Concavifying the Quasiconcave
We 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.
|Date of creation:||Aug 2012|
|Date of revision:|
|Contact details of provider:|| Postal: 1309 East Tenth Street, Room 451, Bloomington, IN 47405-1701|
Web page: http://kelley.iu.edu/bepp/
More information through EDIRC
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
- Ginsberg, William, 1973. "Concavity and quasiconcavity in economics," Journal of Economic Theory, Elsevier, vol. 6(6), pages 596-605, December.
- 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.
- Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680, December.
- Aumann, Robert J, 1975.
"Values of Markets with a Continuum of Traders,"
Econometric Society, vol. 43(4), pages 611-46, July.
- Rosa L. Matzkin & Marcel K. Richter, 1987.
"Testing Strictly Concave Rationality,"
Cowles Foundation Discussion Papers
844, Cowles Foundation for Research in Economics, Yale University.
- Yakar Kannai, 2005. "Remarks concerning concave utility functions on finite sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 333-344, 08.
When requesting a correction, please mention this item's handle: RePEc:iuk:wpaper:2012-10. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Rick Harbaugh)
If references are entirely missing, you can add them using this form.