There has been a strong resurgence of interest in the life cycle model of consumption with a stochastic income stream. In particular, with various degrees of restrictiveness, it is argued that this theory implies that consumption should follow a Markov process. At one extreme with additive quadratic preferences, perfect capital markets and a known constant interest rate, this holds exactly. More generally with additive preferences, expected discounted marginal utilities obey the martingale condition. The purpose of the present note is to point out that such results depend on the existence of interior solutions and that such an assumption is much more restrictive in this framework than it is generally in economics.

With income risk and expected utility maximisation, the utility of each state is weighted by its probability. Suppose that there are some high and low income possibilities and income has a positive mean. The utility function may possess one of two properties: either the marginal utility of consumption in a state remains finite or becomes unbounded as consumption becomes low.