Investment and consumption without commitment
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abstract
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 his/her
successors, he/she can choose the policy that is optimal from his/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 strategy between the successive decision-makers. In
the line of the earlier work by Ekeland and Lazrak we give a precise definition
of equilibrium strategies in the context of the portfolio management problem,
with finite horizon, we characterize it by a system of partial differential
equations, and we show existence in the case when the utility is CRRA and the
terminal time T is small. We also investigate the infinite-horizon case and we
give two different explicit solutions in the case when the utility is CRRA (in
contrast with the case of exponential discount, where there is only one). 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.
