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Working Paper Series
Credit and Self-Employment
WP 09-05
Ahmet Akyol
York University
Kartik Athreya
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Credit and Self–Employment∗
Ahmet Akyol†
Kartik Athreya‡
Working Paper 09-05
April 8, 2009
Abstract
Limited personal liability for debts has long been justified as a tool to promote
entrepreneurial risk taking by providing insurance to the borrower in the event
of low returns. Nonetheless, such limits erode repayment incentives, and so may
increase unsecured borrowing costs. Our paper is the first to evaluate the tradeoff
between credit costs and insurance against failure. We build a life-cycle model
with risky, and repeated, occupational choice in the presence of defaultable debt
contracts. We find that limits to liability can encourage self-employment, and
alter the timing, size, and financing of self-employment projects. We also find
that the positive relationship between wealth and self-employment rates may not
be evidence for credit constraints: We show that such a relationship is present
even when limited liability is eliminated.
JEL Classification: J23, E21, D31.
Keywords: Self-Employment, Bankruptcy
∗
An earlier version of this paper was circulated under the title “Exemptions: Limited Enforcement
Encouraging Entrepreneurship?” We thank Yongsung Chang, Dean Corbae, Cesaire Meh, Pierre
Sarte, Yaz Terajima, Steve Williamson, Eric Young, and especially Marco Basetto, Chris Sleet, and
Randy Wright for comments, as well as seminar participants at the University of Toronto’s 2006 RMM
Conference, the 2007 Midwest Macro Meetings (Cleveland), the 2007 SED Meetings (Prague), the 2007
SAET Meetings (Kos), 2007 Bosphorus Winter Meetings (Istanbul), the Chicago and Cleveland Feds,
Boğaziçi, Virginia, Western Ontario, York, and Seoul National Universities. We also thank Andrea
Waddle for excellent research assistance. The views expressed are those of the authors and do not
necessarily represent those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
All errors are ours.
†
Department of Economics, York University, aakyol@econ.yorku.ca
‡
Federal Reserve Bank of Richmond, kartik.athreya@rich.frb.org
1
1
Introduction
Borrowing constraints are seen as a significant barrier to entrepreneurial activity
in the US. The perception of such constraints has led to the creation of agencies such
as the US Small Business Administration, which channels billions of dollars of credit
to entrepreneurs.1 Moreover, current public policy is crucially premised on the view
that borrowing constraints arise, ultimately, from the risk of borrower default. Striking
evidence for this view is seen in the pervasive use of loan guarantees, rather than outright
grants. The former, after all, could be expected to improve access to credit only if
borrowing constraints arose from default risk. Similarly, recent work of Rosen and
Willen (2002) also argues that credit market frictions may be important; they find that
absent credit constraints, observed self-employment rates in the U.S. are too low to be
justified in a simple risk/return analysis.
If indeed default risk limits credit access, where does it come from? A primary
suspect for small business borrowers is US personal bankruptcy law. As practiced, the
non–waiveable legal right to bankruptcy protections leaves entrepreneurs, and especially
sole proprietors, with no credible way of committing to repay unsecured debts. The
bankruptcy process not only removes all unsecured debt, but also allows, in many
cases, for some wealth to be retained by borrowers. More generally, any legal limit to
liability for debts means that borrowers, especially those with low personal wealth, pose
a risk to lenders that implies greater costs to start a venture.
Given the clear drawbacks such statutes create, why do they exist at all? Here, the
answer is that small business is seen as an inherently high-risk activity where actuarially
fair insurance against failure is difficult, or impossible, to obtain. In the absence of
markets against such risks, bankruptcy and other limits to liability allow borrowers to
partially tailor loan repayment to avoid severe reductions in their standards of living in
the event of poor returns on investment.
The tension between insurance provision and credit access was recognized very early
in US history. The initial political debate on limited liability in general, and bankruptcy
in particular, in the 1700s revolved squarely around balancing access to credit with a
form of insurance against catastrophic failure, and is documented in Moss (2000). In the
end, the provision of insurance was seen as most important, and bankruptcy provisions
1
Throughout the paper, we will use the terms entrepreneurship and self–employment interchangeably. Our focus is on the role played by credit markets in driving risky occupational choices. We
therefore want our definition of self-employment to be broad enough to capture anybody whose primary income arises from a risky business in which they have a large and poorly-diversified interest. In
turn, we do not want to restrict the set of entrepreneurs to only the (much smaller) subset of individuals
who may possess exceedingly productive or innovative project ideas.
2
were viewed by the majority as an aid, not a barrier, to entrepreneurship in the US.
This benign view of limited personal liability continues to have adherents. For example,
Lawless and Warren (2005) argue that strict bankruptcy law stifles entrepreneurial risk–
taking (see also The Economist, 2006).
In addition to credit conditions, those contemplating self-employment must evaluate
its payoff relative to their prospects as paid workers. The potentially important role
played by opportunity costs arising from alternatives to self-employment is suggested to
us by the persistent empirical regularity that entrepreneurship is chosen relatively more
often by those with poor current corporate sector opportunities. Evans and Leighton
(1989), Farber (1999), Rissman (2003), and Fairlie and Krashinsky (2006) each show
that in the data, poor opportunities for “wage” work are important in generating the
switch to self-employment. Specifically, prior job loss, displacement, and high local
unemployment rates are each associated with a heightened likelihood of entrance to
self–employment. Thus, credit conditions can affect individuals’ labor income though
their impact on self-employment decision. This feature motivates a central aspect of
the timing of resolution of uncertainty in our model: Households first learn their productivity in the “paid” work sector and then choose whether or not be an entrepreneur.
Since limits to liability may actually create the credit constraints that other major policies aim to mitigate, it is important to clarify their effect on credit markets
and, in turn, entrepreneurial activity in the US. The main contribution of this paper is
to provide a detailed quantitative evaluation of how US limited liability policy affects
aggregate entrepreneurship rates and unsecured credit conditions, and whether the outcomes are, in turn, desirable from a welfare standpoint. We model occupational choice
over the life cycle, and our analysis emphasizes the role of household-level decisions generating the preceding aggregates. In particular, we measure the role played by liability
policy in influencing credit constraints, risk taking, and self-employment choices over
the entire life cycle.
Our main results are as follows. First, we find that when limited liability is varied
between full liability and current US levels, the insurance provided by the default option largely offsets the disincentives arising from higher credit costs, resulting in minor
changes in self-employment activity. However, when liability is reduced beyond the
current US levels, the associated default risk increases credit costs sufficiently to limit
credit use, which in turn discourages self-employment. Second, we demonstrate that the
positive correlation between wealth and self-employment does not imply the existence
of credit constraints, as it arises primarily from the interaction of risk and life-cycle
savings behavior. Third, changes in liability policy have distributional implications. In
particular, limited liability appears regressive with respect to age. We find that very
3
low liability sharply affects the young, but has only minor effects on the old, primarily
because the latter have accumulated wealth for retirement. In contrast to its effects on
age, liability policy does not appear to affect high- and low-skilled households differentially. Lastly, very low-liability regimes significantly alter the ability of households to
switch occupations in the event of low corporate sector productivity.
Our study is novel along three dimensions. First, our work is novel in using quantitative theory to understand the role that limited liability plays in risk taking, as opposed
to risk sharing of an exogenous income stream.2 Second, by incorporating the “real”
options of entry and exit, we are able to discern the effects of liability policy on not
just the intensive margin (i.e., project size) of self-employment, but also the extensive
margin (i.e., the rate of self-employment). As a result, we overcome the fact that data
on self–employment is by definition censored, capturing only those for whom such a
choice was preferred to an unobserved alternative. Third, our model produces a full
schedule of interest rates for debt, on and off the equilibrium path. This allows us to
overcome the classical problem of the identification of credit demand and supply. As
Berkowitz and White (2004) (footnote 29, p16) acknowledge: “Presumably, firms apply
for the amount of credit they expect lenders to provide, and lenders may tell borrowers
in advance how much they are willing to lend.”
Our work is most closely related to Cagetti and De Nardi (2006), but differs in
two key ways. First, in Cagetti and De Nardi (2006), production is riskless, as the
self-employed know their productivity at the time of borrowing. As a result, limited
commitment to repay debt can only limit borrowing. By contrast, our set-up captures
the original impetus for debt relief of allowing debt repudiation to encourage risky
investment. In our model, therefore, households first borrow, then realize the stochastic
output from the project. Second, we do not derive debt limits by assuming that default
is met by permanent autarky. Rather, we treat the decision not to repay debt as
it is treated in practice, whereby households are forced to surrender wealth above a
threshold in return for debt forgiveness. In turn, our model generates such exchanges
in equilibrium, which are naturally interpretable as “bankruptcies” as they are measured
in the data. Additionally, the presence of equilibrium bankruptcy means that interest
rates will vary with the risk associated with any entrepreneur-project pair. Therefore,
our model captures the empirical regularity that low-wealth borrowers face higher credit
costs, and hence are more credit “constrained.” By contrast, in Cagetti and De Nardi
(2006), all debt is risk-free in equilibrium and, therefore, its price cannot feature a
default premium or vary across borrowers with different default risks. Our work is also
related to Terajima (2004), who studies a general model of occupational choice but
2
See e.g., Athreya (2006), and Chatterjee et al. (2002).
4
abstracts from credit frictions altogether. Similarly, Meh and Terajima (2005) focus
on bankruptcy and home ownership decisions in a model related to Terajima (2004)
where, in sharp contrast to our work, occupations are chosen before any uncertainty is
resolved. Lastly, relative to Polkovnichenko (2003), we study a fully dynamic model of
occupation choice over the life cycle, along with a detailed account of the credit market
and its lending terms.3
The remainder of the paper is organized as follows. Section 2 illustrates the key
tradeoffs that we evaluate. Sections 3 and 4 present and parameterize the model;
Section 5 reports and discusses results. Section 6 concludes.
2
Exemptions and Credit: Intuition from a Static
Model
As a preliminary step to motivate our richer quantitative model, we first provide
intuition for the main forces at work in a simple example.4 Let there be two periods,
0 and 1. In period 0 households with internal wealth a0 sign one-period debt contracts
with lenders and receive “D” units of resources. Contracts are exogenously specified to
resemble debt in that they are completely non-contingent outside of bankruptcy. The
total investment in the project is then of size k = a0 + D. In period one, stochastic
productivity θ is drawn according to the p.d.f. π(θ) from the bounded support [θ, θ] ∈
R++ . Given θ, output is denoted by f (k(a0, D), θ). Given output, the household decides
on whether to repay the debt with interest or file for bankruptcy at cost τ . Lenders
must charge an interest rate that depends on the size of the loan and the exemption
in order to break even, given a gross cost of funds Rf ; we denote the net zero-profit
interest rate by R(D, x). For simplicity, in what follows, let a0 = 0, so that k = D.
There are two thresholds for productivity, θ1 , and θ2 , that determine the payments in
default and non-default states. The threshold θ1 is defined by the level of productivity
that makes output equal to the exemption, given project size k. The threshold θ2
is defined by the level of productivity that makes output exceed the exemption by the
gross-of-interest debt R(D, x)D; i.e. θ2 = {θ|f (D, θ)−x = R(D, x)D}. The household’s
expected utility under a given exemption x, for a given face value of debt D is then:
3
The literature is vast, but other important studies include Evans and Jovanovic (1989), Banerjee
and Newman (1991), Quadrini (2000), Albuquerque and Hopenhayn (2004), Quintin (2003), Krasa et
al. (2004), and Mondragon (2006). In each of the preceding, however, limited commitment can only
limit self-employment, and the questions are then: by whom and by how much?
4
We are grateful to Marco Bassetto and Chris Sleet in what follows.
5
V (D, x) =
Z
θ1
u(f (D, θ)−τ )π(θ)dθ+
θ
Z
θ2
u(x−τ )π(θ)dθ+
θ1
Z
θ
u(f (D, θ)−R(D, x)D)π(θ)dθ
θ2
(1)
Notice that the cutoff points θ1 and θ2 , as well as the payment R, are all functions
of x. Thus, when θ ∈ [θ, θ1 ], the household prefers to default, and can keep all of its
output less the cost of bankruptcy, f (k, θ) − τ . When θ ∈ (θ1 , θ2 ], it consumes x − τ
as it files for bankruptcy, pays τ , and keeps output equal to the exemption. When
θ ∈ (θ2 , θ], the household does not file for bankruptcy, and so repays debt and keeps
the remainder: f (D, θ) − R(D, x)D.
Under the preceding bankruptcy filing behavior of households, a lender who lends
the household D units must receive an expected payoff that satisfies:
f
R D=0
Z
θ1
π(θ)dθ +
θ
Z
θ2
(f (D, θ) − x) π(θ)dθ + R(D, x)D
θ1
Z
θ
π(θ)dθ
(2)
θ2
If bankruptcy is costless, i.e. τ = 0, and households are risk averse, the optimal
exemption x∗τ =0 for any given borrowing level D, is the highest level under which the
lender still recovers Rf D in expectation. This is because a higher exemption weakly
increases both thresholds θ1 and θ2 , and thereby shifts the states in which repayment
occurs to progressively higher productivity states. Risk aversion makes this weakly better for the household, while lenders are indifferent because, by assumption, households
repay an expected payment of Rf D. Therefore, when τ = 0, the optimal exemption x∗
for a loan of size D solves:
x∗ (D) = arg max V (D, x) subject to (2)
x
(3)
In other words, for any given debt level D, when bankruptcy is costless, the borrower
seeks the highest exemption x∗ for which there is an interest rate R(D, x) that allows
lenders to break even. This idea is formalized as follows.
Theorem 1 Assume that the bankruptcy cost, τ , is zero. For an arbitrary borrowing level D, the utility of the risk-averse entrepreneur, V (D, x), is increasing in the
exemption level, x if there exists a payment schedule R(x) that satisfies (2).
Proof. See Appendix.
As a result, within the class of debt contracts studied in this paper, the best static
arrangement for the household then must solve:
6
D ∗ ≡ arg max V (D, x∗ (D)).
D
(4)
An implication of the preceding result is that if households are not given the flexibility to choose exemptions and borrowing simultanously, they may not be able to attain
certain debt levels. That is, under a bounded support for productivity, the lender’s
break-even condition (2) implies that if exemptions rise, the most that an agent can
feasibly borrow falls. That is, if a “high” legal exemption is imposed, bankruptcy will be
ex-post optimal for the household in enough states that the lender will find it infeasible
to recoup a large loan. More precisely, whenever the legal exemption xlegal > x∗ (D ∗ ),
then unless R(D ∗ , xlegal ) exists, the debt level D ∗ will not be feasible. Whenever this occurs, the limited commitment created by a legally imposed exemption will create credit
constraints for households. Moreover, whenever legal exemptions are set at levels that
prevent some households from attaining their desired level of borrowing, their ability
to choose an occupational choice will be compromised.
Another reason why exemptions may hinder household decision making is that
bankruptcy itself is, by all accounts, a costly procedure (see, e.g., Fay, Hurst, White
(2002), Robe et al. (2007), and Athreya (2005)). Costly bankruptcy generates deadweight losses, meaning that even for an arbitrarily given borrowing level, exemptions
should no longer necessarily be made as large as possible. That is, if exemptions are set
such that bankruptcy occurs often in equilibrium, households may be able to do better by repaying fully in more states and filing for bankruptcy less frequently, thereby
avoiding deadweight losses.
Lastly, as noted earlier, two potentially crucial factors in determining credit use
and occupational choice are (i) the relative risk of self-employment to paid work and
(ii) the dynamics of life-cycle earnings. The role of exemptions for self-employment is,
therefore, ultimately a quantitative question, and so we turn next to a fully dynamic
model of unsecured credit and occupational choice over the life cycle.
3
A Dynamic Model of Credit and Self-Employment
3.1
Preferences
The economy is represented by an overlapping generations model of households who
work for J−periods, then retire, and are replaced by the next generation of workers.
Each generation consists of a continuum of ex-ante identical agents who maximize the
expected, additive, and discounted sum of utilities from consumption. Working age is
discrete and is indexed by j = 1, 2, ... J. An agent’s consumption in age−j is denoted
7
by cj . The within–in–the period utility function is given by u(.) where u0 (.) > 0,
u00 (.) < 0. Following work life, households retire and save an amount aJ+1 to provide
consumption in retirement. The valuation of resources at the time of retirement is given
by the function φ(.). We assume that φ0 (.) > 0 and φ00 (.) < 0. Therefore, given the
discount factor βh ∈ (0, 1), households maximize:
E0
J
X
βhj−1u(cj ) + φ(aJ+1 ).
(5)
j=1
An important aspect of our model is that we use the simplest framework possible
to isolate the effect of limited liability on (i) access to credit and (ii) the self-selection
of households into entrepreneurial activity and paid work.5 Our goal is not, for example, to uncover the role of entrepreneurship for inequality, as many previous studies
have done. Therefore, we do not assume “persistence” in entrepreneurial ability across
generations, nor do we assume entry and exit costs or the presence of fixed scales for
project size, as in Quadrini (2000), e.g. We also abstract from complex details of the
tax system.6 Our approach, by contrast, builds in substantial detail to clearly investigate the way in which limited liability, especially as created by bankruptcy law, (i)
alters unsecured credit provision and use,(ii) self-selection into self-employment, (iii)
and entrepreneurial project size.
3.2
Timing, Occupational Choice, and Default
In each period, households choose between being a worker or an entrepreneur.7
At the beginning of each period, a household draws its stochastic productivity in
paid/corporate-sector work, and thereby knows its income under that occupation with
certainty. However, if they choose instead to become self-employed, the household
knows only the probability distribution of entrepreneurial productivity.
With respect to the corporate sector, labor productivity takes on values in a finite
set, i.e., j ∈ {j1 , j2 , ... jN }, for j = 1, ..., J, with each period’s shock drawn according
to the (possibly) age-dependent probability distribution gj (). We denote the mean
5
Throughout the paper, we will use the terms entrepreneurship and self–employment interchange-
ably. Our focus is on the role played by credit markets in driving risky occupational choices. We,
therefore, want our definition of self-employment to be broad enough to capture anybody whose primary income arises from a risky business in which they have a large and poorly-diversified interest. In
turn, we do not want to restrict the set of entrepreneurs to only the (much smaller) subset of individuals
who may possess exceedingly productive or innovative project ideas.
6
See Meh (2005) for the effects of taxes on entrepreneurship.
7
We also rule out “mixing” salaried work and self-employment for self-insurance purposes.
8
level of corporate sector productivity at age-j, for human capital h, by µCorp (j, h).8 The
age-dependence of gj () will allow us to capture the life-cycle path of corporate-sector
labor productivity. The labor income of an agent is then given by n µCorp (j, h).
By contrast, an agent who chooses entrepreneurship operates a stochastic technology
denoted by F (θ, k) whereby gross output depends on the privately observed shock, θ,
and the capital stock, k. The productivity shock θ takes values in {θ1 , ..., θN }, and its
human capital–specific probability density function is given by πh (θ). The variable k
can take values on the set k = [0, k] where k is an endogenous upper bound. To capture
the presence of uninsurable entrepreneurial and labor income risks, insurance contracts
with payments contingent on workers’ and entrepreneurs’ productivities are assumed
to be unavailable. As will be discussed later, to avoid introducing any frictions to
entry and exit, we assume that productivity across salaried work and self–employment
is uncorrelated.
In the model, both entrepreneurial productivity as well as corporate-sector productivity are increasing functions of the human capital index, h. Human capital also
reflects constant (permanent) productivity differences between agents during working
life and is exogenously determined in the first period of the agent’s life.9 . Therefore,
a useful interpretation of h, when it is employed in entrepreneurial activity, is that it
captures the ability of an agent to generate and execute productive ideas for an entrepreneurial project. The advantages enjoyed by those with a general set of skills to
pursue self-employment have been widely documented, most recently in Lazear (2005).
Given the evidence, as well as the difficulties in directly observing the distribution of
“ideas,” we assume at the outset that college–educated agents have higher average productivity than their high school–educated counterparts. Additionally, we discipline our
model by requiring that our benchmark parameterization matches the human capital
distribution among the self-employed, which, in contrast to “ideas”, is something that
is well-measured.
An agent who chooses to become an entrepreneur in period t may finance the project
with any internal wealth a ≥ 0, as well as external borrowing through a bond issue of
face value b ≥ 0. We rule out the possibility of equity issuance by entrepreneurs.10
8
The uncertainty of wage income eliminates the possibility of a non-degenerate wealth distribution
and makes the dynamics of agent’s wealth interesting.
9
Human capital may be interpreted as an agent’s education level, e.g., college graduate vs. noncollege graduate.
10
For instance, entrepreneurs in the model cannot raise funds from “venture capitalists.” There are
two reasons why we make this assumption. First, the goal of the paper is to uncover the tension of
insurance and discounting of risky debt contracts, without resorting to benefits of growth–enhancing
entrepreneurial projects. Second, potential entrepreneurs with growth–enhancing projects are more
9
Default risk on debt depends on the size of the project, the size of the loan, and the
ability of the entrepreneur. Given the observability of educational attainment as a proxy
for the presence of general skills, competition among lenders requires this information
be used. Bond prices, therefore, depend on loan size, internal wealth, and the level of
human capital, and are denoted by q(h, a, b). Given the discount rate, a bond issue
of face value b yields ultimately generates a loan of q(h, a, b)b units of capital. The
entrepreneur’s project size is then simply the sum of the loan and internal funds, and
is denoted by k, where k ≡ q(h, a, b)b + a.11 Once the project size is determined,
the entrepreneur observes the shock θ, whereby output is determined. The agent then
evaluates the default option. If the entrepreneur chooses not to default, the face value
of the loan b must be repaid. If the entrepreneur chooses to default, output and the
productivity shock θ and, hence, output, become publicly observed.
Our treatment of limited liability for the self-employed corresponds most closely to
the structure defined by US Chapter 7, “Fresh Start”, personal bankruptcy policy. In
particular, these provisions determine the maximal wealth that the defaulting household may retain, by means of an “exemption” level. First, denote the sum of current
output and undepreciated capital by F (θ, k), i.e. F (θ, k) ≡ f (θ, k) + (1 − δ)k. We
denote exemptions by x, whereby given output F (θ, k), post-default wealth is simply
min{(F (θ, k), x}. Thus, higher exemptions allow for more wealth to be sheltered while
debts are repudiated.12 Throughout this paper, limited liability will be completely defined by the parameter x. A defaulting household must also pay transactions costs,
denoted by τB .13 Specifically, the payoff to repudiating debts depends on output, which
in turn depends on the human capital level h, the shock value θ, the current wealth
level a, the face value of the loan, b, and the prevailing exemption level, x. Following
the decision to repay debt, the household divides its remaining resources by choosing
current consumption cj and wealth aj+1 for the next period.
If default occurs, lenders seize wealth by using a liquidation technology whereby
they receive any entrepreneurial output (including undepreciated capital) above the
likely to raise capital from venture capitalists, and therefore, can potentially avoid limited commitment
problems in debt contracts.
11
Representing borrowing by nonnegative scalar b is convenient as it makes the capital stock a
nondecreasing function of borrowing for all q ≥ 0.
12
Notice that the exemption level, x, is applied to gross output, whereas bankruptcy law often
has asset-specific exemptions, typically protecting equity in the primary residence of the borrower.
However, case law generally allows the conversion of non-exempt assets (the gross output, F (θ, k))
into exempt assets prior to bankruptcy filing. See, e.g., Huckfeldt (1991) and footnote 3 in Fay, et al.
(2002).
13
These losses include not only legal fees required for filing a bankruptcy petition, but also difficulties
in conducting transactions such as renting a home, car, etc.
10
exemption level. In our model depreciation represents the composite effect of all factors
that transform a given project size in the current period into durable capital stock
tomorrow. Therefore, the correct interpretation of the parameter δ is the following.
Given that we do not explicitly model labor supply by entrepreneurs, or labor hired by
small businesses, the stock of durable capital available one period hence is reduced by
the sum of both physical depreciation and any expenditures on “working capital.” When
productivity and bankruptcy decisions have been made, households choose consumption
and savings. The period then ends. The timing is summarized in Table 1.
In the last period of working-life, the household solves the same problem involving
the occupational choice and the bankruptcy option described above. However, at the
end of the period, the household values wealth carried into retirement according to
the function φ(.). The household then retires, and consumes its wealth. Each retiring
household is then replaced by the next generation, which holds no financial wealth as
it begins its working life. These new households then realize a human capital level h0
drawn from a probability distribution ς(h0 |h) that depends explicitly on parental human
capital.
3.2.1
Can borrowers “opt out” of the bankruptcy option?
A useful feature of our model is that it accomodates the idea that recent financial innovation may now allow entrepreneurs to effectively “opt out” of the bankruptcy
protection provided by the law. Specifically, financial products may now enable households to easily liquefy wealth, which then can be pledged as collateral. An example of a
contractual arrangement that undoes the effects of the bankrupcy code is the reversemortgage contract. The reverse-mortgage contract is equivalent to a secured loan in
which the lender receives some or all of the equity in the entrepreneur’s house in return
for funds lent to the entrepreneur. As a result, an entrepreneur with a house worth
a units can simply convert his equity into funds and run a business of size k = a
14
.
In the model, we capture this possibility: A self-employed agent with internal wealth
a can credibly pledge to repay more than one with less wealth–which is the essence
of posting collateral, and thereby overcoming the limits to commitment created by a
relatively large exemption. Furthermore, as long as the entrepreneur does not default in
any state of the world, he can access secured credit even beyond this level (i.e., k > a)
depending on the exemption level.
14
Thus, the model allows for reverse-mortgage loans even when the exemption level is unbounded,
as is the case for seven states in the U.S.
11
3.2.2
Loan Contracts
Lenders are perfectly diversified and are assumed to observe an agent’s wealth,
human capital, and loan size. The current entrepreneurial productivity shock is not
observable at the time a loan is agreed upon. Competition in financial markets requires
that, in equilibrium, there are zero expected profits on each loan. Let (1+r f ) denote the
gross risk–free interest rate on savings, and let τ denote proportional transactions costs
associated with the intermediation of funds. Therefore, the cost of collecting one unit of
funds by the lender is given by (1 + r f + τ ). Given (1 + r f + τ ), the zero-profit discount
rate on a risk-free loan, which we denote by q f , is given as follows. The zero-profit
discount rate is then simply the reciprocal. That is,
qf =
1
.
1 + rf + τ
(6)
For any given internal wealth and debt level, there is a (possibly empty) set of
realizations for productivity, denoted Λ(h, a, b, q), such that the agent prefers default to
repayment whenever θ ∈ Λ(h, a, b, q). Notice that, in general, the pricing scheme q and
default decision Λ influence each other. In particular, the pricing applied to any given
bond issuance affects the scale of a project and, in turn, ex-post repayment incentives.
The set Λ(h, a, b, q) is restricted by the condition that the face value of debt cannot
be less than the liquidation value of the entrepreneur’s assets. Let σ(h, a, b, q) be the
endogenous probability of default for such an agent, namely
X
σ(h, a, b, q) ≡ Pr(θ ∈ Λ(h, a, b, q)) =
π(θ).
(7)
θ∈Λ(h,a,b,q)
The amount
Ω=
X
θ∈Λ(h,a,b,q))
π(θ) max(0, F (θ, k) − x)
(8)
is the expected amount of output paid to the banks by these defaulting agents, conditional on F (θ, k) − x > 0. For a loan of type (a, b) to a household of with human capital
level h, the lender uses a discounting scheme, q(h, a, b, Λ), in order to break even.15
Given (1 + r f + τ ), the zero–profit condition for the lender of making a loan with
the size qb is given by:
(1 + r f + τ )qb = b(1 − σ(h, a, b, q)) + Ω,
(9)
where the left–hand-side of (9) is the total cost of making loans of size qb, and the
right–hand-side is the sum of revenues from non–defaulting borrowers and recovered
15
This pricing scheme is standard. For applications see Chatterjee et al. (2002), Livshits et al. (2003),
and Athreya (2006).
12
revenues from defaulting borrowers. Expressed in terms of a premium over risk-free
borrowing, we have the following fixed-point condition:
q(h, a, b, Λ) = qlf
"
Ω
(1 − σ(h, a, b, q)) +
b
#
(10)
Given a loan size bq(.), the discount rate is therefore increasing (i.e., q gets smaller)
in the probability of default σ, and decreasing in the recovery of output beyond the
exemption, max{0, F (θ, k) − x}. The mapping from underlying fundamentals to a
“credit supply function” reveals the terms required for loans of arbitrary size, and
endogenously determines borrowing as well as a credit “limit,” whereby the marginal
cost of borrowing goes to infinity.
It is important to emphasize that our model incorporates both secured and unsecured
credit. This distinction between these two types of credit can only be made by inspecting
the corresponding loan prices, i.e., q(.). Notice also that if the default set Λ(h, a, b, q) is
empty, the loan is risk free, and therefore σ(h, a, b, Λ) = 0. This leads (10) to collapse
to q = qlf . For q < qlf , the loan is risky and, therefore, at least partially unsecured.
Furthermore, the above discounting scheme will result in some debt levels not being
observed in equilibrium. However, off-equilibrium discount rates for some debt levels
reflect the fact that, were borrowing to reach certain levels, default likelihoods would
justify the rates.
Notice that limits to liability lower the ability of entrepreneurs to collateralize borrowing. In turn, ex–ante, agents react to generous exemption levels by accumulating
wealth, all else equal, precisely to access secured debt.
3.3
Recursive Formulation
Let the agent’s state vector be denoted S = {a, j, jn , h}. Let V (S) be the value
attained by an agent entering a period with the state vector S. The state vector is
comprised of his current level of assets a, age j, and current corporate-sector wage jn h.
Given the option over whether to be an entrepreneur or worker, V (S) must satisfy
V (S) = max{V e (S), V w (S)},
(11)
where V e denotes the option value of being an entrepreneur and V w denotes the value of
being a worker. Let I1 (S) be the indicator function associated with (11). In particular,
I1 (S) = 1 if V e (S) > V w (S), and zero otherwise. First, we define the Bellman equations
for agents of ages j = 1, 2, ... J − 1. Then, we define the Bellman equations for agents
13
in the last period of their life, i.e. when j = J. We first define the value function for
an entrepreneur and then we define the value function for a worker.
Entrepreneur Given an initial wealth level a, age j and a current corporate–sector
wage level jn h, the agent faces a function q(.) when choosing the face of debt b optimally,
which in turn determines the size of the project according to k = a + q(h, a, b, Λ)b.
Therefore, the ex-ante value of choosing entrepreneurship in the current period, V e (S),
is given by:
V e (S) = max{Eθ W (S, bj , θ)}.
(12)
bj ≥0
Given k, the agent will act optimally for any realization of the productivity shock θ.
In particular, for some realizations of the shock, the agent will choose bankruptcy and
for others, will not. Let W (S, bj , θ) be the maximal value attainable for a household
of age-j whose beginning-of-period state is S, who has chosen to borrow bj units, and
who then receives productivity θ. By definition, W (S, bj , θ) solves
n
o
W (S, bj , θ) ≡ max W B (S, bj , θ), W N B (S, bj , θ) .
(13)
where W B and W N B denote the values of declaring bankruptcy and not doing so,
respectively.
To define the two embedded value functions W B and W N B , note first that the value
of default, W B (S, bj , θ), depends on the size of current output F (θ, kj ), which depends
on the realization of θ and the current-period transactions cost for default, τB . Denote
the price of risk-free bonds (i.e., nonnnegative savings) by q s ≡
1
1+r f
. Letting V (S 0 )
denote the expected continuation value in state S 0 , we then have:
W B (S, bj , θ) = max{u(cj ) + βh EV (S 0 )}
(14)
aj+1
such that
cj + q s aj+1 ≤ min[x, F (θ, kj )] − τB
(15)
kj = aj + q(h, aj , bj , Λ)bj
kj > 0,
cj ≥ 0,
aj+1 ≥ 0,
bj ≥ 0,
(16)
∀ j = 1, J − 1.
(17)
The value of not declaring bankruptcy, W N B (S, b, θ), is given by:
W N B (S, bj , θ) = max{u(cj ) + βh EV (S 0 )}
(18)
aj+1
such that
cj + q s aj+1 ≤ F (θ, kj ) − bj
(19)
kj = a + q(h, aj , bj , Λ)bj
kj > 0,
cj ≥ 0,
aj+1 ≥ 0,
14
bj ≥ 0,
(20)
∀ j = 1, J − 1.
(21)
Entrepreneurs of age-J solve the same discrete optimization problems in (11) − (13)
as entrepreneurs of other ages, with only the modification that the discounted expected
continuation value is given by φ(.). Therefore, the modified objective function of an
age−J entrepreneur choosing bankruptcy is
W B (S, bJ , θ) = max{u(cJ ) + φ(aJ+1 )}.
aJ +1
(22)
Similarly, the value of not declaring bankruptcy, W N B (S, b, θ), is modified for age-J
to read:
W N B (S, b, θ) = max{u(cJ ) + φ(aJ+1 )}.
aJ +1
(23)
Worker Alternatively, the agent may choose to become a worker in the “corporate”
sector, where she faces productivity risk but has access to a technology (and implicitly,
a corporate capital stock) that allows her to produce the consumption good using their
labor alone. To keep matters simple, we assume that workers must hold non–negative
savings. For an agent choosing to be a worker in the current period, the value function
is therefore given by:
V w (S) = max{u(cj ) + βEV (S 0 )}
aj+1
such that
cj + q s aj+1 = n µCorp (j, h) + aj
cj ≥ 0
,
aj+1 ≥ 0,
∀ j = 1, J − 1.
(24)
(25)
(26)
If agents of age-J become workers, their value function satisfies:
V w (S) = max{u(cJ ) + φ(aJ+1)},
aJ +1
(27)
subject to the same constraints as all other workers. With the preceding value functions
in hand, the original comparison to determine occupational choice, in (11), can be made
for all ages j = 1, 2, .., J.
3.4
Equilibrium
Given the individual state space S, we can define X = [0, ∞) × {1, 2, ... J} ×
{j1 , j2 , ...
jN }×{h1 , h2 , ...}×{θ1 , ..., θN }. Let (X, B(X), ω) be a probability space where
B(X) is the Borel σ−algebra on X, and ω is the measure of agents on the state space.
Thus, for each C ∈ B(X), ω(C) is the fraction of agents whose individual states lie in
C. We follow Chatterjee et al. (2002) and Livshits et al. (2003) and fix the risk-free
rate on savings at q f . The individual agent’s policy functions, which solve the dynamic
15
programs in (11) − (27), along with the stochastic process of endowments, induce a
stochastic process for the individual’s state. This process describes the evolution of occupation, borrowing, bankruptcy, and asset holdings according to a transition function
P (x, C), ∀ C ∈ B(X). The transition function in turn implies a stationary probability
measure ω ∗ (C) for all C ∈ B(X), which must satisfy the fixed point property
∗
ω (C) =
Z
P (x, C)dω ∗.
(28)
X
More precisely, the equilibrium is defined as follows:
Definition 2 Given an exemption level, x, a risk–free discount rate on deposits, q f ,
a transaction cost on intermediation, τ , and human capital transitions across generations, ς(.), a recursive (partial) equilibrium for this economy consists of (i) decision rules {I1 (S), b(S), I2 (S; b, θ), c(S; θ), a0 (S; θ)} for occupational choice, borrowing,
bankruptcy, consumption, and savings, respectively, (ii) value functions (11)− (27), and
(iii) a stationary joint distribution ω ∗ of households over asset, wage, and productivity
levels such that
1. Decision rules {I1 (S), b(S), I2 (S; b, θ), c(S; θ), a0 (S; θ)} solve the dynamic programs in (11) − (27).
2. Lenders make zero expected profits on each loan, i.e., the discount rate q is given
by (10).
3. Distributions are stationary and consistent with individual optimal choices as described in (28).
4
Parameters
We parameterize the model in order to generate outcomes consistent with obser-
vations of income, wealth, and occupational choices of US households under current
US liability policy. The full list of parameters is given in Table 2, and the fit of the
benchmark model is given in Figures 1, 2, and 3.
4.1
Preferences
All households have identical within-period preferences of the iso-elastic form:
( 1−ξ
c
if ξ > 1
1−ξ
,
(29)
u(c) =
log(c) if ξ = 1
16
where ξ denotes the coefficient of relative risk aversion. We set ξ equal to 2, as is
standard. The model period corresponds to one calendar year and our benchmark
calibration sets βh = {0.985, 0.970} for college and non-college agents, respectively. We
parameterize the retirement-wealth valuation function such that it has the identical
form, i.e.,
φ(aJ+1 ) = ψh
a1−ρ
J+1
1−ρ
(30)
which, given wealth accumulation patterns during working life, allows us to approximate median wealth among 65-year-old households while eliminating an additional
free parameter. We calibrate the human capital–specific parameter ψh , and set ψh =
{ψc , ψhs } = {15, 7.5} for skilled and unskilled agents respectively. We also set the other
free parameter ρ = 2 to meet our calibration targets.
4.2
Labor Productivity
Our model partitions the population by human capital levels and occupational
choices and, therefore, we closely follow Mondragon (2006) in setting targets. A natural partition for human capital is to allow for two levels of human capital, representing college–educated and high–school educated (non–college) households, respectively.
That is, h = {hc , hhs }. In what follows we use the terms “skilled” and “unskilled” interchangeably with “college-educated”, and “high-school-educated,” respectively. Terajima (2004) measures the fraction of college-educated households to be 35%, and the
high-school educated population at 65%. We match these proportions by assuming that
each child of a college-educated parent attains collegiate education with probability 0.61,
and that each child of the non-college-educated attains collegiate education with probability 0.21. That is, we set ς(h0 = hc |h = hc ) = 0.61 and ς(h0 = hc |h = hhs ) = 0.21.
In terms of the age- and skill-specific path of life-cycle productivity, µCorp (j, h), we
set productivity for college-educated workers by linear interpolation of the estimates of
Hansen (1993). Because agents are assumed to know the corporate sector productivity
prior to choosing self-employment, we will not observe the entire range of such shocks.
Rather, those with particularly low current corporate productivity might exercise the
option of self-employment. Therefore, we set the mean age-profile of high-school educated households µCorp (j, hhs ) such that in equilibrium, we match the average skill premium, given the self-selection of households into the corporate sector. For the volatility
of corporate sector productivity risk, we set the standard deviation of shocks to approximate the unconditional cross-sectional variance of log-earnings as estimated by Hansen
(1993) and others.
17
4.3
Entrepreneurial Production
To parameterize entrepreneurial production, we first follow Polkovnichenko (2003)
and assume that the level of human capital is not occupation–specific and, therefore,
does not depreciate if the agent enters into or exits from entrepreneurship. For the
entrepreneurial production technology, we set F (θ, k) ≡ θk γ + (1 − δ)k.
The production parameter γ is common to all entrepreneurial ventures, and is set
to 0.75, close to the value in Cagetti and De Nardi (2006). A common returns–to–scale
parameter allows us to isolate the tension between the insurance provision and credit
limits created by the ability to shelter wealth while repudiating debts. We set the
effective depreciation rate, δ = 0.11, in line with standard values.
To parameterize production risk, θ, we follow Davis and Willen (2002) and impose
zero correlation between the corporate–sector and entrepreneurial–sector productivity.
The distribution of shocks to the entrepreneurial project varies across human capital
types. The logarithm of θ is given for college-educated households by an intertemporally,
and cross-sectionally, i.i.d. normal random variable with mean µθc = −1.35 and standard deviation σθc = 0.83. For high-school educated households, we have µθhs = −1.44,
and σθhs = 0.83. This parameterization allows us to closely match the respective sizes
of the populations of low and high human capital workers and entrepreneurs, as well as
their relative income and wealth.16
4.4
Credit Markets and Bankruptcy
With respect to the costs of borrowing and the returns to saving, we set the risk-free
rate (1 + r f ) = 1.04, following Mehra and Prescott (1985), which implies a price on
savings of approximately q s = 0.96. With respect to costs of bankruptcy, to capture
various fees and court costs associated with the actual bankruptcy filing procedure, we
assign τB = 0.15, which corresponds to roughly $7,500, given the total of legal fees,
court costs, and other miscellaneous expenses as documented, e.g., in Caher and Caher
(2003).
In order to set the target for default in the model, we proceed as follows. First, Lawless and Warren (2005) report that up to 20% of bankruptcy filings are attributable to
the self-employed, even though their population share has been measured to be as low as
7%. By contrast, in the overall population, the Chapter 7 bankruptcy rate (the form the
model most closely represents) has, over the past decade, averaged approximately 1.25%
of U.S. households annually (see American Bankruptcy Institute: www.abiworld.org).
16
For computation, we discretize the distribution of shocks, following the procedure of Tauchen
(1986), into seven values.
18
However, as Sullivan et al. (1989) argue, the disproportionate share of entrepreneurs in
the pool of bankrupt individuals may be underestimated in the data. Some individuals
list current occupations in categories that suggest that they are currently wage earners,
but list significant amounts of debt for capital equipment, suggesting self-employment.17
Our target, in turn, follows Sullivan et al. (2000), who suggest that the self-employed
file roughly twice as frequently as the general population. Given the range of estimates, we target a default rate between 1% and 2.5%. In addition to using bankruptcy
more frequently, the self-employed also discharge much more debt per filing than other
groups in the population. For debt discharged in bankruptcy, we note that Sullivan et
al. (2000) argue the self-employed discharge roughly twice as much debt as the mean
amount discharged by working households. However, the fact that the self-employed
may well file for default after exiting their business means that the unconditional mean
debt discharged will be informative as well. These considerations lead us to target mean
debts discharged in default in the range of $20,000 to $50,000.
4.5
Limits to Liability
Our central experiment is to study the role played by limited liability on the ability
and willingness of households to enter self-employment. The structure of limited liability in the model is motivated most directly by provisions in US personal bankruptcy
laws that determine the total wealth that a household may retain ex-post bankruptcy.
We select a benchmark level of approximately $50,000 (Rodrı́guez et al., 2002). This
benchmark captures the median exemption level prevailing in the US, as measured in
Athreya (2006). This benchmark is then compared against four alternative exemption
levels ranging in dollar value from $2,500 to $150,000.18
17
For example, Sullivan et al. (2000) note the presence of many who describe themselves as “cooks”
or “restaurant managers,” and then list restaurant equipment as assets.
A second problem in measuring bankruptcy among entrepreneurs is that households can, and do,
switch occupations. In particular, some who report themselves to be workers when filing for bankruptcy
may have accumulated debt during their past activity as an entrepreneur. Using the data from Survey
of Consumer Finance, Rodrı́guez et al. (2002) reports that the probability of being an entrepreneur
conditional being a bankrupt household is only 5.4%. However, there is a one year lag between the
filing for bankruptcy and the response to the SCF.
18
While statutorily, some states in the US have no limits on the wealth that might be exempted,
especially home equity, in practice there are limits. These limits include limits to the size of parcels of
land on which houses sit and the value of homes themselves. Our upper bound corresponds to allowing
households to exempt all wealth up to the value of the median US house as of 2002.
19
4.6
Fit of the Benchmark Model
Our benchmark calibration is able to capture very well nearly all of the preceding
targets. In particular, the model predicts well not only the occupational distribution of
US households, but also their relative earnings, as well as the observed age-profile of selfemployment. Figures 1 and 3 display benchmark outcomes relative to their empirical
counterparts. With respect to default rates and debts discharged in default, Table 3
shows that the default rate in the benchmark case (i.e., x3 ) is 1.53%, while Table 4
shows that the mean debt discharged in default is $30,627.
5
Results
Our central finding is that the tension between the insurance embedded in limited
liability, and the incentive effects arising from higher credit costs is resolved initially in
favor of insurance provision. However, at levels of liability exceeding current US levels,
the impact of limited liability is to decisively increase credit costs, limit credit use, and in
general, obstruct self-employment. We also show, strikingly, that the observed positive
correlation between wealth and self-employment cannot be interpreted as evidence of
credit constraints, as it arises primarily from the interaction of risk and life-cycle savings
behavior. Distributionally, three findings are important. First, limited liability appears
regressive with respect to age. We find that very low liability sharply affects the young,
but has only minor effects on the old, who have accumulated wealth for retirement.
Second, in contrast to its effects on age, liability policy does not appear to affect highand low-skilled households differentially. Third, our findings suggest that very lowliability regimes significantly alter the ability of households to switch occupations in
the event of low corporate sector productivity.
Turning first to the effects of liability policy on the “extensive” margin of selfemployment, our results are as follows. First, and most importantly, liability policy
appears to be relevant for the aggregate rate of self-employment, as reported in Table
3. As liability is decreased from its highest levels, x1 , to its lowest level, x5 , the aggregate entrepreneurship rate decreases by more than two percentage points, from 11.41%
to 9.38%. However, this decrease is not monotone. Notably, as limits to liability fall
from the x1 to x2 , the self-employment rate rises slightly for both skilled and unskilled
households. The initial increase in self-employment occurs despite an increase in the
cost of obtaining credit. Figure 6 makes clear that decreases in liability always result in
more expensive credit. The increased cost of credit has several effects. First, at the “extensive” margin, the fraction of borrowers falls slightly, and monotonically, from 51.88%
20
to 49.19%, as liability falls from x1 to x5 . However, the effects on the “intensive” margin, i.e., the size of debts are much more dramatic and non-monotone. Unconditionally,
the mean level of borrowing initially increases by roughly 5%, from $61,513 to $65,089,
in a move from x1 to x2 . Subsequent decreases in borrower liability lower borrowing
very sharply. In particular, moving from x2 to x5 results in a fall of mean borrowing to
$6,962. This is clear evidence that while limited liability can generate offsetting effects
at low levels, at high levels, credit use is severely restricted.
Do the changes in credit use ultimately affect the size of projects undertaken? From
Table 4 we see that as with the rate of self-employment, project size too is “humpshaped.” Initially, moving from x1 to x2 increases the scale of projects slightly. Further
decreases in liability, however, have much stronger effects. Notably, mean project size
under x5 is approximately 15% smaller than its peak value, which occurs under x1 to
x2 . Therefore, even though the fraction of borrowers and entrepreneurs changes only
modestly with exemption, the most important effects of liability policy occur along the
“intensive” margin, substantially modifying borrowing and, in turn, project size.
The relatively large changes in borrowing and project size are due to sharp increases
in the cost of credit faced by individuals. However, these changes, even though critical
for household decision making, will not be observed in the data. Table 5 shows that the
average interest rate paid by borrowing varies only slightly with limits to liability. In
particular, the average interest rate increases from 6.25% under x1 , to 6.68%, under x3 ,
and then decreases to 6.26%, under x5 . The insensitivity of interest rates to liability
is evidence that equilibrium default risk is also essentially invariant to liability policy.
Notice that default risk, as embedded in loan pricing, increases monotonically as liability
falls. Nontheless, default rates in equilibrium make clear that it is borrowing that
becomes limited to such an extent at low levels of liability that default is rarely observed.
Notice that a reduced-form regression of equilibrium interest rates and measures
of liability will imply, wrongly, that the cost of credit is insensitive to liability policy.
Using such a conclusion to recommend decreases in liability will have the unintended
consequence of discouraging credit use and, in turn, self-employment. Similarly, the
observation that bankruptcy is infrequently used under low personal liability for debts
does not imply that such a policy will have only benign effects on credit provision.
Our equilibrium approach demonstrates directly that liability policy has important
“off-equilibrium” effects on credit pricing that in turn strongly affect outcomes.
In a setting with limited liability, wealth accumulation serves two puposes. First,
for any given loan size, a higher level of wealth implies a lower default probability
and, in turn, a lower interest rate. Second, households can operate self-employment
projects without having to resort to external finance. As before, reductions in liability
21
from very high levels, such as a move from x1 to x2 , lower the average wealth of those
who borrow. In this particular case, the average wealth of the self-employed borrower
falls from $83,431 to $81,721. Further decreases in liability, however, leave households
unwilling to borrow. In turn, for levels of liability below x2 , the wealth position of
borrowers increases sharply, up to $96,458 under x5 . A similar outcome is seen in the
debt-to-capital ratio of self-employment projects. Initially, this ration increases slightly,
from 22.36% to 22.64%, but then drops dramatically to just 3.55% under the lowest
liability level. That is, external finance becomes far less important in the funding of
self-employment.
Under very low liability levels, the reliance of households on internal finance systematically increases the average age of the self-employment households. In short, wealth
takes time to accumulate. As Table 3 shows, the average age of the self-employed
increases substantially from 30.60 years of age to 33.01. The top panel of Figure 4
shows that the entire age-distribution of the self employed shifts to the “right.” Notice
from the figure that while lower liability increases the age of self-employed households,
these effects are very muted prior to retirement. This is very natural and an important
implication of the life cycle. That is, as retirement wealth accumulation leaves most
older households with the ability to internally finance self-employment, the latter are
relatively less sensitive to liability policy on debts.
The preceding makes clear that low liability does introduce credit constraints that
materially alter the timing of self-employment activity. Notice from the bottom panel
of Figure 4, however, that the wealthy always choose self–employment at higher rates,
regardless of liability policy. In particular, the result obtains under the nearly unlimited
liability level x1 , where households can commit to repaying debts in for almost all realizations of entrepreneurial productivity. As seen in Figure 6, almost all households are
able to borrow very large amounts at the risk-free rate. Thus, the observed correlation
of wealth and entrepreneurship (see, e.g., Hurst and Lusardi (2004)) is fundamentally
uninformative about the presence, or extent, of liquidity constraints. Instead, the risks
inherent in self-employment, along with the life-cycle accumulation of wealth deliver a
positive correlation between wealth and self-employment.19
As mentioned at the outset, our model allows us to see what econometricians cannot, and thereby view the precise nature of the self-selection of households into and out
of self-employment. We illustrate the role of liability policy on the ability of households
to enter self-employment in “response” to a low realization of corporate-sector productivity. Occupational choices in our model depend on wealth and corporate-sector
productivity. To isolate the effect of corporate-sector productivity, we first display, in
19
See also Mondragon (2006) for a closely related argument.
22
Figure 5, the probability of entrepreneurship as a function of productivity, conditional
on wealth. In the top panel of the figure, we observe that among low-wealth households,
high levels of liability are associated with relatively high incidence of self-employment
when the outside option to self-employment is poor. This suggests that the relatively
inexpensive credit available under high liability offers those with low corporate-sector
productivity the “safety-valve” of entry into self-employment. However, as household
wealth increases, the extent to which liability alters occupational choice diminishes.
This is seen in the middle and bottom panels of the same figure.
In order to evaluate the welfare implications of liability choice, we focus on the exante indirect utility of a newly-working household. As all households begin life with
zero assets in our model, this criterion avoids complications arising from changes in the
wealth distribution that would bias results arising from expected steady-state welfare.
Even though changes in liability policy induce noticeable changes in the timing, size, and
financing, of self-employment projects, a surprising finding is that the ex-ante welfare
of a newborn household is essentially invariant to the regime chosen. The intuition
is as follows. At high levels of liability (e.g., x1 and x2 ), the insurance benefits of
limiting liability offset the disincentives to self-employment induced by costly credit.
By contrast, at very low levels of liability (e.g., x4 and x5 ), the ability to self-finance via
risk-free savings functions effectively to undo the complications arising from high credit
costs. The latter is seen most clearly in the increased mean age of the self-employed
under low-liability regimes.
6
Conclusion
The main message of this paper is that the limits to liability alter the timing, size,
and financing of self-employment projects, especially at levels that are high relative to
current US practice. High liability, by contrast, appears to generate offsetting effects.
Namely, while cheap credit encourages self-employment, the absence of a generous default option makes such a choice risky. In low liability regimes, while the cost of credit
dominates the insurance benefits, the flexibility of households to switch occupations and
save to overcome high borrowing costs leads to nearly identical allocations. As a result,
welfare remains largely unchanged. We also find that the positive relationship between
wealth and self-employment rates cannot be seen as evidence for credit constraints. In
particular, we show that such a relationship is present even when limited liability is
eliminated.
It is useful to note that our current approach employs inelastic labor and homogeneous risk for self-employment projects. To the extent that availability of limited
23
liability discourages effort, the disincentive effects may be larger than measured here.
Similarly, the insurance provision of limited liability via default may lead households,
if allowed, to choose projects that are inefficiently risky. While beyond the scope of the
current study, these dimensions are worthy of further investigation.
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27
APPENDIX
We provide the proof of the theorem in the paper.
Proof. Taking the partial derivative of (1) with respect to x, using the Leibniz rule,
we get the following:
Z θ2 (x)
∂θ
∂V (D, x)
1
0
= π(θ1 )u f (D, θ1 )
+ u (x)
π(θ)dθ
∂x
∂x
θ1 (x)
∂θ2
∂θ1
− u(x)π(θ1 )
+ u(x)π(θ2 )
∂x
∂x
Z θ
∂R(x)
u0 (f (D, θ) − R(x)D) −
+
Dπ(θ)dθ
∂x
θ2 (x)
∂θ2
.
− u(f (D, θ2) − R(θ2 )D)π(θ2 )
∂x
(31)
Notice that, by definition, when θ = θ1 , we have x = f (θ1 , D). Similarly, when θ = θ2 ,
we have x = f (θ2 , D) − R(x)D. Thus, (31) reduces to
∂V (D, x)
= u0(x)
∂x
Z
θ2 (x)
π(θ)dθ −
θ1 (x)
Z
θ
u0(f (D, θ) − R(x)D)
θ2 (x)
∂R(x)
∂x
Dπ(θ)dθ .
(32)
The lender must still break even after the change in x. Thus, using (2), and taking
the partial derivative we get:
" Z
#
Z θ2 (x)
Z θ
θ1 (x)
∂Rf D
∂
0
=
π(θ)dθ +
(f (D, θ) − x)π(θ)dθ + R(D, x)D
π(θ)dθ
∂x
∂x
θ
θ1 (x)
θ2 (x)
Z θ2 (x)
∂θ2
∂θ1
0=−
− (f (θ1 , D) − x)π(θ1 )
π(θ)dθ + (f (θ2 , D) − x)π(θ2 )
∂x
∂x
θ1 (x)
Z θ
∂θ2
∂R(x)
+
Dπ(θ)dθ − π(θ2 )R(x)D
.
(33)
∂x
∂x
θ2 (x)
Similarly, since x = f (θ1 , D) and x = f (θ2 , D) − R(x)D, (33) reduces to:
0=−
Z
θ2 (x)
π(θ)dθ +
θ1 (x)
Z
θ
θ2 (x)
∂R(x)
Dπ(θ)dθ,
∂x
(34)
which further implies that
R θ2 (x)
π(θ)dθ
∂R(x)
θ (x)
.
D = R1
θ
∂x
π(θ)dθ
θ2 (x)
28
(35)
Substituting (35) into (32), we get
∂V (D, x)
= u0 (x)
∂x
Z
R θ2 (x)
θ2 (x)
θ1 (x)
θ1 (x)
π(θ)dθ − R
θ
θ2 (x)
π(θ)dθ Z
π(θ)dθ
θ
u0 (f (D, θ) − R(x)D)π(θ)dθ ,
θ2 (x)
(36)
which can be re-arranged
∂V (D, x)
=
∂x
Z
"
θ2 (x)
θ1 (x)
π(θ)dθ u0 (x) − R
θ
1
π(θ)dθ
θ2 (x)
Z
θ
u0 (f (D, θ) − R(x)D)π(θ)dθ
θ2 (x)
Notice, however, since u (.) < 0, we have
u0 (x)π(θ)dθ
θ (x)
u0(x) = 2R
>
θ
π(θ)dθ
θ2 (x)
,
(37)
00
Rθ
#
Rθ
θ2 (x)
This completes the proof.
29
u0 (f (D, θ) − R(x)D)π(θ)dθ
Rθ
π(θ)dθ
θ2 (x)
Table 1: Sequence of Events in a Given Period
Entrepreneurs with asset level, aj , decide their project size, kj .
The loan is then determined, qbj = kj − aj .
If no credit is issued, then qbj = 0.
Entrepreneurs decide on bankruptcy as a solution to (13).
If they file for bankruptcy, they keep min{x, F (θ, kj )}.
The payment to the lender in the case of bankruptcy
equals max{0, F (θ, kj ) − x}.
Solvent entrepreneurs pay back their debt, bj , to the intermediary.
b
b
b
Agents observe their wage rate.
They decide to become a worker or an entrepreneur
as a solution to (11).
b
b
b
Period t ends.
b
b
All types of agents between ages 1 to J − 1 choose
current consumption, cj , and next period’s assets, aj+1 .
The idiosyncratic shock to production, θ, gets realized.
Output from the project is determined by F (θ, kj ).
Agents of age J choose their current consumption, cJ
wealth, aJ+1 , and die.
They are replaced with a new generation. The new generation
begins period t + 1
with a probabilistically determined human capital level.
1
Period t begins.
Table 2: Parameters
Preferences
{βc , βhs }
u(c) =
{0.985,0.970}
calibrated
ξ=2
calibrated.
ρ = 2, ψh = {ψc , ψhs } = {15, 7.5}
calibrated.
c1−ξ
1−ξ
φ(aJ+1 ) = ψh
a1−ρ
J +1
1−ρ
Labor Productivity
h = {hc , hhs }
µCorp (j, h)
0
ς(h = hc |h = hc ) = 0.61,
ς(h0 = hc |h = hhs ) = 0.21
calibrated.
calibrated
Hansen (1993).
Entrepreneurial Production
γ
F (.) ≡ θk + (1 − δ)k
γ = 0.75, δ = 0.11
log θ ∼ N(µθh , σΘh )
µΘc = −1.35, σΘc = 0.83.
h = {c, hc}
µΘhs = −1.44, σΘhs = 0.83.
calibrated.
calibrated.
Credit Markets
f
0.96
Mehra and Prescott (1985).
τ
0.0225
calibrated
τB
0.15 ≈ $7,500.
calibrated.
η
0.8
calibrated.
q
Exemptions
x = {x1 , x2 , x3 , x4 , x5 }
x = {2, 500, 25, 000, 50, 000, 100, 000, 150, 000.}
in dollars
xbench = x3 = $50, 000.
calibrated.
Table 3: Aggregates – Extensive Margin
x1
x2
x3
x4
x5
Entrepreneurship
11.41%
11.64%
10.20%
9.29%
9.38%
Entrepreneurship High Sch
9.67%
9.73%
8.03%
7.16%
7.43%
Entrepreneurship College
14.84%
15.40%
14.47%
13.48%
13.23%
Bankruptcy Rate
0.00%
1.30%
1.53%
0.29%
0.00%
Bankruptcy Rate High Sch
0.00%
0.68%
0.74%
0.00%
0.00%
Bankruptcy Rate College
0.00%
2.53%
3.09%
0.85%
0.00%
Prob of Borrowing
51.88%
51.47%
50.24%
50.29%
49.19%
Prob of Borrowing High Sch
58.60%
56.98%
54.93%
58.35%
56.17%
Prob of Borrowing College
38.65%
40.62%
41.01%
34.41%
35.43%
Mean Age of Entrepreneurs
30.60
30.89
32.28
33.16
33.01
Mean Age of HS Entreps
30.87
30.41
31.35
32.94
32.57
Mean Age of Coll Entreps
30.46
31.14
32.75
33.27
33.23
Table 4: Aggregates – Intensive Margin
x1
x2
x3
x4
x5
Project Size uncon.
$179, 936
$182, 486
$179, 709
$160, 113
$159, 376
Project Size High Sch.
$151, 802
$156, 408
$150, 674
$124, 662
$128, 906
Project Size College
$235, 348
$233, 848
$236, 896
$229, 935
$219, 389
Mean Wealth |Borr.
$83, 431
$81, 721
$90, 998
$93, 560
$96, 458
Mean Wealth |Borr, Hs
$69, 943
$69, 178
$78, 451
$77, 007
$79, 127
Mean Wealth |Borr, Coll
$109, 997
$106, 425
$115, 711
$126, 162
$130, 594
Average Debt uncon.
$61, 513
$65, 089
$50, 368
$18, 379
$6, 962
Average Debt High Sch.
$53, 268
$55, 383
$38, 438
$8, 665
$5, 827
Average Debt College
$77, 751
$84, 205
$73, 865
$37, 509
$9, 197
Debt Discharged uncon.
$0
$38, 054
$30, 627
$0
$0
Debt Discharged HS
$0
$25, 973
$18, 375
$0
$0
Debt Discharged Coll.
$0
$61, 848
$54, 759
$0
$0
Table 5: Limited Liability, Borrowing, and Cost of Credit
x1
x2
x3
x4
x5
Debt-to-Capital unc.
22.36%
22.64%
16.52%
6.11%
3.55%
Debt-to-Capital HS
16.32%
18.05%
15.46%
7.19%
2.33%
Debt-to-Capital Coll
25.43%
24.98%
17.06%
5.55%
4.17%
Interest Rate unc.
6.25%
6.62%
6.68%
6.34%
6.26%
Interest Rate HS
6.25%
6.43%
6.47%
6.26%
6.25%
Interest Rate Coll
6.25%
6.99%
7.09%
6.50%
6.27%
Pr(Entrep—Non-College)
Model
Data
0.25
0.2
0.15
0.1
0.05
0
25
30
35
40
age
45
50
55
60
65
50
55
60
65
Pr(Entrep—College)
0.3
Model
Data
0.25
0.2
0.15
0.1
0.05
0
25
30
35
40
age
45
Figure 1: Age Distribution of Self–Employment Rate
4
12
Median Assets of Non-College Households
x 10
Model
Data
10
$
8
6
4
2
0
22
26
31
36
5
2.5
41
age
46
51
56
61
66
51
56
61
66
Median Assets of College Households
x 10
Model
Data
2
$
1.5
1
0.5
0
22
26
31
36
41
age
Figure 2: Median Wealth
46
4
2.8
Income of Non-College workers
x 10
Model
2.6
Data
2.4
2.2
$
2
1.8
1.6
1.4
1.2
1
22
26
31
36
4
5
41
age
46
51
56
61
66
51
56
61
66
Income of College Workers
x 10
Model
Data
4.5
4
$
3.5
3
2.5
2
1.5
1
22
26
31
36
41
age
46
Figure 3: Income of Workers
P(Entrep |age & Non-College)
P(Entrep |age & College)
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
22
26
31
36
41
age
46
51
56
61
66
0
22
P(Entrep |College & Wlth)
0.8
36
41
age
46
51
56
61
66
1
x1
x2
x3
x4
x5
0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
10
31
P(Entrep |Non-Coll & Wlth)
1
0.9
26
30
50
70
Beginning of Period Bench Wlth %-ile
90
0
10
30
50
70
Beginning of Period Bench Wlth %-ile
Figure 4: Self–Employment Rate by Age and Wealth
90
P(Entrep |Coll & 20th %-ile of Wlth)
P(Entrep |Non-Coll & 20th %-ile of Wlth)
1
1
x1
x2
x3
x4
x5
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.3
1
Corporate Sector Shock
3.32
0
0.3
P(Entrep |Coll & 50th %-ile of Wlth)
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1
Corporate Sector Shock
3.32
0
0.3
P(Entrep |Coll & 80th %-ile of Wlth)
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1
Corporate Sector Shock
1
Corporate Sector Shock
3.32
P(Entrep |Non-Coll & 80th %-ile of Wlth)
1
0
0.3
3.32
P(Entrep |Non-Coll & 50th %-ile of Wlth)
1
0
0.3
1
Corporate Sector Shock
3.32
0
0.3
1
Corporate Sector Shock
Figure 5: Corporate Productivity and Occupational Choice
3.32
College, Initial Wealth = $ 25,000
1
Loan Price
0.8
0.6
0.4
0.2
0
−12
−10
−8
−6
-b($)
−4
−2
0
4
x 10
College, Initial Wealth = $ 50,000
1
Loan Price
0.8
0.6
0.4
0.2
0
−12
−10
−8
−6
-b($)
−4
−2
0
4
x 10
College, Initial Wealth = $ 100,000
1
Loan Price
0.8
0.6
x1
x2
x3
0.4
x4
x5
0.2
0
−12
−10
−8
−6
-b($)
Figure 6: Loan Pricing
−4
−2
0
4
x 10
@FedFRASER