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The Mordukhovich Normal Cone and the Foundations of Welfare Economics

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  • M. Ali Khan

Abstract

The statement that Pareto optimal allocations require the equalization of marginal rates of substitution, or in an economy with public goods, require the equalization of the aggregate of the marginal rates in consumption to those in production, is formalized through the use of the Mordukhovich normal cone. Since this cone is strictly contained, in general, in the Clarke normal cone, the results generalize earlier work of Khan and Vohra, Quinzii, Yun, and Cornet. The results are an application of Mordukhovich's 1980 theorem on necessary conditions for optimality in constrained optimization problems involving functions that are not necessarily differentiable or quasi‐concave. As such, the results suggest a distinction between the mathematical programming approach to the “second welfare theorem,” as in the work of Hicks, Lange, and Samuelson, and that based on the separation of sets, as pioneered by Arrow and Debreu.

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  • M. Ali Khan, 1999. "The Mordukhovich Normal Cone and the Foundations of Welfare Economics," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 1(3), pages 309-338, July.
    • Handle: RePEc:bla:jpbect:v:1:y:1999:i:3:p:309-338
    DOI: 10.1111/1097-3923.00014
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    Cited by:

    1. M. Ali Khan, 2007. "Perfect Competition," PIDE-Working Papers 2007:15, Pakistan Institute of Development Economics.
    2. Bonnisseau, J.-M. & Cornet, B., 2008. "Existence of equilibria with a tight marginal pricing rule," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 613-624, July.
    3. Paweł Dziewulski, 2011. "On Time-to-Build Economies with Multiple-Stage Investments," Gospodarka Narodowa. The Polish Journal of Economics, Warsaw School of Economics, issue 9, pages 23-49.
    4. Jean-Marc Bonnisseau & Bertrand Crettez, 2007. "On the Characterization of Efficient Production Vectors," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 31(2), pages 213-223, May.
    5. Jean-Marc Bonnisseau & Bernard Cornet & Marc-Olivier Czarnecki, 2007. "The marginal pricing rule revisited," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(3), pages 579-589, December.
    6. Jean-Marc Bonnisseau & Bernard Cornet, 2006. "Existence of equilibria with a tight marginal," Cahiers de la Maison des Sciences Economiques b06022, Université Panthéon-Sorbonne (Paris 1).
    7. Sjur Didrik Flåm, 2006. "Upward Slopes and Inf-Convolutions," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 188-198, February.
    8. Ceparano, Maria Carmela & Quartieri, Federico, 2019. "A second welfare theorem in a non-convex economy: The case of antichain-convexity," Journal of Mathematical Economics, Elsevier, vol. 81(C), pages 31-47.
    9. repec:hal:journl:halshs-00113332 is not listed on IDEAS
    10. Maria Carmela Ceparano & Federico Quartieri, 2020. "On Pareto Dominance in Decomposably Antichain-Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 68-85, July.
    11. Jean-Marc Bonnisseau & Bernard Cornet & Marc-Olivier Czarnecki, 2007. "The marginal pricing rule revisited," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(3), pages 579-589, December.

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