Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences

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1 The Geneva Papers on Risk and Insurance Theory, 25: 7 28 (2000) c 2000 The Geneva Association Background Risk, Demand for Insurance, and oquet Expected Utility Preferences MEGLENA JELEVA Centre de Recherche en Economic et Statistique, Laboratoire d Economic Industrielle, Paris and EUREQua, Université Paris 1 and C3E Abstract This article deals with demand for insurance with a background risk in a nonprobabilized uncertainty framework, where preferences are represented by a nonadditive model of decision making. The oquet expected utility model that we use generalizes expected utility and allows for a separation of the attitude towards uncertainty and the attitude towards wealth. When the insurable and the background risk are comonotone, the impact of the background risk on the demand for insurance is related to the attitude towards wealth. In contrast, when the two risks are anticomonotone, the attitude towards uncertainty is determinant. In this case, some of the resulting behaviors cannot be explained by the standard expected utility model. Key words: demand for insurance, background risk, nonexpected utility 1. Introduction Demand for insurance has been widely studied under the assumption that agents face a single source of risk, against which an insurance contract is provided. In reality, however, agents face more than one risk, and the assumption that the decisions concerning each risk are taken independently is relevant only when all the risks are independent and insurable (this last assumption corresponds to the completeness of insurance markets). Hirshleifer and Riley [1979] point out the existence of destructive losses that lower the total wealth of society, such as natural catastrophes or macroeconomic fluctuations, implying that full insurance of every potential loss is impossible. Even without referring to such extreme situations, there are examples of risks against which insurance is not possible: risks for which the amounts of loss are difficult to evaluate by the insurance companies, such as losses of production due to a fire in a factory, or the loss of an irreplaceable commodity the value of which for the owner is higher than the market value; macroeconomic risks, such as income risk; etc. Informational asymmetries and search costs might render insurance on some risks too expensive, and the agents will act as if those risks were uninsurable. The existence of uninsurable risks affects the agents risk exposure, so those risks may have an impact on the decisions concerning insurable risks. Insurance purchasing decisions within the context of an incomplete insurance market have been studied by several authors since the 1980s. One of the motivations for research was to explain some of the puzzles of the standard models of insurance purchasing. Indeed, the standard one-risk model is not compatible with the fact that individuals can choose

2 8 MEGLENA JELEVA full coverage even if the insurance premium is unfair. The demand for insurance with background risk was first studied by Doherty and Schlesinger [1983, 1992], Mayers and Smith [1983], Briys, Kahane, and Kroll [1988], and later by Eeckhoudt and Kimball [1992], Eeckhoudt and Gollier [1992], Doherty and Eeckhoudt [1995], and Gollier [1996]. In all these articles, it is assumed that both insurers and agents know precisely the probability distribution of risks, or in other words, that both make their decisions in a context of probabilized uncertainty (called risk, in the sense of Knight [1921]). For the insurers, this assumption seems justified, since they can use data on the sinistrality and the characteristics of their clients to form homogeneous classes of individuals and estimate the accident probabilities of each class. The agents are in a different situation: they have only information based on self-observation and on their own characteristics. The competition between companies may sometimes allow insurees to obtain more information, but their information remains, in most cases, qualitative and imprecise. Consequently, it seems relevant to examine insurance behavior in a nonprobabilized uncertainty framework. In this context and under certain axioms, it is possible to represent preferences using the subjective expected utility model of Savage [1954]. However, experimental studies, the most well-known being that of Ellsberg [1961], showed the existence of situations where choices cannot be explained by an expected utility maximization. Those violations of the expected utility axioms inspired the development of new, more general models of decision making under uncertainty. The oquet expected utility, proposed by Schmeidler [1989], is one of the more widely accepted models in this class. Individuals preferences in this model depend on the one hand on a utility function (which reflects the perception of wealth) and, on the other hand, on a capacity 1 (reflecting the perception of the occurrence of the events). This preferences representation is attractive for at least two reasons: it better represents real choices and allows for a separation between attitude towards uncertainty and attitude towards wealth. Note that the two attitudes are mixed in the expected utility model, where they are both represented by the utility function. Note also that, under probabilized uncertainty, similar models are proposed by Kahneman and Tversky [1979], Quiggin [1982], and Yaari [1987]. These models are known under the denomination of rank-dependant expected utility (RDEU). In this article, we study the impact of a background risk on the insurance purchase behavior in a nonprobabilized uncertainty framework, where the preferences are represented using the oquet expected utility model. The adopted preferences representation makes it possible to determine whether it is the attitude towards uncertainty or the attitude towards wealth that explains the optimal behavior. We also determine the choices, obtained with the standard expected utility model, that can be generalized to the Schmeidler decision model and those that are impossible to explain in the standard model. The results obtained in the literature in the risky context show that the correlation between the insurable and the background risk are determinant for the impact of the background risk on the demand for insurance. In the absence of an objective probability distribution on the set of states, the concept of correlation no longer makes any sense. Thus, to represent the dependence between the two risks, we use the concepts of comonotony and anticomonotony, based directly on the amounts of loss. 2 The case when the two risks are independent is not considered, due to the fact that under nonprobabilized uncertainty, a clear definition of objective independence (which does not depend on preferences) doesn t exist yet.

3 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 9 Section 2 of the article recalls the oquet expected utility model and the definitions of uncertainty aversion. Section 3 gives some results concerning the optimal coverage with a single source of risk in a nonprobabilized uncertainty framework. In Section 4, a background risk is introduced; we consider successively the cases where it is comonotone and anticomonotone with the insurable risk. 2. oquet expected utility The distinction between risk (situations where there exists an objective probability distribution, known by the decision maker) and uncertainty (situations where there is no objective probability distribution, or it is unknown for the decision maker) is due to Knight [1921]. The canonical normative model of decision making under uncertainty is the subjective expected utility model, proposed by Savage. The oquet expected utility model departs from the independence axiom 3 of Savage, which is weakened and replaced by the comonotone independence axiom Preference representation Under nonprobabilized uncertainty, a decision is a mapping, called act, from the set of states into a set of outcomes. Let be a set of states; A the σ -algebra of events on ; C the set of outcomes, assumed bounded (C = [ M, M]); and X the set of acts from into C corresponding to all possible decisions. The symbol is the preference relation on X, where denotes strict preference and denotes indifference. For the preferences representation, it will be necessary to define a class of set functions (the capacities) and to give some of their properties that will be used in the remainder of the article. Definition 1: 1. A mapping ν : A [0, 1] is a capacity if ν( ) = 0, ν( ) = 1, [A, B A, A B] ν(a) ν(b). (1) 2. A capacity ν : A [0, 1] is convex if for all A, B A, ν(a B) ν(a) + ν(b) ν(a B). (2) 3. A capacity ν : A [0, 1] is a probability measure if it is additive. If the preference relation of an agent satisfies the axioms of the oquet expected utility model (CEU), then the functional V ( ) representing his preference relation for an act X is written: 4 V (X) = u(x) dν, (3)

4 10 MEGLENA JELEVA where ν( ) is a capacity and u( ) is a utility function with u(0) = 0 and u (x) >0. The oquet integral that appears in (3) is defined as follows: 0 u(x) dν = {ν[ω : u(x(ω)) c] 1} dc M +M + 0 ν[ω : u(x(ω)) c] dc. (4) So the preferences of a CEU maximizer are characterized by a pair (ν, u) composed of a utility function and a capacity. For a finite set of states, ={ω 1,ω 2,...,ω n }, and an act X such that X (ω i ) = c i, i = 1..n, with c i c i+1, the function V (X) writes V (X) = u(c 1 ) + [u(c 2 ) u(c 1 )]ν(ω 2 ω 3 ω n )+ +[u(c n ) u(c n 1 )]ν(ω n ). (5) The underlying intuition of this preferences representation is the following. The satisfaction corresponding to an act is computed as the utility of the worst outcome, to which are added the successive possible increases of the outcomes weighted by the individual s perception of the likelihood of these increases. It appears immediately that, for an additive capacity, the preference representation function (4) becomes a standard subjective expected utility function. Thus, the Savage subjective expected utility model is a particular case of the CEU model. For a convex capacity, Schmeidler [1986] gives a characterization of the probability measure, appearing in the oquet integral. More precisely, for ν convex, Xdν= min E P X, (6) P core(ν) with core(ν) ={ P L:P(A) ν(a), A A}, (7) where L is the set of all probability distributions over (, A). This result allows us to establish a relation between the CEU model and another model of decision making under nonprobabilized uncertainty: the multiprior model proposed by Gilboa and Schmeidler [1989]. In this model, preferences are characterized by a subjective set of probability distributions and a utility function. The functional V ( ), representing the preference relation for an act X, is written V (X) = min u(x) dp. (8) P P Thus, for a convex capacity, the CEU model is equivalent to the multiprior model, and all the results obtained in applications under CEU remain true for an agent with multiprior preferences. This is no more the case, however, when the capacity is not convex.

5 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 11 Let s now define the concepts of comonotony and anticomonotony, which will be used both in the definitions of uncertainty aversion and in the demand for insurance model. Definition 2: 1. Two acts X, Y X are comonotone if and only if for every ω i,ω j, (X(ω i ) X (ω j ))(Y (ω i ) Y (ω j )) Two acts X, Y X are anticomonotone if and only if for every ω i,ω j, (X(ω i ) X (ω j ))(Y (ω i ) Y (ω j )) 0. The relation between comonotony and covariance has been established by ateauneuf, Kast, and Lapied [1996]. These authors proved that two bounded acts on (,A)are comonotone if and only if their covariance is positive for any probability distribution on (,A). The following two properties of the oquet integral will constitute the main tool for our proofs. Proposition 1: 1. oquet [1953]: For any capacity ν( ) and for any act X, there exists a probability distribution Q ν (X), such that u(x) dν = E Qν (X)u(X). (9) It is important to note that, contrary to the standard subjective expected utility case, the probability distribution Q ν (X) here depends on X. 2. Denneberg [1994]: Let X be a class of comonotone acts (a family of pairwise comonotone acts), and let ν( ) be a capacity. Then there exists a probability measure Q ν such that, for every X X and every u(.) such that u 0, u(x) dν = E Qν u(x). (10) 2.2. Uncertainty aversion: definitions and characterizations Due to the absence of an objective probability distribution over the set of states, the definition of uncertainty aversion has to be based only on the outcomes corresponding to the different acts. A first definition is proposed by Schmeidler [1989]. The underlying idea is that a convex combination of noncomonotone acts smooths the outcomes and makes an uncertaintyaverse decision maker better off.

6 12 MEGLENA JELEVA Definition 3: Schmeidler [1989]: A decision maker is strongly uncertainty averse if, for every pair of acts X, Y and α [0, 1], X Y αx + (1 α)y Y. So a decision maker is strongly uncertainty averse if his preferences are convex. The following results give the characterizations of strong uncertainty aversion. Proposition 2: For a decision maker whose preferences are represented by (ν, u): 1. Schmeidler [1989]: If u(x) = x, the decision maker is strongly uncertainty averse if and only if ν is convex. 2. ateauneuf, Dana, and Tallon [1997]: For any u, the decision maker is strongly uncertainty averse if and only if ν is convex and u is concave. Two definitions are proposed for the weak uncertainty aversion, one by Montesano and Giovannoni [1995] and the other by ateauneuf, Dana, and Tallon [1997]. The first definition is based on a preference for a probabilized uncertainty context, and the second one is based on a preference for a certain outcome. For both of them, an individual with a concave or linear utility function is weakly uncertainty averse if and only if core(ν). It appears that in the CEU model, uncertainty aversion, both weak and strong, is closely related to the properties of the capacity corresponding to individuals beliefs. Note that in the Savage model, the two types of uncertainty aversion are characterized by a concave utility function, which corresponds also to a decreasing marginal utility for wealth in a certainty context. Therefore, it appears that the CEU model allows for a distinction between the attitude towards wealth and the attitude towards uncertainty. It also allows for a distinction between strong and weak uncertainty aversion. 3. Demand for insurance with a single source of risk This section provides conditions for the optimality of full coverage for an agent whose preferences under uncertainty are modeled by the oquet expected utility model. These conditions give another explanation for the fact that, in reality, individuals choose full coverage even if the insurance premium is loaded. The insurance contract is assumed to be linear (the indemnity corresponding to this type of contract is a proportion of the amount of loss). The insurer estimates the probability distribution over the set of potential losses. An agent with initial wealth w faces a risk X : C, with C = [0, L]. The premium corresponding to a contract with a coverage proportion γ is denoted by γ (X). Let P( ) be the probability measure on, estimated by the insurer. The premium is based on the expected value of the indemnity function, including a proportional markup, λ, representing administrative costs. Thus, γ (X) = (1 + λ)γ E P X. (11)

7 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 13 From (3), the value function V γ (X) corresponding to a coverage proportion γ is written V γ (X) = u(w (1 γ)x γ (X)) dν. (12) To determine the optimal level of coverage, we use Proposition 1 and the fact that {W γ } γ [0,1], with W γ = w (1 γ)x γ (X), (13) is a class of comonotone acts. This is now shown. Let γ 1, γ 2 [0, 1]. From Definition 2, W γ1 and W γ2 are comonotone if and only if for every ω 1,ω 2, (W γ1 (ω 1 ) W γ1 (ω 2 ))(W γ2 (ω 1 ) W γ2 (ω 2 )) 0. Replacing W γ1 and W γ2 by their values, the above expression reduces to (1 γ 1 )(1 γ 2 )(X (ω 2 ) X (ω 1 )) 2, which is positive or equals zero for γ i [0, 1], i = 1, 2. By the same means, we obtain that X is comonotone with W γ for every γ. So, using Proposition 1, (12) becomes V γ (X) = E Qν ( X)[u(w (1 γ)x γ (X))], (14) where Q ν ( X) is a probability distribution depending on X and on ν, but not on γ. This form of the functional representing preferences for an insurance contract is used to prove the following results. Proposition 3: A oquet expected utility maximizer with u 0 faces an insurable risk X : [0, L]. The insurance premium is given by (11) and is fair (λ = 0). 1. The necessary and sufficient condition for the optimality of full coverage is E Qν ( X)(X) E P (X), where Q ν ( X) is a probability distribution, depending on ν and X and such that E Qν ( X)( X) = ( X) dν. If ν( ) is convex, then E Qν ( X)( X) ={min E P ( X), P core(ν)}.

8 14 MEGLENA JELEVA 2. The necessary and sufficient condition for the optimality of full coverage for every X on is P core(ν). Proof. From (14), it follows that the optimal level of coverage is the solution of the following optimization problem: max E Q ν ( X)u(w (1 γ)x γ (X)). (15) γ [0,1] The second-order condition is satisfied for all γ [0, 1] because of the concavity of u( ). The first-order condition is E Qν ( X)[(X E P X)u (w (1 γ)x γe P X)]=0. (16) Full coverage (γ = 1) is optimal when u (w E P X)E Q ν ( X)(X E P X) 0, (17) which leads to the condition E Qν ( X) X E P X, (18) which proves point 1 of Proposition 3. The characterization of Q ν ( X) when ν is convex comes from (6). We noticed that X is comonotone with the acts of the family {W γ } γ [0,1]. Then ( X) dν = E Qν ( X)( X). (19) Therefore, the inequality (17) can be written as E P ( X) ( X) dν. (20) It is easy to prove that if P core(ν), the above inequality is true for every X on, which proves point 2 of Proposition 2. If the insurance premium is loaded (λ > 0), that is, if it includes administrative costs or a constant profit, it is possible to prove that the condition for the optimality of full coverage becomes E Qν ( X)(X) (1 + λ)e P (X). (21)

9 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 15 The following example illustrates the difference between points 1 and 2 of the above proposition. Let X i, i = 1,..,n, be the different risks on the house of an agent (fire, flood, etc.) with X i : [0, L]. P is the probability distribution on estimated by an insurance company, is the coverage chosen by the agent for the risk X i. If this agent is a CEU maximizer and P core(ν), then γi = 1, i = 1,..,n. If P / core(ν), then it is possible to have γi = 1 and γ j < 1, j i. which offers contracts against X i, for all i, and γ i The results of Proposition 3 are obtained without referring to attitude towards uncertainty. In any case, it is necessary to mention that, due to Proposition 4, core(ν) if and only if the individual is weakly uncertainty averse. In this section, it appears that, for a given insurable risk, the preferences representation for a proportional insurance contract has the form of an expected utility with respect to a subjective probability distribution. Therefore, when an individual faces a single source of risk, oquet expected utility and Savage subjective expected utility are equivalent. The decision to buy full coverage in this case is caused by the gap between the estimation of the expected losses by the insurer and by the agent. If the estimation of the agent is more pessimistic than that of the insurer, full insurance will be optimal, even with a strictly positive loading rate, which gives a theoretical justification of the observed evidence that agents tend to buy full coverage even with an unfair premium. Notice that this behavior was impossible to explain in the standard insurance models, where agents were assumed to know perfectly the probability distribution of their risk. The particular characteristics of the CEU preferences appear when one addresses the question of the generality of the obtained results. Indeed, if the agent is weakly uncertainty averse and if the probability distribution estimated by the insurer is above the agent s capacity, full coverage is optimal not only for a given loss distribution but also for all risks on the same support. 4. Demand for insurance with a background risk In a probabilized uncertainty framework, the demand for insurance with a background risk has been studied under different aspects. For instance, Mayers and Smith [1983] and Briys, Kahane, and Kroll [1988] assume that preferences are represented by the mean-variance model and that the background risk is related to the possession of risky financial assets. Doherty and Schlesinger [1983] consider agents who are expected utility maximizers and strictly risk averse and who face two risks, one of which is uninsurable. The authors address the question of the optimality of full coverage depending on the correlation between the two risks and the existence or not of loading on the insurance premium. The same kind of questions are addressed by Doherty and Eeckhoudt [1995], who assume that agents have preferences modeled by the dual theory of Yaari [1987]. Both studies leave open the question of the precise impact of the background risk on the amount of insurance when the optimal coverage is partial and when the losses corresponding to the two risks take an infinity of values. The answers to these questions when the two risks are independent or positively dependent are given by Eeckhoudt and Kimball [1992], who use the concept of prudence proposed

10 16 MEGLENA JELEVA by Kimball [1990]. Their results are obtained in a context of probabilized uncertainty, where risk aversion and decreasing marginal utility for wealth are represented by the same function. As will be shown below, the introduction of nonprobabilized uncertainty and the possibility in the CEU model to separate attitudes towards wealth from attitudes towards uncertainty changes some of the results and gives more details about the exact origin of the observed behaviors. The results we obtain when the two risks are comonotone will be compared with those of Quiggin [1991], which deal with the impact on the quantity of risky assets in a portfolio of an increase in risk (in a monotone sense) of the rate of return of this asset. Our study is along the lines of Eeckhoudt and Kimball [1992] in the sense that we try to compare explicitly choices in the presence of a background risk to choices when only the insurable risk is present. We successively consider the two extreme cases for the dependence between the two risks: the case where the risks are comonotone and the case where they are anticomonotone. Fire and production in a factory are examples of comonotone risks; a more severe damage to the building and machinery will induce a longer interruption of activity and a larger decrease in production. An example of anticomonotone risks can be found in farming: weather conditions sometimes have opposite effects on the yield of different crops Comonotone risks An agent faces two risks X and Y, whose losses can take all the values between 0 and L. An insurance contract is available for X. The characteristics of this contract are the same as those in the preceding section. The risk Y is uninsurable. In this framework, for an oquet expected utility maximizer, the functional V γ (X, Y ) representing preferences for an insurance contract is V γ (X, Y ) = u(w (1 γ)x Y γ (X)) dν. Hence, the optimal coinsurance rate is the solution of the following problem: u(w (1 γ)x Y γ (X)) dν. max γ [0,1] In all the following,we denote by γ the optimal coinsurance rate in the absence of the uninsurable risk and by γ the optimal rate when the two risks are present. The following proposition gives the impact of the background risk on the demand for insurance when the two risks are comonotone. Proposition 4: An agent with CEU preferences, characterized by (ν, u) faces two comonotone risks of loss, one of which is uninsurable: 1. If the agent s utility function has the properties u < 0, u > 0, and ( u ) < 0, u ( u ) < 0, then γ <γ. u

11 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET If u(x) = x, then γ = γ. Proof. The proof of point 1 follows Eeckhoudt and Kimball [1992], p Using point 1 of Proposition 1, V γ (X) = E Qν ( X)u(w (1 γ)x γ (X)). (22) Risks X and Y are comonotone and so, for every γ, (1 γ)x Y is comonotone with X. From point 2 of Proposition 1, we have V γ (X, Y ) = E Qν ( X)u(w (1 γ)x Y γ (X)). (23) By defining a function û( ) as û(w) = E Qν ( X)u(w Y ), (23) becomes V γ (X, Y ) = E Qν ( X)û(w (1 γ)x γ (X)). (24) Kihlstrom, Romer, and Williams [1981] prove that the function û( ) keeps many of the properties of u( ). For instance, it is increasing and concave when u( ) is. To prove our result, we need to establish that û is more concave than u. Indeed, a wellknown result from the insurance literature asserts that an agent with a more concave utility function buys more insurance. We define ψ(w,y) 5 as the solution of û (w) = u ( w E Qν ( X)Y ψ(w,y) ). If u > 0, then ψ(w,y)>0, and if we assume that u u û (w) (w) û (w) u u (w) and u u The result follows directly. To prove point 2: If u(x) = x, from (22) and (23), we obtain that V γ (X) = V γ(x,y). are decreasing, then Hence, the first-order conditions determining the optimal coverage with one and two risks are equal, and so we have γ = γ. The results obtained here are similar to those of Eeckhoudt and Kimball [1992] under probabilized uncertainty and expected utility preferences. However, in the CEU model,

12 18 MEGLENA JELEVA conditions on the utility function, appearing in the preceding proposition, do not have the same interpretation as in the expected utility model. Here they refer to attitudes towards wealth in a certain context and no longer to attitude towards uncertainty. For instance, the concavity of the utility function means that an increase in wealth gives more satisfaction to an agent when he is poor than when he is rich. A decreasing u corresponds to the fact that u the desire of an agent to increase his wealth decreases with this wealth. u > 0, together with u < 0, means that the marginal satisfaction decreases faster the larger is the wealth. In the following, we prove that, when a single level of loss is possible for both risks, the decreasing marginal utility for wealth is a sufficient condition for the background risk to raise the demand for insurance. Corollary 1: If =2and u < 0 then γ <γ. Proof. Let ={ω 1,ω 2 }; L and M are the amounts of loss corresponding, respectively, to the insurable and the background risk. If X (ω 1 ) = L and X (ω 2 ) = 0, the comonotony of the risks implies Y (ω 1 ) = M, Y (ω 2 ) = 0. To prove γ >γ when u < 0, it is enough to establish that, for all γ [0, 1], V γ (X, Y ) > V γ(x). The first-order condition, giving γ, is written as ( (1 ν(ω 2 )) L ) γ(x) u (w (1 γ)l M γ (X)) ν(ω 2 ) γ(x) u (w γ (X)) = 0. When only the insurable risk is present, γ is the solution of ( (1 ν(ω 2 )) L ) γ(x) u (w (1 γ)l γ (X)) ν(ω 2 ) γ(x) u (w γ (X)) = 0. For every u concave and every γ [0, 1], u (w (1 γ)l M γ (X)) > u (w (1 γ)l γ (X)). (25) It follows that V γ (X, Y ) > V γ(x), so, γ >γ.

13 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 19 The results of Proposition 4 and its corollary establish that, when the background risk is comonotone with the insurable risk, the impact of this noninsurable risk on the demand for insurance is determined not by the attitude towards uncertainty but by the attitude towards wealth in a certain framework. The independence of the results on the attitude towards uncertainty is justified by the fact that an addition of a comonotone risk doesn t affect the uncertainty faced by the individual in the sense defined in the first section, but only decreases his wealth in all states of nature. Under probabilized uncertainty and with rank-dependent expected utility preferences, Quiggin [1991] proved that if the rate of return X of a risky asset is a monotone spread of the rate of return Y, then an individual with u < 0, u 0 and a probability weighting function f (.) such that f (p) p, p [0, 1], will buy less of the risky asset if its return is X than when its return is Y. Transposed in an insurance context, this result means that an increase in risk in a sense of a monotone spread will increase the demand for insurance of the individuals whose preferences have the foregoing properties. It is shown by Wakker [1990] that under probabilized uncertainty, CEU is equivalent to RDEU (if some conditions are satisfied). Thus, our results should be close to those of Quiggin. Indeed, the conditions on the utility function in this context are similar to ours. In contrast, under our assumptions, we do not need restrictions on the capacity ν, whereas there is a condition on f (.) in the Quiggin results. This difference seems to come from the fact that in our assumptions, the conditions on the relation between the two acts is stronger that a monotone spread Anticomonotone risks In this section, we address the question of the impact of a background risk on the demand for insurance when it is anticomonotone with the insurable risk, that is, when low losses on the background risk correspond to high losses on the insurable risk and vice versa. Automobile and income risks are an example of such risks. In low-income periods, people use their cars less, and hence the risk of accident is lower. When the background risk is anticomonotone with the insurable risk, it will affect not only the wealth of the agent but also the uncertainty he faces. The impact of insurance on the uncertainty can then depend on the amount of loss corresponding to the uninsurable risk. This fact makes it necessary to study different cases for the amounts of loss corresponding to the two risks. To obtain explicit results, we first consider a two states of nature framework. Let ={ω 1,ω 2 }.The amounts of loss corresponding to the two risks are L and M.For the two risks to be anticomonotone, it is necessary to have X (ω 1 ) = L, X (ω 2 ) = 0 and Y (ω 1 ) = 0, Y (ω 2 ) = M. To be able to compare the predictions of the CEU model with those of the subjective expected utility model, we first consider the case where the preferences are represented by the subjective expected utility. As was mentioned in Section 2.1, those preferences are equivalent to CEU preferences with additive capacity. Proposition 5: Assume that the preferences of an individual facing two anticomonotone risks of loss are represented by (P, u), where P is a subjective probability distribution on and u 0. Then

14 20 MEGLENA JELEVA 1. If u < 0, then γ <γ ; 2. If u(x) = x, then γ = γ. Proof. We begin by proving point 1. In the subjective expected utility model, the weighting of the utility of the wealth corresponding to a state of nature is the subjective probability of this state, which doesn t depend on the corresponding wealth. So, the preferences representation functions corresponding to an insurance contract with one or two risks are V γ (X) = P(ω 1 )u(w (1 γ)l γ (X)) + P(ω 2 )u(w γ (X)), V γ (X, Y ) = P(ω 1 )u(w (1 γ)l γ (X)) + P(ω 2 )u(w M γ (X)). γ is the solution of the problem max V γ (X, Y ), γ and γ is the solution of max V γ (X). γ The second-order conditions of both problems are satisfied due to the concavity of u( ), and the first-order conditions are, respectively, V γ (X, Y ) = 0 and V γ (X) = 0. Due to the concavity of u( ), we have V γ (X, Y ) < V γ(x), and so γ <γ. The proof of point 2 is direct. For general CEU preferences (the capacity is not necessarily additive), we have seen that the weights corresponding to the different events depend on the ranking of the outcomes corresponding to the different events. The addition of a noninsurable risk, anticomonotone with the insurable one, may modify the ranking of the wealth in the different states of nature and make it depend on the amount of loss corresponding to the uninsurable risk. This fact makes the comparison between the insurance coverage with one and two risks more complex. First of all, we consider the case where the ranking is preserved, whatever the amount of coverage of the insurable risk is. This case corresponds to M > L. Proposition 6: Assume that an individual with preferences represented by (ν, u) with u < 0 or u(x) = x faces two anticomonotone risks of loss (X and Y ), one of which

15 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 21 is uninsurable. There are two possible states of nature, and the possible amount of loss corresponding to the uninsurable risk is higher than the other (M > L). Then 1. If ν is convex then γ <γ. 2. If ν is concave and ν(ω 2 ) 3. If ν is concave and ν(ω 2 ) (1 ν(ω 1 )) u (w M γ (X)) u (w γ, then γ >γ. (X)), then the result is indeterminate. (1 ν(ω 1 )) < u (w M γ (X)) u (w γ (X)) Proof. For every amount of coverage γ [0, 1], V γ (X, Y ) = ν(ω 1 )u(w (1 γ)l γ (X)) + (1 ν(ω 1 ))u(w M γ (X)), and when only the insurable risk is present, then V γ (X) = (1 ν(ω 2 ))u(w (1 γ)l γ (X)) + ν(ω 2 )u(w γ (X)). (26) To prove point 1, it is enough to establish that, if ν is convex, the following is true for every γ : If u < 0, V γ (X, Y ) V γ (X, Y ) < V γ(x) V γ(x) ( = (ν(ω 1 ) + ν(ω 2 ) 1) L ) γ(x) u (w (1 γ)l γ (X)) (27) γ(x) [(1 ν(ω 1 ))u (w M γ (X)) ν(ω 2 )u (w γ (X))]. When there are two states of nature, a capacity ν is convex if and only if ν(ω 1 ) + ν(ω 2 )<1. From this inequality and from the strict concavity of u( ), it follows that V γ (X, Y ) V γ(x) < 0, and so γ <γ. If u(x) = x, the two first-order conditions do not depend on γ. Then V γ (X, Y ) V γ(x) = (ν(ω 1 ) + ν(ω 2 ) 1)L < 0ifνis convex.

16 22 MEGLENA JELEVA To prove point 2, we see that if ν is concave, then ν(ω 1 ) + ν(ω 2 )>1. Using the same reasoning as above, we establish that γ >γ if (1 ν(ω 1 ))u (w M γ (X)) ν(ω 2 )u (w γ (X)) 0 If the previous inequality doesn t hold, both γ γ and γ <γ are possible which corresponds to point 3 of the proposition. From the above proposition, it follows directly that, for a convex ν( ), if the agent doesn t buy insurance with the single insurable risk, he will still not buy insurance with the background risk. Similarly, if ν( ) is concave, an agent buying full insurance with a single source of risk will, if the conditions of point 2 are satisfied, go on doing so in the presence of a background risk. Remark 1: The results of the foregoing proposition in the case of convex capacity are robust to the introduction of a third state of nature, where there is no loss either for the insurable or for the background risk. To prove this, it is necessary to use the property of the oquet integral, announced in (6). So, when the uninsurable potential loss is higher than the insurable one, an uncertaintyaverse agent, in the sense defined in Section 2.2, will reduce his coverage in the presence of a background risk. On the other hand, if he is an uncertainty lover, 6 he may increase his coverage. Those attitudes put in evidence the fact that attitude towards uncertainty is determinant for the demand for insurance when the two risks are anticomonotone. When the background risk dominates the insurable risk, insurance increases uncertainty by reducing the wealth in the state of nature where it is the lowest (loss on the uninsurable risk) and by increasing it in the case when it is the highest. Consequently, less insurance is desired by the uncertainty-averse agents. They still buy some insurance because of its possible wealth effect. This happens when the insurer is sufficiently more optimistic than the insuree. For uncertainty lovers, the impact of the background risk on the demand for insurance is the result of a trade-off between two contradictory effects, one resulting from the concavity of the utility function and the other from the concavity of the capacity. An agent with a concave utility function will try to increase his wealth in the states where it is the lowest, while an agent with concave capacity will try in contrast to increase the gap between wealth in the different states of nature. The resulting impact on the demand for insurance will depend on the relative power of the two effects. Comparing the results of Proposition 6 with those of Proposition 5, it appears that the CEU model explains optimal attitudes that are impossible in the subjective expected utility framework. Those attitudes can be used for an empirical test of the CEU preferences representation.

17 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 23 Let s now turn to the case where the loss on the insurable risk is higher than that of the uninsurable one. The ranking of the wealth in this case depends on the optimal coverage with a single source of risk. Proposition 7: Assume that an individual with preferences represented by (ν, u), with u < 0 and ν convex, faces two anticomonotone risks of loss, one of which is uninsurable. There are two possible states of nature, and the amount of loss corresponding to the insurable risk, is higher than the other (L > M). Then 1. If γ [0, 1 M L [, then γ >γ. 2. If γ ]1 M L, 1], then γ <γ. 3. If γ = 1 M L, then γ = γ. Proof. Wealth amounts in the two states of nature are not ordered in the same way for every level of coverage. Therefore, the function representing the preferences for an insurance contract is not differentiable everywhere. There is a level of coverage in which it is not differentiable, and it takes two different forms on the two intervals delimited by this amount of coverage. More explicitly, { V1γ (X, Y ) if γ [ 0, 1 M ] L V γ (X, Y ) = V 2γ (X, Y ) if γ [ 1 M L, 1], with V 1γ (X, Y ) = (1 ν(ω 2 ))u(w (1 γ)l γ (X)) + ν(ω 2 )u(w M γ (X)), V 2γ (X, Y ) = ν(ω 1 )u(w (1 γ)l γ (X)) + (1 ν(ω 1 ))u(w M γ (X)). When the agent faces only the insurable risk, the function representing preferences for an insurance contract is continuous and is written as in (26). We have to consider all the possibilities for the optimal coverage in the case of a single risk. Proof of point 1: γ [0, 1 M L [ From (25), in this interval, we have V 1γ (X, Y ) > V γ(x), (28) which proves that in this interval, the background risk raises the demand for insurance. From γ [0, 1 M L [, it follows that, for every γ ]1 M L, 1], V γ (X) < 0. Due to (27), V 2γ (X, Y ) < V γ(x), and we can conclude that γ [0, 1 M [ where it is equal to the local optimum on this L interval. Inequality (28) allows us to conclude that γ >γ.

18 24 MEGLENA JELEVA Proof of point 2: γ ]1 M, 1]. On this interval, from (27), we have L V 2γ (X, Y ) < V γ(x), (29) and the optimal coverage when the agent faces two risks is lower that when only one risk is present. On the other hand, γ ]1 ML ], 1 V γ (X) > 0 if γ [ 0,1 M [. L Knowing that V 1γ (X,Y ) > V γ (X),wehaveγ ]1 M, 1] which proves point 2. L Proof of point 3: If γ = 1 M, using the same arguments as previously, we obtain L V 1γ (X, Y ) > 0 and V 2γ (X, Y ) < 0, from which it follows that γ = γ = 1 M L. Before giving an intuitive justification of the preceding results, let s recall that a smoothing of wealth increases the satisfaction of all strongly uncertainty-averse agents (see Section 2.2). Consider an agent who faces a single source of risk and buys a coverage that belongs to the first interval (γ [0, 1 M [). Thus, in the presence of the background risk, an increase L in his coverage will transfer wealth from the state of nature where it is the highest (loss on the uninsurable risk) to the state of nature where it is the lowest (loss on the insurable risk). So all happens as in the case when the two risks are comonotone. Due to the concavity of the utility function, the agent increases his demand for insurance, but keeps it lower than 1 M, the limit above which the insurance will increase uncertainty. L On the other hand, if γ ]1 M, 1], an increase of the coverage will increase uncertainty, L because with an amount of coverage in this interval, the lowest wealth corresponds to the loss on the background risk. It is difficult to obtain explicit results when the losses corresponding to the two risks take more than two or three values. In this case, a comparison of the two first-order conditions that gives the optimal coverage with one and two risks is no longer evident. In the following, we give a condition on the two risks, which, when the risks take an infinite number of values and when the utility for wealth of an agent is linear, allows us to determine precisely the impact of an uninsurable risk on the demand for insurance. Proposition 8: The two risks that an individual faces take all the values in the interval [0, L] and are anticomonotone, and their sum is comonotone with Y. Then, if the individual has a linear utility function for wealth, the following assertions are true: 1. If ν is convex, γ <γ. 2. If ν is concave, γ >γ.

19 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 25 Proof. As in the preceding propositions, let s first write the functions representing preferences for an insurance contract with one risk and then with two. V γ (X, Y ) = (w (1 γ)x Y γ (X)) dν, V γ (X) = (w (1 γ)x γ (X)) dν. If X + Y is comonotone with Y, then (1 γ)x Y is comonotone with Y, and then with X. It follows from the properties of the oquet integral that it is possible to simplify V γ (X, Y ) and to write it as V γ (X, Y ) = w γ (X) (1 γ) Xdν+ ( Y ) dν. Furthermore, we have V γ (X) = w γ (X) + (1 γ) ( X) dν. Denneberg [1994] proved that, for a convex ν, Xdν< ( X) dν, and for a concave ν, Xdν> ( X) dν. It follows from those two results that V γ (X, Y ) < V γ(x), and thus γ <γ for ν convex and V γ (X, Y ) > V γ(x), and γ >γ for ν concave. The following generalization is possible for the preceding proposition. Proposition 9: The two risks that an individual faces take all the values in the interval [0, L] and are anticomonotone, and there exist β ]0, 1[ such that β X + Y is comonotone with X for β [ β,1] and β X + Y is comonotone with Y for β [0, β]. Then, if the utility for wealth of the individual is linear, the following statements are true:

20 26 MEGLENA JELEVA 1. If ν is convex, γ γ. 2. If ν is concave, γ γ. Proof. After putting γ = 1 β, where γ corresponds to the proportion of coverage, and noting that for all γ [0, 1 β], we have V γ (X, Y ) = V γ(x) for all ν, the proof follows directly by the same reasoning as in the preceding proposition. Note that, when the utility function is linear, there is no more ambiguity about the impact of the background risk on the demand for insurance when the capacity is concave, because this impact is now entirely determined by the attitude towards uncertainty, characterized by the capacity. The results of this last proposition show that to determine, in the general case of continuous risks, the impact of a background risk on the demand for insurance, it is necessary to truncate the interval [0, 1] on which the proportion of coverage is defined in subintervals such that, on each of these intervals, one of the risks dominates the other, in the sense that the ranking of the wealth corresponding to the different states of nature is the same as the ranking of X or Y. This procedure will make it possible to determine the impact of the uninsurable risk on every sub-interval. 5. Concluding remarks The introduction of nonprobabilized uncertainty in the model of demand for insurance with background risk and the preferences representation by the oquet expected utility makes the model closer to the real context in which individuals make their decisions. Comparing the results of our article with those obtained in the standard expected utility model in a risky context, we find that these standard results remain true when the two risks are comonotone. In contrast, when the two risks are anticomonotone, the predicted behavior differs from the predictions obtained in the expected utility framework, both under risk and under nonprobabilized uncertainty. These results can be used in testing, on insurance data or on data generated by laboratory experiments, the predictions of the CEU model versus those of subjective expected utility. In this article, the contract provided for the insurable risk was linear. It could be interesting to extend this study to the case of contracts with deductibles. Under nonprobabilized uncertainty and when preferences are represented by the oquet expected utility model, it was noted that the ranking of wealth in the different states of nature plays an important role in the determination of the impact of the background risk on the demand for insurance. The standard nonlinear contract, as well as the linear one, is rank preserving (if the coverage is not complete, the individual s wealth is always higher with a lower level of loss than with a higher one). This fact suggests that the same kind of results should be obtained when the contract is nonlinear.

21 BACKGROUND RISK, DEMAND FOR INSURANCE AND CHOQUET 27 Acknowledgments I am grateful to M. Cohen, C. Gollier, L. Eeckhoudt, E. Karni, P. Picard, B. Salanié, the participants to FUR X conference, and two anonymous referees for judicious remarks and helpful comments. Notes 1. A capacity is an increasing set function, which, contrary to a probability distribution, is not necessarily additive. 2. The definition of these concepts, their properties, and the way they are related to correlation are given in Section More precisely, this axiom is the Sure Thing principle, which in some sense replaces, in the uncertain context, the von Neumann Morgenstern independence axiom under risk. 4. The set of axioms leading to this preferences representation can be found in Schmeidler [1989] and in Gilboa [1987]. 5. In the expected utility framework, and under probabilized uncertainty, ψ(w,y)corresponds to the precautionary premium defined by Kimball [1990]. Prudence, corresponding to ψ(w,y)>0, is characterized in the expected utility model by u > An agent is an uncertainty lover if his preferences are concave, that is, if for every pair of acts X, Y and α [0, 1], X Y X αx + (1 α)y. In the CEU model, uncertainty appeal is characterized by a concave capacity. References BRIYS, E., KAHANE, Y., and KROLL, Y. [1988]: Voluntary Insurance Coverage, Compulsory Insurance and Risky-Riskless Portfolio Opportunities, Journal of Risk and Insurance, 4, CHATEAUNEUF, A., DANA, R.A., and TALLON, J.M. [1997]: Risk-Sharing Rules and Equilibria with Non- Additive Expected Utilities, to appear in Journal of Mathematical Economics. CHATEAUNEUF, A., KAST, R., and LAPIED, A. [1996]: oquet Pricing for Financial Markets, Mathematical Finance, 6, CHOQUET, G. [1953]: Théorie des capacités, Annales de l Institut Fourier, 5, DENNEBERG, D. [1994]: Non-Additive Measures and Integral. Dordrecht, Holland: Kluwer Academic Publishers. DOHERTY, N. and EECKHOUDT, L. [1995]: Optimal Insurance Without Expected Utility: The Dual Theory and the Linearity of Insurance Contracts, Journal of Risk and Uncertainty, 10, DOHERTY, N. and SCHLESINGER, H. [1983]: Optimal Insurance in Incomplete Markets, Journal of Political Economy, 91, DOHERTY, N. and SCHLESINGER, H. [1992]: Incomplete Markets for Insurance: An Overview, in Foundations of Insurance Economics, G. Dionne (Ed.), Boston: Kluwer Academic Publishers, EECKHOUDT, L. and GOLLIER, C. [1992]: Les Risques Financiers: Evaluation, Gestion, Partage. Ediscience. EECKHOUDT, L. and KIMBALL, M. [1992]: Background Risk, Prudence and the Demand for Insurance, in Contributions to Insurance Economics, G. Dionne (Ed.), Boston: Kluwer Academic Publishers, ELLSBERG, D. [1961]: Risk, Ambiguity and the Savage Axioms, Quarterly Journal of Economics, 75, GILBOA, I. [1987]: Expected Utility with Purely Subjective Non-Additive Probabilities, Journal of Mathematical Economics, 16, GILBOA, I. and SCHMEIDLER, D. [1989]: Maxmin Expected Utility with a Non-Unique Prior, Journal of Mathematical Economics, 18, GOLLIER, C. [1996]: Optimum Insurance of Approximate Losses, Journal of Risk and Insurance, 63,

22 28 MEGLENA JELEVA HIRSHLEIFER, J. and RILEY, J.G. [1979]: The Analytics of Uncertainty and Information An Expository Survey, Journal of Economic Literature, 17, KAHNEMAN, P. and TVERSKY, A. [1979]: Prospect Theory : an Analysis of Decision under Risk, Econometrica, 47, KIHLSTROM, R., ROMER, D., and WILLIAMS, S. [1981]: Risk Aversion with Random Initial Wealth, Econometrica, 49, KIMBALL, M. [1990]: Precautionary Saving in the Small and in the Large, Econometrica, 58, KNIGHT, F.H. [1921]: Risk, Uncertainty and Profit. New York: Houghton Mufflin. MAYERS, D. and SMITH, C. [1983]: The Interdependence of Individual Portfolio Decisions and the Demand for Insurance, Journal of Political Economy, 91, MONTESANO, A. and GIOVANNONI, F. [1995]: Uncertainty Aversion and Aversion to Increasing Uncertainty. Mimeo, Université de Milan. QUIGGIN, J. [1982]: A Theory of Anticipated Utility, Journal of Economic Behavior and Organization, 3, QUIGGIN, J. [1991]: Comparative Statics for Rank-Dependent Expected Utility Theory, Journal of Risk and Uncertainty, 4, SAVAGE, L. [1954]: The Foundation of Statistics. New York: J. Wiley. SCHMEIDLER, D. [1986]: Integral Representation without Additivity, Proceedings of the American Mathematical Society, 97, SCHMEIDLER, D. [1989]: Subjective Probability and Expected Utility without Additivity, Econometrica, 57, SHAPLEY, L.S. [1971]: Cores of Convex Games, International Journal of Game Theory, 1, WAKKER, P. [1990]: Under Stochastic Dominance, oquet Expected Utility and Anticipated Utility are Identical, Theory and Decision, 29, YAARI, M. [1987]: The Dual Theory of oice under Risk, Econometrica, 55,