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114 "Three extensions of the permanentincome hypothesis that we will not discuss are durability of consumption goods, habit formation, and nonexpected utility. For durability, see Mankiw (1982); Caballero (1990a, 1993); Eberly (1994); and Problem 7.6. For habit formation, see Deaton (1992, pp. 2934, 99100) and Campbell and Cochrane (1995). For nonexpected utility, see Weil (1989b, 1990) and Epstein and Zin (1989,1991). Because of tliese and other difficuhies, there has been considerable work on extensions or alternatives to the permanentincome hypothesis. This section touches on some of the issues raised by these theories.* Precautionary Saving and the Growth of Consumption Recall that our derivation of the randomwalk result in Section 7.2 was based on the assumption that utility is quadratic. Quadratic utility requires, however, that marginal utility reaches zero at some finite level of consumption and then becomes negative. It also imphes that the utility cost of a given variance of consumption is independent of the level of consumption. Since the marginal utility of consumption is declining, individuals have increasing absolute risk aversion: the amount of consumption they are willing to give up to avoid a given amount of uncertainty about the level of consumption rises as they become wealthier. These difficulties with quadratic utility suggest that marginal utility falls more slowly as consumption rises; that is, the third derivative of utility is probably positive rather than zero. To see the effects of a positive third derivative, assume that both the real interest rate and the discount rate are zero, and consider again the Euler equation relating consumption in consecutive periods, equation (7.20): u(Cf) = £f[u(Cf+i)]. As described in Section 7.2, if utility is quadratic, marginal utility is linear, and so Et[u(Ct+\)] equals uEtiQi]); thus in this case, the Euler equation reduces to Q = £f[Cf+iJ. But if u"(*) is positive, then u(C) is a convex function of C. Thus in this case £f[u(Cf+i)] exceeds u(£f[Cf+i]). But this means that if Q and EtlQ+i] are equal, £f[u(Cf+i)] is greater than u(Cf), and so a marginal reduction in Q increases expected utility. Thus the combination of a positive third derivative of the utility function and uncertainty about future income reduces current consumption, and thus raises saving. This saving is known as precautionary saving (Leland, 1968). Panel (a) of Figure 7.3 shows the impact of uncertainty and a positive third derivative of the utility function on the expected marginal utility of consumption. Since u"(C) is negative, u(C) is decreasing in C. And since u"{C) is positive, u(C) declines less rapidly as risesthat is, u(C) is convex. If consumption takes on only two possible values, Ca and , each with probability , the expected marginal utility of consumption is the average of marginal utility at these two values. In terms of the diagram, this is shown by the midpoint of the line connecting u{Ca) and u(Cb). As the
uiC) »(Q) [u(Q) + u(Q)]/2 ([ + ]/2) «(Q) (Q+Q)/2 (a) UiC) [u(Q)+u4Cb)]/2 [u(Q) + u(Cj5)]/2 (Q+Q)/2 (b) FIGURE 7.3 The effects of a positive third derivative of the utility function on the expected marginal utility of consumption
For a general utility function, the e + 1 term is replaced by Cu"{C)/u"(C). In analogy to the coefficient of relative risk aversion,  "{ )/ { ), Kimball (1990) refers to Cu"{C)/u"(C) as the coefficient of relative prudence. ""For more on the impact of precautionary saving on the level of aggregate consumption, see Skinner (1988); Caballero (1991); and Aiyagari (1994). diagram sliows, tlie fact that u(C) is convex imphes that this quantity is larger than marginal utihty at the average value of consumption, (C4+ )/2. Panel (b) shows the effects of an increase in uncertainty. When the high value of consumption rises, the fact that u"{C) is positive means that marginal utihty falls relatively little; but when the low value falls, the positive third derivative magnifies the rise in marginal utility. As a result, the increase in uncertainty raises expected marginal utility for a given value of expected consumption. Thus the increase in uncertainty raises the incentive to save. An important question, of course, is whether precautionary saving is quantitatively important. To address this issue, recall that in our analysis of the equity premium we found that the Euler equation for the riskfree asset is F = p H [ ]  ( + l)Var(0)/2 (see [7.40]). For the case of = p, this becomes E[g]  (e + l)Var(0). (7.42) Thus the impact of precautionary saving on expected consumption growth depends on the variance of consumption growth and the coefficient of relative risk aversion.! If both are substantial, precautionary saving can have a large effect on expected consumption growth. If the coefficient of relative risk aversion is 4 (which is toward the high end of values that are viewed as plausible), and the standard deviation of households uncertainty about their consumption a year ahead is 0.1 (which is consistent with the evidence in Dynan, 1993, and Carroh, 1992), (7.42) imphes that precautionary saving raises expected consumption growth by (l/2)(4 + 1)(0.1), or 2.5 percentage points. Finally, the presence of precautionary saving implies that notjust expectations of future income but also uncertainty about that income affects consumption. C. Romer (1990), for example, argues that the tremendous uncertainty generated by the stockmarket crash of 1929 and by the subsequent gyrations of the stock market was a major force behind the sharp faU in consumption spending in 1930, and thus behind the onset of the Great Depression. To give another example, Barsky, Mankiw, and Zeldes (1986) show that the combination of a current tax cut and an offsetting increase in future tax rates reduces households uncertainty about their lifetime aftertax resources. Thus when there is precautionary saving, this change raises current consumption. More generally, Caballero (1990b) observes that, for a given level of expected lifetime resources, uncertainty is likely to be larger when
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