econ483lec03product - 1 Cost, Learning Curves, and Scale...

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1 Cost, Learning Curves, and Scale Economies Berndt Chapter 3 1.1 Economic Theory of Cost and Production the relationship among inputs and output is summarized by a production function y = f ( x 1 ;:::;x n ; A ) y : output f ( ) : production function x i ;i = 1 ;:::n : input variables (capital, labor, . ..) A : technology Cobb-Douglas production function with 3 input variables y = f ( x 1 ;x 2 ;x 3 ; A ) = Ax 1 1 x 2 2 x 3 3 , where 1 2 3 are unknown parameters to be estimated returns to scale: what is the increase in output if all inputs are increased by 100%? Cobb-Douglas: r = 1 + 2 + 3 economies of scale: returns to scale minus 1 Cobb-Douglas: r ± 1 = 1 + 2 + 3 ± 1 cost minimizing behavior a cost function: the minimum possible total cost of producing a given level of output to the prices of the n inputs, the level of output y , and the state of technology A C = C ( p 1 ;:::;p n ;y ; A ) , where p 1 ;:::;p n are input prices of x 1 ;:::;x n total cost: C = C ( p 1 ;:::;p n ;y ; A ) average cost: C=y marginal cost: @C=@y cost minimization minimize P p i x i subject to y = f ( x 1 ;:::;x n ; A ) min x i L = P p i x i + ± [ y ± f ( x 1 ;:::;x n ; A )] , where ± is the Lagrange multiplier FOC: @ L =@x i = p i ± ±f i ( b x 1 ;:::; b x n ; A ) = 0 , for i = 1 ;:::;n @ L =@± = y ± f ( b x 1 ;:::; b x n ; A ) = 0 , where f i i -th input 1
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so we get p i =p n = f i ( b x 1 ;:::; b x n ; A ) =f n ( b x 1 ;:::; b x n ; A ) , for i = 1 ;:::;n 1 then solve for each b x i and substitute it into the expression of total cost C = P
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This note was uploaded on 06/05/2008 for the course ECON 483 taught by Professor Seikkim during the Spring '08 term at University of Washington.

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econ483lec03product - 1 Cost, Learning Curves, and Scale...

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