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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 \(F\), \(h_{1},\dots ,h_{n}\) with \(F>0\) such that \(f(\mathbf x) = F(h_1(x_1)+ \dots +h_n(x_n))\) (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 \(F\), \(h_{1},\dots ,h_{n}\) 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
90B30 Production models

Citations:

Zbl 0858.39013
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References:

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