European Journal of Operational Research 151 (2003) 224–232
www.elsevier.com/locate/dsw
Interfaces with Other Disciplines
Comparative statics under uncertainty: The case
of mean-variance preferences
Andreas Wagener
*
VWL IV, FB 5, University of Siegen, H€olderlinstr. 3, 57068 Siegen, Germany
Received 20 February 2001; accepted 21 February 2002
Abstract
We analyze the comparative statics of optimal decisions under uncertainty when preferences are represented by twomoment, mean-variance utility functions. We relate our findings to concepts for risk attitudes that are familiar from the
expected-utility approach. In the two-parameter approach, a number of plausible comparative static effects already
emerges under the assumption of decreasing absolute risk aversion. To determine comparative static effects of changes
in background risks stronger concepts are needed. Risk vulnerability, temperance and standardness imply, appropriately transferred to the mean-variance framework, the plausible effect that risk taking will be reduced if background
risks increase.
Ó 2002 Elsevier B.V. All rights reserved.
Keywords: Decision analysis; Mean-variance analysis; Utility theory; Choice under uncertainty; Risk preferences
1. Introduction
Analyses of the comparative statics for decision
problems under uncertainty have since long found
the interest of economists. In the past two decades
most of these studies have been undertaken in
the expected-utility (EU) framework. Numerous
studies identified necessary and sufficient restrictions on preferences or on the distribution functions for random payoffs such that optimal
decisions exhibit certain unambiguous and plausible comparative statics when the stochastic and
*
Tel.: +49-271-740-3164; fax: +49-271-740-2732.
E-mail address: [email protected] (A. Wagener).
non-stochastic components of the economic environment change; see Gollier (2001) for an excellent
exposition of that research.
Another venerable and widely applied approach
to risky choices is the two-parameter approach
(also labelled two-moment, mean-variance or
mean-standard deviation approach) which represents preferences towards uncertainty by functions
of the first and second moments of the random
variable of interest (Tobin, 1958; Markowitz,
1970). For long, this approach suffered from the
bad reputation of being consistent with the EU
paradigm only under restrictive and unattractive
restrictions on preferences (Borch, 1969) or probability distributions (Chamberlain, 1983; Owen
and Rabinovitch, 1983). More recently, however,
the two-moment approach seems to be reviving:
0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved.
doi:10.1016/S0377-2217(02)00599-4
A. Wagener / European Journal of Operational Research 151 (2002) 224–232
Meyer (1987), Sinn (1983, 1990), Bar-Shira and
Finkelshtain (1999) and Lajeri and Nielsen (2000)
argue that it is a perfect substitute for the EUapproach for the class of decision problems that
only involve location and scale shifts of random
variables. Many important problems, including the
standard saving, insurance demand, and portfolio
choice problems, possess this structure. 1
The renaissance of the two-parameter approach
has so far focussed on utility-theoretic aspects.
There exist only relatively few studies on comparative statics for the two-parameter approach
(see, e.g., Levy, 1973; Hawawini, 1978; Mayers
and Smith, 1983; and, most recently, Ormiston
and Schlee, 2001). Here, we extend these analyses
by including background risks, an undertaking
that recently has gained much attention in the EUframework. We allow for arbitrary correlations
between primary and background risk. In a simple
generic decision problem (presented in Section 2),
we discuss the comparative statics effects of various parameter changes. In Section 3 we observe
that the assumption of decreasing absolute risk
aversion (DARA) is powerful in being a sufficient
condition for quite many comparative static effects
to possess unambiguous and plausible signs even
in the presence of background risks. Only when it
comes to changes in the background risk itself
does DARA no longer yield any informative insights (Section 4). We derive certain necessary and
sufficient conditions such that agents reduce their
primary risk-taking when they are confronted with
higher background uncertainties. These conditions
are related to concepts such as risk vulnerability
(Gollier and Pratt, 1996), temperance (Eeckhoudt
et al., 1996), or standardness (Kimball, 1993)
1
Clearly, the variance is not an adequate measure to
discriminate between different probability distributions. However, as Grootveld and Hallerbach (1999) argue, the use of risk
measures that surpass the second moment basically entails the
same difficulties as mean-variance analysis (also see Brockett
and Kahane, 1992), while risk measures that only account for
some part of the distribution (such as measures for downside
risk) more often than not possess properties even inferior to the
mean-variance framework. This provides a further argument to
stick to the two-parameter approach at least for practical
purposes.
225
which capture, for different settings, the tempering
effects of background risks in the EU-framework.
Section 5 concludes.
2. The setting
Consider the following individual utility maximization problem where all attainable lotteries are
assumed to differ only by location and scale:
max uðlðaÞ; rðaÞÞ
a2R
ð1Þ
with
yðaÞ ¼ az þ w þ ;
ð2aÞ
lðaÞ ¼ EyðaÞ ¼ alz þ w;
ð2bÞ
r2 ðaÞ ¼ Var yðaÞ
¼ a2 r2z þ r2 þ 2aCov ðz; Þ:
ð2cÞ
Here a is the individualÕs one-dimensional choice
variable, yðaÞ is non-negative random consumption (depending linearly on a), w is a non-negative
constant (wealth), z is a random variable with expected value lz > 0 and variance r2z > 0, and is a
random variable with expectation zero and
positive variance r2 . z and need not be stochastically independent; Cov ðz; Þ 2 R and qðz; Þ :¼
Cov ðz; Þ=ðrz r Þ 2 ½1; 1 denote, respectively, their
covariance and their coefficient of correlation. We
do not restrict a in sign a priori. Some of our results will, however, be valid only if the optimal
value of a is non-negative.
In Eq. (1), u : R Rþ ! R is a two-moment
(mean-standard deviation) utility function. Provided that all lotteries which an agent can choose
from only differ by location and scale parameters,
Meyer (1987), Sinn (1983, 1990) and others have
argued that u is a representation of preferences
that is equivalent to the ordering generated by EU
comparisons. Namely, given a utility function
v : R ! R, the EU for lotteries over y with standardization x ¼ ðy lÞ=r can be written in terms
of the mean l and the standard deviation r of the
distribution of y:
Z b
EvðyÞ ¼
vðl þ rxÞ dF ðxÞ ¼: uðl; rÞ:
ð3Þ
a
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A. Wagener / European Journal of Operational Research 151 (2003) 224–232
The interval ½a; b R with a < b is the support of
x and F is its distribution. Using (3), a series of
equivalences between v and u can be derived (see
Meyer, 1987): 2
0
v ðyÞ > 0 () u1 ðl; rÞ > 0;
ð4aÞ
v00 ðyÞ < 0 () u2 ðl; rÞ < 0;
ð4bÞ
v000 ðyÞ > 0 () u12 ðl; rÞ > 0;
ð4cÞ
v00 ðyÞ < 0 () u
is strictly concave:
u11 < 0; u22 < 0; and u11 u22 u212 > 0:
v00 ðyÞ
0
v ðyÞ
ð4dÞ
0
< 0 () u1 u12 u2 u11 > 0:
ð4eÞ
From (4d) risk aversion of v is mirrored by the
concavity of u. From (4e) DARA of v is equivalent, in terms of u, to the marginal rate of substitution u2 =u1 between mean and standard
deviation being decreasing in l.
Some authors (e.g., Chipman, 1973; Allingham,
1991; Lajeri and Nielsen, 2000) represent preferences by a function of the mean and the variance of
consumption, V ðaÞ r2 ðaÞ. We therefore define a
utility function W : R Rþ ! R such that:
pffiffiffiffi
W ðl; V Þ :¼ uðl; V Þ:
ð5Þ
The properties of W can be easily expressed in
terms of u; we will need this in Section 4. For later
reference we collect some derivatives of rðaÞ here:
orðaÞ ar2z þ Cov ðz; Þ
¼
;
oa
rðaÞ
ð6aÞ
2
Numeral subscripts to u denote partial derivatives. Eq.
(4c), which is not proved by Meyer (1987), can be seen from the
following argument:
u12 ¼
Z
b
a
¼ r
00
xv ðl þ rxÞ dF ðxÞ
Z
a
b
Z x
v000 ðl þ rxÞ
z dF ðzÞ dx
a
where the last equality comes from integration by parts. As the
inner integral is negative for all x due to Ex ¼ 0, v000 > 0 is
necessary and sufficient for u12 > 0.
orðaÞ a2 rz
> 0;
¼
orz
rðaÞ
ð6bÞ
orðaÞ
r
> 0;
¼
or
rðaÞ
ð6cÞ
o2 rðaÞ
r2 r2
¼ 3 z ½1 q2 ðz; Þ P 0;
2
oa
r ðaÞ
ð6dÞ
o2 rðaÞ
r
¼ 3 ½ar2z þ Cov ðz; Þ;
oa or
r ðaÞ
ð6eÞ
o2 rðaÞ
arz 2 2
½a rz þ 2r2 þ 3aCov ðz; Þ;
¼
oa orz r3 ðaÞ
ð6fÞ
o2 rðaÞ
1
¼
½r2 þ aCov ðz; Þ:
oa oCov ðz; Þ r3 ðaÞ ð6gÞ
The FOC of problem (1) is
/ :¼ u1 lz þ u2
orðaÞ
¼ 0:
oa
ð7Þ
We denote the solution of (7) by a . As u is strictly
concave, the second-order condition for a maximum will always be satisfied at a . From (7) and
(6a) a must satisfy
a r2z þ Cov ðz; Þ > 0:
ð8Þ
Hence, a will be positive if, but not only if, z and are negatively correlated or independent.
3. On the role of DARA
We first verify that DARA, as represented by
the RHS of (4e), is (a) equivalent to the idea that
wealthier individuals optimally take higher risks
and (b) implies that a higher expected return raises
risk-taking. Implicit differentiation of (7) yields:
Fact 1 [Ormiston and Schlee, 2001, Prop. 2]
(a) The optimal choice a increases with an increase
in deterministic wealth w if and only if u satisfıes
DARA.
(b) Suppose that a P 0. Then DARA is suffıcient,
but not necessary for the optimal choice a to
be an increasing function of its marginal expected return: oa =olz > 0.
A. Wagener / European Journal of Operational Research 151 (2002) 224–232
Note that, while item (a) in Fact 1 holds independently of the sign of a, this is not true for item
(b) where we the decision maker is required to take
some risk. This is a reasonable assumption for
most applications we have in mind. A graphical
exposition of Fact 1 for the case of a competitive
firm with DARA preferences can be found in
Hawawini (1978).
Next we consider the impact of changes in the
riskiness of the environment on the optimal choice
of a. Three effects are to be considered: changes in
the variance r2z directly associated with the activity
a, changes in the covariance between the returns
on a and the background risk, and changes in the
variance of the background risk r2 (cf. Section 4).
First we consider the direct variance effect. Again,
DARA plays an important role:
Fact 2
(a) Under the DARA-assumption (4e) the optimal
choice a decreases upon an increase in rz if
o2 rðaÞ=ðoa orz Þ P 0.
(b) Suppose that a > 0. Then o2 rðaÞ=ðoa
z Þffi is
por
ffiffiffiffiffiffiffi
non-negative whenever qðz; Þ P 8=9 ¼
0:943.
Proof
(a) Implicit differentiation of (7) yields:
oa
1 orðaÞ
orðaÞ
u22
¼
lz u12 þ
/a orz
oa
orz
2
o rðaÞ
þ u2
oa orz
1
orðaÞ
u1 u22
¼
l
u12 /a z orz
u2
2
o rðaÞ
þ u2
;
oa orz
ð9Þ
where we used (7). Since u2 < 0 and orðaÞ=orz
is positive from (6b), the claim is proven if the
round-bracketed expression is negative. Verify
that (4d) and (4e) together imply u1 u12 =u2 <
u11 < u212 =u22 which, since u12 > 0 > u22 , gives
u12 u1 u22 =u2 < 0.
(b) For a > 0, (6f) is non-negative when the
square-bracketed expression therein is so. This
will always happen when Cov ðz; Þ P 0. If
227
Cov ðz; Þ < 0 rewrite the square-bracketed
term in (6f) as
pffiffiffi
pffiffiffi
½arz 2r 2 þ arz r ð3qðz; Þ þ 2 2Þ P 0:
ð10Þ
This holds if, but not only if, the roundbracketed expression
is non-negative, i.e., if
pffiffiffiffiffiffiffiffi
qðz; Þ P 8=9. From Fact 2a, DARA ensures that a reacts
negatively upon an increase in the variance associated with that choice if such an increase raises
the marginal volatility of the final outcome with
respect to a. DARA is a sufficient, but not a necessary condition here. Moreover, given DARA,
the condition that o2 rðaÞ=ðoa orz Þ be non-negative
is again overly strict; that expression may also be
positive as long as it is not too large.
From (6f), rðaÞ will always possess the curvature property identified in Fact 2a if a ¼ 0. Item
(b) then identifies an undemanding and unneccessarily strict condition for the case a > 0:
qðz; Þ P 0:943. This merely excludes that the
two risks are close to perfectly negatively correlated. Note that item (b) also captures the case of
independent risks where we always have a > 0
from (8). Fact 2 further generalizes findings by
Hawawini (1978) and Ormiston and Schlee (2001)
in the single-risk case (r ¼ 0). There a is always
strictly positive––and DARA thus implies that a
will be reduced if rz increases.
Fact 2 does not contradict the results reported
(and critized as ‘‘misleading’’) by Rothschild and
Stiglitz (1971, p. 70) that ðl; rÞ-preferences imply
that increases in the variance of the returns of a
risky activity lead to an increase in that activity.
Rothschild and Stiglitz (1971) and others identify
ðl; rÞ-preferences with quadratic utility functions v
which involves increasing absolute risk aversion. If
ðl; rÞ-preferences are understood in the sense of
(3), however, then they are capable of representing
DARA––and thus of rationalizing the more intuitive behavioural assumption that higher riskiness
leads to lower risk-taking.
While the case that a < 0 is, in principle, covered by Fact 2, the requirement that (6f) be nonnegative is hardly ever met then; it would necessitate
228
A. Wagener / European Journal of Operational Research 151 (2003) 224–232
that the LHS of (10) is negative. Thus, for a < 0
the comparative statics of a with respect to rz
typically contain two expressions of opposite signs.
Next we consider an increase in the correlation
between z and . Here we find the same rationale as
in Fact 2: If the increase in the parameter raises the
marginal riskiness of the optimal choice a , then
DARA implies that a should be lowered. Formally:
Fact 3. Suppose that a P 0. For DARA preferences the optimal choice a decreases upon an increase in Cov ðz; Þ if Cov ðz; Þ P 0. Otherwise, the
effect is ambiguous.
Proof. Implicit differentiation of (7) yields:
oa
1
a
orðaÞ
lz u12 þ
u22
¼
/a rðaÞ
oa
oCov ðz; Þ
2
o rðaÞ
;
ð11Þ
þ u2
oa oCov ðz; Þ
where a=rðaÞ ¼ orðaÞ=oCov ðz; Þ. By the same
token as in the proof of Fact 2 one shows that the
first term in (11) is negative under DARA (4e).
Hence, due to u22 < 0, oa =oCov ðz; Þ < 0 if, but
not only if, o2 rðaÞ=ðoa oCov ðz; ÞÞ > 0. Combine
(6g) and (8) to see that for Cov ðz; Þ P 0:
ables become more positively correlated. With
negatively correlated random variables the DARA
effects of these parameter changes are partially
offset and, perhaps, overcompensated by a decreasing marginal volatility in the returns to the
risky activity.
In the EU-framework, Gollier and Schlee
(1999) recently analysed how background risks (of
arbitrary dependence structure with the primary
risk) affect the comparative statics of changes in
the marginal or conditional distribution of the
returns to the risky activity. One of the general
observations in that research is that comparative
statics results (or, more precisely, the sets of necessary and sufficient conditions that generate certain behavioural responses) are generally not
preserved upon an addition of background risks.
As the previous paragraph indicates, this observation, in principle, also applies to the twoparameter approach. Yet, given its prominence in
Facts 1–3, DARA turns out to be quite a robust
sufficient condition for plausible comparative
statics in the two-parameter framework even in the
presence of background risks. 3
4. Changes in the background risk
4.1. A necessary condition
Cov2 ðz; Þ
r2 þ aCov ðz; Þ P r2 r2z
¼ r2 ð1 q2 ðz; ÞÞ P 0
––which suffices to render (11) negative. For a
negative covariance (which implies a > 0) this line
of reasoning does not work, however. We now analyse reactions of a on changes in
the background risk, represented by increases in
the variance of . From (7):
oa
1 orðaÞ
orðaÞ
o2 rðaÞ
:
u22 þ u2
¼
lz u12 þ
or
/a or
oa
oa or
ð12Þ
Facts 1–3 confirm the powerful role of DARA
in the two-parameter framework. This is especially
apparent in the single-risk case (Hawawini, 1978;
Ormiston and Schlee, 2001). In the presence of
multiple uncertainties DARA looses some of its
power since additional effects via the correlation of
the random variables prevail: For non-negatively
correlated random variables, DARA is sufficient
to reduce risk-taking when the (direct) variance of
the risky activity increases or if the random vari-
Here, we have orðaÞ=or > 0 from (6c). Again, the
expression in round brackets is negative under (4e).
3
A specific comparison with the findings in Gollier and
Schlee (1999) is difficult since the comparative statics problems
that can be investigated in the two-parameter approach are
confined to changes within the same location/scale family and
do, e.g., not allow for arbitrary first-order stochastic dominance
shifts.
A. Wagener / European Journal of Operational Research 151 (2002) 224–232
Verify from (8) and (6e) that o2 rðaÞ=ðoaor Þ < 0 in
an optimum. Hence, DARA does not suffice under
any circumstances to determine the sign of (12).
Instead we need a stronger condition. Combine
(6a) and (6e) to obtain
o2 rðaÞ
r orðaÞ
:
¼ 2
oa or
r ðaÞ oa
ð13Þ
Use this, the FOC (7) and (6c) to rewrite (12) as:
oa
1 lz r
u1
u2
u12 ¼
u22 :
ð14Þ
/a rðaÞ
or
u2
rðaÞ
A necessary, but insufficient condition for (14) to
be negative is that u22 rðaÞ < u2 for all rðaÞ. We can
extend this to the following.
Fact 4
(a) A necessary condition for an increase in r to induce a decrease in the choice of a is that, for all
ðl; rÞ,
u222 ðl; rÞ < 0:
ð15Þ
(b) (i) If the distribution F of x is symmetric then
u222 < 0 if and only if v0000 < 0.
(ii) If the distribution F of x is negatively skewed
and v0000 < 0 < v000 , then u222 < 0.
Proof
Rb
(a) Verify from (3) that u2 ðl; 0Þ ¼ vðlÞ a
x dF ðxÞ ¼ 0. Hence, u2 being concave in r
(i.e., u222 < 0) is equivalent to u22 ðl; rÞ <
u2 ðl; rÞ=r for all ðl; rÞ > ð0; 0Þ.
(b) Use the defınition of u in (3) to calculate, via
integration by parts,
u222 ðl; rÞ ¼
Z
b
x3 v000 ðl þ rxÞ dF ðxÞ
a
000
¼ v ðl þ rxÞ
Z
Z
x
b
z dF ðzÞ
3
a
b
r
v0000 ðl þ rxÞ
a
Z x
3
z dF ðzÞ dx:
ð16Þ
If F is symmetric, all odd central moments of x
are zero. Hence, the first expression in (16) is
229
zero while the inner integral in the second expression is negative. Thus, u222 < 0 if and only
0000
Rifb v 3 < 0. If F is negatively skewed (i.e., if
x
dF ðxÞ < 0), the first term in (16) will be
a
negative whenever v000 > 0. The second term
will be negative (including the negative sign)
when v0000 < 0. For the mean-variance setting, Eq. (15) provides a simple necessary condition for background
risks to attenuate risk-taking. Moreover, as item
(b) in Fact 4 states, if the distribution of x is nonpositively skewed, condition (15) has a familiar
translation in terms of the EU-framework: The
utility index v satisfying temperance (i.e., v0000 =
v000 > 0) is a prerequisite for lower risk-taking in
the presence of higher risks. For positively skewed
distributions of x an accessible necessary condition
for (12) to be negative unfortunately is not yet
available in terms of the function v. We add three
further comments:
(a) From Chamberlain (1983) and Owen and Rabinovitch (1983) it is known that a necessary
condition for payout distributions to imply
mean-variance utility functions is that they
are jointly elliptical. This rules out asymmetries in distributions, such that the most relevant part of Fact 4b is indeed part (i).
(b) In the EU-framework, v0000 < 0 is known under
the label of temperance, a property which
Kimball (1993); Eeckhoudt et al. (1996), and
Gollier and Pratt (1996) show to be necessary
to have agents reducing their demand for a
risky asset if their independent background
wealth gets riskier in some specifıc sense. Fact
4b, part (i), suggests that u222 < 0 is an indicator for temperance in the mean-variance
framework.
(c) Moreover, note that for symmetric distributions, u222 and u211 are always equal in sign.
Hence, one might defıne a meaningful index
for absolute temperance by u211 =u111 ––which
nicely fıts into the series of two-parametric indexes for risk attitudes that starts with absolute risk aversion u2 =u1 (Meyer, 1987) and
absolute prudence u21 =u11 Lajeri and Nielsen,
2000).
230
A. Wagener / European Journal of Operational Research 151 (2003) 224–232
4.2. A set of sufficient conditions
o
ol
To obtain an interesting set of sufficient conditions for (14) to be negative, we employ the meanvariance function W defined in (5):
Fact 5. If W ðl; V Þ is concave and u satisfies DARA
(4e), then an increase in r reduces optimal risktaking a .
Proof. The second-order derivatives of W can
easily be calculated as Wll ¼ u11 < 0, WVV ¼ 1=
ð4r2 ðaÞÞðu22 ðu2 =rðaÞÞÞ, and WlV ¼ u12 =ð2rðaÞÞ
> 0. For W concave we must have
0 < Wll WVV WlV2
1
u2
2
u11 u22 ¼ 2
u12 :
4r ðaÞ
rðaÞ
ð17Þ
As Wll < 0, this can only hold if WVV < 0. Now
verify from (14) that:
oa
1 lz r u1
u2 u11
u2
u11 u22 ¼
u12
/a rðaÞ u11 u2
or
u1
rðaÞ
1 lz r u1
u2
<
u2 u11 u22 /a rðaÞ u11 u2 12
rðaÞ
< 0:
The first inequality follows from DARA in (4e),
the second from (17). As already noted in the proof, (17) being positive implies WVV < 0. This is equivalent to the
necessary condition for oa =or < 0 that was established in Fact 4.
The set of sufficient conditions in Fact 5 has two
elements: W must be concave and u must satisfy
DARA. These conditions are independent in the
sense that preferences exist that satisfy one condition while violating the other.
At this place it is helpful to employ the notion
of decreasing absolute prudence (DAP), which was
introduced to the EU-framework in Kimball
(1990): A utility function vðyÞ is said to exhibit
DAP if and only if v000 ðyÞ=v00 ðyÞ is decreasing in y.
Recently, Lajeri and Nielsen (2000) have shown
that v exhibits DAP if and only if the associated
mean-standard deviation function u satisfies:
u12 ðl; rÞ
u11 ðl; rÞ
< 0:
ð18Þ
Lajeri and Nielsen (2000, Theorem 2) furthermore
show that, if the stochastics underlying the ðl; rÞapproach are normally distributed, then
W is concave () u satisfies ð18Þ:
ð19Þ
Normality of y will, in particular, hold in our
setting when both z and are normally distributed.
Combining (19) and Fact 5 we obtain 4
Fact 6. Suppose that yðaÞ is normally distributed
for all a. Then (14) is negative whenever v satisfies
DARA and DAP.
Kimball (1993) calls the combination of DARA
and DAP (together with the usual assumptions of
monotonicity and concavity) standard risk aversion. In Proposition 6, Kimball (1993) shows that
standardness is equivalent to the statement that the
addition of any loss-aggravating risk reduces the
optimal level of taking any other, independent
risk. 5
Gollier and Pratt (1996) find that standardness
is sufficient for risk vulnerability: It implies that
the presence of background risk with zero expectation raises the risk aversion to other independent
risks. Fact 6 derives another feature of standardness: With normally distributed risks, standardness suffices to raise aversion against the risk
associated with the activity a when the variance of
4
Fact 6 can be proved without the detour via Fact 5. From
Chipman (1973, Theorem 1) we know that u2 ¼ rðaÞu11 for
normally distributed stochastics. Further differentiation then
yields u12 ¼ rðaÞu111 and u22 ¼ u11 þ r2 ðaÞu1111 . Invoke this in
(14) to see that the round-bracketed expression there is nonpositive iff u111 u1111 u1 =u11 6 0. Now note that u is––in terms
of l––formally equivalent to what Kimball (1993) calls a
derived utility function. Straightforward application of Proposition 4 in Kimball (1993) then yields that u111 u1111 u1 =u11 6 0
will hold whenever v satisfies DARA and DAP.
5
A risk (with possibly non-zero mean) is said to be loss
aggravating if and only if its addition increases expected
marginal utility: Ev0 ðw þ xÞ P v0 ðwÞ for all w > 0. In our
framework an increase in r is always loss aggravating since
ou1 ðlðaÞ; rðaÞÞ=or ¼ u12 ðlðaÞ; rðaÞÞr =rðaÞ > 0 for all ðlðaÞ;
rðaÞÞ.
A. Wagener / European Journal of Operational Research 151 (2002) 224–232
the background risk––which need not be an independent risk!––increases.
Clearly, the assumption of normally distributed
random variables in Fact 6 is restrictive and often
unattractive. Yet it is necessary for Fact 6 since the
equivalence of the concavity of W and DAP for U
in (19) does not hold for non-Gaussian distributions (Eichner and Wagener, 2001). Given that
necessary restrictions on preferences such that the
comparative statics in the presence of simultaneous
risks have unambiguous signs are notoriously
complex, one might often be content to know some
sets of manageable sufficient conditions. Facts 5
and 6 provide two such sets.
5. Concluding remarks
In this paper we trace behavioural reactions for
a ðl; rÞ-specification of preferences back to concepts of risk attitudes familiar from the EUframework. In particular, we see that plausible
comparative statics with respect to background
risks follow from similar restrictions on preferences as in the EU-framework.
At least two directions for future research
emerge from our work quite naturally: First, as the
discussion of the comparative statics impact of
background risks in Section 4 is far from complete,
one might search both for additional conditions for
plausible behavioural responses and for further
clarification of the relations between such conditions and their EU-analogs. Second, one might
wish to extend the analysis to the case where the
second risk is not merely a background risk and
thus functionally separable from the first risk, but
represents another risky activity. For the EUapproach such analyses have been undertaken,
among others, by Tibiletti (1994) and Gollier and
Schlee (1999, Section 4). In both studies the magnitude of relative risk aversion plays an important
role––and it might be interesting to see whether this
is also the case in the two-parameter approach.
Acknowledgements
The author thanks Thomas Eichner and two
referees for valuable comments. Financial support
231
by Deutsche Forschungsgemeinschaft (DFG) is
gratefully acknowledged.
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