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Unformatted text preview: ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Midterm 10/24/08 - Answer Key Question 1: (a)(10 points) L ( x,y, ) = xy + y + ( I- P x x- P y y ) FONCS: y = P x x + 1 = P y This implies y = P x P y ( x + 1). Substitute y into the budget constraint, we get x = 1 2 ( I P x- 1) ,y = P x 2 P y ( I P x + 1) If I P x , the solutions above from Lagrange method are the solutions we want; otherwise, from the shape of the indifference curves, we easily know that x = 0 and y = I P y . (b)(5 points) L ( x,y, ) = P x x + P y y + ( u- xy- y ) By the same FONCS as (a), we get the same relationship between y and x . Substitute y into xy + y = u , x = r P y P x u- 1 ,y = s P x P y u If P y P x u 1, then x c = x and y c = y ; otherwise, x c = 0 and y c = u . (c)(10 points)We need to show that dy c dP x = dy dP x + dy dI x First, consider the RHS of the above equation. If I P x , dy dP x = 1 2 P y = dy dI . So, RHS = 1 4 P y ( I P x + 1). By the duality between the utility maximization and cost minimization problems, I = e ( P,u ) = P x x c + P y y c ....
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This note was uploaded on 01/15/2011 for the course ECO 310 at Princeton.