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On the principle of optimality for nonstationary deterministic dynamic programming

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  • Takashi Kamihigashi

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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Bibliographic Info

Article provided by The International Society for Economic Theory in its journal International Journal of Economic Theory.

Volume (Year): 4 (2008)
Issue (Month): 4 ()
Pages: 519-525

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Handle: RePEc:bla:ijethy:v:4:y:2008:i:4:p:519-525

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References

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  1. LE VAN, Cuong & MORHAIM, Lisa, 2001. "Optimal growth models with bounded or unbounded returns: a unifying approach," CORE Discussion Papers 2001034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. 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.
  3. 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.
  4. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-23, May.
  5. Dana, Rose-Anne & Le Van, Cuong, 2006. "Optimal growth without discounting," Economics Papers from University Paris Dauphine 123456789/433, Paris Dauphine University.
  6. 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.
  7. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
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Citations

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Cited by:
  1. 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.
  2. Takashi Kamihigashi & Kevin Reffett & Masayuki Yao, 2014. "An Application of Kleene's Fixed Point Theorem to Dynamic Programming: A Note," Working Papers 2014-398, Department of Research, Ipag Business School.
  3. Takashi Kamihigashi, 2012. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2012-05, Research Institute for Economics & Business Administration, Kobe University.
  4. Robert A. Becker, 2012. "Optimal growth with heterogeneous agents and the twisted turnpike: An example," International Journal of Economic Theory, The International Society for Economic Theory, vol. 8(1), pages 27-47, 03.
  5. Takashi Kamihigashi, 2012. "Elementary Results on Solutions to the Bellman Equation of Dynamic Programming: Existence, Uniqueness, and Convergence," Discussion Paper Series DP2012-31, Research Institute for Economics & Business Administration, Kobe University.

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