IDEAS home Printed from https://ideas.repec.org/p/kob/dpaper/115.html
   My bibliography  Save this paper

Necessity of Transversality Conditions for Stochastic Problems

Author

Listed:
  • Takashi Kamihigashi

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

Abstract

This paper establishes (i) an extension of Michel's (1990, Theorem 1) necessity result to an abstract reduced-form model, (ii) a generalization of the results of Weitzman (1973) and Ekeland and Scheinkman (1986), and (iii) a new result that is useful particularly in the case of homogeneous returns. These results are shown for an extremely general discrete-time reduced-form model that does not assume differentiability, continuity, or concavity, and that imposes virtually no restriction on the state spaces. The three results are futher extended to a stochastic reduced-form model. The stochastic extensions are easily accomplished since our deterministic model is so general that the stochastic model is in fact a special case of the deterministic model. We apply our stochastic results to a stochastic reduced-form model with homogeneous returns and a general type of stochastic growth model with CRRA utility.

Suggested Citation

  • Takashi Kamihigashi, 2000. "Necessity of Transversality Conditions for Stochastic Problems," Discussion Paper Series 115, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:115
    as

    Download full text from publisher

    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp115.pdf
    File Function: First version, 2000
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Zilcha, Itzhak, 1976. "Characterization by prices of optimal programs under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 173-183, July.
    2. K. Shell, 1968. "Applications of Pontryagin's Maximum Principle of Economics," Working papers 16, Massachusetts Institute of Technology (MIT), Department of Economics.
    3. Montrucchio, Luigi & Privileggi, Fabio, 2001. "On Fragility of Bubbles in Equilibrium Asset Pricing Models of Lucas-Type," Journal of Economic Theory, Elsevier, vol. 101(1), pages 158-188, November.
    4. Manuel S. Santos & Michael Woodford, 1997. "Rational Asset Pricing Bubbles," Econometrica, Econometric Society, vol. 65(1), pages 19-58, January.
    5. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-723, May.
    6. Takashi Kamihigashi, 2002. "A simple proof of the necessity of the transversality condition," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(2), pages 427-433.
    7. Rebelo, Sergio, 1991. "Long-Run Policy Analysis and Long-Run Growth," Journal of Political Economy, University of Chicago Press, vol. 99(3), pages 500-521, June.
    8. Kamihigashi, Takashi, 2003. "Necessity of transversality conditions for stochastic problems," Journal of Economic Theory, Elsevier, vol. 109(1), pages 140-149, March.
    9. William A. Brock & Leonard J. Mirman, 2001. "Optimal Economic Growth And Uncertainty: The Discounted Case," Chapters, in: W. D. Dechert (ed.), Growth Theory, Nonlinear Dynamics and Economic Modelling, chapter 1, pages 3-37, Edward Elgar Publishing.
    10. Yano, Makoto, 1989. "Comparative statics in dynamic stochastic models : Differential analysis of a stochastic modified golden rule state in a banach space," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 169-185, April.
    11. Takashi Kamihigashi, 2000. "A simple proof of Ekeland and Scheinkman's result on the necessity of a transversality condition," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(2), pages 463-468.
    12. Bezalel Peleg & Harl E. Ryder, 1972. "On Optimal Consumption Plans in a Multi-sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 39(2), pages 159-169.
    13. 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.
    14. Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, vol. 69(4), pages 995-1012, July.
    15. King, Robert G. & Plosser, Charles I. & Rebelo, Sergio T., 1988. "Production, growth and business cycles : I. The basic neoclassical model," Journal of Monetary Economics, Elsevier, vol. 21(2-3), pages 195-232.
    16. Ivar Ekeland & José Alexandre Scheinkman, 1986. "Transversality Conditions for Some Infinite Horizon Discrete Time Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 216-229, May.
    17. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-272, March.
    18. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-1445, November.
    19. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
    20. HALKIN, Hubert, 1974. "Necessary conditions for optimal control problems with infinite horizons," LIDAM Reprints CORE 193, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    21. Takekuma, Shin-Ichi, 1992. "Optimal Growth under Uncertainty: A Complete Characterization of Weakly Maximal Programs, 不確実性下の最適成長：弱極大計画の完全特徴付け," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 33(2), pages 169-182, December.
    22. Martin L. Weitzman, 1973. "Duality Theory for Infinite Horizon Convex Models," Management Science, INFORMS, vol. 19(7), pages 783-789, March.
    23. Zilcha, Itzhak, 1978. "Transversality Condition in a Multi-Sector Economy under Uncertainty," Econometrica, Econometric Society, vol. 46(3), pages 515-525, May.
    24. Takashi Kamihigashi, 1998. "Uniqueness of asset prices in an exchange economy with unbounded utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(1), pages 103-122.
    25. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. Ravi Kashyap, 2016. "Solving the Equity Risk Premium Puzzle and Inching Towards a Theory of Everything," Papers 1604.04872, arXiv.org, revised Sep 2019.
    3. Montrucchio, Luigi & Privileggi, Fabio, 2001. "On Fragility of Bubbles in Equilibrium Asset Pricing Models of Lucas-Type," Journal of Economic Theory, Elsevier, vol. 101(1), pages 158-188, November.
    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. 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.
    6. 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.
    7. Mitra, Tapan & Roy, Santanu, 2017. "Optimality of Ramsey–Euler policy in the stochastic growth model," Journal of Economic Theory, Elsevier, vol. 172(C), pages 1-25.
    8. 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.
    9. Tapan Mitra & Santanu Roy, 2022. "Propensity to consume and the optimality of Ramsey–Euler policies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(1), pages 55-89, February.
    10. Takashi Kamihigashi, 2003. "Necessity of the Transversality Condition for Stochastic Models with CRRA Utility," Discussion Paper Series 137, Research Institute for Economics & Business Administration, Kobe University.
    11. Takashi Kamihigashi, 2015. "A Simple No-Bubble Theorem," Discussion Paper Series DP2015-03, Research Institute for Economics & Business Administration, Kobe University.
    12. Gourieroux, C. & Jasiak, J. & Monfort, A., 2020. "Stationary bubble equilibria in rational expectation models," Journal of Econometrics, Elsevier, vol. 218(2), pages 714-735.
    13. Takashi Kamihigashi, 2008. "The spirit of capitalism, stock market bubbles and output fluctuations," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(1), pages 3-28, March.
    14. Buiter, Willem H. & Sibert, Anne C., 2007. "Deflationary Bubbles," Macroeconomic Dynamics, Cambridge University Press, vol. 11(4), pages 431-454, September.
    15. Takashi Kamihigashi, 2002. "A simple proof of the necessity of the transversality condition," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(2), pages 427-433.
    16. Bosi, Stefano & Lloyd-Braga, Teresa & Nishimura, Kazuo, 2021. "Externalities of human capital," Mathematical Social Sciences, Elsevier, vol. 112(C), pages 145-158.
    17. Lars J. Olson & Santanu Roy, 2006. "Theory of Stochastic Optimal Economic Growth," Springer Books, in: Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), Handbook on Optimal Growth 1, chapter 11, pages 297-335, Springer.
    18. Takashi Kamihigashi & Ryonghun Im, 2022. "Two Types of Asset Bubbles in a Small Open Economy," Discussion Paper Series DP2022-15, Research Institute for Economics & Business Administration, Kobe University.
    19. Takashi Kamihigashi, 2006. "Transversality Conditions and Dynamic Economic Behavior," Discussion Paper Series 180, Research Institute for Economics & Business Administration, Kobe University.
    20. Claudio Mattalia, 2003. "Existence of solutions and asset pricing bubbles in general equilibrium models," ICER Working Papers - Applied Mathematics Series 02-2003, ICER - International Centre for Economic Research.

    More about this item

    Keywords

    Transversality condition; Stochastic optimization; Stochastic reduced-form model; Homogeneous returns; Stochastic growth model;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

    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:115. 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.