Theoretical Economics Letters, 2011, 1, 3-7
doi:10.4236/tel.[phone] Published Online May 2011 (http://www.SciRP.org/journal/tel)
The Dynamics of Wealth Inequality under Endogenous
Fertility: A Remark on the Barro-Becker Model with
Heterogenous Endowments
Stefano Bosi1, Raouf Boucekkine2,3, Thomas Seegmuller4
1THEMA and Universit y of C e rgy- P o nt oi se , Cergy, France
2IRES and CORE, Catholic University of Louvain, Louvai n -La -Neuve, Belgium
3GREQAM, Aix-Marseille University, Marseille, France
4CNRS and GREQAM, Paris, France
Received April 13, 2011; revised April 24, 2011; accepted April 27, 2011
Abstract
Implicit in the seminal contribution of Barro-Becker [1], the lack of persistence of inequality in the pre-
sence of endogenous fertility is one of the most striking features of the models à la Barro-Becker. In this
pedagogical note, we show how to uncover and interpret the latter property using standard optimization in
contrast to the dynamic programming under homogeneity usually invoked in this literature.
Keywords: Endogenous Fertility, Heterogeneous Households, Optimal Growth
1. Introduction
Following the paper by Chatterjee [2], several contri-
butions hav e introduced households’ he terogeneity in the
Ramsey model through wealth inequality. The main
focus of this recent literature is the existence of a repre-
sentative consumer summarizing the average behavior of
the economy and the persistence of inequalities during
the transitional dynamics and in the steady state.
These questions have been extensively studied by
Chatterjee, Caselli and Ventura and Garcia- Peñalosa and
Turnovsky [2-4] in the standard setup where households
derive utility from consumption on ly. These papers show
that while wealth inequalities may be reduced, they do
persist in the long-run. In contrast, another much thinner
literature introducing endogenous fertility in line with
Barro and Becker [1,5] has shown that inequalities do
not persist. Implicit in these seminal papers, this striking
property has been demonstrated in a quite general model
using dynamic programming with homogenous functions
by Alvarez [6]. Precisely because it is a general approach
and because it does not rely on easily interpretable Euler
equations, the latter framework is not reader-friendly.
This note makes use of standard optimization and the
resulting Euler equations to provide a simple and intui-
tive reading of the lack of inequality persistence in the
Barro-Becker models. More concretely, we show that
when households smooth consumption over time and
optimally choose the number of children, individual con-
sumptions should be equal after one period. This striking
property essentially derives from the resulting equality
between the marginal benefit and cost of bequest: con-
sumptions are then shown to be independent of the ca-
pital distribution after one period. This in turn implies
that fertility rates and capital held by each household
become also identical after one period. Therefore, intro-
ducing endogenous fertility in the optimal growth model
rules out wealth inequalities after only one adjustment
period. Finally, it is important to no tice that we show our
result for a given sequence of prices over time. This
means that it does not depend on the specification of the
production sector, and occurs under exogenous as well as
endogenous growth.
This note is organized as follows. In the next section,
we present the behavior of heterogeneous households. In
Section 3, we show our result on the loss of hetero-
geneity after one period, while some technical details are
reported in the Appendix.
2. Households’ Behavior
We extend the Barro and Becker model [5] to account
for wealth heterogeneity. We consider an eco- nomy with
H
dynasties of altruistic households, =1,,iH, that

S. BOSI ET AL. 4
differ in their individual wealth. In other words, we focus
on heterogeneity in initial capital endowment 0i k, i.e.
00hj
kk for hj.
As in the seminal setting, each person is assumed to
live for two periods, childhood and adulthood, and has
children at the beginning of his adult period. Parents are
altruistic towards their children, i.e. utility depends on
their own consumption, the nu mber of surviving children
and the utility of each child. The utility of an adult of
type i belonging to the generation born at 1t

, is
given by:


1
=
itititit it
Uuc nnU




 (1)
where it c is the individual consumption giving an
instantaneous utility

it
uc , while it n is the number of
children. We notice that it n



measures the degree of
altruism towards each child, with

,0,1  , and
1it
U is the utility attained by each child. We further
assume:
Assumption 1 The utility function

i
uc is defined
on R, two-times continuously differentiable on R

,
strictly increasing (

>0
i
uc
), strictly concave
(

<0
i
uc
 ) and satisfies


2>1
ii
i
ucuc
uc




.1Noting
 
=
iiii
ccucuc 
, we assume

0< <1
i
c



and

i
c  is non-increasing.
Notice that the utility function

=
ii
uc c

, with
<1

 , satisfies As sum ption 1.2
As stressed by Becker and Barro [1], the recursive
model (1) can be equivalently written as an optimal
growth model where the household of type i maximizes a
dynastic utility:

1
=0
tit it
t
Nuc



 (2)
with it
N the size of the ith subpopulation at period t,
under a sequence of budget constraints:

1=1 itit itt ittit
cnkRkwn 

 (3)
=0,1,t, and given the unequal distribution of initial
capital 0i k, with 00hj kk for hj.
The left-hand side of (3) represents households’ ex-
penditures. In particular, 1it
k represents the bequest per
child, through physical as well as human capital used for
production in the next period. The right-hand side of (3)
represents the disposable income, where t
w is the wage
rate and 1
tt
Rr

  the gross return on capital, with

0,1   the depreciation rate of capital and t
r the
real interest rate. As mentioned in the introduction, we
do not specify the production sector in order to highlight
that our result holds irrespective of the production
technology. Accordingly, we assume that the sequence of
prices


,
tt
wR , =0,1,t, is given.
At adulthood, each household is endowed with one
unit of time that she shares between labor and leisure.
The time cost of rearing children is given by it t nw

,
where

is the constant cost per child in units of time.
Leisure time

=1 0,1
it it
nl   is spent with children,
whereas it l is the individual labor supply at period t.
Since it
n represents the population growth factor of
dynasty i, the size of a dynasty at time t is given by:
1
11 0
=0
==
t
itit itiis
s
NnNNn

  (4)
where 0>0
i
N are given for =1,,iH, and hete-
rogeneous initial population sizes are not excluded, i.e.
00hj
NN

for hj

.
Maximizing utility (2) under the budget constraints (3),
the household i chooses a sequence

1=0
,,
ititit t
knc


of saving, consumption, number of children. The house-
hold behavior m a y be summarized by:3


1
1=
t ittititit it
Rkwncnk 

 (5)




11
=
ititit t
uc ucnR





 (6)
 
11
1=1
1
=
ts
itittitisis
st
it
Nuc wkNuc
n


 




 (7)
with the transversality condition:


11=0
lim titititit
tNucnk






Equation (5) is the household's budget constraint. The
intertemporal trade-offs are summarized by (6) and (7).
The Euler Equation (6) shows how dynasties smooth
consumption over time. Since the marginal utility of
consumption depends on population size it
N and total
bequest 1it it
nk

, which in turn depend on the number of
children, this choice between current and future con-
sumption depends on the fertility rate. Finally, Equation
(7) determines the optimal number of children. The cost
per child in unit of time and in terms of bequest (on the
left-hand side) is equal to the discounted sum of the
marginal utility gains over all the subsequent periods (on
the right-hand side).
3. The “Loss” of Heterogeneity
Using the optimal behavior of the households, we show
now that from period 1 onwards, the individual consum-
ptions it
c, the number of children it n and the wealth
1As it is explicitly shown in Becker and Barro [7], this inequality en-
sures that the second order conditions are satisfied for the house- holds’
utility maximization.
2This utility function is considered in the seminal contribution of an
d
Barro and Becker [5]. 3See the Appendix for details.

S. BOSI ET AL. 5
it
k become equal. As underlined above, this property is
demonstrated without specifying the production sector,
the sequence of prices

,
tt
wR , for =0,1,t, is given.
As explained in the Appendix, Equation (7) can also
be written as:

 



111111
1
1
=
ititt ittit
itittit
ucu cRkwc
nu cwk

 



  
 (8)
Substituting Equation (6), we get:


1
111
1
1=,
for =0,1,, and =1,,
it
itt tt
it
c
cwRw
c
tiH
 




 

(9)
Under Assumption 1, there is a unique solution 1it c

solving this equation. In other words, for all =1,t
and =1, ,iH, we have it t
cc

, with



1
=1
t
tttt t
c
cwRw c


 (10)
From period =1t onwards, the individual consump-
tions are equal. One can see that t
c is increasing in the
time cost per child measured in physical units of the next
period, because the dynasty substitutes investment in
children for future consumption. However, less children
means a lower labor force, which explains that con-
sumption is decreasing in the current wage.
Most importantly for our purpose, individual consum-
ptions get equalized because it c does not depend on
capital holding it k and, therefore, it is independent of
the distribution of capital. Indeed, recall that 1it
k

represents the bequest per child given by a parent living
at period t. Equation (8), represents the intertemporal
trade-off to have children: it equalizes the marginal
benefit to have children (on the left-hand side) to its
marginal cost (on the right-hand side). In particular,

111itt it
uc Rk 

 represents the marginal benefit of
bequest, while

1itit it
nu ck


 is its marginal cost. Tak-
ing into account the optimal choice between current and
future consumption (see (6)), the marginal benefit of
bequest becomes equal to its marginal cost, implying
individual consumption to be independe nt of wealth.
Since consumptions are identical from =1t, the Eu-
ler Equation (6) implies that the fertility rates are also
identical from =1t, i.e. it t
nn

for all =1,t and
=1,,iH. Using (10), we obtain:

11
=
ttt t
nR ucuc




(11)
Because consumptions do not depend on the wealth
distribution and get equalized across dynasties, the
number of child is also equalized for =1,t.
A question now emerges. What can we infer for in-
dividual wealth? We show that capital distribution be-
comes also homogenous from period =1t.
Proposition 1 Under Assumption 1, for =1,t, the
distribution of capital is also homogeneous, i.e. =
it t
kk
for all =1, ,iH.
Proof. Using (5) and (7), we get:
 
1
=1
1
=st is
tit ittis
st
it it
N
Rkcwu c
uc N







 

 (12)
We have =
it t
cc and =
it t
nn for all =1,t, and
1
1=1
=s
is it
t
NN n

. Therefore, Equation (12) writes for
=1t:


1
1
1
111 1=2 =1
1
1
=s
s
its
st
Rkwcnuc
uc



 

 

 (13)
This shows that the capital distribution is homo-
geneous at =1t, i.e. 11i
kk

for all =1,,iH
. From
=1t, the budget constraint (5) becomes:

1
1
=ttt
t
it it
tt
wnc
R
kk
nn



 (14)
Since 111 ==
ij
kkk for all ,=1, ,ij H, we deduce
that =
itjt t
kkk

for all ,=1, ,ij H and =1,t
This proves that the capital distribution is equal from
=1t onwards.
This proposition shows that introducing endogenous
fertility in the optimal growth model with heterogeneous
households has critical implications for the capital dis-
tribution. In contrast to previous contributions with exo-
genous population size starting with Chatterjee [2], ineq-
uality does not persist and the wealth distribution beco-
mes homogenous from =1t. As already emphasized,
this arises from the disjunction between individual con-
sumptions and capital holdings, which is explained by
the fact that households do not only make an inter-
temporal choice between present and future consump-
tions, but also optimally choose the number of their
children.
We now come to some observations on the robu stness
of our findings.
Remark 1. In our analysis, we assume that the
counterpart of having children is a time cost per child
t
w

. Another traditional cost specification is to consider
a cost in terms of the final good, i.e. t w

 
 [1,5].
One may easily see that this alternative specification of
the cost per child does not change our conclusions on the
loss of heterogeneity from =1t.
Remark 2. As already underlined, our result is shown
for a given sequence of prices

,
tt
wR , =0, ,t This
means that it neither depends on the microeconomic

S. BOSI ET AL. 6
foundations of the production sector, nor on the
assumptions on the technology. As a direct implication,
if one introduces a standard convex technology with
constant returns, the economy may obviously converge
to a steady state with a homogenous wealth distribution.
If one introduces a technology leading to endogenous
growth [8], there may exist a unique balanced growth
path where all the individuals have the same wealth
growing at a similar well-defined rate.
4. References
[1] F. Alvarez, “Social Mobility: The Barro-Becker Children
Meet the Laitner-Loury Dynasties,” Review of Economic
Dynamics, Vol. 2, No. 1, 1999, pp. 65-103.
doi:10.1006/redy.1998.0052
[2] R. J. Barro and G. S. Becker, “Fertility Choice in a Model
of Economic Growth,” Econometrica, Vol. 57, No. 2, 1989,
pp. 481-501. doi:10.2307/1912563
[3] G. S. Becker and R. J. Barro, “A Reformulation of the
Economic Theory of Fertility,” Quarterly Journal of Eco-
nomics, Vol. 103, No. 1, 1988, pp. 1-25.
doi:10.2307/1882640
[4] G. S. Becker and R. J. Barro “A Reformulation of the
Economic Theory of Fertility,” The Quarterly Journal of
Economics, Vol. 103, No. 1, 1988, pp. 1-25.
doi:10.2307/1882640
[5] S. Bosi and T. Seegmuller, “Mortality Differential and
Growth: What Do We Learn from the Barro-Becker Mo-
del?” Mathematical Population Studies, Forthcoming , 2010 .
[6] F. Caselli and J. Ventura, “A Representative Consumer
Theory of Distribution,” American Economic Review,
Vol. 90, No. 4, 2000, pp. 909-926.
doi:10.1257/aer.90.4.909
[7] S. Chatterjee, “Transitional Dynamics and the Distribu-
tion of Wealth in a Neoclassical Growth Model,” Journal
of Public Economics, Vol. 54, No. 1, 1994, pp. 97-119.
doi:10.1016/0047-2727(94)90072-8
[8] C. Garcia-Peñalosa and S. J. Turnovsky, “The Dynamics
of Wealth Inequality in a Simple Ramsey Model: A Note
on the Role of Production Flexibility,” Macroeconomic
Dynamics, Vol. 13, No. 2, 2009, pp. 250-262.
doi:10.1017/S1365100508070508

S. BOSI ET AL. 7
Appendix
The optimal behavior of household
We derive the infinite-horizon Lagrangian function
with respect to it
c, it k, it n:


1
=0
1
=0 1
tit it
t
itt ittitititit
t
Nuc
Rkwncn k






 



(15)
in order to obtain the first-order conditions:


1
=t
itit it
Nuc


 (16)
11
=
it ititt
nR



(17)
 
1
1=1
1
=s
ittitisis
st
it
wk Nuc
n


 



 (18)
with the transversality condition:

11=0
lim titititit
tNucnk






Noticing that 1 =
it it it
nNN
, we get from (16) and
(17) a sequence of Euler Equations (6). Using (16) and
(18), we obtain (7). From (18), we also have:






1
11 1
11 =2
=1 1
it ittit
ts
ititis is
st
nwk
Nuc Nuc


 


 
 



Substituting
  
11112
=2
1=
sisisit ittit
st
Nucnw k

 





and using (16) again, we find:




 
11
11 1
1111 12
1=
ititit itittit
itit ittit
NucnNuc wk
Nuc nwk


 


 






Replacing 1 =
it it it
nNN
 and


112 1111
=
ittitt ittit
nwk Rkwc 

 

, we finally ob-
tain the trade-off:



 
 
11
11111
1=
ititittit
ittittit
ucnucwk
ucRkwc

 






