The economist Jagdish Bhagwati explained in one of his public lectures that international trade causes the wage for child labor to increase in developing countries. He then discussed informally that this might lead to more child labor if parents are €œbad€ and less child labor if parents are €œgood€.
A. Suppose that households in developing countries value two goods: €œLeisure time for Children in the Household€ and €œHousehold Consumption.€ Assume that the adults in a household are earning $y in weekly income regardless of how many hours their children work. Assume that child wages are w per hour and that the maximum leisure time for children in a household is E hours per week.
(a) On a graph with €œweekly leisure time for children in the household€ on the horizontal axis and €œweekly household consumption€ on the vertical, illustrate the budget constraint for a household and label the slopes and intercepts.
(b) Now suppose that international trade expands and, as a result, child wages increase to w€².
Illustrate how this will change the household budget.
(c) Suppose that household tastes are homothetic and that households require their children to work during some but not all the time they have available. Can you tell whether children will be asked to work more or less as a result of the expansion of international trade?
(d) In the context of the model with homothetic tastes, what distinguishes €œgood€ parents from €œbad€ parents?
(e) When international trade increases the wages of children, it is likely that it also increases the wages of other members of the household. Thus, in the context of our model, y €”the amount brought to the household by others €” would also be expected to go up. If this is so, will we observe more or less behavior that is consistent with what we have defined as €œgood€ parent behavior?
(f) In some developing countries with high child labor rates, governments have instituted the following policy: If the parents agree to send a child to school instead of work, the government pays the family an amount x. (Assume the government can verify that the child is in fact sent to school and does in fact not work, and assume that the household views time at school as leisure time for the child.) How does that alter the choice set for parents? Is the policy more or less likely to succeed the more substitutable the household tastes treat child €œleisure€ and household consumption?

Graph 8.12: Child Labor and International Trade: Part II
B. Suppose parental tastes can be captured by the utility function u(c,„“) = c0.5„“0.5. For simplicity, suppose further that y = 0.
(a) Specify the parents€™ constrained optimization problem and set up the appropriate Lagrange function.
(b) Solve the problem you have set up to determine the level of leisure the parents will choose for their children. Does w have any impact on this decision?
(c) Explain intuitively what you have just found. Consider the CES utility function (that has the Cobb-Douglas function you just worked with as a special case). For what ranges of ρ would you expect us to be able to call parents €œgood€ in the way that Bhagwati informally defined the term?
(d) Can parents for whom household consumption is a quasilinear good ever be €œgood€?
(e) Now suppose (with the original Cobb-Douglas tastes) that y > 0. If international trade pushes up the earnings of other household members€”thus raising y, what happens to child leisure?
(f) Suppose again that y = 0 and the government introduces the policy described in part A (f). How large does x have to be in order to cause our household to send their child to school (assuming again that the household views the child€™s time at school as leisure time for the child)?
(g) Using your answer to the previous part, put into words what fraction of the market value of the child€™s time the government has to provide in x in order for the family to choose schooling over work for their child?

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