Lecture 15-2005 - An Application of Duality Lecture XV...

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Unformatted text preview: An Application of Duality Lecture XV Introduction and Setup Diewert, W. E. An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function Journal of Political Economy 79(1971): 481-507. The Shephard duality theorem states that technology may be equivalently represented by a production function, satisfying certain regularity conditions, or a cost function, satisfying certain regularity conditions. It is well known that, given fixed factor prices, and an n factor production function satisfying certain regularity conditions, we may derive a (minimum total) cost function under the assumption of minimizing behavior. What is not so well known is that, given a cost function satisfying certain regularity conditions, we may use this cost function to define a production function which in turn may be used to derive our original cost function. There are at least three ways of describing the technology of a sing output, n inputs firm: (i) by means of a production function, (ii) in terms of the firms production possibility setsand (iii) by means of the firms cost function (if the firm purchases the services of factors at fixed prices). Conditions on the Production Function Conditions on the production function f (.): f is a real valued function of n real variables for every x 0. f is finite if x is finite. f (0) = 0, and f is a nondecreasing function in x . f ( x N ) tends to plus infinity for at least one nonnegative sequence of vectors ( x N ). f is continuous form above or f is a right continuous function. Definition: A set X is convex if for every x 1 and x 2 that belongs to X and for every , 0 1, we have x 1 + (1- ) x 2 belongs to X . Definition: A real valued function f defined over a convex set X is concave if for every x 1 and x 2 belonging to X and 0 1, we have ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 1 f x x f x f x +- +- Definition: A real valued function f defined over a set X is quasiconcave if, for every real number y , the set L ( y )=[ x : f ( x )y, x belongs to X ] is a convex set. Lemma: A real valued concave function defined over a convex set X is also quasiconcave. The proof is almost by definition. If x 1 and x 2 both belong Definition: The production possibility sets (or upper contour sets) are defined for every output level y 0 by L ( y )=[ x : f ( x )y, x nonnegative]. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 2 1 1 1 by definition of , f x x f x f x y y x x L y y +- +- +- = L [ y ] 1 x 2 x Conditions on Production Possibility Sets L ( y ): L (0)= for every y >0 L ( y ) is a nonempty closed set which does not contain the origin....
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Lecture 15-2005 - An Application of Duality Lecture XV...

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