Capital Reallocation and Aggregate Productivity∗ Russell W. Cooper† and Immo Schott‡ This Version: June 14, 2016 First Version: December 2013

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.

1

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 #0819682 for financial support. This version adds an endogenous adjustment decision to our December 2013 NBER Working Paper. † Department of Economics, the Pennsylvania State University and NBER, [email protected] ‡ Department of Economics, Universit´e de Montr´eal and CIREQ, [email protected]

1

1 MOTIVATION lies at the heart of the analysis of productivity both within and across countries in Maksimovic and Phillips (2001), Hsieh and Klenow (2009), Bartelsman, Haltiwanger, and Scarpetta (2013) Restuccia and Rogerson (2008) and others. In this paper we consider the cyclical dimension of reallocation in the presence of capital reallocation costs.1 The model is estimated to match both reallocation and aggregate moments. In important empirical contributions, Eisfeldt and Rampini (2006), Kehrig (2011), Kehrig and Vincent (2013), and Osotimehin (2016) 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. In fact, this latter reallocation effect is itself cyclical. 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 (1982) and King, Plosser, and Rebelo (1988). 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. 1

In contrast to Midrigan and Xu (2014) 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 (2014) 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 (2011) constructs dispersion measures based on TFPR estimates. Kehrig and Vincent (2013) find that the dispersion of the cross-sectional distribution of capital productivity is countercyclical as well. Osotimehin (2016) 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

1 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 (2011), Kehrig and Vincent (2013), Osotimehin (2016) and others which point to the quantitative significance of cyclical factor reallocation and its contribution to measures of aggregate productivity. From Olley and Pakes (1996) 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 Thus the covariance reflects the cyclical gains to capital reallocation. Relative to the literature emerging from Krusell and Smith (1998), 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 (2001) 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 (2011) 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 (2012) and Bachmann and Bayer (2013), 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. An exception is Bachmann and Bayer (2013). 6 Specifically, Bloom et al. (2012) 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 5

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 V (A, K) = maxK 0 ,k(ε) u(c) + βEA0 |A V (A0 , K 0 )

(1)

for all (A, K). The constraints are c + K 0 = y + (1 − δ)K,

(2)

Z k(ε)f (ε)dε = K,

(3)

ε

Z 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 (1). 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 (2013) also calibrate treating the Solow residual as aggregate TFP and focus on business cycle, not reallocation moments.

4

2.1 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 < 1 as in Lucas (1978).8 In this frictionless environment, a plants’ optimal capital stock is entirely determined by ε. The assumption of diminishing returns to scale, α < 1, 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 < 1, from spreading capital across production sites.

2.1

Optimal Choices

Within a period, the condition for the optimal allocation of capital across production sites is given by 1

k(ε) = K R

ε 1−α ε ε

1 1−α

.9

(6)

f (ε)dε

Substituting into (4) yields y = AK

α

Z ε

1 1−α

1−α f (ε)dε

.

(7)

ε

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 u0 (c) = βEVK (A0 , K 0 ) so that the marginal cost and expected marginal gains of additional capital are equated. Using (1), 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. (2012) include labor adjustment costs while Bachmann and Bayer (2013) 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 (2011) includes product differentiation at the level of intermediate goods. 1 η α−1 9 . The first order condition implies αAεk(ε)α−1 = η for all ε, where η is the multiplier on (3). It implies k(ε) = αAε R  1−α 1 α−1 Using (3), η = AαK ε 1−α f (ε)dε . Putting these two conditions together yields (6). ε

5

2.2 Aggregate Output and Productivity

2 FRICTIONLESS ECONOMY

" u0 (c) = βEu0 (c0 ) (1 − δ) + A0 αK 0α−1

Z ε

1 1−α

1−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, A0 , 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.10 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: y = AK

Z

α

εξ(ε)α f (ε)dε

(9)

ε

Define a measure of productivity A˜ as A˜ ≡ A

Z

εξ(ε)α f (ε)dε.

(10)

ε

As is well understood from the Olley and Pakes (1996) analysis of productivity, the level of aggregate output will depend on the covariance between the plant-level productivity and the factor allocation. Let R 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: ˜ α y = AK

(11)

A˜ ≡ A(µ + φ).

(12)

where A˜ is given by

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. 10

This builds on Olley and Pakes (1996).

6

3 COSTLY REALLOCATION Researchers interested in measuring TFP from the aggregate data will typically uncover A˜ rather than A. This is the mis-measurement referred to earlier. As the discussion progresses, we will refer to A˜ as the Solow residual, as distinct from aggregate TFP. ˜ The first one is A. The influence of A, aggregate From (10) there are three factors which influence A. TFP, on the Solow residual A˜ is direct and has been central to many studies of aggregate fluctuations. Second, fluctuations in f (ε) influence A˜ 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 ˜ However, the presence of frictions may of capital over plants does not have an independent effect on A. imply that, in a static sense, capital is not efficiently allocated. In that case, even with f (ε) fixed, the ˜ This is the topic of the next section. reallocation process will lead to variations in A.

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.

3.1

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

3.1 The Planner’s Problem

3 COSTLY REALLOCATION

idiosyncratic productivity shocks, parameterized by λ. Such changes can be interpreted as variations in uncertainty.11 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.12 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 (14).13 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 (11), output net of adjustment costs will be proportional to the capital stock. Given the state, the planner makes an investment decision K 0 , determines π, and chooses how much capital to reallocate between adjusting plants, (kj , εj ) ∈ 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) = maxπ,k˜a (k,ε,s),K 0 u(c) + βE[A0 ,Γ0 ,λ0 ,|A,Γ,λ] V (A0 , λ0 ; Γ0 , K 0 )

(13)

subject to the resource constraint (2), amended to include adjustment costs and reflecting the fact that 11

A number of recent papers such as Bloom (2009) and Gilchrist, Sim, and Zakrajˇsek (2014) find that time-varying uncertainty can have effects on aggregate output, while Bachmann and Bayer (2013) contest the importance of these shocks. 12 Other models, such as Bloom et al. (2012), 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. 13 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

3.1 The Planner’s Problem

3 COSTLY REALLOCATION

some but not all plants adjust their capital stocks: 0

c + K = y + (1 − δ)K − K

α

Z

F (π)

F dG(F ),

(14)

0

with

Z

Aεk˜a (k, ε, s)α dΓ(k, ε) +

y=

Z

(k,ε)∈a

Aεk˜n (k, ε, s)α dΓ(k, ε).

(15)

(k,ε)∈n

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 (14) 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 (15). Given π, the amount of capital over all plants must sum to total capital K: Z π

k˜a (k, ε, s)dΓ(k, ε) + (1 − π)

Z

k˜n (k, ε, s)dΓ(k, ε) = K.

(16)

(k,ε)∈n

(k,ε)∈a

As the capital is plant specific, it is necessary to specify transition equations at the plant level. Let i=

K 0 −K K

denote the gross investment rate so that K 0 = (1 − 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 kj0 (k, ε, s) = (1 − δ + i)k˜j (k, ε, s),

(17)

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

Z R(s) ≡ 0.5

r(k, ε, s)dΓ(k, ε).

(18)

(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 Γ0 . 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 (1998) this is an exact solution, not an approximation. The appropriate set of moments, as developed above for an economy without reallocation frictions, ˜ α with A˜ = A(µ + φ).14 For the economy comes from the representation of aggregate output as y = AK with costly adjustment and an endogenous reallocation rate of π, aggregate output, taken from (15), becomes α

y = AK [π(µa + φa ) + (1 − π)(µn + φn )] − K

α

Z

F (π)

F dG(F ),

(19)

0

where µj ≡ ε¯E(k˜j (k, ε, s)α ) and φj ≡ Cov(ε, k˜j (k, ε, s)α ), for j = a, n. Instead of Γ we retain µn and φn in the state vector of (13). This allows us to write the Solow residual as: A˜ = A[π(µa + φa ) + (1 − π)(µn + φn )].

(20)

The two moments, (µn , φn ), contain all the necessary information about the joint distribution of capital and productivity among non-adjusting plants.15 Each period the planner chooses an allocation of capital over plants, which maps into values of µa and φa . Note that by keeping (µn , φn ) 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 (19) precisely because output depends on the assignment of capital to 14 As noted earlier, this decomposition of productivity taken from Olley and Pakes (1996) highlights the interaction between the distribution of productivity and factors of production across firms. Gourio and Miao (2010) 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 (1997) also discuss the role of reallocation for productivity in an aggregate model. 15 The information about capital in plants in F A , captured in µa and φa is not needed in the state vector as capital is freely adjusted within the period.

10

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 (1998) 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 1: 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 (µa , φa ). 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 ˜ with is summarized by µ ˜ and φ, µ ˜ = (1 − π)µn + πµa .

(21)

φ˜ = (1 − π)φn + πφa .

(22)

and

These are convex combinations of the moments from the adjusting and non-adjusting plants, weighted by their respective weights π and 1 − π. Next period’s state variables µ0n and φ0n are given by ˜. µ0n = µ

(23)

˜ φ0n = ρε φ.

(24)

and

Here ρε is the serial correlation of the idiosyncratic shock. For this analysis, a plant is assumed to stay with its current productivity with probability ρε and to draw a new value of productivity with probability 1 − ρε as in Elsby and Michaels (2013).16 Together, (21) - (24) define the law of motion of the joint distribution Γ, allowing us to follow the This uses the following steps: Cov(ε0 , k˜j (k, ε, s)α ) = Cov(ρε ε + (1 − ρε )η, k˜j (k, ε, s)α ) = ρε φ where η is the new draw from the distribution of idiosyncratic shocks. See (27). 16

11

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, φa , while at the same time decreasing µa because α < 1. However, the evolution of ε will create a mismatch between k˜n (k, ε, s) = k and the realization of ε0 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 µn = µ∗ .i Furthermore, the economy converges to a stationary h µa = ∗ ∗ π . Total output in (19) becomes value φ∗ = φ∗ π ρε∗ , so that φ˜∗ = φ∗ ∗ n

a 1−(1−π )ρε

a

1−(1−π )ρε

y=K

4

α



 ∗ ˜ µ ˜ +φ . ∗

(25)

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.

4.1

Parameterization

For this analysis, the parameters are taken from other studies. In section 5.1 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 = 0.6.17 We assume log-utility and a depreciation rate δ = 0.1. Assuming an annual interest rate of 4% implies an annual discount factor β = 0.9615. Aggregate 17

This curvature is 0.44 in Bachmann and Bayer (2013) and 0.4 in Bloom et al. (2012).

12

4 QUANTITATIVE RESULTS: CAPITAL REALLOCATION AND CROSS SECTIONAL 4.2 Effects of Productivity and Dispersion Shocks DISPERSION

productivity takes the form of an AR(1) in logs ln at = ρa ln at−1 + νa,t ,

νa ∼ N (0, σa ),

(26)

where ρa = 0.9 and σa = 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 ρε 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: ( εt =

εt−1 ,

with probability

ln N (0, λt σε ), with probability

ρε

)

1 − ρε

(27)

The parameters of the idiosyncratic shock process are ρε = 0.9 and σε = 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 1 and variance σλ . The stochastic process for λ is given by: ln λt = ρλ ln λt−1 + νλ,t ,

νλ,t ∼ N (0, σλ ).

(28)

We set ρλ = 0.95 and σλ = 0.014. 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-1990 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 1 shows measures of capital reallocation and productivity. The column labeled Et (σarpk ) measures the time series average of the cross sectional standard deviation of the average revenue product ˜ of capital. The column labeled σ(A/A) reports the standard deviation of the SR relative to TFP. Recall ˜ is precisely the part of measured aggregate productivity which is stemming from the from (20) that A/A ˜ allocation of factors: A/A = π(µa + φa ) + (1 − π)(µn + φn ). 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. ˜ and C(σarpk , A) ˜ are the correlations between the SR and, respectively, capital The columns C(R, A) 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 13

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 Et (σarpk ) σ(A/A) stochastic A 0.3602 0.0006 stochastic λ, ρλ = 0.90 0.3616 0.011 stochastic λ, ρλ = 0 0.3613 0.0116

˜ C(σarpk , A) ˜ C(R, A) 0.6510 -0.4733 0.2390 -0.4274 0.5933 -0.9281

Table 1: 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 A/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, Et (σarpk ), 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(A/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, ˜ is countercyclical since the dispersion in productivity is reduced when reallocation increases. C(σarpk , A) Figure 1 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. 14

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(A/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 ρλ = 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 15

5.1 Matching Business Cycles

5 AGGREGATE IMPLICATIONS

Exogenous shock A

Mis−measured TFP

1.04

1.04

1.06

1.03

1.03

1.04

1.02

1.02

1.02

1.01

1.01

1

1

10

20

30

1

10

20

0.9999

10

0.98

φ˜

µ ˜ 1

0.9999

30

20

30

1.06

1.006

1.04

1.004

1.02

1.002

1

Output

10

20

30

0.98

1.04

1.005

1.03

1.03

1

1.02

1.02

0.995

1.01

1.01

0.99

10

20

30

1

10

20

30

20

10

20

30

XS std of MPK

Consumption

1.04

1

10

Total Reallocation

1.008

1

π∗

30

0.985

10

20

30

Figure 1: 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. (2012) 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.

5.1

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. (2012) as well, model economies which induce variations in the 16

5.1 Matching Business Cycles

5 AGGREGATE IMPLICATIONS

Mis−measured TFP

E x oge n ou s s h o c k λ 1.015

1.06

π∗

1.3

1.05

1.2 1.01

1.04

1.1

1.03 1

1.005

1.02

0.9

1.01 1

10

20

30

1

10

20

30

0.8

10

φ˜

µ ˜ 1.005

20

30

Total Reallocation

1.15 1.3 1.2

1

1.1

0.995

1.05

1.1 1 0.9 0.8 0.7

0.99

10

20

30

1

10

Output

20

30

10

Consumption

1.015

20

30

XS std of MPK

1.015

1.04 1.02

1.01 1.01

1 1.005 0.98

1.005 1 1

10

20

30

0.995

0.96 10

20

30

0.94

10

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, ρλ , σλ , ρA , σA ). 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, σε , ρε ) are taken from other sources.18 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 18

Specifically, α = 0.60, σε = 0.2, ρε = 0.9, β = 0.9615, δ = 0.10 throughout. We experimented with estimating (ρε , σε ) and ended up very close to the calibrated values.

17

5.1 Matching Business Cycles

5 AGGREGATE IMPLICATIONS

filtered. The reallocation moments include: 1. C(R, y): Correlation between output and reallocation from Eisfeldt and Rampini (2006).19 2. C(σ, y): Correlation between output and the standard deviation of average revenue product of capital from Kehrig and Vincent (2013).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.21 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

RBC Moments Reallocation Moments All Moments

0.1484 0.754 0.754

RBC Moments Reallocation Moments All Moments

0.500 0.561 0.501

ρλ σλ ρA stochastic A na na 0.782 na na 0.8940 na na 0.894 stochastic λ 0.737 0.063 na 0.954 0.022 na 0.933 0.052 na

σA 0.010 0.003 0.003 na 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, 19

Here reallocation includes sales of property, plant and equipment plus acquisitions. We are grateful to Matthias Kehrig for supplying the data underlying Figure 8 of their paper. 21 The correlation of the cross-sectional standard deviation of plant-level productivity and output, emphasized in Kehrig (2011) 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). 20

18

5.1 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 Data stoch A stoch λ

ρA˜ 0.923 0.779 0.845

˜ std(A) 0.013 0.010 0.013

µ(adj) 0.571 0.577 0.343

C(y, c) 0.858 0.787 0.796

C(y, i) 0.823 0.989 0.989

˜ C(y, A) 0.76 0.711 0.708

C(i, c) 0.66 0.688 0.700

fit na 0.142 0.221

Table 3: RBC Moments Results from simulation of T=20,000. Here C(x, y) are correlations. The variables are: output (y), ˜ µ(adj) represent the consumption (c), investment (i) and the Solow residual (mis-measured TFP) (A). mean rate of adjusting plants. All time series are in logs and have been HP-filtered with λ = 100.

Table 3 shows the traditional business cycle moments. Here A˜ is the Solow residual computed from an aggregate production function. 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 0.571. 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. (2012) 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, 19

5.1 Matching Business Cycles

Case Data stoch A stoch λ

C(R, y) 0.637 0.587 0.231

5 AGGREGATE IMPLICATIONS

C(σ, y) -0.338 -0.338 -0.161

µ(adj) 0.571 0.291 0.329

std(y)/std(R) 0.083 0.302 0.058

std(y)/std(σ) 0.132 1.074 0.162

fit na 58.136 1.008

Table 4: Reallocation Moments Results from simulation of T=20,000. Here C(x, y) are contemporaneous correlations. The variables are: output (y), capital reallocation (R), the standard deviation of the average revenue product of capital (σ). consumption (c), ˜ µ(adj) represent the mean rate of adjusting plants. investment (i) and the Solow residual (mis-measured TFP) (A).

there is enough variation in R and σ that the model can also match the std(y)/std(R) and std(y)/std(σ) moments. While the stochastic A model captures some features of the reallocation process, it simply cannot generate sufficient time series variation in reallocation or the cross sectional dispersion in the average product of capital. As shown in Table 2, the estimated serial correlation in λ is higher and the upper support of the adjustment cost distribution is also slightly higher relative to the estimates matching the RBC moments. The same pattern is seen in the case of aggregate TFP shocks. Table 5 reports moments when both reallocation and business cycle moments are matched. Here there are 11 moments and 3 parameters, as indicated in Table 2. The same pattern appears here as in Table 4. The model with dispersion shocks captures both the business cycle and reallocation moments. The induced variations in the Solow residual mimic quite well the effects of an aggregate TFP shock for business cycle moments. The model with aggregate TFP shocks fails to capture the reallocation moments well. One way to summarize these findings is to ask how much of the variation in the data of the Solow residual is explained by the model of dispersion shocks. The answer is

0.011 0.013

= 0.846. A substantial amount

of the fluctuations in productivity inferred from an aggregate production function are actually induced by capital reallocation alone. Further the model without dispersion shocks is incapable of matching the reallocation process in the data. In particular, as highlighted in the discussion of the impulse response functions, the model with TFP shocks is unable to generate a large response in either reallocation or the standard deviation of average productivity. Put differently, it is an issue of magnitude not the sign of the correlation that is the problem for the model with TFP shocks alone. The parameter estimates appear in the last row of Table 2. The estimated serial correlation in the dispersion shock is similar to the one needed to match the reallocation moments while the upper support of the adjustment cost distribution is close to the one needed to match the extended RBC moments. There appears to be a missing case: allowing both aggregate TFP shocks and dispersion shocks to match both the RBC and reallocation moments. Despite considerable effort, we are unable to find a set of parameters that fits these moments better than the dispersion shocks alone. Since the aggregate TFP model is so far from matching the reallocation moments, particularly the ratios of the 20

5.2 Alternative Models of Adjustment Costs

5 AGGREGATE IMPLICATIONS

standard deviations, perhaps it is not surprising that adding TFP shocks does not improve the fit. Case Data

C(R, y) 0.637

C(σ, y) -0.338

ρA˜ 0.922

˜ std(A) 0.013

µ(adj) 0.571

C(y, c) 0.858

C(y, i) 0.823

˜ C(y, A) 0.760

C(i, c) 0.660

std(y)/std(R) 0.083

std(y)/std(σ) 0.132

fit na

A λ

0.587 0.160

-0.338 -0.227

0.902 0.964

0.003 0.011

0.291 0.347

0.855 0.879

0.988 0.975

0.526 0.408

0.765 0.750

0.302 0.045

1.074 0.184

58.87 1.488

Table 5: All Moments Results from simulation of T=20,000. Here C(x, y) are contemporaneous correlations. The variables are: output (y), capital reallocation (R), the standard deviation of the average revenue product of capital (σ). consumption (c), ˜ µ(adj) represent the mean rate of adjusting plants. investment (i) and the Solow residual (mis-measured TFP) (A).

Figure 3 illustrates the interplay between dispersion shocks and the Solow residual. The simulation is based upon the estimation for the stochastic λ model in Table 2. The red dash-dotted line shows λ, while the green dashed line shows the Solow residual. The fluctuations in the Solow residual are driven by variations in the cross-sectional distribution. The standard deviation of the average productivity of capital is plotted as the solid blue line. These variations are clearly very important for fluctuations in the Solow residual.

5.2

Alternative Models of Adjustment Costs

This sub-section reports results for alternative models of reallocation (adjustment) costs, focusing on the stochastic λ case. Analytics for these cases are presented in the Appendix Section B.1. There is no reestimation in this sub-section, just simulations. The point is to understand how different ways of scaling adjustment costs influence the moments. We find that moment implications in the stochastic λ model are essentially the same. Recall that in the baseline specification, the reallocation cost was scaled by K α . This was motivated in part by tractability, as in (19). Further the scaling also captured a theme in the adjustment cost literature that costs of reallocation should be related to size, as in for example, Cooper and Haltiwanger (2006). The other two specifications we consider are shown in the rows of Table 6. The K case scales the reallocation cost by the aggregate capital stock. This specification introduces an interaction between the reallocation and the accumulation problems of the planner. The case when adjustment costs are scaled by AK α is more like a disruption cost, though only depending on aggregate variables. Again, these costs are independent of the plant state since these costs are incurred before the plant state is known. The table also shows the fit from these simulations using baseline parameters for three models of reallocation costs. Note that for the case when adjustment costs are scaled by K only, the fit is not as good as in the baseline model. In particular, this case does not generate a negative covariance between σ and output.

21

5.2 Alternative Models of Adjustment Costs

5 AGGREGATE IMPLICATIONS

1.4

1.35

1.2

1.3

1

1.25

0.8

Solow Residual

D i sp e r si on / λ

A˜ an d D i sp e r si on

1.2

σ (M R P K ) λ A˜ 1.15

0.6 20

40

60

80

100

120

Time

Figure 3: Simulated Productivity and Dispersion The solid blue line represent the standard deviation of average products of capital, the red dash-dotted line shows λ. The green dashed line shows the Solow residual (right y-axis).

Note also that the baseline and the case when the costs are scaled by AK α are identical. This is simply because in this treatment we are allowing variations in λ only. Finally, because the problem is computed in a different manner, the approximation generates slightly different moments for the baseline. Again, the baseline specification has the advantage of capturing scaling effects and also is more tractable for computing the solution of the planner’s problem. Overall, the finding for the baseline model remains. A model economy with only dispersion shocks is quite capable of reproducing both reallocation moments and those characterizing the aggregate economy. This does not depend on the exact specification of adjustment costs.

22

5.3 Importance of the Cross Sectional Distribution

Case

C(R, y)

C(σ, y)

ρA˜

˜ std(A)

µ(adj)

base: K α AK α K

0.116 0.116 0.296

-0.269 -0.269 0.042

0.950 0.950 0.981

0.014 0.014 0.006

0.342 0.342 0.166

C(y, c)

C(y, i) stochastic 0.473 0.423 0.473 0.423 0.499 0.888

5 AGGREGATE IMPLICATIONS

˜ C(y, A) λ 0.979 0.979 0.966

C(i, c)

std(y)/std(R)

std(y)/std(σ)

fit

0.152 0.152 0.043

0.041 0.041 0.057

0.225 0.225 0.123

2.74 2.74 3.58

Table 6: Moments from Alternative Models of Reallocation Costs Results from simulation of T=20,000. Here C(x, y) are contemporaneous correlations. The variables are: output (y), capital reallocation (R), the standard deviation of the average revenue product of capital (σ). consumption (c), ˜ µ(adj) represent the mean rate of adjusting plants. investment (i) and the Solow residual (mis-measured TFP) (A).

5.3

Importance of the Cross Sectional Distribution

As noted earlier, one of the key findings in Krusell and Smith (1998) is that first moments are all that is needed to approximate a joint distribution. Relatedly, Thomas (2002) and the literature that followed emphasized the near equivalence between the aggregate moments of a model with lumpy investment and the aggregate implications of a real business cycle model at the plant-level. This sub-section returns to that theme in a model that stresses reallocation rather than the accumulation of capital in a setting with stochastic variations in the distribution of idiosyncratic shocks. 5.3.1

The Role of the Covariance in Forecasting

The covariance φ is important for determining the optimal capital allocation because the planner’s problem in (13) includes Γ, the joint distribution of (k, ε). Using its first two moments, µn and φn , the planner can perfectly forecast the evolution of Γ. Knowing Γ0 the planner knows the amount of output produced by non-adjusting plants. We stress the importance of knowing the entire cross-sectional distribution as a regressor in a forecasting rule. In the frictionless version of our model the two moments µ and φ are perfectly correlated. The existence of adjustment costs breaks this perfect correlation. To understand the importance of φ as a carrier of information about the allocation of capital across sites, consider the following example. Assume there are only two levels of ε, ε1 and ε2 , with associated (frictionless) capital levels k(ε1 ) and k(ε2 ). Over time, the two plants will switch their idiosyncratic productivity level and the planner will respond to this switch by reallocating capital across sites so that the optimal allocation is restored. Under this scenario there is no variation in µ or φ over time. Now consider the case where capital adjustment costs are such that after the same switch of idiosyncratic productivity levels the planner does not change the two sites’ capital stocks. This implies that µ, the mean of k α has not changed. The second moment φ, however, capturing the covariance between ε and k will fall. This fall in φ represents an increase in what Eisfeldt and Rampini (2006) have called the “benefits to reallocation”. If the combination of adjustment costs and exogenous shocks is such that changes in the allocation of capital which affect φ differently from µ are important, then omitting φ from the state space will

23

5.3 Importance of the Cross Sectional Distribution

5 AGGREGATE IMPLICATIONS

significantly lower the ability of the planner to correctly predict the effect of contemporaneous reallocation choices on the future output of non-adjusting plants. As was highlighted above, shocks to λ, the variance of idiosyncratic productivity shocks, affect the covariance between ε and k directly. In the presence of such shocks, keeping track of φ ought to be indispensable. In contrast, if the only shocks in the economy are to aggregate TFP, we have seen that the productivity loss from costly reallocation has no cyclical element, and the level effect is quite small. So, omitting φ from the state vector should not significantly affect the planner’s forecasting ability. Case Stochastic A

R2 MFE R2 MFE 0.173 2.25% 0.9989 0.11%

Stochastic λ

0.950 7.09%

additional controls

no

0.985

5.98% yes

Table 7: Forecasting Γ0 with limited set of moments The second and third columns show the R2 and MFE of a regression of output from non-adjusting plants on an intercept and the first moment. In the fourth and fifth rows the aggregate state is added as a regressor.

When solving the dynamic model with capital adjustment costs the planner is ultimately interested in forecasting yn0 , the output from non-adjusting plants next period. As in Krusell and Smith (1998) we can represent this forecast through a linear regression with the moments of Γ as explanatory variables. The correctly specified regression model including both moments is given by yn,t = β0 + β1 µn,t + β2 φn,t + β3 xt + εt ,

(29)

where xt could include any additional information about the aggregate state. Estimation results in βˆ0 = 0, βˆ1 = ε¯, βˆ2 = 1, and βˆ3 = 0 with an R2 = 1. The maximum forecast error (MFE) is zero.23 We now study the importance of φ in predicting output. The planner correctly solves the model, keeping both µ and φ in the state space. We ask: what is the loss of accuracy if φ is omitted from a regression as in (29). The results are shown in Table 7, which evaluates the importance of the higher order moment φ in the different models we consider. The table shows that the model with stochastic A generates a very high R2 and low MFE once we control for TFP. This implies that the loss from omitting φ from the set of regressors is negligible. On the other hand, the introduction of stochastic variation in the distribution of plant-specific shocks, i.e. allowing dispersion shocks, makes the higher-order moment φ indispensable from the forecasting regression. Omitting φ leads to a large drop in the R2 compared to what is normally 23

As discussed in Den Haan (2010) a problem of R2 measures to assess the approximation is that observations generated using the true law of motion (instead of the forecast) are used as the explanatory variable. We construct a series yˆ ˆn which is using only the approximate law of motion. The forecast error is defined as εˆˆt+1 = |yˆˆn,t+1 − yn,t+1 |, and the MFE is the maximum of this series.

24

6 CONCLUSION

obtained in these types of forecasting equations. Also, the MFE goes up to 7% (6% when controlling for λ). Relative to the literature starting with Krusell and Smith (1998), this is an important finding. In particular, this result is distinguished from preceding papers in that we are able to show conditions under which the approximation of the cross sectional distribution requires higher order moments. 5.3.2

The Role of the Covariance in Matching RBC Moments

We now consider a second experiment. Assume that the true data generating process is a heterogeneous plant model with stochastic λ shocks. If we were to treat the moments generated from such a model as targets, how closely could they be replicated by a model with A shocks and homogeneous firms? Such a model would obviously be unable to target the ‘reallocation moments’. We assess the fit of the simple model in Table 8. To keep this exercise as simple as possible, the only parameters to be estimated are ρa and σa . We find that ρa = 0.84156 and σa = 0.012813 fits the data generated by the λ model best. The fit of the model with TFP shocks alone is very good. With the exception of the correlations of output with consumption and investment, the fit is almost perfect. Overall, the simple model with just TFP shocks and without firm heterogeneity can match the aggregate moments generated by a heterogeneous firm model with λ shocks quite well. On the other hand, this model is unable to capture the micro moments, i.e. the dynamics of reallocation and the standard deviation of cross-sectional average products of capital. Case Targets stoch A

ρA˜ 0.796 0.7592

˜ std(A) 0.989 0.9828

C(y, c) 0.708 0.6452

C(y, i) 0.7 0.6261

˜ C(y, A) 0.845 0.8387

C(i, c) 0.013 0.013

fit na 0.021247

Table 8: Representing Heterogeneous Firm Model with Simple RBC Results from simulation of T=20,000. Here C(x, y) are correlations. The variables are: output (y), ˜ All time series are in consumption (c), investment (i) and the Solow residual (mis-measured TFP) (A). logs and have been HP-filtered with λ = 100.

6

Conclusion

The goal of this paper was to understand the productivity gains from capital reallocation in the presence of frictions. To study this we have looked at the optimization problem of a planner facing frictions in capital and shocks to productivity and the distribution of plant specific shocks. The heterogeneity in plant-level productivity provides the basis for reallocation. The frictions in adjustment prevent the full realization of these gains. The model can generate cyclical movements in reallocation and in the cross sectional distribution of the average productivity of capital. 25

6 CONCLUSION

There are three key findings in this paper. The first is the cyclical behavior of reallocation and the distribution of capital productivity. When shocks to the distribution of plant-level shocks are present, then reallocation is procyclical. Further the standard deviation of the cross sectional distribution of average capital productivity is countercyclical, as in Eisfeldt and Rampini (2006) and Kehrig (2011). These effects are quantitatively unimportant when the only shock is to TFP. Second, the covariance of capital and productivity shocks at the plant-level matters for characterizing the planner’s solution in the presence of dispersion shocks. This is important since it is indicative of state dependent gains to reallocation. Our economy is an example of one where moments other than means are needed in the planner’s problem. Third, the model with shocks to the cross sectional distribution of productivity shocks can reproduce many features of the aggregate economy. We found that the model with dispersion shocks alone would ‘explain’ nearly 85% of the times series variation in the Solow residual. A researcher could certainly misinterpret the variations in the Solow residual driven by the reallocation of capital as variations in TFP. In fact, adding a TFP shock to the dispersion shock does not improve the fit of the model.

26

REFERENCES

REFERENCES

References Bachmann, R. and C. Bayer (2013): “‘Wait-and-See’ business cycles?” Journal of Monetary Economics, 60, 704–719. Bartelsman, E., J. Haltiwanger, and S. Scarpetta (2013): “Cross-Country Differences in Productivity: The Role of Allocation and Selection,” American Economic Review. Basu, S. and J. Fernald (1997): “Returns to Scale in U.S. Production: Estimates and Implications,” Journal of Political Economy, 105, 249–83. Bloom, N. (2009): “The Impact of Uncertainty Shocks,” Econometrica, 77, 623–685. Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (2012): “Really uncertain business cycles,” NBER Working Paper #18245. Cooper, R. and J. Ejarque (2000): “Financial intermediation and aggregate fluctuations: a quantitative analysis,” Macroeconomic Dynamics, 4, 423–447. Cooper, R. and J. Haltiwanger (2006): “On the Nature of the Capital Adjustment Costs,” Review of Economic Studies, 73, 611–34. Cui, W. (2014): “Delayed Capital Reallocation,” SSRN Scholarly Paper ID 2482431, Social Science Research Network, Rochester, NY. Den Haan, W. J. (2010): “Assessing the accuracy of the aggregate law of motion in models with heterogeneous agents,” Journal of Economic Dynamics and Control, 34, 79–99. Eisfeldt, A. L. and A. A. Rampini (2006): “Capital reallocation and liquidity,” Journal of Monetary Economics, 53, 369–399. Elsby, M. W. L. and R. Michaels (2013): “Marginal Jobs, Heterogeneous Firms, and Unemployment Flows,” American Economic Journal: Macroeconomics, 5, 1–48. Foster, L., J. C. Haltiwanger, and C. J. Krizan (2001): “Aggregate productivity growth. Lessons from microeconomic evidence,” in New developments in productivity analysis, University of Chicago Press, 303–372. Galindev, R. and D. Lkhagvasuren (2010): “Discretization of highly persistent correlated AR(1) shocks,” Journal of Economic Dynamics and Control, 34, 1260–1276. Gilchrist, S., J. W. Sim, and E. Zakrajˇ sek (2014): “Uncertainty, financial frictions, and investment dynamics,” Tech. rep., National Bureau of Economic Research. 27

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Gourio, F. and J. Miao (2010): “Firm Heterogeneity and the Long-Run Effects of Dividend Tax Reform,” American Economic Journal: Macroeconomics, 2, 131–68. Hsieh, C.-T. and P. Klenow (2009): “Misallocation and Manufacturing TFP in China and India,” Quarterly Journal of Economics. Kehrig, M. (2011): “The Cyclicality of Productivity Dispersion,” The University of Texas at Austin, Job Market Paper. Kehrig, M. and N. Vincent (2013): “Financial Frictions and Investment Dynamics in Multi-Plant Firms,” Working paper. Khan, A. and J. K. Thomas (2008): “Idiosyncratic Shocks and the Role of Nonconvexities in Plant and Aggregate Investment Dynamics,” Econometrica, 76, 395–436. King, R. G., C. I. Plosser, and S. T. Rebelo (1988): “Production, growth and business cycles: I. The basic neoclassical model,” Journal of monetary Economics, 21, 195–232. Krusell, P. and A. Smith (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106, 869–96. Kydland, F. E. and E. C. Prescott (1982): “Time to build and aggregate fluctuations,” Econometrica: Journal of the Econometric Society, 1345–1370. Lucas, R. E. (1978): “On the Size Distribution of Business Firms,” Bell Journal of Economics, 9, 508–523. Maksimovic, V. and G. Phillips (2001): “The Market for Corporate Assets: Who Engages in Mergers and Asset Sales and Are There Efficiency Gains?” The Journal of Finance, 56, 2019–2065. Midrigan, V. and D. Y. Xu (2014): “Finance and Misallocation: Evidence from Plant-level Data,” American Economic Review, 104. Olley, S. G. and A. Pakes (1996): “The Dynamics of Productivity in the Telecommunications Equipment Industry,” Econometrica, 64, 1263–1297. Osotimehin, S. (2016): “Aggregate productivity and the allocation of resources over the business cycle,” Unpublished manuscript. Restuccia, D. and R. Rogerson (2008): “Policy distortions and aggregate productivity with heterogeneous establishments,” Review of Economic Dynamics. Thomas, J. (2002): “Is Lumpy Investment Relevant for the Business Cycle?” Journal of Political Economy, 110, 508–34. 28

A SOLUTION ALGORITHM FOR PLANNER’S PROBLEM

A

Solution Algorithm for Planner’s Problem

This section describes the solution of the planner’s problem in more detail. All exogenous shocks are discretized using the methodology described in Galindev and Lkhagvasuren (2010). The key for the computation is to define a grid for k, which then implies values for µA and φA . The starting point is the non-stochastic environment. Here the planner chooses an allocation of capital over plants whose value is the discounted present value of the implied output. ε¯µ∗ + ΛφA ˜a (k,ε) 1−β π,k

V = max

(30)

We proceed by computing this vector for any non-stochastic value of λ. Using the fact that an adjusting plant with idiosyncratic shock εj > εi must have k(εj ) > k(εi ) we create a grid for capital by interpolating between the vectors for the stationary cases. As a lower bound for the grid the vector where k M IN ≡ k(ε) = 1

(31)

can be used. As an upper bound we use the frictionless benchmark computed in (6). How good is the k-grid? In order to check whether for a given value of λ the vector of capital across plants is indeed optimal we add random Gaussian noise to the policy function. We draw Gaussian i.i.d. shocks from a distribution with N v (0, σG ). Applying 1’000 such perturbations to each of our computed optimal k-vectors we find that the throughout the model simulation the maximum increase in output which can be achieved is in the order of 0.01%. For a given vector k˜ and a realization of ε we can compute the cross-sectional mean and standard deviaR tion of the average products of capital of non-adjusting plants as αεk˜α−1 f (ε)dε. Using the law of motion for ε we can compute the evolution of these two moments analytically as well. Note that we can rewrite the mean of the average product as E(X · Y ), where X = αε and Y = k˜α−1 so that the mean of the average products of plants who draw a random ε next period is E(X 0 Y ) = E(X 0 )E(Y ) + cov(X 0 , Y ) = E(X 0 )E(Y ) since the covariance between X and the random draw of ε is zero. Those plants that do not draw a new ε will have the same mean of average products as in the previous period. The evolution of the mean of nonadjusting plants’ average products is therefore given by ρε E(XY )+(1−ρε ) [E(X 0 ) · E(Y )]. This expression is unaffected by variations in λ. We can simplify the expression further by noting that E(X 0 ) = E(X) and writing E(X) · E(Y ) + ρε cov(X, Y ). For the standard deviation of average products we can proceed in a similar way. Denote Z = X · Y . We have that var(Z) = E(Z 2 ) − E(Z)2 = E(X 2 Y 2 ) − E(XY )2 . The second term has been computed above. The first term E(X 2 Y 2 ) = E(X 2 )E(Y 2 ) − cov(X 2 , Y 2 ) so that var(Z) = E(X 2 )E(Y 2 ) − cov(X 2 , Y 2 ) − E(XY )2 . Define Z 0 = X 0 Y for those plants that do not adjust and draw a new, random ε. For those plants 29

A SOLUTION ALGORITHM FOR PLANNER’S PROBLEM var(Z 0 ) = E(X 2 )E(Y 2 ) − E(XY )2 since the covariance term is zero. The overall variance of non-adjusting plants is therefore given by ρε var(Z) + (1 − ρε ) [E(X 02 )E(Y 2 ) − E(XY )2 ], where the covariance term is again zero. With time-varying λ we draw from an ε distribution that has the same mean as before and a variance that is λ2 times higher. The old variance is E(X 2 )E(Y 2 ) − E(XY )2 and the new variance is E(X 02 )E(Y 2 ) − E(X 0 Y )2 . We have already shown above that E(X 0 Y )2 does not change. Neither does E(Y 2 ). The old variance of average products over the new one is

[µ2 +σ 2 ]E(Y 2 )−E(XY )2 . [µ2 +λ2 σ 2 ]E(Y 2 )−E(XY )2

Capital Reallocation We compute the amount of capital reallocation using a grid vector for capital with J elements. Capital reallocation is always defined to mean the amount of reallocation between the allocation at the beginning and at the end of a period. Suppose we start with a distribution of k and ε so that the two are perfectly synced.24 In what follows we start by assuming that there is only one potential vector of ε, denoted ε1 . We first find the time-invariant part of the vector of capital and ε1 , that is synced. Note that reallocation is equal to zero if 1. a plant’s capital stock cannot be adjusted (probability 1 − π) 2. if the plant’s capital can be adjusted but the plant did not draw a new ε (probability π · (1 − ρ)) The fraction of plants that reallocate is thus given by 1 − ((1 − π) + π(1 − ρ)), or πρ. Next period, after potentially receiving a new draw from the distribution ε1 , a fraction (1 − π)ρ of plants will have received a random matching of ε1 and k, while the remaining part of the ε’s remain synced with capital. Therefore, as long as 0 < ρ < 1 and 0 < π < 1, there will be a fraction x1 of plants in which k and ε1 are synced, and a fraction r1 = 1 − x, in which k and ε are randomly allocated. Note that both x and r are defined at the beginning of a period. The fraction r that is randomly allocated will remain so at the end of the period if capital cannot be adjusted (1 − π). From the above we can derive the law of motion for the fraction x1 as x01 = (1 − (1 − π)ρ) · x1 + π · (1 − x1 ). In the steady state there will be a fraction of plants whose capital is synced equal to xss 1 = while the remaining fraction

(1−π)ρ (1−π)ρ+π

π , (1 − π)ρ + π

will be randomly allocated. This can easily be extended to the case

of M different target distributions of capital. If the index of the new target capital vector is given by k 24

i.e. k is the optimal allocation of capital for a given vector of ε. The k-vector is referred to as a target distribution of capital. It is not important that this actually be the optimal k vector.

30

A SOLUTION ALGORITHM FOR PLANNER’S PROBLEM

and the set of non-target vectors is n = j ∩ k we can generalize the law of motion as follows. x01,k = π + (1 − π)(1 − ρ)x1,k x01,n = (1 − π)(1 − ρ) · x1,n ,

∀n

0 = (1 − π)(ρx1,j + r1,j ), r1,j

∀j.

25 It is easy to see that the new stationary distribution will converge to x1,n = r1,n = 0 and x1,k = xss 1 .

We can now compute the steady state level of capital reallocation. If a plant is within the set x of synced plants, with probability πρ the ε are randomly re-assigned and capital can be adjusted. Reallocation in that case is given by the (weighted) average distance between capital vector and all its permutations. This distance is denoted as d(k, k). With probability 1 − πρ reallocation is zero. We have Rx = 12 πρd(k1,1 , k1,1 ). The multiplication with

1 2

avoids double-counting. The amount of reallocation starting from a distribution

where ε is random across plants is given by Rr = 21 πd(k1,1 , k1,1 ). The steady state level of reallocation is thus x r Rss = xss + (1 − xss 1 ·R 1 )·R 1 ss Rss = πd(k1,1 , k1,1 ) [xss 1 ρ + (1 − x1 )] , 2 ss = 12 πρd(k1,1 , k1,1 ) · By substituting in xss 1 from above R

1 . (1−π)ρ+π

We can obtain the transition path

of reallocation by using the law of motion for x above. The total amount of reallocation in each period depends on the relative weights of the different components of the x and r vectors. It is given by i 1 Xh ˜ 1,j , k1,k ) (x1,j ρ + r1,j )d(k1,j , k1,k ) + x1,j (1 − ρ)d(k R = π 2 j Any plant that receives a new ε draw and reallocates (πρ), reallocates the expected difference between k1,j and k1,k . A fraction π(1 − ρ) will not change its ε but since the target vector of capital has changed, reallocation is given by the expected distance between k1,j and k1,k if the plant was in the randomly allocated set. Finally, if capital can be reallocated but ε does not change, the distance for the previously ˜ k), which is simply the weighted absolute difference between two vectors. ordered vector is given by d(k, If π is allowed to be time-varying we need to replace π with πt in the above formulation. Standard Deviation of MRPK We can use a similar approach to compute the standard deviation of the average revenue product of capital, σM RP K , our measure of misallocation.26 This measure is computed at the end of a period, after new draws of ε are made and new choices of k are implemented. We can first 25

We added the subscript 1 to the vectors x and r because it will facilitate the notation once we introduce time-varying λ. The subscript indicates that capital is taken from the vector j, while the ε distribution is ε1 . 26 We will simply write σ to save on notation.

31

A SOLUTION ALGORITHM FOR PLANNER’S PROBLEM compute the value of σ ss for fixed k and ε vectors. Define the standard deviation of average products for the x . To find the value of the random part we use σ 2 = E(X 2 )−E(X)2 synced part x of the capital vector as σ1,ss

to write 2 2 σ 2 (ε · kss ) = E(ε2 )E(kss ) + Cov(ε2 , kss ) − [E(ε)E(kss ) + Cov(ε, kss )]2 ,

where k will later stand for the average revenue product of k, k α . For any random allocation of ε0 we therefore have that the expected value of σ 2 (ε0 · kss ) is given by PJ 2

0

E(σ (ε · kss )) = E(ε

02

2 )E(kss )

2

− (µE(kss )) −

j=1

Cov(ε0j , kss )2 , J

where J = N ! are the possible permutations of the ε vector. This can be written as 2 E(σ 2 (ε0 · kss )) = E(ε02 )E(kss ) − (µE(kss ))2 −

σ 2 (ε0 )σ 2 (kss ) . N −1

Define the square root of this expression as f (ε, k). We have used E(ε) = µ, as before. We can then write x ss σ1ss = xss 1 σ1,ss + (1 − x1 )f (ε1 , k1,1 ).

Using the laws of motion derived above we can compute the transition path for σ given a different target x be the standard deviation of average products when ε = ε1 and k = k1,k for the vector for capital. Let σ1,k

synced part of the vector. Then the transition path is described by σ0 =

X

x (xj σ1,j + r1,j f (ε1 , k1,j )).

j x Ordered (random) parts x1,k (r1,k ) produce a standard deviation of σ1,k (f (ε1 , k1,k )). Plants that are in sets x x1,n produce σ1,n , while plants in sets r1,n produce f (ε1 , k1,n ). The terms π and ρ only appear indirectly as

part of the laws of motion for the different x and k vectors because σ is computed at the end of a period. Allowing for time-varying λ with fixed k

To see how the laws of motion change when the distribution

of ε is time-varying, first consider the case where λ changes, but the target vector of capital k remains fixed. Denote as e the index of the new ε draw from the #λ different distributions and as f = #λ ∩ e. Define as xi,1 (ri,1 ) the mass of plants that characterized by having a synced (random) allocation of εi and ki,1 = k1 , the fixed target vector of capital. The law of motion can then be written as

32

A SOLUTION ALGORITHM FOR PLANNER’S PROBLEM

x0e,1 = πρ + (1 − ρ)(xe,1 + πre,1 ) x0f,1 = (1 − ρ)(xf,1 + πrf,1 ),

∀f

0 = (1 − π)ρ + (1 − π)(1 − ρ)re,1 re,1 0 = (1 − π)(1 − ρ)rf,1 , rf,1

∀f.

Period reallocation is now given by " # X 1 R = πd(k1 , k1 ) · ρ + (1 − ρ) ri,1 . 2 i The standard deviation of average products of capital is

σ0 =

X

x (xi,1 σi,1 + ri,1 f (εi , ki,1 )).

i

The last expression now has to take into account the different values of σ depending on the distribution of ε. Allowing for time-varying λ and k

Now consider the case where λ and the target vector of capital k

change. Denote as e the index of the new ε draw from the #λ different distributions and as f = #λ ∩ e. The index of the new target capital vector is k i and the set of non-target vectors is ni = j i ∩ k i .27 Define as xi,j (ri,j ) the mass of plants that characterized by having a synced (random) allocation of εi and ki,j i . The law of motion can then be written as

P P x0e,ke = πρ + π(1 − ρ) i j (xe,j i + re,j i ) + (1 − π)(1 − ρ) · xe,ke P P x0f,kf = π(1 − ρ) i j (xf,j i + rf,j i ) + (1 − π)(1 − ρ) · xf,kf , ∀f x0i,j h = (1 − π)(1 − ρ)xi,j h , 0 re,j i

∀(i, j, h 6= i)

x0i,ni = (1 − π)(1 − ρ)xi,ni , ∀(i, n) P = (1 − π)ρ h (xh,j i + rh,j i ) + (1 − π)(1 − ρ)re,j i , 0 rf,j i = (1 − π)(1 − ρ)rf,j i ,

∀(i, j)

∀(i, j).

Period reallocation is now given by 27

Note that there is a target distribution k for each level of λ. This is indicated by the subscript k i

33

B ROBUSTNESS

i 1 XXXh ˜ j h , kki ) . (xi,j h ρ + ri,j h )d(kj h , kke ) + xi,j h (1 − ρ)d(k R = π 2 i j h The standard deviation of average products of capital is

σ0 =

XXX i

B

j

x (xi,j h σi,j h + ri,j h f (εi , ki,j h )).

h

Robustness

B.1

Different specification of adjustment costs

In the model above, capital adjustment costs were scaled by K α . Recall equation (19) which is reprinted here. α

y = AK [π(µa + φa ) + (1 − π)(µn + φn )] − K

α

F (π)

Z

F dG(F ),

(32)

0

This specification of adjustment costs is computationally convenient. If we define the part of the Solow residual A˜ that is coming from plant heterogeneity as B ≡ π(µa + φa ) + (1 − π)(µn + φn ), then we can easily separate the capital reallocation from the accumulation problem, since the resource constraint in the accumulation problem becomes " c + K 0 = K α · AB −

#

F (π)

Z

F dG(F ) + (1 − δ)K.

(33)

0

From the reallocation problem we receive a time series of (AB −

R F (π) 0

F dG(F )), which the planner can

perfectly forecast using the endogenous law of motion for Γ, and the exogenous law of motion for A. An alternative specification of (19) is α

y = AK [π(µa + φa ) + (1 − π)(µn + φn )] − AK

α

Z

F (π)

F dG(F ),

(34)

0

in which case

" c + K 0 = AK α · B −

#

F (π)

Z

F dG(F ) + (1 − δ)K.

(35)

0

The reallocation problem now produces a time series B − forecast.

34

R F (π) 0

F dG(F ) which the planner can perfectly

B.1 Different specification of adjustment costs

B ROBUSTNESS

An third specification of (19) reads Z

α

y = AK [π(µa + φa ) + (1 − π)(µn + φn )] − K

F (π)

F dG(F ).

(36)

0

The adjustment costs are now scaled by aggregate capital. This specification interprets the costs for the Planner of learning a plant’s realization as size-weighted. Since there are now K units of each type of plant, the costs is also scaled up by K. 0

Z

α

c + K = ABK − (1 − δ −

F dG(F ))K. 0

35

F (π)

(37)