Journal of Development Economics 69 (2002) 205 – 226
www.elsevier.com/locate/econbase
A simple model of inequality, occupational choice,
and development
Maitreesh Ghataka,*, Neville Nien-Huei Jiangb
a Department of Economics, University of Chicago, Chicago, IL 60637, USA
b Department of Economics, Vanderbilt University, Nashville, TN 37235, USA
Received 1 February 2000; accepted 1 September 2001
Abstract
We analyze a simple and tractable model of occupational choice in the presence of credit market
imperfections. We examine the effect of parameters governing technology and transaction costs, and
history, in terms of the initial wealth distribution, in determining the long-term wealth distribution
and the level of per capita income of an economy.
D 2002 Elsevier Science B.V. All rights reserved.
JEL classification: D31; D82; O10
Keywords: Wealth inequality; Occupational choice; Poverty traps
1. Introduction
A well-known implication of neoclassical growth theory is that economies that have
similar preferences and technologies converge to the same steady state per capita income.1
In contrast, in development economics, we frequently encounter the idea of poverty traps:
poor individuals and economies tend to remain poor because they start poor. One specific
mechanism leading to the persistence of poverty that has recently received a lot of
attention operates through borrowing constraints.2 Because threats of punishment work
less well against the poor, they face greater borrowing constraints. This in turn prevents
* Corresponding author.
E-mail address: m-ghatak@uchicago.edu (M. Ghatak).
1
See Barro and Sala-i-Martin (1995).
2
See Galor and Zeira (1993), Banerjee and Newman (1993, 1994), Aghion and Bolton (1997), Piketty
(1997), and Mookherjee and Ray (2000).
0304-3878/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 7 8 ( 0 2 ) 0 0 0 5 9 - 7

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them from adopting efficient technologies or choosing profitable occupations, and hence
they remain poor. At the aggregate level, this implies that unlike in neoclassical growth
models, two economies that are identical in terms of all parameters may end up with
different levels of per capita incomes in the steady state if initially they have different
distributions of wealth and hence different sizes of the class of credit rationed. This
argument is often invoked to explain the evidence from cross-country analysis suggesting
that various measures of initial inequality are negatively correlated with growth.3
However, it turns out that the dynamic behavior of an economy in the presence of credit
market imperfections is fairly complicated, and even under strong simplifying assumptions
regarding technology, preferences and market structure, it is difficult to give clear-cut
answers to questions such as when do initial conditions matter, and if they do, what is the
relationship between initial inequality and the steady-state level of per capita income of an
economy. In this paper we try to answer these questions by analyzing a simple and
tractable dynamic model of occupational choice in the presence of credit market
imperfections.
Our paper is closely related to the important contributions of Galor and Zeira (1993)
and Banerjee and Newman (1993). They provide the following insight: in the presence of
credit market imperfections, the current distribution of wealth will determine the
proportion of credit-constrained individuals in the economy, which in turn may affect
equilibrium returns to various occupations in a way that affects the future wealth
distribution through intergenerational transfers. As a result, the transition of the wealth
distribution for the economy as a whole is nonlinear and hence the wealth distribution
dynamics is quite complex. In particular, it is difficult to say much except for multiple
stationary wealth distributions may exist, and that the initial distribution of wealth may
determine which steady-state equilibrium the economy converges to. Banerjee and
Newman (1993) offer some simple examples to show instances of hysteresis. However,
even in these examples, it is not always the case that the greater is the size of the poor
relative to that of the rich in the initial distribution, the lower will be the steady-state level
of income.
We consider a simplified version of the model of Banerjee and Newman (1993). In
particular, we have a simpler occupational structure. It turns out, as a result of this one needs
no more information about the wealth distribution than the proportion of people whose
wealth is below the level needed to start an enterprise. Even though general results in this
class of nonlinear dynamic models of wealth distribution are hard to obtain as demonstrated
by the Banerjee – Newman model, this simplification allows us to characterize precisely all
the steady-state equilibria corresponding to various configurations of parameters governing
technology, preferences and transactions costs. It also allows us to calculate the effect of
changes in parameters of interest and the initial distribution of wealth on steady state per
capita income. However, as a result of this simplification, we lose some of the richness of
the Banerjee – Newman model, which allows for alternative institutional forms associated
with the modern technology that differ in terms of agency costs.
3
See Benabou (1996) for a discussion of the empirical literature as well as other theoretical arguments
consistent with the observed negative relationship between inequality and growth such as those based on political
economy considerations.

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207
Some of our findings are as follows: first, whether hysteresis occurs depends on the size
of the threshold level of wealth needed to start an enterprise relative to the productivity of
the modern and the subsistence technologies. In particular, the larger is the productivity
difference between the modern and subsistence technologies, the greater is the likelihood
of multiple steady states. Second, for parameter values under which initial conditions
matter, the greater is the fraction of the population who are initially poor, the lower is the
steady-state income. Third, while some forms of technological progress can eliminate
poverty traps, all kinds of technological improvements do not necessarily increase steady-
state income. For example, an increase in the productivity of the small scale or subsistence
sector that pushes up wages can act as a drag on the growth of the modern sector.
The plan of the paper is as follows. In Section 2 we analyze the basic model. In Section
3 we extend the basic model, which is nonstochastic, by allowing the saving rate to be
subject to random shocks. In Section 4 we make some concluding remarks and Appendix
A contains some technical proofs.
2. The model
2.1. Demographics and preferences
Consider an economy inhabited by infinitely lived dynasties represented by successive
generations of agents who live for one period. The population is large and its size is
normalized to 1. There is no population growth. There are two goods in the economy,
labor, and some final output which can serve both as a consumption good and a capital
good. In period t a dynasty i is endowed with 1 unit of labor and an initial wealth ai,t. It
earns income by supplying labor and capital and the resulting income yi,t is divided at the
end of the period between consumption ci,t and savings, or bequest to the next generation,
bi,t. Therefore,
ai;tþ1 ¼ bi;t:
Following the literature, we assume that individuals have identical Cobb – Douglas
utility functions over consumption and bequests, with Ui(c
1Às
s
i,t, bi,t)=ci,t
bi,t, where sa(0, 1)
and the budget constraint is yi,t=ci,t+bi,t. This means that the current generation saves a
constant fraction s of its income and leaves it as bequest:
ai;tþ1 ¼ syi;t:
We also assume that all agents are risk-neutral.
In period t, wealth is distributed according to the probability measure kt(Á), and for
convenience, we define
GtðaÞuktððÀl; aÞÞ:
The function Gt is very similar to the distribution function except that it does not
include the measure at point a.

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2.2. Production technologies
There are two production technologies both of which are deterministic. One uses no
capital and one unit of labor to produce w units of output. This will be described as a
subsistence (or agricultural) technology. The other uses I>0 units of capital and two units
of labor (one unit of supervisory labor and one unit of ordinary labor) to produce q units of
output. One supervisor (or entrepreneur) can perfectly monitor one worker spending her
entire labor endowment. This will be described as an entrepreneurial (or industrial)
technology.4
Assumption 1. We assume that this technology is superior in the sense that the net output
of using this technology is greater than were two units of labor using the subsistence
technology. That is,
q À rI > 2w
where r (z1) is the exogenously given gross interest rate.5
2.3. Occupations
There are three possible occupations open to an individual who has inherited wealth ai,t:
(a) Subsistence: The agent earns some income by using her labor endowment to produce w
with the subsistence technology. She puts her inherited wealth in the bank, which
yields rai,t. Therefore, her income is
yS ¼ w þ ra
i;t
i;t :
(b) Worker: The agent works for an entrepreneur for wage income wt (which is determined
endogenously). She puts her inherited wealth in the bank, which yields rai,t. Therefore,
her income is
yW ¼ w
i;t
t þ rai;t :
(c) Entrepreneur: The agent invests an amount I to start a firm and hires one worker to
produce an output q with certainty. Her job is to monitor the worker. The agent’s
income as an entrepreneur is the output of the project less wage and capital costs:
yE ¼ q À w
i;t
t þ rðai;t À I Þ:
4
In contrast in the Banerjee and Newman (1993) model, apart from these two types of technologies, there is
a third one which involves some capital and one unit of labor (‘‘self-employment’’).
5
We can think of the credit market as an international market where the given economy is ‘small’.

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209
2.4. Credit and labor markets
The credit market is subject to transactions costs on the lending side due to imperfect
enforcement of loan contracts.6 This results in credit rationing of the following form: if an
individual’s wealth is below a certain minimum level, she would not get a loan no matter
how high the interest rate she offers. Following Banerjee and Newman (1993), a simple
way to generate this form of credit rationing is as follows: a borrower may default on her
loan (namely, r(IÀa)), but the cost of this action is that she gets caught with some
probability p and then has to pay a fixed nonmonetary cost of F due to imprisonment or
social sanctions. Thus, only those individuals get loans whose wealth satisfies the
incentive compatibility constraint (ICC)7:
ðq À wtÞ À rðI À ai;tÞzq À wt À pF
pF
or; ai;tzI À
:
ð1Þ
r
The lower is an individual’s wealth, the greater is her incentive to default because she
has to borrow a greater amount to start an enterprise, and the level of sanctions against
default is the same for all borrowers. Hence, only those who have a certain minimum
amount of wealth (namely, IÀpF/r) can borrow.8 Without loss of generality, we set p=0 so
that only those who have enough wealth to fully finance their own enterprises are able to
become entrepreneurs.
The wage rate at which entrepreneurs are indifferent between working as wage laborers
and hiring workers is given by:
q À ¯
w þ rðai;t À IÞ ¼ ¯w þ rai;t
q À rI
or; ¯
w ¼
:
2
By Assumption 1, w<w
¯ . Below we show that to ensure labor market equilibrium, the
wage rate w must lie in the interval [w,w
¯ ]. Hence, the occupation of entrepreneurship earns
no less than any other occupation for all wages (and strictly so for all w<w
¯ ). Given the
features of the credit market, only those individuals who own enough capital (azI) can
become entrepreneurs even though everybody else would like to do so. We are going to
6
We are assuming there are no imperfections on the deposit side of the credit market: there is a constant rate
of return of r irrespective of the amount deposited.
7
It is being assumed that even if a borrower gets caught trying to avoid repaying her debt, she gets to
consume her profits.
8
An implication of this form of credit rationing is that the threshold wealth level does not depend on the
wage rate. Otherwise, the threshold wealth level will change with the wage rate. This tends to complicate the
dynamics somewhat, but the basic results are not affected.

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refer to those individuals whose wealth is less than I as capital-constrained, or simply,
poor, and the rest as unconstrained, or rich.
The ICC tells us what fraction of the population is capital-constrained, namely, Gt(I).
Notice that this follows from our assumption that all entrepreneurs are self-financed and
the credit market does not operate as p=0. Otherwise, the relevant fraction of the
population that is capital-constrained would be Gt(IÀpF/r).
For wt<w, labor supply is zero, but for wt=w labor supply jumps to Gt(I) and as wt goes
above w, the supply of labor grows until the wage rate is high enough, namely, w
¯ , such that
entrepreneurs are indifferent between working as wage laborers and hiring workers. Now
we are ready to write down the supply curve of labor:
0 if wt < w
½0; GtðIÞŠ if wt ¼ w
GtðIÞ if wtaðw; ¯wÞ
½GtðIÞ; 1Š if wt ¼ ¯w
1 if wt > ¯w:
Conversely, to derive the demand curve for labor, we notice that for wt>w¯ there is no
demand for labor; as wt falls to w¯, the demand for labor jumps to any value between 0 and
1ÀGt(I). When wt<w¯, the demand for labor is at a maximum, 1ÀGt(I) and continues to
remain so. Therefore, the demand for labor is:
0 if wt > ¯w
½0; 1 À GtðIÞŠ if wt ¼ ¯w
1 À GtðIÞ if wt < ¯w:
From the labor demand and supply schedules we can easily find the equilibrium wage
rate in period t:
1
¯
w if GtðIÞ < 2
1
wt* ¼ ½w; ¯wŠ if GtðIÞ ¼ 2
1
w if GtðIÞ >
:
2
Since each entrepreneur hires exactly one worker, if there are more people who are
capital-constrained (unconstrained), then the competition for entrepreneurs (workers)

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211
among them will drive the equilibrium wage rate down (up) to its lower (upper) bound.
When Gt(I)=1/2, the equilibrium wage rate is indeterminate, and throughout this paper, we
are going to assume that the wage rate is equal to w
¯ in this case.
Notice that on one hand, the equilibrium wage rate depends on the current wealth
distribution but on the other hand, it also influences next period’s wealth distribution
through the savings behavior of currently active agents.
2.5. Dynamics of individual wealth
Consider the factors governing dynasty i’s bequest. First of all, the initial wealth level
of an agent determines her capital income and her occupational choice. Secondly, the
current wage rate is determined by the economy-wide wealth distribution. With the
knowledge of an individual’s occupational choice and that the wage rate can take only two
values (w and w
¯ ), we can write down the difference equations describing the evolution of a
dynasty i’s wealth as:
ai;tþ1ðai;t j wt ¼ wÞ ¼ s½rai;t þ wŠ
if ai;t < I
¼ s½rðai;t À IÞ þ q À wŠ if ai;t z I
ai;tþ1ðai;t j wt ¼ ¯wÞ ¼ s½rai;t þ ¯wŠ
bai;t:
Fig. 1 shows what these difference equations look like. Notice that there are two regimes
of wealth transitions corresponding to the two wage levels. When the wage rate is low, an
agent who is capital-constrained can only choose between being a worker and engaging in
subsistence and in either case, her labor income is w. A fraction s of the sum of her labor
income and her capital income rai,t is left for her next generation. An agent who is not
credit-constrained will strictly prefer to be an entrepreneur and her total income will be
r(ai,tÀI)+qÀw. When the wage rate is high, nobody will engage in subsistence and all
agents will be indifferent between being entrepreneurs and workers.
Fig. 1. Dynasty i’s wealth transitions under different wage regimes.

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Assumption 2. We assume that it is not possible for a dynasty to get arbitrarily rich over time
merely by saving a constant fraction of its income every period and earning interest on it:
sr < 1
Assumptions 1 and 2 will be retained throughout this section.
2.6. Stationary wealth distributions and wages
In this section, we examine the long-run behavior of this economy. If the difference
equations governing the wealth transitions are stable, it would be easy to prove the
existence of a stationary wealth distribution. However, the fact that these difference
equations depend on the wage levels raises the possibility that the process may not be
stable. In particular, the concern here is that the wage rate may change infinitely often. The
following lemma rules out this possibility.
Lemma 1. The wage rate can change at most once.
Proof: Notice that the difference equations are order-preserving. That is, ai,t+1>aj,t+1 if and
only if ai,t>aj,t. Therefore, in order to study the wage dynamics, we can only look at
the wealth dynamics of the dynasty which has the median wealth. Define am
t umax{a:
G
m
t(a)V1/2}. Note that at
is well defined because G(Á) is continuous from below according
to our definition. Then am
which implies w
< I
t zI ZGt ðI ÞV 1
ZG
2
t=w
¯ . Similarly, am
t
t ðI Þ
> 1 which implies w
m<I and a m
2
t=w. Now if wt=w and wt+1=w
¯ , then we must have at
t +1z I.
This implies sðram þ wÞ
t
zI Z swzð1 À srÞI Z sðra þ wÞzI for all azI and we{w,w
¯ }. That
is, once the high-wage rate is reached, there will not be any downward mobility and hence
the high wage will prevail forever. If w
m
t=w
¯ and wt+1=w, then we must have at zI and
a m
t +1< I. This implies sðram þ ¯
wÞ < IZ s ¯
w < ð1 À srÞIZ sðra þ wÞ < I for all a<I and
t
we{w,w
¯ }. That is, there will not be any upward mobility and once the low-wage rate is
reached, it will prevail forever. Therefore, we can conclude that starting with any initial
distribution of wealth, the wage rate can change at most once.
5
Lemma 1 shows that the wage rate is constant in the long run and rules out the
possibility of cycles or chaotic wage dynamics. Once the wage rate switches from low to
high, there will be no downward mobility and so the high wage prevails forever and
similarly, once the wage rate switches from high to low, there will be no upward mobility
and the low-wage prevails forever. As a result, although we have two regimes of the
wealth transition process, there will not be infinite switches from one to the other. Only
one of them will prevail in the long run. However, for the same parameter values, both
wealth transition processes could be candidates for the long-run equilibrium and which
one is arrived at could depend on initial conditions. Together with Assumption 2, which
implies there exists a stationary point for each difference equation, we immediately have:
Proposition 1: Given any initial wealth distribution, there exists a unique stationary
wealth distribution to which it converges.
By Lemma 1 in the long run the wage rate is constant and corresponding to this wage
rate, one of the two possible wealth transition processes will prevail. The difference
equations associated with these processes have unique stationary points and so the wealth

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213
distribution of the economy will converge to a stationary distribution. This stationary
wealth distribution will have all mass concentrated on one point (for the high-wage
equilibrium) or two points (for the low-wage equilibrium) which is a consequence of the
model being nonstochastic. Notice that the Lemma 1 and Proposition 1 do not suggest that
given the parameters of the model there is a unique long-run wage rate, and a
corresponding long-run stationary wealth distribution. Indeed, one of our main goals is
to characterize parameter conditions under which multiple long-run equilibria could exist
and to show which equilibrium the economy converges to depends on initial conditions.
What these results do is to rule out cycles or chaotic behavior. Now we proceed to
characterize how the long-run equilibrium of the economy depends on various parameters
and the initial wealth distribution.
Let aJ(w) be the stationary point of the difference equation describing the wealth
transition of a dynasty engaged in occupation J (where J=S,W,E denotes the three
occupations: subsistence, worker, and entrepreneur) when the wage rate is w. Then we have
sw
aSðwÞ ¼
for all w:
1 À sr
sw
aWðwÞ ¼ 1 À sr
sðq À rI À wÞ
aEðwÞ ¼
1 À sr
sðq À rI Þ
aWð ¯
wÞ ¼ aEð ¯
wÞ ¼
:
2ð1 À srÞ
By Assumption 1, aE(w)>aE(w
¯ )=aW(w
¯ )>aW(w).
Comparing the values of these threshold levels of wealth with I, we can completely
characterize the long-run outcome (in terms of the stationary distribution of wealth, the
equilibrium wage rate and the level of net output) of the economy.
Proposition 2: The initial distribution of wealth matters in determining the stationary
distribution of wealth and the long run equilibrium wage rate if and only if
sw
sðq À wÞzI >
:
1 À sr
Otherwise the economy converges to a high-wage equilibrium (if IVsw/(1Àsr)) or a
subsistence equilibrium (if I>s(qÀw)) irrespective of initial conditions.
Proof: The proof consists of the following two steps
Step 1. The following four cases characterize the steady-state equilibrium of the economy
corresponding to various parameter values:
Case 1. I > sðq À wÞZI > aEðwÞ. This is a situation where the steady-state wealth of
the entrepreneurial class cannot finance the operation of the industrial technology even

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when wages are as low as possible. The only equilibrium in this economy is therefore one
where everyone is engaged in subsistence production irrespective of the initial wealth
distribution G0. As a result the stationary wealth distribution displays no inequality.
Case 2. s( qÀw)zI>sq/(2Àsr)ZaE(w)zI>aE(w¯)=aW(w¯)>aW(w). The condition that
aE(w)zI implies s[r(aÀI)+qÀw]zI bazI. It says when the wage rate is low, offspring
of individuals who are able to start an enterprise in the current period will also be able to
do so in the next period, i.e., there is no downward mobility. Similarly, I>aW(w) implies
s(ra+w)<I ba<I, which means there is no upward mobility when the wage is low. If the
economy starts out with the low-wage rate ( G0(I)>1/2), there will not be any mobility in
either direction. This implies that the wage rate will always be equal to w; the wealth of
those dynasties that are initially capital-constrained will converge to aW(w); the wealth of
those that are not will converge to aE(w); and there will be 1ÀG0(I) firms operating in each
period. Now suppose the economy starts out with the high-wage rate ( G0(I)V1/2). The
condition, I>aE(w
¯ )=aW(w
¯ ), implies w
¯ is not sustainable. There exists a finite s such that
ws=w¯ and ws+1=w. Thereafter the story is the same as above if we take Gs+1(Á) as the initial
wealth distribution in the new low-wage regime. And of course, Gs+1 depends on G0.
Case 3. sq/(2Àsr)zI > sw/(1Àsr)ZaE(w) > aE(w¯)=aW(w¯)zI > aW(w). Again, since
aE(w)>I>aW(w), there is no upward or downward mobility when wage rate is low.
Therefore, if the economy starts out at low-wage rate ( G0(I)>1/2), the story is the same
as in Case 2. However, the condition, aE(w
¯ )=aW(w
¯ )zI, implies s(ra+(( qÀrI)/2))zI bazI.
Hence, when the wage rate is high, people who are not capital-constrained will remain
unconstrained, i.e., there is no downward mobility. Therefore, if the economy starts out
with G0(I)V1/2, the high wage w¯ will last forever. As a result, every dynasty’s wealth will
converge to aE(w
¯ ).
Case 4. sw/(1Àsr)zIZaW(w)zI. The high-wage equilibrium will result irrespective of
G0 because even when wages are low, the steady-state wealth level of the working class
permits them to start a firm. As a result, the unique stationary wealth distribution displays
no inequality.
Step 2. Next we show that the sets of parameter values that correspond to the four cases
analyzed above are mutually exclusive and exhaustive with respect to the set of all
admissible parameter values (i.e., those satisfying Assumptions 1 and 2).
Suppose sw/(1Àsr)VI. This inequality implies [(2Àsr)/(1Àsr)]wV2w+Ir. As a result,
Assumption 1, which guarantees q>2w+Ir, also implies q>[(2Àsr)/(1Àsr)]w, i.e., q/
(2Àsr)>w/(1Àsr). The last inequality in turn implies, upon rearranging, s( qÀw)>sq/
(2Àsr) and sq/(2Àsr)>sw/(1Àsr). Thus, we have the following inequality which is derived
from Assumptions 1 and 2:
sq
sw
sðq À wÞ >
>
2 À sr
1 À sr
which holds so long as sw/(1Àsr)VI. If instead, I<sw/(1Àsr), then Case 4 always applies.
That is, the only possible equilibrium is the high-wage equilibrium.
5
Fig. 2 summarizes the four cases. Proposition 2 has several interesting economic
implications which we discuss below.

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Document Outline

• Introduction
• The model
  • Demographics and preferences
  • Production technologies
  • Occupations
  • Credit and labor markets
  • Dynamics of individual wealth
  • Stationary wealth distributions and wages
• Extension: stochastic model with mobility
  • Constant-wage dynamics
  • Cycles
• Conclusion
• Acknowledgements
• References

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