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,yaInstitute of Social and Economic Research, Osaka University, Japanb

Faculty 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) theoreticallyargued that, in developed economies with imperfect capital markets, inequality retards humancapital accumulation and economic growth.1
While the subsequent studies in this line of research incorporate the fertility decisions of households (cf. Kremer and Chen, 1999; de la Croix and Doepke, 2003), one important aspect of childrearing has been omitted. As formulated by Becker and Lewis (1973), parents face a quantityquality 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.
Corresponding 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).
1
Other 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).
y1

skill 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 unexpected, idiosyncratic income shocks between childbirth and education decisions. Since childbirth isan 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 thissituation, 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…nitetime horizon t

2.1

0.2

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, theaggregate quantity of …nal output produced in period t; denoted as Yt ; isYt = 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 inperiod t. The price of the …nal good is normalized to unity, and physical capital is assumed not todepreciate. Each competitive producer maximizes its own pro…t by taking factor prices as given.
Hence, Kt and Ht maximize Yt

rKt

wt Ht , 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) inperiod t. Accordingly,r = f 0 (kt );where kt = f 0 1 (r)wt = f (kt )f 0 (kt )ktw;

k > 0:

2

The model is an extension of Galor and Weil (2000), who explore the mechanism underlying the demographictransition in the long-term growth process.

2

2.2

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 t1), where Nt is the populationp(i)size of generation t: In the …rst period (childhood), the individual consumes ( + et 1 ) units ofe¤ective labor of her parent p(i) 2 [0; Nt 1 ], wherep(i)> 0 and et 1 > 0 denote the …xed cost

and the education cost, respectively.3 In the second period (adulthood/parenthood), the individualp(i)acquires hit units of e¤ective labor, which is a function of not only et 1 but also ait ; an exogenous,
unexpected, idiosyncratic shock on abilities, health, and so on. More speci…cally,p(i)hit = ait h(et 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 andworking. The adult individual raises nit units of children by spending ( + eit ) 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), cit+1 : It follows that
cit+1 = Rw[hitwhere R

2.3

(3)

1 + r: The preferences of this individual are described byuit = (1where

( + eit )nit ];) ln cit+1 +ln[nit h(eit )];(4)

2 (0; 1) and h(eit ) is the deterministic component of human capital of their children.

Optimization

In period t, the unexpected shock ait happens to parent i between two events: childbirth andeducation investment. The parent decides the number of children by assuming her skill level to bep(i)hit = h(et 1 ). In light of Eqs. (3) and (4), the optimal choice in this circumstance is given by

3nfnit ; eg = arg max (1p(i)) ln[h(et 1 )p(i)( + e)nit ] +oln[nit h(e)] ;The case in which et 1 = 0 is excluded here, because our focus is on developed economies, where education
investment prevails.

3where e denotes the ex-ante optimal level of education. The two …rst-order conditions yield4
p(i)

nit =

+e

h(et 1 ):(5)This fertility choice is assumed to be irreversible.
p(i)Then, by the time an education decision is made, the parent observes ait and thus hit = ait h(et 1 ):As follows from Eqs. (3), (4), and (5), the ex-post optimal (and actually chosen) level of educationiseit = arg max (1where it is assumed that ait > amin

) ln[aitamin (1 + eit = )] +ln h(eit ) ;(6)=( + e) 2 (0; 1) to ensure cit+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
ait =

amin

+ eit +

1h(eit )
h0 (eit )D(eit );

(7)

where D0 (eit ) > 0 8eit > 0; D(e) = 1; limeit !0 D(eit ) = amin ; and limeit !1 D(eit ) = 1: These
properties of D ensure that for any ait > amin ; there exists a unique value eit > 0 satisfying Eq. (7).
Then, applying the Implicit Function Theorem, there exists a single-valued function such thateit = e(ait ) > 0;(8)where e0 (ait ) > 0 8ait > amin , e(1) = e; and limait !amin e(ait ) = 0: Hence, the determining factor in
education investment is ait ; which is equal to the ratio of the ex-post level of human capital to thep(i)to the ex-ante level of human capital, hit =h(et 1 ). The positive relationship of eit with ait indicatesa 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 incapital markets. Let e be the labor cost of education preferred by a child in the absence of theborrowing constraint. Then,e = arg max[wh(e )Rwe ];(9)e is a positive value such that Q(e)h0 (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: Q0 (e) < 0 8e > 0;lime!0 Q(e) = 1; and lime!1 Q(e) = lime!1 [h0 (e)e h(e)] < 0; where the last inequality holds because, on theinterval (0; 1); the di¤erence in the square brackets is negative (by concavity) and decreasing.
44

where 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(ait ) reveal that8
< <efor ait < a ;iiet = e(at ):efor ai a :

(10)

t

In the …rst case above, the education provided by a parent is insu¢ cient for her children— a situationin 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 herparent 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 ait > amin imposed above, it is analytically convenient to consider
that ln(aitamin ) is normally distributed with meant and variancet.

Then, the cumulative

distribution function (cdf) of ait isGt (a)
where a > amin andln(aPr(ait < a) =amin )

t

;

(11)

t

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., ait ? aj for any i 2 [0; Nt ], j 2 [0; N ],and t;

0). Now suppose thatt+2
t =2 = ln(1amin );

so that the mean and the variance of ait are E(ait ) = 1 and var(ait ) = (1(A2)
amin )2 (exprespectively. Thus, on average, no disturbance (ait = 1) occurs, and an increase in

2
t

1);

t creates amean-preserving spread of ait .
5

The existence and the uniqueness of e and a are guaranteed by the properties of h(eit ) and e(ait ) in Eqs. (2)and (8), respectively. In Eq. (9), the future income shock is unexpected and therefore is not taken into account.
6
The last equality follows from the fact that Pr(ln(ait amin ) < ln(a amin )) = Pr(ait < a):5

Proposition 1 Under Eqs. (A1) and (A2); an increase int increases the fraction of children who

are credit-constrained (i.e., eit < e ):Proof. Given the two assumptions, Eq. (11) reveals that an increase int increases Gt (a ), thereby

increasing the fraction of adult individuals with ait < 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 meanpreserving 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 ait+1 has a mean of unity and is independent of the past shocks, the average level ofhuman capital of the working population in period t + 1 isp(i)E(hit+1 ) = E[h(et

)] =

Z 1

h(e(a))dGt (a):

(12)

amin

In addition, E(hi0 ) is obtained from a historically determined distribution of ex-ante human capitalin 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
= E(hit ) 1
Nt

min

a

1+

1

Z 1

e(a)dGt (a)

:

(13)

amin

Let nt be the average number of children per adult individual in period t. In view of Eq. (5), theevolution of the working population is
Nt+1 = nt Nt =+e

Nt E(hit );(14)

where the initial level N0 is given historically. Thus, a lower level of average human capital discourages fertility, leading to slower growth of the working population.
The following analysis examines the case in which e00 (ait )0 for any ait > amin : Namely,

in response to a marginal change in the unexpected shock, richer parents adjust their education

6

Figure 1.

Distributions of Human Capital

investment less sensitively. This case occurs if the function D in Eq. (7) satis…es7
D00 (eit )8eit > 0:0

Proposition 2 Under Eqs. (A2) and (A3); an increase in(A3)

it reduces average human capital E(ht+1 ),output per worker yt+1 ; and the working population Nt+ ; where2:

Proof. Since Eqs. (2) and (A3) reveal that h(e(a)) in Eq. (12) is strictly concave with respect to a;an increase init decreases E(ht+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, ait ; andthe skill level of each of her children. Among the determinants of parental incomes, the key factorp(i)here is the unexpected factor ait ; rather than the predetermined factor et 1 : Ift was small, thep(i)distribution of ex-post human capital in period t, ait h(et 1 ); would largely re‡ect the dispersion of
the latter. Then the number of children chosen in advance would, as mostly expected, be smallenough to allow the parents to provide the education preferred by their children (i.e., eite ): 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 distributionp(i)of ait h(et 1 ) in panel (a) has a higher degree of inequality than that in panel (b), the former re‡ectsp(i)in larger part the variation of et 1 . According to Proposition 2, such higher inequality leads tohigher output per worker— a situation that was overlooked in the previous studies (cf. Footnote 1).
The sign of D00 (eit ); including the third derivative h000 (eit ); is generally ambiguous. For example, D00 (eit ) > 0 if
h (eit ) < 0; and D00 (eit ) = 0 if h(eit ) = (eit ) ; 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:
7

000

7

3

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 macroeconomic 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 indicatesthat mitigating the uncertainty of parental incomes will raise output per worker in the next generation. 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 ona 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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