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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

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File URL: http://hdl.handle.net/10.1007/s00712-012-0270-z
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Article provided by Springer in its journal Journal of Economics.

Volume (Year): 107 (2012)
Issue (Month): 2 (October)
Pages: 101-128

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Handle: RePEc:kap:jeczfn:v:107:y:2012:i:2:p:101-128
Contact details of provider: Web page: http://www.springerlink.com/link.asp?id=108909

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  1. 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.
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  3. Christophe Chamley, 1991. "Externalities and Dynamics in Models of "Learning or Doing"," Boston University - Institute for Economic Development 17, Boston University, Institute for Economic Development.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
  9. Eric W. Bond & Ping Wang & Chong K. Yip, 1993. "A general two-sector model of endogenous growth with human and physical capital: balanced growth and transitional dynamics," Research Paper 9324, Federal Reserve Bank of Dallas.
  10. Danyang Xie, 2002. "Divergence in Economic Performance: Transitional Dynamics with Multiple Equilibria," GE, Growth, Math methods 0210002, EconWPA.
  11. 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 1042-67, December.
  12. Canton, E.J.F., 1996. "Business Cycles in a Two-Sector Model of Endogenous Growth," Discussion Paper 1996-116, .
  13. 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.
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