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Unformatted text preview: ECON 101 – SOLUTIONS TO MIDTERM 1 GARTH BAUGHMAN AND LINGWEN HUANG (1) [30 points] Consider the utility function u ( x 1 ,x 2 ) = x 1 e x 2 , with budget constraint p 1 x 1 + p 2 x 2 = I . (a) Form the Lagrange function associated with this utility maximization problem and derive the firstorder conditions. What are the demand functions for goods 1 and 2. (You need not consider secondorder conditions, and you may assume the solution is interior.) [10 points] Solution. Set the Lagrange function L ( x 1 ,x 2 ,λ ) = x 1 e x 2 + λ ( I p 1 x 1 p 2 x 2 ) . [3 points] The first order conditions are ∂ L ∂x 1 = e x 2 λp 1 = 0 [1 points] (1) ∂ L ∂x 2 = x 1 e x 2 λp 2 = 0 [1 points] (2) ∂ L ∂λ = I p 1 x 1 p 2 x 2 = 0 [1 points] (3) From (1) and (2), we have x 1 = p 2 p 1 . Then from (3), we can solve x 2 = I p 2 p 2 . These are demand functions for goods 1 and 2. [4 points] (b) Formulate the expenditureminimization problem, and solve for the compensated demand functions using the method of Lagrange. [10 points] Solution. Fix any utility level u > 0, the expenditureminimization problem is min x 1 ,x 2 p 1 x 1 + p 2 x 2 s.t. x 1 e x 2 = u. [3 points] Set the Lagrange function as L ( x 1 ,x 2 ,γ ) = p 1 x 1 + p 2 x 2 γ ( u x 1 e x 2 ) . First order condtions are ∂ L ∂x 1 = p 1 γe x 2 = 0 [1 points] ∂ L ∂x 2 = p 2 γx 1 e x 2 = 0 [1 points] ∂ L ∂γ = u x 1 e x 2 = 0 [1 points] 1 2 GARTH BAUGHMAN AND LINGWEN HUANG From these three equations, we can solve x c 1 = p 2 p 1 , [2 points] x c 2 = ln( u ) + ln( p 1 p 2 ) . [2 points] (c) What is the relationship between the demand function and the compensated demand function? Given an answer that holds in general, i.e., that does notdemand function?...
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This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
 Spring '08
 DANNICATAMBAY
 Microeconomics, Utility

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