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Optimal Management and Differential Games in the Presence of Threshold Effects - The Shallow Lake Model

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  • Reddy, P.V.
  • Schumacher, J.M.
  • Engwerda, J.C.

    (Tilburg University, Center for Economic Research)

Abstract

Abstract: In this article we analyze how the presence of thresholds influences multi agent decision making situations. We introduce a class of discounted autonomous optimal control problems with threshold effects and discuss tools to analyze these problems. Later, using these results we investigate two types of threshold effects; namely, simple and hysteresis switching, in the canonical model of the shallow lake. We solve the optimal management and open loop Nash equilibrium solutions for the shallow lake model with threshold effects. We establish a bifurcation analysis of the optimal vector field. Further, we observe that modeling with threshold effects simplifies this analysis. To be precise, the bifurcation scenarios rely on simple rules (inequalities) which can be verified easily. However, the qualitative behavior of the switching vector field is similar to the smooth case.

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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2012-001.

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Date of creation: 2012
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Handle: RePEc:dgr:kubcen:2012001

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Web page: http://center.uvt.nl

Related research

Keywords: Optimal control; Differential games; Threshold effects; Discontinuous dynamics; Shallow lake;

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  1. Stephen Polasky & Aart de Zeeuw & Florian Wagener, 2010. "Optimal Management with Potential Regime Shifts," Tinbergen Institute Discussion Papers 10-111/1, Tinbergen Institute.
  2. Erdlenbruch, Katrin & Jean-Marie, Alain & Moreaux, Michel & Tidball, Mabel, 2010. "Optimality of Impulse Harvesting Policies," IDEI Working Papers 603, Institut d'Économie Industrielle (IDEI), Toulouse.
  3. Kiseleva, T. & Wagener, F.O.O., 2009. "Bifurcations of optimal vector fields in the shallow lake model," CeNDEF Working Papers 09-12, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
  4. Skiba, A K, 1978. "Optimal Growth with a Convex-Concave Production Function," Econometrica, Econometric Society, vol. 46(3), pages 527-39, May.
  5. Michel, Philippe, 1982. "On the Transversality Condition in Infinite Horizon Optimal Problems," Econometrica, Econometric Society, vol. 50(4), pages 975-85, July.
  6. Michael Kopel & Herbert Dawid, 1999. "On optimal cycles in dynamic programming models with convex return function," Economic Theory, Springer, vol. 13(2), pages 309-327.
  7. W.A. Brock & D. Starrett, 2003. "Managing Systems with Non-convex Positive Feedback," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 26(4), pages 575-602, December.
  8. Mäler, K-G. & Xepapadeas, A. & Zeeuw, A.J. de, 2000. "The Economics of Shallow Lakes," Discussion Paper 2000-69, Tilburg University, Center for Economic Research.
  9. 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.
  10. Rod Cross, 2000. "Hysteresis and Emu," Metroeconomica, Wiley Blackwell, vol. 51(4), pages 367-379, November.
  11. Naevdal, Eric, 2006. "Dynamic optimisation in the presence of threshold effects when the location of the threshold is uncertain - with an application to a possible disintegration of the Western Antarctic Ice Sheet," Journal of Economic Dynamics and Control, Elsevier, vol. 30(7), pages 1131-1158, July.
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