Discussion Papers In Economics
And BusinessChild Labor, Corruption, and Development

Toshiki Miyashita
Kohei Okada
Kei Takakura

Discussion Paper 20-20

January 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANChild Labor, Corruption, and Development∗
Toshiki Miyashita,†Kohei Okada‡and Kei Takakura§

Abstract
Employing an overlapping-generations model with endogenous education choice and corruption, we investigate how child labor and corruption influence human capital accumulation anddevelopment. We show that multiple steady-states exist in the economy. One steady-state has ahigh level of human capital, and the other has a low level of human capital. In the steady-statewith a low level of human capital, child labor and corruption exist and welfare is low. In thesteady-state with a high level of human capital, child labor and corruption are diminished andwelfare is high. In addition, we show that it is diﬃcult to steer an economy away from a poverty
trap with child labor and corruption because bureaucrats of the current generation are opposed
to policy changes such as reinforcement of monitoring and penal regulations. However, we canapply the Pareto-improving policy to this poverty trap, for e.g., the government receives funds
from an international organization and distributes them among bureaucrats, which keeps themfrom being corrupt.
Keywords: Child labor, corruption, human capital accumulation, development
JEL Classification: D73, E24, I25, J13∗

We thank Koichi Futagami, Shingo Ishiguro, Tasturo Iwaisako, Kouki Sugawara, Daishoku Kanehara, and theseminar participants at Osaka University and Kyoto University for their helpful comments and suggestions.
†
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan E-mail: mysttsk@gmail.com‡
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan E-mail: koheiokadA337@gmail.com§
Graduate School of Economics, Osaka University Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
E-mail: takakura.kei.0520@gmail.com

1Percentage of children engaged
in child labor

40
35
30
25
20
15
10
5
0
0

20

40

60

80

CPI score
Figure 1: The relationship between Corruption Perceptions Index (CPI) score and percentage ofchildren (aged 5-17 years) engaged in child labor. Source: CPI score: Transparency International.
Percentage of children (aged 5-17 years) engaged in child labor : UNICEF

1

Introduction

In many developing countries, children are engaged in child labor. Sen (2019) reports Bangladesh,Chad, the Republic of the Congo, Ethiopia, India, Liberia, Myanmar, Nigeria, Pakistan, and Somalia to be the countries with the most widespread child labor in the world. For example, in the
Republic of the Congo, although the country has ratified the United Nations Convention on the
Rights of the Child and prohibits child labor, many children are engaged in mining. These children
are employed illegally by informal firms. This is also the case in many other developing countries.
Although the majority of them have ratified the United Nations Convention on the Rights of theChild and prohibit child labor1 , they have failed to expose firms that employ children. However,child labor is not a problem only in developing countries. It was once common in Europe and the
US. According to Hazan and Berdugo (2002), in 1851 England and Wales, 36.6% of all boys and19.9% of girls aged 10-14 worked. However, as the economy developed, child labor diminished inthese countries.
To investigate the diﬀerences between developing and developed countries regarding child labor,
we focus on corruption. Mauro (1995) and Banerjee (1997) reported that corruption, especially1

Liberia does not ratify the United Nations Convention on the Rights of the Child.

2bribery is widespread in developing countries. They emphasize that corruption hinders economic
development. We also show the correlation between corruption and child labor by employing the
Corruption Perceptions Index (CPI). The CPI is an index published annually by Transparency
International since 1995. The CPI currently ranks 180 countries ”on a scale from 100 (very clean)
to 0 (highly corrupt). Figure 1 shows the relationship between CPI and the percentage of children
aged 5-17 years engaged in child labor. There is a statistically significant negative correlation
between CPI and the percentage of children engaged in child labor2 Many past studies, in their
theoretical frameworks, show that corruption hinders economic development such as Blackburn etal.(2006). However, no studies have investigated the simultaneous relationship between corruption,
child labor, and development.
We construct a simple overlapping-generations model with endogenous education choices, childlabor, and corruption. Each individual lives for two periods. In the first period of their lives(childhood), individuals do not make any decisions. Thus, children engage in child labor and study
at school under their parents’ decision. In the second period (adulthood), individuals decide toaccept or reject a contract with an illegal firm. Then, they determine children’s time allocation
between education and child labor, supply their own labor, pay income tax, and consume final
goods. In this model, there are legal firms and only one illegal firm. Legal firms produce final
goods using the eﬀective labor of adult individuals but the illegal firm oﬀers a contract for childlabor with adult individuals. If they accept the contract, the illegal firm employs their children andproduces final goods using their child labor. The government employs bureaucrats and sends themto each firm to expose illegal firms. However, bureaucrats are corrupt and accept bribes from illegal
firms. If bureaucrats are corrupt and accept bribes, they do not expose the illegal firm; thus, childlabor remains. The equilibrium dynamics of this economy are characterized by the level of humancapital. We show that multiple steady-states exist in the economy. One steady-state has a highlevel of human capital, while the other has a low level. In the steady-state with a low level of humancapital, child labor and corruption exist and the welfare level is low. However, in the steady-state2

We estimate simple regressions, in which the percentage of children (aged 5-17 years) engaged in child labor(child labor) as the dependent variable is a function of the CPI score (CPI). The following equation provides simple
estimation results using ordinary least squares:
Child labor = 22.66(7.63) − 0.31(−3.71)CP I,
where the figures in parentheses are the values of t-statistics. The equation above suggests that the percentage ofchildren engaged in child labor has a negative correlation with the CPI score.

3with a high level of human capital, child labor and corruption have been ended and the welfare levelis high. In addition, we investigate how government policy aﬀects the economy. Then, we showthat it is diﬃcult to steer an economy away from a poverty trap with child labor and corruption
because bureaucrats of the current generation are opposed to policy changes such as reinforcementof monitoring and penal regulations. However, we can apply the Pareto improving policy to thispoverty trap, that is, the government borrows funds from an international organization. The model
reveals that if the interest rate is substantially low, the proposed policy is enforceable and thegovernment can repay the subsidy resources.
This study is related to research on child labor and corruption. Dessy (2000), Hazan and Berdugo(2002), and Sugawara (2010) investigated the relationship between child labor, human capital accumulation, and economic development. However, they did not consider the endogenous decision
process of bureaucrats and the contract between an individual and an illegal firm. Blackburn et al.
(2006), Blackburn and Sarmah (2007), Blackburn and Forgues-Puccio (2007), and Akimoto (2018,2019) show that corruption plays an important role in economic development. However, they do not
focus on the relationships between corruption, education, and child labor. Eicher et al. (2009) andVarvarigos and Arsenis (2015) investigated how education and corruption aﬀect economic development. However, they do not refer to the relationship between child labor and corruption. Numerous
studies have shown that child labor and corruption are harmful to economic development. However, no studies have investigated the relationship between child labor, corruption, and economic
development, generating the motivation for our study.
The remainder of this paper is organized as follows: Section 2 presents the basic structure of themodel, and Section 3 analyzes the equilibrium and dynamics. Section 4 analyzes welfare. Section 5
presents the conclusions.

2

The model

2.1

Individuals

There are two types of individuals: households and bureaucrats. Households supply their laborto the private sector and make their children engage in child labor. Conversely, bureaucrats are
employed by the government and work in the public sector. We assume that only households have4

access to the illegal firm; they volunteer their children for hire to the illegal firm because they haveconnections with the illegal firm3 . Hence, bureaucrats cannot make their children engage in childlabor. For simplicity, we assume that children whose parents are households (bureaucrats) become
households (bureaucrats) during any consecutive period. Children born in period t become active
workers in period t + 1.

2.1.1

Households

We consider a two-period overlapping generations model: childhood and adulthood. Both parents
and children have one unit of time. During adulthood, households allocate their children’s time
between child labor lt and schooling et . They devote one unit of time to working in the private
sector and earn income according to their human capital. They also pay income tax and consume
all remaining disposable income. We assume that the population size of households is one and isconstant over time. Households derive their utility from consumption cHt and the child’s humancapital ht+1 . Thus, the utility function of households in generation t isHuH
t = log ct + log ht+1 .

(1)Parents encourage their children to engage in child labor in illegal firms. They collect the income
earned from child labor and use it for their own consumption. Then, the budget constraint ofhouseholds iscH
t = (1 − τ )wt ht + at lt .

(2)

Let wt , ht , at , and τ represent the wage rate in the legal sector, adult human capital, receipts per
child’s working time in the illegal sector, and income tax rate levied on each household, respectively.
For simplicity, we assume that the income tax rate τ is constant over time. The production function

3
We consider the situation where households live in a rural area and bureaucrats live in an urban area. Our model
implicitly assumes that illegal firms employ traﬃckers and gather children in rural areas. Gregory (2017) and Miki etal.(2010) report that traﬃckers link households who supply child labor with an illegal firm. Traﬃckers gather children
by negotiating with their parents in small rural villages for cacao farms in the Ivory Coast.

5

of human capital is

ht+1 = ϕet ,(3)

where ϕ > 0 is a constant parameter. The time constraint of children is

et + lt = 1.

(4)

lt and at are determined by the contract between households and the illegal firm, which will be
explained later. From this contract, cH
t is also determined.
2.1.2

Bureaucrats

We consider the behavior of bureaucrats. We denote the size of the population of bureaucrats as
N . Bureaucrats derive their utility from consumption cB
t . Then, the utility function of bureaucrats
in generation t isBuB
t = log ct .

(5)

We assume that bureaucrats supply one unit of time to engage in the public sector. The government
hires bureaucrats and sends them to each firm to expose illegal firms. If bureaucrats encounter anillegal firm, they decide whether to expose it or set it free by taking the bribe. Then, the budget
constraint of bureaucrats is classified into the following two cases: If bureaucrats, who encounterillegal firms, are not corrupt and do not take bribes, their budget constraint iscB
t = (1 − τ )It ,(6)

where It denotes the bureaucrat’s income from the government. If bureaucrats, who encounterillegal firms, are corrupt and take bribes, their budget constraint iscB
t = (1 − τ )It + bt ,where bt is the amount of the bribe.
6(7)

We assume that corruption is found out at a probability of η ∈ (0, 1) after bureaucrats collectbribes for their own consumption. According to the literature on corruption, such as Varvarigos
and Arsenis(2015), when bureaucrats are corrupt, they suﬀer psychological distress δ log cBt , whereδ ∈ (0, 1). Where bureaucrats are corrupt, cB
t = It + bt holds. Therefore, their expected utility ingeneration t is given byuB
c,t = (1 − η) log[(1 − τ )It + bt ] + η(1 − δ) log[(1 − τ )It + bt ].

(8)

However, if bureaucrats are not corrupt, they only obtain income (1 − τ )It . Therefore, whenbureaucrats are not corrupt, their utility uBn,t in generation t is given byuB
n,t = log(1 − τ )It .

2.2

Private sector

2.2.1

Legal firm

(9)

We assume that the legal sector is perfectly competitive. The number of firms is infinite and one
unit of mass. Legal firms produce final goods Ya,t only from adult labor. The production functionof a legal firm is

Ya,t = ALt ,(10)

where A and Lt represent a technological parameter and the total labor demand for households
belonging to adulthood, respectively. In equilibrium, Lt = ht holds. Hence, the zero-profit condition
yields wt = A.
2.2.2

Illegal firm

We assume that there exists only one illegal firm that hires children from households and produces
final goods Yc,t . We also assume that the government cannot, by itself, distinguish between legal
and illegal firms. Therefore, the government must send bureaucrats to all firms. Once households
oﬀer their children for labor to the illegal firm, the illegal firm enters into an agreement with the7households. The illegal firm determines the amount of labor lt it will require from the child and thewage rate it will pay at . Moreover, if the illegal firm encounters corrupt bureaucrats, it can escapedetection of its child labor practices by bribing them. The production function of an illegal firm is:

Yc,t = Blt ,(11)

where B denotes the technological parameters. Hence, we obtain the illegal firm’s profit as follows:
πt = (B − at )lt .

2.3

Public sector

2.3.1

Government(12)

The government sends bureaucrats to each firm to expose the illegal firms. The government employs
bureaucrats and pays them salaries, which are financed by tax revenues from both households andbureaucrats. Thus, the government s budget constraint is:τ wt ht + τ N It = N It .

(13)

Employing wt = A, we can rewrite (13) as follows:

It =

τ Aht
.
(1 − τ )N(14)

We assume that the government has no means to detect whether bureaucrats are acting honestly ornot. Therefore, if corrupt bureaucrats take a bribe from an illegal firm and report that they cannot
find child labor at a particular firm, the government takes them at their word.

3

Equilibrium

The decision-making process during any period proceeds as follows: First, each household gives birth
to one child and raises them. Second, each household negotiates with the illegal firm to engage theirchild in child labor. Then, the illegal firm makes an oﬀer (at , lt ) to the households. We assume8

that the illegal firm has bargaining power to make oﬀers that satisfy the households’ participation
constraint. If the households decline this oﬀer, then this decision-making process ends with no childlabor in this economy. If households accept the oﬀer, the game unfolds to the corruption phase,i.e., bureaucrats are sent to inspect all firms. If bureaucrats encounter a legal firm, they reportto the government in that way. However, some of them encounter the illegal firm. In this case,negotiations between bureaucrats and the illegal firm begin. The illegal firm oﬀers a bribe to thebureaucrats. If bureaucrats decline this oﬀer, they report to the government, which generates nocorruption and child labor in the economy. However, if bureaucrats accept the bribe, the amountof the bribe is determined and the decision-making process ends.

3.1

Equilibrium path

To see the equilibrium path, we consider the case in which bureaucrats are corrupt. From (8) and(9), bureaucrats are corrupt if:
BuBc,t ≥ un,t ,1⇔ (1 − τ )It + bt ≥ [(1 − τ )It ] 1−δη .

(15)

Next, we consider the determination of the amount of bribe bt . There is a large number of legal
firms and only one illegal firm in the economy. We assume that the total firm size is one. Hence, thenumber of bureaucrats assigned to investigate the illegal firm is N . Let us define the profit that theillegal firm obtains as πt∗ ≡ max{at ,lt } πt , which represents the maximized profit when households
accept the child labor oﬀer. When bureaucrats are corrupt, the illegal firm pays an amount of bribe(bt ) to bureaucrats. Then, its net profit becomes πt∗ − N bt . If bureaucrats expose the illegal firm
without taking the bribe, its profit becomes zero. Conversely, if bureaucrats accept a bribe from anillegal firm, πt∗ −N bt ≥ 0 holds. Here, we assume that bureaucrats have all bargaining power against
the illegal firm,4 that is, bureaucrats can determine the amount of bribe bt on their own terms. A
substantially high amount of bribe makes the illegal firm reject the oﬀer. Therefore, bureaucrats

4This is not an important assumption. We can consider the opposite case in which the illegal firm has all
bargaining power in Appendix A. What is important in this case is that the illegal firm can pay the bribe that thecorrupt bureaucrats may ask for.

9

ask for bribe bt as follows:

max
bt

uB
c,t = (1 − η) log[(1 − τ )It + bt ] + η(1 − δ) log[(1 − τ )It + bt ],s.t. πt∗ − N bt ≥ 0.

(PCI)

(PCI) represents the participation constraint of the illegal firm and (PCI) binds at the optimum.
Therefore, we obtain the optimal bribe amount as follows:

bt =

1 ∗
π .
N t

(16)

The illegal firm and households determine the amount of child labor and the wage rate. Here, we
assume that the illegal firm has bargaining power against the households. Therefore, if the illegal
firm makes an oﬀer (at , lt ) that satisfies the households’ participation constraint, households will
accept the oﬀer. If households oﬀer their children for labor and do not take them to school, they getchild labor income at lt . From (1) to (4), for the households that supply child labor, we can obtainthe following utility uHc,t as:uHc,t = log[(1 − τ )wt ht + at lt ] + log ϕ(1 − lt ).

(17)

When households reject the oﬀer from the illegal firm and do not supply child labor; in that case,lt = 0 holds. Thus, if households do not supply child labor, we can obtain the following utility uHn,t
as follows:uH
n,t = log(1 − τ )wt ht + log ϕ.

(18)

From (12), the illegal firm maximizes the following profit:max (B − at )lt
at ,ltH
s.t. uHc,t ≥ un,t .

(PCH)(PCH) represents the participation constraint of households and binds at the optimum. This opti-

10

mization problem yields the optimum amount of child labor and the wage rate of child labor. By
using wt = A, we obtain the following solutions:√lt = 1 −at =(1 − τ )Aht
,
B

(19)

√
(1 − τ )ABht .

(20)

B
Because 0 ≤ lt ≤ 1 holds, we define h̄ ≡ (1−τ)A such that the level of child labor lt is equal to zero.

Substituting (19) and (20) into (12), we obtain the maximized profit of the illegal firm πt∗ as follows:

πt∗ =

[
]2
√
B − (1 − τ )ABht
B

.

(21)

We note that the maximized profit πt∗ is equal to zero when ht = h̄.
From (14), (15), (16), and (21), bureaucrats are corrupt and households supply child labor when
the following inequality holds:⇔

BuBc,t ≥ un,t ,
(
)2
√
) 1
(
B − (1 − τ )ABht
τ Aht 1−δη
τ Aht
.
≥
−
N{zB
N
|
} | N
{z
}
LHS(ht )(22)

RHS(ht )Let us define the left-hand side of (22) as LHS(ht ) and the right-hand side of (22) as RHS(ht ).
We define ĥ as the value of ht such that (22) holds with equality. LHS(ht ) is decreasing in ht ≤ h̄,and LHS(ht ) is always not less than 0. RHS(ht ) = 0 holds when ht = 0, τNA . Moreover, RHS(ht )is increasing in ht when RHS(ht ) ≥ 0; therefore, when ht ≥ τNA holds. To ensure the existence ofĥ, we assume the following:
Assumption 1.
N≤ B.
τ
Assumption 1 ensures two conditions: one is that LHS(ht ) and RHS(ht ) cross each other only
once, and the other is that the level of bureaucrats’ utility when they are corrupt is always greater
11

B
N

LHS(ht)RHS(ht)

0

ĥ

N
τA

h̄

ht

Figure 2: The determination of ĥ
than 0. For more detail, see Appendix C.
If ht ≥ ĥ holds, then LHS(ht ) ≤ RHS(ht ) holds, as in (22). In this case, corruption and childlabor do not exist in the economy. From (4), (19), and (22), we obtain the education level that√ (1−τ )Aht

households choose as follows:
et =

3.2

B



1

if ht ≤ ĥ

.

(23)

if ht > ĥ

Dynamics

In this section, we derive the dynamics of the model. From (3) and (23), we obtain,

ht+1 =

where Φ(ht ) ≡ ϕ

√




Φ(ht ) if ht ≤ ĥ


ϕ

,

(24)

if ht > ĥ

(1−τ )Aht.
B

The phase diagram is shown in Panels A through D in Figure 3. Two types of steady-state
may exist. The first is the intersection point of ht+1 = Φ(ht ) and the 45 ° line. We define thistype of steady-state as E1 and its human capital level as h∗1 . If there exists a steady-state E1 , child12

labor and corruption coexist in the economy. The other is the intersection point of ht+1 = ϕ and
45 degree line, defined by E2 . We denote the human capital level of E2 as h∗2 . If there exists a
steady-state E2 , child labor and corruption do not exist in the economy. In particular, Panel B in
Figure 3 shows the case of multiple steady-states 5 . From (24), we obtain the steady-state values
as follows:h∗1 =(1 − τ )ϕ2 A
,
B

(25)

h∗2 = ϕ.

(26)

Then, we obtain the following Proposition 1.
B
Proposition 1. Assume that τ > 1 − ϕAholds. Let us define Λ(τ ) and Ω(τ ) as follows:2Λ(τ ) ≡ 1 −)ϕ A
log τ (1−τBN

2log (1−τ )(ϕA−2B)ϕA+B
BN

2

ϕA
log τ N√Ω(τ ) ≡ 1 −log

ϕA+B−2(1−τ )ϕAB
N,.

Then, the following inequality holds:Ω(τ ) < Λ(τ ).
Therefore, we can derive the following results, respectively.
1. If δη < Ω(τ ), the economy converges to the steady-state E1 .
2. If Ω(τ ) ≤ δη ≤ Λ(τ ), there exist multiple steady-states in the economy.
3. If δη > Λ(τ ), the economy converges to the steady-state E2 .
Proof. See Appendix C.
Panel A in Figure 3 shows the case when δη < Ω(τ ), and the economy converges to the steadystate E1 . Panel B in Figure 3 shows the case in which multiple steady-states exist. Panel C in
Figure 3 shows the case in which the economy converges to the steady-state E2 .
5

By setting numeral values: β = 0.5, δ = 0.5, η = 0.5, τ = 0.5, ϕ = 1.4, A = 1.1, B = 1, N = 0.5 into the dynamics,
we can confirm the existence of multiple steady-states.

13

ℎ" #$

ℎ" #$

45°

45°
𝐸.

𝜙
𝜙

Φ ℎ"

Φ ℎ"

𝐸$

𝐸$

0

ℎ"

ℎ.∗ ℎ+

ℎ$∗

0

ℎ" #$

ℎ.∗

ℎ" #$

45°
𝐸.

𝜙

ℎ+

ℎ+

ℎ"

Panel B

Panel A

0

ℎ$∗

ℎ$∗

ℎ.∗

𝐸.

Φ ℎ"

𝜙

ℎ"

0

Panel Cℎ+Panel D

Figure 3: The dynamics of ht

14ℎ.∗ ℎ$∗

45°
Φ ℎ"

ℎ"

𝛿𝜂 ≤ Λ 𝜏
Steady state E1exists.

0

1

Λ 𝜏

𝛿𝜂

Panel A0Λ 𝜏'

𝜏↑ (𝜏' < 𝜏( )Panel B

Λ 𝜏( 1𝛿𝜂Figure 4: The area of δη ≤ Λ(τ ) and the eﬀect of change of τ on Λ(τ )The intuition of Proposition 1 is as follows: if the psychological distress δ or the probability that
corruption is discovered η is small, bureaucrats are corrupt, and this, in turn, leads to the existenceof lower human capital in the steady-state. Moreover, if δ or η take substantially small values, suchas δη ≤ Ω(τ ), there exists only a steady-state in which bureaucrats become corrupt. Panel A of
Figure 3 shows this case. Conversely, if the psychological distress or the probability that corruption
is discovered is large, bureaucrats are not corrupt, and this leads to the existence of higher humancapital in the steady-state. If δ or η take suﬃciently high value, there only exists the steady state inwhich bureaucrats are not corrupt. If δ or η takes middle value, there exist multiple steady state.
Panel A in Figure 4 indicates the area of δη ≤ Λ(τ ), such that a lower human capital steadystate exists in the economy. If δη ≤ Λ(τ ), then there exists a lower human capital steady-state E1 .
This means that the economy may converge to E1 . To discuss the policy that the government steers
the economy away from the poverty trap, we focus on only Λ(τ ). We obtain Lemma 1 as follows:)Band ϕA > 2B, ∂Λ(τLemma 1. Assume that τ > 1 − ϕA∂τ > 0 holds when Λ(τ ) ≤ 1.

Proof. See Appendix D.
)
Since ∂Λ(τ> 0 holds, an increase in τ expands the area of δη ≤ Λ(τ ) in Panel B of Figure 4.
∂τTherefore, higher rates of income tax τ cause a higher incidence of corruption.
B.
From Proposition 1, there exist multiple steady-states for some ranges of δη when τ > 1 − ϕA
B
B
Next, we discuss the case of τ ≤ 1 − ϕA
. First, we obtain Assumption 2 to ensure that 0 < 1 − ϕA.

15

Assumption 2.
B
< 1.
ϕA
Then, we obtain the following Lemma 2.
B
Lemma 2. Given any initial human capital h0 , if τ ≤ 1 − ϕA
, the economy converges to thesteady-state E2 .
B
Proof. From (25) and (27), h∗1 > h∗2 holds when τ ≤ 1 − ϕA
; that is, this corresponds to Panel D

in Figure 3.
B
Panel D in Figure 3 shows the dynamics when τ ≤ 1 − ϕA
. From Panel D in Figure 3, regardlessof any initial human capital h0 , the economy converges to the steady-state E2 , where child laborBand corruption do not exist. If the tax rate τ is suﬃciently low such that τ < 1 − ϕAholds, thehousehold’s disposable income is suﬃciently high such that parents do not supply child labor in thissteady-state.

4

Welfare and government policy

We compare the welfare levels in the steady-states E1 and E2 . To investigate this case, we focus on
case2 of Proposition 1. We discuss the welfare levels of households and bureaucrats, respectively.
Then, we obtain the following proposition:
B
Proposition 2. Assume that τ > 1− ϕAholds. The welfare levels of both households and bureaucrats

in the steady-state E2 are higher than those in the steady-state E1 .
Proof. See Appendix E.
B
From Proposition 1, if the steady-state E1 exists, τ > 1 − ϕAholds. In steady-state E1 , thelevel of human capital is lower than that in steady-state E2 . Lower human capital leads to a lower
household income. This generates child labor and corruption in steady-state E1 . In addition, thewelfare levels of both households and bureaucrats in steady-state E2 are higher than those in steadystate E1 . Therefore, if households have an initial level of human capital h0 that is lower than thethreshold ĥ, the economy suﬀers from child labor, corruption, and low levels of welfare.
16𝛿𝜂 ≤ Λ 𝜏&
Steady state E1exists.

𝛿& 𝜂&
0

Λ 𝜏&

1

𝛿𝜂

Panel A𝛿𝜂 ≤ Λ 𝜏(
𝛿& 𝜂&
0

Λ 𝜏(

𝜏 ↓ (𝜏& > 𝜏()

Λ 𝜏&

1

𝛿𝜂

Panel B𝛿𝜂 ≤ Λ 𝜏&𝛿𝜂↑(𝛿(𝜂( > 𝛿& 𝜂&)
𝛿(𝜂(

𝛿& 𝜂&
0

Λ 𝜏&

1

𝛿𝜂

Panel C
Figure 5: Policy changes
Next, we consider how government policy aﬀects the economy. We assume that the government
can change the values of psychological distress δ and the probability of revelation of corruption η 6 .
We obtain the following proposition.
Proposition 3. Suppose that ϕA > 2B holds, the steady-state E1 exists, and the initial economy isBorin the steady-state E1 . If the government changes the values of δ, η, and τ to satisfy τ ≤ 1 − ϕAδη > Λ(τ ), the economy converges to the steady-state E2 , and the welfare levels of both households
and bureaucrats in E2 are improved from those in E1 .
Proof. See Appendix F.
Let us assume that δ, η, and τ are chosen to satisfy δη ≤ Λ(τ ) before the government enforces achange in policy. We denote the initial values of δ, η, and τ as δ0 , η0 , and τ0 , respectively. As shownin Panel A of Figure 5, if δ0 η0 < Λ(τ ) holds, then the steady-state E1 exists. We assume that the
government decreases the initial tax rate τ0 to τ1 (i.e., τ0 > τ1 ). This change is shown in Panel B
6

Changing the value of the psychological distress δ means that the government can increase the penalty forcorruption. On the other hand, changing the probability of corruption revelation η is that, for example, the government
asks some international organizations to investigate bureaucrats’ income.

17

of Figure 5. If τ1 is suﬃciently low such that δ0 η0 ≥ Λ(τ1 ) holds, the steady-state E1 vanishes andthe economy converges to the steady-state E2 . From Lemma 2, the steady-state E1 also vanishes if
Bτ ≤ 1 − ϕAholds. As shown in Appendix F, the welfare levels of both households and bureaucrats

in E2 are improved from those in E1 if ϕA > 2B holds.
Next, we consider a situation in which the government increases the values of psychologicaldistress δ0 and the probability of revelation of corruption η0 to δ1 and η1 , respectively (i.e., δ1 > δ0and η1 > η0 ). These changes are shown in Panel C of Figure 5. If δ1 η1 is suﬃciently high, suchthat δ1 η1 ≥ Λ(τ0 ) holds, the steady-state E1 vanishes and the economy converges to the steadystate E2 . As shown in Appendix F, the welfare levels of both households and bureaucrats in E2
are improved from those in E1 , regardless of the values of δ and η. Therefore, the government
can improve the welfare levels of future generations by steering the economy away from steadystate E1 . In our model, the improvement in welfare levels is based on the human capital level. If
δη > Λ(τ ) holds, bureaucrats are not corrupt and child labor does not exist in the equilibrium.
B
Conversely, if τ ≤ 1 − ϕAholds, households allocate their children’s time to education. In bothcases, the economy converges to the steady-state E2 . Then, the increase in human capital raises theincome of households and bureaucrats, and their welfare levels also increase. However, government
policies cannot immediately aﬀect the level of human capital because the level of human capital ishistorically given at an initial point in time. Hence, we obtain the following proposition.
Proposition 4. Assume that the economy is initially in a steady-state E1 . The welfare level ofbureaucrats in the initial generation decreases if τ decreases or δη increases. Conversely, the welfare
level of households increases if τ decreases.
Proof. See Appendix G.
Bureaucrats’ income is financed by income tax revenue. Therefore, if the tax rate τ decreases,
bureaucrats’ income decreases and their welfare levels also decrease (see (14)). In addition, bureaucrats are corrupt and receive a bribe in steady-state E1 . Hence, the increase in psychologicaldistress δ and the probability of revelation of corruption η make bureaucrats diﬃcult to be corrupt.
Then, the welfare levels of bureaucrats of the current generation decrease. Therefore, the policies
described in Proposition 3 are not Pareto-improving in this economy.
Hereafter, we discuss the Pareto-improving policy. Let us consider the situation in which the18

government provides a subsidy x to bureaucrats of the current generation and does not charge theresource of this subsidy to the current generation. We assume that x is exogenously given and that it
does not change the government budget constraint. From (6), the budget constraint of bureaucrats
when they are not corrupt and do not take bribes is given bycB
t = (1 − τ )It + x.

(27)

From (7), the budget constraint of bureaucrats when they are corrupt and receive the bribe is givenbycB
t = (1 − τ )It + bt + x.

(28)

Considering subsidy x, we can rewrite the Proposition 1 as follows.
B
Proposition 5. Assume that τ > 1 − ϕAholds and x is exogenously given. There exists a steady-state E1 in this economy if the following inequality holds:[δη ≤ 1 −

log
[
log

τ (1−τ )ϕ2 A2
+x
BN

]

(1−τ )(ϕA−2B)ϕA+B 2+xBN] ≡ Λ̂(x, τ ),(29))
< 0 holds.
where ∂ Λ̂(x,τ∂x

Proof. See Appendix H.
)
From ∂ Λ̂(x,τ< 0, the range of δη > Λ̂(x, τ ) expands in Panel B of Figure 6 if x takes a∂xpositive value. Then, if x is suﬃciently high, such that δ0 η0 > Λ̂(x, τ0 ) holds, the steady-state
E1 vanishes and the economy converges to the steady-state E2 . From Proposition 2, the welfare
levels of both households and bureaucrats in E2 are improved from those in E1 . Therefore, thegovernment can improve the welfare levels of future generations by providing a suﬃciently highsubsidy x to bureaucrats. In addition, this policy increases the welfare levels of bureaucrats inthe current generation because subsidies increase their income. However, the government mustcollect the resources of this subsidy xN . If the government can obtain this resource from aninternational organization and distribute this resource to bureaucrats for the current generation,19𝛿𝜂 ≤ %
Λ 0, 𝜏(Steady state E1exists.

𝛿( 𝜂(%Λ 0, 𝜏( = Λ(𝜏( )

0

1

𝛿𝜂

Panel A𝛿( 𝜂(0 %
Λ 𝑥, 𝜏(𝑥↑(𝑥 > 0)% 0, 𝜏(Λ1𝛿𝜂

Panel B
Figure 6: The eﬀect of x
the economy immediately escapes from the poverty trap along with corruption, child labor, and low
welfare. Therefore, the support of international organizations in developing countries that suﬀer
from corruption and child labor is important for the economic development of these countries.
As shown in Propositions 3 and 4, it is diﬃcult for developing countries to steer the economy
away from a poverty trap through policy change because the welfare levels of bureaucrats of thecurrent generation decrease. However, we show that developing countries can escape the poverty
trap if international organizations support them. As shown in Proposition 5, if international organizations give some funds to the country in the poverty trap in the proposed model, the government
can enforce the Pareto-improving policy by providing subsidies to bureaucrats of the current generation. Therefore, the support of international organizations is necessary for the development ofpoor countries.

5

Conclusion

We constructed an overlapping-generations model with endogenous education choices, child labor,
and corruption. We investigated how child labor and corruption influence human capital accumulation and development. We show that multiple steady-states exist in this economy. One steady-state
has a high level of human capital and the other has a low level of human capital. In the steady-statewith a low level of human capital, child labor and corruption exist and the welfare level of households is low. In the steady-state with a high level of human capital, child labor and corruption are20

diminished and the welfare level is high. In addition, we showed that it is diﬃcult to steer an economy away from a poverty trap with child labor and corruption because bureaucrats of the current
generation are opposed to policy changes. However, we can apply the Pareto improving policy tothis poverty trap, that is, ensure the government receives funds from an international organization.

Appendix
Appendix A: The illegal firm has all bargaining power over bureaucrats
The illegal firm determines the amount of the bribe bt by solving
max πt∗ − N bt
bt

(1 − δη) log((1 − τ )It + bt ) ≥ log(1 − τ )It ,

s.t.

(PC’)

where (PC’) represents the participation constraint of bureaucrats; that is, they never refuse anoﬀer when it is satisfying. Because (PC’) is binding at the optimum, we obtain bt as follows:



0
bt =

if It < 11
[(1 − τ )It ] 1−ηδ − (1 − τ )It.

if It ≥ 1

If corruption and child labor exist in the economy, we obtain the following condition:πt∗ − N bt ≥ 0.

Then, we also obtain the following condition:(1 − τ )It +⇔1
1 ∗πt ≥ [(1 − τ )It ] 1−ηδ ,N

11 ∗π ≥ [(1 − τ )It ] 1−δη − (1 − τ )It .
N

21

From, (14),(15),(16), and (21) we obtain
(

B−

)2
√
(1 − τ )ABht
NB

(
≥

τ Aht
N

)

1
1−δη

−

τ Aht
.
N

This is the same condition as (22).

Appendix B: Assumption 1
First, when Assumption 1 holds, LHS(ht ) and RHS(ht ) intersect only once; therefore, there exists
only one ĥ. LHS(ht ) decreases with respect to ht and is equal to zero at ht = h̄. RHS(ht ) is not
BNN
less than zero when ht ≥ τNA and increases with respect to ht = (1−τ)A ≥ τ A . Therefore, if h̄ ≥ τ A

holds, LHS(ht ) and RHS(ht ) intersect only once. This occurs when (1 − τ ) Nτ < B holds. Second,
we assume that the level of bureaucrats’ utility when they are corrupt is always not less than 0.
This is because if the level of bureaucrats’ utility is smaller than 0, the level of bureaucrats’ utility
when corruption is found is smaller than that when corruption is not found.7 To exclude this casefrom consideration, the level of bureaucrats’ utility must always be less than 0. From (8), the utility
level of bureaucrats when they are corrupt uBc,t is not less than zero if(1 − η) log[(1 − τ )It + bt ] + η(1 − δ) log[(1 − τ )It + bt ] ≥ 0⇔(1 − τ )It + bt ≥ 1.

(30)

Applying (14), (15), (16), and (21), this inequality holds when Nτ ≤ B. Consolidating (1−ττ )N ≤ B
and Nτ ≤ B, we obtain Nτ ≤ B as in Assumption 1.

Appendix C: Proof of Proposition 1
This economy can have two types of equilibrium: one has corruption and child labor, and the otherhas neither. If ht ≤ ĥ (ht > ĥ) holds, then the equilibrium with (without) corruption and childlabor exists. Figure 7 shows various cases that can take place; Panel A, C, and E in Figure 7 show
the relationship between LHS(ht ) and RHS(ht ) while Panel B, D, and F in Figure 7 show the7
The utility level of bureaucrats where corruption is not found is given by log[(1 − τ )It + bt ] and that where
corruption is found is given by (1 − δ) log[(1 − τ )It + bt ]. Therefore, if log[(1 − τ )It + bt ] < 0, then log[(1 − τ )It + bt ] issmaller than the discounted value (1 − δ) log[(1 − τ )It + bt ]. This means that bureaucrats obtain higher utility whencorruption is found.

22

𝑅𝐻𝑆 (ℎ" )𝐿𝐻𝑆(ℎ" )𝑅𝐻𝑆 (ℎ" )𝐿𝐻𝑆(ℎ" )
ℎ$∗

0

ℎ"

ℎ,

ℎ'∗
Panel Aℎ'∗ℎ"

Panel C

45°

ℎ" #$

ℎ,

ℎ$∗

0
ℎ" #$

45°
𝐸'

𝜙

𝜙
Φ ℎ"

Φ ℎ"

𝐸$

0

𝐸$

ℎ$∗

ℎ'∗

ℎ"

ℎ,

ℎ$∗

0

Panel Bℎ,Panel D

𝑅𝐻𝑆 (ℎ" )𝐿𝐻𝑆(ℎ" )

0

ℎ,

ℎ$∗

ℎ"

ℎ'∗

Panel E
ℎ" #$

45°
𝐸'

𝜙

Φ ℎ"

𝐸$

0

ℎ,

ℎ$∗

ℎ'∗

ℎ"

Panel F

Figure 7: Positional relationship of h∗1 , h∗2 , and ĥ
23

ℎ'∗

ℎ"

phase diagrams of the level of human capital ht . Panel A and B in Figure 7 show the case where
only steady-state E1 exists, Panel C and D show the case where there exist multiply steady-states,
and Panel E and F show the case where only steady-state E2 exists.
First, we show that if δη < Ω(τ ), there exists only a steady-state E1 . As shown in Panel B
of Figure 7, all economies converge to the steady-state E1 , where the economy suﬀers from childlabor and corruption if h∗1 ≤ h∗2 ≤ ĥ. As shown in Panel A of Figure 7, this case occurs when
LHS(h∗2 ) > RHS(h∗2 ). From (22) and (27), we can rewrite LHS(h∗2 ) > RHS(h∗2 ) as follows:
LHS(h∗2 ) > RHS(h∗2 ),→ϕA
log τ N√δη < 1 −log

ϕA+B−2(1−τ )ϕABN≡ Ω(τ )

(A1)

B
Then, we must show that if τ > 1− ϕAholds, there exists δη such that (A1) holds; that is, Ω(τ ) > 1.

Ω(τ ) is not less than zero if
ϕA
log τ N√Ω(τ ) = 1 −
[
⇔

log

ϕA+B−2(1−τ )ϕAB
N
)]2

(
B
τ − 1−
ϕA

≥ 0,

≥ 0.

(A2)

B
Note that (A2) holds when τ = 1 − ϕA
. Because the right-hand side of (A1) is always greater thanBzero when τ > 1 − ϕAholds, there must exist δη > 0 such that δη < Ω(τ ) holds.

Second, we show that if Ω(τ ) ≤ δη ≤ Λ(τ ) holds, then there exist multiple steady-states. As
Bshown in Panel D of Figure 7, there exist multiple steady-states if h∗1 ≤ ĥ ≤ h∗2 , that is, τ > 1 − ϕA.

Panel C in Figure 7 states that this case occurs if LHS(h∗1 ) ≥ RHS(h∗1 ) holds at ht = h∗1 , but
LHS(h∗2 ) ≤ RHS(h∗2 ) holds at ht = h∗2 . By using (22), (25), and (27), we can rewrite as follows:
LHS(h∗1 ) ≥ RHS(h∗1 ),2→

δη ≤ 1 −)ϕ A
log τ (1−τBN

2log (1−τ )(ϕA−2B)ϕA+B
BN

24

2

≡ Λ(τ ),

(A3)

and
LHS(h∗2 ) ≤ RHS(h∗2 ),→ϕA
log τ N√δη ≥ 1 −log

ϕA+B−2(1−τ )ϕABN= Ω(τ ).

(A4)

B
. We show this later. Λ(τ ) is greater than or
We can prove that Ω(τ ) < Λ(τ ) for any τ > 1 − ϕAequal to zero if
2Λ(τ ) ≡ 1 −)ϕ A
log τ (1−τBN

2
2

log (1−τ )(ϕA−2B)ϕA+B
BN
[
(
)]2
B
τ − 1−
≥ 0.
ϕA

→

≥ 0,
(A5)

B
Note that (A5) holds when τ = 1 − ϕA
. Because the right-hand side of (A3) is always greater thanB
, and therefore Λ(τ ) > 0, there must exist δη ≥ 0 such that δη ≤ Λ(τ ) holds.
zero when τ > 1 − ϕA
B
Combining the fact that Ω(τ ) < Λ(τ ) for any τ > 1 − ϕAand (A5), there must exist δη such thatΩ(τ ) ≤ δη ≤ Λ(τ ) holds.
Third, we show that if δη > Λ(τ ), there exists only a steady-state E2 . As shown in Panel Fof Figure 7, all economies converge to the steady-state E2 if ĥ < h∗1 ≤ h∗2 . As shown in Panel
E of Figure 7, this case occurs when LHS(h∗1 ) < RHS(h∗1 ). From (22) and (25), we can rewrite
LHS(h∗1 ) < RHS(h∗1 ) as follows:
LHS(h∗1 ) < RHS(h∗1 ),2→

δη > 1 −)ϕ A
log τ (1−τBN

2log (1−τ )(ϕA−2B)ϕA+BBN

2≡ Λ(τ ).

(A6)

B
Because Λ(τ ) is equal to 0 at τ = 1 − ϕAfrom (A5), there exists Λ(τ ) < δη ≤ 1 such that (A6)holds.
B. Rewriting (22), LHS(ht ) ≥ RHS(ht ) if
Finally, we show that Ω(τ ) < Λ(τ ) for any τ > 1 − ϕAtlog τ AhN√δη ≤ 1 −log

Aht +B−2(1−τ )ABht
N

25≡ Ψ(ht , τ ).

(A7)

We define the right-hand side of (A7) as Ψ(ht , τ ). Substituting ht = h∗1 into Ψ(ht , τ ), we obtain2

Ψ(h∗1 , τ ) = 1 −)ϕ A
log τ (1−τBN

2log (1−τ )(ϕA−2B)ϕA+BBN

2= Λ(τ ).

(A8)

= Ω(τ ).

(A9)

As the same way, substituting ht = h∗2 into Ψ(ht , τ ), we obtainϕA
log τ N√Ψ(h∗1 , τ ) = 1 −log

ϕA+B−2(1−τ )ϕABN

2

Because ϕ = h∗2 > h∗1 = (1−τB)ϕ A if Ψ(ht , τ ) is strictly decreasing in ht , we conclude that Ψ(h∗2 , τ ) =Ω(τ ) < Λ(τ ) = Ψ(h∗1 , τ ). Diﬀerentiating Ψ(ht , τ ) with respect to ht , we obtain∂Ψ(ht , τ )
=−
∂ht

[
ht
{

×

1
√

Aht +B−2

log(1−τ )ABht
N

Aht + B − 2

]2

}
√
√
(1 − τ )ABht
Aht − (1 − τ )ABht
τ Aht
√
−
log
.
N
N
Aht + B − 2 (1 − τ )ABht(A10)

From (5), (7), (14), (16), and (21), we have log[(1 − τ )It + bt ] = log√

Aht +B−2(1−τ )ABht.
N

In thetsame way, (5), (8), and (14), we have log(1 − τ )It = log τ Ah
N . Therefore, the curly braces in (A10)can be rewritten as:∂Ψ(ht , τ )
=−
∂ht

×

[
ht
{

1
√

Aht +B−2(1−τ )ABht
N

]2
Aht −

√

(1 − τ )ABht√
log[(1 − τ )It + bt ] −log(1 − τ )It
Aht + B − 2 (1 − τ )ABht

Therefore, we have since bt ≥ 0log(It + bt ) ≥ log It⇒√
Aht + B − 2 (1 − τ )ABht
τ Aht
log
≥ log
.
N
N

26

}
.

Moreover, the coeﬃcient of the second term√(1−τ )ABht
√

Aht −

Aht +B−2(1−τ )ABht

is less than or equal to 1 if√
(1 − τ )ABht√≤1
Aht + B − 2 (1 − τ )ABht
B⇔ ht ≤= h̄.
(1 − τ )A
Aht −B
Note that h̄ represents the value of lt = 0. Because we consider the case in which τ > 1 − ϕA
, wehave h∗2 = ϕ < h̄ holds.

√

Because Assumption 1 holds, log

Aht +B−2(1−τ )ABht
is always greater than or equal to zero forN√
Aht − (1−τ )ABht

any ht . Therefore, because the coeﬃcient of the second term√

Aht +B−2(1−τ )ABht

is not greaterthan one, we have

log
⇒

√
√
Aht − (1 − τ )ABht(1 − τ )ABht
τ Aht
τ Aht
√
≥ log
>
log
N
N
N
Aht + B − 2 (1 − τ )ABht√√
Aht + B − 2 (1 − τ )ABht
Aht − (1 − τ )ABht
τ Aht
√
log
log
−
> 0.
N
N
Aht + B − 2 (1 − τ )ABht

Aht + B − 2

Because the curly brace part of (A10) is greater than 0, the diﬀerentiation of Ψ(ht , τ ) with respectt ,τ )< 0 when ht ≤ h̄ and Ψ(h∗2 , τ ) = Ω(τ ) < Λ(τ ) = Ψ(h∗1 , τ ).
to ht is negative. Therefore, ∂Ψ(h∂htHence Proposition 1 has been proved.

Appendix D: Proof of Lemma 1)
From (A3), we obtain ∂Λ(τas follows∂τ− τ (1−τ ) log∂Λ(τ )=∂τ1−2τ

)ϕ2 A2(1−τ )(ϕA−2B)ϕA+B 2+ (1−τ(ϕA−2B)ϕA
log τ (1−τBN
BN)(ϕA−2B)ϕ+B 2.
[]2(1−τ )(ϕA−2B)ϕA+B 2
log
BN

(A11)

B
2B
2B
τ > 1 − ϕA
can be rearranged as 1 − 2τ < ϕA
− 1, and ϕA > 2B can be rearranged as ϕA− 1 < 0.

By combining them, we obtain 1 − 2τ < 0. In addition, (1−τ )(ϕA−2B)ϕA+B
BN(ϕA−2B)ϕA+B 2 −BN(ϕA−2B)ϕA

holds. We show that

(ϕA−2B)ϕA+B 2 −BN(ϕA−2B)ϕA

27

2> 1 holds if τ <> 1 holds under Assumption 1. Then,we obtain
B2(ϕA − 2B)ϕA + B 2 − BN−1=
(ϕA − 2B)ϕA(ϕA − 2B)ϕA

(

N
1−
B

)
.

(A12)

(ϕA−2B)ϕA−BN
N>
From Assumption 1 and τ ∈ (0.1), N
B < 1 holds. From B < 1, ϕA > 2B, and (A12),(ϕA−2B)ϕA2

1 holds. Hence, τ < (ϕA−2B)ϕA−BN
holds. Then, (1−τ )(ϕA−2B)ϕA+B> 1 holds. Therefore, theBN(ϕA−2B)ϕA2> 1,
first term of the numerator in (A11) is greater than 0. From (27) and (1−τ )(ϕA−2B)ϕA+B
BN
τ (1−τ )ϕ2 A2BN> 1 holds if Λ(τ ) < 1 holds. Therefore, the second term of the numerator of (A11) is)
B
greater than zero when Λ(τ ) < 1. Hence, ∂Λ(τ∂τ > 0 holds if τ > 1 − ϕA , ϕA > 2B, and Λ(τ ) ≤ 1

holds.

Appendix E: Proof of Proposition 2
Let us define the welfare level of the household as uHc (ht ) when child labor exists. Employing (17),(19), (20), and wt = A, we obtain uHc (ht ) as follows:uHc (ht ) = log(1 − τ )Aht + log ϕ.

(A13)

Substituting (27) into (A13), we obtain the welfare level of the household in the steady-state E1(i.e., ht = h∗1 ) as follows:∗uHc (h1 ) = log(1 − τ )2 ϕ3 A2
.
B

(A14)

We define the welfare level of the household as uHn (ht ) when child labor does not exist. From (18)and wt = A, we obtain uHn (ht ) as follows:uHn (ht ) = log(1 − τ )Aht + log ϕ.

(A15)

Substituting (27) into (A15) and rearranging it, we obtain the welfare level of households in thesteady-state E2 (i.e., ht = h∗2 ) as follows:∗2uHn (h2 ) = log(1 − τ )Aϕ .

28

(A16)

B
∗
H ∗
From (A14) and (A16), we find that uHn (h2 ) > uc (h1 ) holds when τ > 1 − ϕA .

Let us define the welfare level of the bureaucrat as uBc (ht ) when child labor exists. From (8),(14), (16), and (21), we obtain uBc (ht ) as follows:{uBc (ht ) = (1 − δη) log

τ Aht [B −+N}√
(1 − τ )ABht ]2
.
BN

(A17)

Substituting (27) into (A17) and rearranging it, we obtain the welfare level of the bureaucrat in thesteady-state E1 (i.e., ht = h∗1 ) as follows:∗uBc (h1 ) = (1 − δη) log(1 − τ )(ϕA − 2B)ϕA + B 2
.
BN

(A18)

We define the welfare level of the bureaucrat as uBn (ht ) when child labor does not exist. From (9)and (14), we obtain uBn (ht ) as follows:uBn (ht ) = log

τ Aht
.
N

(A19)

Substituting (27) into (A19) and rearranging it, we obtain the welfare level of the bureaucrat in thesteady-state E2 (i.e., ht = h∗2 ) as follows:∗uBn (h2 ) = log

Note that δ, η ∈ (0, 1), and δη < 1 hold. Therefore,
we find that uBn (ht ) >

τ ϕA
.
N

(A20)

∗
uB
B ∗
c (h1 )1−δη > uc (h1 ) holds. From (A18) and (A20),

∗
uB
B
B
B ∗
B ∗
c (h1 )1−δη holds when τ > 1 − ϕA . Hence, un (h2 ) > uc (h1 ) holds if τ > 1 − ϕAholds.

Appendix F: Proof of Proposition 3
Let us assume that the government decreases the initial tax rate τ0 to τ1 (i.e., τ0 > τ1 ). As shownB∗
H ∗in Appendix E, if τ > ϕAholds, then uHn (h2 ) > uc (h1 ) holds. From (A14) and (A16), we find that∗∂uHn (h2 )
< 0 holds.
∂τ

H ∗
∗
Therefore, uHn (h2 ) τ =τ1 > uc (h1 ) τ =τ0 holds. From (A14) and (A16), we find∗H ∗that uHn (h2 ) and uc (h1 ) are independent of δ and η.
B∗B ∗As shown in Appendix E, if τ > ϕAholds, then uBn (h2 ) > uc (h1 ) holds. From (A18) and (A20),

29

we find that

∗
∂uB
∗
B ∗
n (h2 )
< 0 holds if ϕA > 2B holds. Therefore, uBn (h2 ) τ =τ1 > uc (h1 ) τ =τ0 holds. As
∂τ

∗
B ∗
shown in Appendix E, regardless of the values of δ and η, uB
n (h2 ) τ =τ1 > uc (h1 ) τ =τ0 holds.

We assume that there exists a steady-state E1 and the initial economy is in the steady-state E1
when τ = τ0 . Then, if the government decreases the initial tax rate τ0 to τ1 such that it satisfies
Bτ1 > 1 − ϕAor δη > Λ(τ1 ), the steady-state E1 vanishes, that is, the dynamics change from that ofPanel B in Figure 3 to that of Panels C and D. Then, the economy converges to the steady-state
E2 . In addition, if the government increases δ or η such that δη > Λ(τ0 ) holds, the steady-state
E1 vanishes, and the economy converges to the steady-state E2 . Hence, if the government changes
τ, δ, η such that multiple steady-states do not exist and the economy converges to the steady-state
E2 , the level of welfare in the steady-state E2 is improved from that of E1 .

Appendix G: Proof of Proposition 4
If the initial economy is in the steady-state E1 , the human capital of this generation is given byh∗1 . Note that even if τ or δη changes, the human capital of this generation h∗1 does not change.
B

From (A17), we obtain ∂uc∂τ(ht )B> 0, ∂uc∂δ(ht )∗ht =h1B< 0, and ∂uc∂η(ht )∗ht =h1ht =h∗1< 0. Therefore, thewelfare level of bureaucrats in the initial generation decreases if τ decreases or δη increases.

Appendix H: Proof of Proposition 5
From (5) and (28), the utility of bureaucrats when they are not corrupt and do not receive the bribe
is given byuB
n,t = log[(1 − τ )It + x].

(A21)

From (5) and (29), the utility of bureaucrats when they are corrupt and receive the bribe is givenbyuB
c,t = (1 − η) log[(1 − τ )It + bt + x] + η(1 − δ) log[(1 − τ )It + bt + x].

30

(A22)

From (A21) and (A22), bureaucrats are corrupt if
BuBc,t ≥ un,t ,1⇔ (1 − τ )It + bt + x ≥ [(1 − τ )It + x] 1−δη .

(A23)

Substituting (16) into (A23), we obtain(1 − τ )It +1
1 ∗πt + x ≥ [(1 − τ )It + x] 1−δη .
N

(A24)

Bureaucrats are corrupt when (A24) holds. From (14), (A24), and (21), bureaucrats are corrupt
and households supply child labor when the following inequality holds:) 1] ( τ Ah
√
1−ηδ
1 [
t
,
Aht + B − 2 (1 − τ )ABht + x ≥+x|N{z} | N {z}\ t)≡LHS(h(A25)\ t)≡RHS(h[ t ) and the right-hand side of (A25) asholds. Let us define the left-hand side of (A25) as LHS(h\ t ). As shown in Appendix C, if LHS(h[ ∗1 ) ≤ RHS(h\ ∗1 ), there exist multiple steady-states.
RHS(h[ ∗1 ) ≤ RHS(h\ ∗1 ) as follows:
From (A25) and (25), we can rewrite LHS(h→[ ∗1 ) ≤ RHS(h\ ∗1 ),LHS(h[])ϕ2 A2log τ (1−τ+xBN[] ≡ Λ̂(x, τ ).
δη ≤ 1 −(1−τ )(ϕA−2B)ϕA+B 2
log
+
x
BN

(A26)

From (A26), we obtain[]log(Υ + x) log(Γ + x)1
∂ Λ̂(x, τ )
,
=−
−
∂τ
[log(Υ + x)]2
Γ+x
Υ+x
2

2

(A27)

2

)ϕ A
where Γ ≡ τ (1−τand Υ ≡ (1−τ )(ϕA−2B)ϕA+B. Then, we obtain
BN
BN

Υ−Γ=

[(1 − τ )ϕA − B]2
> 0.
BN

(A28)

∂ Λ̂(x,τ )Therefore, Υ > Γ holds, and we can obtain log(Υ+x)> log(Γ+x)< holds.
Γ+x
Υ+x . Hence, from (A27),∂τ

31

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