Abstract
We provide several different generalizations of Debreu’s social equilibrium theorem by allowing for asymmetric information and a continuum of agents. The results not only extend the ones in Kim and Yannelis (J Econ Theory 77:330–353, 1977), Yannelis and Rustichini (Stud Econ Theory 2:23–48, 1991), but also new theorems are obtained which allow for a convexifying effect on aggregation (non-concavity assumption on the utility functions) and non-convex strategy sets (pure strategies). This is achieved by imposing the assumption of “many more agents than strategies” (Rustichini and Yannelis in Stud Econ Theory 1:249–265, 1991; Tourky and Yannelis in J Econ Theory 101:189–221, 2001; Podczeck in Econ Theory 22:699–725, 2003).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aliprantis C.D., Burkinshaw O. (1985) Positive Operators. Academic, New York
Arrow J., Debreu G. (1954) Existence of an equilibrium for a competitive economy. Econometrica 22: 265–290
Aumann J.R. (1965) Integrals of set-valued functions. J. Math. Anal. Appl. 12: 1–12
Balder E.J. (2002) A unifying pair of Cournot-Nash equilibrium existence results. J. Econ. Theory 102: 437–470
Balder, E.J.: Existence of Competitive Equilibria in Economies with a Measure Space of Consumers and Consumption Externalities. J. Math. Econ. (forthcoming)
Balder, E.J., Yannelis, N.C.: Equilibrium in random and Bayesian games with a continuum of players in equilibrium theory in infinite dimensional spaces. In: Khan, M.A., Yannelis, N.C. (eds.) Springer, Heidelberg (1991)
Balder E.J., Yannelis N.C. (1993) On the continuity of expected utility. Econ. Theory 3: 625–643
Balder, E.J., Yannelis, N.C.: Bayesian Walrasian equilibrium: a new approach to rational expectations equilibrium. Econ. Theory (submitted)
Castaing C., Valadier M. (1997) Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580. Springer, New York
Cornet, B., Topuzu, M.: Existence of equilibrium for economies with externalities and a measure space of consumers. Econ. Theory (forthcoming)
Da-Rocha, F.M., Topuzu, M.: Cournot-Nash equilibrium in continuum games with non-ordered preferences. J. Econ. Theory (forthcoming)
Debreu G. (1952) A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38: 803–886
Diestel, J., Uhl, J.: Vector measures, mathematical surveys. Am. Math. Soc. Providence 15, (1977)
Dunford, N., Schwartz, J.T.: Linear Operators, vol. I. Interscience, New York (1958)
Haller, H.: Abstract economies with widespread externalities (1993)
Kim T., Yannelis N.C. (1977) Existence of equilibrium in Bayesian games with infinitely many players. J. Econ. Theory 77: 330–353
Nash J.F. (1951) Noncooperative games. Ann. Math. 54: 286–295
Nash J.F. (1950) Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36: 48–49
Podczeck K. (2003) Core and Walrasian equilibrium when agent’s characteristics are extremely dispersed. Econ. Theory 22: 699–725
Podczeck, K.: On the convexity and compactness of the integral of a Banach space valued correspondence, 2006, J. Math. Econ. (in press)
Podczeck, K., Yannelis, N.C.: Equilibrium theory with asymmetric information and with infinitely many commodities. J. Econ. Theory (submitted)
Podczeck, K., Tourky, R., Yannelis, N.C.: Private expectations equilibria (2005)
Rustichini, A., Yannelis, N.C.: What is Perfect Competition. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, vol. 1, pp. 249–265, (1991)
Schmeidler D. (1973) Equilibrium points of non-atomic games. J. Stat. Phys. 7: 295–300
Sun Y., Yannelis, N.C.: Saturation and the integration of Banach valued correspondences, J. Math. Econ. (forthcoming)
Tourky R., Yannelis N.C. (2001) Markets with many more agents than commodities: Aumann’s hidden assumption. J. Econ. Theory 101: 189–221
Yannelis, N.C.: Integration of Banach-valued correspondences. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces. Springer, Heidelberg (1991)
Yannelis N.C. (2002) A Bayesian equilibrium existence theorem. Adv. Math. Econ. 4: 61–72
Yannelis, N.C., Rustichini, A.: Equilibrium points of noncooperative random and Bayesian games. In: Aliprantis, C.D. et al. (eds.) Positive operators, Riesz spaces and economics, vol. 2, pp. 23–48 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Gerard Debreu. A preliminary draft was presented in Paris, in April of 2005. I have benefited from the discussion, comments and questions of Erik Balder, Jean-Marc Bonnisseu, Bernard Cornet and Filipe Martins Da-Rocha and Conny Podczeck. A careful and knowledgeable referee made several useful comments and rescued me from a mishap.
Rights and permissions
About this article
Cite this article
Yannelis, N.C. Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents. Econ Theory 38, 419–432 (2009). https://doi.org/10.1007/s00199-007-0246-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-007-0246-3
Keywords
- Social equilibrium
- Asymmetric information
- Many more players than strategies
- Convexifying effect
- Pure strategy equilibrium