A functional equation approach to the theory of production, technical change, and invariance

Conclusion

The goal of this paper was to reconsider the properties of production and technical change functions using functional equations. More specifically, the properties of associativity and composition, and the existence of inverse and identity transformations were imposed on the technical change functions and it was learned that all of these properties could be satisfied without imposing differentiability. This is a meaningful result; there is no reason to believe that technical change occurs in a smooth, differentiable, manner.

However, when the holotheticity, or invariance, of a production function under technical change was investigated, differentiability was required in order to derive the holothetic technology, givenφ (x, t). This is certainly disappointing for those who desire to impose only the most general and least restrictive conditions, but it provides support for the claim that tractable analysis requires differentiability. Given this apparent necessity of differentiability, it must be concluded that the best method for identifying the holothetic production function, given a particular technical change functionφ, is the solution of a partial differential equation, and Sato (1980, 1981) provides this solution.