Time-Inconsistent Portfolio Investment Problems

The explicit results for the classical Merton optimal investment/consumption problem rely on the use of constant risk aversion parameters and exponential discounting. However, many studies have suggested that individual investors can have different risk aversions over time, and they discount future rewards less rapidly than exponentially. While state-dependent risk aversions and non-exponential type (e.g. hyperbolic) discounting align more with the real life behavior and household consumption data, they have tractability issues and make the problem time-inconsistent. We analyze the cases where these problems can be closely approximated by time-consistent ones. By asymptotic approximations, we are able to characterize the equilibrium strategies explicitly in terms of the corrections to solutions for the base problems with constant risk aversion and exponential discounting. We also explore the effects of hyperbolic discounting under proportional transaction costs