Abstract
We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.
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References
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 (2021)
Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)
Borglin, A., Keiding, H.: Existence of equilibrium actions and of equilibrium: a note on the ’new’ existence theorems. J. Math. Econ. 3, 313–316 (1976)
Carmona, G., Podczeck, K.: Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ. Theory 61, 457–478 (2016)
Cornet, B.: The Gale–Nikaido–Debreu lemma with discontinuous excess demand. Econ. Theory Bull. 8, 169–180 (2020)
Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games I: theory. Rev. Econ. Stud. 53, 1–26 (1986)
Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 886–893 (1952)
Debreu, G.: Market equilibrium. Proc. Natl. Acad. Sci. USA 42, 876–878 (1956)
Debreu, G.: Theory of Value. John Wiley and Sons, New York (1959)
Engelking, R.: General Topology, vol. 6 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin (1989)
Gale, D.: The law of supply and demand. Math. Scand. 3, 155–169 (1955)
Gale, D., Mas-Colell, A.: An equilibrium existence theorem for a general model without ordered preferences. J. Math. Econ. 2, 9–16 (1975)
He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ. Theory 61, 497–513 (2016)
He, W., Yannelis, N.C.: Equilibria with discontinuous preferences: new fixed point theorems. J. Math. Anal. Appl. 450, 1421–1433 (2017)
Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton University Press, Princeton (1974)
Mas-Colell, A.: An equilibrium existence theorem without complete or transitive preferences. J. Math. Econ. 1, 237–246 (1974)
McKenzie, L.W.: On equilibrium in Graham’s model of world trade and other competitive systems. Econometrica 22, 147–161 (1954)
Nikaido, H.: On the classical multilateral exchange problem. Metroeconomica 8, 135–145 (1956)
Reny, P.: On the existence of pure and mixed strategy equilibria in discontinuous games. Econometrica 67, 1029–1056 (1999)
Reny, P.J.: Equilibrium in discontinuous games without complete or transitive preferences. Econ. Theory Bull. 4, 1–4 (2016)
Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2, 345–348 (1975)
Shafer, W.J.: The nontransitive consumer. Econometrica 42, 913–919 (1974)
Shafer, W.J.: Equilibrium in economies without ordered preferences or free disposal. J. Math. Econ. 3, 135–137 (1976)
Sonnenschein, H.: Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. In: Chipman, J., Hurwicz, L., Richter, M., Sonnenschein, H. (eds.) Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York (1971)
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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, 413–420 (2022). https://doi.org/10.1007/s00199-021-01400-6
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DOI: https://doi.org/10.1007/s00199-021-01400-6