1. Suppose the DM has preferences - over three-period consumption streams which are represented by some function V. Suppose more is better than less in each period. Answer each of the following five questions. (a) Suppose you observe ($50, $100, $100) > ($100, $50, $50). Can V be a discounted sum utility? If yes, does the DM exhibit impatience? If yes, can V be an exponential discounted sum utility? Briefly explain. (b) Suppose V(zo, 21, 22) = zo + 0.921 + 0.81(z2]. Which axioms, separability, mono- tonicity, and stationarity, do the DM's preferences - satisfy and which ones do they not satisfy? Briefly explain. (Here [.J denotes the floor operator which gives the output as the greatest integer less than or equal to the input; e.g., (2.4] = 2 or (3.1] = 3.) (c) Suppose V(zo, 21, 22) = Vzo + 0.8\\zi + 0.64\\z, and so the DM is an exponential discounted sum utility maximizer with 6 = 0.8. Suppose the DM has some wealth w = [1.000.000 which she needs to consume over the three periods. Find the DM's optimal allocation of her wealth. Show your workings. (d) Still consider the DM in part (c) but now suppose that the DM can be financially trained so as to increase her o from 0.8 to 0.9. The cost of financial education to the DM is c = 125.000 which she must pay before allocating her wealth over the three periods. Find the DM's maximum utility when she is educated. Should the DM get financial education? Explain. (e) Now consider the DM in part (d) but suppose that her period utility changes from V2 to vz; that is, u(z) = 21/3. How would your answer change to part (d) questions? Explain.2. Suppose that the DM's preferences _ over three periods of consumption is represented by the function V(zo, 21, 22) = In zo + 0.64In z + 0.36 In =2. Suppose that the DM has consistent planning preferences and wants to consume her income w = 25.000 over three periods. Consider the following questions. (a) Suppose the DM states that she is indifferent between committing to consumption plan (2.500, 1.600, 2900) and allocating her income w = 25.000 in period 0 over three periods with a possible revision in period 1. Do you think the DM is naive or sophisticated? Show your workings. (b) Suppose the DM states that she is indifferent between committing to (approxi- mately) consumption plan (62.500, f1.524, 6976) and allocating her income w = 15.000 in period 0 over three periods with a possible revision in period 1. Do you think the DM is naive or sophisticated? Show your workings. (c) Suppose the DM is naive and suppose a policy maker can educate the DM so then she becomes sophisticated. The policy maker cares about the welfare of the DM in period 0 net the cost of education. Would the policy maker provide education to the DM if the cost is c = 650? How about when it is c = 125? Show your workings.3. Consider a threeperiod setting such that there is consumption only in period 1 and period 2. Answer the following questions. Suppose the DM has inmme w = 510.000. (3) (b) Suppose that the DM has self-control preferences. In particular. suppose from the perspective of period 0 the DM's commitment preferences are represented by U (21,33) = 0.6 lnz; +0.3ln 1'; while her temptation preferences are represented by V(|.2-z) = 0.6km; + dqlnzz for some d2 > 0. Suppose you observe that the DM consumes 0.75 share of her wealth to in period 1 and 0.25 share of it in period 2. Find the discount factor :12 . Now suppose that the DM hm additive exibility preferences imtead such that from the perspective of period 0 they are represented with equal probability by either U {9:1, 32) a 0.61n x1 + 0.4111 3:2 or by V{:c1.xg) - 0.811111 + 0.6 Inxg. i. Suppose the DM has to make her consumption decision in period 0 before she observes what utility she has in period 1. What share of her wealth to would the DM optimally consume in period 1 and what share in period 2? ii. Now suppose that by paying an amount it > 0 of money upfront, the DM can delay her consumption decision to period 1 in which her utility function is realized. Find the maximum amount of k the DM would be willing to pay to exercise the option of delay