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


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.


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

JEL Classification

E43 O44 


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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

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