IDEAS home Printed from https://ideas.repec.org/a/eee/dyncon/v34y2010i6p1123-1139.html
   My bibliography  Save this article

Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games

Author

Listed:
  • Van Gorder, Robert A.
  • Caputo, Michael R.

Abstract

Envelope theorems are established for locally differentiable Stackelberg equilibria of a general class of finite horizon differential games with an open-loop information structure. It is shown that the follower's envelope results agree in form with those of any player in an open-loop Nash equilibrium, while those of the leader differ. An unanticipated conclusion is that the costate vector of the leader--but not that of the follower--corresponding to the state vector of the differential game may be legitimately interpreted as the shadow value of the state vector for time-inconsistent open-loop Stackelberg equilibria. Surprisingly, the same cannot be said for time-consistent open-loop Stackelberg equilibria.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:dyncon:v:34:y:2010:i:6:p:1123-1139
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0165-1889(10)00017-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

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

    References listed on IDEAS

    as
    1. A. Michael Spence, 1979. "Investment Strategy and Growth in a New Market," Bell Journal of Economics, The RAND Corporation, vol. 10(1), pages 1-19, Spring.
    2. Jeffrey Ely & Drew Fudenberg & David K. Levine, 2008. "When is reputation bad?," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 10, pages 177-205, World Scientific Publishing Co. Pte. Ltd..
    3. 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.
    4. Jorgensen, Steffen & Kort, Peter M. & Van Schijndel, Geert-Jan C. Th., 1989. "Optimal investment, financing, and dividends : A stackelberg differential game," Journal of Economic Dynamics and Control, Elsevier, vol. 13(3), pages 339-377, July.
    5. repec:ebl:ecbull:v:12:y:2006:i:12:p:1-9 is not listed on IDEAS
    6. Caputo,Michael R., 2005. "Foundations of Dynamic Economic Analysis," Cambridge Books, Cambridge University Press, number 9780521842723.
    7. Morgan, John & Vardy, Felix, 2007. "The value of commitment in contests and tournaments when observation is costly," Games and Economic Behavior, Elsevier, vol. 60(2), pages 326-338, August.
    8. Amitrajeet A. Batabyal & Hamid Beladi, 2008. "A Stackelberg Game Model Of Trade In Renewable Resources With Competitive Sellers," World Scientific Book Chapters, in: Dynamic And Stochastic Approaches To The Environment And Economic Development, chapter 7, pages 103-121, World Scientific Publishing Co. Pte. Ltd..
    9. Yang, Hai & Zhang, Xiaoning & Meng, Qiang, 2007. "Stackelberg games and multiple equilibrium behaviors on networks," Transportation Research Part B: Methodological, Elsevier, vol. 41(8), pages 841-861, October.
    10. Caputo, Michael R., 1992. "Fundamental symmetries and qualitative properties in the adjustment cost model of the firm," Journal of Mathematical Economics, Elsevier, vol. 21(2), pages 99-112.
    11. Solan, Eilon & Yariv, Leeat, 2004. "Games with espionage," Games and Economic Behavior, Elsevier, vol. 47(1), pages 172-199, April.
    12. Koji Shimomura & Danyang Xie, 2008. "Advances on Stackelberg open‐loop and feedback strategies," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(1), pages 115-133, March.
    13. Kenji Fujiwara, 2006. "A Stackelberg Game Model of Dynamic Duopolistic Competition with Sticky Prices," Economics Bulletin, AccessEcon, vol. 12(12), pages 1-9.
    14. Xie, Danyang, 1997. "On Time Inconsistency: A Technical Issue in Stackelberg Differential Games," Journal of Economic Theory, Elsevier, vol. 76(2), pages 412-430, October.
    15. Xie, Jinxing & Ai, Song, 2006. "A note on "Cooperative advertising, game theory and manufacturer-retailer supply chains"," Omega, Elsevier, vol. 34(5), pages 501-504, October.
    16. 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.
    17. 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.
    18. Cremer, Jacques, & Riordan, Michael H, 1985. "A Sequential Solution to the Public Goods Problem," Econometrica, Econometric Society, vol. 53(1), pages 77-84, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Wu, Ruijia & Van Gorder, Robert A., 2018. "Nonlinear dynamics of discrete time multi-level leader–follower games," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 240-250.
    2. 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.
    3. Steffen Jørgensen, 2011. "Intertemporal Contracting in a Supply Chain," Dynamic Games and Applications, Springer, vol. 1(2), pages 280-300, June.
    4. Chen Ling & Shravan Luckraz & Bruno Antonio Pansera, 2023. "Comparative Dynamics in Differential Games: A Note on the Differentiability of Solutions," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1093-1105, March.

    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. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Masahiko Hattori & Yasuhito Tanaka, 2019. "General analysis of dynamic oligopoly with sticky price," Economics Bulletin, AccessEcon, vol. 39(4), pages 2990-2998.
    6. Bolin, Kristian & Caputo, Michael R., 2020. "Consumption and investment demand when health evolves stochastically," Journal of Economic Dynamics and Control, Elsevier, vol. 114(C).
    7. Bertrand Villeneuve & Vanessa Yanhua Zhang, 2013. "Industry Restructuring: A Case for Affirmative Action," Annals of Economics and Statistics, GENES, issue 109-110, pages 179-201.
    8. Chu, Angus C., 2009. "A politico-economic analysis of the European Union's R&D policy," Journal of Macroeconomics, Elsevier, vol. 31(4), pages 582-590, December.
    9. Caputo, Michael R. & Paris, Quirino, 2013. "An intertemporal microeconomic theory of disembodied and price-induced technical progress," Economic Modelling, Elsevier, vol. 33(C), pages 631-640.
    10. Angus Chu, 2010. "Nation states vs. united empire: Effects of political competition on economic growth," Public Choice, Springer, vol. 145(1), pages 181-195, October.
    11. Kenji Fujiwara & Ngo Long, 2011. "Welfare Implications of Leadership in a Resource Market under Bilateral Monopoly," Dynamic Games and Applications, Springer, vol. 1(4), pages 479-497, December.
    12. Baldini, Massimo & Lambertini, Luca, 2011. "Profit taxation and capital accumulation in a dynamic oligopoly model," Japan and the World Economy, Elsevier, vol. 23(1), pages 13-18, January.
    13. Katarzyna Kańska & Agnieszka Wiszniewska-Matyszkiel, 2022. "Dynamic Stackelberg duopoly with sticky prices and a myopic follower," Operational Research, Springer, vol. 22(4), pages 4221-4252, September.
    14. L. Lambertini, 2010. "Oligopoly with Hyperbolic Demand: A Differential Game Approach," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 108-119, April.
    15. Faria João Ricardo & Mixon Franklin G. & Upadhyaya Ashish & Upadhyaya Kamal P., 2019. "Gang Rivalry and Crime: A Differential Game Approach," Review of Law & Economics, De Gruyter, vol. 15(2), pages 1-30, July.
    16. Günther Rehme, 2023. "Capital depreciation allowances, redistributive taxation, and economic growth," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 25(1), pages 168-195, February.
    17. E. Bacchiega & L. Lambertini & A. Palestini, 2010. "On the Time Consistency of Equilibria in a Class of Additively separable Differential Games," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 415-427, June.
    18. Ngo Long, 2011. "Dynamic Games in the Economics of Natural Resources: A Survey," Dynamic Games and Applications, Springer, vol. 1(1), pages 115-148, March.
    19. Benchekroun, Hassan & Ray Chaudhuri, Amrita & Tasneem, Dina, 2020. "On the impact of trade in a common property renewable resource oligopoly," Journal of Environmental Economics and Management, Elsevier, vol. 101(C).
    20. Calzolari, Giacomo & Lambertini, Luca, 2007. "Export restraints in a model of trade with capital accumulation," Journal of Economic Dynamics and Control, Elsevier, vol. 31(12), pages 3822-3842, December.

    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:eee:dyncon:v:34:y:2010:i:6:p:1123-1139. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jedc .

    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.