This paper develops a theory in which households prepare for future education by adjusting the number of children they intend to raise. Income inequality lowers output per worker only if the inequality is attributed in some part to unexpected disturbances after childbirth.

Masao Nakagawa

Yoshiaki Sugimoto

English

Research Papers in Economics

Discussion Paper No. 758 
 
 
 
 
 
 
 
 
 
INCOME DISTRIBUTION AND 
MACROECONOMICS REVISITED: 
THE ROLE OF FERTILITY ADJUSTMENT 
 
 
Masao Nakagawa 
Yoshiaki Sugimoto 
 
 
 
 
 
 
 
 
 
 
 
 
November 2009 
 
 
The Institute of Social and Economic Research 
Osaka University 
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan 
 Income Distribution and Macroeconomics Revisited:
The Role of Fertility Adjustment￿
Masao Nakagawaa and Yoshiaki Sugimotob,y
aInstitute of Social and Economic Research, Osaka University, Japan
bFaculty of Economics, Kagawa University, Japan
November 7, 2009
Abstract
This paper develops a theory in which households prepare for future education by adjusting
the number of children they intend to raise. Income inequality lowers output per worker only if
the inequality is attributed in some part to unexpected disturbances after childbirth.
Keywords: Fertility; Lock-in e⁄ect; Inequality; Education.
JEL Classi￿cation: D10; J13; J24; O15.
1 Introduction
This paper reconsiders the signi￿cance of income distribution in macroeconomics, a topic that has
attracted the attention of economists since the 1990s. They have explored various mechanisms
to explain the negative aspects of inequality. In particular, Galor and Zeira (1993) theoretically
argued that, in developed economies with imperfect capital markets, inequality retards human
capital accumulation and economic growth.1
While the subsequent studies in this line of research incorporate the fertility decisions of house-
holds (cf. Kremer and Chen, 1999; de la Croix and Doepke, 2003), one important aspect of child
rearing has been omitted. As formulated by Becker and Lewis (1973), parents face a quantity-
quality trade-o⁄ of their children. This implies the possibility that, in developed economies, where
￿The authors are grateful to seminar participants at Kagawa and Keio Universities for their helpful comments.
yCorresponding author. 2-1 Saiwai-cho, Takamatsu 760-8523, Japan. Tel.: +81 87 832 1836; fax: +81 87 832
1820. E-mail address: sugimoto@ec.kagawa-u.ac.jp (Y. Sugimoto).
1Other theories using the capital-market imperfection approach include those of Banerjee and Newman (1993),
Aghion and Bolton (1997), Moav (2002), Mookherjee and Ray (2003), and Galor and Moav (2004).
1skill acquisition is advantageous, households in various income classes adjust the number of children
they raise in order to ensure that the children can attain certain education levels. This fertility
adjustment moderates the link between income inequality and skill di⁄erentials.
Based on this motivation, the present research develops a simple theory that allows for unex-
pected, idiosyncratic income shocks between childbirth and education decisions. Since childbirth is
an irreversible investment in the quantity of children (cf. Fraser, 2001; Doepke and Zilibotti, 2005),
locked-in fertility hinders education investment by households that are unexpectedly poor. In this
situation, a mean-preserving spread of such shocks may increase the proportion of credit-constrained
children, leading to lower average human capital, output per worker, and average fertility.
2 The Model
The economy has a small, open, overlapping-generations structure and operates over an in￿nite
time horizon t ￿ 0.2
2.1 Production
In perfectly competitive environments, producers generate a single homogeneous good by using
physical and human capital. Only physical capital is mobile across economic borders. Production
technology is represented as a function F that ful￿lls all of the neoclassical properties. Then, the
aggregate quantity of ￿nal output produced in period t; denoted as Yt; is
Yt = F(Kt;Ht) = F(Kt=Ht;1)Ht ￿ f(kt)Ht; (1)
where Kt and Ht denote the aggregate levels of physical and human capital, respectively, used in
period t. The price of the ￿nal good is normalized to unity, and physical capital is assumed not to
depreciate. Each competitive producer maximizes its own pro￿t by taking factor prices as given.
Hence, Kt and Ht maximize Yt ￿ rKt ￿ wtHt, where r > 0 denotes the stationary interest rate in
the global capital markets, and wt is the wage rate per unit of human capital (e⁄ective labor) in
period t. Accordingly,
r = f0(kt); wt = f(kt) ￿ f0(kt)kt ￿ w;
where kt = f0￿1(r) ￿ k > 0:
2The model is an extension of Galor and Weil (2000), who explore the mechanism underlying the demographic
transition in the long-term growth process.
22.2 Households
A new generation is born at the beginning of each period and lives for three periods. Consider
the life of individual i 2 [0;Nt] of generation t (born in period t ￿ 1), where Nt is the population
size of generation t: In the ￿rst period (childhood), the individual consumes (￿ + e
p(i)
t￿1) units of
e⁄ective labor of her parent p(i) 2 [0;Nt￿1], where ￿ > 0 and e
p(i)
t￿1 > 0 denote the ￿xed cost
and the education cost, respectively.3 In the second period (adulthood/parenthood), the individual
acquires hi
t units of e⁄ective labor, which is a function of not only e
p(i)
t￿1 but also ai
t; an exogenous,
unexpected, idiosyncratic shock on abilities, health, and so on. More speci￿cally,
hi
t = ai
th(e
p(i)
t￿1); (2)
where h is a continuous function such that h0(e) > 0 and h00(e) < 0 8e > 0 and h(0) ￿ 0,
lime!0 h0(e) = 1; and lime!1 h0(e) = 0: The acquired labor is allocated between child rearing and
working. The adult individual raises ni
t units of children by spending (￿+ei
t) units of e⁄ective labor
per child. Supplying the remaining labor to producers brings her wage incomes, that are saved for
consumption during the retirement period (elderhood), ci
t+1: It follows that
ci
t+1 = Rw[hi
t ￿ (￿ + ei
t)ni
t]; (3)
where R ￿ 1 + r: The preferences of this individual are described by
ui
t = (1 ￿ ￿)lnci
t+1 + ￿ln[ni
th(ei
t)]; (4)
where ￿ 2 (0;1) and h(ei
t) is the deterministic component of human capital of their children.
2.3 Optimization
In period t, the unexpected shock ai
t happens to parent i between two events: childbirth and
education investment. The parent decides the number of children by assuming her skill level to be
hi
t = h(e
p(i)
t￿1). In light of Eqs. (3) and (4), the optimal choice in this circumstance is given by
fni
t; ￿ eg = argmax
n
(1 ￿ ￿)ln[h(e
p(i)
t￿1) ￿ (￿ + ￿ e)ni
t] + ￿ln[ni
th(￿ e)]
o
;
3The case in which e
p(i)
t￿1 = 0 is excluded here, because our focus is on developed economies, where education
investment prevails.
3where ￿ e denotes the ex-ante optimal level of education. The two ￿rst-order conditions yield4
ni
t =
￿
￿ + ￿ e
h(e
p(i)
t￿1): (5)
This fertility choice is assumed to be irreversible.
Then, by the time an education decision is made, the parent observes ai
t and thus hi
t = ai
th(e
p(i)
t￿1):
As follows from Eqs. (3), (4), and (5), the ex-post optimal (and actually chosen) level of education
is
ei
t = argmax
￿
(1 ￿ ￿)ln[ai
t ￿ amin(1 + ei
t=￿)] + ￿lnh(ei
t)
￿
; (6)
where it is assumed that ai
t > amin ￿ ￿￿=(￿ + ￿ e) 2 (0;1) to ensure ci
t+1 > 0: This assumption
excludes the possibility that parents unexpectedly have too many children to raise and therefore
cannot invest in education. Then, the ￿rst-order condition is
ai
t =
amin
￿
￿
￿ + ei
t +
1 ￿ ￿
￿
h(ei
t)
h0(ei
t)
￿
￿ D(ei
t); (7)
where D0(ei
t) > 0 8ei
t > 0; D(￿ e) = 1; limei
t!0 D(ei
t) = amin; and limei
t!1 D(ei
t) = 1: These
properties of D ensure that for any ai
t > amin; there exists a unique value ei
t > 0 satisfying Eq. (7).
Then, applying the Implicit Function Theorem, there exists a single-valued function such that
ei
t = e(ai
t) > 0; (8)
where e0(ai
t) > 0 8ai
t > amin, e(1) = ￿ e; and limai
t!amin e(ai
t) = 0: Hence, the determining factor in
education investment is ai
t; which is equal to the ratio of the ex-post level of human capital to the
to the ex-ante level of human capital, hi
t=h(e
p(i)
t￿1). The positive relationship of ei
t with ai
t indicates
a downward adjustment of education in response to a negative income shock.
2.4 Credit Constraint
In the economy considered herein, no one can take out loans because of some imperfections in
capital markets. Let e￿ be the labor cost of education preferred by a child in the absence of the
borrowing constraint. Then,
e￿ = argmax[wh(e￿) ￿ Rwe￿]; (9)
4￿ e is a positive value such that Q(￿ e) ￿ h
0(￿ e)(￿ + ￿ e) ￿ h(￿ e) = 0: The existence and uniqueness of this solution
are ensured by the following results, which are derived from the properties of the function h: Q
0(￿ e) < 0 8￿ e > 0;
lim￿ e!0 Q(￿ e) = 1; and lim￿ e!1 Q(￿ e) = lim￿ e!1[h
0(￿ e)￿ e ￿ h(￿ e)] < 0; where the last inequality holds because, on the
interval (0;1); the di⁄erence in the square brackets is negative (by concavity) and decreasing.
4where we￿ is the payment to the child￿ s teacher. As a result of the ￿rst-order condition h0(e￿) = R;
e￿ is a positive constant over time, and the corresponding disturbance level is such that e(a￿) = e￿
and thus a￿ > amin.5 These results and the properties of e(ai
t) reveal that
ei
t = e(ai
t)
8
<
:
< e￿ for ai
t < a￿;
￿ e￿ for ai
t ￿ a￿:
(10)
In the ￿rst case above, the education provided by a parent is insu¢ cient for her children￿ a situation
in which the borrowing constraint is binding. In contrast, in the second case, the children would
not need further education even without the constraint.
The analysis below is limited to the case in which the child is not credit-constrained if her
parent receives the average income shock. That is, in view of (10),
a￿ ￿ 1: (A1)
This condition is met if the borrowing cost is su¢ ciently large, noting that h0(e(a￿)) = R:
In light of the restriction ai
t > amin imposed above, it is analytically convenient to consider
that ln(ai
t ￿ amin) is normally distributed with mean ￿t and variance ￿t. Then, the cumulative
distribution function (cdf) of ai
t is
Gt(a) ￿ Pr(ai
t < a) = ￿
￿
ln(a ￿ amin) ￿ ￿t
￿t
￿
; (11)
where a > amin and ￿ denotes the cdf of the standard normal distribution.6 This speci￿cation
implies that all members within each generation are exposed to the same risk, and that the shocks
are independent of each other across individuals and time (i.e., ai
t ? a
j
￿ for any i 2 [0;Nt], j 2 [0;N￿],
and t;￿ ￿ 0). Now suppose that
￿t + ￿2
t=2 = ln(1 ￿ amin); (A2)
so that the mean and the variance of ai
t are E(ai
t) = 1 and var(ai
t) = (1 ￿ amin)2(exp￿2
t ￿ 1);
respectively. Thus, on average, no disturbance (ai
t = 1) occurs, and an increase in ￿t creates a
mean-preserving spread of ai
t.
5The existence and the uniqueness of e
￿ and a
￿ are guaranteed by the properties of h(e
i
t) and e(a
i
t) in Eqs. (2)
and (8), respectively. In Eq. (9), the future income shock is unexpected and therefore is not taken into account.
6The last equality follows from the fact that Pr(ln(a
i
t ￿ a
min) < ln(a ￿ a
min)) = Pr(a
i
t < a):
5Proposition 1 Under Eqs. (A1) and (A2); an increase in ￿t increases the fraction of children who
are credit-constrained (i.e., ei
t < e￿):
Proof. Given the two assumptions, Eq. (11) reveals that an increase in ￿t increases Gt(a￿), thereby
increasing the fraction of adult individuals with ai
t < a￿. Hence, the proposition follows from
Eq. (10). ￿
The proposition results from the irreversibility of the fertility decision. Once children are born,
parents have to raise them even if the parental incomes become unexpectedly low. Thus, a mean-
preserving spread of the unexpected shocks, which occur after childbirth, prevents more households
from providing the education level preferred by their children.
2.5 Aggregate Variables
Recalling that ai
t+1 has a mean of unity and is independent of the past shocks, the average level of
human capital of the working population in period t + 1 is
E(hi
t+1) = E[h(e
p(i)
t )] =
Z 1
amin
h(e(a))dGt(a): (12)
In addition, E(hi
0) is obtained from a historically determined distribution of ex-ante human capital
in period 0. Then, Eq. (1) shows that output per worker in period t is expressed as yt = f(k)Ht=Nt;
where, noting the private cost of child rearing in Eq. (6), we have
Ht
Nt
= E(hi
t)
￿
1 ￿ amin
￿
1 +
1
￿
Z 1
amin
e(a)dGt(a)
￿￿
: (13)
Let nt be the average number of children per adult individual in period t. In view of Eq. (5), the
evolution of the working population is
Nt+1 = ntNt =
￿
￿ + ￿ e
NtE(hi
t); (14)
where the initial level N0 is given historically. Thus, a lower level of average human capital dis-
courages fertility, leading to slower growth of the working population.
The following analysis examines the case in which e00(ai
t) ￿ 0 for any ai
t > amin: Namely,
in response to a marginal change in the unexpected shock, richer parents adjust their education
6Figure 1. Distributions of Human Capital
investment less sensitively. This case occurs if the function D in Eq. (7) satis￿es7
D00(ei
t) ￿ 0 8ei
t > 0: (A3)
Proposition 2 Under Eqs. (A2) and (A3); an increase in ￿t reduces average human capital E(hi
t+1),
output per worker yt+1; and the working population Nt+￿; where ￿ ￿ 2:
Proof. Since Eqs. (2) and (A3) reveal that h(e(a)) in Eq. (12) is strictly concave with respect to a;
an increase in ￿t decreases E(hi
t+1): Therefore, the other results follow from Eqs. (13) and (14). ￿
As stated above, the negative e⁄ect of a mean-preserving spread, which widens the income
distribution, results from the concave relationship between the income shock on a parent, ai
t; and
the skill level of each of her children. Among the determinants of parental incomes, the key factor
here is the unexpected factor ai
t; rather than the predetermined factor e
p(i)
t￿1: If ￿t was small, the
distribution of ex-post human capital in period t, ai
th(e
p(i)
t￿1); would largely re￿ ect the dispersion of
the latter. Then the number of children chosen in advance would, as mostly expected, be small
enough to allow the parents to provide the education preferred by their children (i.e., ei
t ￿ e￿): This
fertility adjustment moderates the correlation of skills between parents and their children, thereby
mitigating the negative impact of inequality on future output.
The importance of distinguishing these two factors is shown by Figure 1. While the distribution
of ai
th(e
p(i)
t￿1) in panel (a) has a higher degree of inequality than that in panel (b), the former re￿ ects
in larger part the variation of e
p(i)
t￿1. According to Proposition 2, such higher inequality leads to
higher output per worker￿ a situation that was overlooked in the previous studies (cf. Footnote 1).
7The sign of D
00(e
i
t); including the third derivative h
000(e
i
t); is generally ambiguous. For example, D
00(e
i
t) > 0 if
h
000(e
i
t) < 0; and D
00(e
i
t) = 0 if h(e
i
t) = (e
i
t)
￿; where ￿ 2 (0;1): As clari￿ed later herein, Eq. (A2) is unnecessary as
long as h(e(a)) is strictly concave with respect to a:
73 Concluding Remarks
This article has demonstrated that under some conditions, the contribution of unexpected factors
in income distribution, rather than the degree of inequality itself, is a key determinant of macro-
economic performance. The focus of the present research was on the irreversible fertility decisions
that are made prior to education investment. One may interpret locked-in fertility as an implicit
assumption in the previous literature on inequality, education, and growth. Our result indicates
that mitigating the uncertainty of parental incomes will raise output per worker in the next gener-
ation. In the presence of credit constraints, one useful policy would be to enrich public insurance
systems for health and unemployment.
While the central thesis of the present research is intuitive, the developed theory builds on
a number of simplifying assumptions. In more general environments, predetermined as well as
unexpected factors of income distribution would be relevant to education investment. The relative
importance of these factors should be empirically investigated in the future.
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Income Distribution and Macroeconomics Revisited: The Role of Fertility Adjustment

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Income Distribution and Macroeconomics Revisited: The Role of Fertility Adjustment

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