The Impact of Health-Related Emotions on Belief Formation and Behavior

Abstract

We present a theoretical model of health beliefs and behaviors that explicitly takes into account the emotional impact of possible bad news (i.e., illness), ex-ante in the form of anxiety and ex-post in the form of disappointment. Our model makes it possible to explain (simultaneously) a number of anomalies such as ’low’ testing rates, heterogeneous perceptions of risk levels, underestimation of health risk, ostriches and hypochondriacs, over-use and under-use of health services, patient preference for information when relatively certain of not being ill, yet avoiding information when relatively certain of being ill, etc. Our model matches observed patterns both in health beliefs and health behaviors and irrational health beliefs and behaviors can be characterized as the optimal response under a given structure of emotions and preferences.

Appendix

Proof of Proposition 1

1. and 2. Let us prove that $y^{*}$ increases with $\kappa$. Letting $f\left( \kappa ,y\right) =\kappa \overline{v}_{y}\left( y\right) +\left( 1-p\right) \overline{D}_{y}\left( 0,y\right)$, we have $\frac{\partial y^{*}}{\partial \kappa }=-\frac{\frac{\partial f}{\partial \kappa }}{\frac{\partial f}{\partial y}}$ with $\frac{\partial f}{\partial \kappa }=\overline{v}_{y}\left( y\right) >0$ and $\frac{\partial f}{\partial y}=\frac{\partial ^{2}W\left( y\right) }{\partial y^{2}}<0$, hence $\frac{\partial y^{*}}{\partial \kappa }>0$. Moreover, when $\kappa \rightarrow 0$, then $\kappa \overline{v}_{y}\left( 0\right) +\left( 1-p\right) \overline{D}_{y}\left( 0,0\right) \le 0$ and hence $y^{*}=0$. When $\kappa$ is high enough, then $\kappa \overline{v}_{y}\left( 1\right) +\left( 1-p\right) \overline{D}_{y}\left( 0,1\right) \ge 0$ and hence $y^{*}=1$. The belief $y^{*}=p$ is characterized by a level $\kappa _{p}$ such that $\kappa _{p}\overline{v}_{y}\left( p\right) +\left( 1-p\right) \overline{D}_{y}\left( 0,p\right) =0.$

3. Letting $f\left( p,y\right) =v^{\prime }\left( y\right) +\left( 1-p\right) D_{y}\left( 0,y\right)$. We have $\frac{\partial y^{*} }{\partial p}=-\frac{\frac{\partial f}{\partial p}}{\frac{\partial f}{\partial y}}$ with $\frac{\partial f}{\partial p}=-D_{y}\left( 0,y\right) \ge 0$. Since, by assumption, $\frac{\partial f}{\partial y}=\frac{\partial ^{2}W\left( y\right) }{\partial y^{2}}<0$, we then have $\frac{\partial y^{*}}{\partial p}{>}0.$    $\square$

Proof of Proposition 2

1. Information changes the individual’s level of anxiety from $v\left( 1\right) -v\left( y^{*}\right)$ to $v\left( 1\right) -E\left[ v\left( \widetilde{h}\right) \right]$. Since v is increasing, it reduces the anxiety of individuals with $y^{*}\le v^{-1}\left( E\left[ v\left( \widetilde{h}\right) \right] \right)$ and raises the anxiety levels of individuals with $y^{*}\ge v^{-1}\left( E\left[ v\left( \widetilde{h}\right) \right] \right)$. Since $y^{*}$ increases with $\kappa$ from 0 to 1 (Proposition 1), there exists $\widehat{\kappa }$ for which $y^{*}=v^{-1}\left( E\left[ v\left( \widetilde{h}\right) \right] \right)$.

2. As seen in the proof of 1., if $\kappa >\widehat{\kappa }$, then $v\left( y^{*}\right) >E\left[ v\left( \widetilde{h}\right) \right]$. This implies that for high enough $\kappa ,$ we have $\kappa \left[ E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) \right] <\left( 1-p\right) \overline{D}\left( 0,1\right) \le \left( 1-p\right) \overline{D}\left( 0,y^{*}\right)$ and delayed resolution of uncertainty is preferred. As seen in the proof of 1., if $\kappa \le \widehat{\kappa }$ ,  we have $v\left( y^{*}\right) \le E\left[ v\left( \widetilde{h}\right) \right] ,$ hence $V_{I}\equiv E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -\left( 1-p\right) D\left( 0,y^{*}\right) \ge 0;$ in fact, we have$\ V_{I}>0$ for $\kappa \le \widehat{\kappa }$, because if $E\left[ v\left( \widetilde{h}\right) \right] =v\left( y^{*}\right)$, then $y^{*}\ne 0$ and $\overline{D}\left( 0,y^{*}\right) <0$. Hence there exists $\kappa ^{*}>\widehat{\kappa }$ for which $V_{I}=0$. Let us show that $\frac{\partial V}{\partial \kappa }\left| _{V=0}\right. <0$ where $V=\kappa \left[ E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) \right] -\left( 1-p\right) \overline{D}\left( 0,y^{*}\right)$. We have

$$\begin{aligned} \frac{\partial V}{\partial \kappa }=E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) -\left[ \kappa \overline{v}_{y}\left( y^{*}\right) +\left( 1-p\right) \overline{D}_{y}\left( 0,y^{*}\right) \right] \frac{\partial y^{*} }{\partial \kappa }. \end{aligned}$$

Since either $\frac{\partial y^{*}}{\partial \kappa }=0$ or $v_{y}\left( y^{*}\right) +\left( 1-p\right) D_{y}\left( 0,y^{*}\right) =0$, we have $\frac{\partial V}{\partial \kappa }=E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) <0,$ because $\kappa ^{*}>\widehat{\kappa }.$ The functions V and $V_{I}$ cancel only once at $\kappa ^{*}$ and we have $V_{I}<0$ if and only if $\kappa >\kappa ^{*}$. $\square$

Proof of Proposition 3

• 1. Since $D\le 0$ and v is increasing, if $v\left( y^{*}\right) \le E\left[ v\left( \widetilde{h}\right) \right] ,$ then $V_{I}=E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -\left( 1-p\right) D\left( 0,y^{*}\right) \ge 0$ and immediate resolution of uncertainty is preferred.

• 2. According to Proposition 2, immediate resolution of uncertainty for a given level of objective risk is characterized by $\kappa \le \kappa ^{*}.$ Since for a given objective risk $\left( 1-p\right)$, the level of subjective risk $\left( 1-y^{*}\right)$ decreases with $\kappa$ (Proposition 1); this means that preference for immediate resolution of uncertainty is characterized by a level of perceived risk $\left( 1-y^{*}\right)$ above a given threshold.

• 3. We have $\frac{\partial V_{I}}{\partial p}=v\left( 1\right) -v\left( 0\right) +D\left( 0,y^{*}\right) -\left[ v_{y}\left( y^{*}\right) +\left( 1-p\right) D_{y}\left( 0,y^{*}\right) \right] \frac{\partial y^{*}}{\partial p}$. Since either $\frac{\partial y^{*}}{\partial p}=0$ or $v_{y}\left( y^{*}\right) +\left( 1-p\right) D_{y}\left( 0,y^{*}\right) =0$, we have $\frac{\partial V_{I}}{\partial p}=v\left( 1\right) -v\left( 0\right) +D\left( 0,y^{*}\right) .$ To analyze the impact on the preference for immediate resolution of uncertainty, we analyze the sign of $\frac{\partial V_{I}}{\partial p}$ when $V_{I}=0$. We then have $E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) =\left( 1-p\right) D\left( 0,y^{*}\right)$ and $\frac{\partial V_{I}}{\partial p}$ has the sign of $\left( 1-p\right) v\left( 1\right) -\left( 1-p\right) v\left( 0\right) +v\left( CE_{\tilde{H},v}\right) -v\left( y^{*}\right) =v\left( 1\right) -v\left( y^{*}\right) ,$ which is positive for $y^{*}\in \left] 0,1\right[ .$ For $y^{*}=1$, we have $\frac{\partial V_{I}}{\partial p}=0$ and for $y^{*}=0$, we have $\frac{\partial V_{I}}{\partial p}=v\left( 1\right) -v\left( 0\right) >0.$ If there exists $p^{*}\left( k,d\right)$ such that $V_{I}=0;$ then $V_{I}\ge 0$ on $\left[ p^{*}\left( k,d\right) ,1\right]$. Individuals who test are those for whom $p\ge p^{*}\left( k,d\right)$ and an increase in p increases the testing rate.

• 4. When $p=0$, we have $y^{*}=0$; hence $D\left( 0,y^{*}\right) =0$ and $V_{I}=E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -\left( 1-p\right) D\left( 0,y^{*}\right) =v\left( 0\right) -v\left( 0\right) -D\left( 0,0\right) \ge 0$. Full information is weakly preferred by all individuals. $\square$

Proof of Proposition 4

• 1. Immediate using Inequation (2).

• 2. Immediate since optimal beliefs do not depend upon c nor b.

• 3. Letting $V_{I}=k\left[ E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) \right] -c+\left( 1-p\right) \left[ u\left( b\right) -u\left( 0\right) \right] -\left( 1-p\right) d\overline{D}\left( 0,y^{*}\right)$, we have

  $$\begin{aligned} \frac{\partial V_{I}}{\partial k}=E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) -\left[ k\overline{v}_{y}\left( y^{*}\right) +\left( 1-p\right) d\overline{D}_{y}\left( 0,y^{*}\right) \right] \frac{\partial y^{*}}{\partial k}. \end{aligned}$$

  Since either $\frac{\partial y^{*}}{\partial k}=0$ or $v_{y}\left( y^{*}\right) +\left( 1-p\right) D_{y}\left( 0,y^{*}\right) =0$, we have $\frac{\partial V_{I}}{\partial k}=E\left[ \overline{v}\left( \widetilde{h}\right) \right] -\overline{v}\left( y^{*}\right) ,$ which is negative if $V_{I}=0.$ Analogously, we have

  $$\begin{aligned} \frac{\partial V_{I}}{\partial d}=\left( 1-p\right) \overline{D}\left( 0,y^{*}\right) -\left[ k\overline{v}_{y}\left( y^{*}\right) +\left( 1-p\right) d\overline{D}_{y}\left( 0,y^{*}\right) \right] \frac{\partial y^{*}}{\partial d}. \end{aligned}$$

  Since either $\frac{\partial y^{*}}{\partial d}=0$ or $v_{y}\left( y^{*}\right) +\left( 1-p\right) D_{y}\left( 0,y^{*}\right) =0$, we have $\frac{\partial V_{I}}{\partial d}=\left( 1-p\right) \overline{D}\left( 0,y^{*}\right) ,$ which is negative for $y^{*}\in \left] 0,1\right[ .$

• 4. We have $V_{I}=E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -c+\left( 1-p\right) \left[ u\left( b\right) -u\left( 0\right) \right] -\left( 1-p\right) D\left( 0,y^{*}\right)$ hence, as in the proof of Proposition 3.3, we get $\frac{\partial V_{I}}{\partial p}=\left[ v\left( 1\right) -v\left( 0\right) \right] -\left[ u\left( b\right) -u\left( 0\right) -D\left( 0,y^{*}\right) \right] .$ When $V_{I}=0$, we have $E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -c=-\left( 1-p\right) \left[ u\left( b\right) -u\left( 0\right) -D\left( 0,y^{*}\right) \right]$, hence $\frac{\partial V_{I}}{\partial p}\left| _{V_{I}=0}\right.$ has the sign of $v\left( 1\right) -v\left( y^{*}\right) -c$. When $\left( 1-p\right) \rightarrow 0$, we have $v_{y}\left( 1\right) +\left( 1-p\right) D_{y}\left( 0,1\right) \ge 0,$ hence $y^{*}=1$ and $\frac{\partial V_{I}}{\partial p}\left| _{V_{I}=0}\right.$ is negative for low levels of risk for positive c. More generally, $\frac{\partial V_{I}}{\partial p}$ is positive if and only $y^{*}\le v^{-1}\left[ v\left( 1\right) -c\right]$. Since, as shown in Proposition 1, $y^{*}$ is decreasing in objective risk $\left( 1-p\right)$, we get that $\frac{\partial V_{I}}{\partial p}\left| _{V_{I}=0}\right.$ is positive if and only if the level of objective risk is above a given level.

• 5. When $p\!=\!0$, we have $y^{*}\!=\!0$; hence $D\left( 0,y^{*}\right) \!=\!0$ and we have $E\left[ v\left( \widetilde{h}\right) \right] =v\left( 0\right)$. We then get $V_{I}\equiv E\left[ v\left( \widetilde{h}\right) \right] -v\left( y^{*}\right) -c-\left( 1-p\right) \left[ D\left( 0,y^{*}\right) -u\left( b\right) +u\left( 0\right) \right] =-c+u\left( b\right) -u\left( 0\right)$. Full information is weakly preferred by all individuals if and only if $u\left( b\right) -u\left( 0\right) \ge c$. $\square$