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


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.

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Fig. 1


  1. These data are available on the FEMA website.

  2. Acronym in Dutch.

  3. See also Deryugina and Kirwan (2018).

  4. Their examination of the charity hazard was conducted within-subjects, so contrast effects cannot be ruled out (Greenwald, 1976).

  5. The Klibanoff et al. (2005) smooth model has also been applied in various empirical and theoretical papers in this journal, such as Bajtelsmit et al. (2015), Snow (2011), Conte and Hey (2013) and Qiu and Weitzel (2016).

  6. For simplicity, the extent of relief, \(\theta\), is assumed to be objectively known to the individual, as it is in our experiment.

  7. In our experiment, we examine attitude towards ambiguity due to the existence of multiple non-excludable priors.

  8. See Chakravarty and Roy (2009, pp. 215–216) for a discussion of this assumption. More specifically, the assumption has the potential to confound ambiguity preference parameters. Indeed, values of \(\sigma\) may be impacted by factors like fear and hope (Viscusi & Chesson, 1999). We acknowledge, as do Chakravarty and Roy (2009), that the assumption is a limitation of our study.

  9. It was a design choice to elicit risk and ambiguity attitudes first, then insurance choices. One could argue that subjects have some subconscious motivation to restate their risk and ambiguity preferences in their insurance decisions for some desire to be consistent. We could have studied order effects, although Harrison and Ng (2016) mention that it is likely to be empirically unimportant whether insurance choices or preferences are elicited first. Moreover, we randomly select one decision from either phase one or phase two to be paid, so it was in subjects’ best interest to treat all decisions independently as the only one that they were facing (Papon, 2008). We acknowledge that there are also possible disadvantages of only paying one choice (see e.g., Cubitt et al., 1998).

  10. The rationale for this earnings task was to eliminate the potential for a “house money effect”, where subjects are more risk taking when endowed with a prior monetary gain (Thaler & Johnson, 1990).

  11. For an earlier risk preference elicitation where probabilities are held constant see also Wakker and Deneffe (1996).

  12. The prospects are written: (probability: monetary outcome in currency units, probability: monetary outcome in currency units).

  13. 60,000 CU was endowed to all subjects since every subject answered eight or more questions correctly.

  14. More subjects were randomly assigned to the baseline condition to increase statistical power, given that subjects in this condition faced half as many insurance decisions as in the other versions of the experiment. Subjects were 1.5 times more likely to face the baseline condition than either of the other versions. Otherwise, there was an equal chance subjects would face the other government compensation versions according to the random assignment.

  15. We acknowledge that there are potential confounding variables, given our flood risk context. We do not claim our results are transferrable to other contexts.

  16. In a recruitment flyer individuals were told that in addition to the participation fee, they have ~ 25% chance of earning up to €60 based on their decisions, and a small chance of earning up to €600 based on the one randomly selected subject. In addition to the €15 participation fee, the green card subject earned €599.70. On average the orange card subjects earned €53.73 (min: €0, max: €60).

  17. If subjects were risk neutral, i.e., indifferent between the risky prospects with the same expected value in the risk preference tasks, we flipped a coin to decide which option would decide payment (in the case that the decision line was selected for payment).

  18. This finding is comparable to ours because in our experiment there is one expected loss, which may remove subjects’ entire endowment when flooding probability = 0.001.

  19. It can also be assumed that unobserved subject-specific effects are uncorrelated with government compensation versions, because the versions were randomly assigned across subjects. Our qualitative results are robust to pooled Probit and pooled OLS estimates with clustered standard errors by subject.

  20. As suggested by a reviewer, we investigated in OLS regressions the correlation between risk aversion and ambiguity aversion and the control variables, gender, age, being Dutch, flood risk perceptions and perceptions about government compensation. A Probit model is used when the outcome variable is the stated risk aversion dummy. Males and Dutch subjects are more risk seeking according to risk preferences elicited in the gain domain and regarding the stated risk aversion dummy variable (p-values < 0.05). This gender effect is in line with a large literature base on the impact of gender on risk preferences (Croson & Gneezy, 2009). Moreover, risk preferences can change depending on country-specific variables like cultural factors (Rieger et al., 2015).

  21. We examined different specifications of models 1 to 12, with interaction terms between the order variable and the risk and ambiguity aversion variables, as well as the control variables, gender, age, being Dutch, flood risk perceptions and perceptions about government compensation. The interaction terms are included separately across all specifications to avoid concerns of multi-collinearity that may occur if one were to include all of these terms in the same model. The coefficients on all of these interacting terms are insignificant (p-values > 0.05), which suggests that any learning effects in our experiment are driven by unobserved factors. As an aside, our results are robust to dropping the flood risk and government compensation perceptions variables, which may be perceived as bad controls as they were elicited after the experiment.

  22. The smooth model is commonly used in theoretical examinations of insurance demand under ambiguity or uncertainty of the loss probability (e.g., Alary et al., 2013; Bajtelsmit et al., 2015; Berger, 2016; Brunette et al., 2013; Snow, 2011). Ambiguity aversion increases insurance demand when the probability of loss is more ambiguous according to these studies.

  23. This is true for any insurance coverage level \(\alpha \in (\mathrm{0,1}\)). We consider full insurance is a pure simplification.

  24. An increase in the loading factor (\(\lambda\)) increases the insurance premium of full insurance (\(P(1)\)) and hence reduces the utility of buying full insurance (\(U\left[W-P\left(1\right)\right]\)).

  25. Mossin (1968) and Smith (1968) showed that risk averse \(EU\) maximizers should demand full insurance when \(\lambda =1\), although partial coverage is optimal when \(\lambda >1\). However, in our experiment we observe only full insurance and zero insurance.

  26. An increase in government compensation (\(\theta\)) leads to an increase in final wealth in the loss state (\(W-\left(1-\theta \right)L\)), and the corresponding \(EU\) without insurance.

  27. In our experiment we consider Eq. 16 evaluated at \(\pi =0.5\) and \(\theta =1\) (risky full government compensation), as well as Eq. 16 evaluated at \(\pi =1\) and \(\theta =0.5\) (certain half government compensation).

  28. This holds so long as \(U\left[W-\left(1-{\theta }_{1}\right)L\right]>{\pi }_{2}U\left[W-\left(1-{\theta }_{2}\right)L\right]+\left(1-{\pi }_{2}\right)U\left[W-L\right]\), which always holds in our experiment under risk aversion where we compare \({\pi }_{1}=1\) and \({\theta }_{1}=0.5\) to \({\pi }_{2}=0.5\) and \({\theta }_{2}=1\).

  29. Note that this also holds under the more general condition: \(\sigma =\pi\).


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This research was supported by the Netherlands Organisation for Scientific Research (NWO), Vidi grant number 45214005. The authors would like to thank colleagues at the Institute for Environmental Studies, VU University Amsterdam for helping with the implementation pre-tests. Peter Wakker provided useful suggestions for developing the theoretical framework. We appreciate the comments of Mark Andor on the experiment design.

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Appendix 1 Derivation of the hypotheses

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

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

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:

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

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

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

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


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

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

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

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

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

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

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.Footnote 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 \(WTP>P\left(1\right)\):


More insurance premium loading reduces \(EU\) with full insurance because \(P(1)=Lp\lambda\) is increasing in \(\lambda\).Footnote 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.Footnote 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:

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

When loss occurs, since \({L}_{2}>{L}_{1}\):

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

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.Footnote 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:

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

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 \(\theta >0\).Footnote 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:

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

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 \(WT{P}_{RA,1}<WT{P}_{RA,2}\).Footnote 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 (\(KMM\)):

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

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:

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

If the individual decides not to purchase insurance:

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

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

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

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

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

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

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

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:


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

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

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

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

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

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

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

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

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\(WT{P}_{AA}>WT{P}_{AN}=WT{P}_{RF}>WT{P}_{AS}.\)Footnote 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
figure 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
figure 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

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Robinson, P.J., Botzen, W.J.W. & Zhou, F. An experimental study of charity hazard: The effect of risky and ambiguous government compensation on flood insurance demand. J Risk Uncertain 63, 275–318 (2021).

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  • Ambiguity preferences
  • Charity hazard
  • Economic experiment
  • Flood insurance demand
  • Risk preferences

JEL Classification

  • C91
  • G52