Mathematics and Financial Economics

, Volume 2, Issue 1, pp 57–86 | Cite as

Investment and consumption without commitment

  • Ivar EkelandEmail author
  • Traian A. Pirvu


In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t = 0 can commit her successors, she can choose the policy that is optimal from her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium policy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak (Preprint, 2006) we give a precise definition of equilibrium policies in the context of the portfolio management problem, with finite horizon. We characterize them by a system of partial differential equations, and establish their existence in the case of CRRA utility. An explicit solution is provided for the case of logarithmic utility. We also investigate the infinite-horizon case and provide two different equilibrium policies for CRRA utility (in contrast with the case of exponential discounting, where there is only one optimal policy). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function.


Portfolio optimization Merton problem Equilibrium policies 

JEL Classification


Mathematics Subject Classification (2000)

91B30 60H30 60G44 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of British ColumbiaVancouverCanada
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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