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Concavifying the Quasiconcave

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  • Christopher Connell

    (Department of Mathematics, Indiana University)

  • Eric Rasmusen

    (Department of Business Economics and Public Policy, Indiana University Kelley School of Business)

Abstract

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.

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File URL: http://www.bus.indiana.edu/riharbau/RePEc/iuk/wpaper/bepp2012-10-connell-rasmusen.pdf
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Bibliographic Info

Paper provided by Indiana University, Kelley School of Business, Department of Business Economics and Public Policy in its series Working Papers with number 2012-10.

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Date of creation: Aug 2012
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Handle: RePEc:iuk:wpaper:2012-10

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Related research

Keywords: quasiconcavity; quasiconvexity; concavity; convexity; unique maximum; maximization;

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  1. Yakar Kannai, 2005. "Remarks concerning concave utility functions on finite sets," Economic Theory, Springer, vol. 26(2), pages 333-344, 08.
  2. 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).
  3. Matzkin, Rosa L. & Richter, Marcel K., 1991. "Testing strictly concave rationality," Journal of Economic Theory, Elsevier, vol. 53(2), pages 287-303, April.
  4. 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.
  5. Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
  6. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
  7. Ginsberg, William, 1973. "Concavity and quasiconcavity in economics," Journal of Economic Theory, Elsevier, vol. 6(6), pages 596-605, December.
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Cited by:
  1. Emmanuel Farhi & Iván Werning, 2013. "Estate Taxation with Altruism Heterogeneity," American Economic Review, American Economic Association, vol. 103(3), pages 489-95, May.

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