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econ483lec03product

# 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: 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 [email protected] 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 is the °rst partial derivative of the production function w.r.t. 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 p i b x i cost minimization: Cobb-Douglas minimize P p i x i subject to y = Ax ° 1 1 x ° 2 2 x ° 3 3 min x i L = P p i x i + ± [ y ± Ax ° 1 1 x ° 2 2 x ° 3 3 ] , where ± is the Lagrange multiplier FOC: (1) @ L [email protected] 1 = p 1 ± ° 1 ±Ax ° 1 ° 1 1 x ° 2 2 x ° 3 3 = p 1 ± ° 1 ±x ° 1 1 y = 0 (2) @ L [email protected] 2 = p 2
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econ483lec03product - 1 Cost Learning Curves and Scale...

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