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Differential Models of Production: The Single Product Firm. Lecture XXV. Overview of the Differential Approach. Until this point we have mostly been concerned with envelopes or variations of deviations from envelopes in the case of stochastic frontier models.

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overview of the differential approach
Overview of the Differential Approach
  • Until this point we have mostly been concerned with envelopes or variations of deviations from envelopes in the case of stochastic frontier models.
    • The production function was defined as an envelope of the maximum output level that could be obtained from a given quantity of inputs.
slide3
The cost function was the minimum cost of generating a fixed bundle of outputs based on a vector of input costs.
  • The differential approach departs from this basic formulation by examining changes in optimizing behavior.
slide4
Starting from consumption theory we have
    • We assume that consumers choose the levels of consumption so that these first-order conditions are satisfied.
slide5
The question is then what can we learn by observing changes in these first-order conditions or changes in the optimizing behavior.
slide11
To finish the system, we differentiate the first-order conditions with respect to income, yielding
differential model of production
Differential Model of Production
  • Theil writes the production function in logarithmic space
    • The Cobb-Douglas function then becomes
slide27
Finally, like the demand model, we differentiate the production constraint with respect to output level and input prices.
    • Taking the differential with respect to output level
slide30
Backing up slightly, we start with

Pre-multiplying this matrix equation by F-1 yields

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