Problem 2 Individuals get utility from consumption in 2 periods of life (0 and 1), according to the utility function, u(c0,c1) = ln(c0) + ? ln(c1), 0 < ? < 1. They earn income in both periods and earn interest at a rate of r per period on savings (pay interest r if they borrow). a. Write an expression for the individual's lifetime budget constraint. Solve it with c1 on one side. b. Write an expression for the MRS, based on the utility function above if ? = 0.95. Assume c0 is on the horizontal axis. c. Write an expression for the optimal period zero consumption (c0*) as a function of r, I0, I1 and ? = 0.95. d. Use your solution to part c to solve for the amount of saving (borrowing, which would be <0) of the following person ("Pete") in period 0: Incomes 80 (t=0) and 80 (t=1) facing interest rate r = 0.025. e. Use your solution to part c to solve for the amount of saving (borrowing, which would be <0) of the following person ("Mike") in period 0: Incomes 60 (t=0) and 50 (t=1) facing interest rate r = 0.025.
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The lifetime budget constraint is given by the sum of consumption in both periods, taking into account the interest rate on savings or borrowing. The expression for the lifetime budget constraint is: \(c_0 + \frac{c_1}{1+r} = I_0 + \frac{I_1}{1+r}\) Solving for Show moreā¦
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Barbara lives for two periods and has preferences represented by the following utility function: U(Cā,Cā) = Cā^0.5 + (1/(1+Ī“) Cā^0.5), where the subscripts represent this period (0) and the next period (1) and Ī“ is in millions of dollars. Barbara's wealth is 0.5 million dollars, and she has no labor income. Her rate of time preference Ī“ is 2%. a. Set up and solve Barbara's intertemporal utility maximization problem (i.e., find the optimal amount of consumption in each period) assuming the real interest rate is 2%. How much does she save in period 0? [10 points] b. Now suppose that the interest rate falls to 1%. What happens to the optimal value of Cā? How does the amount Barbara saves in period 0 compare to part (a)? [10 points] c. Explain what has happened to Barbara's savings between parts (a) and (b) in terms of income and substitution effects. [5 points] Bridgett's utility function is . Her initial wealth (in thousands of dollars) is . There are two assets in which Bridgett can invest, X and Y. X is a risk-free asset that returns 5% for sure (i.e., $1 invested in X is worth $1.05 in the next period). Y is a risky asset. With probability 0.7 it has a return of +25% (i.e., $1 invested in Y is worth $1.25 next period), and with probability 0.3 it has a return of -20% (i.e., $1 invested in Y is worth $0.80 next period). [Note: For questions 3 and 4, I'm adding "in thousands of dollars" to make the quantities more realistic. You should use throughout, NOT ] a. If Bridgett must invest all of her wealth in only one asset, which one would she choose? What if she were risk-neutral? [10 points, 5 points for each part of the question] b. Now suppose Bridgett can invest any share of her wealth in X or Y but she cannot borrow. Let be the amount of Bridgett's wealth that is invested in Y. What value of maximizes Bridgett's expected utility? [10 points]
Akash M.
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