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Lecture 25-2005

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

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Unformatted text preview: Differential Models of Production: The Single Product Firm Lecture XXV Overview of the Differential Approach Until this point we have mostly been concerned with envelopes or variations of deviations from envelopes in the case of stochastic frontier models. The production function was defined as an envelope of the maximum output level that could be obtained from a given quantity of inputs. The cost function was the minimum cost of generating a fixed bundle of outputs based on a vector of input costs. The differential approach departs from this basic formulation by examining changes in optimizing behavior. Starting from consumption theory we have We assume that consumers choose the levels of consumption so that these first- order conditions are satisfied. ( 29 ( 29 max . . i i U x U x p x s t p x Y λ ∂ ⇒ = ′ ∂ ≤ The question is then what can we learn by observing changes in these first-order conditions or changes in the optimizing behavior. ( 29 ( 29 ( 29 1 1 2 2 n n U x x p U x p x p U x x λ λ λ ∂ ∂ ∂ = ∂ ∂ ∂ M M ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 2 2 2 2 2 2 1 1 1 2 1 1 2 1 2 1 2 2 2 2 1 2 n n n n n n n n n n n n n n U x U x U x U x U x U x x x x x x x x x p x x p x x p x x p x x p x x p U x U x U x U x U x U x x x x x x x x p x x p x x p x x p x x p x x ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ L L L L L L L L ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 2 1 n n n n n n n n n n n n n n n n n x p U x U x U x U x U x U x x x x x x x x x p x x p x x p x x p x x p x x p p p p p p p p p p λ λ λ λ λ λ λ λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂...
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Lecture 25-2005 - Differential Models of Production The...

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