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Numerical computation of the optimal vector field: Exemplified by a fishery model

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  • Grass, D.

Abstract

Numerous optimal control models analyzed in economics are formulated as discounted infinite time horizon problems, where the defining functions are nonlinear as well in the states as in the controls. As a consequence solutions can often only be found numerically. Moreover, the long run optimal solutions are mostly limit sets like equilibria or limit cycles. Using these specific solutions a BVP approach together with a continuation technique is used to calculate the parameter dependent dynamic structure of the optimal vector field. We use a one dimensional optimal control model of a fishery to exemplify the numerical techniques. But these methods are applicable to a much wider class of optimal control problems with a moderate number of state and control variables.

Suggested Citation

  • Grass, D., 2012. "Numerical computation of the optimal vector field: Exemplified by a fishery model," Journal of Economic Dynamics and Control, Elsevier, vol. 36(10), pages 1626-1658.
  • Handle: RePEc:eee:dyncon:v:36:y:2012:i:10:p:1626-1658
    DOI: 10.1016/j.jedc.2012.04.006
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    References listed on IDEAS

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    1. Jonathan P. Caulkins & Gustav Feichtinger & Dieter Grass & Michael Johnson & Gernot Tragler & Yuri Yegorov, 2005. "Placing the poor while keeping the rich in their place," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 13(1), pages 1-34, July.
    2. Levy, Amnon & Neri, Frank & Grass, Dieter, 2006. "Macroeconomic Aspects Of Substance Abuse: Diffusion, Productivity And Optimal Control," Macroeconomic Dynamics, Cambridge University Press, vol. 10(02), pages 145-164, April.
    3. Peter Kunkel & Oskar von dem Hagen, 2000. "Numerical Solution of Infinite-Horizon Optimal-Control Problems," Computational Economics, Springer;Society for Computational Economics, vol. 16(3), pages 189-205, December.
    4. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-272, March.
    5. Tatiana Kiseleva & Florian Wagener, 2015. "Bifurcations of Optimal Vector Fields," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 24-55, February.
    6. Michel, Philippe, 1982. "On the Transversality Condition in Infinite Horizon Optimal Problems," Econometrica, Econometric Society, vol. 50(4), pages 975-985, July.
    7. Wagener, F. O. O., 2003. "Skiba points and heteroclinic bifurcations, with applications to the shallow lake system," Journal of Economic Dynamics and Control, Elsevier, vol. 27(9), pages 1533-1561, July.
    8. 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.
    9. Hartl, Richard F., 1987. "A simple proof of the monotonicity of the state trajectories in autonomous control problems," Journal of Economic Theory, Elsevier, vol. 41(1), pages 211-215, February.
    10. Anne-Sophie Crépin, 2007. "Using Fast and Slow Processes to Manage Resources with Thresholds," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 36(2), pages 191-213, February.
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    Citations

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    Cited by:

    1. Florian Wagener, 2013. "Shallow lake economics run deep: nonlinear aspects of an economic-ecological interest conflict," Computational Management Science, Springer, vol. 10(4), pages 423-450, December.
    2. M. Chahim & D. Grass & R. F. Hartl & P. M. Kort, 2017. "Product innovation with lumpy investment," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(1), pages 159-182, March.
    3. Fouad El Ouardighi & Gary Erickson & Dieter Grass & Steffen Jørgensen, 2016. "Contracts and Information Structure in a Supply Chain with Operations and Marketing Interaction," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(04), pages 1-36, December.
    4. Dieter Grass, 2015. "From 0D to 1D spatial models using OCMat," Papers 1505.03956, arXiv.org.
    5. repec:spr:joptap:v:170:y:2016:i:1:d:10.1007_s10957-015-0855-0 is not listed on IDEAS
    6. Elke Moser & Dieter Grass & Gernot Tragler, 2016. "A non-autonomous optimal control model of renewable energy production under the aspect of fluctuating supply and learning by doing," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(3), pages 545-575, July.
    7. repec:spr:joptap:v:168:y:2016:i:1:d:10.1007_s10957-015-0747-3 is not listed on IDEAS
    8. repec:spr:annopr:v:238:y:2016:i:1:d:10.1007_s10479-015-2096-x is not listed on IDEAS
    9. Brock, William A. & Engström, Gustav & Grass, Dieter & Xepapadeas, Anastasios, 2013. "Energy balance climate models and general equilibrium optimal mitigation policies," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2371-2396.
    10. El Ouardighi, Fouad & Feichtinger, Gustav & Grass, Dieter & Hartl, Richard & Kort, Peter M., 2016. "Autonomous and advertising-dependent ‘word of mouth’ under costly dynamic pricing," European Journal of Operational Research, Elsevier, vol. 251(3), pages 860-872.
    11. Herbert Dawid & Michel Y. Keoula & Peter M. Kort, 2017. "Numerical Analysis of Markov-Perfect Equilibria with Multiple Stable Steady States: A Duopoly Application with Innovative Firms," Dynamic Games and Applications, Springer, vol. 7(4), pages 555-577, December.
    12. Caulkins, Jonathan P. & Feichtinger, Gustav & Grass, Dieter & Hartl, Richard F. & Kort, Peter M. & Seidl, Andrea, 2017. "Interaction of pricing, advertising and experience quality: A dynamic analysis," European Journal of Operational Research, Elsevier, vol. 256(3), pages 877-885.
    13. Heijnen, P. & Wagener, F.O.O., 2013. "Avoiding an ecological regime shift is sound economic policy," Journal of Economic Dynamics and Control, Elsevier, vol. 37(7), pages 1322-1341.

    More about this item

    Keywords

    Optimal vector field; BVP; Continuation; Multiple optimal solutions; Threshold point;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • 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

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