Concavifying the QuasiConcave

52 Pages Posted: 9 Aug 2011 Last revised: 14 May 2014

Christopher Connell

Indiana University Bloomington - Department of Mathematics

Eric Bennett Rasmusen

Indiana University - Kelley School of Business - Department of Business Economics & Public Policy

Date Written: August 17, 2012

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.

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

JEL Classification: C02

Suggested Citation

Connell, Christopher and Rasmusen, Eric Bennett, Concavifying the QuasiConcave (August 17, 2012). Available at SSRN: https://ssrn.com/abstract=1907180 or http://dx.doi.org/10.2139/ssrn.1907180

Christopher Connell

Indiana University Bloomington - Department of Mathematics ( email )

Rawles Hall, RH 351
Bloomington, IN 47405
United States

Eric Bennett Rasmusen (Contact Author)

Indiana University - Kelley School of Business - Department of Business Economics & Public Policy ( email )

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