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Bifurcations of Optimal Vector Fields

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  • Kiseleva, T.

    (University of Amsterdam)

  • Wagener, F.O.O.

    (University of Amsterdam)

Abstract

We study the structure of the solution set of a class of infinite-horizon dynamic programming problems with one-dimensional state spaces, as well as their bifurcations as problem parameters are varied. The solutions are represented as the integral curves of a multi-valued `optimal' vector field on state space. Generically, there are three types of integral curves: stable points, open intervals that are forward asymptotic to a stable point and backward asymptotic to an unstable point, and half-open intervals that are forward asymptotic to a stable point and backward asymptotic to an indifference point; the latter are initial states to multiple optimal trajectories. We characterize all bifurcations that occur generically in one- and two-parameter families. Most of these are related to global dynamical bifurcations of the state-costate system of the problem.

Suggested Citation

  • Kiseleva, T. & Wagener, F.O.O., 2011. "Bifurcations of Optimal Vector Fields," CeNDEF Working Papers 11-05, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    • Handle: RePEc:ams:ndfwpp:11-05
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    1. Karl-Göran Mäler & Anastasios Xepapadeas & Aart de Zeeuw, 2003. "The Economics of Shallow Lakes," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 26(4), pages 603-624, December.
    2. W. Davis Dechert & Kazuo Nishimura, 2012. "A Complete Characterization of Optimal Growth Paths in an Aggregated Model with a Non-Concave Production Function," Springer Books, in: John Stachurski & Alain Venditti & Makoto Yano (ed.), Nonlinear Dynamics in Equilibrium Models, edition 127, chapter 0, pages 237-257, Springer.
    3. Gernot Tragler & Jonathan P. Caulkins & Gustav Feichtinger, 2001. "Optimal Dynamic Allocation of Treatment and Enforcement in Illicit Drug Control," Operations Research, INFORMS, vol. 49(3), pages 352-362, June.
    4. Hinloopen, Jeroen & Smrkolj, Grega & Wagener, Florian, 2013. "From mind to market: A global, dynamic analysis of R&D," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2729-2754.
    5. Moghayer, S. & Wagener, F.O.O., 2009. "Genesis of indifference thresholds and infinitely many indifference points in discrete time infinite horizon optimisation problems," CeNDEF Working Papers 09-14, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    6. Skiba, A K, 1978. "Optimal Growth with a Convex-Concave Production Function," Econometrica, Econometric Society, vol. 46(3), pages 527-539, May.
    7. Rowthorn, Robert & Toxvaerd, Flavio, 2012. "The Optimal Control of Infectious Diseases via Prevention and Treatment," CEPR Discussion Papers 8925, C.E.P.R. Discussion Papers.
    8. Kiseleva, Tatiana & Wagener, F.O.O., 2010. "Bifurcations of optimal vector fields in the shallow lake model," Journal of Economic Dynamics and Control, Elsevier, vol. 34(5), pages 825-843, May.
    9. Caulkins, Jonathan P. & Feichtinger, Gustav & Johnson, Michael & Tragler, Gernot & Yegorov, Yuri, 2005. "Skiba thresholds in a model of controlled migration," Journal of Economic Behavior & Organization, Elsevier, vol. 57(4), pages 490-508, August.
    10. J. P. Caulkins & G. Feichtinger & D. Grass & G. Tragler, 2007. "Bifurcating DNS Thresholds in a Model of Organizational Bridge Building," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 19-35, April.
    11. Tahvonen, Olli & Salo, Seppo, 1996. "Nonconvexities in Optimal Pollution Accumulation," Journal of Environmental Economics and Management, Elsevier, vol. 31(2), pages 160-177, September.
    12. 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.
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    Cited by:

    1. Hinloopen, J. & Smrkolj, G. & Wagener, F.O.O., 2013. "In Defense of Trusts: R&D Cooperation in Global Perspective," CeNDEF Working Papers 13-05, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    2. Hinloopen, Jeroen & Smrkolj, Grega & Wagener, Florian, 2013. "From mind to market: A global, dynamic analysis of R&D," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2729-2754.
    3. Engelbert Dockner & Florian Wagener, 2014. "Markov perfect Nash equilibria in models with a single capital stock," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(3), pages 585-625, August.
    4. Dieter Grass, 2015. "From 0D to 1D spatial models using OCMat," Papers 1505.03956, arXiv.org.
    5. Reddy, P.V. & Schumacher, J.M. & Engwerda, J.C., 2012. "Optimal Management and Differential Games in the Presence of Threshold Effects - The Shallow Lake Model," Discussion Paper 2012-001, Tilburg University, Center for Economic Research.
    6. 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.
    7. Jeroen Hinloopen & Grega Smrkolj & Florian Wagener, 2016. "R&D Cooperatives and Market Collusion: A Global Dynamic Approach," Tinbergen Institute Discussion Papers 16-048/II, Tinbergen Institute.
    8. Wagener, F.O.O., 2013. "Economics of environmental regime shifts," CeNDEF Working Papers 13-08, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.

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