No one has derived closed-form solutions for consumption with stochastic labor income and constant relative risk aversion utility. A numerical technique is used here to give an accurate approximation to the solution. 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 value of expected future income. The results help explain three important empirical consumption puzzles: excess sensitivity of consumption to transitory income, high growth of consumption in the presence of a low risk-free interest rate, and underspending of the elderly.
Zeldes, Stephen. "Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence." Quarterly Journal of Economics 104, no. 2 (May 1989): 275-98.
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