IDEAS home Printed from https://ideas.repec.org/p/zbw/caseps/200442.html
   My bibliography  Save this paper

The open-loop solution of the Uzawa-Lucas Model of Endogenous Growth with N agents

Author

Listed:
  • Bethmann, Dirk

Abstract

We solve an N 2 N player general-sum differential game. The optimization problem considered here is based on the Uzawa Lucas model of endogenous growth. Agents have logarithmic preferences and own two capital stocks. Since the number of players is an arbitrary fixed number N 2 N, the model?s solution is more general than the idealized concepts of the social planer?s solution with one player or the competitive equilibrium with infinitely many players. We show that the symmetric Nash equilibrium is completely described by the solution to a single ordinary differential equation. The numerical results imply that the influence of the externality along the balanced growth path decreases rapidly as the number of players increases. Off the steady state, the externality is of great importance, even for a large number of players.

Suggested Citation

  • Bethmann, Dirk, 2004. "The open-loop solution of the Uzawa-Lucas Model of Endogenous Growth with N agents," Papers 2004,42, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    • Handle: RePEc:zbw:caseps:200442
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/22215/1/42_db.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Clemhout, Simone & Wan, Henry Jr., 1994. "Differential games -- Economic applications," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 23, pages 801-825, Elsevier.
    2. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    3. Casey B. Mulligan & Xavier Sala-i-Martin, 1993. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(3), pages 739-773.
    4. Caballe, Jordi & Santos, Manuel S, 1993. "On Endogenous Growth with Physical and Human Capital," Journal of Political Economy, University of Chicago Press, vol. 101(6), pages 1042-1067, December.
    5. Casey B. Mulligan & Xavier Sala-i-Martin, 1991. "A Note on the Time-Elimination Method For Solving Recursive Dynamic Economic Models," NBER Technical Working Papers 0116, National Bureau of Economic Research, Inc.
    6. David Levhari & Leonard J. Mirman, 1980. "The Great Fish War: An Example Using a Dynamic Cournot-Nash Solution," Bell Journal of Economics, The RAND Corporation, vol. 11(1), pages 322-334, Spring.
    7. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
    8. Sundaram, Rangarajan K., 1989. "Perfect equilibrium in non-randomized strategies in a class of symmetric dynamic games," Journal of Economic Theory, Elsevier, vol. 47(1), pages 153-177, February.
    9. 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, October.
    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. Dirk Bethmann & Markus Reiß, 2012. "Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations," Journal of Economics, Springer, vol. 107(2), pages 101-128, October.
    2. Neustroev, Dmitry, 2013. "The Uzawa-Lucas Growth Model with Natural Resources," MPRA Paper 52937, University Library of Munich, Germany.
    3. Dirk Bethmann, 2007. "Homogeneity, Saddle Path Stability, and Logarithmic Preferences in Economic Models," Discussion Paper Series 0702, Institute of Economic Research, Korea University.
    4. Mehmet Özer & Çağrı Sağlam, 2016. "Strategic Interaction And Catching Up," Bulletin of Economic Research, Wiley Blackwell, vol. 68(2), pages 168-181, April.

    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. Reiß, Markus & Bethmann, Dirk, 2003. "Transitional Dynamics in the Uzawa-Lucas Model of Endogenous Growth," SFB 373 Discussion Papers 2003,17, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. Engelbert Dockner & Florian Wagener, 2014. "Markov perfect Nash equilibria in models with a single capital stock," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(3), pages 585-625, August.
    3. Hassan Benchekroun & Ngo Van Long, 2001. "Leader and Follower: A Differential Game Model," CIRANO Working Papers 2001s-08, CIRANO.
    4. Koch, Karl-Josef, 1995. "Stability issues of endogenous growth models," Discussion Papers, Series II 270, University of Konstanz, Collaborative Research Centre (SFB) 178 "Internationalization of the Economy".
    5. Klaus Wälde, 1996. "Proof of global stability, transitional dynamics, and international capital flows in a two-country model of innovation and growth," Journal of Economics, Springer, vol. 64(1), pages 53-84, February.
    6. Klaus Waelde, 1994. "Transitional dynamics, convergence and international capital flows in two-country models of innovation and growth," International Trade 9403002, University Library of Munich, Germany, revised 03 Jan 1996.
    7. Dirk Bethmann & Markus Reiß, 2012. "Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations," Journal of Economics, Springer, vol. 107(2), pages 101-128, October.
    8. Jaime Alonso-Carrera, 2001. "More on the dynamics in the endogenous growth model with human capital," Investigaciones Economicas, Fundación SEPI, vol. 25(3), pages 561-583, September.
    9. Xavier Pautrel, 2015. "Abatement Technology and the Environment–Growth Nexus with Education," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 61(3), pages 297-318, July.
    10. Gerhard Sorger, 2005. "A dynamic common property resource problem with amenity value and extraction costs," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(1), pages 3-19, March.
    11. Philip Trostel & Ian Walker, 2006. "Education and Work," Education Economics, Taylor & Francis Journals, vol. 14(4), pages 377-399.
    12. Jaime Alonso-Carrera & Jordi Caballé & Xavier Raurich, 2011. "Sectoral composition and macroeconomic dynamics," UFAE and IAE Working Papers 869.11, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    13. De, Supriyo, 2014. "Intangible capital and growth in the ‘new economy’: Implications of a multi-sector endogenous growth model," Structural Change and Economic Dynamics, Elsevier, vol. 28(C), pages 25-42.
    14. Arantza Gorostiaga & Jana Hromcová & Miguel-Ángel López-García, 2013. "Optimal taxation in the Uzawa–Lucas model with externality in human capital," Journal of Economics, Springer, vol. 108(2), pages 111-129, March.
    15. Jing Dang & Max Gillman & Michal Kejak, 2011. "Real Business Cycles with a Human Capital Investment Sector and Endogenous Growth: Persistence, Volatility and Labor Puzzles," CERS-IE WORKING PAPERS 1128, Institute of Economics, Centre for Economic and Regional Studies.
    16. Thorsten Wichmann, 1995. "Food Consumption and Growth in a Two Sector Economy - A Theoretical Model and Numerical Simulations," Berlecon Research Papers 0001, Berlecon Research.
    17. Gian Maria Milesi-Ferretti & Nouriel Roubini, 1995. "Growth Effects of Income and Consumption Taxes: Positive and Normative Analysis," Working Papers 95-18, New York University, Leonard N. Stern School of Business, Department of Economics.
    18. Yohei Tenryu, 2017. "The role of the private sector under insecure property rights," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 64(3), pages 285-311, September.
    19. Julio Huato, 2023. "Inequality and Growth: A Two-Player Dynamic Game with Production and Appropriation," Papers 2304.01855, arXiv.org.
    20. Xie Danyang, 1994. "Divergence in Economic Performance: Transitional Dynamics with Multiple Equilibria," Journal of Economic Theory, Elsevier, vol. 63(1), pages 97-112, June.

    More about this item

    Keywords

    Value Function Approach; Nash-Equilibrium; Open-loop Strategies; Ordinary Differential Equation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    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:zbw:caseps:200442. 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: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/cahubde.html .

    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.