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Dynamic optimization and Skiba sets in economic examples

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  • W.-J. Beyn, T. Pampel, W.Semmler

Abstract

We discuss two optimization problems from economics. The first is a model of optimal investment and the second is a model of resource management. In both cases the time horizon is infinite and the optimal control variables are continuous. Typically, in these optimal control problems multiple steady states and periodic orbits occur. This leads to multiple solutions of the state-costate system each of which relates to a locally optimal strategy but has its own limiting behavior (stationary or periodic). Initial states that allow different optimal solutions with the same value of the objective function are called Skiba points. The set of Skiba points is of interest, because it provides thresholds for a global change of optimal strategies. We provide a systematic numerical method for calculating locally optimal solutions and Skiba points via boundary value problems. In parametric or higher dimensional systems Skiba curves (or manifolds) appear and we show how to follow them by a continuation process. We apply our method to the models above where Skiba sets consist of points or curves and where optimal solutions have different stationary or periodic asymptotic behavior.

Suggested Citation

  • W.-J. Beyn, T. Pampel, W.Semmler, 2001. "Dynamic optimization and Skiba sets in economic examples," Computing in Economics and Finance 2001 29, Society for Computational Economics.
    • Handle: RePEc:sce:scecf1:29
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    Citations

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    Cited by:

    1. Anne Sophie Crépin, 2003. "Management Challenges for Multiple-Species Boreal Forests," Working Papers 2003.106, Fondazione Eni Enrico Mattei.
    2. Gustav Feichtinger & Vladimir Veliov & Tsvetomir Tsachev, 2004. "Maximum Principle For Age And Duration Structured Systems: A Tool For Optimal Prevention And Treatment Of Hiv," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(1), pages 3-28.
    3. Caulkins, Jonathan P. & Feichtinger, Gustav & Grass, Dieter & Hartl, Richard F. & Kort, Peter M., 2011. "Two state capital accumulation with heterogenous products: Disruptive vs. non-disruptive goods," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 462-478, April.
    4. Anne Sophie Crépin, 2003. "Threshold Effects in Coral Reef Fisheries," Working Papers 2003.107, Fondazione Eni Enrico Mattei.
    5. Charles I. Jones, 2022. "The End of Economic Growth? Unintended Consequences of a Declining Population," American Economic Review, American Economic Association, vol. 112(11), pages 3489-3527, November.
    6. Marschall, Paul, 2001. "Lernen und Lebensstilwandel in Transformationsökonomien," Wirtschaftswissenschaftliche Diskussionspapiere 07/2001, University of Greifswald, Faculty of Law and Economics.

    More about this item

    Keywords

    Skiba sets; multiple steady states; periodic orbits;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity

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