A Diamond-Stiglitz approach to the demand for self-protection

Abstract

The existing research concerning the relationship between risk aversion and prudence and the demand for self-protection assumes that the loss variable follows a Bernoulli distribution, and that changes in the level of self-protection are mean preserving. The analysis here replaces these two very strong conditions with ones which are more general. When doing this, the method of analysis is also significantly modified. This modification includes representing a change in the level of self-protection using the procedure developed by Diamond and Stiglitz (Journal of Economic Theory 8:337-360, 1974) for representing a change in risk. This alternate representation allows the existing findings to be generalized considerably, and also simplifies the analysis.

Appendix

Theorem

$\int_{\rm{a}}^{\rm{b}} { - {\hbox{v}}\prime {\hbox{(W)}} \cdot {\hbox{u}}\prime {\hbox{(W)}}{{\hbox{F}}_{\rm{e}}}{\hbox{(W,e)dW}}} > 0\;( < 0)$for all v′′′(W) > 0 if and only if$\int_{\rm{a}}^{\rm{b}} { - {\hbox{v}}\prime {\hbox{(W)}} \cdot {\hbox{u}}\prime {\hbox{(W)}}{{\hbox{F}}_{\rm{e}}}{\hbox{(W,e)dW}}} > 0\;( < 0)$ (< 0) for all A_v′(W) < 0.

Proof

First the well known direction. v′′′(W) is a necessary condition for A_v′(W) < 0. Therefore when ${\hbox{A}}_{\rm{v}}^{\prime }\left( {\hbox{W}} \right) < 0$ (< 0) for all v′′′(W) > 0, this implies that ${\hbox{A}}_{\rm{v}}^{\prime }\left( {\hbox{W}} \right) < 0$ (< 0) for all persons with A_v′(W) < 0 as well. To demonstrate the other direction, assume that $\int_{\rm{a}}^{\rm{b}} { - {\hbox{v}}\prime {\hbox{(W)}} \cdot {\hbox{u}}\prime {\hbox{(W)}}{{\hbox{F}}_{\rm{e}}}{\hbox{(W,e)dW}}} > 0\;( < 0)$ (< 0) for all A_v′(W) < 0. Pick any person with marginal utility v′(W) such that v′′′(W) > 0. Consider all marginal utility functions of the form (v′(W) + k) for k > 0. The value of $\int_{\rm{a}}^{\rm{b}} { - {\hbox{v}}\prime {\hbox{(W)}} \cdot {\hbox{u}}\prime {\hbox{(W)}}{{\hbox{F}}_{\rm{e}}}{\hbox{(W,e)dW}}} > 0\;( < 0)$ is the same as ${\hbox{A}}_{\rm{v}}^{\prime }\left( {\hbox{W}} \right) < 0$ for all k > 0. The slope of the risk aversion measure for these persons is given by: $\int_{\rm{a}}^{\rm{b}} { - {\hbox{v}}\prime {\hbox{(W)}} \cdot {\hbox{u}}\prime {\hbox{(W)}}{{\hbox{F}}_{\rm{e}}}{\hbox{(W,e)dW}}} > 0\;( < 0)$. Because the second term has k squared in the denominator while the first does not, the second term can be made arbitrarily small in magnitude relative to the magnitude of the first term by choosing k to be large enough. Thus if v′′′(W) > 0, for large enough k, A_v′(W) < 0 for this person. A similar argument is easily made for the case where v′′′(W) < 0 and A_v′(W) > 0. A sufficiently large value for k is one that exceeds $\frac{{(v{\prime}{\prime}{{(W)}^2}}}{{v{\prime}{\prime}{\prime}(W)}} - v{\prime}(W)$ for all W.