Intensity of Preferencesin the Presence of Bivariate Risks
David Crainich
CNRS (LEM, UMR 9221) and Iéseg School of Management
Louis Eeckhoudt
Iéseg School of Management and LEM (UMR 9221)
Olivier Le Courtois
emlyon business school
Abstract: We show that ratios of cross partial derivatives of the utility
measure the intensity of preferences towards bivariate risks in the expected utility model. In order to specify the meaning of these ratios, the definition of an(n1 , n2 )th degree increase in risk is combined with the concept of marginal rateof substitution between two increases in risks of different degrees. An intuitive
explanation of these ratios and an illustration of their usefulness in understanding economic decisions are also provided.
JEL Classification: D81
Keywords: Ratio of cross partial derivatives; (n1 , n2 )th degree increase inbivariate risks; (n1 , n2 )th degree correlation aversion.

1

Introduction
Decision making usually occurs in the presence of multiple risks affecting several aspects of an individual’s well-being. Indeed, risk management strategies(such as precautionary savings, the purchase of insurance contracts, preventive actions, or portfolio choices) are often adopted when agents are exposed torisks associated with attributes of the utility function other than wealth (healthstate, environment, relatives’ wealth or health state,...). Papers dealing withprecautionary saving in bivariate (Courbage and Rey, (2007)) or multivariate(Denuit, Eeckhoudt, and Menegatti, (2011); Jouini, Napp, and Nocetti, (2013);
Courbage, (2014)) settings, with the effects of health risks on portfolio choices(Edwards, (2008); Crainich, Eeckhoudt, and Le Courtois, (2017)), or with investments improving future random health or environmental quality (Denuit,
Eeckhoudt, and Menegatti, (2011); Jouini, Napp, and Nocetti, (2013)) offer illustrations of the way individuals behaving according to the expected utility
model make that kind of decision. Note that non-financial decisions have also
been treated in the literature: the effect of risky life expectancy on the choiceof the intensity of a medical treatment in the presence of therapeutic risk has
for instance been addressed in Bleichrodt, Crainich and Eeckhoudt (2011).
The above-mentioned papers can be classified into two categories: those interested in the direction of changes and those analyzing the intensity of changes
in the decision variable once risks are either introduced or modified. Among thesecond category, Jouini, Napp, and Nocetti (2013) and Crainich, Eeckhoudt,
and Le Courtois (2017) highlight that the trade-offs dictating agents’ decisions
depend on the value of various measures of the intensity of higher order multivariate risk attitude. In both papers, the latter are expressed as ratios wherein
numerators are cross partial derivatives (of different orders) of the utility function and the denominator is the marginal utility of wealth. These measures
share the same denominator because the cost of the economic decision is purely
financial in both contributions. This cost is a forsaken current consumptionthat improves the future value of several attributes in Jouini, Napp, and Nocetti (2013) while it corresponds to reduced expected return when less risky
assets are held in portfolios in Crainich, Eeckhoudt, and Le Courtois (2017).
When the effect of the introduction of multivariate risks on the strength of
precautionary saving is analyzed, the denominators of the intensity measuresof multivariate risk attitudes are second-order cross partial derivatives (Jouini,
Napp, and Nocetti (2013); Courbage (2014)). However, as we illustrate below
with a simple example, the cost of economic decisions can also be expressed in
more general units, such as increases in higher order risks or increases in higher
order correlations. In these cases, the denominators of the intensity measurestake other forms. The purpose of our paper is to provide a comprehensive
framework that encompasses all the measures of the intensity of preferences
towards bivariate risks in the expected utility model.
Our work is based on three important contributions in risk theory: 1) theinterpretation of the signs of successive cross partial derivatives of the utility
function exposed in Eeckhoudt, Rey, and Schlesinger (2007); 2) the conceptof nth order increases in risk à la Ekern (1980); 3) the rate of substitution
between two stochastic changes introduced in Liu and Meyer (2013). Based onthese concepts, we propose an intuitive interpretation of the ratios of partial
derivatives. As will be highlighted in the paper, these ratios will prove useful2

when dealing with risky decisions in bivariate settings.
To provide these interpretations, we first make use of the bivariate preference
ordering introduced by Eeckhoudt, Rey, and Schlesinger (2007). The latter concept generalizes that of correlation aversion (Richard, (1975)), which refers to
the preference for the dissociation between wealth and health1 losses. Richard(1975) indicates that correlation aversion corresponds to u(1,1) < 0 in the expected utility model2 . Its generalization to higher orders is obtained when the
dissociation principle is extended from losses to higher order zero-mean risks
applied to both attributes. In the expected utility model, Eeckhoudt, Rey, andSchlesinger (2007) make the connection between this preference ordering andthe signs of successive cross partial derivatives of the utility function i.e. thesigns of u(2,1) , u(1,2) , u(2,2) ,..., u(n1 ,n2 ) . Doing so, they establish the concept of(n1 , n2 )th degree correlation aversion.
We then apply the concept of nth degree increase in risk (Ekern (1980))
to bivariate cases in order to obtain changes in bivariate distributions corresponding to the preference ordering defined in Eeckhoudt, Rey, and Schlesinger(2007). More precisely, while Ekern (1980) defines an nth degree increase inrisk by keeping the n − 1 first moments of the distribution constant, Denuit etal. (2013) define an (n1 , n2 )th degree increase in bivariate risk as the movementfrom one distribution to another such that the (n1 , n2 )th comoment of these two
distributions differs and such that three constraints are satisfied. Namely, thebivariate distributions must have: 1) the same first n1 moments for the marginal
distribution of the first variable; 2) the same first n2 moments for the marginal
distribution of the second variable; 3) the same first (n1 + n2 − 1) comoments. It
can then be shown that every (n1 , n2 )th degree correlation averse agent dislikes(n1 , n2 )th degree increases in bivariate risks (or, equivalently, (n1 , n2 )th degreeincreases in correlation).
Equipped with this definition, we extend Liu and Meyer (2013) and propose
a general measure of the intensity of preferences towards bivariate risks. Liu′′−u′′′ u′′′and Meyer (2013) show that the existing measures ( −uu′ , u′′ , u′ ,...) of theintensity of preferences towards risk in the univariate setting correspond to a rateof substitution between two increases in risk of different degrees. Specifically,
suppose that G(x) has more nth degree risk than F (x), that H(x) has more
mth degree risk than F (x) and that the agent is indifferent between G(x) and(1 − T )F (x) + T H(x). Liu and Meyer (2013) indicate that T is proportional tomthe ratio uun that corresponds to a measure of the aversion towards increases innth degree risk. As noted by these authors, T is "the weight that the decision
maker is willing to put on H(x) when forming a mixture of F (x) and H(x)to avoid having an nth degree risk increase from F (x) to G(x) instead ". The
same principle, applied to (n1 , n2 )th degree increases in correlation, is used in
our paper in order to provide an intuitive interpretation of the measures of theintensity of bivariate risk preferences. Because the latter measures are expressed
as ratios of cross partial derivatives, they do not depend on utility scales andallow us to compare individual behaviors towards bivariate risks.
The ratios of partial derivatives that we introduce generalize the existingmeasures of the intensity of bivariate preferences proposed by Jouini, Napp,
1 Or any other attribute of the utility function.
2 The following notation is adopted throughout the paper: u(n1 ,n2 ) refers to the nth and nth
2
1
partial derivatives of the utility function with respect to its 1st and 2nd arguments, respectively.

3and Nocetti (2013) and by Crainich, Eeckhoudt, and Le Courtois (2017). The
latter authors exploit the concept of an (n1 , n2 )th degree correlation aversion(Eeckhoudt, Rey, and Schlesinger, (2007)) and extend the technique used by
Crainich and Eeckhoudt (2008) who measure the intensity of downside risk
aversion in an univariate setting. Namely, the intensity of preference towards a
distribution F (x) compared to a distribution G(x) is equivalent to the maximum
amount of money one is willing to sacrifice in order to be exposed to F (x) rather
than to G(x). As a result, the existing measures of the intensity of preferencesin bivariate or multivariate settings are all based on the marginal utility of
wealth. We extend this interpretation by stating that the compensation mighttake other forms, such as changes in the marginal distributions corresponding
to decreases in mth degree risk or changes in the distribution corresponding todecreases in (n1 , n2 )th degree correlation.
Our paper is organized as follows. In Section 1, we introduce a saving decision that must be made in the presence of an environmental risk in order
to illustrate the usefulness of a particular measure of the intensity of bivariate(1,2)
preferences, namely of − uu(1,1) . An explanation of the connection between ourdecision problem and that particular measure of preference towards bivariate
risks is then established. Section 2 generalizes this illustration by adopting anapproach based on the comoments of the distribution. Section 3 provides another generalization based on the concept of an increase in (n1 , n2 )th degreecorrelation. Section 4 concludes.

1

An illustration

Consider the following savings problem. An agent lives two periods during
which he earns an identical income (w). Besides this income, the environment
quality (e) she enjoys also enters in her utility function so that her preferences
are represented by u(w, e). We suppose that the utility of wealth and the utilityof environmental quality are both increasing and concave, so that: u(1,0) > 0,u(0,1) > 0, u(2,0) < 0, and u(0,2) < 0. The agent has the opportunity to transfer
money through savings from period 1 to period 2. To simplify this problem,
suppose that the agent’s utility function is the same across periods and that theinterest rate and the rate of intertemporal preference are both null. Finally, nochange in the environmental quality is anticipated between period 1 and period
2. The agent’s maximization problem is given by:max (u(w − s, e) + u(w + s, e)) .
s(1)

It is straightforward to show that there are no savings (s∗ = 0) at the
optimum. Suppose now that a new environmental program is implemented
between period 1 and period 2. The expected improvement in environmental
quality due to the program is positive, but the final effect of the program isrisky. Suppose that this improvement is given by the random variable q̃ suchthat q̃ = q0 + ǫ̃, with q0 > 0, E(ǫ̃) = 0, and where the variance of ǫ̃ is denotedby σeǫ2 . The agent’s maximization then becomes:max (u(w − s, e) + Eu(w + s, e + q̃)) .
s

4

(2)

The first-order condition that is associated with this program and that defines the new optimal level of savings s∗∗ is given by:u(1,0) (w − s∗∗ , e) + Eu(1,0) (w + s∗∗ , e + q̃) = 0.

(3)

We approximate Eu(1,0) (w + s∗∗ , e + q̃) around (w + s∗∗ , e) with a limited
Taylor expansion:
σ2
Eu(1,0) (w+s∗∗ , e+q̃) ≃ u(1,0) (w+s∗∗ , e)+q0 u(1,1) (w+s∗∗ , e)+ eǫ u(1,2) (w+s∗∗ , e).
2
When s∗∗ is evaluated at s∗ = 0, the first-order condition indicates the extent
to which the implementation of the environmental program modifies savings.
The agent saves (s∗∗ > s∗ = 0) if:
u(1,2) (w, e)
2q0− (1,1)> 2.
σeǫu(w, e)
σ2

Besides, the higher the gap between q0ǫe and the ratio of cross partial deriva(1,2)

tives of the bivariate utility function − uu(1,1) (w,e), the higher the difference be(w,e)∗∗∗
tween s and s . This result obtained in a very simplified model illustrates
that the ratio of cross partial derivatives of the utility function explains decisions made by agents when they are exposed to bivariate risks.
In order to explain this result, we first use the interpretation of the conceptsof correlation aversion and cross-prudence in health - or, equivalently, of theconcepts of (1, 1)th and (1, 2)th degrees increase in correlation aversion - respectively defined by Richard (1975) and Eeckhoudt, Rey, and Schlesinger (2007).
Consider the distributions A1 and B1 in the figure below and suppose that theagents are both (1, 1)th correlation averse (u(1,1) < 0) and (1, 2)th correlation
averse (u(1,2) > 0).
w, ew + s, e

1/2

1/2

A1

B1

1/2

1/2
w + s, e + q0

w, e + q0

Following Richard (1975), it can be shown that A1 ≺ B1 when individuals are(1, 1)th correlation averse. Indeed, using limited Taylor expansions, we obtain:
Eu(B1 ) − Eu(A1 ) ≃ −s q0 u(1,1) (w, e).
This preference towards risk in a bivariate setting reflects the fact that agents
prefer the sure gains (s on wealth and q0 on the environmental attribute) to bedissociated (as in B1 ) rather than associated in the same state of the world (as in
A1 ). In Richard (1975), the two outcome of the distributions A1 and B1 are twostates of the world occurring with equal probability. Similar interpretations can
5be made if the outcomes are two time periods (assuming that there is no discountfactor). Applied to our saving problem, (1, 1)th correlation averse agents do nottransfer of money from period 1 to period 2 in order not to concentrate the gains(the extra wealth due to savings and the expected improvement in environmental
quality) in the same period.
Suppose now that same agents must choose between the distributions C1
and D1 described in the figure below.
w, ew + s, e

1/2

1/2

C1

D1

1/2

1/2
w + s, e + ǫ̃w, e + ǫ̃Using Eeckhoudt, Rey, and Schlesinger (2007), we obtain D1 ≺ C1 since individuals are (1, 2)th degree correlation averse. Using limited Taylor expansions,
it can be shown that:
σ2
Eu(C1 ) − Eu(D1 ) ≃ s eǫ u(1,2) (w, e).
2Agents who are (1, 2)th degree correlation averse prefer to associate the improvement on the first attribute (due to s) with the pain (taking here the formof a zero mean risk ǫ̃) on the second attribute of the utility function. As a result,and once again in order not to concentrate the gains in the same period, therisk associated with the environmental quality in period 2 provides an incentive
to save for agents who are (1, 2)th degree correlation averse.
Consequently, the implementation of the new environmental program has
contradictory effects on correlation averse agents: the positive expected environmental value (q0 ) deters savings as agents are (1, 1)th degree correlation
averse while the environmental risk (ǫ̃) provides an incentive to save as they are(1, 2)th degree correlation averse. The decisions to associate an extra unit ofwealth to e + q0 + ǫ̃ through savings rather than to e through dissavings depends
on the relative values of Eu(B1 ) − Eu(A1 ) and of Eu(C1 ) − Eu(D1 ) i.e. on
whether2q0
u(1,2) (w, e)> 2.
− (1,1)σeǫu(w, e)
which corresponds to condition (1). The final effect of the environmental
change on the saving decision thus depends not only on q0 and ǫ̃ but also on therelative aversions to (1, 1)th degree and (1, 2)th degree correlation. The left-handside of the inequality sign of condition (1) actually corresponds to a marginal
rate of substitution between the improvement in quality q0 and the risk ǫ̃ that
(1,2)
(w,e)
is attached to it. As a result, − uu(1,1) (w,e)
is a measure of the intensity of (1, 2)thdegree correlation aversion.

6

Note that other measures exist. For instance starting from distributions
C1 and D1 , Crainich, Eeckhoudt and Le Courtois (2017) suggest that the sureamount of money which, when subtracted from w in the upper branch of distribution C, restores indifference for (1, 2)th degree correlation averse agents is one
such measure. In that case, the intensity of (1, 2)th degree correlation aversion
is measured relatively to the marginal utility of wealth, i.e. relatively to (1, 0)th
degree correlation aversion. The next section generalizes this idea to describe
the measures of preferences towards bivariate risks.

2

Substitutions of Lotteries

In the previous section, we illustrated the role played by the ratio of crosspartial derivatives of a utility function in explaining a decision made by anagent exposed to a particular risky situation. This kind of trade-off can beextended to any risks. This is what we do in this section where an approach "inthe small" is adopted.
To do so, consider first the lotteries A and B described below:
α̃, φ̃

κ̃, φ̃

1/2

1/2
B

A

1/2

1/2
κ̃, β̃

α̃, β̃

where κ̃, φ̃, α̃, and β̃ are four distinct random variables. The lotteries A and B
respectively generalize the lotteries A1 and B1 described in the previous section
if we assume that κ̃ = w, φ̃ = e, α̃ = w+s and β̃ = e+q0 . Lottery B differs fromlottery A in that the random variables κ̃ and α̃ attached to the first argumentof the utility function are swapped between the two possible states of the world.
As we did in Section 1, we start by computing the difference in expected utility
between the lotteries A and B.
Proposition 2.1. We assume that all the partial derivatives of u are non-null.
We have the following approximation:nE(u(B)) − E(u(A)) ≈

n

2
1 X
E((α̃k − κ̃k )(φ̃h − β̃ h ))1Xuk,h (0, 0).
2
k! h!

(4)

k=1 h=1

Proof. See Appendix.
In the spirit of the utility premium of Friedman-Savage, a decompositionof the gain in expected utility resulting from being exposed to distribution B
instead of distribution A is proposed in Eq. (4). Proposition 2.1 underlines that
this gain is the weighted sum of differences between the moments and comomentsof A and B, where the weights are the cross-derivatives of the utility function.
Note that the two distributions A and B are constructed by associating thesame variables κ̃, φ̃, α̃, and β̃ in two different ways. A and B thus have the same7

marginal distributions and, consequently, the same moments. Besides, when κ̃and α̃ on the one hand, and φ̃ and β̃ on the other, do not have the same higherorder moments (α̃ and κ̃ do not have the same (k)th moment and φ̃ and β̃ do
not have the same (h)th moment), the comoments lower than (k, h)th of A and
B are equal. Consequently, we deduce from Proposition 2.1 that the preference
for B over A is only dictated by the sign of the (k, h)th cross-derivative of theutility function.
For instance, as the difference between distributions A1 and B1 (see Section1) lies in the way the two sure gains attached to each argument of the utility
function (s to the first argument and q0 to the second one) are associated, thepreference for A1 or for B1 depends on the sign of u(1,1) . Likewise, the difference
between distributions C1 and D1 is related to the way a sure gain attached to
the first argument (s) and a risk attached to the second one (ǫ̃) are associated:
the preference for one of these two distribution over the other then depends on
the sign of u(1,2) .
In what follows, we assume that lottery A has more (n1 , n2 )th degree risk
than lottery B in the sense defined below:
Definition 2.2 (More (n1 , n2 )th degree risk). We say that the lottery A has
more (n1 , n2 )th degree risk than the lottery B if3E((α̃k − κ̃k )(φ̃h − β̃ h )) = 0(5)(−1)n1 +n2 −1 E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 )) > 0.

(6)∀(k, h) < (n1 , n2 )andThe link with Ekern’s (1980) definition of an nth order increase in risk isstraightforward: while the last comoment of A and B differs (see Eq. (6)), themoments of each of the two variables are equal and they share the same lower
order comoments (see Eq. (5)).
We now come to the characterization of agents’ preferences towards these
two lotteries.
Definition 2.3 ((n1 , n2 )th degree risk aversion around 0). An agent u is (n1 , n2 )thdegree risk averse around 0 if and only if (−1)n1 +n2 −1 u(n1 ,n2 ) (0, 0) > 0.
This definition generalizes those of (1, 1)th degree and of (1, 2)th degree correlation aversion provided in Section 1. In what follows, being (n1 , n2 )th degreerisk averse and being (n1 , n2 )th degree correlation averse are two terminologies
referring to the same preference.
Proposition 2.4. Consider an agent u who is (n1 , n2 )th degree risk aversearound 0. This agent has the choice between two lotteries A and B, where thelottery A has more (n1 , n2 )th degree risk than the lottery B. Then, B is preferredto A and the difference in utilities between the two lotteries is given by4E(u(B)) − E(u(A)) ≈1 E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 ))
un1 ,n2 (0, 0) > 0.
2
n1 ! n2 !

(9)

3 The case (k, h) < (n , n ) includes the limit subcases (n , h) < (n , n ) with h < n , and
1
2
1
1
2
2
(k, n2 ) < (n1 , n2 ) with k < n1 .
4 If we assume in addition that α̃ and κ̃ are independent from φ̃ and β̃, we deduce that if∀(k, h) < (n1 , n2 )E(α̃k ) = E(κ̃k ) or E(φ̃h ) = E(β̃ h ),8

Proof. See Appendix.
As in the lotteries used by Eeckhoudt, Rey and Schlesinger (2008), the distributions A and B are constructed so that they have equal lower order comoments:
the preference that a (n1 , n2 )th degree risk averse individual has for the latter
distribution over the former thus only results from the difference between theirlast comoment.
Let us now introduce the lotteries C and D:γ̃, υ̃

θ̃, υ̃

1/2

1/2
C

D

1/2

1/2
γ̃, δ̃

θ̃, δ̃

where θ̃, δ̃, γ̃, and υ̃ are independent random variables. The lotteries C and
D respectively generalize the lotteries C1 and D1 described in Section 1 if we
assume that θ̃ = w, υ̃ = e, γ̃ = w + s and δ̃ = e + ǫ̃. Lottery D differs fromlottery C in that the random variables θ̃ and γ̃ attached to the first argumentof the utility function are swapped between the two possible states of the world.
If we assume that the lottery C has more (m1 , m2 )th degree risk than thelottery D and that u is (m1 , m2 )th degree risk averse around 0, then we can use
Proposition 2.4 to obtain:
E(u(D)) − E(u(C)) ≈1 E((γ̃ m1 − θ̃m1 )(υ̃ m2 − δ̃ m2 ))
um1 ,m2 (0, 0) > 0. (10)2
m1 ! m2 !

As we did in the previous section, we can now define the marginal rate of
substitution as the rate at which the agent is willing to exchange the transition
from C to D against that from B to A.
Definition 2.5 (Substitution Rate). The marginal rate of substitution Tu is
defined as follows:
E(u(B)) − E(u(A))
(11)
Tu =
E(u(D)) − E(u(C))
Since the lottery A has more (n1 , n2 )th degree risk than the lottery B and thelottery C has more (m1 , m2 )th degree risk than the lottery D, the substitutionthen

1 E(α̃n1 − κ̃n1 ) E(φ̃n2 − β̃ n2 )
un1 ,n2 (0, 0).
2
n1 ! n2 !
and B is preferred to A provided that
E(u(B)) − E(u(A)) ≈E(α̃n1 − κ̃n1 ) E(φ̃n2 − β̃ n2 ) un1 ,n2 (0, 0) > 0.

(7)

(8)

Further, if we assume that the sign of un1 ,n2 (0, 0) is (−1)n1 +n2 −1 , then assuming the condition in (8) is equivalent to assuming that the sign of the product of moments E(α̃n1 −κ̃n1 ) E(φ̃n2 − β̃ n2 ) is also (−1)n1 +n2 −1 .

9rate becomesTu =E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 )) m1 ! m2 ! un1 ,n2 (0, 0),E((γ̃ m1 − θ̃m1 )(υ̃ m2 − δ̃ m2 )) n1 ! n2 ! um1 ,m2 (0, 0)(12)where we use Eqs. (9) and (10) and where the ratio of moments in this equation
is independent of the utility function u.
Suppose that a potential decision generates both an (n1 , n2 )th increase inbivariate risk (such as the transition from B to A) and an (m1 , m2 )th decreasein bivariate risk (such as the transition from C to D). Such a decision will beapproved by agents as long as:E(u(D)) − E(u(C)) > E(u(B)) − E(u(A)) ⇔un1 ,n2 (0, 0)E((γ̃ m1 − θ̃m1 )(υ̃ m2 − δ̃ m2 )) n1 ! n2 !
<.
um1 ,m2 (0, 0)E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 )) m1 ! m2 !
Finally, the marginal rate of substitution T allows the comparison between
decisions made by two agents u and v, that is, it indicates the agent who isthe more willing to trade an (n1 , n2 )th increase in bivariate risk against an(m1 , m2 )th decrease in bivariate risk:
Proposition 2.6. Consider two decision makers u and v. When the lottery A
has more (n1 , n2 )th degree risk than the lottery B and the lottery C has more
(m1 , m2 )th degree risk than the lottery D, then,Tu ≥ Tv⇔(−1)n1 +n2 −1 u(n1 ,n2 ) (0, 0)(−1)n1 +n2 −1 v (n1 ,n2 ) (0, 0)
≥
m
+m
−1
(m
,m
)
1
2
1
2
(−1)
u
(0, 0)(−1)m1 +m2 −1 v (m1 ,m2 ) (0, 0)(13)

Proof. The proof is a direct consequence of Eq. (12) and of Eq. (6) applied to
A and B, and to C and D, respectively.
Proposition 2.6 establishes that the propensity to substitute changes in bivariate risks depends on the appropriate ratio of cross-derivatives of the utility
function. It generalizes the problem exposed in Section 1 where savings wasshown to simulateously increase (1, 1)th degree risk and decrease (1, 2)th degree risk. In that case, the individual whose (1, 1)th degree correlation aversion
relative to (1, 2)th degree correlation aversion is lower (see condition (1)) has
the higher propensity to save. Proposition 2.6 extends interpersonal comparisons to decisions leading to higher-order changes in bivariate risks. Namely,
it states that individuals whose (n1 , n2 )th degree correlation aversion relativeto (m1 , m2 )th degree correlation aversion is lower are more willing to accept(n1 , n2 )th degree increases in risk in exchange for (m1 , m2 )th degree decreases
in risk.

3

Substitutions in the General CaseIn the previous section, we adopted an approach "in the small" based on preferences that agents have for the comoments of bivariate distributions. We go
one step further in the generalization process in this section by adopting an
10

approach "in the large". The latter is based on the extension to the bivariatecase of the definition of an "nth degree increase in risk" introduced by Ekern(1980). Then, we adapt Liu and Meyer (2013) to the bivariate case in order toprovide a general characterization of the intensity of bivariate preferences.
We consider a two dimensional random variable whose cumulative distribution function is denoted by F . We construct by successive
integrations the
Rxfunction F [k,h] (., .). Indeed, we have: F [k,h] (x, y) = a F [k−1,h] (s, y)ds andRyF [k,h] (x, y) = a F [k,h−1] (x, t)dt, where the initial point is F [1,1] (x, y) = F (x, y).
Following Denuit et al. (2013), we define the concept of more high-orderbivariate degree risks, ("(n1 , n2 )th degree increases in risk"’), which is an extension to the bivariate case of Ekern’s definition of an "nth degree increase inrisk".
Definition 3.1 (More (n1 , n2 )th degree risk). A distribution G has more (n1 , n2 )thdegree risk than a distribution F if and only if, for all (k, h) ≤ (n1 , n2 ),G[k,h] (s, b) = F [k,h] (s, b)∀s ∈ [a, b],(14)

G[k,h] (b, t) = F [k,h] (b, t)∀t ∈ [a, b],(15)∀(s, t) ∈ [a, b]2 .

(16)

and
and also
G(n1 ,n2 ) (s, t) ≥ F (n1 ,n2 ) (s, t)

Equipped with this definition, we extend Ekern (1980)’s result to the caseof bivariate random vectors:
Proposition 3.2. Let X = (X1 , X2 ) and Y = (Y1 , Y2 ) be bivariate random
vectors that are respectively F -distributed and G-distributed. When G has more
(n1 , n2 )th degree risk than F , thenE(X1k X2h ) = E(Y1k Y2h )∀(k, h) < (n1 , n2 )

(17)

and
(−1)n1 +n2 E(X1n1 X2n2 ) ≤ (−1)n1 +n2 E(Y1n1 Y2n1 )(18)

Proof. For brevity, we omit the proof, which is obvious and relies on multiple
integrations by parts.
This proposition states that the comparison of X and Y at order (n1 , n2 ) is
equivalent to the comparison of their non-centered comoment of order (n1 , n2 )when all of their lower-order non-centered moments and comoments are identical. These last constraints are expressed in (5) in the previous section and in(17) in this one. Proposition 3.2 makes the connection with the previous section
where the distributions differ in their last comoments. However, note that there
is no equivalence between (14), (15) and (16) on the one hand and (17) and(18) on the other: if any (n1 , n2 )th degree increase in risk implies that the bivariate distributions considered differ in their last comoment (their lower order
comoments being equal), the opposite is not true. As a result, the approachbased on the concept of (n1 , n2 )th degree increase in risk is more general thanthe comoments approach - based on approximations - that we developed in theprevious section.
We now come to the characterization of agents faced with bivariate choices.
11Definition 3.3 ((n1 , n2 )th degree risk aversion). An agent is (n1 , n2 )th degreerisk averse if and only if(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) > 0∀(s, t) ∈ [a, b]2 .

Then, we can relate the comparisons of (n1 , n2 )th degree risks to the preferences of agents as follows:
Proposition 3.4. G has more (n1 , n2 )th degree risk than F if and only if every(n1 , n2 )th degree risk averter prefers F to G.
Proof. Let X and Y be bivariate random vectors that are F-distributed and Gdistributed, respectively. The proof, which is obvious, relies on the computationofZ bZ bZ bZ bu(s, t) dG(s, t)u(s, t) dF (s, t) −E(u(X)) − E(u(Y)) =
a

a

a

a

via multiple integrations by parts, and on the study of its sign.
Proposition 3.4 in this section is the equivalent of Proposition 2.4 in theprevious section. Both of these propositions refer to preferences that (n1 , n2 )thdegree risk averse agents have towards bivariate distributions. But they say
nothing about the intensity of these preferences or about the choices that two
different (n1 , n2 )th degree risk averse agents might make. The transition fromthe concept of a "direction of preferences" to that of an "intensity of preferences" in bivariate settings is established - as in Sections 1 and 2 - through thecomparison between two changes in distributions.
To do so, we extend to the bivariate case a technique introduced by Liu andMeyer (2013) in the univariate context. Assume that G(x, y) has more (n1 , n2 )thdegree risk than F (x, y) and that H(x, y) has more (m1 , m2 )th degree risk thanF (x, y). What is the value of T such that an agent is indifferent between G(x, y)and (1 − T )F (x, y) + T H(x, y)? The higher T , the more the agent is sensitive
to (n1 , n2 )th degree increases in risk relative to (m1 , m2 )th degree increases inrisk. As a result, T indicates the intensity of preferences in bivariate settings.
It is straightforward to show that for the agent u, the value of T - denoted
Tu - is given by:
Definition 3.5.

Rb Rb

Tu = ab abRRu(s, t)(dF (s, t) − dG(s, t))u(s, t)(dF (s, t) − dH(s, t))a aAs in Liu and Meyer (2013), the proposed measure of intensity of bivariate
risk aversion is a ratio of two expected utility changes. While the Arrow-Prattmeasure of risk aversion is sufficient to deal with the introduction of risks or withchanges in risk "in the small", Ross (1981) has shown that a stronger measure of
the increase in risk aversion was required once increases in risk or once changes
in risk "in the large" were addressed. We now indicate how the generalizationof the Ross measure of "stronger risk aversion" is helpful when higher-order12

increases in bivariate risk "in the large" are considered. The Arrow-Pratt andRoss increases in higher-order bivariate risk aversion are respectively defined asfollows:
Definition 3.6. u is ((n1 , n2 )/(m1 , m2 ))th degree more risk averse than v if,
for all (s, t) ∈ [a, b]2 ,(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t)(−1)n1 +n2 −1 v (n1 ,n2 ) (s, t)≥,(−1)m1 +m2 −1 u(m1 ,m2 ) (s, t)(−1)m1 +m2 −1 v (m1 ,m2 ) (s, t)
whereas u is ((n1 , n2 )/(m1 , m2 ))th degree Ross more risk averse than v if, forall (s, t) ∈ [a, b]2 and for all (w, z) ∈ [a, b]2 ,(−1)n1 +n2 −1 v (n1 ,n2 ) (s, t)(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t)
≥
.
m
+m
−1
(m
,m
)
1
2
1
2
(−1)
u
(w, z)(−1)m1 +m2 −1 v (m1 ,m2 ) (w, z)
These definitions allow the interpersonal comparison of the propensity to
exchange an (n1 , n2 )th increase in bivariate risk against an (m1 , m2 )th decreasein bivariate risk. Combining what precedes, the following Proposition can indeed
be established:
Proposition 3.7. We consider two agents u and v that are each both (n1 , n2 )thdegree risk averse and (m1 , m2 )th degree risk averse. The following statements
are equivalent:(i) u is ((n1 , n2 )/(m1 , m2 ))th degree Ross more risk averse than v on [a, b]2 ,(n1 ,n2 )(m1 ,m2 )

(s,t)
(w,z)
so that there exists λ > 0 such that uv(n1 ,n2 ) (s,t)≥ λ ≥ uv(m1 ,m2 ) (w,z)
for all(s, t) ∈ [a, b]2 and (w, z) ∈ [a, b]2 .

(ii) There exist λ > 0 and φ : [a, b]2 → R such that u = λv + φ and such that(−1)m1 +m2 −1 φ(m1 ,m2 ) (s, t) ≤ 0 and (−1)n1 +n2 −1 φ(n1 ,n2 ) (s, t) ≥ 0 for all(s, t) ∈ [a, b]2 .
(iii) Tu ≥ Tv for all F , G, and H such that G has more (n1 , n2 )th degree risk
than F and H has more (m1 , m2 )th degree risk than F
Proof. See Appendix.
The willingness to trade (n1 , n2 )th increases in risk against (m1 , m2 )th decreases in risk depends on the intensity T of preferences in bivariate settings.
Part (i) of Proposition 3.7 establishes the condition under which an individual whose preferences are represented by the utility function u is more Ross
bivariate risk averse than another individual whose preferences are represented
by v. Parts (ii) of Proposition 3.7 defines the transformation of the bivariate
utility function that preserves this partial order. Part (iii) of the Proposition
underlines that this ordering determines the propensity to substitute two kindsof change in bivariate risks and, as a result, is a measure of the intensity ofpreferences towards risk in the bivariate setting.

13

In Section 2, we adopted an approach "in the small" based on the comparison between comoments of the distribution. In such a case, one can compare the decisions made by two agents by using the Arrow-Pratt concept of an((n1 , n2 )/(m1 , m2 ))th increase in bivariate risk aversion (see Proposition 2.6).
Because an approach "in the large" is adopted in this section, the Propositionindicates that the notion of ((n1 , n2 )/(m1 , m2 ))th increase in bivariate risk aversion in the Ross sense is required in order to compare the preferences expressedby two agents agents exposed to changes in bivariate risk.

4

Conclusion

Decision-makers consider various aspects of their well-being (wealth, health,environment quality,...) when they make choices that modify the risks they
are exposed to. In this paper, we show how the ratios of cross partial derivatives of the utility function measure the intensity of preferences that individuals
display in bivariate settings. When an approach "in the small" based on thecomparison of the comoments of the distributions is adopted, we show that thedecisions agents make depend on their ((n1 , n2 )/(m1 , m2 ))th Arrow-Pratt risk
aversion. Then, we extend Liu and Meyer (2013) to the bivariate case to showthat the propensity to substitute (n1 , n2 )th increases in risk for (m1 , m2 )th decreases in risk "in the large" can be explained by the concept of an "increase in((n1 , n2 )/(m1 , m2 ))th Ross risk aversion". These last ratios constitute generalmeasures of the intensity of preferences towards risk in bivariate settings.

Appendix
Proof of Proposition 2.1
From the definition of the lotteries, we have:def∆ = E(u(B))−E(u(A)) =
or
∆=
Then,

1
1
1
1
E(u(α̃, φ̃))+ E(u(κ̃, β̃))− E(u(κ̃, φ̃))− E(u(α̃, β̃))
2
2
2
2

i 1h
i
1h
E(u(α̃, φ̃)) − E(u(κ̃, φ̃)) −E(u(α̃, β̃)) − E(u(κ̃, β̃)) .
2
2

"n
#
"n
#
n1
n1
1
1
X
X
X
X
1
κ̃k
κ̃k
α̃k
α̃k
1
uk,0 (0, φ̃) −uk,0 (0, φ̃) − E
uk,0 (0, β̃) −uk,0 (0, β̃) ,
∆≈ E
2
k!
k!
2
k!
k!
k=1

k=1

so that

k=1

#
"n
1

X
1
α̃k − κ̃k ∆≈ E
uk,0 (0, φ̃) − uk,0 (0, β̃) .
2
k!
k=1

Developing with respect to the second argument, we obtain:
"n n
#
2
1 X
X
1
α̃k − κ̃k φ̃h − β̃ h∆≈ Euk,h (0, 0) .
2k!
h!
k=1 h=1

which is our result.
14

k=1

Proof of Proposition 2.4
We first cancel the terms in Eq. (4) that are null in Eq. (5). This operation
allows us to write:
E(u(B)) − E(u(A)) ≈1 E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 ))
un1 ,n2 (0, 0),2
n1 ! n2 !

which is equivalent to(19)

E(u(B)) − E(u(A))≈12(−1)n1 +n2 −1E((α̃n1 − κ̃n1 )(φ̃n2 − β̃ n2 ))n1 ! n2 !

!

(−1)n1 +n2 −1 un1 ,n2 (0, 0) .

(20)

Then, we observe that the first bracket is positive thanks to Eq. (6) and that thesecond bracket is positive thanks to the (n1 , n2 )th degree risk aversion around
0 property of u.

Proof of Proposition 3.7
The equivalences are shown by generalizing the arguments in Liu and Meyer(2013).
(i) ⇒ (ii). Using λ defined in (i), we construct φ as follows:φ(s, t) = u(s, t) − λ v(s, t)∀(s, t) ∈ [a, b]2 .

We readily check that(−1)m1 +m2 −1 φ(m1 ,m2 ) (s, t)= (−1)m1 +m2 −1 u(m1 ,m2 ) (s, t) − λ (−1)m1 +m2 −1 v (m1 ,m2 ) (s, t) ≤ 0
and(−1)n1 +n2 −1 φ(n1 ,n2 ) (s, t)= (−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) − λ (−1)n1 +n2 −1 v (n1 ,n2 ) (s, t) ≥ 0.
(ii) ⇒ (iii). From the definition of Tu , we have:Zb Zbau(s, t) [(1 − Tu )dF (s, t) + Tu dH(s, t)] =

Zb Zb
a

a

u(s, t) dG(s, t).

a

Then,
Zb Zb
a

u(s, t) dG(s, t) =

a

Zb Zb
a

(λv(s, t) + φ(s, t)) dG(s, t)

a

and
Zb Zb
a

a

(λv(s, t) + φ(s, t)) dG(s, t) ≤

Zb Zb
a

λv(s, t) dG(s, t) +

a

15

Zb Zb
a

a

φ(s, t) dF (s, t)because φ shows (n1 , n2 )th degree risk aversion. Further, using the definition of
Tv and the fact that φ shows (m1 , m2 )th degree risk taking, we have:Zb Zbaλv(s, t) dG(s, t) +

Zb Zb
a

a

a

Zb Zb

φ(s, t) dF (s, t) ≤λv(s, t) [(1 − Tv )dF (s, t) + Tv dH(s, t)] +

a

a

a

Zb Zb

φ(s, t) [(1 − Tv )dF (s, t) + Tv dH(s, t)] ,aso thatZb Zbaλv(s, t) dG(s, t)+

Zb Zb
a

a

φ(s, t) dF (s, t) ≤

Zb Zb
a

a

u(s, t) [(1 − Tv )dF (s, t) + Tv dH(s, t)] .

a

Recombining the above equalities and inequalities shows that Tu ≥ Tv .
(iii) ⇒ (i). By integration by parts, we can show thatZb Zba(−1)k+h−1 u(k,h) (s, t) (G[k,h] (s, t) − F [k,h] (s, t)) ds dt

a

Zb h

(−1)k+h−1 u(k,h) (s, t) (G[k+1,h] (s, t) − F [k+1,h] (s, t))

a

−

Zb Zb

ib

a

dt

(−1)k+h−1 u(k+1,h) (s, t) (G[k+1,h] (s, t) − F [k+1,h] (s, t)) ds dt.

aaUsing conditions (14) and (15), we obtain:
Zb Zb(−1)k+h−1 u(k,h) (s, t) (G[k,h] (s, t) − F [k,h] (s, t)) ds dt =

a

a

Zb Zb

(−1)(k+1)+h−1 u(k+1,h) (s, t) (G[k+1,h] (s, t) − F [k+1,h] (s, t)) ds dt,aawhich shows that an identical formula prevails at orders k and k + 1 on wealth.
By extension, we have:Zb Zbau(s, t) (dF (s, t) − dG(s, t)) ds dt =

a

Zb Zb
a

(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) (G[n1 ,n2 ] (s, t) − F [n1 ,n2 ] (s, t)) ds dt.

a16

Therefore,Tu ≥ Tv
Rb Rb⇔(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) (G[n1 ,n2 ] (s, t) − F [n1 ,n2 ] (s, t)) ds dt

a a
Rb Rb

≥

(−1)m1 +m2 −1 u(m1 ,m2 ) (s, t) (H [m1 ,m2 ] (s, t) − F [m1 ,m2 ] (s, t)) ds dt

a a

Rb Rb

(−1)n1 +n2 −1 v (n1 ,n2 ) (s, t) (G[n1 ,n2 ] (s, t) − F [n1 ,n2 ] (s, t)) ds dt

a a
Rb Rb

(−1)m1 +m2 −1 v (m1 ,m2 ) (s, t) (H [m1 ,m2 ] (s, t) − F [m1 ,m2 ] (s, t)) ds dt

a aNext, we assume that (i) does not hold: we can find two compact sets [c, d]2
and [e, f ]2 and µ > 0 such thatu(n1 ,n2 ) (s, t)u(m1 ,m2 ) (w, z)< µ < (m ,m )
(n
,n
)
1
2
v
(s, t)
v 1 2 (w, z)
for all (s, t) ∈ [c, d]2 and (w, z) ∈ [e, f ]2 .
Because we assume that v is (n1 , n2 )th and (m1 , m2 )th degree risk averse,
we have:(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) < µ (−1)n1 +n2 −1 v (n1 ,n2 ) (s, t)
for all (s, t) ∈ [c, d]2 and(−1)m1 +m2 −1 u(m1 ,m2 ) (w, z) > µ (−1)m1 +m2 −1 v (m1 ,m2 ) (w, z)
for all (w, z) ∈ [e, f ]2
Then, choosing F̃ , G̃ and H̃ such that G̃ − F̃ > 0 on [c, d]2 , H̃ − F̃ > 0 on[e, f ]2 and such that these two differences are null outside the compact sets, we
can write:
Zb Zb(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) (G̃[n1 ,n2 ] (s, t) − F̃ [n1 ,n2 ] (s, t)) ds dt

<

a

a

µ

Zb Zb
a

(−1)n1 +n2 −1 v (n1 ,n2 ) (s, t) (G̃[n1 ,n2 ] (s, t) − F̃ [n1 ,n2 ] (s, t)) ds dt

a

and
Zb Zb
a

(−1)m1 +m2 −1 u(m1 ,m2 ) (s, t) (H̃ [m1 ,m2 ] (s, t) − F̃ [m1 ,m2 ] (s, t)) ds dt

>

a

µ

Zb Zb
a

(−1)m1 +m2 −1 v (m1 ,m2 ) (s, t) (H̃ [m1 ,m2 ] (s, t) − F̃ [m1 ,m2 ] (s, t)) ds dt

a17

so that
Rb Rb(−1)n1 +n2 −1 u(n1 ,n2 ) (s, t) (G̃[n1 ,n2 ] (s, t) − F̃ [n1 ,n2 ] (s, t)) ds dt

a a
Rb Rb

<

(−1)m1 +m2 −1 u(m1 ,m2 ) (s, t) (H̃ [m1 ,m2 ] (s, t) − F̃ [m1 ,m2 ] (s, t)) ds dt

a a

Rb Rb

(−1)n1 +n2 −1 v (n1 ,n2 ) (s, t) (G̃[n1 ,n2 ] (s, t) − F̃ [n1 ,n2 ] (s, t)) ds dt

a a
Rb Rb

(−1)m1 +m2 −1 v (m1 ,m2 ) (s, t) (H̃ [m1 ,m2 ] (s, t) − F̃ [m1 ,m2 ] (s, t)) ds dt

a awhich is a contradiction. Therefore (i) holds.

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19