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answer6 - Solutions to Problem Set 6 EC720.01 Math for...

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Solutions to 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 consumer chooses c 0 , c 1 , and s to maximize the utility function ln( c 0 ) + β ln( c 1 ) subject to the constraints w 0 c 0 + s and w 1 + (1 + r ) s c 1 . a. With the Lagrangian for the consumer’s problem defined as L ( c 0 , c 1 , s, λ 0 , λ 1 ) = ln( c 0 ) + β ln( c 1 ) + λ 0 ( w 0 - c 0 - s ) + λ 1 [ w 1 + (1 + r ) s - c 1 ] , the first-order conditions can be written as 1 c * 0 - λ * 0 = 0 , β c * 1 - λ * 1 = 0 , and - λ * 0 + λ * 1 (1 + r ) = 0 and the constraints, which will bind at the optimum, can be written as w 0 - c * 0 - s * = 0 and w 1 + (1 + r ) s * - c * 1 = 0 . Combine the budget constraints to obtain w 0 + w 1 1 + r = c * 0 + c * 1 1 + r , which says that the present value of the consumer’s income will equal the present value of his or her consumption over the two periods. Now combine the first-order conditions to obtain 1 c * 0 = β (1 + r ) c * 1 1
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or c * 1 = β (1 + r ) c * 0 . Substitute this last expression into the present value budget constraint to find the desired solution c * 0 = 1 1 + β w 0 + w 1 1 + r , then substitute this solution into the previous expression to obtain c * 1 = β (1 + r ) 1 + β w 0 + w 1 1 + r .
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