Concavifying the Quasiconcave
AbstractWe show that if and only if a real-valued function f is strictly quasiconcave except possibly for a at interval at its maximum, and furthermore belongs to an explicitly determined regularity class, does there exist a strictly monotonically increasing function g such that g o f is strictly concave. Moreover, if and only if the function f is either weakly or strongly quasiconcave there exists an arbitrarily close approximation h to f and a monotonically increasing function g such that g o h is strictly concave. We prove this sharp characterization of quasiconcavity for continuous but possibly nondifferentiable functions whose domain is any Euclidean space or even any arbitrary geodesic metric space. While the necessity that f belong to the special regularity class is the most surprising and subtle feature of our results, it can also be difficult to verify. Therefore, we also establish a simpler sufficient condition for concaviability on Euclidean spaces and other Riemannian manifolds, which suffice for most applications.
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Bibliographic InfoPaper provided by Indiana University, Kelley School of Business, Department of Business Economics and Public Policy in its series Working Papers with number 2012-10.
Date of creation: Aug 2012
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quasiconcavity; quasiconvexity; concavity; convexity; unique maximum; maximization;
Find related papers by JEL classification:
- C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-12-15 (All new papers)
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- Richter, Marcel K. & Wong, K.-C.Kam-Chau, 2004. "Concave utility on finite sets," Journal of Economic Theory, Elsevier, vol. 115(2), pages 341-357, April.
- Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
- AUMANN, Robert J., .
"Values of markets with a continuum of traders,"
CORE Discussion Papers RP
-228, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Yakar Kannai, 2005. "Remarks concerning concave utility functions on finite sets," Economic Theory, Springer, vol. 26(2), pages 333-344, 08.
- Ginsberg, William, 1973. "Concavity and quasiconcavity in economics," Journal of Economic Theory, Elsevier, vol. 6(6), pages 596-605, December.
- Rosa L. Matzkin & Marcel K. Richter, 1987.
"Testing Strictly Concave Rationality,"
Cowles Foundation Discussion Papers
844, Cowles Foundation for Research in Economics, Yale University.
- Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
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