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Unformatted text preview: 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 > and > . (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 > U CC ( C; ) = dU C ( C; ) dC = & & & C 2 < 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 & & > U ( C; ) = dU ( C; ) d = & 1 & & 2 < 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 ! U C ( C; ) = lim ! 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...
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