# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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
##### References:
 [1] Aczél, J.; Maksa, G.: Solution of the rectangular $m×$n generalized bisymmetry equation and of the problem of consistent aggregation, J. math. Anal. appl. 203, 104-126 (1996) · Zbl 0858.39013 · doi:10.1006/jmaa.1996.0369 [2] Chen, B. -Y.: On some geometric properties of h-homogeneous production function in microeconomics, Kragujevac J. Math. 35, No. 3, 343-357 (2011) [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 · doi:10.1016/j.aml.2010.12.038 [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, No. 3, 225-250 (1961) [7] Hicks, J. R.: Theory of wages, (1932) [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, No. 1, 113-125 (2010) · Zbl 1224.62143 [11] Chen, B. -Y.: Pseudo-Riemannian geometry, $\delta$-invariants and applications, (2011)