Differential models of production change in the marginal cost and the multi product firm
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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

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

Lecture XXVI


Change in the marginal cost
Change in the Marginal Cost Cost and the Multi-Product Firm

  • 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




Multiproduct firm
Multiproduct Firm inputs as

  • Expanding the production function to a multiproduct technology


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


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

  • With lecture, allowing for multiple outputs.



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

  • Multiplying through by differentiating the production function by a vector of output levels, holding prices and other outputs constant yieldsγyields

  • Using the tired first-order conditions


Differential models of production change in the marginal cost and the multi product firm

  • With differentiating the production function by a vector of output levels, holding prices and other outputs constant yields



Differential models of production change in the marginal cost and the multi product firm

  • Collecting these equations: yields

    • Differentiating the first-order conditions with respect to ln(z’)

    • Differentiating the first-order conditions with respect to ln(p’)


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



Differential models of production change in the marginal cost and the multi product firm

  • θ yieldsir is the share of the ith input in the marginal cost of the rth product.

  • Summing this marginal cost over all inputs



Introduction of quasi fixed variables
Introduction of Quasi-Fixed Variables yields

  • Expanding the differential model further, we introduce quasi-fixed variables into the production set



Differential models of production change in the marginal cost and the multi product firm