Discussion Paper No. 1033

SIZE-DEPENDENT POLICIESAND EFFICIENT FIRM CREATION

Sakai Ando

June 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Size-Dependent Policies and Efficient Firm Creation∗
Sakai Ando†
June 27, 2018

Abstract
I study the welfare implications of size-dependent firm regulation policies (SDPs)
in the presence of entrepreneurial risks. Although SDP has been considered a sourceof misallocation, I show that, once entrepreneurial risks are taken into account, SDP
can improve efficiency. Quantitatively, I show that, based on French data, removing
the SDP leads to output and welfare loss by 1.5% and 1.3%, respectively, in opposition to the output gain reported by the previous literature that abstracts from risks.
Qualitatively, I solve an optimal non-linear SDP problem and show that the observed
SDP shares certain features with the optimal SDP. The analysis uncovers a novel tradeoff between the inefficiencies of the intensive and extensive margins. In extension, it
is shown that (1) whether SDPs improve efficiency depends on the level of financial
development and (2) capital accumulation and consumption-smoothing motive further
justify SDPs.
JEL: D52, D61, H21, J08, L11, L24, L51
Keywords: misallocation, firm creation, size-dependent policy, financial development,
incomplete markets∗

I thank Yu Awaya, Tania Babina, Pierre-Andre Chiappori, Andres Drenik, Matthieu Gomez, Tomohiro
Hirano, Harrison Hong, Takatoshi Ito, Atsushi Kajii, Jennifer La’O, Misaki Matsumura, Christian Moser,
Andreas Mueller, Emi Nakamura, Yuji Sakurai, Jose Scheinkman, Stephanie Schmitt-Grohe, Jon Steinsson,
Martin Uribe, Michael Woodford, Makoto Yano, Pierre Yared, and all participants in the seminars at Australian National University, Bank of Japan, City University of Hong Kong, Columbia University, Hitotsubashi
University, Kyoto University, Melbourne University, National Taiwan University, Peking University, Purdue
University, University of Tokyo, and Waseda University. This paper is awarded the 18th Moriguchi prize
from Osaka University Institute of Social and Economic Research.
†
Email: sa3016@columbia.edu.

11Introduction

Size-dependent policies (SDPs) that preferentially treat small firms are ubiquitous. For instance, in France, firms that hire more than 50 workers have to pay additional regulatory
costs, such as higher tax rates and more stringent labor regulations. Such SDPs naturally
create the bunching behavior in the firm size distribution as in Fig.1, since some big firms
rationally remain small to save regulatory costs. The literature has focused on the misallocation from the bunching behavior, reporting that the removal of SDPs leads to output gain
by .02 ∼ 4% (Gourio and Roys [2014], Garicano et al. [2016]).
In this paper, I argue that our understanding of SDPs can change drastically if we take
into account the fact that firm creation is risky. Specifically, I show that, once the analysis
incorporates uninsurable entrepreneurial risks, SDPs can generate higher output and welfare,contributing to net efficiency gains despite the remarkable bunching behavior. The key
observation is that, when entrepreneurial risks cause insufficient firm creation relative to thePareto efficient level, SDPs can mitigate the market failure by supporting firm creation. Thus,
SDPs balance the trade-off between the inefficiencies in the intensive (bunching) and extensive
margins (insufficient firm creation). This is the novel trade-off that this paper uncovers and isat the heart of the following quantitative and qualitative analyses that generate the opposite
policy implications to the previous literature.
The key assumption is that entrepreneurial risks deter firm creation. This assumption
is not only intuitive but also has been confirmed by several empirical studies. For instance,
Hombert et al. [2014] study the French unemployment insurance reform in 2002 and find apositive effect of insuring the downside risks of entrepreneurship. Gottlieb et al. [2018] argue
that the Canadian maternity leave policy reform mitigates the downside risks of losing safe
job options and increases entrepreneurship. In addition to these causal studies, we can also
observe the cross-sectional negative correlation between the downside risks and firm creation
as in Fig.2. Thus, in this paper, I take the assumption as given and focus on its implications.
To illustrate the trade-off and quantify the welfare consequence of SDPs, I extend thestandard occupation choice model a la Lucas [1978] and more directly Garicano et al. [2016]
by adding entrepreneurial risks. Specifically, the model has three features. First, firms’
production technology exhibits decreasing returns to scale. This is the same trick as thespan of control model a la Lucas [1978] and generates a well-defined number of firms inequilibrium, the concept necessary to discuss the efficiency of firm creation. Second andmost importantly, agents face uninsurable idiosyncratic entrepreneurial risks in the sense
that, when agents choose to become either entrepreneurs or workers, they are not sure about
their entrepreneurial productivities. Such downside risks lead to insufficient firm creation

2

400
0

200

number of firms

French firm size histogram

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

number of workers
Amadeus, All French firms in 2006

Figure 1: The number of firms by employment size in France. The rise and drop at theemployment size of 50 reflect the endogenous reaction of firms to the regulation threshold
50, above which firms are subject to higher tax and heavier labor regulations. The sample
includes all the firms in Amadeus 2006 with employment size between 31 and 69.
as a result of incomplete markets, opening up the scope for policy interventions. Finally,
entrepreneurs face SDPs modeled as the tax payment based on the number of employees. This
formulation is the same as Garicano et al. [2016] and parsimoniously creates the bunching
behavior by treating small firms preferentially.
The mechanism by which SDPs improve efficiency is based on the two general equilibrium
channels through which SDPs facilitate firm creation. The first channel is market clearing.
Specifically, since big firms pay regulatory costs under the SDPs, they hire fewer employees. The lower labor demand from big firms implies more firm creation since agents areeither entrepreneurs or workers. In other words, SDPs prevent big firms from absorbing
human resources that could otherwise be devoted to entrepreneurship. The second channel
is risk mitigation through wage decline. In particular, since big firms make fewer profits,
entrepreneurship is less attractive. To offset the resulting increase in the supply of salary
workers and reflect weaker labor demand from big firms, wage declines. This is good news
for small firms because they do not face regulations and can enjoy lower labor costs. As aconsequence, the profitability of small firms improves. From the ex-ante point of view, thismeans that entrepreneurs face lower downside risks since, even when businesses end up being
small or unprofitable, they can earn more than in the laissez-faire economy. As a result of
these channels, the economy with SDPs features more firm creation.
3

Correlation between risks and firm creation
Others

0

20

40

60

High income

0

20

40

60

80

0

20

40

60

80

Fear of failure rate
Entrepreneurs (%)

Fitted values

Global Entrepreneurship Monitor 2001 − 2016, 790 country−year pairs

Figure 2: A negative correlation between fear of failure rate and entrepreneurship. Fear of
failure rate is the percentage of 18 − 64 population perceiving good opportunities to starta business who indicate that fear of failure would prevent them from setting up a business.
Entrepreneurship is the percentage of the population of 18 − 65 years old who is eithera nascent entrepreneur or owner-manager of a new business. The plot pools yearly data
2001 − 2016 for all available countries from 28 to 76 depending on years. The definition ofhigh-income countries and others follow the categorization of the World Bank. Data: Global
Entrepreneurship Monitor.
The increase in the number of firms can raise output and welfare in the presence ofentrepreneurial risks. Intuitively, in the laissez-faire economy with entrepreneurial risks,marginal entrepreneurs produce and consume more on average for the compensation of risktakings. Therefore, SDPs that change the occupation of agents from workers to entrepreneurs
can generate positive net output gain, as long as the output loss from bunching is not toolarge. Accordingly, the higher output and the lower entrepreneurial risks can Pareto improve
the laissez-faire economy. Whether the efficiency gain from these channels outweighs the efficiency loss of bunching is a non-trivial question and therefore necessitates the quantification
exercise.
To quantify the welfare impact, I conduct the same counterfactual analysis as Garicano
et al. [2016] except that my model has entrepreneurial risks. Importantly, I use the same specification and the same French data so that I can isolate the pure implication of entrepreneurial
risks. The counterfactual analysis reveals that removing SDPs decreases output by 1.5% andwelfare in wage unit by 1.27%. The result is in sharp contrast to Garicano et al. [2016] thatreport that removing SDPs increases output by .02 ∼ 4% in risk-free settings.
4

In terms of implication, the result not only uncovers the novel trade-off behind SDPs but
also poses a caveat to the misallocation measurement literature in general including Hsieh andKlenow [2009]. Specifically, misallocation measurement exercises typically assume marginal
product equalization in the efficient benchmark and measure the deviation from it. My result
implies that using such benchmark can be misleading since it might be better to have somewedges among marginal products when the economy has uninsurable entrepreneurial risks.
In other words, the trade-off between the inefficiencies of the intensive and extensive margins
uncovered in the analysis highlights a limitation of the core logic of the typical misallocation
measurement that focuses only on the former.
To explore the novel trade-off, I study the non-linear SDP that optimally creates wedges
among firms. More concretely, I extend the piecewise linear taxation schedule used in thequantitative exercise to the set of arbitrary differentiable functions and solve the optimal
SDP problem using mechanism design techniques. The main result is that the optimal SDP
and the observed threshold type SDP share qualitative features; both of them (1) distortmedium-sized firms more than the smallest and biggest and (2) feature a larger number offirms compared to the laissez-faire economy. In addition to these analytical results, I also show
that, calibrated at the same parameter values as the quantification exercise, the optimal SDP
subsidizes small firms and taxes big firms. Thus, the observed SDP has efficiency-enhancing
properties.
Once the novel trade-off is established, I extend my analysis by examining its generality.
A natural question is whether entrepreneurial risks always justify SDPs. I argue that theanswer is negative, especially when the laissez-faire economy creates an excessive number offirms. As an example of excessive firm creation, I study the financial frictions that prevent
firms from expanding to their profit-maximizing sizes.
It is shown that removing SDPs improves efficiency even when there are entrepreneurial
risks if existing firms face severe financial frictions. To be concrete, one can imagine a developing country in which firms cannot expand their employment due to financial frictions.
Since firms cannot hire people, those unemployed have to do businesses on their own. As aresult, one can observe many small businesses in the economy including people engaged infood truck businesses on the streets. I model such intuition and show that severe financial
frictions can lead to too many firms compared to the Pareto efficient level. In this situation,
SDPs should be removed since the laissez-faire economy already generates an excessive number of firms in the first place. In fact, the efficiency gains are doubled under excessive firm
creation since removing SDPs fixes both the intensive and extensive margins.
The analysis of financial frictions implies that whether SDPs improve efficiency depends
on the level of financial development. In financially developed countries where entrepreneurial

5risks cause insufficient firm creation, SDPs should be kept, while in countries with immaturefinancial infrastructure and therefore excessive firm creation, SDPs should be removed.
Finally, I investigate the implications of capital accumulation. To isolate the implicationof inter-temporal decisions in a tractable dynamic environment, I extend the static modelby introducing Kreps-Porteus-Epstein-Zin preference (Kreps and Porteus [1978], Epstein and
Zin [1989]) and capital accumulation a la Krebs [2003]. These modeling techniques allow me
to derive a closed form solution despite the heterogeneous-agent incomplete market dynamic
model.
It is shown that consumption-smoothing motive provides a further justification for keeping
SDPs. In particular, I show that the laissez-faire economy creates insufficient firms, even if
agents are risk neutral. The idea is that, when agents prefer smooth-consumption, they wantto avoid asset fluctuation, so they do not want to take entrepreneurial risks. Such dynamic
trade-off discourages firm creation and therefore justifies SDPs. This result not only confirms
the robustness of the analysis in the static model but also highlights a force specific to the
dynamic environment.
An implication of the dynamic extension is that SDPs should be kept in an economy withpatient agents. This is because patient agents value future consumption more and therefore
have stronger consumption-smoothing motives. Theoretically, this reflects the observationthat consumption-smoothing motive is controlled not just by the inter-temporal elasticity of
substitution but also by the discount factor. Thus, other things being equal, in countries
with patient agents such as ones with the culture of patience or long life-expectancy, it ismore likely that removing SDPs exacerbates the market failure of firm creation.

1.1

LiteratureThis paper contributes to several strands of literature. First, it provides a novel insight into
the conventional wisdom about SDPs. Restuccia and Rogerson [2008], Guner et al. [2008]
and Garicano et al. [2016] measure the output gain from removing SDPs. While the detailsof the models are different across those papers, they share the same feature that the laissezfaire economy without SDPs is efficient. Hence, SDPs decrease output by construction. My
contribution is to show that the introduction of entrepreneurial risks might justify SDPs,
altering the understanding of SDPs being bad. Another important paper in this strand ofliterature is Gourio and Roys [2014], which conduct a counterfactual analysis using a firm
dynamics model with risky TFP and entry cost. Although they obtain an output loss fromremoving SDPs, the aggregate consumption and welfare increase, so their policy implicationis still to remove SDPs. In contrast, I show a case in which SDPs should be kept.
More broadly, my paper poses a caveat to the misallocation measurement exercises based6

on marginal product equalization (Hsieh and Klenow [2009], Hsieh et al. [2013]). In particular,
I show that, with uninsurable entrepreneurial risks, there is a trade-off between the efficienciesof the intensive and extensive margins, so the efficient benchmark does not necessarily feature
marginal product equalization.
The second strand of literature that I contribute studies the efficiency of firm formation.
Kihlstrom and Laffont [1979] and Kanbur [1981] are early references for risky firm formation.
These papers assume both risky occupation choice and non-contingent labor choice so theresulting market failure is not specific to the friction due to risks. To the best of my knowledge, my paper is the first to study the market failure specific to risky occupation choice.
For other environments, Mankiw and Whinston [1986] and Suzumura and Kiyono [1987] areearly references that study the optimality of the firm entry in strategic settings. Jaef [2012]
studies the optimality of entry in the firm dynamics model of Hopenhayn [1992]. All of thememphasize excessive entry, so the results are the opposite of mine.
The third strand of literature that this paper contributes to is the one on financial constraints and entrepreneurship as surveyed by Quadrini [2008] and Buera et al. [2015]. The
literature has studied the impact of financial frictions on the market equilibrium but has notstudied the impact of financial frictions on the efficiency, i.e., the difference between market equilibrium and efficient allocation. I study both in a simple model with a closed form
solution.
Finally, my paper contributes to dynamic macroeconomics modeling by offering a tractable
dynamic general equilibrium model with heterogeneous-agent, incomplete markets and risky
occupation choice. The trick behind the tractability is from Krebs [2003]. Toda [2015] andGottardi et al. [2016] study the efficiency of the descendants of Krebs [2003]. I extend theframework to include risky occupation choice.

2

Model

In this section, I present the baseline model. After discussing the existence and uniquenessof the equilibrium, I use an efficiency analysis to illustrate the market failure that the SDP
mitigates. In particular, I show that the laissez-faire economy without SDP generates aninsufficient number of firms if and only if there are entrepreneurial risks.
There is a continuum of ex-ante identical risk-averse agents indexed by i ∈ [0, 1]. Eachagent is endowed with one unit of indivisible labor that can be spent in running a firm or
working for a firm.
If agent i chooses to be a worker, she receives wage w independent of her entrepreneurial
productivity zi . If she chooses to be an entrepreneur, she observes her entrepreneurial pro7

ductivity zi and then decides the size of the firm measured by the number of employees n to
maximize the profit[π (zi , w) , n (zi , w)] = maxzi f (n) − wn

n≥0 


zi f (n) − wτ n − F

n≤N

(1)

n>N

where f is the production function, n (zi , w) is the associated policy function and (τ, F, N ) ∈[1, ∞) × R × R+ is the SDP modeled as the variable and fixed tax that entrepreneurs haveto pay when they hire more than N workers. Note that once the firm hires more than N
workers, the variable cost τ ≥ 1 applies to not just the net additional workers n − N but also
all the workers n. Therefore, firms have the incentive to shrink the size as in Fig.1 even if
the fixed component is negative F < 0 as long as the total tax payment (τ − 1) wN + F ispositive.
Given the payoffs of the two occupations, each agent i observes the signal si about theentrepreneurial productivity zi and chooses the occupation that maximizes her expectedutility. The joint distribution of the signal and productivity (s, z) is exogenously given anddenoted by G. Formally, each agent i solves
e (si , w) = arg max eE [u (π (zi , w)) |si ] + (1 − e) u (w)e∈[0,1](2)where u is the utility function, and E [·|si ] is the expectation with respect to the productivity
zi ∈ R+ conditional on the observed signal si ∈ R. If the agent chooses e = 1, she becomesan entrepreneur. Accordingly, e = 0 indicates that she becomes a worker.
I make two observations about the occupation choice problem. First, note that the choicevariable e, representing whether to become an entrepreneur, can take a continuous valuee ∈ [0, 1]. This formulation allows agents to take mixed strategies when entrepreneurshipand salary job are indifferent. Second, the occupation choice is risky because the agents
have to decide the occupation before observing the productivity zi . Since the salary job isrisk-free, depending on the realization of the productivity zi , entrepreneurs might consumeless than workers in equilibrium. In this sense, entrepreneurship involves downside risks.
Finally, the wage w clears the labor market
1−Ze (si , w) di =Ze (si , w) n (zi , w) di(3)

where the left-hand side is the aggregate labor supply and the right-hand side is the aggregate
labor demand from firms.
The following definition summarizes the description of the equilibrium as well as other8

objects of interests.
Definition 1. Fix the fundamentals (u, f, G) and the SDP (τ, F, N ). The set of wage,
occupation choice, and production decisions {w, e, π, n} is an equilibrium if it satisfies (1),(2) and (3). The number of firms and the aggregate output in the equilibrium is defined asφ=Ze (si , w) di, Y =Ze (si , w) zi f (n (zi , w)) di.

(4)

The laissez-faire economy is defined as the one with (τ, F ) = (1, 0).
Throughout the paper, I attach LF and SDP to the equilibrium objects whenever the
distinction of the laissez-faire economy and the economy with the SDP needs to be madeexplicit.
To ensure that the equilibrium is well-behaved, I make three assumptions on the fundamentals (u, f, G). First, utility and production functions (u, f ) are strictly increasing,strictly concave and satisfy the Inada condition. As will be clear, the strict concavity ofthe production function ensures the optimal level of firm creation. Second, the joint distribution of the signal and productivity G (s, z) is continuous, has bounded productivity
P (0 < zmin ≤ z ≤ zmax < ∞) = 1 for some (zmin , zmax ), and reflects positively informative
signals, i.e., the conditional distribution G (z|s) first-order stochastically dominates G (z|s0 )whenever s > s0 . The continuity allows the system to adjust continuously, the bounded productivity circumvents the complexity of infinite utility, and the positively informative signal
provides a normalization so that a higher signal is associated with a higher productivity.
Finally, I assume the signal and productivity (si , zi )i are drawn i.i.d. from G. The i.i.d.
assumption makes it possible to invoke the law of large numbers.
I call these assumptions (A). Under these assumptions, the existence and the uniquenesscan be guaranteed as stated in the next proposition. The proof clarifies the structure of theequilibrium.
Proposition 1. Suppose that the fundamentals (u, f, G) satisfy the assumptions (A). An
equilibrium exists and is unique almost surely.
Proof. Given the signal structure, the occupation choice follows a threshold strategy e (si ) =1si ≥s̄ . The “almost surely” qualification reflects the indeterminacy of optimal occupation
for the marginal entrepreneurs {i : si = s̄}. Given the individual occupation choice e (·), themarket equilibrium can be characterized by the equilibrium wage w and the threshold s̄ thatsatisfy the following indifference condition and the market clearing condition
E [u (π (z, w)) |s = s̄] = u (w) ,9(5)

E [n (z, w) 1s≥s̄ ] = Gs (s̄) .

(6)where Eq.(3) invokes the law of large numbers a la Uhlig [1996].
To show the existence and the uniqueness, note that, for a fixed SDP (τ, F, N ), the indifference condition (5) specifies a positive relationship between the wage w and the thresholds̄, while the market clearing condition (6) provides a negative relationship. Both of these
relationships are continuous since n (z, w) jumps at most at one point. Moreover, according
to the market clearing condition, w → ∞ as G (s̄) → 0, and w → 0 as G (s̄) → 1. Therefore,
the two loci must cross each other once.
The proof of the existence and the uniqueness highlights the two forces that govern therelationships between the wage w and the number of firms φ = 1−Gs (s̄). One can graphicallysee them in Fig.3. First, the indifference condition implies that if the wage increases, more
people find it attractive to become workers, so the number of firms declines. This is the labor
supply side intuition from the individual perspective and generates the negative relationshipbetween the wage and the number of firms. Second, the market clearing condition implies
that, if the wage increases, firms’ labor demand declines, so there will be fewer workers,
ending up with more firms. This labor demand side story gives the positive relationshipbetween the wage and the number of firms. Since these two forces bring the system in the
opposite directions, there exists a unique equilibrium irrespective of the specific form of the
productivity and the signal structure G.

2.1

Riskiness of firm creation

Since the key departure from the literature is entrepreneurial risks, I devote this section to
discuss the riskiness of entrepreneurship in the model.
The riskiness of entrepreneurship is defined by how informative the signal s is about the
productivity z. If the signal s is very informative about entrepreneurial productivity z, agentscan accurately forecast future profits when they become entrepreneurs. In contrast, if agents
have uninformative signal s, becoming entrepreneurs involves substantial downside risks.
One way to define informativeness mathematically is to use the information sets. If
the information sets of the two random variables are identical, i.e., the sigma algebras are
identical σs = σz, there are no entrepreneurial risks. In contrast, if the signal is independentof the productivity σs ⊥⊥ σz, agents are called to face full risks. Although this definition has
generality, it does not come with a natural framework to think about the intermediate cases.
Thus, I use a weaker condition to define riskiness.

10

indifference
market clearing

0

1

Figure 3: Equilibrium system on (w, φ) plane. The indifference condition gives a negative
relationship, while the market clearing condition generates a positive relationship.
Definition 2. The riskiness of entrepreneurship for those who observe s is defined as theconditional variance V (z|s) ∈ [0, V (z)]. V (z|s) = 0 corresponds to no risks, while V (z|s) =V (z) > 0 corresponds to full risks.
One can use weaker conditions, defining no risks as the signal structure where each agent
knows the best occupation for sure, i.e., P (π (z, w) ≥ w|s) ∈ {1, 0} for all s, and full risks
as the signal structure where the signal is not informative about the occupation choice, i.e.
P (π (z, w) ≥ w|s) = P (π (z, w) ≥ w) for all s. This definition allows intuitive intermediatecases for some signal structure and is adopted in Appendix E.
In any case, the economy with no risks is characterized by (z̄, w) that satisfies the following
indifference and market clearing conditions
π (z̄, w) = w, E [n (z, w) 1z≥z̄ ] = Gz (z̄) ,(7)and the economy with full risks is characterized by (w, φ) that satisfies another set of indifference and market clearing conditions
Eu (π (z, w)) = u (w) , φEn (z, w) = 1 − φ.

(8)

One can see that, without risks, the utility function disappears from the equilibrium system.
11

With full risks, the equilibrium wage is determined by the indifference condition, and thenumber of firms is determined by the market clearing condition. Graphically, the full-riskcase requires the indifference condition in Fig.3 to be horizontal for the interior number offirms. In terms of the literature, the two cases correspond to Lucas [1978] and Kanbur [1979],
and Garicano et al. [2016] use the former to estimate the efficiency loss due to the SDP.

2.2

Market failure of the laissez-faire economy

Given the riskiness, I can discuss the efficiency of the laissez-faire economy. Once the market
failure of the laissez-faire economy is understood, I describe the mechanism by which the
SDP improves efficiency in section 3.
To discuss the efficiency, I define the planner’s problem who can choose allocations at
each state but faces the same informational constraints as individual agents. Formally,a state of the economy is a collection of the productivity realizations for all agents ω ={zi }i ∈ Ω = [zmin , zmax ][0,1] . The planner chooses the contingency plan of the consumption c = {ci (ω)}i∈[0,1],ω∈Ω , the employment schedule when agents become entrepreneurs{ni (ω)}i∈[0,1],ω∈Ω , and the allocation of occupation e = {e (si )}i . Note that the consumption and employment are contingent on each state, but the occupation has to be measurablewith respect to the signal s since that is the informational constraint agents face. Such informational constraint is standard in the literature of efficiency analysis with informational
frictions. (Angeletos and Pavan [2007])
Given a measurable Pareto weight i 7→ Λi , the planner maximizes the weighted sum of
the expected utilityZ
U (c, Λ) = EΩ u (ci (ω)) dΛi(9)where EΩ denotes the expectation over the state Ω constructed from Gz .1 Since the economy
consists of heterogeneous agents differentiated by the observed signal {si }i , I do not take astand on the Pareto weight such as utilitarian or Rawlsian, and instead, focus on the production side. Note that since the economy does not have aggregate uncertainty and production
in one firm is independent of the productivity of other firms, I can restrict the employment
schedule to be measurable with respect to productivity, n = {n (z)}z . Furthermore, since
the planner maximizes output, the allocation of occupation is a threshold rule e (si ) = 1si ≥s̄
without loss of generality. This is equivalent to choosing the number of firms φ = 1 − Gs (s̄).
Hence, the planner’s problem can be defined as follows.
Definition 3. Fix the fundamentals (u, f, G). A set of consumption, employment schedule,
1

The formal construction relies on Kolmogorov extension theorem.

12

output and number of firms {c, n, Y, φ} is Pareto efficient if it solves

max U (c, Λ) s.t.

c,n,Y,φRci (ω) di = Y




−1

Y = φE [zf (n (z)) |s ≥ Gs (1 − φ)]φ + φE [n (z) |s ≥ G−1 (1 − φ)] = 1(10)∀ω ∈ Ω

s

for some Pareto weight Λ.2
The first constraint requires that individual consumption adds up to the aggregate output.
Since the economy does not have aggregate uncertainty, it implies that the planner can
provide full insurance, ci (ω) = c (ω 0 ) for all ω, ω 0 ∈ Ω. An immediate implication is that theplanner chooses employment schedule n and the number of firms φ to maximize output. The
second constraint illustrates the production technology as a function of (n, φ). There are φfirms and each of them produces E [zf (n (z)) |s ≥ G−1
s (1 − φ)] on average. The conditional
expectation reflects the selection of entrepreneurs s̄ = G−1
s (1 − φ). The third constraint
describes the human resource constraint. φ agents become entrepreneurs and each of themhires E [n (z) |s ≥ G−1
s (1 − φ)] workers, which have to add up to the total population 1.
In general, the planner’s solution depends on the Pareto weight Λ. The production side(n, Y, φ), however, can be uniquely determined independently of the Pareto weight.
Proposition 2. Suppose that the fundamentals (u, f, G) satisfy the assumptions (A). Then,
there is a unique interior solution φP ∈ (0, 1) to the planner’s problem (10).
Proof. See Appendix A.1.
To understand the planner’s trade-off that pins down the interior solution, let Y (φ)
denote the aggregate output as a function of the number of firms, obtained by maximizingout employment schedule {n (z)}z from the last two constraints of (10). To be concrete, let
the production function be Cobb-Douglas f (n) = nα with α ∈ (0, 1). The aggregate output
Y (φ) takes the following formh1

Y (φ) = A (φ) φ1−α (1 − φ)α , A (φ) = E z 1−α |s ≥ G−1
s (1 − φ)

i1−α

.

(11)

One can see the two channels through which the number of entrepreneurs impacts the aggregate output.
The first channel is the average productivity A (φ). It describes a typical concern about entrepreneurship promotion. If the number of entrepreneurs increases φ %, since the marginal
2

Although all variables are functions of states, the assumption of no aggregate uncertainty saves thenotation for aggregate variables.

13

entrepreneur is less productive G−1
s (1 − φ) &, the average productivity declines A (φ) &.
However, the average productivity A (φ) is not the main technological trade-off that pins
down the unique interior solution. After all, the average productivity is bounded A (φ) ∈[zmin , zmax ] and does not even react to the number of firms φ when the signal s is uninformative about the productivity z.
The second channel is the allocation of occupations φ1−α (1 − φ)α . This is the key tradeoff that pins down the unique interior solution. When most agents are workers φ → 0, thesmall number of firms employs a large number of workers. Due to the decreasing returns
to scale assumption on the production technology f , the worker’s marginal product is low.
In this case, dividing the firm into two and shifting some of the workers to higher marginal
product activities raises the aggregate output. However, such firm creation is not costlesssince it requires a decrease in the number of workers engaged in production. In the limit,
if everyone becomes an entrepreneur φ → 1, no workers make production, resulting in 0
aggregate output despite the worker’s marginal product being ∞.
The second channel is present even if the productivity distribution Gz is degenerate or
the signal is not informative s ⊥⊥ z. In these cases, the optimal number of firms is φP = 1−α.
It also differentiates the occupation choice model from the firm dynamics models following
Hopenhayn [1992], in which labor supply is fixed and no entrepreneurs are needed to create
additional firms.
Now I am ready to state the efficiency result. Let φLF and φP be the number of firms in
the laissez-faire economy and the planner’s solution. The following proposition states that
entrepreneurial risks create the market failure of insufficient firm creation. Note that theassumptions on the riskiness G (z|s) affect both the market equilibrium and the planner’s
solution.
Proposition 3. Suppose that the fundamentals (u, f ) satisfy the assumptions (A).
1. If there are no entrepreneurial risks, i.e., V (z|s) = 0 for all s, the laissez-faire economy
is Pareto efficient.
2. If there are entrepreneurial risks, i.e., V (z|s) > 0 for all s, the laissez-faire economy is
not Pareto efficient. In particular, the laissez-faire economy creates insufficient numberof firms φLF < φP .
Proof. See Appendix A.2.
The result is not surprising if one notices that the economy without risks has complete
markets so the analogy of the first welfare theorem holds, while the economy with risks

14features incomplete markets and uninsurable idiosyncratic risks. As is standard in the literature of incomplete markets, the result can be extended to constrained inefficiency as shownin Appendix F.1 and discussed in details in Ando and Matsumura [2017].
The intuition of the results is transparent if one considers a marginal increase in firms
and aggregate output. Without risks, the entrepreneur with productivity z produces π (z, w)
and workers produce w. If one worker changes the occupation to entrepreneur, the net
output gain is π (z, w) − w. In the laissez-faire economy (7), the marginal entrepreneurs are
indifferent to the workers π (z̄, w) = w. Therefore, increasing firms results in zero net output
gain π (z̄, w) − w = 0. With risks, the marginal entrepreneurs produce more than workers on
average in the laissez-faire economy E [π (z, w) |s = s̄] > w. This is because, for risk-averse
agents to become entrepreneurs, they have to be able to produce and consume more thanworkers on average. As a result, switching the marginal workers into entrepreneurs raise
aggregate output by E [π (z, w) |s = s̄] − w > 0.
In the next section, I explain why the SDP can increase the number of firms.

3

MechanismI have shown in the previous section that Pareto improvement is possible only if there areentrepreneurial risks. This section uses the full-risk model to describe how the SDP could
make Pareto improvement in three steps. The same argument carries on for the intermediaterisk case, but I keep the exposition minimal for simplicity. At the end of the section, I provide
discussions of the mechanism.

3.1

SDP can increase the number of firms

The first step is that the SDP can increase the number of firms by (1) preventing big firms
from absorbing potential entrepreneurs and (2) reducing the entrepreneurial risks throughgeneral equilibrium.
To see the first channel, note that the SDP imposes regulatory costs on big firms, sothey hire fewer workers. Since agents are either entrepreneurs or workers, the decline in thenumber of workers implies the number of firms increases in equilibrium. This is a partial
equilibrium impact with fixed wage and is illustrated as the shift of the market clearing
condition in Fig.4.
The second channel relies on the wage decrease in general equilibrium. To see this, notethat the SDP makes entrepreneurship less attractive in partial equilibrium, i.e., π SDP z, wLF ≤π LF z, wLF for all z ∈ [zmin , zmax ]. Since the inequality is strict for big firms, the indifference

15

laissez-faire
SDP
0

1

Figure 4: Indifference condition and market clearing condition.
condition (8) implies that the wage declines wSDP < wLF . Intuitively, since entrepreneurshipis less attractive, agents want to become workers. As a result of the increase in worker supply,
the wage declines. This force is illustrated as the shift of the indifference condition in Fig.4.3
The wage decline mitigates the downside risks of entrepreneurship. This is because,from the ex-ante point of view, even when businesses end up being small or unprofitable,
entrepreneurs do not have to pay the regulatory costs and at the same time can enjoy cheaperlabor costs. The mitigation of the downside risks is illustrated in the right figure of Fig.5. In
partial equilibrium, the profit in the laissez-faire economy is higher for all productivity levels.
However, such profit shift makes entrepreneurship less attractive. To regain equilibrium, thewage has to decline. As a result, the small firms that do not face regulation can hire workers
cheaply, resulting in higher profit.
Both direct regulations on big firms and indirect subsidies to small firms contribute tofirm creation. The former prevents big firms from absorbing potential entrepreneurs, and thelatter reduces the downside risks of entrepreneurship.
3

When the signal s is informative about the productivity z, the lower labor demand from big firms also
pushes down the wage. Mathematically, the indifference condition is not horizontal, so the shift of the market
clearing condition itself causes a wage decline.

16

laissez-faire
SDP PESDP GElaissez-faire
SDP PESDP GEzz

Figure 5: Employment schedule n (z, w) and profit π (z, w) for the laissez-faire economy
and the economy with the SDP. PE denotes partial equilibrium where the wage is fixed atlaissez-faire level wLF . GE means general equilibrium where the wage is wSDP .

3.2

Increase in the number of firms can raise output

The increase in the number of firms can lead to output increase. The intuition can beobtained by the perturbation argument. Suppose the government implements a small valueof SDP around the laissez-faire T = (τ, F ) ≈ (1, 0). Appendix B shows that the marginal
impact on the aggregate output is∂T Y = ∂T φ · (Eπ (z, w) − w) .

(12)

The expression illustrates that the policy impact around the laissez-faire economy is themultiplication of how many firms the policy increases ∂T φ and how much additional output
each firm produces Eπ (z, w) − w. Intuitively, since entrepreneurs produce Eπ (z, w) andworkers w, if an agent changes the occupation from worker to entrepreneur, the net gain is
Eπ (z, w) − w on average.
The former ∂T φ is positive from the previous section 3.1. The latter term Eπ (z, w) − w is
also positive since, in the laissez-faire economy, risk aversion requires entrepreneurs to produce
and consume more in equilibrium for the compensation of the risk-taking. Mathematically,17this is a consequence of the Jensen’s inequality
Eu (π (z, w)) = u (w) ⇒ Eπ (z, w) > w(13)

Thus, the SDP that increases the number of firms can increase the aggregate output.
Although I have used the full-risk model to explain the mechanism, the same argument
holds for all the intermediate cases. As long as there are some risks, the marginal entrepreneurs produce and consume more than workers for the compensation of the residualrisks
E [π (z, w) |s = s̄] > w.
(14)
In fact, even if I extend the model by assuming workers are risky, as long as entrepreneurs
are riskier than workers, the same logic still applies.
However, the output does not increase if there are no risks. When there are no risks, themarginal entrepreneurs consume the same amount as the workers π (z̄, w) = w. As a result,even though the SDP increases the number of firms, it does not lead to output increase. In
this sense, risks are essential for understanding the policy implications.

3.3

Increase in output can lead to Pareto improvement

The increase in output can lead to Pareto improvement if and only if the government returns
the tax revenue. To see this, note that, since agents are ex-ante identical in the full-risk
model (8), the welfare can be measured by the worker’s consumption. As a result, if thegovernment throws away the tax revenue, all agents are worse off due to the lower wage
wSDP < wLF . However, if the government collects enough tax revenue and return it toagents, the disposable wage can increase. How much of the SDP generates tax revenue
or ends up being non-pecuniary labor regulation is an empirical question. I show in thequantification section 4 that only a small amount of tax revenue is sufficient for the SDP tomake Pareto improvement.
In addition to the aggregate output increase, the SDP also improves welfare throughrisk reduction. This channel is not specific to the full risk model but is also at work in theintermediate risk case. However, when agents are heterogeneous, the welfare criterion has toreflect the evaluation of inequality. If the welfare criterion values equality, the SDP raises
welfare through both risk and inequality reduction.
Note that each of the above three steps involves non-trivial assumptions and necessitates
the quantification exercise in section 4. For the first step, the increase in the number offirms hinges on the assumption that the wage does not decline too much. If wage declines

18

to w ≈ 0, firms can hire many workers, so the number of entrepreneurs decreases. In
other words, the general equilibrium impact through wage cannot be larger than the partial
equilibrium impact. For the second step, the output increase is discussed around the laissezfaire economy. This is to illustrate the intuition of the efficiency gain. For the SDP farfrom the laissez-faire, there is an efficiency loss due to the bunching. Therefore, whether the
output increases or not depends on the balance of the two forces and has to be determined
numerically. For the third step, the Pareto improvement requires the government to raise
enough revenue. Although the SDP increases output and reduces risks for each entrepreneur,
the number of entrepreneurs also increases. Since the entrepreneurs need to consume more on
average to obtain the same utility as workers, whether the SDP can increase enough output
to Pareto improve all agents is not obvious. As shown in section 4, it turns out that all ofthese assumptions are satisfied under the standard parametric specifications.

3.4

Discussion about the mechanism

Before moving to the quantification exercise, I provide critical discussions about three featuresof the mechanism.
First, since the mechanism involves the occupation change from workers to entrepreneurs,
one might be concerned about unemployment. Although the model does not address unemployment, I make two heuristic observations. First, there are many countries that have both
SDPs and low unemployment rates. Therefore, at least in the long run, SDPs and unemployment can be considered separate issues. Second, the thought experiment that I am interested
in is the removal of the SDP. Thus, the relevant trade-off is the lower unemployment ratein the short run and the lower aggregate output in the long run. Therefore, the social costassociated with unemployment is not an issue in both the long and short run in the current
context.
Second, the output increase depends on entrepreneurs producing more on average than
workers in the laissez-faire economy, i.e., Eπ LF z, wLF > wLF .4 Hamilton [2000] andMoskowitz and Vissing-Jorgensen [2002] argue that entrepreneurs earn less than workers
on average. Their studies have generated a literature that tries to explain why people choose
to become entrepreneurs in the first place (e.g. Vereshchagina and Hopenhayn [2009]). Morerecent papers, however, revisit the issue and argue that entrepreneurs indeed earn more thanworkers on average. For instance, Levine and Rubinstein [2017] argue that the definition ofentrepreneurs used in the literature may not be appropriate. If entrepreneurs are defined toSince many countries adopt SDPs, it is not easy to empirically test whether Eπ LF z, wLF > wLF holds
in the unobservable laissez-faire economy.
That being said, we can check if a sufficient condition is empiricallysatisfied, assuming Eπ SDP z, wSDP > wSDP implies Eπ LF z, wLF > wLF .
4

19

be incorporated self-employed, the mean return is much higher than salaried workers. Manso[2016] points out that the estimations in the literature might be biased due to the usage ofcross-sectional data to compute the mean of entrepreneurial earnings. Manso [2016] reports
that, once the option value of experimentation is incorporated, the mean lifetime earnings ofentrepreneurs are higher than those of salary workers in the U.S. Recognizing such debate
on the measurement of entrepreneurial return, I explore the implications of the standard
occupation choice model in this paper.
Finally, the Pareto improvement is based on the tax return. Since many SDPs are nonpecuniary such as working hours regulations, there might be no tax revenue in reality. In
the next quantification section 4, I show that, even when all the tax revenue is thrown away,
the economy with SDP generates higher output than the laissez-faire economy. In terms ofwelfare, I show in Appendix C that the economy with SDP generates higher welfare than thelaissez-faire economy with the same amount of non-distortionary regulation costs.

4

Quantification

This section quantifies the welfare impact of removing the SDP. The main result is thatremoving the SDP leads to lower output and welfare. The result of output decline is the
opposite of the previous literature and highlights the potential importance of the channel
described in section 3.
To isolate the role of entrepreneurial risks, I closely follow the setup of Garicano et al.
[2016]. In particular, I impose the parametric assumptions on the fundamentals, with utility
and production functions being constant relative risk aversion and Cobb-Douglasc1−γ − 1, γ > 0, f (n) = nα , α ∈ (0, 1)
u (c) =
1−γ

(15)

and the distribution of the productivity being the power law
gz (z) =1 − βz−βz,
1−βz
1−βz zzmax − zmin

zmin ≤ z ≤ zmax .

(16)This productivity distribution leads to a broken power law of the firm size distribution as
detailed in Appendix D.
I focus on the full risks model (8) and relegate a specific intermediate signal case inAppendix E. In this way, the quantitative exercise in this section is the exact opposite
polar case of Garicano et al. [2016] with the same set of fundamentals. Therefore, one
can transparently see the implications of the entrepreneurial risks without being affected by20

the specific choice of the signal structure.
Formally, the economy with the SDP (τ, F, N ) and the non-distortionary income tax t
has the equilibrium wSDP , φSDP that satisfies the indifference condition
Eu ((1 − t) π (z, w)) = u ((1 − t) w) ,(17)

profit maximization (1), and the market clearing condition in (8). The income tax t is nondistortionary because, when the utility function is CRRA, 1 − t drops out of the equilibrium
system. As a result, all the equilibrium objects except for welfare are independent of theincome tax rate t. To determine the welfare, I set the income tax rate t = t (τ, F ) by
imposing the balanced budget constraint
φSDP E

hn





o

(τ − 1) wSDP n z, wSDP ; τ, F + F 1n(z,wSDP ;τ,F )>N
n





i





o

+ t φSDP Eπ z, wSDP ; τ, F + 1 − φSDP wSDP = 0. (18)
I compare the economy with SDP against the counterfactual laissez-faire economy where(t, τ, F ) = (0, 1, 0). Specifically, I compare aggregate output, welfare, number of firms,
and wage in the two economies. Note that the welfare is given by the disposable income(1 − t) wSDP and wLF .
There are two differences from Garicano et al. [2016]. The first is the existence of entrepreneurial risks, and the second is the welfare criterion. The welfare criterion of Garicano
et al. [2016] is aggregate output minus tax revenue. To make the comparison consistent, I also
report output minus tax revenue in section 4.2. An alternative welfare specification assuming
all the SDP being non-pecuniary is available in Appendix C. Irrespective of the specificationof the welfare criterion, all other equilibrium objects (Y, w, φ) are directly comparable.

4.1

Parameter values

In this section, I specify the parameter values for quantification. The main takeaway is that I
can use the same parameter values as Garicano et al. [2016], and therefore all the differencesin the results are driven by the existence of entrepreneurial risks, not other elements such as
the data used and specifications of the model.
Table 1 summarizes the choice of the parameter values based on the same French data
used by Garicano et al. [2016]. Note that the fixed tax F is negative. This does not mean
the firms at size N get subsidies. Since the variable tax τ applies to all workers, the tax
payment for firms at N is positive (τ − 1) wN + F = .186w > 0.
The main parameter of interest is the SDP (τ, F, N ). The threshold N is determined21

parametervalue

explanation

γ

2

CRRA parameter

α.8

Cobb-Douglas parameter

τ

1.023

marginal tax

F/w

−.941

normalized fixed tax

βz5

power law parameter of Gz

zmax

8.07

productivity upper bound

zmin

1.4

productivity lower bound

Table 1: Parameter values for counterfactual analysis. All parameters are taken from Garicano et al. [2016] except for (γ, zmin ). Wage w is not identified so is normalized to w = 1
without loss of generality.
by the law so there is no other choice than N = 49. The variable and fixed components(τ, F ), as pioneered by Garicano et al. [2016], can be estimated jointly with the power law
parameter βz by using the data of firm size distribution. To be more concrete, I derive theequilibrium firm size distribution n 7→ Gn (n; τ, F, βz ). This firm size distribution follows
a broken power law and its derivation is given in Appendix D. The maximum likelihood
method then fits the empirical firm size distribution data {ni }i by choosing (τ, F, βz ) andother auxiliary parameters. In the estimation, I follow Garicano et al. [2016] to truncate thedata and likelihood function (n ≤ 10 and n ≥ 10, 000) so that the estimation is not impacted
by the two tails that do not fit power law well.
In principle, since I use a different model with entrepreneurial risks, I need to recollect thesame data and redo the same estimation. However, I can use the same estimates as Garicano
et al. [2016] because the implied firm size distributions after truncation are identical. To
see this, recall that the only difference in the equilibrium firm size distributions of the twomodels (7) and (8) is whether low productivity agents become entrepreneurs or not. With norisks, only high productivity agents {i : π (zi , w) ≥ w} become entrepreneurs, while with fullrisks, all productivity levels are possible. When the truncated distribution does not capture
the low productivity region, the fact that the broken power law is invariant to truncation
implies that the two models generate an identical truncated firm size distribution.5
Other parameters follow the calibration and normalization of Garicano et al. [2016]. For
instance, the equilibrium wage is unidentified so is normalized to be 1. The parameter that5

The likelihood function in Garicano et al. [2016] also incorporates measurement error. Although this complication might potentially create discrepancies between the estimates, I show they are numerically negligible
in Appendix D using Amadeus data and the models with and without entrepreneurial risks.

22

Equilibrium objects

Full risks

No risks

Concepts

ln Y LF − ln Y SDP

−1.5%

.02%

outputln φLF − ln φSDP

−8%

−7.2%

number of firms





−.3%

1.3%

output - tax

no−1.25%

NA

welfare

.002%

1.8%

wage

ln Y LF − ln Y SDP − tax
ln wLF − ln (1 − t) wSDP
ln wLF − ln wSDP

Table 2: Results of the counterfactual analysis. The column “Full risks” is the result of thecounterfactual analysis using full-risk model. The one with “No risks” is the result of Garicano
et al. [2016]. Welfare in the model without risks is NA because agents are heterogeneous,so the welfare depends on the Pareto weight. Specifically, big firms and workers lose, whilesmall firms win.
is not in Garicano et al. [2016] is the risk aversion γ. I choose the standard number γ = 2
and discuss the robustness of the results to it in section 4.2. Other parameters (α, zmin , zmax )also follow Garicano et al. [2016]. For the details, see Appendix D.

4.2

Results

The results of the counterfactual analysis are summarized in table 2. The first column shows
the results of the counterfactual analysis based on my model with full entrepreneurial risks.
The second column is a copy from Garicano et al. [2016] which are based on the model withno entrepreneurial risks.
The first row, which is the main result, shows that the economy with SDP produces thehigher output than the laissez-faire economy. This might be surprising if one just sees that themarginal products equalize among all firms in the laissez-faire economy, yet the laissez-faireeconomy produces less than the economy in which the SDP creates wedges in the marginal
products. Fig.6 summarizes the intuition. The red line represents the production possibility
frontier of the economy with the SDP Y SDP (φ) as a function of the number of firms φ. For
each fixed number of firms, the removal of the SDP improves production efficiency so the
production possibility frontier of the laissez-faire economy is located above. Thus, for a given
number of firms, removing SDP does improve resource allocation. However, removing the
SDP exacerbates the market failure of firm creation. In particular, as discussed in section 3,
big firms suck up human resources from entrepreneurial activities and the entrepreneurshipbecomes riskier. In consequence, the number of firms declines as shown in the second row.
What the first row shows is that this efficiency loss in the extensive margin can quantitatively

23

Y( ): aggregate output

LF

SDP

PPF laissez-faire
PPF w. SDP: number of firms
Figure 6: Aggregate output Y (φ). Points A, B, and C are the counterfactual allocation
without the SDP, the allocation in the economy with the SDP and the efficient allocation.
outweigh the gain in the intensive margin.6
The third row of table 2 highlights the implication of the entrepreneurial risks in a stronger
sense. It shows that, even if the laissez-faire economy is given the tax advantage, the economy
with the SDP still produces more when there are entrepreneurial risks. In fact, this output
concept is used as the main welfare criterion in Garicano et al. [2016]. In this sense, myresult is the opposite of Garicano et al. [2016].
The fourth row of table 2 shows that the welfare is also higher in the economy with the
SDP. As discussed in section 3, the result implies that the SDP increases the aggregate output
by a non-trivial amount. Quantitatively, the tax revenue
T = φSDP E

hn





o

(τ − 1) wSDP n z, wSDP ; τ, F + F 1n(z,wSDP ;τ,F )>N

i

(19)

amounts to 1.14% of aggregate output in the current setting. Thus, the result highlights thefirst order importance of the channel highlighted in this paper.
The fifth row confirms the decline in wage. As discussed in section 3, this is an implicationof the specific form of the SDP (τ, F ) and therefore it is not necessarily a robust implicationof SDPs. In fact, I show in section 5 that, if the SDP is allowed to be an arbitrary non-linear
6
This is not the case with Garicano et al. [2016] because the laissez-faire economy is at the top of the
production possibility frontier when there are no risks. Therefore, no matter how the number of firms changes,
the aggregate output always declines.

24

function n 7→ T (n), wage can actually increase. Hence, even if the government throws away
the tax revenue, the SDP can achieve Pareto improvement.
Finally, I argue these results are robust to the change in the risk aversion parameter γ. For
instance, the break-even point for the output Y SDP = Y LF and welfare (1 − t) wSDP = wLF
is γ = .01. This small risk aversion is not surprising if one notes that the output gain fromremoving the SDP in the model with no entrepreneurial risks is only .02%. Similarly, thebreak-even point for Y SDP − tax = Y LF is γ = .7, still way below the standard parametervalue γ = 2.

5

Optimal SDP

The analysis so far has demonstrated that an economy with the threshold type SDP (τ, F ) can
be better than the laissez-faire economy. In other words, an economy with wedges amongfirms’ marginal products can lead to a better outcome than the one with equal marginal
products. A natural question is “what are the optimal wedges implied by SDPs and is thethreshold type SDP (τ, F ) similar to the optimal SDP?” To answer this question, I study
the optimal non-linear SDP and show that the observed threshold type SDP (τ, F ) sharesqualitative features with the optimal SDP.
The optimal SDP is defined as the solution to an optimal taxation problem. Let n 7→ T (n)be the tax imposed on the firms that hire n workers. The government is benevolent andmaximizes the welfare
U = φEu (π (z, w)) + (1 − φ) u (w) .
(20)
Although the expression is the same as utilitarian welfare, any Pareto weight leads to thesame welfare function since the full-risk economy consists of homogeneous agents.
The government faces the market equilibrium constraints and the budget constraint.
The market equilibrium constraints consist of the two equations (8) and the firm’s profit
maximization problem[π (z, n) , n (z, w)] = max zf (n) − wn − T (n) .
n≥0

(21)

Note that the threshold SDP (τ, F ) is a special case where T (n) = {(τ − 1) wn + F } 1n>N .
The budget constraint takes the following formφET (n (z, w)) = T(22)

where T ≥ 0 is exogenously given. When T = 0, the government budget is balanced. Later,25

I conservatively set T > 0 and investigate whether SDPs improve welfare even if resources
are taken out of the economy.
Formally, the optimal taxation problem can be formulated as follows.
Definition 4. Given the budget T ≥ 0, the government chooses (w, φ, π, n, T ) to maximize
wage w subject to (8), (21), and (22).
The optimal SDP problem has a similar structure as Mirrlees [1971] except that it hasa general equilibrium wage. I reformulate it as a mechanism design problem and solve theproblem within the set of differentiable functions.
Proposition 4. The solution to the optimal SDP problem can be obtained as the solution tomaxw,{n(z),π(z)}zw s.t.


Eu (π (z)) − u (w) = 0






π 0 (z) = f (n (z))
n0 (z) ≥ 0





E [zf (n (z)) − wn (z) − π (z)] − T (1 + En (z)) = 0

.

(23)

Proof. See Appendix A.3.
The first constraint is the indifference condition. The second is the envelope condition of
the truth-telling constraint, and the third is the monotonicity constraint that corresponds tothe second order condition. The last constraint is the government budget. Mathematically,
this is an optimal control problem.
It turns out that the optimal SDP and the observed threshold type SDP share somequalitative features. To see it, let (wo , φo , no , π o , T o ) be the solution to the optimal SDP
problem.
Proposition 5. Suppose that the fundamentals (u, f, Gz ) satisfy the assumptions (A), there
are risks V (z) > 0, and the solution does not have bunchingd on (z) > 0, ∀z ∈ (zmin , zmax ) .
dzThen, the optimal SDP distorts medium-sized firms, i.e., for all z ∈ (zmin , zmax ),wo = zmin f 0 (no (zmin )) = zmax f 0 (no (zmax )) < zf 0 (no (z)) .
For a small budget T ≥ 0, the optimal SDP increases the number of firms φo > φLF .
Proof. See Appendix A.4.
26

2.5

60

estimated SDP
optimal SDP
laissez-faire wage

40
20

2

0
-20
1.5

-40
-60

wo
-80
-100
1

estimated SDP
optimal SDP

-120
-140
0
10

z

1

2

10

10

3

10

n

Figure 7: The left panel shows the marginal product as the measure of distortion. The rightpanel shows the optimal SDP as the percentage of the before-tax profit. The wage in theeconomy with the estimated threshold SDP is normalized to be wSDP = 1.
Recall that, as Fig.5 suggests, the bunching firms are most distorted in the sense that
their marginal products are higher than other firms. Moreover, in section 3, we have seenthat the threshold type SDP increases the number of firms. Proposition 5 states that the
optimal SDP shares these features under fairly weak conditions.
To visualize this point, I show a numerical solution in Fig.7. The exercise is based on thesame parameterization as section 4, i.e., (15) and (16), the same parameter values as in table
1, and the same tax revenue T > 0 as (19). Under these conditions, it is straightforward tocheck that no bunching condition is satisfied.
The distortion, measured by the deviation of marginal product equalization, can be seenin the left panel of Fig.7. One can see that, under the optimal SDP, the marginal products
are highest in the middle. This is also the case with the threshold type SDP (τ, F ), under which the bunching firms with the highest productivity sup {z : n (z, w) = N } are most
distorted. One way to understand this property of the optimal SDP is to note the role ofeach firm. Highest productivity firms are socially important since they produce goods efficiently. Lowest productivity firms are also important since they represent the downside riskof entrepreneurship. Thus, the distortion is concentrated on the middle firms.
The left panel of Fig.7 also shows that the wage is higher under the optimal SDP than thethreshold SDP by more than 30%. Therefore, the wage level under the optimal SDP is higher
than the laissez-faire economy wSDP < wLF < wo . This is because the optimal SDP subsidizes
27

5

10

104

estimated SDP
optimal SDP
laissez-faireestimated SDP
optimal SDP
laissez-faire
optimal SDP before-tax

103

2

10

101

70
60
50

100

10-1

40

z

z

Figure 8: Employment and profit schedule under the estimated threshold SDP, optimal SDP
and laissez-faire economy.
small firms as shown in the right panel of Fig.7, and therefore the welfare is higher by the
insurance effect. This feature of the optimal SDP is not unrealistic since there are various
subsidies available for small firms in practice. More importantly, the observation suggeststhat the result wSDP < wLF in section 4 is not a robust implication of SDPs in general
but is a consequence of the unrealistically parsimonious modeling of the SDP (τ, F ) that
abstracts from small firm subsidies. Thus, if the quantification exercise includes subsidies,
the wage could go up and the welfare could be higher. In this sense, the quantification result
is conservative and should be interpreted as saying that the SDP could enhance efficiency
even when the SDP does not subsidize small firms.
For the optimal SDP in the right panel of Fig.7, I make two more observations. First,
the tax paid by the largest firms is smaller than the middle firms in percentage. This is notsaying that the absolute amount of the tax is lower. Actually, the absolute amount of thetax is monotonically increasing in productivity. It is just that the percentage of tax out of
the before-tax profit is lower. The second observation is that the subsidy is not mechanical.
This is where the government budget T matters. If I impose the balanced budget T = 0,
some firms have to be subsidized by construction. But here, I set the tax revenue T to be
the same number as in the previous quantification exercise, which is positive, so the subsidy
is not mechanical. In other words, the mechanism designer does not have to but still chooses
to subsidize the unproductive small firms since it affects ex-ante firm creation.

28

Now let me use Fig.8 to explain that the optimal SDP increases the number of firms. In
the left panel that plots the employment schedule for each productivity level z, one can seethat under the optimal SDP every firm hires fewer than both the laissez-faire and estimated
SDP, so the number of workers declines and firms increases. This is due to both the regulationand higher wage. Thus, the laissez-faire economy generates too few firms and they are too
big.
Finally, the right panel of Fig.8 describes the insurance effect of the optimal SDP. The
purple line is the before-tax profit under the optimal SDP and the red dashed line is theafter-tax profit. The fact that the after-tax profit schedule is flatter reflects insurance. Thisis also true in the threshold type SDP (τ, F ), although the insurance effect of the estimatedthreshold SDP comes from general equilibrium wage effect instead of the direct insurance as
the optimal SDP.

6

Financial frictions

The analysis so far has shown that the insufficient firm creation due to entrepreneurial risks
might justify SDPs. A natural question is whether this is always the case. I argue the answer
is negative by showing that, under severe financial frictions, removing SDPs might improve
efficiency. As a policy implication, whether to remove SDPs depends on the level of financial
development.
In principle, there are three cases where SDPs should be removed. The first case is whenentrepreneurial risks are not quantitatively important. If individual agents obtain a precise
signal si about their entrepreneurial productivity zi , SDPs are harmful as shown by Garicano
et al. [2016] and Gourio and Roys [2014]. The second case is when SDPs are so radical thatthe efficiency loss dominates the gain from more efficient firm creation. In French data, thisis not the case, but data from other countries might find a large estimate of SDP (τ, F ).
Both of these cases rely on the idea that the cost of SDPs is larger than the benefit.
The third case, which is theoretically more interesting, is when there are too many firms
in the laissez-faire economy. This case is different from the previous two because removing
SDPs incurs no cost. Fig.9 illustrates this situation. The SDPs that increase the number offirms not only distort production by creating wedges among firms but also worsen the market
failure of firm creation by adding more firms to the economy that already has excessive firms.
Thus, removing SDPs enhances efficiency through two channels.
Potentially, there are several frictions that could generate excessive firms. As an example,
I argue that severe financial frictions can lead to excessive firm creation in the next section.

29

Y( ): aggregate output

LF

SDP

PPF laissez-faire
PPF w. SDP: number of firms
Figure 9: Aggregate output Y (φ) when market generates too many firms. Points A, B, andC are the market equilibrium without SDPs, the market equilibrium with SDPs, and theunconstrained efficient allocation.

6.1

Financial frictions

One situation that leads to excessive firm creation is severe financial frictions that prevent
firms from expanding their employment to their profit-maximizing sizes. In this section, I
show that, under severe financial frictions, the laissez-faire economy creates more firms than
the Pareto efficient level.
To illustrate this point parsimoniously, I introduce a simple form of financial friction to
the firm’s profit maximization problem as follows.
π (z, w) = max zf (n) − wn s.t. wn ≤ λzf (n)n(24)

where λ−1 > 0 represents the severity of the financial friction. If the financial frictions are notsevere λ > α, the constraint does not bind. As the financial constraint becomes more severe,i.e., λ gets close to 0, the constraint becomes more binding. Such form of financial constraint
has been used in the literature, e.g., Bigio and La’O [2016], and is of particular interest in
this paper since the marginal products among firms are equalized and therefore the resulting
misallocation cannot be understood by the standard marginal product argument
αw, ∀i.
zi f (ni ) = max 1,
λ
0



30



(25)

One way to micro-found the constraint is limited enforcement. Each entrepreneur owes
the workers their salaries. However, due to the limited enforcement, the entrepreneur candivert 1 − λ fraction of the sales and run without paying the salaries. In this case, workers
are not willing to work for the firm if their salaries exceed λ percent of the sales. One canalso attribute λ to institutional immaturity or other non-financial obstacles. After all, theformulation is a reduced-form, so I call it financial friction for simplicity.
With this financial constraint, I conduct the efficiency analysis using the full-risk model.
Specifically, let wLF , φLF be the equilibrium in the laissez-faire economy characterized by(8) and (24), φP be the planner’s solution that solves (10) with the uninformative signals ⊥⊥ z. The following proposition states that under severe financial friction λ ≈ 0, thelaissez-faire economy generates too many firms so that the SDPs that increase the numberof firms exacerbate human resource misallocation.
Proposition 6. Fix CRRA utility, CD production function (u, f ), and an arbitrary risk
distribution Gz with bounded support. Under the severe financial friction, i.e., for small
λ ≥ 0, the laissez-faire economy generates an excessive number of firms
φP < φLF ≤ 1 = lim φLF .
λ→0

Proof. See Appendix A.5.
The proposition imposes particular functional form assumptions for analytical cleanliness,but they can be relaxed at the cost of heavier notation. The inefficiency result can bestrengthened to constrained inefficiency. As is the case for Proposition 3, I relegate the result
to Appendix A.5.
To understand the intuition behind the result, recall that the planner is free from financial
frictions. As a result, the efficient allocation is invariant to the severity of financial frictions
λ. Therefore, the inefficiency follows if the number of firms in market equilibrium can get
arbitrarily close to 1. The intuition comes from the firm side. If financial frictions aresevere λ → 0, due to the cash shortage or limited enforcement, firms cannot expand theiremployment. In the extreme case λ = 0, no firm can hire workers. Since the labor demand
from firms is limited, those who are not employed have to do their own businesses, leading toan increase in the equilibrium number of firms φ → 1. Thus, under severe financial friction
λ ≈ 0, the number of firms in the laissez-faire economy φLF exceeds that of the efficient
allocation φP .
The analysis of the financial friction implies that whether SDPs should be removed or
not depends on the level of financial development. For instance, in developing countries with31

immature financial infrastructure, SDPs should be removed and resources should be devoted
to relaxing financial frictions, even if firm creation is risky.
This implication might be surprising if one considers that entrepreneurial risks and financial frictions are often cited as two big obstacles to entrepreneurship. In fact, Proposition
3 and 6 suggest that the two frictions cause market failure in the opposite directions, i.e.,entrepreneurial risks cause the market to create too few firms while financial frictions cause
too many. There are two ways to reconcile the conventional wisdom and my result.
The first one is about the modeling. If one models the financial friction as the fixed
cost necessary to start a firm, it does suppress firm creation. However, as Quadrini [2008]
discusses, most entrepreneurs can lower the fixed cost by choosing to start small. In fact,
the moment entrepreneurs face financial frictions most is when they want to expand. In thissense, typical financial frictions that entrepreneurs face might be better captured by Eq.(24)than the fixed entry cost.
The second one is to recognize that the conventional wisdom is based on partial equilibrium, and is not suitable to understand the impact of the financial frictions on the generalequilibrium. In partial equilibrium, financial friction reduces profit so people do not want tobecome entrepreneurs. In general equilibrium, however, since firms cannot expand employment due to the financial friction, those unemployed have to do business on themselves. To
rationalize the existence of many more small firms, the wage declines to the level consistent
with such scenario.
The positive correlation between the severity of financial frictions and firm creation is also
consistent with data. Fig.10 plots entrepreneurship activity against the severity of financial
frictions. The entrepreneurship is measured by the percentage of the 18 − 65 population
who is either a nascent entrepreneur or owner-manager of a new business, taken from Global
Entrepreneurship Monitor for 2001 − 2016. The severity of financial frictions is measured by
the value of collateral needed for 100 unit of loans, taken from the World Bank Enterprise
Survey.
One can see that more severe financial frictions are associated with higher entrepreneurial
activities. Fig.11 presents another data consistent with the analysis. It is based on survey
data, asking country experts whether a country has enough entrepreneurship financing. Although the data are subjective, it contains more samples. One can see that higher availabilityof financing is associated with lower entrepreneurial activities.
What these figures do not show is the efficiency of the market equilibrium. In other words,
they suggest financial frictions are associated with more firm creation but do not provide
information about whether the increase is too much or not. In this sense, Proposition 6
provides a normative insight into the empirical regularities.

32

0

10

20

30

40

Financial frictions and firm creation

100

150
200
Collateral value needed to get 100 unit of loan
Entrepreneurs (%)

250

Fitted values

GEM and WB Enterprise survey, 2001−2016, 54 country−year pairs

Figure 10: Entrepreneurship is positively correlated with financial frictions. x axis represents
the collateral value required for 100 unit of loans. y axis represents the percentage of the
18 − 65 population who is either a nascent entrepreneur or an owner-manager of a new
business. The plot pools all available countries and years 2001 − 2016. Sources: World Bank,Enterprise Survey. Global Entrepreneurship Monitor.
As a caveat, financial frictions are not the entire picture. As one can see from Fig.11,
there is a difference between high- and low-income countries unexplained by the financial
frictions. Indeed, as Gollin [2008], Seshadri and Roys [2014], Poschke [2017] discuss, various
explanations can be consistent with the negative correlation between GDP per capita andentrepreneurship. That being said, since the financial friction λ in section 6.1 is a reduced
form formulation, it can also be interpreted as anything that prevents firm’s expansion. In
this sense, the unexplained variation does not necessarily limit the scope of the insights.

7

Capital accumulation

So far, I have developed analyses in static environments. In this section, I extend the analysis
to a tractable dynamic environment with inter-temporal optimization. It is shown that SDPs
are further justified in a dynamic environment, so the analyses in the static environments
are robust.
To study the implication of inter-temporal trade-off in a parsimonious and tractable
dynamic model, I extend the laissez-faire economy by introducing capital accumulation ala Krebs [2003] and Kreps-Porteus-Epstein-Zin preference. (Epstein and Zin [1989], Kreps
33

Access to finance and firm creation
Others

0

20

40

60

High income

1

2

3

4

1

2

3

4

Financing for entrepreneurs 1 insufficient − 9 sufficient
Entrepreneurs (%)

Fitted values

GEM, 2001−2016, 547 country−year pairs

Figure 11: Entrepreneurship is negatively correlated with the availability of financing. x axis
is the survey responses about the availability of financial resources - equity and debt - for
SMEs including grants and subsidies. y axis is the percentage of the 18 − 65 population who
is either a nascent entrepreneur or an owner-manager of a new business. Country year pooleddata. Sources: World Bank, Enterprise Survey. Global Entrepreneurship Monitor.
and Porteus [1978]) The “all-purpose” good a la Krebs [2003] gives dynamic tractability by
invoking a similar capital accumulation structure as Brock and Mirman [1972]. The KrepsPorteus-Epstein-Zin preference isolates the inter-temporal preference from risk aversion sothat I can highlight the implications specific to dynamic environments.

7.1

Equilibrium

This section defines the equilibrium of the laissez-faire economy in a tractable dynamic environment. I derive the closed-form solution of the equilibrium objects and discuss the similarity
to the static model.
The setup of the dynamic economy is a direct extension of the static laissez-faire economy.
There is a continuum of agents facing risky occupational choice and dynamic consumption
maximization problem. The economy has no aggregate uncertainty.
At the beginning of each period, each agent i ∈ [0, 1] is characterized by the (ait−1 , jit−1 , zit )consisting of asset ait−1 > 0, occupation jit−1 ∈ {E, W } where E denotes entrepreneurs andW workers, and productivity zit > 0. The distribution of the state at t is denoted by
St (ait−1 , jit−1 , zit ) and the initial distribution S0 is exogenously given.

34

I make three comments about the state. First, the asset ait is an “all-purpose good”
because it can be used as a consumption good, physical capital and human capital. Second,
for simplicity I assume the productivity {zit }i,t is i.i.d. over i and t, distributed according to
Gz . I assume i.i.d. structure for simplicity, but it is straightforward to extend it to a persistentprocess that corresponds to the model with the intermediate signal in the static environments.
These assumptions give tractability as will be clear later. Third, the convention of the time
subscript is based on the measurability. In other words, the asset and the occupation are
pre-determined at t − 1 before observing the productivity zit . Therefore, in principle, boththe saving and occupation decisions involve risk takings. However, as will be discussed later,
the log inter-temporal preference makes sure the only decision affected by entrepreneurial
risks is the occupation choice.
Within each period t, agents work first, and then make consumption/saving and occupation decisions. For the working part, each agent transforms the all-purpose asset ait−1
into productive capital fI (ait−1 ) to earn income. Each entrepreneur i ∈ Et uses the capital
kit−1 = fI (ait−1 ) to run her firm of productivity zit . In particular, given wage wt , the entrepreneur i solves the profit-maximization problem to obtain πit = π (zit , wt , kit−1 ) by hiring
nit = n (zit , wt , kit−1 ) employees[π (z, w, k) , n (z, w, k)] = max zf (k, n) − wn.

(26)

n≥0

Each worker i ∈ Wt sells his human capital hit−1 = fI (ait−1 ) to earn labor income wt hit−1 .
After obtaining income, each agent i makes consumption-saving decision and chooses
whether to become an entrepreneur or a worker next period. The consumption-saving decision(cit , ait ) has to be chosen from the budget set
Bt (a, E, z) = {(c, a0 ) : k = fI (a) , c + a0 = π (z, wt , k)}
Bt (a, W, z) = {(c, a0 ) : h = fI (a) , c + a0 = wt h} .

(27)

The occupation choice can be a mixed strategy eit ∈ [0, 1] as in the static case, i.e., if anagent chooses eit , she becomes an entrepreneur jit = E with probability eit . Each agent ichooses (cit , ait , eit ) ∈ Bt (ait−1 , jit−1 , zit )×[0, 1] for all states to maximize the recursive utility
h



i

Vit = u−1
(1 − β) uI (cit ) + βuI u−1 (Eu (Vit+1 ))I(28)

where u and uI are intra- and inter-termporal utility functions, and the expectation E istaken with respect to both productivity zit > 0 and the mixed strategy eit ∈ [0, 1].
Finally, the labor market has to clear, i.e., labor demand from entrepreneurs equals to35

human capital supply from workers
Z

eit−1 nit di =Z

(1 − eit−1 ) hit−1 di.

(29)

One can rewrite this equation using the distribution of states St , but I avoid the complicationand define the equilibrium concept using the simple summation over i.
Definition 5. Fix the fundamentals E = (β, u, uI , f, fI , Gz , S0 ). The set of individual choices
and wage {Vit , ait , cit , eit , hit , kit , πit , nit , wt }i,t constitutes an equilibrium if the following conditions are satisfied for all t ≥ 0.
1. At each t, given wt , the profit and the employment decision {πit , nit } by the entrepreneurwith productivity zit solves the profit maximization problem (26).
2. Given {wt }t , the value and policies {Vit , ait , cit , eit }i,t maximize individual welfare (28)subject to (cit , ait , eit ) ∈ Bt (ait−1 , jit−1 , zit ) × [0, 1].
3. At each period t, labor market clears (29).
The equilibrium number of firms at t is pre-determined at t − 1 and can be written as
φt−1 =

Z

eit−1 di.

This is a dynamic general equilibrium model with heterogeneous agents and incomplete
markets, so in general, a closed form solution is not available. However, the following parameterization allows us to study the efficiency of firm creation analyticallyu (c) =

c1−γ − 1, uI (c) = ln c, f (k, n) = k 1−α nα , fI (a) = aθ
1−γ

(30)

where γ > 0, α ∈ (0, 1) and θ ∈ (0, 1). The intra-temporal utility u remains the same as thestatic model. The logarithmic inter-temporal utility uI separates the saving decision fromthe uncertainty about the productivity zit . This is because the optimal saving is a constantfraction of the income and is independent of the potentially risky returns to saving. In thisway, I can make sure that the only risky decision is occupation choice so that the result isdirectly comparable with the static environment. The production function is the standard
Cobb-Douglas form with complementarity between the entrepreneurial capital and the workers’ human capital. As a result, the profit becomes a linear function of entrepreneurial capital
π (z, w, k) = r (z, w) k, and therefore can be interpreted as the return to entrepreneurial capital. The investment function features decreasing returns to scale to ensure a stationary asset
distribution.
36

Given the parameterization, the market equilibrium takes the following form as derived
in Appendix A.6
wtLF =Ez(1−γ)(1−β)(1−α)(1−βθ) (1−α)(1−βθ)(1−γ)(1−β)αα (1 − α)1−α ,

1−α

φLF
=
t
1 − α + αEz

1
1−α



Ez

(1−γ)(1−β)(1−α)(1−βθ)

−

1−βθ
(1−γ)(1−β)

(31)

.

1−β
One can see that the risk aversion 1 − γ is always multiplied by 1−βθ
> 1. To anticipate
the result, this additional term comes from the inter-temporal decision and makes the agentseffectively more risk-averse so that φLF
is further away from the efficient level 1 − α.
t7.2

Efficiency analysis

This section presents the efficiency analysis of the dynamic model. The direction of themarket failure remains the same as in the static economy but the severity is greater due tothe consumption-smoothing motive. A novel insight is that the market failure is more severe
in an economy with more patient agents.
The planner maximizes the welfare by choosing resource allocations at each state ωt ∈tΩt = [zmin , zmax ][0,1] facing the same technology and information constraints as the individual agent. Specifically, given the Pareto weight function i 7→ Λi , the planner maximizes
Rthe welfare Vit (ω) dΛi where Vit (ω) satisfies
h





i

Vit (ωt ) = u−1(1 − β) uI (cit (ωt )) + βuI u−1 EΩt+1 |ωt u (Vit+1 )I(32)

and the expectation EΩt+1 |ωt is taken over the subset of state space Ωt+1 accessible from ωt .
Since the problem reduces to the aggregate production maximization in an economy withno aggregate uncertainty, I omit the notation of the state ω and related measure theoretic
treatment in the following.
The planner’s decision can be described recursively. At each period t, the planner takes asgiven the set of entrepreneurs Et−1 ⊂ [0, 1] and workers Wt−1 = [0, 1] \Et−1 chosen at periodt − 1, and their saving of capital {ait−1 }i . The capital generates entrepreneurial capital
kit−1 = fI (ait−1 ) if stored by an entrepreneur i ∈ Et−1 and human capital hit−1 = fI (ait−1 ) if
stored by a worker i ∈ Wt−1 . The human capital is then reallocated by the planner to makeproduction
Z
Z
Z
Yt =
zit f (kit−1 , nit ) di,
nit di =hit−1 di.
(33)i∈Et−1i∈Et−1i∈Wt−1After the production, the planner chooses the occupation Et , Wt for period t + 1 and divides
37

the aggregate output into consumption today and asset saved for tomorrowZ

cit di +Z

ait di = Yt .

(34)

The key informational assumption is that when the planner chooses the asset ait , the productivity tomorrow zit+1 is not observableait ⊥⊥ zit+1 .

(35)

In summary, the efficient allocation can be defined as follows.
Definition 6. Fix the fundamentals E. The set of individual values, consumption-saving, occupation, and production decisions and the aggregate output {Vit , cit , ait , Et , Wt , kit , hit , nit , Yt }is efficient if it maximizes (32) subject to (34), (35), and (33). The number of firms in theplanner’s solution is defined asZ
φPt−1 =
di.
i∈Et−1

It is immediate to see that the efficient allocation maximizes the aggregate output Yt ateach period given the asset {ait−1 }. Since the planner can control all the allocations, as in
the static case, the planner’s problem can be separated into production and allocation of
consumption.
The following proposition states that the direction of the market failure remains the sameas the static case, but the market failure gets worse due to the inter-temporal consumption
smoothing motive.
Proposition 7. The laissez-faire economy generates an insufficient number of firms if andonly if there are entrepreneurial risks.
φLF< φPt = 1 − α ⇔ V (z) > 0.
tThis is true even when all agents are risk neutral γ = 0.
Proof. See Appendix A.6.
The inefficiency result can be extended to constrained inefficiency as discussed in Appendix F.3.
I make two observations about the statement. First, note that the planner’s solutionP
φt = 1 − α is identical to the static model, making the static and dynamic models directlycomparable. Together with Eq.(31), one can see that the inefficiency in the dynamic environment is worse than the static, and therefore, SDPs are more effective in a dynamic
environment. In this sense, the results in the static analyses are conservative.
38

Second, the last line presents a new insight specific to the dynamic environment. It statesthat the market failure happens even when all agents are risk-neutral. This is not the case in
the static environment. If agents are risk-neutral, both the laissez-faire and planner choose
φ = 1 − α in the static environment, making Pareto improvement impossible. The intuitionis based on the consumption-smoothing motive. Since agents want to smooth consumption,
they want to avoid asset fluctuation. To avoid the asset fluctuation, they can choose thesalary job with a safer return. This inter-temporal motive is conceptually different fromthe static risk aversion at least in the framework of Kreps-Porteus-Epstein-Zin preference.
Therefore, the number of firms is smaller than the efficient level even when all agents arerisk-neutral γ = 0.
In terms of the intuition, observe that the consumption-smoothing motive works similarly as risk aversion. This is not surprising if one realizes that the intra-temporal andinter-temporal preferences both discount the value of fluctuations across a set of states.
Mathematically, they both represent the force of Jensen’s inequality over the states called indifferent ways.
Finally, one non-trivial insight can be obtained about the patience β. As can be seen from
Eq.(31), the more patient agents are β %, the more severe the market failure is φP − φLF %.
This is because patient agents care more about the fluctuation of the future consumption. In
other words, higher patience magnifies consumption-smoothing motive. Therefore, in highpatience economy, removal of SDPs can cause severe human resource misallocation.

8

Conclusion

I have shown that SDPs might enhance efficiency in the presence of entrepreneurial risks.
The analysis not only changes the conventional wisdom about SDPs but also casts a caveat ingeneral to the misallocation measurement exercises based on marginal product equalization.
I have also made the analysis comprehensive by showing that severe financial frictions cancause excessive firm creation even if firm creation is risky. This result implies that developing
countries with severe financial frictions might require different policy prescriptions. Finally,
I have analyzed the implications of dynamic decisions. The analysis not only confirms therobustness of the findings in the static environment but also provides a novel insight into therole of consumption-smoothing motive and patience. These analyses deepen our understanding of SDPs.
I conclude the paper by mentioning three caveats to my analyses. First, there can be
many other justifications for SDPs. For instance, small firms might have some externality
in R&D and employment, although these arguments are highly controversial compared to39

entrepreneurial risks (Biggs [2002], Shane [2008, 2009]). Another justification for SDPs isthat it saves the government administrative costs as discussed in Kaplow [2017]. My analysis
should be considered a complement to these discussions. Second, I have assumed that individual agent maximizes profit, but the importance of non-pecuniary benefit has also been
pointed out in the entrepreneurship literature (Hurst and Pugsley [2011], Gordon and Sarada[2017]). Exploring the implications of behavioral agents that have other objectives than profit
maximization is an interesting future topic. Third, I have focused on SDPs that treat small
firms preferentially. However, some countries treat large firms preferentially to foster international competitiveness. In this case, SDPs are adopted based on a totally different logic
and therefore, their analysis needs a different framework. These points are missing in theanalysis and should be explored in future research.

40

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44

Appendices for Online PublicationA

Proofs of propositions

I use the following lemma.
Lemma 1. (Jensen’s inequality) Suppose X > 0 is a random variable with positive variance
V X > 0 . Then, for any γ > 0,

 1

EX 1−γ 1−γ < EX.

Proof. Since CRRA utility function is strictly concave and increasing when γ > 0, Jensen’s
inequality and taking inverse function imply
 1

Eu (X) < u (EX) ⇔ u−1 (Eu (X)) < EX ⇔ EX 1−γ 1−γ < EX.

A.1

Proof of Proposition 2

The planner’s solution φP maximizes the output.
Y (φ, n) =hmaxn(z)≥0,φ∈[0,1]iφE zf (n (z)) |s ≥ G−1
s (1 − φ)his.t. φ + φE n (z) |s ≥ G−1
s (1 − φ) = 1.
Reformulate the problem using s̄ gives
Y (s̄, n) = maxs̄,n(z)≥0Z

zf (n (z)) 1s≥s̄ dG s.t.

Zn (z) 1s≥s̄ dG = Gs (s̄) .

For each fixed s̄, the sub-problem of choosing n (z) ismaxn(z)≥0Z

zf (n (z)) g (z|s ≥ s̄) dz s.t.

Zn (z) g (z|s ≥ s̄) dz =Gs (s̄).
1 − Gs (s̄)

The objective function is a weighted sum of the strictly concave function of {n (z)}z and theconstraint is a linear function of {n (z)}z . Hence, the unique solution is characterized by the
first order conditions.
Furthermore, let smin , smax ∈ [−∞, ∞] be the smallest and largest values that the signal
45

can take, so that 1s≥smin = 1 and 1s≥smax = 0 for all realizations. When s̄ = smin , the value
is Y (smin , n) = 0 since the only feasible employment schedule is n (z) = 0 for all z. Whens̄ = smax , Y (smax , n) = 0 since the objective function is 0. Since 0 output is the lower boundY ≥ 0, the solution s̄ is not at the boundaries. Therefore, the solution exists and is unique
if there is a unique set of (s̄, n) that solves the first order conditions.
The first order condition can be obtained by taking the derivative of the Lagrangian.
 
0 −1 λ, λ > 0.
n
(z)
=
(f
)

z



E [zf (n (z)) |s = s̄] = λ (1 + E [n (z) |s = s̄])E [n (z) 1 ] = G (s̄)
s≥s̄sCombining the conditions gives

h

 
 
i

0 −1 λF1 (s̄, λ) = E zf (f 0 )−1 λ
−
λ
(f
)
|s
=
s̄
− λ = 0.
z
z
h
i
 

F2 (s̄, λ) = E (f 0 )−1 λ 1s≥s̄ − Gs (s̄) = 0zNote that the integrand of F1 is identical to the value function of the profit maximization
problem with wage λ
0 −1

zf (f )

λ
z

!!

0 −1

− λ (f )λz!

= max zf (n) − λn.
n≥0Thus, F1 gives a strictly positive relationship between s̄ and λ. In contrast, F2 gives a strictly
negative relationship. Hence, there is a unique solution (s̄, n) to the output maximization
problem, and unique solution φP to the original problem.

A.2

Proof of Proposition 3

For the first statement, when there are no risks, the laissez-faire economy is characterized byz̄ LF , wLF , π LF , nLF that satisfy

π z̄ LF , wLF = wLFzf 0 nLF z, wLF



= wLF





 



i



π z, wLF = zf n z, wLFh 

E n z, wLF 1z≥z̄LF = Gz z̄ LF

46







− wLF n z, wLF

Choosing the Pareto weight using these variables.

λ (z) =
 0 1LF)z ≤ z̄1u (π(z,wLF ))z > z̄

u (w


 0

I show that (z̄, w, π, n) is the solution to the planner’s problem. Without loss of generality,ci (ω) = c (zi ). The planner’s solution can be written as

max

Zc,n,Y,z̄Rc (z) dGz = Y

λ (z) u (c (z)) dGz s.t.

Y = E [zf (n (z)) 1z≥z̄ ]E [n (z) 1] = G (z̄)
z≥z̄z

The Lagrangian isL =Z

λ (z) u (c (z)) dGz + µ {E [zf (n (z)) 1z≥z̄ ] − Ec (z) + ν (Gz (z̄) − E [n (z) 1z≥z̄ ])}The FOCs areLc = 0µ
⇒ u0 (c (z)) = λ(z)Ln = 0L = 0z̄⇒ zf 0 (n (z)) = ν⇒ z̄f (n (z̄)) − vn (z̄) = ν







(µ, z̄, v, n) = 1, z̄ LF , w, nLF and c (z) = π LF z, wLF 1z≥z̄LF + w1z<z̄LF satisfies the equilibrium conditions.
For the second statement it suffices to show that the laissez-faire economy does not
maximize output. With risks, the laissez-faire economy is characterized by
 

 

E u zf h

w
z



w
|s = s̄ = u (w)
z
  

w
E h
1s≥s̄ = Gs (s̄)
z

− wh







where h (x) := (f 0 )−1 (x). The FOCs of the planner’s problemmaxs̄,n(z)Zzf (n (z)) 1s≥s̄ dG s.t.

47

Zn (z) 1s≥s̄ dG = Gs (s̄) ,using the Lagrange multiplier λ, is"λE zf h
z

!!

!

#

λ
− λh
|s = s̄ = λ
z
"
!
#
λ
E h
1s≥s̄ = Gs (s̄) .
z

Note that the two systems are different by u. Therefore, the laissez-faire economy can Pareto
improve by producing more and redistribute them.
The two systems also imply w < λ. To see this, note that Jensen’s inequality implies"

λλ = E zf h
z

!!

!

"

#

λ
λ
− λh
|s = s̄ > u−1 E u zf h
z
z

!!

!

λ
− λh
|s = s̄z!#!

.

In addition,
(

λ
∂
zf h
h (λ) < 0,
∂λ
z
0

!!

λ
− λh
z

!)

λ
= −h
z

!

< 0.

so from the market clearing condition, λ and s̄ have to be negatively correlated. Hence,w < λ and s̄P < s̄LF , which then implies φLF < φP .

A.3

Proof of Proposition 4First, I reformulate the optimal taxation problem into a mechanism design problem of choosing a set of menu.

maxw,C(z),Y (z)w s.t.


Eu (zf (C (z)) − wC (z) − Y (z)) = u (w)



0

0

U (C (z) , Y (z) , z) ≥ U (C (z ) , Y (z ) , z) ∀z, zEY (z) − T (1 + En (z)) = 0.

0

(36)

The feasible sets of (21) and (36) are identical. To see this, fix w and choose a feasible
solution (n (z) , T (n)) from (21). Then, (C (z) , Y (z)) = (n (z) , T (n (z))) satisfy
U (C, Y, z) := zf (C) − wC − Y
U (C (z) , Y (z) , z) ≥ U (C (z 0 ) , Y (z 0 ) , z) ∀z, z 048

To see this, pick arbitrary z and z 0 ∈ [zmin , zmax ]. Then,zf (C (z)) − wC (z) − Y (z) =zf (n (z)) − wn (z) − T (n (z))≥zf (n (z 0 )) − wn (z 0 ) − T (n (z 0 ))=zf (C (z 0 )) − wC (z 0 ) − Y (z 0 ) .
The opposite is also true. Given a (C (z) , Y (z)) that satisfies the truth-telling condition(36), the SDP can be constructed by
n





o

T (n) = inf Ȳ : U (C (z) , Y (z) , z) ≥ U n, Ȳ , z ∀z .
T̄By the truth-telling condition,
U (C (z) , Y (z) , z) = U (C (z) , T (C (z)) , z) ≥ U (n, T (n) , z) ∀z, n.
Hence, agents facing T (·) chooses n (z) = C (z).
Next, I reformulate the truth-telling condition into the envelope condition and the monotonicity constraint. Note that the utility function U (C, Y, z) satisfies Spence-Mirrlees condition
f 0 (C)1UY, M RSz = −< 0.
= 0
M RS (C, Y, z) = −UC
zf (C) − w(zf 0 (C) − w)2
Hence, the truth-telling condition is equivalent to
π (z) := U (C (z) , Y (z) , z)
π 0 (z) = Uz (C (z) , Y (z) , z) , C 0 (z) ≥ 0
By substituting Y out, the problem becomesmax

w,n(z),π(z)Eu (π (z)) − u (w) = 0






π 0 (z) = f (n (z))

w s.t. 
n0 (z) ≥ 0



(37)




E [zf (n (z)) − wn (z) − π (z)] − T (1 + En (z)) = 0.

A.4

Proof of Proposition 5

To solve (37) using optimal control theory, rewrite the integral constraints by
B 0 (z) = {zf (n (z)) − wn (z) − π (z) − T (1 + n (z))} g (z) , B (zmin ) = B (zmax ) = 0.
49

I 0 (z) = (u (π (z)) − u (w)) g (z) , I (zmin ) = I (zmax ) = 0.
Wage is constant, so w0 (z) = 0. Monotonicity constraint becomes n0 (z) = L (z) ≥ 0. The
only control variable is L, state variables are π, w, B, I, n, and the co-state variables areµπ , µw , µB , µI , µn . Hamiltonian is
H = wg (z) + µπ f (n) + µw × 0 + µn L + κL
+ µB {zf (n) − wn − π − T (1 + n)} g (z) + µI (u (π) − u (w)) g (z)HL = 0 = µn + κ
Hπ = −µ0π = (−µB + µI u0 (π)) g (z)Hw = −µ0w = (1 − µB n − µI u0 (w)) g (z)HB = −µ0B = 0
HI = −µ0I = 0
Hn = −µn0 = µπ f 0 (n) + µB {zf 0 (n) − w − T } g (z)κL = 0, κ ≥ 0, L ≥ 0.
Boundary conditions are
B (zmin ) = B (zmax ) = I (zmin ) = I (zmax ) = µπ (zmin ) = µπ (zmax )= µw (zmin ) = µw (zmax ) = µn (zmin ) = µn (zmax ) = 0
By reducing the system, the solution (π, w, µB , µI , µπ , µw , I, B, n, µn ) satisfy
π 0 = f (n)w0 = µ0B = µ0I = 0µ0π = (µB − µI u0 (π)) g (z)µ0w = (µB n + µI u0 (w) − 1) g (z)B 0 (z) = {zf (n (z)) − wn (z) − π (z) − T (1 + n (z))} g (z)I 0 (z) = (u (π (z)) − u (w)) g (z)µ0n = −µπ f 0 (n) − µB {zf 0 (n) − w − T } g (z)50B (zmin ) = B (zmax ) = I (zmin ) = I (zmax ) = µπ (zmin ) = µπ (zmax )= µw (zmin ) = µw (zmax ) = µn (zmin ) = µn (zmax ) = 0µn n0 = 0, µn ≤ 0, n0 ≥ 0.
By assumption, n0 > 0. Then, from µ0n = 0, n (z) is determined by
zf 0 (n (z)) =w+Tµπ (z)+1zµB g(z)(38)

In this case, ODE reduced to y = (π, w, µB , µI , µπ , µw , B, I) such that
π 0 = f (n)w0 = µ0B = µ0I = 0µ0π = (µB − µI u0 (π)) g (z)µ0w = (µI u0 (w) + µB n − 1) g (z)B 0 (z) = {zf (n) − wn − π − T (1 + n)} g (z)I 0 (z) = (u (π) − u (w)) g (z)B (zmin ) = B (zmax ) = I (zmin ) = I (zmax ) = µπ (zmin ) = µπ (zmax ) = µw (zmin ) = µw (zmax ) = 0
To show that the medium-sized firms are distorted, it suffices to show µB > 0 > µπ (z)due to 38. To see this, note that from envelope theorem,∂H= −µB E (1 + n (z)) .
∂T
Since higher tax takes more resources, ∂H< 0 so µB > 0. By integrating µ0π , the boundary∂Tcondition impliesµB − µI Eu0 (π (z)) = 0 ⇒ µI > 0.
Now, let’s focus on µ0π . which has to cross zero since µπ (zmin ) = µπ (zmax ) = 0. Since π 0 > 0,µπ must be convex,∂µ0π (z)= −µI u00 (π (z)) π 0 (z) > 0.
∂z
Therefore, µ0π starts from negative and monotonically cross zeros. Hence, µπ < 0.
The fact that the optimal SDP increases the number of firms follows from the distortion.
Note that when T = 0, the laissez-faire economy T (n) = 0 is a feasible choice for the51

government. Therefore, the optimal SDP increases wage wo > wLF . For small T > 0, themarginal products are higher than the laissez-faire wage
zf 0 (n (z)) =wo + Tµπ (z)+1zµB g(z)> wLF .

Hence, employment is lower at each productivity level. Since the entrepreneurs and workers
have to add up to one, it implies that the number of firms increases. φo > φLF .

A.5

Proof of Proposition 6

The equilibrium is characterized by
Eu (π (z, w)) = u (w)π (z, w) = max
znα − wn s.t. wn ≤ λznα .
nφEn (z, w) = 1 − φLet λ∗ = min {α, λ}. The firm’s problem givesn (z, w) =

λ∗ z
w

! 1

1−α

λ∗
, π (z, w) = (1 − λ ) zw(∗!α ) 11−αThe indifference condition leads to1−γ

wLF = (1 − λ∗ )1−α (λ∗ )α Ez 1−α

 1−α
1−γ

The market clearing condition implies that the equilibrium number of firms is
1 − λ∗

φLF =

1



1−γ

1 − λ∗ + λ∗ Ez 1−α Ez 1−α

− 1

.

1−γ

This is a decreasing function of λ∗ satisfying φLF → 1 as λ → 0. Since the planner does not
face any friction, φP does not depend on λ. Hence, for small λ, φP < φLF .

A.6

Proof of Proposition 7

I first solve the laissez-faire economy and then derive the efficient allocation.
(Laissez-faire) Let Vt (a, j, z) be the value function. Guess Vt (a, j, z) = vtj (z) aρ . For

52

instance, the problem for entrepreneurs ishvtE (z) aρ = max
exp (1 − β) ln c + βρ ln rt aθ − c
0



c,e

+β ln

n

1−γ
1−γ
E
W
e Evt+1(z 0 )+ (1 − e0 ) Evt+1(z 0 )
0

o 1 
1−γ

The maximization in terms of c and coefficient matching with respect to a givesc=c = (1 − βθ) rt aθ

1−β1−β
rt aθ , ρ =θ⇒
1 − β + βρ
1 − βθ
a0 = βθrt aθ

The same structure appears in worker’s problem with rt replaced by wt . Since the productivity is i.i.d., the productivity and asset distribution are independent. In the equilibrium where
everyone takes the same strategy eit = φ for all t, the market clearing condition becomes

  1
  1

n (z, w, k) = αz 1−α k = αz 1−α aθww

 h = aθ

α
⇒
wt


 1

1−α

1

Ez 1−α φ = 1 − φ(39)

so wt and rt is time independent. One can also see the value functions are also time independent and v W does not depend on z.
ln v E (z) = (1 − β) ln (1 − βθ) + βθ

1−β
ln βθ
1 − βθ


1−γ
1−β
1−γ
+
ln r (z, w) + β ln φEv E (z 0 )+ (1 − φ) v W
1 − βθ


 1

1−γ

1−β
ln βθ
1 − βθ
1
1−γ  1−γ
1−β
1−γ
+
ln w + β ln φEv E (z 0 )+ (1 − φ) v W
1 − βθ

(40)

ln v W = (1 − β) ln (1 − βθ) + βθ

(41)

By subtracting one from the other, I can simplify the relationship between the two valuefunctions! 1−βr (z, w) 1−βθ WEv (z) =v .
(42)w
The occupation choice gives the indifference condition
1−γ

Ev E (z 0 )

 1

53

1−γ

= vW .

These two conditions lead to the equilibrium condition that the wage has to satisfyr (z, w)
Ew

! (1−β)(1−γ)
1−βθ

= 1.

(43)

Since the return to capital isαα1r (z, w) = (1 − α) α 1−α w α−1 z 1−α ,the equilibrium wage is
1−αmw = (1 − α)

α

α



Ez

(1−γ)(1−β)(1−α)(1−βθ) (1−α)(1−βθ)(1−γ)(1−β)

and the number of firms is given by the market clearing condition
1−α

φm =
1 − α + αEz

1
1−α

(



E z

1
1−α

.
1−βθ
)
 (1−γ)(1−β) − (1−γ)(1−β)
1−βθ(Planner) To derive the optimal number of firms, it suffices to consider output maximization. The problem is recursive, so fix a time period t and the aggregate asset savingR
At := ait di. This is an endogenous variable, but for the purpose of deriving the optimal
number of firms, I can take it as given as will be clear. Since the investment function fI isstrictly concave and zit+1 is not observable at t, it is optimal to choose the same saving acrossagents.
 kit = kt := fI aE, hit := ht := fI aW.
tt
Conditional on the amount of capital (kt , ht ), the problem at t reduces tomax

Yt+1 ,nit ,φt

Yt+1 s.t.

R
φt nit+1 di = (1 − φt ) htR
Yt+1 = φt zit+1 f (kt−1 , nit+1 ) di

By maximizing out nit+1 , the aggregate output can be written as

1

Yt+1 = φ1−α
(1 − φt )α Ez 1−α
t

54

1−α

kt1−α hαt .

.

Hence, the planner’s problem reduces to

max
E W

at ,at ,φt ,kt ,ht

Yt+1 s.t.


W


φt aE

t + (1 − φt ) at = At




1

1−α
(1 − φt )α Ez 1−αYt+1 = φt1−αkt1−α hαt

.


 




 k = f aE , h = f aW
t



I

t

t

I

t



W
By optimizing out aE, the output takes the following formt , atn

Yt+1 = φ1−α
(1 − φt )αto1−θ 

1

Ez 1−α1−α(1 − α)θ(1−α) αθα Aθt .

Hence, the optimal number of firms is φPt = 1 − α. One can see φLF< φPt iff V (z) > 0 sincet
φLF < 1 − α is equivalent to

Ez

1
1−α

(



> E z

1
1−α

 (1−γ)(1−β))1−βθ(1−γ)(1−β)

1−βθ⇔ γ + (1 − θ)

β
> 0,
1−β

which is always true under the parameter assumptions.

B

Derivation of Eq.(12)

Eq.(12) can be derived as follows. The key is that marginal product equalization holds inthe laissez-faire economy.
Y = φE [zf (n (z, w))] .
By taking derivative, noting w = w (T ) and φ = φ (T ) are functions of policies∂T Y = Ezf (n (z, w)) ∂T φ + φE [zf 0 (n (z, w)) ∂T n (z, w)] .
Since the derivative is evaluated at laissez-faire, zf 0 (n (z, w)) = w, and ∂T Y becomes∂T Y = Ezf (n (z, w)) ∂T φ + wφE [∂T n (z, w)]
We can simplify the second term by substituting out the derivative of the market clearing
condition∂T φEn (z, w) + φE [∂T n (z, w)] = −∂T φ.
Hence, the marginal impact can be written as∂T Y = ∂T φEzf (n) − w∂T φ (1 + En) = ∂T φ (Eπ (z, w) − w) .
55

Note that the derivation does not use the indifference condition Eu (π (z, w)) = u (w).

C

Non-pecuniary SDP

If the SDP is non-pecuniary and does not generate tax revenue, the fair comparison is the
economy with non-pecuniary SDP

Eu π SDP z, wSDP

h







π SDP z, wSDP , nSDP z, wSDP

i





= u wSDPzf (n) − w SDP nn≤Nzf (n) − w SDP τ n − F

n>N

= max
n 





φSDP En z, wSDP = 1 − φSDP
and (2) the laissez-faire economy with the same amount of non-distortionary regulation costs






= u (1 − t) wLFi= max zf (n) − wLF n

Eu (1 − t) π LF z, wLF
hπ LF z, wLF , nLF z, wLF






n



φLF EnLF z, wLF = 1 − φLF
where t is chosen to equate the tax revenue in the first economy, i.e., equation (18).
With this specification, the policy implication remains the same.
noln (1 − t) wLF − ln wSDP = −1.27% < 0.

D

MLE estimation

The effective tax rates (τ, F ) are estimated from the firm size distribution. Given the labordemand n (z, w; τ, F ) and the the power law density of productivity z, the model-implied firm size distribution Gn (n; τ, F ) := P n wz ; τ, Fw ≤ n follows a broken power law and canbe fit with the empirical firm size distribution using the maximum likelihood method.
To be more concrete, following the same derivation as Garicano et al. [2016], the likelihood
function is constructed as follows. Given the parametric assumptions f (n) = nα , the demand56

function can be written asn (z, w; τ, F ) =  1zα 1−α




 w
N



  1


 zα 1−αwτzmin ≤ z ≤ z,

z ≤ z ≤ z̄z̄ < z ≤ zmax


  1

z
zα 1−α
N 1−α


⇔
=
N
=

w
w
α


  α
α
z̄N − wN = z̄

z̄α
wτ

1−α

− wτ



z̄α
wτ

 1

1−α

−F



 α

  1


N
⇔ F = N − τ n̄
+ α − 1 , n̄ = z̄α 1−α
w

α

n̄

wτ

Given the density of the productivity gz (z), I can derive the distribution function of ni =n (zi , w) by using the change of variable.




0




1−β

n1−β

min −x1−βn−T n1−β
max

 min
1−β
1−β
−T n̄

n

min
Gn (x) = P (n ≤ x) =  n1−β
−T n1−β

max
min
1−β
n1−β
min −T x
1−β1−βnmin −T nmax













1

x < nmin
nmin ≤ x < N
N ≤ x ≤ n̄



 1

zmin α 1−α

n
=

min

w



 1


zmax α 1−α
n
=

, 

n̄ ≤ x ≤ nmax

max

wτ



1 − β = (1 − α) (1 − βz )





T = τ 1−βz

nmax < x

To reconcile the model in which there are no firms over the region N ≤ n ≤ n̄ with the actual
data (Fig.1), the observed firm size distribution is assumed to be generated with measurement
error, i.e., ñi = n (zi , w) e−σi , i ∼ N (0, 1). The distribution of ñi follows
Gñ (ñ) =P ne−σ ≤ ñ = E [P (n ≤ ñeσ |)]ñ
ln nmin
1
1−β
= 1−β
n
Φ
min
σ
nmin − T n1−β
max

"

!

− T n1−βmax Φñln nmaxσ!

ln nminln Nññ− ñ eΦ (1 − β) σ −− Φ (1 − β) σ −
σ
σ
(
!
!)
n̄
N
ln ñ
ln ñ
− T n̄1−β Φ
−Φ
σ
σ
!
!)#
(
2
2
σ (1−β)ln nmaxln n̄ñ
1−βñ
−T ñ e 2Φ (1 − β) σ −− Φ (1 − β) σ −σσ1−βσ 2 (1−β)2
2

(

!

!)

In addition, to utilize the bunching around N and avoid the deviation of the model from thedata at the extremes, the sample {ñi }i is truncated between ñmin = 10 to ñmax = 10, 000.
Hence, the likelihood function of the parameters θ = (α, nmin , nmax , n̄, β, σ, T ) isQ

L (θ; {ñi }i ) =0
i Gñ (ñi ; θ)Gñ (ñmax ; θ) − Gñ (ñmin ; θ)(44)

The parameters are subject to the constraint specified by the indifference condition. Given

57

1−γ

the CRRA utility parameterization u (c) = c1−γ , the indifference condition is
"

E

F
z
f (n) − n − τ − 1 +
1n>N
w
w




1−γ #= 1.

(45)

By applying the parametric assumptions, this condition can be rewritten as
1−α
1=
α


+

1−γ

β − 1 N 2−γ−β − n2−γ−β
min
1−β
1−β
2 − γ − β nmin − T nmax
Z n̄1−α τ

βz − 1
N 1−α
α

1−βz

−

α

 1−α 1−βz
n̄

τ

α

N 1−α
α

(xN α − N )1−γ x−βz dz(β − 1) T Z nmax 1 − α
F
+ 1−β
τn −
1−β
α
w
nmin − T nmax n̄1−γn−β dn.

Note that since the likelihood is based on truncation, the extreme values (nmin , nmax ) cannot be estimated accurately. Garicano et al. [2016] avoids this problem by fixing (α, nmax ) =(.8, ∞) and maximizing the likelihood L subject to their version of the indifference conditionαnmin = 1−α, so the problem becomes unconstrained maximization of the likelihood (44) over(n̄, β, σ, T ). Similarly, I fix (α, nmax ) = (.8, ñmax ) and maximize the likelihood (44) subject
to the indifference condition (45).
The two models generate almost identical parameter values except for nmin . For nmin ,Garicano et al. [2016] obtain nmin = 4 and my model implies nmin = 1.7796. For otherparameters, the estimates under the two models are listed in the following table.
Table 3: Comparison
Baseline GLV with Amadeus GLV
β
1.848
1.849
1.8
(0.031)(0.028)(0.054)n̄
57.90357.83959.271(0.98)(0.901)(2.051)
τ -1
0.021
0.021
0.023
(0.005)(0.006)(0.008)
σ
0.1
0.099
0.121
(0.014)(0.013)(0.033)
Number of Firms
62549
62549
41067
The first two columns use the same data from Amadeus 2006. The first column assumes
entrepreneurial risks and the second does not. For the purpose of comparison, the thirdcolumn lists the estimates from Garicano et al. [2016] that use FICUS data.
One can see that the first two columns are almost identical. This is not surprising since the58

0

MLE fit excluding truncated area

1

10

MLE fit including truncated area

10

data
estimates

data
estimates

0

10

-1

10

10-1
-2

10

10-2
10-3
10-3

10-4

10-5
101

10-4

102

103

10-5
100

104

101

102

103

104

105

106

Figure 12: MLE fit. The left panel shows the fit on the truncated region. The right panel
shows the fit on the entire region.
only difference between the two estimates is the shape of the indifference condition. Although
the indifference condition affects nmin , since the likelihood is truncated, it does not containmuch information for the estimation of (n̄, β, σ, T ). In fact, if there is no observation error
σ = 0, the truncation makes the likelihood estimation with and without entrepreneurial risks
exactly identical. Therefore, in terms of the estimation, the existence of the entrepreneurial
risks dose not matter.
One can also see that the second and the third columns are not that different. Therefore,
Amadeus 2006 is a good approximation of the administrative FICUS data. Hence, even if I
use FICUS data to estimate all the model parameters, I will get almost the same values as
Garicano et al. [2016].
To see how well MLE estimation fits the data, I refer readers to Figure 8 of Garicano
et al. [2016] for FICUS data and Fig.12 for Amadeus.

EIntermediate signal

In this section, I show one way to introduce intermediate signal. The main finding is thatthe efficiency gain might not be monotonic over riskiness.
To study the intermediate signal, I need to take a stance on the welfare criterion and theparameterization of the signal structure. For the welfare criterion, since the model becomes59

heterogeneous agents whenever there are informative signals, I focus on the aggregate output
instead of focusing on a particular Pareto weight. For the parametric assumption on thesignal, I assume that each individual observes
si = 1π(zi ,w)≥ηw , η ∈ [0, 1] .

(46)

I could choose other signal structures such as log normal, but this signal structure has
at least two several advantages in the current setting. First, the parameter η has a naturalinterpretation, i.e., agents who observe si = 1 know that they can earn at least η% of theirsalary if they are engaged in entrepreneurship. The interpretation makes it clear that η = 0
corresponds to full-risk case, and η = 1 to no-risk case. Specifically, when η = 0, the signal
is uninformative, so the equilibrium is characterized by (8), while when η = 1, the wage
and signal threshold satisfying (7) is an equilibrium. Therefore, η ∈ [0, 1] is a legitimateparameter that connects the full-risk and no-risk cases. Second, the signal structure (46)preserves power law. Since the productivity z follows a power law and the signal gives atruncation
π (zi , w) ≥ ηw ⇔ zi ≥ z (η, w) ,the distribution of the productivity conditional on the signal is still a power law. This allows
me to use the same parameter values as in 4.1.
Under the signal structure (46), the equilibrium system is a function of η. For small η,
there are plenty of agents observing si = 1, so the equilibrium (w, φ) is determined by
E [u (π (zi , w)) |si = 1] = u (w) , φE [n (zi , w) |si = 1] = 1 − φ, P (si = 1) > φ.
Mathematically, the third condition is slack for small η. As η increases, the third condition
becomes tighter, and when it binds, the indifference condition has to be relaxed. Hence, forlarge η,E [u (π (zi , w)) |si = 1] > u (w) , φE [n (zi , w) |si = 1] = 1 − φ, P (si = 1) = φ.
The results are shown in Fig.13. For each η, the wage is normalized to wSDP = 1. At thetwo extremes, the efficiency gain from removing the SDP is −1.5% and .02%. For low value ofη, neither economy reacts to the increase in η. This is because zmin > 0, so for the small valueof η, the information about the productivity is not informative π (zmin , w) ≥ ηw. As η crosses
some threshold, the outputs of both economies start to go up since the average entrepreneurs
are more productive. However, the speed of the output growth might be different, so thedifference might not be monotone. In the current setting, η = 45% is the threshold above
60

10-3

5

0

-5

-10

-15

-20
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 13: Counterfactual analysis with intermediate signal.
which the laissez-faire economy produces higher output. For different signal structure, theefficiency gain can be smoother.

F

Constrained efficiency

F.1

Constrained efficiency with intermediate signal

The constrained planner’s problem is
max
s̄,w

Z ∞
s̄

E [u (π (z, w)) |s] dGs + Gs (s̄) u (w) s.t.

Z ∞s̄E [n (z, w) |s] dGs = Gs (s̄) .

The idea is that the planner’s choice set is restricted to the number of firms φ, or equivalentlys̄. All other allocations, such as employment schedule and consumption, are determined by
the market. This is a stronger result of efficiency and follow the spirit of Diamond [1967].
Denote the solution by φCP . By restricting the functional form, a similar result as unconstrained efficiency can be obtained.
Claim. Fix CRRA utility and Cobb-Douglas production functions (u, f ). The laissez-faireeconomy generates insufficient number of firms φLF < φCP compared to the constrainedplanner iff there are entrepreneurial risks.


Proof. The equilibrium of the laissez-faire economy is characterized by s̄LF , wLF that sat61

isfies

i
i 1
1−γ
1
1 − Gs (s̄) h 1−α
1 − α  h 1−α1−γ
E z |s ≥ s̄ =.
E z |s = s̄Gs (s̄)αThe planner solve
max
s̄

h 1−γ
i
α 1−γα1 − Gs (s̄) Gs (s̄)E z 1−α |s ≥ s̄ +w (s̄)1−γ(1 − α) α 1−α w (s̄) α−1
1−γ
1−γ
!1−α

i
1
1 − Gs (s̄) h 1−αs.t. w (s̄) = α
E z |s ≥ s̄Gs (s̄)Substituting out the constraint with respect to wage, the planner’s problem becomeshi −α(1−γ)

1

h 1−γ
i E z 1−α 1s≥s̄
1 
1−γ

max
E z 1−α 1s≥s̄ 
(1 − α)s̄
1−γGs (s̄)

+Gs (s̄) α1−γ 1hE z 1−α 1s≥s̄i (1−α)(1−γ) Gs (s̄)Taking derivative with respect to s̄ and evaluating the FOC at s̄LF , wLF givesD s̄LF , wLF =

=


Z ∞



Ez|s 

× wLF






 dGs + Gs







s̄LF u0 wLF






− (1 − α)−γ α 1−α wLF

 1−γ

α−1

h

1−γ

E z 1−α |s ≥ s̄LF=:D(s̄LF ,wLF )1 − Gs s̄LF










i



}


{z





w0 s̄LF



1−αγ


1 − Gs (s̄LF )



|
−γ 

 

∂w

 s̄LF







 Gs s̄LF

If D s̄LF , wLF ∂u π z, wLFw0 s̄LF .

< 0, it means that decreasing s̄ increases welfare, so φLF < φCP . Since







w0 s̄LF < 0, it suffices to show D s̄LF , wLF > 0. By applying the indifference condition
and the market clearing condition,
h

1−γ

i

α E z 1−α |s ≥ s̄Gs (s̄)h 1−γi−
D (s̄, w) =1 − Gs (s̄) 1 − α E z 1−α|s = s̄


=

h

1
1−α

i

E z |s ≥ s̄
α 



h 1−γ
i 1
1−α
1−γ
E z 1−α |s = s̄62i
E z |s ≥ s̄ i
− h 1−γ

h

1−γ
1−α

E z 1−α |s = s̄When γ ≥ 1, D s̄LF , wLF > 0. To see this, note that, by lemma 1, the first term is larger
than 1.
ih 1
ih 1
E z 1−α |s = s̄E z 1−α |s ≥ s̄
 h 1−γi 1 >  h 1−γi 1 > 11−γ
1−γ
E z 1−α |s = s̄E z 1−α |s = s̄1−γThe second term is smaller than 1 since z 7→ z 1−α is a decreasing function, so the higher the
signal is the lower the integrand is. When γ ∈ (0, 1), D s̄LF , wLF > 0 still holds. To see
this, note that by Jensen,
h

E z



h

1−γ
1−α

i

|s ≥ s̄ ≤ E z 1−α |s ≥ s̄1−γ

E z 1−α |s = s̄
h

E z

1−γ
1−α

1

 h

i1−γ

i 1

1−γ



|s = s̄h1−γ= E z 1−α |s = s̄

i

⇒

h

1−γ

h

1
1−α

E z 1−α |s ≥ s̄E z|s ≥ s̄

i
i ≤



1

h

E z 1−α |s ≥ s̄i−γi 1 −1
1−γ

≤



h

E z

1
1−α

|s = s̄1−1i1−γ  1−γ



h

1

= E z 1−α |s = s̄iγMultiplying these two equations gives


h

1−γ

E z 1−α |s = s̄
h

i 1

1−γ

1

E z 1−α |s ≥ s̄

i

h

1−γ

i

h

1−γ

i ≤

E z 1−α |s ≥ s̄E z 1−α |s = s̄



h

1

i γ

h

1

i

E z 1−α |s = s̄E z 1−α |s ≥ s̄

≤ 1.

Since all I use is Jensen’s inequality, φLF = φCP when there are no entrepreneurial risks.
The functional form assumption is necessary. If we relax the assumptions to just u0 > 0 >u00 , it is possible to find f, G such that φCP < φLF . For the detail, see Ando and Matsumura[2017].

F.2

Constrained efficiency with financial frictions

The constrained planner solves
max φEu (π (z, w)) + (1 − φ) u (w)
φ,w

s.t.



π (z, w) = maxn znα − wn s.t. wn ≤ λznαφEn (z, w) = 1 − φ63.

The idea is the same as section F.1. The constrained planner can only choose the number offirms and has to let all other allocations be determined by the market.
Claim. Fix CRRA utility and Cobb-Douglas production functions (u, f ) and arbitrary risk
structure Gz with bounded support. If the financial constraints are severe λ ≈ 0, the laissezfaire economy generates an excessive number of firms φCP < φLF .
By substituting out φ using the market clearing condition and applying an increasingtransformation, the objective function can be written asn

U (w) := φ (w) Eπ (z, w)1−γ + (1 − φ (w)) w1−γ

o 1

1−γ

=w


 1
1−γ
1−γ
 Eπ(z,w)+ En (z, w) 
w1−γ

1 + En (z, w)

By taking derivative with respect to w,
1−γ





∂
∂ Eπ(z,w)∂+ ∂w
En (z, w)
En (z, w) ∂11  ∂w
w1−γ∂w
ln U (w) = +−1−γ∂w
w 1 − γ  Eπ(z,w)
1 + En (z, w) 
+ En (z, w)
1−γ
w



∂
The constrained efficient allocation is characterized by ∂w
ln U wCP
FOC at w = wLF gives



1−γ



1



1
1 
1
∂
wLF α−1ln U wLF = LF +
= LF
LF
∂w
w
1 − γ  1 + En (z, w )  w

= 0. Evaluating the

φLF
1−
1−α

(

)

Since φLF → 1 as λ → 0, the FOC at w = wLF can be negative. Hence, wLF > wCP . By themarket clearing condition φCP < φLF .

F.3

Constrained efficiency with capital accumulation

The formulation of the constrained efficiency problem inherits the ex-ante view of the plannerin the static model. In particular, I make the following thought experiment. Suppose allagents are in a symmetric equilibrium in which they face the same lottery and the economy
is in a stationary equilibrium. If, at some period t, a planner shows up and offers a new job
lottery, will everyone prefer it?
Such a thought experiment can be formulated as follows. Let φ = {φt }t be the probabilityof becoming entrepreneurs at each period chosen by the constrained planner. This is the only
choice that the planner can control. Other decisions are made by the market, i.e., each individual i takes φ as given and maximizes the recursive welfare by making consumption/saving

64decision. Formally, the agent i with the state (a, j, z) = (ait−1 , jit−1 , zit ) solves
Vt (a, j, z; φ) =max

(c,a0 )∈Bt (a,j,z)u−1
I [(1 − β) uI (cit ) +i

βuI u−1 (φt Eu (Vt+1 (a0 , E, z 0 ; φ)) + (1 − φt ) Eu (Vt+1 (a0 , W, z 0 ; φ)))(47)

Note that the budget set reflects that the production schedule {π (z, w, k) , n (z, w, k)} is
determined by the market, not by the planner.
Since all the equilibrium objects are functions of the planner’s choice φ, the wage process
w = {wt } also has to be a function of φ. As in the static model, the labor market clearing
conditionZZ
φt−1 nit di = (1 − φt−1 ) hit−1 dispecifies the mapping from the process of the number of the firms φ to wage w, denoted byw (φ). I note that this mapping is implicitly included in the value function Vt (a, j, z; φ).
In summary, the constrained planner can be defined as follows. Fix the fundamentalsnoE = (β, u, uI , f, fI , Gz , S0 ). The path of the number of firms wtCP , φCP
is constrainedttefficient if it solves
max
φ,w

Z

u−1 (φt Eu (Vt (ait , E, zit ; φ)) + (1 − φt ) Eu (Vt (ait , W, ait ; φ))) di

subject to {wt } = w (φ).
Claim. Fix CRRA intra-temporal utility, log inter-temporal utility, and Cobb-Douglas production and asset transformation functions. Then, the laissez-faire economy generates aninsufficient number of firms φLF < φCP .
Proof. The Bellman equations of the constrained planner’s problem after maximizing out
consumption-saving decision are the same, i.e., Eq.(40), (41) and (42) all hold. However, theindifference condition no longer holds. Still, I can derive the constrained planner’s objectivefunction. Note that by (42), the value of the next period can be written as

φEv E (z)1−γ + (1 − φ) v W

11−γ  1−γ





= φE


r (z, w)w

! (1−β)(1−γ)
1−βθ

 1
1−γ



+ 1 − φ

vW .

(48)



By substituting this condition into the worker’s Bellman equation (41), I can solve the65worker’s value function v E in closed formln v W = ln (1 − βθ) +βθ1ln βθ +ln w
1 − βθ
1 − βθ+βr (z, w)
ln φE

1−β 
w

 1
1−γ



! (1−β)(1−γ)
1−βθ

+1−φ

(49)




As a result, the planner maximizes (48) subject to (49) and the market clearing condition(39). Eq.(48) is equivalent to the objective function because the pre-determined asset distribution and the current productivity are independent ait ⊥⊥ zit and the value functions arejmultiplicative with respect to asset Vt (a, j, z) = vt (z) aρ . In other words, the constrainedplanner solvesmax φE
φ,wr (z, w)
w

 1
1−γ



! (1−β)(1−γ)
1−βθ+ 1 − φ



1
1
1

α 1−α Ez 1−α w α−1 φ = 1 − φv W s.t. (49)By taking log of the objective function, the FOC is

U 0 (w) w =






1
1
1−φ−
+
1 − βθ (1 − γ) (1 − β) (1 − α) Since the indifference condition E
faire economy, (see Eq.(43))
U

0



w



LF

r(z)
w



 (1−β)(1−γ)1−βθ(1−β)(1−γ)
E
1−βθ



E

r(z)
w



r(z)
w

 (1−β)(1−γ)
1−βθ (1−β)(1−γ)
1−βθ




+ φ1 − 1 + φ1 − 1.

= 1 holds in the equilibrium of the laissez-

1
φLF
=
1−
.
1 − βθ
1−α
!

w

LF

β
R 0.
Hence, φCP R φLF if and only if φLF Q 1 − α, which is then equivalent to γ + (1 − θ) 1−β
Since the inequality is always positive, the laissez-faire economy always generates insufficientfirms.

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