Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence

Those who study consumption behavior routinely assume that labor income is stochastic and that the utility function exhibits constant relative risk aversion. No one has derived closed form solutions to this problem, however, and therefore we do not know what the resulting consumption function looks like. In this paper, a numerical technique is used to give accurate approximations to the consumption function in multiperiod models with income uncertainty. The resulting consumption function is often dramatically different than the certainty equivalence solution typically used, in which consumption is proportional to the sum of financial wealth and the present discounted value of expected future labor income. The results help explain three important empirical consumption puzzles: the excess sensitivity of consumption to transitory income, the high growth of consumption in periods of low real interest rates, and the under spending of the elderly. In the last section of the paper, the numerical technique is applied to examples in which borrowing constraints are imposed. It is seen that future constraints, which bind only in certain states of the world, can have effects similar to those of current constraints that are binding.