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Solution Midterm Thursday

Solution Midterm Thursday - ECO 310 Fall 2008 Microeconomic...

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ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach midterm 10/23 - Answer Key Question 1: (a)(10 points) L ( x, y, λ ) = ( x - x 0 )( y - y 0 ) + λ ( I - P x x - P y y ) FONCS: y - y 0 = λP x x - x 0 = λP y This implies that y - y 0 x - x 0 = P x P y . Substitute y = y 0 + P x P y ( x - x 0 ) into the budget constraint, we have x = I - P y y 0 + P x x 0 2 P x , y = I + P y y 0 - P x x 0 2 P y Since I P x x 0 + P y y 0 , x x 0 and y y 0 . (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 first order conditions. Substitute y into ( x - x 0 )( y - y 0 ) = u , we have x c = r uP y P x + x 0 , y c = y 0 + 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 0 2 P x , dx dI = 1 2 P x , so RHS= y - y 0 2 P x = I - P x y 0 - P x x 0 4 P x P y . By I = e ( P, u ) = P x x c + P y y c , we have I = P x x 0 + P y y 0 + 2 p uP x P y . Substitute I into RHS, we get RHS=LHS. (d) (10 points) To find Slutsky compensation (4 points) it’s necessary to calculate the initial consumption bundle and, then, find the income that makes this bundle available under the new prices. From part (a) we get
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