On the Propensity to Self-Protect

Abstract

ABSTRACT Defining propensity to self-protect as the maximum amount an individual is willing to pay for a one-unit reduction in the probability of loss, this article studies its basic behavior and its relationship to the individual's degree of risk aversion and the initial loss probability. It is shown that if the initial loss probability is below a threshold, a more risk-averse individual has a higher propensity to self-protect, and the threshold is controlled by individuals' aversion to general downside risk increases and aversion to overall riskiness measured in variance. INTRODUCTION Self-protection is defined as the expenditure on reducing the probability of suffering a loss. Despite its relevance to a wide range of economic issues, [1] self-protection--especially its relationship with an individual's attitude towards risk--has not been adequately understood. In their pioneering work, Ehrlich and Becker (1972) noted that self-protection may be attractive to both risk-averse people and risk lovers and that unlike self-insurance (the expenditure on reducing the severity of loss), self-protection and market insurance can be complements. More recently, Dionne and Eeckhoudt (1985) showed that a more risk-averse individual does not always purchase more self-protection. And Briys and Schlesinger (1990) explained this phenomenon by showing that self-protection in general does not reduce the riskiness of individuals' final wealth. Sweeney and Beard (1992a) went one step further in showing that for a general loss probability function of self-protection spending, it is impossible to characterize th e preferences of an individual who always chooses a higher level of self-protection. In their attempt to verify an interesting intuition that insurance is reducing small chances of bad outcomes and gambling is increasing small chances of good outcomes, McGuire, Pratt, and Zeckhauser (1991) came closest to identifying a relationship between an individual's degree of risk aversion and his or her choice of self-protection. They show that if a less risk-averse individual's optimal choice of self-protection is such that the resulting loss probability is less than a critical switching level, then a more risk-averse individual's optimal choice will be higher than the less risk-averse. [2] From these previous contributions, it is clear that the relationship of an individual's attitude towards risk with his or her propensity to purchase self-protection is not as straightforward as that with market insurance or self-insurance. What they do not imply, however, is that a more primitive and more precise characterization of the relationship between risk aversion and individuals' propensity to purchase self-protection is impossible insofar as one accepts the Expected Utility paradigm. Staying within the Expected Utility framework, one can look at the problem from a slightly different angle. Specifically, instead of investigating the optimal choice of self-protection given an assumed relationship between self-protection spending and the loss probability, one can explicitly consider an individual's willingness (or propensity) to purchase self-protection-the maximum an individual is willing to pay for a given reduction in the probability of loss. [3] So far only scant effort has been made in understanding the problem from this perspective. Under various restrictions, Eeckhoudt, Godfroid, and Gollier (1997) compared the effects of risk-aversion on the risk premium [defined in Pratt (1964)] and on the willingness to pay and conclude that some properties of the risk premium are not shared by the willingness to pay. Eeckhoudt and Godfroid (1998) showed in a quite different context with exponential utility functions that the lower the initial probability of accident, the greater the market value o f a reduction in the probability. The aim of this article is to provide a comprehensive study of individuals' propensities to purchase self-protection. …