A new proof of the index formula for incomplete markets. (English) Zbl 1141.91404
Summary: This paper gives a new proof of the index formula established by T. Momi [J. Math. Econ. 39, No. 3–4, 273–297 (2003; Zbl 1049.91116)] for an economy with incomplete asset markets where the difference between the number of states \((S)\) and the number of assets \((J)\) is an even number. The proof uses a single globally defined homotopy function on the asset pseudo-equilibrium manifold connecting the excess demand of a given economy to the individual excess demand of the unconstrained agent. We show that the asset pseudo-equilibrium manifold is orientable if the number \(S - J\) is even and deduce the index formula from the homotopy invariance theorem for the degree of a map.
MSC:
91B26 | Auctions, bargaining, bidding and selling, and other market models |
91B50 | General equilibrium theory |
47H11 | Degree theory for nonlinear operators |
47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
55M25 | Degree, winding number |
Citations:
Zbl 1049.91116References:
[1] | Brown, D. J.; Demarzo, P. M.; Eaves, B. C., Computing equilibria in the GEI model, Econometrica, 64, 1-27 (1996) · Zbl 0861.90018 |
[2] | Demarzo, P. M.; Eaves, B. C., Computing Equilibria of GEI by Relocalization on a Grassmann Manifold, Journal of Mathematical Economics, 26, 479-497 (1996) · Zbl 0876.90026 |
[3] | Dierker, E., Two Remarks on the Number of Equilibria of an Economy, Econometrica, 40, 951-953 (1972) · Zbl 0259.90005 |
[4] | Dold, A., Lectures on Algebraic Topology (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0234.55001 |
[5] | Duffie, D.; Shafer, W., Equilibrium in incomplete markets I, Journal of Mathematical Economics, 14, 285-300 (1985) · Zbl 0604.90029 |
[6] | Hirsch, M., Differential Topology (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0356.57001 |
[7] | Jongen, H. T.; Jonker, P.; Twilt, F., Nonlinear Optimization in Finite Dimensions: Morse Theory, Chebyshev Approximation, Tranversality, Flows, Parametric Aspects (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0985.90083 |
[8] | Momi, T., The index theorem for a GEI economy when the degree of incompleteness is even, Journal of Mathematical Economics, 39, 273-297 (2003) · Zbl 1049.91116 |
[9] | Munkres, J. R., Elements of Algebraic Topology (1984), Addison-Wesley: Addison-Wesley Redwood City, California · Zbl 0673.55001 |
[10] | Zhou, Y., Genericity analysis on the pseudo-equilibrium manifold, Journal of Economic Theory, 73, 79-92 (1997) · Zbl 0872.90018 |
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