Geometry of quasi-sum production functions with constant elasticity of substitution property

TL;DR

This paper classifies quasi-sum production functions with constant elasticity of substitution, showing their graphs are flat spaces only if they are specific homogeneous functions like generalized ACMS or Cobb-Douglas.

Contribution

It provides a complete classification of quasi-sum production functions with CES property and characterizes when their graphs are flat spaces.

Findings

Graphs have vanishing Gauss-Kronecker curvature only for specific homogeneous functions.

Classifies quasi-sum functions satisfying CES property.

Links geometric properties to economic production functions.

Abstract

A production function $f$ is called quasi-sum if there are strict monotone functions $F, h_1,...,h_n$ with $F'>0$ such that $$f(x)= F(h_1 (x_1)+...+h_n (x_n)).$$ The justification for studying quasi-sum production functions is that these functions appear as solutions of the general bisymmetry equation and they are related to the problem of consistent aggregation. In this article, first we present the classification of quasi-sum production functions satisfying the constant elasticity of substitution property. Then we prove that if a quasi-sum production function satisfies the constant elasticity of substitution property, then its graph has vanishing Gauss-Kronecker curvature (or its graph is a flat space) if and only if the production function is either a linearly homogeneous generalized ACMS function or a linearly homogeneous generalized Cobb-Douglas function.