# Other supply specifications

#### IV.II.2 Other supply specifications

In econometric practice many specifications are used. Here some frequently used models are shortly described as illustrations.

A simple production function may be formulated by the Cobb-Douglas specification

(IV.II.2-1)

where Y denotes output, K is capital, and L is labor. Eq. (IV.II.2-1) can be written in log-linear form as

(IV.II.2-2)

(IV.II.2-3)

This specification can be related to input demand systems by the use elasticities of substitution between input factors defined as

(IV.II.2-4)

(IV.II.2-5)

From this result it also possible to deduce some simple factor demand equations

(IV.II.2-6)

and

(IV.II.2-7)

Now consider the generalized Leontief cost function which states formally that

(IV.II.2-8)

(IV.II.2-9)

(IV.II.2-10)

where dii is obviously a constant term in the equations.

The general Hicks-Allen partial elasticity of substitution is defined by

(IV.II.2-11)

where C is a cost function.

Now (IV.II.2-11) can be applied to the generalized Leontief model which can be shown to yield

(IV.II.2-12)

and

(IV.II.2-13)

Even cross-price elasticities can be obtained for the generalized Leontief cost function by

(IV.II.2-14)

and

(IV.II.2-15)

where

(IV.II.2-16)

A general form of production functions which are nonhomothetic, is defined by the nonhomothetic translog cost function

(IV.II.2-17)

(IV.II.2-18)

which are the so-called homogeneity conditions.

The nonhomothetic translog model can be transformed into a homothetic translog equation if the following restrictions are imposed

(IV.II.2-19)

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