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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

    (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.

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File URL: http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0803.pdf
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Bibliographic Info

Paper provided by Exeter University, Department of Economics in its series Discussion Papers with number 0803.

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Date of creation: 2008
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Handle: RePEc:exe:wpaper:0803

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Keywords: Dynamic Programming; Bellman Equation; Unbounded Returns;

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  1. Manjira Datta & Leonard Mirman & Kevin Reffett, . "Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor," Working Papers 2132846, Department of Economics, W. P. Carey School of Business, Arizona State University.
  2. 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.
  3. 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.
  4. Kevin Reffett & Manjira Datta & Leonard Mirman & Olivier Morand, . "Monotone Methods for Markovian Equilibrium in Dynamic Economies," Working Papers 2133476, Department of Economics, W. P. Carey School of Business, Arizona State University.
  5. Duran, Jorge, 1997. "On dynamic programming with unbounded returns," CEPREMAP Working Papers (Couverture Orange) 9710, CEPREMAP.
  6. Kennan,J., 2001. "Uniqueness of positive fixed points for increasing concave functions on Rn : an elementary result," Working papers 2, Wisconsin Madison - Social Systems.
  7. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, vol. 50(2), pages 326-345, April.
  8. 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.
  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. Wilbur John Coleman II, 1989. "Equilibrium in a production economy with an income tax," International Finance Discussion Papers 366, Board of Governors of the Federal Reserve System (U.S.).
  11. 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.
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