Lecture 25-2005

# Lecture 25-2005 - Differential Models of Production The...

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Differential Models of Production: The Single Product Firm Lecture XXV I. Overview of Differential Approach A. Until this point we have mostly been concerned with envelopes or variations of deviations from envelopes in the case of stochastic frontier models. 1. The production function was defined as an envelope of the maximum output level that could be obtained from a given quantity of inputs. 2. The cost function was the minimum cost of generating a fixed bundle of outputs based on a vector of input costs. B. The differential approach departs from this basic formulation by examining changes in optimizing behavior. 1. Starting from consumption theory we have ( ) ( ) max .. i i Ux p x st px Y λ ⇒= a. We assume that consumers choose the levels of consumption so that these first-order conditions are satisfied. b. The question is then what can we learn by observing changes in these first-order conditions or changes in the optimizing behavior. 2. First note that there are first-order conditions n ( ) () 1 1 2 2 n n x p p x p x ⎡⎤ ⎢⎥ = ⎣⎦ M M 1

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AEB 6184 – Production Economics Lecture XXV Professor Charles Moss Fall 2005 3. Differentiating these n first-order conditions with respect to each price yields () ( ) ( ) ( ) 22 2 2 11 1 1 1 1 2 1 2 1 1 21 1 2 1 2 2 2 2 nn n n Ux xx x xx p x ∂∂ ++ + ∂∂ ∂ + LL L L L L n n n n x 1 1 12 2 1 1 1 00 n n n n n n n n x p x pp p n x p p λ λλ + ⎡⎤ ⎢⎥ =+ ⎣⎦ MM O M L L L L L MMOM L 2 2 n n n p p p p p L O M L a. In matrix space, this equation becomes 2 1 2 2 2 1 2 n n n n n n n UU U n x x x x U U x x x x U x x x x x p x x p p x p =∂∂ = L L O M L L L O M L b. Thus, the matrix mess becomes x UI p p p . 2
AEB 6184 – Production Economics Lecture XXV Professor Charles Moss Fall 2005 4. Differentiating the income constraint yields [] 1 2 12 11 1 22 2 n n n n nn n n x Y x x pp p p Y Y x Y xx x p x p p x x x p x px p ⎡⎤ ⎢⎥ =⇒ = ⎣⎦ ∂∂ = ′′ ⇒= L M L L LL MM O M L x 5. To finish the system, we differentiate the first-order conditions with respect to income, yielding 2

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Lecture 25-2005 - Differential Models of Production The...

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