  This paper considers the Merton portfolio management problem. We are
concerned with non-exponential discounting of time and this leads to time
inconsistencies of the decision maker. Following Ekeland and Pirvu 2006, we
introduce the notion of equilibrium policies and we characterize them by an
integral equation. The main idea is to come up with the value function in this
context. If risk preferences are of CRRA type, the integral equation which
characterizes the value function is shown to have a solution which leads to an
equilibrium policy. This work is an extension of Ekeland and Pirvu 2006.
