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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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.

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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  1. Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm Lecture XXVI

  2. 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

  3. Based on this definition, we define a Firsch price index for inputs as

  4. Completing the single output model

  5. Multiproduct Firm • Expanding the production function to a multiproduct technology

  6. Expanding the preceding proof • Computing the first-order conditions

  7. 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

  8. Again note by the first-order condition • Thus

  9. With

  10. Differentiating with respect to the input prices yields the same result as before

  11. Slightly changing the preceding derivation by differentiating the production function by a vector of output levels, holding prices and other outputs constant yields

  12. Multiplying through by γyields • Using the tired first-order conditions

  13. With

  14. Differentiating the production function with respect to yields

  15. Collecting these equations: • Differentiating the first-order conditions with respect to ln(z’) • Differentiating the first-order conditions with respect to ln(p’)

  16. Differentiating the production function with respect to ln(z’) • Differentiating the production function with respect to ln(p’)

  17. The extended form of the differential supply system is then. • Starting with the total derivative of ln(q) • Premultiplying by F

  18. Note by the results from Barten’s fundamental matrix

  19. θir is the share of the ith input in the marginal cost of the rth product. • Summing this marginal cost over all inputs

  20. Defining the matrix

  21. Introduction of Quasi-Fixed Variables • Expanding the differential model further, we introduce quasi-fixed variables into the production set

  22. Following Livanis and Moss, the differential supply function for this specification becomes

  23. Starting with the input demand system, we add a random disturbance relying on the theory of rational random behavior (RRB, Theil 1975):

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