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

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  • 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)

Abstract

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

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Keywords: Dynamic programming; Bellman equation; Unbounded returns;

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  1. Olivier F. Morand & Kevin L. Reffett, 2002. "Existence and Uniqueness of Equilibrium in Nonoptimal Unbounded Infinite Horizon Economies," Tinbergen Institute Discussion Papers, Tinbergen Institute 02-085/2, Tinbergen Institute.
  2. 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, Elsevier, vol. 103(2), pages 377-410, April.
  3. Le Van, Cuong & Vailakis, Yiannis, 2005. "Recursive utility and optimal growth with bounded or unbounded returns," Journal of Economic Theory, Elsevier, Elsevier, vol. 123(2), pages 187-209, August.
  4. Manjira Datta & Leonard J. Mirman & Olivier F. Morand & Kevin L. Reffett, 2002. "Monotone Methods for Markovian Equilibrium in Dynamic Economies," Tinbergen Institute Discussion Papers, Tinbergen Institute 02-086/2, Tinbergen Institute.
  5. John Kennan, 2001. "Uniqueness of Positive Fixed Points for Increasing Concave Functions on Rn: An Elementary Result," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(4), pages 893-899, October.
  6. Jorge DurÂn, 2000. "On dynamic programming with unbounded returns," Economic Theory, Springer, Springer, vol. 15(2), pages 339-352.
  7. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, Elsevier, vol. 50(2), pages 326-345, April.
  8. Coleman, Wilbur II, 2000. "Uniqueness of an Equilibrium in Infinite-Horizon Economies Subject to Taxes and Externalities," Journal of Economic Theory, Elsevier, Elsevier, vol. 95(1), pages 71-78, November.
  9. Coleman, Wilbur John, II, 1991. "Equilibrium in a Production Economy with an Income Tax," Econometrica, Econometric Society, Econometric Society, vol. 59(4), pages 1091-1104, July.
  10. Streufert, Peter A, 1990. "Stationary Recursive Utility and Dynamic Programming under the Assumption of Biconvergence," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 57(1), pages 79-97, January.
  11. Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, Econometric Society, vol. 71(5), pages 1519-1555, 09.
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