sol_Mock_1_Midterm_101_2011

sol_Mock_1_Midterm_101_2011 - Economics 101 Solutions to...

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Economics 101 Solutions to Mock Midterm 1 February 8, 2011 Soojin Kim This is an hour exam. There are two questions, each worth 30 points. Answer each question in a separate blue book. Make sure to PRINT your name and put your recitation number on each blue book. Give complete answers, and clearly label all diagrams. 1. [30 points] Consider the utility function u ( x 1 ,x 2 ) = x 1 x 2 + x 1 , 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 first-order conditions. What are the demand functions for goods 1 and 2. (You need not consider second-order conditions, and you may assume the solution is interior.) [15 points] Solution The Lagrange and first order conditions are L = x 1 x 2 + x 1 + λ ( I - p 1 x 1 - p 2 x 2 ) L ∂x 1 = x 2 + 1 - λp 1 = 0 (1) L 2 = x 1 - λp 2 = 0 (2) L ∂λ = I - p 1 x 1 - p 2 x 2 = 0 . (3) From (1) and (2), we get x 2 + 1 x 1 = p 1 p 2 , or p 2 x 2 + p 2 = p 1 x 1 , and so from the budget constraint, x * 1 ( p,I ) = I + p 2 2 p 1 x * 2 ( ) = I - p 2 2 p 2 (b) Calculate the indirect utility function. What is the relationship between the indirect utility function and the expenditure function? Use this relationship to solve for the expenditure function. [10 points]
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Solution The indirect utility function is V ( p,I ) = I + p 2 2 p 1 · I - p 2 2 p 2 + I + p 2 2 p 1 = I + p 2 2 p 1 ± I - p 2 2 p 2 + 1 ² = ( I + p 2 ) 2 4 p 1 p 2 . Since V ( p,e ( p, ¯ u )) = ¯ u , ( e ( p, ¯ u ) + p 2 ) 2 4 p 1 p 2 = ¯ u = e ( p, ¯ u ) = 2 p 1 p 2 ¯ u - p 2 .
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sol_Mock_1_Midterm_101_2011 - Economics 101 Solutions to...

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