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112

Complications

This discussion appears to imply that, unless the elasticity of substitution between consumption in different periods is large, increases in the interest rate are unlikely to bring about substantial increases in saving. There are two reasons, however, that the importance of this conclusion is limited.

First, many of the changes we are interested in do not involve just changes in the interest rate. For tax policy, the relevant experiment is usually a change in composition between taxes on interest income and other taxes that leaves govemment revenue unchanged. As Problem 7.5 shows, such a change has only a substitution effect, and thus necessarily shifts consumption toward the future.

Second, and more subtly, if individuals have long horizons, small changes in saving can accumulate over time into large changes in wealth (Summers, 1981a). To see this, first consider an individual with an infinite horizon and constant labor income. Suppose that the interest rate equals the individuals discount rate. From (7.27), this means that the individuals consumption is constant. The budget constraint then implies that the indi-\idual consumes the sum of interest and labor incomes: any higher steady level of consumption implies violating the budget constraint, and any lower level implies failing to satisfy the constraint with equaUty. That is, the individual maintains his or her initial wealth level regardless of its value: the individual is willing to hold any amount of wealth if r = p. A similar analysis shows that if r > p, the individuals wealth grows without bound, and that if r < p, his or her wealth falls without bound. Thus the long-run supply of capita] is perfectly elastic at r = p.

Summers shows that sinhlar, though less extreme, results hold in the case of long bul finite lifetimes. Suppose, for example, that r is slightly larger than p, that the intertemporal elasticity of substitution is small, and that labor income is constant. The facts that r exceeds p and that the elasticity of substitution is small imply that consumption rises slowly over the individuals lifetime. But with a long lifetime, this means that consumption is much larger at the end of life than at the beginning. But since labor income is constant, this in turn implies that the individual gradually builds up considerable savings over the firsl part of his or her Ufe and gradually decumulates them over the remainder. As a result, when horizons are finite but long, wealth holdings may be highly responsive to the interest

Since the stock of wealth in the economy is positive, individuals are on average savers rather than borrowers. Thus the overall income effect of a rise in the interest rate is positive. An increase in the interest rate thus has two competing effects on overall saving, a positive one through the substitution effect and a negative one through the income effect.



The Conditions for Individual Optimization

Consider our usual experiment of an individual reducing consumption in period t by an infinitesimal amount and using the resulting saving to raise consumption in period t + 1. If the individual is optimizing, this change leaves expected utility unchanged regardless of which asset the increased saving is invested in. Thus optimization requires

u(Ct) = --£r[(l + r/i)u(Q+i)] for aU i, (7.28)

where r is the return on asset z. Since the expectation of the product of two variables equals the product of their expectations plus their covariance, we can rewrite this expression as

"(Cf) = rrrffH + rUi]Et[u(Ct.i)] + Covfd + r/i, u(Cf+j)} for all i

1 1+p

(7.29)

where Covf () is covariance conditional on information available at time t.

If we assume that utility is quadratic, u(C) = - then the

marginal utility of consumption is 1 - aC. Using this to substitute for the covariance term in (7.29), we obtain

uiQ) = -- ]£f[l + r/i]Et[u(Cf+i)] - aCovtd + r/i, ,+ )} (7. 1 + p

.30)

Equation (7.30) implies that in deciding whether to hold more of an asset, the individual is not concerned with how risky the asset is: the variance of the assets return does not appear m (7.30). Intuitively, a marginal increase in holdings of an asset that is risky, but whose risk is not correlated with

""Carroll (1992) shows, however, that the presence of uncertainty weakens this conclusion somewhat.

rate in the long run even if the intertemporal elasLicity of substitution is small. i"

7.5 Consumption and Risky Assets

In prartice, individuals can invest in many assets, almost ah of which have uncertain returns. Extending our analysis to account for multiple assets and risk raises some new issues concerning both household behavior and asset markets.



The Consumption CAPM

This discussion takes assets expected returns as given. But individuals demands for assets determine these expected returns. If, for example, an assets payoff is highly correlated with consumption, its price must be driven down to the point where its expected return is high for individuals to hold it.

To see the implications of this observation, suppose that all individuals are the same, and return to the first-order condition in (7.30). Solving this expression for the expected return on the asset yields

Et[l + = 1 7 1 + p)u(Ct) + aCoVtd + )]- (7.31)

Equation (7.31) states that the higher the covariance of an assets payoff with consumption, the higher its expected return must be.

We can simplify (7.31) by considering the return on a risk-free asset. If the payoff to an asset is certain, then the covariance of its payoff with

the overall risk the individual faces, does not increase the variance of the individuals consumption. Thus in evaluating that marginal decision, the individual considers only the assets expected return.

Equation (7.30) implies that the aspect of riskiness that matters to the decision of whether to hold more of an asset is the relation between the assets payoff and consimiption. Suppose, for example, that the individual is given an opportunity to buy a new asset whose expected rettirn equals the rate of return on a risk-free asset that the individual is already able to buy. If the payoff to the new asset is typically high when the marginal utility of consumption is high (that is, when consumption is low), buying one unit of the asset raises expected utility by more than buying one unit of the risk-free asset does. Thus (since the individual was previously indifferent about buying more of the risk-free asset), the individual can raise his or her expected utility by buying the new asset. As the individual invests more in the asset, his or her consumption comes to depend more on the assets payoff, and so the covariance between consumption and the assets return becomes less negative. In the example we are considering, since the assets expected return equals the risk-free rate, the individual invests in the asset until the covariance of its return with consumption reaches zero.

This discussion implies that hedging risks is crucial to optimal portfolio choices. A steel worker whose future labor income depends on the health of the American steel industry should avoid-or better yet, sell short-assets whose retums are positively correlated with the fortunes of the steel industry, such as shares in American steel companies. Instead the worker should invest in assets whose returns move inversely with the health of the U.S. steel industry, such as foreign steel companies or American aluminum companies.



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