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LIMITED ENFORCEMENT AND THE
ORGANIZATION OF PRODUCTION
Erwan Quintin
Research Department
Working Paper 0109
Center for Latin American Economics
Working
Paper 0601
Center for
Latin American
Economics
Working Paper 0201
FEDERAL RESERVE BANK OF DALLAS
Limited Enforcement and the Organization of Production
Erwan Quintin∗
Federal Reserve Bank of Dallas
April 4, 2003
Abstract
This paper describes a dynamic, general equilibrium model designed to assess whether
contractual imperfections in the form of limited enforcement can account for international
differences in the organization of production. In the model, limited enforcement constrains agents to operate establishments below their optimal scale. As a result, economies
where contracts are enforced more efficiently tend to be richer and emphasize large scale
production. Calibrated simulations of the model reveal that these effects can be large and
account for a sizeable part of the observed differences in the size distribution of manufacturing establishments between the United States, Mexico and Argentina.
Keywords: Limited enforcement; Organization of production; Economic development.
JEL classification: L23; O11.
∗
I would like to thank Hugo Hopenhayn, Pat Kehoe, Tim Kehoe, Narayana Kocherlakota, Ed Prescott,
Manuel Santos, Vincenzo Quadrini, as well as seminar participants at the Universidad Torcuato di Tella, Stanford
University, Boston University, Columbia University and the Federal Reserve Bank of Minneapolis for their
valuable comments. I am also grateful to Kathryn Cook for her research assistance. The views expressed are
those of the author and do not necessarily reflect the position of the Federal Reserve Bank of Dallas or the
Federal Reserve System.
1
1
Introduction
Property rights are not effectively enforced in developing nations. For instance, Djankov et al.
(2002) calculate that it takes about 300 days on average to collect on a bad check in Argentina
or Mexico, compared to 50 days in the U.S. In this paper, I assess the importance of the
imperfect enforcement of financing contracts for documented differences in the organization of
production between developing economies and the U.S.
I consider a dynamic version of the industrial organization model proposed by Lucas
(1978). Agents differ in terms of managerial talent and the bequest they receive. Each period,
they can either work as unskilled workers or manage a strictly concave technology that transforms unskilled labor and physical capital into the unique consumption good. Managers need
to finance both their physical capital and payroll. They can write long-term financing contracts with an intermediary, but the contractual framework is imperfect. Specifically, agents
can decide to default on the payment stipulated by the contract in return for the advance of
funds they receive. The intermediary can seek to enforce this payment and are successful with
a fixed, exogenous probability. This probability is my proxy for the efficiency of enforcement
institutions in the economy under consideration.
Because enforcement is limited, some managers are borrowing constrained and operate
establishments at a sub-optimal scale. I measure the impact of the resulting distortions by way
of calibrated numerical simulations. Quite intuitively, rises in the probability of enforcement
result in higher steady state output and labor productivity. Furthermore, production establishments are larger on average and a larger share of employment is found in small establishments
2
in more productive economies, which is consistent with the well-documented fact that developing nations emphasize small scale production (see e.g. Tybout, 2001). In fact, I argue that
the model can generate a distribution of establishments and employment across size categories
that closely resembles its empirical counterparts in Mexico or Argentina. In addition, I find
that wealth and income are more equally distributed in economies where contracts are better
enforced. These findings indicate that a model of development that incorporates contractual
imperfections can account for several distinguishing features of developing economies.
This work builds on a number of studies of environments with limited commitment. Kehoe
and Levine (1993) study the impact of limited commitment on asset trading while Kocherlakota
(1996) considers efficient consumption allocations. Marcet and Marimon (1992) measure the
effect of limited commitment in a stochastic growth model. Monge (1999) studies the impact
of interest rate volatility on firm dynamics in an environment very similar to the one developed
in this paper. Cooley et al. (2000) study the implications of limited enforceability for the
business cycle and the investment policy and dynamics of firms. This paper extends this work
by gauging the quantitative impact of contractual frictions on the organization of production,
and builds upon the work of Banerjee and Newman (1993) and Quadrini (2000) on the impact of
financing constraints on entrepreneurship and wealth accumulation. My work also relates to the
literature on the link between financial intermediation and economic development. McKinnon
(1973) and Shaw (1973) provide some empirical evidence of this link for several nations. King
and Levine (1993) report a correlation between financial and economic development with crosscountry data. More generally, the model is consistent with the correlation between the quality
3
of a nation’s institutions and its level of development, as documented for instance by Hall and
Jones (1998) or Barro (1991).
2
The organization of production in developing economies
Tables 1 shows the distribution of manufacturing establishments and employment in the U.S.,
Mexico and Argentina in 1988.1 Nearly 84 percent of establishments count 10 or fewer employees
in Mexico while fewer than half of establishments have between 1 and 9 employees in the U.S.
Establishments with 50 or more employees account for over 80 percent of all employment in the
U.S., compared to 75 percent in Mexico, and less than half of all employment in Argentina.
Tybout (2000) presents similar data for a larger cross-section of countries. In many
developing countries, including large countries such as India or Indonesia, the majority of
manufacturing employment is found in establishments with fewer than ten employees. Tybout
also presents a survey of existing explanations for this phenomenon. For instance, developing
nations tend to emphasize items that can be produced efficiently on a small scale.2 However,
table 2 presents some disaggregated data for the U.S. and Mexico that indicate that differences
1
The data for Argentina come from the country’s household survey and allow one to estimate the distribution of employment, but not the size distribution of establishments. Also note that throughout this paper, I
concentrate on data from the manufacturing sector because in nations with large informal sectors like Mexico
or Argentina, economic censuses of non-manufacturing establishments are unreliable. Manufacturing censuses
are not immune to this weakness, but because manufacturing establishments tend to be less mobile than other
establishments, they are counted with more precision.
2
To the extent that financing constraints contribute to this feature of the composition of output, the model
I present in this paper could be viewed as a related explanation.
4
Table 1: Size distribution of manufacturing establishments and employment
U.S., 1988
Employment size
Percent of establishments
Percent of employment
0 to 9
48.1
3.3
10 to 49 50 to 249 250+
33.0
15.0
3.9
14.0
30.1 52.6
Mexico, 1988
Employment size
Percent of establishments
Percent of employment
0 to 10 11 to 50
83.6
10.7
11.9
13.0
51 to 250 251+
4.3
1.4
24.7 50.4
Argentina, 1988
Employment size
Percent of employment
0 to 15 16 to 50 51+
37.9
20.7 41.4
Source: County business pattern survey, Census Bureau (U.S.), Economic census, INEGI (Mexico), Permanent household survey, INDEC (Argentina).
in the composition of manufacturing output are unlikely to fully explain observed differences
in the size distribution of establishments.3
Rauch (1991) proposes another explanation for the importance of small scale production in
developing nations. In those countries, many firms do not comply with government regulations
and choose to operate a small scale to avoid detection. The model I describe in this paper
suggests another channel via which the importance of informal economic activities matters
for the organization of production. It is difficult for informal, unregistered firms to write
legally enforceable contracts. Not surprisingly then, informal firms have little access to outside
3
Further disaggregation continues to show that in most sectors establishments are much larger on average
in the U.S. than in Mexico.
5
Table 2: Average employment size of establishments in manufacturing sub-sectors, 1988
Food and kindred products
Tobacco products
Textile mill products
Apparel and other textile products
Lumber and wood products
Paper and allied products
Printing and publishing
Chemicals and allied products
Petroleum and coal products
Stone, clay and glass products
Primary metal industries
Fabricated metal products
Industrial machinery and equipment
Transportation equipment
Instruments and related products
U.S. Mexico
70.79
10.78
323.74
243.84
107.93
25.47
48.37
14.56
21.68
8.49
98.80
18.22
25.23
12.47
69.15
71.73
53.34
44.98
32.69
10.52
108.00
115.30
41.73
28.75
37.66
28.52
177.87
160.69
98.33
38.49
Sources: INEGI and Census Bureau
financing, as documented for instance by Thomas (1992) and Mansell Carstens (1995). These
firms are therefore constrained to operate a suboptimal scale.
3
3.1
The model
Physical environment
Each period a cohort of measure one of agents are born. They live T periods, where T > 2. In
the first T − 1 periods of their life, agents are endowed with a unit of productive time which
they devote to managing productive resources or, instead, to delivering labor. They are also
endowed with an innate level z ∈ (0, 1] of managerial ability. An agent’s managerial ability
6
is public information and remains unchanged as he ages. In addition, the distribution µ of
managerial ability has final support and is constant across cohorts.
An agent of managerial ability z ∈ (0, 1] who, in a given period, manages amounts k ≥ 0
of physical capital and n ≥ 0 of unskilled labor produces quantity
F (k, n; z) = Azk αk nαn + (1 − δ)k
of the unique consumption good, where A, αk , αn > 0 and αk + αn < 1. Throughout this paper,
I will think of a production unit consisting of a manager together with some physical capital
and unskilled labor as an establishment.
At the beginning of each period, agents choose an occupation. Agents who choose to
become managers employ fixed quantities of unskilled labor and physical capital for the duration
of the period. Output is delivered at the end of the period but labor must be paid before
production can start.
All agents have linear preferences and discount consumption flows at a rate of β. What’s
more, a fixed proportion φ < 1 of each cohort leaves a non-negative bequest g > 0 to a single
offspring. The preferences of these altruistic agents can be represented by the following utility
function:
U({ct }Tt=1 , g)
=
T
1−γ
βct
gγ
t=1
where ct is the agent’s consumption at age t. Newly born agents receive their bequest at the
beginning of the first period of their life. Both the altruistic and managerial type of each agent
7
are independent of their parent’s. I impose the following assumption on the degree γ of altruism
to place a bound on each lineage’s wealth:
Assumption 1. γβ −T < 1
The economy also comprises a financial intermediary with access to perfect, outside capital
markets where a risk-free security earns return r ≡ β −1 − 1. One should therefore think of
this economy as a small, open economy. Given the assumed risk free rate and the linearity of
preferences, agents seek to maximize the discounted value of their lifetime income, and altruistic
agents bequeath fraction γβ −(T −1) of that value to their offspring.
3.2
Contracts
Given the timing of production, managers need to finance both the quantity of physical capital
they choose to employ and their payroll before production can start. For this purpose, the intermediary and newly born agents write financing contracts. A financing contract is a sequence
{dt , ρt }Tt=1 that stipulates for every period t of the agent’s life a quantity dt ≥ 0 of consumption
good advanced to the agent at the start of the period, and a net transfer ρt from the agent to
the intermediary at the end of the period. (The total stipulated transfer from the agent to the
intermediary at the end of period t is dt (1 + r) + ρt .)4
The intermediary can commit fully to the terms of the contract, but agents have the
option to renege on the stipulated transfer at the end of any period. When an agent exercises
4
One can interpret the financing contract as a debt contract between a bank and a manager. Alternatively,
the intermediary can be interpreted as a corporation that hires the agent to provide unskilled labor or manage
resources. In this interpretation, the corporation is the proprietor of all output produced and the manager’s net
income sequence is a compensation scheme.
8
this option, they are caught with probability θ ∈ [0, 1] in which case the transfer is enforced.
This probability is my proxy for the efficiency with which contracts are enforced in a given
economy and constitutes the basis of the comparative statics exercise I carry out in this paper.
3.2.1
Self-financing
It will prove useful to first consider the problem agents solve when they must self-finance all
production. Let w > 0 be the wage, i.e. the price of labor. (Throughout, I concentrate my
attention on equilibria in which the wage is constant.) An agent of talent z and age t who in a
given period operates with quantity a ≥ 0 of the consumption good5 solves:
maxn,k≥0 F (k, n; z) − k(1 + r) − nw(1 + r)
subject to:
nw + k ≤ a
Let π(a; z, w) be the corresponding net income while k(a; z, w) and n(a; z, w) are the associated
policies. When δ < 1 the cost of physical capital relative to labor decreases when the financing
constraint is relaxed. This leads to:6
Lemma 2. For any (z, w) ∈ (0, 1] × IR++ ,
k(.;z,w)
n(.;z,w)
is increasing.
Lemma 2 says that all else equal, managers who are more financing constrained choose to
operate their establishment at a lower physical capital to unskilled labor ratio, an observation
5
For notational simplicity, assume that managers can costlessly transform the consumption good into the
capital good.
6
The proof of this result, as all proofs in this paper, is in the appendix.
9
which will help me interpret some of the numerical results of section 4. Now define:
VtS (a; z, w)
= a(1 + r) +
max
T
−1−t
T −1−t
{at+i }i=0
β i max{w(1 + r), π(at+i ; z, w)}
i=0
subject to:
at = a
at+i ≤ at+i−1 (1 + r) + max{w(1 + r), π(at+i−1 ; z, w)} ∀i ≥ 1
With this notation, V1S (b; z, w) is the agent’s maximum discounted lifetime income when all
production must be self-financed and his initial bequest is b ≥ 0. How much they can improve
upon this lower bound depends on what financing contracts can be written.
3.2.2
Competitive contracts
Define, for all t < T , d ≥ 0, ρ ∈ IR and θ ∈ [0, 1] the following value of default function:
S
S
(π(d; z, w) − ρ; z, w) + (1 − θ)Vt+1
(π(d; z, w) + dβ −1; z, w)]
VtD (d, ρ; z, w, θ) = β[θVt+1
This function gives the expected remaining lifetime utility of a manager of age t and ability z
who defaults on a payment ρ due for an advance d of the consumption good, assuming they
are unable to borrow after defaulting. With probability θ, they are caught, in which case they
make the stipulated payment and start the next period with a quantity π(d; z, w) − ρ of the
consumption good. With probability (1 − θ), they are not caught, and start period t + 1 with
10
a quantity π(d; z, w) + dβ −1 , the sum of their net income in period t and the accrued value of
the advance of consumption good.
Consider now an agent of managerial ability z ∈ [0, 1] born with a bequest b ≥ 0. I will
call a financing contract {dt , ρt }Tt=1 feasible when:
bβ −1 + max{wβ −1, π(d1 ; z, w)} − ρ1 ≥ 0
max{wβ −1, π(dt ; z, w)} − ρt ≥ 0
T
(1)
∀1 < t < T
β t−1 ρt ≥ 0
(2)
(3)
t=1
and, for all t < T ,
T
β i−t [max{wβ −1, π(di; z, w)} − ρi ] ≥ VtD (dt , ρt ; z, w, θ)
(4)
i=t
The first two conditions state that the agent’s net payment to the intermediary cannot exceed
his net income in any period of his life. The third equation says that the contract must be
actuarially fair for the intermediary: the expected present value of the payments he receives
must exceed the present value of the quantities of the consumption good advanced. The last
set of conditions is the requirement that in any period of the contract, the agent must be better
off following the terms of the contract than defaulting. The left-hand side of these expressions
is the opportunity cost of defaulting, namely the present value of the remaining income flows
stipulated by the contract. The right-hand side is the expected present value of income after
default, provided agents are excluded from future borrowing.
11
This notion of feasibility can be justified by standard contract-theoretic arguments (see
Kocherlakota, 1996).7 In particular, punishing default with complete exclusion from borrowing
weakens incentive compatibility problems as much as possible, and is therefore optimal.
I assume that the intermediary behaves competitively: among the set of feasible contracts,
the contract most favorable to the agent prevails. In other words, contracts solve:
max −β
T −1
ρT +
T −1
β t−1 (max{wβ −1, π(dt ; z, w)} − ρt )
t=1
subject to (1-4). I will refer to solutions to this problem as competitive contracts. Since net
income is bounded above given the agent’s characteristics, a competitive contract exists for
all agent types. In fact, competitive contracts can be computed by dynamic programming
techniques as in Spear and Srivastava (1987) or Albuquerque and Hopenhayn (1997).
3.2.3
Properties of competitive contracts
Let d∗ (w, z) = arg maxd π(d; w, z) be the optimal advance for an agent of ability z when there
are no contractual imperfections (θ = 1). Absent contractual imperfections, agents become
managers when π(d∗ (w, z); w, z) ≥ w and always operate their technology at its unique optimal
scale. This subsection describes the impact of limited enforcement on occupational choices and
the allocation of resources. The following result will prove useful:8
7
A complete argument is available upon request.
8
This is a version of corollary 1 in Albuquerque and Hopenhayn, 1997.
12
Lemma 3. There exists a competitive contract {dt , ρt }Tt=1 such that
ρ1 = bβ −1 + max{wβ −1, π(d1 ; z, w)} and,
ρt = max{wβ −1, π(dt ; z, w)}
∀1 < t < T
This intuition for this result is simple: postponing consumption weakens incentive compatibility constraints. Note that the consumption strategy implied by this result may exhibit
superfluous patience. It is enough to postpone consumption up to the point where the nodefault constraint no longer binds, which may occur strictly before the last period. But for
simplicity and without any loss of generality, I restrict my attention to contracts that satisfy
the two properties of lemma 3. Under this convention, the value of default in a given period
t < T only depends on the advance d of consumption good and is given by:
S
S
VtD (d; z, w, θ) = β[θVt+1
(0; z, w) + (1 − θ)Vt+1
(π(d; z, w) + dβ −1 ; z, w)]
S
(π(d; z, w) +
Observe that the value of default for a given advance d decreases with t, since Vt+1
dβ −1 ; z, w) does. As a result, agents are more likely to become managers as they become older.
The following result records this observation, and points out that agents born with higher
bequests are also more likely to become managers.
Proposition 4. Assume that {dt , ρt } is a competitive contract for agents of type (b, z). Then,
1. dt+1 > 0 =⇒ dt+1 ≥ dt for all t ∈ {1, T − 2},
2. {dt , ρt } is feasible for agents of type (b , z) for all b ≥ b.
13
In the first part of the proposition, the premise dt+1 > 0 is necessary because it is possible
for an agent of type z to be indifferent between the two occupations for two consecutive period.
It should be obvious that this may only happen when π(d∗ (w, z); w, z) = w. In that case,
an agent could choose to become a manager in a period, but a worker in the next, which
would yield a decreasing advance sequence. But in the generic case, proposition 4 implies that
the optimal occupational profile is characterized by an age threshold. Once agents become
managers, they remain managers for the rest of their life. It also implies that managers run
larger establishments as they grow older. Therefore, establishment growth takes the form of
convergence towards their optimal scale. The second result says that, ceteris paribus, agents
born with larger bequests will be advanced more funds in every period, become managers
earlier and run larger plants at every age. Put another way, proposition 4 says that limited
enforcement disrupts the allocation of resources in two ways. First, talented agents may have to
become unskilled workers because they are unable to borrow enough funds while less talented
but older or wealthier agents become managers. In addition, managers are generally constrained
to operate an establishment at a sub-optimal scale.
I will now characterize the impact of changes in θ on the set of competitive contracts. Let
V ∗ (b, z; w, θ) denote the maximum discounted lifetime income for an agent of type (b, z) given
w and θ, while, as before, V1S (b; z, w) is the lifetime income an agent can obtain without outside
financing. Clearly, V ∗ (b, z; w, θ) ≥ V1S (b; z, w), and V ∗ (b, z; w, θ) − V1S (b; z, w) can be thought
of as measuring the extent to which agents rely on outside financing. When θ = 0, agents who
decide to default are excluded from future borrowing but face no other cost. The next result
says that under those circumstances no outside lending can be supported, as in Bulow and
14
Rugoff (1989). It also makes note of the fact that as θ rises, the availability of outside financing
rises monotonically.
Proposition 5. For all (b, z; w) ∈ IR+ × [0, 1] × IR++ ,
1. V ∗ (b, z; w, 0) = V1S (b; z, w),
2. V ∗ (b, z; w, θ) rises with θ.
I now turn to characterizing steady state equilibria.
3.3
3.3.1
Steady state equilibria
Definition
A steady state equilibrium is a value w > 0 for the unskilled wage, a distribution ν of initial
bequests, and financing contracts for each agent such that almost all financing contracts are
competitive, the labor market clears, and the distribution of bequest is invariant. To make this
more precise, for t ≥ 0 denote by ηt (d; b, z) the mass of agents of age t, initial bequest b and
managerial ability z whose contract stipulates an advance d ≥ of the consumption good. In
equilibrium, labor markets must clear i.e.:
T −1
t=1
n(d; b, z)d(ηt × ν × µ) −
T −1
t=1
15
{(d,b,z):d=0}
d(ηt × ν × µ) = 0
(5)
The first term in equation (5) is the aggregate demand for labor, while the second term gives
the aggregate supply of unskilled labor.9 In addition, the distribution of bequests must be
constant, i.e., for any Borel subset B ⊂ IR+ , the following condition holds:10
(1 − φ)1B {0} + φ
3.3.2
{γβ T −1 V ∗ (b,z;w,θ)∈B}
d(ν × µ) = ν{B}
(6)
Comparative statics
The remainder of this paper is devoted to comparing steady state equilibria in economies that
differ only in the degree θ to which financing contracts can be enforced. For a given θ, the
mapping from unskilled wages to the excess demand for unskilled labor can be decomposed into
two separate mappings. The first maps the wage w to the unique distribution νw of bequests
that satisfies (6). The second operator maps this distribution into the set of corresponding
values for the aggregate excess demand for unskilled labor. In appendix A.2, I show that νw is
the unique fixed point of strongly convergent Markov process. After guessing w and solving for
competitive contracts, both integrals in (5) can then be computed by applying the law of large
numbers.11
It seems natural to conjecture that an increase in the efficiency with which contracts
are enforced has a positive impact on labor productivity. Indeed, let w be the steady state
9
The labor demand function n given a agent’s type and the financing with which they operate is defined in
section 3.2.1.
10
Here, 1B denotes the indicator function corresponding to set B.
11
The basic algorithm I use in my quantitative exercises consists of updating w until this excess demand is
approximately zero. I increase w when there is an excess demand for labor, decrease it otherwise. In all cases,
I also carry out a global grid search over the relevant range of prices to verify that the steady state equilibria I
report are unique.
16
unskilled wage when the probability of enforcement is θ ∈ [0, 1]. Assume now that θ rises
to θ . The set of competitive contracts corresponding to θ and w remains feasible when θ
rises. Therefore, the unique stationary invariant bequest distribution under (w, θ) first order
stochastically dominates the equilibrium distribution of bequests under (w, θ). By proposition 4,
it now follows that there must be an excess demand for labor at w and one would expect the
unskilled wage to increase. In the special case where φ = 0, one can show analytically that
economies where contracts are better enforced are indeed more productive:
Proposition 6. Assume φ = 0.
1. A steady state exists for all θ ∈ [0, 1]
2. Assume that w is a steady state wage given θ ∈ [0, 1]. Then for any θ > θ a steady wage
w exists with w ≥ w.
The opportunity cost of becoming a manager thus tends to rise when θ rises and, consequently, one would expect the equilibrium proportion of managers to decrease and the average
size of establishments to rise. However, changes in θ and the unskilled wage have non-trivial
effects on the quantity of funds managers are able to borrow, both directly and via their impact on the equilibrium distribution of bequests. Furthermore, the optimal employment-size
of establishments decreases with the unskilled wage. The next section describes the result of
these potentially conflicting forces via numerical methods.
17
4
4.1
Quantitative results
Parameters
I begin this section by selecting exogenous parameters so that the economy described in this
paper generates steady state statistics that match the relevant U.S. statistics. I set T = 6 so that
assuming a productive life of 40 years, each period corresponds to 8 years.12 I assume β = 0.79
which implies a yearly real interest rate of roughly 3 percent. As for altruism parameters,
Leitner and Ohlson (2001) find that 35 percent of respondents both of whose parents are dead
report receipt of a positive inheritance in the 1984 Panel Study of Income Dynamics (PSID).
I therefore set φ = 0.35. In that year, the PSID included data on cumulative inheritances. In
present value terms (with a yearly discount rate of 3 percent), the average per capita amount
inherited conditional on receiving a positive inheritance, represents approximately 5 percent of
individuals’ mean lifetime earnings in the PSID sample. When γ = 0.04, my economy matches
this observation.
Turning now to technological parameters, assume first that there are no contractual imperfections (θ = 1). Then αk is the share of capital income in GDP, approximately a third in
the U.S. Similarly, when θ = 1, the managerial share of GDP is 1 − αn − αk . This managerial
share could be set to match the ratio of proprietor’s income to national income, approximately
9% for the 1960-85 period in the U.S. But this ratio is a poor measure of what the managerial
share represents in my model since corporations constitute the leading form of ownership in
12
Recall that agents do not work in the last period of their life. The computational complexity of my exercise
rises quickly as T rises. While raising T up to 6 periods yields noticeably different outcomes, raising T further
appears to affect my results only marginally.
18
the U.S., and corporate data does not enable one to measure the share of value added that
accrues to fixed managerial inputs. I choose instead to use sole proprietorship data for all years
for which data is available from the IRS in the 1979-1992 period. To measure 1 − αn − αk , I
assume that αk is the share of net income that remunerates the sole proprietor’s own capital,
while 1 − αk is the share remunerating their managerial input. I find that for manufacturing
sole proprietorships in the U.S., the ratio
(1−αk )Net income
Value added
averages to 14.8 percent, where value
added is business receipts minus cost of sales and operation, excluding cost of labor. I thus set
αk = 0.33 and 1 − αk − αn = 0.85.13
Naturally, θ = 1 is not an adequate assumption for the U.S., and below I will argue for a
value θ strictly smaller than 1. After making that change however, I find that the capital and
managerial shares of income in steady state do not change much, and, correspondingly, leave
technological parameters unchanged.14 As for δ, the depreciation, rate, I set it to 0.44 (that is,
7 percent yearly) which yields a capital to (yearly) GDP ratio of roughly 3.20, near standard
estimates of this ratio in the U.S.
There remains to specify the distribution of managerial talent, and the degree θ to which
contracts can be enforced. Given other exogenous parameters, I choose those two parameters
to jointly match the U.S. distribution of manufacturing establishments across size categories,
and the average rate of growth of manufacturing establishments. The specific procedure is
described in appendix A.3 and the calibrated distribution of talent is shown in figure 1. Setting
13
Interestingly, Atkeson and Ohanian (1996), using very different calibration arguments, arrive at the same
degree of strict concavity of the production function. In my calibration approach, an implicit assumption is
that different ownership types are not associated with systematically different technological opportunities.
14
Furthermore, the results I present in this paper are not sensitive to even large changes in technological
parameters.
19
θ = 0.72, together with the distribution of managerial talent shown in figure 1, approximately
yields the desired average growth and size distribution of establishments. In the U.S., large
manufacturing establishments represent a small fraction of the total number of establishments,
but account for most employment. For my economy to be consistent with this feature, µ must
assign positive mass to an outlying set of highly talented managers, as shown in figure 1.
I will now ask whether θ can be found so that instead of an economy where the size distribution of establishments resembles the U.S., the model economy generates a distribution similar
to the distribution one observes in Argentina, Mexico or other developing nations. Obviously,
developing economies differ from the U.S. economy in more than one respect. The goal of the
exercise, however, is to focus on the quantitative importance of contractual imperfections.
4.2
Impact of limited enforcement
Figure 2 plots various steady state statistics as a function of the degree θ to which contracts
can be enforced. Economies with better enforcement technologies are richer (aggregate output
rises with θ, panel A) and more productive (w, the marginal product of labor also rises with θ,
panel B.)
As discussed before proposition 5, limited enforcement disrupts the allocation of productive resources by limiting the quantity of funds available to managers. To see this quantitatively,
define the proportion of self-financing for a given contract by s1 =
t−1
st =
i=1 (1
+ r)t−1−i ρi
dt
20
b
d1
and, for t > 1,
This gives the proportion of the funds employed by managers that could be financed with past
income. Let s̄ be the economy wide average of this ratio, in steady state and define the average
proportion of outside financing by 1 − s̄. This measure of average outside financing is plotted
in panel C. Managers operate with more outside financing and, therefore, operate close to their
optimal scale of operation as θ rises. In fact, the average size of establishments rises with θ
(panel D) while, at the same time, the optimal scale of operation of a given establishment falls
since the wage rises. Richer economies, in this model like in the data, emphasize large scale
production. They also operate at a higher capital to output ratio (panel E.) This is largely
because the wage rate rises with θ, but is also due to the force described in lemma 2. Even at
a constant wage, managers who are less financing constraint employ more capital because the
relative price of capital falls with the shadow price of financing.
Note that the average growth rate of plants between their first and second period of
existence (panel F) is relatively stable until roughly θ = 0.7 but falls sharply past that point.
In fact, past θ = 0.9 all establishments are operated at their optimal scale in all periods,
and therefore do not grow. The sharp fall in growth rates around θ = 0.7 coincides with the
tapering off of the average size of establishments. To understand why average size stops rising,
note that when θ goes up the fraction of agent who choose to become managers (which is
inversely related to the average size of establishments) is affected by two forces. The wage goes
up, which all else equal means that fewer agent can profitably operate an establishment, but all
agents have a better access to outside financing. For low values of θ large rises in the wage are
the dominating effect. When θ approaches one, the wage effect becomes small and the average
size of establishments begins to fall, albeit slightly.
21
The bottom two panels of chart 2 describe the impact of limited enforcement on the distribution of bequests and lifetime income. The degree of bequest inequality tends to decrease with
θ as does inequality in lifetime earnings. Intuitively, when the wage is low, the income difference
between workers and managers is large. But the relationship between limited enforcement and
inequality is not monotonic: inequality initially rises with θ. When θ = 0, all production is
self-financed, and even managers of high talent are unable to generate much income, despite
the fact that wages are low. But the availability of outside financing rising steeply with θ, and
the gini coefficient begins to fall very early.
4.3
Size distribution of manufacturing establishments
The model is broadly consistent with the fact that poor economies emphasize low scale production, but I now subject the model to a more demanding quantitative test. When θ is selected
so that the average scale of operation resembles what one observes in Argentina and Mexico,
does the distribution of employment also resemble what one observes in those countries? The
answer turns out to be positive.
When θ = 0.2 the average size of establishments matches the average size of manufacturing
establishments in Mexico in 1988 (approximately 19 employees per establishment.) Table 3
compares the simulated distribution of establishments and employment across size categories
when θ = 0.2 to its empirical counterpart shown in table 1. The model, given its parsimony,
generates distributions that are remarkably close to Mexican data. When θ = 0.1, the average
size of establishments is now about 10 employees per establishment, which is near its empirical
22
Table 3: Size distribution of establishments
θ = 0.72
Employment size
Percent of establishments
Percent of employment
[0,10]
47.4
3.6
(10,50] (50,250] >250
33.3
15.5
3.8
14.6
31.0 50.8
θ = 0.20
Employment size
Percent of establishments
Percent of employment
[0,10]
85.6
18.1
(10,50] (50,250] >250
9.6
3.6
1.2
14.1
24.4 43.4
θ = 0.10
Employment size
Percent of establishments
Percent of employment
[0,15] (15,50] >50
93.7
4.1 2.2
32.3
12.3 55.4
counterpart for Argentina in the mid-80’s.15 The simulated distribution now resembles the data
from Argentina’s 1988 household survey shown in table 1 in that about a third of employment
is found in establishments with 15 employees or fewer.
4.4
Does wealth redistribution alleviate contractual imperfections?
The previous experiments suggest that improving the enforcement of property rights in nations
where they are poorly enforced has the potential to markedly raise income per capita. But
improving legal institutions is costly in nations where the bureaucracy is inefficient, and fiscal
resources are limited. A question of practical interest therefore, is whether wealth redistribution
policies can serve as a partial substitute for investments in property rights enforcement.
15
That average was 10.7 employees in 1985, 8.5 employees in 1993. Argentina carries out an economic census
once a decade.
23
Table 4: Impact of bequest inequality, θ = 0.2
No wealth
Wealth
redistribution redistribution
Wage
0.79
0.80
Output
8.10
8.11
Average size of plants
19.26
18.91
Plants with fewer than 10 emp. (%)
87.83
87.32
Intuitively, since bequest size is assumed independent of managerial talent, bequest inequality potentially amplifies the disruption in the allocation of resources that occurs when
enforcement is limited. Wealthy agents become managers while managers with less inherited
funds, but more talent, are constrained to become unskilled workers. Consider therefore the
following bequest redistribution experiment when θ = 0.2. Assume that bequests are pooled
and redistributed evenly. Assume further (for simplicity and the to give this experiment the
greatest chance to have a large aggregate effect) that this does not affect the bequest policy
of altruistic agents. The results are summarized in the second column of table 4. While the
impact on output is slightly positive, it is very small. That is, even drastic wealth redistribution
policies do very little to alleviate the impact of limited enforcement.16
5
Conclusion
This paper describes and measures the impact of the imperfect enforceability of financing
contracts in a dynamic, general equilibrium model. Limited enforcement distorts the allocation
16
Even a policy such that only agents who choose to become managers receive a bequest raises output and
labor productivity very little. Those results are available upon request.
24
of productive resources across establishments by imposing borrowing constraints on managers.
As a result, labor productivity, output, and the average size of establishments all rise when the
enforcement of contracts improves, while wealth and income inequality fall.
Calibrated numerical simulations indicate that limited enforcement can account for much
of the documented differences in the size distribution of establishments between Latin America
and the U.S. Many developing nations lack a well-functioning, effective judicial system and
property rights are not adequately enforced. The results presented here indicate that these
imperfections have an large impact, particularly on the organization of production. In addition,
I find that wealth redistribution schemes are not a good substitute for improving the degree to
which contracts are enforced.
In more general terms, my results suggest that contractual imperfections are important for
economic development. All the economies I consider operate with the same amounts of human
and physical resources. But in economies where contracts are poorly enforced, the misallocation
of this resources reduces aggregate income by as much as 50 percent. The magnitude of these
effects makes contractual imperfections a promising building block for a theory of total factor
productivity.
25
A
Appendix
A.1
Proofs
A.1.1
Proof of lemma 2
Proof. First order conditions for profit maximization imply
λ+δ+r
αk n
=
αn k
w(λ + 1 + r)
where λ ≥ 0 is the Lagrange multiplier associated with the financing constraint. One easily
shows that λ decreases with a, i.e. that π is concave in a. Since δ < 1, the equality above now
implies that nk decreases with a as claimed.
A.1.2
Proof of lemma 3
Proof. Consider a competitive contract such that, say, feasibility condition (1) does not bind.
Increase ρ1 until the condition binds and decrease ρT until feasibility condition (3) binds. The
advance profile stipulated by the competitive contract remains feasible since the left-hand side
of every equation in feasibility condition (4) rises. Furthermore, since the rates of interest and
time preference coincide, this leaves the agent’s lifetime income unchanged so that the altered
contract is a competitive contract as well.
A.1.3
Proof of proposition 4
Proof. To see the first item, consider without loss of generality the payment profile described
in lemma 3. Since all consumption is postponed until the last period, the opportunity cost
of default grows at the rate of interest β −1 − 1. On the other hand, the value of defaulting
with a given advance of funds decreases as the agent ages. Therefore, the maximum advance
compatible with the no-default constraint is non-decreasing. As for the second item, note that
any contract feasible for a given pair (b, θ) remains feasible when b or θ rises.
A.1.4
Proof of proposition 5
Proof. Let θ = 0 and consider, without loss of generality, contracts that satisfy the conditions of lemma 3. Note thatfeasibility condition (3) must bind at any competitive con−1 −(T −i)
β
ρi . Assume by way of contradiction that dT −1 >
tract so that V ∗ (b, z; w, 0) = Ti=1
26
T −2
i=1
β −(T −2−i) ρi . Then,
VTD−1 (dT −1; z, w, 0) = π(dT −1 , z, w) + dT −1 β −1
> ρT −1 +
T −2
β −(T −2−i) ρi
i=1
= β
T −2
∗
V (b, z; w, 0)
so that feasibility condition (4) is violated in period T − 1. One can then proceed by induction
to show that no net lending can take place in any period of the contract. The second item owes
to the fact that any feasible contract remains feasible when θ rises.
A.1.5
Proof of proposition 6
Proof. Since φ = 0 we have ν{0} = 1, and an agent’s type is summarized by their managerial
talent. By the theorem of the maximum, the set of competitive contract for each type is upperhemi continuous (u.h.c) in w. Because different agents of a given type can be offered different
contracts, and µ has finite support, the left-hand side of equation (5) is the sum of a finite
number of u.h.c, non-empty and convex correspondences, and so is also u.h.c. in w, non-empty
and convex. As w becomes large, this left-hand side can be made negative.
On the other hand, this left-hand side becomes positive when w is small. To see this,
note that in the second period of their life, agents of age 2 can operate with at least financing
of labor and w(1+r)
of
w(1 + r). With financing w(1 + r), managers can finance inputs 1+r
2
2
capital. Therefore,
w(1 + r) 1 + r
,
, z) − w(1 + r)
2
2
α +α
1 + r k n αk
w − w(1 + r)
> Az
2
π(w; z; w) > F (
Because αk < 1 it now follows that π(w; z; w) > w(1 + r) for w small enough. Furthermore,
it should be clear (and simple manipulations of first order conditions show) that as w
0,
r
n → w(1 + r) so that for w small enough, n > 1 + 2 . But this implies that for w small enough,
all agents of age 2 or more become managers, and they all hire more than one unit of labor, so
that the excess demand for labor must be positive.
A standard application of Kakutani’s fixed-point theorem now shows that a value w can
be found such that labor markets clear. Since φ = 0, the distribution of bequests puts all mass
at zero and is trivially invariant. Therefore, we have obtained the desired steady state.
Now consider raising θ to θ . Given θ and w, I have shown that optimal contracts can be
found so as to satisfy (5). When θ rises to θ , holding w fixed, agents become managers earlier,
and employ more workers in every period. Therefore, at w, optimal contracts can be found
that make the left-hand side of equation (5) positive. The same continuity argument as above
now implies that a new steady state wage w must exist with w > w.
27
A.2
Steady state bequest distribution
The excess demand for labor correspondence can be defined as the composition of two mappings.
The first mapping takes the wage w into the unique distribution of bequests satisfying the
third condition of my equilibrium definition. The second maps this distribution into the set of
corresponding values for the aggregate excess demand of labor.
Consider the first mapping. For each w > 0, V ∗ − b is bounded above uniformly across
individual types since µ has compact support. Let V̄ be such an uniform bound and define
g(b, z; w, θ) ≡ γβ T −1 V ∗ (b, z; wθ). Then, for all (b, z) ∈ IR+ × [0, 1],
g(b, z; w, θ) < γβ −(T −1) (V̄ + bβ −1 ).
Since γβ −T < 1 by assumption 1, there exists b̄ ∈ IR+ such that for all b ∈ [0, b̄], g(b, z; w, θ) ∈
[0, b̄]. Now let B[0, b̄] be the set of all Borel measures on [0, b̄] and define Tw : B[0, b̄] → B[0, b̄]
by,
(Tw λ){B} = (1 − φ)1B {0} + φ
g −1 (.,z;w,θ){B}
d(ν × µ).
The operator Tw describes the evolution of the distribution of bequests in this economy under
the assumption that the unskilled wage is fixed at w. Now observe the following:
Remark 7. For all Borel subset B ∈ [0, b̄], and b ∈ [0, b̄], (Tw χb ){B} > 1 − φ or (Tw χb ){B c } >
1−φ
where for all b ∈ IR, χb denotes the point-mass distribution that puts all mass at b while B c
denotes the complement of B in [0, b̄]. To see this, note that from all b ∈ [0, b̄] the Markov
process goes to zero with probability 1 − φ. Since 0 ∈ [0, b̄] = B ∪ B c the result trivially holds.
Theorem 11.12 in Stokey et al. (1989) now implies:
Lemma 8. For all w > 0, the Markov chain defined by Tw is geometrically ergodic. That is,
there exist a distribution νw ∈ B[0, b̄] and a constant 0 < < 1 such that for all b ∈ [0, b̄],
T n χb − νw ≤ n χb − νw .
In the statement of the lemma, . denotes the total variation norm. Finally, the Markov
process defined by Tw satisfies a law of large numbers.
Proposition 9. For w > 0 let {bn }+∞
n=0 be any run of the Markov process defined by Tw . Let f
be a νw -integrable function. Then
1
f (bn ) −→a.s
n i=0
n
f (b)dνw
Proof. Pick any B ∈ [0, b̄] such that νw {B} > 0. For all b, let P be the transition probability
function implied by Tw and let P n be the n-fold composition of this function. Since νw {B} > 0,
lemma 8 and remark 7 imply that for some integer n, P n (0, B) > 0. It must then be the case
that bn ∈ B infinitely often with probability one. The result now follows from Theorem 3 in
Tierney (1994).
28
A.3
Calibration of the distribution of managerial talent
In this appendix, we select θ and specify µ so that, given other exogenous parameters, the average growth rate and the size distribution of establishments match their empirical counterparts
in the U.S. manufacturing sector.
Assume first that θ = 1. In that case, a manager’s talent and the employment-size of the
establishment he operates are monotonically related. Indeed, let w be the equilibrium wage.
First order conditions for unconstrained net income maximization for a manager of talent z
imply:
znαk +αn −1 ∝ w(1 + r),
1
so that employment is linear in z 1−αk −αn . Now observe that the model implies a lower bound
on establishment size in this economy. Indeed, first order conditions for profit maximization
imply that the manager’s net income is given by:
(1 − αk − αn )Azk αk nαn =
1 − αk − αn
nw(1 + r)
αn
An agent will become a manager only provided this income exceeds the unskilled wage, that is:
1 − αk − αn
nw(1 + r) ≥ w(1 + r)
αn
or equivalently,
αn
1 − αk − αn
Thus the distribution of establishments implied by my model given the distribution of talent
will be truncated at 1−ααkn−αn approximately 3.5 employees given the values I set for technological
shares.
n≥
In the Census Bureau’s County Business Patterns survey, manufacturing establishments
are classified in nine employment size categories: 1 to 4 employees, 5 to 9, 10 to 19, 20 to
49, 50 to 99, 100 to 249, 250 to 499, 500 to 999, and 1000 or more employees. Because of
the lower bound implied by my model, I combine the first two size categories. An initial
1
guess for µ is then obtained as follows. Assume that z 1−αk −αn is log-normally distributed with
location and dispersion parameters λ1 and λ2 , respectively. Under the assumption that θ = 1,
the distribution of establishments implied by my model is then log-normally distributed and
left-truncated at 1−ααkn−αn . The two parameters of the distribution can then be selected by
maximum-likelihood so that the implied distribution approximates its empirical counterpart
in the 1988 County Business Pattern Survey. This gives me a continuous guess µ̃ for the
distribution of managerial talent, and a truncation point z ∗ such that agents become managers
only if z > z ∗ . To obtain a discrete guess, I assume that the support of µ consists of 4 points
to the right of z ∗ and forty equally spaced points to the left of z ∗ . Then letting {zi }44
i=1 be the
support of µ, I let µ{zi } = µ̃( zi +z2 i+1 ) − µ̃( zi +z2 i−1 ) for i ∈ {1, 44}, where z0 = 0 ans z45 = 1.
29
The four points to the right of z ∗ are chosen to match the average size of manufacturing
establishments in the U.S. in each of the 4 categories shown in table 1.
This gives me a starting guess for θ and µ. That guess gives an approximately correct
distribution of establishments for the U.S., but, counterfactually, no establishment growth since
θ = 1. Evans (1987) estimates with data from 100 manufacturing industries between 1976 and
1980 that manufacturing firms with 5 years or less of existence grow at yearly rate of roughly
2.6%. Based on this estimate, I arrive at final guess for θ and µ by following the following
iterative method:
1. Given all other parameters, update θ until establishment growth between their first and
second period of existence is (1.026)8 − 1 percent on average in steady state.
2. Update z41 until, in steady state, the fraction of establishments with 1 to 9 employees
matches the fraction shown in table 1 in the U.S.
3. Similarly update z42 , z43 and z44 .
I repeat steps (1-3) until θ and {zi }i≥41 become approximately invariant. With this method, I
arrive at θ = 0.72 and the distribution of managerial talent shown in figure 1.
30
Figure 1: Distribution of managerial talent
0.045
0.04
0.035
0.03
mass
0.025
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
0.4
0.5
z
31
0.6
0.7
0.8
0.9
1
Figure 2: Impact of θ on steady state statistics
A: Aggregate Output
B: Wage
10
1.1
7.5
0.8
5
0.5
0
0.2
0.4
0.6
0.8
1
0
C: Average share of outside financing
60
0.5
50
0.4
40
0.3
30
0.2
20
0.1
10
0
0.2
0.4
0.6
0.8
0.4
0.6
0.8
1
D: Average employment size of establishments
0.6
0
0.2
0
1
0
E: Capital labor ratio
0.2
0.4
0.6
0.8
1
F: Average growth rate of young establishments
1
0.4
0.75
0.2
0.5
0
0.2
0.4
0.6
0.8
0
1
0
0.2
G: Lifetime income gini coefficient
0.4
0.6
0.8
1
0.8
1
H: Bequest gini coefficient
35
78
77
30
76
75
25
74
73
20
0
0.2
0.4
0.6
0.8
72
1
θ
0
0.2
0.4
0.6
θ
32
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