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Unformatted text preview: 1 Economics 11: Microeconomic Theory 1 Professor Christian Hellwig Answer Key of Homework 1 Solution Question 1: (a) Joes budget constraint: M G 10 20 60 + = Lagrangian: ) 10 20 60 ( ) ; , ( 2 / 1 2 / 1 M G M G M G L + + = First order conditions: 20 2 1 ) ; , ( 2 / 1 = = G G M G L 10 2 1 ) ; , ( 2 / 1 = = M M M G L 10 20 60 ) ; , ( = = M G M G L Rearranging the two first order conditions: = 2 / 1 40 1 G = 2 / 1 20 1 M Dividing the first expression by the second: G M G M G M 4 4 2 2 / 1 2 / 1 = = = ( 1 ) Substitute (1) into the budget constraint to find: ( ) G G 4 10 20 60 + = Solving for the optimal * G gives: 1 * = G Substituting the optimal * G back into (1) gives: 4 * = M 2 (b) Use the budget constraint and write M (or G ) as a function of G (or M ): G M 2 6 = ( 2 ) Now substitute this into the utility function: ( ) 2 / 1 2 / 1 2 6 G G U + = Take the first order conditions with respect to G : ( ) 2 6 2 1 2 / 1 2 / 1 = = G G dG dU Rearranging terms: 1 4 2 6 2 ) 2 6 ( ) 2 6 ( 1 2 1 * 2 / 1 2 / 1 2 / 1 2 / 1 = = = = G G G G G G G Substituting the optimal * G back into (2) gives: 4 ) 1 ( 2 6 * = = M (c) As expected, the solutions are the same. (d) Take the natural log of the utility function: ) ln( ) , ( ln 2 / 1 2 / 1 M G M G U + = Set up the Lagrangian: ) 10 20 60 ( ) ln( ) ; , ( 2 / 1 2 / 1 M G M G M G L + + = First order conditions: 20 2 1 ) ; , ( 2 / 1 2 / 1 2 / 1 = + = M G G G M G L 10 2 1 ) ; , ( 2 / 1 2 / 1 2 / 1 = + = M G M M M G L ) ; , ( = = M P G P I M...
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 Spring '08
 cunningham
 Economics

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