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

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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.

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Notes

  1. The related literature includes Laibson et al. [35], Barro [9], O’Donoghue and Rabin [55], Harris and Laibson [24], Angeletos et al. [3], Krusell and Smith [29–31], Krusell et al. [32], Luttmer and Mariotti [37], Maliar and Maliar [39–44], Judd [25], Sorger [61], Gong et al. [23], Chatterjee and Eyigungor [16], Balbus et al. [6, 7], Bernheim et al. [11], among others.

  2. Other methods can be used in the context of models with quasi-geometric discounting. One of them is a “recursive optimization” approach suggested in Strotz [65], and Caplin and Leahy [14]. A possible implementation of this approach is found in the “pseudo-state space/enlarged state space” analysis of Kydland and Prescott [33] and Feng et al. [22]. Recently, in the context of the game theoretic approach, some literature have suggested turning the problem into a stochastic game (e.g., [24], Balbus et al. [6]). Also, in this latter tradition, one could also attempt to apply incentive-constrained dynamic programming methods [60], recursive dual approaches [49, 52, 56], and [17], or set-value dynamic programming methods proposed in Abreu, Pearce and Stachetti [1] (e.g., Balbus and Wozny [6]).

  3. For convergence properties of the envelope condition method (ECM) and its applications, see Maliar and Maliar [46], and Arellano et al. [4].

  4. For the case of the standard geometric discounting, there is a general turnpike theorem that shows that an optimal program of a finite horizon economy asymptotically converges to an optimal program of the corresponding infinite horizon economy under very general assumptions; see Brock and Mirman [13], McKenzie [51], Joshi [26], Majumdar and Zilcha [38], Mitra and Nyarko [54], Becker [10], and Maliar et al. [47]. Turnpike theorems are also known for some dynamic games (see [28] for a survey), but they are not yet established for the economy with quasi-geometric discounting like ours.

  5. In the standard geometric discounting case, there are monotone operators that converge to a limiting stationary solution by iteration on the finite horizon dynamic program; see Coleman [18, 19], Mirman et al. [53], Datta et al. [20], and Feng et al. [22].

  6. See also Barillas and Fernandez-Villaverde [8], Maliar and Maliar [45], Fella [21] and White [67] for extensions and applications of EGM.

  7. Maliar and Maliar [40] shows another example of the model with quasi-geometric discounting that admits a closed-form solution under the assumption of the exponential utility function.

  8. Balbus, Reffett and Wozny [6, 7] suggest a different value-based recursion for the model with qusi-geometric discounting, which under appropriate assumptions delivers a unique SMNE equilibrium using a simple successive approximation scheme.

References

  1. Abreu D, Pearce D, Stachetti E (1986) Optimal cartel equilibria with imperfect monitoring. J Econ Theory 39:251–269

    Article  MathSciNet  MATH  Google Scholar 

  2. Amann H (1976) Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev 18(4):620–709

    Article  MathSciNet  MATH  Google Scholar 

  3. Angeletos M, Laibson D, Repetto A, Tobacman J, Weinberg S (2001) The hyperbolic consumption model: calibration, simulation, and empirical evaluation. J Econ Perspect 15(3):47–68

    Article  Google Scholar 

  4. Arellano C, Maliar L, Maliar S, Tsyrennikov V (2014) Envelope condition method with an application to default risk models (Manuscript)

  5. Aruoba SB, Fernández-Villaverde J, Rubio-Ramí rez J (2006) Comparing solution methods for dynamic equilibrium economies. J Econ Dyn Control 30:2477–2508

    Article  MATH  Google Scholar 

  6. Balbus Ł, Reffet K, Woźny Ł (2015) Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers. Int J Game Theory 44(1):83–112

  7. Balbus L, Reffet K, Wozny L (2012) Existence and uniqueness of time-consistent Markov policies for quasi-hyperbolic consumers under uncertainty (Manuscript)

  8. Barillas F, Fernandez-Villaverde J (2007) A generalization of the endogenous grid method. J Econ Dyn Control 31(8):2698–2712

    Article  MathSciNet  MATH  Google Scholar 

  9. Barro R (1999) Ramsey meets Laibson in the neoclassical growth model. Q J Econ 114(4):1125–1152

    Article  MATH  Google Scholar 

  10. Becker R (2012) Optimal growth with heterogeneous agents and the twisted turnpike: an example. Int J Econ Theory 8(1):24–47

    Article  MathSciNet  Google Scholar 

  11. Bernheim D, Ray D, Yeltekin S (2015) Poverty and self-control. Econometrica (forthcoming)

  12. Bernheim D, Ray D (1986) On the existence of Markov-consistent plans under production uncertainty. Rev Econ Stud 53(5):877–882

    Article  MathSciNet  MATH  Google Scholar 

  13. Brock W, Mirman L (1972) Optimal economic growth and uncertainty: the discounted case. J Econ Theory 4:479–513

    Article  MathSciNet  Google Scholar 

  14. Caplin A, Leahy J (2006) The recursive approach to time inconsistency. J Econ Theory 131(1):134–156

  15. Carroll K (2005) The method of endogenous grid points for solving dynamic stochastic optimal problems. Econ Lett 91:312–320

    Article  Google Scholar 

  16. Chatterjee S, Eyigungor B (2014) Continuous Markov equilibria with quasi-geometric discounting. Working Papers 14–6, Federal Reserve Bank of Philadelphia

  17. Cole H, Kubler F (2012) Recursive contracts, lotteries and weakly concave pareto sets. Rev Econ Dyn 15(4):479–500

    Article  Google Scholar 

  18. Coleman W (1991) Equilibrium in a production economy with an income tax. Econometrica 59(4):1091–1104

    Article  MathSciNet  MATH  Google Scholar 

  19. Coleman W (2000) Uniqueness of an equilibrium in infinite-horizon economies subject to taxes and externalities. J Econ Theory 95(1):71–78

    Article  MathSciNet  MATH  Google Scholar 

  20. Datta M, Mirman L, Reffett K (2002) Existence and uniqueness of equilibrium in distorted dynamic economies with capital and labor. J Econ Theory 103(2):377–410

    Article  MathSciNet  MATH  Google Scholar 

  21. Fella G (2014) A generalized endogenous grid method for non-smooth and non-concave problems. Rev Econ Dyn 17(2):329–344

    Article  Google Scholar 

  22. Feng Z, Miao J, Peralta-Alva A, Santos M (2014) Numerical simulation of nonoptimal dynamic equilibrium models. Int Econ Rev 55:83–110

    Article  MathSciNet  MATH  Google Scholar 

  23. Gong L, Smith W, Zou H (2011) Asset prices and hyperbolic discounting. CEMA Working Papers 486

  24. Harris C, Laibson D (2001) Dynamic choices of hyperbolic consumers. Econometrica 69(4):935–959

    Article  MathSciNet  MATH  Google Scholar 

  25. Judd K (2004) Existence, uniqueness, and computational theory for time consistent equilibria: a hyperbolic discounting example (Manuscript)

  26. Joshi S (1997) Turnpike theorems in nonconvex nonstationary environments. Int Econ Rev 38:245–248

    Article  MathSciNet  MATH  Google Scholar 

  27. Kocherlakota NR (1996) Reconsideration-proofness: a refinement for infinite horizon time inconsistency. Games Econ Behav 15(1):33–54

    Article  MathSciNet  MATH  Google Scholar 

  28. Kolokoltsov V, Yang W (2012) The turnpike theorems for Markov games. Dyn Games Appl 2(3):294–312

    Article  MathSciNet  MATH  Google Scholar 

  29. Krusell P, Smith A (2000) Consumption-savings decisions with quasi-geometric discounting. CEPR discussion paper no. 2651

  30. Krusell P, Smith A (2003) Consumption-savings decisions with quasi-geometric discounting. Econometrica 71:365–375

    Article  MATH  Google Scholar 

  31. Krusell P, Smith A (2008) Consumption-savings decisions with quasi-geometric discounting: the case with a discrete domain (Manuscript)

  32. Krusell P, Kuruşçu B, Smith A (2002) Equilibrium welfare and government policy with quasi-geometric discounting. J Econ Theory 105(1):42–72

  33. Kydland F, Prescott E (1980) Dynamic optimal taxation, rational expectations and optimal control. J Econ Dyn Control 2(1):79–91

    Article  Google Scholar 

  34. Laibson D (1997) Golden eggs and hyperbolic discounting. Q J Econ 112(2):443–477

    Article  MATH  Google Scholar 

  35. Laibson D, Repetto A, Tobacman J (1998) Self-control and saving for retirement. Brookings Pap Econ Act 1:91–172

    Article  Google Scholar 

  36. Leininger W (1986) The existence of perfect equilibria in a model of growth with altruism between generations. Rev Econ Stud LIII:349–367

    Article  MathSciNet  MATH  Google Scholar 

  37. Luttmer E, Mariotti T (2003) Subjective discounting in an exchange economy. J Polit Econ 111(5):959–989

  38. Majumdar M, Zilcha I (1987) Optimal growth in a stochastic environment: some sensitivity and turnpike results. J Econ Theory 43:116–133

    Article  MathSciNet  MATH  Google Scholar 

  39. Maliar L, Maliar S (2003) Solving the neoclassical growth model with quasi-geometric discounting: non-linear Euler-equation methods, IVIE WP AD 2003–23

  40. Maliar L, Maliar S (2004) Quasi-geometric discounting: a closed-form solution under the exponential utility function. Bull Econ Res 56(2):201–206

    Article  MathSciNet  MATH  Google Scholar 

  41. Maliar L, Maliar S (2005) Solving the neoclassical growth model with quasi-geometric discounting: a grid-based Euler-equation method. Comput Econ 26:163–172

    Article  MATH  Google Scholar 

  42. Maliar L, Maliar S (2006a) The neoclassical growth model with heterogeneous quasi-geometric consumers. J Money Credit Bank 38(3):635–654

    Article  MATH  Google Scholar 

  43. Maliar L, Maliar S (2006b) Indeterminacy in a log-linearized neoclassical growth model with quasi-geometric discounting. Econ Model 23(3):492–505

    Article  MATH  Google Scholar 

  44. Maliar L, Maliar S (2007) Short-run patience and wealth distribution. Stud Nonlinear Dyn Econ 11(1), Article 4

  45. Maliar L, Maliar S (2013) Envelope condition method versus endogenous grid method for solving dynamic programming problems. Econ Lett 120:262–266

    Article  MathSciNet  MATH  Google Scholar 

  46. Maliar L, Maliar S (2014) Numerical methods for large scale dynamic economic models. In: Schmedders K, Judd K (eds) Handbook of computational economics, chapter 7, vol 3. Elsevier Science, Amsterdam

    Google Scholar 

  47. Maliar L, Maliar S, Taylor JB, Tsener I (2015) A tractable framework for analyzing a class of nonstationary Markov models. NBER 21155

  48. Manuelli R, Sargent T (1987) Exercises in dynamic macroeconomic theory. Harvard University Press, Cambridge

    Google Scholar 

  49. Marcet A, Marimon R (1998) Recursive contracts, economics Working Papers, European University Institute

  50. Maskin E, Tirole J (2001) Markov perfect equilibrium: I. Observable actions. J Econ Theory 100(2):191–219

    Article  MathSciNet  MATH  Google Scholar 

  51. McKenzie L (1976) Turnpike theory. Econometrica 44:841–865

    Article  MathSciNet  MATH  Google Scholar 

  52. Messner M, Pavoni N, Sleet C (2012) Recursive methods for incentive problems. Rev Econ Dyn 15(4):501–525

    Article  Google Scholar 

  53. Mirman L, Morand O, Reffett K (2008) A qualitative approach to markovian equilibrium in infinite horizon economies with capital. J Econ Theory 139(1):75–98

    Article  MathSciNet  MATH  Google Scholar 

  54. Mitra T, Nyarko Y (1991) On the existence of optimal processes in nonstationary environments. J Econ 53:245–270

    Article  MathSciNet  MATH  Google Scholar 

  55. O’Donoghue T, Rabin M (1999) Doing it now or later. Am Econ Rev 89(1):103–124

    Article  Google Scholar 

  56. Pavoni N, Sleet C, Messner M (2015) The dual approach to recursive optimization: theory and examples, 2014 meeting papers 1267, society for economic dynamics

  57. Peleg B, Yaari M (1973) On the existence of a consistent course of action when tastes are changing. Rev Econ Stud 40(3):391–401

    Article  MATH  Google Scholar 

  58. Phelps ES, Pollak RA (1968) On second-best national saving and game-equilibrium growth. Rev Econ Stud 35:185–199

    Article  Google Scholar 

  59. Pollak RA (1968) Consistent planning. Rev Econ Stud 35:201–208

    Article  Google Scholar 

  60. Rustichini A (1998) Dynamic programming solution of incentive constrained problems. J Econ Theory 78(2):329–354

    Article  MathSciNet  MATH  Google Scholar 

  61. Sorger G (2004) Consistent planning under quasi-geometric discounting. J Econ Theory 118(1):118–129

    Article  MathSciNet  MATH  Google Scholar 

  62. Santos M (1999) Numerical solution of dynamic economic models. In: Taylor J, Woodford M (eds) Handbook of macroeconomics. Elsevier Science, Amsterdam, pp 312–382

    Google Scholar 

  63. Stachurski J (2009) Economic dynamics: theory and computation. MIT Press, Cambridge

    MATH  Google Scholar 

  64. Stokey NL, Lucas RE Jr, Prescott E (1989) Recursive methods in economic dynamics. Harvard University Press, Cambridge

    Google Scholar 

  65. Strotz RH (1955–1956) Myopia and inconsistency in dynamic utility maximization. Rev Econ Stud 23:165–180

  66. Tauchen G (1986) Finite state Markov chain approximations to univariate and vector autoregressions. Econ Lett 20:177–181

    Article  MathSciNet  Google Scholar 

  67. White M (2015) The method of endogenous gridpoints in theory and practice. Working Papers 15-03, University of Delaware

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Acknowledgments

Lilia Maliar and Serguei Maliar acknowledge support from the Hoover Institution and Department of Economics at Stanford University, University of Alicante, Santa Clara University and MECD Grant ECO2012-36719. We thank the editors Edward Prescott and Kevin Reffett for many useful comments and suggestions.

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  1. Office 249, Department of Economics, Stanford University, Stanford, CA, 94305-6072, USA

    Lilia Maliar

  2. Office 316J, Lucas Hall, Santa Clara University, 500 El Camino Real, Santa Clara, CA, 95053, USA

    Serguei Maliar

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Maliar, L., Maliar, S. Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example. Dyn Games Appl 6, 243–261 (2016). https://doi.org/10.1007/s13235-015-0177-8

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