Question1 23-10

Question1 23-10 - ECO 310 - Fall 2008 Microeconomic Theory...

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Unformatted text preview: ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach midterm 10/23 - Answer Key Question 1: (a)(10 points) L ( x;y;& ) = ( x & x )( y & y ) + & ( I & P x x & P y y ) FONCS: y & y = &P x x & x = &P y This implies that y & y x & x = P x P y . Substitute y = y + P x P y ( x & x ) into the budget constraint, we have x = I & P y y + P x x 2 P x ;y = I + P y y & P x x 2 P y Since I P x x + P y y , x x and y y . (b)(5 points)By the duality between the utility maximization and cost minimization problems, we get the same relationship between y and x from the &rst order conditions. Substitute y into ( x & x )( y & y ) = u , we have x c = r uP y P x + x ;y c = y + s uP x P y (c)(10 points)We need to verify the equation dx c dP y = dx dP y + y dx dI dx c dP y = 1 2 q u P x P y , dx dP y = & y 2 P x , dx dI = 1 2 P x , so RHS = y & y 2 P x = I & P x y & P x x 4 P x P y . By I = e ( P;u ) = P x x c + P y y c , we have I = P x x + P y y + 2 p uP x P y . Substitute I into RHS, we get RHS...
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