Precautionary Savings and Borrowing Constraints
The assumption of quadratic preferences is quite restrictive. When preferences are not quadratic, certainty equivalence does not hold, and uncertainty about noninsurable labor income generally affects consumption.
To see this, consider, for example, the first-order condition (10.11) with a constant risk free rate r:
So long as consumers are risk averse (i.e., so long as u′′ < 0), increased uncertainty in the form of an increase in the variance of consumption decreases expected utility. But the effect of increased uncertainty on behavior depends on whether it affects the expected marginal utility of consumers through the first-order condition (10.49).
As long as utility is quadratic, marginal utility is linear in consumption, and u′′′ = 0. Thus, in the case of quadratic utility the variance of consumption has no effect on expected marginal utility and thus no effect on optimal behavior. After all, this is why certainty equivalence holds with quadratic preferences. In the general case, however, for most plausible utility functions, u′′′ > 0. Then marginal utility is convex in consumption, and an increase in uncertainty increases expected marginal utility. Thus, from (10.49), increased variability of future consumption would require an increase in expected future consumption relative to current consumption. Uncertainty leads consumers to defer consumption and thus be more prudent. The effects of prudence on savings were first analyzed by Leland [1968] and subsequently by Sandmo [1970] and Dreze and Modigliani [1972].
However, in general, it is almost impossible to solve for optimal consumption in the presence of prudent behavior. A case that can be solved analytically is the case of constant absolute risk aversion, which implies a utility function of the form
where θ is the coefficient of absolute risk aversion. This case has been analyzed in Caballero [1990].
Nonquadratic preferences and precautionary savings are not the only deviations from the permanent income hypothesis that have been considered in the literature. Other deviations that have been considered are incomplete markets and liquidity (borrowing) constraints, asymmetric information, and departures from full optimization. These extensions have been motivated by empirical weaknesses of the permanent income hypothesis and the consumption CAPM and are surveyed extensively in Attanasio [1999]. Financial frictions (i.e., deviations from perfect financial markets) are discussed in chapter 19.
10.4