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Exchange price equilibria and variational inequalities. (English) Zbl 0709.90013
The Walrasian equilibria of a pure exchange economy are characterized as the solutions of a variational inequality. The theory of variational inequalities is then used to provide a simple proof of the existence of such equilibria under weak assumptions on the aggregate excess demand functions.
The largest part of the paper is devoted to the further characterization of Walrasian equilibria under various monotonicity assumptions on the excess demand functions (i.e. weak, strict, and strong monotonicity). Under weak monotonicity, the set of equilibria is convex; under strict monotonicity, equilibrium is unique; under strong monotonicity, an equilibrium price vector depends continuously on the aggregate excess demand function.
Finally, the connection between Walrasian equilibria and the solution of a convex programming problem is investigated.
Reviewer: P.J.Deschamps

MSC:
 91B50 General equilibrium theory 49J40 Variational inequalities 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C25 Convex programming
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References:
 [1] K.C. Border,Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, 1985). [2] S. Dafermos, ”Traffic equilibrium and variational inequalities,”Transportation Science 14 (1980) 42–54. [3] S. Dafermos, ”The general multimodal traffic equilibrium problem,”Networks 12 (1982) 57–72. · Zbl 0478.90022 [4] S. Dafermos, ”An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47. · Zbl 0506.65026 [5] S. Dafermos, ”Sensitivity analysis for variational inequalities,”Mathematics of Operations Research 13 (1988) 421–434. · Zbl 0674.49007 [6] S. Dafermos and A. Nagurney, ”A network formulation of market equilibrium problems and variational inequalities,”Operations Research Letters 3 (1984) 247–250. · Zbl 0554.90015 [7] S. Dafermos and A. Nagurney, ”Oligopolistic and competitive behavior of spatially separated markets,”Regional Science and Urban Economics 17 (1987) 225–254. [8] S. Dafermos and L. Zhao, ”Variational inequality methods for solving large-scale economic equilibria,” Paper presented at the ORSA/TIMS Meeting (Washington, DC, April 1988). [9] S. Dafermos and L. Zhao, ”General equilibrium and variational inequalities: Existence, uniqueness and sensitivity,” LCDS Report #89-2, Brown University (Providence, RI, 1989). [10] B.C. Eaves, ”On the basic theorem of complementarity,”Mathematical Programming 1 (1972) 68–75. · Zbl 0227.90044 [11] M. Florian and M. Los, ”A new look at the static spatial price equilibrium models,”Regional Science and Urban Economics 12 (1982) 579–597. [12] D. Gabay and H. Moulin, ”On the uniqueness and stability of Nash equilibria in noncooperative games,” in: A. Bensoussan, P. Kleindorfer and C.S. Tapiero, eds.,Applied Stochastic Control of Econometrics and Management Science (North-Holland, Amsterdam, 1980) pp. 271–293. · Zbl 0461.90085 [13] J.M. Grandmont, ”Temporary general equilibrium theory,”Econometrica 45 (1977) 535–572. · Zbl 0355.90014 [14] W. Hildenbrand, ”Law of demand,”Econometrica 51 (1983) 997–1019. · Zbl 0511.90012 [15] S. Karamardian, ”The nonlinear complementarity problem with applications, Part 1,”Journal of Optimization Theory and Applications 4 (1969). · Zbl 0169.51302 [16] D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Applications (Academic Press, New York, 1980). · Zbl 0457.35001 [17] C.E. Lemke, ”Bimatrix equilibrium points and mathematical programming,”Management Science II (1965). · Zbl 0139.13103 [18] A. Mas-Colell,The Theory of General Economic Equilibrium A Differentiable Approach. Econometric Society Publication, Vol. 9 (Cambridge University Press, Cambridge, 1985). [19] L. Mathiesen, ”An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example,”Mathematical Programming 37 (1987) 1–18. · Zbl 0613.90098 [20] H. Scarf (with T. Hansen),Computation of Economic Equilibria (Yale University Press, New Haven, CT, 1973). · Zbl 0311.90009 [21] M. Todd, ”Computation of fixed points and applications,” in:Lecture Notes in Economics and Mathematical Systems, Vol. 124 (Springer, Berlin, 1976). · Zbl 0332.54003 [22] L. Zhao, ”Variational inequalities in general economic equilibrium: Analysis and algorithms,” Ph.D. Thesis, Division of Applied Mathematics, Brown University (Providence, RI, 1988).
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