In economics the generalized-Ozaki (GO) cost function is a general description of the cost of production proposed by Shinichiro Nakamura.
[1] The GO cost function is notable for explicitly considering nonhomothetic technology, where the proportions of inputs can vary as the output changes.
This stands in contrast to the standard production model, which assumes homothetic technology.
By applying the Shephard's lemma, we derive the demand function for input
The GO cost function is flexible in the price space, and treats scale effects and technical change in a highly general manner.
The concavity condition which ensures that a constant function aligns with cost minimization for a specific set of
, necessitates that its Hessian (the matrix of second partial derivatives with respect to
When (FL) hods, it reduces to a non-linear version of Leontief's model, which explains the cross-sectional variation of
In essence, under general conditions, a specific technology can be equally effectively represented by both cost and production functions.
Commonly used forms of production functions, such as Cobb-Douglas and Constant Elasticity of Substitution (CES) functions exhibit homothticity.
can be represented as a positive monotone transformation of a linear-homogeneous function
For a homothetic technology, the cost function can be represented as
From Shephard's lemma, we obtain the following expression for the ratio of inputs
, which implies that for a homothetic technology, the ratio of inputs depends solely on prices and not on the scale of output.
However, empirical studies on the cross-section of establishments show that the FL model (3) effectively explains the data, particularly for heavy industries such as steel mills, paper mills, basic chemical sectors, and power stations, indicating that homotheticity may not be applicable.
[1] Furthermore, in the areas of trade, homothetic and monolithic functional models do not accurately predict results.
This led researchers to explore non-homothetic models of production, to fit with a cross section analysis of producer behavior, for example, when producers would begin to minimize costs by switching inputs or investing in increased production.
CES functions (note that Cobb-Douglas is a special case of CES) typically involve only two inputs, such as capital and labor.
While they can be extended to include more than two inputs, assuming the same degree of substitutability for all inputs may seem overly restrictive (refer to CES for further details on this topic, including the potential for accommodating diverse elasticities of substitution among inputs, although this capability is somewhat constrained).
To address this limitation, flexible functional forms have been developed.
These general functional forms are called flexible functional forms (FFFs) because they do not impose any restrictions a priori on the degree of substitutability among inputs.
These FFFs can provide a second-order approximation to any twice-differentiable function that meets the necessary regulatory conditions, including basic technological conditions and those consistent with cost minimization.
[3] Widely used examples of FFFs are the transcendental logarithmic (translog) function and the Generalized Leontief (GL) function.
A drawback of the GL function is its inability to be globally concave without sacrificing flexibility in the price space.
[2] This limitation also applies to the GO function, as it is a non-homothetic extension of the GL.
In a subsequent study,[5] Nakamura attempted to address this issue by employing the Generalized McFadden function.
For further advancements in this area, refer to Ryan and Wales.
This oversimplifies the reality where technological changes entail significant investments in plant and equipment, thus requiring time, often occurring over years rather than instantaneously.
One way to address this issue will be to resort to a variable cost function that explicitly takes into account differences in the speed of adjustments among inputs.
Production function List of production functions Constant elasticity of substitution Shephard's lemma Returns to scale