Some results on “an income fluctuation problem”

A consumer at each period, given the income available, y, has to decide how much to consume and save. If he consumes c ⩾ 0 units he gets u( c) units of satisfaction or utility, and if x = y − c ⩾ 0 is the amount saved then the available income in the next period is rx + ω k, where ω k is a random variable, and r is an interest factor that is assumed to be known with certainty. Infinite time horizon problems are considered, and it is shown that if 0 < δr < 1, where 0 < δ < 1 is a discount factor, then the limiting policy is optimal. Questions about the behavior of the stock level, such as boundness, are considered, and an example is given that shows that the stock level might converge almost surely to infinity. Finally an economic explanation is given.