Do you trust your insurer? Ambiguity about contract nonperformance and optimal insurance demand

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Do you trust your insurer? Ambiguity about contract nonperformance and optimal insurance demand Richard Peter a,∗, Jie Ying b a b

Department of Finance, University of Iowa, USA Department of Economics and Finance, Southern Illinois University Edwardsville, USA

a r t i c l e

i n f o

Article history: Received 20 May 2018 Revised 30 October 2018 Accepted 4 January 2019 Available online xxx JEL classification: D11 D80 D81 G22 Keywords: Ambiguity Ambiguity aversion Insurance demand Contract nonperformance Comparative statics

a b s t r a c t We study optimal insurance demand for a risk- and ambiguity-averse consumer under ambiguity about contract nonperformance. Ambiguity aversion lowers optimal insurance demand and the consumer’s degree of ambiguity aversion is negatively associated with the optimal level of coverage. A more pessimistic belief and greater ambiguity may increase or decrease the optimal demand for insurance, and we determine sufficient conditions for a negative effect. We also discuss wealth effects and evaluate the robustness of our results by considering several alternative models of ambiguity aversion. Our findings show how ambiguity about contract nonperformance can undermine the functioning of insurance markets, making it a concern for regulators. Caution is required though because demand reactions are only imperfectly informative about the welfare effects of ambiguity about contract nonperformance. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Insurance offers protection for risk-averse consumers against a broad variety of risks including damage to property, liability exposures, healthcare costs, loss of life and loss of income. Despite their usefulness, insurance contracts may fail to perform as intended, which was first pointed out by Doherty and Schlesinger (1990, DS henceforth). Reasons include insurer insolvency and contractual uncertainty on behalf of consumers, who may lack the financial literacy to fully understand their contract. Much of insurance regulation centers around issues of nonperformance, and its behavioral implications for insurance demand are still not well understood. While most of the existing research focuses on the effects of nonperformance in the wealth domain, we consider the case of ambiguity by assuming that the probability of contract nonperformance is only imperfectly known by consumers. We derive several new testable hypotheses. First, we wonder how ambiguity aversion affects optimal insurance demand in such a situation. We find a negative effect because ambiguity aversion makes the consumer more pessimistic about the performance of his insurance contract. Second, we vary several of the exogenous model parameters and provide comparative statics with respect to the degree of ambiguity aversion, the perception of ambiguity and initial wealth. Whereas greater

∗

Corresponding author. E-mail addresses: [email protected] (R. Peter), [email protected] (J. Ying).

https://doi.org/10.1016/j.jebo.2019.01.002 0167-2681/© 2019 Elsevier B.V. All rights reserved.

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ambiguity aversion always reduces the demand for insurance, the effects of a more pessimistic belief, greater ambiguity and initial wealth are more intricate. We identify sufficient conditions under which a more pessimistic belief and greater ambiguity lower insurance demand and that make the wealth effect positive. Third, our findings point out that insurance demand is only imperfectly informative about the underlying welfare effects of ambiguity about contract nonperformance. This is an important caveat for work in the field. Sources of contract nonperformance abound in insurance markets. Premiums are paid upfront, and the promised indemnity in case of a loss may not be observed until the (remote) future. The most important reason for nonperformance is insurer insolvency.1 Even with guarantee funds, insurer insolvency results in uncertainty for the consumer as to the extent and timing of indemnification. The possibility of claims being contested in front of the courts may result in lack of coverage, and similarly, delays in the insurer’s claims handling process, subtleties in the contractual language of the insurance policy or probationary periods can leave consumers uncovered despite having paid a premium to the insurance company.2 Schlesinger (2013) emphasizes the role of the consumer’s perception of contract nonperformance in explaining deviations of his insurance demand behavior. Due to lack of information, consumers will not know the odds of a contract’s future performance at the time of purchase. Tennyson (1997) finds that 59% of consumers are not confident in the financial stability of their insurer and 46% consider bankruptcy of any insurer a serious problem. Trust in the insurance mechanism appears to be limited in the general population. Hence, it is not surprising that the aforementioned examples create significant uncertainty in the consumers’ perception of the reliability of their insurance contracts. Even when taking objective information, the long-term nature of certain insurance contracts can expose consumers to considerable probabilistic uncertainty. Li et al. (2018) use financial strength information from A.M. Best (2015) to find rating transitions for annuity providers. The cumulative average impairment rate of a B++/B rated provider is 20% in 15 years. Future impairment rates of specific provides vary substantially around this average despite the solid initial rating. Siding with Camerer and Weber (1992, page 330), we subsume these considerations under ambiguity because “ambiguity is uncertainty about probability, created by missing information that is relevant and could be known”.3 Since Ellsberg’s (1961) seminal thought experiment, it is widely accepted that individuals are sensitive to ambiguity. Ambiguity aversion has been documented in laboratory experiments (e.g., Chow and Sarin, 2001; Einhorn and Hogarth, 1986), market settings with educated individuals (see Sarin and Weber, 1993), and surveys of business owners and managers (see Chesson and Viscusi, 2003; Viscusi and Chesson, 1999).4 Recent survey evidence of U.S. households confirms the relevance of ambiguity aversion for financial decisions in the field (see Dimmock et al., 2016), and it is persuasive that limited financial literacy (see Lusardi and Mitchelli, 2007) and lack of consumer sophistication (see Li et al., 2018) magnify its effects. Whereas the welfare consequences of ambiguity for ambiguity-averse decision-makers are immediate, its behavioral implications are not, which opens up an interesting area of research. For example, Gollier (2011) provides an instructive counterexample where ambiguity increases the demand for an ambiguous asset for ambiguity-averse investors, contrary to intuition. So while “ambiguityaverse decision-makers prefer acts whose evaluation is more robust to the possible variation in probabilities” (see Klibanoff et al., 2005, page 1852), there are cases in economics where this results in behavior that increases exposure to ambiguity, similar to the possibility of Giffen goods in consumption theory. It is therefore not surprising to see a growing literature on the effects of ambiguity aversion on optimal behavior, where researchers identify conditions that allow for intuitive results. Some examples of this line of research in the context of insurance demand are Alary et al. (2013), Huang et al. (2013), Gollier (2014) and Bajtelsmit et al. (2015). Existing studies on contract nonperformance and insurance demand are by and large based on (subjective) expected utility, which implicitly assumes ambiguity neutrality. DS find that most comparative statics of insurance demand collapse under nonperformance risk: Mossin’s Theorem is violated (see also Mahul and Wright, 2007), an increase in risk aversion does not necessarily raise the optimal level of coverage, and insurance may not be an inferior good under decreasing absolute risk aversion. So unless the premium is actuarially fair, insurance demand may increase or decrease as a result of nonperformance risk. One might thus expect ambiguity to further obfuscate the economic trade-offs associated with the insurance decision. The analysis of nonperformance risk has been extended to the insurer-reinsurer relationship (Bernard and

1 A.M. Best (2016) counts a total of 761 impairments in the period from 1978 to 2015 in the U.S., see http://www.ambest.com/nrsro/FormNRSRO_Ex1_ RatingsImpairment.pdf. 2 Crocker and Tennyson (2002) show that some types of claims are less certain to be paid in full because they are easy to falsify. Tennyson and Warfel (2009) and Asmat and Tennyson (2014) provide evidence of claims underpayment and discuss the effect of the legal framework on the insurers’ settlement and verification practices. Bourgeon and Picard (2014) develop an economic model of insurer “nitpicking”, which reduces the efficiency of insurance contracts and can lead to substantial uncertainty about the performance of the contract. A related issue is that of nonverifiable losses as studied by Doherty et al. (2013). 3 We understand risk as a condition in which the event to be realized is unknown, but the odds of all possible events are perfectly known, either subjectively or objectively. Ambiguity refers to a condition where the probabilities of possible events are not unique (see Iwaki and Osaki, 2014). According to Ghirardato (2004), ambiguity corresponds to situations in which some events do not have an obvious, unanimously agreeable, probability assignment. We provide a more formal distinction in Section 2. 4 For a recent summary of the literature and the boundary conditions of ambiguity aversion in the laboratory, see Kocher et al. (2018). For the prevalence of ambiguity attitudes in a large representative sample, see Dimmock et al. (2015a,b).

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Ludkovski, 2012), to risk management instruments other than insurance (Briys et al., 1991; Schlesinger, 1993), and to heterogeneous beliefs about nonperformance risk (Cummins and Mahul, 2003).5 There is also evidence that contract nonperformance affects actual behavior. Kahneman and Tversky (1979) introduce ‘probabilistic insurance’, which incorporates an element of nonperformance risk but also exhibits probabilistic repayment of the premium. Subjects appear to dislike both probabilistic insurance as well as pure nonperformance risk and adjust insurance demand accordingly, as suggested by various surveys (Kahneman and Tversky, 1979; Wakker et al., 1997; Zimmer et al., 2009) and evidenced in incentive-compatible laboratory experiments (Herrero et al., 2006; Zimmer et al., 2018). What’s more, nonperformance risk reduces welfare of risk-averse individuals directly, because contracts are less efficient, and indirectly, because the compound nature of nonperformance risk induces violations of the reduction of compound lotteries axiom (Harrison and Ng, 2018). There is a small but growing literature that studies the effects of ambiguity about contract nonperformance in insurance. Biener et al. (2017) show that ambiguity about the nonperformance probability raises the marginal willingness to pay at a given level of coverage (their Lemma 3). In the corresponding field experiment ambiguity reduces take-up by 14.5%. Bryan (2018) shows that mandatory rainfall insurance lowers adoption rates of a new farming technology for ambiguityaverse farmers (his prediction 1). He also presents evidence from two randomized controlled trials and documents substantial income losses associated with ambiguity. We complement and extend these papers along several dimensions. First, we allow decision-makers to select a level of coverage to answer the question how optimal demand is affected by the presence of ambiguity about contract nonperformance.6 Second, we provide comparative statics with respect to the degree of ambiguity aversion, the consumer’s perception of ambiguity and initial wealth. This reveals that some conclusions may only generalize under appropriate qualifications. It also helps uncover new economic trade-offs and testable hypothesis, that are not apparent when confining the comparison to ambiguity-averse versus ambiguity-neutral behavior. Third, demand reactions may convey only limited information about welfare. Our paper therefore adds to this stream of research by providing a systematic theoretical foundation, generalizations and new results, including their robustness when it comes to alternative models of ambiguity aversion. While DS are silent about the demand effects of nonperformance risk unless premiums are actuarially fair (see their Table 1), we find that ambiguity aversion and greater ambiguity aversion reduce optimal demand and identify sufficient conditions for a more pessimistic belief and greater ambiguity to reduce optimal demand as well. 2. Model and notation We consider a consumer with initial wealth W, who is subject to a loss of L ∈ (0, W) with known probability p ∈ (0,1). The consumer can purchase insurance against the risk of loss with α ∈ [0, 1] denoting the level of coverage and α L the indemnity in case of a loss. Insurance requires payment of an insurance premium P(α ), which we assume to be an increasing and nonconcave function of the level of coverage, P > 0 and P  ≥ 0, with P (0 ) = 0 (see Appendix B.3 for a discussion). As in DS, insurance contracts are imperfect because they may not perform. This occurs with probability (1 − q ) ∈ (0, 1 ) whereas with probability q the contract works as intended and the indemnity is paid. Three different states of the world are relevant for the analysis. W1 = W − P is the consumer’s final wealth if he does not suffer a loss, in which case potential contract nonperformance is irrelevant. W2 = W − P − L + α L is the consumer’s final wealth when a loss occurs and the insurance contract performs, and W3 = W − P − L is the consumer’s final wealth when he suffers a loss but the contract does not perform. As noted by DS, in hindsight the consumer would have been better off in this last state had he not purchased insurance to save on premium money. For simplicity, we assume no recovery in case of nonperformance so that no indemnity is paid at all.7 In this paper, we focus on the consumer’s perception of contract nonperformance. If consumers knew the probabilities of the various final wealth outcomes for sure, either subjectively or objectively, we would be in a situation of decision-making under risk because uncertainty would be confined to mean uncertainty over outcomes with known probabilities. However, we assume that consumers do not know the probability of contract nonperformance for sure. We are therefore in a situation of decision-making under ambiguity because there is also uncertainty about probability. We model this ambiguity by replacing q with random variable  q, which takes values in the unit interval. We use Neilson’s (2010) simplified second-order expected utility approach to model preferences.8 According to his Theorem 1, we can represent the consumer’s preferences with a vNM utility function u over final wealth, an ambiguity function φ over expected utility levels, and a probability distribution 5 Other theoretical works incorporate taxation (Huang and Tzeng, 2007) or allow default risk to arise endogenously (Biffis and Millossovich, 2012). Recently, Liu and Myers (2016) develop a dynamic model of nonperformance risk and find a negative effect on the demand for microinsurance. 6 Marginal willingness to pay, as studied in Biener et al. (2017), does not inform about optimal demand. Jaspersen (2016) provides an explicit example in Footnote 6 where the individual with lower willingness to pay prefers a contract with higher coverage than the individual with higher willingness to pay. This is also why Alary et al. (2013) study both concepts separately and find conditions specific to each when signing the comparative statics. In Bryan (2018), insurance is mandatory precluding questions of optimal demand. What’s more, his set-up is one of basis risk so there is a chance of receiving an insurance indemnity even if no loss happened. This is a departure from nonperformance risk in the DS sense. 7 This assumption is without loss of generality because the ordering of the different final wealth levels is given by W3 < W2 ≤ W1 for any recovery rate. The last inequality is strict for partial insurance. 8 Neilson’s (2010) approach is a special case of Klibanoff et al. (2005). His representation of preferences takes Anscombe and Aumann (1963) horse and roulette lotteries, applies the Savage (1954) axioms to horse lotteries and the von Neumann and Morgenstern (1944) axioms to roulette lotteries. In Appendices B.1 and B.2, we show that most of our results continue to hold under alternative models of ambiguity aversion.

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of  q, which characterizes the consumer’s second-order belief. We assume that u is increasing and concave, u > 0 and u  < 0, to reflect non-satiation and risk aversion, and that φ is increasing and concave, φ  > 0 and φ   < 0, to reflect non-satiation (in expected utility) and ambiguity aversion. To simplify the exposition of the proofs and derive intuition, we decompose the consumer’s perception of the contract nonperformance probability into an expected component and an additive noise term, based on his belief. If q denotes the ex= q −  . To maintain the interpretation pected value of  q, then ε q is a zero-mean risk, and we can write (1 −  q ) = (1 − q ) + ε  needs to be contained in [ε , ε ] ⊆ [−(1 − q ), q]. We denote by F the cumulative distribution of probabilities, the support of ε  that is induced by the decision-maker’s belief over  function of ε q. This distribution is a measure of how much ambiguity the consumer perceives. If F is fairly concentrated around zero, the consumer is relatively confident about the expected nonperformance probability. If F is very dispersed instead, he has a hard time grasping the exact nonperformance probability. To focus on ambiguity about contract nonperformance, we assume the probability of loss to be unambiguous throughout the analysis. In Appendix B.4, we briefly discuss the demand implications for ambiguity-averse consumers when both the probability of loss and the probability of contract nonperformance are ambiguous. With these specifications, the consumer’s expected utility for a given nonperformance probability is

U (α , ε ) = (1 − p)u(W1 ) + p(q − ε )u(W2 ) + p(1 − q + ε )u(W3 ).

(1)

Ambiguity about contract nonperformance makes the consumer’s expected utility a random variable with values depending . To incorporate the consumer’s attitude towards ambiguity, we express his ex-ante welfare as a φ on the realization of ε weighted expectation over different expected utilities based on his second-order belief, that is,

)]. V (α ) = Eφ [U (α , ε

(2)

The expectation is taken with respect to F. We will study this objective function to understand how ambiguity aversion affects optimal insurance demand in the following section. 3. The optimal level of coverage The optimal level of coverage α ∗ maximizes the consumer’s ex-ante welfare. We characterize an interior solution by the associated first-order condition,

V  (α ∗ ) = E



 φ  [U (α ∗ , ε)]Uα (α ∗ , ε) = 0,

(3)

where subscript α denotes the partial derivative with respect to the coverage level. The consumer’s objective function is globally concave in α ,

V  (α ) = E



 φ  [U (α , ε)]Uα (α , ε)2 + φ  [U (α , ε)]Uαα (α , ε) < 0.

(4)

Due to ambiguity aversion, φ   [U(α ,

ε)] < 0 for every ε, and due to risk aversion and the non-concavity of the premium schedule, Uαα (α , ε ) < 0 for every ε . Concavity of the objective function ensures that the first-order condition characterizes a unique maximum. V (0) > 0 is necessary and sufficient for some insurance to be in demand. We can rearrange it to P  (0 ) <

pqLu (W − L ) , (1 − p)u (W ) + pu (W − L )

(5)

so as long as the marginal increase in premium from purchasing some insurance is not too large, insurance demand is positive. In the special case of a proportional loading factor m ≥ 1 on top of the actuarially fair premium plus a fixed cost K ≥ 0, the premium schedule becomes P (α ) = α pqmL + K, and condition (5) simplifies to

m<

u (W − L )

(1 − p)u (W ) + pu (W − L )

.

(6)

The right-hand side exceeds 1 due to risk aversion, and is larger for rare and more severe losses as well as for more riskaverse consumers (see Peter, 2016). In this case, the expected probability of nonperformance, the level of ambiguity over the nonperformance probability, as well as the consumer’s degree of ambiguity aversion are irrelevant for whether insurance will be in demand. Intuitively, for small levels of insurance demand, the resulting exposure to nonperformance risk is negligible from the consumer’s standpoint. In the sequel, we assume condition (5) to be satisfied so that some insurance is in demand, α ∗ > 0. To analyze how the presence of ambiguity affects the consumer’s trade-off, we decompose Uα (α , ε ) into the marginal utility cost and the marginal utility benefit:





MC(α , ε ) = P  (α ) (1 − p)u (W1 ) + p(q − ε )u (W2 ) + p(1 − q + ε )u (W3 ) , MB(α , ε ) = Lp(q − ε )u (W2 ).

(7)

Insurance obligates the consumer to pay a premium which reduces his final wealth in each state as captured by MC. The indemnity increases final wealth if the loss occurs and the contract performs, corresponding to MB. We rewrite the firstorder condition as follows:

E



   φ  [U (α , ε)]MC (α , ε) = E φ  [U (α , ε)]MB(α , ε) .

(8)

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The trade-off of an ambiguity-neutral consumer is not affected by the uncertainty about the nonperformance probability, and his optimal insurance demand α 0 is characterized by:

Uα (α 0 , 0 ) = −MC (α 0 , 0 ) + MB(α 0 , 0 ) = 0.

(9)

This is the level of coverage studied in DS for a linear premium schedule. To develop some intuition, we investigate how ambiguity about the nonperformance probability affects the marginal utility cost and the marginal utility benefit of insurance. For the former, we derive

MCε

 0    α , ε = P (α 0 ) p u (W3 ) − u (W2 ) > 0,

(10)

where the positive sign is due to risk aversion. The higher ε , the more likely the consumer perceives contract nonperformance to happen. This raises the marginal utility cost of insurance because having incurred the insurance premium is most painful in precisely the state of the world where the insurance contract does not perform. Likewise, the lower ε , the lower the marginal utility cost of insurance. For the marginal utility benefit, we obtain

MBε



 α 0 , ε = −pLu (W2 ) < 0.

(11)

The higher ε , the less likely the consumer perceives the contract to pay the indemnity in case of loss, which lowers the marginal utility benefit of insurance. Likewise, the lower ε , the higher the marginal utility benefit of insurance. So any positive value of ε corresponds to a higher marginal utility cost of insurance relative to MC(α 0 , 0) and a lower marginal utility benefit of insurance relative to MB(α 0 , 0), inducing the ambiguity-averse consumer to lower his demand for insurance compared to α 0 . On the contrary, any negative value of ε corresponds to a lower marginal utility cost of insurance relative to MC(α 0 , 0) and a higher marginal utility benefit of insurance relative to MB(α 0 , 0), inducing the ambiguity-averse consumer to increase his demand for insurance compared to α 0 . Will these opposing effects cancel each other out in equilibrium? We establish in the following proposition that this is not the case because the net effect is negative. All proofs are gathered in Appendix A. Proposition 1. Under ambiguity about the probability of contract nonperformance, ambiguity aversion lowers insurance demand. This result identifies ambiguity about contract nonperformance as a behavioral impediment to insurance demand with the potential to undermine the functioning of insurance markets in the economy. It adds to the literature that shows how ambiguity compromises the effectiveness of financial instruments and financial markets (see Mukerji and Tallon, 2001). Our result extends Biener et al.’s (2017) result on marginal willingness to pay (their Lemma 3) and provides the theoretical underpinning for their Hypothesis 3 on insurance demand. While willingness to pay measures can inform about insurance take-up decisions (i.e., binary choices), they fall short of answering the question how much insurance is optimal (see our Footnote 6 ). Ambiguity-averse decision-makers reduce their exposure to uncertainty about contract nonperformance by choosing a lower level of insurance demand. Due to the binary nature of nonperformance risk in the DS model, we obtain this result without any additional restrictions. Unlike in Gollier (2011) but similar to Corollary 1 in Alary et al. (2013), the intuition that ambiguity-averse decision-makers adjust behavior to be in a situation that is more robust to the possible variation in probabilities applies directly to our problem. To develop some intuition for Proposition 1, we return to the first-order condition (3) and evaluate it at the optimal level of coverage of an ambiguity-neutral consumer, which yields

V  (α 0 ) = E

            φ U (α 0 , ε) Uα (α 0 , ε) = − E φ  U (α 0 , ε) MC (α 0 , ε) + E φ  U (α 0 , ε) MB(α 0 , ε) .



 unconditional marginal cost

(12)

unconditional marginal benefit

From the proof of Proposition 1 we know that φ  [U(α 0 ,

ε)] is increasing in ε if the consumer is ambiguity-averse. Scenarios with ε < 0 receive relatively less weight than scenarios with ε > 0. Said differently, an ambiguity-averse consumer behaves as if he was an expected utility maximizer who overweights scenarios in which the marginal utility cost is higher than the marginal utility benefit of insurance and underweights scenarios in which the marginal utility cost is lower than the marginal utility benefit of insurance. In the spirit of Gollier (2011), we can rewrite the consumer’s first-order expression, evaluated at α 0 , as follows:





) E V  ( α 0 ) = Eφ  U ( α 0 , ε

     0  φ  U (α 0 , ε) 0  ) EQ Uα (α 0 , ε ) , U ( α , ε ) = Eφ  U ( α 0 , ε )] α Eφ  [U (α 0 , ε

(13)

where Q0 denotes a distorted probability measure. The ambiguity-averse consumer behaves in the  same way  as an expected utility maximizer who has distorted his second-order belief.9 The associated distortion factor φ  U (α 0 ,  ε ) /Eφ  U (α 0 , ε) is 0

a Radon-Nikodym derivative describing the change of measure, and EQ is the ambiguity-neutral expectation, which corresponds to the risk-neutral expectation in finance. Our notation captures the endogeneity of this distorted measure because 9

Gollier (2011) refers to this interpretation as “observational equivalence”.

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it depends on the level of coverage selected by an ambiguity-neutral consumer. Under Q0 , the ambiguity-averse consumer perceives contract nonperformance as more likely compared to the expected nonperformance probability under his secondorder belief. The reason is that

       ) , ε  φ  U (α 0 , ε) Cov φ  U (α 0 , ε } = E ) = (1 − q ) + E {1 − q + ε ( 1 − q + ε ,  0  0 )] )] Eφ [U (α , ε Eφ [U (α , ε Q0

(14)

with the second summand being positive because φ  [U(α 0 , ε )] increases in ε . We can therefore interpret the behavior of an ambiguity-averse consumer as that of an expected utility maximizer who perceives the probability of the contract performing as intended as lower than what he would have expected based on his second-order belief. This is why ambiguity aversion lowers insurance demand in the presence of ambiguity about the nonperformance probability. Before we proceed, we make some remarks on the optimality of full insurance (α ∗ = 1), a topic that has received some attention on its own account. When nonperformance risk is absent, full coverage is optimal if and only if insurance is actuarially fair (Mossin, 1968). Under nonperformance risk with total default, DS show that less than full coverage is optimal if the price is actuarially fair.10 As a result, partial insurance is optimal a fortiori when there is ambiguity about the nonperformance probability and consumers are ambiguity-averse, see Proposition 1. So Mossin’s Theorem does not hold in our model. As soon as nonperformance is partial with a sufficiently high recovery rate, full or more-than-full insurance is optimal under ambiguity neutrality (Mahul and Wright, 2007). With the “right” combination of recovery rate, level of ambiguity and degree of ambiguity aversion, optimal insurance demand can then be at full coverage for the ambiguity-averse consumer when the premium is actuarially fair.11 Admittedly, this case is rather contrived and primarily of theoretical interest. 4. Comparative statics We established in Proposition 1 that optimal insurance demand of ambiguity-averse consumers differs from that under subjective expected utility when there is ambiguity about contract nonperformance. In this section, we will extend the analysis and study differences in the consumer’s degree of ambiguity aversion, his perception of contract nonperformance and different wealth levels. While some effects generalize as expected, others require qualifications that we will identify in the sequel. DS find that it is unclear under nonperformance risk whether a more risk-averse consumer buys more coverage, even when using a strong increase in risk aversion in the Ross (1981) sense. The reason is that an increase in coverage reduces the spread between W1 and W2 but widens the spread between W2 and W3 .12 Are better results obtained when it comes to ambiguity aversion? We first provide a formal definition of comparative ambiguity aversion. Definition 1. A decision-maker is more ambiguity-averse than another one if they share the same vNM utility function, the same belief, and if the ambiguity function of the first is more concave than the ambiguity function of the second. This definition is due to Klibanoff et al. (2005) and allows us to use tools from risk theory to operationalize comparative ambiguity aversion. Differences in the degree of ambiguity aversion are commonly observed in experiments (see Berger and Bosetti, 2016) as well as in the field (see Dimmock et al., 2015a), with some people being more sensitive to ambiguity than others. Such differences also prevail for small likelihoods (see Baillon and Emirmahmutoglu, 2018) as in the case of contract nonperformance probabilities, which warrants our analysis. Our next proposition summarizes. Proposition 2. Under ambiguity about contract nonperformance, an increase in the consumer’s degree of ambiguity aversion lowers the optimal level of insurance coverage. Proposition 2 generalizes Proposition 1 because the comparison between an ambiguity-averse and an ambiguity-neutral consumer is a special case of an increase in ambiguity aversion. As for Proposition 1, we do not have to impose additional assumptions to find that decision-makers who are more sensitive to ambiguity adjust behavior to be in a situation that is more robust to possible variation in probabilities. They achieve this by reducing the amount of insurance coverage because it is the insurance contract in our model which introduces ambiguity into their endowment. To develop intuition, we investigate how the consumer’s unconditional marginal cost and unconditional marginal benefit of insurance are affected by an increase in his degree of ambiguity aversion. They are given by

   ψ  [U (α ∗ , ε)]MC (α ∗ , ε) = E k [U (α ∗ , ε)]φ  [U (α ∗ , ε)]MC (α ∗ , ε) ,     )]MB(α ∗ , ε ) = E k [U (α ∗ , ε )]φ  [U (α ∗ , ε )]MB(α ∗ , ε ) . E ψ  [U (α ∗ , ε E



(15)

10 Less than full coverage remains optimal under nonperformance risk with total default and actuarially unfair premiums when the decision-maker is prudent or has quadratic utility (u   ≥ 0). How the levels of coverage with and without nonperformance risk compare to each other with a positive premium loading is, however, an unsettled question. 11 Still, whether the “only if” part of Mossin’s Theorem would hold too is a separate issue because DS show that insurance demand under nonperformance risk is not necessarily lower with a positive loading compared to the actuarially fair case. 12 Eeckhoudt et al. (2017) show that a restricted increase in risk aversion produces the intuitive comparative static result. This notion of comparative risk aversion is even stronger than that of Ross (1981).

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7

The difference to the less ambiguity-averse consumer is captured by k [U(α ∗ , ε )], which is increasing in ε . This reinforces scenarios where ε is high so that the consumer perceives contract nonperformance as more likely relative to scenarios where ε is low and the consumer perceives contract nonperformance as less likely. Consequently, for a more ambiguityaverse consumer scenarios where the marginal cost of insurance is high and its marginal benefit is low are reinforced whereas scenarios where the marginal cost of insurance is low and its marginal benefit is high are attenuated. It is thus optimal to reduce the level of coverage. This intuition extends to the consideration of the observationally equivalent probability distortion of the more ambiguity)] ) and ε  is averse consumer. Due to the fact that k (φ [U(α ∗ , ε )]) is increasing in ε , the covariance between k (φ [U (α ∗ , ε positive. This holds under the physical measure but also with respect to the distorted measure Q∗ with Radon–Nikodym )]/Eφ  [U (α ∗ , ε )]. Expanding this covariance under Q∗ yields derivative φ  [U (α ∗ , ε

CovQ

∗





)] ), ε  = EQ k (φ [U (α ∗ , ε

∗







)] )ε  − EQ k (φ [U (α ∗ , ε

∗





)] ) · EQ {ε } k (φ [U (α ∗ , ε ∗

 φ  [U (α ∗ , ε)]  ∗   k φ [ U ( α , ε ) ] ε ( ) )] Eφ  [U (α ∗ , ε       ∗  φ [U (α , ε )]  φ [U (α ∗ , ε)] ∗   −E k φ [ U ( α , ε ) ] · E ε ( ) )] )] Eφ  [U (α ∗ , ε Eφ  [U (α ∗ , ε    1 )] )φ  [U (α ∗ , ε )], ε  = Cov k (φ [U (α ∗ , ε )] Eφ  [U (α ∗ , ε  2     1 )] )φ  [U (α ∗ , ε )] Cov φ  [U (α ∗ , ε )], ε  . − E k (φ [U (α ∗ , ε  ∗  Eφ [U (α , ε )] =E

(16)

This is positive if and only if

)] )φ  [U (α ∗ , ε )], ε } )], ε } Cov{k (φ [U (α ∗ , ε Cov{φ  [U (α ∗ , ε > ,  ∗  ∗ )] )φ [U (α , ε )]} )] E{k (φ [U (α , ε Eφ  [U (α ∗ , ε

(17)

which, according to Eq. (14), states that the behavior of a more ambiguity-averse consumer can be interpreted as that of an expected utility maximizer who distorts the probability of contract nonperformance more strongly compared to a less ambiguity-averse consumer. The distorted belief consistent with the behavior of the former attributes a higher probability to the event that the contract does not perform relative to the distorted belief consistent with the behavior of the latter. It is thus optimal for a more ambiguity-averse consumer to reduce his optimal demand for insurance. So everything else equal, insurance demand is negatively related to the consumer’s degree of ambiguity aversion, and we can use insurance demand as an inverse indicator for the consumer’s sensitivity to ambiguity. The question dual to the one examined in Proposition 2 is that of a change in the consumer’s perception of ambiguity. As a benchmark, we briefly revisit the comparative statics of changes in risk. A first-order stochastically dominated shift or an increase in risk in the sense of Rothschild and Stiglitz (1970) do not necessarily reduce the demand for the risky asset in the standard portfolio problem (see Rothschild and Stiglitz, 1971). Exploiting the equivalence with the coinsurance problem, these changes do not necessarily raise the demand for insurance. Sufficient conditions that have been proposed in the literature are that relative risk aversion be bounded by unity or that relative prudence be bounded by 2, respectively (see Chiu et al., 2012; Hadar and Seo, 1990). DS show that the relationship between the consumer’s optimal insurance demand and the probability of contract nonperformance is indeterminate. They only obtain a monotonically decreasing relationship when default is total, the premium is actuarially fair and preferences are restricted either to quadratic or exponential utility. This appears to leave little hope for the case of ambiguity. To fix ideas, we first provide two definitions. Definition 2. a) The consumer’s second-order belief becomes more pessimistic if it undergoes a first-order stochastic improvement. b) The consumer perceives greater ambiguity if his second-order belief undergoes an increase in risk in the sense of Rothschild and Stiglitz (1970). Under the first definition, the consumer’s expected probability of contract nonperformance increases as a result of the first-order stochastic shift. The consumer is more pessimistic in the sense that he believes contract nonperformance to happen more frequently. The second definition has been used to study the effects of greater ambiguity on the value of information (Hoy et al., 2014; Snow, 2010), on optimal self-insurance and self-protection (Snow, 2011) and on precautionary saving (Peter, 2019). It is here where Neilson’s (2010) simplified approach offers the greatest benefit because it allows for a tractable characterization of changes in the decision-maker’s perception of ambiguity.13 To sign the comparative statics, we introduce two measures of ambiguity preferences.

13

See Jewitt and Mukerji (2017) for a more general definition of greater ambiguity.

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Definition 3. 

U] a) The consumer’s relative ambiguity aversion is given by R[U] = −U φφ  [[U] .



U] b) The consumer’s relative prudence in ambiguity preferences is given by P[U] = −U φφ  [[U] .

These measures are akin to their counterparts in risk theory. Huang and Tzeng (2017) show their usefulness in a portfolio choice problem. We obtain the following results. Proposition 3. a) A more pessimistic belief reduces optimal insurance demand if relative ambiguity aversion is bounded by unity. b) Under non-negative ambiguity prudence, greater ambiguity reduces optimal insurance demand if relative prudence in ambiguity preferences is bounded by 2. The comparative statics of changes in ambiguity are structurally isomorphic to the comparative statics of risk in the coinsurance problem. Not all ambiguity-averse consumers whose belief becomes more pessimistic or who experience greater ambiguity reduce their optimal insurance demand. They do so if their preferences meet certain restrictions, but might react in the opposite direction otherwise.14 Conversely, an increase in insurance demand is not necessarily indicative of a more optimistic belief or less ambiguity. This is important for empirical and experimental inference because unlike greater ambiguity aversion, insurance demand is no longer a suitable indicator for the level of ambiguity experienced by consumers. This negative result also disqualifies insurance demand as an indicator for the welfare effects of ambiguity unless we know that consumers meet the restriction on ambiguity preferences.15 We provide some intuition for Proposition 3b). A threshold value of 2 for an intensity measure of relative prudence is well known in the risk literature, for example when it comes to the effect of an increase in interest rate risk on saving behavior (see Chiu et al., 2012; Eeckhoudt and Schlesinger, 2008). There is a substitution effect and a precautionary effect, both of which are operative in our setup, too. The substitution effect is negative because greater ambiguity about contract nonperformance compromises the effectiveness of the insurance contract and makes it less attractive. The precautionary effect may be positive or negative. Greater ambiguity makes expected utility riskier. Under prudence in ambiguity preference, the consumer thus experiences an incentive to adjust the level of coverage in such a way as to increase expected utility for precautionary purposes.16 If the consumer perceives contract nonperformance as likely (ε > εˆ), this is achieved by reducing the level of coverage, consistent with the substitution effect. If the consumer perceives contract nonperformance as unlikely (ε < εˆ), this is achieved by increasing the level of coverage resulting in a potentially conflicting positive precautionary effect. The restriction on ambiguity preferences in Proposition 3 ensures that the precautionary effect never dominates when it is positive. Further intuition can be developed by analyzing how greater ambiguity affects the distorted belief that rationalizes the optimal behavior of an ambiguity-averse consumer. The corresponding expected nonperformance probability is ∗

} = (1 − q ) + E Q {1 − q + ε

φ  [U (α ∗ , ε )]} E {ε . )] Eφ  [U (α ∗ , ε

(18)

If the consumer’s prudence in ambiguity preferences is non-negative (i.e., φ    ≥ 0), it holds that Eφ  [U (α ∗ ,  κ )] ≥ )] for  κ being an increase in risk of ε in the sense of Rothschild and Stiglitz (1970) so that the denominator in Eφ  [U (α ∗ , ε (18) increases. The reason is that the second derivative of φ  [U(α ∗ , ε )] with respect to ε is given by φ    [U(α ∗ , ε )]Uε (α ∗ , ε )2 , which is non-negative. But also the numerator in (18) increases with greater ambiguity. The second derivative of εφ  [U(α ∗ , ε)] with respect to ε is given by

2φ  [U (α ∗ , ε )]Uε (α ∗ , ε ) + εφ  [U (α ∗ , ε )]Uε (α ∗ , ε )

2



= −Uε (α ∗ , ε )φ  [U (α ∗ , ε )] P[U (α ∗ , ε )] ·

 εUε (α ∗ , ε ) − 2 . U (α ∗ , ε )

(19)

The two terms outside the curly bracket are both negative. Expected utility is linear in ε so that U (α ∗ , ε ) = U (α ∗ , 0 ) + εUε (α ∗ , ε ). Therefore,

εUε (α ∗ , ε ) U (α ∗ , ε ) − U (α ∗ , 0 ) = < 1, for any ε . U (α ∗ , ε ) U (α ∗ , ε )

(20)

14 A more pessimistic belief lowers insurance demand also for an ambiguity-neutral consumers (i.e., if φ  = 0). This does not contradict with ambiguityneutral decision-makers being insensitive to ambiguity because the behavioral change originates from a higher expected probability of contract nonperformance. 15 The point estimate for the parameterization in Berger and Bosetti (2016) suggests relative ambiguity aversion of 0.53, which implies relative prudence in ambiguity preferences of 1.53 when φ is isoelastic. Little is known about the intensity of relative ambiguity aversion and relative prudence in ambiguity preferences in the field, and further research is much needed. 16 In consumption-saving models under ambiguity, the link between φ    > 0 and precautionary saving breaks down, see Berger (2014) and Peter (2019). So whereas φ    > 0 may or may not induce precautionary savings in those types of models, it always generates an incentive to adjust behavior such as to increase expected utility in our model. Similarly though, the direction of this adjustment depends on the second-order belief, making a further restriction on preferences necessary.

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So the curly bracket in (19) is negative as soon as the consumer’s relative prudence in ambiguity preferences     is bounded by  2. Together with the previous observation, the overall sign of (19) is positive so that E  κφ [U (α ∗ ,  κ )] > E εφ  [U (α ∗ , ε)] . From Proposition 3 we know, however, that the increase in the numerator must outweigh the increase in the denominator for greater ambiguity to induce a lower level of coverage. In such a case, the consumer’s behavior with greater ambiguity can be rationalized as that of an expected utility maximizer whose distorted belief is that nonperformance is more likely compared to the case with less ambiguity. Finally, we vary initial wealth. In the coinsurance problem, risk aversion effects translate into wealth effects. If absolute risk aversion is decreasing (constant, increasing) in wealth, the optimal level of coverage decreases (stays constant, increases) when the consumer’s level of initial wealth increases, see Mossin (1968) and Schlesinger (1981). In the face of nonperformance risk, these comparative statics results are not recouped (see DS). For example, when absolute risk aversion decreases with wealth, insurance may or may not be an inferior good. This indeterminacy will only be exacerbated if we introduce ambiguity. We can still investigate whether ambiguity reinforces or attenuates the wealth effect. According to the implicit function rule we differentiate the consumer’s first-order condition (3) with respect to wealth:

E



   φ  [U (α ∗ , ε)]UW (α ∗ , ε)Uα (α ∗ , ε) + E φ  [U (α ∗ , ε)]UαW (α ∗ , ε) .

(21)

The second term, which is governed by the sign of Uα W , measures the effect of a change in wealth on the consumer’s expected utility trade-off. This effect is known to be indeterminate from DS. The first term measures how a change in initial wealth affects the consumer’s behavioral response to the presence of ambiguity and, more specifically, how an increase in wealth induces him to adjust his level of coverage to mitigate ambiguity about contract nonperformance. We focus on this effect to answer the question whether ambiguity about contract nonperformance has a positive or negative effect on the comparative statics of wealth. To formulate our last result, we state the following definition. 

U] Definition 4. The consumer’s absolute ambiguity aversion is given by A[U] = − φφ  [[U] .

As shown by Klibanoff et al. (2005), this measure corresponds to Definition 1 because the index of absolute ambiguity aversion of a more ambiguity-averse consumer is uniformly higher than that of a less ambiguity-averse consumer. Here is our last result. Proposition 4. Consider a consumer with non-increasing absolute ambiguity aversion and relative ambiguity aversion bounded by unity. Then, ambiguity about contract nonperformance reinforces the wealth effect. The last proposition shows that insurance is less likely to be an inferior good in the presence of ambiguity about contract nonperformance as long as the stated qualifications are met.17 The underlying intuition is simple: If the consumer’s initial wealth increases, his expected utility is higher for any level of insurance coverage. With non-increasing absolute ambiguity aversion the consumer is less ambiguity-averse when expected utility is high than when it is low, and Proposition 2 informs us that it is optimal for the consumer to increase his level of coverage. 5. Conclusion In this paper, we analyze optimal insurance demand if consumers face the possibility of contract nonperformance. Insurance offers protection against risks, but sources of nonperformance abound including insurer default, contested claims, procedural delays, contractual uncertainty and probationary periods. Ambiguity about contract nonperformance can arise from lack of trust in the insurance mechanism, limited financial literacy and sophistication, as well as the long-term nature of some insurance contracts. Existing research primarily focuses on ambiguity-neutral decision-makers and finds that the comparative statics of nonperformance risk are by and large indeterminate (see DS). We, in contrast, study the case of ambiguity aversion and find a variety of clear comparative statics predictions. Using smooth ambiguity preferences, we show that ambiguity aversion lowers the optimal demand for insurance. Furthermore, greater ambiguity aversion lowers insurance demand and so does a more pessimistic belief and the perception of greater ambiguity if the consumer’s relative ambiguity aversion is bounded by unity and his relative prudence in ambiguity preferences is bounded by 2, respectively. We also analyze a partial wealth effect and determine sufficient conditions under which insurance is less likely to be an inferior good. Our main results also hold for alpha-maxmin preference (see Appendix B.1) and a probability weighting model of ambiguity aversion (see Appendix B.2). Depending on the specification of preferences, they continue to hold for ambiguity-averse insurers (see Appendix B.3), and we also offer some thoughts on situations where the loss event is ambiguous, too (see Appendix B.4). The reason why the presence of ambiguity about contract nonperformance admits clear comparative statics for ambiguity-averse consumers whereas nonperformance risk does not, is related to the mechanism through which both operate. In DS, nonperformance risk is reflected in a lower probability of the contract performing as intended, but it also reduces the insurance premium. The latter is a wealth effect which undermines the comparative statics and renders most results indeterminate. In our paper, the focus is on the consumer’s perception of contract nonperformance so we take the insurer’s 17 The estimates in Berger and Bosetti (2016) are consistent with decreasing absolute ambiguity aversion. They also suggest relative ambiguity aversion to be bounded by unity, see Footnote 15 .

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pricing rule as given. As a result our model does not contain the conflicting wealth effect from the original DS model. This assumption appears natural in our context since most insurers may not know their customers’ sensitivity to ambiguity or their perception of ambiguity about contract nonperformance. Our results show that ambiguity aversion has clear demand effects when there is ambiguity about contract nonperformance. These demand effects are not obtained when focusing exclusively on nonperformance risk and risk aversion. The fact that ambiguity aversion lowers optimal demand allows us to identify ambiguity about contract nonperformance as a potential cause of uninsurability. Some have argued that this might be particularly relevant in less developed insurance markets where trust in financial institutions is limited and insurers are less stringently regulated (e.g., Bryan, 2018). The findings in this paper suggest that informational policies that eliminate ambiguity associated with contract nonperformance increase the ex-ante welfare of ambiguity-averse consumers and stimulate demand on the market. For the more likely case of a partial reduction in ambiguity, ex-ante welfare still increases whereas demand might rise or fall, depending on the consumer’s relative prudence in ambiguity preferences. This result is particularly important because it shows that demand reactions are only partially informative about underlying welfare changes, see also Harrison and Ng (2018) on this point. Furthermore, if one believes that ambiguity aversion is a psychological tendency that inhibits “good” decision-making and that expected utility represents the appropriate normative framework, our results show that ambiguity-averse consumers purchase less insurance than they should in the presence of ambiguity about contract nonperformance. In such a situation policy interventions that reduce the consumer’s sensitivity to ambiguity enhance welfare as do policy interventions that eliminate ambiguity altogether. This could be accomplished by the provision of information to consumers or by solvency regulations on the supply side of insurance to guarantee sufficient capital provisioning. There are several avenues for further research. Our model has clear predictions and experimental and empirical followups appear promising. Biener et al. (2017) conducted the first artefactual field experiment with subjects in the Philippines and find that ambiguity about contract nonperformance reduces insurance take-up for ambiguity-averse individuals. This confirms our Proposition 1. It would be interesting to check whether (i) Proposition 1 continues to hold in the field when going beyond take-up and (ii) whether our additional predictions developed in Propositions 2–4 also hold behaviorally. Empirically, it is not trivial to develop convincing proxies for the amount of ambiguity about contract nonperformance experienced by consumers, but contract length or the degree of claims verifiability based on data availability might be interesting starting points.18 On the theory side, one might wonder whether similar demand effects arise for other risk management instruments, like self-insurance or self-protection, when there is ambiguity about their reliability. Whereas self-insurance and (market) insurance behave similarly in most contexts, suggesting the generalizability of our results, the contrary is true for self-protection (see Ehrlich and Becker, 1972), leaving this a topic for future research. We formalized the consumer’s loss exposure as a binary distribution so that insurance demand is collapsed into one single variable, the level of coverage. Researchers have used less restrictive assumptions on the loss distribution to investigate optimal insurance design (see Gollier, 2013, for a survey), and it would be interesting to explore how the possibility of contract nonperformance and ambiguity about its occurrence affect the optimality of deductible insurance. We assume ambiguity as a primitive of the consumer’s choice problem. An important source of nonperformance risk is correlation between the risks of different insurees. In such a situation, contract nonperformance and its perception by consumers arise endogenously from insurance demand, which might generate further implications. Lastly, there are some applications of nonperformance risk in environments with asymmetric information. Agarwal and Ligon (1998) find that high risks can benefit from nonperformance risk when it makes pooling relatively more attractive to low risks. Stephens and Thompson (2015) derive a separating equilibrium in which policyholders reveal their type through their choice of insurer when the insured risk and nonperformance risk are correlated. None of the two studies considers ambiguity, which might yield additional interesting results. Acknowledgments The authors thank Aurélien Baillon, Han Bleichrodt, Jennifer Coats, Georges Dionne, Artem Durnev, Louis Eeckhoudt, Christian Gollier, Florian Kerzenmacher, Wei Li, Wanda Mimra, Alexander Mürmann, Yusuke Osaki, Casey Rothschild and Arthur Snow for helpful comments on an earlier version of this article. We also thank seminar participants of the Department of Finance at the University of Iowa, seminar participants of the Retirement and Savings Institute at HEC Montréal, participants of the CEAR/MRIC Behavioral Insurance Workshop 2016, the 2017 ARIA meeting in Toronto and participants of the 2018 ARIA ASSA session in Philadelphia for valuable comments. Three careful referees provided excellent comments, which helped us improve the exposition and presentation of the results. Any remaining errors are our own. Appendix A. Mathematical proofs A1. Proof of Proposition 1 We insert the optimal level of coverage of an ambiguity-neutral consumer into the first-order condition of a comparable ambiguity-averse consumer and determine the sign. Comparable means that both consumers share the same vNM utility 18

We would like to thank Alexander Mürmann for these suggestions.

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11

function over final wealth and the same second-order belief and only differ in ambiguity attitude. We obtain that

V  (α 0 ) = E

           ) Uα (α 0 ,   ) = Cov φ  U (α 0 , ε) , Uα (α 0 , ε) + Eφ  U (α 0 , ε) EUα (α 0 , ε) φ U (α 0 , 

(22)

from the covariance rule. Expanding the effect of insurance coverage on expected utility yields

) = −(1 − p)P  (α 0 ) · u (W10 ) + p(q − ε )P  (α 0 ) · u (W30 ) )(L − P  (α 0 )) · u (W20 ) − p(1 − q + ε Uα (α 0 , ε = Uα (α

0





(23)

p P  (α 0 )(u (W20 ) − u (W30 )) − Lu (W20 ) , , 0) + ε

where superscript 0 indicates final wealth levels when the level of coverage is given by α 0 . Now Uα (α 0 , 0 ) = 0 by definition  = 0 by definition of ε . As a result, we obtain that EUα (α 0 , ε ) = 0 so that of α 0 and Eε

V  (α 0 ) = Cov

    φ U (α 0 , ε) , Uα (α 0 , ε) .

(24)

To sign the covariance we investigate how φ  [U(α 0 ,

(α 0 ,

ε)] and Uα ε) vary in ε. The consumer’s expected utility strictly decreases in ε because the higher ε , the higher the probability that the contract will not perform. Due to the concavity of φ , the term φ  [U(α 0 , ε)] is then strictly increasing in ε,

      ∂φ  U (α 0 , ε ) = φ  U (α 0 , ε ) Uε (α 0 , ε ) = −φ  U (α 0 , ε ) p(u(W20 ) − u(W30 )) > 0. ∂ε

Per direct computation we obtain that





Uαε (α 0 , ε ) = p P  (α 0 )(u (W20 ) − u (W30 )) − Lu (W20 ) .

(25)

(26)

The square bracket is negative because marginal utility is positive and diminishing. Therefore, Uα (α 0 , ε ) is decreasing in ε . This makes the covariance in Equation (24) negative because its arguments are countermonotonic. Hence, V (α 0 ) < 0, and α ∗ < α 0 follows from the concavity of the objective function. A2. Proof of Proposition 2 Following Klibanoff et al. (2005), we model an increase in the consumer’s degree of ambiguity aversion by replacing his ambiguity function with ψ , where ψ is an increasing and concave transformation of φ , ψ = k(φ ) with k > 0 and k  < 0. Let T denote the objective function of a more ambiguity-averse consumer; his optimal level of coverage α ∗∗ is then characterized by the corresponding first-order condition,

T  (α ∗∗ ) = E



   ψ  [U (α ∗∗ , ε)]Uα (α ∗∗ , ε) = E k (φ [U (α ∗∗ , ε)])φ  [U (α ∗∗ , ε)]Uα (α ∗∗ , ε) = 0.

(27)

The second-order condition is satisfied due to the concavity of ψ and u. To compare the optimal level of coverage of the less ambiguity-averse consumer α ∗ and that of the more ambiguity-averse consumer α ∗∗ , we insert the former into the first-order condition for the latter and determine the sign. Notice that k (φ [U(α , ε )]) is increasing in ε ,

∂ k (φ [U (α , ε )] ) = k (φ [U (α , ε )] )φ  [U (α , ε )]Uε (α , ε ) = −k (φ [U (α , ε )] )φ  [U (α , ε )] p(u(W2 ) − u(W3 )) > 0. ∂ε (28) As a result, it holds for any ε ∈ (ε , ε ) that

k (φ [U (α , ε )] ) < k (φ [U (α , ε )] ) < k (φ [U (α ,  )] ).

(29)

The optimal level of coverage of the less ambiguity-averse consumer

V  (α ∗ ) = E





φ  [U (α ∗ , ε)]Uα (α ∗ , ε) =

 ε ε

α∗

is obtained from his first-order condition,

φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) = 0.

(30)

φ  [U(α ∗ , ε)] is strictly positive for any ε ∈ [ε, ε] and Uα (α ∗ , ε) is strictly decreasing in ε by the proof of Proposition 1. For the integral to be zero, it follows that Uα (α ∗ , ε ) must change sign on [ε , ε ]; due to strict monotonicity, this can only happen once. If εˆ ∈ (ε , ε ) denotes the unique null of Uα (α ∗ , ε ), we obtain  εˆ ε

φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) > 0 and

 ε εˆ

φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) < 0.

(31)

Combining this with (29) yields

T  (α ∗ ) = =

 ε ε

 εˆ ε

k (φ [U (α ∗ , ε )] )φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) k (φ [U (α ∗ , ε )] )φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) +

 ε εˆ

k (φ [U (α ∗ , ε )] )φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF ( )

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< k =k



 

φ [U (α ∗ , εˆ )] φ [U (α ∗ , εˆ )]

 

εˆ ε

  ε  φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε ) + k φ [U (α ∗ , εˆ )] φ [U (α ∗ , ε )]Uα (α ∗ , ε )dF ( ) εˆ

ε

φ  [U (α ∗ , ε )]Uα (α ∗ , ε )dF (ε )

ε    ∗  ∗ = k φ [U (α , εˆ )] V (α ) = 0.

(32)

So T (α ∗ ) is strictly negative, and it is optimal for the more ambiguity-averse consumer to decrease his level of insurance below α ∗ , that is α ∗∗ < α ∗ . A3. Proof of Proposition 3 The consumer’s conditional first-order expression at the optimal level of coverage is:

g(ε ) = φ  [U (α ∗ , ε )]Uα (α ∗ , ε ).

(33)

) = 0. As shown by Rothschild and Stiglitz (1970) and more generally by We recoup the first-order condition as Eg(ε  affect Ekern (1980), the sign of subsequent derivatives of g(ε ) informs about how risk changes in the distribution of ε this expectation. The first and second derivative of g(ε ) are:

g (ε ) =

φ  [U (α ∗ , ε )]Uε (α ∗ , ε )Uα (α ∗ , ε ) + φ  [U (α ∗ , ε )]Uαε (α ∗ , ε ), g (ε ) = φ  [U (α ∗ , ε )]Uε (α ∗ , ε )2Uα (α ∗ , ε ) + 2φ  [U (α ∗ , ε )]Uε (α ∗ , ε )Uαε (α ∗ , ε ),

(34)

because Uεε (α ∗ , ε ) = Uαεε (α ∗ , ε ) = 0. We also define the following function:

Uε (α ∗ , ε )Uα (α ∗ , ε ) . U (α ∗ , ε )Uαε (α ∗ , ε )

h (ε ) =

(35)

It is positive for ε < εˆ, zero for ε = εˆ and negative for ε > εˆ.19 The derivative of h(ε ) is



h ( ε ) =

Uε (α ∗ , ε )Uαε (α ∗ , ε ) · U (α ∗ , ε )Uαε (α ∗ , ε ) − Uε (α ∗ , ε )Uα (α ∗ , ε ) U (α ∗ , ε ) Uαε (α ∗ , ε ) 2

2



.

(37)

Function h(ε ) allows us to rewrite the first derivative of g(ε ) as follows,



 φ  [U (α ∗ , ε )] Uε (α ∗ , ε )Uα (α ∗ , ε ) · − 1 , φ  [U (α ∗ , ε )] U (α ∗ , ε )Uαε (α ∗ , ε )   = −φ  [U (α ∗ , ε )] Uαε (α ∗ , ε ) R[U (α ∗ , ε )] ·h(ε ) − 1 ,

  

g (ε ) = −φ  [U (α ∗ , ε )]Uαε (α ∗ , ε ) −U (α ∗ , ε )

<0

<0

≤1

and similarly for the second derivative of g(ε ),



φ  [U (α ∗ , ε )] Uε (α ∗ , ε )Uα (α ∗ , ε ) g (ε ) = −φ  [U (α ∗ , ε )]Uε (α ∗ , ε )Uαε (α ∗ , ε ) −U (α ∗ , ε )  · −2 φ [U (α ∗ , ε )] U (α ∗ , ε )Uαε (α ∗ , ε ) 





<0

>0

 <0







= −φ  [U (α ∗ , ε )] Uε (α ∗ , ε ) Uαε (α ∗ , ε ) P[U (α ∗ , ε )] ·h(ε ) − 2 .



(38)

 ≤2



(39)

The signs of g (ε ) and g  (ε ) are determined by the signs of the curly brackets in Eqs. (38) and (39). If h(ε ) is bounded by unity, the restrictions on relative ambiguity aversion and relative prudence in ambiguity preferences stated in Proposition 3 ensure that both g (ε ) and g  (ε ) are non-positive. To show this, assume that we could find ε  with h(ε  ) > 1. This ε  would have to be below εˆ because h(ε ) ≤ 0 for ε ≥ εˆ. Now h(ε ) is continuous, and for it to connect continuously from h(ε  ) > 1 to h(εˆ ) = 0, there needs to be a ε  ∈ (ε  , εˆ ) with h(ε  ) = 1, h(ε ) ≥ 1 for ε ∈ [ε  , ε   ], and h(ε ) > 1 for some ε ∈ [ε  , ε  ] with positive Lebesgue measure. On [ε  , ε   ], the square bracket in (37) is non-negative and strictly positive for some ε in the interval so that h (ε ) is also non-negative on [ε  , ε   ] and strictly positive for some ε in the interval. The fundamental theorem of calculus then yields

h(ε  ) − h(ε  ) =

19

 ε ε

h  ( ε )d ε > 0 ,

(40)

We can interpret h(ε ) if we rewrite it as



h (ε ) =

−

Uε (α ∗ , ε ) U (α ∗ , ε )

  Uαε (α ∗ , ε ) − for ε = εˆ. Uα (α ∗ , ε )

(36)

It compares the decay rate of expected utility with respect to the nonperformance probability with the decay rate of the first-order expression with respect to the nonperformance probability.

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which is a contradiction because h(ε  ) = 1 and h(ε  ) > 1. Therefore, h(ε ) ≤ 1 for all ε . But then the restrictions imposed on relative ambiguity aversion and relative prudence in ambiguity preferences imply g (ε ) ≤ 0 and g  (ε ) ≤ 0 for all ε , which completes the proof. A4. Proof of Proposition 4 We conclude that

E



 φ  [U (α ∗ , ε)]UW (α ∗ , ε)Uα (α ∗ , ε)    φ [U (α ∗ , ε)] ∗   [U (α ∗ , ε ∗   = −E −  U ( α , ε ) φ ) ] ) U ( α , ε ) α φ [U (α ∗ , ε)] W   )]UW (α ∗ , ε ), φ  [U (α ∗ , ε )]Uα (α ∗ , ε ) , = −Cov A[U (α ∗ , ε

(41)

where the second equality follows from the consumer’s first-order condition (3). Under non-increasing absolute ambiguity aversion, the first term in the covariance is an increasing function of ε . To see this, note that

UW (α ∗ , ε ) = (1 − p)u (W1∗ ) + p(q − ε )u (W2∗ ) + p(1 − q + ε )u (W3∗ ) > 0, and that



(42)



UW ε (α ∗ , ε ) = p u (W3∗ ) − u (W2∗ ) > 0,

(43)

where the asterisk indicates final wealth levels for α =

α∗.

Consequently, we obtain that

d (A[U (α ∗ , ε )]UW (α ∗ , ε ) ) = A [U (α ∗ , ε )] Uε (α ∗ , ε ) UW (α ∗ , ε ) + A[U (α ∗ , ε )] UW ε (α ∗ , ε ) > 0.      dε ≤0

<0

>0

>0

(44)

>0

Whether the second term in the covariance is increasing or decreasing in ε depends on the sign of g (ε ), which was introduced in the proof of Proposition 3 and shown to be non-positive if relative ambiguity aversion is bounded by unity. Consequently, the covariance is negative and the overall sign of the partial wealth effect is positive. Appendix B. Robustness and Extensions B1. Alpha-maxmin expected utility Smooth ambiguity preferences are popular because they disentangle ambiguity and ambiguity attitude. Many alternative models of decision-making under ambiguity exist in the literature, and we will examine the robustness of our findings in two other frameworks. We start with Ghirardato et al.’s (2004) model, in which the consumer’s objective function is a weighted average of the worst case and the best case. If all weight is attached to the worst case, maxmin expected utility emerges as a special case (Gilboa and Schmeidler, 1989).20 We use β ∈ [0, 1] to denote the consumer’s degree of pessimism to avoid confusion with the level of insurance coverage. The consumer’s objective function is then

V (α ) = β min U (α , ε ) + (1 − β ) max U (α , ε ) = β U (α , ε ) + (1 − β )U (α , ε ) = U (α , θ ). ε ∈[ε ,ε ]

ε ∈[ε ,ε ]

(45)

θ = βε + (1 − β )ε combines the effects of ambiguity and ambiguity aversion. V is concave, and for ambiguity to have no effect on the consumer, it must be that θ = 0 (ambiguity neutrality). Then, the optimal level of insurance coverage is α 0 as in Equation (9). Ambiguity aversion obtains for θ > 0 and renders decisions that are behaviorally equivalent to those made by an expected utility maximizer who perceives the probability of nonperformance as elevated. Our previous results imply that α ∗ < α 0 in this case, confirming Proposition 1. Furthermore, an increase in β increases θ and lowers insurance demand, confirming Proposition 2. The alpha-maxmin model does not allow for clear predictions when it comes to changes in the consumer’s perception of ambiguity. The reason is that a change in risk can leave the support of the distribution unchanged, expand it on one end, or expand it on both ends. Adapting the terminology developed by Meyer and Ormiston (1985), a change in risk can be weak, “semi-strong”, or strong depending on how it affects the support of the underlying distribution. For a given β , the effect of such stochastic changes on insurance demand is nil in the first case, positive (negative) in the second case if it is the lower (upper) end of the support which is expanded, or indeterminate in the third case.21 Wealth effects are similar to those derived in DS, but under a distorted probability distribution. So our main results carry over to alpha-maxmin expected utility while some comparative statics require different qualifications. 20 Maxmin utility is also a special case of Klibanoff et al.’s (2005) model for an exponential ambiguity function as the degree of ambiguity aversion tends to infinity. The general alpha-maxmin model is, however, not implied by our previous analysis, which warrants its separate investigation. 21 The interpretation of β is not independent of the support of the consumer’s belief. See Huang and Tzeng (2017) for a recent application to portfolio choice.

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B2. A probability weighting model Snow and Warren (2005) and Snow (2011) use a probability weighting model to analyze decision-making under ambiguity. Probability weighting models include rank dependent expected utility (Quiggin, 1982), the decision weighting model of Kahn and Sarin (1988), and cumulative prospect theory (Tversky and Kahneman, 1992). Let ξ denote the consumer’s probability weighting function, which is an increasing function of cumulative probabilities with ξ (0 ) = 0 and ξ (1 ) = 1. We rule out overinsurance (i.e., α ≤ 1) so that W3 < W2 ≤ W1 with the last inequality strict for partial insurance (see Footnote 7 ). For a given ε , the consumer’s welfare is

ξ ( p(1 − q + ε ))u(W3 ) + [ξ ( p) − ξ ( p(1 − q + ε ))]u(W2 ) + [ξ (1 ) − ξ ( p)]u(W1 ),

(46)

so that his expected ex-ante welfare is given by the following utility objective:

) )[u(W3 ) − u(W2 )]. V (α ) = u(W1 ) + ξ ( p)[u(W2 ) − u(W1 )] + Eξ ( p(1 − q + ε

(47)

Ambiguity neutrality means that ambiguity does not affect the consumer’s objective function. For this to hold whatever the level of ambiguity, ξ must be linear. Now consider an ambiguity-averse consumer; for any level of ambiguity to reduce ) ) > ξ ( p(1 − q )).22 But if ξ is convex, ambiguity about the his objective function, ξ must be convex so that Eξ ( p(1 − q + ε probability of contract nonperformance reduces optimal insurance demand, confirming Proposition 1. Likewise, an increase in the consumer’s degree of ambiguity aversion due to a convex transformation of ξ lowers insurance demand, confirming Proposition 2. Changes in the consumer’s perception of ambiguity defined in Definition 3 always reduce the demand for insurance without any additional assumptions, and wealth effects correspond to those derived in DS under a distorted probability distribution. So all our results continue to hold under a probability weighting model, and some are even simplified. B3. Some remarks on the supply side In the main text we assume an increasing and non-concave premium schedule that is independent of the consumer’s perception of ambiguity. This includes the case of a risk- and ambiguity-neutral insurer on a perfectly competitive market. In such a situation, the premium is based on the actuarial value of the policy such that P (α ) = α pqmL + K with m ≥ 1 being a proportional loading factor and K ≥ 0 a fixed cost. Do we still obtain such a premium schedule if there is ambiguity about the nonperformance probability on behalf of the insurer and if the insurer is sensitive to ambiguity as well (see Cabantous, 2007; Cabantous et al., 2011)? We answer this question based on the three criteria of decision-making under ambiguity that were presented thus far. We model ambiguity about the nonperformance probability from the insurer’s perspective as (1 −  q ) = (1 − q ) +  τ where  τ is a zero-mean risk based on the insurer’s second-order belief. This belief may be different from the consumer’s. We assume the insurer maximizes the expected profit,

 = (

τ ) = P − α p( q −  τ )mL − K.

(48)

If the insurer is ambiguity-neutral and the market is perfectly competitive, the premium reduces to the expression at the end of the previous paragraph. To incorporate ambiguity aversion, let χ denote the insurer’s ambiguity function with χ  > 0 and χ   < 0. If the insurer has other assets in place, denoted by A, and the market is perfectly competitive, the premium is implicitly defined by

E χ ( A + P ∗ − α p( q −  τ )mL − K ) = χ (A ),

(49)

so that the insurer is just indifferent between offering and not offering the policy. In other words, the insurer’s participation constraint is binding under perfect competition. Then, the premium can be written as P ∗ = α pqmL + K + ρ , where ρ > 0 is a strictly positive ambiguity premium. The insurer’s ambiguity aversion raises the premium. To determine slope and curvature of the premium schedule, we apply the implicit function rule to (49). This yields



E dP ∗ τχ = pmL q − dα Eχ 



> 0,

(50)

where the argument of χ  is suppressed to simplify notation. The premium schedule is increasing in the level of coverage. The second derivative is

d2 P ∗ = dα 2

 pmL 2 Eχ 



−Eτ

2

χ  Eχ  + 2E τ χ  E τχ −

τχ Eχ  E Eχ 

2 

.

(51)

The first and third term in the curly bracket are positive. The middle term is non-negative if the insurer is not prudent in ambiguity preferences (χ    ≤ 0). Then, the premium schedule is convex in the level of coverage and all our results go 22 Any twice continuously differentiable weighting function that is increasing, convex and satisfies ξ (0 ) = 0 and ξ (1 ) = 1 necessarily implies global underweighting of probabilities, that is ξ (p) ≤ p for all p. This disqualifies the descriptively attractive case of inverse S-shaped weighting functions.

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through. Another way to analyze the curly bracket is to rewrite it as follows:





E τ χ  E τ χ  − E τ 2 χ  Eχ  +

 E τχ   Eχ E τ χ  − Eχ  E τχ .  Eχ

(52)

Similar techniques as in Appendices 3 and 4 of Peter et al. (2017) show that the sign of first round bracket is governed by the slope of relative ambiguity aversion whereas the sign of the second round bracket is governed by the slope of absolute ambiguity aversion.23 If absolute ambiguity aversion is non-increasing and relative ambiguity aversion is non-decreasing, the overall sign is negative resulting in a concave premium schedule. As such, a necessary condition for a non-concave premium schedule is that either absolute ambiguity aversion is increasing, in which case relative ambiguity aversion is too, or that relative ambiguity aversion is decreasing, in which case absolute ambiguity aversion is too. In the alpha-maxmin model, we let γ ∈ [0, 1] denote the insurer’s degree of pessimism and [τ , τ ] the support of  τ . The premium is then determined from

γ min (A + (τ ) ) + (1 − γ ) max (A + (τ ) ) = A + (η ) = A, τ ∈[τ ,τ ]

τ ∈[τ ,τ ]

(53)

with η = γ τ + (1 − γ )τ , so that P ∗ = α p(q − η )mL + K. From the insurer’s perspective, η < 0 represents ambiguity aversion because scenarios with a higher probability of having to pay the indemnity reduce expected profits. The premium is higher than the premium charged by an ambiguity-neutral insurer but it is still linearly increasing in the level of coverage so that all our results go through.24 Finally, we can apply the probability weighting model to the insurer’s pricing decision. For the insurer there are only two relevant states of the world, the one where it pays the indemnity and the one where it does not. The insurer’s profit is lower in the first one, which occurs with probability p(q −  τ ), than in the second one, which occurs with probability (1 + p) + p(1 − q +  τ ). If ζ denotes the insurer’s probability weighting function, the premium is determined by requiring that

E ζ ( p( q −  τ ) )(A + P − α mL − K ) + (1 − Eζ ( p(q −  τ ) ))(A + P − K ) = A + P − α mL · Eζ ( p(q −  τ ) ) − K = A,

(54)

so that = α mL · Eζ ( p(q −  τ ) ) + K. For ambiguity to reduce the insurer’s expected profit, ζ needs to be convex. Then, ambiguity raises the premium charged by an ambiguity-averse insurer. The premium is, however, still a linear function of coverage and our results from the main text continue to hold. The same qualification as in Footnote 24 applies. Ambiguity aversion on behalf of the insurer can result in concavity of the premium schedule under the smooth model but not under the other two models. Non-concavity of the premium schedule is a sufficient condition for the validity of the first-order approach but not necessary. Loosely speaking, as long as the premium schedule is not too concave, all our results are upheld. Under alpha-maxmin expected utility and the probability weighting model the insurer’s ambiguity aversion results in an increase in the proportional loading factor. As long as insurance is not a Giffen good, the general equilibrium effect of ambiguity about the nonperformance probability will thus be two-fold: The increase in the loading factor due to the insurer’s ambiguity aversion lowers insurance demand, and the consumer’s ambiguity aversion lowers insurance demand even further, see Proposition 1. P∗

B4. Ambiguity about the loss probability To focus on ambiguity about the nonperformance probability, we assumed throughout the paper that the probability of loss is unambiguously known by the consumer. We briefly discuss the case when both the probability of loss and the probability of contract nonperformance are ambiguous. It is instructive to consider the polar case of a known nonperformance probability and an ambiguous loss probability first. One can show that Alary et al.’s (2013) Corollary 1 generalizes to a situation with known nonperformance risk, meaning that ambiguity aversion and greater ambiguity aversion raise optimal insurance demand in the presence of ambiguity about the loss probability. Therefore, our Propositions 1 and 2 do not generalize to the case with both probabilities being ambiguous. Rather, there is a positive effect on insurance demand arising from the ambiguity about the loss probability and a negative effect on insurance demand arising from the ambiguity about the nonperformance probability. One can also find cases, in which both effects neutralize each other so that the optimal insurance demand of an ambiguity-averse consumer coincides with that of his ambiguity-neutral counterpart, despite the fact that only the welfare of the former is affected by the presence of ambiguity. This reinforces our point that demand reactions may not inform about welfare. References Agarwal, V., Ligon, J.A., 1998. Insurer contract nonperformance in a market with adverse selection. J. Risk Insur. 65 (1), 135–150. Alary, D., Gollier, C., Treich, N., 2013. The effect of ambiguity aversion on insurance and self-protection. Econ. J. 123 (573), 1188–1202.

τ χ  < 0 follows from ambiguity aversion. E Depending on their informational endowment, consumers might be able to back out the insurer’s perceived level of ambiguity from the observed market prices. Then, to be consistent the consumer’s perception of ambiguity should be such that it is compatible with the one implied by the insurer’s pricing. 23 24

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