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 γ yields 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’)

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

21.  Note by the results from Barten’s fundamental matrix

22.  θ i r is the share of the i th input in the marginal cost of the r th product.  Summing this marginal cost over all inputs

23.  Defining the matrix

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

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

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