PRECAUTIONARY SAVING AND RISK AVERSION An Anticipated Utility Approach

One of the surprises that came with the development and applications of expected utility theory to economics was the realization that risk aversion alone does not give rise to precautionary saving. As modeled by Leland (1968), precautionary saving was interpreted as an increase in saving in the current period if future income has the same mean but becomes less certain. Leland found that existence of precautionary saving requires that the third derivative of the utility function is positive, or equivalently, that the marginal utility function is convex. Leland’s finding may be interpreted as an evidence that our intial understanding of precautionary saving was lacking. But it could also imply that the way risk is modeled in expected utility theory, solely through declining marginal utility, is unsatisfactory. The development of new theories of decision making under uncertainty by Machina (1982) and Quiggin (1982) provides a suitable framework to examine this possibility. This examination is the subject of our note. The third derivative condition and its modification under the generalizations of expected utility theory can be illuminated by the following simple example. The consumer selects his first and second period consumption ci and cl, g iven his income in both periods y, and y2. y, is known with certainty, but y, is a random variable, being v2 e or & + c with probability l/2. For simplicity we assume a separable utility u, identical for both periods, zero interest rate, and no time preference. Under expected utility, the consumer solves