  The paper [12] examines a concept of equilibrium policies instead of optimal
controls in stochastic optimization to analyze a mean-variance portfolio
selection problem. We follow the same approach in order to investigate the
Merton portfolio management problem in the context of non-exponential
discounting, a context that give rise to time-inconsistency of the decision
maker. Equilibrium policies are characterized in this context by means of a
variational method which leads to a stochastic system that consists of a flow
of forward-backward stochastic differential equations and an equilibrium
condition. An explicit representation of the equilibrium policies is provided
for the special cases of power, logarithmic and exponential utility functions.
