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Numerical solution of optimal control problems with constant control delays

  • Ulrich Brandt-Pollmann

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

  • Ralph Winkler

    ()

    (Center of Economic Research (CER-ETH) at ETH Zürich)

  • Sebastian Sager

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

  • Ulf Moslener

    (Centre for European Economic Research (ZEW), Mannheim, Germany)

  • Johannes P. Schlöder

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

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Paper provided by CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich in its series CER-ETH Economics working paper series with number 06/59.

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Length: 26 pages
Date of creation: Oct 2006
Date of revision:
Handle: RePEc:eth:wpswif:06-59
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  2. Boucekkine, Raouf & Germain, Marc & Licandro, Omar, 1997. "Replacement Echoes in the Vintage Capital Growth Model," Journal of Economic Theory, Elsevier, vol. 74(2), pages 333-348, June.
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  4. Raouf Boucekkine & Omar Licandro & Christopher Paul, . "Differential-Difference Equations in Economics: On the Numerical Solution of Vintage Capital Growth Models," Computing in Economics and Finance 1996 _036, Society for Computational Economics.
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  10. Fabrice Collard & Omar Licandro & Luis A. Puch, . "The short-run dynamics of optimal growth models with delays," Working Papers 2004-05, FEDEA.
  11. Patrick Asea & Paul J. Zak, 1997. "Time-to-Build and Cycles," UCLA Economics Working Papers 767, UCLA Department of Economics.
  12. Boyer, Marcel, 1978. "A Habit Forming Optimal Growth Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 19(3), pages 585-609, October.
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  14. Winkler, Ralph & Brandt-Pollmann, Ulrich & Moslener, Ulf & Schlöder, Johannes, 2005. "On the Transition from Instantaneous to Time-Lagged Capital Accumilation: The Case of Leontief Type Production Functions," ZEW Discussion Papers 05-30, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
  15. de la Croix, David & Licandro, Omar, 1997. "Life expectancy and endogenous growth," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 1997029, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
  16. El-Hodiri, Mohamed A & Loehman, Edna & Whinston, Andrew B, 1972. "An Optimal Growth Model with Time Lags," Econometrica, Econometric Society, vol. 40(6), pages 1137-46, November.
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