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Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case

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Author Info
Juan Pablo RincÛn-Zapatero
Carlos RodrÌguez-Palmero

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Abstract

We study the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns. We introduce a new approach based both on consideration of a metric on the space of all continuous functions over the state space, and on the application of some metric fixed point theorems. With appropriate conditions we prove uniqueness of solutions with respect to the whole space of continuous functions. Furthermore, the paper provides new sufficient conditions for the existence of solutions that can be applied to fairly general models. It is also proven that the fixed point coincides with the value function and that it can be approached by successive iterations of the Bellman operator. Copyright The Econometric Society 2003.

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Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 71 (2003)
Issue (Month): 5 (09)
Pages: 1519-1555
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Handle: RePEc:ecm:emetrp:v:71:y:2003:i:5:p:1519-1555

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  1. Ryoji Hiraguchi, 2004. "Some Foundations for Multiplicative Habits Models," Economics Working Paper Archive 516, The Johns Hopkins University,Department of Economics. [Downloadable!]
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  2. Takashi Kamihigashi, 2007. "On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming," Discussion Paper Series 200, Research Institute for Economics & Business Administration, Kobe University. [Downloadable!]
  3. Matkowski, Janusz & Nowak, Andrzej S., 2008. "On Discounted Dynamic Programming with Unbounded Returns," MPRA Paper 12215, University Library of Munich, Germany. [Downloadable!]
  4. Vailakis, Yiannis & Martins-da-Rocha, V. F., 2008. "Existence and Uniqueness of a Fixed-Point for Local Contractions," Economics Working Papers (Ensaios Economicos da EPGE) 677, Graduate School of Economics, Getulio Vargas Foundation (Brazil). [Downloadable!]
  5. Juan Rincón-Zapatero & Carlos Rodríguez-Palmero, 2007. "Recursive utility with unbounded aggregators," Economic Theory, Springer, vol. 33(2), pages 381-391, November. [Downloadable!] (restricted)
  6. Richard M. H. Suen, 2009. "Bounding the CRRA Utility Functions," Working Papers 200902, University of California at Riverside, Department of Economics, revised Feb 2009. [Downloadable!]
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