1. Lump-sum vs. Proportional Tax: (s7q1) Consider the two-period model with consumption and leisure decisions. Assume that a representative individual has the following utility function: U = u(ci, n) + 0.9u(c2, n). (1) and u(ct, nt) = ct - nz 2 where and c and nt denote consumption and labor supply of the individual (potentially ex- ceeding 1) for t = {1,2}. Assume the real wage rate (in terms of the consumption good) is equal to 1.1 in the first period and 1 in the second period. The individual pays a labor income tax in both periods, which is equal to 7 in both periods. He starts his life with no assets and can freely save and borrow in the first period. Let a denote his savings at the end of the first period. He receives interest income r for his savings at the beginning of the second period. At the end of the second period, he leaves no assets behind, i.e. no savings in the second period. (a) Write down the budget constraint for each period and obtain the life-time budget con- straint. (b) Set up the Lagrangian for the utility maximization problem and obtain the first order conditions. (c) What is equilibrium interest rate, r? (d) What value of 7 maximizes the present discounted value of government tax revenue? (e) Suppose that the present discounted value of government spending is equal to 0.48. What value of 7 generates this tax revenue and maximizes consumer utility? (f) Suppose that the present discounted value of government spending is equal to 0.48 and the government replaces the labor income tax in both periods with a lump-sum tax T = 0.48 in the first period. Is the consumer better off? Clearly show your work.