Concavifying the Quasi-Concave

19 Pages Posted: 18 Aug 2015

See all articles by Christopher Connell

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, 2015

Abstract

This is a 1/3 length math version of a previous paper with the same title. We revisit a classic question of Fenchel from 1953: Which quasiconcave functions can be concavified by a monotonic transformation? While many authors have given partial answers under various assumptions, we offer a complete characterization for all quasiconcave functions without a priori assumptions on regularity. In particular, we show that if and only if a real-valued function f is strictly quasiconcave, (except possibly for a flat interval at its maximum) and furthermore belongs to a certain explicitly determined regularity class, there exists a strictly monotonically increasing functiong such that g\smallcircle f is strictly concave. Our primary new contribution is determining this precise minimum regularity class.

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. Under the assumption of twice differentiability, we also establish simpler sufficient conditions for concavifiability on arbitrary Riemannian manifolds, which essentially generalizes those given by Kannai in 1977 for the Euclidean case. Lastly we present the approximation result that if a function f is either weakly or strongly quasiconcave then there exists an arbitrarily close strictly concavifiable approximationh to f.

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

JEL Classification: C61

Suggested Citation

Connell, Christopher and Rasmusen, Eric Bennett, Concavifying the Quasi-Concave (August 17, 2015). Kelley School of Business Research Paper No. 15-60. Available at SSRN: https://ssrn.com/abstract=2645925 or http://dx.doi.org/10.2139/ssrn.2645925

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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HOME PAGE: http://rasmusen.org

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