The impact of ambiguity and prudence on prevention decisions

Abstract

Most decisions concerning (self-)insurance and self-protection have to be taken in situations in which (a) the effort exerted precedes the moment uncertainty realizes, and (b) the probabilities of future states of the world are not perfectly known. By integrating these two characteristics in a simple theoretical framework, this paper derives plausible conditions under which ambiguity aversion raises the demand for (self-)insurance and self-protection. In particular, it is shown that in most usual situations where the level of ambiguity does not increase with the level of effort, a simple condition of ambiguity prudence known as decreasing absolute ambiguity aversion (DAAA) is sufficient to give a clear and positive answer to the question: Does ambiguity aversion raise the optimal level of effort?

Appendix

Proof of Proposition 4

This proof is based on the following Lemma, that can be found in Gollier (2001).²¹

Lemma 2

Let $\phi$ be a twice differentiable, increasing and concave function: $\mathbb {R} \rightarrow \mathbb {R}$. Consider a probability vector $(q_1,\ldots , q_n) \in \mathbb {R}^n_+$ with $\sum _{\theta =1}^n q_\theta =1$, and a function $f: \mathbb {R}^n \rightarrow \mathbb {R}$, defined as

$$\begin{aligned} f(U_1,\ldots , U_n)=\phi ^{-1}\left\{ \sum _{\theta =1}^n q_\theta \phi \left\{ U_\theta \right\} \right\} . \end{aligned}$$

Let T be the function such that $T(U)=-\frac{\phi '\{U\}}{\phi ''\{U\}}$. Function f is concave in $\mathbb {R}^n$ if and only if T is weakly concave in $\mathbb {R}$.

First, remark that programme (10) is convex if

$$\begin{aligned} V(e)=\phi ^{-1}\left\{ \mathrm {E}_\theta \phi \left\{ p(e,\tilde{\theta })\mathrm {E}u({\tilde{w}}_{2}^b(e,\tilde{\theta })) +\left[ 1-p(e,\tilde{\theta })\right] \mathrm {E} u({\tilde{w}}_{2}^g(\tilde{\theta }))\right\} \right\} \end{aligned}$$

is concave in e.

The proof is given in the case where ambiguity is concentrated on the type of states. The two other cases follow trivially.

Self-insurance $(p(e,\theta )=p(\theta )$ for all levels of e): consider two scalars $e_1$ and $e_2$, and let $U_{j\theta }$ denote the second period expected utility conditional on $\theta$, for a level of effort $e_j: U_{j\theta }=p(\theta )\mathrm {E}u({\tilde{w}}_{2}^b(e_j)) +\left[ 1-p(\theta )\right] \mathrm {E} u({\tilde{w}}_{2}^g)$. Under the notations above, $V(e_j)=f(U_{j1},\ldots ,U_{jn})$. Then, under concavity of u and $w_{2}^b$, and for any $(\lambda _1,\lambda _2)\in \mathbb {R}^2_+$ such that $\lambda _1+\lambda _2=1$, we have:

$$\begin{aligned} \lambda _1 \mathrm {E} u({\tilde{w}}_{2}^b(e_1))+\lambda _2 \mathrm {E} u({\tilde{w}}_{2}^b(e_2))\le \mathrm {E} u(\lambda _1 {\tilde{w}}_{2}^b(e_1)+\lambda _2 {\tilde{w}}_{2}^b(e_2))\le \mathrm {E} u({\tilde{w}}_{2}^b(\lambda _1 e_1+\lambda _2 e_2)). \end{aligned}$$

Multiplying the first and the third parts of this chain of inequalities by $p(\theta )$ and adding $\left[ 1-p(\theta )\right] \mathrm {E}u({\tilde{w}}_2^g)$ yields:

$$\begin{aligned} \lambda _1 U_{1\theta }+\lambda _2 U_{2\theta }\le U_{\lambda \theta }\equiv p(\theta )\mathrm {E}u({\tilde{w}}_{2}^b(e_\lambda ))+\left[ 1-p(\theta )\right] \mathrm {E} u({\tilde{w}}_{2}^g) \end{aligned}$$

for all $\theta$, where $e_\lambda =\lambda _1 e_1+\lambda _2 e_2$. Because f is increasing in $\mathbb {R}^n$ if $\phi$ is increasing, this implies:

$$\begin{aligned} V(e_\lambda )= f(U_{\lambda 1},\ldots ,U_{\lambda n}) \ge f(\lambda _1U_{11}+\lambda _2U_{21},\ldots ,\lambda _1U_{1n}+\lambda _2U_{2n}). \end{aligned}$$

On the other side, if $-\phi '/\phi ''$ is concave, by Lemma 2 we have:

$$\begin{aligned} f(\lambda _1U_{11}+\lambda _2U_{21},\ldots ,\lambda _1U_{1n}+\lambda _2U_{2n})\ge & {} \lambda _1f(U_{11},\ldots ,U_{1n})+\lambda _2f(U_{21},\ldots ,U_{2n})\\= & {} \lambda _1 V(e_1)+\lambda _2 V(e_2). \end{aligned}$$

Combining these two results yields $V(\lambda _1 e_1+\lambda _2 e_2)\ge \lambda _1 V(e_1)+\lambda _2 V(e_2)$ implying that V is concave in e.

Self-protection $(w_{2}^b(e)=w_{2}^b$ for all levels of e): in this case, the proof is similar but $U_{j\theta }$ is now given by $U_{j\theta }=p(e_j,\theta )\mathrm {E} u({\tilde{w}}_{2}^b)+\left[ 1-p(e_j,\theta )\right] \mathrm {E} u({\tilde{w}}_{2}^g)$, and we exploit the convexity of $p(e,\theta )$ in e to obtain $\lambda _1 U_{1\theta }+\lambda _2 U_{2\theta }\le U_{\lambda \theta }$. $\square$

Proof of Proposition 7

• 1. Imagine that the set of risky second period wealth in the bad states of the world can be ranked according to FSD:

  $$\begin{aligned} {\tilde{w}}_2^b(e^*,\theta _1) \succeq _{FSD}\dots \succeq _{FSD}{\tilde{w}}_2^b(e^*,\theta _n). \end{aligned}$$

  (14)

  From the definition of FSD, this is the case if and only if for every function f weakly increasing, we have:

  $$\begin{aligned} \mathrm {E} f({\tilde{w}}_2^b(e^*,\theta _1))\ge \dots \ge \mathrm {E} f({\tilde{w}}_2^b(e^*,\theta _n)). \end{aligned}$$

  (15)

  Given that $u'>0$, it is, therefore, clear that $U(e^*,\theta )=p\mathrm {E} u({\tilde{w}}_2^b(e^*,\theta ))+(1-p) \mathrm {E} u({\tilde{w}}_2^g)$ is decreasing in $\theta$, and that $U_e(e^*,\theta )=\mathrm {E} g({\tilde{w}}_2^b(e^*,\theta ))$ will be increasing in $\theta$ if and only if g is decreasing. Now remark that this will be the case if, by ordering (without loss of generality) the different bad states such that $w_{2,s}^b\ge w_{2,s+1}^b$, we have:

  $$\begin{aligned} g(w_{2,s}^b(e^*,\theta ))= & {} u'(w_{2,s}^b(e^*,\theta ))\frac{\partial w_{2,s}^b(e^*,\theta )}{\partial e}\nonumber \\\le & {} u'(w_{2,s+1}^b(e^*,\theta ))\frac{\partial w_{2,s+1}^b(e^*,\theta )}{\partial e}=g(w_{2,s+1}^b(e^*,\theta ))\quad \quad \quad \end{aligned}$$

  (16)

  for all $\theta$, and all s. It is then easy to see that inequality (16) will be satisfied if $u''\le 0$ and $\frac{\partial w_{2,s}^b(e^*,\theta )}{\partial e}\le \frac{\partial w_{2,s+1}^b(e^*,\theta )}{\partial e}$, meaning that the marginal benefit of self-insurance is higher for higher losses.

• 2. Imagine that the set of risky second period wealth in the bad states of the world can be ranked according to SSD:

  $$\begin{aligned} {\tilde{w}}_2^b(e^*,\theta _1) \succeq _{SSD}\cdots \succeq _{SSD}{\tilde{w}}_2^b(e^*,\theta _n). \end{aligned}$$

  (17)

  From the definition of SSD, this is the case if and only if for every non-decreasing, concave function f, we have:

  $$\begin{aligned} \mathrm {E} f({\tilde{w}}_2^b(e^*,\theta _1))\ge \cdots \ge \mathrm {E} f({\tilde{w}}_2^b(e^*,\theta _n)). \end{aligned}$$

  (18)

  Given that $u'>0$ and $u''>0$ under risk aversion, we know that $U(e^*,\theta )=p\mathrm {E} u({\tilde{w}}_2^b(e^*,\theta ))+(1-p) \mathrm {E} u({\tilde{w}}_2^g)$ is decreasing in $\theta$. Similarly, $U_e(e^*,\theta )=\mathrm {E} g({\tilde{w}}_2^b(e^*,\theta ))$ will be increasing in $\theta$ if and only if $g'\le 0$ and $g''>0$. Since we know that $g(w_{2,s}^b(e^*,\theta ))= u'(w_{2,s}^b(e^*,\theta ))\frac{\partial w_{2,s}^b(e^*,\theta )}{\partial e}$, this will be the case if

  $$\begin{aligned}&u'\big (\lambda w_{2,i}^b(e^*,\theta )+(1-\lambda ) w_{2,j}^b(e^*,\theta )\big ) \Bigg [ \lambda \frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e} +(1-\lambda ) \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e}\Bigg ]\nonumber \\&\quad \le \lambda \Bigg [ u'(w_{2,i}^b(e^*,\theta ))\frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e}\Bigg ]+ (1-\lambda ) \Bigg [ u'(w_{2,j}^b(e^*,\theta ))\frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e}\Bigg ]\nonumber \\ \end{aligned}$$

  (19)

  for all $\theta$, all $i,j\in s$, and all $\lambda \in [0,1]$. It is then easy to see that inequality (19) is satisfied with an equality if $u''= 0$, and with a strict inequality if $\frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e}= \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e}$ and the agents exhibits risk prudence: $u'''>0$. More generally, the risk prudence property implies that:

  $$\begin{aligned} u'\big (\lambda w_{2,i}^b(e^*,\theta )+(1-\lambda ) w_{2,j}^b(e^*,\theta )\big )\le \lambda u'\big (w_{2,i}^b(e^*,\theta )\big ) + (1-\lambda ) u'\big (w_{2,j}^b(e^*,\theta )\big ), \end{aligned}$$

  (20)

  and it is, therefore, possible to form a chain of inequalities with expressions (19) and (20) to show that g is convex if

  $$\begin{aligned}&u'(w_{2,i}^b(e^*,\theta )) \Bigg [ \frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e} - \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e} \Bigg ]\nonumber \\&\quad \ge u'(w_{2,j}^b(e^*,\theta )) \Bigg [ \frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e} - \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e} \Bigg ]. \end{aligned}$$

  (21)

  A sufficient condition for this expression to be satisfied is, therefore, that $\frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e}\ge \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e}$ whenever $w_{2,i}^b(e^*,\theta )\le w_{2,j}^b(e^*,\theta )$, and vice versa.

$\square$