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An Extension of the Sard-Smale Theorem to Domains with an Empty Interior

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  • Accinelli Elvio
  • Covarrubias Enrique

Abstract

A stumbling block in the modelling of competitive markets with commodity and price spaces of infinite dimensions, arises from having positive cones with an empty interior. This issue precludes the use of tools of differential analysis, ranging from the definition of a derivative, to the use of more sophisticated results needed to understand determinacy of equilibria and, more generally, the structure of the equilibrium set. To overcome these issues, this paper extends the Preimage Theorem and the Sard-Smale Theorem to maps between spaces that may have an empty interior.

Suggested Citation

  • Accinelli Elvio & Covarrubias Enrique, 2013. "An Extension of the Sard-Smale Theorem to Domains with an Empty Interior," Working Papers 2013-23, Banco de México.
    • Handle: RePEc:bdm:wpaper:2013-23
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    References listed on IDEAS

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    1. Kehoe, Timothy J & Levine, David K, 1985. "Comparative Statics and Perfect Foresight in Infinite Horizon Economies," Econometrica, Econometric Society, vol. 53(2), pages 433-453, March.
    2. Balasko, Yves, 1997. "Equilibrium analysis of the infinite horizon model with smooth discounted utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 783-829, May.
    3. Darrell Duffie & Chi-Fu Huang, 2005. "Implementing Arrow-Debreu Equilibria By Continuous Trading Of Few Long-Lived Securities," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 4, pages 97-127, World Scientific Publishing Co. Pte. Ltd..
    4. Balasko, Yves, 1975. "Some results on uniqueness and on stability of equilibrium in general equilibrium theory," Journal of Mathematical Economics, Elsevier, vol. 2(2), pages 95-118.
    5. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    6. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    7. Magill, Michael J. P. & Shafer, Wayne J., 1990. "Characterisation of generically complete real asset structures," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 167-194.
    8. Chichilnisky, Graciela & Heal, Geoffrey, 1996. "On the existence and the structure of the pseudo-equilibrium manifold," Journal of Mathematical Economics, Elsevier, vol. 26(2), pages 171-186.
    9. Mas-Colell, Andreu & Zame, William R., 1991. "Equilibrium theory in infinite dimensional spaces," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 34, pages 1835-1898, Elsevier.
    10. Balasko, Yves, 1997. "The natural projection approach to the infinite-horizon model," Journal of Mathematical Economics, Elsevier, vol. 27(3), pages 251-263, April.
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    More about this item

    JEL classification:

    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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