IDEAS home Printed from https://ideas.repec.org/p/kob/dpaper/200.html
   My bibliography  Save this paper

On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming

Author

Listed:
  • 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 -∞+∞.

Suggested Citation

  • 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.
    • Handle: RePEc:kob:dpaper:200
    as

    Download full text from publisher

    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp200.pdf
    File Function: First version, 2007
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura, 2006. "Handbook on optimal growth (volume 1)," Post-Print halshs-00101345, HAL.
    2. Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July.
    3. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-723, May.
    4. David Gale, 1967. "On Optimal Development in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 1-18.
    5. Rose-Anne Dana & Cuong Le Van, 2006. "Optimal growth without discounting," Post-Print halshs-00101355, HAL.
    6. Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), 2006. "Handbook on Optimal Growth 1," Springer Books, Springer, number 978-3-540-32310-5, June.
    7. W. A. Brock, 1970. "On Existence of Weakly Maximal Programmes in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 37(2), pages 275-280.
    8. 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.
    9. Rose-Anne Dana & Cuong Le Van, 2006. "Optimal Growth Without Discounting," Springer Books, in: Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), Handbook on Optimal Growth 1, chapter 1, pages 1-17, Springer.
    10. repec:dau:papers:123456789/433 is not listed on IDEAS
    11. 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.
    12. Rose-Anne Dana & Cuong Le Van, 2006. "Optimal growth without discounting," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00101355, HAL.
    13. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
    14. Mordechai I. Henig, 1985. "The Principle of Optimality in Dynamic Programming with Returns in Partially Ordered Sets," Mathematics of Operations Research, INFORMS, vol. 10(3), pages 462-470, August.
    15. Le Van Cuong & Dana Rose-anne, 1988. "Note on the bellman equation of the overtaking criterion (a)," CEPREMAP Working Papers (Couverture Orange) 8820, CEPREMAP.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Agnieszka Wiszniewska-Matyszkiel & Rajani Singh, 2020. "When Inaccuracies in Value Functions Do Not Propagate on Optima and Equilibria," Mathematics, MDPI, vol. 8(7), pages 1-25, July.
    2. 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.
    3. Takashi Kamihigashi & Cuong Le Van, 2015. "Necessary and Sufficient Conditions for a Solution of the Bellman Equation to be the Value Function: A General Principle," Post-Print halshs-01159177, HAL.
    4. 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.
    5. Ronaldo Carpio & Takashi Kamihigashi, 2015. "Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Infinite-Horizon Dynamic Programming in Discrete Time," Discussion Paper Series DP2015-11, Research Institute for Economics & Business Administration, Kobe University.
    6. Ronaldo Carpio & Takashi Kamihigashi, 2016. "Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Deterministic Dynamic Programming in Discrete Time," Discussion Paper Series DP2016-04, Research Institute for Economics & Business Administration, Kobe University.
    7. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
    8. Takashi Kamihigashi, 2014. "An order-theoretic approach to dynamic programming: an exposition," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 13-21, April.
    9. Takashi Kamihigashi, 2013. "Ergodic chaos and aggregate stability: A deterministic discrete-choice model of wealth distribution dynamics," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 45-56, March.
    10. 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, March.
    11. 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, March.
    12. Luis A. Alcala, 2016. "On the time consistency of collective preferences," Papers 1607.02688, arXiv.org, revised Jul 2018.
    13. Ronaldo Carpio & Takashi Kamihigashi, 2019. "Fast Value Iteration: An Application of Legendre-Fenchel Duality to a Class of Deterministic Dynamic Programming Problems in Discrete Time," Discussion Paper Series DP2019-24, Research Institute for Economics & Business Administration, Kobe University.
    14. Masayuki Yao, 2016. "Recursive Utility and the Solution to the Bellman Equation," Discussion Paper Series DP2016-08, Research Institute for Economics & Business Administration, Kobe University.
    15. Takashi Kamihigashi & Masayuki Yao, 2015. "Infnite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-32, Research Institute for Economics & Business Administration, Kobe University.
    16. Takashi Kamihigashi & Masayuki Yao, 2016. "Infinite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle and a Penalty Method," Discussion Paper Series DP2016-05, Research Institute for Economics & Business Administration, Kobe University, revised May 2016.
    17. Takashi Kamihigashi & Masayuki Yao, 2015. "Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-15, Research Institute for Economics & Business Administration, Kobe University.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Banerjee, Kuntal, 2017. "Suppes–Sen maximality of cyclical consumption: The neoclassical growth model," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 51-65.
    2. Julio Dávila, 2018. "Internalizing fertility and education externalities on capital returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(2), pages 343-373, August.
    3. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
    4. Takashi Kamihigashi, 2015. "A Simple No-Bubble Theorem for Deterministic Sequential Economies," Discussion Paper Series DP2015-38, Research Institute for Economics & Business Administration, Kobe University.
    5. Evstigneev, Igor & Taksar, Michael, 2009. "Dynamic interaction models of economic equilibrium," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 166-182, January.
    6. Kamihigashi, Takashi, 2018. "A Simple optimality-based no-bubble theorem for deterministic sequential economies with strictly monotone preferences," Mathematical Social Sciences, Elsevier, vol. 91(C), pages 36-41.
    7. Janusz Matkowski & Andrzej Nowak, 2011. "On discounted dynamic programming with unbounded returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 455-474, April.
    8. Takashi Kamihigashi, 2015. "A Simple No-Bubble Theorem for Deterministic Dynamic Economies," Discussion Paper Series DP2015-24, Research Institute for Economics & Business Administration, Kobe University.
    9. Anna Jaśkiewicz & Janusz Matkowski & Andrzej Nowak, 2014. "On variable discounting in dynamic programming: applications to resource extraction and other economic models," Annals of Operations Research, Springer, vol. 220(1), pages 263-278, September.
    10. Takashi Kamihigashi, 2016. "A Simple Optimality-Based No-Bubble Theorem for Deterministic Sequential Economies," Discussion Paper Series DP2016-22, Research Institute for Economics & Business Administration, Kobe University.
    11. Lee H. Endress & Sittidaj Pongkijvorasin & James Roumasset & Christopher Wada, 2013. "Intergenerational Equity with Individual Impatience in an OLG Model of Optimal and Sustainable Growth," Working Papers 2013-9, University of Hawaii Economic Research Organization, University of Hawaii at Manoa.
    12. 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.
    13. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, April.
    14. Endress, Lee H. & Pongkijvorasin, Sittidaj & Roumasset, James & Wada, Christopher A., 2014. "Intergenerational equity with individual impatience in a model of optimal and sustainable growth," Resource and Energy Economics, Elsevier, vol. 36(2), pages 620-635.
    15. M. Ali Khan & Tapan Mitra, 2005. "On choice of technique in the Robinson–Solow–Srinivasan model," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(2), pages 83-110, June.
    16. Kamihigashi, Takashi, 2005. "Necessity of the transversality condition for stochastic models with bounded or CRRA utility," Journal of Economic Dynamics and Control, Elsevier, vol. 29(8), pages 1313-1329, August.
    17. Suen, Richard M. H., 2009. "Bounding the CRRA Utility Functions," MPRA Paper 13260, University Library of Munich, Germany.
    18. 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.
    19. Fleurbaey, Marc & Michel, Philippe, 2003. "Intertemporal equity and the extension of the Ramsey criterion," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 777-802, September.
    20. Le Van, Cuong & Cagri Saglam, H., 2004. "Optimal growth models and the Lagrange multiplier," Journal of Mathematical Economics, Elsevier, vol. 40(3-4), pages 393-410, June.

    More about this item

    Keywords

    Bellman equation; Dynamic programming; Principle of optimality; Value function;
    All these keywords.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:kob:dpaper:200. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Office of Promoting Research Collaboration, Research Institute for Economics & Business Administration, Kobe University (email available below). General contact details of provider: https://edirc.repec.org/data/rikobjp.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.