Back to Fundamentals: Equilibrium in Abstract Economies

Abstract

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)

Topological connectedness and behavioral assumptions on preferences: a two-way relationship

“…For a finite choice set, this proposition shows that the quotient topology with respect to the symmetric part of a continuous binary relation is discrete, hence for any connected set in the original space, all of its the elements must be indifferent to each other. As such this direction is a dead end and substantial rethinking is needed, perhaps along the lines of the literature stemming from the application of convex geometry recently developed in the imaginative contribution of Richter and Rubinstein (2015); see Edelman and Jamison (1985) for an early survey.…”

Endowments, Exclusion, and Exchange

“…Richter and Rubinstein (2015) introduced the notion of a "primitive equilibrium," which does not rely on budget sets or endowments. The possible absence of personal endowments precludes the application of standard price equilibrium definitions to our setting.…”

“…The possible absence of personal endowments precludes the application of standard price equilibrium definitions to our setting. Richter and Rubinstein (2015) introduced the notion of a "primitive equilibrium," which does not rely on budget sets or endowments. Instead, they observed that equilibria induce an ordering of goods, from more to less desirable.…”

“…Instead, they observed that equilibria induce an ordering of goods, from more to less desirable. Exclusion core allocations identified by Algorithm 2, which orders goods based on the step in which the good is assigned, satisfy Richter and Rubinstein's (2015) equilibrium definition.…”

Embedding convex geometries and a bound on convex dimension

“…In fact, there are several other equivalent formulations (for surveys, see Stern [16], Adaricheva and Czédli [2], Czédli [6] and Adaricheva and Nation [3]). Finally, we note that convex geometries have proven useful in economics in the study of choice (Johnson and Dean [10,11,12], Koshevoy [14]) and of abstract economic equilibrium (Richter and Rubinstein [15]).…”