Precautionary Saving and Prudence

Core Idea

Precautionary saving is extra saving caused by uncertainty about future income. In the Euler equation, saving depends on expected future marginal utility. If marginal utility is convex, which holds when the third derivative of utility is positive (u‴ > 0), then more uncertainty raises expected marginal utility of future consumption, and the consumer saves more today. Quadratic utility has u‴ = 0, which is why Hall's random-walk model shows no precautionary motive; common utility functions such as log and constant relative risk aversion (CRRA) have u‴ > 0. Miles Kimball called this property prudence and measured it by −u‴/u″.

How It's Best Learned

Work a two-period example with log utility. Compare a consumer whose second-period consumption is certain with one facing a fifty-fifty spread around the same mean, and compute expected marginal utility in each case. Then draw the convex marginal-utility curve and the chord between the two outcomes, so the role of convexity is visible.

Common Misconceptions

• Precautionary saving is the same thing as risk aversion. (Risk aversion depends on concave utility, u″ < 0; precautionary saving depends on convex marginal utility, u‴ > 0. Quadratic utility is risk averse but has no precautionary motive.)
• Uncertainty affects saving only if people expect income to fall on average. (A mean-preserving increase in risk, with no change in expected income, raises saving for a prudent consumer.)
• Precautionary savers keep accumulating wealth without limit. (In buffer-stock models with impatient consumers, saving targets a modest wealth buffer and stops growing once it is reached.)

Explainer

Hall's random-walk result rests on quadratic utility. That assumption is convenient, but it hides one channel through which uncertainty affects saving. Once we drop it, the consumer facing risky future income generally saves more than one facing certain income with the same mean. That extra saving is precautionary saving.

Begin with the Euler equation with r equal to the discount rate: u′(C_t) = E_t[u′(C_{t+1})]. The consumer saves until the marginal utility of consuming today equals the expected marginal utility of consuming tomorrow. Now ask how uncertainty about C_{t+1} affects the right-hand side. If u′ is a straight line, as with quadratic utility, the expectation of u′(C_{t+1}) equals u′ of the expected value. Spreading out outcomes changes nothing. This is called certainty equivalence.

If u′ is convex, Jensen's inequality gives E[u′(C_{t+1})] > u′(E[C_{t+1}]). Uncertainty raises expected future marginal utility. To restore the Euler equation, the consumer must raise u′(C_t), which means lowering current consumption, which means saving more. Convex marginal utility is the same as a positive third derivative, u‴ > 0. Log utility and CRRA utility both have this property. A numerical check with log utility: if next-period consumption is a sure 100, expected marginal utility is 0.01; if it is 50 or 150 with equal chance, it is about 0.0133.

Miles Kimball, in a 1990 paper, named this property prudence and proposed −u‴/u″ as its measure, by analogy with the Arrow-Pratt measure of risk aversion, −u″/u′. The two concepts differ. Risk aversion concerns dislike of risk; prudence concerns how risk changes the desire to save. Quadratic utility is risk averse and not prudent.

Precautionary saving matters for several reasons. It helps explain why households hold liquid wealth well before retirement. Combined with impatience and borrowing limits, it yields buffer-stock behavior, studied by Angus Deaton and Christopher Carroll: households build a modest cushion, spend from it in bad years, and rebuild it in good years. It also implies that consumption grows faster when income is more uncertain, because people consume less now and more later. Finally, it gives a reason why social insurance, which reduces income risk, can lower private saving.