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Federal Reserve Bank of Chicago
Entrepreneurship, Frictions, and
Wealth
Marco Cagetti and Mariacristina De Nardi
REVISED July, 2006
WP 2005-09
Entrepreneurship, Frictions, and Wealth*
Marco Cagetti
Federal Reserve Board
Mariacristina De Nardi
Federal Reserve Bank of Chicago, NBER, and University of Minnesota
*We are grateful to Gadi Barlevy, Marco Bassetto, Francisco Buera, Jeff Campbell, V. V.
Chari, Narayana Kocherlakota, Per Krusell, Ellen McGrattan, Kulwant Rai, Victor R´
ıos-Rull,
Emmanuel Saez, Nancy Stokey, Jenni Schoppers, Kjetil Storesletten, four anonymous referees,
and many seminars participants for helpful comments. We gratefully acknowledge financial
support from NSF grants SES-0317872 and SES-0318014. De Nardi thanks the University
of Minnesota Grant-in-Aid and Marco Cagetti the Bankard Fund for Political Economy for
funding. The views herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Chicago, the Federal Reserve Board, the Federal Reserve System, or the NSF.
1
Abstract
This paper constructs and calibrates a parsimonious model of occupational choice that allows
for entrepreneurial entry, exit, and investment decisions in presence of borrowing constraints.
The model fits very well a number of empirical observations, including the observed wealth
distribution for entrepreneurs and workers. At the aggregate level, more restrictive borrowing
constraints generate less wealth concentration, and reduce average firm size, aggregate capital,
and the fraction of entrepreneurs. Voluntary bequests allow some high-ability workers to
establish or enlarge an entrepreneurial activity. With accidental bequests only, there would
be fewer very large firms, and less aggregate capital and wealth concentration.
J.E.L. Classification: E21, E23, J23,
Keywords: Entrepreneurship, wealth inequality, borrowing constraints, bequests
2
I. Introduction
Although many empirical studies argue that potential and existing entrepreneurs face borrowing constraints, there has been so far little work on how these constraints affect aggregate
capital accumulation and wealth inequality through entrepreneurial choices. Do these financial
constraints hamper aggregate capital accumulation and if so, how big is this effect? What effect do these constraints have on wealth inequality: do they exacerbate it or mitigate it? These
are potentially important forces to understand the consequences of policy reforms that affect
the tightness of these borrowing constraints, such as changes in the leniency of bankruptcy
laws and in the degree of enforcement of property rights.
In this paper we analyze the role of borrowing constraints as determinants of entrepreneurial
decisions (entry, continuation, investment, and saving), and their effects on wealth inequality
and aggregate capital accumulation, in a framework that matches the observed wealth inequality very closely. In presence of borrowing constraints, the decision to invest, the fraction
of entrepreneurs, and the size distribution of firms depend on the distribution of assets in the
economy. Because of this interaction, it is key to perform such an analysis in a model that
matches well the extreme concentration of wealth observed in the data.
We find that more restrictive borrowing constraints generate less inequality in wealth
holdings, but also reduce average firm size, the number of people engaging in entrepreneurial
activities, and aggregate capital accumulation. Our results also indicate that voluntary bequests are an important channel allowing some high-ability workers to establish or enlarge
an entrepreneurial activity. If there were only accidental bequests there would be fewer very
large firms, and less aggregate capital, but also less wealth concentration.
These findings are based on a quantitative life-cycle model with altruism across generations
and entrepreneurial choice, in an environment in which debt repayment cannot be perfectly
3
enforced. The amount that entrepreneurs can borrow depends on their observable characteristics, and the entrepreneurs’ assets act as collateral for their debts. Since the implicit rate of
return for entrepreneurs is higher than the rate for workers, and consistently with the data,
entrepreneurs have a higher saving rate. We calibrate the parameters of the model to match
key moments of the data and discuss the implications of the model and its components for
entrepreneurial choice and wealth inequality. We show that our model with entrepreneurial
choice matches very well the observed distribution of wealth, for both entrepreneurs and nonentrepreneurs.
This paper is related to the quantitative literature on wealth inequality. (See Cagetti and
De Nardi (2005) for a comprehensive survey.) The most closely related works are the ones by
De Nardi (2004), Quadrini (2000), and Casta˜eda, D´
n
ıaz-Gim´nez, and R´
e
ıos-Rull (2003).
De Nardi (2004) evaluates the importance of bequest motives and intergenerational transmission of ability to explain wealth dispersion in a life-cycle model and shows that a realistically calibrated bequest motive can raise wealth concentration and bring it closer to the
observed data. Her model does not allow for entrepreneurial choice, and falls short of explaining the extreme concentration of wealth in the hands of the richest 1% of the population.
Quadrini (2000) shows that a model that incorporates individual specific technologies
(entrepreneurs) and financial frictions can generate more wealth inequality than that implied
by a precautionary motive, for a given process of individual ability, or “labor” income. His
model relies on exogenous stochastic processes for both entrepreneurial ability and the scale
of the project. We improve upon Quadrini’s framework by using a very parsimonious model
and by allowing for endogenous choice of the amount of capital invested by the entrepreneur
in the firm. We also study how financial frictions and channels affecting the intergenerational
transmission of wealth affect wealth inequality and aggregate output.
4
Casta˜eda, D´
n
ıaz-Gim´nez, and R´
e
ıos-Rull (2003) adopt a dynastic model with idiosyncratic shocks and reconstruct an exogenous labor income process (which also includes most
of business income) that matches earnings and wealth dispersion. The resulting labor and
entrepreneurial income process implies very large earnings risk for the highest income earners. This large risk associated to high income realizations is the driving force that, in their
framework, generates a large saving rate for the richer households, which is the fundamental mechanism driving the extreme amount of wealth observed in the hands of the richest
few. Compared to Casta˜eda et al., we endogenize and model explicitly the entrepreneurial’s
n
investment decision, and hence entrepreneurial income. In our framework the main driving
force that allows the model to match the observed wealth inequality is given by potentially
high rates of return from entrepreneurial investment coupled with borrowing constraints, or
the observation that one needs money to make money.
Section II first documents the relationship between wealth and entrepreneurship, and then
surveys the evidence that entrepreneurs are borrowing constrained. Section III describes the
model and our calibration procedure. Section IV discusses the role of entrepreneurship and
voluntary bequests in generating large wealth concentration, and studies the aggregate effects
of changing the borrowing constraints. Section V inspects further the mechanisms at work in
our model and compares their observable implications to those in the observed data. Section
VI concludes.
5
II. Wealth, entrepreneurship, and borrowing constraints
We first document the relationship between wealth and entrepreneurship, and we then survey
the empirical evidence on the effects of borrowing constraints on entrepreneurial choice.1
A. Who are the rich households?
Wealth holdings are massively concentrated in the hands of a small fraction of households,
and this wealth concentration is much larger than the one documented for labor earnings
and total income. This observation begs the question of which saving motives generate the
amplification in the concentration of wealth with respect to the one in income.
When looking at the data it is clear that there is a tight relationship between being an
“entrepreneur” and being rich. We begin by documenting this relationship, using different
definitions of entrepreneurship, and we then discuss alternative ways of acquiring wealth.
The SCF asks several questions that we can use to classify a household by its occupational
status:
1) “Do you work for someone else, are you self-employed, or what?”
2) “Do you (and your family living here) own or share ownership in any privately-held
businesses, farms, professional practices or partnerships?”
3) “Do you (or anyone in your family living here) have an active management role in any
of these businesses?”
1
Whenever possible we use data from the Survey of Consumer Finances (SCF). Unlike other data sets,
the SCF oversamples rich households and thus provides important advantages. First, it gives a better picture
of the concentration of wealth and of the asset holdings of richer households, which include a large share of
entrepreneurs. Second, as shown by Curtin, Juster, and Morgan (1989), the total wealth implied by the SCF
is very close to the total wealth implied by aggregate data; the SCF can thus be used to calibrate aggregates
(for instance, the share of entrepreneurial wealth and the percentage of entrepreneurs) in a general equilibrium
model such as the one developed in this paper.
6
Table 12 shows the fraction of people in a given occupation and the total fraction of
aggregate net worth that they hold. The first line refers to people that declare that they are
either business owners or self-employed (that is, who answer yes to either question (1) or (2)).
This group makes up for about 17% of the population, and owns more than half of the total
net worth. The second line refers to all households that own privately held business, but do
not necessarily manage them (that is, who answer yes to question (2)), while the third one
focuses on the business owners that effectively manage their own business(es) (that is, who
answer yes to question (3)). The fourth line refers to those that report being self-employed
(yes to question (1)) and the fifth line to those that are both self-employed and business
owners with an active management role (yes to questions (1), (2) and (3)). The self-employed
business owners are 7.6% in the population, and yet hold 33% of the total net worth. The
key message of this table is that, regardless of the specific definition of entrepreneurship used,
entrepreneurs are a relatively small fraction of the population and hold a large fraction of the
total net worth.
Table 2 documents wealth concentration in the Unites States: the households in the top
1% of the wealth distribution hold about 30% of total net worth, and those in the top 5% hold
more than half of the total. Table 3 reports the fraction of various definitions of entrepreneurs
in the corresponding wealth quantile of the overall wealth distribution. A whopping 81% of
those that belong to the top 1% of the wealth distribution declare that they are either selfemployed or business owners. All business owners are 76% of the richest 1% of households,
while the fraction of the business owners that actively manage their own business(es) is 65%,
hence some of the business owners are “investors” that own a business that is managed and
2
All of the statistics that we report here use data from the 1989 wave of the SCF. The data for the 1992
and 1995 waves are similar. The results are available from the authors upon request.
7
run by someone else. The self-employed make up for 62% of the households in the top 1% of
the wealth distribution, while the self-employed business owners are 54%. The overall message
of this table is that most rich people are entrepreneurs.
Table 4 reports mean and median asset holdings by occupational status. Regardless of the
specific definition of entrepreneurship, entrepreneurs are much richer than non entrepreneurs.
The business owners, however, tend to be richer than the self-employed. Not surprisingly,
the poorest are those that declare being “self-employed”, but not “business owners”; some
of these households might be the low-wage workers that turn to self-employment for lack of
better opportunities3 , or people that are self-employed as a hobby. Interestingly, the business
owners that do not have an active management role in the business are very rich, and are
likely to use the business as an investment opportunity.
We have seen that many of the rich people are entrepreneurs, but who are the others, and
how did they become rich? Unfortunately the SCF provides only very coarse classifications by
occupation, for example lumping together managers, professionals, singers, performers, etc,
and thus provides very little data to answer this question. The other nationally representative
samples miss the very rich. We study the Forbes magazine list of the 400 richest people in
the United States. While this is a very restricted sample, it certainly focuses on the rich.
According to this data set, of the 400 wealthiest American people in various years, 61% to
80% were self-made (typically by individuals that started a firm), while the rest inherited
the family’s fortune, which was typically originated by one or more businesses started by
3
Rissman (2003) documents that in the National Longitudinal Survey of Youth (NLSY) more than one
quarter of all younger men experience some period of self-employment, and many of them return to wage work.
She argues that for these workers self-employment is a low-income alternative to wage work and provides an
alternative source of income for unemployed workers. Rissman also finds that young men are more likely to
become self-employed when their wage opportunities are more limited, as in periods of economic downturns.
8
one of their parents or grandparents.4 Extremely few entries in this list were people such as
entertainers or sportsmen, who acquired their wealth through high incomes without starting
as entrepreneurs. By cross-comparing the 2004 list with the one for the top 100 “celebrities”
for the same year (also compiled by Forbes), we find that only 3 of the top 100 “celebrities”
make it to the list of the top 400 richest Americans: George Lucas, Oprah Winfrey, and Steven
Spielberg. Interestingly, Steven Spielberg put up $33 million for 22% of his upstart studio in
1994, and thus used a significant amount of his own money to start his empire.
B. Entrepreneurship and borrowing constraints
To estimate the severity of borrowing constraints on entrepreneurial entry and continuation
decisions one would want to know how much potential and existing entrepreneurs would like
to borrow, at what interest rate, and how much they are actually able to borrow, and at what
price. Unfortunately such data are not available.
Many papers have used a variety of data sets and methodologies to indirectly estimate the
severity of borrowing constraints for entrepreneurs. Among these works, Evans and Jovanovic
(1989) and Buera (2006) estimate structural models of entrepreneurship, and find evidence
of borrowing constraints; Gentry and Hubbard (2004) and Eisfeldt and Rampini (2005) also
argue that costly external financing has important implications for investment and saving
decisions. Holtz-Eakin, Joulfaian, and Rosen (1994) study the effects of receiving a bequest
on both potential and existing entrepreneurs. They find that the receipt of a bequest (and
thus an increase in wealth) increases the probability of starting a business. They also find that
existing sole-proprietors who receive a bequest not only are more likely to stay in business,
4
The fraction of heirs in the Forbes 400 list was 39% in 2004 (our computations), while it varies between
20% and 30% in other years according to Smith (2001). This fraction is quite volatile due to the small sample
size of this list.
9
but also experience a substantial increase in the enterprise’s receipts.
More recently, Hurst and Lusardi (2004) have disputed the relevance of borrowing constraints to entrepreneurial entry. They estimate that the probability of entering entrepreneurship as a function of initial wealth is first flat over a large range of the wealth distribution,
and it then increases for the richest workers. We will show that a model of entrepreneurial
choice with borrowing constraints is capable of generating this type of entry probabilities as a
function of one’s own wealth. We will also discuss that the lack of borrowing constraints to entrepreneurial entry does not imply lack of borrowing constraints on entrepreneurial investment
after entry.
The need to accumulate assets in presence of borrowing constraints may also generate high
saving rates among entrepreneurs (or households planning to become entrepreneurs). Using
different data sets, Gentry and Hubbard (2004) and Quadrini (1999) show higher saving rates
for entrepreneurs than for the rest of the population, and Buera (2006) shows higher saving
rates also in the years before entry into entrepreneurship.
To provide more evidence on the existence of borrowing constraints we also look at the
data on entrepreneurs using their collateral for their business, and on entrepreneurs declaring
that they have been turned down for credit, or that they did not apply for credit because they
thought that they would be turned down.
The SCF asks explicitly about whether some of the debts are explicitly collateralized with
the entrepreneur’s own private assets. These numbers are just an indication, because they
include the use of only personal assets (other than the business itself) and do not indicate
the relation between the amount borrowed and the size of the business, nor the amount of
borrowing desired by the entrepreneur. Among the self-employed business owners, 29% declare
that they currently use their own personal assets as collateral to finance their business. Within
10
this group, the median ratio of personal collateral to business value is 21%, the top decile is
77%, and the top 5% is 100%. These fractions do not change significantly across quantiles of
the wealth distribution, thus suggesting that many businesses do need to put up collateral in
order to borrow, regardless of their size.
Among the self-employed business owners 18% report that they have been turned down
for credit, and 9% state that they thought of applying, but changed their mind because they
thought they might be turned down.
The severity of borrowing constraints potentially depends on bankruptcy laws. Berkowitz
and White (2004) show that the higher exemption levels on personal bankruptcy, the higher
the probability of being denied credit and the smaller the amount of loans made. This suggests that higher exemptions lower the incentive to repay, and thus generate more stringent
borrowing constraints.
III. The model
A. Demographics
We adopt a life-cycle model with intergenerational altruism. To make the results quantitatively
interesting, we need short time periods. To make the model computationally manageable, we
have to keep the number of stages of life small. To reconcile these two necessities, we adopt
a modeling device introduced by Blanchard (1985) and generalized by Gertler (1999) to a
life-cycle setting.
Households go through two stages of life, young and old age. A young person faces a
constant probability of aging during each period (1 − πy ), and an old person faces a constant
probability of dying during each period (1 − πo ). When an old person dies, his offspring enters
11
the model, carrying the assets bequeathed to him by the parent. Appropriately parameterized,
this framework generates households for which the average lengths of the working period and
the retirement period are realistic. Our model period is one year.
There is a continuum of households of measure 1. The households are subject to idiosyncratic shocks, but there is no aggregate uncertainty, as in Bewley (1977).
B. Preferences
The household’s utility from consumption is given by
c1−σ
.
1−σ
The households discount the future
at rate β, and, in addition, they discount the utility of their offspring at rate η.
To study the role of bequests, our model nests life-cycle and fully altruistic households as
two extreme cases. In the purely life-cycle version of the model individuals put no weight on
the utility of their descendants (η = 0). In the perfectly altruistic version, individuals care
about their descendants as much as themselves (η = 1). We assume exogenous labor supply.
C. Technology
Each person possesses two types of ability, which we take to be exogenous, stochastic, positively correlated over time, and uncorrelated with each other. Entrepreneurial ability (θ) is
the capacity to invest capital more or less productively. Working ability (y) is the capacity to
produce income out of labor.
Entrepreneurs can borrow and invest capital in a technology whose return depends on
their own entrepreneurial ability: those with higher ability levels have higher average and
marginal returns from capital. When the entrepreneur invests k, the production is given
by θk ν , where ν ∈ [0, 1]. Entrepreneurs thus face decreasing returns from investment, as
their managerial skills become gradually stretched over larger and larger projects (as in Lucas
12
(1978)). Hence, while entrepreneurial ability is exogenously given, the entrepreneurial rate of
return from investing in capital is endogenous and is a function of the size of the project that
the entrepreneur implements.
There is no within-period uncertainty regarding the returns of the entrepreneurial project.
The ability θ is observable and known by all at the beginning of the period. We therefore
ignore problems arising both from partial observability and costly state verification and from
diversification of entrepreneurial risk. The simplification is adopted to focus only on the effect
of the borrowing constraint.
Workers can save (but not borrow) at a riskless, constant rate of return.
Many firms are not controlled by a single entrepreneur and are not likely to face the
same financing restrictions that we stress in our model. Therefore, as in Quadrini (2000),
we model two sectors of production: one populated by the entrepreneurs and one by “nonentrepreneurial” firms. The non-entrepreneurial sector is represented by a standard CobbDouglas production function:
α
F (Kc , Lc ) = AKc L1−α
c
(1)
where Kc and Lc are the total capital and labor inputs in the non-entrepreneurial sector and
A is a constant. In both sectors, capital depreciates at a rate δ.
D. Credit market constraints
As in Albuquerque and Hopenhayn (2004), Kehoe and Levine (1993), Marcet and Marimon (1992), and Cooley, Marimon, and Quadrini (2005), the borrowing constraints are
endogenously determined in equilibrium and stem from the assumptions that contracts are
imperfectly enforceable.
Imperfect enforceability of contracts means that the creditors will not be able to force the
13
debtors to fully repay their debts as promised, and that the debtors fully repay only if it is
in their own interest to do so. Since both parties are aware of this feature and act rationally,
the lender will lend to a given borrower only an amount (possibly zero) that will be in the
debtor’s interest to repay as promised.
In particular, we assume that the entrepreneurs who borrow can either invest the money
and repay their debt at the end of the period or run away without investing it and be workers
for one period. In the latter case, they retain a fraction f of their working capital k (which
includes their own assets and borrowed money), and their creditors seize the rest.
In the absence of market imperfections, the optimal level of capital is only related to
technological parameters and does not depend on initial assets. In our framework, instead,
the higher is the amount of an entrepreneur’s own wealth invested in the business, the larger is
the amount that the entrepreneur would loose in case of default, the lower the temptation to
default, and the larger is the sum that creditor is willing to lend to the entrepreneur. Hence,
the entrepreneur’s assets act as collateral, although the loan need not be fully collateralized.
As a result, not all potentially profitable projects receive appropriate funding. Households
with little wealth can borrow little, even if they have high ability as entrepreneurs. Since
the entrepreneur forgoes his potential earnings as a worker, he will choose to become an
entrepreneur only if the size of the firm that he can start is big enough; that is, he is rich
enough to be able to borrow and invest a suitable amount of money in his firm.
E. Households
At the beginning of each period, before making any economic decisions, the current ability
levels are known with certainty, while next period’s levels are uncertain.
Each young individual starts the period with assets a, entrepreneurial ability θ, and worker
14
ability y and chooses whether to be an entrepreneur or a worker during the current period.
An old entrepreneur can decide to keep the activity going or retire, while a retiree cannot
start a new entrepreneurial activity. We allow entrepreneurs to remain active when old to
capture the fact that, while most workers retire before age 65, entrepreneurs often continue
their activity until much later.
The young’s problem
The young’s state variables are his current assets a, working ability y, and entrepreneurial
ability θ. His value function is
V (a, y, θ) = max{Ve (a, y, θ), Vw (a, y, θ)},
(2)
where Ve (a, y, θ) is the value function of a young individual who manages an entrepreneurial
activity during the current period. In order to invest k, the young entrepreneur borrows
(k − a) from a financial intermediary at the interest rate r, which is the risk-free interest rate
at which people can borrow and lend in this economy. Consumption c is enjoyed at the end
of the period. We have
Ve (a, y, θ) = max{u(c) + βπy EV (a′ , y ′ , θ′ ) + β(1 − πy )EW (a′ , θ′ )}
′
(3)
a′ = (1 − δ)k + θk ν − (1 + r)(k − a) − c
(4)
u(c) + βπy EV (a′ , y ′ , θ′ ) + β(1 − πy )EW (a′ , θ′ ) ≥ Vw (f · k, y, θ)
(5)
a≥0
(6)
k ≥ 0.
(7)
c,k,a
15
The expected value of the value function is taken with respect to (y ′ , θ′ ), conditional on (y, θ),
F (y ′ , θ′ |y, θ) is a first-order Markov process, and W (a′ , θ′ ) is the value function of the old
entrepreneur at the beginning of the period, before he has decided whether he wants to stay
in business or retire.
The function Vw (a, y, θ) is the value function for the young who chooses to be a worker
during the current period. We have
Vw (a, y, θ) = max{u(c) + βπy EV (a′ , y ′ , θ′ ) + β(1 − πy )Wr (a′ )}
′
c,a
(8)
subject to eq. (6) and
a′ = (1 + r)a + (1 − τ ) w y − c,
(9)
where w is the wage and τ is a proportional payroll tax used to finance old-age social security.
We explicitly model old-age social security because it is a very important program affecting
life-cycle saving decisions.
When the worker becomes old, he is retired, and Wr (a′ ) is the corresponding value function.
The old’s problem
The old entrepreneur can choose to continue the entrepreneurial activity or retire. The old
person’s state variables are therefore his current assets a, his entrepreneurial ability θ, and
whether he was a retiree or an entrepreneur during the previous period.
The value function of an old entrepreneur is
W (a, θ) = max{We (a, θ), Wr (a)},
(10)
where We (a, θ) is the value function for the old entrepreneur who stays in business, and Wr (a)
16
is the value function of the old, retired person.
We denote with η the weight on the utility of the descendants. If η = 0, the household
behaves as a pure life-cycle; if η = 1 the household behaves as a dynasty. We have:
We (a, θ) = max{u(c) + βπo EW (a′ , θ′ ) + ηβ(1 − πo )EV (a′ , y ′ , θ′ )}
′
c,k,a
(11)
subject to eq. (4), eq. (7), and
u(c) + βπo EW (a′ , θ′ ) + ηβ(1 − πo )EV (a′ , y ′ , θ′ ) ≥ Wr (f · k).
(12)
The offspring of an entrepreneur is born with ability level (θ′ , y ′ ). The expected value of the
offspring’s value function with respect to y ′ is computed using the invariant distribution of
y, while the one with respect to θ′ is conditional on the parent’s θ and evolves according to
the same Markov process that each person faces for θ while alive. This is justified by the
assumption that the offspring of an entrepreneur inherits the parent’s firm.
A retired person (who is not an entrepreneur) receives pensions and social security payments (p) and consumes his assets. His value function is
Wr (a) = max{u(c) + βπo EWr (a′ ) + ηβ(1 − πo )EV (a′ , y ′ , θ′ )}
′
c,a
(13)
subject to eq. (6) and
a′ = (1 + r)a + p − c.
(14)
The expected value of the child’s value function is taken with respect to the invariant distribution of y and θ.
17
F. Equilibrium
Let x = (a, y, θ, s) be the state vector for an individual in our economy, where s distinguishes
young workers, young entrepreneurs, old entrepreneurs, and old retired. From the decision
rules that solve the maximization problem and the exogenous Markov process for income
and entrepreneurial ability, we can derive a transition function which provides the probability
distribution of x′ (the state next period) conditional on x.
A stationary equilibrium is given by
a risk-free interest rate r, wage rate w, and tax rate τ ,
allocations c(x), a(x), occupational choices, and investments k(x),
and a constant distribution of people over the state variables x: m∗ (x)
such that, given r, w, and τ the following hold:
• The functions c, a, and k solve the maximization problems described above.
• The capital and labor markets clear. Entrepreneurs use their own labor. The total
labor supplied by the workers equals the total labor employed in the non-entrepreneurial
sector. The total savings in the economy equal the sum of the total capital employed in
the non-entrepreneurial and in the entrepreneurial sectors.
• The wage and interest rates are given by the marginal products of each factor of production, and the rate of return from investing in capital in the non-entrepreneurial sector
must equate the risk-free rate that equates savings and investment.
• The social security budget constraint is balanced period by period: τ is chosen so that
total labor income taxes equal total old-age social security payments.
18
• The distribution m∗ is the invariant distribution for the economy.
G. Calibration
The empirical definition of entrepreneurship that we use for the calibration must be consistent
with the notion of entrepreneur in our framework. In our model an entrepreneur runs his own
business, invests his own wealth in it, has a potentially high return from investing his business,
and faces borrowing constraints to start or expand his firm. Our entrepreneur is not simply a
manager in a firm, is not an “investor” (who does not have a key role in managing the firm),
and is not a person working on his own because he is virtually unemployable in any other firm.
For this reason we use the SCF data to classify as entrepreneurs the households who declare
that they are self-employed, that they do own a business (or a share of one), and that they
have an active management role in it. Our definition thus eliminates managers (who are not
likely to think of themselves as self-employed) and the business owners that do not manage
the business that they own. It is thus likely to eliminate (at least part) of “reverse causation”:
people that for example are rich and acquire business for investment or as a hobby, but do not
have an active management role in it. By taking the intersection of the self-employed and the
active business owners, our definition is also likely to eliminate the self-employed households
that either mostly invest their (possibly considerable) human capital in the business, but very
little physical capital; or that are self-employed only because their wage opportunities are very
poor. Although for different reasons, none of these households are entrepreneurs in the sense
of our model, nor are they likely to be borrowing constrained to start a profitable business.
Our general calibration strategy is to reduce the number of parameters that we use to
match the data as much as possible. We thus divide our parameters in two sets. The first
set of parameters can either be easily estimated from the data without using our model (for
19
example the length of young and old age), or has been estimated by many previous studies
(for example risk aversion). We use the second set of parameters to match some relevant
moments of the data.
Table 5 lists the parameters of the model. The first panel of the table shows the set of
parameters that we take from other studies and do not use to match moments of the data.
We take the coefficient of relative risk aversion to be 1.5, a value close to those estimated
by, among others, Attanasio et al. (1999). As is standard in the business cycle literature,
we choose a depreciation rate δ of 6%. The share of income that goes to capital in the nonentrepreneurial sector is 0.33, and the scaling factor A is normalized to 1. The probability
of aging and of death are such that the average length of the working life is 45 years, and
the average length of the retirement period is 11 years. The logarithm of the income process
y for working people is assumed to follow an AR(1). We take its persistence to be 0.95, as
estimated by Storesletten, Telmer, and Yaron (2004). The variance is chosen to match the
Gini coefficient for earnings of 0.38, the average found in the Panel Study of Income Dynamics
(PSID). The matrix Py is transition matrix for the discretized labor income process. We
assume that the income and the entrepreneurial ability processes evolve independently. (See
appendix A1 for exact values of the income and ability processes and a discussion of the effects
of assuming positive correlation between entrepreneurial and working abilities.) The social
security replacement rate is 40% of average income, net of taxes. (See Kotlikoff, Smetters,
and Walliser 1999.) In the baseline case we set η = 1 (perfect altruism) and then study the
no-altruism case.
The second panel of table 5 lists the remaining parameters of the model: β, θ, Pθ , ν, and
f and their corresponding values in the baseline calibration. We consider a very parsimonious
calibration and allow for only two values of entrepreneurial ability: zero (no entrepreneurial
20
ability) and a positive number. This implies that the transition matrix Pθ is a two-by-two
matrix. Since its rows have to sum to one, this gives us two parameters to calibrate, corresponding to the persistence of each of the two ability states (see appendix A1 the actual
values used). We also have to choose values for ν, the degree of decreasing returns to scale
to entrepreneurial ability, and f , the fraction of working capital the entrepreneur can keep in
case he defaults. This gives us a total of six parameters to calibrate to the data.5
We use these six parameters to pin down the following moments generated by the model:
the capital-to-output ratio, the fraction of entrepreneurs in the population, the fraction of
entrepreneurs exiting entrepreneurship during each period, the fraction of workers becoming
entrepreneurs during each period6 , the ratio of median net worth of entrepreneurs to that of
workers, and the wealth Gini coefficient. It should be noted that the Gini coefficient is just a
summary of wealth inequality. A model can match the Gini coefficient for wealth while at the
same time doing a very poor job of matching the overall wealth distribution. For example, a
high Gini coefficient can be generated either by having too many people holding no wealth,
or by having just a few people holding a lot of it.
Given the features matched in the calibration, we analyze how well the model matches the
overall distribution of wealth and the distributions of wealth for entrepreneurs and workers.
We use the implications of the model in this respect as a check of the validity of our model.
We then study the role of borrowing constraints and voluntary bequests.
5
Note that we do not impose an exogenous minimum firm size or investment level, nor start-up costs. We
experimented adding a fixed start-up cost and a minimum firm size (both on the order of $5,000–20,000), but
doing so had no significant impact on our numerical results.
6
Both in the model and in the data, entry and exit rates refer only to people that were in the model (or
survey) in both periods and transitioned from one occupation to the other; they do not include people that
die while running an entreprise, nor people that start their enterprise at the beginning of their economic life.
For this reason, entry, exit, and the steady-state fraction of entrepreneurs are not linked by the identity that
would hold in an economy with infinitely-lived agents.
21
IV. Results
We first study the two versions of our model (one without and one with entrepreneurs) and
discuss their ability to reproduce the observed inequality in wealth. We also highlight the key
intuition of the underlying saving behavior and its implications for wealth concentration.
We then study the effect of borrowing constraints and voluntary bequests on both inequality and aggregate capital accumulation.
The first row in table 6 displays the aggregate capital-output ratio and several statistics
on the wealth distribution in the United States. The notion of capital that we use includes
residential structures, plant, equipment, land, and consumer durables, and it implies a capitaloutput ratio of about 3 for the period 1959–92 (Auerbach and Kotlikoff 1995). (The ratio of
average wealth to average income is also about 3.) The data pertaining to the distribution of
wealth come from the 1989 SCF. The waves for other years are similar.
In the other rows of the table, we report the corresponding statistics generated by the
simulations of various versions of our model economy.
A. The model without entrepreneurs
The second row of table 6 refers to the model economy without entrepreneurs. In this run,
we assign zero entrepreneurial ability to everyone and change the household’s discount factor to match the same capital-to-output ratio. All other parameters, including the general
equilibrium prices, are the same as in the benchmark economy.
These results thus refer to a model economy with labor earnings risk and a simplified lifecycle structure. As we can see from the table, this model economy produces a distribution of
wealth that is much less concentrated than that in the data and that, in particular, does not
22
explain the emergence of the large estates that characterize the upper tail of the distribution
of wealth. Figure 1 compares the data on the distribution of wealth (SCF, 1989 in thousands
of dollars) with the one implied by the model without entrepreneurial choice. While the data
on wealth display a fat tail, in the model without entrepreneurial choice all households hold
less than $1.1 million.
B. The model with entrepreneurs
The third row of table 6 refers to the benchmark economy with entrepreneurs. In our baseline simulation the equilibrium interest rate r is 6.5%, the share of total wealth held by
entrepreneurs is 29%, compared with 33% in the data, and the degree of decreasing returns to
scale to the entrepreneurial technology is 0.88, which is a value consistent with those estimated
by Burnside, Eichenbaum, and Rebelo (1995) and Basu and Fernald (1997).
This parameterization matches the distribution of wealth very well both for the overall
population (figure 2) and for that of the entrepreneurs (figure 4).
Figure 3 compares the wealth distributions generated by the model for entrepreneurs and
workers. Figure 4 shows the wealth distribution for the subpopulation of entrepreneurs for
the model and the data. These pictures reveal two important features of the baseline model.
First, and consistently with the data, the distribution of wealth for the population of entrepreneurs displays a much fatter tail than the one for workers. Second, contrary to the
model without entrepreneurial choice, the baseline model generates distributions of wealth for
both entrepreneurs and non-entrepreneurs with a significant mass of people who have more
than $1.1 million. In the model, the non-entrepreneurs in the right tail of the wealth distribution are former entrepreneurs or descendants of entrepreneurs who have not continued the
business of the parents.
23
In order to explain entrepreneurial behavior, figure 5 displays the saving rate7 for people
who have the highest ability level as workers during the current period. The solid line refers
to the people who get the high entrepreneurial ability level during the current period, while
the dash-dot line refers to those who get the low entrepreneurial ability draw. Given the same
asset level (and potential earnings as workers), the people with high entrepreneurial ability
have a much higher saving rate.
Those with low entrepreneurial ability (who are thus workers) exhibit the buffer-stock
saving behavior highlighted by Carroll (1997): if their assets are low, they save because they
are experiencing a high ability level as workers and want to build up their buffer-stock. If
their assets are high enough, they dissave, and the richer they are, the higher their rate of
dissaving. In this simulation, the asset level at which the saving rate goes from positive to
negative is below $1 million.
The people with high entrepreneurial ability become entrepreneurs only if their wealth
is above a certain level, denoted in the graph by a vertical line. The saving rate of those
with high entrepreneurial ability who do not own enough assets to become entrepreneurs is
higher than the one for the workers because ability is persistent, and the workers with high
entrepreneurial ability save to have a chance to start a business in the future. In this region,
the distance between the solid line and the dash-dot line is solely due to the higher implicit
rate of return from saving that one could obtain becoming an entrepreneur in the future: all
households become workers in this range and earn the same income, but the desire to become
entrepreneurs generates a higher saving rate for those who have such ability.
The saving rate of those with high entrepreneurial ability and enough assets to become
7
The saving rate in the graph is defined as assets in a given period minus assets in the previous period,
divided by total income during the period.
24
entrepreneurs is positive and considerably higher than that for workers. The return on the
entrepreneurial activity is high, and the entrepreneur would like to increase the size of the
firm by borrowing capital. However, the borrowing constraint limits the size of the firm. In
order to expand the business, the entrepreneur must in part self-finance the increase in capital.
The combination of higher returns from the business together with the budget constraint thus
generates a very high saving rate for entrepreneurs. As the firm expands, the returns decrease.
Therefore, the saving rate will also eventually decrease. (We truncate the axis of the graph
for easier readability.)
With only one positive level of entrepreneurial ability (as we assume in our calibration) and
in absence of borrowing constraints, there would be only one optimal firm size. Figure 6 shows
how in our framework borrowing constraints can generate a large amount of heterogeneity in
the firm size distribution. The distribution generated by the model exhibits high dispersion
and a fat tail; the tail is generated by the entrepreneurs who have remained in business for a
long period (and have possibly inherited the firm from the parents) and have thus had time
to save and increase the size of their firms.
C. The borrowing constraints
In this section, we examine the effect of changing the tightness of the borrowing constraints.
To make the constraints more stringent, we increase f , the fraction of working capital that
cannot be seized by creditors, from 0.75 to 0.85. The more the entrepreneur can appropriate
in case of default, the stronger the incentive to default for a given collateral level, and the
less the creditor is willing to lend. This increase in f could be interpreted as less efficient
enforcement of property rights by the courts, or as more lenient bankruptcy laws.
Figure 7 shows the maximum amount of investment (including one’s own assets and bor25
rowed funds) for a young entrepreneur who has the highest ability level as a worker as a
function of his own assets. The solid line refers to the baseline model, while the dash-dot line
refers to the model with more restrictive borrowing constraints (and nonrecalibrated β). In
both economies the entrepreneurs with few assets cannot borrow. The amount of collateral
necessary to borrow a positive amount in the two economies coincides at low levels of assets.
The entrepreneur with the lowest ability level as a worker must have at least $10,000 in order
to borrow some funds; this amount increases to $86,000 for the entrepreneur with the highest
ability level as a worker. This happens because a more able worker is better off in case of
default; therefore, he has to provide more collateral. The key difference in the two economies
is that richer entrepreneurs can borrow and invest less in the economy with more restrictive
borrowing constraints. For this reason they need more initial assets to implement a project
of a given size, and it takes them longer to become rich and own and run a large firm. If the
entrepreneur is rich enough, he is unconstrained.
The first two lines of table 7 report, respectively, selected statistics of the U.S. data and
the of the baseline calibration. The third line of table 7 reports the effects of more restrictive
borrowing constraints. The capital-to-output ratio drops drastically, from 3.0 to 2.7, and the
fraction of entrepreneurs falls from 7.5% to 6.9% as fewer high-ability individuals can now
borrow and start a firm. The decrease in the fraction of entrepreneurs happens despite an
increase of the equilibrium interest rate from 6.5% to 7.5%, which makes it easier (and faster)
for savers with high entrepreneurial ability to accumulate enough capital to start a business.
An increase in the tightness of the borrowing constraint, as seen in figure 7, forces entrepreneurs, and in particular rich ones, to borrow less and run smaller firms. They make
fewer total profits and save less, and, as a result, they are poorer. The distribution of wealth
becomes less concentrated; for instance, the share of total net worth held by the richest 1%
26
decreases from 31% in the baseline calibration to 24%, and the share of total net worth held
by entrepreneurs decreases from 29% to 25%.
Hence, as the collateral requirements rise, wealth inequality falls, but this comes at the
expense of lower capital accumulation and output.
D. Bequests
In the baseline economy households are altruistic toward their offspring; therefore, the total
amount of bequests includes both voluntary and accidental bequests due to life-span risk. We
use our model to study what happens to entrepreneurial choice and to wealth inequality when
households do not care about their descendants and all bequests are accidental.
The fifth line of table 7 displays how the aggregates change when we set to zero the
degree of intergenerational altruism. The absence of the voluntary bequest motive reduces
the incentives to accumulate capital and run larger and larger firms. On the one hand, younger
people are bequeathed less wealth, and in presence of borrowing constraints, this means that
young potential entrepreneurs have fewer resources to start and increase their businesses. On
the other hand, the equilibrium interest rate increases to 9.3%, thus allowing more high-ability
individuals to use the increased proceedings from their earnings to start a business activity.
As a result, the fraction of entrepreneurs is roughly unchanged.
The effects on aggregate capital accumulation are large: in absence of a voluntary bequest
motive to save, the total capital of the economy would decrease from 3.0 to 2.5. The concentration of wealth would also drop substantially: the Gini coefficient of inequality would go from
0.8 to 0.7, and the fraction of wealth held by the richest 1% from 31% to 21%. As also shown
by De Nardi (2004), voluntary bequests are fundamental in explaining the concentration of
wealth.
27
In this model economy, voluntary bequests provide rich entrepreneurs with an additional
incentive to save and also generate the intergenerational transmission of large fortunes (and
firms) across generations.
To better understand the role of voluntary bequests, we run another experiment (last line
of the table), in which we increase the discount factor β to .882 (up from .867 in the baseline
calibration) to match a capital-output ratio of 3.0. The fraction of entrepreneurs increases
compared to the baseline model, from 7.5% to 7.9%. This effect is mainly due to the increase
in the household’s discount factor (β). In this calibration, households have no bequest motive,
but are more patient. This implies that the younger households accumulate more wealth than
in the baseline model, while the old decumulate faster, and thus keep less wealth, because
of the lack of altruism. More people of working age become entrepreneurs, and the old have
fewer incentives to continue and expand the entrepreneurial activity and pass to their offspring
less wealth and smaller firms. This reduces the number and the size of large firms. For these
reasons, the wealth concentration generated by this experiment is lower than the one in the
benchmark economy and in the actual data; for instance, the share of total net worth held by
the richest 1% drops to 28%, down from 31% in the baseline economy.
V. Inspecting the model’s mechanisms
Recent literature has cast doubt on the relevance of borrowing constraints to entrepreneurial
entry (Hurst and Lusardi 2004) and on the size of the returns to entrepreneurship (Moskowitz
and Vissing-Jørgensen 2002).
High wealth inequality in our model is generated by the combination of occupational choice
in presence of borrowing constraints and high potential returns to entrepreneurship.
We check here if the observable implications generated by our model are consistent with
28
the observed data that are the focus of these two papers.
A. Borrowing constraints
The main finding of Hurst and Lusardi (2004) is that the probability of entering entrepreneurship is almost flat over a large portion of the wealth distribution and it then increases for the
richest workers. Based on this finding one might (erroneously) conclude that entrepreneurs
are not borrowing constrained.
There are two main points worth discussing.
The first point is that the results of our paper do not depend in important ways from
the fact that the financial constraint affect the decision to become an entrepreneur. Rather,
it is the greater incentive to save after entry that is crucial for the results. We have run
and calibrated versions of the model in which people retain labor income upon entering entrepreneurship, and in which, therefore, the entry decision does not depend on wealth, but
only on entrepreneurial ability. This version of the model produces numbers that are quantitatively very close to the other version, and all of the conclusions that we draw in the paper
remain the same.
The second point is that our model of occupational choice with borrowing constraints
produces entry decisions that are consistent with Hurst and Lusardi’s finding. To check on
this we proceed as follows. We generate many samples of households from our model, each
of which is of the same size of Hurst and Lusardi’s sample. We then use each sample to
estimate a probit regression, according to which the probability of entering entrepreneurship
is a function of a fifth order polynomial in the household’s own wealth, controlling for income,
age, and previous entrepreneurial status.8 We finally use the estimated probit coefficients
8
We do not need to condition on education, gender, marital status, and race, as such dimensions of het-
29
from all of these samples to construct 95% confidence intervals for the estimated probability
of entry as a function of wealth (we fix all other controls at their mean).
The two panels in figure 8 plot Hurst and Lusardi’s estimated function (dashed line), and
the confidence intervals (between the two solid lines) generated by two versions of our model.
The left panel refers to our benchmark model. Two features are worth noticing: first, the
entry probabilities implied by the benchmark model are lower than in Hurst and Lusardi’s
sample. This makes sense since the relevant notion of entrepreneurship for our model (7.5% of
households are entrepreneurial households) is more restricted than the one in the Hurst and
Lusardi’s sample (they do not report the exact number, but our calculations with the PSID
bound it between 11% and 13%). If the relevant fraction of “entrepreneurs” in the population
is higher, so is the entry probability.
Second, both the Hurst and Lusardi’s estimates and our confidence intervals are consistent
with an entry probability that is a convex function of wealth. The intuition is linked to the
endogeneity of both wealth and entry into entrepreneurship. In presence of borrowing constraints a worker with high entrepreneurial ability is likely to save to enter entrepreneurship.
As a result, when observing a cross-section of people, we are not very likely to observe many
high ability potential entrepreneurs among the poor. On the other hand, several of the rich
workers are those workers that have high ability as entrepreneurs and that have been saving
to accumulate enough wealth to enter enter entrepreneurship. For this reason, if we give an
extra dollar to someone in the lower part of the distribution, this person is not very likely to
enter entrepreneurship, while if we give a dollar to someone that is wealthier, he is more likely
to be around the entry threshold, and thus to enter entrepreneurship. Our estimates for this
definition of entrepreneurship, however, predict a steeper positive relationship at low levels of
erogeneity are absent from our model.
30
wealth than the one estimated by Hurst and Lusardi.
As we discussed, there is a lot of heterogeneity among the households that report being
“self-employed” or “business owners”, and those that are called “entrepreneurs” in the data
(and in Hurst and Lusardi’s paper as well) are not necessarily all entrepreneurs in the sense
of our model. The second panel in figure 8 estimates the same regression in the case in
which a subset of the simulated agents that are workers for the purposes of our model9 are
classified as self-employed workers, and counted as entrepreneurs for the purpose of comparing
our results with the survey data.10 While stylized, this experiment is very instructive: the
function estimated by Hurst and Lusardi now falls within our 95% model-generated confidence
interval. Given that we assume only one type of entrepreneurial ability, and that none of the
calibrated parameters were chosen to match this aspect of the data, it is remarkable how our
model is not inconsistent with flat entry probabilities over large sections of wealth holdings.
Consistently with our findings, Buera (2006) estimates a model of entrepreneurial choice,
and finds that allowing for a slightly more general formulation of entrepreneurial heterogeneity
can do an even better job of matching the estimated entry probability.
B. Returns from private business ownership
Moskowitz and Vissing-Jørgensen’s (2002) computations cast doubt on the assumption that
entrepreneurs face potentially high rates of returns. Their computations are complex because
9
Notice that being an “entrepreneur” in our model is a statement about the household’s production function
and rate of return from saving. It is not a statement on who the household employer is, nor concerning where
they work, nor the flexibility of hours worked and so on.
10
We choose this fraction so that the “entrepreneurs” (including the true ones in the sense of our model) in
our model-generated data are about 12% (a number similar to the one in Hurst and Lusardi’s sample). We
assume all workers across the wealth distribution have a constant probability of becoming “non-entrepreneurial
self-employed,” and that this probability is uncorrelated to all other characteristics. We also assume that the
probability of exiting the “non-entrepreneurial self-employed” status is the same as exiting entrepreneurship,
but the results are not very sensitive to this assumption.
31
their goal is to compare aggregate returns to private and public equity, and they thus need to
adjust their computations for firms entry and exit.
Given that our goal is to compare returns to entrepreneurship in the data and in our
model, and that our framework explicitly deals with entrepreneurial mobility, we compare the
cross-sectional distribution of returns to entrepreneurship in a given period in our model and
in the data. Consistently with our model, we compute this distribution of returns for the
self-employed business owners (who are a subset of all of those who hold private equity) using
the 1989 SCF wave (Table 8). These returns are computed as entrepreneurial income divided
by business net worth.
Interestingly, we find that the size of the returns to private equity crucially hinges on
how income from the firm is divided between entrepreneurial wages and return to capital. It
is well known that this split is in practice arbitrary and likely to depend on tax incentives
and possibly other considerations. In the SCF data, if one does not include the self-reported
wages and salaries, such rate of return is 3% for the entrepreneurs at the 50th percentile, and
143% for those at the top 10%. If, as an extreme case, all wages and salaries received by
the entrepreneurs are included in the computation of such return, the corresponding numbers
become 40% and 520%, which are far bigger numbers.
In our model, it is not clear how one should compute wages for the entrepreneurs. One
could take the view that their labor income is the shadow one, meaning the one that they
could make if they were to work as workers (which is not observed in the SCF data). But
one could equally plausibly assume that entrepreneurial profits should be computed as the
amount of entrepreneurial capital times the rate of return from capital in our economy (which
is 6.5%), while the rest of the entrepreneurial income is due to entrepreneurial talent, and
should thus attributed to entrepreneurial wages. Given the arbitrariness of this split, we
32
believe the returns to be compared in our model and the data should be computed by using
total income from the entrepreneurial business activity, both in our model and in the data.
Given that our model, by design, abstracts from many aspects of entrepreneurial choice,
the distribution of returns is less disperse in our model than in the data. We thus compare
the median distribution of returns, which is 49% in our model, compared to 40% in the data.
Our median entrepreneurial return is thus only a little higher in our model than in the SCF
data.
There are two reasons why our computed return is overstated compared to the one computed in the SCF data. First, households tend to underreport income. Research by Internal
Revenue Service (IRS) (1990) computes business income underreporting for the 1985 to 1992
period ranging from 28% to 40%. The SCF data are not collected for tax purposes, and it is
possible that the households underreport less to the SCF than to the IRS. To be conservative, we compute the implied median return in our economy for 10% to 20% business income
underreporting. The corresponding median returns become 43% and 36% respectively.
Second, the computed return from our model does include capital gains (which, for the
purpose of our model, are indistinguishable from other entrepreneurial income), while, due
available data limitations, the returns computed from the SCF do not include capital gains.
We thus conclude that entrepreneurial returns in our model are consistent with those
measured in the data.
VI. Conclusions
We developed and solved numerically a model of occupational choice, wealth accumulation,
and bequests in which entrepreneurs face an endogenous borrowing constraint that limits the
amount that they can borrow. The entrepreneur’s wealth acts as collateral, so the richer the
33
entrepreneur, the higher the amount that he can borrow.
A very parsimonious parameterization of our model generates a wealth distribution that
matches the one observed in the data, both for entrepreneurs and for workers. It also produces
returns from entrepreneurship, and household’s entry probabilities into entrepreneurship as a
function of one’s wealth, that are consistent with the ones measured in the data. None of the
parameters of the model were chosen to obtain these results.
The key mechanism is that many entrepreneurs face potentially high rates of returns but
are constrained in the amount that they can borrow. To expand their firm, these entrepreneurs
keep saving. In doing so, they become richer and richer. The most successful dynasties share
their fortunes with their children, some of which will keep the family firm going, thus expanding
the dynasty’s fortune even more.
We show that the tightness of borrowing constraints and voluntary bequests are main
forces in determining the number of entrepreneurs, the size of their firms, the overall wealth
concentration in the population, and the aggregate capital accumulation.
These results have implications for policy analysis, such as subsidized loans to entrepreneurs
and estate taxes. Subsidized loans would make it cheaper for the entrepreneurs to borrow,
but would also change their incentives to default, making the effects of this policy a priori ambiguous. Taxing bequests may decrease inequality, while at the same time reduce the amount
of entrepreneurial wealth that could be used as collateral, and thus affect both the number of
entrepreneurs and the total capital of the economy, as shown by Cagetti and De Nardi (2004).
We have assumed that an agent can exploit one’s own entrepreneurial ability only by
starting and developing a business. In presence of borrowing constraints the entrepreneur
might want to sell his idea or project to another, potentially less constrained, party. In many
situations, however, markets for ideas or projects are very limited. Potential explanations
34
for this, first expressed in Arrow’s work (1962), are that informational problems may prevent
potential buyers from evaluating the entrepreneur’s project, and also that the innovating
entrepreneur may have problems in appropriating the returns from his idea because it might
be too complex to write or enforce a contract specifying the usage of the idea and the payments
for information. In our model there are two sectors: entrepreneurial and non-entrepreneurial
firms. Only part of the productive and inventive activity is generated by the constrained
sector, while the rest is generated by non-entrepreneurial firms, which face no borrowing
constrains, and where it does not matter who develops the idea and who implements it. We
leave all of these issues for future research.
35
A1. Income and entrepreneurial ability processes
We assume that the income process is lognormal and AR(1). We approximate it with a five
point discrete Markov chain, using the method described in Tauchen and Hussey (1991).
The resulting grid points y for the income process (normalized to an average of 1) are
0.2468 0.4473 0.7654 1.3097 2.3742
and the transition matrix Py is
0.7376 0.2473 0.0150 0.0002 0.0000
0.1947 0.5555 0.2328
0.0113 0.2221 0.5333
0.0001 0.0169 0.2328
0.0000 0.0002 0.0150
0.0169 0.0001
0.2221 0.0113 .
0.5555 0.1947
0.2473 0.7376
We assume that the entrepreneurial ability process is uncorrelated with the income process.
The two values for ability θ are 0 (meaning no entrepreneurial ability) and a positive value
(0.514), and the transition matrix Pθ is
0.964 0.036
0.206 0.794
.
A. Correlation between abilities
We have so far assumed that working ability (y) and entrepreneurial ability (θ) are uncorrelated. It is difficult to measure such correlation in the data. While many entrepreneurs
are high ability individuals who would have high earnings if employed by a company, others
36
successful entrepreneurs may do poorly if they were to work for a corporation.
One important piece of evidence in favor of our specification is that we replicate the income
of entrepreneurs prior to starting their businesses fairly well. Using the PSID, one can compare
household previous labor incomes for individuals that subsequently decide to either enter
entrepreneurship or remain workers in a given period. Hurst and Lusardi (2004) report that
the labor earnings over the previous five years of those that enter entrepreneurship in a given
period is 1.32 times the labor income during the previous 5 years of those that choose to remain
workers in the same period. Our simulations reproduce this feature: the ratio of the incomes
for the two groups (entrants and non-entrants) is 1.35. This correlation arises endogenously
in our model. Because of borrowing constraints high entrepreneurial-ability workers save to
reach their constrained optimal firm size at entry. Among the high entrepreneurial ability
workers, those that receive high labor earnings realizations can save more and are thus more
likely to accumulate enough capital and to enter entrepreneurship.
Moreover, as a robustness check we also study the effects of allowing for positive correlation
between these two ability processes. To make this comparison as clean as possible we keep the
marginal distributions and transition probabilities as in the baseline case. We then assume
that the two processes have a positive correlation of 0.4. All other parameters are as in the
baseline economy, except for the discount factor β, which we recalibrate to obtain the same
capital-income ratio as in the benchmark. In this economy, the ratio of labor income over the
previous five periods for entrants and non-entrants becomes 1.59, which is higher than the one
observed in the data. The resulting wealth distribution is close to the one in our benchmark
economy: the richest 1% hold 33% of the total wealth, and the richest 5% hold 63%. The
main difference is that in this economy the number of entrepreneurs decreases to 5.2%. This
happens because in presence of positive correlation between the two dimensions of abilities,
37
people with high entrepreneurial ability tend to have a higher option value of remaining in the
non-entrepreneurial sector, and are thus less less likely to become entrepreneurs. It is worth
noting that this feature does not significantly affect the right tail of the wealth distribution
and the saving and investment behavior of the richest entrepreneurs. As we have already
mentioned, this tail is composed of the few entrepreneurs who have remained successful for
several periods and who have therefore managed to grow their business.
38
A2. The algorithm
The algorithm proceeds as follows.
• Construct a grid for the state variables. The maximum asset level is chosen so that it
is not binding for the household’s saving decisions.
• Fix a tax rate τ , an interest rate r and a wage rate w. Taking these as given, solve for
the value functions using value function iteration.
• Construct the transition matrix M . Compute the associated invariant distribution over
states, starting from a guess for π and iterating on π ′ = M π ′ until (π ′ − π) is smaller
than a given convergence criterion.
• Compute total savings and total capital invested in the entrepreneurial sector implied
by the invariant distribution. Total capital invested by the non-entrepreneurial sector is given by the difference between total savings and total capital invested by the
entrepreneurs.
• Compute r and w implied by the above quantities and the non-entrepreneurial aggregate
production function, update the wage and interest rate used to solve the problem, and
iterate until convergence on the factor prices is reached.
• Compute the social security system imbalance and iterate on τ until outlays equal revenues.
The computation of the value functions is nonstandard because of the endogenous borrowing constraints. For each state x, the endogenous borrowing constraint specifies a maximum
ˆ
ˆ
amount k(x) that an entrepreneur can borrow. The specific function k depends, however, on
39
the value functions themselves. In the algorithm we exploit the fact that, for a given set of
˜
state variables, if an entrepreneur runs away with a given level of capital k, he would also run
˜
ˆ
away with any k + ǫ, where ǫ ≥ 0. We adopt the following algorithm: initialize k(x) = kmax ,
the maximum investment level in the economy. We solve the value functions, iterating until
convergence, conditional on this borrowing constraint. For each value of x, we compare the
value function associated with remaining an entrepreneur and repaying the debt with the value
function associated with default; we find the maximum level of investment (and borrowing)
ˆ
for which the entrepreneur would not default and set the new k(x) to this new value, and
compute again the value functions conditional on this updated constraint. This procedure is
ˆ
iterated until k does not change across iterations.
ˆ
Because we do not constrain the k(x) functions to be decreasing when we iterate on them,
we are not imposing convergence. Together with the initialization of these functions at the
maximum possible level of borrowing, this implies that if the model has more than one solution,
and if the algorithm converges monotonically, then we converge to the “best” solution, that
is, the one that allows for the most borrowing in the economy. In all of our simulations the
algorithm did converge monotonically.
40
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45
TABLE 1
Percentage of entrepreneurs (according to various definitions) in the
population and corresponding share of total wealth held.
Percent in population
Share of total wealth
Business owners or self employed
16.7
52.9
All business owners
13.3
48.8
Active business owners
11.5
41.6
All self-employed
11.1
39.0
Self-employed business owners
7.6
33.0
46
TABLE 2
U.S. Wealth distribution
Fraction of people, top
Total net worth held
1%
5% 10%
20%
30%
54% 67%
81%
47
TABLE 3
Fraction of entrepreneurs (according to various definitions) in a given
wealth percentile of the overall U.S. wealth distribution
Wealth percentile, top
1%
5%
10%
20%
Business owners or self-employed
81%
68%
54%
39%
76%
62%
49%
36%
65%
51%
42%
30%
62%
47%
38%
26%
39%
32%
22%
All business owners
Active business owners
Self-employed
Self-employed business owners
54%
48
TABLE 4
Median and mean net worth (in thousands of dollars) for various groups of
people
median
mean
47
189
Business owners or self-employed
172
599
All business owners
205
695
Bus owners but not active mgmt
293
768
Business owners not self employed
179
470
All self-employed
169
665
Self-employed (active) business owners
265
829
Self-employed and not business owners
36
224
Whole population
49
TABLE 5
Parameters of the model
Fixed Parameter
Value
Source(s)
σ
1.5
Attanasio et al. (1999)
δ
0.06
Stokey and Rebelo (1995)
α
0.33
Gollin (2002)
A
1.0
Normalization
πy
0.978
See text.
πo
0.911
See text.
Py
See text.
Storesletten et al. (2004)
p
40% of average yearly income
Kotlikoff, Smetters, and Walliser (1999)
η
1.0
Perfect Altruism
Calibrated
Parameter
Value
β
0.865
θ
[0, 0.51]
Pθ
See text
ν
0.88
f
75%
50
TABLE 6
Comparing data and models with and without entrepreneurs
Capital-
Percentage wealth in top
output
Wealth
ratio
Gini
entrepreneurs 1% 5%
20%
40%
54
81
94
4
20
58
95
31
60
83
94
U.S. data
3.0
0.8
7.55%
30
Baseline model without entrepreneurs
3.0
0.6
0.0%
Baseline model with entrepreneurs
3.0
0.8
7.50%
51
TABLE 7
The role of borrowing constraints and voluntary bequests
Capitaloutput
ratio
Percentage wealth in the top
Interest Wealth
rate
Gini
entrepreneurs
1%
5%
20%
40%
-
0.8
7.55%
30
54
81
94
7.50%
31
60
83
94
24
49
75
91
7.55%
21
45
73
90
7.9%
28
57
81
94
U.S. data
3.0
Baseline with entrepreneurs
3.0
6.5%
0.8
More stringent borrowing constraints: f = 0.85
2.7
7.5%
0.7
6.90%
No altruism: η = 0, only involuntary bequests
2.5
9.3%
0.7
η = 0, recalibrated β = .88
3.0
6.4%
0.8
52
TABLE 8
Distribution of rates of returns for self-employed business owners.
25th
50th
75th
90th
Only income from business
0%
3%
25%
143%
Including wages and salaries
10%
40%
125%
520%
Notes. — Returns: entrepreneurial income divided by business net worth. First line: entrepreneurial income includes only income or loss from business. Second line: also includes
wages and salaries received by the business owner.
53
0.09
0.08
Fraction of people
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
Positive wealth, in thousands of dollars
5000
Figure 1: Distribution of wealth, conditional on wealth being positive, for the whole population. Dash-dot line: data; solid line: model without entrepreneurs.
54
0.09
0.08
Fraction of people
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
Positive wealth, in thousands of dollars
5000
Figure 2: Distribution of wealth, conditional on wealth being positive, for the whole population. Dash-dot line: data; solid line: baseline model with entrepreneurs.
0.09
0.08
Fraction of people
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
Positive wealth, in thousands of dollars
5000
Figure 3: Distribution of wealth, conditional on wealth being positive, in the baseline model
with entrepreneurs. Solid line: workers; dash-dot line: entrepreneurs.
55
0.09
0.08
Fraction of people
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
Positive wealth, in thousands of dollars
5000
Figure 4: Distribution of the entrepreneurs’ wealth, conditional on wealth being positive.
Dash-dot line: data; solid line: baseline model.
0.4
0.3
0.2
Saving rate
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
0
500
1000 1500 2000 2500 3000 3500 4000 4500
Wealth, in thousands of dollars
Figure 5: Saving rate for highest-ability workers. Solid line: with high entrepreneurial ability; dash-dot line: with no entrepreneurial ability; vertical line: asset level at which highentrepreneurial-ability individuals enter entrepreneurship.
56
0.05
Fraction of firms
0.04
0.03
0.02
0.01
0
500
1000 1500 2000 2500 3000 3500 4000 4500
Firm size, in thousands of dollars
Figure 6: Firm size distribution, baseline model with entrepreneurs.
7000
Maximum investment
6000
5000
4000
3000
2000
1000
0
0
500
1000 1500 2000 2500 3000 3500 4000 4500
Own assets, in thousands of dollars
Figure 7: Maximum investment. Solid line: baseline; dash-dot line: more restrictive borrowing
constraints.
57
7
Probability of entrepreneurial entry
Probability of entrepreneurial entry
7
6
5
4
3
2
1
0
0
6
5
4
3
2
1
100
200
300
Wealth, in thousands of dollars
400
500
0
0
100
200
300
Wealth, in thousands of dollars
400
500
Figure 8: Probability of entering entrepreneurship as a function of own wealth as estimated
by Hurst and Lusardi (dashed line), and confidence interval generated by two versions of the
model (solid lines). Left panel: benchmark model. Right panel: benchmark with a small
fraction of “non-entrepreneurial self-employed”.
58
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Marco Bassetto with Thomas J. Sargent
WP-05-07
Rigid Prices: Evidence from U.S. Scanner Data
Jeffrey R. Campbell and Benjamin Eden
WP-05-08
Entrepreneurship, Frictions, and Wealth
Marco Cagetti and Mariacristina De Nardi
WP-05-09
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