Firm Dynamics and the Granular Hypothesis

Transcription

1 Firm Dynamics and the Granular Hypothesis Vasco. Carvalho Basile Grassi arch, 21, 2014 [Preliminary. Link the the latest version] Abstract Building on the standard firm dynamics setup of Hopenhayn (1992), we develop a quantitative theory of aggregate fluctuations arising from idiosyncratic shocks to firm level productivity. This allows us to generalize the theoretical results in Gabaix (2011) to account for persistent micro-level shocks, optimal size decisions as well as endogenous firm entry and exit. We then use our model to provide a quantitative evaluation of Gabaix s granular hypothesis and find that it yields aggregate fluctuations of the same order of magnitude as a standard representative-firm real business cycle model. A calibration of our model to the US economy with a large number of firms leads to sizable aggregate fluctuations: the standard deviation of aggregate TFP (respectively output) is 0.8% (respectively 1.7%). We use this calibration to explore firms comovement over the business cycle. The model predicts that the differential growth between large and small firms is pro-cyclical as it is in the data. Keywords: Firm Dynamics; Granular Hypothesis; Firm Size Distribution; Aggregate Fluctuations The authors would like to acknowledge helpful comments from Axelle Arquié, Florin Bilbiie, Gian Luca Clementi, Antoine Coste, Jan Eeckhout, Xavier Gabaix, Wouter den Haan, Hugo Hopenhayn, Jean Imbs, Boyan Jovanovic, orten Ravn, Edouard Schaal and Rob Shimer. The authors thank seminar participants at CREi, CREST-INSEE, Essex, LSE, NYU, PSE, UCL and the Conference in Honor of Lars Hansen at the University of Chicago. Part of this work was made during Basile Grassi s stay at the NYU Economics department and he is grateful for their hospitality. He also thanks the Fulbright commission for financial support. University of Cambridge and Centre de Recerca en Economia Internacional, Universitat Pompeu Fabra and Barcelona GSE. vcarvalho@crei.cat. Paris School of Economics, Université Paris I Panthéon Sorbonne, and Centre de Recherche en Économie et STatistique. basile.grassi@gmail.com. 1

2 1 Introduction When large firms represent a disproportionately large share of the economy, aggregate fluctuations may arise from idiosyncratic shocks to these firms. This hypothesis to which we refer as the granular hypothesis as proposed in Gabaix (2011) - opens the possibility of doing away with aggregate shocks, instead tracing back the origins of aggregate fluctuations to micro shocks hitting a small number of large firms. While this hypothesis has proven influential in the literature, it remains largely an accounting result - concerning the aggregation of firm level variability in static environments whose relation to the extant theory of firm dynamics is left unspecified. Partly as a result, the granular hypothesis remains a possibility result whose quantitative relevance has not yet been established. In this paper we cast the granular hypothesis in a standard firm dynamics setting. Building on the setup of Hopenhayn (1992), we develop a quantitative theory of aggregate fluctuations arising only from idiosyncratic shocks to firm level productivity. This allows us to generalize the theoretical results in Gabaix (2011) to account for persistent micro-level shocks, optimal size decisions as well as endogenous firm entry and exit. Further, we provide a quantitative evaluation of this hypothesis and find that it yields aggregate fluctuations of the same order of magnitude as a standard representativefirm real business cycle model. A calibration of our model to the US economy with a large number of firms leads to sizable aggregate fluctuations: the standard deviation of aggregate TFP (respectively output) is 0.8% (respectively 1.7%). Our setup follows Hopenhayn (1992) closely. Firms differ in their idiosyncratic productivity level, which here is assumed to follow a discrete arkovian process. Incumbents have access to a decreasing returns to scale technology using labor as the only input. They produce a unique good in a perfectly competitive market. They face an operating cost at each period, which in turn generates endogenous exit. The crucial difference relative to Hopenhayn (1992)- and much of the large literature that follows from it - is that we do not rely on the traditional continuum of firms assumption in order to characterize the law of motion for the firm size distribution. Instead, we characterize the law of motion for any finite number of firms and show that, generically, the distribution of firms across productivity levels is time-varying in a stochastic fashion. As is well known, this distribution is the aggregate state variable in this class of models. 2

3 An immediate implication of out findings is therefore that aggregate productivity, aggregate output and factor prices are themselves stochastic. In a nutshell, we show that the standard workhorse model in the firm dynamics literature once the assumption regarding a continuum of firms is dropped already contains in it a theory of the aggregate fluctuations. We then show that whether standard firm dynamics models can deliver a quantitatively relevant theory of the business cycle depends on the rate at which aggregate fluctuations vanish as a function of the number of firms. We offer a characterization of this decay rate under parametric conditions on the firm size distribution. In particular, we show analytically that, as the number of firms increases, the rate at which aggregate volatility decays is slower than what a simple central limit argument would predict. This result relies on the fat-tailedness of the firm size distribution as in Gabaix (2011). Therefore, our results generalize the latter by taking into account the persistence of the idiosyncratic shock process, firm entry, exit and optimal size decisions. In particular, we show that a low persistence of the idiosyncratic productivity process increases the rate of convergence of the aggregate volatility. As mentioned above, quantitatively we are able to generate aggregate fluctuations of the same magnitude as the standard real business cylce model but arising from idiosyncratic firm-level shocks only. A calibration of our model to the US economy with a large number of firms leads to sizable aggregate fluctuations: the standard deviation of aggregate TFP (respectively output) is 0.8% (respectively 1.7%). Our framework also allows us to study the macro and micro dynamic effects of a negative productivity shock on the largest firm. Such a shock is contractionary at the aggregate level. By construction, the decrease in the productivity of the largest firm induces a smaller aggregate productivity, since the latter is shown to be an average of the idiosyncratic productivities. However, this shock induces the largest firm to reduce its size and to cut its labor demand. The latter pushes the wage down. Other firms benefit from this reduction in their labor cost. On average, their productivity remains the same while their cost decreases, so they expand. Overall, a negative shock on the largest firm is contractionary at the aggregate level but expansionary for the typical firm in the economy. The upshot of this theory is that the business cycle, in our setting, is led by the large firms dynamics. In particular, if a small number of large firms face higher productivity 3

4 than usual, this will lead to an aggregate expansion. To the econometrician, it would thus seem as if large firms are more cyclically sensitive, in that they co-move more with the aggregate echoing the recent findings in oscarini and Postel-Vinay (2012). The latter find that the correlation between the differential net growth rate of large versus small firms is negatively correlated with the unemployment rate. We find a similar result on a simulated representative panel drawn from our calibrated model. Though our results are consistent with oscarini and Postel-Vinay (2012) findings, causality is reversed : in our setting, it is not that large firms are more cyclically sensitive. Rather it is the business cycle itself that reflects large firms dynamics. The paper relates to two distinct literatures: an emerging literature on the microorigins of aggregate fluctuations and the more established firm dynamics literature. Gabaix (2011) describes the granular hypothesis and shows the possibility result that we extend to our framework. Other papers studying the micro-origins of aggregate fluctuations are Acemoglu et al. (2012), di Giovanni and Levchenko (2012), Carvalho (2010) and Carvalho and Gabaix (2013) 1. Relative to this literature, we contribute by grounding the granular hypothesis in a standard firm dynamics setting, extending the existing theoretical results to this setting and providing a first attempt at quantification. This paper is also related to the firm dynamics literature: Hopenhayn (1992), Campbell (1998), Veracierto (2002), Khan and Thomas (2003, 2008), Bachman and Bayer (2009). Some papers have studied aggregate fluctuations in an entry/exit framework as Lee and ukoyama (2008), Clementi and Palazzo (2010) and Bilbiie et al. (2012). However they restrict their analysis to common, aggregate, shocks. Relative to this literature, we show that its standard workhorse model once the assumption regarding a continuum of firms is dropped and the firm size distribution is fat tailed already contains in it a theory of the business cycle. We show this both theoretically and quantitatively. The paper is organized as follows. Section 2 derives the model. Section 3 shows our theoretical results on the rate of decay of aggregate fluctuations. Section 4 explains the calibration, the origin of fat-tailed firm size distribution and exposes our first quantitative results. Section 5 displays the impulse response to a negative shock on the largest firm. Section 6 looks at the cyclicality of large versus small firm in our model. Finally, section 7 concludes. 1 Some empirical evidence can be found in di Giovanni, Levchenko and ejean (2012). 4

5 2 odel We analyze a standard firm dynamics setup (Hopenhayn, 1992) with a finite but possibly large number of firms. We show how to solve for and characterize the evolution of the firm size distribution without relying on the usual law of large numbers assumption. We prove that in this setting the firm size distribution does not converge to a stationary distribution, but instead fluctuates stochastically around it. As a result, we show that aggregate prices and quantities are not constant over time as the continuum assumption in Hopenhayn (1992) - repeatedly invoked by the subsequent literature - does not apply. Because of this crucial property, we are able to show that the standard firm dynamics setup is able to generate aggregate fluctuations with only idiosyncratic, firm-level shocks. In this section we first describe the economic environment. As is standard in this class of models, this involves specifying a firm-level productivity process, the incumbents problem and the entrants problem. We then show how to characterize the law of motion of the productivity distribution. 2.1 odel Setup The setup follows Hopenhayn (1992) closely. Firms differ in their productivity level, which here is assumed to follow a discrete arkovian process. Incumbents have access to a decreasing returns to scale technology using labor as the only input. They produce a unique good in a perfectly competitive market. They face an operating cost at each period, which in turn generates endogenous exit. There is a large but finite number of potential entrants that differ in their productivity. To operate next period, potential entrants have to pay an entry cost. The economy is closed in a partial equilibrium fashion by specifying a labor supply function that increases with the wage. Productivity Process We assume a finite but potentially large number of idiosyncratic productivity levels. The productivity space is thus described by a S-uple Φ := {ϕ 1,...,ϕ S } such that ϕ 1 <... < ϕ S. A firm is in state (or productivity state) k when its idiosyncratic productivity is equal to ϕ k. Each firm s (log) productivity level is assumed to follow 5

6 a arkov chain with a transition matrix P. We denote F(. ϕ) as the conditional distribution of the next period s idiosyncratic productivity ϕ given the current period s idiosyncratic productivity ϕ. 2 Incumbents Problem The only aggregate state variable of this model is the distribution of firms on the set Φ. We denote this distribution by a (S 1) vector µ t giving, for each level of productivity, the number of firms at this productivity level at t. For the current setup description, we abstract from explicit time t notation, but will return to it when we characterize the law of motion of the aggregate state. Given an aggregate state µ, and an idiosyncratic productivity level ϕ, the incumbent solves the following static profit maximization problem: π (µ,ϕ) = ax{exp(ϕ)n α w(µ)n c f } n where n is the labor input, w(µ) is the wage for a given aggregate state µ, and c f is the operating cost to be paid every period. It is easy to show that π is increasing in ϕ and decreasing in w for a given aggregate state µ. The output of a firm is then ( ) y(µ,ϕ) = exp(ϕ) 1 α 1 α 1 α. Inwhat follows, thesize of a firmwill refer to its output α w(µ) level if not otherwise specified. The timing of decisions for incumbents is standard and described as follows. The incumbent first draws its idiosyncratic productivity ϕ at the beginning of the period, pays the operating cost c f and then hires labor to produce. It then decides whether to exit at the end of the period or to continue as an incumbent the next period. We denote the present discounted value of being an incumbent for a given aggregate state µ and idiosyncratic productivity level ϕ by V(µ, ϕ), defined by the following Bellman equation: { V(µ,ϕ) = π (µ,ϕ)+max 0,β µ Λ ϕ Φ } V(µ,ϕ )F(ϕ ϕ)γ(dµ µ) 2 That is, given a productivity level ϕ s the distribution F(. ϕ s ) is given by the s th -row vector of the matrix P. Throughout we assume that F(. ϕ) is decreasing in ϕ. 6

7 where β is the discount factor, Γ(. µ) is the conditional distribution of µ, tomorrow s aggregate state and F(. ϕ) is the conditional distribution of tomorrow s idiosyncratic productivity given today s idiosyncratic productivity of the incumbent. The second term on the right hand side of the value function above encodes an endogenous exit decision. As is standard in this framework, this decision is defined by a threshold level of idiosyncratic productivity given an aggregate state. Formally, since the instantaneous profit is increasing in the idiosyncratic productivity level and F(. ϕ) is decreasing in ϕ, there is a unique index s (µ), for each aggregate state µ, such that (i) for ϕ ϕ s (µ) the incumbent firm continues in operationnext period and, conversely (ii) for ϕ ϕ s (µ) 1 firm decides to exit next period. After studying the incumbents problem, we now turn to the problem of potential entrants. Entrants Problem We formulate the entry decision as a discretized version of Clementi and Palazzo (2010). There is an exogenously given, constant and finite number of prospective entrants. Each potential entrant has access to a signal about their potential productivity next period, should they decide to enter today. To do so they have to pay a sunk entry cost which, in turn, leads to an endogenous entry decision which is again characterized by a threshold level of initial signals. Formally, entrants signals are distributed according G = (G q ) q [1...S], a discrete distribution over Φ. There is a total of potential entrants every period, so that the.g q gives the number of potential entrants for each signal level ϕ q. If a potential entrant decides to pay the entry cost c e, then she will produce next period with a productivity level drawn from F(. ϕ q ). Given this we can define the value of a potential entrant with signal ϕ q for a given the aggregate state µ as V e (µ,ϕ q ): V e (µ,ϕ q ) = β V(µ,ϕ )F(dϕ ϕ q )Γ(dµ µ) µ Λ ϕ Φ A prospective entrant pay the entry cost and produce the next period if the above value is greater or equal to the entry cost c e. As in the incumbent exit decision, this now induces a threshold level of signal, e (µ), for a given aggregate state mu such that (i) 7

8 for ϕ q ϕ e (µ) the potential entrant starts operating next period and, conversely (ii) for ϕ q ϕ e (µ) 1 the potential entrant decides not to do so. For simplicity, henceforth we assume that the entry cost is normalized to zero: c e = 0 which in turn implies that ϕ e (µ) = ϕ s (µ). Labor arket and Aggregation Following Clementi and Palazzo (2010), we assume that the supply of labor at a given wage w is given by L s (w) = L()w γ with γ > 0. We assume that the parameter in the labor supply function is a function of, the number of potential entrants. This assumption is necessary because in what follows we will be interested in characterizing the behavior of aggregate quantitites and prices as we let increase. If total labor supply were to remain fixed across these economies while the total demand for labor market increases, the wage would increase mechanically. We therefore make this assumption to abstract from this effect of increasing on the equilibrium wage. To find equilibrium wages, we now derive the aggregate labor demand in this economy. To do this, note that if Y t is the aggregate output, i.e. the sum of all individual incumbents output, then Y t = A t (L d t )α where L d t is the aggregate labor demand, the sum of all period t incumbents labor demand. A t, the aggregate total factor productivity, is given by: A t = ( Nt i=1 exp(ϕ i t ) 1 1 α ) 1 α where ϕ i t is the productivity level at date t of the ith firm among the N t operating firms at date t. This can be rewritten by aggregating all firms that have the same productivity level: A t = ( S s=1 1 α µ s t exp(ϕ s ) 1 α) 1 = (B.µ t ) 1 α where B is the (S 1) vector of parameters (exp(ϕ 1 ) 1 1 α,...,exp(ϕs ) 1 1 α ). As discussed above, the distribution of firms µ t across the discrete state space Φ = {ϕ 1,...,ϕ S } is a (S 1) vector equal to (µ 1 t,...,µs t ) such that µs t is equal to the number of operating firms in state s at date t. By the same argument, it is easy to show that aggregate 8

9 ( ) 1 labor demand is given by L d αa 1 α (w t ) = t w t. Note that the model behaves as a one factor model with aggregate TFP A t. Themarketclearingconditionthenequateslaborsupplyandlabordemand, i.e. L s (w t ) = L d (w t ). Given date t productivity distribution µ t, we can then solve for the equilibrium wage to get: w t = ( ) α 1 B 1 α.µ γ(1 α)+1 t 1 α L() From this expression, note that the wage is fully pinned down by the distribution µ t. That is, given a current period distribution of firms across productivity levels we can solved for all equilibrium quantities and prices. We are left to understand how this distribution evolves over time, i.e. how to solve for for Γ(. µ t ), which is addresses in the following section. (1) 2.2 Aggregate State Dynamics and Uncertainty In this section, we first show how to characterize the law of motion for the productivity distribution, the aggregate state in this economy. We will prove that, generically, the distribution of firms across productivity levels is time-varying in a stochastic fashion. An immediate implication of this result is that aggregate productivity A t and aggregate prices are themselves stochastic as they are simply a function of this distribution. Additionally, we then show that the characterization of the stationary firm productivity distribution offered in Hopehayn obtains only as a limiting result of our setup, as the number of firms goes to infinity. Law of otion of the Productivity Distribution In the standard setting with a continuum of firms, Hopenhayn (1992) shows that by appealing to a law of large numbers, the law of motion for the productivity distribution is in fact deterministic. In the current setting, with a finite number of incumbents, a similar argument cannot be made. We now show how to characterize the law of motion for the productivity distribution. In order to build intuition for our general result below, we start by exploring a simple example where, for simplicity, we ignore entry and exit of firms. Assume there are only 9

10 Figure1: Whythevectorµ t+1 followsamultinomialdistribution. Toppanel: continuum of firms case, unique possible outcome. Bottom panel: finite number of firms, one (of many) possible outcome. three levels of productivity (S = 3) and four firms. At time period t these firms are distributed according to the bottom-left panel of figure 1, i.e. all four firms produce with the intermediate level of productivity. Further assume that these firms have an equal probability of 1/4 of going up or down in the productivity ladder and that the probability of staying at the same intermediate level is 1/2. That is, the second row of the transition matrix P in this simple example is given by (1/4, 1/2, 1/4). First note that, if instead of four firms we had assumed a continuum of firms, the law of large number would hold such that at t+1 there would be exactly 1/4 of the (mass of) firms at the highest level of productivity, 1/2 would remain at the intermediate level and 1/4 would transit to the lowest level of productivity (top panel of figure 1). This is not the case here, since the number of firms is finite. For instance, a distribution of firms such as the one presented in the bottom-right panel of figure 1 is possible with a positive probability. Of course, many other arrangements would be possible outcomes. Thus, in this example, the number of firms in each productivity bin at t+1 follows a multinomial distribution with a number of trials of 4 and an event probability vector (1/4,1/2,1/4). In this simple example, all firms are assumed to have the same productivity level at time t. It is easy however to extend this example to any initial arrangement of firms 10

11 over producivity bins. This is because, for any initial number of firms at a given productivity level, the distribution of these firms across productivity levels next period follows a multinomial. Therefore, the total number of firms in each productivity level next period, is simply a sum of multinomials, i.e. the result of transitions from all initial productivity bins. ore generally, for S productivity levels, an (endogenous) finite number of incumbents, N t making optimal employment and production decisions and and accounting for entry and exit decisions, the following theorem holds. Theorem 1 The number of firms at each productivity level at t+1, given by the (S 1) vector µ t+1, conditional on the current vector µ t, follows a sum of multinomial distributions and can be expressed as: µ t+1 = m(µ t )+ǫ t+1 (2) where ǫ t+1 is a random vector with mean zero and a variance-covariance matrix Σ(µ t ) and m(µ t ) = (Pt ) (µ t +G) S Σ(µ t ) = (G s +µ s t).w s s=s (µ t) where P t is the transition matrix P with the first (s (µ t ) 1) rows replaced by zeros. W s = diag(p s,. ) P s,. P s,. where P s,. denotes the s-row of the transition matrix P. Proof. The distributionoffirmsµ t acrossthediscrete statespace Φ = {ϕ 1,...,ϕ S }is a (S 1)vectorequalto(µ 1 t,...,µs t )suchthatµs t isequaltothenumberofoperatingfirms in state s at date t. The next period s distribution of firms across discrete state space Φ = {ϕ 1,...,ϕ S } is the sum of the evolution of incumbents and successful entrants. In what follows, we define two conditional distributions. First, the distribution of incumbent firms at date t+1 conditional on the fact that incumbents were in state s at date t is denoted as f t+1..,s This (S 1) vector is such that for each state k in {1,...,S}, ft+1, k,s the k th element of f.,s t+1 gives the number of incumbents in state k at t+1 which were in state ϕ s at t. 11

12 Similarly, let us define g.,s t+1 the distribution of successful entrants at date t + 1 given that they received the signal ϕ s at date t. This (S 1) vector is such that for each state k in {1,...,S}, g k,s t+1, the kth element of g.,s t+1 gives the number of entrants in state k at t+1 which received a signal ϕ s at t. The period t+1 firms distribution is the sum of all these conditional distributions and thus the vector µ t+1 satisfies: µ t+1 = S s=s (µ t) f.,s t+1 + S s=s (µ t) g.,s t+1 (3) Note that f.,s t+1 and g.,s t+1 are now multivariate random vectors implying that µ t+1 also is a random vector. Atdatet+1fors s (µ t ), f.,s t+1 followsamultinomialdistributionwithtwo parameters: the integer µ s t and the (S 1) vector P s,. where P s,. is the s th row vector of the matrix P. For s s (µ t ) 1, f.,s t+1 is equal to zero. Similarly, at date t + 1 for s s (µ t ), g.,s t+1 follows a multinomial distribution with two parameters: the integer.g q and the (S 1) vector P q,.. For s s (µ t ) 1, g.,s t+1 is equal to zero. Recall that the mean and variance-covariance matrix of a multinomial distribution, (m,h), isrespectively the(s 1)vector mhandthe(s S)matrixH = diag(h) h h. So let us define W s = diag(p s,. ) P s,. P s,.. From the right hand side of equation 3, using the fact that the f.,s t+1 and g.,s t+1 follow multinomials, µ t+1 has a mean m(µ t ) and a variance-covariance matrix Σ(µ t ) where m(µ t ) := Σ(µ t ) := S s=s (µ t) S s=s (µ t) [µ st.p s,. +G s.p s,. ] (G s +µ s t ).W s = (P t ) (µ t +G) where P t is the transition matrix P with the first (s (µ t ) 1) rows replaced by zeros. Equation 3 can be rewritten in a simple way as the sum of its mean and a zero-mean shock: µ t+1 = m(µ t )+ǫ t+1 12

13 where ǫ t+1 = S s=s (µ t) [ f.,s t+1 µs t.p ] S s,. + s=s (µ t) [ g.,s t+1 Gs.P s,. ] i.e ǫ t+1 is the demeaned version of µ t+1. This gives us the result stated in the theorem. After taking into account the dynamics of incumbent firms and entry/exit decisions, the law of motion of the aggregate state -the firm distribution over productivity levelsis remarkably simple: tomorrow s firm distribution is an affine function of today s distribution up to a stochastic term that reshuffles firms across productivity levels. It is easy to understand this characterization by recalling our simple example economy above without entry and exit. Given the state transition probabilities in this simple case we should, for example, observe that on average the number of firms remaining at the intermediate level of productivity is twice that of those transiting to the highest level of productivity. This is precisely what the affine part of equation 2 captures: the m(µ t ) term reflects these typical transitions, which are a function of the P matrix only. However, with a finite number of firms, in any given period there will be stochastic deviations from these typical transitions as we discuss above. In the theorem, this is reflected in the reshuffling shock term, ǫ t+1, in the law of motion 2. How important this reshuffling shock is for the evolution of the firm distribution is dictated by the variance-covariance matrix Σ(µ t ) which, in turn, is a function of the transition matrix P, the current firm distribution µ t and, in the general case with entry and exit, the signal distribution available to potential entrants. Limit Economies The above characterization - in particular, Equation 2 - is instructive of the differences of the current setup relative to a standard Hopenhayn economy. The latter corresponds to the case where all of the relevant firm dynamics are encapsulated by the affine term m(µ t ). Appealing to a law of large numbers argument with a finite number of firms is therefore equivalent to imposing the variance-covariance matrix to be zero so that the aggregate state µ t becomes non-stochastic. That is, the reshuffling shock, ǫ t+1, would be absent. In this section we show that this assumption is justified as the number of firms goes to infinity. That is, we show that the characterization of the stationary 13

14 firm productivity distribution offered in Hopehayn obtains as a limiting result of our setup, as the number of firms goes to infinity. In order to show this, we first make an assumption 1 about the labor supply function. Assumption 1 The labor supply, for a given wage, is a linear function of the number of potential entrants: L() = L. This assumption ensures that the equilibrium wage is independent of the number of potential entrants. To see this note that given the labor market equilibrium condition (equation 1), under assumption 1 the equilibrium wage is now a function of ˆµ t := µt, the normalized productivity distribution across productivity levels. Given that is a parameter of the model, we can therefore use µ t or ˆµ t interchangeably as the agggregate state variable. We find it convenient to work with the latter in what follows. In the following limit characterization we will be taking the number of potential entrants to infinity. Notice that this implies that the number of incumbents, N t, also goes to infinity, since the latter scales with the former. To see this note that letting e be the (S 1) unit vector, then N t = e.µ t = e.ˆµ t. Indeed one can see from equation 2 that µ t scales with and so does N t. We can now state the following corollary of theorem 1 which shows that the Hopenhayn framework is nested in ours as the number of potential entrants goes to infinity. Corollary 1 Let us define ˆµ t := µt infinity we have: for any t. Under assumption 1, as goes to ˆµ t+1 = (P t ) (ˆµ t +G) (4) where Pt is the transition matrix P where the first s (µ t ) 1 rows are replaced by zero, and where s (µ t ) is the threshold of the exit/entry rule when there is no aggregate uncertainty i.e the variance-covariance of the ǫ t+1 is zero. Proof. See appendix B.1. Therefore, as the number of potential entrants,, goes to infinity, the law of motion for the distribution of firms across productivity levels is deterministic and its evolution 14

15 is given by equation 4. An immediate consequence of this is that under appropriate conditions on the transition matrix P, a stationary distibution exists and is given by: ˆµ = (I P ) 1 P G (5) the matrix P as the matrix P where the first (s (ˆµ) 1) rows are replaced by zeros. This is the analogue in the current setting to Hopenhayn s (1992) stationary distribution. Taking stock, we have derived a law of motion for any finite number of firms and shown that, generically, the distribution of firms across productivity levels is time-varying in a stochastic fashion. An immediate implication of this is that aggregate productivity A t is itself stochastic as it is simply a function of this distribution. The latter corollary implies that only in the continuum case does the distribution converge to a stationary distribution and hence in this limiting case there are no aggregate fluctuations. As such, whether the model considered above can be taken as a quantitatively relevant theory of the business cycle, depends on the rate at which aggregate fluctuations vanish as a function of the number of firms. We characterize this decay rate under some conditions in the next section. 3 Aggregate Fluctuations In this section, we explore a tractable parameterization of the stationary distribution and characterize analytically the rate of decay of aggregate fluctuations. We first show some preliminary results and define a fat-tail distribution in our context. Then we turn to our theoretical result concerning the rate of decay of aggregate volatility. Finally, we also briefly study the implications of firm level disturbances on the persistence of aggregate variable. 3.1 Preliminary Results and Definitions We first show how the tail of the productivity distribution is related to its counterpart in the case of infinite number of firms. We then specialize our assumption on the class of firm-level productivity processes. We then explain why the firm-size distribution is 15

16 simply a relabelling of the productivity distribution. Finally we define what we mean by a fat-tail distribution. We start by stating a lemma that shows that, for a large enough number of firms, the right tail of the firm productivity distribution at each date t is well approximated by the right tail behavior of the stationary distribution ˆµ. Formally, we show that the tail of firm productivity distribution at any point in time, ˆµ t converges in probability to the tail of the stationary productivity distribution, ˆµ. Lemma 1 Assuming that the economy starts at date t = 0 with the stationary firm productivity distribution ˆµ. Then at each period t countercumulative of the firm productivity distribution ˆµ t converges in probability to the countercumulative stationary distribution ˆµ as, the number of potential entrants, goes to infinity. Proof See appendix B.2. The proof first defines the deviation of µ t from µ making use of equation 2 derived in the previous section. It then defines the associated (discrete) counter-cumulative distribution function (CCDF). Finally, it shows that the CCDF of the productivity distribution at any given date t converges in probability to the countercumulative stationary productivity distribution. Additionally, the proof also shows that for a given history of reshuffling shocks and a fixed, the distance between the countercumulative productivity distribution at any given date t and the countercumulative stationary distribution is governed by the firm-level transition probability matrix P. Heuristically, this is stating that for any finite number of firms, the productivity distribution hovers around the stationary distribution and how far it can be from the latter is a function of persistence of the stochastic productivity process. For the remainder of this section, we restrict the class of transition probability matrices P to be that given by the Rouwenhorst (1995) method (see Kopecky and Suen, 2010 for a formal discussion). This is a method to discretize an AR(1) process, such that the markov chain is entirely characterized by three parameters, ρ, the first order autocorrelation of the corresponding AR(1), σ e, its conditional variance and ϕ S, the upper bound of the discretized state space. We follow Kopecky and Suen, (2010) in calling such transition matrices Rouwenhorst matrices. Specifically in the proofs of the theorems below we make use of the fact that, for this class of transition probability matrices, 16

17 the first-order autocorrelation and conditional variance associated to the markov chain P are exactly ρ and σ e. 3. Assumption 2 The transition probability matrix P is a Rouwenhorst matrix. Thus far we have been making use of the aggregate state variable, µ t, the firm-level productivity distribution. As typical in this class of models, the firm-size distribution maps one to one to µ t. To see this note that the output of a firm with productivity ( ) α level ϕ s at time t is given by: yt s = exp( ϕs ) α 1 α 1 α w t. Therefore, at time t, the number of firms of size yt s is given by µs t so that the firm size distribution is given by µ t. We also define the latent size of a potential entrant with signal ϕ s, as the output level that would have obtained if an operating firm had productivity equal to ϕ s. By the same argument as above, the number of such firms would be given by G s. Therefore the latent size distribution is given by the vector.g. In what follows, following up on Gabaix (2011), we restrict our attention to power law distributions which we now define formally following Acemoglu et al. (2012). Definition 1 A sequence of (S 1)-vector (E ) N has a power law tail if there exist a constant β > 1, a slowly varying function L(.) satisfying lim t L(t)t δ = and lim t L(t)t δ = 0 for all δ > 0, and a sequence of positive numbers c = O(1) such that, for all N and all k < S, we have P(k) = c k β L(k) where P(k) = S s=k+1 E s, is the counter-cumulative function. In our context, when we say that µ t has a power law tail with parameter ξ, we are applying the definition 1 with E =.ˆµ t and β = ξ. Simililarly we say that the latent size distribution of potential entrant has a power law tail with parameter ζ. 3 For completeness we describe this class of transition matrices in detail in the appendix A 17

18 3.2 Rate of Decay of Aggregate Volatility We are now ready to state theorem 2 describing the rate of decay of the aggregate volatility, under the assumption of power law distribution. Theorem 2 Let the firm size distribution have a power law tail with parameter ξ and the latent size distribution of potential entrants has a power law tail with parameter ζ. Under assumption 2, if ξ/(2ρ) < 1 and ζ/(2ρ) < 1 then [ σ t ( Y ( ) t+1 u N I 2ρ/ξ ( ) ) = Ω σ t w N E 2ρ/ζ ] 1/2 + t (6) Y t (N t ) 2 2ρ/ξ N t (N t ) 2 2ρ/ζ N t where Ω() means that, given two sequences of positive real numbers {a } N and {b } N, a n = Ω(b n ) if liminf a /b > 0 and where u and w are non degenerate random variables; σ is the standard deviation of exp(e t i /(1 α)) and N t, N I t,ne t are respectively the number of incumbents at t, surviving time t incumbents and actual entrants in period t. Proof See appendix B.3. Theorem 2 characterizes the conditional volatility of aggregate output growth in this economy, i.e. the volatility of the growth rate of output conditional on the time t state. As announced before, we are assuming that both the stationary incumbent size distribution and the stationary latent size distribution of potential entrants are power law distributed. As a result of Lemma 1 we know that, as goes to infinity, the time t firm size distribution is also power law distributed. The assumptions on ξ and ζ ensure that these distributions are sufficiently fat tailed. The Ω notation means that aggregate growth volatility scales with N t, the number of time t incumbents, at rate that is no faster than the rate of the variable in the right hand side parenthesis. The latter reflects the separate contribution of (i) surviving time t incumbents (i.e. the first term in parenthesis) and that of (ii) time t entrants that start producing at t+1 (i.e. the second term in parenthesis). The key conclusion of the theorem is therefore that, for ξ/(2ρ) < 1 and ζ/(2ρ) < 1, the aggregate volatility scales at a slower rate than 1/ (N t ). Recall that the latter would be the rate of decay of aggregate volatitlity implied by a standard diversification 18

19 argument relying on standard central limit theorems. Such a theorem does not apply in a world of fat tailed distributions. In this way, the theorem generalizes Gabaix (2011) to an environment where firms are making endogenous decisions about their optimal scale (along with exit and entry decisions) while being faced with potentially persistent idiosyncratic productivity shocks. ore generally, the theorem implies that the large extant literature that builds on the Hopenhayn (1992) in order to understand firm-level dynamics and the firm size distribution potentially contains in it a theory of aggregate fluctuations. To gain an understanding of how slow this convergence to the continuum case is, we find it useful to abstract from the contribution of entry an exit. This can be justified by the fact that at any time period, both in quantitative versions of the model (as we show below) and in the data, entry/exit dynamics are of second order in accounting for aggregate fluctuations. Thus, focusing on the first term alone for N I t N t : σ t ( Y ( ) t+1 u 1/2 ) = Ω σ Y t (N t ) 1 ρ/ξ This approximation is useful to isolate the role played by mean-reversion of the firm-level productivity in shaping the rate of decay of aggregate volatility. As the expression makes clear, the higher is the persistence of the firm-level productivity process, the slower is the rate of decay of aggregate volatility. To understand this, recall our previous discussion on the role of productivity transition matrix P in determining how distant can the µ t distribution can be from its stationary counterpart µ. Given asssumption 2, this role is now pinned down by a single persistence parameter ρ. Therefore, higher persistence, i.e. a higher ρ, implies that any time t deviation from the stationary distribution is likely to persist at time t+1. Intuitively, high persistence implies that time t large firms are reverting to the average size in a slower fashion. So in the high persistence case, a given large firm would be larger on average next period relative to the low persistence case. Its impact on the aggregate is then higher, i.e. its contribution to the aggregate volatility is higher than in the low persistence case. 4 4 The persistence parameter ρ actually plays a dual role: (i) as we have just discussed, it influences the out of stationary equilibrium dynamics; (ii) it also influences directly the tail behavior of the stationary distribution as equation 5 shows. 19

20 3.3 Aggregate Persistence (Preliminary) We now turn our attention to the persistence of aggregate output growth and how it scales with the number of firms. The theorem below characterizes the conditional expectation of aggregate growth rate as a function of model parameters. Recall that, for a generic AR(1) process with persistence parameter ρ the conditional expectation of aggregate growth rate is given by ρ 1, so that the latter simply reflects the first-order autocorrelation of the underlying process. Theorem 3 Let the firm size distribution have a power law tail with parameter ξ and the latent size distribution of potential entrants has a power law tail with parameter ζ. Under assumption 2 and for 1 < ξ < 2 and 0 < ζ /ρ < 1 then: ( ) Tt+1 1 Y t+1 Y t E t [ Tt+1 T t ] α γ(1 α)+1 T t NI t k Z ρ t Z t NX t Xt v/īt N t Ī t N t Ī t N 1 1/ξ t ( ) N I 1/ξ t + u/īt N t N 1 ρ/ζ t ( N E t N t ) ρ/ζ N t, N I t,ne t and N X t are respectively the number of incumbents at t, surviving time t incumbents, entrants in period t and exiters in period t. Zt, Z ρ t, X t,īt are time dependent variables proportional to, respectively, the average size of surviving time t incumbents, the average size at t+1 of surviving time t incumbents, the average size of exiters at t and the average size of all time t incumbents. v and u are non-degenerate random variables and k is a constant. Proof See appendix B.4. The theorem expresses the conditional expectation of aggregate growth rate as the sum of the contribution of entry, exit and surviving incumbent dynamics. To understand how the conditional expectation of aggregate growth rate scales with the number of firms we again find it convenient to focus on a particular case where we abstract from entry and exit dynamics 5 : [ ] Tt+1 E t ρ 1 v/ Z t T t (N t ) 1 1/ξ 5 Formally, we assume N I t N t, N E t N X t 0 and k 1 20

21 As expected, the conditional expectation of aggregate growth rate of aggregate output converges to ρ 1 in the limit. Under the usual central limit theorem arguments this convergence should take place at rate 1/ N. The above equation states that, under a fat tailed firm size distribution, this rate of convergence is slower and governed by the tail of the firm size distribution. 4 Simulations (Preliminary) The model is solved using the algorithm described in section C.1. A discussion of the source of the fat-tailedness of the firm size distribution follows. Finally, we compute some business cycle statistics. 4.1 Calibration TocalibratethemodeltotheUSeconomy, wefirstsetthevalueofsomedeepparameters and the functional form associated with the distribution of potential entrants and the labor supply function. The span of control parameter α is set at 0.8. This value is chosen to be on the lower end of recent estimates, such as Basu and Fernald (1997) and Lee (2005). The discount factor β is set at 0.95 so that the implied annual gross interest rate is 4%, a value in line with the business cycle literature. The entry cost is normalized to be zero to keep the mathematics as simple as possible. Finally, the potential entrant distribution is such that the latent size distribution of potential entrant is a generalized pareto distribution with tail parameter ζ, scale parameter λ and zero as location parameter. As we discussed above, we assume that the labor supply scaling parameter is a linear function of the number of potential entrants: L() = L. The labor supply elasticity parameter, γ, is chosen to be 2. We jointly calibrate the operating cost c f, the first order autocorrelation ρ and variance σ e of the productivity process 6, the constant L, the parameter of the latent size distribution of potential entrants ζ, λ, and, the upper bound of the idiosyncratic producivity space ϕ S. The parameter are chosen such that some statistics of the stationary equilibrium (the one when goes to infinity) match their empirical counterpart. These 6 The productivity process is a discretization of a AR(1) process with autocorrelationρ and variance of shocks σe. 2 To do so, we follow the Tauchen (1986) method. 21

22 Statistic odel Data References Entry Rate BDS 2011 Entrants relative size BDS 2011 aximum size Larger Category in BDS is Tail index of Firm size dist Gabaix (2011) Table 1: Targets for the calibration procedure 0.35 Incumbents labor share BDS 2011 model Entrants labor share of total entrants employement BDS 2011 model Figure 2: Incumbent and entrants labor share by size clase (Sources: BDS and odel) 22

23 statistics are the entry rate, the entrant relative size, the maximum size, the incumbent and entrant size distribution, and finally the tail index of the firm size distribution. Except form the latter, these statistics are computed using the Business Dynamics Statistcs (BDS) in The target for the tail index of the firm size distribution is the one estimated by Gabaix (2011). Table 1 and figure 2 summarize the result of this calibration procedure. The corresponding moments in the model are computed for the stationary deterministic equilibrium. All the calibrated parameters are described in table 2. The model is able to match most moments, except the entry rate, which is too low in our baseline calibration. As the right panel of figure 2 shows the entry rate is too low in our baseline calibration because we are not able to match the number of small entrants in our model. This can be address by assuming a more sophisticated distribution for entrants. The first order autocorrelation and the conditional standard deviation of the productivity process are in line with previous estimates as in Lee and ukoyama (2008) or in Castro et al (2013). 4.2 The Tail of the Firm Size Distribution The calibration strategy is such that the model matches the fat tail of firm size distribution documented in Gabaix (2011). However the reason why the model is able to do so is different from the usual explanation provided in the firm dynamics literature. In this section, we show numerically that the fat tail comes from the high persistence and the fat tail of the potential entrants distribution rather than Gibrat s law or imitation. Power law distribution can come from proportional random growth also called Gibrat s law, as described in Gabaix (2009). The literature initiated by Luttmer (2007, 2010, 2012) build models that can generate fat-tailed distribution along a balanced growth path based on technology diffusion and spillover of technology by entrants. The flow of ideas literature (Lucas 2009, Lucas and oll 2011 and Alvarez et al. 2008) also experiments growth and fat-tailed distribution along a balanced growth path. In our framework, the fat-tailed distribution comes from a high persistence and large entrants. The persistence of the productivity process has an effect on the tail because with high persistence, firms benefit longer from a good shock and remain larger for a longer time than in a low persistence case. At the stationary equilibrium, there are 23

24 Parameters Value Description ρ 0.98 Autocorrelation of firms level shocks σ e 0.22 Std of idio shocks S 300 Number of productivity levels [ϕ 1,..,ϕ S ] [ 5.13,..,5.13] Productivity grid γ 2 Labor Elasticity α 0.8 Production function c f Operating cost c e 0 Entry cost β 0.95 Discount rate Number of potential entrants L Parameters of the labor supply function G Generalized Pareto(ζ,λ,0) Entrant s distr. of exp(ϕ) ζ 0.76 Tail parameter of the distr. G λ 0.70 Scale parameter of the distr. G ζ(1 α) Tail parameter of the entrants size distr. Table 2: Baseline calibration tail 2.5 tail ρ ζ (1 α) Figure 3: The tail of the stationary firm size distribution as a function of ρ (left) and ζ(1 α), the tail of the potential entrant distribution G (right) 24

25 tail ζ (1 α) ρ Figure 4: The tail of the stationary firm size distribution as a function of both ρ and ζ(1 α), the tail of the potential entrant distribution G more large firms and thus the tail is fatter. This can be seen in the left panel of figure 3: as the persistence ρ increases, the tail parameter decreases which means that the tail becomes fatter. The fatter the distribution (i.e the smaller the tail index) of potential entrants, the fatter the stationary firm size distribution as one can see on the right panel of figure 3. This is pretty intuitive, since bigger entrants also implies bigger firms and thus a fatter tail of the firm size distribution. A high persistence and a fat-tailed distribution are both necessary to get fat-tailed firm size distribution at the stationary equilibrium. This can be seen in figure 4 which plots the tail of the stationary firm size distribution as a function of both the persistence of the productivity process and the tail of the potential entrant distribution. The tail of the firm size distribution gets close to one only for high persistence and fat tail of the potential entrants distribution. 25