Analysis of Production Functions by Lie Theory of Transformation Groups: Classification of General Ces Functions
In my earlier papers [6, 7], the class of the most general types of constant-elasticity of substitution (CES) functions was derived as a general solution of a second-order partial differential equation which defines the elasticity of factor substitution. It is shown that this class of CES (nonhomothetic) functions has more meaningful economic applicability than the ordinary type of homothetic CES functions. However, one disadvantage of the general family is that it contains a large number of different types, actually an infinite number, and that the production functions are, in general, not expressible in explicit forms. Thus, those who are accustomed to thinking of the production function as an explicit relationship between inputs and outputs, may consider the general family of nonhomothetic CES functions to be very strange. Of course, this implicitness aspect presents no serious problems both in theory and in estimation, for the concept of the production function is simply the relationship, explicit or implicit, between the inputs and the maximum level of output resulting from them.
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