An experimental study of charity hazard: The effect of risky and ambiguous government compensation on flood insurance demand

Abstract

This paper examines the problem of “charity hazard,” which is the crowding out of private insurance demand by government compensation. In the context of flood insurance and disaster financing, charity hazard is particularly worrisome given current trends of increasing flood risks as a result of climate change and more people choosing to locate in high-risk areas. We conduct an experimental analysis of the influence on flood insurance demand of risk and ambiguity preferences and the availability of different forms of government compensation for disaster damage. Certain and risky government compensation crowd out demand, confirming charity hazard, but this is not observed for ambiguous compensation. Ambiguity averse subjects have higher insurance demand when government compensation is ambiguous relative to risky. Policy recommendations are discussed to overcome charity hazard.

Appendices

Appendix 1 Derivation of the hypotheses

Without government compensation the individual will choose a level of $\alpha$ to maximize his/her Expected Utility ($EU$):

$$EU=pU\left[W-P\left(\alpha \right)-\left(L-V\left(\alpha \right)\right)\right]+\left(1-p\right)U[W-P\left(\alpha \right)]$$

(5)

Assuming the individual anticipates the government will provide compensation, $\theta$ ($0<\theta <1)$, to pay for a proportion of the uninsured damage, Eq. 5 is modified to:

$$EU=pU\left[W-P\left(\alpha \right)-\left(1-\theta \right)\left(L-V\left(\alpha \right)\right)\right]+\left(1-p\right)U[W-P\left(\alpha \right)]$$

(6)

If the individual decides not to purchase insurance, her/his ${EU}_{NI}$ ($EU$ with no insurance) is:

$${EU}_{NI}=pU\left[W-\left(1-\theta \right)L\right]+\left(1-p\right)U[W]$$

(7)

We denote willingness-to-pay for full insurance ($\alpha =1$) as $WTP$, defined by:

$${EU}_{NI}=U[W-WTP]$$

(8)

Under linear $U\left(\bullet \right)$ (risk neutrality), ${EU}_{NI}$ is equal to the utility of the expected value:

$$U[W-WT{P}_{RN}]={EU}_{NI}=U[p[W-\left(1-\theta \right)L]+\left(1-p\right)[W]]$$

(9)

Under concave $U\left(\bullet \right)$ (risk aversion), ${EU}_{NI}$ is less than the utility of the expected value:

$$U[W-WT{P}_{RA}]={EU}_{NI}<U[p[W-\left(1-\theta \right)L]+\left(1-p\right)[W]]=U[W-WT{P}_{RN}]$$

(10)

Under convex $U\left(\bullet \right)$ (risk seeking), ${EU}_{NI}$ is greater than the utility of the expected value:

$$U[W-WT{P}_{RS}]=E{U}_{NI}>U[p[W-\left(1-\theta \right)L]+\left(1-p\right)[W]]=U[W-WT{P}_{RN}]$$

(11)

Assuming $p$, $\theta$, $\alpha$ and $\lambda$ remain constant across levels of risk aversion, we can infer from Eqs. 9, 10, and 11 that $U\left[W-WT{P}_{RA}\right]<U\left[W-WT{P}_{RN}\right]<U[W-WT{P}_{RS}]$, hence $WT{P}_{RS}<WT{P}_{RN}<WT{P}_{RA}$. That is, willingness-to-pay for full insurance increases with the degree of risk aversion.²³

H1: Willingness-to-pay for full insurance is positively related to the degree of risk aversion.

We assume that the individual will purchase full insurance if her/his $EU$ with full insurance is greater than $EU$ without insurance, therefore $WTP>P\left(1\right)$:

$${EU}_{NI}=U[W-WTP]<U\left[W-P\left(1\right)\right]$$

(12)

More insurance premium loading reduces $EU$ with full insurance because $P(1)=Lp\lambda$ is increasing in $\lambda$.²⁴ Consequently, the gap between $EU$ with full insurance and $EU$ without insurance becomes smaller. There is a critical value of $\lambda$ where $EU$ without insurance becomes greater than the $EU$ with full insurance, and so the individual chooses not to insure, i.e., $WTP<P\left(1\right)$.

H2: Willingness-to-pay for full insurance is negatively related to the loading factor.²⁵

Consider two risks of loss, ${R}_{1}({p}_{1},{L}_{1})$ and ${R}_{2}({p}_{2},{L}_{2})$, i.e., a loss ${L}_{1}$ (${L}_{2}$) occurs with probability ${p}_{1}$ (${p}_{2}$). The two risks have the same expected value, but ${L}_{2}>{L}_{1}$ and ${p}_{2}<{p}_{1}$. That is, ${R}_{1}$ and ${R}_{2}$ have equal mean but ${R}_{2}$ has higher variance than ${R}_{1}$. Under concave $U\left(\bullet \right)$ (risk aversion), without insurance $EU$ across the two scenarios are given by:

$${EU}_{NI}^{1}={p}_{1}U\left[W-\left(1-\theta \right){L}_{1}\right]+\left(1-{p}_{1}\right)U\left[W\right]=U\left(W-WT{P}_{RA}^{1}\right)$$

(13)

$${EU}_{NI}^{2}={p}_{2}U\left[W-\left(1-\theta \right){L}_{2}\right]+\left(1-{p}_{2}\right)U\left[W\right]=U\left(W-WT{P}_{RA}^{2}\right)$$

(14)

When loss occurs, since ${L}_{2}>{L}_{1}$:

$$U\left[W-\left(1-\theta \right){L}_{1}\right]>U\left[W-\left(1-\theta \right){L}_{2}\right]$$

(15)

For both ${R}_{1}$ and ${R}_{2}$, the alternative utility without loss is the same ($U\left[W\right]$). Also, $U\left[W\right]>U\left[W-\left(1-\theta \right){L}_{1}\right]>U\left[W-\left(1-\theta \right){L}_{2}\right]$. Under concave $U\left(\bullet \right)$ (risk aversion), ${EU}_{NI}^{1}>{EU}_{NI}^{2}$, hence $U\left(W-WT{P}_{RA}^{1}\right)>U\left(W-WT{P}_{RA}^{2}\right)$ and ${WTP}_{RA}^{1}<{WTP}_{RA}^{2}$.

H3: Willingness-to-pay for full insurance is negatively related to the probability of loss for risk averse individuals, holding the expected value of loss constant.

Consider again Eq. 8. Higher levels of government compensation increases the $EU$ without insurance because the share of uninsured loss $L$ becomes lower.²⁶ Consequently, $U[W-WTP]$ increases and $WTP$ decreases. This result follows from the charity hazard highlighted in the introduction section.

H4: Willingness-to-pay for full insurance is negatively related to government compensation.

Assuming an objective probability of receiving government compensation equal to $\pi$, $EU$ is given by:

$$EU=\pi \left\{pU\left[{W}_{LG}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}+\left(1-\pi \right)\left\{pU\left[{W}_{L}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}$$

(16)

We define ${W}_{LG}=W-P\left(\alpha \right)-\left(1-\theta \right)\left(L-V\left(\alpha \right)\right)$ as final wealth in the loss with government compensation state. Similarly, let ${W}_{L}=W-P\left(\alpha \right)-\left(L-V\left(\alpha \right)\right)$ be final wealth in the loss state without government compensation, and ${W}_{NL}=W-P\left(\alpha \right)$ be final wealth in the no loss state. Note that ${W}_{LG}>{W}_{L}$ when $\theta >0$.²⁷ There is a risk of government compensation $G(\pi ,\theta )$ in Eq. 16, i.e., the individual receives government compensation $\theta$ with probability $\pi$ in the event of a loss. Assume two risks of government compensation, ${G}_{1}({\pi }_{1},{\theta }_{1})$ and ${G}_{2}({\pi }_{2},{\theta }_{2})$ with the same expected value, but ${\theta }_{2}>{\theta }_{1}$ and ${\pi }_{2}<{\pi }_{1}$. Under concave $U\left(\bullet \right)$ (risk aversion), without insurance $EU$ across the two scenarios are given by:

$$\begin{aligned}E{U}_{NI,1}&={\pi }_{1}\left\{pU\left[W-\left(1-{\theta }_{1}\right)L\right]+\left(1-p\right)U\left[W\right]\right\}+\left(1-{\pi }_{1}\right)\left\{pU\left[W-L\right]+\left(1-p\right)U\left[W\right]\right\}\\&=p\left\{{\pi }_{1}U\left[W-\left(1-{\theta }_{1}\right)L\right]+\left(1-{\pi }_{1}\right)U\left[W-L\right]\right\}+\left(1-p\right)U[W]\\&=U[W-WT{P}_{RA,1}]\end{aligned}$$

(17)

$$\begin{aligned}E{U}_{NI,2}&={\pi }_{2}\left\{pU\left[W-\left(1-{\theta }_{2}\right)L\right]+\left(1-p\right)U\left[W\right]\right\}+\left(1-{\pi }_{2}\right)\left\{pU\left[W-L\right]+\left(1-p\right)U\left[W\right]\right\}\\&=p\left\{{\pi }_{2}U\left[W-\left(1-{\theta }_{2}\right)L\right]+\left(1-{\pi }_{2}\right)U\left[W-L\right]\right\}+\left(1-p\right)U[W]\\&=U[W-WT{P}_{RA,2}]\end{aligned}$$

(18)

Comparing ${\pi }_{1}U\left[W-\left(1-{\theta }_{1}\right)L\right]+\left(1-{\pi }_{1}\right)U\left[W-L\right]$ with ${\pi }_{2}U\left[W-\left(1-{\theta }_{2}\right)L\right]+\left(1-{\pi }_{2}\right)U\left[W-L\right]$, because ${\theta }_{2}>{\theta }_{1}$, we can infer that $W-\left(1-{\theta }_{2}\right)L>W-\left(1-{\theta }_{1}\right)L$ and $U\left[W-\left(1-{\theta }_{2}\right)L\right]>U\left[W-\left(1-{\theta }_{1}\right)L\right]$. Moreover, $W-L<W-\left(1-{\theta }_{1}\right)L<W-\left(1-{\theta }_{2}\right)L$, therefore $U\left[W-L\right]<U\left[W-\left(1-{\theta }_{1}\right)L\right]<U[W-\left(1-{\theta }_{2}\right)L]$. Similar to H3, the concavity of $U\left(\bullet \right)$ implies that $E{U}_{NI,1}>E{U}_{NI,2}$, or equivalently $U[W-WT{P}_{RA,1}]>U[W-WT{P}_{RA,2}]$, hence $WT{P}_{RA,1}<WT{P}_{RA,2}$.²⁸

H5: Willingness-to-pay for full insurance is negatively related to the probability of government compensation for risk averse individuals, holding the expected value of government compensation constant.

Under ambiguous government compensation, decisions can be made in accordance with the second order $EU$ function, i.e., the Klibanoff et al. smooth model value ($KMM$):

$$KMM=E\left\{\varphi \left\{\overline{\pi }\left\{pU\left[{W}_{LG}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}+\left(1-\overline{\pi }\right)\left\{pU\left[{W}_{L}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}\right\}\right\}$$

(19)

In our experiment, under ambiguous government compensation, there are two possible objective probability distributions regarding $\overline{\pi }$, either probability 1 is assigned to $\{pU\left[{W}_{LG}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\}$, or probability 1 is assigned to $\left\{pU\left[{W}_{L}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}$. Evaluation of the insurance decision is then given by:

$$KMM={\sigma }_{1}\varphi \left\{pU\left[{W}_{LG}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}+{\sigma }_{0}\varphi \left\{pU\left[{W}_{L}\right]+\left(1-p\right)U\left[{W}_{NL}\right]\right\}$$

(20)

If the individual decides not to purchase insurance:

$${KMM}_{NI}={\sigma }_{1}\varphi \left\{U\left[W\right]\right\}+{\sigma }_{0}\varphi \left\{pU\left[W-L\right]+\left(1-p\right)U\left[W\right]\right\}=E\left\{\varphi \left\{EU(\overline{\pi })\right\}\right\}$$

(21)

Under linear $\varphi \left(\bullet \right)$ (ambiguity neutrality):

$$\varphi {\{U\left[W-WT{P}_{AN}\right]\}=KMM}_{NI}=\varphi \left\{E\left(EU\left(\overline{\pi }\right)\right)\right\}$$

(22)

Under concave $\varphi \left(\bullet \right)$ (ambiguity aversion):

$$\varphi \left\{U\left[W-WT{P}_{AA}\right]\right\}={KMM}_{NI}<\varphi \left\{E\left(EU\left(\overline{\pi }\right)\right)\right\}=\varphi \{U\left[W-WT{P}_{AN}\right]\}$$

(23)

Under convex $\varphi \left(\bullet \right)$ (ambiguity seeking):

$$\varphi \left\{U\left[W-WT{P}_{AS}\right]\right\}={KMM}_{NI}>\varphi \left\{E\left(EU\left(\overline{\pi }\right)\right)\right\}=\varphi \left\{U\left[W-WT{P}_{AN}\right]\right\}$$

(24)

We can infer from Eqs. 22, 23, and 24 that $\varphi \{U\left[W-WT{P}_{AA}\right]\}<\varphi \{U\left[W-WT{P}_{AN}\right]\}<\varphi \{U\left[W-WT{P}_{AS}\right]\}$, therefore $WT{P}_{AS}<WT{P}_{AN}<WT{P}_{AA}$. That is, willingness-to-pay for full insurance increases with the degree of ambiguity aversion under ambiguous government compensation.

H6: Willingness-to-pay for full insurance is positively related to the degree of ambiguity aversion when government compensation is ambiguous.

Under risky full government compensation (Eq. 16 evaluated at $\pi =0.5$ and $\theta =1$), and without insurance $EU$ becomes:

$${EU}_{NI,RF}=0.5U\left[W\right]+0.5\left\{pU\left[W-L\right]+\left(1-p\right)U\left[W\right]\right\}=U\left[W-WT{P}_{RF}\right]$$

(25)

Assuming $\sigma =\left(\mathrm{0.5,0.5}\right)$, under ambiguous full government compensation, without insurance $KMM$ becomes:

$${KMM}_{NI}=0.5\varphi \left\{U\left[W\right]\right\}+0.5\varphi \left\{pU\left[W-L\right]+\left(1-p\right)U\left[W\right]\right\}=\varphi \{U\left[W-WTP\right]\}$$

(26)

Under linear $\varphi \left(\bullet \right)$ (ambiguity neutrality), the individual is a (subjective) $EU$ maximizer:

$$U\left[W-WT{P}_{RF}\right]=E{U}_{NI,RF}={KMM}_{NI}=\varphi \left\{U\left[W-WT{P}_{AN}\right]\right\}$$

(27)

Under concave $\varphi \left(\bullet \right)$ (ambiguity aversion):

$$U\left[W-WT{P}_{RF}\right]=E{U}_{NI,RF}>{KMM}_{NI}=\varphi \{U\left[W-WT{P}_{AA}\right]\}$$

(28)

Under convex $\varphi \left(\bullet \right)$ (ambiguity seeking):

$$U\left[W-WT{P}_{RF}\right]=E{U}_{NI,RF}<{KMM}_{NI}=\varphi \{U\left[W-WT{P}_{AS}\right]\}$$

(29)

We can infer from Eqs. 27, 28, and 29 that $\varphi \left\{U\left[W-WT{P}_{AA}\right]\right\}<\varphi \left\{U\left[W-WT{P}_{AN}\right]\right\}=U\left[W-WT{P}_{RF}\right]<\varphi \left\{U\left[W-WT{P}_{AS}\right]\right\}$, therefore$WT{P}_{AA}>WT{P}_{AN}=WT{P}_{RF}>WT{P}_{AS}.$²⁹

H7: Willingness-to-pay for full insurance is higher under ambiguous full government compensation vs. risky full government compensation for ambiguity averse individuals.

H6 and H7 are robust to other ambiguity theories like Maxmin $EU$ (Gilboa & Schmeidler, 1989), because an ambiguity averse individual following Maxmin $EU$ will consider the minimal $EU$ under ambiguous full compensation, which is $EU$ under no government compensation. Note that Maxmin $EU$ is a special case of the Klibanoff et al. (2005) smooth model, where $\varphi$ places all of the weight on the worst $EU$. Online Resource 1 provides a welfare evaluation of the insurance decision over the experimental parameters involved in our study using simulations that illustrate our hypotheses numerically.

Appendix 2 Descriptive statistics and coding of variables

Table 8 Descriptive statistics and coding of the dependent and independent variables

Fig. 2

Distributions of risk and ambiguity preferences in the gain and loss domain with the MPL tasks. Higher values represent more risk and ambiguity aversion; 1 means a switch from left to right in the first row (very risk or ambiguity seeking) and 11 or 10 means a subject never switches (very risk or ambiguity averse, respectively); risk neutral = 6 for risk aversion loss domain, and = 7 for risk aversion gain domain; the ambiguity neutral switching point is on decision line 6 in the loss domain, and decision line 5 in the gain domain

Fig. 3

Distribution of stated risk preference. Higher values represent more risk aversion

Appendix 3 Results from additional analyses

Table 9 Random effects Probit regression of variables of influence on flood insurance purchases with risk and ambiguity preferences elicited in the loss domain