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HW1key - Econ 3102 Intermediate Macroeconomics Homework#1...

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Econ 3102: Intermediate Macroeconomics Homework #1 Answer Key 1 Problems (65 points) Exercise 1 (10 points) First note that the utility function is unde°ned when C = 0 or = 0 , so we need only focus on the cases in which C > 0 and ‘ > 0 . (a) Derive the marginal utility of consumption. Is it decreasing in C ? Answer: The marginal utility of consumption is: U C ( C; ‘ ) = dU ( C; ‘ ) dC = ° C > 0 U CC ( C; ‘ ) = dU C ( C; ‘ ) dC = ° ° ° C 2 ± < 0 Since U CC is strictly less than zero, we know that marginal utility is a decreasing function in C . (b) Derive the marginal utility of leisure. Is it decreasing in leisure? Answer: The marginal utility of leisure is: U ( C; ‘ ) = dU ( C; ‘ ) d‘ = 1 ° ° > 0 U ‘‘ ( C; ‘ ) = dU ( C; ‘ ) d‘ = ° ² 1 ° ° 2 ³ < 0 Since U ‘‘ is strictly less than zero, we know that marginal utility is a decreasing function in . (c) Does this utility function satisfy the Inada conditions? Show your answer. Answer: Yes. Given the marginal utilities, it is straightforward to verify that lim C ! 0 U C ( C; ‘ ) = lim ! 0 U ( C; ‘ ) = + 1 lim C ! + 1 U C ( C; ‘ ) = lim ! + 1 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; ‘ ) = C 1 = 3 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 .
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