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On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming

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  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

Abstract

This note studies a general nonstationary infinite-horizon optimization problem in discrete time. We allow the state space in each period to be an arbitrary set, and the return function in each period to be unbounded. We do not require discounting, and do not require the constraint correspondence in each period to be nonempty-valued. The objective function is defined as the limit superior or inferior of the finite sums of return functions. We show that the sequence of time-indexed value functions satisfies the Bellman equation if and only if its right-hand side is well defined, i.e., it does not involve -∞+∞.

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File URL: http://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp200.pdf
File Function: First version, 2007
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Bibliographic Info

Paper provided by Research Institute for Economics & Business Administration, Kobe University in its series Discussion Paper Series with number 200.

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Length: 9 pages
Date of creation: Jan 2007
Date of revision:
Handle: RePEc:kob:dpaper:200

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Keywords: Bellman equation; Dynamic programming; Principle of optimality; Value function;

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References

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  1. Le Van, C. & Morhaim, L., 2000. "Optimal Growth Models with Bounded or Unbounded Returns : a Unifying Approach," Papiers d'Economie Mathématique et Applications 2000.64, Université Panthéon-Sorbonne (Paris 1).
  2. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-23, May.
  3. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
  4. 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.
  5. Dana, Rose-Anne & Le Van, Cuong, 2006. "Optimal growth without discounting," Economics Papers from University Paris Dauphine 123456789/433, Paris Dauphine University.
  6. Brock, William A, 1970. "On Existence of Weakly Maximal Programmes in a Multi-Sector Economy," Review of Economic Studies, Wiley Blackwell, vol. 37(2), pages 275-80, April.
  7. McKenzie, Lionel W., 2005. "Optimal economic growth, turnpike theorems and comparative dynamics," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 2, volume 3, chapter 26, pages 1281-1355 Elsevier.
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Citations

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Cited by:
  1. Takashi Kamihigashi, 2013. "Elementary Results on Solutions to the Bellman Equation of Dynamic Programming:Existence, Uniqueness, and Convergence," Discussion Paper Series DP2013-35, Research Institute for Economics & Business Administration, Kobe University, revised Dec 2013.
  2. Takashi Kamihigashi & Kevin REFFETT & Kevin REFFETT, 2014. "Partial Stochastic Dominance," Discussion Paper Series DP2014-23, Research Institute for Economics & Business Administration, Kobe University.
  3. Takashi Kamihigashi, 2011. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2011-23, Research Institute for Economics & Business Administration, Kobe University.
  4. Robert Becker, 2011. "Optimal Growth with Heterogeneous Agents and the Twisted Turnpike: An Example," Caepr Working Papers 2011-008, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
  5. Takashi Kamihigashi, 2013. "An Order-Theoretic Approach to Dynamic Programming: An Exposition," Discussion Paper Series DP2013-29, Research Institute for Economics & Business Administration, Kobe University, revised Nov 2013.

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