Child labor and the interaction between the quantity and quality of children.

• Author: Fan, C. Simon

1. Introduction

   Just as dogs were raised to hunt for their masters before they were pets, so in early traditional China children were raised as a source of income ... (Cheung 1972, p. 641). Motivated by substantial empirical evidence on the negative correlation between fertility rate and income, Becker and Lewis (1973) formulated the idea that parents obtain utility from both the quality and the quantity of their children. Depending on the elasticity of substitution between quantity and quality of children in parents' utility function, this framework yields the important theoretical implication that individuals may spend more on the quality improvement rather than on the increase of the quantity of their children as their incomes rise. (1) In fact, the Becker-Lewis model has become a pillar of population and family economics.

   This article attempts to provide a simple extension of the Becker-Lewis model by introducing child labor into this framework. The model implies that the negative correlation between fertility and income can be obtained with much less reliance on the property of parents' utility function if child labor is considered. In particular, the model illustrates that, if both the quantity and the quality of children enter symmetrically into parents' utility function, without child labor, fertility may be a normal good so that it increases with parental income. However, when the role of child labor is taken into account, we obtain the opposite result: As parental incomes rise, fertility decreases and children are better educated.

   The intuition for why child labor affects the interaction between the quantity and quality of children can be explained as follows. When children's earnings are sufficiently high relative to their cost, raising children is cheap but sending them to school is expensive. So, even if the quantity and the quality of children enters symmetrically into parents' utility function, parents may regard the quantity of children as a necessity and the quality of children as a luxury due to their price difference. As people demand more luxuries and fewer necessities when they become richer, fertility will decrease and children's educational attainment will increase when parental incomes rise.

   This article also complements the recent theoretical literature on child labor started with Basu and Van (1998) in several aspects. First, it shows that fertility tends to increase with the wage rate of child labor, which provides an explanation for the empirical evidence on the close correlation between high fertility rate and high child labor productivity (e.g., Levy 1985; Weiner 1991). Second, it implies that the relative wage of child labor and parental income are both crucial in determining whether parents will send some of their children to work. Third, this article yields some interesting policy implications. In particular, it suggests that government intervention and the law not only directly affect the supply of child labor but also influence parents' decisions on fertility, which indirectly determine children's labor market participations.

2. Theoretical Antecedents

   This article is related to the literature on fertility. For example, Willis (1973) extends Becker and Lewis (1973) by showing that the constraints of both time and material resources affect parents' choices of the quantity and quality of children. Based on a dynasty model developed by Barro and Becker (1989), Zhang and Zhang (1997) show that, in the steady state, fertility and wage rates are negatively related if bequests are operative. Galor and Weil (2000) develop a model in which an economy evolves from a Malthusian regime, where technological progress is slow and population growth is high, into a post-Malthusian regime, where technological progress rises and fertility rate is low.

   This article is also related to the literature on child labor. For example, Basu and Van (1998) and Bardhan and Udry (1999) analyze the causes and consequences of child labor in a model of multiple equilibria in the labor market. They show that coordination failure may result in high rates of both fertility and children's labor market participation. So they suggest that banning child labor can solve the problem of coordination failure and eliminate child labor. Baland and Robinson (2000) demonstrate the inefficiency of child labor and show that a legal restriction of child labor can reduce fertility. They also show that banning child labor, either a marginal ban or a complete ban, can lead to an outcome of Pareto improvement in a general equilibrium framework. Hazan and Berdugo (2002) assume that time is the only input required in raising children and they show that technological progress leads to a decrease in both child labor and fertility. Basu (2000) illustrates the intriguing effects of government policies on child labor by showing that a minimum-wage law does not necessarily lead to a reduction of child labor.

3. Child Labor, Fertility, and Human Capital

   An individual (i.e., an adult) obtains utility from three sources: her consumption of material goods (c), the total number of children (n), and the number of educated children (s). We assume that an adult's utility function takes the following form:

   (1) V [equivalent to] ln(c) + [beta]ln(s) + [delta]ln(n),

   where [beta] and [delta] are positive coefficients. In the standard Becker-Lewis model, s stands for the human capital of each child. The above formulation modifies the Becker-Lewis model in such a way that it will be able to account for the empirical evidence of the unequal treatment of parents toward their children, which has been widely observed in the phenomenon of child labor. (2) Meanwhile, the assumption that the utility function is log-linear allows us to derive closed-form solutions without loss of generality. In fact, log-linear utility functions are most commonly used in the related existing literature. (3)

   The cost of raising a child is constant and is denoted by [pi]. Due to the altruism within the family, each child's consumption is also related to the level of her parents' consumption. Thus, each child's consumption, [c.sup.k], is

   (2) [c.sup.k] = [pi] + [alpha]c,

   where [alpha] is a positive coefficient that measures the extent of parents' altruism toward their children. (4) Meanwhile, the cost of education is fixed and is denoted by b, b [greater than or equal to] 0.

   For simplicity, we assume that, if a child goes to school, she cannot work at all. So a child's cost of education includes not only the financial cost but also the opportunity cost of working. An adult's income and a child's wage rate are denoted by w and [w.sub.c], respectively. (Note that the Becker-Lewis model is the special case of the current model in which [w.sub.c] = 0.) Then, a household's budget constraint is

   (3) c + n([pi] + [alpha]c) + sb = w + (n - s)[w.sub.c].

   A parent maximizes her utility subject to Equation 3. For clarity of exposition, we analyze the optimization problem here by assuming that an adult makes decisions in two stages. (5) In the first stage, she chooses the number of her offsprings. In the second stage, when her infants grow into school-aged children who have the capacity of studying and working, she makes decisions on her children's education and household consumption. To solve this problem, we first treat n as a parameter. Meanwhile, for simplicity, we assume that the constraint, s [less than or equal to] n, is not binding until the discussions in the last part of this section. Then, from the first-order conditions, we get

   (4) c = 1/(1 + [beta]) (w + [nw.sub.c] - n[pi])/(1 + n[alpha]),

   and

   (5) s = [beta]/(1 + [beta]) (w + [nw.sub.c] - n[pi])/(b + [w.sub.c]).

   Inserting Equations 4 and 5 into Equation 1, we get

   (6) V = ln[1/(1 + [beta]) (w + [nw.sub.c] - n[pi])/(1 + n[alpha])] + [beta]ln[[beta]/(1 + [beta]) (w + [nw.sub.c] + n[pi])/(b + [w.sub.c])] + [delta]ln(n).

   We assume the solution to Equation 6 is interior. (We will consider the biological constraint of fertility and its implications in the last part of this section.) Then, the first-order condition of Equation 6 is

   (7) dV/dn = (1 + [beta])([w.sub.c] - [pi])/w + n([w.sub.c] - [pi]) - [alpha]/1 + n[alpha] + [delta]/n = 0.

   Like the Becker-Lewis model, this article intends to examine the relationship between fertility and parental income and between parental income and the average level of children's human capital. In this model, the average level of children's human capital, h, is defined as

   h [equivalent to] s/n.

   Then, from Equation 7, we get the following proposition.

   PROPOSITION 1. (i) As parents' income increases, fertility will decrease (i.e., dn/dw

   (8) [w.sub.c] > [pi].

   (ii) As parents' income increases, a higher proportion of children will receive an education (i.e., dh/dw >0) if [w.sub.c] > [pi].

   PROOF. (i) Totally differentiating Equation 7 with respect to n and w and rearranging, we get

   (9) dn/dw = (1 + [beta])([w.sub.c] - [pi])/[w + n([w.sub.c] - [pi])].sup.2]V''.

   Note that the second-order condition, V'', must be negative when n is at its optimal solution and the solution is interior. So, if [w.sub.c] > [pi] so that [w.sub.c] - [pi] > 0, we have

   dn/dw

   (ii) From Equation 5, we get

   (10) ds/dw = [differential]s/[differential]w + [differential]s/[differential]n dn/dw = [beta]/(1 + [beta])(b + [w.sub.c]) + [beta]/(1 + [beta]) ([w.sub.c] - [pi])/(b + [w.sub.c]) dn/dw.

   So when [w.sub.c] > [pi] so that dn/dw

   dh/dw = n ds/dw - s dn/dw/[n.sup.2]

   = 1/[n.sup.2]{[beta]n/(1 + [beta])(b + [w.sub.c]) + [[beta]/(1 + [beta]) n([w.sub.c] - [pi])/(b + [w.sub.c]) - s] dn/dw}

   = 1/[n.sup.2]{[beta]n/(1 + [beta])(b + [w.sub.c]) + [[beta]/(1 + [beta]) n([w.sub.c] - [pi])/(b + [w.sub.c]) - [beta]/(1 + [beta])(w + [nw.sub.c] - n[pi])/(b + [w.sub.c])]dn/dw}

   = 1/[n.sup.2][[beta]n/(1 + [beta])(b + [w.sub.c]) - [beta]/(1 + [beta]) w/(b + [w.sub.c] dn/dw] > 0.

   QED.

   From Proposition 1, clearly, we have the following corollary.

   ...