GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.77

Perverse effects of a ban on child labour
in an overlapping generations model

Kouki Sugawara, Atsue Mizushima, and Koichi Futagami

August 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Perverse effects of a ban on child labour in an overlapping
generations model∗
Kouki Sugawara †, Atsue Mizushima ‡, Koichi Futagami §August 19, 2009

Abstract
Despite the International Programme on the Elimination of Child Labour, child labour
remains particularly common in developing countries. Why has the Programme failed toachieve the expected outcome? To address this question, we construct a two-period overlapping generations model with the detection probability in regard to the ban on child labour,which depends on government expenditure. By analyzing this model, we show that whether
the ban on child labour has a suppression effect or not depends on the human capital levelof parents. We also demonstrate the human capital dynamics in the model and show that
there exist multiple equilibria: one is the poverty trap, which has a higher incidence of childlabour and a low level of human capital; the other is the equilibrium that has a lower level ofchild labour and a high level of human capital. Finally, we incorporate public education into
the model and obtain the expenditure rate for monitoring child labour and education, which
minimizes the level of child labour.
JEL classification:

I21, I28, O11

Keywords: ban on child labour, detection probability, human capital∗

We are grateful to Ken-ichi Hashimoto, Jyunichi Itaya, Noritaka Kudoh, Tadashi Morita, Yoshiyasu Ono,
Yasuyuki Sawada and seminar participants at Osaka University and Hokkaido University, and the spring meetingof the Japanese Economic Association at Kyoto University. All remaining errors are naturally our own.
†
Graduate School of Economics, Osaka university, 1-7 Machikaneyama, Toyonaka 560-0043, Japan, E-mail:dg025sk@mail2.econ.osaka-u.ac.jp‡
Department of Economics, Otaru University of Commerce, Midori 3-5-21, Otaru, 047-850, JAPAN, E-mail:mizushima@res.otaru-uc.ac.jp§
Faculty of Economics, Osaka university, 1-7 Machikaneyama, Toyonaka 560-0043, Japan, E-mail:futagami@econ.osaka-u.ac.jp1

1

Introduction

Because child labour damages children’s psychological and physical growth and restricts theiropportunity to obtain an education, various regulations or conventions have been established inorder to eliminate child labour. Convention 138 of the International Labour Organization (ILO)
provides a definition of child labour. Under the Convention, child labour should be restricted tolight work that does not prevent a child from being educated in school. One hundred and fifty
member nations have signed Convention 138, and each individual nation passes laws to preventthe problem and makes an international treaty with other nations. This includes child labour
regulation, that is, governments monitor firms/persons that employ children and fine them fordoing so. For example, in India, the government issues fines of between Rs 10,000 and Rs 20,000
per year for firms, whereas in Thailand, the government issues fines of up to 200,000 Baht or
imposes a one-year prison sentence for a person or a firm that employs children who are fifteen
years old or less.1
Despite these regulations, in developing countries, many children are involved in economic
activities. In percentage terms, the proportion of children aged between 5 and 14 years who
work amount to 19 % in Asian and Pacific countries, 16 % in Latin American countries, 29 % in
Caribbean countries, 29 % in Sub-Saharan African countries, and 15 % in Middle Eastern andNorth African countries. (See the Statistical Information and Monitoring Programme on Child
Labour (SINPOC).)
We consider why the regulation of child labour does not achieve its expected outcome. To
address this question, this paper introduces the detection probability of a government that investigates child labour. For this purpose, we set up a two-period overlapping generations modelwith human capital, in which we assume that the detection probability depends on government
expenditure. Then, we examine how changes in government expenditure affect the individual
decision making of households in relation to child labour. In the model, parents are the decision
makers for all members in a family. Parents derive their utility from consumption in their old age,and the level of human capital of their children. However, they rely not only on their own income
but also on labour income from their children. Thus, they allocate their children’s endowed time
between child labour and education, that is, they must send their children to workplaces, forexample factories or farms.
1

In the United States, the Department of Labor fined the Wal-Mart Store $135,540 for violating the youth
employment provisions of the Fair Labor Standards Act in 2005.

2

Numerous studies have examined the incidence of child labour. For example, Gupta (2000)
examined the wage determination of a child worker. He showed that firms enforce lower wage
rates for children, which results in an increased number of child labourers when the firms havethe bargaining power over the child wage. Rogers and Swinnerton (2008) examined issues on
exploitation child labour by farms. They showed that the policy that reduces exploitative childlabour is Pareto improving.2 In contrast to these studies, in our work, we focus our attention on
the ban on child labour and examine how the ban on child labour affects the incidence of childlabour.
Our analysis builds primarily on previous research by Basu and Van (1998) and Basu (2005).
Assuming substitutability in production between child labour and adult labour, Basu and Van(1998) showed that there exist multiple equilibria in the labour market: in one equilibrium,
children work, whereas in the other equilibrium, they do not work. They showed that regulation offirms simply increases the volume of child labour and leads the economy to the worse equilibrium.
Basu (2005) also showed that an increase in the fine for employing children leads to an increase inchild labour. We also assume substitutability in production between child and adult labour, andthen incorporate the detection probability in regard to the ban on child labour into the present
model. In contrast with the results of the above-mentioned authors, we show that whether thegovernment’s regulation increases the incidence of child labour depends on the human capitallevel of parents.
The crux of the problem of child labour from an economic development perspective is thatit restricts the opportunity for children to have an education and prevents the economy fromdeveloping. For example, using Zambian data, Jensen and Nielsen (1997) showed that poverty
forces households to keep their children away from school. In addition, using data from Peru,
Patrinos and Psacharopoulos (1997) showed that even if they do not work, many children still
will not go to school at all. Furthermore, SIMPOC data indicate that there is a higher incidenceof child labour in developing countries compared with developed countries (ILO (2002)). In thispaper, we present a theoretical model that supports these empirical results and examine theeffects of public education on the incidence of child labour.
This paper is organized as follows. Section 2 develops the model. Section 3 characterizes
2

The Convention concerning the Prohibition and Immediate Action for the Elimination of the Worst Forms ofChild Labour, known in short as the Worst Forms of Child Labour Convention, was adopted by the ILO in 1999.
By ratifying this Convention 182, a country commits itself to taking immediate action to prohibit and eliminate
the worst forms of child labour. For analysis of the Worst Forms of Child Labour, see Basu and Chau (2004),
Dessy and Pallage (2005) and Sugawara (2008).

3

the equilibrium and analyzes the impacts of government regulation on child labour. Section 4analyzes the dynamics of this economy. Section 5 states some extensions. Section 6 concludes
the paper.

2

The Model

Consider an infinite-horizon economy composed of households, perfectly competitive firms and agovernment. A new generation of a unit mass referred to as generation t is born in period t − 1.
Time is indexed by t = 1, 2, · · · . Generation t ≥ 1 is composed of a continuum of Nt units wholive for two periods, childhood and adulthood.

Firms
We assume that there are two sectors: the illegal sector and the legal sector, both of which
produce the same goods in the economy. A fraction of firms, p ∈ (0, 1), operates in the illegal
sector, whereas 1 − p firms operate in the legal sector. In what follows, we express the status offirms by using indexes I and L, where I denotes the firms of the illegal sector and L denotes thoseof the legal sector. In the illegal sector, firms employ children illegally and produce goods byusing child labour and the human capital of adult agents according to the following production
technology:
ytI = wc lt + wA ht ,where wc , lt , wA and ht , respectively, represent the productivity of children, the incidence of childlabour, the productivity of adults and the level of human capital of adult agents. We assume
that individuals have full information about the types of firms. Accordingly, parents who want
their children to work send their children to the illegal sector.
In the legal sector, firms employ only the human capital of adult agents and produce goods
according to the following production technology:
ytL = wA ht .
We assume that the productivity of adults is the same in both sectors. Thus, adult agentsare indifferent between working in the legal sector or in the illegal sector. Then, a fraction,p ∈ (0, 1), of adults work in the illegal sector and 1 − p work in the legal sector. We also assume
that wc < wA ; that is, the wage rate of adults is higher than that of children. Because the firms
in the legal sector do not hire children, children work only in the illegal sector.
4Government
We assume that the government has the authority to ban child labour. The government levies
income tax on households and uses the funds for the ban on child labour. The government enforces
the child labour ban through monitoring. In each period, the government randomly selects firms
and monitors them to inspect whether they employ children. For example, the Ministry of Labour
and Social Security in the Republic of Turkey carried out such a program enforcing the supervisionof child labour from 1994. Accordingly, the number of working children aged fifteen years oldor less decreased between 1997 and 1999 in Turkey (OECD 2003). We denote the probability ofdetection of child labour as π(g) (where 0 ≤ π(g) ≤ 1), where g shows the expenditure required
for the government to carry out the inspection program. The probability of detection also has
the following properties: π 0 (g) > 0, π 00 (g) < 0, π(0) = 0, limg→∞ π(g) = 1, limg→0 π 0 (g) = ∞,and limg→∞ π 0 (g) = 0. If a firm employs children, it is certain that the authority will find thisout.The effective probability of detection is decided by pπ(gt ), because only the illegal sector
employs children. When the government finds incidents of child labour by monitoring firms, thedetected firms are fined an amount equivalent to the output produced by the children. Notingthat the aggregate amount of fines is π(g)wc lt , the budget constraint of the government is asfollows:
g = Tt + pπ(g)wc ltd ,where Tt shows lump-sum transfers between households and the government. When Tt > 0, it
indicates a per capita lump-sum tax, whereas when Tt < 0, it indicates a per capita lump-sum
subsidy. When we assume that child labour is distributed uniformly in the illegal sector, theldequilibrium of the child labour market becomes pt = lt . Therefore, the budget constraint of thegovernment is as follows:
Tt = gt − π(gt )wc lt .

(1)

Households
An agent is endowed with one unit of labour in childhood and adulthood, respectively. Parents(generation t − 1) allocate the time endowment of their children (generation t) between working
time, lt , and schooling time, et . In each period, the sum of the time spent working and the timespent in schooling must be unity and each time commitment must lie between zero and unity:
lt + et = 1, 0 ≤ lt ≤ 1, 0 ≤ et ≤ 1.

5(2)

Adult agents of generation t + 1 are also endowed with one unit of time. They supply their wholetime and human capital to firms and earn the effective wage income, wA ht .
Following Galor and Weil (2000), Holtz-Eakin, Lovely and Tosun (2000) and Tabata (2003),
we assume that the time spent in schooling increases the agents’ level of human capital. For
simplicity, we assume that agents born in period t obtain human capital as follows:
ht+1 = (δ + γet )σ , δ > 0, γ > 0, 0 < σ < 1,(3)

where γ and σ express, respectively, the parameters of efficiency of education and the adjusted
elasticity of the human capital with respect to an educational level. When et = 0, the level ofhuman capital is δ σ . Therefore, δ expresses the adjusted innate ability of human beings.
In what follows, we distinguish between families by using indexes i = c, f . i = c indicates afamily that can obtain wage income from child labour. It indicates that, despite the ban on childlabour, the government cannot completely find out the child labour of each family. Thus, somefamilies can evade the ban and receive wage income from child labour. i = f indicates a family
that does not obtain a wage income from child labour. It indicates that because the government
has found out the child labour, the family cannot receive the wage income from their children
working in the illegal sector.
We assume that adult agents consume all of their family income; that is, the wage income
from their own labour and that from child labour. Consequently, the budget constraints become
as follows:(
cti,t+1 =wA ht+1 − Tt+1 ,wA ht+1 + wc lt+1 − Tt+1 ,if i = f,if i = c.

(4)Each agent of generation t has preferences over his or her consumption in each state, cti,t+1 , andthe level of human capital of his or her children, ht+2 . The probability of the agent’s consumption
becoming ctf,t+1 is π(gt+1 ), which is the level of probability that the government will find out thechild labour of each family. Meanwhile, the agent’s consumption is ctc,t+1 with a probability of
1 − π(gt+1 ). We assume that each agent has the following expected lifetime utility function:
Ut = π(g) log ctf,t+1 + (1 − π(g)) log ctc,t+1 + β log ht+2 ,(5)where β ∈ (0, 1) indicates the degree of altruism towards their children.
As parents are the decision makers of families, they determine the allocation of their children’stime by maximizing (5). Hence, the optimization problem of each parent of generation t can be6expressed as follows:
max Ut+1 = π(g) log(wA ht+1 − Tt+1 ) + (1 − π(g)) log(wA ht+1 + wc lt+1 − Tt+1 )lt+1+ β log(δ + γ(1 − lt+1 ))σ ,s.t.

0 ≤ lt+1 ≤ 1.

The first-order condition with respect to lt+1 is given by:∂Ut+1(1 − π(g))wc
βσγ=−
S 0, (with equality if 0 < lt+1 < 1). (6)∂lt+1
wA ht+1 + wc lt+1 − Tt+1 δ + γ(1 − lt+1 )The first term on the right-hand side of (6) represents the marginal benefit of child labour. The
second term on the right-hand side of (6) represents the marginal cost of child labour.

3

Equilibrium and the effect of the ban on child labour

In this section, we first examine how parents allocate the time of their children, given the humancapital level of the parents and the detection probability of child labour. Then, we examine howchanges of government expenditure affect the incidence of child labour.
By using (1) and (6), we obtain the following relationship between child labour and humancapital:lt+1 (g; ht+1 ) = lt+1


1,


 (1 − π(g))w (γ + δ) − βγσ(w hc
A t+1 − g),=γwc (βσ(1 + π(g)) + (1 − π(g)))
0,where: h̃ ≡(1−π(g))wc ( δ−βγσ)+gβγσ, and:wAh̄ ≡(1−π(g))wc ( δ+γ
)+g
βγσ
.
wA

if h < h̃,
if h̃ ≤ h < h̄,(7)

if h̄ ≤ h,
As can be seen from (7), the time

allocation of child labour depends on the human capital level of the parents, ht+1 , and thegovernment expenditure level, g. When the level of human capital is sufficiently low, that is,
ht+1 < h̃, parents cannot earn sufficient income and must rely on child labour to assist theirfamily. Therefore, the parents force their children to work instead of obtaining an education. In
contrast, when the level of human capital is sufficiently high, that is, h̄ ≤ ht+1 , parents can earnsufficient income and they do not need to send their children to a factory or farm. Therefore, theparents send their children to school to obtain an education. Figure 1 depicts this relationshipand shows that child labour, lt+1 , decreases with an increase in the human capital level of parents,
ht+1 , when ht+1 is in the internal [h̃, h̄].
7

Next, let us examine how changes in government expenditure influence the decision makingof parents. We can state the following lemma for this effect.
Lemma 1 An increase in government expenditure raises the level of child labour if and only if
ht+1 < ĥ, where ĥ ≡(βσ(1+π(g))+(1−π(g)))+(1−βσ)g−2wc δ+γγ
π 0 (g)(1−βσ)wA.

Proof. Differentiating lt+1 with respect to g, we have:∂lt+1 (g; ht+1 )=∂g(1 − βσ)βγσg + βγσ( βσ(1+π(g))+(1−π(g)) − βσ[2wc (δ + γ) + γ(1 − βσ)wA ht+1 ]
π 0 (g)γwc [βσ(1 + π(g) + (1 − π(g)]2.

As the denominator takes a positive value, the numerator determines the sign of this derivative.
The sign of the numerator is positive if and only if ht+1 < ĥ; that is, an increase in government
expenditure raises the level of child labour.
This lemma indicates that government expenditure has two effects on child labour. One is an“income effect”. Increases in government expenditure decrease the family income because theymust either raise the level of lump-sum taxes or reduce the level of lump-sum transfers due tothe balanced budget requirement of the government. This increases the burden on families that
rely on child labour. Therefore, this effect has a positive impact on the incidence of child labour.
The other effect is a “substitution effect”. As an increase in detection probability decreases the
expected income from child labour, it decreases the benefit of child labour. Therefore, this effecthas a negative impact on the incidence of child labour. The impact of increases in lump-sum
taxes imposes a much heavier burden on the families with a lower level of human capital than on
the family with higher levels of human capital. Thus, the positive income effect dominates thenegative substitution effect.
We can classify the effects of the government expenditure on child labour into the following
three cases according to the levels, h̃, ĥ and h̄. To compare ĥ with h̃ and h̄, it is convenient to defineβσ(1+π(g))+(1−π(g))
the following functions: H̃(g) ≡ (1 − π(g))(1 − βσ)wc δ−βγσ− 2wc δ+γ
βγσ , Ĥ(g) ≡γ ,π 0 (g)and H̄(g) ≡ (1 − π(g))(1 − βσ)wc δ+γβγσ . It can be easily seen that the order of h̃, ĥ and h̄ is thesame as the order of H̃(g), Ĥ(g) and H̄(g).
Noting that π(0) = 0, limg→∞ π(g) = 1, limg→0 π 0 (g) = ∞, and that limg→∞ π 0 (g) = 0, we
can easily obtain that: H̃ 0 (g) < 0, H̃(0) = (1 − βσ)wc δ−βγσ
βγσ , and limg→∞ H̃(g) = 0, and, alsothat H̄ 0 (g) < 0, H̄(0) = (1 − βσ)wc δ+γβγσ , and limg→∞ H̄(g) = 0. Furthermore, it is obvious that8H̃(g) < H̄(g). On the other hand, asĤ 0 (g) =

´
³
1
0
2
00
−((π
(g))
(1
−
βσ)
−π
(g)(βσ(1
+
π(g))
+
(1
−
π(g))
,
{z
}|
{z
}
(π 0 (g))2 |
(−)

(+)

the sign of Ĥ(g) is ambiguous. However, we can show that limg→0 Ĥ(g) = 0 and limg→∞ Ĥ(g) =∞. Therefore, intersection points of the graphs of H̃(g) and Ĥ(g) and those of the graphs Ĥ(g)and H̄(g) exist. We assume that there is a unique intersection point of the graphs of H̃(g) andĤ(g). Let gA denote the value at which H̃(gA ) = Ĥ(gA ). Similarly, let gB denote the value atwhich Ĥ(gB ) = H̄(gB ). These relationships are depicted in Figure 2.
Figure 3 depicts the case of H̃(g) < Ĥ(g) < H̄(g) that shows the regime gA < g < gB . In
this regime, the income effect of government expenditure dominates the substitution effect if thelevel of human capital is low, and vice versa. Figure 4 depicts the case of H̃(g) > Ĥ(g) thatshows the regime gB < g. In this regime, the substitution effect always dominates the income
effect. Figure 5 depicts the case of Ĥ(g) > H̄(g) that shows the regime g < gA . In this regime,
the income effect always dominates the substitute effect. The following proposition formalizes
this observation.
Proposition 1 When the government expenditure is sufficiently small (large), an increase in thegovernment expenditure decreases (increases) the incidence of child labour.
When the government expenditure, g, is sufficiently small, each family can receive the lumpsum transfers from the government. Thus, parents prefer that their children obtain an education
to them working. In contrast, because limg→0 π 0 (g) = ∞, the marginal effect of the detection
probability is sufficiently large when the government expenditure is small. Thus, when thegovernment expenditure is small, the negative substitution effect dominates the positive income
effect, and vice versa.

4

Dynamics of human capitalIn this section, we examine the dynamics of human capital. Using (2), (3) and (7), the dynamics
of the per capita human capital is represented as follows:σδ ≡ Γ1 (ht )ht+1 = Γ(ht ) = Γ2 (ht )(δ + γ)σ ≡ Γ3 (ht )9if ht ≤ h̃,
if h̃ < ht ≤ h̄,
if h̄ < ht .

(8)

³
where: Γ2 (ht ) ≡wc βσ(1+π(g))(δ+γ)+βγσ(wA ht −g)
wc (βσ(1+π(g))+(1−π(g))´σ. We can easily show that Γ02 (h) > 0 andΓ002 (h) < 0. The steady state is expressed as the stationary level of the per capita human capital,h, such that h = Γ(h). There can be several patterns of dynamics. To avoid unnecessary
complexity, we focus on some interesting cases, which are described in Figures 6 and 7.
Figure 6 shows the case where the following condition holds:³´³ βγσ ´
wA δ σ −g< (1−π(g))wc <δ − βγσÃ! σ

wc (1 + π(g))( δ+γ
γ )−g
(βγσ)

σ
1−σ

1
1−σ

wA (σ

1
1−σ

−σ

σ
1−σ

1−σ

−wc βσ(1+π). (9))

In this case, there are three steady states. As δ σ < ĥ, the graph of Γ1 (ht ) intersects with the
45-degree line once. We denote this intersection as E1 and define the level of h at E1 as h∗ .
In addition, the appendix proves that the graph of Γ2 (ht ) has intersections with the 45-degreeline. Thus, we denote these intersections as E2 and E3 , respectively, and define the levels of h
at E2 and E3 as h∗∗ and h∗∗∗ . It is easily confirmed that E1 and E3 are the stable steady-state
equilibria. E1 is the steady state characterized by low income and low education. E3 is thesteady state characterized by high income and high education. E2 is the unstable steady-state
equilibrium and shows the threshold level of human capital. When the initial level of humancapital is lower (higher) than h∗∗ , the economy will converge to E1 (E3 ). Therefore, the initial
per capita human capital h0 is crucial for the destiny of the economy.
On the other hand, Figure 7 shows the case where the following condition holds:³ βγσ ´
(1 − π(g))wc < (wA δ σ − g).
δ − βγσ(10)

In this case, as δ σ > h̃, only the graph of Γ2 (ht ) intersects with the 45-degree line. Thus, thedynamical system in Figure 7 has a unique steady-state equilibrium, E3 , characterized by highincome and high education.
Our concern is to examine how changes in government expenditure affect the dynamics of theeconomy. For this purpose, we focus our attention on the economy that has multiple equilibria.
The following proposition shows the condition that derives the poverty trap equilibrium; that is,
the steady state characterized by low income and low education.
Proposition 2 Suppose that wwAc > δ−βγσ
δ σ βγσ , the poverty trap equilibrium exists when government
expenditure is sufficiently large.
βγσ
Proof. Suppose Ω1 (g) ≡ (1−π(g))wc (wA δ σ −g) and Ω2 (g) ≡ ( δβγσ), the poverty trap equilibrium

exists if and only if Ω1 (g) < Ω2 (g), as on the left-hand side of (9). Noting that Ω1 (0) < Ω2 (0)10holds when wwAc > δ−βγσ
δ σ βγσ holds and that limg→∞ π(g) = 1, then: Ω1 (g) = Ω2 (g) at the pointg ∗ ≡ δ σ . Therefore, Ω1 (g) > Ω2 (g) exists when g ∗ < g holds.
Proposition 2 shows that if the relative wage rate differential between an adult and a child islarge, that is, if the wage rate of a child is low in comparison to the adult wage, a higher amountof government expenditure can lead the economy to the poverty-trap equilibrium. When thewage rate of children is sufficiently low, the income effect dominates the substitution effect, and,thus, the level of government expenditure has a significant impact on individual decision making
in relation to child labour. When the government expenditure, g, is sufficiently small, each family
can receive lump-sum transfers from government. On the other hand, each family has to pay tax(see Proposition 1) when the government expenditure is sufficiently large. Therefore, it imposesa great tax burden on each family when the level of government expenditure is sufficiently large.
This effect leads to a higher incidence of child labour and moves the economy into the poverty
trap.
As shown by Lemma 1, when the level of human capital is smaller (higher) than the criticallevel of human capital ĥ, an increase in government expenditure increases (decreases) the levelof child labour. It is convenient to analyze the facts in three cases: (i) ĥ < h̃, (ii) h̃ < ĥ < h̄, and(iii) h̄ < ĥ. Figures 8, 9 and 10 respectively show the cases (i), (ii) and (iii). These figures showthat changes in the level of child labour have a substantial impact on the dynamical system of
the economy. When the level of human capital is smaller than the critical level, ĥ, the dynamicalsystem shifts downward, and vice versa. Therefore, we show that government expenditure havepossibility to expand the level of human capital to take off the poverty trap.

5

Education Policy

In this section, we incorporate the education policy of the government into the model. Most
developing countries attempt to improve the efficiency of education in parallel with controllingchild labour. For example, India, Nicaragua and The Philippines have implemented return-toschool programs, Brazil and Mexico have provided financial support for education, and Egypt
and The Philippines have improved educational infrastructure. Thus, we examine how public
education policy can increase the efficiency of education. The budget constraint of the governmentis rewritten as follows:
π(g)wc lt + Tt = g + u,

11

(11)

where u shows the level of public education. The government increases the efficiency of education
as follows: γ 0 (u) > 0, γ 00 (u) < 0 and γ(0) = γ. Then, the human capital accumulation equation
becomes as follows:
ht+1 = (δ + γ(u)et )σ .

(12)

We assume that the government divides its revenue between expenditure on monitoring of childlabour and public education. Thus, we have the following constraints:
g = φ(π(g)wc lt + Tt ),(13)

u = (1 − φ)(π(g)wc lt + Tt ),(14)

where φ denotes the share of the government expenditures devoted to monitoring. ¿From (13)and (14), we have the following relationship:
u=

1−φ
g.
φ

(15)

Taking π(g), u and ht+1 as given, agents maximize (5) subject to (2), (4), (11), (12) and (13).
Solving this problem in a similar way to that employed in section 2, we obtain the level of childlabour as follows:
lt+1 (g; ht+1 ) = lt+1
1,
if h < h̃∗ , (1 − π(g))wc (γ( 1−φ g) + δ) − βγ( 1−φ g)σ(wA ht+1 − 1 g)φφφ=
, if h̄∗ < h < h̃∗ ,
1−φ

g)w
(βσ(1
−
π(g))
+
(1
−
π(g)))
γ(

c
φ



0,
if h̄∗ < h,(1−π(g))wc (where h̃∗ ≡

1−φ
g)
φ
)+g
1−φ
βσγ(
g)
φ

δ−βσγ(wA(1−π(g))wc (and h̄∗ ≡

1−φ
g)
φ
)+g
1−φ
βσγ(
g)
φ

(16)

δ+γ(

wA

.

We examine how the increase in public education affects the level of child labour. The
following proposition shows how a change in the ratio φ affects the incidence of child labour.
t+1
t+1
Proposition 3 There exists a unique φ∗ ∈ (0, 1) such that ∂l∂φ
< 0 ∀φ ∈ [0, φ∗ ] and ∂l∂φ>

0 ∀φ ∈ [φ∗ , 1].
Proof. Differentiating lt+1 with respect to φ, we have the following:
Ω(φ)
∂lt+1
=
,
1−φ
0
2
∂φ
gwc γ (g)(−1 + φ) φ2 (1 − π(g) + βσ(1 + π(g)))
φ

2
2
0 1−φ
2 0 1−φwhere: Ω(φ) ≡ −(g 2 βγ 0 ( 1−φφ g)σ − (1 − π(g))wc δ)φ + 2g βγ ( φ g)σφ − g γ ( φ g)σ. As thedenominator is positive, the numerator determines the sign of these derivatives. As Ω(0) =12

−g 2 γ 0 ( 1−φφ g)σ < 0 and Ω(1) = wc δ(1 − π(g)) > 0, noting that Ω(φ) is a quadratic function withrespect to φ, we have a unique φ∗ ∈ (0, 1) such that Ω(φ) = 0. It follows that Ω(φ) < 0 ∀φ ∈ [0, φ∗ ]and Ω(φ) > 0 ∀φ ∈ [φ∗ , 1]. This proves the proposition.
This proposition shows that there exists a ratio of expenditure, φ∗ , between monitoring childlabour and education, that achieves the minimum level of child labour. This ratio does not dependon the level of human capital. As stated in section 3, an increase in government expenditure has
a positive “income effect” and a negative “substitution effect” on the incidence of child labour.
In addition to these effects, it has a negative “education effect” on the incidence of child labour
because public education increases the marginal cost of child labour. When the government
expenditure ratio, φ, is small, the “substitution effect” and the “education effect” dominate the“income effect” because the government spends more on public education than on monitoringchild labour. When the government expenditure ratio, φ, is large, the “income effect” dominates
the “substitution effect” and the “education effect” because the government spends less on public
education than on the monitoring of child labour. Therefore, we obtain the expenditure rate, φ,that minimizes the level of child labour.
The dynamics of this economy is given as (8). To examine how public education policy affects
the dynamics, we analyze its effects on inequalities (9) and (10), taking public education into
account. Note that inequality (9) (resp. (10)) holds if and only if the efficiency of education,γ, is small (resp. large). The poverty trap does not appear if the efficiency of education is
sufficiently large, as Proposition 2 shows. Therefore, an inequality can change from (9) to (10)if the government increases the efficiency of education. Thus, public education can help the
economy escape from the poverty trap of the low-education and low-development equilibrium.

6

Conclusion

In this paper, we have analyzed how government expenditure on monitoring to detect firms
illegally employing children affects the incidence of child labour. For this purpose, we set up
a two-period overlapping generations model. In the model, there exist two sectors: one is thelegal sector that produces output by using the human capital of adult agents, and the other isthe illegal sector, which produces output by using the human capital of adult agents and childlabour. We assumed that the government bans child labour and monitors firms to prevent childlabour. However, despite the monitoring, the government will not detect all instances of childlabour. We also assumed that the detection probability in relation to child labour depends on the13

level of government expenditure. Then, we examined how the change in government expenditureaffected the incidence of child labour.
In the first part of the paper, we analyzed the above basic model and showed that government
intervention in relation to the ban on child labour has perverse effects when the level of humancapital is sufficiently low, that is, an increase in the government expenditure on monitoring anddetection increases the level of child labour. In the second part of this paper, we extended ourmodel by taking account of public education to enhance the efficiency of education. By analyzing
the model, we obtained the expenditure ratio between monitoring child labour and educationthat minimizes the level of child labour. Public education also provides a possible means bywhich the economy can escape from the poverty trap.

Appendix
Appendix
In this appendix, we show that there exist three steady states that satisfy h = Γ(h) if and only
if inequalities (9) hold. First, we consider the property of Γ1 (h). As shown in Figure 5, whenδ σ < ĥ, the graph of Γ1 (h) intersects with the 45-degree line once. ¿From the definition of ĥ, theβγσinequality δ σ < ĥ holds when wA δ σ − gt+1 ( δ−βγσ) < (1 − π(gt+1 ))wc .

Next, let us consider the property of Γ2 (h). ¿From (8), the relation h = Γ2 (h) is rearranged
as:
1

hσ =

wc βσ(1 + π(gt+1 ))(δ + γ) + βγσ(wA h − gt+1 ).
wc (βσ(1 + π(gt+1 )) + (1 − π(gt+1 ))

(17)

1

Ah
In addition, we define the value of h that satisfies σ1 h1− σ = wc (βσ(1+π(gβγσw, as h0 .
t+1 ))+(1−π(gt+1 ))01t+1 ))(δ+γ)+βγσ(wA h −gt+1 )If h0 σ = wc βσ(1+π(gholds, then the graphs on the left-hand side andwc (βσ(1+π(gt+1 ))+(1−π(gt+1 ))1right-hand side have two intersections. ¿From the above equation, the condition σ1 h01− σ >βγσwA h0wc (βσ(1+π(gt+1 ))+(1−π(gt+1 )) holds when:wc (1 − π(gt+1 )) < βσ³¡ wc (1 + π(gt+1 ))( δ+γ − gt+1 )1−σ ¢´σγ− (1 + π(gt+1 )) .
σwA

(18)

Thus, the graph of Γ2 (h) intersects with the 45-degree line twice if and only if (18) holds. Therefore, noting that Γ2 (ĥ) = δ σ , the graph of Γ(h) has three intersections with the 45-degree line if
and only if inequalities (9) hold.

14

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16

lt+1
1

0

h̃

h̄

ht+1

Figure 1: The incidence of child labour

H(g)
Ĥ(g)

H̄(g)

H̃(g)

gA gB
Figure 2:

17

g

lt+1
1

0

h̃

ĥ

h̄

ht+1

Figure 3: The effect of a ban on child labour in the case where h̃ < ĥ < h̄
lt+1
1

0

ĥ h̃

h̄

ht+1

Figure 4: The effect of a ban on child labour in the case where ĥ < h̃ < h̄
lt+1
1

0

h̃

h̄

ht+1

Figure 5: The effect of a ban on child labour in the case where h̃ < h̄ < ĥ

18

ht+1
45 − degree lineΓ3 (ht )(δ + γ)σE3Γ2 (ht )Γ1 (ht )
δσ

0

E2

E1

h∗ h̃ h∗∗

h∗∗∗

h̄

ht

Figure 6: Dynamics of human capital in the case where δ σ < h̃

19

ht+1
45 − degree lineΓ3 (ht )(δ + γ)σE3Γ2 (ht )Γ1 (ht )
δσ

0

h∗∗∗

h̃

h̄

ht

Figure 7: Dynamics of human capital in the case where δ σ > h̃

20

ht+1
45 − degree lineE30

(δ + γ)σ

E3

δσ

0

E2

E20
E1

h̄

ĥ h̃

ht

Figure 8:ht+1
45 − degree line(δ + γ)σ
E3

E2
δσ

0

E1

h∗ h̃ h∗∗ĥFigure 9:

21

h∗∗∗

h̄

ht

ht+1
45 − degree lineE30

(δ + γ)σδσ0h∗ h̃ h∗∗h∗∗∗Figure 10:22h̄ ĥ ht