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The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents

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  • Bethmann, Dirk

Abstract

We solve an 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 , 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, 2008. "The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents," Journal of Macroeconomics, Elsevier, vol. 30(1), pages 396-414, March.
  • Handle: RePEc:eee:jmacro:v:30:y:2008:i:1:p:396-414
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    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. Casey B. Mulligan & Xavier Sala-i-Martin, 1993. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," The Quarterly Journal of Economics, Oxford University Press, pages 739-773.
    3. 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.
    4. 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.
    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. 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, December.
    7. 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.
    8. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
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    Cited by:

    1. Neustroev, Dmitry, 2013. "The Uzawa-Lucas Growth Model with Natural Resources," MPRA Paper 52937, University Library of Munich, Germany.
    2. Dirk Bethmann & Michael Kvasnicka, 2007. "Uncertain Paternity, Mating Market Failure, and the Institution of Marriage," SFB 649 Discussion Papers SFB649DP2007-013, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Dirk Bethmann & Markus Reiß, 2012. "Simplifying numerical analyses of Hamilton–Jacobi–Bellman equations," Journal of Economics, Springer, vol. 107(2), pages 101-128, October.

    More about this item

    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

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