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On some geometric properties of quasi-sum production models. (English) Zbl 1243.39019
Summary: A production function $f$ is called quasi-sum if there are continuous strict monotone functions $F$, ${h}_{1},\cdots ,{h}_{n}$ with $F>0$ such that $f\left(𝐱\right)=F\left({h}_{1}\left({x}_{1}\right)+\cdots +{h}_{n}\left({x}_{n}\right)\right)$ (cf. J. Aczél and G. Maksa [J. Math. Anal. Appl. 203, No. 1, 104–126 (1996; Zbl 0858.39013)]). A quasi-sum production function is called quasi-linear if at most one of $F$, ${h}_{1},\cdots ,{h}_{n}$ is a nonlinear function. For a production function $f$, the graph of $f$ is called the production hypersurface of $f$. In this paper, we obtain a very simple necessary and sufficient condition for a quasi-sum production function $f$ to be quasi-linear in terms of the graph of $f$. Moreover, we completely classify quasi-sum production functions whose production hypersurfaces have vanishing Gauss-Kronecker curvature.
##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 39B22 Functional equations for real functions 91B38 Production theory, theory of the firm (economics) 90B30 Production models