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.

Several recent studies have suggested that empirical rejections of the permanent income/life cycle model may be due to the existence of liquidity constraints. This paper tests the permanent income hypothesis against the alternative hypothesis that consumers optimize subject to a well-specified sequence of borrowing constraints. Implications for consumption in the presence of borrowing constraints are derived and then tested using time-series/cross-section data on families from the Panel Study of Income Dynamics. The results generally support the hypothesis that an inability to borrow against future labor income affects the consumption of a significant portion of the population.

Several recent studies have suggested that empirical rejections of the permanent income/life cycle model might be due to the existence of liquidity constraints. This paper tests the permanent income hypothesis against the alternative hypothesis that consumers optimize subject to a well-specified sequence of borrowing constraints. Implications for consumption in the presence of borrowing constraints are derived and then tested using time-series/cross-section data on families from the Panel Study of Income Dynamics. The results generally support the hypothesis that an inability to borrow against future labor income affects the consumption of a significant portion of the population.