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On a parameterized system of nonlinear equations with economic applications

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  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

  • Yang, Z.F.

Abstract

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder’s fixed point theorem and Ky Fan’s coincidence theorem. An economic application with constrained equilibria is also discussed.

Suggested Citation

  • Talman, A.J.J. & Yang, Z.F., 2012. "On a parameterized system of nonlinear equations with economic applications," Other publications TiSEM 8233343d-0b60-428d-a20b-6, Tilburg University, School of Economics and Management.
  • Handle: RePEc:tiu:tiutis:8233343d-0b60-428d-a20b-6bcefbc8542f
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    References listed on IDEAS

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    1. P. J. J. Herings & G. A. Koshevoy & A. J. J. Talman & Z. Yang, 2004. "General Existence Theorem of Zero Points," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 375-394, February.
    2. Weddepohl, Claus, 1987. "Supply-constrained equilibria in economies with indexed prices," Journal of Economic Theory, Elsevier, vol. 43(2), pages 203-222, December.
    3. van der Laan, Gerard, 1980. "Equilibrium under Rigid Prices with Compensation for the consumers," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 21(1), pages 63-73, February.
    4. Dai, Y. & van der Laan, G. & Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron," Discussion Paper 1989-52, Tilburg University, Center for Economic Research.
    5. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
    6. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
    7. Ning Sun & Zaifu Yang, 2001. "On Fair Allocations and Indivisibilities," Cowles Foundation Discussion Papers 1347, Cowles Foundation for Research in Economics, Yale University.
    8. Ning Sun & Zaifu Yang, 2001. "Perfectly Fair Allocations with Indivisibilities," Cowles Foundation Discussion Papers 1318, Cowles Foundation for Research in Economics, Yale University.
    9. Jean-Jacques Herings & Dolf Talman & Zaifu Yang, 1996. "The Computation of a Continuum of Constrained Equilibria," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 675-696, August.
    10. Dehez, Pierre & Dreze, Jacques H., 1984. "On supply-constrained equilibria," Journal of Economic Theory, Elsevier, vol. 33(1), pages 172-182, June.
    11. Schalk, S., 1999. "Equilibrium theory : A salient approach," Other publications TiSEM 69a1bde7-60ee-425f-8013-2, Tilburg University, School of Economics and Management.
    12. Laan, Gerard van der, 1982. "Simplicial approximation of unemployment equilibria," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 83-97, January.
    13. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.
    14. Kurz, Mordecai, 1982. "Unemployment equilibrium in an economy with linked prices," Journal of Economic Theory, Elsevier, vol. 26(1), pages 100-123, February.
    15. Cox, Charles C, 1980. "The Enforcement of Public Price Controls," Journal of Political Economy, University of Chicago Press, vol. 88(5), pages 887-916, October.
    16. Talman, A.J.J. & Yamamoto, M., 2004. "Contimuum of zero points of a mapping on a compact, convex set," Other publications TiSEM 45f27466-e81f-412f-a975-4, Tilburg University, School of Economics and Management.
    17. Herings, P. Jean-Jacques, 1998. "On the existence of a continuum of constrained equilibria1," Journal of Mathematical Economics, Elsevier, vol. 30(3), pages 257-273, October.
    18. van der Laan, Gerard, 1984. "Supply-constrained fixed price equilibria in monetary economies," Journal of Mathematical Economics, Elsevier, vol. 13(2), pages 171-187, October.
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