Lecture 26-2005 - Differential Models of Production: Change...

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm Lecture XXVI I. Change in the Marginal Cost A. Shares of Marginal Cost 1. Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price ( ) 1, ii i pq z in C z θ == L . 2. Based on this definition, we define a Firsch price index for inputs as () 1 ln ln n i dP dp = = B. Completing the single output model () ( ) ln 1 ln ln C dd z z γ ψ ⎛⎞ ⎛∂ d P =− + ⎜⎟ ⎝⎠ ( ) () () 2 2 ln 11 1 ln ln C z z ψγ =+ ∂∂ II. Multiproduct Firm A. Expanding the production function to a multiproduct technology ( ) ,0 hqz = ( ) 1 , 1 ln m r r z = = − B. Expanding the preceding proof ( 1 ,, n i ) L qp q h q ρρ = z 1. Computing the first-order conditions ( ) 01 , ln ln Lq i n qq ρ = = K which implies that the optimum values of each input can be formulated as a function of the output level and input prices: ( ) ,1 , z p i K n p and the optimal cost function: 1 n i Cp q z p C z = 1
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AEB 6184 – Production Economics Lecture XXVI Professor Charles Moss Fall 2005 2. Now we replicate some of the steps from the preceding lecture, allowing for multiple outputs a. Taking the differential of the first-order condition with respect to each output () () () ( ) 22 1 ln ln , ln , , 0 ln ln ln ln ln ln ln ln n j i ii j ri r r ij q qh q z h q z h q z pq zq z z qq ρ ρρ = ∂∂ −− = (i) Again note by the first-order condition ( ) , ln i hqz q = (ii) Thus ( ) 1 ln ln ln , , 0 ln ln ln ln ln ln ln n j i j rr r i q q z zz z q z = r h q z = (iii) With i f pq C = and ( ) ( ) * ln ln ln ln q F HF H γ ιγ ′′ = (iv) This result is similar to the result presented in the last
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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 26-2005 - Differential Models of Production: Change...

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