The economics of self-protection

Abstract

Self-protection is a costly activity that reduces the probability of an unfavorable outcome. Even the simplest model with a binary risk of loss and expected utility of final wealth produces interesting comparative statics that are by no means trivial. This article provides a selective survey of the economics of self-protection. It puts particular emphasis on the contributions made by members of the European Group of Risk and Insurance Economists and research published in the Geneva Risk and Insurance Review. The article provides a conceptual framework to catalog existing models of self-protection, discusses the tension between risk aversion and downside risk aversion, reveals the role of probability thresholds, surveys extensions to non-expected utility, and highlights the recent surge in two-period models. Ideas for future research directions are also developed.

Appendices

Appendix A: The top ten papers on self-protection

Table 1 ranks papers about self-protection published in economics journals by Google Scholar citations. I am fully aware that Google Scholar citations do not correlate perfectly with quality. Furthermore, citations accumulate over time, which introduces a bias in favor of old papers. At the same time, readership generally increases over time, and researchers tend to cite contemporary authors and colleagues, which favors new papers. I am shamelessly admitting that I resort to Google Scholar citations because they are readily available. Whoever is offended by my list will hopefully accept my sincerest apologies.²¹

Table 1 The top ten most cited papers on self-protection in economics

Appendix B: Increasing risk and self-protection

Dionne and Eeckhoudt (1988) consider a marginal increase in $\ell$ that keeps expected final wealth unchanged. I refer to this change as a compensated increase in loss severity. Expected final wealth is given by $w_0-e^*-p^* \ell$. A one-unit increase in $\ell$ needs to be accompanied by a $p^*$-unit increase in $w_0$ for expected final wealth to stay constant before the adjustment in self-protection occurs. The behavioral effect of a compensated increase in $\ell$ follows from the implicit function rule by signing the following expression:

$$\begin{aligned} \Delta= &\, {} p^* \cdot \left\{ -p'(e^*) \left[ u'(w^*_N)-u'(w^*_L)\right] -\left[ p^* u''(w^*_L)+(1-p^*)u''(w^*_N)\right] \right\} \\{} & {} + \left\{ -p'(e^*) u'(w^*_L) + p^* u''(w^*_L) \right\} . \end{aligned}$$

Solve first-order condition 3 for $-p'(e^*)$ and substitute in $\Delta$, let $\alpha = -(u'(w^*_L)-u'(w^*_N))^2+(u''(w^*_N)-u''(w^*_L))(u(w^*_N)-u(w^*_L))$ and $\beta =u'(w^*_L) u'(w^*_N)$. I then obtain

$$\begin{aligned} \Delta = \frac{1}{u(w^*_N)-u(w^*_L)} \cdot \left[ \alpha \cdot (p^*)^2 - \alpha \cdot p^* + \beta \right] , \end{aligned}$$

which is quadratic in $p^*$. Because of $\beta >0$, I find immediately that $\Delta$ is positive if $p^*$ is close to zero or close to one. In other words, a compensated increase in loss severity raises optimal self-protection if the variance of final wealth is small.

Further analysis is possible on the sign of $\alpha$. Let

$$\begin{aligned} v(w) = \frac{1}{\ell } \int _{-\ell }^0 u(w+t) \, \textrm{d}t \end{aligned}$$

denote the indirect utility function for a risk that is uniformly distributed between $-\ell$ and zero. The fundamental theorem of calculus yields

$$\begin{aligned} \ell \cdot v'(w) = \int _{-\ell }^0 u'(w+t) \, \textrm{d}t = u(w)-u(w-\ell ), \end{aligned}$$

and likewise for $v''(w)$ and $v'''(w)$. I rearrange $\alpha \ge 0$ to

$$\begin{aligned} - \frac{u''(w^*_N)-u''(w^*_L)}{u'(w^*_N)-u'(w^*_L)} \ge - \frac{u'(w^*_N)-u'(w^*_L)}{u(w^*_N)-u(w^*_L)}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} -\frac{v'''(w^*_N)}{v''(w^*_N)} \ge - \frac{v''(w^*_N)}{v'(w^*_N)}. \end{aligned}$$

If u has constant absolute risk aversion (CARA), then v has CARA as well so that $\alpha = 0$ and $\Delta > 0$ regardless of the size of $p^*$. If u has DARA, then v inherits DARA from u (Nachman 1982), and $\alpha > 0$. In this case, $\Delta < 0$ cannot be ruled out.

In the DARA case, one can study the discriminant of $\Delta$, which is given by $\alpha (\alpha -4\beta )$. The discriminant is negative if $4\beta > \alpha$. In this case $\Delta > 0$ for all values of $p^*$, and a compensated increase in loss severity always raises self-protection. DARA implies prudence so that

$$\begin{aligned} \frac{1}{2} (u'(w^*_L)+u'(w^*_N)) > \frac{1}{\ell } (u(w^*_N)-u(w^*_L)), \end{aligned}$$

see Lemma 1 in Eeckhoudt and Gollier (2005). Let $\eta = \ell /w^*_N$ be shorthand for the share of riskless final wealth that the loss puts at risk. Suppose that

$$\begin{aligned} \eta ^2 \cdot \left( -w^*_N \frac{v'''(w^*_N)}{v''(w^*_N)} \right) \cdot \left( -w^*_N \frac{v''(w^*_N)}{v'(w^*_N)} \right)\le & {} 4; \end{aligned}$$

(5)

it is straightforward to show that (5) implies $4\beta > \alpha$ if u is prudent. For example, if relative prudence of v is bounded by 2 and relative risk aversion of v is bounded by one, condition (5) is satisfied. Given that v is an indirect utility function, the same restrictions on u do not guarantee that (5) holds. If u and $-u'$ are risk vulnerable (Gollier and Pratt 1996), relative risk aversion and relative prudence of v are larger than those of u.

To dig deeper, I assume power utility, $u(w)=w^{1-\gamma }/(1-\gamma )$ for $\gamma \ne 1$ and $u(w) =\ln (w)$ for $\gamma =1$. In this case, $\alpha > 0$ because u satisfies DARA. The sign of the discriminant of $\Delta$ coincides with the sign of $\alpha -4\beta$. I rewrite $w^*_L = w^*_N(1-\eta )$ and find that the sign of $\alpha -4\beta$ is the same as that of

$$\begin{aligned} -\left( (1-\eta )^{-\gamma }+1\right) ^2+\frac{\gamma }{1-\gamma } \left( -1+(1-\eta )^{-\gamma -1}\right) \left( 1-(1-\eta )^{1-\gamma }\right) \quad \quad \text {for }\gamma \ne 1, \end{aligned}$$

and

$$\begin{aligned} -\left( (1-\eta )^{-1}+1\right) ^2 -\left( -1+(1-\eta )^{-2}\right) \ln (1-\eta ) \quad \quad \text {for }\gamma =1. \end{aligned}$$

Figure 3 illustrates. Consistent with condition (5), the discriminant of $\Delta$ is only positive if risk aversion is high and the loss puts a significant share of final wealth at risk. In particular, it is positive for $\eta >88\%$ if $\gamma =0.5$, for $\eta >65.4\%$ if $\gamma =2$, and for $\eta > 45\%$ if $\gamma =5$. For $\eta$ below these values, the discriminant of $\Delta$ is negative, $\Delta$ is uniformly positive, and the effect of a compensated increase in loss severity on self-protection is positive for any value of $p^*$.

Fig. 3

Sign of $\alpha -4\beta$ for power utility

If the discriminant of $\Delta$ is positive, there are two roots between zero and one,

$$\begin{aligned} p_1 = \frac{1}{2} \left( 1 - \sqrt{1-4\beta /\alpha }\right) \quad \quad \text { and } \quad \quad p_2 = \frac{1}{2} \left( 1 + \sqrt{1-4\beta /\alpha }\right) . \end{aligned}$$

A compensated increase in loss severity then raises self-protection for $p^* \in (0,p_1) \cup (p_2,1)$, and lowers self-protection for $p^* \in (p_1, p_2)$. Given that $p_1+p_2=1$, one can equivalently say that the effect is positive if and only if the variance of final wealth, normalized by $\ell ^2$, is less than $p_1(1-p_1)$. The following proposition summarizes these insights.

Proposition

A compensated increase in loss severity raises self-protection if and only if the variance of the final wealth distribution, normalized by $\ell ^2$, is below $\beta /\alpha$. For CARA utility, this condition is always satisfied because $\alpha =0$. For DARA utility, condition (5) is sufficient because it implies $\beta /\alpha > 0.25$. For power utility, it is satisfied if risk aversion is low and the loss puts a small enough share of final wealth at risk. Otherwise, it is satisfied if the loss probability is far enough away from one half.