Consider an individual who lives for three periods and whose preferences are represented by the utility function: U (cl, 02,03) 2 log (cl) + 6 [310g (Cg) + )32 log (C3)]. Suppose this individual has access to a perfect capital market. That is, he can borrow or lend unlimited amounts at the going interest rate, which is given by (1 + 7'). Let wt denote the consumer's income in period t =1, 2, 3. You may assume that: w1+w2+w3=1. In this problem, the parameter 6 is an additional discount factor that is applied to every future period. 1. Set up and solve this individual's utility maximization problem under the assumption that ,8 = 1, (1+ 1') = 1, and 6 = 1. Find consumption in each of the three periods. Does the consumer smooth consumption completely? If not, explain why not. 2. Consider now the same problem but assume that 6 < 1. You may continue to assume that ,8 = 1 and (1 + r) = 1. How much does he consume in each period? We will refer to this solution as the consumer's "?rst period plan" 3. Consider, now, the continuation problem. That is, consider the same individual, but assume that he has already lived one period. From the vantage point of period 2, the individual's preferences are represented by: U (C2, 63) = 103 (C2) + (5)3103; (63)- Note that the individual discounts utility one period ahead by 63 in both cases. What is the consumer's "period two plan"? 4. What is the consumer's "period three plan"?

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