Optimal saving under general changes in uncertainty: a nonexpected utility maximising approach.

I. Introduction

There has been an extensive discussion of optimal consumption-saving behavior of expected utility maximizing risk averse individuals[6; 7]. There are, however, two limitations of such works. First, the widely used time additive von Neumann Morgenstern (VNM) preferences may not be suitable for analyzing choice problems in a dynamic context. Since for this class of preferences the coefficient of relative risk aversion turns out to be the reciprocal of the elasticity of intertemporal substitution, these preferences fail to distinguish between the importance of intertemporal substitution and risk aversion in determining the optimal choice for the individual decision maker. Secondly, in analyzing the comparative static effect of an increase in risk, the increase in risk has been usually captured by the mean preserving spread of the distribution of the underlying random variable. But, since the mean of the distribution is stipulated to be unchanged, the mean preserving spread, undoubtedly, provides a restrictive characterization of an increase in risk.

The limitation of the VNM preferences has motivated researchers to look for an alternative framework to analyze dynamic choices under uncertainty. It was Selden[8; 9] who developed a nonexpected utility maximizing approach by proposing the Ordinal Certainty Equivalent (OCE) preferences to distinguish between intertemporal substitution and risk aversion. Since then a number of other authors have further examined die implications of the nonexpected utility maximizing framework. Not surprisingly, in the literature of nonexpected utility maximizing analysis a considerable attention has been given to the individual saving decision under capital risk. In a clear departure from the expected utility maximizing analysis, under the nonexpected utility maximizing approach, optimal saving tends to be determined by the elasticity of intertemporal substitution as well as the risk aversion parameter.

However, even in the nonexpected utility maximizing framework, the increase in capital risk has usually been characterized in terms of a mean preserving spread of the random rate of return. It has been shown by Selden[9] and Weil[10] that the effect of an increase in capital risk on the level of saving depends only on the elasticity of intertemporal substitution and not on the risk aversion parameter. The question remains whether or not the irrelevance of risk aversion result is robust. Does this result hold for more general characterizations of increases in risk when the mean of the distribution of the random rate of return does not remain unchanged as stipulated under the mean preserving spread? In this paper we consider general increases in risk and examine their effects on the optimal saving with OCE preferences. Specifically, we consider the shifts of the distribution of the random return that are characterized by Stochastic Dominance relationships that allow for mean returns to change.(1)

The plan of the paper is as follows. In section II we develop the basic nonexpected utility maximizing model of the saving-consumption decisions for an individual with OCE preferences. We also review the standard effect of a mean preserving spread of the rate of return on the level of saving. Section III contains a brief description of the stochastic dominance characterization of an increase in risk. In section IV we present the main results regarding the effect on saving of an increase in risk characterized by a First Degree Stochastic Dominance (FSD) and a Second Degree Stochastic Dominance (SSD) shift of the distribution function of the rate of return. We show that even under such general shifts of the distribution function, the qualitative effect of an increase in risk on optimal saving depends on the elasticity of intertemporal substitution and not on risk aversion. In section V we examine the issue of the relevance of risk aversion for characterizing an increase in risk. Concluding remarks are made in section VI.

II. The Model

Following the standard two period models of saving-consumption [7] we consider an individual who has an income of W in period 1. Saving in period 1 amounts to

[S.sub.1] = W - [C.sub.1] (1) where [C.sub.1] is the level of consumption in period 1. [S.sub.1] generates an income of (W - [C.sub.1])R for period 2 where R = 1 + rate of return on saving. The random rate of return R [element of] [R.sub.1,R.sub.2] has the continuous distribution function F(R, [theta] where [theta] is a shift parameter. Since the individual does not have any other source of income in period 2, the random consumption, [C.sub.2] in period 2 is given by [C.sub.2] = (W - [C.sub.1)R = [S.sub.1]R. (2)

The individual has OCE preferences a la Selden [8). Thus the individual maximizes

U([C.sub.2]) + [beta]U([C.sub.2]) (3) where U(.) is a concave utility function, [beta] the utility discount factor. [C.sub.2] is the certainty equivalent level of period 2 consumption, i.e., the nonstochastic level of consumption which provides utility equal to the expected utility of the random consumption, [C.sub.2] i.e.,

V([C.sub.2]) = E[V([C.sub.2])]

or

[C.sub.2] = [V.sup.-1]{E[V([C.sub.2])]} (4)

where V(.) is a strictly concave function. Note that in view of (4) the objective functional in (3) is not an expected utility functional since it is not linear in probabilities. The first order condition for maximization of (3) subject to (2) and (4) is given by

[Mathematical Expression Omitted]

In order to examine the importance of intertemporal substitution and risk aversion in the optimal consumption-saving decision we consider the following familiar specifications of the U and V functions [1]:

  U([C.sub.1]) = [C.sup.1-[alpha].sub.1]/(1 - [alpha]), [alpha] > 0    (6)
  V([C.sub.2]) = [C.sup.1-[epsilon].sub.2]/(1 - [epsilon]), [epsilon] > 0   (7)

The above specifications can distinguish between the intertemporal substitution parameter and the risk aversion parameter. Specifically [alpha] is the reciprocal of the elasticity of intertemporal substitution for consumption and f the coefficient of relative risk aversion. Note that if [alpha] = [epsilon], our nonexpected utility maximizing framework reduces to familiar expected utility maximizing framework. The first order condition (5) now reduces to

[C.sup.-[alpha] = [beta]E[[RC.sup.-[epsilon].sub.2]][C.sup.([epsilon]-[alpha]).sub.2]. (8)

Simplification of (8) leads to the following expressions for optimal consumption and optimal saving in period 1, [C.sub.1] = W/[1 + [[beta].sup.1/[alpha]][R.sup.(1-[alpha])]/[alpha]] (9) and [S.sub.1] = W[([[beta].sup.1/[alpha]][R.sup.(1-[alpha])/[alpha])/(1 + [beta].sup.1/[alpha]] [R.sup.(1-[alpha]])/[alpha])] (10) where [R = [E([R.sup.1-[epsilon]])].sup.1/(1-[epsilon]) (11) is the certainty equivalent interest rate.

For future reference we can now briefly demonstrate the effect of a mean preserving increase in the capital risk on the level of saving. As is well known[10] for risk averse individuals ([epsilon] > 0), a mean preserving spread of the rate of return distribution lowers R. From (11) it is straightforward to show that

[Mathematical Expressions Omitted]

Thus we have the conventional result of the nonexpected utility maximizing framework[9; 10] as summarized in the following proposition.

Proposition 1. The direction of the effect of a mean preserving increase in capital risk on the level of optimal saving depends only on the elasticity of intertemporal substitution and not on the degree of risk aversion; if the elasticity of intertemporal substitution is large with [alpha] < I (small with [alpha] > 1) a mean preserving increase in the rate of return risk leads to a smaller (larger) saving.

III. Stochastic Dominance

In this section we briefly discuss the general concepts of stochastic dominance. Let Re [RI, R2] be a random variable with a continuous distribution function. For any two distribution functions F and G the following stochastic dominance relationships are defined.

First Degree Stochastic Dominance (FSD): F dominates G in the sense of First Degree Stochastic Dominance if and only if, [Mathematical Expressions Omitted] with strict inequality holding at least once over the domain of the distribution functions.

Second Degree Stochastic Dominance (SSD): F dominates G in the sense of Second Degree Stochastic Dominance if and only if

[Mathematical Expressions Omitted]

with strict inequality holding at least once over the domain of the distribution functions.

In order to characterize an increase in risk in terms of a stochastic dominance relationship we follow Hadar and Russell [3]. Let F(R, [theta]) denote the continuous distribution function for the random variable R where [theta] is a shift parameter. An increase in risk associated with R is captured by a parametric shift in the distribution function F so that the original distribution function stochastically dominates the shifted one. Thus we can distinguish between an FSD and an SSD characterization of an increase in risk.

If the original distribution is to dominate the shifted distribution function (thereby showing an increase in risk underlying the new shifted distribution) in the sense of FSD then the distribution function must be shifted upward at least for some R as a result of an increase in 0. In other words 

[F[theta]([R, [theta]) [greater than or equal to] 0 (13) (where the strict inequality needs to hold for some R in die interior of [R.sub.1],[R.sub.2])

To characterize an increase in risk by a similar SSD shift, let us define,

[Mathematical Expressions Omitted]

For the original distribution function to dominate the shifted one in the sense of SSD, H(R, [theta] must be shifted upward at least for some R, in response to an increase in [theta]. Thus [H.sub.[theta]](R, [theta]) [greater than or equal to] 0 (15) (where the strict inequality needs to hold for some i? in the interior of [R 1, R2

The SSD shift of the distribution function is more general than a mean preserving spread. It is well known[2] that if a distribution function F dominates the distribution function G in the sense of SSD then the mean under F is no less than the mean under G. A mean preserving spread is a special case of the SSD shift that assures that die two means are equal.

The relationship between an SSD shift, an FSD shift and a mean preserving spread can be further clarified by the following insight. An SSD shift can be viewed as a combination of an FSD shift of the kind characterized in (13) and a mean preserving spread. As it has been shown by Hadar and Seo[4] if the distribution function F undergoes an FSD shift and the shifted distribution, say, T is subject to a further mean preserving spread, the resulting distribution G is dominated by F in the sense of SSD as in (15).

IV. Stochastic Dominance and the Certainty Equivalent Interest Rate

When an increase in risk is characterized by a shift of the distribution function that is represented by a stochastic dominance relationship, the level of utility under the stochastically dominated distribution must fall.(2) In order to note such implication of stochastic dominance we can derive the indirect utility function [phi](W, R) by substituting (6), (7), (9), (10), (11) in (3) and noting

[C.sub.2] = [S.sub.1]R.

The indirect utility is given by

[phi](W, R) = ([W.sup.1-[alpha]][1 + [[beta].sup.1/[alpha]][R.sup.(1-[alpha])/[alpha]] [alpha/(1 - [alpha])

It is clear that [phi] is increasing in R. Thus for an FSD or an SSD shift to capture an increase in risk, the shift must lower R. In this section we demonstrate that the FSD and SSD shifts of the distribution function of R unambiguously lower the certainty equivalent interest rate, R

FSD Shift of the Distribution Function

The certainty equivalent interest rate is given by

R = {E([R.sup.1-[epsilon]]}.sup.1/(1-[epsilon]) = [Z([theta])].sup1/(1-[epsilon])

where

[Mathematical Expressions Omitted]

Integrating (19) by parts we obtain

[Mathematical Expressions Omitted]

Differentiating (20) with respect to [theta] we get

[Mathematical Expressions Omitted]

Now, differentiation of (18) yields

[Mathematical Expressions Omitted]

For an FSD shift that captures an increase in risk, [F.sub.[theta]] [greater than or equal to] 0 and consequently dR / d[theta] < 0. Thus an increase in risk that is represented by an FSD shift lowers R. Hence from (12) it follows that the sign of the comparative static effect on optimal saving of an FSD shift of the distribution function that captures an increase in capital risk depends only on the elasticity of intertemporal substitution and not on the coefficient of risk aversion. The above result is summarized in Proposition 2.

Proposition 2. The direction of the effect of an increase in capital risk, that is characterized by an FSD shift of the distribution of the rate of return, depends only on the elasticity of intertemporal substitution; if the elasticity of intertemporal substitution is large, with [alpha] < 1 (small, with [alpha] > 1), an increase in such risk leads to a smaller (larger) saving regardless of the degree of risk aversion.

SSD Shift of the Distribution Function

To examine the effect of an SSD shift on R, we obtain from (21) using integration by parts 

[Mathematical Expressions Omitted]

But for the random variable, R,

[Mathematical Expressions Omitted]

Differentiating (24) with respect to [theta] we get

[Mathematical Expressions Omitted]

Also, differentiation of (14) with respect to [theta] yields

[Mathematical Expressions Omitted]

Using (25) and (26) we obtain from (23)

[Mathematical Expressions Omitted]

Consequently,

[Mathematical Expressions Omitted]

Since for an SSD shift that captures an increase in risk, [H.sub.[theta]] [less than or equal to] and [eta]' ([eta]) [less than or equal to] 0, dR /d [eta] < 0.

In view of (12) we can thus conclude that when an increase in capital risk is captured by a generalized shift of the distribution function, such as an SSD shift, the effect on optimal saving depends only on the elasticity of intertemporal substitution and not on risk aversion. These results are summarized in the following proposition.

Proposition 3. The direction of the effect of an increase in capital risk that is characterized by an SSD shift of the distribution of the rate of return depends only on the elasticity of intertemporal substitution; if the elasticity of intertemporal substitution is large, with [alpha] < 1 small, with [alpha] > 1), an increase in such risk leads to a smaller (larger) saving regardless of the degree of risk aversion.

Note that for a mean preserving spread [eta]'([theta]) = 0 and thus (28) yields the familiar result that for a mean preserving spread, dR/d[theta] < 0.

V. Is Risk Aversion Irrelevant for Ordering Uncertain Prospects?

A Case for Higher Order Stochastic Dominance

In the previous section we have shown that the effect on optimal saving of an FSD or an SSD shift of the distribution function does not depend on the coefficient of risk aversion. Does this imply that risk aversion plays no role in analyzing the effect of an increase in risk on the optimal behavior of a nonexpected utility maximizer with OCE preferences? It turns out that the measure of risk aversion plays a very critical role in characterizing a change in risk that may be represented by higher order stochastic dominance relationships. In this section assuming a lognormal distribution of the random return, R, we explore the role played by the risk aversion parameter, [epsilon], when the distribution function undergoes a shift that is more general than the FSD or the SSD shift.

The distribution function of R is given by

[Mathematical Expression Omitted]

with

   E(R) = [e.sup.[mu]+[sigma.sup.2]([theta]/2]   (30)
   Var(R) = [e.sup.2[mu]+[sigma.sup.2]([theta])][[e.sup.[sigma.sup.2]([theta])]
- 1].   (31)

It is well known that while any change in [sigma.sup.2] affects all the moments of the distribution, the consequent shifts cannot be captured by an FSD or an SSD shift. Moreover, it will be demonstrated that an increase in [sigma.sup.2] can increase or decrease the level of indirect utility [phi](W, R) in (17), depending on the value of [epsilon].(3) Thus in order to characterize an increase in capital risk due to an increase in the shift parameter [theta], we first note that

[Mathematical Expression Omitted]

Consequently,

logR = [mu] + (1 - [epsilon])[sigma.sup.2]([theta])/2.

Thus,

dR/d[theta] = [R(1 - [epsilon])/2]d[sigma.sup.2]/d[theta]. (33)

We now consider alternative representations of an increase in capital risk associated with an increase in [theta]. First, for an increase in [sigma.sup.2] to capture an increase in capital risk, with d[sigma.sup. 2]/d[theta] > 0 and [character no conversion] [phi]/[character no conversion] R > 0, dR/d[theta] must be negative which in turn is possible if and only if [epsilon] > 1. Alternatively, a reduction in [sigma.sup.2] with d[sigma.sup.2]/d[theta] < 0, can represent an increase in risk by lowering the indirect utility [phi] (W,R), if and only if, [epsilon] is less than 1. These results are summarized in proposition 4.

Proposition 4. For a lognormal distribution of R where log R N ([mu], [sigma.sup.2]) an increase (a decrease) in [sigma.sup.2] can capture a general increase in risk if and only if the coefficient of relative risk aversion [epsilon] > 1 ([epsilon] < 1).

The intuition behind these results is simple. An increase in [sigma.sup.2] increases both the variance and the mean of R. In spite of the increase in the mean, strongly risk averse individuals (with [epsilon] > 1) will consider the prospect with higher [sigma.sup.2] more risky, and hence utility reducing. A reduction in [sigma.sup.2], on the other hand, lowers the variance of R; but since it also reduces the mean return, individuals with lower risk aversion ([epsilon] < 1) do not prefer the distribution with lower [sigma.sup.2]. Thus for the stochastic dominance shifts of the distribution function that are more general than the mean preserving spread and the FSD shift or the SSD shift, the value of the risk aversion coefficient, [epsilon] becomes crucially important to characterize the notion of an increase in risk.

VI. Concluding Remarks

Nonexpected utility maximizing models are useful in distinguishing between the roles played by the elasticity of intertemporal substitution and, the coefficient of risk aversion in analyzing the optimal intertemporal decision made by a rational decision maker. It has, however, been found that in an intertemporal model of consumption-saving with OCE preferences, the effect of a mean preserving spread of the random return on the optimal level of saving or consumption depends only on the elasticity of intertemporal substitution and not on the risk aversion parameter. In this paper we have examined the robustness of this result under more general characterization of an increase in risk. We have shown that even under an increase in risk that is characterized by a First Degree or a Second Degree Stochastic Dominance shift of die distribution function, the effect on optimal saving does not depend on the risk aversion coefficient. However, the irrelevance of risk aversion in analyzing the optimal decision of a nonexpected utility maximizer with OCE preferences should not be overemphasized. We have shown that the coefficient of risk aversion plays a very fundamental role in characterizing an increase in risk that can be represented by higher order stochastic dominance shifts of the distribution function.

Finally, it must be noted that like most of the works in the area of nonexpected utility maximization with OCE preferences, the analysis of this paper may be restrictive as it uses the utility functions with constant relative risk aversion. Since the stochastic dominance analysis is an extremely powerful analytical tool for ranking uncertain prospects for more general classes of utility functions, we intend to extend the analysis of this paper to include other types of utility functions.

References

[1.] Barsky, Robert B., "Why Don't the Prices of Stocks and Bonds Move Together?" American Economic Review, December 1989, 1132-45. [2.] Hadar, Josef and William R. Russell. "Stochastic Dominance and Diversification." Journal of Economic Theory, September 1971, 288-305. [3.] _____ and _____. "Applications in Economic Theory and Analysis," in Stochastic Dominance, edited by G. A. Whitmore and M. C. Findlay. Lexington: Lexington Books 1977, pp. 295-333. [4.] _____ and Tae Kun Seo, "The Effects of Shifts in a Return Distribution on Optimal Portfolios." International Economic Review, August 1990, 721-36. [5.] _____ and _____, "General Changes in Uncertainty." Southern Economic Journal, January 1992, 671-81. [6.] Levhari, D. and T. N. Srinivasan, "Optimal Saving Under Uncertainty." Review of Economic Studies, April 1969, 153-63. [7.] Sandmo, A., "The effect of Uncertainty on Saving Decisions." Review of Economic Studies, July 1970, 353M-60. [8.] Selden, Larry, "A New Representation of Preferences over |Certain x Uncertain' Consumption Pairs: The Ordinal Certainty Equivalent Hypothesis." Econometrica. September 1978, 1045-60. [9.] _____, "An OCE Analysis of the Effect of Uncertainty on Saving under Risk Independence." Review of Economic Studies, January 1979, 73-82. [10.] Weil, Philippe, "Nonexpected Utility in Macroeconomics." Quarterly Journal of Econmics, February 1990, 29-42.

(1.) Recently, Hadar and Seo [5] have considered such general changes in uncertainty in the context of a competitive firm operating under output-price uncertainty. (2.) The formal equivalence results between stochastic dominance and expected utility comparison are well known [2]. (3.) A mean preserving spread of the distribution of the random return represents an increase in [sigma.sup.2] with compensating reduction in [mu] so that the mean of the distribution remains unchanged. Here in order to capture higher order stochastic dominance shift, we are considering a parametric shift of the distribution function that causes a pure change in [sigma.sup.2] without any compensating change [mu].