Question

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 one unit of time. She earns the wage w per unit of labor 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.

Suppose that the government levies a proportional tax on labor income  , where 0 <  < 1. The revenues from the tax on labor income are rebated lump-sum to the households. Let T denote the lump sum transfer that the representative consumer receives. So the consumerís budget constraint is:

C = (1  )wN + (1 + r) A + T

1. (a) (4 points) Write the budget constraint relating consumption with leisure and

   use it to derive the relative price of leisure in terms of consumption.

2. (b) (4 points) Write the representative consumerís problem as a constrained maxi- mization then transform it into a simpler unconstrained maximization.

3. (c) (4 points) Derive an equation that implicitly deÖnes the optimal labor supply N* of the household as a function of (w;;A;r;T).

   Assume from now on that the householdís preferences are

   U(C;L)=ln(C)+ ln(L)

4. (d) (4 points) Derive the optimal labor supply N* of the household as a function of

   (w; ;A; r; T).

5. (e) (6 points) Assess how N* responds to both the tax rate  and the transfer T.

   Explain these responses referring to the income and substitution e§ects.

6. (f) (4 points) What is the e§ect 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)?

7. (g) (6 points) Accounting for the impact of the tax revenue, wN, on transfers, T, assess how N* responds to the tax rate . Explain this response referring to the income and substitution e§ects.

Answer 1