This paper investigates the income inequality generated by a job-search process when di¤erent cohorts of homogeneous workers are allowed to have di¤erent degrees of impatience. Using the fact the average wage under the invariant Markovian distribution is a decreasing function of the time preference (Cysne, 2004), I show that the Lorenz curve and the between-cohort Gini coe¢ cient of income inequality can be easily derived in this case. An example with arbitrary measures regarding the wage o¤ers and the distribution of time preferences among cohorts provides some quantitative insights into how much income inequality can be generated, and into how it varies as a function of the probability of unemployment and of the probability that the worker does not find a job offer each period.

Rubens P. Cysne

English

Research Papers in Economics

Income Inequality in a Job-Search Model
With Heterogeneous Time Preferences￿
Rubens Penha Cysney
November 6, 2004
Abstract
This paper investigates the income inequality generated by a job-
search process when di⁄erent cohorts of homogeneous workers are al-
lowed to have di⁄erent degrees of impatience. Using the fact the av-
erage wage under the invariant Markovian distribution is a decreasing
function of the time preference (Cysne (2004)), I show that the Lorenz
curve and the between-cohort Gini coe¢ cient of income inequality can
be easily derived in this case. An example with arbitrary measures re-
garding the wage o⁄ers and the distribution of time preferences among
cohorts provides some quantitative insights into how much income in-
equality can be generated, and into how it varies as a function of the
probability of unemployment and of the probability that the worker
does not ￿nd a job o⁄er each period.
1 Introduction
A condition for a stochastic dynamic model to be of some use in the under-
standing of income distribution is that it delivers an endogenous distribution
of incomes. Within this class of models, job-search models are certainly
among the simplest.
￿JEL Primary Field: D31.
yEPGE/FGV and, in 2004, a Visiting Scholar at the Department of Economics of the
University of Chicago. E mail: rpcysne@uchicago.edu
1Pissarides (1974) uses a job-search model to compare the income distri-
bution1 of two cohorts2 with di⁄erent degrees of risk aversion. This author
argues that cohorts with higher risk aversion will have a lower reservation
wage and, because the wages of employed workers will have a greater range,
the income distribution will be worse.
Pissarides does not explicit derive the stationary distribution generated by
the interactions of the transition functions implicit in his model. Moreover,
his work concentrates only on comparing (between two cohorts) within-cohort
inequalities, not formally deriving a measure of inequality among cohorts.
In this work I complement Pissaride￿ s analysis by pursuing answers to
three di⁄erent questions. First, instead of dealing with heterogenous degrees
of risk aversion, I allow consumers in di⁄erent cohorts to have di⁄erent time-
preference parameters. Second, in each cohort I concentrate the analysis
on the invariant Markovian distribution of wages, rather than on the cross-
sectional distribution. Third, I analyze inequality among cohorts, rather
than within cohorts.
The inequality among cohorts is generated by the fact that more patient
workers have higher reservation wages and higher average wages as well (see
Cysne (2004) for a demonstration of this fact).
The paper proceeds as follows. Section 2 is used for the presentation
of the general model, fully characterizing a the job-search problem within a
cohort. Section 3 is dedicated to the derivation of the invariant-distribution
between-cohort Gini coe¢ cient of income inequality. Section 4 is used to
make quantitative assessments of the model, based on an example with arbi-
trary measures for the wage o⁄ers and for the distribution of time-preferences
among cohorts . Section 5 concludes.
2 The Model
The basic model presented here is the same as the one presented in Cysne
(2004), which in turn draws on Stokey and Lucas￿ s (1989) version of McCall￿ s
(1970) model. The givens of the model are the distribution of wage o⁄ers
(the same for all workers and along all cohorts) obtained by the workers, the
distribution of the time-preference parameter among cohorts of workers, the
1More rigorously, we are dealing here with wage inequality. However, transfers and
capital income usually represent only a small fraction of most households￿total income.
For the United States, for instance, following the 1992 SCF (Survey of Consumer Finances),
transfers and capital income account in average for only around 28% of the total income of
the households surveyed. This percentage tends to be even lower in developing countries.
2In this work, a cohort is de￿ned as a group of homogenous workers/consumers.
2probability of layo⁄s and the probability that, each period, the worker does
not get any job o⁄er next period.
In the measurable space
￿
[0;1);B[0;1);L
￿
; B[0;1); standing for the borelians
in [0;1) and L for the Lebesgue measure, consider a continuum of cohorts, in-
dexed by j; each cohort with a large number of homogenous workers: Cohorts
di⁄er only in terms of their time-preference parameter. The distribution of
time preferences ￿j 2 [0;1) along these cohorts is determined by:
￿j = H
￿1(j) , H
0(:) > 0 (1)
In equation (1), j has a uniform distribution in [0;1) and H stands for
a cumulative probability distribution of a random variable taking values
in [0;1):The isomorphism (1) allows us to put di⁄erent probability mea-
sures m in the space where the time-preference parameters take value (also
([0;1);B[0;1))). For instance, if H is the cumulative distribution function of a
Beta (s;v) random variable, then the ￿
0
js will be distributed as a Beta (s;v).
Note that having H0(:) > 0 allows us to identify each cohort j with its time
preference parameter ￿j:
For 0 < D < 1; consider also the second measurable space
￿
￿;B[0;D];p
￿
and, in this space, the measure q induced by the wage function w: ￿ !
[0;D]: In the induced space
￿
[0;D];B[0;D];q
￿
; denote by F(t) the distribution
function that (q￿a:e: -uniquely) determines the measure q : F(t) = p(w ￿ t):
By assumption, in each cohort, there are two states regarding the con-
sumer￿ s optimization problem: w and 0: State \w" corresponds to a job o⁄er
of w at hand, and state \0" to no job o⁄er. In state w the worker can accept
or turn down the o⁄er. If he accepts it, by assumption he stays employed
with that wage till he is laid o⁄ (which happens with probability ￿): If he
does not accept the o⁄er or if he gets no o⁄er, he remains in state 0: Being in
state zero the only thing he can do is wait again for a job o⁄er next period,
which happens with probability (1 ￿ ￿). The individual is not allowed to
search while in his job. The job o⁄ers are drawn from [0;D] according to the
measure q: q is known by all workers in all cohorts. Workers are not allowed
to borrow or to lend. Their consumption ct is equal to their income wt in
each period. Consumers in cohort j maximize the expected present value of
their consumption E
￿P1
t=0 ￿
t
jct
￿
: From now one I will only use the index j
when necessary.
With v(w) stating for the value function, ￿ w; the reservation wage is de-
termined by (Cysne (2004)):
￿ w =
￿(1 ￿ ￿)
1 ￿ ￿(1 ￿ ￿)
Z
[ ￿ w;D]
(w
0 ￿ ￿ w)dF(w
0) (2)
3Following the analysis in Stokey and Lucas (1989, c. 10), the reservation wage
￿ w(j) divides [0;D] into two regions: the acceptance region A(￿) = [￿ w(￿);D]
and the non-acceptance region Ac(￿) = [0; ￿ w(￿)]:
Consider a new measure ￿t in ([0;D],B[0;D]); representing the state of the
state o⁄er of workers of a certain cohort j (the j is omitted), in period t:
This measure converges strongly to an invariant measure (Cysne (2004)). For
C ￿ A this invariant measure reads:
￿(C) =
(1 ￿ ￿)q(C)
￿ + (1 ￿ ￿)q(A)
in which case the average wage in each cohort j is:
wA(j) =
Z
[ ￿ w(j);D]
(1 ￿ ￿)wdF(w)
￿ + (1 ￿ ￿)q(A(j))
(3)
where ￿ w(j) follows (2). As shown in Cysne (2004), wA is an increasing
function of the time preference parameter and, given (1), of j as well.
3 Income Inequality
The existence of di⁄erent time preferences between cohorts leads to a between-
cohort income inequality. The Proposition below provides an expression for
the Gini coe¢ cient as a function of the exogenous measure to be put into
the time preference parameters.
Proposition 1 The between-cohort Gini coe¢ cient of income inequality as-
sociated with the problem outlined above is given by:
G = 1 ￿ 2
Z 1
0
dm(￿j)
Z ￿j
0
wa(u)dm(u) (4)
Proof. The Lorenz curve L(j) plots the percentage of total income earned
by the economic agents of a certain cohort, when these cohorts are ordered
from those with lower average income to those with higher average income.
The Gini coe¢ cient (G) of income inequality is then, by de￿nition, given as
a function of the Lorenz curve, by
G = 1 ￿ 2
Z
[0;1)
L(j)dj: (5)
A crucial point concerning the calculation of the Gini coe¢ cient in this
model, is that, as shown by Cysne (2004), there is an isomorphism linking
4the average wage wA and the time-preference parameter ￿ (because w0
A(￿) >
0): Since (1) also de￿nes an isomorphism, between j and ￿j, there is (by
composition) an isomorphism between the measure of the population j and
the average wage wA(￿j(j)); with w0
A(j) > 0: This fact implies that, by
ordering the population by ￿j we are, automatically, also ordering it by
income, as required by the construction of the Lorenz curve.
Keeping (1) in mind, the measure of cohorts with time preference equal or
less than a certain constant a 2 [0;1) is given by: P(￿j ￿ a) = P(H￿1(j) ￿
a) = P(j ￿ H(a)) = H(a): This is also equivalent to the measure of people
with income less or equal than wA(￿j): The proportion of income earned by
the j% poorer workers of the cohort (the Lorenz curve), therefore, can be
written as 1
￿ wA
R ￿j
0 wA(￿j)dm(￿j); where ￿ wA =
R 1
0 wA(￿j)dm(￿j): (4) follows
trivially from (5).
4 Quantitative Assessments
This is as far as one can go without specifying the measures q (of wage o⁄ers)
and m (of the distribution of time preferences among cohorts ). In order to
get some idea of the numbers generated by the analysis developed here, from
now on I will assume that measure q is the Lebesgue measure in [0;1] (this
measure allows for closed-form solutions to the reservation wage, the average
wage, and the Gini coe¢ cient), and that m has a density with respect to the
Lebesgue measure given by the density of the Beta (114:5;1:01) distribution.
The parameters of this distribution have been chosen in order to make the
(monthly) average equal to 114:5=(114:5+1:01) = 0:9913: This corresponds
to an yearly average time-preference of 0:900: Some numerical results are
presented in the example below.
Example 1 This example assumes that in each period the workers face a
probability ￿ of layo⁄, a probability ￿ of not ￿nding a wage o⁄er and measures
q and m speci￿ed as above: Using (2), (3) and (4), the Gini coe¢ cient reads:
G = 1 ￿ 2
Z 1
0
dm(￿j)
R ￿j
0
(1￿￿)(1￿
h
1￿u(￿￿￿)
u(1￿￿) ￿
q
(
1￿u(￿￿￿)
u(1￿￿) )2￿1
i2
)
2
h
￿+(1￿￿)(1￿
1￿u(￿￿￿)
u(1￿￿) +
q
(
1￿u(￿￿￿)
u(1￿￿) )2￿1)
idm(u)
R 1
0
(1￿￿)(1￿
h
1￿u(￿￿￿)
u(1￿￿) ￿
q
(
1￿u(￿￿￿)
u(1￿￿) )2￿1
i2
)
2
h
￿+(1￿￿)(1￿
1￿u(￿￿￿)
u(1￿￿) +
q
(
1￿u(￿￿￿)
u(1￿￿) )2￿1)
idm(u)
The values of both the Gini coe¢ cients, for di⁄erent values of theta and
alpha, are shown in Table 1 below:
5Table 1
Gini Coe¢ cient
Theta
0 0:1
Alpha 0 0:0253 0:0094
0:2 0:0270 0:0094
Note in Table 1 that the between-cohort inequality decreases when the proba-
bility of layo⁄ increases. An increase in ￿ has two e⁄ects. First, it decreases
the reservation wages of all cohorts, thereby making it more likely that low
wage o⁄ers are taken by workers in some cohorts. Second, having workers
unemployed more frequently, the average wages in all cohorts decrease, im-
poverishing all cohorts at the same time, and decreasing inequality.
The sensibility of the income distribution to the probability of not ￿nding
a job o⁄er, though, in the range of values of theta and alpha in which we
derived the results of Table 1, was of a di⁄erent sign and very small. The
basic di⁄erence is that even though an increase in alpha, as well as an increase
in theta, impoverishes all cohorts, it has a di⁄erent e⁄ect on the mass of
workers in each cohort with a wage equal to zero.
A last comment with respect to the results in Table 1 is that, even though
the heterogeneity of the time-preference parameters leads to income inequality
among cohorts, the quantitative impact of this e⁄ect is very small.
5 Conclusions
In this paper I have investigated income inequality among cohorts of ho-
mogenous workers, when each cohort is characterized by a di⁄erent degree
of impatience. The work can be seen as being complementary to Pissarides￿ s
(1974), who used a job-search model to study within-cohort inequalities.
Using the result obtained by Cysne (2004), that the average wage in each
cohort based on the invariant Markov measure is a decreasing function of im-
patience, I have shown how to derive the Lorenz curve and the between-cohort
Gini coe¢ cient of income inequality. Next, I have provided an example, based
on arbitrary measures concerning the job o⁄ers and the distribution of time
preferences among the di⁄erent cohorts, in order to provide a quantitative
assessment of the problem. The numbers obtained indicate that the degree of
inequality is decreasing in the probability of unemployment and increasing in
the probability that workers in each cohort do not ￿nd job o⁄ers next period,
when unemployed. The probability of unemployment has shown to have a
greater impact over the generation of income inequality than the probabil-
6ity of ￿nding job o⁄ers. Finally, the inequality generated by the job-search
process, based solely on heterogeneous degrees of impatience, has been shown
to be very small, with Gini coe¢ cients ranging from 0.94% to 2.70%.
7References
[1] Cysne, Rubens Penha (2004): "A Search-Theoretic Explanation For the
Negative Correlation Between Labor Income and Impatience". Working
Paper, EPGE/FGV and Department of Economics, The University of
Chicago.
[2] McCall, John, J.: (1970), ￿Economics of Information and Job Search".
Quarterly Journal of Economics, 84: 113-126.
[3] Pissarides, C. (1974): ￿ Risk, Job Search, and Income Distribution". The
Journal of Political Economy, Vol. 82, No. 6. (Nov. - Dec., 1974), pp.
1255-1267.
[4] Stokey, N. L., Lucas Jr., Robert, and Edward C. Prescott (Contributor):
(1989) ￿Recursive Methods in Economic Dynamics￿ . Harvard University
Press.
8

INCOME INEQUALITY IN A JOB-SEARCH MODEL WITH HETEROGENEOUS TIME PREFERENCES?

http://www.anpec.org.br/encontro2004/artigos/A04A051.pdf

INCOME INEQUALITY IN A JOB-SEARCH MODEL WITH HETEROGENEOUS TIME PREFERENCES?

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