PP&E or Acquisitions. 4. std(y)/std(r): The ratio of the standard deviation of output relative to the standard deviation of reallocation. 5. std(y)/std(σ): The ratio of the standard deviation of output relative to the (time series) standard deviation of the (cross sectional) standard deviation of the average product of capital. The first two of these moments were emphasized in our motivation as representing the importance of cyclical reallocation. 2 The RBC moments are the traditional correlations between output, consumption and investment. They also include the properties of the Solow residual. These are mainly taken from Thomas (2002) and are annual and HP filtered. 22 Case B ρ λ σ λ ρ A σ A stochastic A RBC Moments 0.484 na na 0.782 0.00 Reallocation Moments 0.754 na na 0.8940 0.003 All Moments 0.754 na na 0.894 0.003 stochastic λ RBC Moments 0.500 0.737 0.063 na na Reallocation Moments 0.56 0.954 0.022 na na All Moments 0.50 0.933 0.052 na na Table 2: Parameter Estimates The parameters were chosen to minimize the squared percentage difference between the simulated and data moments. The estimates are shown in Table 2. Two cases are explored. In the first only aggregate TFP (A) is stochastic. In the second, only the dispersion shock (λ) is stochastic. For each of these cases, 9 Here reallocation includes sales of property, plant and equipment plus acquisitions. 20 We are grateful to Matthias Kehrig for supplying the data underlying Figure 8 of their paper. 2 The correlation of the cross-sectional standard deviation of plant-level productivity and output, emphasized in Kehrig (20) is not matched. Instead, the focus here is on the standard deviation of the average product of capital which reflects both the underlying distribution of shocks and endogenous reallocation. 22 The correlation between consumption and investment comes from Cooper and Ejarque (2000). 8

5. Matching Business Cycles 5 AGGREGATE IMPLICATIONS parameter estimates and moments are presented for three sets of targeted moments: (i) the RBC moments, (ii) reallocation moments and (iii) a combination of reallocation and RBC moments. These are presented in Tables 3, 4 and 5 respectively. Case ρã std(ã) µ(adj) C(y, c) C(y, i) C(y, Ã) C(i, c) fit Data 0.923 0.03 0.57 0.858 0.823 0.76 0.66 na stoch A 0.779 0.00 0.577 0.787 0.989 0.7 0.688 0.42 stoch λ 0.845 0.03 0.343 0.796 0.989 0.708 0.700 0.22 Table 3: RBC Moments Results from simulation of T=20,000. Here C(x, y) are correlations. The variables are: output (y), consumption (c), investment (i) and the Solow residual (mis-measured TFP) (Ã). µ(adj) represent the mean rate of adjusting plants. All time series are in logs and have been HP-filtered with λ = 00. Table 3 shows the traditional business cycle moments. Here an aggregate production function. Ã is the Solow residual computed from The positive co-movement of consumption, investment, output and the Solow residual is well documented. Both of the models broadly match these standard business cycle properties. These properties are not surprising in the presence of TFP shocks. Interestingly, these same patterns emerge for dispersion shocks although the fit is not as good. Specifically, the model with TFP shocks alone does a very good job of matching these features of the data as well as the frequency of capital reallocation of 0.57. The estimated serial correlation for the TFP shock is around 0.8, close to the estimated persistence of aggregate TFP from plant-level data of 0.76 reported in Cooper and Haltiwanger (2006). But this estimate is substantially lower than the serial correlation of the Solow residual, which is traditionally used as a measure of TFP. In contrast, the fit of the RBC moments is not quite as good when the only shock is to the dispersion of the idiosyncratic shock distribution. That said, as emphasized as well by Bloom et al. (202) for their environment, the case with dispersion shocks does replicate many of the features of aggregate fluctuations including: (i) a procyclical and persistent Solow residual, (ii) positive correlations between output, consumption and investment. To do so, the stochastic λ case underpredicts the fraction of adjusting plants. Note that the serial correlation of the Solow residual is 0.845 although there is no aggregate TFP shock. This serial correlation is induced by the dynamics of the cross-sectional distribution, i.e. the time series variation in φ. Table 4 reports simulated and data moments, focusing on cyclical reallocation. Here the moments include the procyclical reallocation, and the countercyclical standard deviation of the average product of capital as well as the variability of R and σ relative to y. For these moments, the case of stochastic λ does considerably better than the stochastic A specification: the fit is almost 60 times worse in the stochastic A case. In particular, the case with dispersion shocks alone captures: (i) the procyclical reallocation, (ii) the serial correlation in output and (iii) the dynamics of output and reallocation. As suggested by Figure 2, 9