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The Effect of Ambiguity Aversion on Insurance and Self‐protection

The Effect of Ambiguity Aversion on Insurance and Self‐protection Abstract In this article, we derive a set of simple conditions such that ambiguity aversion always raises the demand for self‐insurance and the insurance coverage, but decreases the demand for self‐protection. We also characterise the optimal insurance design under ambiguity aversion and exhibit a case in which the straight deductible contract is optimal as in the expected utility model. Almost all models used in insurance economics up to now have been based on subjective expected utility theory. Therefore, this literature has assumed that ambiguity over probabilities does not matter for decisions. This assumption is inconsistent with many experimental results (Camerer, 1995). The most famous observation illustrating violation 6 of the subjective expected utility theory is the Ellsberg (1961) paradox. The paradox can be explained by ambiguity aversion, which can be thought of as an aversion to any mean‐preserving spread in the space of probabilities. For example, ambiguity‐averse agents prefer the lottery that yields a specific gain with a probability 1/2 to another lottery, in which the probability of earning the same gain is uncertain, but with a subjective mean of 1/2 (e.g. if the agent believes that the probability is either 1/4 or 3/4 with equal probability); see for example, Halevy (2007) for recent experimental evidence. This psychological trait differs from risk aversion, which is an aversion to any mean‐preserving spread in the pay‐off of the lottery. Since the seminal work by Gilboa and Schmeidler (1989), decision theorists have proposed various decision models that exhibit a form of ambiguity aversion (for a literature survey, see Etner et al., 2012). In this article, we study the effect of ambiguity aversion on insurance and self‐protection decisions using the theory of ambiguity axiomatised by Klibanoff et al. (2005). This theory captures the idea that mean‐preserving spreads in probabilities reduce the welfare of ambiguity‐averse agents. Also, this theory permits us to separate the effect of ambiguity aversion from that of risk aversion. Therefore, our results permit us to examine whether the effect of ambiguity aversion on insurance and self‐protection differs from that of risk aversion. Several papers have studied the effect of risk aversion on insurance and self‐protection within the subjective expected utility model. A well‐known result is that risk aversion increases the demand for insurance, that is, it raises the coverage rate, and it reduces the straight deductible. Indeed, in the case of coinsurance, this result is a direct consequence of the well‐known result by Pratt that risk aversion decreases the demand for the risky asset. Similarly, it can be shown that risk aversion always increases the demand for self‐insurance (Ehrlich and Becker, 1972). In contrast, the effect of risk aversion on self‐protection is not clear, as first shown by Dionne and Eeckhoudt (1985).1 Intuitively, ambiguity aversion reinforces risk aversion. Under this intuitive view, ambiguity aversion should raise the demand for insurance and it should have an ambiguous effect on self‐protection. Following Gollier (2011), we show that ambiguity aversion is not equivalent to an increase in risk aversion. For example, Gollier shows that ambiguity aversion may raise the demand for an ambiguous risky asset in a simple one‐risky‐one‐risk‐free asset portfolio model. In general, the effect of ambiguity aversion on the optimal decision is complex but strongly connected to the comparative statics analysis of the effect of risk.2 In this article, we mostly consider specific ambiguous contexts where the ambiguity is concentrated on one state of nature. That is, we assume that conditional to the information that this state does not occur, the distribution of final wealth is unambiguous. This restrictive structure of ambiguity allows us to get simple results regarding the effect of ambiguity aversion. Our results are driven by the observation that the behavioural effect of ambiguity aversion is as if an expected utility maximiser would use a more pessimistic distribution of his beliefs when he acts. And it happens that the comparative statics of a change in subjective probability is simpler than that of risk aversion within the subjective expected utility model. We in turn derive interpretable conditions so that ambiguity aversion increases self‐insurance and the demand for insurance but decreases self‐protection. Finally, we examine the robustness of the celebrated Arrow (1971) result to the introduction of ambiguity aversion. Within the subjective expected utility model, Arrow showed that the optimal insurance contract has a straight deductible when the insurance tariff is based on the actuarial value of the contract. We show that under our specific structure of ambiguity the straight deductible contract remains optimal under ambiguity aversion. The only effect of ambiguity aversion is to reduce the level of the optimal deductible. The article is organised as follows. In Section 1., we introduce the model of ambiguity aversion with the basic full insurance problem. We then study the self‐insurance problem in Section 2. and the self‐protection problem in Section 3.. In Section 4., we examine the problem of the optimal insurance design. Section 5. concludes. 1. Full Insurance We consider an agent who faces an uncertain final wealth that can take values w1<w2<…<wn ⁠. The wealth distribution is represented by a vector [p1(θ),…,pn(θ)] that belongs to the simplex of Rn ⁠. It is ambiguous in the sense that it depends upon a parameter θ that can take values in Θ. The ambiguity takes the form of a probability distribution for θ. Let F: Θ→ [0,1] denote the cumulative distribution function describing this parametric uncertainty. Let Eθg(θ)=∫Θg(θ)dF(θ) denote the expectation operator with respect to the parametric uncertainty. Suppose that the true value of θ is known, and let U(θ) denote the expected utility reached for that specific θ. It is defined as follows: U(θ)=∑s=1nps(θ)u(ws),(1)u being the vNM utility function. We say that the agent is ambiguity neutral if he evaluates his welfare ex ante by the expected value EθU(θ) ⁠, that is, if he uses expected utility with the mean state probabilities p¯s=Eθps(θ) ⁠. This agent is indifferent to any mean‐preserving spread in state probabilities. In accordance with the resolution of the Ellsberg paradox, let us assume alternatively that the agent dislikes mean‐preserving spreads in state probabilities, that is, that he is ambiguity averse. Following the work by Klibanoff et al. (2005) (KMM hereafter), let us assume that the agent evaluates his welfare ex ante by the certainty equivalent of the random variable U(θ) computed with an increasing and concave valuation function ϕ. The concavity of ϕ expresses ambiguity aversion, that is, an aversion to mean‐preserving spreads in the random state probabilities ps(θ) ⁠. The ex ante welfare is equal to the certainty equivalent of U(θ) using the concave function ϕ: ϕ−1{Eθϕ[U(θ)]}. In this Section, we study the willingness to pay (hereafter WTP) P for risk elimination under ambiguity aversion, defined by: u(w¯−P)=ϕ−1{Eθϕ[U(θ)]}, where w¯ is equal to ∑s=1nps¯ws ⁠. The special case of a linear ϕ function corresponds to ambiguity neutrality. In that case, the WTP for the elimination of risk is denoted P0 ⁠, which is defined by u(w¯−P0)=EθU(θ) ⁠. Observe that under ϕ concave, namely under ambiguity‐aversion, we have ϕ[u(w¯−P)]=Eθϕ[U(θ)]≤ϕ[EθU(θ)]=ϕ[u(w¯−P0)]. Therefore, P0 is always less than P under ϕ concave. In other words, ambiguity aversion always raises the WTP for risk elimination. The intuition is that eliminating the risk also eliminates all the ambiguity associated with the risk. Therefore, this is no surprise that ambiguity averse agents are willing to pay an extra premium for risk elimination. 2. Self‐insurance Self‐insurance is a technique aimed at increasing wealth in a specific state i against a cost incurred in all states. We assume that the ambiguity is concentrated on that state i in the sense that the distribution of final wealth conditional on the information that ‘the state is not i’ is unambiguous. This means that for all θ ∈ Θ and all s ≠ i, ps(θ)=[1−pi(θ)]πs,(2) where πs is the probability of state s conditional on the state not being i, with Σs≠iπs=1 ⁠. This structure of ambiguity is without loss of generality when there are only two states of nature. It is restrictive when there are more than two states of nature. Without loss of generality, we assume in this article that pi is increasing in θ. 2.1. Willingness to Pay for Self‐insurance We first examine the willingness to pay for an infinitesimal insurance for state i. Let P(ε) denote the WTP for an increase in wealth in state i : Eθϕ{pi(θ)u[wi+ε−P(ε)]+[1−pi(θ)]∑s≠iπsu[ws−P(ε)]}=Eθϕ[U(θ)]. Fully differentiating this equation with respect to ε yields P’(0)=u’(wi)Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)]{pi(θ)u’(wi)+[1−pi(θ)]Eu’(w−i)},(3) where Eu’(w−i)=Σs≠iπsu’(ws) is the expected marginal utility conditional on the state not being i. We are interested in determining the effect of ambiguity aversion on the willingness to pay for insurance. Using (3) with ϕ’≡1 implies that the marginal WTP for insurance under ambiguity neutrality equals p¯iu’(wi)p¯iu’(wi)+(1−p¯i)Eu’(w−i).(4) Under ambiguity aversion, we can define a distorted probability of state i as follows: p̂i=Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)].(5) We can then rewrite (3) as P’(0)=p̂iu’(wi)p̂iu’(wi)+(1−p̂i)Eu’(w−i).(6) The effect of ambiguity aversion on the marginal WTP for insurance in favour of state i is equivalent to a change in the probability of state i from p¯i to p̂i ⁠. It is easy to check that an increase in the distorted probability p̂i always raises the marginal WTP. Thus, ambiguity aversion raises the marginal WTP to insure state i if and only if it raises the distorted probability of state i. Let us define ψ1 as the certainty equivalent of w−i ⁠: Eu(w−i)=u(ψ1). It is easy to check that p̂i is larger than p¯i if and only if wi is smaller than ψ1 ⁠. Observe that U(θ)=pi(θ)u(wi)+[1−pi(θ)]u(ψ1) is decreasing in θ in that case. Because ϕ’ is decreasing under ambiguity aversion, the covariance rule (Kimball, 1951) implies that p̂i=Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)]≥Eθpi(θ)Eθϕ’[U(θ)]Eθϕ’[U(θ)]=Eθpi(θ)=p¯i. Thus, ambiguity aversion has the effect to increase the distorted probability of the state to insure if the corresponding state wealth is smaller than the certainty equivalent wealth in other states and to reduce the distorted probability of the state to insure otherwise. This concludes the proof of the following Proposition: 1 Proposition Consider the decision to self‐insure state i in which ambiguity is concentrated. Ambiguity aversion raises the marginal WTP to self‐insure state i if wealth in state i is smaller than the certainty equivalent wealth ψ1 ⁠, and it reduces it otherwise. There is a simple intuition to this result. When state i has a wealth smaller than the certainty equivalent ψ1 ⁠, raising wi reduces the dispersion of U(θ). Therefore, it is positively valued by all ambiguity‐averse agents. The opposite result holds when wealth in state i is larger than ψ1 ⁠. Note that in the two‐state case where a loss L occurs in state i, we simply have wi=w−L<ψ1=w ⁠, so that ambiguity aversion raises the marginal WTP to self‐insure state i. 2.2. Optimal Self‐insurance We now examine the impact of ambiguity aversion on the optimal insurance in favour of state i. It is convenient to consider first a general model in which the expected utility U(e, θ) is a function of a decision e and of an unknown parameter θ. Under ambiguity neutrality, the agent selects the level e that maximises her unconditional expected utility EθU(e,θ) ⁠. It yields the following first‐order condition (hereafter subscripts with decision variables denote derivatives): EθUe(e*,θ)=0.(7) From now on, e* denotes the solution of this equation and is interpreted as the optimal prevention effort under ambiguity neutrality. Under ambiguity aversion, the objective of the decision‐maker is to maximise ϕ[W(e)]=Eθϕ[U(e,θ)] where W(e) could be interpreted as the certainty equivalent utility in the same way than the certainty equivalent wealth in the expected utility theory. Assuming that U is concave in e, the optimal level of effort is increased by ambiguity aversion if W’(e*) is positive. We have that ϕ’[W(e*)]W’(e*)=EθUe(e*,θ)ϕ’[U(e*,θ)].(8) As a simple application of the covariance rule, we obtain the following preliminary result. 1 Lemma In the decision problem maxeEθϕ[U(e,θ)] with Uee≤0 ⁠, ambiguity aversion increases (decreases) the optimal decision e for any distribution of θ if and only if U(e*,θ) and Ue(e*,θ) are anti‐co‐monotonic (co‐monotonic), where e* is defined by (7). Proof If U(e*,θ) and Ue(e*,θ) are anti‐co‐monotonic, we have that EθUe(e*,θ)ϕ’[U(e*,θ)]≥EθUe(e*,θ)Eθϕ’[U(e*,θ)]=0. The first inequality comes from the fact that ϕ’ is decreasing, which implies that U(e*,θ) and ϕ’[Ue(e*,θ)] are co‐monotonic. The second equality is due to (7). This shows that W’(e*)≥0 ⁠, so that the optimal prevention under ambiguity aversion is less than e* ⁠, the optimal effort under ambiguity neutrality. This proves sufficiency. For necessity, suppose that the optimal effort under ambiguity aversion is larger than e* for all distributions of θ. Because W is concave in e, this implies that W’(e*) is positive, or that EθUe(e*,θ)ϕ’[U(e*,θ)] is non‐negative. Because of (7), this means that Ue(e*,θ) and ϕ’[U(e*,θ)] must covary positively for all distributions of θ. Chateauneuf et al. (1994) have demonstrated that two random variables have a positive covariance under any probability measure on the set of states of nature if and only if they are co‐monotonic. Applying this result, we obtain that Ue(e*,θ) and ϕ’[U(e*,θ)] must be co‐monotonic, or that Ue(e*,θ) and U(e*,θ) must be anti‐co‐monotonic. A symmetric proof can be written for the bracketed part of the Lemma when U(e*,θ) and Ue(e*,θ) are co‐monotonic. The optimal effort is increased by ambiguity aversion if the marginal value of effort (expressed by Ue ⁠) is larger for less favourable priors, that is, for θ yielding a smaller expected utility U. Indeed, ambiguity‐averse agents put more weight on these unfavourable priors than ambiguity‐neutral agents. We apply this result to the case of the optimal insurance of state i. More specifically, suppose that one can undertake ex ante an effort e that raises wealth in state i to wi(e)−e, and that reduces wealth in all other state s ≠ i to ws−e ⁠. The problem can be written as follows: U(e,θ)=pi(θ)u[wi(e)−e]+[1−pi(θ)]∑s≠iπsu(ws−e). To make the problem interesting, we assume that wi’(e) is larger than unity. The condition Uee≤0 is satisfied under wi″(e)≤0 ⁠. If e* is the optimal demand for insurance of the ambiguity‐neutral agent as defined by (7), we can redefine the certainty equivalent wealth level ψ1 conditional on s ≠ i in this self‐insurance context as follows: ∑s≠iπsu(ws−e*)=u(ψ1−e*).(9) Suppose first that wi(e*) is smaller than ψ1 ⁠. It implies that U(e*,θ) is decreasing in θ. Observe also that Ue(e*,θ)=pi(θ)u’[wi(e*)−e*][wi’(e*)−1]+[pi(θ)−1]∑s≠iπsu’(ws−e*). The two terms in the right‐hand side of the above equation are increasing in θ, independent of the value of wi(e*) relative to ψ1 ⁠. Thus, Lemma 1 implies that the optimal self‐insurance effort is increased by ambiguity aversion when the wealth level in the insurable state is smaller than the certainty equivalent wealth level in the other states. Because U and Ue are co‐monotonic when wi(e*) is larger than ψ1, the opposite result holds in that case. This proves the following Proposition. 2 Proposition Consider the decision to self‐insure state i in which ambiguity is concentrated. Ambiguity aversion raises the optimal self‐insurance effort if wealth in state i is smaller than the certainty equivalent wealth ψ1 and it reduces it otherwise. This result generalises a result in Snow (2011), who considers the special case with only two states of nature. As mentioned above in such a case, ψ1 is equal to the initial wealth and the result holds true. Indeed, the intuition of our result is the same as for Proposition 1. When the insurable state has a low wealth level, self‐insurance reduces the dispersion of conditional expected utility levels. Ambiguity‐averse agents will therefore invest more in self‐insurance. A special case of this result is obtained in the case of the demand for insurance, where e is reinterpreted as the insurance premium. Suppose that there are two states of nature. In the no‐loss state, initial wealth is w, whereas it is only w − L in the loss state i. For each dollar of premium, the policyholder receives an indemnity k if and only if a loss occurs. If a premium e is paid ex ante, final wealth will be equal to wi(e)−e and ws−e ⁠, respectively, in the loss state and in the no‐loss state, with wi(e)=w−L+ke and ws=w ⁠. Assuming that the indemnity k can never exceed the loss, we have that wi(e*) is always smaller than the certainty equivalent wealth ψ1=w ⁠. This yields the following Corollary. 1 Corollary Consider the standard coinsurance problem with two states of nature. Ambiguity aversion always raises the insurance coverage rate. We can infer from Gollier (2011) that this result does not hold under a general structure of ambiguity as considered in Section 2.. 3. Self‐protection Self‐protection is a technique aimed at reducing the probability of a specific state i at a cost incurred in all states. As in the previous section, we assume that the ambiguity is concentrated on that state i in the sense of (2). 3.1. Willingness to Pay for Self‐protection We first examine the impact of ambiguity aversion on the willingness to pay for a marginal investment in self‐protection. Suppose that one can reduce the probability of state i by ε in all possible scenarios θ. The distribution of wealth conditional on s ≠ i is supposed to be unaffected by this investment. These two ceteris paribus assumptions mean that the self‐protection effort affects the risk but has no impact on the degree of ambiguity. The WTP for this action is denoted P(ε), which is defined as follows: Eθϕ{[pi(θ)−ε]u[wi−P(ε)]+[1−pi(θ)+ε]∑s≠iπsu[ws−P(ε)]}=Eθϕ[U(θ)]. Straightforward computations lead to P’(0)=[∑s≠iπsu(ws)−u(wi)]Eθϕ’[U(θ)]Eθ{pi(θ)u’(wi)+[1−pi(θ)]∑s≠iπsu’(ws)}ϕ’[U(θ)].(10) This can be rewritten as follows: P’(0)=u(ψ1)−u(wi)p̂iu’(wi)+(1−p̂i)u’(ψ2),(11) in which p̂i is the distorted probability defined by (5), and ψ1 and ψ2 are, respectively, the certainty equivalent and the precautionary equivalent wealth level conditional on s ≠ i: u(ψ1)=∑s≠iπsu(ws)andu’(ψ2)=∑s≠iπsu’(ws).(12) We hereafter assume that the wealth level in state i is smaller than the certainty equivalent ψ1, so that the marginal WTP for self‐protection is positive. We see from (11) that the marginal WTP is the ratio between the marginal utility benefit of self‐protection and the marginal utility cost of the corresponding effort. The marginal utility benefit is measured by the expected utility difference when s = i and when s ≠ i , whereas the utility cost of self‐protection is measured by the ambiguity‐distorted expected marginal utility p̂iu’(wi)+(1−p̂i)u’(ψ2) ⁠. Equation (11) corresponds to the marginal WTP of an ambiguity‐neutral agent with distorted beliefs p̂i ⁠. Thus, as in the previous section, the effect of ambiguity aversion is to transform probability p¯i=Eθpi(θ) into a distorted probability p̂i ⁠. We have shown in the previous Section that p̂i is larger than p¯i if and only if wi is smaller than ψ1 ⁠. On the other hand, it is immediate from (11) that the marginal WTP for self‐protection of state i is decreasing in p̂i if and only if wi is smaller than the precautionary equivalent wealth ψ2 ⁠. Let us assume that the utility function of the agent exhibits decreasing absolute risk aversion (DARA). From Kimball (1990), we know that this means that the precautionary equivalent ψ2 is smaller than the certainty equivalent ψ1 ⁠. Let us first consider a situation in which the wealth level wi in state i is smaller than ψ2≤ψ1 ⁠. In that case, p̂i is larger than p¯i and the marginal WTP is decreasing in p̂i ⁠. This implies that ambiguity aversion has a negative impact on the marginal WTP for self‐protection. The intuition is that ambiguity aversion induces the agent to put more weight on scenarios θ with a large probability pi(θ) ⁠, since U(θ) is decreasing with θ when wi<ψ1 ⁠. This raises the distorted probability p̂i above p¯i ⁠. However, this has the effect of raising the marginal utility cost p̂iu’(wi)+(1−p̂i)u’(ψ2) of the ex ante effort, since wi<ψ2 ⁠. The marginal utility benefit of self‐protection expressed by u(ψ1)−u(wi) being unaffected by ambiguity aversion, we can conclude that ambiguity aversion reduces the marginal WTP of self‐protecting states whose wealth level is below the precautionary equivalent wealth. The opposite result prevails when wi is in between ψ2 and ψ1 ⁠. This is because the increased distorted probability p̂i now implies a reduction in the utility cost of effort. These results are summarised in the following Proposition. 3 Proposition Consider the decision to self‐protect state i in which ambiguity is concentrated. Suppose that the agent is DARA, and that the wealth level wi in state i is less than the certainty equivalent wealth level ψ1 ⁠. Ambiguity aversion reduces the marginal WTP to self‐protect state i if wi is smaller than the precautionary equivalent wealth ψ2 ⁠, and it raises it if wi∈[ψ2,ψ1] ⁠. 3.2. Optimal Self‐protection We now consider a specific prevention model in which the effort e affects the probability pi(e,θ) of state i in which ambiguity is concentrated. We define expected utility conditional on θ as follows: U(e,θ)=pi(e,θ)u(wi−e)+[1−pi(e,θ)]∑s≠iπsu(ws−e).(13) We assume that the probability of state i is differentiable with respect to e, with pie(e,θ)≤0 ⁠. We also assume that the second‐order condition is satisfied.3 Let e* denote the optimal effort of self‐protection for the ambiguity‐neutral agent. Suppose without loss of generality that pi(e*,θ) is increasing in θ. Using Lemma 1, we need to determine whether U and Ue are co‐monotonic or anti‐co‐monotonic to determine the impact of the ambiguity aversion on the optimal effort. Redefining the certainty and the precautionary equivalents ψ1 and ψ2 as u(ψ1−e*)=∑s≠iπsu(ws−e*) and u’(ψ2−e*)=∑s≠iπsu’(ws−e*), we can rewrite the expected utility conditional on θ as U(e*,θ)=pi(e*,θ)u(wi−e*)+[1−pi(e*,θ)]u(ψ1−e*),(14) and its derivative with respect to e as Ue(e*,θ)=pie(e*,θ)[u(wi−e*)−u(ψ1−e*)]−{pi(e*,θ)u’(wi−e*)+[1−pi(e*,θ)]u’(ψ2−e*)}.(15) As earlier, suppose that wi is smaller than the certainty equivalent ψ1 ⁠. This means that we focus on a state satisfying the natural property that a reduction in its probability raises expected utility. In other words, from (14), wi≤ψ1 implies that the conditional expected utility is decreasing in θ. The sensitiveness of Ue to changes in θ is more difficult to evaluate at that degree of generality, as can be seen from (15). As stated earlier in this Section, let us now assume that the degree of ambiguity is not affected by the effort. In other words, let us assume at this stage that pie is independent of θ. This situation is illustrated by the dashed lines in Figure Figure Fig. 1.., with θ1<θ2 ⁠. Under this condition, the first term on the right‐hand side of (15) is independent of θ. What remains on the right‐hand side of the equation is decreasing in θ if wi is smaller than ψ2 ⁠. Using Lemma 1 yields the following Proposition. Fig. 1. Open in new tabDownload slide The Probability of State i in the Self‐protection Model Fig. 1. Open in new tabDownload slide The Probability of State i in the Self‐protection Model 4 Proposition Consider the decision to self‐protect state i in which ambiguity is concentrated. Suppose that the agent is DARA, and that the wealth level wi in state i is less than the certainty equivalent wealth level ψ1 ⁠. If effort does not affect the ambiguity of state i, that is, if pie(e*,θ) is independent of θ, ambiguity aversion reduces the optimal self‐protection effort if wi is smaller than the precautionary equivalent wealth ψ2 ⁠, and it raises it if wi is larger than ψ2 ⁠. The intuition of this result is the same as for Proposition 3. When the wealth level in the state whose probability is reduced by the effort is smaller than the precautionary equivalent wealth, the increase in the distorted probability p̂i that is induced by ambiguity aversion raises the marginal utility cost of effort, thereby reducing the optimal effort. Let us now relax the assumption that the effort level has no impact on the degree of ambiguity. Suppose alternatively that an increase in effort raises the degree of ambiguity. Because we assume that pi(e*,θ) is increasing in θ, this means that pie(e*,θ) is increasing in θ. An illustration of this is given by the two plain lines in Figure Figure Fig. 1... The dashed lines correspond to a situation in which the effort does not affect the degree of ambiguity of state i. The plain lines correspond to a situation in which more effort raises the degree of ambiguity of state i since the probabilities of state i become more dispersed when effort is increased. Intuitively, this should reinforce the negative impact of ambiguity aversion on effort. This can be checked by observing that the first term on the right‐hand side in (15) is decreasing in θ, as is the second term when wi≤ψ2 ⁠. 5 Proposition Under the conditions of Proposition 4, if effort raises the ambiguity of state i, that is, if pie(e*,θ) is increasing in θ, then ambiguity aversion reduces the optimal self‐protection effort if wi is smaller than the precautionary equivalent wealth ψ2 ⁠. This result is related to Snow (2011), who exhibits a case in which ambiguity aversion raises the optimal self‐protection effort.4 Using a two‐state model, Snow restricts the functional form describing the effect of self‐protection on the probability of accident. Using our notation, Snow assumes that, for all (e, θ), pi(e,θ)=pi(θ)ρ(e) where ρ(e) captures the effect of the self‐protective effort, with ρ(e) > 0 and ρ’(e)<0 ⁠. This is a special case in which the proportional effect of effort on the loss probability is the same for all θ. Under this condition, Snow shows that the self‐protection effort is increased by ambiguity aversion. Because pi(e,θ)=pi(θ)ρ(e) implies that pie is decreasing in θ, Snow’s result is not inconsistent with our Proposition 5 in which we instead assume that pie is increasing in θ.5 The practical implication of these results depends upon whether the effort has a positive or a negative effect on the ambiguity surrounding the probability of the loss state. In many situations, it appears more natural that self‐protection would reduce both risk and ambiguity, consistent with Snow’s assumption. However, a typical case where pie(e*,θ) should instead increase in θ is when ambiguity affects the effectiveness of self‐protection but not the baseline risk. In that case, more effort should naturally induce more ambiguity. An illustrative example is when a person with a well‐known disease accepts a treatment that has never been tested before. 4. Optimal Insurance Design In this Section, we explore the problem of the optimal insurance contract design. As first proved by Arrow (1971), any insurance contract is dominated by a contract with a straight deductible. As it is intuitive, the deductible contract optimises the risk transfer to the insurer for any insurance budget level, since it provides indemnities where the marginal utility of wealth is the largest. We are interested here in determining whether this result is robust to the introduction of ambiguity aversion. As a particular case of the specific structure of ambiguity considered in Sections 3. and 3, we limit the analysis to the case in which the ambiguity is concentrated in the probability of the no‐loss state. We have wi=w and ws=w−Ls where Ls is the loss in state s. Let p(θ) denote the ambiguous probability that a loss occurs. Conditional on the occurrence of a loss, the level of loss is a random variable L which is independent of θ. That is, the distribution of the severity of the loss is unambiguous. An insurance contract stipulates an indemnity I(Ls) for each possible loss level Ls ⁠. To any such indemnity schedule, there is an insurance premium which is proportional to the actuarial value of the policy: P=p¯(1+τ)EI(L), where as before p¯=Eθp(θ) is the expected probability of accident and τ is the loading factor. Conditional on θ, the policyholder’s expected utility is written as follows: U(θ)=[1−p(θ)]u(w−P)+p(θ)Eu[w−L+I(L)−P]. The ambiguity‐averse policyholder selects the indemnity schedule I(Ls) that maximises its ex ante welfare which is measured by Eθϕ[U(θ)] ⁠. By the concavity of u and ϕ, this maximisation is well‐behaved, and its first‐order conditions are necessary and sufficient. We first prove the following Proposition. 6 Proposition Suppose that the policyholder is ambiguity averse and that the ambiguity is concentrated on the probability of the no‐loss state. Under this specification, the optimal insurance contract contains a straight deductible d: I(Ls)=max(0,Ls−d) ⁠. Proof Suppose by contradiction that the contract I0 that maximises Eθϕ[U(θ)] is not a straight deductible contract. It yields an insurance premium P0=p¯(1+τ)EI0(L) ⁠. Then, let d0 denote the deductible which yields the same insurance premium than P0 ⁠: p¯(1+τ)E[max(0,L−d0)]=P0. However, we know from Arrow (1971) that this alternative contract with a straight deductible d0 dominates any other insurance schedule as I0 ⁠: Eu[w−min(L,d0)−P0]≥Eu[w−L+I0(L)−P0]. It implies that for all θ, we have [1−p(θ)]u(w−P0)+p(θ)Eu[w−min(L,d0)−P0]≥[1−p(θ)]u(w−P0)+p(θ)Eu[w−L+I0(L)−P0]. Because the expected utility conditional on θ is larger with the straight deductible d0 than with contract I0 for all θ, the former necessary yields a larger ex ante welfare Eθϕ[U(θ)] ⁠. This is a contradiction. We now examine the impact of ambiguity aversion on the optimal deductible. Corollary 1 answered this question in the special case of only one possible loss, since in that case a proportional coinsurance or a straight deductible is formally equivalent. With more than one possible loss level, the decision problem can thus be rewritten as follows: maxdEθϕ[U(d,θ)],(16) with U(d,θ)=[1−p(θ)]u[w−P(d)]+p(θ)Eu[w−min(L,d)−P(d)] and P(d)=p¯(1+τ)Emax(0,L−d) ⁠. We suppose that p¯(1+τ) is smaller than unity, so that reducing the deductible makes the policyholder wealthier in high loss states. Let d* denote the optimal deductible under ambiguity neutrality. By Lemma 1, and because U is concave in d,6 ambiguity aversion reduces the optimal deductible if U(d*,θ) and Ud(d*,θ) are co‐monotonic. Suppose that p is increasing in θ, so that U(d*,θ) is decreasing in θ. We have Ud(d*,θ)=−P’(d*){[1−p(θ)]u’(w−P*)+p(θ)Eu’[w−min(L,d*)−P*]}−p(θ)[1−F(d*)]u’(w−d*−P*),(17) where P*=P(d*) and F is the cumulative distribution function of L. This equality can be rewritten as follows: Ud(d*,θ)=−P’(d*)[1−p(θ)]u’(w−P*)+p(θ)[1−F(d*)]×{p¯(1+τ)Eu’[w−min(L,d*)−P*]−u’(w−d*−P*)}.(18) This is obviously decreasing in θ, since P’(d*)<0 ⁠, Eu’[w−min(L,d*)−P*]≤u’(w−d*−P*) and p¯(1+τ)≤1 ⁠. This proves the following Proposition. 7 Proposition Under the specification of Proposition 6, the optimal deductible is reduced by ambiguity aversion. The intuition for this result can be obtained from the observation that the effect of ambiguity aversion is here equivalent to an increase in the implicit loss probability defined by p̂=Eθϕ’[U(d*,θ)]p(θ)Eθϕ’[U(d*,θ)]. Because U(d*,θ) and p(θ) are negatively correlated and because ϕ’ is decreasing, we have that p̂ is larger than Eθp(θ)=p¯ ⁠, under ambiguity aversion. We assumed in this Section that the ambiguity is concentrated on the no‐loss state. We finally show that the optimal insurance schedule is in general not a straight deductible for other structures of ambiguity. This is done with a numerical counter‐example. Suppose that the agent has an initial wealth w = 3 and that there are three possible loss levels L = (0, 1, 2). The ambiguity is structured as follows. There are two equally likely probability distributions for L with support (0, 1, 2): the good distribution is p(θ = 1) = (1/2, 1/2, 0), whereas the bad distribution is p(θ = 2) = (1/4, 1/4, 1/2). Observe that, under our terminology, the ambiguity is here concentrated on the large loss L = 2. The loading factor of the insurance premium is τ = 0.2. The policyholder’s utility function is u(c) = −1/c, and we calibrate ambiguity aversion with ϕ[u]=−e−100u ⁠. Using Mathematica, we solve numerically the following problem: maxI1∈[0,1],I2∈[0,2],P12ϕ[12u(3−P)+12u(2+I1−P)]+12ϕ[14u(3−P)+14u(2+I1−P)+12u(1+I2−P)] subject to P=1.2(38I1+14I2). We obtain I1*=0.407 and I2*=1.630 ⁠. Because I2*−I1*>L2−L1 ⁠, we see that the optimal contract is not a straight deductible, that is, there is no d such that I*(Ls)=max(0,Ls−d) for all Ls ⁠. Final wealth is larger in state with L = 2 than in state with L = 1. This is done to compensate for the relatively large ambiguity associated with the large loss. We conclude that the result expressed in Proposition 6 is not robust to the relaxation of the condition that ambiguity is concentrated on the no‐loss state. 5. Conclusion In this article, we have shown that ambiguity aversion tends to raise the incentive to insure and to self‐insure but to decrease the incentive to self‐protect. The intuition for our results is that ambiguity aversion has an effect which is behaviourally equivalent to the effect of more pessimism under subjective expected utility theory. Since the effect of a change in probability in expected utility models is usually different from that of a change in risk aversion, this explains why the effect of ambiguity aversion differs from that of risk aversion. We have also exhibited a case where the optimal insurance contractual form contains a straight deductible as in the expected utility model. This article has thus generalised the analysis of some standard problems in insurance economics to ambiguous risks. Some technical difficulties remain however. In particular, it could be interesting to consider more general ambiguous probability distributions, ambiguity‐averse insurers and other forms of ambiguity‐sensitive preferences. At a more conceptual level, we finally observe that if the distinction between (self‐)insurance and self‐protection has classically been done in the insurance economics literature, the case for maintaining such a distinction may become more problematic under conditions of ambiguity. Footnotes 1 " An activity of self‐protection consists in investing to reduce the probability of an accident. In fact, no general result can be obtained to sign the effect of risk aversion on self‐protection (Sweeney and Beard, 1992). To sign this effect, it is necessary to specify the value of the probability of loss and to make further assumptions on the utility function (Dachraoui et al., 2004; Eeckhoudt and Gollier, 2005). 2 " Gollier (1995) summarises the main findings on this comparative statics analysis in the case of a portfolio choice problem, which is further examined in Abel (2002). Strong restrictions on the utility function are required to guarantee that ambiguity aversion reduces the optimal risk exposure. 3 " In this model, the condition Uee≤0 requires some specific restrictions on functions p ( Jullien et al., 1999). 4 " See Treich (2010) for a related result and discussion but in a state‐dependent model. See also Berger (2011) who examines a model of self‐protection and self‐insurance under ambiguity aversion but in a two‐period model. 5 " Here is a sketch of the proof of Snow’s (2011) result. Let us assume wi=w−L ⁠, ws=w and pi(e,θ)=pi(θ)ρ(e) ⁠. We need to show that Ue(e*,θ) increases in θ in this specific model. It can easily be shown that this holds if and only if f(e*)=ρ’(e*)[u(w−L−e*)−u(w−e*)]−ρ(e*)[u’(w−L−e*)−u’(w−e*)] is positive. Note that the first term of f(e*) is positive, while the second term is negative, illustrating that the effect is in general indeterminate when pie is decreasing in θ. Nevertheless, observe that the first‐order condition of this problem simply reduces to [Eθp(θ)]f(e*)=u’(w−e*) ⁠, which implies f(e*)>0 ⁠. 6 " See Schlesinger (1981). References Abel , A.B. ( 2002 ). ‘ An exploration of the effects of optimism and doubt on asset returns ’, Journal of Economic Dynamics and Control , vol. 26 ( 7–8 ), pp. 1075 – 92 . Google Scholar Crossref Search ADS WorldCat Arrow , K.J. ( 1971 ). Essays in the Theory of Risk Bearing , Chicago, IL : Markham Publishing Company . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Berger , L. ( 2011 ). ‘Does ambiguity aversion raise the optimal level of effort? A two period model’ , Working Paper, University of Brussels . Camerer , C. ( 1995 ). ‘Individual decision making’, in ( J.H. Kagel and A.E. Roth, eds.), Handbook of Experimental Economics , pp. 587 – 703 , Princeton, NJ : Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Chateauneuf , A. , Kast , R. and Lapied , A. ( 1994 ). ‘Market preferences revealed by prices: non‐linear pricing in slack markets’, in ( B. Munierand M.J. Machina, eds.), Models and Experiments in Risk and Rationality , pp. 289 – 306 , Dordrecht : Kluwer Academic Publishers . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Dachraoui , K. , Dionne , G., Eeckhoudt , L. and Godfroid , P. ( 2004 ). ‘ Comparative mixed risk aversion: definition and application to self‐protection and willingness to pay ’, Journal of Risk and Uncertainty , vol. 29 ( 3 ), pp. 261 – 76 . Google Scholar Crossref Search ADS WorldCat Dionne , G. and Eeckhoudt , L. ( 1985 ). ‘ Self‐insurance, self‐protection and increased risk aversion ’, Economics Letters , vol. 17 ( 1–2 ), pp. 39 – 42 . Google Scholar Crossref Search ADS WorldCat Eeckhoudt , L. and Gollier , C. ( 2005 ). ‘ The impact of prudence on optimal prevention ’, Economic Theory , vol. 26 ( 4 ), pp. 989 – 94 . Google Scholar Crossref Search ADS WorldCat Ehrlich , I. and Becker , G.S. ( 1972 ). ‘ Market insurance, self‐insurance and increased risk‐aversion ’, Journal of Political Economy , vol. 80 ( 4 ), pp. 623 – 48 . Google Scholar Crossref Search ADS WorldCat Ellsberg , D. ( 1961 ). ‘ Risk, ambiguity, and the Savage axioms ’, Quarterly Journal of Economics , vol. 75 ( 4 ), pp. 643 – 69 . Google Scholar Crossref Search ADS WorldCat Etner , J. , Jeleva , M. and Tallon , J.‐M. ( 2012 ). ‘ Decision theory under uncertainty ’, Journal of Economic Surveys , vol. 26 ( 2 ), pp. 234 – 70 . Google Scholar Crossref Search ADS WorldCat Gilboa , I. and Schmeidler , D. ( 1989 ). ‘ Maximin expected utility with non‐unique prior ’, Journal of Mathematical Economics , vol. 18 ( 2 ), pp. 141 – 53 . Google Scholar Crossref Search ADS WorldCat Gollier , C. ( 1995 ). ‘ The comparative statics of changes in risk revisited ’, Journal of Economic Theory , vol. 66 ( 2 ), pp. 522 – 36 . Google Scholar Crossref Search ADS WorldCat Gollier , C. ( 2011 ). ‘ Portfolio choices and asset prices: the comparative statics of ambiguity aversion ’, Review of Economic Studies , vol. 78 ( 4 ), pp. 1329 – 44 . Google Scholar Crossref Search ADS WorldCat Halevy , Y. ( 2007 ). ‘ Ellsberg revisited: an experimental study ’, Econometrica , vol. 75 ( 2 ), pp. 503 – 36 . Google Scholar Crossref Search ADS WorldCat Jullien , B. , Salanié , B. and Salanié , F. ( 1999 ). ‘ Should more risk averse agents exert more effort? ’, Geneva Papers on Risk and Insurance Theory , vol. 24 ( 1 ), pp. 19 – 28 . Google Scholar Crossref Search ADS WorldCat Kimball , A.W. ( 1951 ). ‘ On dependent tests of significance in the analysis of variance ’, The Annals of Mathematical Statistics , vol. 22 ( 4 ), pp. 600 – 2 . Google Scholar Crossref Search ADS WorldCat Kimball , M.S. ( 1990 ). ‘ Precautionary savings in the small and in the large ’, Econometrica , vol. 58 ( 1 ), pp. 53 – 73 . Google Scholar Crossref Search ADS WorldCat Klibanoff , P. , Marinacci , M. and Mukerji , S. ( 2005 ). ‘ A smooth model of decision making under ambiguity ’, Econometrica , vol. 73 ( 6 ), pp. 1849 – 92 . Google Scholar Crossref Search ADS WorldCat Schlesinger , H. ( 1981 ). ‘ The optimal level of deductibility in insurance contracts ’, Journal of Risk and Insurance , vol. 48 ( 3 ), pp. 465 – 81 . Google Scholar Crossref Search ADS WorldCat Snow , A. ( 2011 ). ‘ Ambiguity aversion and the propensities for self‐insurance and self‐protection ’, Journal of Risk and Uncertainty , vol. 42 ( 1 ), pp. 27 – 43 . Google Scholar Crossref Search ADS WorldCat Sweeney , G. and Beard , T.R., ( 1992 ). ‘ Self‐protection in the expected‐utility‐of‐wealth model: an impossibility theorem ’, Geneva Papers on Risk and Insurance Theory , vol. 17 ( 2 ), pp. 147 – 58 . Google Scholar Crossref Search ADS WorldCat Treich , N. ( 2010 ). ‘ The value of a statistical life under ambiguity aversion ’, Journal of Environmental Economics and Management , vol. 59 ( 1 ), pp. 15 – 26 . Google Scholar Crossref Search ADS WorldCat Author notes " We thank the editor and two referees for useful comments. We also thank seminar participants at the University of Technology Sydney, the University of Auckland, the NHH at Bergen and the World Risk and Insurance Economics Congress (25–29 July 2010 in Singapore). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) Grant Agreement no. 230589 and from the Chair ‘Risk markets and value creation’ financed by SCOR at IDEI. © 2013 The Author(s). The Economic Journal © 2013 Royal Economic Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Economic Journal Oxford University Press

The Effect of Ambiguity Aversion on Insurance and Self‐protection

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Oxford University Press
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© 2013 The Author(s). The Economic Journal © 2013 Royal Economic Society
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0013-0133
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1468-0297
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10.1111/ecoj.12035
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Abstract

Abstract In this article, we derive a set of simple conditions such that ambiguity aversion always raises the demand for self‐insurance and the insurance coverage, but decreases the demand for self‐protection. We also characterise the optimal insurance design under ambiguity aversion and exhibit a case in which the straight deductible contract is optimal as in the expected utility model. Almost all models used in insurance economics up to now have been based on subjective expected utility theory. Therefore, this literature has assumed that ambiguity over probabilities does not matter for decisions. This assumption is inconsistent with many experimental results (Camerer, 1995). The most famous observation illustrating violation 6 of the subjective expected utility theory is the Ellsberg (1961) paradox. The paradox can be explained by ambiguity aversion, which can be thought of as an aversion to any mean‐preserving spread in the space of probabilities. For example, ambiguity‐averse agents prefer the lottery that yields a specific gain with a probability 1/2 to another lottery, in which the probability of earning the same gain is uncertain, but with a subjective mean of 1/2 (e.g. if the agent believes that the probability is either 1/4 or 3/4 with equal probability); see for example, Halevy (2007) for recent experimental evidence. This psychological trait differs from risk aversion, which is an aversion to any mean‐preserving spread in the pay‐off of the lottery. Since the seminal work by Gilboa and Schmeidler (1989), decision theorists have proposed various decision models that exhibit a form of ambiguity aversion (for a literature survey, see Etner et al., 2012). In this article, we study the effect of ambiguity aversion on insurance and self‐protection decisions using the theory of ambiguity axiomatised by Klibanoff et al. (2005). This theory captures the idea that mean‐preserving spreads in probabilities reduce the welfare of ambiguity‐averse agents. Also, this theory permits us to separate the effect of ambiguity aversion from that of risk aversion. Therefore, our results permit us to examine whether the effect of ambiguity aversion on insurance and self‐protection differs from that of risk aversion. Several papers have studied the effect of risk aversion on insurance and self‐protection within the subjective expected utility model. A well‐known result is that risk aversion increases the demand for insurance, that is, it raises the coverage rate, and it reduces the straight deductible. Indeed, in the case of coinsurance, this result is a direct consequence of the well‐known result by Pratt that risk aversion decreases the demand for the risky asset. Similarly, it can be shown that risk aversion always increases the demand for self‐insurance (Ehrlich and Becker, 1972). In contrast, the effect of risk aversion on self‐protection is not clear, as first shown by Dionne and Eeckhoudt (1985).1 Intuitively, ambiguity aversion reinforces risk aversion. Under this intuitive view, ambiguity aversion should raise the demand for insurance and it should have an ambiguous effect on self‐protection. Following Gollier (2011), we show that ambiguity aversion is not equivalent to an increase in risk aversion. For example, Gollier shows that ambiguity aversion may raise the demand for an ambiguous risky asset in a simple one‐risky‐one‐risk‐free asset portfolio model. In general, the effect of ambiguity aversion on the optimal decision is complex but strongly connected to the comparative statics analysis of the effect of risk.2 In this article, we mostly consider specific ambiguous contexts where the ambiguity is concentrated on one state of nature. That is, we assume that conditional to the information that this state does not occur, the distribution of final wealth is unambiguous. This restrictive structure of ambiguity allows us to get simple results regarding the effect of ambiguity aversion. Our results are driven by the observation that the behavioural effect of ambiguity aversion is as if an expected utility maximiser would use a more pessimistic distribution of his beliefs when he acts. And it happens that the comparative statics of a change in subjective probability is simpler than that of risk aversion within the subjective expected utility model. We in turn derive interpretable conditions so that ambiguity aversion increases self‐insurance and the demand for insurance but decreases self‐protection. Finally, we examine the robustness of the celebrated Arrow (1971) result to the introduction of ambiguity aversion. Within the subjective expected utility model, Arrow showed that the optimal insurance contract has a straight deductible when the insurance tariff is based on the actuarial value of the contract. We show that under our specific structure of ambiguity the straight deductible contract remains optimal under ambiguity aversion. The only effect of ambiguity aversion is to reduce the level of the optimal deductible. The article is organised as follows. In Section 1., we introduce the model of ambiguity aversion with the basic full insurance problem. We then study the self‐insurance problem in Section 2. and the self‐protection problem in Section 3.. In Section 4., we examine the problem of the optimal insurance design. Section 5. concludes. 1. Full Insurance We consider an agent who faces an uncertain final wealth that can take values w1<w2<…<wn ⁠. The wealth distribution is represented by a vector [p1(θ),…,pn(θ)] that belongs to the simplex of Rn ⁠. It is ambiguous in the sense that it depends upon a parameter θ that can take values in Θ. The ambiguity takes the form of a probability distribution for θ. Let F: Θ→ [0,1] denote the cumulative distribution function describing this parametric uncertainty. Let Eθg(θ)=∫Θg(θ)dF(θ) denote the expectation operator with respect to the parametric uncertainty. Suppose that the true value of θ is known, and let U(θ) denote the expected utility reached for that specific θ. It is defined as follows: U(θ)=∑s=1nps(θ)u(ws),(1)u being the vNM utility function. We say that the agent is ambiguity neutral if he evaluates his welfare ex ante by the expected value EθU(θ) ⁠, that is, if he uses expected utility with the mean state probabilities p¯s=Eθps(θ) ⁠. This agent is indifferent to any mean‐preserving spread in state probabilities. In accordance with the resolution of the Ellsberg paradox, let us assume alternatively that the agent dislikes mean‐preserving spreads in state probabilities, that is, that he is ambiguity averse. Following the work by Klibanoff et al. (2005) (KMM hereafter), let us assume that the agent evaluates his welfare ex ante by the certainty equivalent of the random variable U(θ) computed with an increasing and concave valuation function ϕ. The concavity of ϕ expresses ambiguity aversion, that is, an aversion to mean‐preserving spreads in the random state probabilities ps(θ) ⁠. The ex ante welfare is equal to the certainty equivalent of U(θ) using the concave function ϕ: ϕ−1{Eθϕ[U(θ)]}. In this Section, we study the willingness to pay (hereafter WTP) P for risk elimination under ambiguity aversion, defined by: u(w¯−P)=ϕ−1{Eθϕ[U(θ)]}, where w¯ is equal to ∑s=1nps¯ws ⁠. The special case of a linear ϕ function corresponds to ambiguity neutrality. In that case, the WTP for the elimination of risk is denoted P0 ⁠, which is defined by u(w¯−P0)=EθU(θ) ⁠. Observe that under ϕ concave, namely under ambiguity‐aversion, we have ϕ[u(w¯−P)]=Eθϕ[U(θ)]≤ϕ[EθU(θ)]=ϕ[u(w¯−P0)]. Therefore, P0 is always less than P under ϕ concave. In other words, ambiguity aversion always raises the WTP for risk elimination. The intuition is that eliminating the risk also eliminates all the ambiguity associated with the risk. Therefore, this is no surprise that ambiguity averse agents are willing to pay an extra premium for risk elimination. 2. Self‐insurance Self‐insurance is a technique aimed at increasing wealth in a specific state i against a cost incurred in all states. We assume that the ambiguity is concentrated on that state i in the sense that the distribution of final wealth conditional on the information that ‘the state is not i’ is unambiguous. This means that for all θ ∈ Θ and all s ≠ i, ps(θ)=[1−pi(θ)]πs,(2) where πs is the probability of state s conditional on the state not being i, with Σs≠iπs=1 ⁠. This structure of ambiguity is without loss of generality when there are only two states of nature. It is restrictive when there are more than two states of nature. Without loss of generality, we assume in this article that pi is increasing in θ. 2.1. Willingness to Pay for Self‐insurance We first examine the willingness to pay for an infinitesimal insurance for state i. Let P(ε) denote the WTP for an increase in wealth in state i : Eθϕ{pi(θ)u[wi+ε−P(ε)]+[1−pi(θ)]∑s≠iπsu[ws−P(ε)]}=Eθϕ[U(θ)]. Fully differentiating this equation with respect to ε yields P’(0)=u’(wi)Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)]{pi(θ)u’(wi)+[1−pi(θ)]Eu’(w−i)},(3) where Eu’(w−i)=Σs≠iπsu’(ws) is the expected marginal utility conditional on the state not being i. We are interested in determining the effect of ambiguity aversion on the willingness to pay for insurance. Using (3) with ϕ’≡1 implies that the marginal WTP for insurance under ambiguity neutrality equals p¯iu’(wi)p¯iu’(wi)+(1−p¯i)Eu’(w−i).(4) Under ambiguity aversion, we can define a distorted probability of state i as follows: p̂i=Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)].(5) We can then rewrite (3) as P’(0)=p̂iu’(wi)p̂iu’(wi)+(1−p̂i)Eu’(w−i).(6) The effect of ambiguity aversion on the marginal WTP for insurance in favour of state i is equivalent to a change in the probability of state i from p¯i to p̂i ⁠. It is easy to check that an increase in the distorted probability p̂i always raises the marginal WTP. Thus, ambiguity aversion raises the marginal WTP to insure state i if and only if it raises the distorted probability of state i. Let us define ψ1 as the certainty equivalent of w−i ⁠: Eu(w−i)=u(ψ1). It is easy to check that p̂i is larger than p¯i if and only if wi is smaller than ψ1 ⁠. Observe that U(θ)=pi(θ)u(wi)+[1−pi(θ)]u(ψ1) is decreasing in θ in that case. Because ϕ’ is decreasing under ambiguity aversion, the covariance rule (Kimball, 1951) implies that p̂i=Eθpi(θ)ϕ’[U(θ)]Eθϕ’[U(θ)]≥Eθpi(θ)Eθϕ’[U(θ)]Eθϕ’[U(θ)]=Eθpi(θ)=p¯i. Thus, ambiguity aversion has the effect to increase the distorted probability of the state to insure if the corresponding state wealth is smaller than the certainty equivalent wealth in other states and to reduce the distorted probability of the state to insure otherwise. This concludes the proof of the following Proposition: 1 Proposition Consider the decision to self‐insure state i in which ambiguity is concentrated. Ambiguity aversion raises the marginal WTP to self‐insure state i if wealth in state i is smaller than the certainty equivalent wealth ψ1 ⁠, and it reduces it otherwise. There is a simple intuition to this result. When state i has a wealth smaller than the certainty equivalent ψ1 ⁠, raising wi reduces the dispersion of U(θ). Therefore, it is positively valued by all ambiguity‐averse agents. The opposite result holds when wealth in state i is larger than ψ1 ⁠. Note that in the two‐state case where a loss L occurs in state i, we simply have wi=w−L<ψ1=w ⁠, so that ambiguity aversion raises the marginal WTP to self‐insure state i. 2.2. Optimal Self‐insurance We now examine the impact of ambiguity aversion on the optimal insurance in favour of state i. It is convenient to consider first a general model in which the expected utility U(e, θ) is a function of a decision e and of an unknown parameter θ. Under ambiguity neutrality, the agent selects the level e that maximises her unconditional expected utility EθU(e,θ) ⁠. It yields the following first‐order condition (hereafter subscripts with decision variables denote derivatives): EθUe(e*,θ)=0.(7) From now on, e* denotes the solution of this equation and is interpreted as the optimal prevention effort under ambiguity neutrality. Under ambiguity aversion, the objective of the decision‐maker is to maximise ϕ[W(e)]=Eθϕ[U(e,θ)] where W(e) could be interpreted as the certainty equivalent utility in the same way than the certainty equivalent wealth in the expected utility theory. Assuming that U is concave in e, the optimal level of effort is increased by ambiguity aversion if W’(e*) is positive. We have that ϕ’[W(e*)]W’(e*)=EθUe(e*,θ)ϕ’[U(e*,θ)].(8) As a simple application of the covariance rule, we obtain the following preliminary result. 1 Lemma In the decision problem maxeEθϕ[U(e,θ)] with Uee≤0 ⁠, ambiguity aversion increases (decreases) the optimal decision e for any distribution of θ if and only if U(e*,θ) and Ue(e*,θ) are anti‐co‐monotonic (co‐monotonic), where e* is defined by (7). Proof If U(e*,θ) and Ue(e*,θ) are anti‐co‐monotonic, we have that EθUe(e*,θ)ϕ’[U(e*,θ)]≥EθUe(e*,θ)Eθϕ’[U(e*,θ)]=0. The first inequality comes from the fact that ϕ’ is decreasing, which implies that U(e*,θ) and ϕ’[Ue(e*,θ)] are co‐monotonic. The second equality is due to (7). This shows that W’(e*)≥0 ⁠, so that the optimal prevention under ambiguity aversion is less than e* ⁠, the optimal effort under ambiguity neutrality. This proves sufficiency. For necessity, suppose that the optimal effort under ambiguity aversion is larger than e* for all distributions of θ. Because W is concave in e, this implies that W’(e*) is positive, or that EθUe(e*,θ)ϕ’[U(e*,θ)] is non‐negative. Because of (7), this means that Ue(e*,θ) and ϕ’[U(e*,θ)] must covary positively for all distributions of θ. Chateauneuf et al. (1994) have demonstrated that two random variables have a positive covariance under any probability measure on the set of states of nature if and only if they are co‐monotonic. Applying this result, we obtain that Ue(e*,θ) and ϕ’[U(e*,θ)] must be co‐monotonic, or that Ue(e*,θ) and U(e*,θ) must be anti‐co‐monotonic. A symmetric proof can be written for the bracketed part of the Lemma when U(e*,θ) and Ue(e*,θ) are co‐monotonic. The optimal effort is increased by ambiguity aversion if the marginal value of effort (expressed by Ue ⁠) is larger for less favourable priors, that is, for θ yielding a smaller expected utility U. Indeed, ambiguity‐averse agents put more weight on these unfavourable priors than ambiguity‐neutral agents. We apply this result to the case of the optimal insurance of state i. More specifically, suppose that one can undertake ex ante an effort e that raises wealth in state i to wi(e)−e, and that reduces wealth in all other state s ≠ i to ws−e ⁠. The problem can be written as follows: U(e,θ)=pi(θ)u[wi(e)−e]+[1−pi(θ)]∑s≠iπsu(ws−e). To make the problem interesting, we assume that wi’(e) is larger than unity. The condition Uee≤0 is satisfied under wi″(e)≤0 ⁠. If e* is the optimal demand for insurance of the ambiguity‐neutral agent as defined by (7), we can redefine the certainty equivalent wealth level ψ1 conditional on s ≠ i in this self‐insurance context as follows: ∑s≠iπsu(ws−e*)=u(ψ1−e*).(9) Suppose first that wi(e*) is smaller than ψ1 ⁠. It implies that U(e*,θ) is decreasing in θ. Observe also that Ue(e*,θ)=pi(θ)u’[wi(e*)−e*][wi’(e*)−1]+[pi(θ)−1]∑s≠iπsu’(ws−e*). The two terms in the right‐hand side of the above equation are increasing in θ, independent of the value of wi(e*) relative to ψ1 ⁠. Thus, Lemma 1 implies that the optimal self‐insurance effort is increased by ambiguity aversion when the wealth level in the insurable state is smaller than the certainty equivalent wealth level in the other states. Because U and Ue are co‐monotonic when wi(e*) is larger than ψ1, the opposite result holds in that case. This proves the following Proposition. 2 Proposition Consider the decision to self‐insure state i in which ambiguity is concentrated. Ambiguity aversion raises the optimal self‐insurance effort if wealth in state i is smaller than the certainty equivalent wealth ψ1 and it reduces it otherwise. This result generalises a result in Snow (2011), who considers the special case with only two states of nature. As mentioned above in such a case, ψ1 is equal to the initial wealth and the result holds true. Indeed, the intuition of our result is the same as for Proposition 1. When the insurable state has a low wealth level, self‐insurance reduces the dispersion of conditional expected utility levels. Ambiguity‐averse agents will therefore invest more in self‐insurance. A special case of this result is obtained in the case of the demand for insurance, where e is reinterpreted as the insurance premium. Suppose that there are two states of nature. In the no‐loss state, initial wealth is w, whereas it is only w − L in the loss state i. For each dollar of premium, the policyholder receives an indemnity k if and only if a loss occurs. If a premium e is paid ex ante, final wealth will be equal to wi(e)−e and ws−e ⁠, respectively, in the loss state and in the no‐loss state, with wi(e)=w−L+ke and ws=w ⁠. Assuming that the indemnity k can never exceed the loss, we have that wi(e*) is always smaller than the certainty equivalent wealth ψ1=w ⁠. This yields the following Corollary. 1 Corollary Consider the standard coinsurance problem with two states of nature. Ambiguity aversion always raises the insurance coverage rate. We can infer from Gollier (2011) that this result does not hold under a general structure of ambiguity as considered in Section 2.. 3. Self‐protection Self‐protection is a technique aimed at reducing the probability of a specific state i at a cost incurred in all states. As in the previous section, we assume that the ambiguity is concentrated on that state i in the sense of (2). 3.1. Willingness to Pay for Self‐protection We first examine the impact of ambiguity aversion on the willingness to pay for a marginal investment in self‐protection. Suppose that one can reduce the probability of state i by ε in all possible scenarios θ. The distribution of wealth conditional on s ≠ i is supposed to be unaffected by this investment. These two ceteris paribus assumptions mean that the self‐protection effort affects the risk but has no impact on the degree of ambiguity. The WTP for this action is denoted P(ε), which is defined as follows: Eθϕ{[pi(θ)−ε]u[wi−P(ε)]+[1−pi(θ)+ε]∑s≠iπsu[ws−P(ε)]}=Eθϕ[U(θ)]. Straightforward computations lead to P’(0)=[∑s≠iπsu(ws)−u(wi)]Eθϕ’[U(θ)]Eθ{pi(θ)u’(wi)+[1−pi(θ)]∑s≠iπsu’(ws)}ϕ’[U(θ)].(10) This can be rewritten as follows: P’(0)=u(ψ1)−u(wi)p̂iu’(wi)+(1−p̂i)u’(ψ2),(11) in which p̂i is the distorted probability defined by (5), and ψ1 and ψ2 are, respectively, the certainty equivalent and the precautionary equivalent wealth level conditional on s ≠ i: u(ψ1)=∑s≠iπsu(ws)andu’(ψ2)=∑s≠iπsu’(ws).(12) We hereafter assume that the wealth level in state i is smaller than the certainty equivalent ψ1, so that the marginal WTP for self‐protection is positive. We see from (11) that the marginal WTP is the ratio between the marginal utility benefit of self‐protection and the marginal utility cost of the corresponding effort. The marginal utility benefit is measured by the expected utility difference when s = i and when s ≠ i , whereas the utility cost of self‐protection is measured by the ambiguity‐distorted expected marginal utility p̂iu’(wi)+(1−p̂i)u’(ψ2) ⁠. Equation (11) corresponds to the marginal WTP of an ambiguity‐neutral agent with distorted beliefs p̂i ⁠. Thus, as in the previous section, the effect of ambiguity aversion is to transform probability p¯i=Eθpi(θ) into a distorted probability p̂i ⁠. We have shown in the previous Section that p̂i is larger than p¯i if and only if wi is smaller than ψ1 ⁠. On the other hand, it is immediate from (11) that the marginal WTP for self‐protection of state i is decreasing in p̂i if and only if wi is smaller than the precautionary equivalent wealth ψ2 ⁠. Let us assume that the utility function of the agent exhibits decreasing absolute risk aversion (DARA). From Kimball (1990), we know that this means that the precautionary equivalent ψ2 is smaller than the certainty equivalent ψ1 ⁠. Let us first consider a situation in which the wealth level wi in state i is smaller than ψ2≤ψ1 ⁠. In that case, p̂i is larger than p¯i and the marginal WTP is decreasing in p̂i ⁠. This implies that ambiguity aversion has a negative impact on the marginal WTP for self‐protection. The intuition is that ambiguity aversion induces the agent to put more weight on scenarios θ with a large probability pi(θ) ⁠, since U(θ) is decreasing with θ when wi<ψ1 ⁠. This raises the distorted probability p̂i above p¯i ⁠. However, this has the effect of raising the marginal utility cost p̂iu’(wi)+(1−p̂i)u’(ψ2) of the ex ante effort, since wi<ψ2 ⁠. The marginal utility benefit of self‐protection expressed by u(ψ1)−u(wi) being unaffected by ambiguity aversion, we can conclude that ambiguity aversion reduces the marginal WTP of self‐protecting states whose wealth level is below the precautionary equivalent wealth. The opposite result prevails when wi is in between ψ2 and ψ1 ⁠. This is because the increased distorted probability p̂i now implies a reduction in the utility cost of effort. These results are summarised in the following Proposition. 3 Proposition Consider the decision to self‐protect state i in which ambiguity is concentrated. Suppose that the agent is DARA, and that the wealth level wi in state i is less than the certainty equivalent wealth level ψ1 ⁠. Ambiguity aversion reduces the marginal WTP to self‐protect state i if wi is smaller than the precautionary equivalent wealth ψ2 ⁠, and it raises it if wi∈[ψ2,ψ1] ⁠. 3.2. Optimal Self‐protection We now consider a specific prevention model in which the effort e affects the probability pi(e,θ) of state i in which ambiguity is concentrated. We define expected utility conditional on θ as follows: U(e,θ)=pi(e,θ)u(wi−e)+[1−pi(e,θ)]∑s≠iπsu(ws−e).(13) We assume that the probability of state i is differentiable with respect to e, with pie(e,θ)≤0 ⁠. We also assume that the second‐order condition is satisfied.3 Let e* denote the optimal effort of self‐protection for the ambiguity‐neutral agent. Suppose without loss of generality that pi(e*,θ) is increasing in θ. Using Lemma 1, we need to determine whether U and Ue are co‐monotonic or anti‐co‐monotonic to determine the impact of the ambiguity aversion on the optimal effort. Redefining the certainty and the precautionary equivalents ψ1 and ψ2 as u(ψ1−e*)=∑s≠iπsu(ws−e*) and u’(ψ2−e*)=∑s≠iπsu’(ws−e*), we can rewrite the expected utility conditional on θ as U(e*,θ)=pi(e*,θ)u(wi−e*)+[1−pi(e*,θ)]u(ψ1−e*),(14) and its derivative with respect to e as Ue(e*,θ)=pie(e*,θ)[u(wi−e*)−u(ψ1−e*)]−{pi(e*,θ)u’(wi−e*)+[1−pi(e*,θ)]u’(ψ2−e*)}.(15) As earlier, suppose that wi is smaller than the certainty equivalent ψ1 ⁠. This means that we focus on a state satisfying the natural property that a reduction in its probability raises expected utility. In other words, from (14), wi≤ψ1 implies that the conditional expected utility is decreasing in θ. The sensitiveness of Ue to changes in θ is more difficult to evaluate at that degree of generality, as can be seen from (15). As stated earlier in this Section, let us now assume that the degree of ambiguity is not affected by the effort. In other words, let us assume at this stage that pie is independent of θ. This situation is illustrated by the dashed lines in Figure Figure Fig. 1.., with θ1<θ2 ⁠. Under this condition, the first term on the right‐hand side of (15) is independent of θ. What remains on the right‐hand side of the equation is decreasing in θ if wi is smaller than ψ2 ⁠. Using Lemma 1 yields the following Proposition. Fig. 1. Open in new tabDownload slide The Probability of State i in the Self‐protection Model Fig. 1. Open in new tabDownload slide The Probability of State i in the Self‐protection Model 4 Proposition Consider the decision to self‐protect state i in which ambiguity is concentrated. Suppose that the agent is DARA, and that the wealth level wi in state i is less than the certainty equivalent wealth level ψ1 ⁠. If effort does not affect the ambiguity of state i, that is, if pie(e*,θ) is independent of θ, ambiguity aversion reduces the optimal self‐protection effort if wi is smaller than the precautionary equivalent wealth ψ2 ⁠, and it raises it if wi is larger than ψ2 ⁠. The intuition of this result is the same as for Proposition 3. When the wealth level in the state whose probability is reduced by the effort is smaller than the precautionary equivalent wealth, the increase in the distorted probability p̂i that is induced by ambiguity aversion raises the marginal utility cost of effort, thereby reducing the optimal effort. Let us now relax the assumption that the effort level has no impact on the degree of ambiguity. Suppose alternatively that an increase in effort raises the degree of ambiguity. Because we assume that pi(e*,θ) is increasing in θ, this means that pie(e*,θ) is increasing in θ. An illustration of this is given by the two plain lines in Figure Figure Fig. 1... The dashed lines correspond to a situation in which the effort does not affect the degree of ambiguity of state i. The plain lines correspond to a situation in which more effort raises the degree of ambiguity of state i since the probabilities of state i become more dispersed when effort is increased. Intuitively, this should reinforce the negative impact of ambiguity aversion on effort. This can be checked by observing that the first term on the right‐hand side in (15) is decreasing in θ, as is the second term when wi≤ψ2 ⁠. 5 Proposition Under the conditions of Proposition 4, if effort raises the ambiguity of state i, that is, if pie(e*,θ) is increasing in θ, then ambiguity aversion reduces the optimal self‐protection effort if wi is smaller than the precautionary equivalent wealth ψ2 ⁠. This result is related to Snow (2011), who exhibits a case in which ambiguity aversion raises the optimal self‐protection effort.4 Using a two‐state model, Snow restricts the functional form describing the effect of self‐protection on the probability of accident. Using our notation, Snow assumes that, for all (e, θ), pi(e,θ)=pi(θ)ρ(e) where ρ(e) captures the effect of the self‐protective effort, with ρ(e) > 0 and ρ’(e)<0 ⁠. This is a special case in which the proportional effect of effort on the loss probability is the same for all θ. Under this condition, Snow shows that the self‐protection effort is increased by ambiguity aversion. Because pi(e,θ)=pi(θ)ρ(e) implies that pie is decreasing in θ, Snow’s result is not inconsistent with our Proposition 5 in which we instead assume that pie is increasing in θ.5 The practical implication of these results depends upon whether the effort has a positive or a negative effect on the ambiguity surrounding the probability of the loss state. In many situations, it appears more natural that self‐protection would reduce both risk and ambiguity, consistent with Snow’s assumption. However, a typical case where pie(e*,θ) should instead increase in θ is when ambiguity affects the effectiveness of self‐protection but not the baseline risk. In that case, more effort should naturally induce more ambiguity. An illustrative example is when a person with a well‐known disease accepts a treatment that has never been tested before. 4. Optimal Insurance Design In this Section, we explore the problem of the optimal insurance contract design. As first proved by Arrow (1971), any insurance contract is dominated by a contract with a straight deductible. As it is intuitive, the deductible contract optimises the risk transfer to the insurer for any insurance budget level, since it provides indemnities where the marginal utility of wealth is the largest. We are interested here in determining whether this result is robust to the introduction of ambiguity aversion. As a particular case of the specific structure of ambiguity considered in Sections 3. and 3, we limit the analysis to the case in which the ambiguity is concentrated in the probability of the no‐loss state. We have wi=w and ws=w−Ls where Ls is the loss in state s. Let p(θ) denote the ambiguous probability that a loss occurs. Conditional on the occurrence of a loss, the level of loss is a random variable L which is independent of θ. That is, the distribution of the severity of the loss is unambiguous. An insurance contract stipulates an indemnity I(Ls) for each possible loss level Ls ⁠. To any such indemnity schedule, there is an insurance premium which is proportional to the actuarial value of the policy: P=p¯(1+τ)EI(L), where as before p¯=Eθp(θ) is the expected probability of accident and τ is the loading factor. Conditional on θ, the policyholder’s expected utility is written as follows: U(θ)=[1−p(θ)]u(w−P)+p(θ)Eu[w−L+I(L)−P]. The ambiguity‐averse policyholder selects the indemnity schedule I(Ls) that maximises its ex ante welfare which is measured by Eθϕ[U(θ)] ⁠. By the concavity of u and ϕ, this maximisation is well‐behaved, and its first‐order conditions are necessary and sufficient. We first prove the following Proposition. 6 Proposition Suppose that the policyholder is ambiguity averse and that the ambiguity is concentrated on the probability of the no‐loss state. Under this specification, the optimal insurance contract contains a straight deductible d: I(Ls)=max(0,Ls−d) ⁠. Proof Suppose by contradiction that the contract I0 that maximises Eθϕ[U(θ)] is not a straight deductible contract. It yields an insurance premium P0=p¯(1+τ)EI0(L) ⁠. Then, let d0 denote the deductible which yields the same insurance premium than P0 ⁠: p¯(1+τ)E[max(0,L−d0)]=P0. However, we know from Arrow (1971) that this alternative contract with a straight deductible d0 dominates any other insurance schedule as I0 ⁠: Eu[w−min(L,d0)−P0]≥Eu[w−L+I0(L)−P0]. It implies that for all θ, we have [1−p(θ)]u(w−P0)+p(θ)Eu[w−min(L,d0)−P0]≥[1−p(θ)]u(w−P0)+p(θ)Eu[w−L+I0(L)−P0]. Because the expected utility conditional on θ is larger with the straight deductible d0 than with contract I0 for all θ, the former necessary yields a larger ex ante welfare Eθϕ[U(θ)] ⁠. This is a contradiction. We now examine the impact of ambiguity aversion on the optimal deductible. Corollary 1 answered this question in the special case of only one possible loss, since in that case a proportional coinsurance or a straight deductible is formally equivalent. With more than one possible loss level, the decision problem can thus be rewritten as follows: maxdEθϕ[U(d,θ)],(16) with U(d,θ)=[1−p(θ)]u[w−P(d)]+p(θ)Eu[w−min(L,d)−P(d)] and P(d)=p¯(1+τ)Emax(0,L−d) ⁠. We suppose that p¯(1+τ) is smaller than unity, so that reducing the deductible makes the policyholder wealthier in high loss states. Let d* denote the optimal deductible under ambiguity neutrality. By Lemma 1, and because U is concave in d,6 ambiguity aversion reduces the optimal deductible if U(d*,θ) and Ud(d*,θ) are co‐monotonic. Suppose that p is increasing in θ, so that U(d*,θ) is decreasing in θ. We have Ud(d*,θ)=−P’(d*){[1−p(θ)]u’(w−P*)+p(θ)Eu’[w−min(L,d*)−P*]}−p(θ)[1−F(d*)]u’(w−d*−P*),(17) where P*=P(d*) and F is the cumulative distribution function of L. This equality can be rewritten as follows: Ud(d*,θ)=−P’(d*)[1−p(θ)]u’(w−P*)+p(θ)[1−F(d*)]×{p¯(1+τ)Eu’[w−min(L,d*)−P*]−u’(w−d*−P*)}.(18) This is obviously decreasing in θ, since P’(d*)<0 ⁠, Eu’[w−min(L,d*)−P*]≤u’(w−d*−P*) and p¯(1+τ)≤1 ⁠. This proves the following Proposition. 7 Proposition Under the specification of Proposition 6, the optimal deductible is reduced by ambiguity aversion. The intuition for this result can be obtained from the observation that the effect of ambiguity aversion is here equivalent to an increase in the implicit loss probability defined by p̂=Eθϕ’[U(d*,θ)]p(θ)Eθϕ’[U(d*,θ)]. Because U(d*,θ) and p(θ) are negatively correlated and because ϕ’ is decreasing, we have that p̂ is larger than Eθp(θ)=p¯ ⁠, under ambiguity aversion. We assumed in this Section that the ambiguity is concentrated on the no‐loss state. We finally show that the optimal insurance schedule is in general not a straight deductible for other structures of ambiguity. This is done with a numerical counter‐example. Suppose that the agent has an initial wealth w = 3 and that there are three possible loss levels L = (0, 1, 2). The ambiguity is structured as follows. There are two equally likely probability distributions for L with support (0, 1, 2): the good distribution is p(θ = 1) = (1/2, 1/2, 0), whereas the bad distribution is p(θ = 2) = (1/4, 1/4, 1/2). Observe that, under our terminology, the ambiguity is here concentrated on the large loss L = 2. The loading factor of the insurance premium is τ = 0.2. The policyholder’s utility function is u(c) = −1/c, and we calibrate ambiguity aversion with ϕ[u]=−e−100u ⁠. Using Mathematica, we solve numerically the following problem: maxI1∈[0,1],I2∈[0,2],P12ϕ[12u(3−P)+12u(2+I1−P)]+12ϕ[14u(3−P)+14u(2+I1−P)+12u(1+I2−P)] subject to P=1.2(38I1+14I2). We obtain I1*=0.407 and I2*=1.630 ⁠. Because I2*−I1*>L2−L1 ⁠, we see that the optimal contract is not a straight deductible, that is, there is no d such that I*(Ls)=max(0,Ls−d) for all Ls ⁠. Final wealth is larger in state with L = 2 than in state with L = 1. This is done to compensate for the relatively large ambiguity associated with the large loss. We conclude that the result expressed in Proposition 6 is not robust to the relaxation of the condition that ambiguity is concentrated on the no‐loss state. 5. Conclusion In this article, we have shown that ambiguity aversion tends to raise the incentive to insure and to self‐insure but to decrease the incentive to self‐protect. The intuition for our results is that ambiguity aversion has an effect which is behaviourally equivalent to the effect of more pessimism under subjective expected utility theory. Since the effect of a change in probability in expected utility models is usually different from that of a change in risk aversion, this explains why the effect of ambiguity aversion differs from that of risk aversion. We have also exhibited a case where the optimal insurance contractual form contains a straight deductible as in the expected utility model. This article has thus generalised the analysis of some standard problems in insurance economics to ambiguous risks. Some technical difficulties remain however. In particular, it could be interesting to consider more general ambiguous probability distributions, ambiguity‐averse insurers and other forms of ambiguity‐sensitive preferences. At a more conceptual level, we finally observe that if the distinction between (self‐)insurance and self‐protection has classically been done in the insurance economics literature, the case for maintaining such a distinction may become more problematic under conditions of ambiguity. Footnotes 1 " An activity of self‐protection consists in investing to reduce the probability of an accident. In fact, no general result can be obtained to sign the effect of risk aversion on self‐protection (Sweeney and Beard, 1992). To sign this effect, it is necessary to specify the value of the probability of loss and to make further assumptions on the utility function (Dachraoui et al., 2004; Eeckhoudt and Gollier, 2005). 2 " Gollier (1995) summarises the main findings on this comparative statics analysis in the case of a portfolio choice problem, which is further examined in Abel (2002). Strong restrictions on the utility function are required to guarantee that ambiguity aversion reduces the optimal risk exposure. 3 " In this model, the condition Uee≤0 requires some specific restrictions on functions p ( Jullien et al., 1999). 4 " See Treich (2010) for a related result and discussion but in a state‐dependent model. See also Berger (2011) who examines a model of self‐protection and self‐insurance under ambiguity aversion but in a two‐period model. 5 " Here is a sketch of the proof of Snow’s (2011) result. Let us assume wi=w−L ⁠, ws=w and pi(e,θ)=pi(θ)ρ(e) ⁠. We need to show that Ue(e*,θ) increases in θ in this specific model. It can easily be shown that this holds if and only if f(e*)=ρ’(e*)[u(w−L−e*)−u(w−e*)]−ρ(e*)[u’(w−L−e*)−u’(w−e*)] is positive. Note that the first term of f(e*) is positive, while the second term is negative, illustrating that the effect is in general indeterminate when pie is decreasing in θ. Nevertheless, observe that the first‐order condition of this problem simply reduces to [Eθp(θ)]f(e*)=u’(w−e*) ⁠, which implies f(e*)>0 ⁠. 6 " See Schlesinger (1981). References Abel , A.B. ( 2002 ). ‘ An exploration of the effects of optimism and doubt on asset returns ’, Journal of Economic Dynamics and Control , vol. 26 ( 7–8 ), pp. 1075 – 92 . Google Scholar Crossref Search ADS WorldCat Arrow , K.J. ( 1971 ). Essays in the Theory of Risk Bearing , Chicago, IL : Markham Publishing Company . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Berger , L. 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Google Scholar Crossref Search ADS WorldCat Sweeney , G. and Beard , T.R., ( 1992 ). ‘ Self‐protection in the expected‐utility‐of‐wealth model: an impossibility theorem ’, Geneva Papers on Risk and Insurance Theory , vol. 17 ( 2 ), pp. 147 – 58 . Google Scholar Crossref Search ADS WorldCat Treich , N. ( 2010 ). ‘ The value of a statistical life under ambiguity aversion ’, Journal of Environmental Economics and Management , vol. 59 ( 1 ), pp. 15 – 26 . Google Scholar Crossref Search ADS WorldCat Author notes " We thank the editor and two referees for useful comments. We also thank seminar participants at the University of Technology Sydney, the University of Auckland, the NHH at Bergen and the World Risk and Insurance Economics Congress (25–29 July 2010 in Singapore). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) Grant Agreement no. 230589 and from the Chair ‘Risk markets and value creation’ financed by SCOR at IDEI. © 2013 The Author(s). The Economic Journal © 2013 Royal Economic Society

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The Economic JournalOxford University Press

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