ENTRY, EXIT AND MISALLOCATION FRICTIONS JOB MARKET PAPER 1

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1 ENTRY, EXIT AND MISALLOCATION FRICTIONS ROBERTO N. FATTAL JAEF UNIVERSITY OF CALIFORNIA, LOS ANGELES JOB MARKET PAPER 1 Abstract. Frictions that misallocate resources across heterogeneous firms may lead to large losses in output, productivity and welfare. What do these frictions imply for the decisions of firms to enter and exit the economy, and for the size distribution of firms? Would their aggregate effects be mitigated or reinforced by the consideration of these additional margins of adjustment? I address these questions introducing factor misallocation frictions into a general equilibrium model that features firm dynamics and endogenous entry and exit decisions. I find that when distortions display the property of subsidizing low productivity firms at the expense of high productivity ones, entry and exit decisions have a large offsetting effect over the static misallocation costs of distortions on long run productivity and output. Nevertheless, in spite of the mild long run effects, I find large welfare gains associated with the withdrawal of misallocation frictions arising from transition dynamics. My findings suggest that although distortions to productive efficiency cannot account for much of the observed differences in productivity across countries, there are substantial welfare gains to be obtained by policies that remove these type of distortions 1. Introduction An increasingly accepted thesis in development economics is that distortions to the allocation of resources across heterogeneous producers can have substantial negative effects on aggregate productivity, output, and welfare. The thesis is supported by a large and growing literature that provides evidence of such distortions in developing countries, as well as quantifies their implications for aggregate output 2. Largely unexplored however, is the interaction between misallocation frictions and firm dynamics, and their implications for entry and exit decisions of firms. Are the aggregate consequences of these frictions underestimated by abstracting from these important margins of adjustment?, or are their detrimental effects overstated? I am greatly indebted to Francisco Buera and Ariel Burstein for guidance and support along all stages of this project. I have also benefited from insightful comments by Andy Atkeson, Hugo Hopenhayn and Lee Ohanian, as well as helpful suggestions by Javier Cravino, Venky Venkateswaran and seminar participants at UCLA s student macro lunch. All errors are my own. 1 Please, check the following URL for the latest draft of the paper: 2 I describe the literature in detail in the next section of the paper. 1

2 ENTRY, EXIT AND MISALLOCATION FRICTIONS 2 I address these questions in this paper introducing factor misallocation frictions into a model of firm dynamics with endogenously determined number of firms. I characterize analytically the response in entry and exit, and quantify the productivity, income and welfare gain to be obtained from a hypothetical reform that eliminates all distortions. I find that a large part of the productivity increase delivered by the reversal of the misallocation costs is offset by firms entry and exit decisions. Welfare gains, on the other hand, are significantly higher in the economy with endogenous entry and exit once transitional dynamics are taken into account. The model in the paper builds on Hopenhayn (1992) [6] and Luttmer (2007) [9], which derive a stationary size distribution of firms from shocks to idiosyncratic productivity. I model misallocation frictions in the form of firm specific revenue taxes and subsidies, as in Restuccia and Rogerson (2008) [10] and Hsieh and Klenow (2009) [7], although in my model idiosyncratic distortions vary stochastically over time in accordance with the stochastic process for productivity and a time-independent conditional distribution of distortions. Entry occurs after payment of a sunk cost in units of labor, after which the entrant gets a productivity draw from an arbitrary ex-ante distribution. Exit arises endogenously, as a result of fixed costs of operation, and exogenously, as a consequence of an exogenous death shock. I first show analytically how entry and exit decisions respond to misallocation distortions in three special cases that are nested in the full quantitative model. In the first case, I abstract from firm dynamics and assume exit is exogenous, so that the selection margin is shut down by assumption. In this setup, I show that idiosyncratic distortions have no impact on the measure of new firms in the economy, and that the static misallocation effect is the sole source of variation in aggregate productivity. The stationary distribution of firms is given by the ex-ante distribution of productivities prior to entry, and the extent by which productivity falls or increases in this economy is determined by the degree of resource misallocation, receiving no offsetting or reinforcing force from general equilibrium considerations. Secondly, I study an economy still with exogenous exit but with productivity dynamics, driven by deterministic idiosyncratic productivity growth. As in Luttmer (2010) [8], a Pareto distribution of firms across productivities arises in the stationary equilibrium of the model, induced by the distribution of firms across ages. I show that entry does respond to misallocation frictions when interest rates are positive, since distortions imply an asymmetric effect on the entrant s time-series expected profits relative to the cross-sectional average profitability of incumbents. Moreover, if, as found in the data, distortions favor low productivity firms at the expense of high productivity ones, I show that entry increases in a distorted long run equilibrium, partially offsetting the negative effect of resource misallocation on aggregate productivity.

3 ENTRY, EXIT AND MISALLOCATION FRICTIONS 3 Firm dynamics play the key role of differentiating entering firms expected profits from those of the average incumbent. To illustrate the mechanism, consider the case where distortions subsidize firms below a certain productivity, and tax those above. Since firms start small and grow large as their productivity increases, profits are initially boosted by the subsidy, and hindered later on by the tax. Because of discounting, the latter effect gets dampened. As a result, the expected profit of a new firm increases relative to the profit of the average incumbent, leading to higher entry. Next, I abstract from firm dynamics and introduce a fixed cost of production in order to characterize the effect of misallocation distortions on exit decisions. The effect of selection on aggregate productivity operates through the following channels: changes in the composition of active firms, the fraction of firms that engage in production and the total mass of producers. Assuming productivity is distributed Pareto, I show that subsidies to low productivity firms and taxes to high productivity ones lead to fewer exit, which increases the fraction of active firms and thereby aggregate productivity. The response in the mass of firms is in general ambiguous. However, I show that for entry costs that are sufficiently larger than the fixed cost of production, misallocation distortions expands the total number of firms, further increasing aggregate productivity. The magnitude by which entry and exit decisions affect aggregate variables is a quantitative question. To explore it, I consider a calibrated version of the model that matches salient features of the US employment-based size distribution, and patterns about the dynamics of new firms. I specify distortions following Restuccia and Rogerson (2008), and split firms into low and high productivity in terms of the median productivity in the economy, making the former group be subject to a revenue subsidy and the latter group to a tax. In the first set of experiments I concentrate on the implications of misallocation frictions for long run output and productivity. I solve for the stationary equilibrium of the model considering a wide range of distortion rates and ask: 1) how much would productivity improve if all taxes and subsidies were to be withdrawn, keeping constant the number of firms and for a given distribution of productivities? 2) how large would the gains once the response in entry and exit decisions is taken into account? 3 I find that the gains from reverting the efficiency costs of resource misallocation are to a large degree offset by entry and exit decisions. For levels of distortion whose removal delivers a productivity gain of 35% from factor reallocation, the change in the number of firms induced by entry and exit reduces this gain to just 22%. At low to middle values of distortions, productivity could even fall when moving to a frictionless equilibrium, because the response in the number of firms more than offset the reallocation gain. 3 I chose to perform a counterfactual where distortions are removed in order to facilitate the comparison of my result with those of Hsieh and Klenow (2009), who consider an experiment of this nature in their paper.

4 ENTRY, EXIT AND MISALLOCATION FRICTIONS 4 The mild effects of misallocation frictions on long run output and productivity raise the concern of whether welfare gains are also overstated when abstracting from entry and exit. To address this concern, I solve for the transition path of the economy from a distorted long run equilibrium towards a frictionless one, and compute the permanent consumption compensation required to make the household indifferent between inhabiting the two economies. In stark contrast with the long run comparison of output, I find that welfare gains are up to 60% higher in the economy with endogenous entry and exit. Transition dynamics reconcile the discrepancy between long run comparisons of output and welfare because it contemplates the welfare gain that takes place in the transition, as the economy depreciates the inefficiently high stock of firms and reduces the amount resources devoted to entry, increasing consumption. Throughout my analysis, I have made assumptions that have a direct impact over the magnitude with which the forces described here are at work. Of particular importance is the assumption about the technology to create new firms, which I have assumed to be linear in the labor input. An alternative that has been followed by several researchers is to assume that the efficiency of entry is determined by the ability of the members of the household in creating firms. This specification adds curvature to the cost of expanding the number if firms, which may be important in my analysis by weakening the offsetting force implied by entry in my model. I consider an extension of my model where the technology to create firms is subject to an increasing marginal cost, and find the parameter governing the curvature in entry to be important for the sensitivity of entry. The rest of the paper is organized as follows. Section 2 discusses related research, Section 3 presents the model, and characterizes a stationary equilibrium. Section 4 provides analytical results in the simple version of the model that I solve analytically. In Section 5, I calibrate parameter values and perform the quantitative experiments I mentioned above. Section 7 offers a sensitivity analysis to relevant parameters in the model, and discusses some extensions. In Section 6 I conclude. Additional proofs are provided in the appendix 2. Literature Review The paper relates to an increasing literature that highlights the role of factor misallocation frictions as an explanation for cross-country differences in productivity and income per worker. Hsieh and Klenow (2009) study the degree of factor misallocation in India and China and find evidence of large deviations from the efficient allocation. Restuccia and Rogerson (2008) find quantitatively that idiosyncratic taxes and subsidies that positively correlate with the underlying distribution of physical productivities can lead to large losses in long run productivity and output. These papers, however, have considered models that abstract from changes in entry and exit decisions of firms in response to misallocation distortions,

5 ENTRY, EXIT AND MISALLOCATION FRICTIONS 5 which could potentially exacerbate or mitigate the aggregate effects. The contribution of my work is to fill this gap, and provide an analysis of the mechanisms by which misallocation frictions distort entry and exit decisions, as well as a quantification of the relevance of this margins of adjustment. Guner, Ventura and Xu (2008) [5] study the macroeconomic implications of size depending policies in a model with an endogenous number of entrepreneurs. The key differences between my work and theirs is that I consider productivity dynamics, and I focus on an entry and exit process that allows for changes not only on the fraction of firms that engage in production every period, but on the mass of firms itself. As I show in the paper, these two margins critically affect the mechanism by which misallocation frictions distort entry decisions. Atkeson and Burstein (2010) [1] find that the general equilibrium response in product innovation could undermine the welfare gains of trade liberalizations. My work shares the result that consideration of the general equilibrium response in entry is important for aggregate outcomes, although I focus on distortions that are broader in scope, and find a discrepancy between long run effects on income and welfare costs. 3. The Economic Environment The model economy is similar to that in Hopenhayn (1992) and Luttmer (2007), who characterize a long run equilibrium where a stationary size distribution of firms arises as a result of shocks to idiosyncratic productivity. The innovation will be to incorporate firmspecific revenue taxes and subsidies and characterize the response of the economy to such distortions. The production side of the model is composed of a continuum of heterogeneous producers of intermediate inputs that are buffeted with idiosyncratic productivity shocks and are subject to firm-specific sales taxes and subsidies. The industry experiences entry and exit, the latter driven by the combination of idiosyncratic shocks and fixed costs of production, as well as exogenous exit shocks. The final good sector combines intermediate inputs to produce consumption goods, under a CES technology. There is no capital in the model and labor is inelastically supplied, so the representative consumer s problem is reduced to determining sequences of consumption subject to an intertemporal budget constraint Intermediate and Final Goods Producers. There is a perfectly competitive representative firm that produces the consumption good under a constant elasticity of substitution technology of the form: Q t = [ˆ ] θ q t (ω) θ dmt (ω) where M t (ω) is the mass of operating firms in the economy for each intermediate input of variety ω, and θ denotes the elasticity of substitution. Taking prices as given, firms in this

6 ENTRY, EXIT AND MISALLOCATION FRICTIONS 6 sector maximize ˆ π t = Q t p t (ω)q t (ω)dm t (ω) subject to the production technology. I choose the final good to be the numeraire, so p t (ω) denotes the price of intermediate input ω in units of the final good, and q t (ω) represents the quantity demanded of such variety. Optimization outcomes are standard, and imply the following expression for the demand function: ( ) θ pt (ω) q t (ω) = Q t Notice that the elasticity of substitution θ, is also the price elasticity of demand, and that the demand schedule displays a unit elasticity with respect to total real expenditure on the final good. Intermediate inputs are produced by a continuum of heterogeneous producers that operate a constant returns to scale technology, and sell their goods in monopolistically competitive markets. Labor is the only factor of production in this sector, and e ω is the idiosyncratic productivity and sole source of heterogeneity across firms. The production function, then, is given by: P t q t (ω) = (e ω ) 1 ρ 1 lt (ω) I model idiosyncratic distortions as taking the form of a firm specific revenue tax rate τ ω, which can be positive or negative depending on the firm being subject to a tax or a subsidy. I assume there exists a time invariant function that relates distortions to productivity, Γ(ω), so that firm dynamics are driven by the stochastic evolution of idiosyncratic productivity and the corresponding draw from Γ (ω). Taking the entire demand function and the wage rate as given, the intermediate producer chooses employment and prices in order to solve the following static profit maximization problem: πt v (ω) = max (1 τ ω) p t (ω) q t (ω) w t l t (ω) l t(ω),p t(ω) subject to the production technology. The resulting optimal labor demand and pricing rule are: ( ) θ θ 1 Q t (3.1) l t (ω) = e ω (1 τ ω ) θ θ (3.2) p t (ω) = θ (θ 1) w θ t w t [e ω ] 1 (1 τω ) These two equations reflect the distortionary effect of revenue taxes on employment and pricing decisions. Given final output and the wage rate, subsidized firms will be oversized

7 ENTRY, EXIT AND MISALLOCATION FRICTIONS 7 relative to the frictionless counterpart, whereas prices will be lower. The opposite occurs for firms that face positive tax rates. Plugging back the optimal choices of labor and prices into the objective function, I get the following expression for the indirect profit function: π v t (ω) = (θ 1) θ θ Q t wt e ω (1 τ ω ) θ 3.2. Entry and Exit. There are endogenous and exogenous sources of firm exit in the intermediate goods sector. The former arises as a consequence of fixed costs of production for producers that stay in operation. This, combined with a stochastic evolution of idiosyncratic productivity and distortions, gives rise to firms endogenously exiting the market. Furthermore, I assume there is an exogenous death-shock, occurring with probability δ, that pushes the firm to exit regardless of its current valuation. The latter is not essential for the existence of an equilibrium with entry and exit, although it will allow me to isolate the effect of distortionary taxation on entry decisions while abstracting from the endogenous selection margin. Producers currently in operation with productivity ω confront the following dynamic problem: Vt o (ω) = πt v (ω) w t f c + R t (1 δ) E t [V t+1 (ω ) ω] V (ω) = max xt(ω) {0, Vt o (ω)} where Vt o (ω) is the value of an existing firm with productivity ω, R t is the real interest factor and x t (ω) is an indicator function that encodes the status of the firm, being equal to one for all firms that stay in the market, and taking the value of zero for those that exit 4. Entry is costly in this economy, requiring a sunk cost of f e units of labor. In return, the entrepreneur gets a productivity draw from an arbitrary distribution G(ω), and decides whether to start production or exit immediately. I assume there is an infinite pool of potential entrants, so that the following free entry condition must hold in equilibrium w t f e = R t (1 δ) V e,t+1 ˆ V e,t+1 = V t+1 (ω) dg(ω) Free entry, then, requires that the expected discounted value of a new firm gets equalized to the entry cost. Notice that as part of the entry and exit process, there will be entrepreneurs that face the entry cost but never get to produce, as they are hit by the exogenous exit shock 4 Without imposing further structure about the stochastic process for productivity, the value of the fixed cost and the function that relates distortions to productivity, I cannot establish the properties of the set of firms that exit. I will impose such assumptions later in the paper, and clarify their implication for the properties of the exit decision.

8 ENTRY, EXIT AND MISALLOCATION FRICTIONS 8 during the time lag between entry and production, or optimally decide to do so once they learn the quality of their production technology Stochastic Process for Productivity. The source of firm dynamics in my model is given by the stochastic evolution of idiosyncratic productivity. I parametrize the stochastic process for ω taking a discrete-time random walk approximation to a Brownian Motion with drift µ, and variance σ 2. Following Stokey (2008) [11], I assume that given current productivity ω, next period s productivity could give an upward jump of size h, with probability p, or it could jump downwards, also in amount h, with probability (1 p). The discrete time approximation of the drift and variance of the process is µ t = (2p 1) h σ 2 t = 4p (1 p) h 2 The appeal of the discrete time random walk approximation of the Brownian Motion is that it easily maps my model to that in Luttmer (2007) and Luttmer (2010), who characterize the shape of the stationary size distribution of firms and identify restrictions on the drift and variance of the process so that there actually exists a stationary distribution in equilibrium. This will turn out to be extremely helpful at the time of calibrating the probability and size of the jump to match features of the US size distribution of firms. Another advantage of the binomial specification is that it easily nests two special cases of interest that I will study later to characterize analytically the effect of distortions on entry and exit. First, by setting p = 1, I will be considering an economy where firm dynamics follow from deterministic productivity growth at rate µ. Second, by letting h be equal to zero I would be considering an economy with no firm dynamics, where the size distribution of firms is entirely given by the distribution of productivities upon entry. Given entry and exit decisions and a specification the stochastic process of idiosyncratic productivity, I can describe the law of motion for the distribution of firms across productivities. Let M t (ω) be the mass of firms in period t with productivity less than or equal to ω. Then, (3.3) M t+1 (ω ) = (1 δ)pm t (ω h) + (1 δ) (1 p) M t (ω + h) + (1 δ) M e,t [G(ω ) G(ω t )] The expression establishes that a fraction (1 δ) p of firms with productivity less than or equal to ω h survives the exit exogenous exit shock and transit to a productivity level that is less than or equal to ω. A fraction (1 δ) (1 p) of the mass of firms with productivity between ω and ω + h survives the exit shock and jump downward to have productivity less

9 ENTRY, EXIT AND MISALLOCATION FRICTIONS 9 than or equal to ω. There is also an inflow of new firms to this group which is given by the fraction of entrants whose productivity lies between ω and the exit productivity cutoff Household s Problem. There is an infinitely lived representative household in the economy who seeks to maximize lifetime utility from consumption. I assume there are two asset markets open for consumption smoothing. One is a market to trade a risk-free bond that promises a payment of r t units of the numeraire in the following period; and the other is a market to trade shares on a mutual fund that has ownership of all firms in the economy, both incumbents and recently entered but currently idle ones. Then, on any given period the household chooses consumption c t, bond holdings B t+1 and shares of the mutual fund ψ t+1 to maximize: max β t [log(c t )] {c t,b t+1,ψ t+1 } t=0 t=0 subject to the budget constraint and a time-resource constraint for labor supply: c t + Λ t ψ t+1 + B t+1 = w t L t + (1 + r t )B t + (d t + Λ t ) ψ t + T t L t L Here Λ t denotes the value of a share of the mutual fund at time t, and T t represents a lump-sum transfer received from (or made to) the government to ensure a balanced budget. The mutual fund pays dividends d t equal to total profits from incumbent firms, minus the entry costs: d t = [ˆ π(ω)dm t (ω) w t f e M e,t ] First order conditions give the standard asset pricing equations: ( ) σ Ct Λ t = β [d t+1 + Λ t+1] 1 = β C t+1 ( Ct C t+1 ) σ (1 + r t ) Notice that the Euler equation for bond holdings pins down the interest rate factor, R t = (1 + r t ), with which firms discount future profits in the value functions Definition of Equilibrium. An equilibrium in this economy is: 1) a sequence of consumption, bond holding and asset shares for the household {C t, B t+1, ψ t+1 } t=0, 2) sequences of prices, labor demands, value functions and exit cutoffs for the producers of intermediate goods, {p t (ω), l t (ω), V t (ω), ω t } t=0 4) a sequence of demand functions for each intermediate 5 I have not characterized yet the nature of the exit decision, although I am anticipating that it will be given by a unique productivity cutoff ω, above which firms stay in operation and below which they exit. I will later impose assumptions that ensures that this is the case both in a distorted and in a frictionless world.

10 ENTRY, EXIT AND MISALLOCATION FRICTIONS 10 inputs {q t (ω)} t=0, 5) a sequence of measures of firms{m t(ω)} t=0 and its law of motion (equation 3.3), ( 7) a sequence of entrants {M e,t } t=0, and 7) a sequence of aggregates {w t, Q t, r t } ; such that: a) given wages, interest rates and 5, 1 solves household s optimization problem, b) given 7 and 4, 2 solves incumbent s dynamic optimization problem, c) given p t (ω), 4 solves the final good sector s profit maximization problem, d) M e,t is such that the free entry condition is satisfied in every period, and e) markets clear in every period: ˆ L = [l t (ω) + f c ] dm t (ω) + f e M e,t C t = Q t x t+1 = 1 B t+1 = Characterization of the Stationary Equilibrium. I now focus on a stationary equilibrium of the model where quantities and prices are constant over time, and there is a stationary distribution of firms across productivities 6. The objects to be determined in equilibrium are final output, the wage rate, the measure of entering firms and the exit rule. All other variables can be recovered once I have solved for these four, given the stationary distribution. In this situation, incumbent firms value function become V o (ω) = (θ 1) θ θ Q w eω (1 τ ω ) θ wf c + β (1 δ) E [V (ω ) ω] V (ω) = max x(ω) {0, V o (ω)} where notice that I have replaced the interest factor by the subjective discount factor of the household, as implied by the steady state version of the bond holding Euler equation. A convenient rescaling of the value function is to change its units and express it in units of labor, dividing both sides of the Bellman equation by the wage rate. Denoting υ (ω) = V (ω), w we get: υ o (ω) = (θ 1) θ θ Q w θ eω (1 τ ω ) θ f c + β (1 δ) E [υ (ω ) ω] υ (ω) = max x(ω) {0, υ o (ω)} When expressed in units of labor, it is clear that all information about aggregates that incumbents need to know to determine whether to stay or exit the market is summarized in Q w θ. Then, given a value for this ratio I can determine firms exit decision and solve for the cross-section of value functions in units of labor. Averaging across these values according to the distribution of productivity upon entry, G(ω), I can solve for the value of an entrant in 6 All time subscripts are removed when referring to stationary equilibrium objects

11 ENTRY, EXIT AND MISALLOCATION FRICTIONS 11 units of labor: ˆ υ e = υ (ω) dg (ω) The underlying ratio Q would be consistent with an equilibrium if it satisfies the free w θ entry condition in units of labor: Q w θ f e = β (1 δ) υ e Hence, free entry together with firms value functions determine the equilibrium ratio and the optimal exit rule x (ω), independently from other aggregates and other market clearing conditions in the model. Imposing stationarity in the law of motion for the distribution of firms across productivities, equation 3.3, delivers the stationary distribution as a function of the exit decision and the mass of entrants. I have already solved for the former, but still have to pin down the latter. However, I can aggregate individual decision rules according to the stationary distribution of firms per-unit of entrant, which is fully determined by the stochastic process for productivity and the exit decision, already known objects. Denoting such distribution with M (ω) = M(ω) Q M e, and making use of the knowledge about the equilibrium value of, I w θ can solve for each intermediate producer s labor demand from equation 3.1, aggregate across firms using the stationary distribution per unit of entrant, and compute aggregate labor demand in production per entrant: (3.4) L p = ˆ Ω = ( θ 1 θ ) θ Q w θ Ω e ω (1 τ ω ) θ d M (ω) Ω is a statistic of the distribution that is sufficient for summarizing the aggregate properties of production labor demand. I shall refer to Ω as after-tax labor productivity per unit of entrant. misallocation frictions have a direct impact over this statistic through the distribution of (1 τ ω ) θ, and through the effect of entry and exit on the stationary distribution of firms per-unit of entrant in a distorted long run equilibrium. Notice that the integral aggregates over the space of idiosyncratic productivities only, although after-tax productivity is also a function of the idiosyncratic distortion. Since I have restricted the specification of distortions to yield a one-to-one mapping with idiosyncratic productivity, the relevant distinguishing element of the firms state space is their idiosyncratic productivity. Thus, for the sake of preserving the clarity of notation, I have avoided to make explicit the integration over the space of distortions. The knowledge about the exit decision and the distribution of firms per-unit of entrant allows me also to calculate the aggregate demand of labor for fixed costs of production and

12 ENTRY, EXIT AND MISALLOCATION FRICTIONS 12 aggregate labor demand for entry: L fc = f c ˆ d M (ω) Imposing labor market clearing: L fe = f e L = M e [ L p + L fc + L fe ] determines the equilibrium number of new firms. Output, wages and other aggregates of Q interest can be inferred from the equilibrium values of, M w θ e and the distribution of productivities. For instance, to untangle the wage and total production of the final good from the ratio Q w θ, I can appeal to the equation for aggregate prices that follows from the final good producer s optimization problem, recalling that I have chosen the final good to be the numeraire: (3.5) 1 = get: [ˆ p(ω) 1 θ dm(ω) Substituting away intermediate inputs price from equation 3.2 and solving for the wage I (3.6) w = ˆ Ω w = (θ 1) θ ] 1 1 θ ( 1 Me Ω w) e ω (1 τ ω ) d M (ω) where Ω w is a statistic of the distribution that is relevant for the determination of the wage. Hereafter, I shall refer to this statistic as after-tax wage productivity per unit of entrant. Notice that Ω w differs from Ω only in how distortions take part of the expression, but would be identical in a frictionless economy with no taxes and subsidies. Unlike differences in productivity, which have a direct impact on the firms marginal cost of production, differences in revenue taxes translates into different mark-ups that the firms set over marginal cost. Although the wage undoes the inefficiency implied by a common mark-up on competitive equilibrium s allocation, it no longer does when mark-ups are idiosyncratic to the firm. Therefore, the difference between Ω w and Ω is a reflection of the inefficiency of the competitive equilibrium s allocation implied by the participation of idiosyncratic revenue taxes and subsidies. Combining equations 3.4 and 3.6 I can solve for final output: (3.7) Q = M 1 ( Ωw ) θ e L p Ω

13 ENTRY, EXIT AND MISALLOCATION FRICTIONS 13 Wage and labor after-tax productivity have opposite effects on the economy s aggregate productivity. To understand this, consider the expression for aggregate labor demand in production per-unit of entrant, 3.4. The equation reveals that, for a given amount of final output Q, the higher the after-tax average labor productivity, Ω, the higher is labor demand in production. But since the amount of final output is given at some value Q, it has to be that the economy is being less productive. Then, it justifies Ω being in the denominator of equation 3.7. Similarly, a higher value of after-tax average wage productivity implies higher wages in the economy. Since higher wages reduce labor demand in production, for a given value of Q, the economy would be producing a given amount of final goods with lower labor input requirement, which can only be attained through higher productivity. Hence, Ω w shows up in the numerator of equation 3.7. Summarizing, they key pair of equations that characterize an equilibrium are the free entry condition and labor market clearing. Then, for a given theory of the size distribution I can aggregate individual decisions and recover the aggregates of interest, such as total employment and final output. A key question I address in this paper is whether different theories of the size distribution, when interacted with idiosyncratic distortions, have different implications for entry and exit decisions, and thereby for productivity, output and welfare Productivity, Output and Welfare. Having characterized a stationary equilibrium, I now propose a decomposition of output and productivity that isolates the direct effects of entry and exit. To this aim, I decompose aggregate productivity into the following terms: 1) a misallocation effect, 2) the entry effect, and 3) a selection effect. The misallocation effect captures the productivity loss that arises from the inefficient allocation of factors of production, under a given distribution of firms across productivities. This is the source of productivity loss that underlies the counterfactual experiments in Hsieh and Klenow (2009) for China and India, and Restuccia and Rogerson s (2008) quantitative study. With firm dynamics and endogenous exit, misallocation frictions place the economy into a transition path towards a new long run equilibrium where the entire distribution of firms across productivities is subject to change. The last two terms in the decomposition are meant to capture these dynamic effects of idiosyncratic distortions. Two steps must be taken in deriving the misallocation component of aggregate productivity. First, I must introduce notation clarifying which is the distribution of firms across which factors are being distributed. I refer with M CM (ω) to a distribution that is held constant when introducing or withdrawing idiosyncratic distortions. Secondly, I must transform the distribution of firms into a proper density function so as to isolate the misallocation effect from changes in the mass of firms. For this purpose, I define M CM (ω) = M CM (ω) M CM to be the probability density function of productivities, where M CM is the total mass of firms under the distribution M CM (ω). Then, I define the misallocation effect on aggregate productivity

14 ENTRY, EXIT AND MISALLOCATION FRICTIONS 14 to be the ratio of wage to labor after-tax average productivities under the density M CM (ω), ( Ωw CM ) θ. Ω CM The selection effect is composed of two terms. The first one contemplates the change in the long run distribution of firms across productivities induced by the response in exit. This change will imply an adjustment to both wage and labor after-tax average productivity, Ω which I am going to capture with the ratios w Ω and. The sole difference between Ω w CM Ω CM numerator and denominator is the underlying productivity density functions across which I am allocating factors of production. Then, if the ratio were to be less (greater) than one, it would be implying an extra loss (gain) of productivity that is due to the endogenous response in the long run distribution of firms 7. In addition, selection has an effect over aggregate productivity by determining the fraction of firms that stay in operation in a distorted equilibrium. If misallocation frictions where to reduce the exit rate, as in the example of the previous paragraph, it would expand the range of intermediate inputs available to the final good producer and increase aggregate productivity through this channel. I shall refer to this component of the misallocation effect with the inverse of the exit rate, M M e. Taken together, the overall effect of selection on ( Ωw / Ωw CM ) θ aggregate productivity will be given by ( ) 1 M M e. ( Ω/ ΩCM) Finally, there is the entry effect, which reflects the additional productivity gain or loss followed by changes in entry decisions. All terms being considered, final output and aggregate productivity can be written as Q = M 1 e ( ) 1 M M e ( Ωw / Ωw CM ) θ ( Ω/ ΩCM) ( Ωw CM ) θ Ω CM L p ) ( ) 1 1 (3.8) T F P = M M ( Ω w / Ω w θ ) CM ( Ω w θ CM e }{{} ) Me ( Ω/ Ω CM Ω CM Entry }{{}}{{} Selection Mis-Allocation Notice that this is a model-based measure of productivity, which does not necessarily map into how researchers construct measures of TFP from the data. First, it assumes that the expenditure incurred in creating firms are expensed, and not capitalized as some form of investment in physical capital. Hence, firms are not considered a measured factor of production and show up in total factor productivity instead. Second, even under this assumption, there is the question of whether changes in productivity that are due to changes 7 The argument assumes that in a distorted economy it is still going to be the case that it is the low productivity firms that exit the market. Although this is necessarily the case in a frictionless equilibrium, distortions could be such that they revert the ordering of profitability across productivities. As it will be the case below, I will restrict my analysis to specifications of distortions for which this is not the case.

15 ENTRY, EXIT AND MISALLOCATION FRICTIONS 15 in the number of firms are captured in statistical agencies measures of real output. In the interpretation of my model were there is a single final good that produces aggregating over differentiated intermediate inputs, the price of the final good is equal to one at all times, so the model s real GDP can be mapped into what we observe in the data. However, on the downside, it has the implication that CPIs and PPIs diverge as the number of firms grow large (REVISE THIS!!!!!). Lastly, there is a stand to be taken as to what is the composition of the labor input at the time of deflating final output, stemming from the three sources of labor demand in my model: production of intermediate goods, fixed costs of production and creation of firms. In my current definition of TFP, it is from the former that I have inferred aggregate productivity. Alternatively, it could be argued that cross-country comparisons of productivity based on national income accounts data do not distinguish between sources of labor demand and nets out the entire labor force from total production to infer TFP. In such a case, the equation for TFP would become: ( ( ) T F P L = M 1 e M 1 Ω w / Ω ) w θ ( ) CM Ω w θ CM L }{{} ( Me Ω/ Entry Ω ) p CM Ω CM }{{} L }{{}}{{} Labor -Share Selection Mis-Allocation The new term, which I denominate the labor-share effect, captures productivity gains or loses that arise from the reallocation of labor in or out of positive value added activities. Of course, if the employment share of each source of labor demand did not change in a distorted long run equilibrium, then this distinction would be irrelevant for the discussion. Although my analytical results refer to the model-based definition of TFP, I also take into account the alternative definition T F P L in the quantitative analysis. I will show that looking at one measure of the other does not substantially change the message of the paper. 4. Analytical Results The general model presented in the previous section nests three special cases of interest for the purpose of characterizing the impact of misallocation frictions on entry and exit. The simplest version is one where there are no firm dynamics, idiosyncratic productivity being fixed upon entry, and there are no fixed costs of production. This economy does not have an active exit margin, but still has positive entry in equilibrium due to exogenous exit. I will show how idiosyncratic distortions affect the measure of new firms in a long-run distorted equilibrium. I then allow for idiosyncratic productivity to grow deterministically over time, while exit is still exogenous, in order to identify a possible interaction between the specification of taxes and subsidies and the non-stationarity of firm-level productivity in shaping the response in entry. Finally, I bring endogenous exit to the analysis, considering

16 ENTRY, EXIT AND MISALLOCATION FRICTIONS 16 a case where productivity is fixed upon entry, as in the first simplification, but with fixed costs of production. The goal in this case is to characterize the response in exit and the number of firms in isolation of interaction effects coming from productivity dynamics. The appeal of these three economies is that they admit analytical solutions, providing a deeper understanding of the mechanisms in the model prior to the quantitative analysis No Firm Dynamics, Exogenous Exit. Setting the fixed cost of production and the size of the jump in the binomial process to zero, the model becomes one where there are no firm dynamics, and exit is exogenous. Uncertainty is resolved upon entry, and the crosssectional distribution of productivities is identical to the ex-ante distribution G (ω). Then, the exogeneity of the exit decision and the permanent nature of idiosyncratic productivity reduces the number of unknowns to be solved for in equilibrium to just Q and M w θ e. As in the general model, these two aggregates are pinned down from the free entry condition and labor market clearing. The value of a firm with productivity ω is equal to the perpetuity of variable profits, and is given by ( ) 1 (θ 1) Q υ (ω) = 1 β (1 δ) θ θ w θ eω (1 τ ω ) θ where recall that υ (ω) is the value of the firm in units of labor. Then, free entry implies: ( ) β (1 δ) (θ 1) Q f e = Ω 1 β (1 δ) θ θ w θ e ˆ Ω e = e ω (1 τ ω ) θ dg (ω) The term Ω e denotes the ex-ante expectation of after-tax labor productivity implied by the ex-ante distribution G (ω). As I show below, this term plays a key role in characterizing the impact of idiosyncratic distortions on entry in this economy with no firm dynamics and exogenous exit. In terms of labor market clearing, average labor demand in production and average labor demand for entry costs are given by: L p = ( θ 1 θ ) θ Q w θ Ω L e = δ 1 δ f e Notice that here, contrary to how I presented the model in the previous section, I chose to work with average productivities and average labor demands, rather than aggregates per-unit of entrant. This is immaterial, since the steady state total number of firms is proportional to the measure of entrants, with a factor of proportionality that is independent from endogenous

17 objects: ENTRY, EXIT AND MISALLOCATION FRICTIONS 17 (1 δ) (4.1) M = M e δ Imposing labor market clearing pins down the equilibrium number of firms: [ ] δ L = M L p + f e (1 δ) Regarding the definitions of final output and aggregate productivity, the selection margin is completely exogenous here, as implied by equation 4.1. Therefore, there is only an entry effect and a misallocation effect potentially responding to idiosyncratic distortions. following proposition, however, states that regardless of the specification of distortions in the function Γ (ω), the number of firms does not change relative to its value in a frictionless economy. Proposition 1. Let Γ (ω) be any function that relates taxes and subsidies to productivities, with e ω (1 τ ω ) > 0 for all ω; let f c = 0 and h = 0; and let the superscript f denote variables in the frictionless equilibrium. Then: Q = M = M f L p = (L p ) f ( Ωw ) θ Ω ( ) 1 Q f Ω f The The result of the proposition is that the only channel through which idiosyncratic distortions affect the aggregate economy s output and productivity is a misallocation effect, with no reinforcing or offsetting effect from lower or higher entry and with no changes in aggregate demand for production labor. The key element leading to the neutrality of entry is that the ex-ante and the ex-post average after-tax labor productivities are identical. This implies that the required change in Q to satisfy the free-entry condition exactly offsets changes in w θ average labor demand due to the effect of distortions on average after-tax labor productivity, leaving the economy s average labor demand unchanged. Labor market clearing, in turn, imposes that if average employment does not change, then number of firms cannot change either. Further details about the proof are in the appendix. Another implication of the neutrality of entry is that there are no transition dynamics from a frictionless steady state to a distorted one. Therefore, in this case, the long run welfare costs of distortions are fully captured by steady state comparisons of income. I will come back to this feature when I perform a welfare cost analysis in the quantitative section of the paper.

18 ENTRY, EXIT AND MISALLOCATION FRICTIONS A Particular Specification of Distortions and Productivity. The further characterize the effect of misallocation frictions on aggregate productivity, I impose a particular structure to the the function that relates distortions with productivities, that I will carry to following sections in the analytical results and the quantitative part. Following Restuccia and Rogerson (2008), I split firms into those whose productivity is below and above the median productivity of the economy, and assume that those that are below (above) the median are subsidized (taxed) at a common rate τ. Regarding the distribution of productivities G (ω), I assume e ω to be distributed Pareto, with shape parameter and truncation point e 0. Under this assumptions, it is easy to compute the average after-tax labor and wage productivity in the economy: [ Ω = (1 + τ) θ + [ (1 τ) θ (1 + τ) θ] ] 0.5 ( 1) ( 1) [ Ω w = (1 + τ) + [ (1 τ) (1 + τ) ] ] 0.5 ( 1) ( 1) where the participation of 0.5 in the expression obeys to the fact that I am splitting taxed and subsidized firms according to the median productivity. Considering these statistics in the expression for the misallocation term in aggregate productivity, I get: T F P = ( ) 1 1 [ 1 + [ 1 + [ (1 τ) ] 1 (1+τ) [ ] (1 τ) θ 1 (1+τ) θ ] 0.5 ( 1) θ ] 0.5 ( 1) The equation reflects the severity of the misallocation effect as a function of the tax/subsidy rate, the fraction of taxed/subsidized firms and the Pareto s shape parameter. In particular, notice that in the extreme cases where all firms were taxed or all firms were subsidized, then the distortions would have no effect on productivity, income and welfare Deterministic Idiosyncratic Productivity Growth, Exogenous Exit. I now introduce firm dynamics into the economy of the previous section, in the form of deterministic idiosyncratic productivity growth, and preserve the assumption of zero fixed costs of production. I abstract from ex-ante heterogeneity by making all newly born firms start with the same level of productivity, although a stationary size distribution arises provided there is an equilibrium with positive entry. As shown in Luttmer (2010), such an equilibrium exists if idiosyncratic productivity grows at a rate that is lower than the exogenous exit rate. Notice that this economy is nested in the general model. First, let G (ω) be degenerate at ω 0 = 0, so that all new firms start with productivity equal to e ω 0 = 1. Then, set the probability of the upward jump to one, so that the variance of the discrete random walk 8 The intuition for why distortions have no welfare costs when all firms are taxed and subsidized is the same as for why the equilibrium with monopolistic competition is still efficient: the wage rate undoes the effect of distortions on allocations.

19 ENTRY, EXIT AND MISALLOCATION FRICTIONS 19 goes to zero, and idiosyncratic productivity grows deterministically at rate µ. Finally, for simplicity, consider the case were t is small. Then, idiosyncratic productivity grows with the firm s age in the form: e ωa = e µa There is a single dimension of heterogeneity across firms in this model arising from differences in age. Thus, I only need to characterize the age distribution of firms in order to be able to solve, through a change of variable, for the distribution of productivities. To determine the age distribution, notice that for each firm that enters the economy there is a fraction e δa that is still alive at age a. fraction is equal to: f(a) = δe δ Relative to the total number of firms, this where I have used the fact the steady state number of firms is proportional to the number of entrants in the form M = M e δ Then, f(a) is a proper probability distribution function that integrates to 1. After a change of variable, I can solve for the density function of the productivity distribution to get: g (ω) = δ µ (eω ) (1+ δ µ) which is a Pareto distribution with shape parameter δ µ. Therefore, this model with deterministic growth and exogenous exit endogenously delivers a cross-sectional distribution of productivities that is Pareto, just as I have assumed before for the model with no firm dynamics and with ex-ante heterogeneity. Having characterized the cross-sectional distribution of productivities, average labor demand in production is given by: ( ) θ θ 1 Q L p = Ω θ w θ ˆ δ Ω = µ (eω ) (1+ δ µ) (1 τω ) θ dg (ω) Letting ρ be the subjective discount factor of the household 9, the value function of a newly borned firm is: (θ 1) Q υ e = θ θ w θ (ρ + δ) Ω e ˆ Ω e = (ρ + δ) e (ρ+δ µ)a (1 τ a ) θ da 9 Since I am looking at a timing of the model where t is small, I wanted to distinguish the notation for the subjective discount factor in this case from the one in discrete time. Of course, they relate with each other in that 1/β = (1 + ρ).

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