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

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

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• Completing the single output model

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Multiproduct Firm

• Expanding the production function to a
  multiproduct technology

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

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• 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

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• Again note by the first-order condition
• Thus

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• With

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

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

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• Multiplying through by ? yields
• Using the tired first-order conditions

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• With

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• Differentiating the production function with
  respect to yields

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

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

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

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• Note by the results from Bartens fundamental
  matrix

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

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• Defining the matrix

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Introduction of Quasi-Fixed Variables

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

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

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