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Finance and Economics Discussion Series: 2008-14 Screen Reader version ^♣

Firm Dynamics with Infrequent Adjustment and Learning

Eugénio Pinto¹

February 2008

Keywords: Adjustment costs, learning, young firms

Abstract:

We propose an explanation for the rapid post-entry growth of surviving firms found in recent studies. At the core of our theory is the interaction between adjustment costs and learning by entering firms about their efficiency. We show that linear adjustment costs, i.e., proportional costs, create incentives for firms to enter smaller and for successful firms to grow faster after entry. Initial uncertainty about profitability makes entering firms prudent since they want to avoid incurring superfluous costs on jobs that prove to be excessive ex post. Because higher adjustment costs imply less pruning of inefficient firms and faster growth of surviving firms, the contribution of survivors to growth in a cohort's average size increases. For the cohort of 1988 entrants in the Portuguese economy, we conclude that survivors' growth is the main factor behind growth in the cohort's average size. However, initial selection is higher and the survivors' contribution to growth is smaller in services than in manufacturing. An estimation of the model shows that the proportional adjustment cost is the key parameter to account for the high empirical survivors' contribution. In addition, firms in manufacturing learn relatively less initially about their efficiency and are subject to larger adjustment costs than firms in services.

JEL Classification: E24, L11, L16

1 Introduction

In recent years there has been renewed interest in explaining patterns of firm dynamics, with new longitudinal datasets confirming heterogeneities between firms of different size and age. In particular, small and young (surviving) firms tend to grow faster and have higher failure rates than large and old firms, so that job destruction due to plant exit and job creation due to the scaling-up of firm size decrease with age.² Moreover, entering plants tend to be small, but survivors grow rapidly after entry and are the main factor behind the shift to the right of a cohort's size distribution.³ These patterns differ significantly across countries and sectors, suggesting that technological differences are important, but that country specific factors also matter.⁴

This paper proposes an explanation for the leading role of survivors' growth in post-entry firm dynamics based on the interaction between adjustment costs and a learning-about-efficiency mechanism. Following a literature that uses adjustment costs to account for some dynamic properties of firms' labor demand, such as Campbell and Fisher (2000), we show that adjustment costs can impact the lifetime dynamics of firms' labor demand in a way consistent with the data. To implement our theory, we use a standard model of firm dynamics with passive learning. In order to check the empirical fit of our model, we also assume that inefficient firms are pruned from the market, although the predictions of our theory hold even in the absence of a selection mechanism (e.g. when exit is not allowed).

Our contribution is twofold. First, we contribute to the empirical literature by introducing a decomposition of the change in a cohort's average size into a survivor component and a non-survivor component. Given the emphasis on survivors' growth, our measure allows a quick assessment of how well a particular theory matches the data in that respect. We apply our decomposition to the 1988 cohort of entrants in the Portuguese economy, using the Quadros de Pessoal dataset. Similarly to Cabral and Mata (2003), we find that growth of survivors is the main force behind the change in the cohort's average firm size. However, we also find that growth of survivors is especially intense in the initial years after entry and that there are significant cross-sector differences in terms of our decomposition. In particular, initial exit rates are smaller and the survivors' contribution to changes in size is higher in manufacturing than in services.

Second, we contribute to the theoretical literature by introducing linear adjustment costs into a model of Bayesian learning about efficiency. Our assumption of linear or proportional costs is justified by the finding of high inaction rates in employment adjustment, in varying degrees across sectors. Our model builds on Jovanovic (1982) by adding proportional costs that apply not only to regular labor adjustment, but also to job creation at entry and job destruction at exit. We show that proportional adjustment costs create incentives for firms to start smaller and, if successful, grow faster after entry. We prove this analytically in a simplified model in which there is no exit of firms. This result shows that proportional costs can generate firm growth without selection. When firms are allowed to exit, selection intensifies the effects of adjustment costs on firm growth, while costs to adjustment reduce exit rates. Therefore, adjustment costs increase the contribution of surviving firms to growth in the cohort's average size.

All that is needed for firm growth under linear adjustment costs is the existence of a learning environment that generates a stochastic process for perceived efficiency with both persistence and decreasing uncertainty in age.⁵ The intuition for why firms grow faster and display smaller exit rates under proportional adjustment costs is that initial uncertainty about true profitability makes entering firms prudent; that is, they enter small and "wait and see" since they want to avoid incurring superfluous entering/hiring costs and firing/shutdown costs on jobs that prove to be excessive ex post. This implies that surviving firms will grow faster, even though adjustment costs imply that there are fewer firms exiting the market and therefore less pruning of inefficient firms.

The assumption that entering firms face a Bayesian learning problem concerning their efficiency is standard in selection theories and has been advanced as an explanation for the high rates of exit, job creation, and job destruction among young firms. The initial literature on adjustment costs used a (strictly) convex specification in an attempt to explain the sluggishness in input responses to aggregate shocks. However, the assumption that costs of adjustment are linear is now standard in dynamic labor demand models, following a number of studies since the late 1980s that have documented the importance of inaction in employment adjustment at the micro level.⁶ Since strictly convex costs imply smooth adjustments over time, whereas linear costs imply immediate adjustment when it occurs, allowing for strictly convex costs, instead of linear costs, in the context where they also apply at entry and exit, would bias our analysis and eventually make our argument stronger. In the case of hiring/entering costs, entering firms would prefer to start smaller and adjust gradually to their optimal size, even if their perceived productivity remained unchanged or learning was absent. For firing/exiting costs, firms experiencing large declines in perceived productivity would adjust downwards in several steps, a scenario that would make firms start smaller to attenuate its effects. Therefore, by avoiding a bias towards firm growth, our decision to assume linear costs is conservative and permits a simplification of the methods employed to measure the effects of adjustment costs.⁷

To assess our model quantitatively, we calibrate and estimate a version of the model with finite learning horizon and positive dispersion in entry size. We conclude that linear costs are the key element to account for the high empirical contribution of survivors to changes in a cohort's average size. A calibration/estimation for the manufacturing and services cohorts also suggests that firms in manufacturing learn relatively less initially about their efficiency and are subject to substantially larger adjustment costs than firms in services.

This paper is related to the literature on both adjustment costs and firm dynamics. Within the literature on adjustment costs, the paper is associated with theories that use linear adjustment costs to explain certain aspects of the dynamic behavior of labor demand and job flows. Well-known examples are Bentolila and Bertola (1990), Hopenhayn and Rogerson (1993), and Campbell and Fisher (2000). Bentolila and Bertola (1990) and Hopenhayn and Rogerson (1993) analyze the effects of proportional firing (and hiring) costs on the dynamics of hiring and firing decisions, and on average labor demand. Both papers conclude that high firing costs make hiring and firing adjustments more sluggish, but they disagree on the implications of that for long-run employment. Campbell and Fisher (2000) use proportional costs of job creation and job destruction to explain the higher aggregate volatility of job destruction found in the U.S. manufacturing sector. These costs imply that in reaction to aggregate wage shocks employment changes at contracting firms are larger than employment changes at expanding firms. What is new in our paper is the assumption that adjustment costs apply equally to the entry/exit decisions and the hiring/firing decisions.⁸

Within the literature on firm dynamics the paper is connected with theories that attempt to explain the stylized facts on the lifecycle dynamics outlined above. The two main explanations for these facts are theories based on selection of firms and theories based on financing constraints.⁹ Selection theories stress the tendency for firms that accumulate bad realizations of productivity to exit the market and for firms that accumulate good realizations to survive and expand. This implies a composition bias towards larger and more efficient firms as smaller, inefficient, and slow-growing firms gradually exit the industry. Representative papers of selection theories are Jovanovic (1982), Hopenhayn (1992), Ericson and Pakes (1995), and Luttmer (2007). In all cases productivity realizations are exogenous, except in Ericson and Pakes (1995) where they are to some extent endogenous.

Meanwhile, theories employing financing constraints argue that some imperfection in financial markets causes young firms to have limited access to credit, forcing them to enter at a suboptimally small scale. As firms get older and survive, they establish creditworthiness and build up internal resources that enable them to expand to their optimal size. Important contributions to this literature are those of Cooley and Quadrini (2001), Albuquerque and Hopenhayn (2004), and Clementi and Hopenhayn (2006). In Cooley and Quadrini (2001) a transaction cost on equity and a default cost on debt imply that equity and debt are not perfect substitutes, so that size depends positively on the amount of equity. In Albuquerque and Hopenhayn (2004) and Clementi and Hopenhayn (2006) lenders introduce credit constraints because of limited liability of borrowers and limited enforcement of debt contracts and because of asymmetric information on the use of funds or the return on investment, respectively.

Cabral and Mata (2003) analyze whether these two theories are consistent with the evolution of a cohort's size distribution in the Portuguese manufacturing sector. They find that, as the cohort ages, the firm size distribution shifts to the right largely due to growth of surviving firms rather than exit of small firms. In addition, they find that in the first year after entry younger business owners are associated with smaller firms but that is no longer the case once the cohort gets to age seven. Assuming that age is a proxy for the entrepreneur's initial wealth, the authors conclude that the age-size effect supports the idea of financially constrained firms starting at a suboptimal size and present a model with financing constraints capturing this effect.

More recently, Angelini and Generale (forthcoming) use survey and balance sheet information for Italian manufacturing firms to analyze the impact of financing constraints on the evolution of the firm size distribution. They find that financially constrained firms tend to be small and young, although this does not have a significant effect on the overall firm size distribution. Moreover, they find that financing constraints decrease firm growth, with this effect being entirely due to small firms. In particular, being young and financially constrained does not have any additional effect. Based on these results and the fact that young firms grow faster than old firms, the authors conclude that financing constraints are not the main factor behind the evolution of the firm size distribution. In line with their argument, this paper interprets the facts presented in Cabral and Mata (2003) and other cross-sector evidence as the result of the interaction between adjustment costs and learning about efficiency.¹⁰

To our knowledge, this work is the first to suggest adjustment costs as an explanation for differences in firm dynamics by age. The paper by Cabral (1995) is nearest to this paper. In his model, firms must pay a proportional sunk cost to increase their production capacity. He argues that, in a model with Bayesian learning, a proportional capacity cost would make small entering firms grow faster than large entering firms. The reason is that small entrants are those whose initial profitability signals were not good, so their exit probabilities are higher, and therefore they choose to invest more gradually. Unlike our model, Cabral's model depends on the existence of selection. Also, by analyzing a size-growth relationship, his model is not able to explain why some large entering firms also grow substantially.

The paper is organized as follows. In section 2, we present evidence of firm dynamics for a cohort of entering firms. In section 3, we build the general model, obtain optimality conditions, and provide heuristic arguments explaining the effects of adjustment costs. In section 4, using a simplified version of the model we analytically prove the effect of linear adjustment costs on survivors' growth. In section 5, we calibrate and estimate a finite learning horizon version of the model and quantify the contribution of adjustment costs to firm dynamics. Section 6 concludes. All proofs are left for an appendix.

2 Firm Dynamics in a Cohort of Entering Firms

There is a well established literature on the identification and explanation of differences in behavior between young and old firms. In this section, we analyze firm dynamics in a cohort of entering firms. We use Quadros de Pessoal, a database containing information on all Portuguese firms with paid employees. This dataset originates from a mandatory annual survey run by the Ministry of Employment, which collects information about the firm, its establishments, and its workers. All economic sectors except public administration are included. The panel we have access to covers the period 1985-2000. Information refers to March of each year from 1985 through 1993, and to October of each year since the reformulation of the survey in 1994. On average the dataset contains 250,000 firms, 300,000 establishments, and 2,500,000 workers in each year.

The literature on firm dynamics typically finds that young firms grow faster than old firms. Using kernel density estimates of the firm size distribution in a cohort of entrants, Cabral and Mata (2003) argue graphically that the cohort's evolution is mostly due to growth of survivors rather than exit of small firms. Their analysis points to the need for a measure of the contribution of survivors versus non-survivors to the growth in a given cohort's average size. To accomplish this, we propose a decomposition of the cohort's cumulative growth that will later allow an assessment of the empirical relevance of adjustment costs. We consider the following decomposition:

$\begin{multline*} \frac{1}{N\left( S_{\tau}\right) }\sum_{i\in S_{\tau}}l_{i,\tau}-\frac {1}{N\left( S_{0}\right) }\sum_{i\in S_{0}}l_{i,0}=\underset{\text{Survivor Component}}{\underbrace{\frac{1}{N\left( S_{\tau}\right) }\sum_{i\in S_{\tau}}l_{i,\tau}-\frac{1}{N\left( S_{\tau}\right) }\sum_{i\in S_{\tau}% }l_{i,0}}}~+\ \underset{\text{Non-Survivor Component}}{\underbrace{\frac{N\left( D^{\tau }\right) }{N\left( S_{0}\right) }\left( \frac{1}{N\left( S_{\tau}\right) }\sum_{i\in S_{\tau}}l_{i,0}-\frac{1}{N\left( D^{\tau}\right) }\sum_{i\in D^{\tau}}l_{i,0}\right) }}% \end{multline*}$

where $\tau$ is the firm's age, $l_{i,\tau}=\ln\left( L_{i,\tau}\right)$ is log-employment at firm

in period $\tau$ , $S_{\tau}$ is the set of age- $\tau$ surviving firms, $D^{\tau}$ is the set of age- $\tau$ non-surviving firms, so that $\left\{ S_{\tau},D^{\tau}\right\}$ is a partition of $S_{0}$ , and $N\left( X\right)$ is the number of firms in set

.¹¹

In general, the growth in a cohort's average size can originate from growth of survivors or from smaller initial size of non-survivors. Any theory of firm dynamics should consider both these sources of growth. Our measure allows a check on whether a particular theory can explain the key source of growth in a cohort's average size. The survivor component compares the current average size of period $\tau$ survivors with their initial average size, so that it measures how much survivors have grown. The non-survivor component compares the average initial size of period $\tau$ non-survivors with the average initial size of period $\tau$ survivors, so that it measures how relatively small non-survivors were initially.

We can obtain a similar decomposition for employment-weighted moments. The weighted decomposition contains information about the entire distribution of employment, not just its cross-sectional mean, and is affected both by within- and between-firm growth. Therefore, the weighted decomposition would be more relevant for assessing a richer model that considers the reallocation of employment shares between firms within the cohort. In the results that follow we focus on the unweighted decomposition because it analyzes within-firm growth, which in our model is the most relevant statistic to assess the effect of adjustment costs on the incentives for firms to grow.¹²

We can also produce a decomposition based on the cohort's annual growth instead of the cohort's cumulative growth. However, the annual version of the above decomposition is more sensitive to two aspects that would complicate the analysis in the paper. First, the annual survivor component is significantly affected by the business cycle, especially after the first few years of life. To control for this, we would need to somehow remove the cyclical part of the survivor component. Second, as the age of the cohort increases, the annual survivor component becomes increasingly sensitive to downsizing and exit by some survivors that become technologically outdated and consequently relatively less efficient. To fully consider this aspect of the data would force us to introduce additional parameters into the model that we present in section 3. Therefore, we believe that by employing a decomposition based on the cohort's cumulative growth we avoid having to adjust the analysis for these two aspects, and instead focus on how intense is survivor's growth while learning-about-efficiency effects are significant.

Table 1: 1988 Firm Cohort: All Economy
Year	CEx	AvEmp	CGrEmp	SComp
1988		1.11
1989	15.6	1.27	15.2	69.5
1990	24.4	1.36	24.8	70.4
1991	30.8	1.43	31.2	69.7
1992	35.4	1.46	34.2	69.3
1993	40.0	1.46	34.6	68.9
1994	43.4	1.47	35.3	69.1
1995	46.7	1.48	36.1	68.7
1996	49.9	1.49	37.4	67.2
1997	52.7	1.51	39.6	68.5
1998	55.5	1.52	40.7	68.3
1999	58.5	1.54	43.0	68.9

Notes: CEx is the cumulative exit rate, $100 \times N(D^{\tau}) / N(S_{0})$ ; AvEmp is the mean of log-employment among survivors, $N(S_{\tau})^{-1}\sum_{i \in S_{\tau}}l_{\tau }$ ; CGrEmp is the cumulative log-growth rate (in %) of employment among survivors, $100\times[N(S_{\tau})^{-1}\sum_{i \in S_{\tau}}l_{\tau} - N(S_{0})^{-1}\sum_{i \in S_{0}}l_{0}]$ ;

is the survivor component (in %), $100\times[N(S_{\tau})^{-1}\sum_{i \in S_{\tau}}l_{\tau} - N(S_{\tau })^{-1}\sum_{i \in S_{\tau}}l_{0})] / [N(S_{\tau})^{-1}\sum_{i \in S_{\tau}% }l_{\tau} - N(S_{0})^{-1}\sum_{i \in S_{0}}l_{0}]$ .

In table 1, we present the evolution of exit rates and the share of firm growth due to the survivor component in the 1988 cohort of entering firms for the overall economy.¹³ In 1988 there were entering firms. The exit rate is very high initially but tends to decrease as firms get older.¹⁴ However, ten years after entry only $41.5\%$ of the initial entrants remain active. There is significant growth in the cohort's average size, especially in the first few years, which is mostly due to the growth of survivors rather than to the exit of small inefficient firms: survivors' growth contributes around $69\%$ to the growth in the cohort's average size.¹⁵

Table 2: 1988 Firm Cohort: Summary Characteristics by Sector
Sector	$ESh_{88}$	$CEx_{89}$	$CEx_{92}$	$CEx_{99}$	$AvEmp_{88}$	$CGrEmp_{89}$	$CGrEmp_{92}$	$CGrEmp_{99}$	$SComp_{89-99}$
All	100.0	15.6	35.4	58.5	1.11	15.2	34.2	43.0	69.0
Manu	41.8	14.6	35.9	58.9	1.58	17.4	38.7	45.5	82.8
Serv	20.1	17.1	36.6	58.0	0.99	11.7	30.8	40.2	61.7

Notes: ESh is the employment share of the sector in the overall economy cohort; CEx, CGrEmp, SComp are as defined in table 1.

Table 2 presents similar evidence on cohort dynamics for the manufacturing and services sectors.¹⁶ We include the employment shares of each sector in the 1988 cohort of entering firms, which are close to shares in the overall economy. Although manufacturing has a much higher employment share than services, the number of entering firms in services surpasses that of manufacturing ( 6074 and 4834, respectively). Both sectors display a cumulative exit rate around $58\%$ by 1999. However, initial differences in exit rates are more significant, with manufacturing displaying the smallest values, and services displaying the highest values. In terms of initial size, manufacturing has the largest entrants, and services the smallest. Although manufacturing has the largest entrants, it exhibits more growth in average employment and a larger contribution of survivors to that growth than services.

We perform two robustness checks on the previous findings. First, we redo our calculations using establishments rather than firms as the unit of analysis. For the 1988 establishment cohort, we obtain similar results, although exit rates and the survivor component are higher than in the case of firms. Second, we examine an alternative cohort to make sure our results are not driven by business cycle conditions. The Portuguese economy experienced an expansion between 1986 and 1991, a period of slow growth with a recession between 1992 and 1994, and another weaker expansion between 1995 and 2000. The growth rates of real GDP were $6.4\%$ in 1989, $1.1\%$ in 1992, and $4.3\%$ in 1995, so that the 1991 cohort did not face as favorable a macroeconomic environment as the 1988 cohort. However, the results for the 1991 cohort are, in all dimensions, very similar to those presented above. The results for the 1994 cohort are also very similar, but with slightly smaller values for the survivor component in the first few years after entry.¹⁷

Table 3: 1988 Firm Cohort: Characteristics of Labor Adjustment by Sector
Sector	$N30_{89}$	$NA_{89}$	$P30_{89}$	$N30_{93}$	$NA_{93}$	$P30_{93}$
All	7.9	43.0	13.7	13.7	45.3	17.1
Manu	10.8	31.5	20.7	20.9	33.4	24.6
Serv	7.3	47.7	11.8	11.3	50.3	15.0

Notes: N30 is the fraction of firms with an adjusted growth rate of employment, conditional on survival, in the interval $(-30\%,0\%)$ ; NA is the fraction of firms that do not adjust employment, conditional on survival; P30 is the fraction of firms with an adjusted growth rate of employment, conditional on survival, in the interval $(0\%,30\%)$ .

In table 3, we provide evidence on characteristics of labor adjustment in the 1988 cohort of entering firms. We use the distribution of the adjusted growth rate conditional on survival. Following Davis and Haltiwanger (1992), the adjusted growth rate in period $\tau$ is defined as $100\times\left( L_{\tau}-L_{\tau-1}\right) /\tilde{L}_{\tau-1}$ , where $\tilde{L}_{\tau -1}=\frac{1}{2}\left( L_{\tau}+L_{\tau-1}\right)$ . The table shows that the incidence of inaction is very high, increases with age, and is higher in services than in manufacturing. This may reflect technology-induced differences in adjustment costs, or job indivisibilities affecting to a larger extent the services sector for having a higher share of small firms. The table also shows that the large majority of firms have adjustment rates within the $(-30\%,30\%)$ interval. A high rate of inaction and small adjustment is usually considered consistent with the presence of linear or proportional adjustment costs. An additional fact is the left skewness of the 1989 distributions, showing that survivors tend to grow initially, especially in manufacturing. This skewness is less evident in 1993, suggesting that adjustment patterns are different in initial years. The evidence on inaction justifies our assumption of linear adjustment costs in the model that we present next.

3 A Model of Learning with Linear Adjustment Costs

3.1 Assumptions and Solution

In this section, we introduce linear adjustment costs into a model of Bayesian learning about efficiency. We derive conditions for optimal employment over time and present heuristic arguments about the effects of adjustment costs on the path of employment. Our model is based on Jovanovic (1982), adding adjustment costs and using a different specification for the idiosyncratic shock.

We assume an industry with competitive output and input markets. Current profits of a representative firm are defined by

$\displaystyle \Pi\left( L,\theta\right) =F\left( L\right) \theta-wL$ , $\displaystyle %$

where $F\left( L\right) \theta$ is the production function;

is the amount of labor input; $\theta$ is a productivity shock; and

is the wage rate. The output price is normalized to unity, so that all monetary values are expressed in units of the output price. Given the competitive environment, the firm treats

as a constant.

Concerning technology we make the following assumption.

Assumption 1 The two components of the production function satisfy: (a) $F:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is $C^{2}$ , $F^{\prime}>0$ , $F^{\prime\prime}<0$ , $F\left( 0\right) =0$ , $F^{\prime }\left( 0^{+}\right) =\infty$ , and $F^{\prime}\left( \infty\right) =0$ . (b) Letting $\tau$ denote the firm's age and 0 the period in which the firm enters, the stochastic process of $\theta$ is defined by

$\begin{subequations}\begin{gather}\theta_{\tau}=\xi\left( \eta_{\tau}\right) \text{, \ \ \ \ \ }\eta_{\tau}% =\mu+\varepsilon_{\tau}\text{, \ \ \ \ \ }\mu=\mu_{0}+\mu_{1}\text {, \ \ \ \ \ }\tau=0,1,\dots\\ \varepsilon_{\tau}\sim N\left( 0,\sigma^{2}\right) \text{, \ \ \ \ \ }\mu _{0}\sim N\left( \bar{\mu},\sigma_{\mu_{0}}^{2}\right) \text{, \ \ \ \ \ }% \mu_{1}\sim N\left(0,\sigma_{\mu_{1}}^{2}\right) \text{,}\end{gather}\end{subequations}$

where $\mu_{0}$ , $\mu_{1}$ , $\left\{ \varepsilon_{\tau}\right\} _{\tau\geq 0}$ are mutually independent, $\xi:\mathbb{R}\rightarrow\mathbb{R}_{++}$ is $C^{1}$ , $\xi^{\prime}>0$ and $\xi\left( -\infty\right) =\nu_{1}\geq0$ , $\xi\left( \infty\right) =\nu_{2}<\infty$ .

Part (a) basically ensures a well defined interior optimum. In some of the analyses below, we will further specialize by assuming that is Cobb-Douglas. Meanwhile, part (b) establishes that in each period productivity is stochastic with a constant mean over the firm's lifetime. The productivity component, $\mu$ , is made of two parts: $\mu_{0}$ , which is observed before entry, and $\mu_{1}$ , which is never directly observed by the firm. Intuitively, $\mu_{0}$ can be thought of as indexing ex ante efficiency, measuring initial technology choice, while $\mu_{1}$ indexes ex post productivity, measuring how well a firm performs within its technology choice.

The introduction of $\mu_{0}$ is essential to obtain a non-degenerate distribution of firms' entry size, allowing an analysis of the contribution of survivors to growth in the cohort's average size. In contrast, the absence of $\mu_{0}$ in Jovanovic's (1982) model generates a degenerate distribution of firms' entry size. Under this scenario, for any period after entry, survivors and non-survivors have the same average initial size implying a value of $100\%$ to our measure of the survivors' component. By assuming $\sigma_{\mu_{0}}>0$ , we avoid this aspect of Jovanovic's model.

Before entry the firm knows the parameters governing the stochastic process of $\theta$ , i.e., $\bar{\mu}$ , $\sigma_{\mu_{0}}^{2}$ , $\sigma_{\mu_{1}}^{2}$ and $\sigma^{2}$ , and learns its ex ante productivity, $\mu_{0}$ , after paying a research cost, . After entry, the firm will learn about its specific ex post productivity, $\mu_{1}$ , over time as it observes the realizations of productivity, $\theta$ . In particular, the firm forecasts period- $\tau$ productivity based on the ex ante efficiency parameter $\mu_{0}$ and on the past realizations of productivity, $\left\{ \theta _{s}\right\} _{s=0}^{\tau-1}$ . Similarly to Zellner (1971), a firm with age $\tau$ has the following Bayesian posterior distribution for $\mu$ at the beginning of period $\tau$ :

$\begin{subequations} % latex2html id marker 7693 % \begin{gather}\left. \mu\righ... ...ounter{temp1}{\value{theorem}} \addtocounter{theorem}{1}% \par\end{subequations}$

(2a)

Assumption 2 A potential entering firm, at the beginning of period 0, takes the following actions: (i.a) Research cost and ex ante productivity: the firm pays a fixed cost , associated with the process of initial research, after which it observes a realization of ex ante productivity, $\mu_{0}$ . (i.b) Entry decision and entry cost: based on the idiosyncratic realization of $\mu_{0}$ , the firm chooses whether to enter the industry or not. In case of entry, the firm pays for acquiring the (exogenously determined) capital stock. (i.c) Initial employment and production decisions: conditional on entering the industry, the firm chooses how much labor to use and how much output to produce in period 0. A firm of age $\tau>0$ takes the following actions: (ii.a) Update of posterior productivity: at the beginning of period $\tau$ , the firm updates its posterior expectation of $\theta_{\tau}$ , $\theta_{\tau}^{\ast}$ , based on the observation of $\theta_{\tau-1}=\xi\left( \eta_{\tau-1}\right)$ at the end of period $\tau-1$ . (ii.b) Exit decision: given the new posterior productivity estimate, $\theta_{\tau}^{\ast}$ , and employment from last period, $L_{\tau-1}$ , the firm chooses whether to stay or exit the industry. In case of exit, the firm sells the capital stock for the value initially paid, (no depreciation). (ii.c) Employment and production decisions: conditional on staying, the firm chooses how much labor to use and how much output to produce in the current period. At the end of period $\tau$ , the firm observes the productivity realization, $\theta_{\tau}$ , and the process repeats itself again until the firm decides to leave the industry.¹⁸

In the absence of adjustment costs, while deciding whether to stay one more period or to exit, the firm compares the expected profit in case it stays, , with the opportunity cost of doing so, , the value it would recover by selling the (exogenously determined) capital initially acquired, i.e.,

$\displaystyle V\left( \theta_{\tau}^{\ast},\tau\right) =\max_{L_{\tau}}\left\{ \Pi\left( L_{\tau},\theta_{\tau}^{\ast}\right) +\beta E_{\tau}\left[ \max\left\{ W,V\left( \theta_{\tau+1}^{\ast},\tau+1\right) \right\} \right] \right\} %$

(3)

where

represents expected profits conditional on staying in period $\tau$ .

At entry, we have $\Omega_{0}\equiv\mu_{0}$ , and in equilibrium expected profits must compensate for the cost of acquiring capital, i.e., $V^{EN}\left( \theta_{0}^{\ast}\right) >W$ . Since markets are competitive and there is no friction in the entry and exit processes, in equilibrium the research cost equals expected gains at the research phase, i.e., $E(V^{EN}\left( \theta_{0}^{\ast}\right) )=I$ . If $E\left( V^{EN}\right) >I$ more firms will initiate research and later enter the industry, causing a decrease in output price until equality is restored. A strictly positive fixed research cost, , is essential to avoid the extreme situation where trial research is so high that only the highest productivity firms enter and survive. Because there is no reliable capital stock variable in Quadros de Pessoal, we do not make the capital decision endogenous to the model. Instead, we assume that firms are homogeneous along the capital dimension and face the same opportunity cost of remaining in activity, .

Up to this point, the only differences between our model and Jovanovic (1982) are that in the latter model the efficiency parameter implicitly affects the cost function and the cohort's entry size distribution is degenerate. Therefore, without adjustment costs there would be no intertemporal linkages in our model aside from the exit decision. As in Jovanovic, because is strictly increasing in $\theta^{\ast}$ , the exit decision is characterized by an age-dependent exit threshold. For values of $\theta_{\tau}^{\ast}$ above or equal to that threshold, the firm would stay and choose employment to maximize current period profits. For values of $\theta_{\tau}^{\ast}$ below that threshold, the firm would leave the industry, since its expected profitability is below the opportunity cost. The increasing confidence the firm puts in $\theta_{\tau}^{\ast}$ as it grows older implies that the exit threshold is increasing with age. This is the driving force underlying Jovanovic's result that the size distribution and the survival probability increase with age.

We now introduce linear adjustment costs into the model. The adjustment cost for continuing firms, $C^{S}$ , is defined as

$\displaystyle C^{S}\left( L_{\tau},L_{\tau-1}\right) =P~\left\vert L_{\tau}-L_{\tau -1}\right\vert$

where

is the cost per unit of adjustment. Since this is a model with endogenous entry and exit of firms, we consider that this cost also applies to the entry and exit decisions, so that the costs for entering and exiting firms, $C^{EN}$ and $C^{EX}$ respectively, are given by

$\displaystyle C^{EN}\left( L_{0}\right) =P~L_{0}$ , $\displaystyle C^{EX}\left( L_{\tau}\right) =P~L_{\tau}$ . $\displaystyle %$

With adjustment costs, the problem now becomes,

$\begin{multline} V^{S}\left( L_{\tau-1},\theta_{\tau}^{\ast},\tau\right) =\max_{L_{\tau}% }\left\{ \left[ \Pi\left( L_{\tau},\theta_{\tau}^{\ast}\right) -C^{S}\left( L_{\tau},L_{\tau-1}\right) \right] +\right. \ \left. \beta E_{\tau}\left[ \max\left\{ V^{EX}\left( L_{\tau}\right) ,V^{S}\left( L_{\tau},\theta_{\tau+1}^{\ast},\tau+1\right) \right\} \right] \right\} \text{,} % \end{multline}$

for all periods after entry ( $\tau\geq1$ ) in which the firm remains in the industry, and

$\displaystyle V^{EN}\left( \theta_{0}^{\ast}\right) =\max_{L_{0}}\left\{ \left[ \Pi\left( L_{0},\theta_{0}^{\ast}\right) -C^{EN}\left( L_{0}\right) \right] +\beta E_{0}\left[ \max\left\{ V^{EX}\left( L_{0}\right) ,V^{S}\left( L_{0},\theta_{1}^{\ast},1\right) \right\} \right] \right\}$ , $\displaystyle %$

(4)

for the entry period, where $V^{EX}$ , the value of exiting, is defined as

$\displaystyle V^{EX}\left( L_{\tau}\right) =W-C^{EX}\left( L_{\tau}\right)$ . $\displaystyle %$

Note that contrary to the case without adjustment costs, the previous period employment is a state variable for the current period optimization problem. Also, in $V^{EN}$ and in $V^{EX}$ the costs of hiring at entry and firing at exit are taken into account.

In general, we could allow for asymmetry among the cost parameters in $C^{S}$ , $C^{EN}$ , and $C^{EX}$ . However, asymmetries between the cost of regular firing and the cost of firing at exit or between the cost of regular hiring and the cost of hiring at entry lead to biases in entry and exit decisions. For example, if the per unit regular hiring cost is higher than the per unit entry hiring cost, then firms will hire more workers at entry in order to save on expected future higher hiring costs. Similarly, if the per unit regular firing cost is smaller than the per unit exit firing cost, then firms facing the prospect of exit will fire workers before exiting the industry, saving on expected future higher exit firing costs. To avoid these biases, throughout the paper we assume symmetry between the parameters in $C^{S}$ , $C^{EN}$ , and $C^{EX}$ . A more interesting distinction is between firing and hiring costs. We will see below that the conclusion of the paper is immune to asymmetries between the costs of adding and subtracting workers.

In solving the firm's problem, we consider a two-step optimization procedure where the firm first chooses optimal employment in each of three possible scenarios, and then selects the scenario with the highest pay-off. More precisely,

$\displaystyle V^{S}\left( \cdot\right) =\max\left\{ V^{SD}\left( \cdot\right) ,V^{SN}\left( \cdot\right) ,V^{SU}\left( \cdot\right) \right\}$ , $\displaystyle %$

where $V^{SD}$ and $V^{SU}$ are obtained by maximizing the objective function in (4) over $L_{\tau}\leq L_{\tau-1}$ and $L_{\tau}\geq L_{\tau-1}$ , respectively, and $V^{SN}$ is obtained by choosing $L_{\tau}=L_{\tau-1}$ in (4). Although the adjustment cost function introduces a non-differentiability of the objective function at the frontiers between adjustment and non-adjustment, the usual properties of the value function $V^{S}$ and its associated optimal exit policy function hold

Proposition 3 Let $V^{S}$ be defined as in (4). Then: (a) There exists a unique value function $V^{S}\left( L_{\tau -1},\theta_{\tau}^{\ast},\tau\right)$ satisfying (4) that is bounded, continuous in $\left( L_{\tau-1},\theta_{\tau}^{\ast}\right)$ , and strictly increasing in $\theta_{\tau}^{\ast}$ . (b) There exists a unique optimal exit policy function $\chi_{\tau}^{\ast}\left( L_{\tau -1},\theta_{\tau}^{\ast}\right) =\mathbf{1}\left( \theta_{\tau}^{\ast }<\theta^{EX}\left( L_{\tau-1},\tau\right) \right)$ , where $\theta ^{EX}\left( L_{\tau-1},\tau\right)$ is a unique continuous function in $L_{\tau-1}$ .

Proof. See appendix A.¹⁹

$\qedsymbol$

In contrast, the non-differentiability of the objective function generates an inaction region in the employment policy, within which optimal employment does not vary with changes in productivity.

Proposition 5 For any period $\tau>0$ , if the firm adjusts upwards, optimal employment satisfies

$\displaystyle \left[ F^{\prime}\left( L_{\tau}^{\ast}\right) \theta_{\tau}^{\ast }-w\right] +\sum_{s=1}^{\infty}E_{\tau}\beta^{s}\left\{ \tilde{\chi}% _{\tau+s}^{\ast}\left( -P\right) +\hat{\chi}_{\tau+s}^{\ast}\left[ F^{\prime}\left( L_{\tau+s}^{\ast}\right) \theta_{\tau+s}^{\ast}-w\right] \right\} =P$ , $\displaystyle %$

(5)

whereas if the firm adjusts downwards optimal employment satisfies

(6)

In period 0, the firm enters the industry if $V^{EN}\left( \theta_{0}% ^{\ast}\right) \geq W$ , in which case optimal employment satisfies

$\displaystyle \left[ F^{\prime}\left( L_{0}^{\ast}\right) \theta_{0}^{\ast}-w\right] +\sum_{s=1}^{\infty}E_{0}\beta^{s}\left\{ \tilde{\chi}_{s}^{\ast}\left( -P\right) +\hat{\chi}_{s}^{\ast}\left[ F^{\prime}\left( L_{s}^{\ast }\right) \theta_{s}^{\ast}-w\right] \right\} =P$ . $\displaystyle %$

(7)

$L_{\tau+s}^{\ast}$ is the optimal employment in period $\tau+s$ , and $\tilde{\chi}_{\tau+s}^{\ast}$ , $\hat{\chi}_{\tau+s}^{\ast}$ are functions of the optimal exit decision, $\chi_{\tau+j}^{\ast}$ , in periods $\tau+1$ to $\tau+s$ , such that $\tilde{\chi}_{\tau+s}^{\ast}$ equals one when the firm has remained in the industry until period $\tau+s-1$ , but decides to exit in period $\tau+s$ , and $\hat{\chi}_{\tau+s}^{\ast}$ equals one when the firm is still in the industry in period $\tau+s$ .

Proof. See appendix A.

$\qedsymbol$

Equations (6), (7) and (8) are marginal conditions, similar to the smooth pasting conditions in the (S, s) model literature, and they state that if the firm adjusts then the marginal adjustment cost must equal the expected present discounted value of the marginal revenue product for all future periods in which the firm is still in the industry, minus the increase in the exit cost when the firm decides to exit. This is the discrete-time analog of the continuous-time result present in Nickell (1986) and Bentolila and Bertola (1990), adjusted for the fact that now we also have an exit decision. Because the firm will not change employment if the marginal cost of adjustment exceeds its marginal benefit for the first unit of adjustment, proportional costs imply inaction in the employment decision of the firm.

Although the results in proposition 6 do not allow a formal proof of the effects on firm growth of adjustment costs in this general model, the following corollary of proposition 6 enables us to make qualitative heuristic statements about those effects.

Corollary #.:

For any period $\tau\geq0$ , the marginal benefit of one additional unit of labor, that is, the LHS of expressions (6), (7), and (8), can be recursively represented as

$\begin{multline} MB\left( L_{\tau-1},\theta_{\tau}^{\ast},\tau\right) =\left( F^{\prime }\left( L_{\tau}^{\ast}\right) \theta_{\tau}^{\ast}-w\right) +\beta E_{\tau}\left[ \chi_{\tau+1}^{\ast}\left( -P\right) +\right. \ \left. \left( 1-\chi_{\tau+1}^{\ast}\right) MB\left( L_{\tau}^{\ast },\theta_{\tau+1}^{\ast},\tau+1\right) \right] % \end{multline}$

where $L_{\tau}^{\ast}=L_{\tau}^{\ast}\left( L_{\tau-1},\theta_{\tau}^{\ast }\right)$ , $\chi_{\tau-1}^{\ast}=\chi_{\tau-1}^{\ast}\left( L_{\tau -1},\theta_{\tau}^{\ast}\right)$ are the optimal employment and exit decisions.

Proof. See appendix A.

$\qedsymbol$

3.2 Linear Adjustment Cost and Firm Growth

As we have seen above, in the absence of adjustment costs, optimal employment is determined solely to maximize current period profits, so that $F^{\prime}\left( L^{\ast}\right) \theta^{\ast}=w$ . Therefore, firms' growth is essentially a by-product of a selection mechanism: those firms that are inefficient, and therefore small, exit, while those firms that are efficient survive and grow. There is an additional source of positive growth when the frictionless employment decision rule, $L^{\ast}\left( \theta^{\ast}\right)$ , is convex in $\theta^{\ast}$ . Because of Jensen's inequality and because $\theta_{\tau}^{\ast}$ is a Martingale, surviving firms will grow over time: $E_{\tau}\left[ L^{\ast}\left( \theta_{\tau+1}^{\ast}\right) \right] >L^{\ast}\left[ E_{\tau}\left( \theta_{\tau+1}^{\ast}\right) \right] =L^{\ast}\left( \theta_{\tau}^{\ast}\right)$ . However, $L^{\ast}$ will not be convex in $\theta^{\ast}$ for general $F\left( L\right)$ .²⁰

Figure 1: Proportional Hiring/Entry Cost
$Figure 1: Proportional Hiring/Entry Cost. Data plotted as a curve. X axis displays $\theta _{\tau }^{\ast }$, Y axis displays $MB_{\tau }$. The figure plots a hypothetical function $MB_{\tau }$ against $\theta _{\tau }^{\ast }$ for a given $L_{\tau -1}$. Two vertical lines drawn at $\theta ^{SD}$ and $% \theta ^{SU}$ identify the frontiers between adjustment and non-adjustment. For $\theta _{\tau }^{\star }$ smaller than $\theta ^{SD}$ the firm destroys jobs, i.e., $L_{\tau }^{\ast }\left( \theta _{\tau }^{\ast },L_{\tau -1}\right) <L_{\tau -1}$, and $MB_{\tau }=0$. For $\theta _{\tau }^{\ast }$ higher than $\theta ^{SU}$ the firm creates jobs, i.e., $L_{\tau }^{\ast }\left( \theta _{\tau }^{\ast },L_{\tau -1}\right) >L_{\tau -1}$, and $% MB_{\tau }=P^{H}$. For $\theta _{\tau }^{\star }$ higher than or equal to $% \theta ^{SD}$ but smaller than or equal to $\theta ^{SU}$, the firm does not adjust, i.e., $L_{\tau }^{\ast }\left( \theta _{\tau }^{\ast },L_{\tau -1}\right) =L_{\tau -1}$, and $MB_{\tau }$ is increasing in $\theta _{\tau }^{\ast }$, but contained in the interval from $0$ to $P^{H}$.$

In arguing heuristically about the impact of the proportional cost on firm growth we use the property that $MB_{\tau}$ is weakly increasing in $\theta_{\tau}^{\ast}$ , and that $L_{\tau}^{\ast}$ is locally weakly increasing in $\theta_{\tau}^{\ast}$ . Because it is not immediately obvious why firing and hiring costs should give similar incentives for firm growth, we analyze separately these two costs.²¹ We present in figure 1 the case where there is a hiring cost, $P^{H}>0$ , and no firing cost, $P^{F}=0$ . This figure assumes a given $L_{\tau-1}$ . For that specific value of $L_{t-1}$ , $\theta^{SU}$ and $\theta^{SD}$ are the frontiers between non-adjustment and upward and downward adjustment, respectively. Therefore, if $\theta_{\tau}^{\ast}\in\left[ \theta^{SD},\theta^{SU}\right]$ there will be no adjustment and the marginal benefit of an additional unit of labor (represented by the dashed line) is contained in the interval $\left[ 0,P^{H}\right]$ . To simplify the argument, we consider a firm whose sequence of productivity draws is such that in every period it has a perceived productivity equal to the unconditional mean of $\theta^{\ast}$ , even though the firm's uncertainty over next period $\theta^{\ast}$ decreases with age.

Case 1: Hiring Cost: $P^{H}>0$ , $P^{F}=0$

Because the firm starts at the hiring margin, we must have $MB_{0}=P^{H}$ at entry, and $MB_{\tau}\in\left[ 0,P^{H}\right]$ , for all subsequent periods, $\tau=1,2,\dots$ , with the two extremes of the interval representing firing and hiring of workers, respectively. Consider first a situation where exit is not allowed. Under this assumption, (9) would become

$\displaystyle MB_{\tau}=\left( F^{\prime}\left( L_{\tau}^{\ast}\right) \theta_{\tau }^{\ast}-w\right) +\beta E_{\tau}MB_{\tau+1}%$

For the entry period, we have $MB_{0}\left( \theta_{0}^{\ast}\right) =P^{H}$ , which implies that the firm will start smaller when $P^{H}>0$ than when $P^{H}=0$ .²² Since $MB_{1}\left( \theta_{1}^{\ast }\right) \in\left[ 0,P^{H}\right]$ , $E_{0}\left( MB_{1}\right) <P^{H}$ and thus we must have $pF^{\prime}\left( L_{0}\right) -w>0$ , for all $\beta\in\left( 0,1\right)$ , if $P^{H}>0$ . In the following period, firms will adjust upwards as frequently with $P^{H}>0$ as when $P^{H}=0$ , because they start at the hiring margin and $E_{0}\theta_{1}^{\ast}=\theta_{0}^{\ast}%$ , even though they might have smaller magnitudes of adjustment due to the hiring cost. The proportional hiring cost implies that firms will adjust downwards only if $\theta_{1}^{\ast}<\theta_{1}^{SD}$ , so that there is a region of inaction when $P^{H}>0$ that is not present when $P^{H}=0$ . That is, firms hire fewer workers initially because the resulting smaller probability of having to fire them, and therefore wasting the initial hiring cost, compensates for the expected decrease in profits this period. Consequently, in period 1 more firms will hire than fire, and this tendency towards growth in young firms will persist for several periods.

The Bayesian learning mechanism implies both persistence and a reduction in variance with age in the Markov process associated with $\theta^{\ast}$ . The effect of persistence, that is, the fact that $E\left( \theta_{\tau+1}^{\ast }\mid\theta_{\tau}^{\ast},\tau\right) =\theta_{\tau}^{\ast}$ , was analyzed in the previous paragraph. The reduced uncertainty in the posterior estimate of productivity will be reflected in a smaller inaction region as firms accumulate information on realized productivity; that is, $\theta^{SU}$ decreases with $\tau$ . This causes an increase in $E_{\tau}\left( MB_{\tau +1}\right)$ for those firms already at the hiring margin, which must be balanced by an increase in $L_{\tau}^{\ast}$ for the right hand side of (9) to remain equal to $P^{H}$ . As firms become more certain about their true productivity they are more willing to adjust to their long run optimal size. Because most firms are at the hiring margin, this will cause a further increase in average size.

Consider now the possibility of exit. In this case, the uncertainty reduction as the firm ages implies a decrease in the exit probability, and a further increase in the future-periods component of in (9). Consequently, $L_{\tau}^{\ast}$ needs to increase further in order to offset that.²³ On the other hand, the smaller exit probability implies less pruning of inefficient slow-growing firms as a cohort ages, which tends to make growth in average firm size smaller. Therefore, we will have less cohort growth due to non-survivors and more cohort growth due to survivors, so that survivors' contribution to average firm growth in the cohort should increase when exit is allowed.

Case 2: Firing Cost: $P^{F}>0$ , $P^{H}=0$

In this case we have $MB_{0}=0$ , $MB_{\tau}\in\left[ -P^{F},0\right]$ , $\tau=1,2,\dots$ . Assume first that exit is not allowed. The intuition is the same as in case 1. In comparison with $P^{F}=0$ , when $P^{F}>0$ firms start smaller and subsequently hire more frequently than they fire. As firms age, the reduction in variance of $\theta^{\ast}$ causes an increase in $E_{\tau}\left( MB_{\tau +1}\right)$ , which must be compensated by an increase in $L_{\tau}^{\ast}$ for firms at the hiring margin. When exit is possible, those effects become more intense, since the exit probability will decrease as firms age.

From the heuristic intuition we have just given it becomes clear that proportional hiring and proportional firing costs reinforce each other in creating incentives for firms to grow. In the end, our assessment of the relevance of linear adjustment costs for firm growth will depend on how well a pure selection model can fit the empirical evidence, and on how much adjustment costs improve the fit. Before we move into a quantitative assessment, we present analytical results for a simple version of the general model.

4 Model with One Period Learning Horizon and No Exit

In this section, we analyze a model where firms' efficiency is revealed after the first period of life and where firms' lifetime horizon is know with certainty at entry. We assume that firms live for $\bar{T}$ periods, $\bar{T}\in\left\{ 2,\dots,\infty\right\}$ , and that no exit is allowed prior to age $\bar{T}$ . These two simplifications allow us to determine the effect of linear adjustment costs on firm growth.

The introduction of adjustment costs implies an additional expected operating cost for entering firms. Therefore, the equilibrium price must increase to generate higher expected future profits that compensate for the costs incurred while adjusting to optimal size. As a consequence, pre-entry pruning of inefficient firms should increase while post-entry pruning should decrease. This is optimal from a social point of view, since with higher adjustment costs there should be less experimentation in order to save in unrecoverable costs. Therefore, the assumption that exit is exogenous is not critical for the results in this section. Since adjustment costs attenuate post-entry pruning, even if exit was endogenous to the model, the relative contribution of survivors to growth in the cohort's average size would increase through this channel. By eliminating any exit prior to $\bar{T}$ we focus only on the incentives for survivors to grow.

To formulate the problem, we use the fact that once the firm learns its true efficiency in period 2, it will adjust once and for all to its long run employment level, and it will not adjust in any of the following periods.²⁴ Therefore, assuming that upon exit at age $\bar{T}$ the firm recovers its initial investment net of exit costs, the optimization problem in period 2 is

$\displaystyle V^{S}\left( L_{1},\theta_{2}^{\ast}\right) =\max_{L_{2}}\left\{ \delta\left( \bar{T}\right) \Pi\left( L_{2},\theta_{2}^{\ast}\right) -C^{S}\left( L_{2},L_{1}\right) +\beta^{\bar{T}-1}V^{EX}\left( L_{2}\right) \right\}%$

(8)

where $\delta_{\bar{T}}\equiv\sum_{s=0}^{\bar{T}-2}\beta^{s}=\left( 1-\beta^{\bar{T}-1}\right) /\left( 1-\beta\right)$ , and $C^{S}$ and $V^{EX}$ are defined as above. In period 1, we then have

$\displaystyle V^{EN}\left( \theta_{1}^{\ast}\right) =\max_{L_{1}}\left\{ \Pi\left( L_{1},\theta_{1}^{\ast}\right) -C^{EN}\left( L_{1}\right) +\beta E_{1} \left[ V^{S}\left( L_{1},\theta_{2}^{\ast}\right) \right] \right\} .$

Finally, the equilibrium price is determined by the condition that potential entrants break even, i.e., $E_{0}\left[ V^{EN}\left( \theta_{1}^{\ast }\right) \right] =I$ .

We examine the impact of adjustment costs on the log growth rate of employment, rather than the standard growth rate, in order to attenuate the effect of Jensen's inequality on firm growth.²⁵ In this simple model, the inaction region of optimal employment can be expressed as an interval: $\Theta ^{SN}=\left[ \theta^{SD},\theta^{SU}\right]$ . Therefore, the average log growth rate between period 1 and period 2, conditional on $\theta_{1}^{\ast}$ , is defined as

$\begin{multline*} g\left( \theta_{1}^{\ast}\right) =E\left[ \ln\left( L_{2}^{\ast}\right) -\ln\left( L_{1}^{\ast}\right) \right] =\ \int_{\nu_{1}}^{\theta^{SD}}\left\{ \ln\left( L_{2}^{\ast SD}\right) -\ln\left( L_{1}^{\ast}\right) \right\} dF_{\theta_{1}^{\ast}}\left( \theta_{2}^{\ast}\right) +\int_{\theta^{SU}}^{\nu_{2}}\left\{ \ln\left( L_{2}^{\ast SU}\right) -\ln\left( L_{1}^{\ast}\right) \right\} dF_{\theta_{1}^{\ast}}\left( \theta_{2}^{\ast}\right) \end{multline*}$

where $\Theta\equiv\left[ \nu_{1},\nu_{2}\right]$ is the support of the distribution of $\theta_{2}^{\ast}$ , and $\theta^{SD}\left( L_{1}^{\ast }\right)$ and $\theta^{SU}\left( L_{1}^{\ast}\right)$ are the frontiers between non-adjustment and downward and upward adjustment, respectively. Depending on the specific value of $\theta_{1}^{\ast}$ and the magnitude of the adjustment cost parameters, we might have $\theta^{SD}\left( L_{1}^{\ast }\right) =\nu_{1}$ and/or $\theta^{SU}\left( L_{1}^{\ast}\right) =\nu_{2}$ . However, in the results that follow, we assume that $\theta_{1}^{\ast}$ and the adjustment cost parameters are such that both downward adjustment and upward adjustment occur with positive probability, i.e., $\theta^{SD}\left( L_{1}^{\ast}\right) >\nu_{1}$ and $\theta^{SU}\left( L_{1}^{\ast}\right) <\nu_{2}$ . Since we assume exogenous exit, we ignore the indirect effect of adjustment costs that works through changes in the equilibrium price, and implicitly assume that the research cost,

, adjusts to maintain an equilibrium. This indirect price effects influence average firm size in both periods, but are of second order importance for the average log-growth rate.²⁶

Optimal employment in period 2 is determined by

$\begin{displaymath} L_{2}^{\ast}\left( L_{1},\theta_{2}^{\ast}\right) =\left\{ \... ...{SD}\left( L_{1}\right) >\theta_{2}^{\ast}% \end{array}\right. \end{displaymath}$

where the frontiers of adjustment are defined as

$\displaystyle \theta^{SU}\left( L_{1}\right) \equiv\frac{w+\frac{\beta^{\bar{T}-1}P^{F}% }{\delta_{\bar{T}}}+\frac{P^{H}}{\delta_{\bar{T}}}}{F^{\prime}\left( L_{1}\right) }\text{, ~~~}\theta^{SD}\left( L_{1}\right) =\frac {w+\frac{\beta^{\bar{T}-1}P^{F}}{\delta_{\bar{T}}}-\frac{P^{F}}{\delta _{\bar{T}}}}{F^{\prime}\left( L_{1}\right) }\text{.}%$

Note that the numerator of $\theta^{SU}$ equals the pro-rated per-period cost of adding another worker, including the wage, the marginal hiring cost, and the discounted cost of firing the worker after period $\bar{T}$ . The numerator of $\theta^{SD}$ has a similar interpretation, as the benefit of shedding a worker.

We then have the following result concerning the effects of changes in $P^{H}$ and $P^{F}$ on the cohort's average log growth rate of employment.

Proposition 6 Assuming that $F\left( L\right)$ is Cobb-Douglas and that $\theta^{SD}\left( L_{1}^{\ast}\right) >\nu_{1}$ and $\theta^{SU}\left( L_{1}^{\ast}\right) <\nu_{2}$ :²⁷(a) The marginal effect of $P^{H}$ on $g\left( \theta_{1}^{\ast}\right)$ , assuming $P^{F}$ is zero, is positive for a high enough value of $\bar{T}$ . (b) The marginal effect of $P^{F}$ on $g\left( \theta_{1}^{\ast}\right)$ , assuming $P^{H}$ is zero, is positive for all $\bar{T}$ .

Proof. See appendix A.

$\qedsymbol$

Consider first the hiring cost. In the proof, we show that an increase in $P^{H}$ decreases both $L_{1}^{\ast}$ and $L_{2}^{\ast SU}$ . The impact of $P^{H}$ on the growth rate depends on two opposing effects. First, while in the case of $L_{2}^{\ast SU}$ the cost of hiring can be equally spread out over $\bar{T}-1$ periods with certainty, in the case of $L_{1}^{\ast}$ it will be spread out over either $\bar{T}$ periods or one period, depending on whether the firm learns in period 2 that it has overhired. Therefore, ex ante a proportionately greater part of $P^{H}$ is attached to period 1 in the case of $L_{1}^{\ast}$ than in the case of $L_{2}^{\ast SU}%$ , affecting more $L_{1}^{\ast}$ than $L_{2}^{\ast SU}$ . This explains the positive effect on growth of $P^{H}$ for $\bar{T}=\infty$ . Second, the hiring cost on $L_{1}^{\ast}$ can possibly be spread out over $\bar{T}$ periods, while the hiring cost on $L_{2}^{\ast SU}$ can only be spread out over $\bar{T}-1$ periods. This affects more $L_{2}^{\ast SU}$ than $L_{1}^{\ast}$ , and explains why the effect of $P^{H}$ on growth is not necessarily positive for finite $\bar{T}$ . However, as $\bar{T}$ increases the first effect dominates so that $P^{H}$ decreases $L_{1}^{\ast}$ more than $L_{2}^{\ast SU}$ and growth increases.²⁸

With respect to $P^{F}$ there is always a positive effect on growth, independently of the lifetime horizon. This occurs because an increase in $P^{F}$ decreases $L_{1}^{\ast}$ and increases $L_{2}^{\ast SD}$ . This positive effect always dominates the uncertain effect due to the fact that $L_{2}^{\ast SU}$ also decreases with $P^{F}$ .

When there are both hiring and firing costs and these costs are identical ( $P^{H}=P^{F}=P$ ), then an increase in has a positive effect on $g\left( \theta_{1}^{\ast}\right)$ , for sufficiently high $\bar{T}$ , where the required $\bar{T}$ is lower than in item (a) of proposition 8.

5 Calibration/Estimation Under Finite Learning Horizon

In the previous two sections, we developed heuristic and some formal arguments about the effect of adjustment costs on the incentives for firms to start smaller and grow faster after entry. In this section, we assess the contribution of adjustment costs to explain some of the basic facts on firm dynamics found in section 2, both for the overall economy and for the manufacturing and services sectors. To accomplish this, we perform a calibration/estimation of the model using computational methods.

To simulate the infinite learning horizon model we follow the suggestion of Ljungqvist and Sargent (2004) and consider an approximation where firms live forever, but learn their ex post true productivity component, $\mu_{1}$ , with certainty at some age .²⁹

We assume that is Cobb-Douglas, i.e., $F\left( L\right) =L^{\alpha}$ , $\alpha\in\left( 0,1\right)$ . Under this assumption, when adjustment is costless, optimal employment is given by