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Abstract

We characterize an equilibrium development process driven by the interaction of the distribution of wealth with credit constraints and the distribution of entrepreneurial skills. When efficient entrepreneurs are relatively abundant, a "traditional" development process emerges in which the evolution of macroeconomic variables accord with empirical regularities and income inequality traces out a Kuznets curve. If, instead, efficient entrepreneurs are relatively scarce, the model generates long-run "distributional cycles" driven by the endogenous interaction between credit constraints, entrepreneurial efficiency and equilibrium wages. Copyright 2000 by The Review of Economic Studies Limited
Enterprise, Inequality and Economic Development
Huw Lloyd–Ellis Dan Bernhardt
CREFE and
Department of Economics Department of Economics
University of Toronto University of Illinois Urbana–Champaign
150 St. George Street 463 Commerce West
Toronto, M5S 3G7 Champaign, IL 61820
Accepted: February 1999
Abstract
We characterize an equilibrium development process driven by the interaction of the distribution of
wealth with credit constraints and the distribution of entrepreneurial skills. When efficient entre-
preneurs are relatively abundant, a traditional’ development process emerges in which the evo-
lution of macroeconomic variables accord with empirical regularities and income inequality traces
out a Kuznets curve. If, instead, efficient entrepreneurs are relatively scarce, the model generates
long–run ‘distributional cycles’’ driven by the endogenous interaction between credit constraints,
entrepreneurial efficiency and equilibrium wages.
Keywords: Economic development, income distribution, credit constraints, endogenous cycles.
JEL Classification Numbers: E0, O1, O4.
This paper has benefitted from the comments of David Andolfatto, Charles Beach, Jim Bergin, V.
V. Chari, Mick Devereux, Burton Hollifield, Arthur Hosios, Tom McCurdy, Aloysius Siow, Robert
Townsend, Dan Usher, and seminar participantsat Brock, Cornell, Minnesota, Northwestern, Penn-
sylvania, Queen’s, Rutgers, Toronto, UBC, and Waterloo. All remaining errors and omissions are
our own. Funding from the Social Sciences and Humanities Research Council of Canada is grate-
fully acknowledged.
1. Introduction
Research in developing economies typically identifies two key obstacles to the creation and expan-
sion of small and medium sized enterprises: a lack of access to credit, especially to finance working
capital, and a relative scarcityof entrepreneurial skills.
1
This paper characterizes an equilibrium de-
velopment process driven by the interaction of the distribution of wealth with credit constraints and
the distribution of entrepreneurial efficiency. When efficient entrepreneurs are relatively abundant,
a ‘traditional’ development process emerges (e.g. Lewis 1954 and Fei and Ranis 1966), in which
the evolution of macroeconomic variables accord with key empirical regularities and income in-
equality traces out a Kuznets curve. If, instead, efficient entrepreneurs are relatively scarce, our
model generates long–run distributional cycles’ driven by the endogenous interaction between
credit constraints, entrepreneurial efficiency and equilibrium wages.
Our model features single–period lived agents who value both consumptionand bequests to their
children. Individuals, who differ in both entrepreneurial efficiency and inherited wealth, choose
whether to work as entrepreneur, as wage laborers in industry or in subsistence agriculture. An
entrepreneurs uses his inherited wealth as collateral in return for a loan to pay for start–up costs and
working capital. Potential moral hazard in debt repayment limits borrowing. He hires labor at the
prevailing equilibrium wage, which he pays out of earned revenues, receiving the balance as profit.
Initially, few agents are efficient enough to become entrepreneurs and those who do operate on a
smallscale. Withlittlecompetitionforlabor, theequilibriumwageisjustsufficientto enticeworkers
away from the rural sector. Over time, the distribution of wealth grows in the first–order stochastic
sense, permitting the extent and scale of entrepreneurial activity to expand, so that profits rise and
See, for example, Levy (1993) and Fidler and Webster (1996).
1
income inequality worsens. When wages are low, even relatively inefficient projects are worth
undertaking if sufficient capital can be employed wealth rather than entrepreneurial efficiency is
the primary determinant of economicactivity. Consequently, even though entrepreneurialefficiency
is uncorrelated across generations, wealth inequality persists.
Eventually, competition amongst entrepreneurs for workers bids up the equilibrium wage, so
that profits eventually fall, and income and wealth inequality start to decline. At this point, the
skewnessof the efficiencydistributioniscrucialindeterminingwhethergrowth continuesunabated,
or whether cycles appear as the economy matures. With decreasing returns to scale, past wage
increases effectively transfer wealth from the rich with low marginal products to the poor with high
marginal products, (the ‘productivity effect’’). However, if entrepreneurial efficiency is skewed so
that low cost entrepreneurs are scarce, then this transfer reduces the supply of entrepreneurs coming
from the rich by more than it increases that coming from the poor (the ‘‘enterprise effect’’).
If the productivity effect dominates, the supply of entrepreneurs and their demand for labor still
rise. Wages are bid up further, output expands and the distributions of wealth and income grow
in the second–order stochastic sense. As wages rise, less efficient agents prefer to enter the labor
force (though they could profitably become entrepreneurs) and production becomes increasingly
efficient. Gradually, entrepreneurial efficiency replaces wealth as the primary determinant of oc-
cupational choice, so that wealth becomes less persistent along family lineages. This benchmark
process matches several empirical regularities associated with development :
Market–clearing wages remain low initially, but then rise monotonically.
2
There is continuous rural–urban migration and increasing participation in manufacturing.
2
The labor share of GNP and of value–added in manufacturing rises with per capita GNP.
3
The capital–output ratio initially rises, but levels off and may decline as wages rise.
4
Average firm size grows and then declines. The dispersion in firm sizes also grows and then falls.
Equilibrium growth rates are high in early stages, decline as workers leave the rural sector, then
increase after all workers leave subsistence, before declining again as the economy matures.
Income inequality rises in the Lorenz dominance sense during the early phases of development,
then gradually falls as the economy matures.
5
If, instead, the enterprise effect dominates, then the redistribution of inheritances caused by past
wage increases actually causes the supply of entrepreneurs and their demand for labor to decline.
Wages fall and aggregate output contracts. The resulting redistribution towards the rich increases
the supply of entrepreneurs and their demand for labor, so that the economy’s decline is reversed.
The process continues in this fashion, generating endogenous ‘distributional cycles’’ in economic
Young (1993) finds that the recent rapid growth of East Asian NICs was largely due to the accumulation of capital and increased
participation in the manufacturing sector rather than total factor productivity growth.
The prediction holds for a cross–section of countries (Lloyd–Ellis and Bernhardt, 1998) and for the industrial revolution in
Europe(Mathias and Postan, 1978). Labor share ofGNP also rose in the US,but labor shareofmanufacturing value–added hasbeen
constant for the last 100 years.
See Maddison (1989) and Kim and Lau (1994) for related evidence.
This Kuznets relationship holds for a cross–section of market economies (Paukert, 1973, Ahluwalia, 1976, Lydall 1979, Sum-
mers, Kravis and Heston 1984). Although inequality declined in most developed countries during the twentieth century (Fields and
Jakubsen, 1994), studies of the UK and US that include nineteenth century data find support for Kuznets’ hypothesis (e.g. Lindert
and Williamson, 1980).
3
activity which persist into the long run. Wages evolve procyclically so that income inequality wors-
ens during recessions and improves during economic booms.
Our model is most closely related to those of Banerjee and Newman (1993), and Aghion and
Bolton (1997).
6
Both models feature one–period lived, risk–neutral agents who ex ante differ only
in their inheritances but not their efficiency, and whose entrepreneurs face ex post production risk.
Banerjee and Newman have thesame moral hazard problemin credit marketsas we do, and consider
an environment in which entrepreneurs employ workers and capital in one of two exogenously–
specified combinations. Their analysis focuses on how the long–run distribution of wealth is related
to technology parameters: they find that because of the non–convexities in the feasible technology
choices of entrepreneurs, the long–run distributions of wealth may depend on initial conditions.
In Aghion and Bolton, agents either invest in a fixed–size, risky project, lending any remaining
wealth or borrowing if necessary; or they earn a safe, low income and lend. Limited liability and
the dependence of the success probabilityon non–contractable effort induces credit–rationing based
on inherited wealth. Equilibrium between borrowers and lenders determines a market interest rate
which, in contrast to ours, varies with the distribution of wealth. As wealth accumulates, demand
for credit declines and supply rises, so that interest rates fall and, although it may initially rise,
wealth inequality eventually falls. Neither model features a fully–specified labor market, ex ante
differences in entrepreneurial efficiency or working capital, so they are silent with respect to the
evolution of key macroeconomic aggregates.
A crucial distinction between our model and theirs is the timing of the revelation of information
regarding entrepreneurial efficiency. In our economy, agents learn their potential efficiency before
Other relatedcontributions tothe literature include GreenwoodandJovanovic (1990), Galor andZeira (1993)andPiketty(1997).
4
choosing occupations, and we abstract from ex post production risk.
7
Thistiming captures, to some
extent, the feature that if there were multiple production periods within a lifetime, agents would
condition actions on their revealed efficiency. As a result of this timing, both potential efficiency
and inherited wealth affect agent decision–making occupational choice, capital employed and
labor hired — and consequently the paths of macroeconomic aggregates. The impacts of inherited
wealth and efficiency are distinct and vary as the economy develops. While in initial stages, wealth
is the primary determinantof occupation because wealthy agents can invest in capital and profitably
exploit cheap labor on a grander scale; in later stages, entrepreneurial efficiency matters more both
because fewer agents are wealth constrained and because higher wages reduce the profitability of
large scale production. The consequence for the dynamics of income and wealth inequality is that
theyfirst riseandpersistalongfamilylineages,andthenfalland areless persistent alonglineages. If
the efficiency distribution is skewed, the importance of entrepreneurial efficiency relative to wealth
is cyclical; rising during booms and falling during recessions.
Section 2 of our paper lays out the economic environment and characterizes the periodic equilib-
rium of the economy. Sections 3 details the mechanisms driving the economy through each phases
of the development cycle and characterizes the movements of key macroeconomic variables and
the evolution of inequality over this cycle. Section 4 shows how a highly skewed distribution of
entrepreneurial efficiency can lead to endogenous cycles. All proofs are in the appendix.
Productionrisk would play no real role if, as in Banerjee and Newman, it is bounded in such a way that default never occurs (see
Lloyd–Ellis and Bernhardt, 1998). In contrast, production risk plays a central role in Aghion and Bolton because of the possibility
of default.
5
2. The Model
There are countably many time periods, . The economy is populated by a contin-
uum of family lineages of measure one. Each agent is active for one period, then reproduces one
agent. An agent’s endowment consists of a bequest inherited from his parent. Agents have identical
preferences over consumption, and bequests to their children, represented by
(1)
The utilityfunction is homogeneous of degree one, strictly increasing in both arguments and strictly
quasi–concave. The assumption that the preference is for the bequest itself, not for the offspring’s
utility, simplifies the analysis and captures the idea of a tradition for bequest–giving (see Andreoni
1989). Homogeneity can be relaxed considerably without changing the results.
At time a new production technology comes into existence. Entrepreneurs who pay the
necessary start–up costs (discussed below), combine labor, , and capital, , to produce a single
consumption good according to the common production function,
(2)
Theproductionfunctionisstrictlyincreasinginbotharguments,strictlyconcaveandquasi–homothetic.
Capital and labor are complements and third–order derivatives are negligible, so the production
function is well approximated by a second–order Taylor series expansion.
Agents are distinguished by two characteristics: their initial wealth inheritances, , and their
personal costs of undertaking a project, . Project start–up costs are drawn from a time–invariant
6
distribution, , with support and a linear density
(3)
where Start–up costs reflects innate entrepreneurial efficiency and are uncorrelated
with inherited wealth. Linearity can be relaxed considerably without changing our results.
Entrepreneurscanborrow to finance their investments. However, the capitalmarket is limitedby
a moral hazard problem as in Sappington (1983) and Banerjee and Newman (1993). Specifically,
there exists an alternative activity which yields a safe gross return of . Competition amongst
lenders then drives the interest on loans down to . Entrepreneurs can borrow , but they must put
up their inheritance as collateral. After production they can abscond, losing , but escaping the
repayment obligation, . If absconders are apprehended, which they are with probability , they
can hide their income from the authorities, but they receive a punishment which imposes on them an
additivedisutility of . Withhomogeneouspreferences, borrowers wouldrenegeif ,
where is a positive preference parameter. Recognizing this, lenders only make loans that satisfy
where . Thescaleof the projectis therefore limited by an agent’s inheritance,
start–up cost and the degree of market completeness summarized by .
Givenhis type , each agent chooses his occupation takingthe equilibriumwage and his own
potential profits as given. If his inheritance is large enough, he can become an entrepreneur in the
manufacturing sector. Both project set–up costs and productive capital employed must be financed
out of bank loans. Alternatively, agents can work in the manufacturing sector at the prevailing
equilibrium wage rate, , and invest their inheritance in the alternative activity, earning . Finally,
agents may prefer to remain inruralareas and receivethesubsistence income, . Themanufacturing
sector is located in urban areas, where agents incur a cost of living equal to . With homoge-
7
neous preferences, one can interpret this cost as a disutility of labor incurred by entrepreneurs and
workers, but not by subsisters.
The occupational choice of an agent of type determines his lifetime income given the
equilibrium obtaining that period, . His total lifetime wealth equals
(4)
where if the agent lives in an urban area and otherwise.
2.2 Optimal behavior
An agent with wealth maximizes subjectto yielding optimal linear
consumption and bequest policies, and . We assume that the pre–industrial economy
has reached a steady state with constant bequests defined by
Agents who are efficient enough become entrepreneurs. Wages are paid out of end–of–period
revenues. After paying the start–up cost, , an entrepreneur chooses capital to maximize profits
subject to the constraint that capital must be financed out of the remainder of his loan, .
Thus, the net profits earned by a type entrepreneur equal
s.t.(5)
Profitmaximizationyieldscapitalandlabordemandfunctions, and
,where denotes unconstrained levels.
8
We assume that most agents are initially borrowing
constrained, i.e. where is sufficiently small, and is the
reservation wage below which potential workers prefer to remain in subsistence.
The assumption of quasi–homotheticity implies that the demand for labor by a constrained entrepreneur increases linearly with
firm size.
8
For an agent with inherited wealth to undertake a project, he must draw a start–up cost that
is less than . Even if such an agent can afford to become an entrepreneur there may still
exist a marginal set–up cost level, , defined implicitly by at which
the agent would be indifferent between working and becoming an entrepreneur. We assume that
so that some projects are always undertaken.
9
The start–up cost of an agent with
inherited wealth who is just willing and able to undertake a project at time is given by
(6)
Lemma 1: is increasing and concave in , decreasing in and .
2.3 Macroeconomic Equilibrium
Let denote the time distribution of inheritances. Then, integrating the optimal decisions of
agents over their types, yields the following aggregates:
Enterprise(7)
Start-up costs(8)
Labor force(9)
Capital invested(10)
Output(11)
Net income(12)
Aggregatesubsistence isgiven by . Theseaggregatesaretime–varying
This condition is sufficient for the limiting distribution to be ergodic.
9
functions of the wage because they also depend on the distribution of inherited wealth.
Acompetitiveequilibriumforan economywithinheritancedistribution isa vector
such that:
Given the wage , an agent of type selects his occupation to maximize utility.
Type entrepreneurs choose capital and labor to maximize profits subject to .
Markets clear: where if .
Two kinds of equilibria can arise. In a dual economy equilibrium, the supply of entrepreneurs
and their labor is insufficient to draw all agents out of subsistence and the wage settles at its reser-
vation level, .Inanadvanced economy equilibrium the supply of entrepreneurs and their labor
demand are high enough that the surplus labor is exhausted and the wage is bid up above .The
following Lemma is useful for understanding how the economy evolves in such an equilibrium:
Lemma 2: Aggregate income, , is equal to the area below the upper envelope generated by the
supply and demand curves for labor.
At time , a fraction of agents that inherit remain in subsistence. A proportion
of agents with inheritance realize a low enough start–up cost, , to undertake
their projects. Let be the value of such that . Then the distribution of
income conditional on inherited wealth and the equilibrium obtaining at time can be represented
by the cumulative distribution function,
if or
if
if
if
otherwise
(13)
10
Here isthelowersupportonprofits and is
theuppersupport. The unconditionaldistributionof incomesistherefore
An agent’s final wealth is simply the sum of the return on his inheritance and his lifetime income,
net of any urban living costs. An agent of type bequests
Note that an agent’s inheritance depends not only on his parent’s inheritance and cost realization,
but also on the past distribution of wealth via its effect on the equilibrium wage. The distribution
of wealth evolves according to an endogenous non–stationary probability transition function, so the
unconditional distribution of bequests is
(14)
where .
A firm’s ‘size’ corresponds to the amount of capital it employs. The distribution of firm size
conditional on inherited wealth is
if
if
otherwise,
(15)
where is the inheritance level such that all wealthier entrepreneurs are unconstrained. Thus,
the time distribution of firm sizes is
3. A Lewis–Kuznets Development Process
In this section, we detail the dynamic evolution of the economy when entrepreneurial ability is
uniformly distributed ( . The results hold more generally as long as the distribution is not
too skewed towards high start–up costs (i.e. is not too large). The implications of more skewed
distributions for the development process are discussed in Section 4.
11
3.1 The Phases of Economic Development
Proposition 1: Following the time introduction of a new production technology to the long–
run pre–industrial economy, the economy passes through four distinct phases of development:
Phase 1 (The Dual Economy, ): Wages remain at . Incomes and wealths grow in the
first–order stochastic sense.
Phase 2 (The Transition, ): Wages begin to rise, but incomes and wealths continue to
grow in the first–order stochastic sense.
Phase 3 (Advanced Economic Development, ): Wages rise, and incomes and wealths grow
in the second–order stochastic sense.
Phase 4 (Long Run): Wages converge and the distributions of incomes and wealths converge to
unique limiting distributions, and , which are independent of the initial distribution.
At , agents with sufficiently low start–up costs migrate to urban areas to become entre-
preneurs. They employ additional agents at the prevailing wage. If is sufficiently low, a dual
economy equilibrium obtains: Surplus labor remains in the rural sector earning the subsistence in-
come, , and the equilibrium wage settles at . At the end of their lives, workers and farmers
bequeath . However, entrepreneurs bequeath more than , so that the distribution of inheritances
at , , dominates in the first–order stochastic sense.
The stochastic increase in wealth encourages more generation agents to engage in entre-
preneurial activities and on a greater scale. The associated increase in labor demand draws migrants
from the rural sector, output rises and the distribution of income at , dominates that at in
the first–order stochastic sense. Althoughmost agents still receive or less, entrepreneurial profits
increase in the first–order stochastic sense. The development process continues in this fashion as
12
long as there is surplus labor in the economy and the wage remains at .
Eventually, the supply of subsistence labor is exhausted. At some date , an advanced economy
equilibrium obtains: increased competition for workers by entrepreneurs bids the wage rate above
.
10
Although wages rise, profits may still rise due to the relaxation of financing constraints. So
long as the bequests of the richest lineages do not decline, the distribution of inheritances continues
togrow in the first–orderstochasticsense for several periodsafter . During thistransitional phase,
labor demand rises and supply declines, so that both wage and aggregate income rise. However,
with strict concavity in production, the largest entrepreneurs eventually achieve the optimal scale
so their incomes start to decline. Eventually, this declining income offsets the rising lineage wealth,
so that at some time , , and the next phase of development begins.
To understand how the economy evolves in the advanced phase, suppose the distribution of in-
heritances at date , , dominates in the second–order stochastic sense. Since the
production function is strictly concave and quasi–homothetic, a unit transfer of wealth from rich to
poor entrepreneurs raises labor demand. Although some high cost projects become undesirable, the
uniform distribution of entrepreneurial efficiencies ensures that the redistribution of wealth does
not reduce the supply of entrepreneurs. Hence, the equilibrium wage and aggregate incomes rise.
The children of entrepreneurs are more likely to be entrepreneurs themselves, but receive lower
profits than their parents. Conversely, thechildren of workers are predominantly workers and expe-
rience an increase in income. As a result,thedistributionofincome evolves in themanner illustrated
in Figure 1. Formally, there exists a such that and and
(16)
A sufficient condition for
, is given in the appendix in Lemma A4.
13
where the inequality is strict on an interval of positive measure. We say that distribution is
more X–Dispersed than distribution
Since average income also increases, the distribution of income must grow in the second–order
stochastic sense. Similar results hold for wealth so that the distribution of inheritances, ,
dominates in the second–order stochastic sense. Since first–order stochastic dominance im-
plies second–order stochastic dominance, the result follows by induction from period .
The economy cannot growwithoutbound. In particular, there exists an efficient state in which all
agents who wish to become entrepreneurs can do so at the unconstrained optimum scale given the
equilibrium wage rate. The associated market clearing condition, implicitly
defines an upper bound for the equilibrium wage, . Since the wage increases monotonically with
time and is bounded, it must converge to some long–run value . It follows that the
transitionaldynamics governing the distribution of wealth convergeto a stationaryMarkov process,
. This processsatisfies the Monotone Mixing Condition(Hopenhayn and Prescott, 1992),
so that the distributions of wealth and income converge to unique limiting distributions.
The limiting distributions are non–degenerate: wealth disparities continue to exist, but do not
persist forever between lineages. Since the limiting distributions are independent of the initial dis-
tributionof inheritances, , economiesthat start out with more unequal distributions, while they
may not follow the same cycle of development, will end up with the same long–run distribution.
Ifthe shareofwealthbequeathed is sufficientlyhigh, thentheefficientstate is eventuallyreached.
In this case, the conditional distribution of income is independent of inherited wealth:
, and inequality is low because the wage is at its highest possible level, . If the share
14
of wealth bequeathed is very low, the economy never achieves enough momentum to leave the dual
economy phase (i.e. ). Consequently, the wage remains at , production is inefficient
because most entrepreneurs are constrained, wealth and income inequality remain very high and
even though the economy satisfies the mixing condition, wealth is very persistent along lineages.
3.2 The Macrodynamics of Development
We now detail the evolution of key macroeconomic variables over the economy’s development
cycle. We illustrate the results using the following specifications for preferences and technology:
(17)
(18)
To emphasize the robustness of the results, we allow for some skewness in the distribution of start–
up costs (i.e. ). Figure 2 illustrates the time paths for key aggregate variables.
11
Growth rates of wages and aggregate income are non–monotonic. After leaving the dual econ-
omy phase, wage growth first accelerates because, with decreasing returns to scale, the demand
for labor from low wealth entrepreneurs expands more rapidly than the reduction in demand from
high wealth entrepreneurs. In general, the growth rate in aggregate income first rises then declines
gradually in the dual phase, rises again when the wage starts to increase, and then declines as the
economy develops further. The first growth spurt occurs because as some lineages become wealthy,
the supply of entrepreneurs increases rapidly, offsetting the decline in the marginal return to wealth.
Parameter values are
The numerical example tracks the entire distribution of wealth on a discretized support. The associated code is
available from the authors.
15
The growth rate then declines as agents become less constrained and the marginal return to their
wealth falls. Output growth rises again when wages rise due to the effective redistribution of wealth
towards poorer agents who are more productive on the margin. As the economy develops further,
because there is no engine for long–run growth, wage increases decline and growth rates fall.
Proposition 2 (Occupations): During the dual phase the rate of enterprise, and labor force
participation, , grow over time. There exists a such that if the slope of the marginal
product curve for labor is sufficiently small, , then during later phases the rate of enterprise,
, still increases but wage laboring, , declines.
We have already described the mechanism driving the increase in enterprise and labor force partic-
ipation in the early phases of the cycle. In the advanced phase, second–order stochastic growth is
sufficient to cause labor supply to fall and labor demand to rise. In general, whether the equilibrium
number of laborers rises or falls depends on the wage elasticities of supply and demand. If the wage
elasticity of demand is sufficiently high ( sufficiently small), then the number of entrepreneurs
increases over time.
12
Proposition 3 (Factor Shares): There exists a and a such that if the optimal firm–
level demand for labor is sufficiently great, , and the slope of the marginal product of
labor curve is sufficiently small, , then the labor share of value–added in manufacturing
rises monotonically throughout the development cycle.
During the dual phase, the manufacturing labor force expands and the wage is constant. If the
wage elasticities of supply and demand are high enough (e.g. when the sufficient conditions on
The qualitative implications for the wage rate, aggregate income and inequality do not depend on this elasticity.
16
primitives in Proposition 3 hold), then the labor share of value–added rises as the manufacturing
sector expands. In the advanced phase, when the equilibrium labor force may decline, continued
growth in the labor share requires that the equilibrium wage, , rises sufficiently rapidly relative
to aggregate income, . Again, if the wage elasticities of labor demand and supply are sufficiently
high, the increments to aggregate profits are always proportionately less than the increments in the
aggregate wage bill.
Proposition 4 (Investment): During the early phases both aggregate working capital, , and
aggregate start–up capital, , grow monotonically. However, during the advanced phase these
variables may evolve non–monotonically.
In the dual phase, first–order stochastic growth in wealth causes both aggregate working capital,
, and aggregate start–up costs, , to grow. Later in the development cycle, the time
paths of these aggregates depend on the net effect of two opposing forces: the second–order sto-
chastic increase in inheritances causes the demand for start–up and working capital to rise for a
given wage; however, the rising wage induces less efficient agents to become workers, causing the
equilibrium level of capital to fall. If the wage elasticity of demand for capital is high enough,
13
then the aggregate working capital stock falls (see Figure 2). Similarly, aggregate start–up costs,
, decline as long as the wage elasticity of the supply of entrepreneurs is sufficiently high.
Proposition 5: The distribution of firm sizes, , becomes increasingly X–Dispersed during
phases 1 and 2, before becoming gradually less X–Dispersed in the advanced phase.
That is, if labor and capital are sufficiently strong complements, the supply of entrepreneurs is sufficiently wage elastic and the
fraction of constrained entrepreneurs is small enough.
17
During the dual phase, the wealth of agents drawing a particular cost level, stochastically increases
with each generation. Hence, the level of capital employed by them (firm size) also stochastically
increases. Although during later phases, the rising wage implies that average firm size eventually
falls, the dispersion of firm sizes evolves in a way similar to that of incomes and wealths. Figure
2 depicts the time paths of the average and variance of firm sizes for the parameterized economy.
Recall that the rate of enterprise, , increases throughout the process, but that the aggregate capital
stock first falls and then rises during the advanced stages. The evolution of the average firm size
reflects this. Thevariancerises, reachingapeak duringthedualstage and then decliningthroughout
the advanced stage. If maximum efficiency is attained, the variance falls to zero because all entre-
preneurs produce at the optimal scale, . Otherwise, the limiting distribution of firm sizes is
non–degenerate and the variance converges to a positive value.
Proposition 6 (The Lorenz Curve):
(a) There exists a time period , such that for all , income inequality increases in the
Lorenz dominance sense.
(b) There exists a and a such that if the optimal firm level demand for labor is
sufficiently great, and the slope of the marginal product of labor curve is sufficiently
small, , then for all , income inequality declines in the Lorenz dominance sense.
The first panel of Figure 3 depicts a typical Lorenz curve, , during the dual stage of de-
velopment.
14
The linear segment corresponds to the traditional sector, the linear segment
corresponds to wage laborers and the convex segment corresponds to entrepreneurs. Inequality
TheLorenzcurveis
where . Our analysis here is influenced by Bourgignon (1990).
18
increases in the Lorenz dominance sense if the entire Lorenz curve shifts to the right (i.e. the curve
shifts from to ). Sufficient conditions are that (1) the growth rate in per capita in-
come exceeds the growth rate in the income of migrating agents; and, (2) the largest entrepreneurs
profit, , increases faster than mean income, . Both conditions are trivially true in the
first period and hold subsequently as long as the marginal product of capital is sufficiently high and
is sufficiently low.
During the advanced stages of economic development, the Lorenz curve no longer includes the
segment corresponding to the traditional sector. The maximum income in the economy falls and
the mean rises, so the slope at declines. For inequality to decline unambiguously, the wage must
grow more quickly than per capita income. This will be the case if the elasticitiesof the demand for,
and supply of, labor are sufficiently large (i.e. the sufficient conditions in Proposition 6 (b) apply),
because this ensures that a wage increase has a large negative impact on aggregate profits. Figure
2 illustrates the implied relationship between the Gini coefficient
15
and per capita income.
A key factor driving these development dynamics is that an agent’s ability to borrow is limited
by his inherited wealth. As the value of is increased (see Figure 4), the development cycle occurs
more rapidly, but its nature remains much the same. Income inequality rises to a high value much
more rapidly before declining. As the borrowing constraint is relaxed further the economy reaches
the efficient state. For , the economy jumps immediately to the efficient state
and the degree of income inequality reflects only variations in entrepreneurial efficiency.
Twice the area between the Lorenz curve and the
line.
19
4. Economic Development when Efficient Entrepreneurs are Scarce
A key problem facing developing economies is the relative scarcity of high skilled entrepreneurs.
As Fidler and Webster (p. 25, 1996) point out ‘Poorly developed business skills are a binding
constraint to enterprise growth, even more than lack of access to credit in many cases. Entrepre-
neurs’ complaints about lack of access to credit often mask technical and managerial inadequacies
...’’. This section considers the implications for the development process when we incorporate the
feature that efficient entrepreneurs are a relatively scarce resource — i.e. the distribution of entre-
preneurial efficiency is skewed towards high cost entrepreneurs: .
During the dual–economy phase, the qualitative nature of our results are independent of the
shape of the distribution of entrepreneurial efficiency. However, in later phases, when the wage rate
is endogenous, the nature of the development path is sensitive to the skewness of this distribution:
If the distribution of entrepreneurial efficiency is sufficiently skewed towards high cost projects,
then the economy exhibits endogenous long–run cycles in production. Wages evolve pro–cyclically
so that income inequality falls during booms and rises during recessions.
To understand the mechanism driving this result, consider the special case where no working
capital is used in production and the skill density is highly skewed ( ). Then the relationship
between the rate of enterprise and wealth for any given wage is
(19)
Considertwowealthlevels and andtheagentswhoinheritthem(seeFigure 5). If
then a unit transfer of wealth from the rich agents to the poor agents unambiguously increases
the supply of entrepreneurs. If, however, (i.e. both wealth levels lie on the convex
20
segment) then such a transfer decreases the supply of entrepreneurs from the high wealth group by
more than it increases the supply from the low wealth group.
Past wage increases result in many such transfers from rich to poor agents. If the support of the
distribution of inheritances is such that the measure of agents with wealth greater than is
small and if the increase in mean inheritance is sufficiently small, then a second–order stochastic
increase in the distribution of inherited wealth, can cause the aggregate supply of entrepreneurs at
the wage to decline. If this is the case at each and every wage, the supply curve for labor must
shift inward.
With no capital in production and , the demand for labor from entrepreneurs with wealth
is proportional to the supply of entrepreneurs:
(20)
Hence, a second–order stochastic increase in the distribution of wealth reduces the aggregate de-
mand for labor at each wage. From Lemma 2, both the wage and aggregate income must decline in
equilibrium as a consequence of this ‘‘enterprise effect’.
As the wage falls, wealth starts to be redistributed away from poorer lineages and back towards
richer lineages. By a converse argument to that given above, this causes an increase in the supply
of entrepreneurs and an increase in their demand for labor, even if the inheritances fall slightly. By
Lemma 2, the equilibrium wage and aggregate income rise and the economy’s decline is reversed.
As wages rise, wealth is redistributed towards the centre and the process begins all over again.
When the production function exhibits decreasing returns to working capital, the wealth transfer
due to rising wages in the past represents a transfer from agents with low marginal products to those
withhighmarginalproducts. This‘productivityeffect’tendstoincreasethesupplyofentrepreneurs
21
and the demand for labor. However, so long as the enterprise effect dominates, which it will if
is large enough, the redistribution of wealth resulting from past wage increases can still cause a
decline in the equilibrium wage and reduce aggregate income. The economy continues to evolve in
this fashion so that the long run path exhibits recurrent ‘distributional cycles’ in economic activity.
These cycles are intimately related to the interaction between capital market imperfections, the
skewness of the distribution of entrepreneurial efficiency and the labor market equilibrium. As
is increased (left panel of Figure 6), so that relatively efficient entrepreneurs become increasingly
scarce, the economy moves from no cycles, to cycles of gradually increasing amplitude relative to
the mean. Eventually, efficient entrepreneurs become so scarce that the economy cannot escape
from the first development phase. Relaxing the borrowing constraints (i.e. raising , holding
constant as in the right panel of Figure 6) causes the cycles to decline and eventually disappear.
Thus, both the severity of capital market imperfections and the skewness of entrepreneurial ef-
ficiency offer a potential explanation for differences in the amplitude and persistence of long–run
cycles across economies. Moreover, these factors may help us to understand differences in the long
run variability of factor shares and income inequality across economies. For example, as Blan-
chard (1997) documents, factor shares have been more variable in European than in Anglo–Saxon
economies (the U.S., the U.K. and Canada); one potential explanation is that credit–market im-
perfections are more severe in Europe. Cross–country evidence on cyclical movements in income
inequality is mixed (Deninger and Squire, 1996). However, Beach (1976) finds that U.S. income
inequality tends to fall during booms and rise during recessions.
5. Concluding Remarks
In the model described here, the distribution of entrepreneurial efficiency is time invariant so the
22
only source of economic growth comes from the reduced importance of borrowing constraints as
agents accumulate wealth. If the distribution of start–up costs were to improve over time, this
would reinforce the results of Section 3. As Figure 6 suggests, in the long–run cyclical economy,
an improving distribution of skills would eventually eliminate the cycles.
Our analysiscouldbeextendedinseveral directions. The impacts oftheshocks ofthemid–1970s
varied considerably across economies according to their stage of development. Some economies
have yet to recover to their 1973 levels of per capita income and many have experienced large in-
creases in inequality. When such shocks are incorporated into our model, the time taken to recover
depends crucially on the interaction between the distribution of entrepreneurial efficiency and the
extent of credit market imperfections. Relatedly, the recent financial crisis in Asia can be repre-
sented by an increase in potential entrepreneurial moral hazard leading to a decline in the ratio of
lending to wealth. Our model can be used to trace out the consequences. Models such as that de-
scribed here can also be used to study and evaluate the impacts of macroeconomic policy. Townsend
(1997) argues that such models are crucial to the systematic formation of development policies.
23
Appendix
Lemma A1:
For constrained firms:
For unconstrained firms
Lemma A2:
LemmaA3: Let beimplicitlydefinedby . Total differentiation revealsthat
Also and
Proof of Lemma 1: Let denote the inheritance below which the marginal entrepreneur is
constrained on the extensive margin, so that Let denote the inheritance
level above which marginal entrepreneurs are unconstrained, so that . Then if
: , ;
: , ;
: .
Since for all ,itfollowsthat .
Proof of Lemma 2 : Aggregate income is given in (12). This can written as
(A1)
From Lemma A2, , so that this can be reduced to
(A2)
24
But and so
(A3)
However, and so,
(A4)
which is the area under the upper envelope created by the supply and demand curves.
Theorem 1 (Hadar and Russell, 1971): Let where ;
on an interval. If FSD (first–order stochastically dominates) then .
Theorem 2 (Hadar and Russell, 1971): Let where , for
some and .If SSD (second–order stochastically dominates) then
.
Lemma A4: Suppose that the increase in the urban population between time and is
sufficient to absorb all immigrant entrepreneurs,
(A5)
Then, if the distribution of wealth grows in the first–order stochastic sense, migration occurs in the
rural–to–urban direction only.
Proof: See Lloyd–Ellis and Bernhardt (1998).
Proof of Proposition 1:
Phase 1 (The Dual Economy): Suppose that for some , FSD and .The
25
resulting change in the distribution of income for any equals
(A6)
But , from Lemma A3. Hence, from Theorem 1, FSD . Analo-
gous results hold for thedistribution of final wealth and, hence, bequests: FSD .Since
FSD , the first part of Proposition 1 follows by induction.
Phase 2 (The Transition): Suppose that for some period , , and FSD
.Since and it must be that .
For incomes , decompose the change in the distribution of income as follows:
(A7)
Since and FSD , thefirst termmust be negative. Since
and from Lemma A3, the second term is positive. Hence, the sign of the
expression is, in general, ambiguous. However, differentiating with respect to ,for
the change in the slope of the distribution function is:
(A8)
From Lemma A3, and . Hence, using Theorem
1, must increase with on .If , this implies that the cumulative
distribution functions do not intersect. It follows that FSD . A similar decomposition
26
can be carried out for the distribution of inheritances:
(A9)
Since and it follows that increases with . Hence,
implies FSD .Since FSD , the proposition follows by induction.
Phase 3(Advanced Economic Development): Suppose that for some , SSD . Since,
from Lemma A4, , the change in the supply of entrepreneurs equals
(A10)
Since where the inequality is strict on a set of positive measure, Theorem 2 implies
that the supply of entrepreneurs increases for a given wage. In turn, the increase in the supply of
entrepreneurs implies that the supply of labor must fall. Since , the shift in the
demand schedule for labor equals
(A11)
where . By The-
orem 2, the demand for labor rises, so that and (Lemma 2). The change in
the slope of the income distribution function is given by (A8). Since , the second term
is positive. Since SSD and, from Lemma A3, , Theorem 2 implies
that the first term is also positive. Hence, increases with . Hence,
implies the c.d.f.’s intersect only once:
if(A12)
if
27
Since , it follows that SSD . An analogous argument establishes that
SSD .Since SSD , the proposition follows by induction.
Phase 4 (The Long Run): The wage increases monotonically and is bounded above by . Hence,
it must converge to some . In particular, for all there must exist a such that
(A13)
Since the l.h.s. decreases over time and the r.h.s. increases, if this inequality holds for it must
hold for all . Using the differentiability of in , we have:
(A14)
Using Lemma A4 this can be expressed as
(A15)
But and ,andso
(A16)
Hence, forall , Thesameanalysisholdsforthedistribution
of inherited wealth, so that the evolution of the distributionof inheritances converges to a stationary,
monotone Markov process, . Hopenhayn and Prescott (1992) detail conditions for this
class of Markov processes that ensure the limiting distribution is unique and invariant. Thus, given
any initial distribution at time , , the associated sequence converges:
(A17)
Proof of Proposition 2: and , are both expected values of increasing functions of
28
and, hence, by Theorem 1, grow monotonically during the dual economyphase. In the later phases,
when the wage rises, decompose the equilibrium change in the labor force into:
(A18)
The slope of the individual firm’s labor demand curve, , increases with (see Lemma A1). A
valueof thatissufficientlyclose to , (thus preservingtheconcavityof the productionfunction)
ensures that .
Proof of Proposition 3: The slope of labor supply is equal to minus the slope of entrepreneurial
supply: Using Lemma A2, the greater is , the greateris the slope
of the labor supply curve. From Lemma A1, as , . Ceteris paribus, if these
slopes are sufficiently large, then the increment in the equilibrium wage proportionately exceeds
the increment in per capita income.
Proof of Proposition 4: , and are all expected values of increasing functions
of and, hence, by Theorem 1, must grow monotonically during the dual economy phase.
Proof of Proposition 5: During the dual phase, the change in the distribution of firm sizes is
(A19)
where
(A20)
Since and , this expression is negative. Thus, by
Theorem 1, FSD implies that FSD . Since the lower support on firm sizes
29
is fixed at and a strictly positive measure of firms are of this size, the
distribution of firm sizes must exhibit increasing X–Dispersion throughout this phase.
During the later phases, the upper support on firm size is . Since the wage increases over
time, the upper support on firm size falls. The lower support on firm size is .
Both ,and rise, so must also rise over time. The c.d.f. for the distribution of firm sizes for
is given by
(A21)
which is linear in . Hence, the distributions and intersect only once on .
Proof of Proposition 6(a): Since migration is strictly positive and increases, lies above and
to the left of . The horizontal component of exceeds that of because increases and its
slope is lower since rises. The slope of exceeds the slope of ,if
(A22)
Since increases, the horizontal component of exceeds that of . Hence, if (A22) holds,
the fact that is steeper then implies that must lie below and to the left of .
Now consider the change in the convexity of the Lorenz curve segment :
(A23)
During the dual economy stage of development, if FSD then Now
(A24)
Now and so, since FSD ,thefirsttermispositive.Also,
30
since and ,
(A25)
Hence, the second term is positive, so that .Since ,
(A26)
It follows that if the slope at C is increasing, the Lorenz curves cannot intersect on this segment.
This is the case if
(A27)
If this holds throughout the dual phase, then is the period in which the profit of the richest en-
trepreneur ceases to grow faster than mean income. Hence, for all , the slope of the Lorenz
curve at must decline.
Proof of Proposition 6(b): Consider the percentile–income combination at which
and intersect: Since for all , it must be
true that for any , . Given that ,thisimpliesthat
(A28)
But this says that the slope of the Lorenz curve must be less at time than at time forall .
Since the Lorenz curves meet at ,itfollowsthat
(A29)
For we know that , so the above argument cannot hold. Consider, now,
the change in the density function for any . This can be decomposed into:
31
(A30)
As in Lemma A3, the first two terms must be positive and, as in Proposition 6(a), the third term
must also be positive. Hence, the Lorenz curve at must be less convex than at :
(A31)
Thus, the Lorenz curves intersect at most once. If the supply and demand curves for labor are so
wage elastic that , then the Lorenz curves cannot intersect at all.
32
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List of Figures for ‘Enterprise, Inequality and Economic Development’’ by Huw
Lloyd–Ellis and Dan Bernhardt (RES 4110)
Figure 1: Declining X–Dispersion
Figure 2: The Macrodynamics of Development
Figure 3: Evolution of Lorenz Curves
Figure 4: Relaxing the Collateral Requirement
Figure 5: Occupational Choice with a Skewed Distribution of Skills
Figure 6: Endogenous Distributional Cycles
1
1
0
w
w
1
2
(y)
(y)
1
2
2
1
y
Investment
0 5 10 15 20 25
Generation
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Fraction of Output
Start-Up
Capital
Working
Capital
Firm Size Distribution
0 5 10 15 20 25
Generation
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Consumption Units
Optimal Firm Size
Average Firm Size
Variance
Kuznets’ Curve
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Per Capita Income
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gini Coefficient
Wages and Per Capita Income
0 5 10 15 20 25
Generation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Consumption Units
Per Capita
Income
Wage
Occupations
0 5 10 15 20 25
Generation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Population
Farmers
Entrepreneurs
Wage Laborers
Income Shares
0 5 10 15 20
Generation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Value Added
Profits
Wage Income
0
1
p
p
0
1
1
1
A
A’
B
B’
C
C
B
B’
Z
Z
Dual Economy Advanced Economy
0 2 4 6 8 10 12 14 16
Generation
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Per Capita Income
0
0.05
0.1
0.15
b
(b,w)
x(w)
x(w)
m
m
0
25 30 35 40 45 50
Generation
1.25
1.3
1.35
1.4
1.45
1.5
Per Capita Income
Relaxing Collateral Requirements
0
.01
.02
.04
1.0
a
Increasing Skewness
15 20 25 30 35 40 45 50
Generation
1.2
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
1.38
1.4
Per capita income
m=0.8
m=.85
m=.95
a
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This paper analyzes the role of wealth distribution in macroeconomics through investment in human capital. It is shown that in the presence of credit markets' imperfections and indivisibilities in investment in human capital, the initial distribution of wealth affects aggregate output and investment both in the short and in the long run, as there are multiple steady states. This paper therefore provides an additional explanation for the persistent differences in per-capita output across countries. Furthermore, the paper shows that cross-country differences in macroeconomic adjustment to aggregate shocks can be attributed, among other factors, to differences in wealth and income distribution across countries.
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