The Catastrophe Map of a Two Period Production Model with Uncertainty

Author(s)
Pascal Stiefenhofer

Affiliation(s)

Department of Statistics, University College London, London, UK;Department of Mathematics, University of Sussex, East Sussex, UK.

Department of Statistics, University College London, London, UK;Department of Mathematics, University of Sussex, East Sussex, UK.

ABSTRACT

This paper shows existence and efficiency of equilibria of a two period production model with uncertainty as a consequence of the catastrophe map being smooth and proper. Its inverse mapping defines a finite covering implying finiteness of equilibria. Beyond the extraction of local equilibrium information of the model, the catastrophe map renders itself well for a global study of the equilibrium set. It is shown that the equilibrium set has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere implying connectedness, simple connectedness, and contractibility.

Cite this paper

P. Stiefenhofer, "The Catastrophe Map of a Two Period Production Model with Uncertainty,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 114-121. doi: 10.4236/am.2013.48A016.

P. Stiefenhofer, "The Catastrophe Map of a Two Period Production Model with Uncertainty,"

References

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[1] G. Debreu, “Theory of Value,” New York, Wiley, 1959.

[2] G. D. K. Arrow, “Existence of an Equilibrium for a Com petitive Economy,” Econometrica, Vol. 22, No. 3, 1954, pp. 265-290.

[3] Y. Balasko, “Economic Equilibrium and Catastrophe The ory: An Introduction,” Journal of Mathematical Eco nomics, Vol. 46, No. 3, 1978, pp. 557-569.

[4] E. Dierker, “Topological Methods in Walrasian Econom ics,” Vol. 92, Springer-Verlag, Berlin, 1974. doi:10.1007/978-3-642-65800-6

[5] Y. Balasko, “The Equilibrium Manifold: Postmodern De velopments in the Theory of General Economic Equilib rium,” The MIT Press, Cambridge, 1988.

[6] E. Jouini, “The Graph of the Walras Correspondence: The Production Economies Case,” Journal of Mathematical Economics, Vol. 22, No. 2, 1993, pp. 139-147.

[7] G. Fuchs, “Private Ownership Economies with a Nite Num ber of Equilibria,” Journal of Mathematical Economics, Vol. 1, No. 2, 1974, pp. 141-158.

[8] S. Smale, “Global Analysis and Econmics IV: Finitness and Stability with General Consumption Sets and Produc tion,” Journal of Mathematical Economics, Vol. 1, No. 2, 1974, pp. 107-117.

[9] T. Keho, “An Index Theorem for General Equilibrium Models with Production,” Econometrica, Vol. 48, No. 5, 1980, pp. 1211-1232.

[10] T. Keho, “Regularity and Index Theoy for Econmies with Smooth Production Technologies,” Econometrica, Vol. 51, No. 4, 1983, pp. 895-918.

[11] G. Debreu, “Smooth Preferences,” Econometrica, Vol. 40, No. 4, 1972, pp. 603-615.

[12] K. Binmore, “Mathematical Analysis,” 2nd Edition, Cam bridge University Press, Melbourne, 1999.

[13] V. Guillemin and A. Pollack, “Differential Topology,” Prentice Hall, Upper Saddle River, 1974.

[14] M. Hirsch, “Differential Topology,” Springer, New York, 1972.

[15] J. Lee, “Introduction to Smooth Manifolds,” Springer, New York, 2000.