Consider again the end-of-chapter problem 8.9 about the impact of international trade on child labor in the developing world.
A: Suppose again that households have non-child income Y , that children have a certain weekly time endowment L, and that child wages are w in the absence of trade and w′ > w with trade.
(a) On a graph with child leisure hours on the horizontal axis and household consumption on the vertical, illustrate the before and after trade household budget constraints.
(b) Suppose that tastes over consumption and child leisure were those of perfect complements. Illustrate in your graph how much a household would be willing to pay to permit trade—i.e. how much would a household be willing to pay to increase the child wage from w to w′?
(c) If the household paid the maximum it was willing to pay to cause the child wage to increase, will the child work more or less than before the wage increase?
(d) Re-draw your graph, assume that the same bundle (as at the beginning of part (b)) is optimal, but now assume that consumption and leisure are quite (though not perfectly) substitutable. Illustrate again how much the household would be willing to pay to cause the wage to increase.
(e) If the household actually had to pay this amount to get the wage to increase, will the child end up working more or less than before trade?
(f) Does your prediction of whether the child will work more or less if the household pays the maximum bribe to get the higher wage depend on how substitutable consumption and child leisure are?
(g) Can you make a prediction about the relative size of the payment the household is willing to make to get the higher child wage as it relates to the degree of substitutability of consumption and child leisure? Are “good” parents willing to pay more or less?
B: Suppose that the household’s tastes over consumption and leisure can be represented by the CES utility function u(c, ℓ) = (αc−ρ + (1−α) ℓ−ρ) −1/ρ.
(a) Derive the optimal household consumption and child leisure levels assuming the household has non-child weekly income Y, the child has a weekly time endowment of L, and the child wage is w.
(b) Verify your conclusion from end-of-chapter problem 8.9 that parents are neither “good” nor “bad” when Y = 0 and ρ = 0; i.e. parents will neither increase nor decrease child labor when w increases.
(c) If international trade raises household income Y, what will happen to child labor in the absence of any change in child wages? Does your answer depend on how substitutable c and ℓ are?
(d) When α = 0.5 and w = 1, does your answer depend on the household elasticity of substitution between consumption and child leisure?
(e) How much utility will the household get when α = 0.5 and w = 1?
(f) Derive the expenditure function for this household as a function of w and u. What does this reduce to when α = 0.5? (Hint: You can assume Y = 0 for this part.)
(g) Suppose non-child income Y = 0, child time is L = 100, α = 0.5, ρ = 1 and w is initially 1. Then international trade raises w to 2. How does the household respond in its allocation of child leisure?
(h) Using your expenditure function, can you determine how much the household would be willing to pay to cause child wages to increase from 1 to 2? If it did in fact pay this amount, how would it change the amount of child labor?
(j) Are your calculations consistent with your predictions in (f) and (g) of part A of the question?