Topological Properties of the Catastrophe Map of a General Equilibrium Production Model with Uncertain States of Nature

Author(s)
Pascal Stiefenhofer

ABSTRACT

This paper shows existence and efficiency of equilibria of a production model with uncertainty, where production is modeled in the demand function of the consumer. Existence and efficiency of equilibria are a direct consequence of the catastrophe map being smooth and proper. Topological properties of the equilibrium set are studied. 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. The set of economies with discontinuous price systems is shown to be of Lebesgue measure zero.

This paper shows existence and efficiency of equilibria of a production model with uncertainty, where production is modeled in the demand function of the consumer. Existence and efficiency of equilibria are a direct consequence of the catastrophe map being smooth and proper. Topological properties of the equilibrium set are studied. 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. The set of economies with discontinuous price systems is shown to be of Lebesgue measure zero.

Cite this paper

Stiefenhofer, P. (2014) Topological Properties of the Catastrophe Map of a General Equilibrium Production Model with Uncertain States of Nature.*Applied Mathematics*, **5**, 2719-2727. doi: 10.4236/am.2014.517259.

Stiefenhofer, P. (2014) Topological Properties of the Catastrophe Map of a General Equilibrium Production Model with Uncertain States of Nature.

References

[1] Debreu, G. (1959) Theory of Value. Wiley, New York.

[2] Arrow, K. and Debreu, G. (1954) Existence of an Equilibrium for a Competitive Economy. Econometrics, 22, 265-290.

http://dx.doi.org/10.2307/1907353

[3] Balasko, Y. (1988) The Equilibrium Manifold: Postmodern Developments in the Theory of General Economic Equilibrium. The MIT Press, Cambridge, Massachusetts.

[4] Jouini, E. (1993) The Graph of the Walras Correspondence. The Production Economies Cas. Journal of Mathematical Economics, 22, 139-147.

http://dx.doi.org/10.1016/0304-4068(93)90043-K

[5] Fuchs, G. (1974) Private Ownership Economies with a Finite Number of Equilibria. Journal of Mathematical Economics, 1, 141-158.

http://dx.doi.org/10.1016/0304-4068(74)90005-6

[6] Stiefenhofer, P. (2011) Equilibrium Structure of Production Economies with Uncertainty: The Natural Projection Approach. Discussion Papers in Economics, University of York, York, 7.

[7] Stiefenhofer, P. (2013) The Catastrophe Map of a Two Period Production Model with Uncertainty. Applied Mathematics, 4, No. 8A.

[8] Debreu, G. (1972) Smooth Preferences. Econometrica, 40, 603-615.

http://dx.doi.org/10.2307/1912956

[9] Lee, J.M. (2004) Introduction to Topological Manifolds. Springer, New York.

[10] Hirsch, M. (1972) Differential Topology. Springer Verlag, New York.

[11] Lee, J.M. (2000) Introduction to Smooth Manifolds. Springer, New York.

[12] Guillemin, V. and Pollack, A. (1974) Differential Topology. Prentice Hall, Upper Saddle River.

[1] Debreu, G. (1959) Theory of Value. Wiley, New York.

[2] Arrow, K. and Debreu, G. (1954) Existence of an Equilibrium for a Competitive Economy. Econometrics, 22, 265-290.

http://dx.doi.org/10.2307/1907353

[3] Balasko, Y. (1988) The Equilibrium Manifold: Postmodern Developments in the Theory of General Economic Equilibrium. The MIT Press, Cambridge, Massachusetts.

[4] Jouini, E. (1993) The Graph of the Walras Correspondence. The Production Economies Cas. Journal of Mathematical Economics, 22, 139-147.

http://dx.doi.org/10.1016/0304-4068(93)90043-K

[5] Fuchs, G. (1974) Private Ownership Economies with a Finite Number of Equilibria. Journal of Mathematical Economics, 1, 141-158.

http://dx.doi.org/10.1016/0304-4068(74)90005-6

[6] Stiefenhofer, P. (2011) Equilibrium Structure of Production Economies with Uncertainty: The Natural Projection Approach. Discussion Papers in Economics, University of York, York, 7.

[7] Stiefenhofer, P. (2013) The Catastrophe Map of a Two Period Production Model with Uncertainty. Applied Mathematics, 4, No. 8A.

[8] Debreu, G. (1972) Smooth Preferences. Econometrica, 40, 603-615.

http://dx.doi.org/10.2307/1912956

[9] Lee, J.M. (2004) Introduction to Topological Manifolds. Springer, New York.

[10] Hirsch, M. (1972) Differential Topology. Springer Verlag, New York.

[11] Lee, J.M. (2000) Introduction to Smooth Manifolds. Springer, New York.

[12] Guillemin, V. and Pollack, A. (1974) Differential Topology. Prentice Hall, Upper Saddle River.