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111

(7.27)

tax treatment of interest income would increase saving, and thus increase growth. But if consumption is relatively unresponsive to the rate of return, such policies would have little effect. Understanding the impact of rates of return on consumption is thus important.

The Interest Rate and Consumption Growth

We begin by extending the analysis of consumption under certainty in Section 7.1 to allow for a nonzero interest rate. This largely repeats material in Section 2.2; for convenience, however, we quickly repeat that analysis here.

Once we allow for a nonzero interest rate, the individuals budget constraint is that the present value of lifetime consumption cannot exceed initial wealth plus the present value of lifetime labor income. For the case of a constant interest rate and a lifetime of T periods, this constraint is

where r is the interest rate and where all variables are discounted to period 0.

When we allow for a nonzero interest rate, it is also useful to allow for a nonzero discount rate. In addition, it simplifies the analysis to assume that the instantaneous utility function takes the constant-relative-risk-aversion form used in Chapter 2: uiQ) = C}" 1(1 - ), where is the coefficient of relative risk aversion (the inverse of the elasticity of substitution between consumption at different dates). Thus the utility function, (7.1), becomes

where p is the discount rate.

Now consider our usual experiment of a decrease in consumption in some period, period f, accompanied by an increase in consumption in the next period by 1 -i- r times the amount of the decrease. Optimization requires that a marginal change of this type has no effect on lifetime utility. Since the marginal utilities of consumption in periods t and f -i-1 are Cf 1(1 + pY and CJiKl + pY, this condition is

;)"(--(1-- -2

We can rearrange this condition to obtain



The Interest Rate and Saving in the Two-Period Case

.Although an increase in the interest rate causes the path of consumption to be more steeply sloped, it does not necessarily foUow that the increase reduces initial consumption and thereby raises saving. The complication is that the change in the interest rate has not only a substitution effect, but also an income effect. Specifically, if the individual is a net saver, the increase in the interest rate allows him or her to attain a higher path of consumption than before.

The qualitative issues can be seen in the case where the individual lives for only two periods. For this case, we can use the standard indifference-curve diagram shown in Figure 7.2. Assume, for simplicity, that the indi-N-idual has no initial wealth. Thus in (Ci, Ci) space, the individuals budget constraint goes through the point ( Yi): the individual can choose to consume his or her income each period. And the slope of the budget constraint is -(1 -I- r): giving up one unit of first-period consumption allows the individual to increase second-period consumption by 1 \- r. When r rises, the budget constraint continues to go through ( ?) bul becomes steeper; thus it pivots clockwise around (Yi, 2).

In Panel (a), the individual is initially at the point ( , 2); that is, saving is initially zero. In this case the increase in r has no income effect-the individuals initial consumption bundle continues to be on the budget constraint. Thus first-period consumption necessarily falls, and so saving necessarily rises.

In Panel (b), Cl is initially less than 1, and thus saving is positive. In this case the increase in r has a positive income effect-the individual can now afford strictly more than his or her initial bundle. The income effect acts to decrease saving, whereas the substitution effect acts to increase it. The overall effect is ambiguous; in the case shown in the figure, saving does not change.

Finally, in Panel (c) the individual is initially borrowing. In this case both the substitution and income effects reduce first-period consumption, and so saving necessarily rises.

This analysis imphes that once we allow for the possibihty that the real interest rate and the discount rate are not equal, consumption need not be a random walk: consumption is rising over time if r exceeds p and falling if r is less than p. In addition, if there are variations in the real interest rate, there are variations in the predictable component of consumption growth. Mankiw (1981), Hansen and Singleton (1983), Hall (1988b), Campbell and Mankiw (1989a), and others therefore examine how much consumption growth responds to variations in the real interest rate. For the most part they find that it responds relatively little, which suggests that the intertemporal elasticity of substitution is low (that is, that is high).



FIGURE 7.2 The interest rate and consumption choices in the two-period case



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