Discussion Papers In Economics
And BusinessChild mortality, child labor, fertility, anddemographics *Kei Takakura

Discussion Paper 20-13

September

2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANChild mortality, child labor, fertility, and demographics ∗Kei Takakura†
September 10, 2020

Abstract
In this study, we analyze how an improvement in child mortality aﬀects fertility, childlabor, and investments education investments. We consider an overlapping generations
model, in which skilled and unskilled workers coexist. Improvement in child mortality
has diﬀerent eﬀects on skilled and unskilled workers. We study three alternative policies
of increasing the proportion of skilled workers in the economy: improvement in childmortality, a ban on child labor, and child education. The ban on child labor meansthat the government enforces a law that prohibits a household from supplying childlabor. The model reveals that improvements in child mortality and a ban on childlabor can decrease the proportion of skilled workers and the average income in theeconomy. However, the child education policy, which supports both skilled and unskilledworkers’ investments in the education of their children by building schools, increases
the proportion of skilled workers and the average income in the economy.

Keywords: Child mortality, Child labor, Fertility, Education, Health, Overlapping generations model
JEL Classiﬁcation: D1, I1, I2, J1, O1∗

I would like to deeply thank to Koichi Futagami and Kohei Okada for their valuable comments throughout
this study. I also would like to thank Tatsuro Iwaisako, Tetsuo Ono, Kouki Sugawara, Akihisa Shibata,Katsuya Takii, Takeshi Murooka, Hirokazu Ishise, Shuhei Kitamura, and the seminar participants at Osaka
University, Musashi University, and University of Strasbourg for their useful comments. Without theircomments and instructions, I would not have completed this paper. All remaining errors are mine.
†
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 5600043, Japan. E-mail: takakura.kei.0520@gmail.com

1.

Introduction

Many countries have experienced signiﬁcant improvements in child mortality over the past
several decades (UNIGME, 2019). However, extant empirical studies indicate that the least
developed countries, such as the sub-Saharan African countries, still have high child mortality. UNIGME (2019) states it is impossible for these developing countries to reduce under-ﬁve
mortality by 2030 as per the SDG targets. Moreover, many theoretical and empirical studies
have elucidated that these countries show features such as high fertility, low investments in
education, high mortality, abundant child labor, and low income.
In this study, we clarify how an improvement in child mortality aﬀects the behaviorsof heterogeneous individuals and demographics by focusing on fertility, child labor, and investments in education. While existing studies (e.g., Dessy 2000; Hazan and Berdugo 2002;
Kalemli-Ozcan 2002, 2003, 2008; Strulik 2004; Azarnert 2006; Fioroni 2010) also analyze thisissue in developing countries, they do not consider the heterogeneity of individuals. Since
this study considers the heterogeneity among individuals, that is, the presence of both skilled
and unskilled individuals, an improvement in child mortality has diﬀerent eﬀects on these
diﬀerent individuals. This study thus reveals that an improvement in child mortality causes
skilled workers to decrease fertility and the child labor supply and increase investments in
education. Conversely, a reduction in child mortality makes unskilled individuals increase
both fertility and the child labor supply. These opposite behaviors aﬀect demographics andcan decrease the proportion of skilled workers in the economy. Hence, if there is a signiﬁcant
improvement in child mortality that increases the proportion of unskilled individuals, ourmodel can explain why poor countries still have high fertility and abundant child labor, especially in sub-Saharan African countries, although these countries have experienced progressin terms of child mortality.
Speciﬁcally, we consider the heterogeneity of individuals according to Fan and Zhang(2013). Skilled workers can freely choose investments in education for their children, while1unskilled workers are less productive in educating their children, that is, they face the upperbound for the investments in education to their children. We also incorporate child mortality
into the proposed model, similarly to Chakraborty (2004). Here, the mortality in adulthoodand old age improves through public health expenditure by the government. Several studies(e.g., Kalemli-Ozcan 2002, 2003, 2008; Strulik 2004; Sato and Yamamoto 2005; Azarnert
2006; Fironi 2010; Strulik and Weisdorf 2014) incorporate child mortality into their models. For instance, Kalemli-Ozacan (2002, 2003, 2008) introduces child mortality, which isconstant over time, and analyzes how its uncertainty aﬀects economic growth. Sato andYamamoto (2005) assume that the survival of children is deterministic and analyze how an
improvement in child mortality aﬀects the demographic transition. Strulik (2004) assumes
that child mortality depends on the per capita income. Azarnert (2006) assumes that childmortality depends on parents’ investments in education and Fioroni (2010) assumes it positively depends on parents’ human capital. These two latter studies show that, if parents havea suﬃciently high human capital, they have better knowledge about health and are willingto improve their children’s health. Strulik and Weisdorf (2014) assume that child mortality
depends on the geographical location and food expenditure per child. In contrast with thesestudies, the method of introducing child mortality in the present study simpliﬁes the analysis
as follows. The government levies a lump-sum tax on each household. The collected taxes
reduce child mortality through public health investment by the government, which includes
new medical facilities, sanitation improvement, disease control, and inoculation programs.
Further, we examine three alternative policies: a policy that improves child mortality, aban on child labor, and an education policy. Strulik (2004) states that the ban on child labor
is not eﬀective in escaping from economic stagnation, compared to compulsory schooling andimprovement in health policies. In his model, compulsory schooling and a health policy canmake parents invest in education for their children, which leads to sustained economic growth.
However, even if the ban on child labor is executed by the government, parents can choose
not to invest in the education of their children because such investments in education are still2

costly. Conversely, Dessy (2000) and Hazan and Berdugo (2002) assert that the prohibitionof child labor is eﬀective and is the only way to escape from the poverty trap. However,these two studies do not consider the existence of child mortality. The setup of our modelis closer to that of Strulik (2004) rather than those of Dessy (2000) and Hazan and Berdugo(2002); however, it is diﬀerent from Strulik’s (2004) in that we consider the heterogeneity of
individuals by paying attention to the diﬀerential wages and abilities that constrain parents’investments in the education of their children. Since both skilled and unskilled parents caninvest in education in the proposed model, the eﬀect of the health policy diﬀers from theresults of Strulik (2004). The health policy in the proposed model spurs economic stagnation by lowering the per capita output. The model also shows that the ban on child labor
policy is not suﬃcient for economic growth. However, the education policy, which supports
both skilled and unskilled workers’ investments in the education of their children by building
schools, increases the proportion of skilled workers and the average income in the economy.
The government executes this education policy, as well as its public health expenditure, byusing its tax revenue.
The ﬁndings are as follows. First, an improvement in child mortality makes skilled workers have higher quality input regarding their children, thereby reducing child labor and thenumber of children. Conversely, an improvement in child mortality makes unskilled workers
have more children, thus increasing the child labor supply. Second, skilled parents have alower fertility rate and child labor supply than unskilled ones. We ﬁnd that if the upperbound of the investments in education for unskilled parents decreases, the diﬀerences between fertility rates and child labor supplies become large between the two groups. Third,
an improvement in child mortality can decrease the proportion of skilled individuals and theaverage income in the economy. Fourth, the ban on child labor also reduces the proportionof skilled workers and the average income in the economy. Finally, the child education policy
is the most promising among the three, since it increases the ratio of skilled workers, as wellas the average income in the economy. As a result, the ﬁndings are partly related to those of3

Strulik (2004), while the ban on child labor policy goes against the ﬁndings of Dessy (2004)
and Hazan and Berdugo (2002).
The remainder of this paper is organized as follows. Section 2 develops the basic modeland considers how an improvement in child mortality aﬀects individuals’ fertility, child labor
supply, and investments in education. Section 3 focuses on population dynamics and analyzes its steady state. Section 4 examines three alternative policies: a ban on child labor,
an improvement in child mortality, and child education. We demonstrate that only the childeducation policy increases the proportion of skilled workers and the average income in theeconomy. Section 5 provides several concluding remarks.

2.

The model

2.1.

Environments

We consider a three-period overlapping generations model: childhood, adulthood, and oldage. Time is discrete and inﬁnite. As we consider a small open economy, the interest rate isequal to the world interest rate. Childhood consists of two sub-periods: early childhood andschool age. Children may die between early childhood and school age1 . During childhood,individuals do not make any decisions. During adulthood, individuals give birth to theirchildren, raise and educate them, and determine the supply of child labor. During old age,
individuals retire and consume only. An individual dies at the beginning of school age withprobability 1 − π c ∈ [0, 1) and lives through school age with probability π c ∈ (0, 1].
The economy consists of two sectors: a traditional sector and a modern one. Production
takes place in either or in both sectors. Both sectors produce a single good and the same
output, but employ diﬀerent factors. The traditional sector employs unskilled workers andchild labor from both unskilled and skilled households, while the modern sector employs
skilled workers and physical capital.
1

See, for example, Azarnert (2004) and Fironi (2010).

4

The government imposes a lump-sum tax on each individual at the end of adulthood. Collected taxes are used for public health expenditure to improve medical facilities, sanitation,disease control, and inoculation programs. Public health expenditure improves child healthand, thus, reduces child mortality.

2.2.

Production

Because the output level in the traditional sector is based only on unskilled workers and childlabor, the production function of the traditional sector is

Ytu = wu Lut .

(1)

Let Ytu , wu , and Lut represent production in this sector, the wage rate of the unskilled labor,
and the amount of unskilled labor, respectively. Lut includes child labor from both skilled
and unskilled households. Conversely, production in the modern sector employs skilled laborand physical capital. The production function of the skilled sector is

Yts = F (Kt , Lst ).

(2)

Let Yts , Kt , and Lst express production in its sector, physical capital, and the amount ofskilled labor, respectively. We assume the production function of the skilled sector satisﬁes
all neoclassical characteristics. The ﬁrm’s proﬁt maximization problem is

max F (Kt , Lst ) − r̄Kt − wts Lst ,{Kt ,Lst }(3)

where r̄ and wts are the world interest rate and the wage rate for the skilled labor. Maximizing
the objective function (3) with respect to Kt and Lst yields the following conditions:r̄ = f ′ (kt ),5(4)

wts = f (kt ) − f ′ (kt )kt ,(5)twhere kt ≡ K. Here, kt and wts are constant. These values are denoted as k̄ and ws and we
Lst

also assume ws > wu .2

2.3.

Individuals

We consider the behavior of individuals. Each individual i ∈ {s, u} derives his/her utility
from the number of surviving children (πtc nit ), rearing (nit ) children, children’s leisure (1−ℓic,t ),the investments in education for each child (eit ), his/her own consumption in adulthood (cit ),and his/her future consumption in old age (cit+1 ). The utility function of the individual ibelonging to generation t is

uit = γ[log(πtc nit ) + ϕ log(1 − ℓic,t ) + β log(eit )]+ (1 − γ)[log(cit ) + δ log(cit+1 )],γ, ϕ, β, δ ∈ [0, 1),(6)where γ stands for the extent to which each individual values her children relative to his/her
own lifetime consumption. ϕ represents the weight on the leisure of each child. β represents
the weight of investments in education for the surviving children. Component γβ log(eit )represents either the warm glow of giving (Andreoni 1989) or a preference of having higherquality children (Becker 1960). Since ϕ and β are bellow one, the individual values her utility
from having surviving children more than the surviving children’s leisure and investments in
education. This assumption indicates that having a family is more important than investingin education for children and children’s leisure.3 δ is the discount rate that gives priority topresent consumption over future consumption.
Each individual and child have one unit of time. During adulthood, the individual
gives birth to children, educates surviving children, and determines the child labor supply.

2
3

See, for example, Galor and Zeira (1993) and Hazan and Berdugo (2002).
See, for example, Efrlich and Lui (1991), Zhang et al. (2001), and Hashimoto and Tabata (2016).

6

The individual then collects the income earned from child labor and uses it for his/her own
consumption. The individual pays a lump-sum tax τ to the government. Here, all individual’s
incomes are called family income, being described as (1 − znit − eit πtc nit )ws + dℓic,t πtc nit wu − τ .
The individual also divides his/her family income into present consumption and savings.
Hence, the budget constraint of individual i at time t is given by:

cit = (1 − znit − eit πtc nit )wi + dℓic,t πtc nit wu − τ − sit ,cit+1 = (1 + r̄)sit ,

(7)
(8)

ē ≥ eut ,(9)where z is the time to bear and raise children.4 Following Becker and Lewis (1973), Rosenzweig and Wolpin (1980), Galor and Weil (2000), and Moav (2005), we assume that this time
cost z does not vary with family size. d ∈ (0, 1) indicates that children can only provide
dℓic,t units of parents’ labor because children’s physical ability is inferior to that of adults
and they can only work in the traditional sector. We assume that unskilled parents are lessproductive in terms of education and always choose the upper bound and their marginalcontribution of their time to children’s education is substantially lower than that of skilledparents, that is, eut = ē > 0.5 When parents school their children at home, they must possess
basic abilities such as writing, reading, and arithmetics. We assume that only skilled parents
have these abilities. This assumption and its implications originate from empirical evidences
as follows. Lynch (2000) and Ramey and Ramey (2010) show that, in the USA, mothers
who had a higher education level spend much more time educating their children and givethem an overall better education than mothers who did not have a high education level.
Ballantine (2001) also supports the stylized fact that better educated parents spend moretime educating their children from the sociological side.

4
5

See, for example, Becker (1965).
See, for example, Moav (2005) and Fan and Zhang (2013) for the theoretical background.

7

Maximizing (6) subject to (7) and (8) for skilled workers yields the following solutions:δ(1 − γ)(ws − τ )≡ ss ,(1 − γ)(1 + δ) + γ
γ(ϕ + β − 1)(ws − τ )nst =,[(1 − γ)(1 + δ) + γ](dπtc wu − zws )ϕ(dπtc wu − zws ),ℓsc,t = 1 −(ϕ + β − 1)dπtc wuβ(dπtc wu − zws )est =,(ϕ + β − 1)πtc ws(1 − γ)(ws − τ )cst =≡ cs .
(1 − γ)(1 + δ) + γ
sst =

(10)
(11)
(12)
(13)
(14)

Maximizing (6) subject to (7) to (9) for unskilled workers yields the following solutions:δ(1 − γ)(wu − τ )≡ su ,(1 − γ)(1 + δ) + γ
γ(1 − ϕ)(wu − τ ),nut =[(1 − γ)(1 + δ) + γ][z − (d − ē)πtc ]wu
ϕ[z − (d − ē)πtc ]ℓuc,t = 1 −,(1 − ϕ)dπtc(1 − γ)(wu − τ )cut =≡ cu .
(1 − γ)(1 + δ) + γ
sut =

(15)
(16)
(17)
(18)

Because all variables are required to be non-negative, the following conditions must be satisﬁed:

dπtc wu − zws > 0,(19)z − (d − ē)πtc > 0,

(20)

ws > τ,(21)

wu > τ,(22)

ϕ + β > 1.

(23)

8

We explore how an improvement in child mortality aﬀects skilled and unskilled workers.
Hereafter, we make the following assumption on educational investments of unskilled workers:
Assumption 1.
0 < ē < minInequality ē <β(dπtc wu −zws )(ϕ+β−1)πtc ws{ β(dπ c wu − zws )t(ϕ + β − 1)πtc w

}
,d .
s

(24)

ensures that skilled parents spend more time educating theirchildren than unskilled ones, while inequality ē < d ensures that an improvement in childmortality always makes unskilled workers increase the number of their children. As a result,
we obtain the following proposition.
Proposition 1. An improvement in child mortality makes skilled workers decrease fertility,
the child labor supply, and increase investments in education. However, unskilled workers
increase both their fertility and the child labor supply.
Proof. We can obtain the partial derivatives of (11), (12), (13), (16), and (17) with respect
to πtc as follows:∂nstγ(ϕ + β − 1)(ws − τ )dwu
=
−
< 0,
∂πtc
[(1 − γ)(1 + δ) + γ](dπtc wu − zws )2∂ℓsc,t
ϕzws
=
−
< 0,
∂πtc
(ϕ + β − 1)dwu (πtc )2
∂est
βz
=
> 0,
c
∂πt
(ϕ + β − 1)(πtc )2γ(wu − τ )(1 − ϕ)(d − ē)
∂nut
=
> 0,
∂πtc
[(1 − γ)(1 + δ) + γ](1 − γ + γwu )wu [z − (d − ē)πtc ]2∂ℓuc,t
ϕz
=
> 0.
c
∂πt
(1 − ϕ)d(πtc )2

(25)
(26)
(27)
(28)
(29)

Since skilled workers freely choose to invest in education, an improvement in child mortality
makes them have better educated children and this eﬀect dominates the budget and time
constraints of skilled individuals. Higher education increases the cost of having children

9and, thereby, reduces the number of children because investments in education are timeconsuming. Hence, skilled workers are inclined to have a smaller family, which makes children’s contributions to the family income smaller and, thus, reduces the child labor supply.
Conversely, since unskilled workers have an upper bound for investments in the education of
their children, an improvement in child mortality contributes to having more children, whichleads to a larger family. As child survival and fertility rates increase, the income from childlabor also increases, compensating for the higher disutility of making their children work.
Therefore, unskilled agents increase the child labor supply.
In addition, we compare the fertility rates and child labor supply between skilled andunskilled workers. Then, we obtain the following proposition.
Proposition 2. Skilled parents have lower fertility rates and child labor supply compared tounskilled ones:

nut > nst ,(30)ℓuc,t > ℓsc,t .

(31)

10

Proof. We examine the sign of diﬀerential fertility nut − nst by using (11) and (16):γ(1 − ϕ)(wu − τ )γ(ϕ + β − 1)(ws − τ )−,[(1 − γ)(1 + δ) + γ][z − (d − ē)πtc ]wu [(1 − γ)(1 + δ) + γ](dπtc wu − zws )γ[(1 − ϕ)(wu − τ )(dπtc wu − zws ) − (ϕ + β − 1)(ws − τ )(z − dπtc + ēπtc )wu ]=,[1 − γ)(1 + δ) + γ][z − (d − ē)πtc ](dπtc wu − zws )wu

nut − nst =(Here, we employ Assumption 1: ē < est .)>γ[(1 − ϕ)(wu − τ )(dπtc wu − zws ) − (ϕ + β − 1)(ws − τ )(z − dπtc + est πtc )wu ],[1 − γ)(1 + δ) + γ][z − (d − ē)πtc ](dπtc wu − zws )wu(Here, we employ (13) and ws > wu .)γ[(1 − ϕ)(wu − τ )(dπtc wu − zws ) − (ϕ + β − 1)(ws − τ )(z − dπtc )wu − β(ws − τ )(dπtc wu − zws )][1 − γ)(1 + δ) + γ][z − (d − ē)πtc ](dπtc wu − zws )wuγ{(1 − ϕ)[(wu − τ )(dπtc wu − zws ) + (ws − τ )(dπtc − z)wu ] + βz(ws − τ )(ws − wu )},=[1 − γ)(1 + δ) + γ][z − (d − ē)πtc ](dπtc wu − zws )wu>> 0.

In the last inequality, we employed (19), (21), (22), (23), and ws > wu . Similarly, we examine
the sign of child labor supply ℓuc,t − ℓsc,t employing (12) and (17):ϕ(dπtc wu − zws ) ]ϕ[z − (d − ē)πtc ] [−1−,(1 − ϕ)dπtc(ϕ + β − 1)dπtc wu
ϕ{[−z + (d − ē)πtc ](ϕ + β − 1)wu + (dπtc wu − zws )(1 − ϕ)}=,(1 − ϕ)(ϕ + β − 1)dwu πtcℓuc,t − ℓsc,t = 1 −(Here, we employ Assumption 1: ē < est .)>ϕ{[−z + (d − est )πtc ](ϕ + β − 1)wu + (dπtc wu − zws )(1 − ϕ)},(1 − ϕ)(ϕ + β − 1)dwu πtc(Here, we employ (13) and ws > wu .)ϕ[(ϕ + β − 1)(dπtc − z)wu − βwu (dπtc wu − zws ) + (1 − ϕ)(dπtc wu − zws )],(1 − ϕ)(ϕ + β − 1)dwu πtc
ϕ(ϕ + β − 1)(ws − wu )z,=(1 − ϕ)(ϕ + β − 1)dwu πtc>> 0.

In the last inequality, we employed (23) and ws > wu .
11

If Assumption 1: ē < est is satisﬁed, the fertility rate of unskilled workers is always greaterthan that of skilled workers. Further, the child labor supply of unskilled workers is always
greater than that of skilled workers. Therefore, condition ē < est is suﬃcient to determine
the signs of nut − nst and ℓuc,t − ℓsc,t . Since unskilled workers cannot provide their children witha suﬃcient level of investments in education and their wage rate is small in contrast to thatof skilled workers, unskilled workers have a large family and depend on child labor income.

3.

Population dynamics

The total public health expenditure is determined by the total tax revenue, that is,

Gt = τ Lt ,where Lt denotes the number of adult workers in the economy. We deﬁne public healthservices per family as follows:g = τ.

(32)

Because the lump-sum tax per family is constant over time, g also becomes constant. We
assume that the child survival probability is a monotonically increasing function of publichealth services. Therefore, children’s survival probability function is given by

πtc = π c (gt ) = π c (τ ) ≡ π c ,

12

(33)

where π ′ (gt ) > 0, π ′′ (gt ) < 0, limgt →0 π(gt ) = 0, limgt →∞ π(gt ) < 1, and limgt →∞ π(gt ) < ∞.
Substituting (33) into (11), (12), (13), (16), and (17) yields
γ(ϕ + β − 1)(ws − τ )≡ ns ,[(1 − γ)(1 + δ) + γ](dπ c wu − zws )ϕ(dπ c wu − zws )ℓsc,t = 1 −≡ ℓsc ,
(ϕ + β − 1)dπ c wuβ(dπ c wu − zws )est =≡ es ,(ϕ + β − 1)π c ws
γ(1 − ϕ)(wu − τ )≡ nu ,nut =[(1 − γ)(1 + δ) + γ][z − (d − ē)π c ]wu
ϕ[z − (d − ē)π c ]ℓuc,t = 1 −≡ ℓuc .
(1 − ϕ)dπ c
nst =

(34)
(35)
(36)
(37)
(38)

We now derive population dynamics. Let λt and p represent the proportion of skilledworkers at time t and the probability of a child whose parent is skilled or unskilled to become
skilled, respectively. We further assume that p is a function of investments in education byparents: ps = p(es ) ∈ (0, 1) and pu = p(ē) ∈ (0, 1).6 These probability functions are strictlyincreasing with investments in education and are concave. Since unskilled parents are lessproductive in terms of investments in education based on Assumption 1, ps > pu always
holds. The total number of workers at time t is π c ns λt Lt + π c nu (1 − λt )Lt . The total numberof skilled workers at time t + 1 is ps π c ns λt Lt + pu π c nu (1 − λt )Lt . Therefore, the proportionof skilled workers at time t + 1 is
ps π c ns λt Lt + pu π c nu (1 − λt )Lt
π c ns λt Lt + π c nu (1 − λt )Lt
ps ns λt + pu nu (1 − λt ).
=ns λt + nu (1 − λt )λt+1 =(43)

We consider whether population dynamics (43) has at least one stable steady state. Then,
we obtain the following proposition.
6de la Croix and Michel (2002), Anger and Heineck (2010), and Becker (2009) state that family backgroundand parental academic achievements and human capital have substantial eﬀects on an individual’s educational
attainment based on empirical evidence. See, for example, Cremer and Pestrieau (2006), Mookherjee andNapel (2007), and Fan and Stark (2008) for the theoretical background of this assumption.

13

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1

t+1 =

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ps ns t + pu nu (1ns t + nu t

t)

ps

⇤

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Figure 1: The transition process of λt when ns < nu holds.
Proposition 3. Population dynamics (43) has a stable steady state that satisﬁes pu < λ∗ <ps . Moreover, this steady state is given by1λ∗ =(1 + pu )nu − ps ns − (Φ) 2,2(nu − ns )(44)

where Φ = [(1 + pu )nu − ps ns ]2 − 4pu (nu − ns )nu > 0.
Proof. We can obtain the ﬁrst and second order derivatives of (43) as follows:∂λt+1(ps − pu )nu ns
= s
> 0,
∂λt
{n λt + nu (1 − λt )}2(45)∂ 2 λt+1
2nu ns (ps − pu )(nu − ns )
=
> 0.
∂λ2t
{ns λt + nu (1 − λt )}3(46)

When λt = 0, then λt+1 =pu . Conversely, when λt = 1, then λt+1 =ps . Therefore, we obtain
population dynamics as per in Figure 1. Hence, the steady state that satisﬁes pu < λ∗ < psis globally stable.
We consider the characteristics of the steady state and how an increase in fertility rates, the14

probability for children to become skilled, and an improvement in child mortality aﬀects thesteady state.
Proposition 4. In the steady state, we have the following inequalities:
∂λ∗
∂λ∗
∂λ∗
< (≥)0,> 0,< (≥)0.
∂z
∂ē
∂τ

(47)

Proof. We can obtain the partial derivatives of equation (43) in the steady state with respect
to z, ē, and π c , respectively, as follows:s−(1 − λ∗ )(λ∗ − pu ) ∂n
+ λ∗ (ps − λ∗ ) ∂nns λ∗ ∂p
∂λ∗
∂z
∂z
∂z
=
< (≥)0,
1
∂z
(Φ) 2
u

u

(48)

u

−(1 − λ∗ )(λ∗ − pu ) ∂n
+ (1 − λ∗ )nu ∂p
∂λ∗
∂ē
∂ē
=
> 0,
1
∂ē
(Φ) 2
u

(49)
s

∂p
−(1 − λ∗ )(λ∗ − pu ) ∂n
+ (ps − λ)λ∗ ∂n+ ns λ∗ ∂π
∂λ∗
c
∂π c
∂π c
=
< (≥)0,
1
c
∂π
(Φ) 2
s

u

s

u

s

u

u

c

u

s

(50)

s

∂p
> 0, ∂n< 0, ∂p< 0, ∂n< 0, ∂p> 0, ∂π> 0, ∂n> 0, ∂n< 0, ∂π
where ∂nc > 0.
∂z
∂z
∂z
∂ē
∂ē
∂τ
∂π c
∂π c

Therefore, we obtain the following inequalities:
∂nu
∂ns
∂ps
+ (ps − λ∗ )λ∗< −ns λ∗
,
∂z
∂z
∂z
∂nu
∂ns
∂ps
(1 − λ∗ )(λ∗ − pu ) c − (ps − λ∗ )λ∗ c > ns λ∗ c .
∂π
∂π
∂π

−(1 − λ∗ )(λ∗ − pu )

(51)
(52)

An increase in the time cost to bear and rear children and the health policy leading to an
improvement in child mortality negatively aﬀect the proportion of skilled agents in the steadystate.
The amount of time, (z), represents parents’ opportunity cost of the time spent away fromwork to raise their children. If z increases, skilled parents choose to decrease investments in
education (es ) and increase the child labor supply (ℓsc ) to compensate for these opportunity
costs. As such, skilled parents increase their number of children (ns ). Conversely, an increase15

in z makes it diﬃcult for unskilled parents to bear and rear their children. As a result,they choose to decrease their number of children (nu ). If the eﬀect on the probability (ps )dominates the eﬀect of skilled and unskilled fertility rates, as in inequality (51), an increasein the time cost can decrease the proportion of skilled workers in the steady state. An
increase in the upper bound of investments in education (ē) indicates that unskilled parents
provide better education to their children. Hence, they require more time for education andthey have to decrease nu . Moreover, the increase in ē positively aﬀects the probability (pu ).
Therefore, the increase in ē positively aﬀects the proportion of skilled agents in the steadystate. As per Proposition 1, an improvement in child mortality increases nu and es anddecreases ns . An increase in (es ) positively aﬀects the probability (ps ). If the eﬀect on thefertility rates (nu , ns ) dominates the inﬂuence on the probability (ps ), as in inequality (52),the government’s health policy can decrease the proportion of skilled agents in the steadystate.

4.

Policy implications

We examine three alternative policies: a ban on child labor, improvement in child mortality,
and child education. Then, we examine whether these policies can increase the ratio of skilledworkers.

4.1.

Improvement in child mortality policy and a ban on childlabor policy

We investigate how the ban on child labor aﬀects skilled and unskilled individuals, assumingthat the government imposes the abolition of child labor, that is, ℓic = 0.7 We employ thefollowing assumption to ensure that both skilled and unskilled individuals choose investmentsin the education of their children as an interior solution because we focus on the case where

7

See, for example, Strulik (2004).

16

the ban decreases the investments in the education of skilled individuals to the level wherees < ē holds and ē is no longer the upper bound for unskilled individuals.
Assumption 2.
ē >

βz
.
(1 − β)π c(53)

Similarly to Section 2, maximizing (6) subject to (7) and (8) for skilled workers yields thefollowing solutions:δ(1 − γ)(ws − τ )≡ ssg ,
(1 + δ)(1 − γ) + γ
γ(1 − β)(ws − τ )≡ nsg ,
nsg,t =[(1 + δ)(1 − γ) + γ]zws
βzesg,t =≡ eg ,(1 − β)π c(1 − γ)(ws − τ )csg,t =
≡ csg ,
(1 + δ)(1 − γ) + γ
ssg,t =

(54)
(55)
(56)
(57)

where the subscript g denotes that the government enforces the ban on child labor. Moreover,
the optimal solutions for unskilled workers are given byδ(1 − γ)(wu − τ )≡ sug ,
(1 + δ)(1 − γ) + γ
γ(1 − β)(wu − τ )nug,t =≡ nug ,[(1 + δ)(1 − γ) + γ]zwu
βz≡ eg ,
eug,t =
(1 − β)π c(1 − γ)(wu − τ )cug,t =≡ cug .
(1 + δ)(1 − γ) + γ
sug,t =

(58)
(59)
(60)
(61)

Therefore, once the ban on child labor is executed, both skilled and unskilled workers choose
the same amount of investments in the education of their children as an interior solution.
Then, we obtain the following proposition, which is in contrast with Proposition 1.
Proposition 5. When the government imposes a ban on child labor, an improvement inchild mortality no longer aﬀects the fertility rates of skilled workers and unskilled workers.
17Moreover, it makes skilled and unskilled workers decrease their investments in the educationof their children.
Proof. We can obtain the partial derivatives of (55), (59), and (60) with respect to π c asfollows:
∂nsg
∂nug
=
= 0,
∂π c
∂π c
∂esg
∂eug
βz
=
=−
< 0.
c
c
∂π
∂π
(1 − β)(π c )2

(62)
(63)

Equation (62) follows from budget constraint (7), that is, an improvement in child mortality
increases the cost of education for children, which in turn makes parents decrease the numberof children. The marginal beneﬁt of an improvement in child mortality and the marginal
disutility of decreasing the number of children neutralize each other due to the characteristicsof the log utility function (6), which implies unitary elastic demand for children with respect
to the cost of having children. The negative sign of (63) also follows from budget constraint(7), that is, an improvement in child mortality increases the cost of education for children,which in turn makes parents decrease investments in the education of their children. This
result comes from the characteristics of the utility function (6) that parents always prioritize
an improvement in child mortality over investments in education because β is bellow one.
Skilled and unskilled parents give birth to their children, expecting them to contribute tofamily income before the government enforces the ban on child labor. However, once the ban isin place, they can no longer obtain the rewards from child labor. Therefore, an improvementin child mortality gives parents a heavier burden of increased surviving children and lesseducation for each child. 8
We next examine how the ban on child labor aﬀects fertility rates and clarify how the8See, for example, Kalemli-Ozcan (2002, 2003, 2008). The ﬁndings here correspond to those of KalemliOzcan (2008).

18

diﬀerence in the fertility rates between skilled and unskilled workers change. The ﬁndings,
which are in contrast to Proposition 2, can be summarized as follows.
Proposition 6. The ban on child labor overturns the fertility diﬀerence between skilled andunskilled workers. Moreover, it makes skilled workers increase the number of children andunskilled workers decrease it.

nug < nsg ,(64)

ns < nsg ,(65)

nug < nu .

(66)

Proof. By using equations (55) and (59), we obtainγ(1 − β)(ws − τ )γ(1 − β)(wu − τ )−,[(1 + δ)(1 − γ) + γ]zws [(1 + δ)(1 − γ) + γ]zwu
γ(1 − β)τ (ws − wu )=,[(1 + δ)(1 − γ) + γ]zws wu

nsg − nug => 0.

Assumptions 1 and 2 yield the following condition9 :(1 − β)dπ c wu > ϕzws .

(67)

Assumptions 1 and 2 ensure that es > esg holds. Moreover, we can derive the value of es − eg from (36)βc uand (56) as follows: es − eg = πc (ϕ+β−1)w− ϕzws ].
s (1−β) [(1 − β)dπ w
919

Then, we obtain the signs of nsg − ns and nu − nug from (34), (37), (55), and (59).
γ(ϕ + β − 1)(ws − τ )γ(1 − β)(ws − τ )−,[(1 − γ)(1 + δ) + γ](dπ c wu − zws ) [(1 + δ)(1 − γ) + γ]zws
γ(ws − τ )=[ϕzws − (1 − β)dπ c wu ],[(1 − γ)(1 + δ) + γ](dπ c wu − zws )zws

ns − nsg =(Here, we employ (67).)< 0.

γ(1 − β)(wu − τ )γ(1 − ϕ)(wu − τ )−,[(1 − γ)(1 + δ) + γ][z − (d − ē)π c ]wu [(1 + δ)(1 − γ) + γ]zwu
γ(wu − τ )=[(1 − β)dπ c − ϕz + z − (1 − β)(z + ēπ c )],[(1 + δ)(1 − γ) + γ](z − dπ c + ēπ c )zwu

nu − nug =(Here, we employ (24).)>γ(wu − τ )β(dπ c wu − zws )
c
[(1
−
β)dπ
−
ϕz
+
z
−
(1
−
β)(z
+
)],
[(1 + δ)(1 − γ) + γ](z − dπ c + ēπ c )zwu
(ϕ + β − 1)ws

(Here, we employ ws > wu and (23).)γ(wu − τ )[(1 − β)dπ c − ϕz + z − (1 − β)z + β(dπ c − z)],[(1 + δ)(1 − γ) + γ](z − dπ c + ēπ c )zwu
γ(wu − τ )=[(1 − β)dπ c − ϕz + βz + β(dπ c − z)],[(1 + δ)(1 − γ) + γ](z − dπ c + ēπ c )zwu>

(Here, we employ ws > wu , (19), and (67).)> 0.

The abolition of child labor signiﬁcantly decreases unskilled parents’ incentives to bear andrear their children more than skilled parents because the former have lower incomes than thelatter. Therefore, the ban on child labor overturns the relative magnitude of fertility rates
between skilled and unskilled parents. However, skilled parents increase their fertility ratesince they decrease investments in education and they reallocate their time to having a larger20t+1 ,

g,t+1<latexit sha1_base64="d8L5Euoo/Kx5N6ngqMnk+GRLR8Q=">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</latexit>

t+1 =

t

<latexit sha1_base64="75RXNDrAD2hDeaC4b+xk9hYuB6M=">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</latexit>

t+1 =

1

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ps ns t + pu nu (1ns t + nu tt)<latexit sha1_base64="ghvlZoTOpW87GrZvaHEuGkzH2Gk=">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</latexit><latexit sha1_base64="brNLuCiOFjQy133+IQzmu0dg/P8=">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</latexit>

ps⇤<latexit sha1_base64="3S7fmsno2vvfjnp4a0uRz64HQ/o=">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</latexit><latexit sha1_base64="OnDPyy/N1OoxUFNxezeNdcGaSlg=">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</latexit><latexit sha1_base64="En0A3X3haLoWLphBKeT9WbJIrGg=">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</latexit>

pu

⇤
g

g,t+1 = pg
<latexit sha1_base64="L5/+go2u9M/iO3+tSI9z18fD+Lw=">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</latexit>

O<latexit sha1_base64="fj1t5Z5VnjI3M5gaQlQxucSKTCo=">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</latexit><latexit sha1_base64="En0A3X3haLoWLphBKeT9WbJIrGg=">AAACcHichVG7SgNBFD1Z3/GRqE3AwmhQxCLMRkGxEmwsfcUENIbddYxD9uXuJhAXf8AfsLAxgoj4GTb+gEU+QexUsLHw7mZBVNQ7zMyZM/fcOTOj2rpwPcZaMamjs6u7p7cv3j8wOJRIDo9su1bN0Xhes3TLKaqKy3Vh8rwnPJ0XbYcrhqrzglpdCfYLde64wjK3vIbNS4ZSMcWB0BSPqNKuTqn7Srmy58+elJMZlmVhpH8COQIZRLFmJa+xi31Y0FCDAQ4THmEdClxqO5DBYBNXgk+cQ0iE+xwniJO2RlmcMhRiqzRWaLUTsSatg5puqNboFJ26Q8o0ptgDu2Ev7J7dskf2/mstP6wReGnQrLa13C4nTlObb/+qDJo9HH6q/vTs4QCLoVdB3u2QCW6htfX147OXzaWNKX+aXbIn8t9kLXZHNzDrr9rVOt84R5w+QP7+3D/Bdi4rz2Vz6/OZ5Vz0Fb0YwyRm6L0XsIxVrCFP5x7hDBdoxp6llDQuTbRTpVikGcWXkGY/AOjJju8=</latexit>⇤g<latexit sha1_base64="OnDPyy/N1OoxUFNxezeNdcGaSlg=">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</latexit>

pu<latexit sha1_base64="3S7fmsno2vvfjnp4a0uRz64HQ/o=">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</latexit>⇤<latexit sha1_base64="brNLuCiOFjQy133+IQzmu0dg/P8=">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</latexit>

t,

ps

g,t

<latexit sha1_base64="Yy5KEsdwHyyuimejSFBMa6i3duw=">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</latexit>

1<latexit sha1_base64="ghvlZoTOpW87GrZvaHEuGkzH2Gk=">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</latexit>

Figure 2: The transition process of λt where ns < nu holds and the transition process of λg,twhen the ban on child labor is executed and nsg > nug holds.
family.
When the government imposes a ban on child labor, population dynamics becomes
λg,t+1 = p(eg ) ≡ pg by employing (55), (56), (59), and (60). We can obtain the ﬁgure of
population dynamics as per Figure 2, which indicates that the ban on child labor policy∂λ∗decreases the proportion of skilled workers in the economy. Moreover, ∂πgc < 0 holds in thesteady state because an improvement in child mortality decreases both skilled and unskilledparents’ investments in education based on (63). Therefore, once the ban on child labor is
implemented, an improvement in child mortality can no longer increase the proportion ofskilled workers in the economy; however, it strictly decreases not only this proportion but
also the average income in the economy. This result can be demonstrated as follows. We
deﬁne ytp as the average income per person in the economy: ytp = λt (ws −wu )+wu . Therefore,a decrease in the proportion of skilled workers reduces the average income.

21

4.2.

Child education policy

Here, we examine how the child education policy aﬀects the ratio of skilled workers
and the average income in the economy. The child education policy can be interpreted as
the provision of schools, which supports investments in the education of children from bothskilled and unskilled. Although unskilled parents do not possess basic abilities, such as
writing, reading, and arithmetic, they can help their children master these abilities at school.
The government executes this education policy by dividing the total tax revenue (τ Lt ) into
education policy and public health expenditure.
The government expends a fraction θ ∈ (0, 1) of the total tax revenue for the provisionof schools and the remaining fraction (1 − θ) for improving child mortality. We assume
the following functions for the child survival probability: π c (θ) ≡ π c [(1 − θ)τ ]: for the
probability of a child whose parent is skilled to become skilled: ps (θ) ≡ p[es (θ), θτ ]: andfor the probability of a child whose parent is unskilled to become skilled: pu (θ) ≡ p(ē, θτ ).
Because ps (θ) and pu (θ) are strictly increasing with investments in education and concave,s(θ)
as in Section 3, these probability functions also satisfy the following properties: ∂p>0∂es (θ)
s

u

(θ)
(θ)
and ∂p= ∂p> 0. We obtain the following results instead of (34), (36), and (37):∂(θτ )∂(θτ )γ(ϕ + β − 1)(ws − τ ),[(1 − γ)(1 + δ) + γ][dπ c (θ)wu − zws ]β[dπ c (θ)wu − zws ]es (θ) =,(ϕ + β − 1)π c (θ)wsγ(1 − ϕ)(wu − τ )nu (θ) =.
[(1 − γ)(1 + δ) + γ][z − (d − ē)π c (θ)]wu
ns (θ) =

(68)
(69)
(70)

The fertility rate of skilled parents increases with the ratio of child education expenditures(i.e., dndθ(θ) > 0). However, investments in education for skilled parents’ children decreaseswith θ (i.e., dedθ(θ) < 0). Conversely, the fertility rate of unskilled workers decreases withu

θ (i.e., dndθ(θ) < 0). We derive the population dynamics under this model in an analogous

22manner to (43) in Section 3:

λt+1 (θ) =ps (θ)ns (θ)λt (θ) + pu (θ)nu (θ)[1 − λt (θ)].
ns (θ)λt (θ) + nu (θ)[1 − λt (θ)](71)

Then, we obtain the following proposition.
ss](θ),θτ ]Proposition 7. We assume that dps (θ) = ∂p[e∂es(θ),θτdes (θ) + ∂p[e∂(θτd(θτ ) > 0. An increase(θ))

in the ratio of child education expenditure has a positive eﬀect on the proportion of skilledworkers and the average income in the steady state.
Proof. We consider population dynamics based on (71) in the steady state. Then, thefollowing equation is satisﬁed in the steady state:[nu (θ) − ns (θ)][λ(θ)]2 + {−[1 + pu (θ)]nu (θ) + ps (θ)ns (θ))}λ(θ) + pu (θ)nu (θ) = 0.

(72)

Diﬀerentiating (72) with respect to θ yields
dnu (θ) dns (θ)dλ(θ)−][λ(θ)]2 + 2[nu (θ) − ns (θ)]λ(θ)
dθ
dθ
dθ
u
s
dpu (θ) u
dn
(θ)
dp
(θ)
dns (θ)+ {−n (θ) − [1 + pu (θ)]+
ns (θ) + ps (θ)
}
dθ
dθ
dθ
dθ
dnu (θ)dλ(θ) dpu (θ) u+n (θ) + pu (θ)= 0.
+ {−[1 + pu (θ)]nu (θ) + ps (θ)ns (θ)}
dθ
dθ
dθ

[

Therefore, we obtain the following comparative static result:
dλ(θ)
=
dθ
u
u
s
s
−[1 − λ(θ)][λ(θ) − pu (θ)] dndθ(θ) + [1 − λ(θ)]nu (θ) dpdθ(θ) + [ps (θ) − λ(θ)]λ(θ) dndθ(θ) + ns (θ)λ(θ) dpdθ(θ)1[Φ(θ)] 2
> 0,

(73)
u

u

s

s

(θ)−p (θ)n (θ)−[Φ(θ)]
where λ(θ) ≡ [1+p (θ)]n2[nu (θ)−ns (θ)]

12and Φ(θ) ≡ {[1 + pu (θ)]nu (θ) − ps (θ)ns (θ)}2 −4pu (θ)[nu (θ) − ns (θ)]nu (θ).
23,

An increase in the ratio of child education expenditure has two opposite eﬀects on the probs

s

s

s

] de (θ)(θ),θτ ] d(θτ )ability of skilled workers (i.e., dpdθ(θ) = ∂p[e∂es(θ),θτ· dθ + ∂p[e∂(θτ· dθ ). One is the nega(θ))

tive eﬀect on investments in education through an increase in the child mortality rate (i.e.,des (θ)< 0), while the other is the positive eﬀect on investments in education through schooldθ)> 0). Assumption dps (θ) > 0 ensures that the positive eﬀect of an
education (i.e., d(θτdθincrease in the ratio of child education expenditure dominates the eﬀect of a decrease in investments in education through the child mortality rate increase. From (73), all four terms in
the numerator take positive values. Since these four terms positively inﬂuence the proportionof skilled workers and average income in the economy, the child education policy is the mostpromising among the three policies.

5.

Concluding remarks

We considered a three-period overlapping generations model to analyze how child mortalityaﬀects fertility, child labor, and investments in education in an economy where skilled andunskilled workers coexist. In the proposed model, an improvement in child mortality has
diﬀerent eﬀects on the diﬀerent worker types. A ban on child labor and an improvementin child mortality can decrease the proportion of skilled workers and the average income inthe economy. However, a child education policy, which supports both skilled and unskilledworkers’ investments in the education of their children by building schools, can increase theproportion of skilled workers and the average income in the economy.

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