## 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

Zbl 0858.39013
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### References:

 [1] Aczél, J.; Maksa, G., Solution of the rectangular $$m \times n$$ generalized bisymmetry equation and of the problem of consistent aggregation, J. math. anal. appl., 203, 104-126, (1996) · Zbl 0858.39013 [2] Chen, B.-Y., On some geometric properties of $$h$$-homogeneous production function in microeconomics, Kragujevac J. math., 35, 3, 343-357, (2011) · Zbl 1289.91107 [3] Vilcu, A.D.; Vilcu, G.E., On some geometric properties of the generalized CES production functions, Appl. math. comput., 218, 124-129, (2011) · Zbl 1231.91278 [4] Vilcu, G.E., A geometric perspective on the generalized cobb – douglas production functions, Appl. math. lett., 24, 777-783, (2011) · Zbl 1208.91076 [5] Cobb, C.W.; Douglas, P.H., A theory of production, Amer. econom. rev., 18, 139-165, (1928) [6] Arrow, K.J.; Chenery, H.B.; Minhas, B.S.; Solow, R.M., Capital-labor substitution and economic efficiency, Rev. econom. stat., 43, 3, 225-250, (1961) [7] Hicks, J.R., Theory of wages, (1932), Macmillan London [8] Allen, R.G.; Hicks, J.R., A reconsideration of the theory of value, pt. II, Economica, 1, 196-219, (1934) [9] B.-Y. Chen, Classification of $$h$$-homogeneous production functions with constant elasticity of substitution, Tamkang J. Math. (in press). [10] Losonczi, L., Production functions having the CES property, Acta math. acad. paedagog. nyházi. (N.S.), 26, 1, 113-125, (2010) · Zbl 1224.62143 [11] Chen, B.-Y., Pseudo-Riemannian geometry, $$\delta$$-invariants and applications, (2011), World Scientific Hackensack, NJ · Zbl 1245.53001
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