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Problem 1: [25 points]. Ebenezer Scrooge is an expected utility maximizer who three for two periods: a = o,1. He consumes a single consumption good

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Problem 1: [25 points]. Ebenezer Scrooge is an expected utility maximizer who three for two periods: a = o,1. He consumes a single consumption good 1:. Hr. Scrooge's utility as a function of his consumption in each period {era and 3:1, respectively) is giyen by U 2 etc\") + E[u(x1)], where Mart} = . At the start of period I], Mr. Scrooge is endWed with w :2- 0 units ofthe good. which he can either consume in period I] or same until period 1. Let 3 denote the amount of the good he saves in 2 period [1 for consumption in period 1. The return on his savings in period 1 is uncertain. With probability 2: Hr. Scrooge will receive Be in period 1, and with probability (1 :nr) he will receiye ds , where D :2- d E In. Mr. Scrooge has to make the sayings decision before this uncertainty is realized. 1) [5 points] Set up Mr. Scrooge's expected utility maximization problem and write out the first order conditions that characterize the solution. Are the second order conditions satisfied (explain or show)? 2) [5 points] Solve Mr. Scrooge's expected utility maximization problem for the optimal amount of savings s*. Find the maximized value of his expected utility. 3) [7 points] How does Mr. Scrooge's optimal amount of savings depend on the probability of having high return, it? Provide both the intuitive explanation and the analytical proof. Suppose that Mr. Scrooge's has an alternative saving strategy: instead of saving s* with the uncertain return, in period 0 he can put the amount of the good y > O under his mattress and consume this very same amount in period 1 ( note that he does not have an option of saving now). 4) [4 points] Set up Mr. Scrooge's expected utility maximization problem and solve it for the optimal amount of y*. 5) [4 points] Suppose It = 1/2 and d = 0. For what values of D should Mr. Scrooge put his money under the mattress instead of saving it (note we do not allow him to save a part of money and put the other part under the mattress at the same time)

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