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

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

Completing the single output model

Multiproduct Firm Expanding the production function to a multiproduct technology

Expanding the preceding proof  Computing the first-order conditions

 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

Again note by the first-order condition Thus

With

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

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

Multiplying through by γ yields Using the tired first-order conditions

With

Differentiating the production function with respect to yields

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

 Differentiating the production function with respect to ln(z’)  Differentiating the production function with respect to ln(p’)

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

 Note by the results from Barten’s fundamental matrix

 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

 Defining the matrix

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

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

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