Journal of Convex Analysis 30 (2023), No. 4, 1115--1138
Copyright Heldermann Verlag 2023
On the Stability and Evolution of Economic Equilibrium
Alejandro Jofré
CMM & DIM, Universidad de Chile, Santiago, Chile
ajofre@dim.uchile.cl
Ralph T. Rockafellar
Dept. of Mathematics, University of Washington, Seattle, U.S.A.
rtr@uw.edu
Roger J-B Wets
Dept. of Mathematics, University of California, Davis, U.S.A.
rjbwets@ucdavis.edu
In an economic model of exchange of goods, an equilibrium of market prices and resultant holdings of the agents can, in response to small additions and subtractions of goods, uniquely reconstitute itself with slightly adjusted prices and holdings. That stability property of equilibrium is shown to persist even when individual agents are only interested in some of the goods and prefer zero quantities of others, as long as some good is indispensable to all agents and able to serve then effectively as money. Assumptions about marginal utility that entail concavity rather than just quasi-concavity of utility functions assist in establishing this and lead to a new vision of equilibrium where prices and holdings are not static. Instead, they evolve continuously in time according to a utility-based law in the form of a one-sided differential equation. Broad possibilities are opened up for dynamic modeling in which the changes in holdings that require ongoing readjustments could be driven by consumption, production, taxation or subsidy, among other influences. The goods can then be more than commodities destined for immediate disposal. Because the equilibrium at any moment has a past and a future, money, in particular, can carry value as a good and naturally enter that way into preferences.
In these developments, tools of convex analysis, and variational analysis beyond, are employed to extend and reorient stability results in the theory of economic equilibrium that previously had to rely on differential analysis alone.
Keywords: Walrasian economic equilibrium, market stability, evolutionary dynamics, evolution equation, concave utility functions, monotone mappings, variational inequalities, variational analysis.
MSC: 91B50, 46N10.
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