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A production function $f$ is called quasi-sum if there are strict monotonefunctions $F, h_1,...,h_n$ 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 thatthese functions appear as solutions of the general bisymmetry equation and theyare related to the problem of consistent aggregation.  In this article, first we present the classification of quasi-sum productionfunctions satisfying the constant elasticity of substitution property. Then weprove that if a quasi-sum production function satisfies the constant elasticityof substitution property, then its graph has vanishing Gauss-Kroneckercurvature (or its graph is a flat space) if and only if the production functionis either a linearly homogeneous generalized ACMS function or a linearlyhomogeneous generalized Cobb-Douglas function.

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