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A simple proof of the nonconcavifiability of functions with linear not-all-parallel contour sets

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  • Reny, Philip J.

Abstract

Consider a real-valued function that, on a convex subset of a real vector space, is continuous on line segments and has convex contour sets. Inspired by a compelling intuitive argument due to Aumann (1975), we provide a simple proof that no strictly increasing transformation of such a function can be concave unless all contour sets are parallel, i.e., unless for every pair of contour sets, either their affine hulls are disjoint or one of their affine hulls contains the other.

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  • Reny, Philip J., 2013. "A simple proof of the nonconcavifiability of functions with linear not-all-parallel contour sets," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 506-508.
    • Handle: RePEc:eee:mateco:v:49:y:2013:i:6:p:506-508
    DOI: 10.1016/j.jmateco.2013.10.006
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    References listed on IDEAS

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    1. Aumann, Robert J, 1975. "Values of Markets with a Continuum of Traders," Econometrica, Econometric Society, vol. 43(4), pages 611-646, July.
    2. Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
    3. Monteiro, Paulo Klinger, 2010. "A Class of Convex Preferences Without Concave Representation," Revista Brasileira de Economia - RBE, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil), vol. 64(1), March.
    4. Simon Grant & Atsushi Kajii & Ben Polak, 2000. "Temporal Resolution of Uncertainty and Recursive Non-Expected Utility Models," Econometrica, Econometric Society, vol. 68(2), pages 425-434, March.
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