A Dynamic Model of Entrepreneurship With Borrowing Constratints

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A Dynamic Model of Entrepreneurship with

Borrowing Constraints: Theory and Evidence

Francisco J. Buera

Northwestern University

August 2005

Abstract

Do nancial constraints have persistent eects on the creation of busi-

nesses, or should we expect that forward-looking individuals always save

their way out of credit constraints? This paper studies the interaction

between savings and the decision to become an entrepreneur in a multi-

period model with borrowing constraints to answer this question. The

model has a simple threshold property: able individuals who start with

wealth above a threshold save to become entrepreneurs, while those who

start below this threshold remain wage earners forever. The theory gen-

erates a well-dened transition from wage earners to entrepreneurs amajor focus of recent empirical work. The probability of b ecoming an

entrepreneur as a function of wealth is increasing for low wealth levels

as predicted by standard static models - but it is decreasing for higher

wealth levels. US data are used to study the qualitative and quantitative

predictions of the dynamic model. Welfare cost of borrowing constraints

are found to be signicant, around 6% of lifetime consumption. The size

of poverty traps is small.

I thank Fernando Alvarez, Gary Becker, Pierre-Andr Chiappori, Marty Eichenbaum,Xavier Gine, James Heckman, Boyan Jovanovic, Joe Kaboski, Alex Monge, va Nagypl,Juan Pablo Nicolini, Robert Townsend and Ivn Werning for valuable comments and sugges-tions. I also beneted from comments of participants at seminars at UCLA, the Universityof Chicago, Universidad Torcuato Di Tella, Stanford University, Universitat Pompeu Fabra,Cem, University of Maryland, Boston University, Northwestern University, Purdue Univer-sity, the NBER Working Group on Entrepreneurship and University of Pennsylvania. E-mail:[email protected]

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1 Introduction

There is wide support for the view that nancial frictions aect the dynamicsof small rms. Investment by smaller rms is more sensitive to cash-ows and

aggregate shocks, age and size of rms are positively correlated, and the exit

hazard rate of rms is decreasing with size and age.1 A recent theoretical

literature has studied alternative models of rm dynamics in environments with

nancial constraints and has shown how they help to interpret the empirical

evidence.2 Still, a fundamental question remains open. How are rms that

require capital investment created in the rst place? Do nancial constraints

have persistent eects on the creation of businesses, or should we expect that

forward-looking individuals always save their way out of credit constraints? I

present a simple theory of this process and provide answers to these questions.I study the optimal saving decision of individuals facing a choice between

working for a wage or starting a business in a continuous-time framework. 3 The

optimal decision to save is shown to have a simple threshold property. Able in-

dividuals who start with wealth above the threshold but below what is needed

to start a protable business save to become entrepreneurs; able individuals

who start with wealth below the threshold have low savings and remain work-

ers forever. In this sense, individuals with wealth below the threshold are in a

poverty trap and nancial frictions can have persistent eect on the number

of businesses. Furthermore, it is shown that the threshold wealth above which

able individuals nd it optimal to save to become entrepeneurs is a decreas-ing function of ability. Indeed, there is threshold ability such that individuals

above it are never in a poverty trap, regardless of their initial wealth. In this

sense, the size of poverty traps is bounded and nancial frictions have only a

transient eect for suciently able individuals.

By studying the interaction between nancial frictions and the creation of

businesses, this paper closely relates to the recent development literature (e.g.,

the earlier work by Banerjee and Newman (1993), Galor and Zeira (1993)).4 In

1 See Caves (1998), Hubbard (1998) and Stein (2001) for recent surveys of this literature.2 See Albuquerque and Hop enhayn (2004), Clementi and Hopenhayn (forthcoming), Cooley

and Quadrini (2001) and Hopenhayn and Vereshachagina (2005).3

The continuous-time framework allows a simple characterization of the savings problemdespite the non-convexity introduced by the occupational choice. I extend earlier work bySkiba (1978) and Brock and Dechert (1983) to study the dynamics of wealth and occupationalchoice.

4 See also the theoretical contributions by Aghion and Bolton (1997), Lloyd-Ellis and Bern-hardt (2000), Caselli and Gennaioli (2002), Matsuyama (2000, 2003), Mokeerjee and Ray(2002, 2003), Galor and Moav (forthcoming); and the recent quantitative evaluation of these

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these papers, borrowing constraints aect productivity and the distribution of

income by restricting agents from protable occupations that require capital,

such as entrepreneurship. Individuals that start poor are doomed to remain

poor. A limitation of these analyses is that they rely on strong assumptions.

Generations are assumed to live for a single time period and the evolution of

wealth is determined by a warm-glow bequest motive that is not forward-looking.

This paper sheds light on the robustness of these results to environments where

savings are forward-looking.

As prevously discussed, in a model with forward-looking savings, initial con-

ditions can have a persistent eect for individuals that start suciently poor.

At rst glance, lessons drawn from myopic models of the evolution of wealth

seem robust. There is an important caveat, though. In forward-looking models

poverty traps are a function of the protability of the business opportunities.More able entrepreneurs are less likely to be in a poverty trap and the most

able are never in a poverty trap, regardless of their wealth. Ultimately, it is a

quantitative question whether initial conditions play a signicant and persistent

role.

The size of poverty traps and the welfare cost of borrowing constraints turn

out to be especially sensitive to the span of control and the capital intensity

of the entrepreneurial technology. They are larger the higher is the span of

control or the more capital intensive is the entrepreneurial technology. These two

parameters determine the returns to scale to the factors that need to be nanced.

If the entrepreneurial technologies are fairly linear with respect to these factors,then the scale at which entrepreneurs obtain the surplus from operating their

technology will be very large. This implies that the savings needed to overcome

borrowing constraints involve too large a sacrice. Forward looking individuals

will choose to have low savings and remain workers in the long run. Summing

up, within the range of parameter values used in the literature,5 poverty traps

can be quantitatively important.

Ultimately, the relevance of these arguments relies on the empirical evidence

supporting its micro underpinnings. This route is taken by a complementary

empirical literature. An important focus of this literature is the study and inter-

pretation of the observed relationship between entry into entrepreneurship andwealth.6 A positive relationship between entry into entrepreneurship and wealth

models by Gine and Townsend (2004) and Jeong and Townsend (2005).5 In the case of the returns to scale, this set is unpleasantly large.6 Of particular relevance for this paper is the work by Evans and Leighton (1989), Evans

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is often seen as evidence in favor of borrowing constraints. Indeed, this is the

basic implication of a simple static model (e.g., Evans and Jovanovic (1989)).

Nevertheless, as argued by many authors, it is important to understand the

endogenous determination of wealth to be able to interpret this correlation.

For example, people with no entrepreneurial ability tend to save less and are

also less likely to start businesses. The observed correlation between entry and

wealth partially reects the endogenous determination of wealth as opposed to

a causal eect of liquidity7 . This paper directly addresses this question by

modeling the endogenous determination of wealth and entrepreneurship. Inter-

estingly enough, novel predictions arise from the analysis of a dynamic model

of wealth determination and entrepreneurship.

The analysis of the dynamic model provides three novel predictions that are

compared to data: (1) individuals who eventually become entrepreneurs havehigher savings rates than individuals who expect to remain workers; (2) the

growth rate of consumption of new entrepreneurs is higher than that of workers

and old entrepreneurs; (3) the probability of becoming an entrepreneur as a

function of wealth is increasing for low wealth levels as predicted by standard

static models - but it is decreasing for higher wealth levels.

The rst prediction summarizes nicely the essential ingredient of the model:

workers can save up to overcome borrowing constraints to become entrepreneurs.

As discussed above, depending on their relative skills, some workers choose to

never become entrepreneurs, either because it is not productively ecient or

because the savings required to overcome the constraints require too large asacrice. Others nd that becoming an entrepreneur will have a large enough

surplus and consequently save more. The second prediction follows from the fact

that new entrepreneurs are still constrained with respect to their capital level

and consequently have a higher marginal product of capital than the market

interest rate.

The last prediction follows from the fact that wealth and entrepreneurship

are jointly determined in the model. For low wealth levels, entry into entrepre-

neurship increases with wealth because it relaxes the borrowing constraint, just

as predicted by standard static models. For high wealth levels, however, en-

and Jovanovic (1989), Holtz-Eakin et. al. (1994), Glenn and Hubbard (2001), and Hurst andLusardi (2004) for the US and Paulson and Townsend (2004) for Thailand.

7 See Holtz-Eakin et. Al. (1994) for an early discussion of how the endogeneity of wealthmay aect the interpretation of the entry and wealth relationship. In a recent paper, Hurstand Lusardi (2004) present evidence against wealth being a bad indicator of liquidity forindividuals becoming entrepreneurs.

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try into entrepreneurship and wealth become negatively related. This negative

relationship reects the fact that over time individuals with high entrepreneur-

ial skills are selected out of the pool of workers and that this selection eect

increases with wealth. Intuitively, if an individual is rich and still works for a

wage, then it is unlikely that he has a high entrepreneurial skill. This prediction

contrasts sharply with the conventional wisdom in the empirical literature on

borrowing constraints and entrepreneurship.

The analysis of the model thus suggests that using data on saving rates,

consumption growth, and the transition into entrepreneurship as a function of

wealth would allow the estimation of the model. This is done by minimizing

the distance between the statistics describing the behavior of saving rates of

entrepreneurs and the dynamics of entrepreneurship in the U.S. data and the

same statistics from the simulated model. In other words, the model is estimatedby an Indirect Inference procedure as described by Gourieroux and Monfort

(1996).

At the estimated parameter values, welfare costs of borrowing constraints

are substantial. Consumption must be increase by 6% to make individuals in-

dierent between the economy with borrowing constraints and the one with

perfect capital markets. At the same time, poverty traps tend not to be im-

portant for the US economy. This is because the entrepreneurial technology is

estimated to have sharp decreasing returns to the variable factors that need to

be nanced.8 Welfare costs arise mainly because it take long for extremely able

entrepreneurs to operate their businesses at their unconstrained scale. Mostprotable individuals eventually get to start businesses in the estimated model.

It is important to note that the model is able to t the relationship between

the transition into business ownership and wealth, specially the fact that only

for households at the top of the wealth distribution is there a strong and posi-

tive relationship between household wealth and business entry (see Hurst and

Lusardi (2004)). Throught the lens of the dynamic model, the data is consis-

tent with borrowing constraints playing a substantial role on the dynamics of

entrepreneurship, although the eect of borrowing constraints on the creation

of businesses appear to be transitory.

This paper is also motivated by and complements a recent literature thatstudies the quantitative implications of models of occupational choice and bor-

rowing constraints. Numerical solutions of related models are studied by Quadrini

8 Note that this could be either because the span of control of (mainly small) businesses inthe data is low or because they operate labor-intensive technologies.

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(1999) and Cagetti and De Nardi (2003). These authors have shown that mod-

els featuring entrepreneurship and nancial frictions help to explain the thick

right tale of the wealth distribution in the U.S. In a similar vein, Cagetti and

De Nardi (2004), Li (2002), and Meh (forthcoming) quantify the eect of vari-

ous policies in models featuring entrepreneurs and credit contraints for the US

economy. Besides providing an analytical characterization of the dynamic of

wealth in this type of models, this paper illustrates how the conclusions of these

models can be quite sensitive to the parameters describing the technology of

entrepreneurs.

The rest of the paper is organized as follows. Section 2 describes the indi-

viduals dynamic occupational choice problem and Section 3 characterizes its

solution. Section 4 examines simple numerical examples to understand the im-

portance of poverty traps and the welfare cost resulting from borrowing con-straints implied by the model. Section 5 presents the implications of the model

for the average relationship between wealth and the likelihood of a transition

to entrepreneurship. Section 6 confronts the predictions of the model and mea-

sures the welfare cost of borrowing constraints by estimating the model using

U.S. data. Section 7 concludes and discusses directions for future research.

2 The Model Economy

The model is set in continuous time. Households are endowed with entrepre-

neurial ability, e, and initial wealth, a0. In each instant of their life, they havethe option to work for a wage, w, and invest their wealth at a constant interest

rate, r, or to work and invest in an individual specic technology with produc-

tivity e, i.e., to become entrepreneurs. If households decide to be entrepreneurs

they must devote all their labor endowment to run their businesses, i.e. occu-

pations are indivisible. This captures a fundamental non-convexity: households

are more productive by specializing in one activity. Households are only allow

to borrow up to a fraction of their wealth.

2.1 Preferences

Agents preferences over consumption proles are represented by the time sep-

arable utility function

U(c) =

Z10

etu (c (t)) dt (1)

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where t is the age of the individual and is the rate of time preference. The

utility function over consumption, u (c), is strictly increasing and strictly con-

cave.9

The innite horizon is a convenient analytical assumption. The theory

should be understood as describing the life-cycle of an individual. Under this

interpretation, = + p, where is the rate of time preferences and p is the

constant rate at which agents die. When studying the quantitative implications

of the theory, a discrete time version of the model with nite horizon will be

simulated and estimated.

2.2 Resource Constraints and Technologies

Agents start their lives with wealth a0. At any time t 0, their wealth, a (t),evolves according to the following law of motion

_a (t) = y (a (t)) c (t) t 0, (2)

where y (a (t)) is the income of the agent with wealth a (t), and _a refers to@a(t)@t

.10 The shape of the income function depends on occupational choices as

follows.

If agents choose to be wage earners, they will sell their labor endowment for

a wage w and invest their wealth at a rate of return r. In this case, their income

y (a) is

yW (a) = w + ra, (3)

where ra is the return on their wealth a. I refer to w as the wage, but it should

be understood that wages are individual-specic. Formally, w = wl, where l

are the eciency units that an individual can supply and w is the price of an

eciency unit of labor.

If individuals run a business they must devote their entire labor endowment

to operate the business. Their revenue is given by the function, f(e; k), where e

is the agent-specic ability and k is the amount of capital invested in the busi-

ness. 11 f(e; k) is assumed to be strictly increasing in both arguments, homo-

geneous of degree 1, and strictly concave in capital, fe

(e; k) > 0, fk

(e; k) > 0,

9 In a life-cycle interpretation of the model t is the age of the agents and = + p,where is the rate of time preferences and p is the constant rate at which agents die. T hisinterpretation of the m odel is used when studying the quantitative implications of the theory.

10 For simplicity of exposition, I drop the time as an argument of the dierent functions.11 The production function should be interpreted as the reduced form of a more general

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fkk (e; k) < 0. Inada conditions are assumed to hold, limk!0 fk (e; k) = 1

and limk!1 fk (e; k) = 0. A higher entrepreneurial ability is associated with

a higher marginal product of capital, fek (e; k) > 0, also f (0; k) = 0 and

lime!1 f(e; k) = 1.

The amount of capital that agents can invest in their businesses is con-

strained by their wealth. To focus the analysis on the interaction between

individual savings and occupational choice, I choose a simple specication of

borrowing constraints. In particular, I assume that the value of an individuals

business assets, k, must be less than or equal to the value of their wealth, k a.

If wealth exceeds the value of desired business assets, the remaining wealth is

invested at the rate r.12

Therefore, the income of an entrepreneur solves the following static prot

maximization problem:

yE (e; a) = maxka

ff (e; k) + r (a k)g . (4)

Note that the scale of the business equals the individuals wealth, a, as long

as wealth is lower than the unconstrained scale of the business, ku (e). The

unconstrained scale is the solution to the unconstrained prot maximization

problem, i.e.,

ku (e) = arg maxk

ff(e; k) rkg .

This function is strictly increasing. Inada conditions are necessary to guarantee

that this function is well dened for all e.

technology requiring capital and labor,

f (e; k) = maxn

~f (e;k;n) wn,

where n are the eciency units of labor employed and w is the price of an eciency unitof labor. When calibrating the model and when discussing the predictions of the model fortechnologies with dierent capital intensities, the more general notation will be used.

12 When studying the quantitative implications of the theory, I allow for entrepreneurs toinvest up to a fraction of their wealth, i.e., k a with 1.

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2.3 Consumers Problem

Agents choose proles for consumption, c (t), wealth, a (t), occupational choice,and business assets, k (t), to solve

maxc(t),a(t),k(t)0

Z10

etu (c (t)) dt

s:t:

_a (t) = y (a (t)) c (t)

y (e; a (t)) = max

yE (e; a (t)) ; yW (a (t))

.

As is implicitly recognized in the statement of the problem, the occupational

decision is a static one. That is, given current wealth,a

, agents choose to beentrepreneurs if their income as entrepreneurs, ye (e; a), exceeds their income as

wage earners, yw (a), i.e., ye (e; a) yw (a).

This can be expressed as a simple policy function. Dene e to be the ability

at which individuals are just indierent between being wage earners and being

entrepreneurs conditional on being able to borrow at the interest rate r.13 Able

individuals (individuals with ability above e) decide to be entrepreneurs if their

current wealth is higher than the threshold wealth a (e), a a (e),where a (e)

solves

f (e; a (e)) = w + ra (e) .

Intuitively, agents of a given ability choose to become entrepreneurs if theyare wealthy enough to run their businesses at a protable scale. Alternatively,

agents of a given wealth a choose to become entrepreneurs if their ability is high

enough. Both ability and resources determine the occupational decision.

Given the optimal static decision, the dynamic program is equivalent to a

standard capital-accumulation problem subject to a production function of the

form

y (e; a) =

8>:

w + ra ifa 2 [0; a (e))

f(e; a) ifa 2 [a (e) ; ku (e))

f(e; ku (e)) + r (a ku (e)) ifa 2 [ku (e) ; 1)

.

13 e solvesmaxk

f (e; k) rk = w.

The left hand side of this equation is well dene, increasing, continuous and take the valuezero for e = 0 and goes to innity as e goes to innity.

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a

y(e,a)

a

yE(e,a)

yW(a)

w

ku

Figure 1: Technologies Available to Households

This technology is given by the upper envelope of the wage earner technology,

yw (a), and the entrepreneurial technology, ye (e; a). Figure 1 describes these

technologies. Notice that this production function is not concave. The return

to capital increases if individuals invest more than a (e).

Necessary conditions for the wealth accumulation problem are given by

1. the Euler equation,

u00 (c) c

u0 (c)

_c

c=

8>:

r ifa 2 [0; a (e))

fk (e; a) ifa 2 [a (e) ; ku (e))

r ifa 2 [ku (e) ; 1)

, (5)

stating that the marginal rate of substitution should equal the marginal

rate of transformation;

2. the law of motion for wealth,

_a = y (e; a) c, (6)

describing the evolution of the individual wealth;

3. and the tranversality condition,

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limt!1

etu0 (c (t)) a (t) = 0, (7)

stating that value of wealth should converge to zero.

In the present case, these conditions are only necessary. As in any non-

convex problem, there are solutions to the rst order conditions that correspond

to global minima or to local maxima that are not global maxima. Also, there

can be multiple maxima. In section 4, I analyze the optimal accumulation path

under this technology.

I conclude this section by noting that:

Remark 1: The model is homogeneous of degree 1 in (a ;w ;e) :

Exploiting this property, I normalize all the variables in the model by the

wage. When studying the behavior of entrepreneurs in the data, this also sug-gests that wealth to wage ratios are the relevant measure of resources available

to individuals.

3 The Evolution of Individual Wealth

This section characterizes the evolution of individual wealth. The main re-

sults are: (a) There exists a threshold wealth level, as (e), such that individuals

with initial wealth below the threshold, a0 < as (e), follow a path associated

with decreasing wealth, converging to a zero-wealth steady state where they

are wage-earners. Meanwhile, households with initial wealth above the thresh-old, a0 as (e), save to become entrepreneurs and converge to a high-wealth

entrepreneurial steady state. (b) The function as (e) is strictly decreasing in

entrepreneurial ability and there exists a minimum ability, ehigh, such that in-

dividuals with ability above ehigh save to become entrepreneurs regardless of

their initial wealth. (c) The threshold as (e) is increasing in the discount rate

and decreasing in the intertemporal elasticity of substitution.

3.1 Analysis of the Phase Diagram

As is standard in the analysis of deterministic dynamic models in continuous

time, this section proceeds by analyzing the phase diagram of the system of or-dinary dierential equations (5) and (6) given the agents rst order conditions.

The analysis is restricted to the case of r < .14

14 Buera (2004) shows that r < in a stationary equilibrium of a general equilibrium versionof the model where non-altruisitc individuals die at a constant rate, their wealth is inherit by

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a

c

a ass

a = 0.

I III

II IV VI

V

c = 0

.

css

c

w

Figure 2: Phase Diagram of Households Problem (r < )

By setting _c = 0 in (5) an equation that describes the set of wealth-consumption

pairs, (a; c), for which consumption is constant over time is obtained. In the case

fk (e; a) > , there exists a unique solution to this equation.15 In particular,

there exists a unique value of wealth, ass 2 (a,ku), solving the equation

fk (e; a) = 0:

This gives the rightmost vertical curve in the (a; c) space. I label it _c = 0 in

Figure 2. To the left of this curve, the return to capital exceeds the rate of

time preference, therefore consumption increases over time. To the right of this

curve the opposite is true.

Additionally, note that the locus a = a divides the space between points for

which consumption decreases (to the left) and points for which consumption

increases (to the right). At a agents switch occupations. Thus, the relevant

return to their wealth changes from being low, r < , to being high, fk (e; a) > a single descendant, and a corporate sector demand labor and capital services as in Cagettiand De Nardi (2004). Otherwise, the wealth of all individuals would drift upward.

15 If fk (e; a) < there is no steady state with positive wealth. In this case, it is optimal forindividuals to desacumulate wealth. Individuals with a (0) > a(e) start being entrepreneursbut they eventually become workers.

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(to the right of a). This can we seen by inpecting the Euler equation (see

Equation (5)).

Similarly, by setting _a = 0 in equation (6), I obtain an equation that de-

scribes the locus of points where wealth is constant. These correspond to points

for which consumption equals income. Above this curve consumption exceeds

income and therefore wealth decreases over time. Below, consumption is less

than income thus wealth decreases over time. This curve is labeled as _a = 0 in

Figure 2.

Trajectories in Region III move to the northwest, those in Region IV move

to the northeast, in Region V they move to the southwest, while those in Region

VI travel to the southeast.

Combination of wealth and consumption (a; c) in Region I will follow trajec-

tories going to the southwest. These are points for which consumption exceedsincome, and wealth is not high enough for the entrepreneurial technology to be

protable, therefore the relevant return to savings is given by the interest rate

that is lower than the rate of time preference. Wealth and consumption pairs

in Region II also correspond to pairs with low wealth level, and low return to

savings, therefore consumption will tend to decrease. But for pairs in Region

II, since consumption is lower than income, wealth increases over time.

Of all these trajectories, only the ones converging to the points (ass; css)

and (0; w) satisfy the transversality condition and do not exhibit jumps in nite

time. For example, trajectories in region I above the one converging to the

point (0; w) will eventually hit the y axis above w and will be associated with adiscontinuous consumption path in nite time.

To identify the trajectories that converge to the two steady states, it is

helpful to view the phase diagram of this model as the combination of the phase

diagram of the standard neoclassical growth model (regions III, IV, V and VI)

and the phase diagram of the saving problem of a worker with no borrowing

constraints and r < (regions I and II).

From the analysis of the standard growth model it is known that in a neigh-

borhood of (ass; css), there exists a single trajectory converging to this steady

state (the stable path). In a similar fashion, from the savings problem with

r < , it is known that, locally, there exists a single trajectory passing throughthe point (0; w).

Since the problem is not concave, there is not a unique path starting from

a given level of initial wealth that satises the necessary conditions. Also, the

necessary conditions are not sucient: there may be many trajectories that

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aa ass

a = 0.

I III

II IV VI

V

a* a*

c = 0

.

c

Figure 3: Trajectories Satisfying the Necessary Conditions (Intermediate Abil-ity)

start from a given level of initial wealth and satisfy the necessary conditions.

But most of them are not optimal.

For instance, for levels of initial wealth close to a there exist at least two ini-

tial levels of consumption associated with trajectories that satisfy the transver-

sality condition and do not exhibit jumps in nite time. The one with high

consumption will lead in nite time to the low-wealth steady state. The tra-

jectory associated with low initial consumption will eventually lead to the high

wealth entrepreneurial steady state.

Moreover, for a0 = a there exist many other initial consumption levels that

eventually converge to one of these steady states after cycling around the point

(a; c). Figure 3 illustrates the trajectories satisfying the necessary conditions

for the case of intermediate ability (see Proposition 1).

The next result characterizes the possible congurations of the trajectories

in the phase diagram: for agents with high entrepreneurial ability only the tra-jectory converging to the low wealth worker steady state cycles around the point

(a; c); for agents with low entrepreneurial ability only the trajectory converging

to the high-wealth steady state cycles around the point (a; c); for individuals

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with intermediate entrepreneurial ability Figure 3 is the relevant case.16

Proposition 1: There are three possible congurations of the trajectories sat-isfying the necessary conditions:

1. Only the trajectory converging to the (ass; css) steady state cycles around

the point (a; c).

2. Both the trajectory converging to the (0; w) steady state and the trajec-

tory converging to the (ass; css) steady state cycle around the point (a; c)

(intermediate case).

3. Only the trajectory converging to the (0; w) steady state cycle around the

point (a; c).

The next step is to discriminate among all of the paths satisfying the neces-

sary conditions for optima. In order to do this, I use results from Skiba (1978)

and Brock and Dechert (1983) developed to analyze optimal control problems

when the Hamiltonian function is not concave in the state variable (wealth a

in the present context). An executive summary of their results is the following:

(a) whenever there are multiple candidate paths the ones with extreme initial

value for the co-state (or consumption in our application) are optimal; (b) if

the optimal co-state as a function of the state jumps (exhibits discontinuities),

then it will jump down.

Applying Skiba (1978) and Brock and Dechert (1983) I prove the main re-sult of this section: given an ability level e, households with low initial wealth

will follow a path converging to a zero wealth worker steady state, and house-

holds with high initial wealth will follow a path converging to a high wealth

entrepreneurial steady state.

Proposition 2: Depending on the conguration of the trajectories satisfying

the necessary conditions (see Proposition 1), the optimal trajectories are:

1. In the rst case of Proposition 1, for all levels of initial wealth it is optimal

for agents to follow the trajectory converging to the (0; w) steady state.

2. In the intermediate case of Proposition 1, there will a single initial wealth,

as (e), such that individuals starting with wealth level, as (e), will be indif-

ferent between following the trajectory converging to the (0; w) steady state

16 All the proofs are in the appendix.

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or the trajectory converging to the (ass; css) steady state. Agents with ini-

tial wealth to the left of as (e) prefer to follow the trajectory converging to

the (0; w) steady state. The converse holds for agents starting with wealth

to the right of as (e).

3. In the third case of Proposition 1, for all levels of initial wealth it is optimal

for agents to follow the trajectory converging to the (ass; css) steady state.

Intuitively, households with low initial wealth require a larger investment in

terms of forgone consumption to save up toward the ecient scale. Thus, they

prefer to have a lower but smoother consumption prole as wage earners. Figure

4 illustrates the optimal trajectories in the intermediate ability case (for e low

enough as > a, implying that there are individuals that start as entrepreneurs,but choose to eat their wealth an eventually become workers).

This proposition tells us that the typical policy function for consumption will

be discontinuous. For agents with low initial wealth, it is optimal to start with

relatively high consumption, but one that is decreasing. For agents with high

initial wealth it is optimal to start with a relatively low consumption, but which

is increasing. Moreover, there is a unique threshold on initial wealth that divides

individuals into these two groups. I refer to this threshold as the poverty trap

threshold. The poverty trap threshold is a function of entrepreneurial ability.

This characterization implies the following corollary.

Corollary to Proposition 2: (a) The saving rate of individuals who eventuallybecome entrepreneurs is higher than the saving rate of individuals who remain

wage earners. (b) The growth rate of consumption increases after individuals

become entrepreneurs.

This suggests two obvious tests for the model that are performed in section

7.

The next result states that the threshold as (e) is decreasing in the agents

entrepreneurial ability. It also tells us that there is a minimum ability elow and

a maximum ability ehigh such that nobody with ability lower than elow is an

entrepreneur in the long run and everybody with ability higher than ehigh is an

entrepreneur in the long run.

Proposition 3: If as (e) a (e), the poverty trap, as (e), is strictly decreas-

ing in the agents ability, i.e.,

@as (e)

@e< 0.

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a

c

aa

ss

c = 0

a = 0.

as

.

Figure 4: Optimal Trajectories (Intermediate Ability)

Moreover, there exits a strictly positive ability level, elow, such that

Vw (a; elow) Ve (a; elow) for all a

and a nite ability level, ehigh, such that

Ve (a; ehigh) Vw (a; ehigh) for all a,

where Vw (a; e) (Ve (a; e)) is the value of the problem if the trajectory with high

(low) initial consumption is selected.

The intuition of this result is straightfoward. For individuals that are still

workers, entrepreneurial ability only aects those who plan to follow the entre-

preneurial path. In other words, the more protable it is to be an entrepreneur,

the less likely an individual will be stuck in a poverty trap. The ability to save

gives an upper bound on the importance of borrowing constraints.

Next, the relationship between various parameters of the model and the

poverty trap threshold is discussed.

Proposition 4: The following is true in a neighbourhood of r = such that

r < : the poverty trap, as, is increasing in the discount rate, , and decreasing

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in the intertemporal elasticity of substitution, 1

. Moreover, for e close to elow,

the poverty trap, as, increases with the interest rate, while for e close to ehigh,

it decreases with the interest rate.

Poverty traps are more likely the lower the intertemporal elasticity of sub-

stitution and the higher the discount rate, since these two parameters aect the

cost of a high savings plan. Interestingly, the eects of the interest rate and a

given workers wage rate are ambiguous. A higher interest rate and a higher

wage make it easier for able individuals to save the capital needed to start a

protable business, but, at the same time, a higher interest rate and higher

wage raise the cost of being in business and make the option of being a wage

earner relatively more attractive.

In the next section, I give a quantitative assessment of the ability of savings

to overcome borrowing constraints to entrepreneurship.

4 A Numerical Example

This section studies a calibrated version of the dynamic model to understand

the potential of borrowing constraints to generate signicant poverty traps and

welfare cost when foward-looking saving decisions are incorporated into the

analysis.

4.1 Parametrization

I choose standard CES preferences

u (c) =c1

1

where is the reciprocal of the intertemporal elasticity of substitution.

Following Lucas (1978), I propose a constant return to scale technology on

the entrepreneurial ability, capital and labor,

~f (~e;k;n) = ~e

kn11

where represents the share of payments going to the variable factors, capitaland labor, that are paid to capital, and the share of payments going to the

entrepreneur (it is also referred to as the span of control parameter (Lucas

1978)). Note that I only need to specify the reduced form for output net of

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labor cost as a function of capital and entrepreneurial ability, once labor is

chosen optimally,

f(~e; k) = maxl

n~e

kn11

wno

= A~e1k,

where w is the price of an eciency unit of labor, A = (1 (1 ) (1 ))h(1)(1)

w

i (1)(1)1(1)(1)

,

= (1)1(1)(1) .

It convenient to normalize the ability index to correspond to the prots an

entrepreneur would make if unconstrained, e = maxk

A~e1k rk

. Note

that when holding x the unconstrained prots, e, and the level of capital, k,

the prots are decreasing in the parameter . The larger the share of capitalthe lower the prots of an entrepreneur holding constant the scale and the

unconstrained surplus. This implies that the larger the parameter the larger

will be the initial wealth required by an individual of ability e to prefer saving

in order to become an entrepreneur, i.e. @as(e;)@

> 0. Also, the scale at which

an individual nds it protable to start a business will be larger, @a(e;)@

> 0.

The bite of borrowing constraints will be larger the larger is (see Figure 7 for

an illustration of this result).

I assume individuals live and work for 40 periods. As mentioned when

introducing the model, the continuous time framework with innite horizon was

chosen to obtain an clearer analytical characterization of the problem. Whensimulating and estimating the model, it is more convenient and natural to work

with the discrete and nite horizon version of the theory.

4.2 Simulated Poverty Traps and Welfare Cost of Borrow-

ing Constraints

I need to specify ve parameters, , , r, and . The rst four parameters

can be set with little controversy. I choose = 1:5, a reasonable value for the

reciprocal of the intertemporal elasticity of substitution. I set the time period

to be 1 year, and, correspondingly, I let = r = 0:02 to reect the average

market return to wealth. The depreciation of business assets, , is set to 0:08.

It is less obvious how to choose a reasonable value for the curvature of the

entrepreneurial technology, . On the one hand, recent evidence from micro data

suggests to use a relatively low value for this parameter: Evans and Jovanovic

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

1

2

3

4

5

6

7

8

9

Unconstrained Profits/Wages (e/w)

Wealth/wages(a/w)

Entrepreneurs right away

entrepreneurs with delaynever

entrepreneurs

a

as

Figure 5: Poverty Trap, = 0:5.

(1989) estimate a static version of the above model using household data on

self-employment and nd = 0:39; Cooper and Haltiwanger (2000) estimate

= 0:6 using panel data on plants from the Longitudinal Research Dataset. On

the other hand, recent numerical exercises evaluating the eect of tax policies

using models of entrepreneurship have calibrated their model using values of

above 0:8 (e.g. Cagetti and De Nardi (2003), Li (2002)). I set = 0:5 as thebenchmark case.

Figure 5 illustrates the poverty trap threshold, as (e), as a function of the

ability of the entrepreneur. On the horizontal axis, I measure ability as the

prots an individual would make if operating at the unconstrained scale relative

to her wage, a measure of the returns to entrepreneurship. In the vertical axis, I

measure wealth relative to the wage.17 Individuals with ability and initial wealth

to the southwest of this curve, but with relative ability higher than one, are in

a poverty trap. Even though they could run protable businesses if operating

at a unconstrained scale, the cost in terms of an uneven consumption prole is

too large. I also plot the current wealth and returns combinations such thatindividuals are indierent today between starting a business and working for a

17 As discussed earlier, this is a natural normalization of the data since the parametrizedmodel is homogeneous of degree 1 in the opportunity cost, w, the wealth, a, and the entre-preneurial ability.

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

5

10

15

20

25

30

Years

a. Time to entry and time to reap half the unconstrained returns, = 0.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.2

0.4

0.6

0.8

1

Unconstrained Profits/Wages

Fraction

ofConsumption b. Consumption Equivalent Compensation for People that Start with Zero Wealth

time to enter

time to reap half the returns

no savings

savings

Figure 6: Figure: Time to Enter and Wealfare Cost, = 0.5

wage, i.e., the function a (e). This curve would be the poverty trap threshold

if individuals were not allowed to save. The dierence between the two curves

gives a measure of the role of savings.18 Initial wealth no longer fully determines

whether individuals become entrepreneurs. Rather, even individuals who start

with low wealth could save to become entrepreneurs. Nevertheless, individuals

that could earn up to 35% more as unconstrained entrepreneurs remain workers

if they start with zero wealth!

In Figure 6.a, I plot the time required to start a business starting from the

poverty trap threshold, as, as a function of ability. There, I also plot the time

required to earn half the unconstrained returns starting from the poverty trap

threshold, as. Even people that could earn 100% more income as unconstrained

entrepreneurs require ve years to save the capital required to become entrepre-

neurs and almost ten years to operate at a scale at which they make half the

unconstrained returns. The delay to entry and to operate at a protable scale

suggest important welfare losses.These welfare losses are illustrated in Figure 6.b. There I plot the fraction

18 Notice that for low ability individuals, a (e) < as (e), implying that these individuals willnever transit from being workers to being entrepreneurs. This is an illustration of the rstcase of Proposition 2.

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by which the path of consumption must be increased to make an individual of

a given ability, who is born with zero wealth, indierent between living in the

economy with no credit and in the economy with perfect capital markets. The

lower curve is the welfare cost associated with an economy where individuals are

allowed to save, while the upper ones are the welfare costs in a static model. As

the time to entry suggests (Figure 6.a), there are potentially enormous welfare

losses due to borrowing constraints. Interestingly enough, the ability of individ-

uals to cope with nancial frictions by saving is less eective than their ability to

ameliorate the eect of missing insurance markets. Aiyagari (1994) found that

the welfare cost of missing insurance markets are reduced by half when agents

can save to smooth consumption. In the case of nancial frictions, not even

for extremely productive entrepreneurs (those earning over 100% more income

as entrepreneurs) is it the case that the welfare costs of borrowing constraintsare reduced by at least 50 % when they are able to save in anticipation of their

future business opportunity.

As I mentioned when discussing the choice of parameter values, there is no

consensus regarding the value for . It is therefore important to understand the

sensitivity of these numerical results to the choice of. This is done in gures 7.a

(poverty traps) and 7.b (welfare costs). The size of poverty traps and the welfare

cost of borrowing constraints increase dramatically if we choose a value of

above 0:6 while they tend to be unimportant if the entrepreneurial technology is

characterized by strong decreasing returns to scale, below 0:4. This ambiguity

further motivates the study of U.S. data on savings rates and entrepreneurialdynamics to sharpen the understanding of the underlying parameters of the

model that is done in section 7.

Before turning to study the data on savings and entrepreneurial dynamics,

the next section explores the predictions of the model for the aggregate rela-

tionship between entry into entrepreneurship and wealth, the focus of a large

empirical literature that studies the importance of borrowing constraints to en-

trepreneurship.

5 Entry into Entrepreneurship and Wealth

Most of the empirical literature on entrepreneurship and borrowing constraints

has focused on measuring the eect of wealth on the likelihood that an individ-

ual becomes an entrepreneur. A positive relationship is often seen as evidence

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

2

4

6

8

1012

= 0.3

= 0.5

= 0.6

Wealth/Wage

a. Poverty Traps for Different Values of

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.2

0.4

0.6

0.8

1

= 0.3

= 0.5

= 0.6

Unconstrained Profits/WagesFractionofConsumption

b. Welfare Cost for Different Value of

Figure 7: Sensitivity to

in favor of the direct eect of borrowing constraints on entrepreneurship. Un-

derstandably, many researchers have expressed the concern that wealth and

unobservable ability may be positively correlated, making such interpretation

problematic. In this section I address these issues.The likelihood that an individual becomes an entrepreneur as a function

of current wealth, conditional on age, P (transitionja; t), is a non-monotonic

function of wealth. The fraction of individuals that enter entrepreneurship is

an increasing function of wealth for low wealth levels, as in a static model (e.g.

Evans and Jovanovic (1989)), and a decreasing function of wealth for high wealth

levels. Moreover, with time the transition probability is compressed from below

and above. The intuition for this result is extremely simple. When looking

at transitions, we are considering the agents that are working today. But this

is a selected sample. In particular, these are individuals that did not nd it

protable to start a business by the current period, i.e. they are relatively lowability individuals. Moreover, this selection increases with the wealth of the

agents. If somebody is rich and hasnt started a business yet, then he must be

a bad entrepreneur. Next, I give a formal statement of this result (see Figure 8

for an illustration this result).

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P(transition|a,t,)

P(transition|a,t)

t>t

Frac

tion

th

atent

er

aahigh(t)alow(t)

Figure 8: Transition into Entrepreneurship as a Function of Wealth and Age

Proposition 5: There exists an age t, an increasing function alow (t) and a

decreasing function ahigh (t), satisfying alow (t) ahigh (t), a positive constant

> 0, and neighborhoods N(ahigh (t) ; ) and N(alow (t) ; ) such that:

1. for all t < t

P (transitionja; t) = 0 8a 2 [0; alow (t)] [ [ahigh (t) ; 1);

and

P (transitionja; t) > 0 8a 2 N(ahigh (t) ; ) and a ahigh (t)

P (transitionja; t) > 0 8a 2 N(alow (t) ; ) and a alow (t)

2. For all t t

P (transitionja; t) = 0 all a.

This proposition suggests a way to obtain information about the importance

of borrowing constraints. In particular, it suggests a way to obtain a lowerbound on the unconstrained scale of businesses, i.e., the unconstrained scale of

the least ecient business for a xed wage and borrowing constraints coecient,

. Indeed, if we were to observe that the decreasing portion of the transition

into entrepreneurship occurs at very high wealth levels we would infer that

24


25/41

the entrepreneurial technology is close to linear, large , or that borrowing

constraints are very tight, low .

Remark 2: The minimum wealth level such that nobody with wealth higher

than this makes the transition to entrepreneurship is a lower bound on the un-

constrained scale of the least ecient business in operation, i.e. ahigh (t)

ku (e) = a (e).19

Next, I consider a straightforward implication of this result.

An important concern of the empirical literature on entrepreneurship and

borrowing constraints is to measure the causal eect of wealth on the likelihood

that an individual starts a business. It is often believed that the actual corre-

lation between entry into entrepreneurship and wealth overestimates the causal

eect (e.g. Holtz-Eakin et. al. (1994)). It turns out that in a dynamic model,

the opposite is true. From proposition 5 we know that, for high wealth levels,wealth close to ahigh, the transition to entrepreneurship as a function of wealth

is decreasing. But the eect of a wealth transfer is still strictly positive. Thus,

the observed eect of wealth on entrepreneurship underestimates the eect of a

wealth transfer. This is the content of the next remark.

Remark 3: For 0 < t < t

@P (transitionja;t; ~a)

@a 1 (i.e., we allow borrowing), this inequality becomes ahigh (t) ku (e).A large ahigh (t) can be rationalized by a large scale of businesses (large ) or by a tightborrowing constraint (low ).

25


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U.S. data and those same statistics calculated from the simulated model. I be-

gin by discussing the main features of the savings rates of entrepreneurs and the

dynamics of entrepreneurship in two U.S. datasets.

6.1 Empirical Evidence

I use a yearly panel for the period 19841995 from the Panel Study of Income

Dynamics (PSID) with rich information on occupational choice, ownership of

businesses, and the wealth of U.S. households; and a quarterly rotating panel

(19841999) from the Consumer Expenditure Survey (CEX) providing consump-

tion data and information on occupational choice. Since the model provides a

theory of the initial transition into entrepreneurship, I estimate the model with

data on the savings behavior and the dynamics of entrepreneurship for younghouseholds (those that are up to 31 years old). Therefore, unless otherwise no-

tice, all statistics are calculated for household that are up to 31 years old in the

initial period. The data used is described in the data appendix.

Following the recent literature (see Gentry and Hubbard (2001) and Hurst

and Lusardi (2004)), an entrepreneur in the data is identied as someone who

reports to own a business. Unfortunately, this information is not available for

the CEX. In the case of the CEX, an entrepreneur is identied as someone who

reports to be self-employed. Whenever it is possible, results are shown for both

denitions.

The rst and second column of Table 1 report the main facts regarding thebehavior of savings rates and consumption growth of entrepreneurs as measured

by the ownership of a business and the self-employed status of the head of the

household20 . Here I sketch a summary of the data.

Individuals save more prior to starting a business. Among the

young households (age 31), those that became business owners between

periods t and t + 1 save 7% more in the previous 5 years (between t 5

and t) than households that neither owned a business in t nor in t + 1

(see rst row of Table 1). Related, individuals becoming business owners

between t 5 and t save 26% more between these years than those that

do not own businesses in t 5 and t. This is refered to as the savings ratedierential during entry in Table 221 .

20 All the moments are calculated using individuals characteristics as controls (sex, maritalstatus and education of the head). The value of the dierent moments should therefore beinterpreted as the one for a single white male with college education.

21 Although the rst is a more appropiate measure of savings in anticipation of entry, it

26


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Business owners have higher saving rates than non-business own-

ers, and their saving rates decrease sharply with age. Household

that own a business in t 5 and t and are up to 31 years old in period t 5

save 26% more than households that neither own a business in t 5 nor in

t: Among mature households (those that are between 32 and 41 years old

in t 5), business owners save just 10% more than non-business owners.

Individuals becoming self-employed have higher consumption

growth, both relative to workers and to individuals that are al-

ready self-employed. The growth in consumption between t1 and t of

individuals becoming self-employed between t 1 and t is 7% higher than

that of those who are workers in both periods. The consumption growth

of households that are already self-employed is not particularly high.

The following fact is illustrated in Figure 9.

Poor individuals and extremely rich individuals are less likely to

start businesses. Among the young, the probability that a household

that is not a business owner in period t 5 starts a business in period

t is mostly constant (around 10%) for the rst 3 wealth to wage ratio

quartiles, increases sharply to 20% for wealth to wage ratios in the 90th

to 95th percentiles, and then decreases for wealth to wage ratios in the

top 5 percentiles.

Related datasets have been studied by other authors. Using the 1983-1989

panel from the Survey of Consumer Finances, Gentry and Hubbard (2001) also

report business owners having larger savings rates than non-business owners,

and this eect being stronger for younger households as oppose to middle-aged

households. They also nd that households save more while becoming business

owners. In their sample, the median household that becomes a business owner

between t 6 and t save 30% more between t 5 and t than households that are

neither business owners in t 6 nor in t. Unfortunately, given the short SCF

panel they cannot study the savings behavior in anticipation of the entry decision

as is done in the current paper. More in line with my ndings on relatively

low savings rates in anticipation of entry, Hurst and Lusardi (2004), also usingthe PSID, nd that previous wealth changes are weakly (and even negatively)

correlated with future entry decision. They also study the relationship between

cannot always be calculated in the simulated model since for some parameter values all en-trepreneurs enter in less than 5 years.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

wealth to wage ratio

Fraction

starting

a

business

50th75th 75th 90th 90th 95th 95th 98th

model

data

Figure 9: Wealth to wage ratios and the Transition into Business Ownership,PSID and Estimated Model. The relationship in the data corresponds to asemiparametric regression using the method described in Robinson (1988).

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Moments Data ModelBusiness Self-employed

Savings rate dierentials of entrepreneursvis-a-vis workers

Prior to entry 0.07 -0.06 0.30(0.05) (0.05)

During entry * 0.26 0.19 0.34(0.04) (0.04)

After entry

young * 0.26 0.30 0.25

(0.08) (0.05)mature * 0.10 0.11 0.11(0.05) (0.06)

Consumption growth dierential entrepreneursvis-a-vis workers

During entry 0.09 0.24(0.04)

After entry 0.01 0.09(0.03)

Table 1: Data and Simulated Moments (see Data Appendix for details). Themoments that are signaled with * are the ones targeted when estimating themodel.

wealth and the likelihood of becoming a business owner and nd that the positive

eect of wealth is stronger for the top percentiles of the wealth distribution.

They do not report evidence on the negative relationship between wealth and

entry into entrepreneurship for extremely wealthy individuals, as measured by

their wealth to wage ratios.

6.2 Structural Estimation

The structure of the model is characterized by two preferences parameters,

and , one technological parameter, , the borrowing constraint coecient, ,

and the distribution of ability, g (e). To close the model, the interest rate, r, and

initial distribution of wealth, h (a0), also need to be specied. The parameters

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are chosen using a combination of calibration and structural estimation. As in

Section 5, I choose values for the preferences parameters, the interest rate and

the depreciation rate following the standard practices: = 1:5, = r = 0:02

and = 0:08. The distribution of initial wealth is chosen to correspond to the

distribution of wealth to income ratios of non-business owners that are up to 26

years old in the PSID. This leaves the technology parameter, , the borrowing

constraint coecient, , and the distribution of ability to be jointly estimated

by minimizing the distance between the moments in the data and the moments

in the model. The details of the algorithm are described in the appendix. An

heuristic discussion of the identication of the models follows.

There are eight parameters (, , and six probabilities describing the abil-

ity distribution) that are estimated by matching nine moments (three moments

related to the savings behavior of entrepreneurs vis-a-vis workers and the entryrates for six wealth to income groups22 ). Given values for and , the distrib-

ution of ability is identied from the fraction of entrants at each wealth group.

To do this mappping, we need to use the information on the initial distribu-

tion of wealth and the dynamic of wealth implied by the model. Obviously,

the shape of the distribution of ability for ability levels such that individuals

never transition into entrepreneurship cannot be identied, i.e., for e such that

a(e) < as (e). The parameters and are then identied by matching the rela-

tionship between wealth to wage ratios and entry for large values of the wealth

to wage ratios as suggested by Remark 2, and the saving rate dierential. In

particular, I use the information on how fast business owners converge to thesteady state scale of their businesses as measure by the decrease in their saving

rate dierential with tenure.

Table 2 reports the parameter estimates and their standard errors. An econ-

omy where entrepreneurs face strong decreasing returns to scale, = 0:36, and

tight borrowing constraints, = 1:01, best ts the data in the rst column of

Table 1. The distribution of ability turned out to be bimodal with 40% of the

protable entrepreneurs (those with relative ability ew

> 1) having low returns

(they could earn 5% more income as unconstrained entrepreneurs) and 30%

facing large returns to entrepreneurship (they could earn three times as much

income as unconstrained entrepreneurs)23

. The estimate for is in line with22 The wealth to income groups are: the bottom 25th percentile, 25th to 50th percentiles,

50th to 75th percentiles, 75th to 90th percentile, 90th to 95th percentile, and the top 5percentile

23 For the estimated economy, since all transitions intro entrepreneurship are permanent,46% of individuals are entrepreneurs and the probability of e = 1 is 0.24 . This is an artifact

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the previous estimates by Evans and Jovanovic (1989) also using micro data for

the U.S., while the estimate for is relatively low.

Parameters Estimate

Technology, 0.36(0.01)

Borrowing Constraint, 1.01(0.03)

Probability (e j e>1)e = 1.20 0.39

1.05 0.02

1.50 0.233.00 0.325.00 0.05

Table 2: Parameter Estimates

Welfare cost of borrowing constraints are large (see Table 3). For the me-

dian among the productive entrepreneurs (individuals with e > 1) welfare costs

amounts to 22% of lifetime consumption. The average welfare cost for the econ-

omy are 6% of lifetime consumption24 . But, because there are strong decreasing

returns to scale, individuals can undo most of the welfare cost of borrowing con-straints by internally nancing their businesses. This is specially true for very

able individuals. In the same direction, the size and economic signicance of

poverty traps are low. Only individuals that would increase their income by

just 5% decide not to save to start a business (12% of the population fall in this

category).

The third column of Table 1 and Figure 4 reports the moments from the sim-

ulated model at the estimated parameters. The model does a good job of tting

the relationship between the transition into business ownership and wealth, spe-

cially the fact that only for households at the top of the wealth distribution is

there a strong and positive relationship between household wealth and business

of not modelling exit and reentry of businesses.24 To calculate the average welfare cost, I force the distribution of ability to match the frac-

tion of entrepreneurs in the data. This is done by shifting the mass of the ability distributionfrom ability values with e > 1 to values with e = 1, in a proportional way so that the distrib-ution of ability conditional on e > 1 is not aected (and therefore not aecting the shape ofthe transition function (see Figure 9)).

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Ability Wealth

zero wealth50th perc.wealth/ wage

75th perc.wealth/ wage

90th perc.wealth/ wage

1.05 5.00 4.94 4.86 4.711.20 16.12 14.99 13.35 9.811.50 24.48 22.07 18.62 12.113.00 40.96 34.42 26.20 18.625.00 52.08 39.10 31.02 23.52

Mean Welfare Costs = 6.24

Table 3: Welfare Costs by Ability (rows) and Initial Wealth (columns) Measured

as a Fraction of Lifetime Consumption (%)

entry (see Hurst and Lusardi (2004)). Savings rates of entrepreneurs, both

young and mature, are matched but the model substantially overpredicts the

savings rates prior to entry. The same is true for the growth rate of consumption

of entrepreneurs. One the one hand, the behavior of savings rates and consump-

tion growth suggest that borrowing constraints are not very important. But the

relationship between wealth and the transition into entrepreneurship suggests

the opposite conclusion. The estimation is the result of a compromise between

these sets of moments.

7 Conclusions

The motivation of this paper can be summarize by two questions. (1) Are bor-

rowing constraints limiting entrepreneurship of any signicance once we take

into account that individuals can save to overcome them? (2) Is the interpreta-

tion of the evidence on the importance of borrowing constraints to entrepreneurs

aected by modelling the dynamic aspects of the problem? This paper provides

an armative answer to both questions. Individuals facing large returns to en-

trepreneurship will remain workers in the long run provided returns to scale are

not too decreasing. The prediction of a hump-shaped relationship between the

transition into entrepreneurship and wealth is a striking illustration of impor-

tance to model the dynamics of the problem.

This is an important rst step. Nevertheless, the model abstract from po-

tentially relevant dimensions that require further research.

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For instance, while the initial wealth of individuals was allowed to be cor-

related with the ability of entrepreneurs, it was assumed that the ratio of the

wealth to the labor income is uncorrelated with the ratio of the entrepreneurial

ability relative to the ability as a worker. In other words, it was not allowed

that individuals that are relatively able entrepreneurs are also wealthy relative

to their opportunity cost. If able individuals also happen to start rich the cost

of imperfect credit markets will tend to be smaller.

As well, in this paper the uncertainty faced by entrepreneurs is not mod-

eled. As suggested by the large amount of exit in the data this is an important

dimension. In this lines, it would be extremely interesting to measure the role

of uncertainty relative to initial resources in determining entrepreneurship, and

the welfare cost associated with each potential market failure. These impor-

tant questions are left for future research. Recent work by Hopenhayn andVereshcabina (2005) is an important step in this direction.

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A Data

I use a yearly panel for the period 19841995 from the Panel Study of In-come Dynamics (PSID) with rich information on occupational choices, owner-

ship of businesses and the wealth of US households; and a quarterly rotating

panel (19841999) from the Consumer Expenditure Survey (CEX) providing

consumption data and information on occupational choices.

In the case of the PSID, I create a 7 years panel pooling the households in

the 1984-1990 and 1989-1995 samples. This gives a panel with two observations

for wealth (1984 and 1989 in the 1984-1990 subsample, 1989 and 1994 in the

1989-1995 subsample), and yearly observations on the ownership of businesses

and income. Using the pooled panel, I construct savings rates between the

rst and the fth years, savings15 = wealth5wealth1P

5t=1 incomet , wealth to income ratios

= wealthtincomet

(the relevant measure of wealth in the model) and business ownership

histories. Wealth corresponds the sum of net equity in a main home, other

real estate, vehicles, farm/business, stocks, savings accounts and other assets,

less debt; income equals total family money income plus food stamps minus

federal income taxes paid; and business ownership status is determined by the

question Do you (or anyone in your family living there) own part or all of

a farm or business? For comparison purposes, I also use information about

the self-employ status of the head of the household as an alternative proxy for

entrepreneurship.

I restrict the sample to the households that are at least 22 and at most 61years old in the rst period, that are working in the 1, 2, 6 and 7th periods

(these are the periods for which business ownership information is used) and

I drop the observations with savings rates below and above the 1st and 99th

percentiles respectively. These restrictions leaves 5354 observations that are

used to calculate the moments reported in Table 1.

In the case of the CEX I use the quarterly Interview component for the

1984-1999 period. From this dataset I use information on non-durable con-

sumption, self-employed status of the head of the household and demographic

characteristics. After applying similar restrictions to the PSID sample, 5545

observations of household that are up to 31 years old are left to calculate themoments reported in Table 1.

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B Algorithm

Given values for the preferences and technology parameters (, , ; ), theborrowing constraint coecient (), the interest rate (r), the individual decision

problem is solved for each ability level in the ability grid,

e = f1; 1:05; 1:1; 1:2; 1:5; 2; 3; 5; 7; 10g, by backward induction from the last pe-

riod, T = 40. Given the policy functions and 10000 values for the initial wealth

drawn from the distribution of wealth to income ratio of non-business own-

ers that are up to 26 years old in the PSID, 10000 histories are generated for

each value of ability, fxt (e)gTt=1. Where the initial wealth to income ratio

takes values on the grid a0 = f0; 0:17; 0:49; 1; 1:84; 2:87; 8:03g with probabili-

ties h (a0) = f0:25; 0:25; 0:25; 0:15; 0:05; 0:03; 0:02g. Then, by stacking the data

for individuals at dierent ages I obtained a simulated (unbalanced) panel of400000 observations (10000 forty years observations, fxt (e)g

Tt=1, 10000 thirty-

nine years observations, fxt (e)gTt=2, ...) that is used to calculate the statistics

for each value of ability. Given the distribution of ability, a vector of dimension

10, the statistics from the model are calculated. The algorithm then search over

values of the returns to scale parameter, , the borrowing constraint coecient,

, and the distribution of ability to minimized the weighted distance between

the statistics calculated using the PSID data and the data from the simulated

model. The weights are given by the inverse of the covariance matrix of the

moments from the PSID data.

A important simplication is introduced by the fact that ability is xed overthe life of an individual, implying that the individual decision problem is inde-

pendent of the ability distribution. For given values of and , a gradient based

method (matlab routine fmincon.m) is used to minimize over the distribution of

ability subject to the constraint that at most six ability values get to be assigned

a positive probability. Then, grid search is used to minimized over and .

C Proof of the Results in the Paper

Proof of Proposition 1:. For suciently high e the rst case arises since the

trajectories to the left ofa are not aected by ability, e. Similarly for sucientlylow e (e.g. e = e) we have the third case. For intermediate value of ability

the intermediate case arises. The only thing that need to be proved is that a

case where neither the trajectory converging to the (0; w) nor the trajectory

converging to the (ass; css) cycle around the point (a; c) is not possible. This is

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proven by contradiction.

Assume that a case where neither the trajectory converging to the (0; w) nor

the trajectory converging to the (ass; css) cycle around the point (a; c) is possi-

ble. Then for values of initial wealth close to a0 = 0 the trajectory converging

to the (ass; css) steady state is optimal since for a0 = 0 the plan that states

forever at the point (0; w) corresponds to a zero of the Hamiltonian function

therefore the plan starting with lower consumption and converging eventually

to the (ass; css) steady state is preferred.25 Similarly, for values of initial wealth

close to a0 = ass the trajectory converging to the (0; w) steady state is optimal.

But this contradicts Theorem 2 in Brock and Dechert (1983).

Proof of Proposition 2:. The rst step is to rule out the trajectories

that circle around (a; c). That these trajectories are not optimal follows from the

Hamiltonian function being strictly convex in the initial Lagrange multiplier, i.e.optimal trajectories are among the trajectories that start with extreme values

for Lagrange multiplier and therefore consumption. Let a (a) be the rst

point at which the upper (lower) trajectory crosses the _a = 0 locus. Then, the

result is a direct application of Theorem 2 in Brock and Dechert (1983) and

the fact that at point a the trajectory converging to the (0; w) steady state is

optimal and at the point a the trajectory converging to the (ass; css) steady

state is the optimal one. This last observation follow from the fact that at the

point a the trajectory converging to the (ass; css) steady state path through

the _a = 0 curve, a zero of the Hamiltonian function with respect to the Lagrange

multiplier. A similar argument applies for a = a.Proof of Proposition 3. The threshold is implicitly dened by the

following equation

Ve (as; e) Vw (as; e) = 0

where Vw is the value of following the upper trajectory and Ve is the value

associated with the lower trajectory in Figure 3. For as < a Vw does not

depend on e while Ve is a strictly increasing function of e therefore as is strictly

decreasing in e. Clearly, for e < e Vw (a; e) > Ve (a; e) for all a. Therefore, there

is an inmum ability elow e such that Vw (a; e) Ve (a; e) for all a. Similarly,

25 The Hamiltonian function of this problem is

H(a0; 0) = maxc

fu (c) + 0 (y (e; a0) c)g .

This function gives the value of following a path that satises the necessary conditions (SeeSkiba (1978) and Brock and Dechert (1983)). Note that the Hamiltonian is a strictly convexfunction of 0. Thus @H@0 = _a = 0 correspond to a global minima.

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since Vw (0; e) is independent ofe, as long as u (c) is not bounded, there exist a

supremum (nite) ability ehigh such that Ve (0; ehigh) > Vw (0; ehigh), and from

Theorem 2 in Brock and Dechert (1983), Ve (a; ehigh) > Vw (a; ehigh) for all a.

Proof of Proposition 4. If r = , the poverty traps threshold solve

w + ras = ce (a), where ce (:) is the policy function for consumption associated

with the stable path of the entrepreneurial problem. If both, or , increases,

then, ce (a) increases and, therefore, as locally increases. Ife is close to ehigh, a

decrease in r has no eect on Vw (e) while it decreases the value ofVe (e) since

individuals are saving to become entrepreneurs. For e close to elow the opposite

is true.

The following assumption is needed for the next result.

Assumption A.1: The policy function, a (a0; t ; e), is strictly increasing andcontinuous in the entrepreneurial ability, e, for a0 as (e).

This is a natural assumption. It requires that present consumption does not

increase too much when ability increases. In terms of assumptions about prefer-

ences, it needs that the preferences between consumption today and tomorrow

are not too bias toward present consumption. For example, it will true in a two

period model if preferences are homothetic.

Proof of Proposition 5. The transition probability conditional on wealth,

a, and age, t, is given by the integral over all agents with ability high enough

such that they nd it protable to start a business by t + , e > a1 (a + ~a; ),

but not so high for them to be already entrepreneurs, e < a1 (a). Wherea (e; ) solves the equation a

a (e; ) ; ; e

= a (e). Formally

P (transitionja;t; ~a; ) =

Re2Ew(a;t) ; e>a1(a+~a;)

g (e) h (a0 (a;t;e)) de

1 P (entrepreneur ja; t)(8)

where Ew (a; t) is the support of ability conditional on wealth a, age t and

currently being a worker of those individuals that will eventually become entre-

preneurs, Ew (a; t) = fe 2 R+ : 9a0 for which a (a0; t ; e) = a, e < a1 (a) and

e a1s (a)g. Then, to proof this result we need to characterize the lowest and

highest wealth levels, alow (t) and ahigh (t), such that this set is non-empty.That there exist an increasing function alow (t) such that for a < alow (t) the

set Ew (a; t) is empty follows trivially from the fact that the wealth of individuals

that will eventually become entrepreneurs increases overtime and that a0 0.

The function ahigh (t) = a (emin (t)) where emin (t) = inffe 2 [elow; 1] :

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a (as (e) ; t ; e) = a (e)g. Note that emin (t) is a strictly increasing function of

t since for e close to elow (as (elow) = a (elow)) we know that a (as (e) ; t ; e) >

a (e) and, by continuity, we know that for the minimum root of the equation

a (as (e) ; t ; e) = a (e) the function a (as (e) ; t ; e) is decreasing and crosses the

function as (e) from below. Thus, the function ahigh (t) is then decreasing since

it is the composition of a strictly decreasing function with an increasing function.

The maximum age such that the set Ew (a; t) is non-empty for some wealth

level is dene by

t = infft 2 R++ : a (as (e) ; t ; e) a (e) 8e eming

Age t is nite since a (e) < ass (e) 8e emin.

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A Dynamic Model of Entrepreneurship With Borrowing Constratints

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