The paper studies pure exchange economies with infinite dimensional commodity spaces in the setting of Riesz dual systems. Several new concepts of equilibrium are introduced. An allocation (x1, …, xm) is said to be a) an Edgeworth equilibrium whenever it belongs to the core of every n-fold replication of the economy; and b) an ε-Walrasian equilibrium whenever for each ε > 0 there exists some price p not equal to 0 with p∙ω = 1 (where ω = Σωi is the total endowment) and with x ≥i xi implying p times x ≥ p∙ωi – ε. The major results of the paper are the following:
Theorem I: Edgeworth equilibria exist.
Theorem II: An allocation is an Edgeworth equilibrium if and only if it is an ε-Walrasian equilibrium.
Theorem III: If preferences are proper, then every Edgeworth equilibrium is a quasi-equilibrium.
In [6], Guha gave a complete characterization of path independent social decision functions which satisfy the independence of irrelevant alternatives condition, the strong Pareto principle, and UII, i.e., unanimous indifference implies social indifference. These conditions necessarily imply that a path independent social decision function is neutral and monotonic. In this paper, we extend Guha’s characterization to the class of neutral monotonic social functions. We show that neutral monotonic social functions and their specializations to social decision functions, path independent social decision functions, and social welfare functions can be uniquely represented as a collection of overlapping simple games, each of which is defined on a nonempty set of concerned individuals.
Moreover, each simple game satisfies intersection conditions depending on the number of social alternatives; the number of individuals belonging to the concerned set under consideration; and the collective rationality assumption.
We also provide a characterization of neutral, monotonic and anonymous social decision functions, where the number of individuals in society exceeds the (finite) number of social alternatives, that generalizes both the representation theorem of May [10] and the representation theorems of Ferejohn and Grether [5].
The existence of equilibria is established in an overlapping generations exchange economy, where each generation lives for two periods and the commodity space is the positive cone of an infinite dimensional Riesz space. In particular, we establish the existence of equilibria in the stochastic overlapping generations model, i.e., we establish the existence of equilibria when the commodity space in each period is L∞ equipped with the Mackey topology τ(L∞, L1).