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

• Author: Basu, Parantap

1. 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.

2. 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...