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Lecture 14-2005 - Comparative Statics and Duality of the...

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Comparative Statics and Duality of the Cost Function Lecture XIV I. Comparative Statics A. Comparative statics with respect to changes in input prices. 1. The most common results of the comparative statics with respect to input prices involve intuition about derived demand functions. a. From the primal approach, we expect the demand functions for each input to be downward sloping with respect to input prices. b. Starting from the cost function: ( ) ( ) 2 * , , i i j j c w y x w y w w w = By Shephard’s lemma. c. In addition, we know that if i j = then by the concavity of the cost function in input prices: ( ) ( ) 2 * , , 0 i i i i c w y x w y w w w = d. In addition, we know that by Young’s theorem, the Hessian matrix for the cost function is symmetric: ( ) ( ) 2 2 , , i j j i c w y c w y w w w w = e. The Hessian matrix for the cost function is also singular. (I) Euler’s Theorem: Euler’s theorem is based on the definition of homogeneity: ( ) ( ) r f tx t f x = Differentiating both sides with respect to t and applying the chain rule yields: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 N i r i i i N r i i i f tx tx rt f x tx x f tx x rt f x tx = = = = (II) Letting 1 t = then yields ( ) ( ) 1 N i i i f x x rf x x = = (III) Coupling this result with the observation that if a function is homogeneous of degree r , then its derivative is homogeneous of degree . We know that the input demand functions are homogeneous of degree zero in prices. Thus, 1 r 1
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AEB 6184 – Production Economics Lecture XIV Professor Charles B. Moss Fall 2005 ( ) ( ) * 2 1 1 , , 0 N N i j j j j j i j x w y c w y w w w w w = = = = (IV)
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