On the uniqueness of the solution of the cost minimization problem with generalized Sato production function

Abstract

Whenever a firm is maximizing its profit, it necessarily has to minimize its cost. Thus, the cost minimization problem is one of the central problems in the theory of the firm. When presenting this problem, the majority of microeconomic textbooks use very well-known production functions, such as Leontief, Cobb-Douglas, or other CES production functions. The goal of this paper is to analyze the cost minimization problem with the generalized Sato production function. The generalized
Sato production function is one of the non-standard production functions with variable elasticity of substitution. First, we show that the generalized Sato production function is continuous, strictly monotone, strictly quasiconcave and that a positive amount of output requires positive amounts of some of the inputs. Next, by using mathematical programming we show that the cost minimization problem with generalized Sato production function has a unique solution. This result is very important since it implies the existence of the corresponding cost function and conditional input demands.