by kristian estevez - university of...
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1
ESSAYS ON CHILD LABOR, PRODUCTIVITY, AND TRADE
By
KRISTIAN ESTEVEZ
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2010
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© 2010 Kristian Estevez
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To my family, for the love and support they provide
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ACKNOWLEDGMENTS
I would like to thank Elias Dinopoulos, whose guidance paved the way for this
research. I would also like to thank Steven Slutsky, Richard Romano, Mark Rush,
James Seale, and the Department of Economics at the University of Florida for
suggestions and advice that have proved to be invaluable. This research would also
not have been possible were it not for the groundwork laid by Kaushik Basu, Erik
Edmonds, Nina Pavcnik, Kenneth Swinnerton, Carol Ann Rogers, and all others who
have worked tirelessly to research ways to end child labor. Lastly, I would like to thank
my wife for reading this dissertation more times than is probably healthy.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 7
LIST OF FIGURES .......................................................................................................... 8
ABSTRACT ..................................................................................................................... 9
CHAPTER
1 THE ECONOMICS OF CHILD LABOR ..................................................................... 11
Supply of Child Labor.............................................................................................. 12 Demand for Child Labor .......................................................................................... 13
2 NUTRITIONAL EFFICIENCY WAGES AND CHILD LABOR .................................... 15
The Model ............................................................................................................... 16 Household Decision ......................................................................................... 17
Production ........................................................................................................ 22 Modern sector ............................................................................................ 23
Agrarian sector ........................................................................................... 24 Steady-State Equilibrium .................................................................................. 27
Comparative Statics ................................................................................................ 29 Foreign Direct Investment ................................................................................ 29 Trade Sanctions ............................................................................................... 31
Education Improvements .................................................................................. 32 Migration ........................................................................................................... 33
Subsidies .......................................................................................................... 35 Child wage subsidies ................................................................................. 35 Education subsidies ................................................................................... 37
Conclusion .............................................................................................................. 39
3 CHILD LABOR AND FIRM HETEROGENEITY ........................................................ 41
The Basic Model ..................................................................................................... 43 Consumer Demand .......................................................................................... 44
Production ........................................................................................................ 45 Child Labor Demand ........................................................................................ 49 Firm Value ........................................................................................................ 50 Solving the Benchmark Case ........................................................................... 51 Aggregation ...................................................................................................... 52 Free Entry and Exit ........................................................................................... 54
Solving the Model when 1 ................................................................................. 59
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Free Entry and Exit ........................................................................................... 60
Enforcement ..................................................................................................... 62 Traditional industry case ............................................................................ 63
Modern industry case ................................................................................. 64 Intra-industry Trade ................................................................................................ 65
Free Entry and Exit ........................................................................................... 66 Trade Liberalization .......................................................................................... 69 Trade Liberalization in Traditional and Modern Sectors ................................... 71
Conclusion .............................................................................................................. 72
INCIDENCE OF CHILD LABOR IN A NORTH-SOUTH MODEL OF TRADE ................ 73
The Model ............................................................................................................... 74
Consumption .................................................................................................... 75 Production in the North ..................................................................................... 76
Exporting Firms in the North ................................................................................... 81
North-South Free Trade Equilibrium ....................................................................... 83 Production in the South .................................................................................... 83
Firm Value for Southern Firm ........................................................................... 85 Free-Entry Condition for Northern Firms .......................................................... 86 Share of Firms .................................................................................................. 86
Incidence of Child Labor ................................................................................... 88 Comparative Statics ................................................................................................ 90
Increase in Child-Labor Enforcement, S .................................................. 90
One Time Increase in the Population in the South, SL ..................................... 91
Increase in the Rate of Imitation, I ................................................................ 92
Trade Costs ...................................................................................................... 93 Conclusion .............................................................................................................. 96
APPENDIX
A FIRST-ORDER CONDITIONS .................................................................................. 98
B PROOF OF UNIQUE STEADY-STATE EQUILIBRIUM ............................................ 99
C SIMULATION WITH ENDOGENOUS RATE OF IMITATION ................................. 101
LIST OF REFERENCES ............................................................................................. 102
BIOGRAPHICAL SKETCH .......................................................................................... 105
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LIST OF TABLES
Table page 2-1 Summary of Comparative Statics Results .......................................................... 38
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LIST OF FIGURES
Figure page 2-1 Steady-state equilibrium ..................................................................................... 28
2-2 Increase in FDI ................................................................................................... 30
2-3 Welfare among households with an increase in FDI ........................................... 31
2-4 Welfare among households with trade sanctions ............................................... 32
2-5 Welfare of households with emigration of skilled workers .................................. 34
2-6 Child wage subsidies .......................................................................................... 36
3-1 Steady-state equilibrium ..................................................................................... 56
3-2 Increase in enforcement when 1 ................................................................... 63
3-3 Increase in enforcement when 1 ................................................................... 64
3-4 Effect of trade in the steady-state equilibrium ..................................................... 70
4-1 Autarky equilibrium ............................................................................................. 79
4-2 Trade equilibrium ................................................................................................ 83
4-3 Increase in enforcement ..................................................................................... 91
4-4 Increase in population ........................................................................................ 92
4-5 Price indices and trade costs .............................................................................. 95
4-6 Child labor and trade costs ................................................................................. 96
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
ESSAYS ON CHILD LABOR, PRODUCTIVITY, AND TRADE
By
Kristian Estevez
December 2010
Chair: Elias Dinopoulos Major: Economics
The problem of children working around the world is not a new phenomenon, but
rather a legacy of poverty that is slowly being eradicated as incomes inch up in
developing economies. While the incidence of child labor has been on the decline,
awareness of the issue has grown in part due to globalization. This has led to debates
as to the best way to cure the problem once and for all. Chapter 1 describes what leads
to child labor, briefly reviews the economic literature of the last 20 years, and
summarizes the policy prescriptions resulting from the research.
Chapter 2 develops a dynamic, overlapping generations general-equilibrium model
of a small open economy where the demand and supply of child labor are analyzed.
There are two goods: a modern good produced by skilled labor and capital, and an
agrarian good produced by unskilled adult labor, child labor, and land. The model
predicts that an increase in foreign direct investment (FDI) and improvements in
education will decrease the incidence of child labor. Emigration of skilled (unskilled)
workers will reduce (increase) the supply of child labor, while trade sanctions will reduce
the demand for child labor. Child wage subsidies have an ambiguous effect on the
incidence of child labor, while education subsidies are effective in reducing child labor.
10
Chapter 3 examines the role of firm heterogeneity in the demand for child labor.
The effect of child labor enforcement and trade liberalization will depend on how a firm’s
productivity parameter affects the relative productivity between adult and child workers.
When the productivity elasticity of adult and child labor are equal, all firms choose the
same proportion of child workers, and only an increase in enforcement will reduce the
demand for child labor. When the productivity elasticity of child labor is higher (lower)
than that of adult labor, trade liberalization will result in a decrease (increase) in the
demand for child labor.
Last, Chapter 4 studies how international trade affects the incidence of child labor
in a North-South model of trade. Innovating firms in the North are heterogeneous and
differ in their marginal costs, while imitating firms in the South are homogeneous and
may use child labor in production. The incidence of child labor depends not only on
domestic factors, such as the relative wage of adult and child labor in the South, but
also on the endogenous rate of innovation in the North and the exogenous rate of
imitation by Southern firms. Reductions in trade costs decrease the number of
Southern firms and will lower the demand for child labor. An increase in the exogenous
rate of imitation by Southern firms will reduce the total number of varieties of the
differentiated good and decrease the demand for child labor, while an increase in the
population in the South will increase the demand for child labor.
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CHAPTER 1 THE ECONOMICS OF CHILD LABOR
Over the last decade, the incidence of child labor has been declining steadily
worldwide. However, the number of children classified as economically active (over 191
million as of 2006)1 is still too high and highly concentrated in the poorest nations. The
number of economically active children accounts for 14% of the children in the world,
but in sub-Saharan Africa and Asia, the number is closer to 25% and 17%, respectively
(ILO 2006b). Furthermore, many of the children employed outside the agricultural
sector work in unsafe and sometimes hazardous conditions.
In recent years, the increase of globalization has raised awareness of the problem
of child labor in the industrialized world. The International Labour Organization (ILO)
passed Convention 29 in 1930, which prohibits all forms of forced and compulsory
labor. In 1973, the ILO passed Convention 138, which sets a minimum age for children
depending on the type of work. Light work, meant as work that does not significantly
detract from schooling, is limited to children 13 years or older, while “hazardous work” is
limited to children 18 and older. Many countries where child labor is most visible, such
as India, Nepal, and Thailand, have national laws limiting or banning the use of child
labor. Many others have compulsory education laws to ensure that children are
receiving an education but that do not outlaw the use of child labor outside compulsory
schooling.
Unfortunately, laws outlawing child labor and compulsory education laws have
shown to have a minimal effect in low-income countries (Krueger 1996). The passage
1 See International Labor Organization (ILO) 2006a. For a comprehensive survey of the child labor
literature, see Basu (1999); Rogers and Swinnerton (2001); and Brown, Deardorff, and Stern (2003).
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of compulsory education laws is endogenous to the current state of child labor, and it
has been shown that these laws are usually passed following a decline in child labor,
not before it. Even programs that are meant to discourage child labor by providing
incentives to poor families to replace work with schooling have proved to be ineffective
due to the difficulty in monitoring compliance.
Supply of Child Labor
The main justification for government intervention to eliminate child labor is that of
externalities. The social returns to education have been shown to exceed private
returns, so a situation in which children work rather than attend school is not socially
optimal. In this case, eliminating child labor maximizes social welfare. The best way to
achieve that objective eludes policymakers. Most economists would agree that
economic growth that reduces poverty is guaranteed to end child labor, but child labor in
itself is what prevents economic growth in the poorest countries. Not surprisingly,
Krueger (1996) found a strong correlation between a country’s per capita GDP and the
employment rate of 10- to 14-year-olds.
Most theoretical models make the assumption that parents are the sole decision
makers with regard to children’s educational opportunities. The education of children
can thus be treated as an asset: a means of increasing future income at the expense of
present consumption. Poverty is one of the causes of child labor, but not necessarily
the main one. Basu and Van (1998) initiated the theoretical investigation into the
incidence of child labor when they assumed: 1) families would not send their children to
work if the family’s income without child labor was above some subsistence level; 2)
adult labor and child labor were perfect substitutes in production. In their model, they
proved the possible existence of multiple stable equilibria: an equilibrium with low
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wages where children worked and competed with unskilled adults, and another with
high wages and no child labor. The model suggests that a short-term ban on child labor
might be used effectively to jolt the economy to the favorable equilibrium with no child
labor, but it will only be successful if the ban increases adult wages sufficiently.
Lately, a greater importance has been placed on the role of credit market
imperfections in developing countries. Baland and Robinson (2000) viewed child labor
as a means for low-income households to transfer future income to the present when
borrowing was not available. Parents thus weighed their children’s future income
against the forgone income incurred from education. Ranjan (2001) and Jafarey and
Lahiri (2002) also focused on the lack of available credit as a reason why parents resort
to sending their children to work. Ranjan (2001) used an overlapping generations
model where households differ in their talent level and found that the incidence of child
labor increases as credit availability decreases and as income inequality increases.
Jafarey and Lahiri (2002) also found that the incidence of child labor decreases as
access to credit markets increases.
Demand for Child Labor
In the developed world, it has been debated whether trade policies are effective at
lowering the incentive for firms to use child labor. Staunch advocates against the use of
child labor believe that countries with lax labor standards should be sanctioned.
Unfortunately, proof of the use of child labor in production is difficult to find, and in those
countries where child labor is most rampant, existing laws against child labor often go
unenforced. Some economists worry that trade sanctions might increase the incidence
of child labor by punishing unskilled adult workers in the export sector, reducing the
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income of low-skilled households and possibly forcing children into more dangerous
work (Maskus 1997).
While studies have examined the supply of child labor from households, the
demand for child labor has not received the same amount of attention. This is partly
due to the lack of firm data, which makes empirical research difficult. The few empirical
studies, such as Busse and Braun (2004), tend to use macro-level data to find a
relationship between child labor and trade openness. Busse and Braun find that an
increase in trade openness is generally associated with a decrease in the incidence of
child labor, but the effect disappears after controlling for income. This suggests that the
method by which trade liberalization decreases child labor is through increasing
parental incomes, which then decreases the supply of child labor.
The theoretical models that have examined the demand for child labor have
focused on the effects of trade liberalization, trade sanctions, and foreign direct
investment. Gupta (2000) built a bargaining model with an efficiency wage function that
determines the productivity of children given the wage paid to the child. In that model,
parents were assumed to be selfish with regard to the child’s interest and are able to
bargain with firms over the child wage and the efficiency wage that is paid to the child in
the form of food. Dinopoulos and Zhao (2007) explored how trade liberalization affects
the demand for child labor in a model with efficiency wages. They find that both trade
liberalization and FDI that increases the output of the modern good decrease the
incidence of child labor.
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CHAPTER 2 NUTRITIONAL EFFICIENCY WAGES AND CHILD LABOR
This chapter builds a theoretical model that examines both the supply and demand
of child labor to examine the various policy options available to combat the problem.
Economists have mostly examined the issue of child labor and globalization through the
use of theoretical models due to the difficulty of acquiring data for empirical studies.
The few published empirical papers have focused on household surveys in small
regions in developing countries, but it is uncertain whether their results are applicable
elsewhere. For instance, Edmonds and Pavcnik (2005) found that globalization led to
an increase in the price of rice in Vietnam, which decreased the incidence of child labor
even though child labor is used heavily in the production of rice. On the other hand,
Kruger (2007) found that globalization had the opposite effect, increasing the incidence
of child labor in the coffee sector in Brazil even though globalization led to an increase
in the price of coffee beans and in the wages in that sector.
Gupta (2000) and Dinopoulos and Zhao (2007) published papers that focus
predominately on the demand for child labor. Both studies used child nutritional
efficiency wages, a practice continued in this chapter, which allows for the child wage to
be fixed. In Dinopoulos and Zhao (2007), market imperfections exist such that there is
an underemployment of children, and the income that guardians receive from sending
their child to work is exogenous. Unfortunately, this assumption appears to be highly
unrealistic, given that one of the main results in that paper, the effect of subsidies, has
been shown to have a significant impact on the supply of child labor. This paper
endogenizes the decision that parents make about whether to educate or employ their
children and thus allows for the supply of child labor to depend on the returns that
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parents receive from sending their children to work. The premium that parents receive
is endogenously determined and allows the model to analyze the effect that policies
have on both the supply and demand of child labor.
The model shows that policies enacted to reduce the incidence of child labor must
carefully explore both the supply and demand components of child labor. A policy like
child wage subsidies, while meant to reduce the supply of child labor, will also increase
the demand for child labor by reducing the cost of hiring one unit of child labor. This
can result in an increase in the overall incidence of child labor. Education subsidies
given to unskilled households are a better policy that will reduce the supply of child
labor without affecting demand. This result is supported by Schultz (2004) and
Ravallion and Wodon (2000). Trade sanctions, which reduce the demand for products
made with child labor, will reduce the demand for child labor. Child wage subsidies,
which in Dinopoulos and Zhao (2007) cause an increase in the incidence of child labor,
have an ambiguous effect when one accounts for the reduced supply of child labor.
This chapter is organized as follows. Section 2 describes the dynamic general
equilibrium model, starting with the characterization of the child schooling decision
made by parents and concluding with a description of the two production sectors in the
economy. Section 3 solves for the steady-state equilibrium, and Section 4 analyzes the
effect of domestic and international policies on the incidence of child labor. Simulations
are also included to examine welfare and distributional effects of the various policies.
Section 5 concludes with some final remarks.
The Model
The model is a dynamic, overlapping-generations model that endogenizes the
incidence of child labor. The model has two homogeneous goods: an agrarian good
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that is produced using land and unskilled adult and child labor, and a modern good that
uses skilled labor and capital in its production. The productivity of skilled workers
depends on their innate ability, which is assumed to differ among households. Perfect
competition in the production sector guarantees that adult workers are paid their
marginal revenue product of labor. The cost of one unit of child labor is split between
the amount given to children in the form of meals, which affects their productivity, and
the parental premium given to parents for the employment of their child.
This paper builds on two recent theoretical papers in the child-labor literature.
Ranjan (2001) uses differing talent levels to differentiate households, assuming that a
household’s talent remains constant across generations. This chapter assumes that
households are differentiated by ability levels, which determines the skilled wage if the
individual attended school as a child. It is also assumed that the ability of households is
constant across generations. Dinopoulos and Zhao (2007) utilize child nutritional
efficiency wages to fix the child wage. This leads to the adult skilled wage being fixed
and is used to derive the demand for child labor. A key difference between this paper
and that of Dinopoulos and Zhao is that in this paper the parental premium is
endogenously determined by bargaining between parents and firms.
Household Decision
Household income is the primary reason that parents resort to sending their child
to work. This is referred to as the “luxury” axiom in Basu and Van (1998) since
educating a child is considered an unaffordable luxury to poor families. In this model,
households are differentiated by their innate ability level, which subsequently
determines their adult wage if they attended school as children.
18
For notational convenience, the population of each generation is normalized to 1.
A family consists of one adult and one child, so the overall population in the economy is
2. The ability of each family follows a uniform distribution, where the range of abilities is
0,1 . Parents are assumed to know their child’s ability because it is the same as
their own. The assumption that parents and children have the same ability is for
notational simplification, while the assumption that parents are aware of their child’s
ability is a plausible one. Children sent to work receive some form of education before
they become old enough to work, whether it is in primary schooling or home schooling,
and parents are able to gauge their child’s aptitude in these early stages.
It is assumed that parents care about the future well-being of their children as well
as the family’s current consumption of a modern and agrarian good. This assumption is
a standard one used in the child-labor literature2. Let tV be the parent’s utility function
at time t :
1( , )t Xt Yt tV U C C V (2-1)
where is the level of altruism that the parent has toward his child’s future utility, and
( , )Xt YtU C C represents the family’s current consumption of the agrarian and modern
good, respectively. For simplicity, it is assumed that all families have identical
preferences. Writing Equation 2-1in terms of prices and income gives the following
indirect utility function:
, 1( , )t Xt Yt t tV Z p p I V (2-2)
2 See Basu (1998), Ranjan (2001), and Jafarey and Lahiri (2001).
19
Income is dependent on the child schooling decision and the household’s ability
level, where income at time t for any family is equal to:
if parent sends child to work
if parent sends child to school
ct
t
b wI
b
(2-3)
where b is the parent’s income, ctw is the child wage paid in kind to child workers at
time t , and ctw is the parental income from sending his/her child to work at time t ,
where will be referred to as the parental premium. It will be assumed that children
are fed at school if parents choose not send them to work, and if children are sent to
work, the firm will pay children in-kind by providing them food. The amount of food that
they provide will determine the productivity of the child, as will be discussed in the
production section in this chapter.
To simplify the model and to allow for the supply of child labor to be determined
explicitly, a Cobb-Douglas specification is used to represent the parent’s utility from
current consumption3:
1( , )U x y x y (2-4)
This leads to the following indirect utility function from Equation 2-2:
( , )rI
Z p IP
(2-5)
where 1(1 )r and 1
x yP p p is the price index.
It is necessary to examine in the steady-state equilibrium both the child-schooling
decision of parents who are skilled workers and those who are unskilled. A household
3 The results of the model hold generally for any homothetic utility function where income enters linearly.
A possible extension of the model would be to incorporate a utility function in which the marginal utility of income decreases as income increases, which would allow for income effects in the determination of the supply of child labor.
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is characterized by two factors: the parent’s skill level and the household’s ability level,
, which is constant across generations. This allows for a given household’s child-
schooling decision to be written in the form ( )iV , where ,i H L corresponds to
whether the parent is skilled ( i H ) or unskilled ( i L ).
A skilled parent’s child-schooling decision is summarized by the following
equation:
( ) ((1 ) ) ( ), ((1 ) ) ( )H H H H C LV Max Z w V Z w w V (2-6)
where the first part represents the parent’s utility if he sends his child to school, and the
second part represents the parent’s utility if he sends his child to work. To find the
critical ability level that makes a skilled parent indifferent between sending the child to
school versus work, we equalize Equation 2-6 using the Cobb-Douglas specification in
Equation 2-4:
((1 ) ) ((1 ) ) ( ) ( )H C H H LZ w w Z w V V
(2-7)
( )(1, ) (0, )Cr w
V VP
(2-8)
Let H represent the critical ability level that solves Equation 2-8. For all skilled
households with H , parents will chose to send their child to school. For all skilled
households with H , parents will opt to send their children to work.
Similarly, an unskilled parent’s child schooling decision is summarized by:
( ) ( ) ( ), ( ) ( )L L H L C LV Max Z w V Z w w V (2-9)
21
where the first part once again corresponds to the parent educating his/her child, and
the second part to sending the child to work. Equalizing to find the critical ability level,
L , yields:
( ) ( ) ( ) ( )L C L H LZ w w Z w V V (2-10)
Using the Cobb-Douglas specification, Equation 2-4, gives us the same equation as the
one for skilled parents, Equation 2-8.
Since Equation 2-8 represents the child schooling for both skilled and unskilled
households, H L , and the parent’s decision over whether to educate his child or not
is independent of whether the parent is educated himself. The ability level that solves
Equation 2-8 is represented by *
H L , where critical values will be denoted with
an asterisk.
In the steady-state equilibrium, values of the endogenous variables must remain
constant. To solve for * in the steady-state equilibrium, we can use the corresponding
value functions. In the steady-state, it must be true that the first term in Equation 2-6
solves an educated parent’s maximization problem since skilled workers are going to
choose to educate their child. Likewise, the second term in Equation 2-9 must solve an
unskilled parent’s maximization problem. The following must therefore be true in the
steady-state equilibrium:
1 1
( ) ((1 ) ) (1 )1 1
H H H
rV Z w w
P
(2-11)
1 1( ) ( ) ( )
1 1L L C L C
rV Z w w w w
P
(2-12)
22
Substituting Equations 2-11 and 2-12 into Equation 2-8, we can solve for the
critical ability level * that determines the supply of child labor:
( )
(1 ) ( )1
CH L C
r w rw w w
P P
*
1
0, 1L c
H
w w
Maxw
(2-13)
Families with ability level * educate their children, while families with ability level
* send their children to work.
Since a uniform distribution of abilities is assumed, and the population of children
is normalized to 1, the supply of child labor is equal to the critical ability level:
*
*
0
1
( ) 0, 1L c
S S
H
w w
C f d C Maxw
(2-14)
As the unskilled wage, Lw , and the parent’s income from sending their child to
work, , increases, the supply of child labor also increases. As the skilled wage, Hw , or
the level of altruism, , increases, the supply of child labor decreases.
Production
The production sector is characterized by perfect competition, which ensures that
factors are paid their marginal productivities. Capital complements skilled labor in the
production of a modern good, while land complements unskilled adult and child labor in
the production of an agrarian good. The production functions in both sectors are
represented by constant returns to scale technologies of Cobb-Douglas form.
23
Modern sector
Skilled labor and capital are used in the production of the modern good. The
productivity of a skilled worker will depend on his ability level. Using specific sector
capital which is fixed in the modern sector allows for the analysis of foreign direct
investment and its effect on the returns to education and the parental schooling
decision.
The production of the modern good is described by the following Cobb-Douglas
production function:
1( , )Y F H K H K (2-15)
where *
1
(1 )H d
4 is the total human capital stock of skilled workers and 1 is
the productivity of a skilled worker given his ability. The price of the modern good will
act as the numeraire. The profit function for a firm producing the modern good is:
1
Y Y YH K w H r K (2-16)
Firms maximize Equation 2-16 with respect to the employment of skilled workers and
sector-specific capital, yielding the following first-order conditions:
1
0YH
d Kw
dH H
(2-17)
(1 ) 0YK
d Kr
dK H
(2-18)
4 For an evaluation of the integral, see equation (33).
24
The wage paid to skilled workers and the rental of capital are given by Equations 2-17
and 2-18. Since a skilled worker with ability has productivity equal to 1 , his
income will be equal to (1 ) Hw .
Agrarian sector
Output in the agrarian sector is determined by the amount of unskilled labor, both
adult and child, and the amount of land available. Studies by the International Labor
Organization (2006a) have found that the majority of children who forgo schooling tend
to work in rural settings, so the use of land as a complement to child labor is warranted.
The use of nutritional efficiency wages, not unlike that used in Stiglitz (1976), describes
how the productivity of child laborers is dependent on the amount of food given to them
in the form of meals. The nutritional efficiency function, which determines the
productivity of children, ( )ch w , is an increasing and concave function with respect to the
consumption of food (the in-kind child wage), and it is bounded from above. (i.e., there
is a limit to how productive children can be, and since it is assumed that child labor is
always less productive than adult unskilled labor, 0 ( ) 1c
cw
h w
.)
Gupta (2000) developed a model where the productive efficiency of child labor
depends on the amount of food their employer gives them. He found that when
employers maximize their profits, this leads to the common efficiency wage equation
that fixes the child wage. Dinopoulos and Zhao (2007) utilize nutritional efficiency
wages for children along with efficiency wages for skilled adults to analyze the effects of
globalization and domestic policies on the demand for child labor.
The production of the agrarian good is determined by the following production
function:
25
1( , ) ( )cX G L C L h w C T (2-19)
where ( )ch w is the nutritional efficiency function of a child worker; is a child
equivalent scaling constant that equates how one unit of adult unskilled labor
corresponds with one unit of child labor; α is a productivity parameter; and C , L , and T
are the amount of child labor, adult unskilled labor, and land, respectively. Firms in the
agrarian sector maximize their profit with respect to land, adult unskilled labor, child
labor, and the child wage paid to children in the form of meals:
1( ) (1 )X X c L C Tp L h w C T w L w C r T (2-20)
Although children are paid Cw in the form of food, firms have to pay the premium,
cw , to parents, which makes the total cost of one unit of child labor equal to (1 ) Cw .
Maximizing Equation 2-20 yields the following first-order conditions:
1
0( )
XX L
c
d Tp w
dL L h w C
(2-21)
1
( ) (1 ) 0( )
XX c c
c
d Tp h w w
dC L h w C
(2-22)
1
'( ) (1 ) 0( )
XX c
c c
d Tp h w C C
dw L h w C
(2-23)
(1 ) 0( )
XX T
c
d Tp r
dT L h w C
(2-24)
If we combine Equations 2-22 and 2-23, we get the standard result in the
nutritional efficiency wage literature:
**
*
( )'( ) c
c
c
h wh w
w (2-25)
26
This leads to the child wage, *
cw , being fixed in the steady-state equilibrium for a
given nutritional efficiency function. For agrarian firms to maximize profits, they must
pay child workers a wage that equates their marginal productivity of labor to their
average productivity.
Combining Equations 2-21, 2-22, and 2-25 solves for the adult unskilled wage in
terms of the child wage, the parental premium, and the child’s productivity:
**
*
(1 )
( )
cL
c
ww
h w
(2-26)
To determine the relationship between the fixed child wage and the rental of land, we
combine Equations 2-21 and 2-24 to determine the relative rental of land in proportion
to the unskilled adult wage:
*( )
(1 )
cT
L
L h w Cr
w T
(2-27)
Equations 2-21 and 2-27 lead to the zero-profit condition in terms of the unskilled adult
wage, the productivity parameter, , and the price of the agrarian good, Xp :
1 * 1(1 )X L Tp w r
(2-28)
This zero-profit condition, along with Equation 2-26, determines the rental of land:
1 * 1* 1 1
*
(1 )(1 )
( )
cT X
c
wr p
h w
(2-29)
And, using Equation 2-27, the demand for child labor, DC :
1 1 1* *1 1 1 1
*( ) (1 )
( )
D
X c c
c
LC p h w w T
h w
(2-30)
27
The demand for child labor is increasing in the amount of land in the agrarian sector and
the price of the agrarian good, and is decreasing in the amount of adult unskilled labor
and in the parental premium.
Substituting Equations 2-17 and 2-26 in the household schooling decision,
Equation 2-14, the supply of child labor can be derived in terms of the parental premium
and the parameters of the model:
* * 1
* 1
(1 )1
( )
S c c
c
w w HC
h w K
(2-31)
The supply of child labor is increasing in the parental premium and in the supply of adult
skilled workers and is decreasing in the amount of capital in the modern sector and in
the price of the modern good.
Steady-State Equilibrium
In the steady-state equilibrium, 1t tC C for all 0t . The amount of child labor at
any time t has to be in the range 0,1C . Children who work become unskilled
laborers in the next period, while children who attend school become skilled laborers
working in the modern sector. The supply of unskilled workers is equal to the quantity
of child labor in the previous generation, 1t tL C , while the amount of skilled workers in
efficiency units is:
1
1 2
1 123(1 )
2 2t
t tt
C
C CH d
(2-32)
Substituting these values into Equations 2-30 and 2-31 and writing the equations
in terms of the inverse supply and demand of child labor in the steady-state equilibrium
yields:
28
1
*1
* 1 1
*
1 ( )( ) 1
( )
D cX c c
c
h wp h w w T C
h w
(2-33)
1 *
1 ** *
2
1 ( )
( )( )13 2
2
S c
cc c
K C h w
h wh w wC C
(2-34)
These equations not only determine the incidence of child labor in the steady-state
equilibrium, but they also ensure an interior equilibrium, 0,1C .
Figure 2-1. Steady-state equilibrium
As 0C , the demand for child labor goes to infinity because the scarcity of unskilled
labor drives the unskilled wage, and the parental premium, upward. The same holds as
1C . In this case, most of the population is employed in the agrarian sector, and the
marginal productivity of a unit of skilled labor goes to infinity. As shown in Figure 2-1,
the parental premium and the incidence of child labor in the steady-state are determined
by the intersection of Equations 2-33 and 2-34.
C 1
* CD
C*
CS
29
Comparative Statics
In this section, the comparative statics are computed to show how globalization
and domestic policies affect the incidence of child labor. The paper first examines how
an increase in foreign direct investment can impact the incidence of child labor before
exploring the effects of domestic policies. When applicable, simulations were
conducted to analyze the effect of the different policies on welfare. The parameters
used in the simulations were .75 , .3p , .5 , .2cw , ( ) .7ch w , .6 , 10K ,
.9T , .5 , and 1 5. Using these figures, the incidence of child labor is roughly
21% of the child population, and the parental premium, , is 2, meaning that parents
receive .4cw for sending their child to work, which is a little less than half of the adult
unskilled wage.
Foreign Direct Investment
Globalization can impact an economy by allowing an additional influx of foreign
capital and investment. In this model, foreign direct investment impacts the parent’s
schooling decision by increasing the marginal product of skilled labor. The increase in
the skilled wage, Hw , shifts the supply of child labor leftward, as shown in Figure 2-2.
This results in a decrease in the incidence of child labor and an increase in the parental
premium.6 An interesting observation is that the increase in the parental premium not
5 Since the population is normalized to 1, the amount of land and capital can be thought of the land per
capita and the capital per capita, respectfully. The values of K and L were calculated using statistics from the Philippines, where K is an approximation of the total capital divided by the population, and T is the amount of usable land (in square miles) divided by the population. The other values were arbitrarily assigned, but changes in these values do not qualitatively impact results.
6 Davis and Voy (2007) and Edmonds and Pavcnik (2005) have studied the relationship between FDI and
trade openness with the incidence of child labor while controlling for endogenous factors. They also find a negative relationship between child labor and foreign direct investment.
30
only increases the family’s income from sending the child to work, but it also increases
the adult unskilled wage through the relationship in Equation 2-26.
Figure 2-2. Increase in FDI
Consequently, an increase in foreign direct investment not only has the benefit of
directly decreasing the incidence of child labor, it also increases the incomes of poor
families. This result can better be seen by comparing the indirect utility of households,
Equation 2-2, before and after the increase in foreign direct investment in Figure 2-3.
The increase in capital reduces the incidence of child labor from 22% to 19%
(since a uniform distribution of abilities is assumed), and all households are better off
than previously. Skilled households with the highest abilities benefit the most from an
increase in foreign direct investment since the higher-skilled wage benefits workers with
the highest productivity. The results on the incidence of child labor depend on the fact
C
*
0
0
SC
*
0C
1
SC
*
1C
*
1
DC
31
that capital is used only to produce the modern good. If capital were used in the
production of both goods, then the results would be ambiguous.
Figure 2-3. Welfare among households with an increase in FDI
Trade Sanctions
Internationally, trade sanctions have been recommended as a way of punishing
countries that use child labor in the production of traded goods. By reducing the
international demand for the good in question, trade sanctions attempt to lower the
international demand, which corresponds to a drop in the price of the agrarian good in
the model. The fall in Xp lowers the demand for child labor, Equation 2-33, and lowers
the incidence of child labor in the steady-state equilibrium. However, families with low
ability may be punished because sanctions reduce nominal incomes by decreasing the
parental premium and the adult unskilled wage.
32
Figure 2-4. Welfare among households with trade sanctions
As shown in Figure 2-4, the effect of trade sanctions on unskilled household utility
is ambiguous due to the fact that the lower agrarian price reduces the price level and
can increase real income. Whether the decrease in the price of the agrarian good
negates the fall in unskilled households’ incomes depends on the relative demand for
the agrarian good. Skilled nominal wages fall due to the increase in skilled workers, but
real incomes may rise due to the decrease in the price of the agrarian good.
Education Improvements
One way governments can increase child enrollment in schools is to improve the
efficiency of the education system, which makes skilled workers more productive. By
increasing the marginal productivity of skilled workers, the incomes of skilled workers
and the returns to education will decrease the supply of child labor. This can be
modeled by changing Equation 2-32, the amount of skilled adult labor in terms of
efficiency units, to:
33
*
1
(1 )H d
(2-35)
where represents improvements in education that increase the productivity of skilled
workers. The supply of child labor then becomes:
* 1*
* 1
(1 ) 11
( )
S cc
c
w HC w
h w K
(2-36)
which is unambiguously less than the supply of child labor in Equation 2-31. Similar to
the case of foreign direct investment, an increase in the education efficiency parameter,
, will shift the supply of child labor leftward, leading to an increase in the parental
premium and a decrease in the incidence of child labor. Welfare effects are also
similar, but there are greater gains for adult skilled households due to the increase in
productivity.
Migration
Emigration of skilled workers is common in developing countries as wages for
skilled workers are higher in developed economies. Here, the paper examines how this
migration affects the incidence of child labor. First, assume that the skilled workers who
migrate are those with the highest abilities since they would benefit the most from
moving. Let ( ,1) represent the skilled worker with the lowest ability who decides to
relocate. Therefore, the effective units of skilled labor in Equation 2-32 becomes:
22
1 122(1 )
2 2t
t tt
C
C CH d
(2-37)
which is unambiguously smaller than Equation 2-32 since 1 . Replacing Equation 2-
37 in the supply of child labor equation yields:
34
12
2
1 1
**
* 1
22
2 2(1 ) 1
1( )
t t
S cc
c
C C
wC w
h w s K
(2-38)
which is less than Equation 2-34 and represents a decrease in the supply of child labor
in the steady-state equilibrium. Like the case of foreign direct investment, emigration of
skilled labor causes the supply of child labor to shift leftward, reducing the incidence of
child labor in the steady-state and increasing the current income of unskilled families.
When skilled labor migrates, a void of skilled labor is left in the modern sector
while the amount of capital remains fixed. This increases the marginal productivity of
skilled workers and thus the skilled wage. The increase in the returns to education
reduces the number of parents who are willing to forgo sending their child to school. As
shown in Figure 2-5, the welfare of unskilled households is unchanged, but the welfare
of skilled households (assuming .9 ) increases.
Figure 2-5. Welfare of households with emigration of skilled workers
35
Subsidies
Last, the paper examines how two different types of subsidies affect the
employment of children in the economy. The first type analyzes financial assistance
given directly to child workers in the form of meals (Dinopoulos and Zhao, 2007). The
second type, which has been empirically tested, deals with subsidies given directly to
low-income families to encourage them to send their children to school.
Child wage subsidies
The child wage subsidy is assumed to come from an exogenous source, which
might include foreign aid from developed countries and aid from non-governmental
organizations. If the subsidy were financed by the government, we would then have to
examine the scope of government and the way in which the subsidy is financed. A
direct subsidy given to children in the form of meals effectively changes an agrarian
firm’s profit maximizing problem, Equation 2-20, to:
1( ) (1 )W
X X c L C Tp L h w s C T w L w C r T
(2-39)
where Ws is the value of the wage subsidy. When the agrarian firms maximize their
profits with respect to the amount of child labor and the wage paid to child labor in terms
of food, the standard nutritional efficiency wage equation becomes:
**
**
'( )1
( )
W
c c
W
c
h w s w
h w s
(2-40)
This child wage subsidy increases the average productivity while decreasing the
marginal productivity. This causes firms to lower the child wage that they pay in terms
of food, changing the steady-state equations to:
11
** **1 11**
( ) (1 )( )
DD W
X c c W
c
CC p h w s w T
h w s
(2-41)
36
1
2
****
** 1
13 2
(1 ) 1 21
( )
S S
S ccW
c
C Cw
C wh w s K
(2-42)
The demand for child labor, Equation 2-41, increases, while the supply of child labor,
Equation 2-42, decreases, as shown in Figure 2-6. The effect of the child wage
subsidies on the incidence of child labor is ambiguous since the increase in the parental
premium is countered by a decrease in the child wage. Child wage subsidies lead to a
decrease in the adult unskilled wage, and its effect on the skilled wage depends on
whether the level of child labor changes or not.
Figure 2-6. Child wage subsidies
This result differs from that found in Dinopoulos and Zhao (2007). In that paper,
the supply of child workers is perfectly elastic. This amounts to the supply of child labor
being represented by a horizontal line at the exogenous parental premium. The child
wage subsidy would therefore only increase the demand for child labor, leading to an
C
*
0
0
SC
*
0C
1
SC
*
1
0
DC
1
DC
37
increase in child labor in the agrarian sector. With land instead of skilled labor in the
agrarian sector and an endogenous supply of child labor, the opposite holds true. The
increase in the average productivity of child laborers decreases the adult unskilled
wage, which therefore increases the relative returns to education and decreases the
supply of child labor. This leads to an ambiguous change in the incidence of child
workers.
Education subsidies
Some countries have used education subsidies to reduce the incidence of child
labor. Schultz (2004) examined a Mexican program called Progressa, in which
households in a randomly selected low-income locality were given income subsidies if
they sent their children to school. This resulted in an increase in average schooling for
children in the localities that received the subsidy compared with similar localities that
did not. Likewise, Ravallion and Wodon (2000) examined a similar education subsidy in
Bangladesh and found that although increases in school enrollments came mostly at the
expense of child leisure, the education subsidy did have a significant effect on reducing
the incidence of child labor.
To incorporate an education subsidy into the model, it is necessary to look back to
the supply of child labor equation, Equation 2-13, and add the subsidy, Es , that parents
would receive if they send their child to school. The household maximization problem
becomes:
*
1
0, 1
E
L c
H
sw w
Maxw
(2-43)
38
The education subsidy becomes an opportunity cost to parents who send their child to
work. This changes the supply of child labor equation to:
1 * *
1 * * ** *
2
1 ( ) ( )
( ) ( ( )( )13 2
2
ES c c
c c cc c
K C h w s h w
h w w h wh w wC C
(2-44)
An education subsidy will cause a leftward shift of the child-labor supply curve and
therefore will have an outcome similar to an increase in FDI. Unskilled family income
will benefit twice: once through a direct increase in household income caused by the
education subsidy, and then through an indirect increase in the unskilled wage caused
by the decrease in child workers.
Table 2-1. Summary of Comparative Statics Results
Supply of
Child Labor
Demand for
Child Labor
Incidence of
Child Labor
Welfare of Unskilled
Households
Domestic Policies
Education
Improvement
Decreases Unchanged Decreases Increases
Migration of Skilled
Workers
Decreases Unchanged Decreases Increases
Child Wage
Subsidies
Decreases Increases Ambiguous Increases
Education
Subsidies
Decreases Unchanged Decreases Increases
Trade Policies
Foreign Direct
Investment
Decreases Unchanged Decreases Increases
Trade Sanctions Unchanged Decreases Decreases Ambiguous
Table 2-1 summarizes the comparative statics results and the effects that policies
have on the welfare of unskilled households. As shown, most policies that reduce the
incidence of child labor will lead to an increase in the welfare of unskilled households,
even though some of these policies reduce the wage of unskilled workers.
39
Conclusion
Child labor is a major problem in developing countries, but one that looks to be in
decline around the world. Still, some forms of child labor might always exist as long as
parents fail to sustain their family using only their income, and as long as firms have
access to this cheap form of labor. The only way to eradicate the problem truly is to
ensure that families can sustain adequate incomes without child labor earnings, and
that there are high rewards for schooling so that families can escape the vicious circle of
poverty that plagues parts of the developing world.
This paper develops a dynamic general-equilibrium model of child labor that
incorporates the parental schooling decision, which determines the supply of child, labor
and the profit-maximizing conditions of private firms, which determine the demand for
child labor. The use of child nutritional efficiency wages allows for the development of
an active market for child labor that is dependent on the skilled and unskilled wages in
both sectors, the amount of capital and land in the economy, and parental preferences
toward educating their children. This allows us to study the impact of domestic and
foreign policy and its effects on both the demand for and supply of child labor.
Increases in foreign direct investment increase the returns to education and lead
to a decrease in the incidence of child labor. In the long run, this increases the human
capital stock in future generations and leads to higher sustained economic growth. This
finding is consistent with similar works by Dinopoulos and Zhao (2007). This paper
differs from Dinopoulos and Zhao in regard to the impact of child wage subsidies. While
Dinopoulos and Zhao find that child wage subsidies increase the incidence of child labor
by increasing their complement of production in the agrarian sector, this paper finds that
child wage subsidies increase the incomes of unskilled households but have an
40
ambiguous effect on the incidence of child labor. Finally, this paper shows that
education subsidies can unambiguously decrease the incidence of child labor by giving
families a monetary incentive to send their children to school.
41
CHAPTER 3 CHILD LABOR AND FIRM HETEROGENEITY
The phenomenon of child labor is a stubborn problem that continues to plague the
least-developed countries and contributes to the perpetual cycle of poverty from which
many nations have been unable to break free. Although the incidence of child labor has
been steadily declining over the last decade, it still remains staunchly prevalent in the
poorest nations. The research on the causes of child labor has grown over the last
twenty years but has been principally rooted in one side of the story, namely the
decisions of households that determine the supply of child labor7.
Edmonds and Pavcnik (2005b), Dinopoulos and Zhao (2007), and Kis-Katos
(2007) are a few examples of recent theoretical papers that have examined child labor
from the demand side. This allows for the analysis of how trade liberalization, FDI, and
other global factors affect the demand for child labor. These papers assume that
children work in sectors that produce goods that are traded in an inter-industry trade
setting. In doing so, they assume that the wages paid to children (or to the family of the
children) reflect the productivity of the child worker. A study conducted by the
International Labour Organization (2007) on child wages and productivity reveals that
differences in adult and child wages are not reflected in their productivity differences.
Even in sectors where children were nearly as productive as their adult counterparts,
the child wage was anywhere from one-sixth to one-fourth the wage paid to adult
unskilled workers (ILO 2007).
7 Basu and Van (1998), Ranjan (2000), and Jafarey and Lahiri (2002) are just a few notable papers that
have examined the supply of child labor from the household perspective. For a comprehensive survey of the child labor literature, see Brown, Deardorff, and Stern (2003).
42
The aim of this paper is to analyze the short-run demand for child labor in the
presence of firm heterogeneity and intra-industry trade. The model developed by Melitz
(2003) will be used as the foundation for the model in this chapter, which will
endogenously determine the cutoff productivity level needed for a firm to enter an
industry and the export cutoff productivity level that makes it profitable for a firm to
export its good. As will be shown, sector characteristics, particularly how firm
productivity affects the relative productivity of child and adult workers, will determine
whether trade liberalization can remedy or exacerbate the incidence of child labor.
Intra-industry trade, while not as prevalent in developing countries as in
industrialized countries, is still a significant source of trade between similar developing
nations and therefore must be examined with regard to child labor.8 Heterogeneous
firms engaging in intra-industry trade in developing countries tend to be located in
sectors characterized by significant amounts of child labor. Balassa (1998) examined
the role of firm heterogeneity in developing countries. He looked at the level of intra-
industry trade in Latin America and noted that intra-industry trade was prevalent among
similar South American countries in sectors such as textiles; fabricated metal goods;
and paper, clay, and glass products. Kucera (2002) concluded that child labor
employed in the export sector was mostly located in textiles, apparel, craft production,
and other light manufacturing in developing countries, most of the industries where
intra-industry trade is dominant in the developing economies studied by Balassa (1998).
8 Baland and Robinson (2000) note the need to consider firm heterogeneity before concluding that a ban
on child labor will always lead to a Pareto improvement. Hummels and Klenow (2005) examine the extent of the extensive-margin in 126 countries and show that in export variety makes up a large percentage of exports for large developing countries like China (.70) and India (.44).
43
Empirical work into the demand for child labor is relatively lacking compared with
the amount of work done on household factors. Unfortunately, the difficulty of acquiring
firm level-data on child labor, necessary to test the results of this paper’s model, is
considerable, and even if it were possible to obtain these data, the survey’s reliability
would be questionable since firms have a notable disincentive to disclose information on
their use of child labor and the wages paid to their workers. Using macro-level data,
Kucera (2002), Busse and Braun (2004), and Davis and Voy (2007) have empirically
found a relatively weak relationship between trade liberalization and child labor after
accounting for changes in income. The use of macro-level data does not allow the
demand for child labor to vary by sector, which this paper examines. It is therefore
necessary to rely on theory to analyze the impacts of trade liberalization on the demand
for child labor, and to account for the fact that the relative productivity of child workers
differs among sectors.
This chapter is organized as follows. Section 2 outlines the closed model and
solves the benchmark case where the productivity elasticity of adult and child labor are
equalized. Section 3 describes how the model differs when adult labor and child labor
differ in their productivity elasticity and shows how this might affect the ability of
enforcement to reduce the demand for child labor. Section 4 introduces trade and
shows how trade liberalization affects the demand for child labor in three cases. Last,
Section 5 summarizes the policy implications and offers concluding remarks.
The Basic Model
The model presented below is based on Marc Melitz’s (2003) intra-industry trade
model. For a firm to enter the market, it first incurs a fixed entry cost that allows it to
conduct research and development (R&D). Once R&D has taken place, firms discover
44
how productive they are in manufacturing a unique variety. Firms then choose the
optimal amount of child labor (in proportion to the amount of adult workers) given their
productivity level by maximizing their expected firm value. The use of child labor in
production lowers a firm’s marginal cost, which increases the profits earned each
period, but has the trade-off of exposing the firm to additional risk each period. This
trade-off results in an interior equilibrium where the proportion of child labor is greater
than or equal to 0 but is bounded from above.
Last, firms decide whether their expected firm value will be able to cover their fixed
cost of production. If a firm’s expected value exceeds its fixed production cost, then it
will choose to produce. If a firm’s expected value is less than the fixed production cost,
then the firm will choose to exit the industry. In the steady-state equilibrium, there exists
a unique cutoff productivity level such that firms with productivity equal to or above that
threshold will choose to produce, and firms with productivity below the threshold will exit
the market.
Consumer Demand
The preferences of a representative consumer are given by a C.E.S. utility function
over a continuum of goods, i :
1
( )
i
i iU q d
(3-1)
which will be maximized subject to the representative consumer’s budget constraint.9
Assuming that 0,1 , this yields a demand function for each variety i ,:
9 The Lagrangian and the corresponding first-order conditions are shown in Appendix A.
45
( ) ( )i iq Z p (3-2)
where is the Lagrangian multiplier defined in Appendix A,
1
( )
i
i iZ q d
and
1
1
measures the elasticity of substitution between any two varieties. Total
expenditure for a given variety is calculated by multiplying the demand for a variety by
its price. Aggregating over all varieties nets total revenue, which must equal total
expenditure in the steady-state equilibrium:
1( )
i
i iR Z p d
(3-3)
The price index,
1
1
1( )
i
i iP p d
, is a weighted average of the price of all
varieties. Since total revenue equals the aggregate quantity, Q , times the aggregate
price, we can rewrite Equation 3-2 as:
1
( ) ( )( )
( )
i
i ii
i i
p R pq Q
Pp d
(3-4)
The relative quantity demanded of two goods will therefore be dependent on their
relative price:
1
1
( ) ( )
( ) ( )
i i
j j
p q
p q
(3-5)
Production
Adult labor and child labor are assumed to be perfect substitutes in production,
where 0 0,1b is an adult-scaling constant similar to the one used in Dinopoulos and
46
Zhao (2007). The supplies of both types of labor are assumed to be fixed and perfectly
inelastic in order to examine the short-run demand for child labor. Although the
employment of children reduces a firm’s marginal cost since the wage of children
relative to their productivity is less than that of adult labor, the trade-off that firms face of
additional exposure to risk ensures that firms will not want to employ only children to
produce the differentiated good. The quantity produced by each firm is a function of the
amount of labor hired and the firm’s productivity parameter, i :
0i i i i iq a b l (3-6)
where the amount of child labor, ic , demanded by a firm is proportional to the amount of
adult labor, il :
i i ic a l (3-7)
The productivity elasticity for adult labor is equal to unity for all firms. A firm with a
productivity parameter that is 10% greater than a rival firm will have adult labor that is
also 10% more productive. For child labor, the productivity elasticity is equal to the
parameter . Since different sectors might have different s, whether is greater or
less than unity will critical in how a policy will affect the incidence of child labor.
When 1 , the relative productivity of adult and child workers will be constant for
all firms. This will be referred to as the benchmark case to compare against the cases
when is greater and less than unity. When 1 , referred to as the traditional-
industry case, child labor is relatively more productive compared with adult labor when
working in firms with a higher productivity parameter. A traditional sector is one in
which better technology simplifies work for all workers, thereby reducing any
47
productivity advantage that adult labor might have over child workers (e.g., certain
textiles).
Alternatively, when 1 , referred to as the modern-industry case, better
technology can lead to productivity gains that worsen the relative productivity of child
laborers. This can be due to technology that complicates the production process,
resulting in a need for workers with greater ability and experience. In this scenario,
child labor becomes less productive relative to adult labor in firms with higher
productivity, and in extreme cases ( 0 ), the productivity of children might actually
decline when working for more firms with higher productivity.
The wage of adult labor will act as the numeraire, and the child wage will be
exogenously determined and equal to 1Cw . Empirical evidence conducted by the
International Labour Organization (2007) shows that the wage of child labor is generally
not reflected by their productivity and is always less than the corresponding adult wage
after accounting for productivity differences. The total cost function for a firm is:
0
11 i C
i C i i
i i i
a wTC a w l q
a b
(3-8)
The marginal cost, i , of a given firm depends on the productivity parameter drawn by a
firm, i , and the proportion of child labor, ia , chosen:
0
1 i Ci
i i i
a wMC
a b
(3-9)
Since empirical evidence (ILO 2007) has shown that the reason why firms employ child
labor is that it is cheaper than adult labor after accounting for productivity differences, it
will be assumed that:
48
0
1C
i i
w
b (3-10)
This assumption ensures that firms that use a higher proportion of child labor will have
lower marginal costs10.
Maximizing per-period profits, Equation 3-11, with respect to price yields the
standard profit maximizing price which is a constant markup over the firm’s marginal
cost, Equation 3-12:
1
0
( ) 1 ( )i i C i
i i i
p a w pR Q
P a b P
(3-11)
( )1
ii ip
(3-12)
A firm’s per-period profit and revenue can then be calculated as a function of the firm’s
marginal cost, the price index, and aggregate expenditure (equal to total revenue):
1
( )( ) ( ) 1 i i
i i
r Rr
P
(3-13)
11( )
( ) i ii
pr R R
P P
(3-14)
The per-period revenue and profit of a firm increase as the price index and total
expenditure increase and as marginal cost, i , decreases. Since marginal cost is
decreasing in the proportion of child labor and the productivity parameter, firms with
higher productivity and/or higher proportion of child labor will earn higher per-period
10 This assumption places a lower bound on the productivity elasticity of child labor,
0
ln
ln
i C
i
w
b
49
revenue and profit. From Equation 3-5, the price ratio of two firms depends on the
relative quantities produced. We can rewrite it as:
( )
( )
i i
j j
q
q
(3-15)
All else equal, firms with lower marginal costs will produce more output.
Child Labor Demand
The demand for child labor, ic , by a firm with productivity i is determined by the
firm’s output:
0
i ii i i
i i i
a qc a l
a b
(3-16)
1
00
1i i Ci
i i ii i i
a a wRc
P a ba b
(3-17)
Taking the derivative of Equation 3-17 with respect to ia yields:
1
0
2 21
0 000 0
1 1i i i C i i C C i i
i i i i i i ii i ii i i i i i
dc a w a a w w bR
da P a b a ba ba b a b
(3-18)
The assumption in Equation 3-10 ensures that Equation 3-18 is positive for all firms. As
the proportion of child laborers increases, the total amount of child labor hired by that
firm also increases. This is due not just to the direct increase in the proportion of child
labor, but also because the increase in ia lowers a firm’s marginal cost and increases
the output produced.
50
Firm Value
As in Melitz (2003), all firms face a probability of receiving a negative shock each
period that might force the firm to exit. A firm’s probability of death, iD , will depend on
the proportion of child labor employed by the firm:
1
ii
aD
(3-20)
where 0,1 represents the strength of child labor enforcement, and firms that hire
no child labor face a probability of death equal to 0 11
. Firms that choose to
employ a higher proportion of child labor are more likely to be “caught” by law
enforcement and are therefore more likely to face a negative shock. In this sense,
strength of enforcement can not only represent government action to catch employers
of child labor, but it may also incorporate the success of consumer groups advocating
against the purchase of goods produced using child labor.
A firm’s market value, or expected total profit, is equal to their per-period profit, i ,
divided by the probability of death faced each period, and then subtracting a one-time
fixed production cost equal to Pf :
1
0
1 1 1i c
P P
i ii i i
a wRV f f
a aP a b
(3-21)
Firms maximize their market value with respect to the proportion of child labor, taking
aggregate revenue and aggregate prices as given:
11
20
2
0 00
1 1 110i C C i i i C
i
i i i i i i i ii i i
R P a w w b a wdVa
da a a b a ba b
(3-22)
51
0 01 1i C i i i i C i
i
a w a b b w
a
(3-23)
The optimal proportion of child labor, ia , that maximizes a firm’s value will be
determined by Equation 3-23 and depends on a firm’s productivity parameter, i , and
the exogenous parameters of the model: *( )i ia . When 1 , the productivity
parameter drops from both sides of 3-23, and the optimal proportion of child labor will
be the same for all firms regardless of their productivity parameter: *
0i
i
da
d .
When 1 (the traditional-sector case), the RHS of Equation 3-23 becomes
greater than the LHS as the productivity parameter increases. Therefore, the more
productive firms will choose a higher proportion of child labor compared to firms with
lower productivity. The inverse is true when 1 (modern-sector case). Firms with
high productivity will not benefit as much from child labor compared to firms with low
productivity and therefore will choose a lower proportion of child labor. The rest of this
section will detail the solution for the benchmark case ( 1 ).
Solving the Benchmark Case
When 1 , the optimal proportion of the child labor equation becomes:
0 01 1 1C C
aw ab b w
a
(3-24)
where a is the proportion of child labor chosen by all firms, which solves Equation 3-24
and is therefore the average proportion of child labor in the industry. The marginal cost
for a firm with productivity i is:
52
0
1 1
1
Ci i
i
aw
ab
(3-25)
An increase in the child wage, Cw , or the strength of enforcement, , reduces the
optimal proportion of child labor that firms chose and increases firms’ marginal costs.
Similarly, an increases in the elasticity of substitution, , raises the optimal proportion
of child labor and reduces the marginal costs of all firms.
Aggregation
Aggregate price is equal to a weighted average of firm prices:
1
11
0
( )
Max
iP p Mu d
(3-26)
where M is the mass of firms in the steady-state equilibrium and ( )u is the ex-ante
distribution of the productivity parameter with the range 0, Max . Aggregate price can
also be expressed as:
1111
1
0
( )
Max
i
AMP u d
(3-27)
where 0
1
1
CawA
ab
. The price index can be summarized as a function of the average
productivity, :
1
1
1
0
( )
Max
i u d
(3-28)
and the mass of firms in the steady-state equilibrium, M :
53
1
111
AMP M p
(3-29)
The rest of the aggregate variables are then calculated using Equation 3-29. Total
quantity is equal to:
1
1
0
( )
Max
iQ M q u d
(3-30)
The relationship between the relative quantities of two firms in Equation 3-15 can
be used to relate the quantity produced by any firm to the output of the firm with
average productivity, :
1
11
1
0
( )
Max
q MQ u d M q
(3-31)
Total revenue and aggregate per-period profit, , can similarly be found as functions of
the mass of firms and average productivity:
R M r
(3-32)
R
M
(3-33)
Aggregate per-period profit can also be used to solve for the average per-period profit,
defined by , which is equal to the per-period profit of the firm with average
productivity:
M
(3-34)
The demand for child labor, derived in Equation 3-16, is simplified to:
54
1
01i i
a Ac Q
Pab
(3-35)
where A depends on a as shown in Equation 3-27. An increase in the proportion of
child labor used by firms, a , will increase the demand for child labor by all firms.
Free Entry and Exit
For firms to enter the market, they first incur a fixed entry fee, Ef , needed to
conduct R&D. Firms then draw a productivity parameter from a given distribution, g
, with a continuous cumulative distribution G . Let * be the productivity parameter
that corresponds to the firm whose firm value is equal to zero:
*10PV f
a
(3-36)
The expected value of firms with * will be greater than zero, so those firms will
remain in the industry; firms with * will have a negative expected value and will
choose to exit. The probability of successful entry is denoted by Ep :
*1Ep G (3-37)
The average productivity in an industry will be a function of the cutoff productivity level,
* , and the probability distribution, ( )g :
*
1
1
* 1
*
1( )
1
Max
g dG
(3-38)
55
An increase in the cutoff productivity level, * , will increase average productivity, ,
since the least productive firms are forced to exit. The average firm value, equal to the
value of the firm with average productivity, is equal to:
1
P
rV f
a
(3-39)
From Equation 3-36, the value of the cutoff firm must equal zero. We can substitute
Equation 3-36 into Equation 3-39 to yield the average firm value in terms of the cutoff
productivity level:
1
*
*1PV f
(3-40)
Equation 3-40 is the zero-value cutoff condition (ZVC). It shows the relationship
between the cutoff productivity level, * , and the average firm value, V .
Potential entrants have the probability Ep of obtaining a productivity parameter
equal to or above the cutoff level. The expected value of a potential entrant is:
*1Ep V G V (3-41)
Setting the expected firm value equal to the fixed entry cost yields the free-entry
condition (FE):
*1
EfVG
(3-42)
When the cutoff probability increases, it decreases the probability of successful entry. A
higher average firm value is then necessary for the expected firm value to equal the
fixed cost of entry.
56
Setting the free-entry condition equal to the zero-value cutoff condition will
determine the cutoff productivity level and the average firm value in the steady-state
equilibrium. As shown in Appendix B, this will result in a unique equilibrium. Since the
ZVC and FE conditions are independent of the proportion of child labor, a , changes in
the average proportion of child labor, Equation 3-23, have no effect on the cutoff
productivity level and average productivity in the steady-state equilibrium.
Figure 3-1. Steady-state equilibrium
Aggregate variables must remain constant over time in the steady-state
equilibrium. In the steady-state equilibrium, the mass of firms that exit each period is
equal to the number of successful entrants, the cutoff productivity parameter and
average firm value remain constant, and the incidence of child labor is unchanged in the
long-run. The number of successful entrants has to replace the number of firms exiting
the market each period:
FE
*
V
V
ZVC
57
1E E
ap M M
(3-43)
where EM is equal to the number of potential entrants to the market. The total amount
of adult labor in the R&D sector, EL , is then equal to the number of firms that attempt to
enter each period, EM , times the fixed entry cost, Ef :
1
E E E E
E
a ML M f f M
p
(3-44)
Total income of adult workers in the R&D sector is equal to the aggregate per-
period profit earned by firms. Total income received by all workers, which must be
equal to total expenditure, is found by summing the income from all forms of labor:
1E P C E P CE L L w C L L aw (3-45)
Last, total revenue must equal total profits plus payments to the factors of production:
1 1P C e P C CR L aw L L aw L w C E (3-46)
Average revenue can be written as a function of the average firm value solved in
equilibrium by the free-entry and zero-value cutoff conditions:
Pr V f (3-47)
As average firm value increases, so will average per-period revenue. The mass of firms
in the steady-state equilibrium is found by dividing total revenue, equal to total income
of all factors, by the revenue of the average firm. The mass of firms in equilibrium
depends on the endogenous demand for child labor, C :
1C
P
L w CRM
r aV f
(3-48)
58
The price index and aggregate quantity are equal to:
1
1
*
MP
(3-49)
*
1
1
CC
L w CQ L w C
PM
(3-50)
The amount of adult labor hired for R&D, adult labor hired for production, and child labor
hired for production are equal to:
EL M (3-51)
PL L M (3-52)
PC aL a L M (3-53)
Equalizing Equations 3-48 and 3-53 will determine the mass of firms in the steady-state
equilibrium and the incidence of child labor:
1 1 1c c
c P c
L aw L awM
aw V f aw a
(3-54)
1
c
aLC
aw
(3-55)
In the benchmark case, the incidence of child labor is determined solely by the
parameters of the model ( L , Cw , and ) and the average proportion of child labor,
determined by Equation 3-24. The derivative of Equation 3-55 with respect to a is
strictly greater than zero, so an increase in the average proportion of child labor in an
industry will increase the incidence of child labor.
59
From Equation 3-48, the mass of firms, M , increases with the demand for child
labor because child labor causes a rise in aggregate expenditure and revenue. An
increase in the number of firms, though, increases the demand for adult labor to
conduct R&D, removing adult labor from production and reducing the demand for child
labor. These two opposing effects stabilize the demand for child labor in the steady-
state equilibrium in the benchmark case.
Solving the Model when 1
This section will explore the steady-state equilibrium when the productivity
elasticity of child labor differs from that of adult labor. The equation that determines the
optimal level of child labor is rewritten below for convenience:
0 01 1i C i i i i C i
i
a b w a b w
a
(3-56)
Unlike the case where 1 , the optimal proportion of child labor for a firm will now
depend on the productivity level of the firm. Similar to the benchmark case, an increase
in the child wage or the strength of enforcement will reduce the optimal proportion of
child labor chosen by all firms.
When the productivity elasticity of child labor is greater than that of adult labor, i.e.,
1 , firms that draw a higher productivity parameter will choose a higher proportion of
child labor, so 0i
i
da
d . Alternatively, when the productivity elasticity of child labor is
less than that of adult labor, i.e., 1 , firms that draw a higher productivity parameter
will choose a lower proportion of child labor, and 0i
i
da
d .
60
Since the proportion of child labor depends on a firm’s productivity parameter, we
can rewrite a firm’s marginal cost as:
1 i i C
i
i i i i
a w
a
(3-57)
where Equation 3-10 ensures that 0i
i
d
d
for all firms. The price index is still equal to
the weighted average price of all producing firms:
1
111
MP M p
(3-58)
where average marginal cost, , is equal to:
1
11
0
( )
Max
i i Mu d
(3-59)
The aggregate equations from the benchmark case, Equations 3-30 through 3-33,
remain unchanged.
Free Entry and Exit
The value of the firm with average productivity, , is equal to:
1
P
rV f
a
(3-60)
while the value of the cutoff firm is equal to zero:
*
*
1P
rf
a
(3-61)
Combining Equations 3-60 and 3-61 yields the zero-value cutoff condition (ZVC):
61
1
**
*1P
aV f
a
(3-62)
which depends solely on the cutoff productivity level, * . The free-entry condition (FE)
is the same as in the benchmark case:
*1
EfVG
(3-63)
Firms that enter the market and draw a productivity parameter less than * will choose
to exit the market. Equalizing the free-entry and zero-value cutoff conditions determines
the cutoff productivity level and the average firm value in the steady-state equilibrium.
The average proportion of child labor is a function of the cutoff productivity level:
*
*
*
1
1
Max
i ia a g dG
(3-64)
In the traditional-industry case, where 1 , the average proportion of child labor
increases as the cutoff productivity level increases, so *
0da
d ; the inverse is true in the
modern-industry case, and *
0da
d .
Aggregate variables must once again be constant in the steady-state equilibrium
so the number of successful entrants must equal the number of firms exiting the market.
The rest of the aggregate variables are the same as in the benchmark case. The
demand for child labor is equal to:
* *
PC a L a L M (3-65)
62
Equations 3-48 and 3-65 determine the mass of firms and the incidence of child labor in
the steady-state equilibrium:
*
*
1 1c
P
L a wM
aV f a
(3-66)
*
*
1
c
a LC
a w
(3-67)
Since 1Cw and 1 , an increase in the average proportion of child labor will always
lead to an increase in the incidence of child labor, and 0dC
da .
Enforcement
The parameter measures the risk of using child labor in production. Firms that
use a high proportion of child labor are more likely to draw suspicion from authorities or
face repercussions from consumers; as a result, these firms face greater risk compared
with those that use little to no child labor11. In the benchmark case, 1 , the
proportion of child labor, a , decreased as the strength of enforcement increased. The
derivative of the total demand for child labor, Equation 3-55, with respect to a was
always greater than zero, so that a decrease in a always reduced the incidence of child
labor. The case is more complicated when 1 . Equation 3-64 determines the
relationship between the cutoff productivity level, * , and the average proportion of child
labor, a . This is shown in Figure 3-2 for the traditional-industry case ( 1 ). The
11
Nike, Levi’s, and Firestone are just a few large corporations that have faced public backlashes due to their connections with firms that employ children. In all cases, the large multinational corporations severed ties with the subcontracted firm accused of employing child labor.
63
relationship between the average proportion of child labor, a , and the incidence of child
labor in Equation 3-67 has also been added to Figure 3-2.
Traditional industry case
Figure 3-2. Increase in enforcement when 1
An increase in the level of child-labor enforcement, , decreases i ia for all
firms (except those where 0ia ). More enforcement also shifts the zero-value
FE
*
V
V
ZVC
a
*a
a
C
C a
64
condition curve upward, increasing the cutoff productivity level. Increasing enforcement
in a traditional sector will therefore have ambiguous effects on the average proportion of
child labor, a , and on the incidence of child labor, C .
Modern industry case
Figure 3-3. Increase in enforcement when 1
In the modern-industry example, 1 , an increase in enforcement, , not only
increases the zero-value cutoff condition, leading to an increase in the cutoff
productivity level, but it also decreases the proportion of child labor used by all firms.
FE
*
V
V
ZVC
a
*a
a
C
C a
65
The increase in the cutoff productivity level causes the least productive firms to exit,
which reduces the average proportion of child labor. Both effects lower the demand for
child labor in the steady-state equilibrium.
Intra-industry Trade
This section introduces trade into the basic model. Firms have the option of
exporting their good to n identical countries. (Therefore, the total number of countries is
equal to 1n ). If a firm exports, it faces an additional fixed cost of production equal to
Xf for each country to which it exports and incurs an iceberg per-unit trade cost equal
to 1 . Per-period profit from exporting to each country is:
i
X i i
i
pA Rr
P P
(3-68)
Maximizing per-period profit with respect to price yields the profit-maximizing price for
an exported good:
X i
i
Ap
(3-69)
The revenue from exporting to each country is proportional to the revenue earned from
selling domestically:
1
1X i
X i D i
pr R r
P
(3-70)
where D ir is the revenue from selling to the domestic market and is determined by
Equation 3-14. The combined revenue of a firm that exports is:
11i D i X i D ir r nr n r (3-71)
66
Similarly, the profit from exporting to each country will also be proportional to domestic
profits:
1
X i D i
X i
r r
(3-72)
and a firm’s combined per-period profit is equal to:
11i D i X i D in n (3-73)
An exporting firm will have the following firm value:
11 11 iD i X i
P X P X
i i
n rnV f nf f nf
a a
(3-74)
As in the closed model, firms will maximize their expected value with respect to the
proportion of child labor, which yields the same equation as in Equation 3-23. Thus, a
firm’s decision to export does not affect its child-labor decision.
Free Entry and Exit
Setting firm value equal to zero yields the zero-value cutoff for an exporting firm:
*
*
10
X X
X
X
rV f
a
(3-75)
The productivity level ensuring that Equation 3-75 holds is *
X . The zero-value cutoff for
a firm that only sells domestically is derived from Equation 3-61:
*
*
1D
P
rf
a
(3-76)
where the productivity level ensuring that Equation 3-76 holds is * . Substituting
Equation 3-76 into Equation 3-75 yields the following relationship between *
X and * :
67
* *
* *
X XX
PD X
r a f
fr a
(3-77)
The probability of successful entry is equal to *1Ep G . The probability that
a firm has a productivity level that will make it profitable to export is:
*
*
1
1
X
X
Gp
G
(3-78)
The mass of exporting firms is equal to X XM p M , so the total mass of varieties
available to consumers in any country, TM , is equal to the number of domestic firms
plus the number of exporting firms:
T XM M nM (3-79)
The weighted average productivity of firms that export is:
*
1
1
* 1
*
1( )
1
Max
X
X X
X
g dG
(3-80)
and the weighted average productivity of all firms is:
1
1 1XX X
T
M M M
M M
(3-81)
The aggregate variables are the same as those in the closed case, only
substituting total average productivity, T . Average per-period profit, revenue, and firm
value are:
XD ex Xr r np r (3-82)
XD ex Xnp (3-83)
68
1 1XD X
P ex X
X
r rV f np f
a a
(3-84)
We can compare the firm revenue that meets the domestic cutoff productivity level with
the firm revenue that meets the export cutoff productivity level:
1* *1
**
X X X
D
r
r
(3-85)
This expresses the export cutoff productivity level, *
X , as a function of the domestic
cutoff productivity level, * :
1
* * 1
* *
* *
X X X
X
P
f a
f a
(3-86)
Due to the added costs of exporting, the export cutoff productivity parameter will always
be greater than the domestic cutoff productivity parameter. Therefore, only the most
productive firms will export. The average firm value, Equation 3-84, can be described
completely in terms of the domestic cutoff level. Setting this equal to zero defines the
zero-value cutoff condition (ZVC):
1 1
* *
* ** *1 1
X X X
P ex X
XX
a aV f np f
a a
(3-87)
The average productivity of exporting firms, X , is a function of the export cutoff level,
*
X , while *
X is itself a function of * (from Equation 3-86). The free-entry condition
(FE) is the same as in the autarky case:
*1
EfVG
(3-88)
69
The intersection of the ZVC and FE conditions determines the average firm value and
the cutoff productivity level in the steady-state equilibrium. The mass of incumbent
firms each period is equal to:
*
* *
1 1c
P
L a wM
v f a a
(3-89)
and the incidence of child labor is denoted by:
*
*
1
c
a LC
a w
(3-90)
Similar to the autarky case, the incidence of child labor increases only if there is a rise in
the average proportion of child labor, a , which still depends on the cutoff productivity
level, * (with the exception of the benchmark case).
Trade Liberalization
Trade liberalization can be shown through an increase in the number of trading
partners, n ; a decrease in the fixed cost of exporting, Xf ; or a reduction in iceberg
trade costs, . All three methods of showing trade liberalization, although they affect
the mass of firms and varieties differently, have the same effect on the incidence of
child labor in the steady-state equilibrium. Trade liberalization has no impact on the
free-entry condition, Equation 3-88, but it will shift the zero-value condition, Equation 3-
87, upward, shown in Figure 3-4.
As in the Melitz (2003) model, trade liberalization will always cause an increase in
the domestic cutoff productivity level, * , forcing the least productive firms to exit and
increasing the average firm value, V . In the benchmark case, 1 , an increase in the
cutoff productivity level does not affect the child-labor decision of firms and therefore
70
does not have any impact on the average proportion of child labor. The increase in the
average firm value will decrease the mass of firms in the steady-state equilibrium;
however, from Equation 3-90, trade liberalization will have no effect on the incidence of
child labor.
Figure 3-4. Effect of trade in the steady-state equilibrium
The sectoral allocation of adult labor in R&D and production remains the same
since the number of potential entrants, EM , is constant in all cases even though the
mass of firms in the steady-state equilibrium, M , falls. Since the cutoff productivity
level increases with trade liberalization, the probability of successful entry decreases.
The price level, which is a weighted average of the prices of producing firms, decreases
after trade liberalization since the firms that charge the highest prices are forced to exit.
The least productive firms have the highest marginal costs and charge the highest
prices due to price being a constant mark-up over marginal cost. As in Melitz (2003), a
FE
*
V
V
ZVC
ZVC’
* ’
71
decrease in the price level increases the real wage of adult and child labor. Since trade
liberalization increases the ex-ante firm value of a potential entrant, this raises the
demand for adult workers in the R&D sector and leads to an increase in the relative
wage.
Trade Liberalization in Traditional and Modern Sectors
When 1 , trade liberalization will affect the average proportion of child labor.
Since the mass of potential entrants, EM , remains fixed, the number of adult workers in
the R&D sector, and thus the number of adult workers in production, will also remain
unchanged. The incidence of child labor, C , which is proportional to the amount of
adult labor in production, will then depend solely on how trade liberalization affects the
average proportion of child labor, a .
In the traditional-sector case, 1 , trade liberalization causes the exit of low-
productivity firms, which rely the least on child labor. It also increases the quantity
produced by productive firms. Both effects will increase the average proportion of child
labor that firms use. The decrease in the price index increases real wages, which, in a
more general model, would affect the supply of child labor ambiguously. This ambiguity
(see Basu and Van 1998) results from opposing income and substitution effects derived
from the parental choice of sending one’s child to work.
In the modern-industry case, 1 , trade liberalization reduces the demand for
child labor. Unlike in a traditional industry, the more productive firms use relatively little
child labor in production. Trade liberalization, by increasing the cutoff productivity level,
forces the least productive firms to exit the industry. This will decrease the average
proportion of child labor and reduce the incidence of child labor. As in the traditional-
72
sector case, trade liberalization increases real wages, whose effects the child-labor
supply depend on the corresponding substitution and income effects.
Conclusion
Although the incidence of child labor has been declining around the world over the
last half century (ILO 2006), the percentage of child laborers in many of the poorest
countries remain high. The culprit, as many economists will agree, is extreme poverty
that takes away a family’s opportunity to educate its children. However, it is important to
recognize that firms employing cheap child labor to earn higher profits also play a large
role. While completely eliminating the demand for child labor is not realistic, demand in
sectors exposed to trade may be reduced via targeted trade policies that take into
account the relationship between child labor and productivity.
In industries where the productivity gap between adult workers and child labor
decreases as productivity increases, trade liberalization can increase the demand for
child labor. Trade sanctions, which increase trade costs and are generally
accompanied by negative welfare effects, can lower the cutoff productivity level, which
can reduce the demand for child labor in the steady-state equilibrium. Trade
liberalization can be successful in industries where the more productive firms see little
use in employing child labor since the gap between adult and child workers is large. A
reduction in trade costs in this case will result in an exit of firms whose workforce is
largely composed of children, which can reduce the incidence of child labor in the
steady-state equilibrium.
73
CHAPTER 4 INCIDENCE OF CHILD LABOR IN A NORTH-SOUTH MODEL OF TRADE
Supporters of globalization argue that increased exposure to trade increases
incomes of low-skilled households, thereby reducing the incidence of child labor when
insufficient income is the main cause. Trade liberalization opponents in developing
countries argue that globalization opens the doors for foreign firms to take advantage of
cheap labor, increasing the demand for and incidence of child labor. While household
factors such as insufficient income, lack of access to credit markets, and effectiveness
of schooling have been examined and empirically researched, less work has explained
the relationship between trade liberalization and the incidence of child labor.
Davis and Voy (2007) and Busse and Braun (2004) are two examples of recent
works that empirically examine the relationship between foreign direct investment and
the incidence of child labor. Davis and Voy (2007) use openness to trade, defined by a
country’s trade volume as a percentage of GDP, to regress against child labor.
Controlling for a host of country factors, they find a significant negative relationship
between child labor and globalization; however, this relationship vanishes once they
control for income. This suggests that when globalization is able to increase the income
of low-skilled households, it can reduce the incidence of child labor. What has not been
examined, due to the complexity of gathering sufficient data, is how trade liberalization
affects the demand for child labor from domestic and foreign firms.
North-South trade models have been used to explain many facets of trade but
have not been used to analyze the trade implications on child labor. These models
explain the product cycle of manufactured goods, where they are first innovated and
produced in the North, but over time production shifts to the imitating South. As such,
74
the South exogenously absorbs innovation from firms in the North. Reductions in trade
costs tend to increase the rate of innovation in the North and increase the total number
of varieties. For a summary on the North-South trade literature, see Chui et al. (2002).
The model in this paper incorporates firm heterogeneity, as in Melitz (2003) and
Melitz and Ottaviano (2008), in a standard North-South trade model. Firms in the North
will differ in their marginal costs. The free-entry condition in the North will then
endogenize the rate of innovation. Southern firms are assumed to be homogeneous,
but their costs will depend on their use of child labor in production. Child labor, while
reducing a firm’s marginal cost, has the trade-off of increasing the probability of
receiving a negative shock.
This paper is organized as follows: Section 2 solves for the autarky equilibrium in
the North, characterizing the critical cost parameter necessary for a firm to enter the
market. Section 3 expands the model to allow Northern firms to export to the South.
Section 4 develops the traditional North-South trade model where Southern firms imitate
products from the North and can use child labor in production to reduce their costs.
Section 5 shows the comparative statics and describes how changes in the exogenous
rate of imitation, population increases in the South, and increases in trade costs and
enforcement affect the incidence of child labor. The paper then presents concluding
remarks.
The Model
This model assumes that firms are heterogeneous in the North, while firms in the
South that imitate Northern products share a common marginal cost that depends on
the use of child labor in production. This allows for the examination of how trade
liberalization affects the demand for child labor in the South. As in Krugman (1979), the
75
rate of imitation is assumed to be exogenous so that the model can be solved explicitly,
but simulations that endogenize this variable are shown to reinforce the results of the
model.
Consumption
Consumer utility will be based on a quadratic utility model, similar to that found in
Melitz and Ottaviano (2008), where utility depends on the consumption of an agricultural
good, z , and differentiated varieties of a manufactured good, iq , where i :
2
21 1( , )
2 2i i i i
i i i
U z q z q di q di q di
(4-1)
A representative consumer maximizes utility subject to a budget constraint:
i i i
i
E z p q q di
(4-2)
This leads to the following Lagrangian and its corresponding first-order conditions:
2
21 1
2 2i i i i i i
i i i i
z q di q di q di z p q q di E
(4-3)
1 0d
dz
(4-4)
0i i i
i
dq Q p q
dq
(4-5)
0i i i
i
dz p q q di E
d
(4-6)
where i
i
Q q di
, and the price of the agricultural good is the numeraire. This leads to
the following inverse demand function for a differentiated variety:
i ip q Q when 0iq (4-7)
76
Aggregating over all symmetric consumers leads to the following market demand
function for a specific variety:
i i
L L MLq p p
M M
(4-8)
where M is the number of consumed varieties, L is the number of consumers, and p
is the average price, or the price index, equal to:
i i
i
p d
pM
(4-9)
The maximum price that a firm can charge is the price such that demand is driven to
zero, 0iq
Max M
p pM M
(4-10)
Production in the North
Production of the numeraire good, z , follows a constant returns to scale
production function and differs in the North and South. In the North, production of the
numeraire good is equal to the amount of adult labor, Zl , in that sector:
N
Zz l (4-11)
This leads to a unitary wage in the North, 1Nw . In the South, the numeraire good is
produced using only adult labor:
S
Zz l (4-12)
where is assumed to be less than 1. Therefore, the wage of adult labor in the South
is equal to and less than the adult wage in the North, 1S Nw w . The rest of this
section will solve for the autarky steady-state equilibrium in the North.
77
Firms in the differentiated-good sector incur a fixed entry cost to conduct research
and development. Once R&D is conducted, firms discover part of their marginal cost of
producing the good, ic . Adult labor is the only factor used in the production of this
manufactured good and is equal to:
1i i il c q (4-13)
where the random cost, ic , is found after the R&D process. The marginal cost of a firm
is 1 ic . The profit of a firm with cost parameter ic is:
1i i i i i ip q l p c q (4-14)
Maximizing Equation 4-14 with respect to quantity yields the profit-maximizing quantity
produced by this firm:
1i i i
Lq c p
(4-15)
Let *c be the cutoff cost level that corresponds to the maximum possible price that
drives the quantity demanded to zero:
*1Max M pp c
M
(4-16)
This cutoff cost level is a function of the parameters of the model and the average price
in the differentiated goods market. Firms with *
ic c earn positive profits and remain in
the market, while firms with *
ic c earn negative profits and exit the industry. As the
average price, p , increases, the cutoff cost level, *c , will also increase.
The quantity, price, revenue, and per-period profit of a firm with productivity ic are
functions of the firm’s productivity parameter and *c :
78
* 1i i i i i i
Mp c p q c c q c
M M L L
(4-17)
*12
2i i ip c c c (4-18)
*
2i i i
Lq c c c
(4-19)
* * * *24
i i i i
Lr c c c c c
(4-20)
2
*
4i i i
Lc c c
(4-21)
Firms in the North are assumed to face an exogenous probability of death, N , so that a
firm’s expected value, iV , is equal to:
2
*
0
14
tN
i i i iNt
LV c c c
(4-22)
In the steady-state equilibrium, the expected firm value from conducting research
and development must be equal to the fixed cost of R&D. This is the free-entry
condition in the North:
* *
2*
0 04
c c
i i
i i i EN N
c LdG c c c dG c f
(4-23)
This equation determines the cutoff cost level in the steady-state equilibrium, *c , which
then determines the price index, p , from Equation 4-16. This can best be illustrated in
Figure 4-1, which shows the necessary *c so that the expected firm value equals the
fixed entry cost:
79
Figure 4-1. Autarky equilibrium
If the cutoff cost level is above *c , the probability of entry is high, and the ex-ante
firm value exceeds the fixed entry cost. This induces firms to enter, lowering the price
index and driving out firms with high marginal costs until the cutoff cost level converges
to *c . If the cutoff cost level is below *c , the ex-ante firm value is not large enough to
cover the fixed cost of entry. This reduces the number of firms willing to enter the
market until the cutoff marginal cost rises to *c .
To simplify the analysis, a uniform distribution of the cost parameter is assumed.
The cumulative distribution and probability distribution functions are:
ii Max
cG c
c (4-24)
1
i Maxg c
c (4-25)
where 0, Max
ic c . Solving the free-entry condition, Equation 4-23, for the cutoff cost
level yields:
fE
Cost parameter
E (V)
c*
Expected Firm Value,
Entry Cost
80
1
3* 12 N Max
Ec fc
L
(4-26)
To ensure an interior solution, * Maxc c , it is assumed that the population is large
enough so that the following condition holds:
2
12 NMax Efc
L
(4-27)
With the cutoff cost determined by Equation 4-27, the price index and the average
marginal cost in the North are equal to:
* *
*
*
0 0
* *
23
142
c c
i i i ip dG c c c dG cc
pG c G c
(4-28)
*
*
0
* 2
c
i ic dG cc
cG c
(4-29)
The number of firms in this autarky equilibrium in the North can then be derived from
Equation 4-16:
* * *
****
1 1 4 1
3 411
4
c c cM
ccc pc
(4-30)
Note that the derivative of Equation 4-30 with respect to *c is negative, so that an
increase in the cutoff cost variable will decrease the number of firms in the steady-state
equilibrium. Average per-period profit is equal to:
* *
2*
3*
0 0
** * 124
c c
i i i i N Max
E
dG c L c c dG cc L f c
cG c G c
(4-31)
81
The number of new firms each period has to equal the number of firms exiting in
the steady-state equilibrium:
N
EM M (4-32)
Welfare in the North, shown by Equation 4-1 [Do you mean utility? 4-1 is defined as
utility.], can be written in terms of a consumer’s indirect utility:
* 21
2
0
1 11
2 2
c
i i iW p p c p G cM
(4-33)
A decrease in the cutoff cost level, *c , reduces the price level and increases the number
of varieties available to consumers. Both these effects mean that per capita welfare
rises as *c falls.
Exporting Firms in the North
This section analyzes the steady-state equilibrium when Northern firms have the
option to export their goods to the South. It will be assumed that the demands for the
agricultural and differentiated goods are identical in both the North and South. The
population in the North is NL , the population in the South is SL , and N SL L L is the
world population. Firms that trade incur an additional iceberg trade cost of 1 . The
per-period profit from exporting to the South for a Northern firm with ic is:
2 2
* *1 1 14 4
s SS
i i i
L Lc c c c
(4-34)
The cutoff export cost level can be defined as a function of the domestic cutoff cost
level:
* 11Ex
cM pc
M
(4-35)
82
Firms with a low cost parameter, Ex
ic c , will export to the South and produce for the
domestic market, firms with *,Ex
ic c c will produce solely for the domestic market, and
firms with *, Max
ic c c will not produce for either market.
The firm value of an exporting firm is:
2 2
* *11
4
SEx N Si i
i i i iN NV c c L c c L c c
(4-36)
A potential entrant into the market has to take into account the probability that it will
have a cost parameter low enough that will allow it to be an exporter. The free-entry
condition will then determine the cutoff cost level:
*
2 2* *
0 0
14 4
Exc cN S
i i i i EN N
L Lc c dG c c c dG c f
(4-37)
The extreme free-trade case occurs when 1 and results in *Exc c , where all
Northern firms choose to export. The larger the trade cost, 1 , the smaller the chance
that a firm will have a cost parameter low enough to allow it to export, and the lower the
per-period profits from exporting.
An autarky case occurs when * 1
0Exc
c
. If 1Maxc , no firms in the
North have an incentive to export to the South. Figure 4-2 compares the steady-state
equilibrium in the autarky case, denoted by EA, and the free-trade steady-state
equilibrium when 1 , denoted by ET:
83
Figure 4-2. Trade equilibrium
As trade costs are reduced, the ex-ante firm value from entering the market
increases. This will decrease the cutoff cost level, *c , while simultaneously increasing
the cutoff export cost level, * 1
Exc
c
. Compared with the autarky case, by
reducing *c , trade raises the welfare of Northern consumers by lowering the average
price of differentiated goods and increasing the number of varieties available for
purchase.
North-South Free Trade Equilibrium
This section examines the equilibrium where Southern firms can imitate Northern
products. Southern firms are assumed to be homogeneous and face a probability of
death that depends on their use of child labor in production. This section assumes no
trade costs, or 1 .
Production in the South
Similar to firms in the North, the profit of a Southern firm is:
fE
Cost Parameter
EA (V)
c*A
ET (V)
c*T
Expected Value,
Entry Cost
84
S S Sp c q (4-38)
where represents the proportion of child labor used in production. The derivative of a
Southern firm’s marginal cost with respect to the proportion of child labor is negative,
0sdc
d . Note that if a Southern firm chooses not to use any child labor, its marginal
cost equals the adult wage, which is 1 . Therefore, Southern firms always have a
cost advantage over Northern firms, even if they choose not to employ children.
The profit-maximizing quantity and price charged by a Southern firm is a function
of the cutoff cost level in the North determined by Equation 4-16:
*12
S S SLq c c c
(4-39)
*11
2
S S Sp c c c (4-40)
Since Southern firms imitate products produced by the North, the lowest price that
Northern firms can charge is equal to its marginal cost. If the wage gap between the
North and South is small, this will force firms in the South to charge a price slightly lower
than the Northern firm’s marginal cost. To assure a high-wage gap that will result in
Southern firms choosing their monopolist price, the monopolist price for a Southern firm
has to be less than the smallest marginal cost for a Northern firm. Therefore, a
sufficient condition to ensure that a high-wage gap exists is that
1
3* 12
1 1N Max
Ec fc
L
.
This means that the North-South wage gap, 1 Sw ,
has to be greater than the cutoff marginal cost level in the North:
85
1
3* 12 N Max
Ec fc
L
(4-41)
The larger the combined populations of the North and South, L , the lower the wage gap
has to be so that Southern firms can charge their monopolist prices. The per-period
profit of a Southern firm is then equal to:
2
*14
S S SLc c c
(4-42)
Firm Value for Southern Firm
Firms in the South maximize their firm value with respect to the optimal quantity of
child labor, :
2
*14
S S
S S
S S
c LV c c
(4-43)
where S is a Southern firm’s probability of receiving a negative shock that will put it
out of business. As a firm’s proportion of child labor employed increases, the so-called
probability of death for the firm also increases. The optimal proportion of child labor is
found by taking the derivative of Equation 4-42 with respect to and setting it equal to
zero:
2* *
22 1 1 0
4
S S SS S S
S
dV L dc dc c c c
d d d
(4-44)
*1 2 0S S
S Sd dcc c
d d
(4-45)
86
Since 0Sdc
d while 0
Sd
d
, and assuming that the second-order derivatives are
2
20
Sd c
d and
2
20
Sd
d
,there exists a unique so that Equation 4-44 holds for a given
*c . Note that an increase in *c increases the first term in Equation 4-44, which
decreases the proportion of child labor used by all Southern firms, or *
0d
dc
.
Free-Entry Condition for Northern Firms
The free-entry condition for Northern firms, like in the autarky case, will determine
the cutoff cost level:
*
2*
04
c
i i EN I
Lc c dG c f
(4-46)
1
3
*12 N I Max N
Ec fc
L
(4-47)
where Northern firms face an exogenous probability of imitation, I , in addition to the
Northern probability of death, N . With a fixed probability of imitation, the number of
Southern firms depends on the number of Northern firms in the steady-state equilibrium.
As such, firms in the South act as oligopolists, which is why they can charge their
monopolist prices.
Share of Firms
The total number of firms (and varieties), TM , is equal to the total number of firms
in the North, NM , plus the total number of firms in the South, SM . Let represent the
share of varieties located in the North:
87
N
T
M
M (4-48)
The share of Southern varieties is then:
1S
T
M
M (4-49)
In the steady-state equilibrium, the number of Northern firms that are imitated each
period must equal the number of new Southern firms. This must also equal the number
of Southern firms exiting each period. The number of Northern firms that are imitated
each period is equal to:
I N I TM M (4-50)
while the number of Southern firms that exit each period is equal to:
1S S S TM M (4-51)
In the steady-state equilibrium, the number of Northern firms that have their
product imitated must equal the number of new Southern entrants, S
EM , which itself
must equal the number of Southern firms that exit each period:
I N S S S
EM M M (4-52)
Therefore, in the steady-state equilibrium, the share of Northern firms must be:
*S
I S
(4-53)
This depends on the endogenous rate of death in the South, which comes from the
child-labor decision of Southern firms, Equation 4-44, and the exogenous rate of
imitation, I .
88
The price index depends on the shares of firms located in the North and South.
The price of Northern products comes from Equation 4-28, and the price of Southern
varieties is determined by Equation 4-40:
* *
**
0
2
12 31 1
4 2
ciN S S
Si
T
c cM dG c M p
c ccp
M
(4-54)
The cutoff cost level, *c , and the share of firms located in the North, , determine the
price index in the steady-state equilibrium. An increase in *c will increase the price
index, but by an amount less than the change in *c :
*
* *
21 0
4
d c p d p
dc dc
(4-55)
Note that with the assumption of a high-wage gap between the North and South,
the average price charged by firms in the North will always be higher than that charged
by Southern firms. Therefore, an increase in the share of Northern firms, all else equal,
will increase the price index. The last step is then to calculate the total number of
varieties:
1* * **
*
*
1 1 131 1 1
4 21
S
Tc c c cc
M cc p
(4-56)
Incidence of Child Labor
The quantity produced by a Southern firm is equal to:
*12
S SLq c c
(4-57)
so the number of children being employed by each Southern firm, Cl , is equal to:
89
*
0
11 2
C A SLl l c c
b
(4-58)
where 0b is an adult scaling constant, so that one unit of child labor produces the same
amount as 0b units of adult labor. 01 b
is therefore the proportion of output produced
using child labor. The total incidence of child labor, CL , is computed by multiplying
Equation 4-58 by the total number of Southern firms, 1S TM M :
*
* *
*0 0
111 1 1
1 2 1 2 1
C S T ScL L
L c c M c cb b c p
(4-59)
The demand for child labor with respect to the cutoff cost level then has the
following property:
* * * *
*
2**
0
1 2 1 1 1 11
1 2 1
S S
C
d pc p c c c c c
dcdL L
dc b c p
(4-60)
The derivative of Equation 4-59 will depend on the sign of the following equation:
* * * *
*1 2 1 1 1 1 0S Sd p
c p c c c c cdc
(4-61)
Note that when all firms are located in the North, 0 , the sign of Equation 4-61
becomes:
2
*
*8 1 0
CSdL
Sign c cdc
(4-62)
When 1 , the sign of Equation 4-61 becomes:
*2
*4 1 1 0
CS SdL
Sign c c cdc
(4-63)
90
Since 1 and 1Sc , Equation 4-63 is less than zero as well. Since an increase in
increases the price index, an increase in will always monotonically reduce Equation
4-61. Therefore, an increase in the cutoff marginal cost level will, all else equal, reduce
the incidence of child labor, or *
0CdL
dc .
Comparative Statics
This section examines how changes in some of the parameter values affect the
incidence of child labor. Graphs showing the results of simulations with an endogenous
rate of imitation are shown when applicable, but they reinforce the results of the closed
model. See Appendix C for more details on the simulation.
Increase in Child-Labor Enforcement, S
An increase in child-labor enforcement is a policy that increases S for all .
Using a probability of death formula:
1
S
(4-64)
an increase in child-labor enforcement is characterized by an increase in the parameter
. From Equation 4-44, this decreases the proportion of child labor that every Southern
firm chooses but does not affect the cutoff cost level in the North. Although *c is not
affected, the fall in raises the marginal cost of Southern firms, Sc , which
increases the price they charge. An increase in enforcement affects the allocation of
firms between the North and South. The total number of firms decreases, but the share
of firms located in the North rises.
91
Figure 4-3. Increase in enforcement
The decrease in the number of firms in the South reduces the incidence of child
labor. An increase in child-labor enforcement raises the average price for the
differentiated goods and lowers the number of varieties available in both the North and
South, reducing welfare in both countries. The reduction of welfare in the South, at
least in the short run, can partially explain the reluctance of officials in developing
countries to clamp down on child labor.
One Time Increase in the Population in the South, SL
An increase in the South’s population raises the ex-ante value of all firms by
increasing the profit earned each period. The increased competition for resources
decreases the cutoff cost level in the North. This causes the high-cost firms to exit,
reduces the price level, and increases the total number of firms in the steady-state
equilibrium. Since the allocation of firms between the North and South remains
unchanged, the number of Southern firms increases, which also raises the incidence of
child labor. The lower price level and the increase in total varieties increases welfare in
both the North and South.
92
Figure 4-4. Increase in population
Increase in the Rate of Imitation, I
An increase in the exogenous rate of imitation by Southern firms decreases the
ex-ante firm value for Northern firms from Equation 4-47, increasing *c . The increase in
the exogenous rate of imitation also reduces the share of firms located in the North:
2
0S
I I S
d
d
(4-65)
The number of firms in both the North and South decreases with the increase in the
cutoff cost level from Equation 4-56, reducing the total number of firms in the steady-
state equilibrium:
1* **
*1 13
1 1 14 2
S
Tc c cc
M c
(4-66)
The derivative of Equation 4-66 with respect to the exogenous rate of imitation is less
than zero, 0T
I
dM
d , since
*
0I
dc
d and 0
I
d p
d . From Equation 4-60, the increase in
the cutoff cost level *c reduces the incidence of child labor due to the decrease in the
93
total number of Southern firms in the steady-state equilibrium. Social welfare of
households in both the North and South falls due to the increase in the price index and
the lower number of varieties.
Trade Costs
This section assumes that all firms face iceberg trade costs, equal to 1 , when
exporting their products. The per-period profit from exporting, expected firm value, and
free-entry condition for firms in the North are shown in the previous section. The
exporting profit for a Southern firm is:
S S S
X p c q (4-67)
The profit-maximizing quantity, price, and profit for a Southern firm from exporting are:
*12
NS S S
X
Lq c c c
(4-68)
*11
2
S S S
Xp c c c (4-69)
2
*14
NS S
X
Lc c
(4-70)
Total profit and expected firm value for a Southern firm are then:
2 2
* *1 14 4
S NS S SL L
c c c c
(4-71)
2 2
* *11 1
4
S S S N S S
ESV L c c L c c f
(4-72)
All other equations remain the same as those in the free-trade case.
Compared with free trade ( 0 ), the addition of trade costs decreases per-period
profits and the firm value of both Northern and Southern firms. As shown in the
previous section, the addition of trade costs for Northern firms splits them into those that
94
produce domestically and export and those that produce for the domestic market only.
The export marginal cost cutoff level for firms in the North is equal to:
* 1Ex
cc
(4-73)
and the expected firm value for a Northern firm is equal to:
*
2 2* *
0 0
11
4
Exc c
N S
i i i i iN IV L c c dG c L c c dG c
(4-74)
Setting Equation 4-74 equal to the fixed cost of entry and solving for *c determines the
cutoff cost level for Northern firms. An increase in trade costs, , increases the cutoff
marginal cost level and decreases the cutoff export cost level.
Trade costs create a price divergence in the North and South. Unlike their
Northern counterparts, all Southern firms choose to export their goods and pass on their
trade costs. The price index in the North is:
* *
**
0
2
12 31 1
4 2
ciN S S
Si XN
T
c cM dG c M p
c ccp
M
(4-75)
while the price index in the South is:
*
*
0
1
12 31 1
4 2
ExciN S S
SX i X ExS
XT
c cM dG c M p
c ccp
M
(4-76)
where N
XM is the number of exporting Northern firms, equal to *
ExNc
Mc
, and X
represents the share of Northern firms that export relative to the total number of firms,
TM .
95
An interesting result of this model is that while an increase in trade costs raises the
price index in the North, the price index in the South also rises but to a much lesser
extent due to a shift in consumption toward domestic goods. The increase in trade
costs raises the cutoff cost level and reduces the export cutoff level in the North. It also
increases the price charged by Southern firms.
Figure 4-5. Price indices and trade costs
The total number of firms in the North increases, but the share of total firms that exports
to the South, X , decreases. Therefore, the South consumes a greater proportion of
goods produced domestically, but the goods they import from the North are more costly.
96
Figure 4-6. Child labor and trade costs
A rise in trade costs not only shifts the production of goods to the South relative to
the North, but it also increases the mass of firms in the South. The quantity produced
by each firm also increases due to the higher cutoff cost level, so the incidence of child
labor rises. The numbers of available products in the North and South increases, but
higher prices limit any welfare gains in the North.
Conclusion
This paper examines how trade liberalization affects the incidence of child labor in
a North-South model of trade. Since the supply of child labor is assumed to be perfectly
elastic, the demand for child labor will determine the number of children working in the
steady-state equilibrium. Firms in the North are assumed to differ in their cost of
producing a differentiated good, which endogenizes the rate of innovation in the North,
while the cost of Southern firms depends on their use of child labor in production. The
rate of innovation in the North, along with the exogenous rate of imitation in the South,
determines how the share of output is allocated between the North and South.
97
A rise in the cost cutoff level, caused by a decrease in the size of the population
in either the North or South or by an increase in trade costs, increases the number of
firms located in the South and raises the output produced by each firm. Since the
increase in the cost cutoff level does not affect the proportion of child labor that
Southern firms employ, the demand for child labor increases. A reduction in trade costs
can therefore help lower the incidence of child labor in the steady-state equilibrium. An
increase in the exogenous rate of imitation by Southern firms can similarly work to
decrease the incidence of child labor by reducing the total number of firms located in the
South.
98
APPENDIX A FIRST-ORDER CONDITIONS
The utility optimization problem of the representative consumer is given by:
1
( ) ( ) ( )
i i
i i i i iq d p q d R
(A-1)
which results in the following first-order conditions:
1
1( ) ( ) ( ) 0
i
i i i i
i
dq d q p
dq
(A-2)
1
1( ) ( ) ( ) 0
j
j j j j
j
dq d q p
dq
(A-3)
( ) ( ) 0
i
i i i
dR p q d
d
(A-4)
99
APPENDIX B PROOF OF UNIQUE STEADY-STATE EQUILIBRIUM
The intersection of the zero-value condition and the free-entry condition results in
a unique equilibrium. From Equations 3-40 and 3-42:
ZVC:
1*
*1PV f
(B-1)
FE: *1
EfVG
(B-2)
Setting the two conditions equal yields:
1*
*
*1 1E
P
fG
f
(B-3)
To prove a unique equilibrium, we need to show that the RHS is monotonic and
always downward sloping so that it intersects the LHS once. The derivative of the RHS
with respect to the cutoff productivity level is:
1*
1 1** * *
* *
* * **
1
1 1 11
gG g
G
(B-4)
This equation simplifies to:
1*
*
*
*
1 1
0
G
(B-5)
Since this equation is always negative, the RHS in (B3) is always downward sloping.
Furthermore, the RHS goes to infinity as the cutoff productivity goes to zero, and since it
100
is always downward sloping and does not converge to a positive number, it must
intersect only once with the LHS of Equation B-3.
101
APPENDIX C SIMULATION WITH ENDOGENOUS RATE OF IMITATION
The simulations conducted assume that the rate of imitation is determined
endogenously by the free-entry conditions in the North and South. It is necessary to
add an exogenous probability of successfully imitating a product, 0,1 . Therefore,
the ex-ante firm value in the South is equal to:
2
*14
s s
s s s
Es s
c LV c c f
(C-1)
The free-entry condition in the South will actually determine the cutoff cost level in the
North:
1
2* 4
1s s
sEfc cL
(C-2)
The free-entry condition in the North will then determine the endogenous cost of
imitation:
3
*
12
I
Max n
E
L c
c f
(C-3)
The rest of the equations are the same as those in Chapter 4. For the simulations, the
following values are given for the exogenous parameters:
1000L , 10 , .5 , .2s , 5000n s
E Ef f , 0 .75b , .3sw and .2cw
An increase in the cutoff cost level, *c , in the steady-state equilibrium, which decreases
the demand for child labor, will increase the endogenous rate of imitation, all else equal.
102
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BIOGRAPHICAL SKETCH
Kristian Estévez was born in Miami, Florida, to a Cuban father and Ecuadoran
mother. He grew up in Little Havana and graduated from G. Holmes Braddock Senior
High School in 2001. He received his Bachelor of Science degree in economics from
the University of Florida in 2005 and earned his Ph.D. in economics in 2010. His fields
of specialization are international trade, public economics, and game theory, and his
research focuses on trade policies, income inequality, and other issues affecting
developing countries.