Lecture 14 - Lecture 14: Cost Functions 1 Contingent Demand...

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1 Lecture 14: Cost Functions 1 Contingent Demand for Inputs z Contingent demand functions for all of the firms inputs can be derived from the cost function 2 Shephard’s lemma z the contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that input’s price Contingent Demand for Inputs Shephard’s lemma: q w v L q w v C q w v k c = = ) , , , ( ) , , ( ) ( λ 3 Intuition: If price of labor (w) increases by 1, by how much will cost increase if nothing else changes? by l (the number of workers) v v , , w q w v L w q w v C q w v l c = = ) , , , ( ) , , ( ) , , (
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2 Contingent Demand for Inputs z Example: Production of automobiles s = A(r l ) 0.5 4 The Lagrangian expression for cost minimization of producing s is L = vr + w l + λ [ s - A(r l ) 0.5 ] The first-order conditions for a minimum are L / r = v - λ 0.5 A r -0.5 l 0.5 = 0 L / l = w - λ 0.5 A r 0.5 l -0.5 = 0 L / ∂λ = s - A(r l ) 0.5 = 0 Contingent Demand for Inputs z Dividing the first equation by the second gives us r w w 5 l v = Hence s = A (w/v) 0.5 l l= s / A (w/v) 0.5 = s A -1 (v/w) 0.5 r= s / A (v/w) 0.5 =
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Lecture 14 - Lecture 14: Cost Functions 1 Contingent Demand...

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