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Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations


  • Dirk Bethmann


  • Markus Reiß



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

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    References listed on IDEAS

    1. 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.
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    3. 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.
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    5. 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.
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    7. 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.
    8. 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.
    9. 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.
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    12. 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.
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    More about this item


    Transitional dynamics; Value function approach; Symmetric equilibrium; Initial value problem; U-shaped growth rates; C61; O41;

    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


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