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Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example

Author

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  • Lilia Maliar

    (Stanford University)

  • Serguei Maliar

    (Office 316J, Lucas Hall, Santa Clara University)

Abstract

The literature that conducts numerical analysis of equilibrium in models with hyperbolic (quasi-geometric) discounting reports difficulties in achieving convergence. Surprisingly, numerical methods fail to converge even in a simple, deterministic optimal growth problem that has a well-behaved, smooth closed-form solution. We argue that the reason for nonconvergence is that the generalized Euler equation has a continuum of smooth solutions, each of which is characterized by a different integration constant. We propose two types of restrictions that can rule out the multiplicity: boundary conditions and shape restrictions on equilibrium policy functions. With these additional restrictions, the studied numerical methods deliver a unique smooth solution for both the deterministic and stochastic problems in a wide range of the model’s parameters.

Suggested Citation

  • Lilia Maliar & Serguei Maliar, 2016. "Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example," Dynamic Games and Applications, Springer, vol. 6(2), pages 243-261, June.
    • Handle: RePEc:spr:dyngam:v:6:y:2016:i:2:d:10.1007_s13235-015-0177-8
    DOI: 10.1007/s13235-015-0177-8
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    Cited by:

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    2. Lilia Maliar & Serguei Maliar & John B. Taylor & Inna Tsener, 2020. "A tractable framework for analyzing a class of nonstationary Markov models," Quantitative Economics, Econometric Society, vol. 11(4), pages 1289-1323, November.
    3. Balbus, Łukasz & Reffett, Kevin & Woźny, Łukasz, 2018. "On uniqueness of time-consistent Markov policies for quasi-hyperbolic consumers under uncertainty," Journal of Economic Theory, Elsevier, vol. 176(C), pages 293-310.
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    5. Łukasz Balbus & Anna Jaśkiewicz & Andrzej S. Nowak, 2020. "Equilibria in Altruistic Economic Growth Models," Dynamic Games and Applications, Springer, vol. 10(1), pages 1-18, March.

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    More about this item

    Keywords

    Hyperbolic discounting; Quasi-geometric discounting; Time inconsistency; Markov perfect equilibrium; Markov games; Turnpike theorem; Neoclassical growth model; Endogenous gridpoints; Envelope condition;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth
    • H63 - Public Economics - - National Budget, Deficit, and Debt - - - Debt; Debt Management; Sovereign Debt
    • P16 - Political Economy and Comparative Economic Systems - - Capitalist Economies - - - Capitalist Institutions; Welfare State

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