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On coincidence of feedback and global Stackelberg equilibria in a class of differential games

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  • Martín-Herrán, Guiomar
  • Rubio, Santiago J.

Abstract

This paper shows for a class of differential games that the global Stackelberg equilibrium (GSE) coincides with the feedback Stackelberg equilibrium (FSE), although the GSE assumes that the leader/regulator announces at the initial time the regulatory instrument rule she will follow for the rest of the game, while in the FSE, the regulator at any time chooses the optimal level of the regulatory instrument rate. This coincidence is based on the fact that the FSE is calculated using dynamic programming what implies that although the regulator chooses the regulatory instrument rate level that maximizes social welfare, the first-order condition for the maximization of the right-hand side of the Hamilton-Jacobi-Bellman equation implicitly defines a rule for the regulatory instrument. Then, as the regulatory instrument rule defined by the FSE implements the efficient outcome as the GSE does, the rules defined by both equilibria must be the same. In the second part of the paper, we check that this is the case for two examples. The first is an operations research model, while the second is an economic model. The first example fits in a linear-state differential game structure, while the second example presents a linear-quadratic specification. In both cases the regulatory instrument rules for both equilibria (GSE and FSE) are calculated and identical expressions are obtained.

Suggested Citation

  • Martín-Herrán, Guiomar & Rubio, Santiago J., 2021. "On coincidence of feedback and global Stackelberg equilibria in a class of differential games," European Journal of Operational Research, Elsevier, vol. 293(2), pages 761-772.
    • Handle: RePEc:eee:ejores:v:293:y:2021:i:2:p:761-772
    DOI: 10.1016/j.ejor.2020.12.022
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    as
    1. Martín-Herrán, Guiomar & Taboubi, Sihem, 2015. "Price coordination in distribution channels: A dynamic perspective," European Journal of Operational Research, Elsevier, vol. 240(2), pages 401-414.
    2. Anshuman Chutani & Suresh P. Sethi, 2012. "Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 615-643, August.
    3. Xepapadeas, A. P., 1992. "Environmental policy, adjustment costs, and behavior of the firm," Journal of Environmental Economics and Management, Elsevier, vol. 23(3), pages 258-275, November.
    4. S. J. Rubio, 2006. "On Coincidence of Feedback Nash Equilibria and Stackelberg Equilibria in Economic Applications of Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 203-220, January.
    5. Benchekroun, Hassan & Van Long, Ngo, 2002. "On the multiplicity of efficiency-inducing tax rules," Economics Letters, Elsevier, vol. 76(3), pages 331-336, August.
    6. Wirl, Franz, 2012. "Global warming: Prices versus quantities from a strategic point of view," Journal of Environmental Economics and Management, Elsevier, vol. 64(2), pages 217-229.
    7. Rubio, Santiago J. & Escriche, Luisa, 2001. "Strategic pigouvian taxation, stock externalities and polluting non-renewable resources," Journal of Public Economics, Elsevier, vol. 79(2), pages 297-313, February.
    8. Feenstra, T.L. & Kort, P.M. & de Zeeuw, A.J., 2001. "Environmnetal policy instruments in an international duopoloy with feedback investment strategies," Other publications TiSEM 35079c82-8638-48b2-b04c-6, Tilburg University, School of Economics and Management.
    9. Jorgensen, Steffen & Taboubi, Sihem & Zaccour, Georges, 2003. "Retail promotions with negative brand image effects: Is cooperation possible?," European Journal of Operational Research, Elsevier, vol. 150(2), pages 395-405, October.
    10. Karp, Larry, 1992. "Efficiency Inducing Tax for a Common Property Oligopoly," Economic Journal, Royal Economic Society, vol. 102(411), pages 321-332, March.
    11. De Giovanni, Pietro & Karray, Salma & Martín-Herrán, Guiomar, 2019. "Vendor Management Inventory with consignment contracts and the benefits of cooperative advertising," European Journal of Operational Research, Elsevier, vol. 272(2), pages 465-480.
    12. Alain Haurie & Jacek B Krawczyk & Georges Zaccour, 2012. "Games and Dynamic Games," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8442, February.
    13. Karp, Larry & Livernois, John, 1992. "On efficiency-inducing taxation for a non-renewable resource monopolist," Journal of Public Economics, Elsevier, vol. 49(2), pages 219-239, November.
    14. Karray, Salma & Martín-Herrán, Guiomar, 2009. "A dynamic model for advertising and pricing competition between national and store brands," European Journal of Operational Research, Elsevier, vol. 193(2), pages 451-467, March.
    15. Akihiko Yanase, 2009. "Global environment and dynamic games of environmental policy in an international duopoly," Journal of Economics, Springer, vol. 97(2), pages 121-140, June.
    16. Daniel Cohen & Philippe Michel, 1988. "How Should Control Theory Be Used to Calculate a Time-Consistent Government Policy?," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 55(2), pages 263-274.
    17. Colombo, Luca & Labrecciosa, Paola, 2019. "Stackelberg versus Cournot: A differential game approach," Journal of Economic Dynamics and Control, Elsevier, vol. 101(C), pages 239-261.
    18. Martín-Herrán, Guiomar & Sigué, Simon Pierre & Zaccour, Georges, 2011. "Strategic interactions in traditional franchise systems: Are franchisors always better off?," European Journal of Operational Research, Elsevier, vol. 213(3), pages 526-537, September.
    19. Martín-Herrán, Guiomar & Rubio, Santiago J., 2018. "Second-best taxation for a polluting monopoly with abatement investment," Energy Economics, Elsevier, vol. 73(C), pages 178-193.
    20. Benchekroun, Hassan & van Long, Ngo, 1998. "Efficiency inducing taxation for polluting oligopolists," Journal of Public Economics, Elsevier, vol. 70(2), pages 325-342, November.
    21. Bergstrom, Theodore C. & Cross, John G. & Porter, Richard C., 1981. "Efficiency-inducing taxation for a monopolistically supplied depletable resource," Journal of Public Economics, Elsevier, vol. 15(1), pages 23-32, February.
    22. Wirl, Franz, 2014. "Taxes versus permits as incentive for the intertemporal supply of a clean technology by a monopoly," Resource and Energy Economics, Elsevier, vol. 36(1), pages 248-269.
    23. Tahvonen, Olli, 1996. "Trade with Polluting Nonrenewable Resources," Journal of Environmental Economics and Management, Elsevier, vol. 30(1), pages 1-17, January.
    24. Feenstra, Talitha & Kort, Peter M. & de Zeeuw, Aart, 2001. "Environmental policy instruments in an international duopoly with feedback investment strategies," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1665-1687, October.
    25. Martina Stimming, 1999. "Capital-accumulation games under environmental regulation and duopolistic competition," Journal of Economics, Springer, vol. 69(3), pages 267-287, October.
    26. Basar, Tamer & Haurie, Alain & Ricci, Gianni, 1985. "On the dominance of capitalists leadership in a Feedback-Stackelberg solution of a differential game model of capitalism," Journal of Economic Dynamics and Control, Elsevier, vol. 9(1), pages 101-125, September.
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