Tax on labor income - Consider a one-period economy where the representative 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

Question 1:

Part(a) Write the budget constraint relating consumption with leisure and use it to derive the relative price of leisure in terms of consumption.

Part(b) Write the representative consumer's problem as a constrained maximization then transform it into a simpler unconstrained maximization.

Part(c) Derive an equation that implicitly defines 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) = ln(C) + y ln(L)

Part (d) Derive the optimal labor supply N* of the household as a function of (w; a; r).

Part(e) What is the effect of a rise in the interest r on labor supply? Does it matter if a is positive (agent is a lender) or negative (agent is a borrower)? Why?

1. (18 points) Tax on labor income - Consider a one-period economy where the rep- resentative consumer has a utility function U(CL) 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. D (b) (3 points) Write the representative consumer's problem as a constrained maxi- mization then transform it into a simpler unconstrained maximization. e) (3 points) Derive an equation that implicitly defines the optimal labor supply Nof the household as a function of w,a,r). Assume from now on that the household's preferences are UCL) --In()+yIn(L) (a) (3 points) Derive the optimal labor supply N* of the household as a function of (war). (e) (6 points) What is the effect of a rise in the interest r on labor supply? Does it matter if a is positive (agent is a lender) or negative (agent is a borrower)? Why