Economics

1. (18 points) Tax on labor income - Consider a one-period economy where the rep- resentative consumer has a utility function U (C, L) over consumption C and leisure L. Assume preferences satisfy the standard properties we assumed in class. The consumer has an endowment of H units of time. The household earns the wage w per hour supplied to the market and has wealth a which yields an interest rate r, so her income is partly coming from labor, partly from capital. So the consumer's budget constraint is: C=wN+(1+r)a (a) (3 points) Write the budget constraint relating consumption with leisure and use it to derive the relative price of leisure in terms of consumption. (b) (3 points) Write the representative consumer's problem as a constrained maxi- mization then transform it into a simpler unconstrained maximization. (c) (3 points) Derive an equation that implicitly denes the optimal labor supply N* of the household as a function of (w,a,r). Assume from now on that the household's preferences are U(C, L) = 111(0) + 7111(L) (d) (3 points) Derive the optimal labor supply N* of the household as a function of (w, a, r). (e) (6 points) What is the effect of a rise in the interest 1' on labor supply? Does it matter if a is positive (agent is a lender) or negative (agent is a borrower)? Why