Problem 1. Time inconsistent preferences. In this exercise, we reconsider the topic of choice over time, with the twist that consumers have timeinconsistent preferences, as introduced in lecture. We assume three periods, t = 0, t = 1, and t = 2. We will call this timeinconsistent agent Jim. To make things simpler, assume that Jim only receives income in period 0, that is, MO > 0, M1 = M2 = 0. He earns perperiod interest 7" on each dollar saved. We denote Mi the income saved from period 1, i.e, Mi 2 (1 + r) (M0 7 c0) . Similarly, Mi : (1 + 7') (Mi C1) . We assume that Jim's utility function in each period is the following no 2 u(C0, C1, C2) = \"yln(C0) + 1 +T +1- ln(C1) + (1 1 rum) 1 1+1" U1 : u(C1, C2) : *yln(C1) + ln(C2) U2 : u(C2) : yln(C2) Assume \"y > 1. This is meant to capture the fact that Jim derives more utility from today's consumption \"disproportionately\" . 1. In this sort of inter-temporal problems, you need to start from the last period and work backward. In period 2 Jim receives Mi in income. How much ice cream will Jim consume in period 2'? [Remember, period 2 is the last period, any ice cream that the agent does not consume in the last period is wasted. Therefore, the agent maximizes ln (C2) s.t. C2 3 Mi] 2. Let us now go back to period 1. In period 1 Jim has income Mi and has to decide how much ice cream to consume, and how much money to save for period 2. Argue that this leads to the budget constraint C2 +r Cl+l SMi. 3. Now that we have derived the budget constraint, consider maximization problem of Jim in period 1: I 13113.6? 'yln(C1) + 1 + T ln(C2) (1) C12 ,. .t.