Concave Risk Aversion and Strongly More Risk Averse1
August 2012
Liqun Liu
Private Enterprise Research Center
Texas A&M University
College Station, TX 77843
[email protected]
Jack Meyer
Department of Economics
Michigan State University
East Lansing, MI 48824
[email protected]
Abstract: The Arrow-Pratt (A-P) definitions of absolute and relative risk aversion dominate the
discussion of risk aversion and defining “more risk averse”. Ross notes, however, that being A-P
more risk averse is not sufficient for addressing many important comparative static questions.
Consequently he introduces “a new and stronger measure for comparing two agents’ attitudes
towards risk…”. Ross does not provide a corresponding measure of risk aversion. This paper
proposes “concave risk aversion” as an alternative to absolute risk aversion and uses it to
characterize the Ross definition of strongly more risk averse on bounded intervals. Other
applications of the concave risk aversion are also presented.
Key Words: Concave risk aversion, Arrow-Pratt risk aversion, Strongly more risk averse
JEL Classification Codes: D81
1
Support from the Private Enterprise Research Center at Texas A&M University is gratefully acknowledged. We
thank Louis Eeckhoudt and Stephen Ross for helpful comments and suggestions on an earlier draft. This paper was
presented at the Conference in Honor of Louis Eeckhoudt, Toulouse, France, July 13, 2012. We thank Ed Schlee and
other conference participants for valuable comments.
1
1. Introduction
Almost fifty years ago, Arrow (1974) and Pratt (1964) (A-P) provide definitions of
absolute and relative risk aversion for expected utility maximizing decision makers with utility
functions u(x). These definitions have dominated the discussion of risk aversion since that time.
In addition to measuring the intensity of risk aversion, absolute risk aversion, Au(x) =
‐u'' x
u' x
, is
also used to define when one decision maker is A-P more risk averse than another. Au(x) 
Av(x) for all x is known to be necessary and sufficient for a variety of interesting comparative
static statements. It is safe to say that the A-P definition for more risk averse has been used in
literally hundreds of published papers in economic research over the past fifty years.
A little more than thirty years ago, Ross (1981) observes that even though one decision
maker being A-P more risk averse than another is sufficient for some comparative static
statements, a stronger definition of more risk averse is needed to allow other interesting
comparative static questions to be addressed. Ross formulates a stronger definition of more risk
averse, and provides three equivalent ways to characterize what it means for one decision maker
to be strongly more risk averse than another. He then shows that this stronger order is sufficient
for addressing these additional comparative static questions.
The focus in Ross’s analysis is on comparing risk aversion across decision makers based
on properties of their utility functions, not on providing a measure of risk aversion. Unlike A-P,
Ross does not begin his analysis with a measure of risk aversion, nor does he ever provide a risk
aversion measure that characterizes the strongly more risk averse order. No measure of risk
aversion comparable to absolute risk aversion is presented. Perhaps as a result of this, the Ross
2
definition of strongly more risk averse has not been used as often, nor has it achieved the
prominence of the A-P definition.
The analysis here discusses a “new” measure of risk aversion, termed concave risk
aversion. This definition is similar in structure and simplicity to the A-P measure of absolute
risk aversion. This definition is new only in the sense that it has not been formally proposed in
this particular form, nor supported to the extent it is supported here. This measure is similar to
one used by Pratt (1990) when defining the more partial-risk averse order over decision makers.
The concave risk aversion measure for utility function u(x) is defined by Cu(x) =
‐u'' x
u' a
, where
“a” denotes the left end of the closed interval [a, b] containing the supports of all random
alternatives under consideration. Cu(x) depends both on the utility function u(x) and the interval
in which supports lie. Alternatively, if without loss of generality one assumes that marginal
utility of the lowest possible outcome is normalized to be equal to one, then Cu(x) is simply the
second derivative of this normalized utility function. Hence, the term concave risk aversion.
Using the magnitude of the second derivative of u(x) to measure risk aversion is hardly a
new idea. As Arrow points out, however, using only u''(x) is problematical in that a vonNeumann-Morgenstern u(x) is unique to a positive linear transformation, and thus u''(x) (and
u'(x)) can always be scaled by an arbitrary positive constant. This scaling ambiguity is solved by
Arrow and Pratt when they divide u''(x) by u'(x) to form the absolute and relative risk aversion
measures. It is similarly solved here by dividing by u', but now the denominator is u' evaluated
at a particular point and therefore u' takes on a fixed value; that is, the denominator in C(x) is a
constant. With this definition of concave risk aversion, it is natural to say that one decision
3
maker is more concave risk averse than another on [a, b] if and only if Cu(x)  Cv(x) for all x in
[a, b].
Cu(x) completely represents risk preferences on [a, b] and is a risk aversion measure that
can serve as an alternative to or substitute for the absolute risk aversion measure of Arrow and
Pratt. That is, if Cu(x) is the concave risk aversion measure for utility function u(x), this same
measure is obtained from all utility functions of the form v(x) = a + b·u(x) for b > 0.
Furthermore, integrating Cu(x) twice recovers the utility function up to a positive linear
transformation. Another way to emphasize the fact that Cu(x) is a risk aversion measure in the
same sense as Au(x) is to observe that if one knows Cu(x), Au(x) is easily obtained and vice
versa. Other familiar concepts can be related to Cu(x). The signs of the first two derivatives of
C(x), for instance, define prudence and temperance.
The value of risk aversion measure Cu(x), or any risk aversion measure, lies in the way
that simple assumptions concerning the risk aversion measure reflect interesting behavior. The
fundamental and perhaps surprising finding concerning this concave risk aversion measure is that
more concave risk averse on [a, b] is both necessary and sufficient for strongly more risk averse
on [a, b]. That is, C(x) not only is a measure of risk aversion that completely represents risk
preferences, this measure also corresponds directly to the behavior associated with Ross’s
strongly more risk averse order. Because the relationship is necessary and sufficient, concave
risk aversion also provides an additional way to characterize that order.
An important feature of the analysis here is the assumption that is made concerning the
possible location of the supports of the random alternatives. Recall that Ross assumes that these
supports can be anywhere along the line. The analysis here replaces that assumption with the
4
requirement that these supports lie in a bounded interval denoted [a, b]. This interval can be as
large as desired. This change in assumptions concerning the possible locations of the supports of
random alternatives is of greater consequence than was anticipated. This same point is also
made by Pratt (1990).
There are several reasons for assuming that the supports of the random alternatives lie in
a bounded interval. One was pointed out by Ross who notes that if v'(x) goes to zero as x goes to
infinity, then no other utility function can be strongly more risk averse than v(x), thereby
eliminating many utility functions from the set of pairs of utility functions that can be ordered
using strongly more risk averse. Machina and Neilson (1987) point to this fact when they
assume that “the supports of all relevant distributions lie within the appropriate bounded
interval.” A second reason for making this assumption, and the main reason for the analysis
here, is that it allows concave risk aversion to be defined in the way we define it, and it allows
more concave risk averse to be shown to be equivalent to Ross’s strongly more risk averse for
these bounded intervals. It is worth noting, however, that for many economic decisions there are
natural bounds on outcome variables, especially non-negativity restrictions. In addition, there
may be empirical support for bounds which are even tighter than those imposed by theory. For
the annual gross nominal return on a diversified U. S. stock portfolio, for instance, a = .5 and
b = 1.5 has empirical support. The analysis here shows that bounds such as these can be used to
make strongly more risk averse a more useful tool of analysis.
Ross gives three characterizations of strongly more risk averse, and shows that these are
equivalent. More concave risk averse is related to strongly more risk averse in the following
way. Utility function u(x) is strongly more risk averse than v(x) on [a, b] if and only if
5
Cu(x)  Cv(x) for all x in [a, b]. Thus, for bounded intervals, more concave risk averse is
equivalent to Ross’s three characterizations of the strongly more risk averse order. This is one of
the major findings of the paper, and is summarized in Theorem 2.
It is also the case that the risk premium concept used by Arrow, Pratt and Ross can be
replaced by one that is much more general, and the change in risk whose effect is measured by
the risk premium can also be specified differently. Subtracting a risk premium is a special first
degree stochastic dominant worsening of the random alternative, and considering any first degree
stochastic dominant worsening of the random alternative instead is a useful generalization. Also,
rather than locally increasing the riskiness of the outcome variable and approximating this
increase by an increase in the variance as A-P do, an increase in risk as defined by Rothschild
and Stiglitz (R-S) (1970) can be considered. Ross models an increase in risk in this way, and we
do as well.
The analysis here shows that the rate of substitution between an arbitrary R-S risk
increase and an arbitrary first degree stochastic dominant change increases as the decision maker
becomes more concave risk averse. This extension generalizes results of Arrow, Pratt, and Ross.
The first degree stochastic dominant extension here is different from a related extension
presented by Machina and Neilson. This rate of substitution measure defined here provides an
additional way to characterize strongly more risk averse for bounded intervals and is the second
major finding of the paper. This finding is summarized in Theorem 3.
One of the reasons for defining measures of risk aversion is to pose questions concerning
how the choice between cumulative distribution functions (CDF) F(x) and G(x) depends on the
various properties of risk aversion. The various forms of stochastic dominance follow this
6
pattern. Another important finding of this paper is that the comparison of the expected utility
from random alternatives F(x) and G(x) can be written as a very simple function of C(x) and no
other property of the utility function. Of course, this expected utility difference can also be
written as a function of only A(x), but doing so leads to an expression which is not simple, and
has proven to be cumbersome enough that it has not been used as a tool of analysis. For C(x),
one can very directly pose questions concerning the effect of risk aversion on the ordering of
random alternatives, and brief stochastic dominance example is used to illustrate this. Writing
expected utility changes as a function of C(x) is useful not only in a stochastic dominance
context, but also in determining comparative static findings in decision models where expected
utility is maximized.
The paper is organized as follows. The next section formally defines the concave risk
aversion measure C(x) and shows that u(x) more concave risk averse than v(x) on [a, b] is
equivalent to u(x) strongly more risk averse than v(x) on [a, b]. Also in this section is discussion
of the rate of substitution between arbitrary R-S risk increases and first degree stochastic
dominant changes, culminating in Theorem 3 which provides an additional way to characterize
the strongly more risk averse order.
Next, in section 3, some of the uses of the definition of both concave risk aversion and
more concave risk averse on an interval [a, b] are discussed. Several things are mentioned.
First, concave risk aversion provides a convenient way to determine whether u(x) is more
strongly risk averse than v(x) on [a, b] for specific utility functions u(x) and v(x). In addition, a
general way is given to define families of utility functions such that by changing a single
parameter, one member is identified as more strongly risk averse than another. Also in this
7
section, an “in the small” discussion of the risk premium is presented to gain a better
understanding of what concave risk aversion measures, and what the strongly more risk averse
order represents. Finally, two results are presented that can be viewed as tools that are available
when doing comparative static analysis. The uses of these tools are briefly illustrated, but
complete discussion is deferred to a companion paper, Liu and Meyer (2012).
2. Concave Risk Aversion and Strongly More Risk Averse on [a, b]
In order to be able to make additional comparative static statements, Ross (1981)
introduces a definition that yields “a new and stronger measure for comparing two agent’s
attitudes towards risk…”. This definition, a definition of strongly more risk averse, is given
below. In this definition and the theorem which follows, Ross assumes that random alternatives
have supports that are “defined on the line.”
Definition 1: (Ross) For u(x) and v(x) that are increasing and risk averse, u(x) is strongly more
risk averse than v(x) if and only if inf
u'' x
v'' x
 sup
u' x
v' x
for all x on the real line.
After providing this definition, Ross shows that strongly more risk averse is characterized by
three equivalent conditions. His theorem provides the working tools for using and applying the
strongly more risk averse order.
Theorem 1: (Ross) The following three conditions are equivalent.
i) There exists a  > 0 such that
u'' x
v'' x

u' y
v' y
for all x and y.
8
ii) There exists a  > 0 and function (x) with '(x)  0 and ''(x)  0 such that
u(x) = ·v(x) + (x) for all x.
iii) For all random
and  such that E(|x) = 0, Eu( + ) = Eu( - u) and Ev( + ) = Ev( -
v) imply that u  v.
Ross characterizes the definition of what it means for decision maker with utility u(x) to
be strongly more risk averse than another with utility v(x) in these three ways. Ross does not,
however, provide a measure of risk aversion that corresponds to this strongly more risk averse
order. That is, from Definition 1 and Theorem 1, conditions are available that can be used to
determine the implications of u(x) strongly more risk averse than v(x), and to test whether u(x) is
more strongly risk averse than v(x), but there is no measure of risk aversion, similar to absolute
risk aversion, that provides a simple way to perform this test or to characterize this order. No
simple property of a utility function was presented such that its level indicates when one utility
function is strongly more risk averse than another. It is precisely such a measure that the
definition of the concave risk aversion measure provides for the strongly more risk averse order.
Definition 2: The concave measure of risk aversion for utility function u(x) for points in [a, b] is
C(x) =
‐ u'' x
u' a
.
Following Ross, it is assumed that u'(x) > 0 and u''(x) < 0 on the line and hence on [a, b]. This
implies that the denominator in the C(x) expression, u'(a), is the largest value that u'(x) attains in
the interval [a, b].
9
Pratt (1990) defines the "partial-risk premium", which is the amount a decision maker
would pay to eliminate a risk  in the presence of a random
where E(|w) = 0. Using the
usual Taylor's series approximation procedures, Pratt then relates the size of this partial-risk
premium to the ratio of the second to first derivative of utility evaluated at different points, and
the size of the risk. Turning from "in the small" to "in the large", Pratt then defines one decision
maker as more partial-risk averse than another if this partial- risk premium is always larger.
Most relevant to the discussion here, Pratt shows that being more partial-risk averse is equivalent
to having a more negative second derivative of utility when the utility functions are each scaled
so that marginal utility at the left hand end point is one. Thus, Pratt characterizes the partial-risk
averse order using the concave risk aversion measure in Definition 2. Pratt does not however
suggest that concave risk aversion be used as a risk aversion measure. In fact, Pratt indicates that
locally, the measure of partial-risk aversion and the measure of absolute risk aversion are the
same.
The first and primary reason that defining the concave measure of risk aversion is useful
is that Cu(x)  Cv(x) for all x in [a, b] is both necessary and sufficient for u(x) to be strongly
more risk averse than v(x) on [a, b]. This use of C(x) allows strongly more risk averse to be
defined in a manner very similar to the way A-P more risk averse is defined. The next theorem
formally states this result.
Theorem 2: The following four conditions are equivalent.
i) There exists a  > 0 such that
u'' x
v'' x

u' y
v' y
for all x and y in [a, b].
10
ii) There exists a  > 0 and a function (x) with '(x)  0 and ''(x)  0 such that
u(x) = ·v(x) + (x) for all x in [a, b].
iii) For all random
and  such that E(|x) = 0, Eu( + ) = Eu( - u) and Ev( + ) = Ev( -
v) imply that u  v when ,  and +  all have supports in [a, b].
iv) Cu(x)  Cv(x) for all x in [a, b].
Proof: That i), ii) and iii) are equivalent has been established by Ross for supports on the real
line and these same equivalencies hold with the finite interval restriction. What remains is to
connect iv) with the other three. That i) implies iv) is presented first.
Assume that i) holds, i.e.,
Letting y = a implies that
u'' x
v'' x
‐u'' x
u' a


u' y
for all x and y in [a, b].
v' y
‐v'' x
for all x in [a, b].
v' a
This of course is Cu(x)  Cv(x) for all x in [a, b]. Thus i) implies iv) has been demonstrated.
Now assume that iv) holds, i.e.,
‐u'' x
u' a

‐v'' x
v' a
Integrating both sides from a to y implies that
holds for all x in [a, b].
u'(y)
u'(a)

Combining the above two inequalities and letting  =
u'' x
v'' x

u' y
v' y
v'(y)
v'(a)
u'(a)
v'(a)
for all y in [a, b].
, one obtains
for all x and y in [a, b].
Thus iv) implies i) has been demonstrated.
11
QED
This theorem shows that the concave risk aversion measure C(x) can be used to
characterize the strongly more risk averse order on bounded intervals [a, b]. It is well known
that strongly more risk averse implies A-P more risk averse. Concave risk aversion can help
explain the difference between these two orders. C(x) is the rate of change of D(x) =
‐u' x
u' a
,
while the absolute risk aversion measure A(x) is the percentage rate of change of this same
function. That is, C(x) = D'(x) while A(x) = -D'(x)/D(x). Because at x = a, Du(a) = Dv(a), for all
u(x) and v(x), when Cu(x)  Cv(x), Au(x) and Av(x) must satisfy Au(x)  Av(x) as well. Of
course, this is very obvious if u'(a) = 1 is assumed.
C(x) also allows a possible  and (x) to be identified that are consistent with
characterizations i) and ii) provided by Ross. The minimum possible value for  is  =
u' a
v' a
,
and corresponding to this minimum value for  is a (x) function with the properties '(a) = 0
and ''(x) = - u'(a)[Cu(x) - Cv(x)]. This implies that for this (x), Cu(x) = Cv(x) -
'' x
u' a
. These
possible values for  and (x) are used in the final section of the paper when providing one
parameter families of utility functions such that one member is strongly more risk averse than
another.
Characterization iii) concerning the tradeoff between the risk premium and a risk increase
can be made more general and more useful by altering how an increase in risk is represented, and
by considering the rate of substitution between this risk increase and an arbitrary rather than
12
specific first degree stochastic dominant (FSD) change. Rather than making x riskier by adding
 to x, assume instead that x is described by CDF F(x) and this increase in risk is represented by
changing F(x) to G(x), where G(x) satisfies the Rothschild and Stiglitz (1970) definition of an
increase in risk from F(x). This alteration is nothing more than a change in notation from
random variables to CDFs.
For the risk premium however, rather than assuming a uniform leftward shift in the CDF
for x resulting from the subtraction of  from x, the CDF for x is changed from F(x) to H(x),
where F(x) dominates H(x) in the first degree. What remains is defining the rate of substitution
between the risk increase from F(x) to G(x) and the FSD reduction from F(x) to H(x). This rate
of substitution is then used to measure the intensity of the decision maker’s reaction to the risk
increase, and hence to measure the intensity of the decision maker's risk aversion. The theorem
that is presented next is very general and reflects the power of the strongly more risk averse
assumption.
For any increase in risk from F(x) to G(x) and for any FSD change from F(x) to H(x), let
Tu(F(x), G(x), H(x)) =
b
u
a
b
a u
x d F x ‐G x
x d F x ‐H x
=
b
u'
a
b
a u'
x G x ‐ F x dx
x H x ‐ F x dx
The magnitude of Tu measures the relative sizes of the utility losses that occur because x
becomes riskier, the numerator, and because x becomes smaller in an FSD sense, the
denominator. If T is larger for one utility function than another, and this is true for all increases
in risk to G(x), and for all FSD changes to H(x), then this is a clear indication that the one
decision maker is more risk averse than the other. That is, a larger reduction in the size of the
outcome variable is always required to compensate for or match the utility loss from a risk
13
increase, no matter how risk is increased and no matter how the size of the random outcome is
decreased.
Both the numerator and denominator in the definition of T are a change in utility caused
by a change in the random alternative. Forming the ratio of these two changes in utility is
defining a rate of substitution of one change for the other. This same thing is true for the Pratt
risk premium. Pratt shows that to a local approximation, a reduction in wealth by one unit
changes utility by -u'(x), and a one unit increase in variance changes utility by u''(x)/2. The ratio
of these two changes in utility gives a rate of substitution of a small risk introduction for a
reduction in the certain outcome, and reflects the decision maker's willingness to pay for
avoiding the introduction of risk. Of course, for the risk premium this rate of substitution is the
A-P measure of absolute risk aversion.
While T and the A-P absolute risk aversion are similar in that they can be viewed as rates
of substitution, they are different in many other ways. First, the definition of T does not rely on a
local approximation. The changes represented can be large, and the ratio presented is not an
approximation of any sort. Second, T does not assume the initial, before change, setting is
certainty, but instead allows any random starting point represented by F(x). Finally, the
definition of T uses a representation of a decrease in the outcome variable and an increase in its
riskiness that is completely general. In light of this substantial increase in generality it is quite
surprising, to us at least, that Theorem 3 is true. In addition, this theorem is "if and only if", and
therefore it shows that T can be used to provide another characterization of strongly more risk
averse on the interval [a, b].
14
Theorem 3: u(x) is more concave risk averse than v(x) on [a, b] if and only if Tu  Tv for all
F(x), G(x) and H(x) such that G(x) is riskier than F(x), and F(x) first-degree stochastically
dominates H(x).
Proof: “only if”
b
Tu 
b


ud
(
F

H
)


a
b
  u (G 2  F 2 )dx
ud ( F  G )
a
b
a
b
Tv 
a

1
u( a )
b


 vd ( F  H ) 
a
b
u ( H  F ) dx
b
  v (G 2  F 2 )dx
vd ( F  G )
a
b
a
a
x
x
a
a
v ( H  F )dx
 u 1(a )  u (G 2  F 2 )dx
a
b

a
u ( H  F )dx
b

 v(1a )  v (G 2  F 2 )dx
1
v ( a )
a
b

a
v ( H  F )dx
where F 2 ( x)   F ( y )dy , G 2 ( x)   G ( y )dy , and G 2 ( x)  F 2 ( x) for all x in [a, b] with the
strict inequality holding for at least some x in (a, b). When u is more concave risk averse than v,
the numerator of Tu is larger than the numerator of Tv . Further,
u ( x) v ( x)

for all x  [a, b] ,
u (a ) v (a )
which implies that the denominator of Tu is smaller than the denominator of Tv . Hence, Tu  Tv .
“if”
b
b
a
b
a
b
a
a
 ud ( F  G )   vd ( F  G)
 ud ( F  H )  vd ( F  H )
Suppose
for all F(x), G(x) and H(x) such that G(x) has
more risk than F(x), and F(x) first-degree stochastically dominates H(x). This is equivalent to
(*)

b
a
u (G 2  F 2 )dx

b
a
u ( H  F )dx


b
a
v (G 2  F 2 )dx

b
a
v ( H  F )dx
.
To show that u must be more concave risk averse than v, proof by contradiction is used. Note
that u being more concave risk averse than v is equivalent to, from Theorem 2,
15
u ( x) u (a) u ( y )


v ( x) v (a ) v ( y )
for all x, y  [a, b] .
Assume that u is not more concave risk averse than v. Then there exist some x, y  [a, b] and
  0 , such that
u ( x )
u ( y )

v ( x )
v ( y )
,
which implies, due to continuity, that there exist [a1 , b1 ]  (a, b) and [a2 , b2 ]  (a, b) such that
(**)
u ( x )
u ( y )

v ( x )
v ( y )
for all x  [a1 , b1 ] and all y  [a2 , b2 ] .
Choose F, G and H such that
2
2
G  F  0 x  [a1 , b1 ]
 2
2
G  F  0 x  [a1 , b1 ]
 H  F  0 x  [a2 , b2 ]
.

 H  F  0 x  [a2 , b2 ]
Then, from (**),

b

b
a
a
b
u (G 2  F 2 )dx    v (G 2  F 2 )dx
a
b
u ( H  F )dx    v ( H  F )dx
,
a
which contradicts (*).
QED
3. Using More Concave Risk Averse on [a, b]
The focus in this section is on using the more concave risk averse characterization to
increase our understanding of the strongly more risk averse order. The analysis shows that for
some questions, C(x) is a very useful characterization. For other questions this may not be the
case. As a preface to this discussion, it is important to recognize that the three characterizations
of the strongly more risk averse order provided by Ross have been available for more than thirty
16
years. As a consequence, many implications of assuming that one decision maker is strongly
more risk averse than another are already known. This seems to be especially true for
comparative static analysis where u(x) = ·v(x) + (x) for  > 0 and '(x)  0 and ''(x)  0
proves to be very useful in demonstrating a variety of theorems. Concave risk aversion often
does provide an alternate and sometimes simpler way to demonstrate these known findings.
As an illustration of a use of C(x), the discussion here begins with two very familiar
families of utility functions and determines when one member is strongly more risk averse than
another. Pratt (1990) presents the same findings. First, suppose the utility functions under
consideration are all constant absolute risk averse (CARA); that is, each takes the form u(x) =
‐1 ‐x
e , where  > 0. Using the definition of C(x) and straightforward manipulation shows that

C(x)  C(x) for all x in [a, b] if and only if


e ‐
x‐a
for all x in [a, b] where  >  is
assumed, and  and  each are risk aversion parameters in the CARA utility form. Since the
right hand side is increasing in x, for this inequality to hold for all x in [a, b] it must hold at b.
Thus the restriction can be changed to


e ‐
b‐a
and also written as (b – a) 
ln  ‐ ln 
‐
The latter form emphasizes the relationship between the location of the supports of the random
alternatives and the difference in risk aversion levels for CARA functions for one CARA
decision maker to be strongly more risk averse than the another on [a, b]. The inequality also
verifies the known result that no CARA function is strongly more risk averse than another if the
interval [a, b] is allowed to increase without bound. It is interesting to note that the length of the
.
17
interval [a, b] is important, but its placement is not. This is a usual feature of constant absolute
risk averse utility functions.
A similar analysis using constant relative risk averse (CRRA) utility functions leads to
the requirement that (ln b – ln a) 
ln  ‐ ln 
‐
where now  and  are the relative risk aversion
parameters in a utility function of the form u ( x) 
x1
. For these CRRA utility functions it is
1
also the case that no function is strongly more risk averse than the other as the interval increases
in length without bound. For CRRA utility functions, the placement of the interval as well as its
length both matter; that is, now the restriction for x in [a, b] differs from that for x in [a+, b+].
Each of the above two examples indicates that placing bounds on the interval in which
the supports of random alternatives lie can make the strongly more risk averse order a more
applicable and useful assumption. Thus, empirical evidence concerning [a, b], and especially the
value for a, can be useful in doing comparative static analysis. Sometimes it is the case that the
value for a is determined by theoretical considerations as well. Requiring a  0 is a common
assumption. These two examples also indicate that for the two most often used families of utility
functions, their properties are such that only under quite restrictive conditions (on the location
and the length of the interval) is one member strongly more risk averse than another. Because of
this, C(x) is used next to identify other families of utility functions where a single parameter
change does lead to one utility function being strongly more risk averse than another on intervals
of any size.
Ross gives an example of two utility functions such that one is strongly more risk averse
than the other. Analysis of his example leads to a general procedure for finding families of
18
utility functions indexed by parameter  such that one member is strongly more risk averse than
the other if and only if u  v. Consider families of utility functions indexed by  > 0 and of
the form u(x) = x + u·w(x) for any w(x) satisfying w'(x)  0 and w''(x)  0. For this function
u'(x) = 1 + u·w'(x) and u''(x) = u·w''(x). Thus, Cu(x) =
‐u w'' x
1
u ·w' a
. Forming the ratio
Cu(x)/Cv(x) allows one to verify that u(x) is more strongly risk averse than v(x) on any interval
[a, b] if and only if u  v. Another way to demonstrate this same result is to note that if u(x)
and v(x) are two members of this family of utility functions, the one utility function can always
be written as a function of the other taking the general form u(x) =
u
u
v(x) + (1 - )x. Thus,
v
v
if u  v, this transformation satisfies characterization ii), and u(x) is strongly more risk averse
than v(x). This procedure allows one to begin with any increasing and concave function w(x)
and use it to define a one parameter family of utility functions where the parameter orders the
members by the strongly more risk averse criterion. Since w(x) can be any increasing and
concave function, any number of families of such utility functions can be created.2
The discussion so far has focused on when u(x) and v(x) are such that one is strongly
more risk averse than the other on [a, b]. Strongly more risk averse on [a, b] is necessary and
sufficient for the “in the large” results concerning the risk premium or any FSD change required
as equivalent variation for a risk change. These results are presented in Theorems 1 or 2, and 3.
Strongly more risk averse on [a, b] is also necessary and sufficient for a number of other
2
In their study of the relationship between changes in risk aversion and changes in prudence ( - u'''(x)/u''(x)),
Eeckhoudt and Schlesinger (1994) also work with these families of utility functions. They point out that, using the
parameterization of the current paper, prudence remains the same but the A-P risk aversion increases as u increases.
Note also that these families of utility functions include the so-called “linex” family, where w(x) takes the form of
any CARA utility function, as a special case (Denuit, Eeckhoudt and Schlesinger 2011).
19
comparative static findings. Many of these were presented by Ross and the literature which
followed, and are not replicated here. The next discussion deals with the “in the small” results
associated with risk premiums. A more general risk premium is defined and its relationship to
C(x) is determined. This analysis provides an interpretation of C(x) as an intensity measure of
risk aversion.
Arrow and Pratt, in their argument relating A(x) to the risk premium, use analysis that
can be extended to the case of a random starting position. Doing so helps in the understanding of
the meaning of C(x). A-P consider two different changes to fixed starting point x. First, a small
introduction of risk at x is made by adding z to x with E( z ) = 0. The utility change from the
introduction of this risk is given by [Eu(x + z ) – u(x)]. A-P show that this is approximately
equal to u''(x)·z2/2 for small z . This value is negative when u''(x) < 0. The second change that
A-P consider is constructed to give a utility change that is equivalent to that caused by the risk
increase. This change to x is obtained by subtracting an amount (x) from x. This reduction in
the size of x results in a utility change equal to [u(x - (x)) – u(x)] which is approximated by
u'(x)(-(x)). Equating the sizes of the two utility changes gives the very familiar equation:
(x) =
1 ‐u'' x
2
·
u' x
·z2 = (1/2)A(x)z2.
Because Arrow and Pratt assume that the starting point x is nonrandom, both the
introduction of risk and the subtraction of the risk premium must each take place at the same
point x. When there is the possibility that x can take on values in the interval [a, b], the locations
where these two changes occur can be decoupled. Consider an increase in risk that occurs at x1.
The change in utility from this risk increase is approximately u''(x1)·z2/2 for the small risk
20
increase z . Similarly, when the risk premium is subtracted at x2, the utility change from this
size decrease is approximately equal to u'(x2)(-(x2)) . Equating these utility changes makes it
clear that  now depends on where each of these changes occurs, that is,  depends on both x1
and x2. Solving as before gives (x1, x2) =
1 ‐u'' x1
·
·z2.
2 u' x2
Using this discussion, C(x) then has the following interpretation. For a small increase in
risk at x in interval [a, b] and with a risk premium subtracted at the left end point of this interval,
a, (x, a) = (1/2)C(x)·z2. Thus, C(x) has exactly the same interpretation “in the small” as does
A(x). Both measure the decision maker’s reaction to a small risk, and the difference between the
two is where the subtraction of the risk premium takes place.
The paper concludes with two tools useful for comparative static analysis. First is a
rewriting of the comparison of expected utility from a pair of random alternatives so that C(x)
enters the calculation in a very simple and direct way. Let F(x) and G(x) denote the CDFs
representing any pair of random alternatives. By using integration by parts twice, one has
EFu(x) - EGu(x)=
=
b
u
a
b
u'
a
x d F x ‐G x
x G x ‐ F x dx
= u'(b)[F -G] -
b
u''
a
x G2 x ‐ F 2 x dx
In this expression, F is the mean of F(x) and F2(x) =
Now rewrite u'(b) as u'(b) =
b
u''
a
x
F
a
s ds. Similar notation holds for G(x).
x dx + u'(a). Substituting this into the line above and
normalizing so that u'(a) = 1, yields expression (1).
EFu(x) - EGu(x) = [F - G] +
b
C
a
x G ‐ F ‐ G x ‐ F 2 x dx
(1)
21
Of course, whether F(x) or G(x) yields the higher level of expected utility depends on properties
the CDFs and of utility. This expression shows, however, that C(x) summarizes the relevant
properties of the utility function u(x), and C(x) enters this calculation in a very simple and direct
way. Expression (1) proves useful in many comparative static questions, and for space reasons
most of that analysis is deferred to another paper, Liu and Meyer (2012). One very simple
illustration, however, is provided.
Consider the usual stochastic dominance question. The general or generic form of this
question in this context is when is F(x) preferred or indifferent to G(x) by all utility functions
with concave risk aversion measures C(x) satisfying given restrictions? The only restriction
imposed on C(x) by u'(x)  0 and u''(x)  0 is that C(x)  0 and
b
C
a
x dx = 1 -
u' b
u' a
 1. The
latter restriction results from the requirement that marginal utility be nonnegative. With only
these restrictions on C(x) one can verify that the standard second degree stochastic dominance
condition arises; that is, F(x) is preferred or indifferent to G(x) by all utility functions with
concave risk aversion measures C(x) satisfying C(x)  0 and
b
C
a
x dx  1 if and only if G2(x) 
F2(x) for all x. The "if" part follows immediately by noting that G2(x)  F2(x) implies F  G
and then applying (1), whereas the analysis showing the "only if" part first notes that when C(x)
= 0, F  G must hold. Next, when F  G, the "worst" C(x) is the most risk averse possible i.e.
where
b
C
a
b
C
a
x dx = 1. When
b
C
a
x dx = 1, expression (1) reduces to EFu(x) - EGu(x) =
x G x ‐ F 2 x dx and the second degree stochastic dominance condition follows
immediately by the usual argument of proof by contradiction.
22
A new form of stochastic dominance arises if a different restriction is imposed on C(x).
b
C
a
Consider for example, assuming that C(x) satisfies the stronger restriction:
x dx  .5. This
restriction places an upper bound on the total amount of concave risk aversion, but not where in
[a, b] that risk aversion occurs. Using expression (1), again C(x) = 0 implies that F  G must
hold. For the case where F  G, the worst case is again the most risk averse possible. Using
b
C
a
x dx = .5, (1) reduces to EFu(x) - EGu(x) = (.5)[F - G] +
b
C
a
x G x ‐ F 2 x dx. From
this one can observe that G2(x)  F2(x) is no longer required and the size of the violation allowed
is directly related to the difference in the mean values of the two alternatives. There are a large
number of other potential and interesting restrictions that can be placed on risk aversion measure
C(x), including placing both a lower and upper bound on
b
C
a
x dx or restricting its slope. The
implications of such restrictions can be explored in a similar fashion.
The second tool that is also useful in comparative static analysis is the following theorem.
This theorem indicates that assuming that one decision maker is strongly more risk averse than
another allows the first derivative and second derivative effects for each decision maker to be
determined separately and compared in a particular manner.
Theorem 4: Cu(x)  Cv(x) for all x in [a, b] implies that
for any M(x)  0 and N(x)  0 for all x in [a, b].
b
‐u'' x M x dx
a
b
a u' x N x dx

b
‐v'' x M x dx
a
b
a v' x N x dx
23
‐u'' x
Proof: First, observe that
b
‐v''
a
x M x dx
v' a
or
u' a
b
‐u''
a

‐v'' x
v' a
x M x dx 
for all x in [a, b] implies that
u' a
v' a
b
‐v''
a
b
‐u''
a
x M x dx

u' a
x M x dx for any M(x)  0. In addition,
as argued in the proof of Theorem 2, Cu(x)  Cv(x) for all x in [a, b] implies that
u' x
u' a

v' x
v' a
for all x in [a, b]. This then implies that
b
u'
a
x N x dx 
u' a
v' a
b
v'
a
x N x dx for any N(x)  0. Combining these two findings, one
concerning the numerator and one the denominator, we have
b
‐v'' x M x dx
a
b
a v' x N x dx
b
‐u'' x M x dx
a
b
a u' x N x dx

for any M(x)  0 and N(x)  0.
QED
A specific case of Theorem 4 was used in the demonstration of Theorem 3. The finding
in Theorem 4 is also directly relevant to the discussion of the effects of background risk and is
why strongly more risk averse is preserved under background risk. Another use for this result
arises when doing an “in the small” version of Ross’s risk premium relationship. Suppose
is
random with support in [a, b]. A change in risk could occur at any x in this interval or even at all
x in the interval. Let ̃ satisfy E( ̃ |x) = 0, and , ̃ and
+ ̃ have supports in [a, b]. For small ̃ ,
the utility change from each of these risk increases, one for each x, is given by [u''(x)·z2(x)/2]
and the total expected utility change is then given by Ex[u''(x)·z2(x)/2]. In Ross's analysis, a risk
24
premium of a fixed size of  is subtracted at every x, with the total expected utility change being
Ex[u'(x)·(-)]. Therefore, =Ex[-u''(x)·z2(x)/2]/ Ex[u'(x)], an expression which is a positively
weighted sum of -u''(x) in the numerator and a positively weighted sum of u'(x) in the
denominator. Theorem 4 then indicates that if one decision maker is strongly more risk averse
than the other, the “in the small” relationship for  holds as well. Theorem 4 likely has many
other applications especially in the discussion of background risk and is presented as a tool for
comparative static analysis.3
4. Conclusion
The work presented in this paper does two main things. First, the usefulness of the
strongly more risk averse partial order proposed by Ross is increased by adding several ways to
characterize this order and determine its implications. Second, the concave risk aversion
measure is identified as a legitimate and useful measure of risk aversion. This measure of risk
aversion represents risk preferences in the same manner as does the absolute risk aversion.
Concave risk aversion can allow one to discover findings that are difficult to discover when
using absolute risk aversion as the risk aversion measure. One important new finding is that on
an interval [a, b], Cu(x)  Cv(x) characterizes the strongly more risk averse order. The stochastic
dominance example provided in section 3 illustrates, however, that C(x) as a measure of risk
aversion can be useful even without its relationship to the Ross strongly more risk averse order.
3
For a summary of the issues in this line of research, see Gollier (2001).
25
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L. Eeckhoudt, H. Schlesinger, Increases in prudence and increases in risk aversion, Economics
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