  We consider the problem of optimizing lifetime consumption under a habit
formation model, both with and without an exogenous pension. Unlike much of the
existing literature, we apply a power utility to the ratio of consumption to
habit, rather than to their difference. The martingale/duality method becomes
intractable in this setting, so we develop a greedy version of this method that
is solvable using Monte Carlo simulation. We investigate the behaviour of the
greedy solution, and explore what parameter values make the greedy solution a
good approximation to the optimal one.
