IDEAS home Printed from https://ideas.repec.org/a/kap/jeczfn/v107y2012i2p101-128.html
   My bibliography  Save this article

Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations

Author

Listed:
  • Dirk Bethmann
  • Markus Reiß

Abstract

We introduce a simple method for computing value functions. The method is demonstrated by solving for transitional dynamics in the Uzawa and Lucas endogenous growth model. We use the value function approach to solve both the social planner’s optimization problem in the centralized economy and the representative agent’s optimization problem in the decentralized economy. The complexity of the Hamilton–Jacobi–Bellman equations is significantly reduced to an initial value problem for one ordinary differential equation. This approach allows us to find the optimal controls for the non-concave Hamiltonian in the centralized case and to identify the symmetric equilibrium in the decentralized case. Copyright Springer-Verlag 2012

Suggested Citation

  • Dirk Bethmann & Markus Reiß, 2012. "Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations," Journal of Economics, Springer, vol. 107(2), pages 101-128, October.
    • Handle: RePEc:kap:jeczfn:v:107:y:2012:i:2:p:101-128
    DOI: 10.1007/s00712-012-0270-z
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00712-012-0270-z
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00712-012-0270-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Benhabib Jess & Perli Roberto, 1994. "Uniqueness and Indeterminacy: On the Dynamics of Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 63(1), pages 113-142, June.
    2. Erik Canton, 2002. "Business cycles in a two-sector model of endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 19(3), pages 477-492.
    3. Dirk Bethmann, 2007. "A Closed-form Solution of the Uzawa-Lucas Model of Endogenous Growth," Journal of Economics, Springer, vol. 90(1), pages 87-107, January.
    4. Casey B. Mulligan & Xavier Sala-i-Martin, 1993. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," The Quarterly Journal of Economics, Oxford University Press, vol. 108(3), pages 739-773.
    5. Bond, Eric W. & Wang, Ping & Yip, Chong K., 1996. "A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Balanced Growth and Transitional Dynamics," Journal of Economic Theory, Elsevier, vol. 68(1), pages 149-173, January.
    6. Xie Danyang, 1994. "Divergence in Economic Performance: Transitional Dynamics with Multiple Equilibria," Journal of Economic Theory, Elsevier, vol. 63(1), pages 97-112, June.
    7. Caballe, Jordi & Santos, Manuel S, 1993. "On Endogenous Growth with Physical and Human Capital," Journal of Political Economy, University of Chicago Press, vol. 101(6), pages 1042-1067, December.
    8. Boucekkine, R. & Ruiz-Tamarit, J.R., 2008. "Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model," Journal of Mathematical Economics, Elsevier, vol. 44(1), pages 33-54, January.
    9. Brunner, Martin & Strulik, Holger, 2002. "Solution of perfect foresight saddlepoint problems: a simple method and applications," Journal of Economic Dynamics and Control, Elsevier, vol. 26(5), pages 737-753, May.
    10. Chamley, Christophe, 1993. "Externalities and Dynamics in Models of "Learning or Doing."," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 34(3), pages 583-609, August.
    11. Bethmann, Dirk, 2008. "The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents," Journal of Macroeconomics, Elsevier, vol. 30(1), pages 396-414, March.
    12. Casey B. Mulligan & Xavier Sala-i-Martin, 1991. "A Note on the Time-Elimination Method For Solving Recursive Dynamic Economic Models," NBER Technical Working Papers 0116, National Bureau of Economic Research, Inc.
    13. 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)

    Citations

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


    Cited by:

    1. Constantin Chilarescu, 2018. "The effect of externality on the transitional dynamics: the case of Lucas model," Economics Bulletin, AccessEcon, vol. 32767(3).
    2. Bella, Giovanni & Mattana, Paolo & Venturi, Beatrice, 2017. "Shilnikov chaos in the Lucas model of endogenous growth," Journal of Economic Theory, Elsevier, vol. 172(C), pages 451-477.

    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. Brito, Paulo & Venditti, Alain, 2010. "Local and global indeterminacy in two-sector models of endogenous growth," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 893-911, September.
    2. Reiß, Markus & Bethmann, Dirk, 2003. "Transitional Dynamics in the Uzawa-Lucas Model of Endogenous Growth," SFB 373 Discussion Papers 2003,17, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    3. Bella, Giovanni & Mattana, Paolo & Venturi, Beatrice, 2017. "Shilnikov chaos in the Lucas model of endogenous growth," Journal of Economic Theory, Elsevier, vol. 172(C), pages 451-477.
    4. Jaime Alonso-Carrera, 2001. "More on the dynamics in the endogenous growth model with human capital," Investigaciones Economicas, Fundación SEPI, vol. 25(3), pages 561-583, September.
    5. Arantza Gorostiaga & Jana Hromcová & Miguel-Ángel López-García, 2013. "Optimal taxation in the Uzawa–Lucas model with externality in human capital," Journal of Economics, Springer, vol. 108(2), pages 111-129, March.
    6. Pedro Garcia-Castrillo & Marcos Sanso, 2000. "Human Capital and Optimal Policy in a Lucas-type Model," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 3(4), pages 757-770, October.
    7. Benhabib, Jess & Gali, Jordi, 1995. "On growth and indeterminacy: some theory and evidence," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 43(1), pages 163-211, December.
    8. Ladron-de-Guevara, Antonio & Ortigueira, Salvador & Santos, Manuel S., 1997. "Equilibrium dynamics in two-sector models of endogenous growth," Journal of Economic Dynamics and Control, Elsevier, vol. 21(1), pages 115-143, January.
    9. Dirk Bethmann, 2007. "Homogeneity, Saddle Path Stability, and Logarithmic Preferences in Economic Models," Discussion Paper Series 0702, Institute of Economic Research, Korea University.
    10. Chilarescu, Constantin, 2011. "On the existence and uniqueness of solution to the Lucas–Uzawa model," Economic Modelling, Elsevier, vol. 28(1), pages 109-117.
    11. Garcia-Belenguer, Fernando, 2007. "Stability, global dynamics and Markov equilibrium in models of endogenous economic growth," Journal of Economic Theory, Elsevier, vol. 136(1), pages 392-416, September.
    12. De, Supriyo, 2014. "Intangible capital and growth in the ‘new economy’: Implications of a multi-sector endogenous growth model," Structural Change and Economic Dynamics, Elsevier, vol. 28(C), pages 25-42.
    13. Posch, Olaf & Trimborn, Timo, 2013. "Numerical solution of dynamic equilibrium models under Poisson uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2602-2622.
    14. Olaf Posch & Timo Trimborn, 2010. "Numerical solution of continuous-time DSGE models under Poisson uncertainty," Economics Working Papers 2010-08, Department of Economics and Business Economics, Aarhus University.
    15. Alfred Greiner & Willi Semmler, 1996. "Multiple steady states, indeterminacy, and cycles in a basic model of endogenous growth," Journal of Economics, Springer, vol. 63(1), pages 79-99, February.
    16. Kejak, Michal, 2003. "Stages of growth in economic development," Journal of Economic Dynamics and Control, Elsevier, vol. 27(5), pages 771-800, March.
    17. Paolo Mattana & Kazuo Nishimura & Tadashi Shigoka, 2012. "Homoclinic Bifurcation and Global Indeterminacy of Equilibrium in a Two-Sector Endogenous Growth Model," Springer Books, in: John Stachurski & Alain Venditti & Makoto Yano (ed.), Nonlinear Dynamics in Equilibrium Models, edition 127, chapter 0, pages 427-451, Springer.
    18. Shiro Kuwahara, 2017. "Multiple steady states and indeterminacy in the Uzawa–Lucas model with educational externalities," Journal of Economics, Springer, vol. 122(2), pages 173-190, October.
    19. Constantin Chilarescu, 2018. "The effect of externality on the transitional dynamics: the case of Lucas model," Economics Bulletin, AccessEcon, vol. 32767(3).
    20. Klaus Waelde, 1994. "Transitional dynamics, convergence and international capital flows in two-country models of innovation and growth," International Trade 9403002, University Library of Munich, Germany, revised 03 Jan 1996.

    More about this item

    Keywords

    Transitional dynamics; Value function approach; Symmetric equilibrium; Initial value problem; U-shaped growth rates; C61; O41;
    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:kap:jeczfn:v:107:y:2012:i:2:p:101-128. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.