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5 Borrowing Constraint and Ricardian Equivalence [30 Points] Consider a representative consumer who lives for three periods, he or she has an exogenous endowment stream given by y1, y2, y3 and can borrow and lend at a given interest rate i. Assume that he/she starts out with no wealth (that is, fo = 0). The discount factor is B = 1. The consumer's problem is: max u(c1) + u(c2) + u(c3) C1, C2,C3 subject to C1 + b1 = y1 C2 + b2 = y2 + (1 + i) . by C3 = 13 + (1 + i) . b2 where by is savings in Period 1. 1. Derive the inter-temporal budget constraint from the period constraints given above. 2. For simplicity, further assume that i = 0 and the utility function, u(c), is concave. The income stream is as follows: y1 = 2.2, yz = 1 and y3 = 2.8. a. Solve for the optimal consumption plan for the three periods, c;, where t = 1, 2, 3. b. How much does the consumer save during Period 3, the last period of his life? 3. Further assume that there is a government in this economy. It has an exogenous expenditure to finance in Period 2, that is 92 = 0.9, and 91 = 93 = 0. Suppose that 1) The government can use lump-sum tax to finance the expenditure 92 ; 2) The government can also borrow and lend in the international market at a given interest rate i = 0. It has to repay all the debt in Period 3. Denote the lump sum tax in each period by 7t. Moreover, assume that 7: 2 0 , t = 1, 2, 3. a. Write down the government's inter-temporal budget constraint. b. Solve for the household's optimal consumption plan for the three periods, c;, where t = 1, 2,3 . 4. Further assume that the consumer cannot borrow from the market for the whole three periods. And the government intends to maximize the life time utility of the consumer. (It is sufficient to justify your answers in words.) a. Find the government's optimal lump sum tax in Period 3, 75. b. Find the the household's optimal consumption plan for the three periods, cf, where t = 1, 2,3 5. Further assume that the government cannot borrow from the international market either. (It is suffi- cient to justify your answers in words.) a. Find the government's optimal lump sum tax in Period 3, 75. b. Find the the household's optimal consumption plan for the three periods, c;, where t = 1, 2,3