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Monotone concave operators: An application to the existence and uniqueness of solutions to the Bellman equation

Author

Listed:
  • Cuong Le Van

    (University of Paris 1, CNRS, Centre d’ Economie de la Sorbonne, Paris School of Economics)

  • Lisa Morhaim

    (University of Paris 1, CNRS, Centre d’ Economie de la Sorbonne, Paris School of Economics, Institute Math´ematique de Bourgogne)

  • Yiannis Vailakis

    (Department of Economics, University of Exeter)

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 (1964) and 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.

Suggested Citation

  • Cuong Le Van & Lisa Morhaim & Yiannis Vailakis, 2008. "Monotone concave operators: An application to the existence and uniqueness of solutions to the Bellman equation," Discussion Papers 0803, Exeter University, Department of Economics.
  • Handle: RePEc:exe:wpaper:0803
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    File URL: http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0803.pdf
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    References listed on IDEAS

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    1. Morand, Olivier F. & Reffett, Kevin L., 2003. "Existence and uniqueness of equilibrium in nonoptimal unbounded infinite horizon economies," Journal of Monetary Economics, Elsevier, vol. 50(6), pages 1351-1373, September.
    2. Le Van, Cuong & Vailakis, Yiannis, 2005. "Recursive utility and optimal growth with bounded or unbounded returns," Journal of Economic Theory, Elsevier, vol. 123(2), pages 187-209, August.
    3. 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.
    4. Manjira Datta & Leonard Mirman & Olivier Morand & Kevin Reffett, 2002. "Monotone Methods for Markovian Equilibrium in Dynamic Economies," Annals of Operations Research, Springer, vol. 114(1), pages 117-144, August.
    5. Peter A. Streufert, 1990. "Stationary Recursive Utility and Dynamic Programming under the Assumption of Biconvergence," Review of Economic Studies, Oxford University Press, vol. 57(1), pages 79-97.
    6. 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, September.
    7. 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.
    8. Coleman, Wilbur John, II, 1991. "Equilibrium in a Production Economy with an Income Tax," Econometrica, Econometric Society, vol. 59(4), pages 1091-1104, July.
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    Cited by:

    1. Massimo Marinacci & Luigi Montrucchio, 2017. "Unique Tarski Fixed Points," Working Papers 604, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.

    More about this item

    Keywords

    Dynamic Programming; Bellman Equation; Unbounded Returns;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

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