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Non-linear strategies in a linear quadratic differential game

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Author Info
Colin Rowat (University of Birmingham)

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Abstract

We study non-linear Markov perfect equilibria in a two agent linear quadratic differential game. In contrast to the literature owing to Tsutsui and Mino (1990), we do not associate endogenous subsets of the state space with candidate solutions. Instead, we address the problem of unbounded-below value functions over infinite horizons by use of the `catching up optimality' criterion. We present sufficiency conditions for existence based on results in Dockner, Jorgenson, Long and Sorger (2000). Applying these to our model yields the familiar linear solution as well as a condition under which a continuum of non-linear solutions exist. As this condition is relaxed when agents are more patient, and allows more efficient steady states, it resembles a Folk Theorem for differential games. The model presented here is one of atmospheric pollution; the results apply to differential games more generally.

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Paper provided by EconWPA in its series GE, Growth, Math methods with number 0502001.

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Length: 31 pages
Date of creation: 01 Feb 2005
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Handle: RePEc:wpa:wuwpge:0502001

Note: Type of Document - pdf; pages: 31. University of Birmingham Department of Economics Discussion Paper 05-05
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Related research
Keywords: differential game; non-linear strategies; catching up optimal; Folk Theorem;

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Find related papers by JEL classification:
C61 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Optimization Techniques; Programming Models; Dynamic Analysis
C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
H41 - Public Economics - - Publicly Provided Goods - - - Public Goods
Q00 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - General - - - General

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Tsutsui, Shunichi O., 1996. "Capacity constraints and voluntary output cutback in dynamic Cournot competition," Journal of Economic Dynamics and Control, Elsevier, vol. 20(9-10), pages 1683-1708. [Downloadable!] (restricted)
  2. Maria Luisa Gota & Luigi Montrucchio, 1999. "On Lipschitz continuity of policy functions in continuous-time optimal growth models," Economic Theory, Springer, vol. 14(2), pages 479-488. [Downloadable!] (restricted)
  3. Lockwood, Ben, 1996. "Uniqueness of Markov-perfect equilibrium in infinite-time affine-quadratic differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 20(5), pages 751-765, May. [Downloadable!] (restricted)
  4. Wirl, Franz & Dockner, Engelbert, 1995. "Leviathan governments and carbon taxes: Costs and potential benefits," European Economic Review, Elsevier, vol. 39(6), pages 1215-1236, June. [Downloadable!] (restricted)
  5. Tsutsui, Shunichi & Mino, Kazuo, 1990. "Nonlinear strategies in dynamic duopolistic competition with sticky prices," Journal of Economic Theory, Elsevier, vol. 52(1), pages 136-161, October. [Downloadable!] (restricted)
  6. Feichtinger, Gustav & Wirl, Franz, 1993. "A Dynamic Variant of the Battle of the Sexes," International Journal of Game Theory, Springer, vol. 22(4), pages 359-80.
  7. Seierstad, Atle & Sydsaeter, Knut, 1977. "Sufficient Conditions in Optimal Control Theory," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(2), pages 367-91, June. [Downloadable!] (restricted)
  8. Dockner Engelbert J. & Van Long Ngo, 1993. "International Pollution Control: Cooperative versus Noncooperative Strategies," Journal of Environmental Economics and Management, Elsevier, vol. 25(1), pages 13-29, July. [Downloadable!] (restricted)
  9. Karp, Larry, 1996. "Depreciation erodes the Coase Conjecture," European Economic Review, Elsevier, vol. 40(2), pages 473-490, February. [Downloadable!] (restricted)
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  10. Karl-Göran Mäler & Anastasios Xepapadeas & Aart de Zeeuw, 2003. "The Economics of Shallow Lakes," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 26(4), pages 603-624, December. [Downloadable!] (restricted)
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  11. Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July. [Downloadable!] (restricted)
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  12. Colin Rowat, 2005. "Non-Linear Strategies in a Linear Quadratic Differential Game," Discussion Papers 05-05, Department of Economics, University of Birmingham.
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  13. Dockner, Engelbert J. & Sorger, Gerhard, 1996. "Existence and Properties of Equilibria for a Dynamic Game on Productive Assets," Journal of Economic Theory, Elsevier, vol. 71(1), pages 209-227, October. [Downloadable!] (restricted)
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  1. Colin Rowat, 2005. "Non-linear strategies in a linear quadratic differential game," GE, Growth, Math methods 0502001, EconWPA. [Downloadable!]
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