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Unformatted text preview: University of Minnesota Department of Economics Econ 3102: Intermediate Macroeconomics Problem Set 1 Answer Key 1 Problems (65 points) Exercise 1 (10 points) Consider the utility function U ( C, ) = logC + (1- ) log . (a) Derive the marginal utility of consumption. Is it decreasing in C? Answer: The marginal utility of consumption is: U C ( C, ) = C (b) Derive the marginal utility of leisure. Is it decreasing in leisure? Answer: The marginal utility of leisure is: U ( C, ) = 1- (c) Does this utility function satisfy the Inada Conditions? Show your answer. Answer: Yes. Given the marginal utilities, it is straightforward to verify that U C (0 , ) = + U ( C, 0) = + , Exercise 2 (15 points) Consider a representative consumer whose preferences over consumption and leisure are given by the utility function U ( C,l ) = C 1 / 3 l 2 / 3 . Assume that she has a total of h = 18 hours which she can use for leisure or she can work for the wage rate w = 6. Finally, assume that she enjoys a dividend income = 36 and has to pay taxes T = 24. Determine the optimal consumption bundle ( C * , * ). Answer: You can do this in a variety of ways; here Ill use the MRS = w trick. Since the MRS equals the ratio of the marginal utilities, we get that MU /MU C = w ; given that the marginal utilities are just the partial derivatives of the utility function, we rearrange and get U ( C, )- wU C ( C, ) = 0 or 2 3 C 1 / 3 - 1 / 3 = 6 1 3 C- 2 / 3 2 / 3 . (1.1) Multiply across by - 1 / 3 C- 2 / 3 to get C = 3 . (1.2) Now plug (2) in the budget constraint: 3 = 6(18- ) + 36- 24 . (1.3) Working this out, we get...
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