Abstract
Chapter 3 covers Topology and convex optimization. In the Chap. 2 we introduced the notion of the scalar product of two vectors in ℜn. More generally if a scalar product is defined on some space, then this permits the definition of a norm, or length, associated with a vector, and this in turn allows us to define the distance between two vectors. A distance function or metric may be defined on a space, X, even when X admits no norm. More general than the notion of a metric is that of a topology. This notion allows us to define the idea of continuity of a function as well as analogous ideas for a correspondence. We then introduce three powerful theorems, the Brouwer Fixed Point Theorem for a function, Michael’s Selection Theorem, and the Browder Fixed Point Theorem for a correspondence.
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Further Reading
Shafer, W., & Sonnenschein, H. (1975). Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics, 2, 345–348.
The reference for the Kuhn-Tucker Theorems is:
Kuhn, H. W., & Tucker, A. W. (1950). Non-linear programming. In Proceedings: 2nd Berkeley symposium on mathematical statistics and probability, Berkeley: University of California Press.
A useful further reference for economic applications is:
Heal, E. M. (1973). Theory of economic planning. Amsterdam: North Holland.
The classic references on fixed point theorems and the various corollaries of the theorems are:
Brouwer, L. E. J. (1912). Uber Abbildung von Mannigfaltikeiten. Mathematische Annalen, 71, 97–115.
Browder, F. E. (1967). A new generalization of the Schauder fixed point theorem. Mathematische Annalen, 174, 285–290.
Browder, F. E. (1968). The fixed point theory of multivalued mappings in topological vector spaces. Mathematische Annalen, 177, 283–301.
Fan, K. (1961). A generalization of Tychonoff’s fixed point theorem. Mathematische Annalen, 142, 305–310.
Knaster, B., Kuratowski, K., & Mazerkiewicz, S. (1929). Ein Beweis des Fixpunktsatze fur n-dimensionale Simplexe. Fundamenta Mathematicae, 14, 132–137.
Michael, E. (1956). Continuous selections I. Annals of Mathematics, 63, 361–382.
Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36, 48–49.
Schauder, J. (1930). Der Fixpunktsatze in Funktionalräumn. Studia Mathematica, 2, 171–180.
A useful discussion of the relationships between the theorems can be found in:
Border, K. (1985). Fixed point theorems with applications to economics and game theory. Cambridge: Cambridge University Press.
Hildenbrand, W., & Kirman, A. P. (1976). Introduction to equilibrium analysis. Amsterdam: Elsevier.
The proof of the Browder fixed point theorem by KKM is given in:
Yannelis, N., & Prabhakar, N. (1983). Existence of maximal elements and equilibria in linear topological spaces. Journal of Mathematical Economics, 12, 233–245.
Applications of the Fan Theorem to show existence of a price equilibrium can be found in:
Aliprantis, C., & Brown, D. (1983). Equilibria in markets with a Riesz space of commodities. Journal of Mathematical Economics, 11, 189–207.
Bergstrom, T. (1975). The existence of maximal elements and equilibria in the absence of transitivity (Typescript). Washington University in St. Louis.
Bergstrom, T. (1992). When non-transitive relations take maxima and competitive equilibrium can’t be beat. In W. Neuefeind & R. Riezman (Eds.), Economic theory and international trade, Berlin: Springer.
Shafer, W. (1976). Equilibrium in economies without ordered preferences or first disposal. Journal of Mathematical Economics, 3, 135–137.
The application of the idea of the Nakamura number to existence of a voting equilibrium can be found in:
Greenberg, J. (1979). Consistent majority rules over compact sets of alternatives. Econometrica, 41, 285–297.
Schofield, N. (1984). Social equilibrium and cycles on compact sets. Journal of Economic Theory, 33, 59–71.
Strnad, J. (1985). The structure of continuous-valued neutral monotonic social functions. Social Choice and Welfare, 2, 181–195.
Finally there are results on existence of a joint political economic equilibrium, namely an outcome \((\overline{p}, \overline{t}, \overline{x}, \overline{y})\), where \((\overline{p}, \overline{x})\) is a market equilibrium, \(\overline{t}\) is an equilibrium tax schedule voted on under a rule \({\mathcal{D}}\) and \(\overline{y}\) is an allocation of publicly provided goods.
Konishi, H. (1996). Equilibrium in abstract political economies: with an application to a public good economy with voting. Social Choice and Welfare, 13, 43–50.
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Schofield, N. (2014). Topology and Convex Optimisation. In: Mathematical Methods in Economics and Social Choice. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39818-6_3
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DOI: https://doi.org/10.1007/978-3-642-39818-6_3
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