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.
For a given output
y
t
m
pi
C
bij=bji
\sumibij=1
i,j=1,..,m
i
xi
p
pi
pj
Several notable special cases can be identified:
byi=by
i
xi
y
by=0
byi=0
i
xi
p
bti=bt
i
When (HT) holds, the GO function reduces to the Generalized Leontief function of Diewert,[2] A well-known flexible functional form for cost and production functions. When (FL) hods, it reduces to a non-linear version of Leontief's model, which explains the cross-sectional variation of
xi
In economics, production technology is typically represented by the production function
f
y
m
y=f(x)
p
y
C(p,y)
The duality theorems of cost and production functions state that once a well-behaved cost function is established, one can derive the corresponding production function, and vice versa. For a given cost function
C(p,y)
f
In essence, under general conditions, a specific technology can be equally effectively represented by both cost and production functions. One advantage of using a cost function rather than a production function is that the demand functions for inputs can be easily derived from the former using Shephard's lemma, whereas this process can become cumbersome with the production function.
Commonly used forms of production functions, such as Cobb-Douglas and Constant Elasticity of Substitution (CES) functions exhibit homothticity. This property means that the production function
f
h
y=f(x)=\phi(h(x))
where
h(λx)=λh(x)
λ>0
\sigma
For a homothetic technology, the cost function can be represented as
C(p,y)=c(p)d(y)
where
d
c
i
j
| xi | = | |
| xj |
| \partialc(p)/\partialpi | |
| \partialc(p)/\partialpj |
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 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.
Furthermore, in the areas of trade, homothetic and monolithic functional models do not accurately predict results. One example is in the gravity equation for trade, or how much will two countries trade with each other based on GDP and distance. 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. Widely used examples of FFFs are the transcendental logarithmic (translog) function and the Generalized Leontief (GL) function. The translog function extends the Cobb-Douglas function to the second order, while the GL function performs a similar extension to the Leontief production function.
A drawback of the GL function is its inability to be globally concave without sacrificing flexibility in the price space. 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.[6]
Moreover, both the GO function and the underlying GL function presume immediate adjustments of inputs in response to changes in
p
y
Constant elasticity of substitution