Advertisement

Mathematics and Financial Economics

, Volume 4, Issue 1, pp 29–55 | Cite as

The golden rule when preferences are time inconsistent

  • Ivar EkelandEmail author
  • Ali Lazrak
Article

Abstract

We investigate the classical Ramsey problem of economic growth when the planner uses non-constant discounting. It is well-known that this leads to time inconsistency, so that optimal strategies are no longer implementable. We then define equilibrium strategies to be such that unilateral deviations occurring during a small time interval are penalized. Non-equilibrium strategies are not implementable, so only equilibrium strategies should be considered by a rational planner. We show that there exists such strategies which are (a) smooth, and (b) lead to stationary growth, as in the classical Ramsey model. Finally, we prove an existence and multiplicity result: for logarithmic utility and quasi-exponential discount, there is an interval I such that, for every k in I, there is an equilibrium strategy converging to k. We conclude by giving an example where the planner is led to non-constant discount rates by considerations of intergenerational equity.

Keywords

Time inconsistency Markov strategies Ramsey models Nash equilibria Intergenerational equity Implicit differential equation 

JEL Classification

E43 O44 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Carr J.: Applications of Centre Manifold Theory. Springer, NY (1981)zbMATHGoogle Scholar
  2. 2.
    Dasgupta P.: On some alternative criteria for justice between generations. J. Public Econ. 3, 405–423 (1974)CrossRefGoogle Scholar
  3. 3.
    Dasgupta P.: On some problems arising from professor Rawls’ conception of distributive justice. Theory Decis. 4, 325–344 (1974)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ekeland, I., Lazrak, A.: Being Serious About Non-Cmmitment. http://arxiv.org/abs/math/0604264. Accessed 12 April 2006
  5. 5.
    Ekeland, I., Lazrak, A.: Equilibrium policies when preferences are time inconsistent. http://arxiv.org/abs/0808.3790. Accessed 27 August 2008
  6. 6.
    Frederick S., Loewenstein G., O’Donoghue T.: Time discounting and time preference: a critical review. J. Econ. Lit. 40, 351–401 (2002)CrossRefGoogle Scholar
  7. 7.
    Harris C., Laibson D.: Dynamic choices of hyperbolic consumers. Econometrica 69, 935–957 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Karp L.: Global warming and hyperbolic discounting. J. Public Econ. 89, 261–282 (2005)CrossRefGoogle Scholar
  9. 9.
    Karp L.: Non-constant discounting in continuous time. J. Econ. Theory 132, 557–568 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Karp L., Fuji T.: Numerical analysis of a non-constant pure rate of time preference: a model of climate policy. J. Environ. Econ. Manag. 56, 83–101 (2008)zbMATHCrossRefGoogle Scholar
  11. 11.
    Karp L., Lee I.H.: Time-consistent policies. J. Econ. Theory 112, 353–364 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Krusell P., Smith A.: Consumption-savings decisions with quasi-geometric discounting. Econometrica 71(1), 365–375 (2003)zbMATHCrossRefGoogle Scholar
  13. 13.
    Phelps E.S.: The indeterminacy of game-equilibrium growth. In: Phelps, E.S. (ed.) Altruism, Morality and Economic theory, pp. 87–105. Russell Sage Foundation, New York (1975)Google Scholar
  14. 14.
    Phelps E.S., Pollak R.A.: On second-best national saving and game-equilibrium growth. Rev. Econ. Stud. 35, 185–199 (1968)CrossRefGoogle Scholar
  15. 15.
    Ramsey F.P.: A mathematical theory of saving. Econ. J. 38(152), 543–559 (1928)CrossRefGoogle Scholar
  16. 16.
    Sumaila U., Walters C.: Intergenerational discounting: a new intuitive approach. Ecol. Econ. 52, 135–142 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Canada Research Chair in Mathematical EconomicsUniversity of British ColumbiaVancouverCanada
  2. 2.Sauder School of BusinessUniversity of British ColumbiaVancouverCanada

Personalised recommendations