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probset6 - Problem Set 6 EC720.01 Math for Economists...

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Problem Set 6 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 22 1. The Permanent Income Hypothesis The permanent income hypothesis describes how a forward-looking consumer optimally saves or borrows to smooth out his or her consumption in the face of a fluctuating income stream. This problem formalizes the permanent income hypothesis using a two-period model. So consider a consumer who lives for two periods, earning income w 0 during period t = 0 and w 1 during period t = 1. Let c 0 and c 1 denote his or her consumption during periods t = 0 and t = 1 and let s denote his or her amount saved (or borrowed, if negative) during period t = 0. Suppose that savings earn interest between t = 0 and t = 1 at the constant rate r . Then the consumer faces the budget constraints w 0 c 0 + s (1) at t = 0 and w 1 + (1 + r ) s c 1 (2) at t = 1. Finally, suppose that the consumer’s preferences are described by the utility function ln( c 0 ) + β ln( c 1 ) , (3) where the discount factor lies between zero and one: 0 < β < 1. a. Find the values for c * 0 , c * 1 , and s * that solve the consumer’s problem – choose c 0 , c 1 , and s to maximize the utility function in (3) subject to the constraints in (1) and (2) – in
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