Lecture 15-2005

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

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An Application of Duality Lecture XV I. 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. A. “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.” 1. “It is well known … that, given fixed factor prices, and an factor production function satisfying certain regularity conditions, we may derive a (minimum total) cost function under the assumption of minimizing behavior. n 2. “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.” B. “There are at least three ways of describing the technology of a sing output, inputs firm: (i) by means of a production function, (ii) in terms of the firm’s production possibility sets…and (iii) by means of the firm’s cost function (if the firm purchases the services of factors at fixed prices).” n II. Conditions on the Production Function A. Conditions on the production function ( ) . f : 1. f is a real valued function of real variables for every . n 0 x f is finite if x is finite. 2. ( ) 0 f = 0 , and f is a nondecreasing function in x . 3. ( ) n f x tends to plus infinity for at least one nonnegative sequence of vectors () n x . 4. f is continuous form above or f is a right continuous function. 5. f is quasiconcave over Ω . B. Definition: A set X is convex if for every 1 x and 2 x that belongs to X and for every , , we have λ 0 ≤λ≤ 1 ( ) 1 1 2 x x λ+− λ belongs to X . C. Definition: A real valued function f defined over a convex set X is concave if for every 1 x and 2 x belonging to X and 01 , we have ( ) ( ) ( ) ( ) 12 1 11 2 f xx f x f x λλ λ ⎡⎤ +− ⎣⎦ 1

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AEB 6184 – Production Economics Lecture XV Professor Charles Moss Fall 2005 D. Definition: A real valued function f defined over a set X is quasiconcave if, for every real number , the set y ( ) ( ) :, L yx f x y x X =≥ is a convex set. L [ y ] 1 x 2 x E. Lemma: A real valued concave function defined over a convex set X is also quasiconcave. The proof is almost by definition. If 1 x and 2 x both belong to a level set , then () Ly ( ) ( ) ( ) ( ) () ( 12 1 2 11 1 by definition of , fx x f x f x yy x x L y λλ λ ⎡⎤ +− ⎣⎦ ≥+ = ) y 2
AEB 6184 – Production Economics Lecture XV Professor Charles Moss Fall 2005 F. Definition: The production possibility sets (or upper contour sets) are defined for every output level by 0 y ( ) ( ) : , nonnegative Ly x f x yx =≥ . III. Conditions on the Production Possibilities Sets A. Conditions on Production Possibility Sets ( ) : 1.

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## This note was uploaded on 07/15/2011 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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

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