On risk aversion with two risks

Journal of Mathematical Economics 31 Ž1999. 239–250

On risk aversion with two risks Israel Finkelshtain a

a,)

, Offer Kella

b,1

, Marco Scarsini

c,2,3

Department of Agricultural Economics, The Hebrew UniÕersity, P.O. Box 76-100, RehoÕot, Israel b Department of Statistics, The Hebrew UniÕersity, Mount Scopus, Jerusalem 91905, Israel c Dipartimento di Scienze, UniÕersita` D’Annunzio, Viale Pindaro 42, Pescara I-65127, Italy Received 1 May 1997; accepted 1 September 1997

Abstract We consider necessary and sufficient conditions for risk aversion to one risk in the presence of another non-insurable risk. The conditions Žon the bivariate utility function. vary according to the conditions imposed on the joint distribution of the risks. If only independent risks are considered, then any utility function which is concave in its first argument will satisfy the condition of risk aversion. If risk aversion is required for all possible pairs of risks, then the bivariate utility function has to be additively separable. An interesting intermediate case is obtained for random pairs that possess a weak form of positive dependence. In that case, the utility function will exhibit both risk aversion Žconcavity. in its first argument, and bivariate risk aversion Žsubmodularity.. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Risk aversion; Bivariate risk aversion; Submodularity; Concavity; Positive dependence

1. Introduction An economic agent is risk averse if, given the choice between a random amount of money X and its expected value Ew X x, she always prefers the second to the first. Risk aversion was studied by Arrow Ž1974. and Pratt Ž1964. within the framework of expected utility theory. Let u be the utility function of our economic )

Corresponding author. E-mail: [email protected]. E-mail: [email protected]. 2 E-mail: [email protected]. 3 This work was carried out when the author was visiting the Department of Statistics, Hebrew University of Jerusalem. 1

0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 5 8 - X

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agent, let w be her initial wealth and, for a random amount of money Z, let p satisfy uŽ w q E w Z x y p . s E uŽ w q Z . , where Ew Z x is the expected value of Z. The risk premium p depends on w and FZ Žthe distribution function of Z .. Whenever it is positive, the agent is ready to pay in order to eliminate risk from her scenario. The agent is risk averse if p ) 0 for all w g R and all random variables Z. Arrow and Pratt proved that risk aversions of different agents can be equivalently compared through their risk premiums or their utility functions, and therefore risk aversion Žin the sense of positive p . corresponds to the concavity of the utility function u. In their framework risk can always be completely removed Žby paying the risk premium., namely, full insurance is always available. Later, several authors have examined the case of partial insurance. For example, see Kihlstrom et al. Ž1981., Ross Ž1981., Jewitt Ž1986, 1987., Pratt and Zeckhauser Ž1990., Pratt Ž1988, 1990., and Gollier and Pratt Ž1996.. In this case, the initial wealth W is itself a random variable and the best that can be done is to replace W q Z with W q Ew Z x y p , where the risk premium p now depends on the joint law of ŽW,Z .. In almost all of these models the risks are additive and the utility function depends only on one argument. Pratt Ž1988. is an exception, in that he briefly considers multi-attribute utilities, whose arguments are one insurable and one uninsurable risk. He is able to prove a preservation result of decreasing risk aversion when the two risks are independent. Grant et al. Ž1992a,b. have provided conditions under which a multivariate choice problem can be reduced to a univariate problem concerning preferences over money. In many situations these conditions are not satisfied Žmainly when the exchange rates are not fixed and known. and therefore bona fide multivariate utilities have to be considered. Multi-attribute utilities have been studied by Kihlstrom and Mirman Ž1974., who extended the results of Arrow and Pratt and proved that it is possible to compare multi-attribute utility functions in terms of risk aversion only when the different utility functions represent the same preference ordering. This is quite a severe restriction for the applicability of multivariate comparison results. Grant Ž1995. somehow weakened this restriction by allowing risk premia to be random Žalbeit positive.. A different approach was taken by de Finetti Ž1952. and, independently, by Richard Ž1975., who defined a bivariate concept of risk aversion, which is completely independent of that of Arrow–Pratt, and is based on a comparison of bivariate lotteries having the same marginals Žsee also Epstein and Tanny Ž1980... Given two lotteries L1 s

½

Ž x1 , x 2 . Ž y1 , y 2 .

w.p. 1r2 w.p. 1r2

L2 s

½

Ž x1 , y2 . Ž y1 , x 2 .

w.p. 1r2 w.p. 1r2

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an agent who is bivariately risk averse will prefer L2 over L1 for all x 1 F y 1 , x 2 F y 2 . Bivariate risk aversion is equivalent to submodularity of the utility function, namely, for all x, y g R 2 , uŽ x k y . q uŽ x n y . F uŽ x . q uŽ y . , where x k y is the maximum of x and y Žcomponentwise., and x n y is the minimum of x and y Žcomponentwise.. Mosler Ž1982, 1984, 1987., and Scarsini Ž1985a,b, 1988. have generalized the concept of bivariate risk aversion to the multivariate case and have provided stochastic dominance conditions for bivariate and multivariate risk averters. In this article, the two approaches to risk aversion in the bivariate case will be connected and necessary and sufficient conditions will be provided for the preference of always insuring an insurable risk X, in the presence of an uninsurable risk Y, when the agent maximizes the expected utility Ew uŽ X,Y .x. We will show that the above mentioned preference is verified for all pairs of risks Ž X,Y . if and only if the utility function u admits an additive representation uŽ x, y . s g Ž x . q hŽ y ., with g concave. On the other hand, if we require the preference only for pairs of random variables that exhibit a particular Žweak. form of positive dependence, then a necessary and sufficient condition is that u is concave in its first argument and submodular. This fact has a natural interpretation in terms of risk aversion. When the two risks are weakly positively dependent, the agent prefers to insure the insurable risk if and only if she is risk averse in that risk and bivariately risk averse in both risks. These results generalize a result stated by Pratt Ž1988. according to which an agent always prefers to insure against X, in the presence of another independent risk Y if and only if her utility function is concave in the first risk. In the case of additive risks our results and Pratt’s collapse to the assumption that only concavity of the univariate utility function is required. The above results can be easily generalized to the multivariate case, where one risk can be insured, and several others are uninsurable. An example of possible applications of our results will be given in Section 3, where the demand for fair insurance will be considered.

2. Main results We will prove necessary and sufficient conditions for the following inequality E u Ž X ,Y . F E u Ž E w X x ,Y . ,

Ž 1.

under different conditions on the joint distribution of Ž X,Y .. In the following theorems, whenever taking expectations, conditional expectations, or correlations, it is implicitly assumed that the appropriate expectations Žor

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second moments. exist. Thus, for the sake of brevity, this will not be explicitly mentioned in the statements of the theorems. Following Landsberger and Meilijson Ž1994. Žand references cited therein. we will say that Ž X,Y . are co-monotone if any of the following equivalent conditions hold. Ži. P X F x 4 F P Y F y4 if and only if  X F x 4 ;  Y F y4 with probability one. Žii. The joint distribution function of Ž X,Y . is given by P  X F x ,Y F y 4 s min Ž P  X F x 4 , P  Y F y 4 . .

Ž 2.

Žiii. There exists some random variable Z, on a possibly different probability space, and non-decreasing real valued functions f, g such that Ž f Ž Z ., g Ž Z .. and Ž X,Y . have the same joint distribution. Theorem 1. The following statements are equiÕalent. (a) For any pair of independent random Õariables X and Y, inequality (Eq. (1)) holds. (b) The function u is concaÕe in its first argument. The result of Theorem 1 was stated in Pratt (1988), who noticed that concaÕity (as well as other properties) is preserÕed under mixture. We will see what happens if we expand the class of distributions, by allowing some dependence between X and Y. In particular we will choose a Õery general form of positiÕe dependence, which includes both independence and maximal dependence (co-monotonicity) as special cases. Theorem 2. The following statements are equiÕalent. (a) For any distribution of (X,Y), such that E[X < Y] is non-decreasing in Y, inequality (Eq. (1)) holds. (b) The function u is submodular and concaÕe in its first argument. (c) For any (X,Y) s (f (X1 , . . . ,X n ), g(Y1 , . . . ,Yn )), where n is an arbitrary positiÕe integer, {(X i ,Yi )< 1 F i F n} are independent pairs of co-monotone random Õariables, and for each argument f,g are either both non-decreasing or both non-increasing, or one of them is constant, inequality (Eq. (1)) holds. (d) For eÕery co-monotone (X,Y) inequality (Eq. (1)) holds. (e) For eÕery (X,Y) whose correlation coefficient is one, inequality (Eq. (1)) is satisfied. (f) For eÕery x 1 F x 2 and y1 F y2 , and for eÕery 0FlF1

l u Ž x 1 , y 1 . q Ž 1 y l . u Ž x 2 , y 2 . F l u Ž xl , y 1 . q Ž 1 y l . u Ž xl , y 2 . ,

Ž 3.

where xl s l x 1 q Ž1 y l. x 2 . The strongest aspect of Theorem 2 is Žb. implies Ža.. The rest serves simply to show that Žb. is a weak requirement in the sense that consideration of only a small subset of Ž X,Y . pairs Ži.e., in Že., those with perfect positive correlation. is needed to imply Žb..

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In our next result we will show that trying to satisfy inequality ŽEq. Ž1.. for all possible random pairs Ž X,Y . results in a substantially stronger condition on u. Theorem 3. The following statements are equiÕalent. (a) Inequality (Eq. (1)) holds for eÕery random (X,Y). (b) u(x,y) s g(x) q h(y) where g(x) is concaÕe. (c) Inequality (Eq. (1)) holds wheneÕer either (X,Y) or (X,y Y) are co-monotone. (d) Inequality (Eq. (1)) holds for eÕery (X,Y) of which absolute correlation is one. The above theorem shows that risk aversion to one risk in the presence of every other possible risk is equivalent to separability of the bivariate utility function Žand, of course, to concavity of the component that represents risk aversion.. This form of risk aversion is extremely strong, since it precludes the possibility of hedging with negatively correlated risks. This explains the equivalence to separability of the utility function. The case examined in Theorem 2 was more restrictive, but, in a sense more reasonable. It is possible to interpret condition Ža. of Theorem 2 as risk aversion to one risk, when the other risk does not allow any hedging, since the two risks are positively dependent Žin a very mild way.. It is remarkable that this form of risk aversion is equivalent to concavity in the first argument and submodularity of the bivariate utility function. The meaning of concavity is clear, and submodularity represents preference for hedging, whenever the marginal distributions are fixed. The three theorems show that if we enlarge the class of random pairs Ž X,Y . for which inequality ŽEq. Ž1.. holds, then the class of utility functions, that satisfy it, becomes smaller and smaller. There exists a sort of duality between the distribution of Ž X,Y . and the utility function u. If separability is achieved by the utility function, then the random variables can have any joint distribution, whereas if separability is achieved by the random variables Žthrough their independence., then the utility can have any form, provided it is concave in its first argument. The case examined in Theorem 2 is an interesting middle way, where no separability is assumed, either for the joint distribution of the random variables, or for the bivariate utility function. However, in this case both the joint distribution and the utility function are required to satisfy certain constraints that suggest that bivariate risk aversion Žrepresented by submodularity of the utility function. is the counterpart to weak positive dependence of the random variables. Remark 4. In Theorem 2, in each of the six statements, if we restrict Ž X,Y . to belong to A = B where A and B are open intervals Žpossibly rays or the whole real line. and also restrict the domain of u to be A = B, the proof will not change at all and hence, under this restriction, the theorem remains unchanged. Remark 5. Whenever Ž X,Y . are positive regression dependent Žsee Lehmann Ž1966.. then Ew f Ž X .< Y x is non-decreasing in Y for every non-decreasing f. Thus,

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it is clear that Ew X < Y x from condition Ža. in Theorem 2 is non-decreasing in Y. Note that in this case if u is concave in its first argument and submodular, then E u Ž f Ž X . ,Y . F E u Ž E f Ž X . ,Y .

Ž 4.

for every non-decreasing f. If in addition u is non-decreasing in its first argument, then E u Ž f Ž X . ,Y . F E u Ž f Ž E w X x . ,Y .

Ž 5.

for every non-decreasing concave f. It is interesting to note that two examples of Ž X,Y . which are positive regression dependent are the normal and Bernoulli cases discussed in Ža. and Žb. of Corollary 8 below. Remark 6. It is worthwhile pointing out that if f is concave, and h is non-decreasing, then uŽ x , y . s f x q hŽ y .

Ž 6.

is concave in x and submodular. If in addition f ŽP. is non-decreasing, then for every concave non-decreasing function g ŽP. the same conclusion holds for functions of the form uŽ x , y . s f g Ž x . q hŽ y . .

Ž 7.

The family of functions defined by Eq. Ž6. has been considered in the literature Že.g., Jewitt Ž1987... Remark 7. It has been brought to our attention that Rinott and Samuel-Cahn Ž1991., in a completely unrelated area needed and proved the following inequality: E max Ž g 1 Ž Z . , g 2 Ž Z . . G E max Ž g 1 Ž Z . , E g 2 Ž Z .

.

Ž 8.

for every random variable Z, non-decreasing g 1 and non-increasing g 2 . Since this inequality is valid if and only if Ew < g 1Ž Z . y g 2 Ž Z .
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Proof. Ža. In the normal case Ew X < Y x s aY q b, where a s CovŽ X,Y .rVarŽ Y . G 0. Hence, Ew X < Y x is non-decreasing in Y and the result follows from part Ža. of Theorem 2. Žb. It is easy to check that CovŽ X,Y . G 0 is equivalent to p 11 p 00 G p 01 p 10 , where pi j s P w X s i,Y s j x, which in turn is equivalent to p 10 p 11 E w X < Y s 0x s F s E w X < Y s 1x . Ž 9. p 00 q p 10 p 01 q p 11 Hence, as for the normal case, this result also follows from part Ža. of Theorem 2. I Remark 9. We note that it is known that for both Ža. and Žb. of Corollary 8 non-negative correlation is equivalent to association, that is, the requirement that CovŽ w Ž X,Y ., c Ž X,Y .. G 0 for all non-decreasing w and c . We also note that by applying part Žc. of Theorem 2 it is easy to construct more complicated distributions for which Eq. Ž1. holds while using these two cases as building blocks.

3. The demand for fair financial insurance In this section we apply our main results to the demand for fair insurance. Under the assumption of a single commodity, a well known proposition in the theory of insurance demand states the equivalence between the concavity of the utility function and the optimality of full purchasing of an actuarially fair insurance for every risk. Perhaps slightly less known is the fact that for a concave utility function of one commodity, if the insurance contract is actuarially unfair, strictly less than full purchasing is optimal Žsee e.g., Kreps Ž1990., p. 92.. Should these statements generalize to more complex situations, such as the purchasing of reimbursement health or unemployment insurance, where risks in other arguments of the utility are present? In general, this is a difficult question, and we will investigate it for situations where these risks and wealth exhibit the positive dependence assumed in Theorem 2. We will show that, under this hypothesis, the purchasing of full actuarially fair financial insurance is equivalent to aversion to income risk Ždefinition 3.1.. This gives our definition of risk aversion additional economic context and intuition. Let an agent face a risk of potential monetary loss L with P w L g A x s 1, where A is an open subinterval of Ž0,`.. Assume that EwyL < Y x is non-decreasing in Y. As an example consider the case of potential income loss due to the deterioration of the state of health. Suppose that the agent is offered an actuarially fair insurance contract against the potential loss L. That is, the insurance premium equals the expected indemnity payments a EL, where a g w0,1x denotes the share of wealth chosen to be insured. Using insurance terminology, 1-a is the individual’s co-insurance. Of course, certain restrictions, such as perfectly competitive insur-

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ance industry, negligible overhead costs and risk neutral underwriters, are required for the existence of actuarially fair insurance contracts. In the following proposition we assume that such contracts exist and focus on the individual’s behavior. We will denote the individual’s initial wealth by W0 and risks in other arguments of the utility by Y. Proposition 10. Let the utility function u Žwhose arguments are ŽW0 y L,Y .. be submodular and concave in its first argument. If EwyL < Y x is non-decreasing in Y and if an actuarially fair insurance contract is available, then each of the conditions of Theorem 2 is equivalent to the optimality of purchasing of full insurance Ž a s 1.. Proof. The payoff of an individual who purchases a percent of coverage is given by W0 y Ž1 y a . L y a EL, whose expected value is W0 y EL, which is independent of a . Therefore, EU Ž W0 y Ž 1 y a . L y a EL,Y . F EU Ž W0 y EL,Y .

Ž 10 .

where the right side is precisely the value corresponding to the purchasing of full insurance Ž a s 1.. I It should be noted that the restriction that a g w0,1x is not moot. For a ) 1, the first argument of U and Y will not exhibit positive dependence, and therefore we cannot insure that the inequality ŽEq. Ž10.. holds in this case. If U is only a function of its first argument, this restriction can be lifted and the conclusion would be that a s 1 is optimal among all a .

4. Future research In this paper we, focused exclusively on the question of risk aversion the presence of background risks in other argument of the utility function. A clear avenue for the continuation of our research is the analysis of comparative risk aversion in the same circumstances. That is, in future research one may search for conditions that assure that one individual is more risk averse than another, or identify variations that make a single individual more risk averse. To this end, some of the concepts and methods that were developed in the literature with additive background risks may be found useful. A particular example is the notion of vulnerability, recently proposed by Gollier and Pratt Ž1996.. These authors provided an interesting analysis of the situation examined in Theorem 1, namely, when X and Y are independent. They ask the following question: When does adding an unfair background risk to wealth makes a risk-averse individual behave in a more risk-averse way with respect to any other

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independent risk? Their suggested answer to this question is the notion of vulnerability, defined formally by Ew Y x F0 ´y

E uY Ž x q Y . E uX Ž x q Y .

Gy

uY Ž x . uX Ž x .

for all x.

Ž 11 .

The idea of risk vulnerability may be generalized to bivariate von Neumann– Morgenstern utility function as following: for all y Ew Y x F0 ´y

E ux xŽ x, yqY . E uxŽ x, yqY .

Gy

ux xŽ x, y. uxŽ x, y.

for all x.

Ž 12 .

The characterization of the class of bivariate utility function that exhibit the above property is an interesting subject for future research.

Acknowledgements We thank two referees for their helpful comments. Support from the Lady Davis Fellowship Trust is gratefully acknowledged.

Appendix A. Proof of Theorem 1. Ža. ´ Žb. Choose Y s y a.s., for some y g R4 . Then Eq. Ž1. becomes E uŽ X , y . F uŽ Ew X x , y . ,

Ž 13 .

which is true for every X if and only if u is concave in its first argument. Žb. ´ Ža. Since u is concave in its first argument, then for any independent X and Y, E u Ž X ,Y . s E E u Ž X ,Y . < Y x F E u Ž E w X < Y x ,Y . s E u Ž E w X x ,Y . . I

Ž 14 .

Proof of Theorem 2. The equivalence between Žc. and Žd. is clear, since Žd. is a special case of Žc. and Žc. can be derived from Žd. by sequentially conditioning on all but one Zi , each time applying Žd..Thus, we will first show that Žd. ´ Že. ´ Žf. ´ Žb. ´ Žd.. We then argue that Žd., Žb. ´ Ža. and that Ža. ´ Žf. which will complete the argument. Žd. ´ Že. Unit correlation is equivalent to positive linear dependence, in the sense that there is some Z such that X s aZ q b,Y s cZ q d and a,c are positive. In particular Ž X,Y . are co-monotone, hence this implication is obvious. Že. ´ Žf. For every choice of x 1 F x 2 and y 1 F y 2 we can take some z 1 - z 2 and 0 F a g R 2 and b g R 2 such that Ž x i , y i . s az i q b for i s 1,2. We then let Z s z 1 with probability l, Z s z 2 with probability 1 y l and apply Že..

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Žf. ´ Žb. Setting y s y 1 s y 2 in Eq. Ž3. gives the concavity of uŽP, y ., for every fixed y. Setting l s 1r2 one obtains u w x 2 , y 2 x y u Ž x 1 q x 2 . r2, y 2 F u Ž x 1 q x 2 . r2, y 1 y u w x 1 , y 1 x .

Ž 15 .

Applying Eq. Ž15. to the sequence z kn s x1 q Ž krn.Ž x 2 y x 1 . gives n n n n u z kq 1 , y 2 y u z k , y 2 F u z k , y 1 y u z ky1 , y 1

Ž 16 .

for k s 1, . . . ,n y 1, and upon summation u w x 2 , y 2 x y u x 1 q Ž x 2 y x 1 . rn, y 2 Fu x 2 y Ž x 2 y x 1 . rn, y 1 y u w x 1 , y1 x .

Ž 17 .

Since uŽP, y . is concave for every fixed y, it is also continuous, hence, letting n ™ ` gives uw x 2 , y 2 x y uw x 1 , y 2 x F w x 2 , y1 x y u w x 1 , y1 x .

Ž 18 .

for every x 1 F x 2 and y 1 F y 2 . Hence u is submodular. Žb. ´ Žd. For a fixed y g B, let uq 1 Ž x , y . slim

u Ž x q h, y . y u Ž x , y .

hx0

Ž 19 .

h

be the right derivative of uŽP, y . evaluated at x. Its existence is assured as uŽP, y . Ž . is concave. Also, uq 1 x,P is non-increasing, for every fixed x, as a limit of functions having this property Žfrom submodularity, in particular see Eq. Ž18... We Ž . Ž . also know that uq 1 x, y is a supergradient of u P, y at x, i.e., u Ž x 2 , y . y u Ž x 1 , y . F uq 1 Ž x1 , y . Ž x 2 y x1 .

Ž 20 .

for every x 1 g A, x 2 g A and y g B. In particular, u f Ž Z . , g Ž Z . y u E f Ž Z . , g Ž Z . Fuq 1 E f Ž Z. ,gŽ Z. yE f Ž Z .

Ž f Ž Z. Ž 21 .

.

for every random variable Z and every non-decreasing f and g such that Ew f Ž Z .x exists and is finite. Let z be such that sup  z < f Ž z . - E f Ž Z .

4 F z F inf  z < f Ž z . ) E f Ž Z . 4 ,

Ž 22 .

so that f Ž z F Ew f Ž Z .x for every z - z and f Ž z . G Ew f Ž Z .x for z ) z . Since w w Ž .x Ž .x uq 1 E f Z , g P is a non-increasing function on R,

Ž uq1

E f Ž Z . , g Ž Z . y uq 1 E f Ž Z. ,gŽ z .

.Ž f Ž Z. yE f Ž Z. . F0 Ž 23 .

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w w Ž .x Ž .xŽ f Ž Z . y Ew f Ž Z .x.. Therehence, the right side of Eq. Ž21. is F uq 1 E f Z ,g z fore u f Ž Z . , g Ž Z . Fu E f Ž Z . , g Ž Z . q uq 1 E f Ž Z. ,gŽ z . yE f Ž Z .

Ž f Ž Z. Ž 24 .

.

and upon taking expected values in Eq. Ž24. the proof of this part complete. Žd.,Žb. ´ Ža. This follows from E uX ,Y . s E E u Ž X ,Y . < Y

F E u Ž E w X < Y x ,Y . F E u Ž E w X x ,Y . Ž 25 .

where the first inequality follows from the concavity of u in its first argument, while the second follows from the fact that Ž Ew X < Y x,Y . are co-monotone. Ža. ´ Žf. Let PŽ X,Y . s Ž x 1 , y 1 .4 s 1 y PŽ X,Y . s Ž x 2 , y 2 .4 s l with Ž x 1 , y 1 . Ž F x 2 , y 2 .. Then either y 1 s y 2 , in which case Ew X < Y s y 1 x s Ew X < Y s y 2 x s Ew X x, or Ew X < Y s y 1 x s x 1 F x 2 s Ew X < Y s y 2 x. Thus Ew X < Y x is non-decreasing in Y and Žf. follows from Ža.. Thus, the theorem is proved. I Proof of Theorem 3. It is clear that Žb. ´ Ža. ´ Žc. ´ Žd., hence it suffices to show that Žd. ´ Žb.. To begin, first note that Žd. holds if and only if for every random variable Z Žwith finite second moment., every a g R 2 and b g R 2 , Eq. Ž1. holds for Ž X,Y . s aZ q b. Since Žd. holds, we must have, from the equivalence of parts Žf. and Žb. in Theorem 2, that uŽ x, y . is concave in x and that both uŽ x, y . and uŽ x,y y . are submodular in Ž x, y .. Therefore, u is both submodular and supermodular, which implies that for every x 1 F x 2 and every y 1 F y 2 u Ž x 2 , y1 . y u Ž x 1 , y1 . s u Ž x 2 , y 2 . y u Ž x 1 , y 2 . .

Ž 26 .

This also implies that this equality holds for every x 1 , x 2 , y 1 , y 2 Žnot necessarily satisfying x 1 F x 2 and y 1 F y 2 .. Thus, we can write u Ž x , y . y u Ž 0, y . s u Ž x ,0 . y u Ž 0,0 .

Ž 27 .

and by taking g Ž x . s uŽ x,0. y uŽ0,0. and hŽ y . s uŽ0, y . we have that uŽ x, y . s g Ž x . q hŽ y .. The concavity of g follows immediately since Eq. Ž1. implies that Ew g Ž X .x F g Ž Ew X x. for every random X. I

References Arrow, K.J., 1974. Essays in the Theory of Risk-Bearing. North-Holland, New York. de Finetti, B., 1952. Sulla preferibilita. ` Giornale degli Economisti e Annali di Economia 11, 685–709. Epstein, L.G., Tanny, S.M., 1980. Increasing generalized correlation: a definition and some economic consequences. Can. J. Econ. 13, 16–34. Gollier, C., Pratt, J.W., 1996. Risk vulnerability and the tempering effect of background risk. Econometrica 64, 1109–1123.

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