Abstract
We study the Walrasian objection mechanism in the framework of economies with a measure space of agents and a separable Banach space of commodities whose positive cone has a non-empty interior. We provide several characterizations of Walrasian objections and use them to study the bargaining set of the economy, as defined in Mas-Colell (J Math Econ 18(2):129–139, 1989). Our main result shows that whenever the measure space of agents is saturated, every non-competitive allocation can be blocked with a Walrasian objection. This implies that the bargaining set, the core and the set of competitive allocations are equivalent solution concepts.
Similar content being viewed by others
References
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)
Aliprantis, C.D., Brown, D.J., Burkinshaw, O.: Existence and Optimality of Competitive Equilibria. Springer, Berlin (1990)
Anderson, R.M., Trockel, W., Zhou, L.: Nonconvergence of the Mas-Colell and Zhou bargaining sets. Econometrica 65(5), 1227–1239 (1997)
Anderson, R.M., Duanmu, H., Khan, M.A., Uyanik, M.: Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context. Econ. Theory 73(2), 387–412 (2022)
Armstrong, T.E., Richter, M.K.: The core-Walras equivalence. J. Econ. Theory 33(1), 116–151 (1984)
Centrone, F., Martellotti, A.: Coalitional extreme desirability in finitely additive exchange economies. Econ. Theory Bull. 4(1), 17–34 (2016)
Cheng, H.H.: The principle of equivalence. In: Khan, M.A., Yannellis, N. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, vol. 1, pp. 197–221. Springer (1991)
Cornet, B.: The Gale–Nikaido–Debreu lemma with discontinuous excess demand. Econ. Theory Bull. 8(2), 169–180 (2020)
Donnini, C., Graziano, M.G.: The Edgeworth’s conjecture in finitely additive production economies. J. Math. Anal. Appl. 360(1), 81–94 (2009)
Dutta, B., Ray, D., Sengupta, K., Vohra, R.: A consistent bargaining set. J. Econ. Theory 49(1), 93–112 (1989)
Fajardo, S., Keisler, H.J.: Model theory of stochastic processes. Bull. Symb. Logic 10(1), 110–112 (2004)
Florenzano, M.: General Equilibrium Analysis: Existence and Optimality Properties of Equilibria, 1st edn. Springer, New York (2003)
Gabszewicz, J.J., Mertens, J.-F.: An equivalence theorem for the core of an economy whose atoms are not “too’’ big. Econometrica 39(5), 713–721 (1971)
Geanakoplos, J.: The bargaining set and nonstandard analysis. Technical report, Harvard Univ Cambridge Mass (1978)
Greinecker, M., Podczeck, K.: Liapounoff’s vector measure theorem in Banach spaces and applications to general equilibrium theory. Econ. Theory Bull. 1(2), 157–173 (2013)
Greinecker, M., Podczeck, K.: Edgeworth’s conjecture and the number of agents and commodities. Econ. Theory 62(1), 93–130 (2016)
Grodal, B.: A second remark on the core of an atomless economy. Econometrica 40(3), 581–583 (1972)
Grodal, B.: The equivalence principle. In: Derigs, U. (ed.) Optimization and Operation Research. Encyclopedia of Life and Support Systems (EOLSS), pp. 248–273. University of Cologne, Cologne (2009)
Hara, C.: The anonymous core of an exchange economy. J. Math. Econ. 38(1–2), 91–116 (2002)
He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ. Theory 61(3), 497–513 (2016)
Hervés-Estévez, J., Moreno-García, E.: On restricted bargaining sets. Int. J. Game Theory 44(3), 631–645 (2015)
Hervés-Estévez, J., Moreno-García, E.: A limit result on bargaining sets. Econ. Theory 66(2), 327–341 (2018)
Hoover, D.N., Keisler, H.J.: Adapted probability distributions. Trans. Am. Math. Soc. 286(1), 159–201 (1984)
Jang, H.S., Lee, S.: Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences. J. Math. Econ. 90, 57–64 (2020)
Kakutani, S.: Construction of a non-separable extension of the Lebesgue measure space. Proc. Imp. Acad. 20(3), 115–119 (1944)
Keisler, H.J., Sun, Y.: Why saturated probability spaces are necessary. Adv. Math. 221(5), 1584–1607 (2009)
Khan, M.A., Sagara, N.: Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Ill. J. Math. 57(1), 145–169 (2013)
Khan, M.A., Sagara, N.: Maharam-types and Lyapunov’s theorem for vector measures on locally convex spaces with control measures. J. Convex Anal. 22, 647–672 (2015)
Khan, M.A., Sagara, N.: Relaxed large economies with infinite-dimensional commodity spaces: the existence of Walrasian equilibria. J. Math. Econ. 67, 95–107 (2016)
Khan, M.A., Yannelis, N.C.: Equilibria in markets with a continuum of agents and commodities. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, pp. 233–248. Springer, Berlin (1991a)
Khan, M.A., Yannelis, N.C.: Equilibrium Theory in Infinite Dimensional Spaces, vol. 1. Springer, Berlin (1991b)
Lee, S.: Competitive equilibrium with an atomless measure space of agents and infinite dimensional commodity spaces without convex and complete preferences. Hitotsubashi J. Econ. 54(2), 221–230 (2013)
Maharam, D.: On homogeneous measure algebras. Proc. Natl. Acad. Sci. U.S.A. 28(3), 108–111 (1942)
Martellotti, A.: Finitely additive economies with free extremely desirable commodities. J. Math. Econ. 44(5–6), 535–549 (2008)
Martins-da Rocha, F.: Equilibria in large economies with a separable Banach commodity space and non-ordered preferences. J. Math. Econ. 39(8), 863–889 (2003)
Mas-Colell, A.: An equivalence theorem for a bargaining set. J. Math. Econ. 18(2), 129–139 (1989)
Mas-Colell, A., Zame, W.R.: Equilibrium theory in infinite dimensional spaces. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of Mathematical Economics, vol. 4, pp. 1835–1898. Elsevier (1991)
Mehta, G., Tarafdar, E.: Infinite-dimensional Gale–Nikaido–Debreu theorem and a fixed-point theorem of Tarafdar. J. Econ. Theory 41(2), 333–339 (1987)
Noguchi, M.: Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space. J. Math. Econ. 27(1), 1–21 (1997)
Ostroy, J.M., Zame, W.R.: Nonatomic economies and the boundaries of perfect competition. Econometrica 62, 593–633 (1994)
Podczeck, K.: Markets with infinitely many commodities and a continuum of agents with non-convex preferences. Econ. Theory 9(3), 385–426 (1997)
Podczeck, K.: Core and Walrasian equilibria when agents’ characteristics are extremely dispersed. Econ. Theory 22(4), 699–725 (2003)
Podczeck, K.: On the convexity and compactness of the integral of a Banach space valued correspondence. J. Math. Econ. 44(7), 836–852 (2008)
Podczeck, K., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets. Econ. Theory 73(2), 413–420 (2022)
Rustichini, A., Yannelis, N.C.: What is perfect competition? In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces. Springer, Berlin (1991)
Schjødt, U., Sloth, B.: Bargaining sets with small coalitions. Int. J. Game Theory 23(1), 49–55 (1994)
Schmeidler, D.: A remark on the core of an atomless economy. Econometrica 40(3), 579 (1972)
Shitovitz, B.: Oligopoly in markets with a continuum of traders. Econometrica 41(3), 467–501 (1973)
Shitovitz, B.: The bargaining set and the core in mixed markets with atoms and an atomless sector. J. Math. Econ. 18(4), 377–383 (1989)
Sun, Y., Yannelis, N.C.: Saturation and the integration of Banach valued correspondences. J. Math. Econ. 44(7), 861–865 (2008)
Sun, X., Zhang, Y.: Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces. Econ. Theory 58(1), 161–182 (2015)
Suzuki, T.: A coalitional production economy with infinitely many indivisible commodities. Econ. Theory Bull. 4(1), 35–52 (2016)
Suzuki, T.: Fundamentals of General Equilibrium Analysis, vol. 6. World Scientific, Singapore (2020)
Tourky, R., Yannelis, N.C.: Markets with many more agents than commodities: Aumann’s “hidden’’ assumption. J. Econ. Theory 101(1), 189–221 (2001)
Urbinati, N.: A convexity result for the range of vector measures with applications to large economies. J. Math. Anal. Appl. 470(1), 16–35 (2019)
Vind, K.: A third remark on the core of an atomless economy. Econometrica 40(3), 585 (1972)
Vohra, R.: An existence theorem for a bargaining set. J. Math. Econ. 20(1), 19–34 (1991)
Yamazaki, A.: Bargaining sets in continuum economies. In: Maruyama, T., Takahashi, W. (eds.) Nonlinear and Convex Analysis in Economic Theory, pp. 289–299. Springer (1995)
Yannelis, N.C.: On a market equilibrium theorem with an infinite number of commodities. J. Math. Anal. Appl. 108(2), 595–599 (1985)
Yannelis, N.C.: Integration of Banach-valued correspondence. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, pp. 2–35. Springer, Berlin, Heidelberg (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Relevant improvements to this paper were suggested by the Referees, to whom I am especially grateful. I also wish to thank M. G. Graziano, M. LiCalzi and the Editor for their useful comments and their suggestions.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Urbinati, N. The Walrasian objection mechanism and Mas-Colell’s bargaining set in economies with many commodities. Econ Theory 76, 45–68 (2023). https://doi.org/10.1007/s00199-022-01454-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-022-01454-0
Keywords
- Walrasian objections
- Bargaining set
- Infinite dimensional commodity spaces
- Saturation property
- Lyapunov’s theorem