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Lecture 02-2005

# Lecture 02-2005 - Definition and Properties of the...

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1 Definition and Properties of the Production Function: Lecture II I. Overview of the Production Function – Chambers A. “The production function (and indeed all representations of technology) is a purely technical relationship that is void of economic content. Since economists are usually interested in studying economic phenomena, the technical aspects of production are interesting to economists only insofar as they impinge upon the behavior of economic agents.” (Chambers p. 7). B. “Because the economist has no inherent interest in the production function, if it is possible to portray and to predict economic behavior accurately without direct examination of the production function, so much the better. This principle, which sets the tone for much of the following discussion, underlies the intense interest that recent developments in duality have aroused.” (Chambers p. 7). 1. The point of these two statements is that economists are not engineers and have no insights into why technologies take on any particular shape. We are only interested in those properties that make the production function useful in economic analysis, or those properties that make the system solvable. 2. There are several interpretations of the dual. Let use briefly discuss one concept. Assume that we are interested in analyzing production of some crop (say cotton). a. One approach would be to estimate a production function, say a Cobb-Douglas production function in two relevant inputs: 12 yx x α β = b. Given this production function, we could derive a cost function by minimizing the cost of the two inputs subject to some level of production: 11 2 2 , min .. xx wx st y x x αβ + = Forming the Lagrangian of this optimization problem, we have ( ) 1 2 1 2 22 0 0 0 Lw x w x x L w L w L x λ =+ +− =− = = = Taking the first two first-order conditions together we have

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AEB 6184 – Production Economics Lecture II Professor Charles B. Moss Fall 2005 2 11 2 2 12 21 1 2 L xw x w x x L wx w x ⇒= = Substituting this relationship into the final first-order condition yields () 1 * 22 2 1 2 0, , ww L yx x x w w y y α β αβ λ + +  ⇒− =⇒ =   By substituting this relationship back into the previous condition with respect that solves 1 x as a function of 2 x , we have 1 * 2 112 1 ,, w xwwy y w + + = Substituting both of these optimal relationships (output conditional input demand curves) back into the cost function yields 1 2 1 1 2 Cww y w y w y y w w ++ + + +   =+  c. Thus, in the end, we are left with a cost function that relates input prices and output levels to the cost of production based on the economic assumption of optimizing behavior.
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Lecture 02-2005 - Definition and Properties of the...

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