Discussion Papers In Economics
And Business

A Note on Sufficient Conditions of Cross Risk
Vulnerability

Yusuke Osaki
Discussion Paper 06-17

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Note on Sufficient Conditions of Cross Risk
Vulnerability

Yusuke Osaki
Discussion Paper 06-17

June 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Note on Sufficient Conditions of Cross Risk
Vulnerability∗
Yusuke Osaki†
JSPS Research Fellow and Osaka UniversityAbstract
This note gives sufficient conditions of cross risk vulnerability introducedby Malevergne and Rey (2005), which is the equivalent condition to guaranteethat an unfair non–monetary background risk makes decision makers more riskaverse. The sufficient conditions determined by this note expand the results
for univariate utility function into bivariate utility functions.
JEL Classification: D81.
Keywords: Background risk, cross risk vulnerabiilty, risk aversion.

∗I would like to thank Masamitsu Ohnishi for his valuable comments and discussions. The
financial support from JSPS Fellowships for Young Scientists, is greatly acknowledged.
†Addreess: Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toyonaka, Osaka 560–0043, Japan. E–mail: osakiyusuke@srv.econ.osaka-u.ac.jp

1Introduction

In recent two decades, a considerable number of studies has been focused on theeffects of background risks on optimal decisions under uncertainty. Since changes
in risk aversion have unambiguous comparative statics predections in most models
with uncertainty, many authors derived conditions to guarantee that backgroundrisks make decision mekers more risk averse. A landmark concerning this topic
is the “risk vulnerability” introduced by Gollier and Pratt (1996), which is theweakest condition among them. As the risk vulnerability is a technical condition,
several authors determined sufficient conditions of risk vulenrability which havesome intuitive interpretations. The following two conditions are well known: (i) thestandard risk aversion derived by Eeckhoudt and Kimball (1992), (ii) the decreasing
and convex risk aversion derived by Gollier and Pratt (1996). An excellent survey
on this topic can be found in Part IV in Gollier (2001), the interestend readers cansee it.
Most studies concerning background risks in optimal choices under uncertainty,
consider decision makers with univariate von–Neumann Morgenstern utility functions. However, there are many situations in which decision makers have multi–
dimensional utility functions. For exapmle, utilities of decision makers are not only
depend on finacial wealth, but also health status in the context of health economics.
We can list many examples of decision making with multi–dimension, like environmental status, social status, and so on. Hence, we need to understand the effectsof background risks on optimal choices with multi–dimension under uncertainty. In
their recent intersting paper, Malevergne and Rey (2005) generalize risk vulnerability to the case that decision makers have bivariate utiity functions, and they call it“cross risk vulenerability”. The purpose of this note gives the sufficient conditionsof cross risk vulnerability, which are correspond to the well–known sufficient conditions of risk vulnerability above mentioned. A best way to understand the effects ofbackground risks in multi–dimensional decisions under uncertainty is a generalization of results in one–dimensional as possible as we can, baecause we obtain many
understaings regarding them in last two decades. One importance of our results
comes from this point of view.

2

Preliminary Analysis

First of all, we give some preliminary analyses to derive sufficient conditions ofcross risk vulnerability introduced by Malevergne and Rey (2005). Our setting is
identical with them, and therefore we borrow their notaions. Let us consider adicision maker (DM) with a bivariate (von Nuemann Morgenstern) utility function1

U : R × R → R. We impose standard assumptions of the utility functions U : (i)U1 , U2 > 0, (ii)U11 , U22 ≤ 0, and (iii) U12 ≤ 0. Here subscripts denote partial andcross derivatives of corresponding arguments, i.e. Ui denotes the partial derivative of
the i–th argument, and Uij denotes the cross derivative of the i– and j–th arguments.
The assumption (ii) and (iii) mean risk and correlation aversion, respectively. The
notion of correlation aversion was initiated by Epstein and Tanny (1980). In a
recent paper, Eeckhoudt, et al. (2005) characterized some desiable properties ofcross derivatives of bivariate utility functions using preferences over the fifty–fifty
lottery suggested by Eeckhoudt and Schlensinger (2005). As Malevergne and Rey(2005), the first and second arguments stand for wealth and health, respectively.
We define the cross derived utility function toward health background risk ²̃ as
V (x, y) = E[U (x, y + ²̃)].

(1)The health background risk ²̃ is a random variable followed by the distributionfunction F : [², ²] → [0, 1]. For the sake of simplicity, we assume that the distibutionfunction F is differentiable, that is the density function f (²) = F 0 (²) exists. The
cross derived utility function is a slight modification of derived utility function for
univariate utility functions defined by Kihlstrom, et al. (1981) and Nachman (1982).
The Arrow–Pratt (absolute) risk aversions toward a wealth risk of the utility function
U and V are given asA(x, y) := −U11 (x, y)
V11 (x, y), and AV (x, y) := −.
U1 (x, y)
V1 (x, y)(2)

Following Malevergne and Rey (2005), we define the cross risk vulnerability:
Definition 2.1. The utility function U is cross risk vulnerable, if any unfair healthbackground risks make the DM in a more risk averse manner:
E[²̃] ≤ 0 ⇒ AV (x, y) ≥ A(x, y), ∀(x, y).

(3)

As a slight generalization of the notion introduced by Kimball (1990), we define
two precautionary premiums toward a zero–mean health background risk:E[U1 (x, y + ²̃)] = U1 (x, y − ΨH ) = U1 (x − ΨW , y).

(4)

We call ΨH a precautionary premium in health, and ΨW a precautionary premiumin wealth, respectively. It is obvious that both the precoutionary premium ΨH andΨW are the functions of x, y, ²̃, and U , but we suppress them in order to simplify
the notation. Using the Arrow–Pratt approximation, we obtain the precautionary
premium in health and wealth as follows:µ¶U122 (x, y)
11H(5)−E[²̃2 ] = P H (x, y)E[²̃2 ],Ψ '2U12 (x, y)22WΨ

1'2µU122 (x, y)−U11 (x, y)¶1E[²̃2 ] = P W (x, y)E[²̃2 ].
2(6)P H and P W can be interpreted as Kimball (absolute) cross prudence in healthand wealth toward a health risk. As recalling that U12 and U11 is positive, this(approximation) equality imlies that
ΨH(W ) ≥ (≤) 0 ⇔ U122 (x, y) ≥ (≤) 0.

(7)

Eeckhoudt, et al. (2005) justified the nonnegativity of U122 , and they call it crossprudence in wealth.

3Main Result

In the remainder of the paper, we give three sufficient conditions of cross risk vulnerabulity, thier remarks, and proofs.

3.1

Propositions and their Remaks

Proposition 3.1. If the Arrow–Pratt risk averison is decreasing and convex in thesecond argument, i.e. A2 (x, y) ≤ 0 and A22 (x, y) ≥ 0, then a utility function U iscross risk vulenrable.
Proposition 3.2. If the Arrow–Pratt risk aversion is decreasing in the second argument, and the Kimball prudence in halth is decreasing in the first argument, i.e.
A2 (x, y) ≤ 0 and P1H (x, y) ≤ 0, then a utility function U is cross risk vulenrable.
Proposition 3.3. If the Arrow–Pratt risk aversion is decreasing in the first andsecond argument, and the Kimball prudence in wealth is decreasing in the firstargument, i.e. A1 (x, y) ≤ 0, A2 (x, y) ≤ 0 and P1W (x, y) ≤ 0, then a utility function
U is cross risk vulenrable.
Proposition 3.1 was already obtained by Malevaergne and Rey (2005) in a complete different way. They obtained it as a corollary of proposition 1 and 3 in theirpaper. In contrast to them, we prove it in a direct way. While they did not give acomment, Proposition 3.1 is a natural generalization of the result that is a sufficientcondition of risk vulenerablity in the case of univariate utility functions, which isdetermined by Gollier and Pratt (1996). Indeed, our proof provides for a different
proof of their result. As our proof is similar to Eeckhoudt and Kimball (1992) which
derived a sufficient condition of risk vulenerablity in the case of univariate utility
functions, Proposition 3.2 and 3.3 can be viwed as a natural generalization of theresult derived by Eeckhoudt and Kimball (1992). This implies that the conditionsof the proposition 3.2 and 3.3 correspond to the standard risk aversion in the caseof bivariate utility functions, which is introduced by Kimball (1993).
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3.2

ProofsThe proofs of proposition 3.2 and 3.3 are same except for some minor points, hence
we omit the proof of proposition 3.3.
Proof of propsition 3.1: Let us define the following function for ² ∈ [², ²]
U1 (x, y + ²)f (²)fˆ(²) := R ².
U(x,y+²)f(²)d²1²(8)

R²Since fˆ(²) is positive for all ² ∈ [², ²] and F̂ (²) = ² fˆ(²)d² = 1, F̂ : [², ²] → [0, 1] canbe viewed as a distribution function defined over the compact support [², ²]. The
correlation aversion implies that U1 (x, y + ²) ≥ U1 (x, y + ²0 ) for all ², ²0 ∈ [², ²] with² ≤ ²0 . Hence, we have thatf (²0 )U1 (x, y + ²0 )f (²0 )f (²0 )fˆ(²0 ).
≥⇔≥f (²)
U1 (x, y + ²)f (²)
f (²)
fˆ(²)

(9)

The equation (9) means that F dominates F̂ in the sense of monotone likelihood
ration dominance. The monotone likelihood ratio dominance is a strong stochastic
dominance of the first–order stochastic dominance. Since the expectation of thehealth background risk is non–positive, we have Ê[²̃] ≤ E[²̃] ≤ 0. For details of
stochastic dominance, the intersted readers are refereed to e.g. Shaked and Shanthikumar (1994) or Müller and Stoyan (2002). Using this distirubtion function, wehaveZ ²VA (x, y) =A(x, y + ²)fˆ(²)d²(10)²= Ê[A(x, y + ²̃)],(11)

where Ê denotes the expectation operator with respect to the distribution function
F̂ . By the equation (11), we obtain that
AV (x, y) = Ê[A(x, y + ²̃)](12)

≥ A(x, y + Ê[²̃])(13)≥ A(x, y).

(14)

The first inequaltiy follows from the Jensen’s inequality, the second inequaliy follows
from Ê[²̃] ≤ 0 and A2 ≤ 0.
¤Proof of propsition 3.2: The health background risk ²̃ decomposes to ²0 and ²̃1 , i.e.
d²̃ = ²0 +²̃1 , where ²0 = E[²̃] ≤ 0 and E[²̃1 ] = 0. By the definition of the precautionary
premium in health ΨH , we have that
V1 (x, y) = U1 (x, y + ²0 − ΨH ),4(15)V11 (x, y) = U11 (x, y + ²0 − ΨH ) − U12 (x, y + ²0 − ΨH )ΨHx ,(16)HH
where ΨH
with respect to x, i.e. ΨHx denotes the partial derivative of Ψx = ∂Ψ /∂x. Using the equation (15) and (16), we obtain that

V11 (x, y)
U11 (x, y + ²0 − ΨH ) − U12 (x, y + ²0 − ΨH )ΨHx−= −H
V1 (x, y)
U1 (x, y + ²0 − Ψ )U11 (x, y + ²0 − ΨH ) U12 (x, y + ²0 − ΨH ) H= −+Ψ
U1 (x, y + ²0 − ΨH )U1 (x, y + ²0 − ΨH ) x
U11 (x, y + ²0 − ΨH )≥ −U1 (x, y + ²0 − ΨH )U11 (x, y)≥ −= A(x, y).
U1 (x, y)

(17)
(18)
(19)
(20)

As U1 > 0, U12 < 0, and P1H ≤ 0 is equivalent to ΨHx ≤ 0, we have that the second
term in the equation (18) is non–negative. Hence, we have the first inequality. The
second inequality follows from A2 ≤ 0 and y > y + ²0 − ΨH .
¤4Concluding Remarks

In this note, we give sufficient conditions of cross risk vulnerability. These conditions
are a generealization of the well–known sufficient conditions of risk vulnerability:
standard risk aversion and decreasing and convex risk aversion. Since most decisions under uncertainty are multi–dimensional, we need to understand the effectsof background risks in multi–dimensional decisions under uncertainty. A best way
to understand them generalize those of one–dimesional decisions under uncertainty.
This paper can be regarded as one of a series of studies. We have many future reserches to understand the effects of background risks in multi–dimensional optimal
choices under uncertainty, since we have a great number of understandings regarding
them in one–dimension.

5

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