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Differential Models of Production: The Single Product FirmPowerPoint Presentation

Differential Models of Production: The Single Product Firm

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The differential approach departs from this basic formulation by examining changes in optimizing behavior.

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.

- The cost function was the minimum cost of generating a fixed bundle of outputs based on a vector of input costs.

- Starting from consumption theory we have bundle of outputs based on a vector of input costs.
- We assume that consumers choose the levels of consumption so that these first-order conditions are satisfied.

- The question is then what can we learn by observing changes in these first-order conditions or changes in the optimizing behavior.

- Differentiating the income constraint yields in these first-order conditions or changes in the optimizing behavior.

- To finish the system, we differentiate the first-order conditions with respect to income, yielding

- Putting each of the bits into order, we have Barten’s fundamental matrix equation:

- Solving Barten’s fundamental matrix equation yields fundamental matrix equation:

Differential Model of Production fundamental matrix equation:

- Theil writes the production function in logarithmic space
- The Cobb-Douglas function then becomes

- The Lagrange formulation for the logarithmic production function becomes

- As in the differential demand model, everything has to end up as a share equation, therefore

- Returning to the earlier expression up as a share equation, therefore

- A slightly more ominous form of this expression is then up as a share equation, therefore

- Logarithmically differentiating with respect to the output level, ln(z) , yields

- Given level, ln(

- Which becomes yields

- 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

- Taking the differential with respect to input the natural logarithm of input prices yields

- Putting these bits together logarithm of input prices yields

- Backing up slightly, we start with logarithm of input prices yields
Pre-multiplying this matrix equation by F-1 yields

- Next, multiplying the first term by a special form of the identity matrix F-1F = I yields

- Using a similar process identity matrix

- The matrix-Barten’s equation then becomes identity matrix

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