IDEAS home Printed from https://ideas.repec.org/a/spr/dyngam/v2y2012i3p313-334.html
   My bibliography  Save this article

The Envelope Theorem for Locally Differentiable Nash Equilibria of Discounted and Autonomous Infinite Horizon Differential Games

Author

Listed:
  • Chen Ling
  • Michael Caputo

Abstract

The envelope theorem is extended to cover the class of discounted and autonomous infinite horizon differential games that possess locally differentiable Nash equilibria. The theorems cover open-loop and feedback information structures and are applied to an analytically solvable linear-quadratic game. The linear-quadratic structure permits additional insight into the theorems that is not available in the general case. With open-loop information, for example, the costate variable is shown to uniformly overstate the shadow value of the state variable, but the growth rates of the two are identical. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Chen Ling & Michael Caputo, 2012. "The Envelope Theorem for Locally Differentiable Nash Equilibria of Discounted and Autonomous Infinite Horizon Differential Games," Dynamic Games and Applications, Springer, vol. 2(3), pages 313-334, September.
    • Handle: RePEc:spr:dyngam:v:2:y:2012:i:3:p:313-334
    DOI: 10.1007/s13235-012-0045-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s13235-012-0045-8
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s13235-012-0045-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Caputo, Michael R., 2007. "The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games," Games and Economic Behavior, Elsevier, vol. 61(2), pages 198-224, November.
    2. Miller, Marcus & Salmon, Mark, 1985. "Dynamic Games and the Time Inconsistency of Optimal Policy in Open Economies," Economic Journal, Royal Economic Society, vol. 95(380a), pages 124-137, Supplemen.
    3. Caputo,Michael R., 2005. "Foundations of Dynamic Economic Analysis," Cambridge Books, Cambridge University Press, number 9780521842723.
    4. Groot, Fons & Withagen, Cees & de Zeeuw, Aart, 2003. "Strong time-consistency in the cartel-versus-fringe model," Journal of Economic Dynamics and Control, Elsevier, vol. 28(2), pages 287-306, November.
    5. 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.
    6. Van Gorder, Robert A. & Caputo, Michael R., 2010. "Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 34(6), pages 1123-1139, June.
    7. Caputo, Michael R., 1996. "The Envelope Theorem and Comparative Statics of Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 13(2), pages 201-224, April.
    8. 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.
    9. List, John A. & Mason, Charles F., 2001. "Optimal Institutional Arrangements for Transboundary Pollutants in a Second-Best World: Evidence from a Differential Game with Asymmetric Players," Journal of Environmental Economics and Management, Elsevier, vol. 42(3), pages 277-296, November.
    10. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, September.
    11. Caputo, Michael R., 1998. "A dual vista of the Stackelberg duopoly reveals its fundamental qualitative structure," International Journal of Industrial Organization, Elsevier, vol. 16(3), pages 333-352, May.
    12. Engwerda, J.C., 2000. "Feedback Nash equilibria in the scalar infinite horizon LQ-Game," Other publications TiSEM 58ccf964-4ca1-4d67-9a68-a, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Caputo, Michael R. & Ling, Chen, 2013. "The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 37(10), pages 1982-1994.
    2. Van Gorder, Robert A. & Caputo, Michael R., 2010. "Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 34(6), pages 1123-1139, June.
    3. Victoria Umanskaya & Charles Mason & Edward Barbier, 2007. "Dynamic Game of Transboundary Pollution Regulation and Strategic Abatement," Energy and Environmental Modeling 2007 24000063, EcoMod.
    4. Caputo, Michael R., 2007. "The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games," Games and Economic Behavior, Elsevier, vol. 61(2), pages 198-224, November.
    5. Jaakkola, Niko, 2019. "Carbon taxation, OPEC and the end of oil," Journal of Environmental Economics and Management, Elsevier, vol. 94(C), pages 101-117.
    6. Insley, Margaret & A. Forsyth, Peter, 2019. "Climate games: Who’s on first? What’s on second?," L'Actualité Economique, Société Canadienne de Science Economique, vol. 95(2-3), pages 287-322, Juin-Sept.
    7. Santiago J. Rubio, 2002. "On The Coincidence Of The Feedback Nash And Stackelberg Equilibria In Economic Applications Of Differential Games," Working Papers. Serie AD 2002-11, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    8. Michael Caputo & Chen Ling, 2015. "Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria," Dynamic Games and Applications, Springer, vol. 5(1), pages 1-25, March.
    9. Charles F. Mason, 2021. "Transboundary Externalities and Reciprocal Taxes: A Differential Game Approach," Strategic Behavior and the Environment, now publishers, vol. 9(1-2), pages 27-67, July.
    10. Charles F. Mason & Victoria I. Umanskaya & Edward B. Barbier, 2018. "Trade, Transboundary Pollution, and Foreign Lobbying," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 70(1), pages 223-248, May.
    11. Margaret Insley & Tracy Snoddon & Peter A. Forsyth, 2018. "Strategic interactions and uncertainty in decisions to curb greenhouse gas emissions," Working Papers 1805, University of Waterloo, Department of Economics, revised 06 Jan 2018.
    12. Shuhua Chang & Suresh P. Sethi & Xinyu Wang, 2018. "Optimal Abatement and Emission Permit Trading Policies in a Dynamic Transboundary Pollution Game," Dynamic Games and Applications, Springer, vol. 8(3), pages 542-572, September.
    13. George E. Halkos & George J. Papageorgiou, 2012. "Pollution Control Policy: A Dynamic Taxation Scheme," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 6(1), pages 14-37, March.
    14. G. Zaccour, 2003. "Computation of Characteristic Function Values for Linear-State Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 183-194, April.
    15. Simon Hoof, 2020. "On a class of linear-state differential games with subgame individually rational and time consistent bargaining solutions," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 5(1), pages 79-97, December.
    16. Benchekroun, Hassan & Ray Chaudhuri, Amrita, 2014. "Transboundary pollution and clean technologies," Resource and Energy Economics, Elsevier, vol. 36(2), pages 601-619.
    17. Yuankan Huang & Takehiro Inohara, 2023. "Stable Markov perfect equilibria in the asymmetric differential-game duopoly with a renewable resource," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(1), pages 45-63, April.
    18. Chatelain, Jean-Bernard & Ralf Kirsten, 2016. "Countercyclical versus Procyclical Taylor Principles," EconStor Preprints 129796, ZBW - Leibniz Information Centre for Economics.
    19. Levine, Paul & Pearlman, Joseph, 2011. "Computation of LQ Approximations to Optimal Policy Problems in Different Information Settings under Zero Lower Bound Constraints," Dynare Working Papers 10, CEPREMAP.
    20. Benchekroun, H. & Ray Chaudhuri, A., 2010. "'The Voracity Effect' and Climate Change : The Impact of Clean Technologies," Discussion Paper 2010-97, Tilburg University, Center for Economic Research.

    More about this item

    Keywords

    Envelope theorem; Differential games; Open-loop Nash equilibria; Feedback Nash equilibria;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:dyngam:v:2:y:2012:i:3:p:313-334. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.