Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context

This paper provides four theorems on the existence of a free-disposal equilibrium in a Walrasian economy: the first with an arbitrary set of agents with compact consumption sets, the next highlighting the trade-offs involved in the relaxation of the compactness assumption, and the last two with a countable set of agents endowed with a weighting structure. The results generalize theorems in the antecedent literature pioneered by Shafer–Sonnenschein in 1975, and currently in the form taken in He–Yannelis 2016. The paper also provides counterexamples to the existence of non-free-disposal equilibrium in cases of both a countable set of agents and an atomless measure space of agents. One of the examples is related to one Chiaki Hara presented in 2005. The examples are of interest because they satisfy all the hypotheses of Shafer’s 1976 result on the existence of a non-free-disposal equilibrium, except for the assumption of a finite set of agents. The work builds on recent work of the authors on abstract economies, and contributes to the ongoing discussion on the modelling of “large” societies.

Technical Note Alternative Theorem: Equilibrium in Abstract Economies

We prove a new theorem for the existence of equilibrium in discontinuous games in which the players’ preferences are neither complete nor transitive. Our result is an alternative version of Shafer and Sonnenschein ([1975] J. Math. Econ. 2, 345–348), He and Yannelis ([2016] Econ. Theory 61, 497–513), Reny ([2016] Econ. Theory Bull.) and Carmona and Prodczeck ([2016] Econ. Theory 61, 457–478).

On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems

Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of L -majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP).

Back to Fundamentals: Equilibrium in Abstract Economies

We propose a new abstract definition of equilibrium in the spirit of competitive equilibrium: a profile of alternatives and a public ordering (expressing prestige, price, or a social norm) such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from convex geometry. We apply the concept to a variety of convex economies and relate it to Pareto optimality. The “magic” of linear equilibrium prices is put into perspective by establishing an analogy between linear functions in the standard convexity and “primitive orderings” in the abstract convexity. (JEL I11, I18, J44, K13)

On Equilibrium for Abstract Economies in GFC-Spaces

In this paper, the GFC-KKM mapping is introduced and GFC-KKM theorems are established in GFC-spaces. As applications, a fixed point theorem and maximal element theorem are obtained. Our results unify, improve and generalize some known results in recent reference. Finally, equilibrium existence theorems for qualitative games and abstract economies are yielded in GFC-spaces.