problem-set-4-solutions

problem-set-4-solutions - Ecn 100 - Intermediate...

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Ecn 100 - Intermediate Microeconomic Theory University of California - Davis November 3, 2010 John Parman Problem Set 4 - Solutions This problem set will be not be graded and does not need to be turned in. It will help you prepare for the Chapter 18 material that will be on the second midterm. It is highly recommended that you also use the old midterms on Smartsite to get additional practice with the new material. 1. Production Functions For the first two production functions given below, derive expressions for the marginal product of x , the marginal product of y and the technical rate of substitution, graph three isoquants and determine whether the technology exhibits increasing, constant or decreasing returns to scale. For the last production function, you only need to graph the isoquants and determine whether the technology exhibits increasing, constant or decreasing returns to scale. (a) f ( x,y ) = 5 x 3 4 y 3 4 (b) f ( x,y ) = 2 x + 3 y 1 2 (c) f ( x,y ) = min ( x, 3 y ) For production function (a): MP x = df ( x,y ) dx = 5 3 4 x - 1 4 y 3 4 MP y = df ( x,y ) dy = 5 3 4 x 3 4 y - 1 4 TRS = - MP x MP y = 5 3 4 x - 1 4 y 3 4 5 3 4 x 3 4 y - 1 4 = - y x f ( λx,λy ) = 5 ( λx ) 3 4 ( λy ) 3 4 = λ 3 4 λ 3 4 5 x 3 4 y 3 4 f ( λx,λy ) = λ 6 4 f ( x,y ) From the above algebra, we can see that increasing inputs by a factor of λ will increase output by a factor greater than λ , so this production function exhibits increasing returns to scale. As for the isoquants, they will be standard convex isoquants. We know this because the marginal products of both inputs are positive, guaranteeing downward sloping isoquants, and the technical rate of substitution gets smaller in magnitude as we
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2 Problem Set 4 - Solutions move down and to the right, telling us that the isoquants get flatter as we move to the right. For production function (b):
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problem-set-4-solutions - Ecn 100 - Intermediate...

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