Chichilnisky, Graciela (1993): Topoloy and economics: the contributions of S. Smale. Published in: Topology to Computation, Proceedings of the Smalefest (1993): pp. 147161.

PDF
MPRA_paper_8485.pdf Download (2MB)  Preview 
Abstract
90 Classical problems in economics are concerned with the solutions of several simultaneous nonlinear optimization problems, one for each consumer or producer, all facing constraints posed by the scarcity of resources. Often their interests conflict, and it is generally impossible to find a single realvalued function representing the interests of the whole of society. To deal with this problem, John Von Neumann introduces the theory of games. He also defined and established the existence of a general economic equilibrium, using topological tools [Von Neumann, 1938]. The work of Stephan Smale follows this tradition. He uses topological tools to deepen and refine the results on existence and other properties of another type of economic equilibrium, the Walrasian equilibrium (Walrus [187477]), as formalized by Kenneth J. Arrow and Gerard Debreu [1954], and of noncooperative equilibrium in game theory as formalized by Nash (1950). This article aim to show that topology is intrinsically necessary for the understanding of the fundamental problem of conflict resolution in economics in its various forms and to situate Smale's contribution within this perspective. The study of conflicts of interests between individuals is what makes economics interesting and mathematically complex. Indeed, we now know that the space of all individual preferences, which define the individual optimization problems, is topologically nontrivial, and that its topological complexity is responsible for the impossibility of treating several individual preferences as if they were one, i.e., aggregating them (Chichilnisky, 1980; Chichilnisky and Heal, 1983). Because it is not possible, in general, to define a single optimization problem, other solutions are sought. This article will develop three solutions, discussed below.
Because of the complexity arising from simultaneous optimization problems, economics differs from physics where many of the fundamental relations derive from a single optimization problem. The attempts to find solutions to conflicts among individual interests led to three different theories about how economies are organized and how they behave. These are general equilibrium theory, the theory of games, and social choice theory. Each of these theories leads naturally to mathematical problems of topological nature. Steve Smale has contributed fruitfully to the first two theories: general equilibrium theory and the theory of games. I will argue that his work is connected also with the third approach, social choice theory, by presenting in Section 4 results which link closely, and in unexpected ways, two seemingly different problems: the existence of a general equilibrium and the resolution by social choice of the resource allocation conflict in economics (Chichilnisky, 1991).
Item Type:  MPRA Paper 

Original Title:  Topoloy and economics: the contributions of S. Smale 
Language:  English 
Keywords:  social choice; game theory; general equilibrium; Stephan Smale; Steve Smale; optimization; optimization problems; topology; topological; preferences; conflict resolution 
Subjects:  D  Microeconomics > D5  General Equilibrium and Disequilibrium D  Microeconomics > D7  Analysis of Collective DecisionMaking > D71  Social Choice ; Clubs ; Committees ; Associations C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis 
Item ID:  8485 
Depositing User:  Graciela Chichilnisky 
Date Deposited:  27 Apr 2008 04:14 
Last Modified:  08 Oct 2019 05:03 
References:  AROW, K. AND G. DEBREU [1959] "Existence of an equilibrium for a competitive economy," Econometrica 22, 264290. K. J. Arrow and F. H. Hahn, "General Competitive Analysis," HoldernDay, San Francisco, 1971. Brown, D. and G. M. Heal (1979) "Equity Efficiency and Increasing Returns", Review of Economic Studies 46, 571585. Chichildeky, G. (1980). Social choice and the topology of preference spaces. Advances in Mathematics, 37, 165176. CHICHILNISKY, G. [1981], "Intersecting families of sets: a topological characterization," Working Paper No 166, Essex Economic Papers, Univ. of Essex. CHICHILNISKY, G. [1990], "General equilibrium and social choice with increasing returns," Annals of Operations Research 23, 289297. CHICHILNISKY, G. [1991], "Markets, arbitrage and social choice," Working paper, Columbia University. CHICHILNISKY, G. [1992], "Limited arbitrage necessary and sufficient for the existence of a competitive equilibrium," Working paper, Columbia University. CHICHILNISKY, G.AND G. M. HEAL [1982], "Necessary and sufficient conditions for a resolution of the social choice paradox," J. of Econ. Theory 31(1), 6887. CHICHILNISKY, G. AND G. M. HEAL, [1984, revised 1993], "Existence of a cometitive equilibrium in Sobolev Spaces without bounds on short sales," Journal of Economic Theory. DEBREU, G. [1952], "A social equilibrium existence theorem," Proceedings of the National Academy of Sciences. G. Debreu, Economics with a finite set of equilibria, Econometrics 38 (1970). DEBREU, G. [1993], "Stephan Smale and the economic theory of general equilibrium," this colume. EAVES, B. C. [1971], "On the basic theorem of complementarity," Mathematical programming, 1(1), 6875. EAVES, B. C. [1972], " Homotopies for computation of fixed points," Mathematical Programming, 3(1). GUESNERIE, R. [1975], "Pareto optimality in nonconvex economies," Econometrica 43, 129. HIRSCH, M. [1963], "A proof of the nonretractiblity of a cell onto its boundary," Proceedings of the American Mathematical Society 14, 264265. HIRSCH, M. AND S. SMALE [1979], "On algorithms for solving f(x)=0," Communications on Pure and Applied Mathematics 32, 281312. LEMKE, C.E. [1965], "Bimatrix equilibrium points and mathematical programming," Management Science 11(7). LEMKE, C. E. "Recent results on complemtarity problems," Math Dept., Rensselaer Polytechnic Institute. NASH, J. F. [1950], "Equilibrium points in nperson games," Proceedings of the National Academy of Sciences (USA) 36, 4849. SMALE, S. [1973], "Global analysis and economies I, Pareto optimum and a generalization of Morse Theory," in Dnamical Systems, M. Peixoto, ed., Academic Press, New York. S. Smale, (1974), "Global analysis and economics, IIA, extension of a theorem of Debreu," J. Math. Econ. 1, 114. S. Smale, (1974b) "Global analysis and economics, III, Pareto optima and price equilibrium," J. Math. Econ. 1, 107117. S. Smale, (1974c) "Global analysis and economics, IV, finiteness and stability of equilibria with general consumption sets and production," J. Math. Econ. 1, 19127. S. Smale, (1974d) "Global analysis and economics, V, Pareto theory with constraints," J. Math. Econ. 1, 213221. S. Smale, (1975), "Optimizing several functions," Manifold Tokyo 1975, Univ. of Tokyo Press. Tokyo, pp. 6975. S. Smale, (1976a), "Dynamics in general equilibrium theory," American Economic Review 66, 288294. S. Smale, (1976b), "A converngent progress of prce adjustment and global newton method," J. of Math. Econ. 3, 107120. S. Smale, (1976c), "Exchange processes with price adjustment," J. of Math. Econ. 3, 211226. S. Smale, (1976d), "Glbal analysis and economics VI, geometric analysis of Pareto optimum and price equilibrium under classical hypothesis," J. of Math. Econ. 3, 114. S. Smale, (1977), "Some synamical questions in mathematical economics," Colloques Internationaux du Centre National de la Recherche Scientifiqu, N. 259: Systems Dynamiques et Modeles Economiques. S. Smale, (1978), "An approach to the analysis of dynamic processes in economic systems," Equilibrium and Disequilibrium in Economic Theory G. Schwodiauer, ed. D. Reidel Publishing Co., Boston, pp. 363367. S. Smale, (1980), "The prisoner's dilemma and dynamical systems associated to noncooperative games," Econometrica 48, 16171634. VON NEUMANN, J. [1938], "Uber ein okonomisches gleichungsystem und ein verallgemainerung des brouwerschen fixpunktatzes," Erebnisse eines Mathematischen Seminars, K. Mengen , ed., Vienna. VON NEUMANN, J. [194546], "A model of general equilibrium" (translated into English by G. Morgenstern), Review of Economic Studies 13(1) 33, 19. WALRAS, L. [187477], Elements d'economie politique pure, L. Corbaz and Company, Lausanne. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/8485 