Consider an 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 representative firm has a production technology given by Y = zf (K; N) where Kis the fixed capital input and N is labor input.Suppose that the government levies a proportional tax, ; where 2 (0; 1) on consumption: So the taxes paid by the representative consumer are C. The revenuesfrom the taxes collected by the government are rebated lump-sum to consumers. LetT R denote the lump sum transfer that the representative consumer receives.

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Consider an economy where the representative consumer has a utility function u (C, _) over consumption C and leisure L. Assume preferences satisfy the standard proper- ties we assumed in class. The consumer has an endowment of H units of time. The representative firm has a production technology given by Y = =/ (K, N) where K is the fixed capital input and / is labor input. Suppose that the government levies a proportional tax, T, where 7 E (0, 1) on con- sumption. So the taxes paid by the representative consumer are TC. The revenues from the taxes collected by the government are rebated lump-sum to consumers. Let TR denote the lump sum transfer that the representative consumer receives. (A) Write and explain the budget constraint for the representative consumer in this economy. (B) ) Find the relative price of leisure in terms of consumption. (C) Set up the consumer's problem. (D) The general formula for the optimality condition of the consumer's problem is (Relative price of leisure in terms of consumption) MUC - MUL where MUC and MUL are the marginal utility of consumption and the marginal utility of leisure respectively. Use this expression and the relative price of leisure in terms of consumption that you found above to write a set of equations that implicitly allow you to find the optimal labor supply N', optimal consumption, "", and optimal leisure, L', as a function of (w, T, TR, D). Assume for the remaining parts of this exercise that the utility function is given by u (C, L) - CL . 0