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Article Excerpt
BECAUSE IT DOES not distinguish between aversion to risk and aversion to intertemporal substitution, the traditional theory of precautionary saving based on intertemporal expected utility maximization is a framework within which one cannot ask questions that are fundamental to the understanding of consumption in the face of labor income risk. For instance, how does the strength of the precautionary saving motive vary as the elasticity of intertemporal substitution changes, holding risk aversion constant? Or, how does the strength of the precautionary saving motive vary as risk aversion changes, holding the elasticity of intertemporal substitution constant?
Moreover, one might wonder whether some of the results from an intertemporal expected utility framework continue to hold once one distinguishes between aversion to risk and resistance to intertemporal substitution. For instance, does decreasing absolute risk aversion imply that the precautionary saving motive is stronger than risk aversion regardless of the elasticity of intertemporal substitution? And under what conditions does the precautionary saving motive decline with wealth?
To make sense of these questions, and to gain a better grasp on the channels through which precautionary saving may affect the economy, (1) we adopt a representation of preferences based on the Selden and Kreps-Porteus axiomatization, which separates attitudes toward risk and attitudes toward intertemporal substitution (see, e.g., Selden 1978, 1979, Kreps and Porteus 1978, Hall 1988, Epstein and Zin 1989, Farmer 1990, Weil 1990, Hansen, Sargent, and Tallarini 1999).
The existing literature on the theory of precautionary saving under Selden/Kreps-Porteus preferences is not very extensive. Barsky (1986) addresses some of the aspects of this theory in relation to rate-of-return risks in a two-period setup and Weil (1993) analyzes a parametric infinite-horizon model with mixed isoelastic/constant absolute risk aversion preferences (which do not allow one to look at the consequences of decreasing absolute risk aversion), but as of yet there has not been any systematic treatment of precautionary saving under Selden/Kreps-Porteus preferences.
Under intertemporal expected utility maximization, the strength of the precautionary saving motive is not an independent magnitude but is linked to other aspects of risk preferences. In that case, the absolute prudence -v"'/v" of a von Neumann-Morgenstern second-period utility function v measures the strength of the precautionary saving motive (see Kimball 1990b), and there is an identity linking prudence to risk aversion under additively time- and state-separable utility:
v"'(x)/v"(x) = a(x) - a'(x)/a(x), (1)
where
a(x) = - v"(x)/v'(x) (2)
is the Arrow-Pratt measure of absolute risk aversion. Similarly, the coefficient of relative prudence -xv"'(x)/v"(x) satisfies
- xv"'(x)/v"(x) = [gamma] + [epsilon](x), (3)
where
[gamma](x) = - xv"(x)/v'(x) = xa(x) (4)
is relative risk aversion, and
[epsilon](x) = xa'(x)/a(x) (5)
is the elasticity of absolute risk tolerance, (2) which approximates the wealth elasticity of risky investment.
The purpose of this paper is to determine what links exist--in the more general Selden/Kreps-Porteus framework that allows for risk preferences and intertemporal substitution to be varied independently from each other--between the strength of the precautionary saving motive, risk aversion, and intertemporal substitution.
In Section 1, we set up the model. In Section 2, we derive a local measure of the precautionary saving motive for small risks. Section 3 deals with large risks: it performs various comparative statics experiments to provide answers to the questions we asked in our first two paragraphs. The conclusion discusses the implications of our main results.
1. SETUP
This section describes our basic model and characterizes optimal consumption and saving decisions.
1.1 The Model
We use essentially the same two-period model of the consumption-saving decision as in Kimball (1990b), except for departing from the assumption of intertemporal expected utility maximization. (3) We assume that the agent can freely borrow and lend ar a fixed risk-free rate and that the constraint that an agent cannot borrow against more than the minimum value of human wealth is not binding at the end of the first period. Since the interest rate is exogenously given, all magnitudes can be represented in present-value terms--so that, without loss of generality, the real risk-free rate can be assumed to be zero. We also assume that labor supply is inelastic, so that labor income can be treated like manna from heaven. Finally, we assume that preferences are additively rime separable.
The preferences of our agent can be represented in two equivalent ways. First, there is the Selden ordinal certainty equivalence representation,
U([c.sub.1]) + U ([v.sup.-1](Ev([[??].sub.2]))) ----= U([c.sub.1]) + U(M([[??].sub.2])),
where [c.sub.1] and [c.sub.2] are first- and second-period consumption, u is the first-period utility function, U is the second-period utility function for the certainty equivalent of random second-period consumption (computed according to the atemporal von Neumann-Morgenstern utility function v), (4) E is ah expectation conditional on all information available during the first period, and M is the certainty equivalent operator associated with v:
M([[??].sub.2]) = [v.sup.-1](Ev(([[??].sub.2])).
According to this representation, the utility our consumer derives from the consumption lottery ([c.sub.1], [[??].sub.2]) is the sum of the felicity provided by [c.sub.1] and the felicity provided by the certainty equivalent M([[??].sub.2]) of [[??].sub.2] Obviously, v plays no role, and M is an identity, under certainty. These functions thus capture pure risk preferences.
Second, there is the Kreps-Porteus representation,
U([c.sub.1]) + [phi](Ev(([[??].sub.2])),
where nonlinearity of the function [phi] indicates departure from intertemporal expected utility maximization. This formulation expresses total utility as the nonlinear aggregate of current felicity and expected future felicity. (5)
These two representations are equivalent as long as v is a continuous, monotonically increasing function, so that M([[??].sub.2]) is well defined whenever Ev([[??].sub.2]) is well defined. The link between the two representations is that
[phi](x) = U([v.sup.-1](x)). (6)
Thus, as can be seen by straightforward differentiation, [phi] is concave if -U"/U' [greater than or equal to] -v"/v', while [phi] is convex if -U"/U' [less than or equal to] -v"/v'. We mainly use the Selden representation--which is more intuitive--but, when more convenient mathematically, we use the Kreps-Porteus representation.
1.2 Optimal Consumption and Saving
Our consumer solves the following problem:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where w is the sum of initial wealth, first-period income, and the mean of second-period income; x is "saving" out of this whole sum; and [??] is the deviation of second-period income from its mean.
The first-order condition for the optimal level of saving x is
u'(w - x) = U'(M(x + [??]))M'(x + [??]), (8)
where M' is defined by
M'(x + [??]) = [partial derivative]/[partial derivative]x M(X + [??]). (9)
To guarantee that the solution to (8) is uniquely determined, we would like the marginal utility of saving,
U'(M(x + [??]))M'(x + [??]),
to be a decreasing function of x. (6) In the main body of the paper, we simply assume that this condition holds; it is equivalent to the reasonable assumption that first-period consumption is a normal good. Appendix A gives a deeper treatment, proving that the marginal utility of saving is decreasing under plausible restrictions on preferences.
Equation (8) and the assumption of a decreasing marginal utility of saving imply that the uncertainty represented by y will cause additional saving if
U'(M(x + [??]))M'(x + [??]) > U'(x), (10)
that is, if the risk [??] raises the marginal utility of saving.
As in Kimball (1990b), we can study the strength of precautionary saving effects by looking at the size of the precautionary premium [theta]* needed to compensate for the effect of the risk y on the marginal utility of saving. The precautionary premium [theta]* is the solution to the equation
U'(M(x + [theta]* + [??]))M'(x + [theta]* + y) = U'(x), (11)
or equivalently, using the Kreps-Porteus representation of preferences, the solution to the equation
[phi]'[Ev(x + [theta]* + [??])]Ev'(x + [theta]* + [??]) = d/[v(x)]v'(x). (12)
[FIGURE 1 OMITTED]
To be more specific, [theta]* is the compensating Kreps-Porteus precautionary premium. It generalizes the compensating von Neumann-Morgenstern precautionary premium [psi]* that is defined (see Kimball 1990b) as the solution to
Ev'(x + [psi]* + [??]) = v'(x). (13)
As illustrated in Figure 1, the Kreps-Porteus precautionary premium [theta]", which obviously depends (among other things) on x, measures at each point the rightward shift of the marginal utility of saving curve due to the risk [??]. It enables us to isolate the effect of uncertainty from the other (purely intertemporal) factors that determine optimal savings under certainty.
Also, as shown in Kimball (1990b), Appendix B, the compensating precautionary premium [theta]* is the rightward shift of the consumption function due to [??]. The closely related equivalent precautionary premium (7) [theta] gives the leftward shift of the consumption function due to eliminating the risk [??]. Hence, the extra consumption due to eliminating [??], or equivalently the reduced consumption and increased saving due to the presence of [??] is equal to [[integral].sup.w.sub.w-[theta]] ([omega], 0) d[omega], where c'(w, 0) is the marginal propensity to consume out of wealth in the absence of uncertainty. If the marginal propensity to consume is constant in the absence of uncertainty, the increment to saving from y will be directly proportional to the size of the equivalent precautionary premium.
2. SMALL RISKS: A LOCAL MEASURE OF THE STRENGTH OF THE PRECAUTIONARY SAVING MOTIVE
In Appendix B, we prove that, for a small risk [??] with mean zero and variance [[sigma].sup.2],
[theta]*(x) = a(x)[1 + s(x)[epsilon](x)] [[sigma].sup.2]/2 + o([[sigma].sup.2]), (14)
where
s(x) = -U'(x)/xU"(x) (15)
denotes the elasticity of intertemporal substitution for the second period utility function U(x), (8) 0([[sigma].sup.2]) collects terms going to zero faster than [[sigma].sup.2], a(x) is the absolute risk aversion of v defined in (2), and [epsilon](x) denotes the elasticity of absolute risk tolerance defined in (5).
Therefore, the local counterpart for Kreps-Porteus preferences to the concept of absolute prudence defined for intertemporal expected utility preferences by Kimball (1990b) is
a(x)[1 + s(x)[epsilon](x)].
Similarly, the local counterpart for Kreps-Porteus preferences to relative prudence is
[??](x) = [gamma](x)[1 + s(x)[epsilon](x)], (16)
where [gamma](x) = xa(x) is relative risk aversion as above. Therefore, in the more general framework of Kreps-Porteus preferences, the strength of the precautionary saving motive is determined both by attitudes toward risk and attitudes toward intertemporal substitution.
Three important special cases should be noted:
(i) For intertemporal expected utility maximization, s(x) = 1/[gamma](x), so that [??](x) = [gamma](x) + [epsilon](x)--which is the expression given in (3).
(ii) For constant relative risk aversion, [gamma](x) = [gamma] and [epsilon](x) = 1, so that [??](x) = [gamma][1 + s(x)].
(iii) For constant relative risk aversion [gamma](x)= [gamma], and constant (but in general distinct) elasticity of intertemporal substitution (9) s(x) = s, and [??](x) = [gamma] [1 -4- s].
From (16), the local condition for positive precautionary saving is simply
[epsilon](x) [greater than or equal to] - 1/s(x), (17)
or, by calculating [epsilon](x) from (5) and moving one piece to the right-hand side of the equation:
-xv"'(x)/v" [greater than or equal to] [gamma](x) - [rho](x), (18)
where
[rho](x) = 1/s(x)
denotes the resistance to intertemporal substitution. The right-band side of (18) is zero under intertemporal expected utility maximization, in which case (18) reduces to the familiar condition v"'(x) [greater than or equal to] 0.
Intriguingly, equation (18) shows that when one departs from intertemporal expected utility maximization, quadratic risk preferences do not in general lead to the absence of precautionary saving effects that goes under the name of "certainty equivalence."
Though the local measure [??](x) cannot be used directly to establish global results, equation (14) does suggests several principles that are valid globally, as the next section shows. First, if one holds the outer intertemporal utility function U fixed, a change in risk preferences v which raises both risk aversion and its rate of decline tends to strengthen the precautionary saving motive. Second, decreasing absolute risk aversion (which implies [epsilon](x) [greater than or equal to] 0) guarantees that the precautionary saving motive is stronger than risk aversion. Third, if one holds risk...
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