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On the origin of r-concavity and related concepts

Author

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  • Tamás L. Balogh
  • Christian Ewerhart

Abstract

In a less widely known contribution, Béla Martos (1966, Hungarian Academy of Sciences) introduced a generalized notion of concavity that is closely related to what is nowadays known as r-concavity in the operations research literature, and that is identical to what is nowadays known as ρ-concavity in the economics literature. The present paper aims at making the original contribution accessible to a wider audience and illustrating its importance from a modern perspective. To this end, we offer a translation of those parts of Martos (1966) that are directly related to generalized concavity. Reviewing the virtues of r-concavity and ρ-concavity, we find a surprisingly short proof of the univariate Prékopa-Borell theorem. We also survey a number of applications of the considered concepts in operations research and economics.

Suggested Citation

  • Tamás L. Balogh & Christian Ewerhart, 2015. "On the origin of r-concavity and related concepts," ECON - Working Papers 187, Department of Economics - University of Zurich.
  • Handle: RePEc:zur:econwp:187
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    References listed on IDEAS

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    1. Zahra Mashayekhi Zadeh & Esmaile Khorram, 2012. "Convexity of chance constrained programming problems with respect to a new generalized concavity notion," Annals of Operations Research, Springer, vol. 196(1), pages 651-662, July.
    2. Anderson, Simon P. & Renault, Regis, 2003. "Efficiency and surplus bounds in Cournot competition," Journal of Economic Theory, Elsevier, vol. 113(2), pages 253-264, December.
    3. Caplin, Andrew & Nalebuff, Barry, 1991. "Aggregation and Social Choice: A Mean Voter Theorem," Econometrica, Econometric Society, vol. 59(1), pages 1-23, January.
    4. David P. Myatt & Chris Wallace, 2005. "Production Targets and Free Disposal in the Private Provision of Public Goods," Economics Series Working Papers 231, University of Oxford, Department of Economics.
    5. Y. X. Zhao & S. Y. Wang & L. Coladas Uria, 2010. "Characterizations of r-Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 186-195, April.
    6. Gupta, Somesh Das, 1980. "Brunn-Minkowski inequality and its aftermath," Journal of Multivariate Analysis, Elsevier, vol. 10(3), pages 296-318, September.
    7. Moyes, Patrick, 2003. "Redistributive effects of minimal equal sacrifice taxation," Journal of Economic Theory, Elsevier, vol. 108(1), pages 111-140, January.
    8. Simon Cowan, 2012. "Third-Degree Price Discrimination and Consumer Surplus," Journal of Industrial Economics, Wiley Blackwell, vol. 60(2), pages 333-345, June.
    9. Iñaki Aguirre & Simon Cowan & John Vickers, 2010. "Monopoly Price Discrimination and Demand Curvature," American Economic Review, American Economic Association, vol. 100(4), pages 1601-1615, September.
    10. Caplin, Andrew & Nalebuff, Barry, 1991. "Aggregation and Imperfect Competition: On the Existence of Equilibrium," Econometrica, Econometric Society, vol. 59(1), pages 25-59, January.
    11. Simon Cowan, 2007. "The welfare effects of third-degree price discrimination with nonlinear demand functions," RAND Journal of Economics, RAND Corporation, vol. 38(2), pages 419-428, June.
    12. Horst, R., 1984. "On the convexification of nonlinear programming problems: An applications-oriented survey," European Journal of Operational Research, Elsevier, vol. 15(3), pages 382-392, March.
    13. Ewerhart, Christian, 2014. "Cournot games with biconcave demand," Games and Economic Behavior, Elsevier, vol. 85(C), pages 37-47.
    14. AVRIEL, Mordecai, 1973. "Solution of certain nonlinear programs involving r-convex functions," LIDAM Reprints CORE 129, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. DavidP. Myatt & Chris Wallace, 2009. "Evolution, Teamwork and Collective Action: Production Targets in the Private Provision of Public Goods," Economic Journal, Royal Economic Society, vol. 119(534), pages 61-90, January.
    16. An, Mark Yuying, 1998. "Logconcavity versus Logconvexity: A Complete Characterization," Journal of Economic Theory, Elsevier, vol. 80(2), pages 350-369, June.
    17. Harvey J. Greenberg & William P. Pierskalla, 1971. "A Review of Quasi-Convex Functions," Operations Research, INFORMS, vol. 19(7), pages 1553-1570, December.
    18. Hirofumi Uzawa, 1962. "Production Functions with Constant Elasticities of Substitution," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 29(4), pages 291-299.
    19. Christian Ewerhart, 2013. "Regular type distributions in mechanism design and $$\rho $$ -concavity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(3), pages 591-603, August.
    20. Mares, Vlad & Swinkels, Jeroen M., 2011. "Near-optimality of second price mechanisms in a class of asymmetric auctions," Games and Economic Behavior, Elsevier, vol. 72(1), pages 218-241, May.
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    More about this item

    Keywords

    Generalized concavity; r-concavity; ρ-concavity; nonlinear optimization; economic applications;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • D0 - Microeconomics - - General
    • Y8 - Miscellaneous Categories - - Related Disciplines

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