The Comparative Statistics of Self-Protection
Sweeney, George H., Beard, T. Randolph, Journal of Risk and Insurance
The expenditure of resources to modify the probabilities of suffering losses is called "self-protection."(1) As noted by Briys and Schlesinger (1990), expenditures on self-protection do not merely trade income in one state of the world for income in another as market insurance contracts do. Rather, self-protection reduces income in all states, shifting the support of the wealth distribution to the left. Hence, increases in self-protection spending do not in general lead to less risky income prospects in the sense of stochastic dominance.(2) Consequently, the willingness of agents to engage in self-protection fails to parallel agents' willingness to buy market insurance in several important respects. Both risk-loving and risk-averse agents may buy self-protection, and more risk-averse agents will not generally spend more on self-protection than less risk-averse agents.(3) This characteristic of self-protection makes its analysis considerably more difficult than that of insurance. In addition, analysis of self-protection choice must account for the fact that any agent's optimal level of self-protection spending depends on an exogenous, technical" relationship between spending and loss probability. This relationship summarizes the agent's opportunities to reduce loss probabilities and is a critical component in determining optimal self-protection choice.
The purpose of this article is to present a set of comparative static results for the simplest and most widely-used model of self-protection choice. Interest focuses on the effects of changes in initial wealth and potential losses on optimal self-protection expenditures.(4) The goal is to seek results of a general character which do not depend on the particular form of the technical relationship between spending and loss probability, nor on restrictive characterizations of agents' von Neumann - Morgenstern utility functions for wealth. Further, the main findings will be stated in terms of the characteristics of agents' attitudes towards risk in a relatively straight-forward fashion, and a set of results useful for evaluating economic problems that involve self-protection will be derived.
Optimal Self-Protection Choice
Consider the simplest model of self-protection choice in which an agent faces the potential of losing a fixed sum of money L. Letting [y.sub.o] be the agent's initial wealth, s her spending on self-protection, p(s) the probability of suffering the loss L given self-protection expenditure s > 0, and u(.), the agent's von-Neumann Morgenstern utility function for money, write the agent's expected utility given self-protection spending level s as v(s,L,yo), where
v(s,L,[y.sup.o]) = p(s)u([y.sup.o]-s-L) + [1 - p(s)]u([y.sub.o]-s). (1)
Letting y = [y.sub.o] - s, and assuming the required differentiability of p(-) and u(-), the optimal level of self-protection spending s* is given by the firstorder condition
-p'(s*)[u(y) - u(y-L)l - [1 - p(s*)]u'(y) - p(s*)u'(y-L) [less than or equal to] 0, (2)
with equality holding if s* > 0. The second-order necessary condition for an interior solution requires
-p"(s*)[u(y) - u(y-L)] + 2p'(s*)[u'(y) - u'(y - L)]
+ [1 - p(s*)]u"(y) + p(s*)u"(y-L) < 0. (3)
The first-order condition given by (2) has an immediate conventional interpretation in terms of costs and benefits: the (expected) marginal benefit of self-protection spending (-p'(s*)[u(y) - u(y-L)]) should be equal to the (expected) marginal "cost" of such spending ([1-p(s*)]u'(y) + p(s*)u'(y-L)) at the optimal choice s*.
Comparative Static Results for Changes in Initial Wealth
The initial goal is evaluating the effect of changes in the agent's initial wealth [y.sub.o] on optimal self-protection spending s*. If the purchase of selfprotection is identified with the purchase of an economic good such as a burglar alarm, security guard service, or smoke detector, then the issue of how changes in (initial) wealth [y. …