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Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies

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  • Jiuqiang Liu

    (College of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
    Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA)

Abstract

For an exchange economy with a continuum of traders and a finite-dimensional commodity space under some standard assumptions, Aumann showed that the core and the set of competitive allocations (two most important solutions) coincide and Mas-Colell proved that the bargaining set and the set of competitive allocations coincide. However, in the case of exchange economies with a finite number of traders, it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we establish the equivalence of the fuzzy core, the fuzzy bargaining set (or Aubin bargaining set), and the set of competitive allocations in a finite coalition production economy with an infinite-dimensional commodity space under some standard assumptions. We first derive a continuous equivalence theorem and then discretize it to obtain the desired equivalence in finite economies.

Suggested Citation

  • Jiuqiang Liu, 2022. "Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3379-:d:917381
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    References listed on IDEAS

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    1. Ezra Einy & Diego Moreno & Benyamin Shitovitz, 2005. "The bargaining set of a large economy with differential information," Studies in Economic Theory, in: Dionysius Glycopantis & Nicholas C. Yannelis (ed.), Differential Information Economies, pages 541-552, Springer.
    2. Sondermann, Dieter, 1974. "Economies of scale and equilibria in coalition production economies," Journal of Economic Theory, Elsevier, vol. 8(3), pages 259-291, July.
    3. Jiuqiang Liu, 2017. "Existence of competitive equilibrium in coalition production economies with a continuum of agents," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 941-955, November.
    4. Konrad Podczeck, 2003. "Core and Walrasian equilibria when agents' characteristics are extremely dispersed," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 22(4), pages 699-725, November.
    5. Mas-Colell, Andreu, 1986. "The Price Equilibrium Existence Problem in Topological Vector Lattice s," Econometrica, Econometric Society, vol. 54(5), pages 1039-1053, September.
    6. Liu, Jiuqiang, 2017. "Equivalence of the Aubin bargaining set and the set of competitive equilibria in a finite coalition production economy," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 55-61.
    7. Hildenbrand, Werner, 1970. "Existence of Equilibria for Economies with Production and a Measure Space of Consumers," Econometrica, Econometric Society, vol. 38(5), pages 608-623, September.
    8. Mas-Colell, Andreu, 1989. "An equivalence theorem for a bargaining set," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 129-139, April.
    9. Jiuqiang Liu & Huihui Zhang, 2016. "Coincidence of the Mas-Colell bargaining set and the set of competitive equilibria in a continuum coalition production economy," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 1095-1109, November.
    10. Zame, William R, 1987. "Competitive Equilibria in Production Economies with an Infinite-Dimensional Commodity Space," Econometrica, Econometric Society, vol. 55(5), pages 1075-1108, September.
    11. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    12. BEWLEY, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," LIDAM Reprints CORE 122, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Rustichini, Aldo & Yannelis, Nicholas C., 1991. "Edgeworth's conjecture in economies with a continuum of agents and commodities," Journal of Mathematical Economics, Elsevier, vol. 20(3), pages 307-326.
    14. Gerard Debreu, 1963. "On a Theorem of Scarf," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 30(3), pages 177-180.
    15. Podczeck, Konrad & Yannelis, Nicholas C., 2008. "Equilibrium theory with asymmetric information and with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 141(1), pages 152-183, July.
    16. SONDERMANN, Dieter, 1974. "Economies of scale and equilibria in coalition production economies," LIDAM Reprints CORE 172, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. Tourky, Rabee & Yannelis, Nicholas C., 2001. "Markets with Many More Agents than Commodities: Aumann's "Hidden" Assumption," Journal of Economic Theory, Elsevier, vol. 101(1), pages 189-221, November.
    18. HILDENBRAND, Werner, 1970. "Existence of equilibria for economies with production and a measure space of consumers," LIDAM Reprints CORE 79, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    19. Noguchi, Mitsunori, 1997. "Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space," Journal of Mathematical Economics, Elsevier, vol. 27(1), pages 1-21, February.
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