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Unformatted text preview: 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 , c 1 , and s to maximize the utility function ln( c ) + ln( c 1 ) subject to the constraints w c + s and w 1 + (1 + r ) s c 1 . a. With the Lagrangian for the consumers problem defined as L ( c ,c 1 ,s, , 1 ) = ln( c ) + ln( c 1 ) + ( w c s ) + 1 [ w 1 + (1 + r ) s c 1 ] , the firstorder conditions can be written as 1 c * * = 0 , c * 1 * 1 = 0 , and * + * 1 (1 + r ) = 0 and the constraints, which will bind at the optimum, can be written as w c * s * = 0 and w 1 + (1 + r ) s * c * 1 = 0 . Combine the budget constraints to obtain w + w 1 1 + r = c * + c * 1 1 + r , which says that the present value of the consumers income will equal the present value of his or her consumption over the two periods. Now combine the firstorder conditions to obtain 1 c * = (1 + r ) c * 1 1 or c * 1 = (1 + r ) c * . Substitute this last expression into the present value budget constraint to find the desired solution c * = 1 1 + w + w 1 1 + r , then substitute this solution into the previous expression to obtain c * 1 = (1 + r ) 1 + w + w 1 1 + r ....
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
 Fall '09
 IRELAND
 Utility

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