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

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.