ENTRY, EXIT AND MISALLOCATION FRICTIONS
ROBERTO N. FATTAL JAEF
UNIVERSITY OF CALIFORNIA, LOS ANGELES
JOB MARKET PAPER1
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 output2. 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: HTTP://sites.google.com/site/rfjucla
2
I describe the literature in detail in the next section of the paper.
1
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.
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.
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,
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.
3.1. 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:
"ˆ
# θ
θ−1
θ−1
Qt =
qt (ω) θ dMt (ω)
where Mt (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
ENTRY, EXIT AND MISALLOCATION FRICTIONS
6
ˆ
sector maximize
π t = Qt −
pt (ω)qt (ω)dMt (ω)
subject to the production technology. I choose the final good to be the numeraire, so pt (ω)
denotes the price of intermediate input ω in units of the final good, and qt (ω) represents
the quantity demanded of such variety. Optimization outcomes are standard, and imply the
following expression for the demand function:
qt (ω) =
pt (ω)
Pt
!−θ
Qt
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 eω is the idiosyncratic
productivity and sole source of heterogeneity across firms. The production function, then,
is given by:
1
qt (ω) = (eω ) ρ−1 lt (ω)
I model idiosyncratic distortions as taking the form of a firm specific revenue tax rate τω ,
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:
πtv (ω) = max (1 − τω ) pt (ω) qt (ω) − wt lt (ω)
lt (ω),pt (ω)
subject to the production technology. The resulting optimal labor demand and pricing rule
are:
!θ
(3.1)
lt (ω) =
θ−1
θ
(3.2)
pt (ω) =
wt
θ
1
(θ − 1) [eω ] θ−1 (1 − τω )
Qt ω
e (1 − τω )θ
wtθ
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
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:
πtv
(θ − 1)θ−1 Qt ω
θ
(ω) =
θ−1 e (1 − τω )
θ
θ
wt
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:
Vto (ω) = πtv (ω) − wt fc + Rt (1 − δ) Et [Vt+1 (ω 0 ) |ω]
V (ω) = maxxt (ω) {0, Vto (ω)}
where Vto (ω) is the value of an existing firm with productivity ω, Rt is the real interest factor
and xt (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 exit4.
Entry is costly in this economy, requiring a sunk cost of fe 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
wt fe = Rt (1 − δ) Ve,t+1
ˆ
Ve,t+1 = Vt+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
4Without
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.
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.
3.3. 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) h2
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 Mt (ω) be the mass of firms in period t with productivity less than or equal to ω.
Then,
(3.3)
Mt+1 (ω 0 ) = (1 − δ)pMt (ω 0 − h)
+ (1 − δ) (1 − p) Mt (ω 0 + h) + (1 − δ) Me,t [G(ω 0 ) − G(ω t )]
The expression establishes that a fraction (1 − δ) p of firms with productivity less than or
equal to ω 0 − h survives the exit exogenous exit shock and transit to a productivity level that
is less than or equal to ω 0 . A fraction (1 − δ) (1 − p) of the mass of firms with productivity
between ω 0 and ω 0 + h survives the exit shock and jump downward to have productivity less
ENTRY, EXIT AND MISALLOCATION FRICTIONS
9
than or equal to ω 0 . There is also an inflow of new firms to this group which is given by the
fraction of entrants whose productivity lies between ω 0 and the exit productivity cutoff5.
3.4. 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 rt 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 ct , bond holdings Bt+1 and shares of the mutual fund
ψt+1 to maximize:
∞
X
max
{ct ,Bt+1 ,ψt+1 }∞
t=0
β t [log(ct )]
t=0
subject to the budget constraint and a time-resource constraint for labor supply:
ct + Λt ψt+1 + Bt+1 = wt Lt + (1 + rt )Bt + (dt + Λt ) ψt + Tt
Lt 5 L
Here Λt denotes the value of a share of the mutual fund at time t, and Tt represents a
lump-sum transfer received from (or made to) the government to ensure a balanced budget.
The mutual fund pays dividends dt equal to total profits from incumbent firms, minus the
entry costs:
"ˆ
#
dt =
π(ω)dMt (ω) − wt fe Me,t
First order conditions give the standard asset pricing equations:
Ct
Λt = β
Ct+1
!σ
[dt+1 + Λt+1 ]
!σ
Ct
1=β
(1 + rt )
Ct+1
Notice that the Euler equation for bond holdings pins down the interest rate factor, Rt =
(1 + rt ), with which firms discount future profits in the value functions.
3.5. Definition of Equilibrium. An equilibrium in this economy is: 1) a sequence of consumption, bond holding and asset shares for the household {Ct , Bt+1 , ψt+1 }∞
t=0 , 2) sequences
of prices, labor demands, value functions and exit cutoffs for the producers of intermediate
goods, {pt (ω), lt (ω) , Vt (ω), ω t }∞
t=0 4) a sequence of demand functions for each intermediate
5I
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.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
10
∞
inputs {qt (ω)}∞
t=0 , 5) a sequence of measures of firms{Mt (ω)}t=0 and its law of motion (equation 3.3), ( 7) a sequence of entrants {Me,t }∞
t=0 , and 7) a sequence of aggregates {wt , Qt , rt } ;
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 pt (ω), 4
solves the final good sector’s profit maximization problem, d) Me,t is such that the free entry
condition is satisfied in every period, and e) markets clear in every period:
ˆ
L =
[lt (ω) + fc ] dMt (ω) + fe Me,t
Ct = Qt
xt+1 = 1
Bt+1 = 0
3.6. 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 productivities6. 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−1 Q ω
e (1 − τω )θ − wfc + β (1 − δ) E [V (ω 0 ) |ω]
θθ
wθ−1
V (ω) = maxx(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 w(ω) ,
we get:
(θ − 1)θ−1 Q ω
o
υ (ω) =
e (1 − τω )θ − fc + β (1 − δ) E [υ (ω 0 ) |ω]
θ
θ
θ
w
υ (ω) = maxx(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
. Then, given a value for this ratio I can determine firms’ exit decision and solve for the
wθ
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
6All
time subscripts are removed when referring to stationary equilibrium objects
ENTRY, EXIT AND MISALLOCATION FRICTIONS
11
ˆ
units of labor:
υe =
υ (ω) dG (ω)
The underlying ratio wQθ would be consistent with an equilibrium if it satisfies the free
entry condition in units of labor:
fe = β (1 − δ) υe
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
c (ω) = M (ω) , and making use of the knowledge about the equilibrium value of Q , I
with M
Me
wθ
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:
Q
wθ
(3.4)
c
L
p
=
θ−1
θ
!θ
Qb
Ω
wθ
ˆ
b
Ω
=
c (ω)
eω (1 − τω )θ dM
b is a statistic of the distribution that is sufficient for summarizing the aggregate properΩ
b as after-tax labor productivity per unit
ties of production labor demand. I shall refer to Ω
of entrant. misallocation frictions have a direct impact over this statistic through the distribution of (1 − τω )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
ENTRY, EXIT AND MISALLOCATION FRICTIONS
aggregate labor demand for entry:
12
ˆ
b
L
fc
= fc
c (ω)
dM
b =f
L
fe
e
Imposing labor market clearing:
h
c +L
b +L
b
L = Me L
p
fc
fe
i
determines the equilibrium number of new firms. Output, wages and other aggregates of
interest can be inferred from the equilibrium values of wQθ , Me and the distribution of productivities. For instance, to untangle the wage and total production of the final good from
the ratio wQθ , 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:
"ˆ
# 1
1−θ
1−θ
(3.5)
1=
p(ω) dM (ω)
Substituting away intermediate inputs price from equation 3.2 and solving for the wage I
get:
(3.6)
1
(θ − 1) b w θ−1
Me Ω
w=
θ
ˆ
bw =
c (ω)
Ω
eω (1 − τω )θ−1 dM
b w is a statistic of the distribution that is relevant for the determination of the wage.
where Ω
Hereafter, I shall refer to this statistic as after-tax wage productivity per unit of entrant.
b w differs from Ω
b only in how distortions take part of the expression, but
Notice that Ω
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.
b w and Ω
b is a reflection of the inefficiency of the competiTherefore, the difference between Ω
tive 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)
1
θ−1
Q = Me
θ
(Ωbw ) θ−1
b
Ω
Lp
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,
Ωw 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.
3.7. 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 MCM (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
MCM (ω)
f
from changes in the mass of firms. For this purpose, I define M
to be the
CM (ω) = MCM
probability density function of productivities, where MCM is the total mass of firms under
the distribution MCM (ω). Then, I define the misallocation effect on aggregate productivity
ENTRY, EXIT AND MISALLOCATION FRICTIONS
14
f
to be the ratio of wage to labor after-tax average productivities under the density M
CM (ω),
θ
(ΩewCM ) θ−1
eCM .
Ω
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,
w
which I am going to capture with the ratios eΩew and e Ωe . The sole difference between
ΩCM
ΩCM
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 firms7.
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
M
with the inverse of the exit rate, M
. Taken together, the overall effect of selection on
e
1
θ−1
θ
(Ωew /ΩewCM ) θ−1
aggregate productivity will be given by
.
(Ωe/ΩeCM )
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
1
θ−1
Q = Me
(3.8)
M
Me



θ
θ
1 Ω
w /Ω
w
w
θ−1
θ−1
e
e
e
Ω
(
)
(
)
CM
 M θ−1
  CM
 Lp
Me
eCM
Ω
(Ωe/ΩeCM )

θ 
θ 
θ−1
θ−1
1
w
w
w
e
e
e
1 
  ΩCM

M θ−1 Ω /ΩCM
θ−1 



T F P = |M{z
e




}
e CM
e Ω
e CM
Me
Ω
Ω/
Entry
|
{z
Selection
}|
{z
}
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
7The
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.
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:

θ 
1
w ew
θ−1
e
Ω /ΩCM θ−1 
 M


Me } 

| {z
e
e
Me
Ω/Ω

ew
Ω
θ
θ−1

Lp
e
L
Ω
CM
CM
|{z}
Entry
|
{z
}|
{z
} 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 PL in the quantitative analysis. I will show that looking at one
measure of the other does not substantially change the message of the paper.
T F PL =
1
θ−1



CM



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
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.
4.1. 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 wQθ and Me . 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
!
(θ − 1)θ−1 Q ω
1
e (1 − τω )θ
υ (ω) =
θ
θ
1 − β (1 − δ)
θ
w
where recall that υ (ω) is the value of the firm in units of labor. Then, free entry implies:
β (1 − δ)
(θ − 1)θ−1 Q e
fe =
Ωe
1 − β (1 − δ)
θθ
wθ
ˆ
e
Ωe = eω (1 − τω )θ dG (ω)
!
e denotes the ex-ante expectation of after-tax labor productivity implied by
The term Ω
e
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:
f =
L
p
θ−1
θ
!θ
Qe
Ω
wθ
δ
fe
1−δ
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
f =
L
e
ENTRY, EXIT AND MISALLOCATION FRICTIONS
17
objects:
(1 − δ)
Me
δ
Imposing labor market clearing pins down the equilibrium number of firms:
(4.1)
M=
"
δ
c +f
L=M L
p
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. The
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 eω (1 − τω ) > 0 for all ω; let fc = 0 and h = 0; and let the superscript f denote variables
in the frictionless equilibrium. Then:
M = Mf
Lp = (Lp )f
θ
(Ωew ) θ−1
Q= e
Ω
ef
Ω
1
θ−1
Qf
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 wQθ to satisfy the free-entry condition exactly offsets changes in
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.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
18
4.1.1. 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
eω to be distributed Pareto, with shape parameter η and truncation point e0 . Under this
assumptions, it is easy to compute the average after-tax labor and wage productivity in the
economy:
h
i
(η−1)
η
θ
θ
θ
e
η
Ω=
(1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
(η − 1)
h
i
(η−1)
η
ew =
(1 + τ )θ−1 + (1 − τ )θ−1 − (1 + τ )θ−1 0.5 η
Ω
(η − 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:
TFP =
η
η−1
!
1
θ−1
1+
(1−τ )θ−1
(1+τ )θ−1
1+
(1−τ )θ
(1+τ )θ
− 1 0.5
(η−1)
η
− 1 0.5
(η−1)
η
θ
θ−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 welfare8.
4.2. 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 ω0 = 0, so that all new firms start with productivity equal to eω0 = 1. Then, set the
probability of the upward jump to one, so that the variance of the discrete random walk
8The
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.
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: eωa = eµa
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. Relative to the total number of firms, this
fraction is equal to:
f (a) = δe−δ
where I have used the fact the steady state number of firms is proportional to the number
of entrants in the form
Me
M=
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:
δ
δ
g (ω) = (eω )−(1+ µ )
µ
which is a Pareto distribution with shape parameter µδ . 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:
f =
L
p
ˆ
θ−1
θ
!θ
Qe
Ω
wθ
δ ω −(1+ µδ )
(e )
(1 − τω )θ dG (ω)
µ
Letting ρ be the subjective discount factor of the household9, the value function of a newly
borned firm is:
(θ − 1)θ−1
Q
υe =
Ω̃e
θ
θ
θ
w (ρ + δ)
ˆ
Ω̃e = (ρ + δ) e−(ρ+δ−µ)a (1 − τa )θ da
e
Ω
9Since
=
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).
ENTRY, EXIT AND MISALLOCATION FRICTIONS
20
where I have multiplied and divided by (ρ + δ), so that (ρ + δ) e−(ρ+δ)a has the interpretation
of a time-series age distribution. Importantly, the time-series age distribution is different
from the cross-sectional one, provided the interest rate is positive. The reason is that from
the perspective of a given firm looking forward into the future, the time-series distribution
reflects the current period valuation of the fraction of time the firm will spend at each age
which, with positive interest rates, is subject to a discount. This force, absent in the model
with constant idiosyncratic productivity, plays a critical role in characterizing the response
in entry, since it drives an asymmetry in the effect of misallocation frictions on entering
firms’ average profitability relative to that of the average incumbent.
The characterization of the equilibrium follows from the free entry condition and labor
market clearing, which pins down the ratio wQθ and the mass of entrants Me .
The next proposition proves that, with a positive interest rate, the measure of entrants
does change in a distorted long run equilibrium. Letting κe = Ω̃˜fe and κ = ΩΩ̃˜f denote the
Ωe
ratios of time-series and cross-sectional after-tax average labor productivity in a distorted
and a frictionless economy, the proposition also shows that if κe > κ, entry is higher in a
distorted equilibrium, offsetting the drop in productivity due to the misallocation effect.
Proposition 2. Consider an economy with p = 1, fc = 0, ∆t → 0 and µ < δ. Then, for any
given specification of distortions Γ (ω) such that (1 − τω )θ > 0 for all ω, the mass of firms
in a distorted equilibrium relative to a frictionless counterpart is
1+A
M
i
= hκ
f
M
+
A
κe
ne δ
f.
(Lep )
M
> (<)0 and entry has an offsetting (reinforcing) effect on aggregate
If κκe < (>)1, then M
f
productivity.
where10 A =
Corollary. If ρ = 0, then κκe = 1, M = M f and distortions have only a misallocation effect
on aggregate productivity and income.
The key implication of the proposition is that what drives the change in the number of
firms in the economy is the differential response in the time-series ratio of after-tax average
labor productivity relative to the cross-sectional one. The corollary states that for such
differential response to exist, the interest rate has to be strictly positive. Otherwise, the
economy with firm dynamics behaves exactly as the same as the economy with no firmdynamics and ex-ante heterogeneity.
10Clearly,
if A were to be a very large number, then the sensitivity of the number of firms to κκe would be
minimal. However, notice that A defines a ratio between the fraction of total employment devoted to entry
relative to the fraction devoted to production. Sensible parametrization of this share gives A to be fairly
low. I will come back to this when I discuss the calibration of the quantitative model.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
21
Before addressing the question of what features of the specification of distortions determine
whether κκe is greater or smaller than one, it is useful to go over the logic of the equilibrium
to gather intuition about how is it that this ratio is so critical for the response in entry.
Suppose I start off the economy at the efficient long run equilibrium. Assume, in turn, that
I introduce a profile of distortions described by the function Γ (ω), while I keep constant the
ratio of aggregates wQθ at its frictionless level. The value of an entering firm in the distorted
economy relative to the frictionless equilibrium is:
υe
Ω̃e
= κe
f =
υe
Ω̃fe
That is, for given wages and final output, the change in the value of a new firm is given by
the ratio of the time-series the after-tax average labor productivity. The equilibrium value of
final output and wages must adjust accordingly so that the free entry condition is satisfied.
Given that entry costs are equal in the two economies, equating the free entry conditions
shows that the required adjustment in wQθ is governed by κe :
Q
Q
=
wθ
wθ
(4.2)
f
1
κe
Considering this relationship in the equation for average labor demand, I get:
e
L
p
=
θ−1
θ
!θ Q
wθ
f
Ω̃
κe
which, when expressed relative to average labor demand in a frictionless economy becomes:
e
κ
L
p
f =
e
κe
Lp
This ratio makes clear that if the time-series and the cross-sectional after tax average
labor productivities respond differently to idiosyncratic distortions, it will have an impact
on the economy’s average employment size and, through labor market clearing, on the total
number of firms.
4.2.1. A Particular Specification of Distortions. I now restrict to a specification of distortions that tax/subsidize firms at a common rate τ depending on the position of the firm’s
productivity relative to the median productivity in the economy, as in the previous subsection. The goal is to show that a key property of the profile of distortions that determines
whether κκe is higher or lower than one is its correlation with idiosyncratic productivity. If,
as found in the data11, distortions are such that low-productivity firms are subsidized and
high productivity firms are taxed, I show that entry increases in a distorted equilibrium,
offsetting the misallocation effect on aggregate productivity.
11Evidence
on the properties of the correlation between idiosyncratic distortions and the distribution of
productivities can be found in Hsieh and Klenow (2009), and Bartelsman, Haltiwanger and Scarpetta (2009)
ENTRY, EXIT AND MISALLOCATION FRICTIONS
22
For the given specification of distortions, the time-series and cross-sectional average aftertax labor productivities are given by:
e =
Ω
h
i
δ−µ i
δ h
(1 + τ )θ + (1 − τ )θ − (1 + τ )θ 0.5 δ
δ−µ
h
i
ρ+δ−µ i
(ρ + δ) h
(1 + τ )θ + (1 − τ )θ − (1 + τ )θ 0.5 δ
(ρ + δ) − µ
The resemblance of the first equation to that in the simplified model in the previous section
is not a coincidence. It follows from the two being a moment of the Pareto distribution, the
shape parameter being here equal to µδ . The difference, however, is that if the interest rate is
positive, the time-series average productivity is higher than the cross-sectional one. In other
words, with distortions that subsidize low-productivity firms and with positive interest rates,
κ
< 1. With positive interest rates, firms apply a discount to the moment of time after
κe
which productivity would have grow enough to have entered the taxation region. Therefore,
form the perspective of the time-series distribution of productivities, the fraction of time
spent by an entering firm being subject to a positive tax is lower than the fraction of firms
that are being taxed in the cross-sectional distribution, implying that the profitability of the
former get less affected than the average profitability of the latter.
I now sharpen the results in proposition 2 exploiting the properties of the specification of
distortions that I have adopted.
f =
Ω
e
Proposition 3. Consider an economy with p = 1, fc = 0, ∆t → 0,µ < δ and ρ > 0.
Let firms with productivity ω < ωmedian be subsidized at rate τ, and firms with productivity
ω > ωmedian be taxed at the same rate. Then
1+A
M
i
= hκ
f
M
+
A
κe
where A =
ne δ
f,
L
( ˆp )
and
1+
1−τ
1+τ
θ
− 1 0.5
δ−µ
δ
κ
<1
=
θ
ρ+δ−µ
κe
δ
1 + 1−τ
−
1
0.5
1+τ
so Entry has an offsetting effect on Aggregate Productivity.
A few comments are in order before concluding the section. First, notice that an inherent
feature of the deterministic idiosyncratic productivity growth economy is the age dependence
of productivity and the realization of the distortion. By construction, the model forces the
firm to go through a period of taxation in the late ages its lifecycle, when profits are more
heavily discounted. Alternatively, I could have reversed the correlation of distortions with
ENTRY, EXIT AND MISALLOCATION FRICTIONS
23
productivity, and made young firms be subject to a tax. In that case, it is straightforward to see that entry would fall, and reinforce the misallocation effect. The motivation
for emphasizing the case of subsidies for low-productivity firms is twofold. First, from an
empirical standpoint, Bartelsman, Haltiwanger and Scarpetta (2008) and Hsieh and Klenow
(2009) find evidence favoring a positive correlation between the revenue tax rate and physical
productivity. Second, from an abstract point of view, Restuccia and Rogerson (2008) shows
quantitatively that the productivity loses due to misallocation are maximized when tax rates
are increasing in productivity.
4.3. No Firm Dynamics, Endogenous Exit. The simplifications of the general model
that I have considered until now were aimed at highlighting the forces by which misallocation
frictions affect entry decisions. In this section I propose a simplification that allows me to
isolate the effect of idiosyncratic distortions on exit. The simplification consists on abstracting from firm dynamics, setting h = 0, and imposing a positive fixed cost of production
fc . In this economy, the ex-ante and the cross-sectional distribution of productivities are
identical, given by the distribution G (ω) and exit decisions x (ω). Regarding the profile of
distortions Γ (ω) , I inherit the same specification as in the previous two sections, and also
extend the assumption that G (ω) is a Pareto distribution with shape parameter η.
Consider the value function of an active firm with productivity ω:
(θ − 1)θ−1 Q ω
e (1 − τω )θ − fc
θθ
wθ
υ (ω) = max {0, υ o (ω)}
υ o (ω) =
It is readily seen that if τω = 0 for all ω, then the exit decision takes the form of a cutoff
rule, with a unique productivity ω separating firms into those that stay in operation and
those that exit the market. With non-zero taxes and subsidies, it may be the case that
the exit set is empty, or that the cutoff productivity is not unique. This is easy to see in
the case where distortions impose a flat subsidy for low productivity firms and a tax for
high-productivity ones where, depending on the tax/subsidy rates, there could be exit at
the lower and the upper end of the productivity distribution. To preserve the nature of the
exit decision in the frictionless equilibrium, I rule out empty exit sets or multiple cutoffs by
assumption, adjusting the support of the distribution, the size of the fixed costs of production
and the magnitude of the tax/subsidy rates to ensure that this is the case.
The cutoff productivity can be solved imposing zero profits for the marginal producer:
eω =
fc θ θ
(1 + τ )θ wQθ (θ − 1)θ−1
Clearly, for given wages and final output, subsidizing the least-productivity firms reduces
the exit cutoff, allowing more low-quality firms to stay in operation. Free entry, in turn,
ENTRY, EXIT AND MISALLOCATION FRICTIONS
24
implies that:
β (1 − δ)
[1 − G(ω)]
fe =
1 − β (1 − δ)
"ˆ
dG(ω)
(θ − 1)θ−1 Q ω
e (1 − τω )θ − fc
θ
θ
θ
w
1 − G(ω)
!
ω
#
Given the Pareto distribution of productivities and the specification of distortions, I can
e
re-write the free entry condition in terms of the average after-lax labor productivity12 Ω
e
θ−1
β (1 − δ)
Qe
ω −η (θ − 1)
fe =
(e )
Ωe − f c
θ
1 − β (1 − δ)
θ
wθ
"
#
h
i
(η−1)
η
eω (1 + τ )θ + (1 − τ )θ − (1 + τ )θ 0.5 η
=
(η − 1)
e
Ω
e
where [1 − G(ω)] = (eω )−η . Notice that selection implies a trade-off for the ex-ante value
of an entrant: on one hand, a higher cutoff increases average productivity ex-ante, but on
the other hand it reduces the probability of staying in operation ex-post. The next result
solves the system of equations determining the equilibrium values of wQθ and the exit productivity cutoff and proves that wQθ increases and the exit cutoff decreases when introducing
misallocation frictions.
Proposition 4. Consider the economy with h = 0, fc > 0 and G (ω) ∼ P areto (η). Let
firms with productivity ω < ωmedian be subsidized at rate τ, and firms with productivity
ω>
1
η−1
1+η
(η−1)
η
1
η
ωmedian be taxed at the same rate. Also, let Ψ (τ ) =
Ψ (0) =
1
η−1
1−τ
1+τ
θ
− 1 0.5
η−1
η
, where
and Ψ0 (τ ) < 0. Then:
(1)
θθ
Q
=
wθ
(θ − 1)θ−1
"
(4.3)
ω
#1 "
e =
η
1 − β (1 − δ)
β (1 − δ)
fc
fe
!1 "
η
#1
η
fc
β (1 − δ)
1 − β (1 − δ)
#1
η
fe
1
(1 + τ )ρη Ψ(τ )
!1
η
1
Ψ (τ ) η
(2)
∂
Q
wθ
≥0
∂τ
∂ (eω )
≤0
∂τ
Proof. See appendix
The first result exhibits more clearly the trade-off involved in determining equilibrium aggregates. On one hand, a subsidy that reduces that exit cutoff increases the value of an
12To
preserve the comparability of results across the three simplified models, I keep a separate notation
for ex-ante and ex-post average productivity, although in the economies with no firm dynamics there is no
distinction between the two.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
25
entrant by reducing the probability of exit. This force is reflected by the term (1 + τ )ρη in
the denominator of the right hand side in the equation for wQθ . On the other hand, it reduces
the average profitability ex-post, which is captured in the term Ψ (τ ), a decreasing function
of the distortion. The second result in the proposition establishes that for the given specification of distortions, the second effect outweights the first so aggregates have to increase to
satisfy the free entry condition13. With aggregates going up, and a subsidy that protects the
marginal firm, it follows directly that the exit productivity cutoff falls in a distorted long
run equilibrium.
In addition to free entry and zero-profit for the marginal producer; there is the labor
market clearing condition that pins down the mass of firms:
h
e +L
e +L
e
L=M L
p
fc
e
i
(θ − 1)θ−1 Q ω −η e
(e ) Ω
p =
θθ
wθ
h
i
(η−1)
θ
θ
θ
ω
η
e (1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
e
L
e =
Ω
η
(η − 1)
e = f (eω )−η
L
fc
c
h
i
fe
e =
L
1 − (eω )−η + δ
e
(1 − δ)
Notice that with endogenous exit, all sources of average labor demand are subject to change
as a consequence of adjustments in the exit productivity cutoff.
Given the knowledge of firm’s exit decisions, the question is how they impact on the
aggregate economy. Recall the expression that decomposes aggregate productivity:

1
1 
θ−1
T F P = M θ−1 
[1 − G (ω)]
θ 
θ  
θ−1
θ−1
w
w ew
e
e
Ω /ΩCM
  ΩCM






e
e
e
Ω
Ω/Ω
CM
CM
The term in the first bracket represents the direct effect of changes in exit decisions on
aggregate productivity, for a given mass of firms and a given degree of misallocation at the
frictionless equilibrium’s stationary distribution of productivities. Written in terms of the
cutoff, it becomes:


1

[1 − G (ω)] θ−1

13The
e w /Ω
ew
Ω
CM
e Ω
e
Ω/
CM
θ
θ−1



= (eω )−
(η−1)
ρ−1
result holds regardless of the value of η in the case where distortions tax and subsidize half the firms
in the economy. More generally, it is also true for any case where the fraction of taxed firms is greater
than or equal to 0.5. For values below this, then the result depends on the parametrization of the shape
parameter of the Pareto. However, it holds true for all values of the Pareto parameter that guarantee a finite
variance of the distribution of productivities. The reason why the fraction of taxed firms matters is that
this fraction determines the extent by which average entrant’s profit fall and, therefore, the likelihood that
it will outweight the positive effect of a lower exit probability.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
26
Expectedly, this term is decreasing in the exit threshold, since lower exit leads to higher
productivity through an expansion in the number of active firms in the economy. Therefore,
the result that misallocation frictions reduce the equilibrium fraction of exiting firms implies
that aggregate productivity increases through this channel, partially offsetting the negative
misallocation effect.
The remaining component of the selection effect on aggregate productivity is the response
in the mass of firms in the distorted long run equilibrium. As it was the case in the previous
simplifications of the model, labor market clearing dictates that the response in the mass
of firms is determined by the change in the average demand for labor. With a higher
fraction of firms staying in the market, there is one component of average labor demand
which unambiguously increases in a distorted equilibrium, which is average labor demand
for fixed costs. Average labor demand due to entry decreases, as a result the change in exit
decisions, since fewer firms need to be created to recompose those that are exiting. The
reduction in exit also increases average labor demand in production, although in this case
the overall effect depends on the change in wQθ and the reduction in after-tax average labor
productivity implied by the distortions. In the next proposition I establish that if entry costs
are sufficiently larger than fixed costs of production, then total average labor demand falls
and the mass of firms goes up in a distorted long run equilibrium.
Proposition 5. Consider the economy with h = 0, fc > 0 and G (ω) ∼ P areto (η), with
η > 1. Let firms with productivity ω < ωmedian be subsidized at rate τ, and firms with
productivity ω > ωmedian be taxed at the same rate. Then:
M > Mf ⇔
Proof. See Appendix
fe
fc
> θ (1 − δ)
Although the proposition does not unambiguously determine the direction of the change
in the number of firms, it establishes a lower bound on the ratio of entry to fixed costs
of production so that the mass of firm increases, further reinforcing the offsetting effect of
selection. In order to get a sense of how tight is the lower bound, one can recognize that the
ratio ffec determines the exit cutoff in the frictionless equilibrium, as shown in equation 4.3,
and thereby the exit rate in the economy. The firm exit rate in the U.S. economy is in the
order of 10%, and the model is generating a 2.5% exit rate from the exogenous death shock14.
Thus, for given parameter values of the Pareto distribution and the elasticity of substitution,
I can find the ratio ffec that makes the economy attain the 10% target. I find that for a wide
range of values of the elasticity of substitution and the Pareto shape parameter, in particular
14I
will discuss further properties of the U.S. firm size distribution and the dynamics of firms later in the
calibration section
ENTRY, EXIT AND MISALLOCATION FRICTIONS
27
those that I pick in the calibration below, the ratio of entry and fixed costs is large enough
to satisfy the condition in the proposition.
5. Quantitative Analysis
In this section I propose a calibration of the general model to quantify the relative magnitudes of the entry, exit and misallocation effects on long run aggregate productivity, income
and welfare. Then, I compute transition dynamics in the model to calculate the welfare
gains of a hypthetical reform that eliminates idiosyncratic distortions, and compare them to
welfare gains resulting from long-run effects only, and to those from a model with constant
number of firms and exogenous exit.
5.1. Calibration. I must choose parameter values for the elasticity of substitution θ, the
subjective discount factor of the household β, the size of the labor force L, and the set of
parameters governing the process of firm dynamics, entry and exit: entry and fixed operation
costs fe and fc , the size and probability of the jump in the binomial process h and p, and
the exogenous exit rate δ. For calibration purposes, I will be considering the economy with
no distortions. Table 1 summarizes the parameter values.
Table 1. Parameter Values and Calibration Targets
Parameter
Value
Target
ρ
3
Hsieh and Klenow (2009), Broda and Weinstein (2006)
β
1
1.05
Interest Rate of 5%
δ
0.025
Employment-Based Exit Rate of Large Firms of 2.5%
p
0.467
Slope Log of Right Tail of Empl. Based Size Distribution= -0.2
h
0.25
Std Dev. of Employment Growth of Large Firms
G(ω)
ω0 = 0
Size of Entrants = 6% of Median Incumbent (Luttmer 2010)
fc
fe
0.1
Exit Rate of 5%
I pick the first block of parameter values as follows. For the elasticity of substitution I
set θ = 3, which lies in the middle of estimates of substitutability found in the trade and
industrial organization literature15, and is the value adopted by Hsieh and Klenow (2009). I
choose the discount factor so that β1 − 1 equals a real interest rate of 5%, and normalize the
size of the labor force to be equal to 1.
My strategy to pin down micro-level parameter values is to target features of the employmentbased firm size distribution and the patterns of firm dynamics in the U.S. I set the size of
the jump to h = 0.25, to target the cross-sectional standard deviation of employment growth
rates among large firms in the US, following the practice in Atkeson and Burstein (2010).
15See
Broda and Weinstein (2006) [3] for a range of estimates of the elasticity of substitution for US imports
at a 4-digit disaggregation level.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
28
I pick the probability of the upward jump in the binomial process, p, to match the properties of the right tail of the U.S. employment based size distribution. Luttmer (2010)
highlights the linearity of the right tail of the US establishment and firm size distribution
across employment, as well as the stationarity of the distribution over time, using various
sources of US micro-data16. Restricting to the employment-weighted size distribution from
the Business Dynamics Statistics, the fraction of total employment accounted for by firms
with 5000 employees or more is on average about 30% for the period 1980-2005. The same
statistic for firms with 1000 employees or more amounts to 42%. Since the median employment is 500 employees, employment levels in the range 1000-5000 employees are sufficiently
large to have reached the linear portion of the right tail of the size distribution. Therefore,
I choose the jump probability to target the slope of the linear function that maps the logarithm of employment into the logarithm of the fraction of employment accounted for by
firms larger than a given size, restricting to the range of 1000-5000 employees17. Given the
numbers in the data, the slope of such function is equal to -0.2.
Regarding the probability of the exogenous exit shock, recall that this parameter participates in the overall exit rate of the economy, but specifically determines the fraction of large
firms that exit, which are those that would not choose to exit endogenously in the model.
Looking at the Small Business Administration (SBA) data on employment-weighted exit
rates for large firms, I choose a value of δ = 0.025.
For the distribution of productivities at entry, I assume that G (ω) is degenerate at ω0 = 0,
so that all firms start with the same productivity and employment size, and spread out over
time according to the stochastic process. Normalizing the units of employment so that the
median employment level is 500 employees, as in the data, new firms’ employment will be
6% of the median employment in the economy, in line with the findings for these relative
statistics in Luttmer (2007). Finally, I normalize fe = 1 and set fc = 0.1, which implies an
exit rate18 of 5%.
Given the calibration of the frictionless economy, I have to specify a functional form for
the profile of distortions Γ (ω). For consistency with the analytical results, I consider a
specification with identical characteristics. That is, there is a flat subsidy for all firms with
productivity lower than or equal to ωmedian , and a flat tax rate for firms whose productivity
16County
Business Patterns Database, statistics from the Small Business Administration and the Business
Dynamics Statistics from the census.
17The reason to focus on the slope of right tail in logs is that employment units are just a normalization in
the model
18In a model with no firm dynamics, the ratio of entry and fixed costs pins down the exit rate of the economy.
With firm dynamics, however, the exit rate is a function not only of those relative costs, but also the variance
of the stochastic process.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
29
are above the median. I assume that subsidy and tax rates are the same19, and I experiment
with different values throughout the experiments below.
5.2. Entry, Exit and Aggregate Productivity. I start evaluating the quantitative magnitude of the offsetting forces provided by entry and exit on aggregate productivity and
income. To do so, I solve for the stationary equilibrium of the economy at various levels
of the tax/subsidy rate and perform the following counterfactual experiments: how much
would productivity increase from the removal of misallocation frictions, holding fixed entry
and exit decisions and the distribution of firms across productivities? This counterfactual
is in the spirit of Hsieh and Klenow (2009), and Restuccia and Rogerson (2008). Then, I
ask the same question but allowing entry, exit and the productivity distribution to adjust.
The difference in the counterfactual productivity gain will be given by the offsetting force
of entry and exit, and the magnitude of the difference will be reflecting the strength of the
offsetting effect.
First, I focus on the aggregate implications of misallocation frictions for the theoretical
measure of aggregate productivity, which has been the object of analysis in the analytical
results. The goal is to verify the validity of the forces identified in the simplified models
once I move to the fully fledged stochastic general equilibrium model, and to quantify the
overall effects for this perfect measure of efficiency. Then, I consider a measure of total factor
productivity that considers the entire labor force as the measured input of production, more
closely resembling the practices in development accounting studies. Notice that once I do so,
given the inelasticity of labor supply, the income and productivity effects become identical
with each other.
5.2.1. Theoretical Aggregate Productivity. Figure 1 illustrates the results of the counterfactual experiments under the model-based definition of aggregate productivity. The blue line,
T F PCM , corresponds to the case where the productivity distribution, entry and exit are
held fixed at their distorted equilibrium’s levels and idiosyncratic distortions are removed.
The red line, T F P , represents the case where the endogenous response in entry and exit are
taken into account. Values correspond to levels in the corresponding frictionless economy
relative to the distorted equilibrium.
19The
common rate assumption is without loss of generality for the purpose of my analysis. Restuccia and
Rogerson (2008), who feature physical capital stock in their model, can pin down the subsidy rate, for a
given tax rate, in order to ensure that the capital stock is constant across long-run equilibriums.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
30
Figure 1: The Offsetting Effect, Theoretical Aggregate Productivity
Eliminating distortions brings strict productivity improvements on aggregate productivity
measured by T F PCM , as the resource misallocation problem is resolved. For tax rates that
get as high as 45%, the productivity gain accruing from the reversion of the misallocation
effect reach up to 35%. However, solving for the general equilibrium response of entry and exit
decision drastically changes the pattern of the response in aggregate productivity. For low
to middle distortion rates, lower entry and higher exit more than offsets the static efficiency
gain from labor reallocation, leading to an overall reduction in aggregate productivity in
the frictionless equilibrium. For large levels of distortions the offset is only partial but still
substantial, turning a 35% productivity gain from reallocation into a 8% increase, about
80% weaker.
The fact that welfare could be even lower in a frictionless relative to a distorted long run
equilibrium may seem puzzling, given the Pareto optimality of the undistorted competitive
equilibrium’s allocation20. However, it should be regarded as a sign of the importance of
transitional dynamics in assessing the welfare effects of misallocation frictions, and the inaccuracy of welfare comparisons based on steady state outcomes. I will come back to the role
of transition dynamics for welfare costs below.
20It
can be shown that with inelastic labor supply, the inefficiency of the monopoly pricing is undone by the
wage, preserving the equilibrium optimality of the competitive equilibrium’s allocation.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
31
I now turn to decomposing the effects on aggregate productivity according to the decomposition proposed above, which I reproduce here for convenience:
TFP =

θ 
1
θ−1
w
w
e
e
1 

M θ−1 Ω /ΩCM
θ−1 

M
e

| {z
}  Me
e
e
Ω/Ω
CM
Entry
|
{z




}|
ew
Ω
θ
θ−1
CM

CM



{z
}
e
Ω
Selection
Mis-Allocation
To idea is to verify that the forces highlighted in the analytical results are at work in the
quantitative model, and quantify the contribution of each effect. The left panel in figure 2
reproduces figure 1, while the middle and third graphs show how the aggregate productivity
gain implied by T F PCM turn into T F P , by subsequently adding the selection and the entry
effect.
Figure 2: Decomposition of Theoretical Productivity
To implement the decomposition numerically, I proceed as follows. First, I keep constant
the number of entrants and isolate the components of the selection effect. In the middle
graph, the dashed green line (Cutoff-Prod) represents the combination of the misallocation
effect and its reinforcement implied by the change in exit decisions, summarized by the term
θ
(Ωew /ΩewCM ) θ−1
. As we can see, the productivity gain gets amplified through this channel, as
(Ωe/ΩeCM )
low productivity firms are pushed out of the market, increasing average productivity in the
economy.
The solid green line constitutes the remaining component of the selection effect, namely
the change in the mass of firms per unit of entrant. Since more firms are exiting in the
undistorted equilibrium, the ratio has a negative contribution on aggregate productivity. The
overall selection effect, for a given mass of entrants, is depicted by the dashed red line (T F P −
Endo), which shows that selection has an offsetting effect over the aggregate productivity,
ENTRY, EXIT AND MISALLOCATION FRICTIONS
32
undermining the gain that is obtained from the removal of misallocation frictions. Notice
that selection, in itself, is enough to make the offsetting effect more than outweight the
misallocation effect for low to middle values of the distortion rate; and is capable of eroding
about 30% of the 35% productivity gain that arises from removing a tax rate of 45%.
The third graph to the right brings the entry effect into the analysis. As we can see
from the light blue line, entry also acts as an offsetting force on aggregate productivity,
reducing the overall counterfactual productivity gain even further, and turning the dashedred line into the solid red one. The strength of the entry effect can be appreciated from the
difference between these two lines. Consideration of entry decisions revert about 80% of the
productivity gain for the case of a 45% tax, and it enchances the set of distortions for which
entry and exit effects more than offset the misallocation effect.
The analysis of the decomposition reveals that the economic forces highlighted in the
analytical results in the context of simplified models extend to the case where all margins
are active simultaneously. There is one novel interaction, however, that could not be captured
in the simple models where selection and firm dynamics were not in place at the same time.
The interaction has to do with the observation that the entry effect is non-monotone in the
degree of distortion, as shown by the corresponding line in the third panel to the right in
figure 2. The source of this non-monotonicity is that at larger tax rates, the ex-post median
productivity in the economy is reduced relative to the entry point of new firms. Since I am
adjusting the distribution of subsidies and taxes so that, for each specification, half of the
firms in the economy are subsidized and half are taxed, the possibility of new firms being
taxed is now more likely at the early years of their lifespan, reducing the gap between the
ratio of time-series and cross-sectional average labor productivities. Therefore, the incentive
to increase the measure of entering firms in the distorted equilibrium is weakened through
this channel.
5.2.2. Measured Aggregate Productivity. I now turn to discuss the results of my experiments
adopting a measure of aggregate productivity that, more closely resembling the practices
in development accounting studies, does not distinguish between the composition of the
labor force when determining the input of production from which total factor productivity
is inferred. As mentioned in earlier sections, this has the implication of adding a new
component to the ingredients of aggregate productivity, which is the change in the share of
the labor force that is allocated to produce final goods. As a reminder, measured aggregate
productivity is now given by:
T F PL =

θ 
1
θ−1
w
w
e
e
1 

M θ−1 Ω /ΩCM
θ−1 

Me } 

| {z
e
e
Me
Ω/Ω
CM
Entry
|
{z
Selection

ew
Ω
θ
θ−1

Lp
L
CM
|{z}
}|
{z
} Labor -Share
Mis-Allocation



CM
e
Ω



ENTRY, EXIT AND MISALLOCATION FRICTIONS
33
Notice that by contemplating changes in the share of labor in production, steady state
comparisons of productivity are essentially comparisons of income. Therefore, whether as
part of total factor productivity or as a measured input of production, changes in Lp are a
component of the sources of change in aggregate output that has to be taken into account.
Figure 3: The Offsetting Effect in the Alternative Measure of TFP
In Figure 3, I reproduce the outcome of the same counterfactual experiments as figure 1
contemplating the change in the share of production labor. The figure shows that although
the magnitude of the offsetting forces are weakened through this channel, they are still
substantial. For tax rates of up to 30%, that would have led to a 10% increase in output
having kept constant the mass of firms and the productivity distribution, the offsetting
effect more than outweights the misallocation effect, implying that output actually falls in
the frictionless equilibriums. The combined effect of entry, exit and changes in the share of
production labor is capable reverting a 35% of the counterfactual productivity gain that is
obtained under a tax rate of 45%.
The weakening of the offsetting forces that arises when considering the adjustments in
the fraction of production employment reflects that the economy is allocating labor out of
production purposes in order to attain an equilibrium with a higher number of firms.
5.3. Welfare Analysis. An important feature of the long run comparison of income was
that there existed levels of distortions for which welfare in a frictionless long run equilibrium
could be lower than in an equlibrium with distortions. Of course, this cannot be true for
the lifetime welfare of the household, since the undistorted competitive equilibrium was
Pareto optimal. Therefore, additional sources of welfare cost must be taking place during
the transition path from one equilibrium to the other. In this section I perform a welfare cost
calculation that takes into account transitional dynamics, and compare it to those resulting
from a long run comparison of income, and to those arising in an economy with constant
number of firms. I will show that accounting for transition dynamics yields sizeable welfare
gains from removing distortions, larger than in any of the other cases.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
34
I adopt a standard metric of welfare cost, given by the permanent consumption compensation that must be given to the household in a distorted stationary equilibrium to make
it be indifferent between inhabiting such economy, or one with no distortions. Although
the measure captures the welfare gains that result from removing distortions, it can also be
interpreted as a welfare cost of misallocation frictions, in terms of a foregone improvement
in economic well-being that is incurred living in a distorted economy.
f
Let W∞ (τ ) denote welfare in a stationary equilibrium with distortions at rate τ , W∞
be
the welfare in a stationary frictionless equilibrium, and WT r (τ ) be the lifetime welfare of
transiting from a distorted stationary equilibrium to an equilibrium with τ = 021:
W∞ (τ ) =
f
=
W∞
log [Q (τ, Mτ (ω))]
1−β
log [Q (0, Mf (ω))]
1−β
h
i
τ →f
t
(ω)
WT r = Σ∞
t=0 β log Qt 0, Mt
Here Q (τ, Mτ (ω))and Mτ (ω) are final output and the stationary distribution of an equilibrium with tax rate τ , Q (0, Mf (ω)) denotes final output and Mf (ω) the stationary distribution for an economy with no distortions, and Qt (τ, Mt (ω)) is final output along a transition
from a distorted long run equilibrium to a frictionless one, with the distribution of firms
evolving according to Mtτ →f (ω).
Given these definitions, welfare costs are given by22:
f
ΘLR (τ ) = e(1−β)[W∞ −W∞ (τ )] − 1
ΘT r (τ ) = e(1−β)[WT r −W∞ (τ )] − 1
where the superscript identifies which experiment I am considering.
The left panel in Figure 4 illustrates welfare costs for the same range of idiosyncratic
distortions considered in the analysis of productivity, both from a long run perspective and
accounting for transition paths. The divergence in the magnitude of welfare costs are stark.
While from a long run perspective welfare could even rise for intermediate values of taxes
and subsidies, distortions will always reduce the welfare metric that contemplates transition
21In
order to minimize the introduction of new notation, I avoid defining welfare costs for the case of constant
number of firms. It is straightforward to see that in such case, welfare costs are given exactly by differences
in productivity, which I have discussed extensively in previous sections.
LR
22Straightforward algebra shows that this expressions solve the following equation: log [Θ (τ )] + W (τ ) =
∞
1−β
f
W∞
ENTRY, EXIT AND MISALLOCATION FRICTIONS
35
dynamics23. Large distortionary rates have a powerful damaging power over the economy’s
well-being, twice as large as those resulting from long run comparisons of welfare.
Figure 4: Welfare Cost Calculation: Transition Dynamics, Long Run and Constant Mass
The right panel adds to the picture the welfare costs implied by the economy with constant
number of firms. As explained in previous sections, the offsetting effects of entry and exit
explain the difference between the long run welfare costs in this economy and those of the
economy with a given mass of firms. Notably, once transition dynamics are taken into
account the pattern gets reversed and welfare costs end up being higher in the economy
where entry and exit respond endogenously to the withdrawal of distortions.
To shed light on the origins of welfare loss in the economy with entry and exit, I study
the properties of aggregate output along a transition path from a stationary equilibrium
with τ = 0.4 to an undistorted one, and compare it with that of the economy with constant
number of firms24. In the latter case, however, notice that transition dynamics are trivial,
since the economy’s output jumps immediately to the level consistent with the long run
frictionless equilibrium. The left panel of figure 5 shows graphs for final output, while the
right panel decomposes output into productivity and production labor for the economy with
endogenous entry and exit.
23Recall
that the competitive frictionless equilibrium of the economy is efficient, in spite of the distortionary
effect of monopoly pricing. The reason is that the real wage exactly undoes the distortion of the mark-up,
without distorting any other margin in the economy. Had labor supply being endogenous, for example, then
the efficiency of the equilibrium would have been lost. See Bilbiie, Ghironi and Melitz (2008) [2] for a full
discussion of conditions of the model that preserve the efficiency of the equilibrium.
24Recall that under the assumptions of the model of section 3.1 the optimal response of the economy is to
not change the number of firms when introducing or removing distortions. Therefore, whenever I refer to
the model with constant number of firms it should be though of as exactly the model of section 3.1, with
the arbitrary distribution of productivities chosen to be the same as the endogenously determined one in the
distorted long run equilibrium of the economy with entry and exit, for each given value of τ .
ENTRY, EXIT AND MISALLOCATION FRICTIONS
36
Figure 5: Transition Dynamics. Removing Distortions of τ = 0.4
On impact, final output increases about 10% more in the full economy than in the economy with constant number of firms. In both cases the inefficiency of the allocation of labor
is resolved immediately, although the economy with endogenous selection experiences a reduction in the fraction of active firms. However, output ends up increasing by more in the
former since there is a reallocation of labor out firm creation and into final good’s production, as the economy lets the inefficiently high stock of firms with which it started the period
depreciate over time25. Through this channel, final output overshoots in the earlier years,
and it converges from above to a steady state value that is only 10% higher than its level at
the original distorted long run equilibrium. The fact that output eventually crosses the long
run level of the economy with constant number of firms and stabilizes below it is another
manifestation of the offsetting effect of higher exit and lower entry in the undistorted long
run equilibrium.
6. Concluding Remarks
In this paper I proposed a model of resource misallocation to re-evaluate the implications
of misallocation frictions for aggregate productivity, output and welfare. The focus of my
model was on three features that were largely neglected in the literature of misallocation:
firm dynamics, entry and exit.
The findings of my paper suggest that as a mechanism for understanding cross-country
differences in income and productivity, misallocation frictions lose most of its explanatory
power once I consider adjustments at the level of entry and exit of firms. However, from a
normative point of view, I find large welfare costs associated with these distortions, arising
mostly from transitional dynamics.
The paper’s approach about the form adopted by the policies or frictions that led to
misallocation was very stylized, modelling them as idiosyncratic revenue taxes. Even though
25Recall
that Me ≥ 0, so the economy is hitting the constraint in the first periods of the transition
ENTRY, EXIT AND MISALLOCATION FRICTIONS
37
the resulting tractability allowed me to provide sharp characterizations of the response of the
economy to such policies and enabled me to make quantitative predictions, the large welfare
gains associated with their removal calls for a deeper investigation of what the sources of
misallocation really are. Several candidates have already being explored in the literature,
such as financial frictions and firing costs 26, although none of these seem to be accounting
for most of the misallocation that has been documented in the data.
The model incurred in additional simplifications that were shown to have a direct impact
on its mechanisms. One such simplification consisted on the technology of entry, treated
here as a linear function of labor. Some researchers have modelled entry as a result of
an occupational choice decision where agents select themselves into working for a wage or
undertaking entrepreneurial activities according to their talent. In this environment, it costs
more than just one unit of labor to switch a worker out of the labor force and start a new
firm. In the formulation of my model, the entry cost has the interpretation of a sunk cost
that is required to set up the firm, the profitability of which is determined ex-post from
a random productivity draw. The two set-ups share the property that not any “idea” is
profitable enough to become a firm, although they differ in how costly it is to set it up.
Further evidence allowing to more concretely identify the properties of the process of firm
creation would be very valuable for all models where the number of producers is determined
endogenously.
26
See, for instance, Veracierto (2001) [12], and Buera and Shin (2010) [4]
ENTRY, EXIT AND MISALLOCATION FRICTIONS
38
References
[1] A. Atkeson and A. T. Burstein. Innovation, firm dynamics, and international trade.
Journal of Political Economy, 118(3):433–484, 06 2010.
[2] F. O. Bilbiie, F. Ghironi, and M. J. Melitz. Monopoly power and endogenous product
variety: Distortions and remedies. NBER Working Papers 14383, National Bureau of
Economic Research, Inc, Oct. 2008.
[3] C. Broda and D. E. Weinstein. Globalization and the gains from variety. The Quarterly
Journal of Economics, 121(2):541–585, May 2006.
[4] F. J. Buera and Y. Shin. Financial frictions and the persistence of history: A quantitative
exploration. NBER Working Papers 16400, National Bureau of Economic Research, Inc,
Sept. 2010.
[5] N. Guner, G. Ventura, and X. Yi. Macroeconomic implications of size-dependent policies. Review of Economic Dynamics, 11(4):721–744, October 2008.
[6] H. A. Hopenhayn. Entry, exit, and firm dynamics in long run equilibrium. Econometrica,
60(5):1127–50, September 1992.
[7] C.-T. Hsieh and P. J. Klenow. Misallocation and manufacturing tfp in china and india*.
Quarterly Journal of Economics, 124(4):1403–1448, 2009.
[8] E. G. Luttmer. Models of Growth and Firm Heterogeneity. Annual Review of Economics, Vol. 2, pp. 547-576, 2010, 2010.
[9] E. G. J. Luttmer. Selection, growth, and the size distribution of firms. The Quarterly
Journal of Economics, 122(3):1103–1144, 08 2007.
[10] D. Restuccia and R. Rogerson. Policy distortions and aggregate productivity with
heterogeneous establishments. Review of Economic Dynamics, 11(4):707 – 720, 2008.
[11] N. L. Stokey. The Economics of Inaction: Stochastic Control Models with Fixed Costs.
Princeton University Press, 2008.
[12] M. Veracierto. Employment flows, capital mobility, and policy analysis. International
Economic Review, 42(3):571–95, August 2001.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
39
Appendix A. Technical Appendix
A.1. Proof of Proposition 1. Here I prove the result that with no firm dynamics, the
number of firms does not change in response to idiosyncratic distortions
Free entry in this economy establishes that:
β (1 − δ)
(θ − 1)θ−1 Q e
fe =
Ωe
1 − β (1 − δ)
θθ
wθ
ˆ
e
Ωe = eω (1 − τω )θ dG (ω)
!
Take the ratio of the free entry condition in a distorted and a frictionless economy, and
denote variables in the latter with the superscript f . Then
Qe
Q f ef
(A.1)
Ωe
Ω
=
e
wθ
wθ
Now, consider the expression for aggregate labor demand in production in the distorted
economy
!θ
Q
θ−1
(1 − δ) e
Ω
Lp =
θ Me
θ
δ
(w)
e and that in steady state M =
where I am using that Ω = M Ω
result from free entry we have:
Lp
=
Me
θ−1
θ
!θ Q
wθ
f
(1−δ)
Me .
δ
Combining with the
ef
(1 − δ) e Ω
Ω ee
δ
Ωe
The key implication of the assumption of no firm dynamics and exogenous exit is that both
e , and the cross sectional
the entrants assessment of its expected productivity after entry, Ω
e
average productivity are identical, because both are driven from the same distribution G (ω).
e and Ω
e cancel out in the expression above, and I am left with:
Then, the terms Ω
e
Lp
=
Me
θ−1
θ
!θ Q
wθ
f
(1 − δ) e f
Ωe
δ
which says that labor demand in production, per-unit of entrant is identical in a frictionless
and in the distorted economy.
Now, proving that entry does not change is straightforward given this result. From labor
market clearing, we have:
Lp
L = Me
+ ne
Me
Since the term in brackets on the right hand side is equal to that in the frictionless economy,
it follows that the number of new entrants in the distorted equilibrium cannot change with
respect to its undistorted counterpart. This proves the first part of the proposition.
ENTRY, EXIT AND MISALLOCATION FRICTIONS
40
That aggregate labor demand in production is also unchanged by idiosyncratic distortions
f
Lp
Lp
follows from the above result that M
, in conjunction with the finding that Me =
=
Me
e
f
Me .
Given that the mass of firms and employment in production do not change with distortions,
it follows directly that relative outputs are entirely driven by relative average productivities.
A.2. Proof of Proposition 4. Recall the expressions for the exit cutoff and the free entry
condition:
fc θ θ
eω =
(1 + τ )θ wQθ (θ − 1)θ−1
β (1 − δ)
(θ − 1)θ−1 Q e
fe =
Ωe − f c
(eω )−η
1 − β (1 − δ)
θθ
wθ
h
i
(η−1)
η
θ
θ
θ
ω
e
η
Ωe =
e (1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
(η − 1)
"
To simplify notation, define Π =
second gives
#
(θ−1)θ−1 Q
.
θθ
wθ
Then, replacing the first equation in the
"
#
h
i
(η−1)
η
β (1 − δ)
θ
θ
θ
η
Πη fc−(η−1) (1+τ )θη
fe =
−1
θ (1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
[1 − β (1 − δ)]
(η − 1) (1 + τ )
1
η−1
Defining Ψ (τ ) =
1+η
1−τ
1+τ
θ
− 1 0.5
η−1
η
=
η
(η−1)(1+τ )θ
θ
h
θ
β (1 − δ)
Πη fc−(η−1) (1 + τ )θη Ψ (τ )
[1 − β (1 − δ)]
which, after solving for Π and backing out
proposition:
Q
θθ
=
wθ
(θ − 1)θ−1
"
#1 "
η
Q
wθ
1 − β (1 − δ)
β (1 − δ)
transforms into the first equation in the
#1
η
(η−1)
η
fc
1
η
fe
1
(1 + τ )ρη Ψ(τ )
!1
η
Plugging back in the expression for the cutoff, we get:
eω =
fc
fe
!1 "
η
β (1 − δ)
1 − β (1 − δ)
#1
η
i
(1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
free entry becomes:
fe =
θ
1
Ψ (τ ) η
which corresponds to the second equation in proposition 4.
It is straightforward to see that Ψ (0) = 1 and Ψ0 (τ ) < 0. Then, given the sign of this
derivative it follows directly that the exit threshold is a decreasing function of the distortion,
i.e.
∂ (eω )
≤0
∂τ
(η−1)
η
ENTRY, EXIT AND MISALLOCATION FRICTIONS
41
∂ ( Qθ )
w
To prove that ∂τ
≥ 0, it suffices to prove that the term (1 + τ )ρη Ψ (τ ) is decreasing in
τ . Denote this term with J (τ ) . Then:
(
∂J (τ )
θη
=
(1 + τ )θη−1 1 + η
∂τ
η−1
η−1
"
1−τ
1+τ
θ
#
− 1 0.5
η
(1 − τ )θ
θη
θη 0.5
θ−1
(1 + τ )
(1 − τ )
+
−
η−1
(1 + τ )
(1 + τ )θ
"
η−1
η
)
#
Grouping terms:

"

∂J (τ )
1
θη
=
(1 + τ )θη
1+η

∂τ
η−1
(1 + τ )
"
1−τ
1+τ
θ
#
− 1 0.5
η−1
η
#
"
#
η−1
0.5 η
(1 − τ )θ 
θ−1
−
(1 − τ )
+
(1 + τ ) 
(1 + τ )θ

Searching for conditions under which this is <0, we get:


"
"
1
1+η
 (1 + τ )
1−τ
1+τ
θ
#
− 1 0.5
η−1
η
#
"
#
η−1
0.5 η
(1 − τ )θ 
θ−1
−
(1 − τ )
+
<0
(1 + τ ) 
(1 + τ )θ

⇔
η−1
η−1
1−τ
(1 − τ )θ
θ−1
η
η
(1 + τ )
1+η
− 1 0.5
< 0.5
(1 − τ )
+
1+τ
(1 + τ )
Taking common factors in the RHS:
"
"
θ
#
"
"
θ
#
θ−1
θ−1
(1 + τ )
1+η
1−τ
1+τ
#
− 1 0.5
η−1
η
"
#
< 0.5
η−1
η
"
θ−1
(1 − τ )
#
(1 − τ )
1+
(1 + τ )
#
Moving (1 − τ )θ−1 to the LHS and distributing terms:
1+τ
1−τ
θ−1
+ η0.5
< 0.5
η−1
η
η−1
η
(1 − τ )
1+τ
−
(1 + τ )
1−τ
+ 0.5
η−1
η
θ−1
η0.5
η−1
η
(1 − τ )
(1 + τ )
grouping terms:
0.5
η−1
η
"
#
η−1
(1 − τ ) (1 − τ )
(η − 1) − 1 < η0.5 η − 1
θ−1
(1 + τ )
(1 + τ )
η−1
Notice that the function on the right hand side, η0.5 η , is strictly increasing in η and is
greater than 1 for all27 η > 1. Thus, the right hand side is a positive term.
The left hand side may be non-positive or non-negative depending on the value of η and
τ . However, notice that conditional on being positive, it is a decreasing function of τ. Also,
conditional on being negative, it is an increasing function of τ . Therefore, it suffices to prove
that the inequality holds for two relevant cases: τ = 0 and η such that LHS is ≥ 0, and
τ = 1 and η such that LHS is ≤ 0.
27Recall
that the Pareto distribution requires η > 1 for it to have a well defined mean
ENTRY, EXIT AND MISALLOCATION FRICTIONS
42
Setting τ = 0 and solving I get
η−1
1
< 0.5 η
2
which is true for all η in the relevant range (i.e. η > 1).
On the other hand, setting τ = 1:
0 < η0.5
η−1
η
−1
which, as argued above, is always true for all η > 1. Thus, this proves that
∂(
Q
wθ
)
∂τ
>0
A.3. Proof of Proposition 5. Characterizing the response in the mass of firms amounts
to establishing whether average labor demand goes up or down in a distorted long run
equilibrium. The set of equations that will allow me to determine this are the three sources
of average labor demand in the economy, and the free entry condition. In all cases I would
have already substituted out the exit cutoff as a function of aggregates from the zero profit
(θ−1)
Q
, in
condition for the marginal firm. As in the previous proposition, I define Π = (θ−1)θθ
wθ
order to simplify notation.
Average labor demand is given by:
h
i
(η−1)
η
θ(η−1)
θ
θ
θ
η
−(η−1)
e
η
Lp = (θ − 1) Π
f
(1 + τ )
(1 + τ ) + (1 − τ ) − (1 + τ ) 0.5
(η − 1) c
e = f −(η−1) Πη (1 + τ )θη
L
fc
c
e
L
e
(1 + δ)
− (1 + τ )θη Πη fc−η fe
+ fe
=
1−δ
1−δ
h
Πη =
(1 + τ )θη
η
(η−1)(1+τ )θ
η
(η−1)
i
−1
h
i
(1 + τ )θ + (1 − τ )θ − (1 + τ )θ 0.5
(η−1)
η
Πf
η
−1
where the equation for average labor demand for entry is using that in steady state, the law
of motion for the mass of firms becomes:
(1 − δ)
M = Me
[G (ω) + δ]
The last equation, which establishes a relationship between equilibrium aggregates in a
distorted and a frictionless economy, follows from the fact that the value of an entrant, in
units of labor, must be equal to each other in the two economies.
Notice that both in average labor demand
in production, and in the
relationship between
(1+τ )θ +[(1−τ )θ −(1+τ )θ ]0.5
equilibrium aggregates, there is a term
(1+τ )θ
(η−1)
η
that take place in the
ENTRY, EXIT AND MISALLOCATION FRICTIONS
43
two. I summarize this term with the following function:
"
"
Φ (τ ) = 1 +
1−τ
1+τ
θ
#
− 0.5
(η−1)
η
#
e d , I get:
Denoting total average labor demand with L
"
e
L
d
θη
η
= Π (1 + τ )
fc−(η−1)
#
(θ − 1) η
fe
1+δ
Φ (τ ) + 1 −
+ fe
η−1
fc (1 − δ)
1−δ
The key of the proof is to establish whether this term increases or decreases in a distorted
equilibrium. If it decreases, then labor market clearing allows for a higher mass of firms to
exit in equilibrium.
Replacing Πη as a function of aggregates in the frictionless economy I get:
e =
L
d
fc−(η−1)
f
Π
η
η
η−1
−1
"
#
(θ − 1) η
fe
1+δ
h
i
Φ (τ ) + 1 −
+ fe
η
η−1
fc (1 − δ)
1−δ
Φ (τ ) − 1
η−1
In a frictionless equilibrium, total average labor demand is equal to:
ff
L
=
d
fc−(η−1)
f η
"
Π
#
fe
1+δ
(θ − 1) η
+1−
+ fe
η−1
fc (1 − δ)
1−δ
e <L
e f if and only if:
Therefore, I will get that L
d
d
fc−(η−1)
f
Π
η
η
η−1
−1
#
"
fe
1+δ
(θ − 1) η
h
i
Φ
(τ
)
+
1
−
+
f
e
η
η−1
fc (1 − δ)
1−δ
Φ (τ ) − 1
η−1
#
"
fe
1+δ
(θ − 1) η
<
Π
+1−
+ fe
η−1
fc (1 − δ)
1−δ
Cancelling terms, the condition becomes:
f η
η
η−1
−1
fc−(η−1)
h
η
Φ (τ )
η−1
h
i
−1
<
i
(θ−1)η
fe
+ 1 − fc (1−δ)
η−1
h
i
(θ−1)η
fe
Φ
(τ
)
+
1
−
η−1
fc (1−δ)
Cross multiplying terms and cancelling when possible28, I get:
fe
(θ − 1) + 1 −
fc (1 − δ)
!
<0
which leads to the result in the proposition that the inequality holds if and only if
fe
> θ (1 − δ)
fc
28There
are no negative terms in any side of the inequality, so I should not worry about changes in the sense
of the inequality
ENTRY, EXIT AND MISALLOCATION FRICTIONS
Then, from labor market clearing, it follows that
ef
M
L
= ed
f
M
Ld
which implies completes the proof.
44