Journal of Convex Analysis 19 (2012), No. 4, 975--997
Copyright Heldermann Verlag 2012
Parametric Stability of Solutions in Models of Economic Equilibrium
Asen L. Dontchev
Mathematical Reviews, Ann Arbor, MI 48107-8604, U.S.A., Bulgaria
ald@ams.org
Ralph Tyrrell Rockafellar
Dept. of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A.
Recent results about strong metric regularity of solution mappings are applied to a challenging situation in basic mathematical economics, a model of market equilibrium in the exchange of goods. The solution mapping goes from the initial endowments of the agents to the goods they end up with and the supporting prices, and the issue is whether, relative to a particular equilibrium, it has a single-valued, Lipschitz continuous localization. A positive answer is obtained when the chosen goods are not too distant from the endowments. A counterexample is furnished to demonstrate that, when the distance is too great, such strong metric regularity can fail, with the equilibrium then being unstable with respect to tiny shifts in the endowment parameters, even bifurcating or, on the other hand, vanishing abruptly.
The approach relies on passing to a variational inequality formulation of equilibrium. This is made possible by taking the utility functions of the agents to be concave and their survival sets to be convex, so that their utility maximization problems are fully open to the methodology of convex analysis. The variational inequality is nonetheless not monotone, at least in the large, and this greatly complicates the existence of a solution. Existence is secured anyway through truncation arguments which take advantage of a further innovation, the explicit introduction of "money" into the classical exchange model, with money-tuned assumptions of survivability of the agents which are unusualy mild. Those assumptions also facilitate application to the model of refinements of Robinson's implicit mapping theorem in variational analysis.
Keywords: Strong metric regularity, solution stability, economic equilibrium, ample survivability, nonmonotone variational inequalities, convex analysis, variational analysis, Robinson's theorem.
MSC: 49J53, 47J07, 49J40, 49K40, 90C31
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