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Monotone Concave Operators: An application to the existence and uniqueness of solutions to the Bellman equation

  • Cuong Le Van

    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

  • Lisa Morhaim

    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne, IMB - Institut de Mathématiques de Bourgogne - CNRS : UMR5584 - Université de Bourgogne)

  • Yiannis Vailakis

    (School of Business and Economics - Exeter university)

We propose a new approach to the issue of existence and uniqueness of solutions to the Bellman equation, exploiting an emerging class of methods, called monotone map methods, pioneered in the work of Krasnosel'skii-Zabreiko (1984). The approach is technically simple and intuitive. It is derived from geometric ideas related to the study of fixed points for monotone concave operators defined on partially order spaces.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number hal-00294828.

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Date of creation: 28 Jul 2008
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Handle: RePEc:hal:cesptp:hal-00294828
Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00294828
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  1. Duran, Jorge, 1997. "On Dynamic Programming with Unbounded Returns," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 1997033, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
  2. Coleman, Wilbur John, II, 1991. "Equilibrium in a Production Economy with an Income Tax," Econometrica, Econometric Society, vol. 59(4), pages 1091-1104, July.
  3. Olivier F. Morand & Kevin L. Reffett, 2001. "Existence and Uniqueness of Equilibrium in Nonoptimal Unbounded Infinite Horizon Economies," Working papers 2001-02, University of Connecticut, Department of Economics.
  4. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, vol. 50(2), pages 326-345, April.
  5. Coleman, Wilbur II, 2000. "Uniqueness of an Equilibrium in Infinite-Horizon Economies Subject to Taxes and Externalities," Journal of Economic Theory, Elsevier, vol. 95(1), pages 71-78, November.
  6. LE VAN, Cuong & VAILAKIS, Yiannis, 2002. "Recursive utility and optimal growth with bounded or unbounded returns," CORE Discussion Papers 2002055, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  7. Datta, Manjira & Mirman, Leonard J. & Reffett, Kevin L., 2002. "Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor," Journal of Economic Theory, Elsevier, vol. 103(2), pages 377-410, April.
  8. repec:dgr:uvatin:20020086 is not listed on IDEAS
  9. Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, vol. 71(5), pages 1519-1555, 09.
  10. Streufert, Peter A, 1990. "Stationary Recursive Utility and Dynamic Programming under the Assumption of Biconvergence," Review of Economic Studies, Wiley Blackwell, vol. 57(1), pages 79-97, January.
  11. Kennan,J., 2001. "Uniqueness of positive fixed points for increasing concave functions on Rn : an elementary result," Working papers 2, Wisconsin Madison - Social Systems.
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