Consider a decision-maker who lives for two periods, 1 and 2. Their lifetime utility is given...

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Consider a decision-maker who lives for two periods, 1 and 2. Their lifetime utility is given by c+202, where c, is consumption in period i (i = 1,2). The decision-maker's income in period 1 is r =100. The second-period income is denoted by X2 The decision-maker allocates their first-period income between consumption and savings. If they choose to save k 0 in period 1, their consumption will equal 100 - k in period 1 and X2 + k in period 2 (that is, the net interest rate is zero). (a) Suppose that X = 100. Find the optimal value of k. For the rest of the problem, suppose that X2 is a random variable equal to 160 with probability 0.25 and 80 with probability 0.75. (b) Find the expected value and the variance of X2. (c) Will the optimal value of k be greater or less than what you found in (a)? (You can either find the optimal value of k numerically or answer the question without computing it, like we did in class for the precautionary savings problem). (d) Does the decision-maker's second-period utility index exhibit prudence? Does this finding agree with what you found in (c)? Consider a decision-maker who lives for two periods, 1 and 2. Their lifetime utility is given by c+202, where c, is consumption in period i (i = 1,2). The decision-maker's income in period 1 is r =100. The second-period income is denoted by X2 The decision-maker allocates their first-period income between consumption and savings. If they choose to save k 0 in period 1, their consumption will equal 100 - k in period 1 and X2 + k in period 2 (that is, the net interest rate is zero). (a) Suppose that X = 100. Find the optimal value of k. For the rest of the problem, suppose that X2 is a random variable equal to 160 with probability 0.25 and 80 with probability 0.75. (b) Find the expected value and the variance of X2. (c) Will the optimal value of k be greater or less than what you found in (a)? (You can either find the optimal value of k numerically or answer the question without computing it, like we did in class for the precautionary savings problem). (d) Does the decision-maker's second-period utility index exhibit prudence? Does this finding agree with what you found in (c)?

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a To find the optimal value of k we need to maximize the decisionmakers lifetime utility function wh... View the full answer