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

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Unformatted text preview: Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm Lecture XXVI Change in the Marginal Cost Shares of Marginal Cost Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price ( 29 1, i i i p q z i n C z = = L Based on this definition, we define a Firsch price index for inputs as ( 29 ( 29 ( 29 ( 29 1 ln ln n i i i d P d p = = Completing the single output model ( 29 ( 29 ln 1 ln ln C d d z d P z =- + ( 29 ( 29 ( 29 2 2 ln 1 1 1 ln ln C z z = + Multiproduct Firm Expanding the production function to a multiproduct technology ( 29 , h q z = ( 29 ( 29 1 , 1 ln m r r h q z z = = - Expanding the preceding proof Computing the first-order conditions ( 29 ( 29 1 , , n i i i L q p q h q z = =- ( 29 ( 29 ( 29 ( 29 , , 1, ln ln i i i i L q h q z p q i n q q =- = = K Now we replicate some of the steps from the preceding lecture, allowing for multiple outputs. Taking the differential of the first-order condition with respect to each output ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 ln , ln ln ln ln ln , , ln ln ln ln ln i i i r i r n j j r i r i j q h q z p q z q z q h q z h q z z q z q q = - -- = Again note by the first-order condition Thus ( 29 ( 29 , ln i i i h q z p q q = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29...
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Lecture 26-2005 - Differential Models of Production: Change...

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