In this case, although consumption is approximately proportional to mcome, the constant of proportionality is less than 1; that is, consumption is on average less than permanent mcome. As Friedman describes, there are various ways of extending the basic theory to make it consistent with this result. One is to account for turnover among generations and long-run growth: if the young generally save and the old generally dissave, the fact that each generahon is wealthier than the previous one implies that the youngs saving is greater than the olds dissaving.

wliere the last line uses the assumption that the mean of transitory income IS zero.

Thus the permanent-income hypothesis predicts that the key determinant of the slope of an estimated consumption function, b, is the relative variation in permanent and transitory income. Intuitively, an increase in current income is associated with an increase in consumption only to the extent that it reflects an increase in permanent income. When the variation in permanent income is much greater than the variation in transitory income, almost all differences in current income reflect differences in permanent income; thus consumption rises nearly one-for-one with current income. But when the variation in permanent income is small relative to the variation m transitory income, little of the variation in current income comes from variation in permanent income, and so consumption rises little with current mcome.

This analysis can be used to understand the estimated consumption functions in Figure 7.1. Across households, much of the variation in income reflects such factors as unemployment and the fact that households are at different points in their life cycles. As a result, the estimated slope coef-hcient is substantiaUy less than 1, and the estimated intercept is positive. Over time, in contrast, almost all of the variation in aggregate income reflects long-run growth-thai is, permanent increases in the economys resources. Thus the estimated slope coefficient is close to 1, and the estimated mtercept is close to zero.

Now consider the differences between blacks and whites. The relative variances of permanent and transitory income are similar in the two groups, and so the estimates of b are similar. But blacks average incomes are lower than whites; as a result, the estimate of a for blacks is lower than the estimate for whites (see [7.9]).

To see the intuition for this result, consider a member of each group whose income equals the average income among whites. Since there are many more blacks with permanent incomes below this level than there are with permanent incomes above it, the blacks permanent income is much more likely to be less than his or her current income than more. As a result, blacks with this current income have on average lower permanent income; thus on average they consume less than their income. For the white, in contrast, his or her permanent income is about as likely to be more than current income as it is to be less; as a result, whites with this current income on average have the same permanent income, and thus on average they consume

E[U] = E

a>0. (7.10)

We will assume that the individuals wealth is such that consumption is always in the range where marginal utility is positive. As before, the individual must pay off any outstanding debts at the end of his or her life. Thus the budget constraint is again given by equation (7.2), Xf=i Cf < Ao - SLi r-To describe the individuals behavior, we use the Euler-equation approach that we employed in Chapters 2 and 4. Specifically, suppose that the individual has chosen first-period consumption optimally given the information available, and suppose that he or she will choose consumption in each future period optimally given the information then available. Now consider a reduction in Ci of dC from the value the individual has chosen and an equal increase in consumption at some future date from the value he or she would have chosen. If the individual is optimizing, a marginal change of this type does not affect expected utility. Since the marginal utihty of consumption in period 1 is 1 ~ aCi, the change has a utility cost of (1 ~ aC\)dC. And since the marginal utility of period-f consumption is 1 - aCt, the change has an expected utility benefit of Ei[l ~ aCt]dC. where fiH denotes expectations conditional on the information available in period 1. Thus if the individual is optimizing,

1 ~ ] = £i[l - flCtJ, for f = 2,3,...,r. (7.11)

Since £i[l ~ aQ] equals 1 - aEilQ], this implies

Ci=£i[Cf], for t = 2,3,....r. (7.12)

The individual knows that his or her hfetime consumption will satisfy the budget constraint, (7.2), with equaUty. Thus the expectations of the two

their mcome. In sum, the permanent-income hypothesis attributes the different consumption pattems of blacks and whhes to the different average incomes of the two groups, and not to any differences in tastes or culture.

7.2 Consumption under Uncertainty: The Random-Walk Hypothesis

Individual Behavior

We now extend our analysis to account for uncertainty. Continue to assume that both the interest rate and the discount rate are zero. In addition, suppose that the instantaneous utility function, is quadratic. Thus the individual maximizes

7.2 Consumption under Uncertainty: Tlie Random-Walk Hypothesis 317

sides of tlie constraint must be equal:

XEilQ] = Ao+XEi[Yt]. (7.13)

f=] f=i

Equation (7.12) implies that the left-hand side of (7.13) is TQ. Substituting this into (7.13) and dividing by T yields

1 /

(7.14)

\ f=i /

That is, the individual consumes 1 / of his or her expected lifetime resources.

Implications

Equation (7.12) implies that £i [ C2 ] equals Q. More generally, reasoning analogous to what we have just done implies that in each period, expected next-period consumption equals current consumption. This imphes that changes in consumption are unpredictable. By the definition of expectations, we can write

C, = Eti[Ct] + et, (7.15)

where is a variable whose expectation as of period t - 1 is zero. Thus, since £f i[Cf] = Cf i, we have

Ct = Ct-i + et. (7.16)

This is Halls famous result that the life-cycle/permanent-income hypothesis implies that consumption follows a random walk (Hall, 1978). The intuition for this result is straightforward: if consumption is expected to change, the individual can do a better job of smoothing consumption. Suppose, for example, that consumption is expected to rise. This means that the current marginal utility of consumption is greater than the expected future marginal utility of consumption, and thus that the individual is better off raising current consumption. Thus the individual adjusts his or her current consumption to the point where consumption is not expected to change.

In addition, our analysis can be used to find what determines the change in consumption, c. Consider for concreteness the change from period 1 to period 2. Reasoning parallel to that used to derive (7.14) implies that C2 equals 1/( - 1) of the individuals expected remaining lifetime resources: