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Necessity of the Transversality Condition for Stochastic Models with Bounded or CRRA Utility

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  • Takashi Kamihigashi

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

Abstract

This paper shows that the standard transversality condition (STVC) is nec-essary for optimality in stochastic models with bounded or constant-relative-risk- aversion (CRRA) utility under fairly general conditions. We consider an infinite-horizon stochastic maximization problem that takes a general form of a multi-sector growth model with a single consumption good. We show that the STVC is necessary if utility is bounded or logarithmic. We also show that the STVC is necessary in the case of non-logarithmic CRRA utility as long as lifetime utility is finite at the optimum. These results apply to various stochastic growth models, including real business cycle models with endoge-nous labor supply. Since unbounded utility functions that do not belong to the CRRA class are rather rare in applications, our results provide a fairly complete set of solutions regarding necessity of the STVC in practice.

Suggested Citation

  • Takashi Kamihigashi, 2004. "Necessity of the Transversality Condition for Stochastic Models with Bounded or CRRA Utility," Discussion Paper Series 162, Research Institute for Economics & Business Administration, Kobe University, revised Oct 2004.
    • Handle: RePEc:kob:dpaper:162
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    Cited by:

    1. Jean-Bernard Chatelain & Kirsten Ralf, 2012. "The Failure Of Financial Macroeconomics And What To Do About It," Manchester School, University of Manchester, vol. 80, pages 21-53, September.
    2. Takashi Kamihigashi, 2015. "Multiple interior steady states in the Ramsey model with elastic labor supply," International Journal of Economic Theory, The International Society for Economic Theory, vol. 11(1), pages 25-37, March.
    3. Karim Azizi & Nicolas Canry & Jean-Bernard Chatelain & Bruno Tinel, 2013. "Government Solvency, Austerity and Fiscal Consolidation in the OECD: A Keynesian Appraisal of Transversality and No Ponzi Game Conditions," Working Papers hal-00818474, HAL.
    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. Jean-Bernard Chatelain & Bruno Tinel & Karim Azizi & Nicolas Canry, 2012. "Are the No-Ponzi Game and the Transversality Conditions Relevant for Public Debt? A Keynesian Appraisal," Post-Print hal-00686788, HAL.
    6. Takashi Kamihigashi, 2006. "Transversality Conditions and Dynamic Economic Behavior," Discussion Paper Series 180, Research Institute for Economics & Business Administration, Kobe University.
    7. Adam, Klaus & Merkel, Sebastian, 2019. "Stock Price Cycles and Business Cycles," CEPR Discussion Papers 13866, C.E.P.R. Discussion Papers.
    8. Vasilev, Aleksandar, 2015. "The welfare effect of at income tax reform: the case of Bulgaria. Technical Appendix," EconStor Research Reports 124168, ZBW - Leibniz Information Centre for Economics.
    9. 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.
    10. Gourieroux, C. & Jasiak, J. & Monfort, A., 2020. "Stationary bubble equilibria in rational expectation models," Journal of Econometrics, Elsevier, vol. 218(2), pages 714-735.
    11. Takashi Kamihigashi, 2017. "Regime-Switching Sunspot Equilibria in a One-Sector Growth Model with Aggregate Decreasing Returns and Small Externalities," Studies in Economic Theory, in: Kazuo Nishimura & Alain Venditti & Nicholas C. Yannelis (ed.), Sunspots and Non-Linear Dynamics, chapter 0, pages 125-146, Springer.
    12. Takashi Kamihigashi, 2016. "A Generalization of Fatou's Lemma for Extended Real-Valued Functions on σ-Finite Measure Spaces: With an Application to Infinite-Horizon Optimization in Discrete Time," Discussion Paper Series DP2016-37, Research Institute for Economics & Business Administration, Kobe University, revised Jan 2017.
    13. Takashi Kamihigashi, 2015. "A Simple No-Bubble Theorem," Discussion Paper Series DP2015-03, Research Institute for Economics & Business Administration, Kobe University.
    14. 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.
    15. 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.
    16. 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.
    17. Travis D. Nesmith, 2005. "Solving stochastic money-in-the-utility-function models," Finance and Economics Discussion Series 2005-52, Board of Governors of the Federal Reserve System (U.S.).
    18. 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.
    19. Fu, Wentao & Le Riche, Antoine, 2021. "Endogenous growth model with Bayesian learning and technology selection," Mathematical Social Sciences, Elsevier, vol. 114(C), pages 58-71.

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    More about this item

    Keywords

    Transversality condition; Stochastic optimization; Stochastic growth; CRRA; Real business cycle;
    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

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