In economics the generalized-Ozaki (GO) cost function is a general description of the cost of production proposed by Shinichiro Nakamura. 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.
The GO function For a given output y {\displaystyle y} , at time t {\displaystyle t} and a vector of m {\displaystyle m} input prices p i {\displaystyle p_{i}} , the generalized-Ozaki (GO) cost function C ( ) {\displaystyle C()} is expressed as
Here, b i j = b j i {\displaystyle b_{ij}=b_{ji}} and ∑ i b i j = 1 {\displaystyle \sum _{i}b_{ij}=1} , i , j = 1 , . . , m {\displaystyle i,j=1,..,m} . By applying the Shephard's lemma, we derive the demand function for input i {\displaystyle i} , x i {\displaystyle x_{i}} :
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 p {\displaystyle p} , necessitates that its Hessian (the matrix of second partial derivatives with respect to p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} ) being negative semidefinite. Several notable special cases can be identified:
Homothticity (HT): b y i = b y {\displaystyle b_{yi}=b_{y}} for all i {\displaystyle i} . All input levels ( x i {\displaystyle x_{i}} ) scale proportionally with the overall output level ( y {\displaystyle y} ). Homogeneity of (of degree one) in output (HG): b y = 0 {\displaystyle b_{y}=0} in addition to HT. Factor limitationality (FL): b y i = 0 {\displaystyle b_{yi}=0} for all i {\displaystyle i} . None of the input levels ( x i {\displaystyle x_{i}} ) depend on p {\displaystyle p} . Neutral technical change (NT): b t i = b t {\displaystyle b_{ti}=b_{t}} for all i {\displaystyle i} . When (HT) holds, the GO function reduces to the Generalized Leontief function of Diewert, 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 x i {\displaystyle x_{i}} when variations in input prices were negligible:
Background
Cost- and production functions In economics, production technology is typically represented by the production function f {\displaystyle f} , which, in the case of a single output y {\displaystyle y} and m {\displaystyle m} inputs, is written as y = f ( x ) {\displaystyle y=f(x)} . When considering cost minimization for a given set of prices p {\displaystyle p} and y {\displaystyle y} , the corresponding cost function C ( p , y ) {\displaystyle C(p,y)} can be expressed as:
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 ) {\displaystyle C(p,y)} , the corresponding production function f {\displaystyle f} can be obtained as (a more rigorous derivation involves using a distance function instead of a production function) :
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.
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