RISK PREMIUMS AND BENEFIT MEASURES FOR
GENERALIZED-EXPECTED-UTILITY THEORIES
JOHN QUIGGIN AND ROBERT G. CHAMBERS
1. Introduction
Over the past …fteen years, the theory of choice under uncertainty has undergone
radical change. The pivotal contribution was Machina’s (1982) demonstration that
a large class of preferences could be locally approximated by expected-utility functionals and that global preferences inherited properties, such as risk aversion, of the
local utility functions. Less progress has been made, however, in developing tools
relating to non-local properties of preferences such as the absolute and relative risk
premiums used in expected-utility theory.
During this same period, however, the literature on choice under certainty made
substantial progress in developing new techniques for characterizing preferences
and technologies using the concepts of distance (Färe, 1988) and bene…t functions
(Luenberger, 1992). In particular, Luenberger (1992, 1994) introduced the bene…t
function and demonstrated its usefulness in characterizing preferences and Paretoefficient outcomes. It is natural, therefore, to ask whether these techniques can be
informatively applied to problems of choice under uncertainty.
This paper shows that a wide range of standard tools for the analysis of economic
problems involving uncertainty, including risk premiums, certainty equivalents and
the notions of absolute and relative risk aversion, can be developed and applied
without making speci…c assumptions on functional form beyond the basic requirements of monotonicity, transitivity, continuity, and the presumption that individuals prefer certainty to risk. In particular, individuals are not required to display
probabilistic sophistication, in the sense of Machina and Schmeidler (1992). Our
approach relies on the distance and bene…t functions to characterize preferences
relative to a given state-contingent vector of outcomes, and then derives results
directly from the properties of these functions.
After introducing our notation, we start by de…ning a concept of risk aversion
from which mean values and a generalized concept of subjective probabilities can
be derived. The distance and bene…t functions are then used to derive absolute and
relative risk premiums and to derive conditions under which preferences display constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA).
An immediate by-product of this discussion is a result characterizing preferences
displaying both CARA and CRRA. This result is then used to suggest several ‡exible functional speci…cations of preferences satisfying both properties. Finally, a
generalization of the notion of Schur-concavity, following Chew and Mao (1995), is
presented. It is shown that if preferences are generalized Schur concave, the absolute and relative risk premiums are generalized Schur convex, and the certainty
equivalents are generalized Schur concave.
Date : Forthcoming, Journal of Risk and Uncertainty.
1
2
JOHN QUIGGIN AND ROBERT G. CHAMBERS
2. Notation
We adopt a state-contingent approach, as in the work of Hirshliefer (1965) and
Yaari (1969). That is, we are concerned with preferences over random variables
represented as mappings from a state space to an outcome space Y <, or, in
the analysis of measures of relative risk aversion, Y <++ , where <++ denotes the
positive real numbers. Our focus is on the case where is a …nite set {1, ...S }, and
the space of random variables is Y S <S . (No major difficulties arise with the
extension to general measure spaces.) We make particular use of the unit vector 1 =
(1, 1, ... 1). Preferences over Y S are given by a total ordering denoted notationally
by .
A preference function is a mapping W : Y S → < such that W (y) > W (y0 )
if and only if y y0 . W is assumed everywhere continuous and nondecreasing.
Two alternative function representations of preferences provide the bulwarks of our
analysis. The …rst is a particular version of Luenberger’s (1992) bene…t function
for the preference structure. The bene…t function, B : < Y S → <, is de…ned for
g ∈ <S by:
B (w, y) = max{ ∈ < : W (y
g) w}
if W (y g) w for some and ∞ otherwise. To maximize comparability
with previous work on the expected-utility model, we con…ne attention to the case
where g = 1.1 The second function representation, which is only relevant when y
is restricted to be nonnegative, is the Shephard (1953)-Malmquist (1953) distance
S
function D : <+
< → <+ de…ned by:
D(y, w ) = sup{ > 0 : W (y/) w }
S
y ∈ <+
.
The properties of both of these functions are well known (Luenberger, 1992;
Chambers, Chung, and Färe, 1996; Färe, 1988) and are summarized for later use
in the following lemmata:
Lemma 1: B (w, y) satis…es:
a) B (w, y) is nonincreasing in w and nondecreasing in y;
b) B (w, y + 1) = B (w, y) + , ∈ < (the translation property);
c) B (w, y) 0 ⇔ W (y) w , and B (w, y) = 0 ⇔ W (y) = w ; and
d) B (w, y) is jointly continuous in y and w in the interior of the region < Y S
where B (w, y) is …nite.
Lemma 2: D(y, w ) satis…es:
a) D(y, w ) is nonincreasing in w and nondecreasing in y;
b) D(y, w ) = D(y, w ), > 0;
c) D(y, w ) 1 ⇔ W (y) w, and D(y, w ) = 1 ⇔ W (y) = w ;
d) D(y, w ) is continuous in w and y jointly in the interior of the region Y S <
where D(y, w ) is …nite.
These properties are fairly self-explanatory, but two prove particularly important
in what follows: The …rst is the fact that, by 1.c and 2.c, both the bene…t function
and the distance function are alternative representations of the preference structure.
1 The bene…t function in this case corresponds to the translation function introduced by Blackorby
and Donaldson (1980).
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 3
The second is the translation property of the bene…t function (1.b) and the corresponding homogeneity property (2.b) of the distance function. These properties
make it very easy for us to deduce the relationship between the certainty equivalent
and the various risk premia that we consider. The following lemma characterizes
arbitrary functions satisfying the translation property:
Lemma 3: A continuous function f : Y S → < satis…es the translation property if
and only if it can be expressed as
f (y) = g (y
M in{y1 , ..., yS }1) + M in{y1 , ..., yS },
with g : Y S → < continuous.2
Proof : By the translation property f (y + 1) = f (y) + , ∈ <. Take =
M in {y1 , ..., yS } to establish necessity. Sufficiency is obvious.
Remark: Lemma 3 is obvious. Its importance is that it offers a natural way to
generate functions satisfying the translation property. In turn, this permits us to
generate whole families of preference functionals satisfying constant absolute risk
aversion. A number of other normalizations, for example the mean of y, would
suffice in Lemma 3 for the case where the domain of the function is simply <S .
For any y and W , we de…ne the certainty equivalent
e(y) = inf {c > 0 : W (c1) W (y)}.
In the expected utility case, e(y) = u 1 (E [u(y)]). From the previous de…nitions,
it is now obvious that:
Lemma 4: e(y) = 1/D(1, W (y)) = B (W (y), 0)
2.1. Measures of risk aversion. The basic de…nition of risk aversion is a preference for certainty over risk, usually interpreted to mean that a random variable
y is less preferred than the certainty of receiving the expected value E [ y]. Thus
far in this paper, however, the expected value has not been de…ned and it has not
been assumed that the decision-maker is probabilistically sophisticated.
The most common approach to these issues is to derive probabilities from the
Savage axioms, then de…ne the expected value and risk aversion in terms of probabilities. In the present paper, following the approach developed by Yaari (1969),
the corresponding concepts are de…ned simultaneously.
De…nition 1: A decision-maker is weakly risk-averse if there exists a vector ∈
<S+ , with i = 1 and
W (E [y]1) W (y),
∀y
where E [y] = i yi and E [y]1 is the state-contingent outcome vector with E [y]
occurring in every state of nature. A decision maker is risk-neutral if equality holds
for every y and risk-averse otherwise. A decision maker is strictly risk-averse if
strict inequality holds whenever y =
6 E [ y ] 1.
Weak risk aversion in this sense is equivalent to requiring a non-negative absolute
risk premium
2 Note
S
<+
that, by virtue of the de…nition of f , we are only concerned with the behavior of g on
S , that is, on the boundary of the non-negative orthant.
<++
4
JOHN QUIGGIN AND ROBERT G. CHAMBERS
r (y) = max{c : W ((E [y]
c)1) W (y)} = B (W (y), E [y]1).
Hence, a non-negative absolute risk premium for a weak risk averter simply re‡ects,
via property 1.c of the bene…t function, the fact that E [y]1 is always at least as
desired as y. The absolute risk premium is thus recognizable as a special case of
Luenberger’s (1996) compensating bene…t. More precisely, it is the compensating
bene…t for moving from y to E [y]1 and is equal to the negative of the equivalent
bene…t for moving from E [y]1 to y.
From the translation property of the bene…t function (1.b) we deduce the wellknown fact that the absolute risk premium is the difference between the expected
value and the certainty equivalent:
r (y) = E [y] + B (W (y), 0) = E [y]
e(y).
Moreover, it follows immediately that:
r (c1) = 0
for all and c.
Alternatively, when all state-contingent net returns are nonnegative, the preference for certainty can be expressed as requiring the existence of a positive scalar
v (y) 1 such that
W (E [y]1/v (y)) = W (y).
Following Arrow (1965) and Pratt (1964), we de…ne the relative risk premium as
v (y) = sup{ > 0 : W (E [y]1/) W (y)} = D(E [y]1, W (y)).
Using Lemma 4 and Lemma 2.b, it follows immediately that:
v (y) = E [y]D(1, W (y)) = E [y]/e(y).
Finally, from the viewpoint of an individual risk-averse decision-maker with preference function W , it is reasonable to say that if E [y] = E [y0 ] and W (y0 ) W ( y), then y0 is regarded as more risky than y. Whenever this is true, we may
de…ne the conditional absolute risk premium B (W (y0 ), y) and the conditional relative risk premium D(y, W (y0 )) and conclude that if E [y] = E [y0 ], y0 is more
risky than y in this sense if and only if the conditional absolute risk premium is
non-positive and the conditional relative risk premium is less than or equal to one.
2.2. Probabilistic sophistication. The vector is not necessarily a probability
vector in the sense of the Savage axioms, so risk aversion does not imply probabilistic
sophistication. For example, in the case where all the entries of are equal to 1/S ,
preferences are probabilistically sophisticated if and only if W is symmetric. Two
examples illustrate the point that an individual can be both risk-averse in the sense
de…ned above and ambiguity-averse in the sense of Ellsberg (1961).
Example 1: Consider the state space arising from the possibility of betting on
a roulette wheel or the stock-market index. We de…ne the states of nature as (R,
Up), (R, Down), (B, Up), (B, Down) where
R = Roulette wheel stops on Red
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 5
B = Roulette wheel stops on Black
Up = Stock market rises
Down = Stock market falls
Suppose the individual is risk-averse with vector = {1/4, 1/4, 1/4, 1/4}. That
is, the individual will reject all even-money bets on either the roulette wheel or the
stock market index. For initial wealth and any stake , consider the prospects
yR = { + , + , , };
that is, a bet that the roulette wheel will stop on Red
yU = { + , , + , };
that is, a bet that the stock market will go up. Assuming yU is seen as more
ambiguous than yR , the hypothesis of risk aversion is consistent with yR yU .
Example 2: Consider the Ellsberg problem of drawing one ball from an urn with
99 numbered balls of which 1-33 are red, and 34-n are black and n-99 are yellow
for some unknown n. The state space may be considered as consisting of S = 66 99 elements, each corresponding to a value of n and a number for the ball drawn.
The usual behavior in the Ellsberg problem is consistent with risk-aversion relative
to the probability vector with all elements equal to 1/S .
The de…nition of risk aversion does not require that the vector is unique.
Uniqueness can be assured in two main ways. First, smoothness of the preference
functional W ensures that, in a neighborhood of E [y]1, the individual is approximately risk neutral. We have:
Lemma 5: If W is smooth, the vector is unique.
Uniqueness may also be imposed if the individual is probabilistically sophisticated so that the Savage axioms (or the modi…ed axioms of Machina and Schmeidler) imply that preferences may be characterized by a unique vector . Then if
the individual is risk-averse relative to any 0 , they must also be risk-averse relative
to . Hence, we can con…ne attention to the expected value E [y] = i yi . For
the remainder of this paper, we will focus on the case when is unique and will
drop the associated subscript.
2.3. Certainty equivalent representations of preferences. Since any monotonic
transformation of W represents the same ordering on Y S , a monotonic transformation of W will leave the risk premium unchanged. Conversely, any two preference
functions that have the same risk premiums represent the same preferences and are,
therefore, related by a monotonic transformation. A particularly appealing representation is the mean-value or certainty equivalent representation, obtained from
any continuous monotonic W by setting
W (y) = e(y) =
B (W (y), 0) = 1/D(1, W (y)).
The idea of representing preferences under uncertainty as mean values has previously been used by Chew (1983), developing the work of Hardy, Littlewood, and
Polya (1952). As Chew shows, the expected utility certainty equivalent u 1 (E [u(y)])
may be regarded as a quasilinear mean value. In particular, the case of expected
utility with logarithmic utility may be represented by the geometric mean and may
6
JOHN QUIGGIN AND ROBERT G. CHAMBERS
be monotonically transformed into a preference function W in Cobb-Douglas form.
To illustrate this point, let u(y) = log(y), so that
E [u(y)] =
X
s log(ys )
s∈
and
e(y)
=
=
=
1
(E [u(y)])
!
X
exp
s log(ys )
u
Y
s∈
ys s
s∈
In the case of equal probabilities, where s = 1/S ∀s, this is simply the geometric
mean, and the relative risk premium is the ratio of the arithmetic to the geometric
mean. More generally, any expected utility function with a constant coefficient of
relative risk aversion corresponds to the geometric mean of order and may be
monotonically transformed into a preference function W in CES utility functional
form.
These observations suggest the possibility of ‡exibly representing large classes
of preferences using ‡exible functional representations familiar from the literature
on producer and demand theory. In particular, it is natural to consider possibilities such as the translog and generalized Leontief functional forms. Since these
functional forms are not, in general, additively separable, they will not generally
correspond to expected-utility preferences.
2.4. Constant absolute and relative risk aversion. The bene…t and distance
functions yield simple characterizations of constant absolute and relative risk aversion. We de…ne:
De…nition 2: W displays constant absolute risk aversion (CARA) if, for all y, t,
r (y + t1) = r (y)
De…nition 3: W displays constant relative risk aversion (CRRA) if, for all y, t,
v(ty) = v(y)
Replacing the equalities in De…nitions 2 and 3 with appropriate inequalities
yields concepts of decreasing absolute risk aversion (DARA), increasing absolute
risk aversion (IARA), decreasing relative risk aversion (DRRA), and increasing
relative risk aversion (DRRA). As Yaari (1969) observes, however, the de…nition
of DARA and related concepts obtained in this way is one of a family of closely
related concepts. For example, different speci…cations of the bene…t function would
give rise to different de…nitions of DARA. Alternatively, DARA conditions may be
expressed in terms of asset demands. Dybvig and Lippman (1983) show that the
asset demand condition proposed by Yaari is equivalent to the usual de…nition of
DARA for risk-averse expected utility preferences. However, this may not be true
for more general preferences (Machina 1982).
Result 1: W displays CARA if and only if W is a continuous nondecreasing transformation of a function satisfying the translation property. W displays CRRA if and
only if W is homothetic, i.e., a monotonic transformation of a linearly homogeneous
function.
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 7
Proof : Suppose …rst that the preference structure satis…es CARA. By the fact
that:
r (y) = E [y]
e(y)
it follows immediately that e(y + 1) = e(y) + . Lemma 1.a and 1.d and Lemma 4
establish necessity. To go the other way, let W (y) = F (f (y)) where F is continuous
and nondecreasing and f is continuous, monotonic, and satis…es the translation
property. By de…nition, it follows that:
r (y + 1) =
=
=
max{c : F (f ([E y + c] 1)) F (f (y + 1))}
max{c : f ([E y + c] 1) f (y + 1)}
max{c : f ([E y c] 1) f (y)} = r (y).
The proof of the second part of the result is exactly parallel using the distance
function instead of the bene…t function.
Remark: The property that characterizes preference structures with CARA has
been referred to as translatability by Blackorby and Donaldson (1980) and as BDtranslation homotheticity by Chambers and Färe (1997) who derive it as a special
case of preference structures which can be expressed by translating a single at-leastas-good set in an arbitrary direction.
Corollary 1.1:
(i) A utility function displays CARA if and only if ∀ y, y0
B (W (y + k 1), y0 ) = B (W (y), y0 )
k
(ii) A utility function displays CRRA if and only if ∀ y, y0 , t > 0
D(W (ty), y0 ) = D(W (y), y0 )/t
Proof : Necessity follows from Result 1. For sufficiency, take y0 = E [y]1. As Result 1 and Corollary 1.1 show, constant absolute or relative risk aversion
implies that all indifference curves have the same shape, except for a …xed or proportional shift respectively. However, in the absence of further restrictions, such
as the requirement that W be an expected-utility function, constant absolute or
relative risk aversion does not impose any restrictions on the shape of indifference
curves.
The conditions for constant absolute and relative risk-aversion may be expressed
simply in terms of the partial derivatives of the certainty equivalent e(y).
Result 2: If W is a differentiable preference structure:
(i) W displays CARA if and only if, for all y
Oy e(y) 1 = 1
(ii) W displays CRRA if and only if, for all y,
Oy e(y) y = e(y)
Proof : (i) The stated condition is satis…ed if and only if
8
JOHN QUIGGIN AND ROBERT G. CHAMBERS
e(y + b1) = e(y) + bOy e(y) 1 = e(y) + b
which, from De…nition 2, is equivalent to CARA.
(ii) The stated condition is satis…ed if and only if
e(ty) = e(y) + (t
1)Oy e(y) y = e(y) + (t
1)e(y) = te(y)
which, from De…nition 2, is equivalent to CRRA. For general W , let
w (y) = ∂W (c1)/∂c evaluated at c = e(y)
In an expected-utility framework, w (y) would be interpreted as the marginal
utility of income at the certainty equivalent. From Result 2, it is easy to derive the
following conditions for CARA and CRRA in terms of the partial derivatives of W .
Corollary 2.1: For any differentiable preference function W
(i) W displays CARA if and only if, for all y
Oy W (y) 1 = w (y)
(ii) W displays CRRA if and only if, for all y,
Oy W (y) y = e(y)w (y)
From the results of Arrow (1965) and Pratt (1964) it is obvious that, except for
the trivial case of risk neutrality, expected-utility preferences cannot display both
CARA and CRRA. However, it is possible for more general preferences to display
both CARA and CRRA. In fact, we have the following parallel to results derived
in the context of social choice theory by Roberts (1980)3 :
Result 3: A preference function exhibits both CRRA and CARA if and only if it
can be expressed as a nondecreasing monotonic transformation of
f (y) = g (y
M in{y1 , ..., yS }1) + M in{y1 , ..., yS },
where g is positively linearly homogeneous, nondecreasing and continuous.
Proof : From Result 1 and Lemma 3, a preference function can exhibit CRRA if
and only if the certainty equivalent assumes the general form:
e(y) = g (y
M in{y1 , ..., yS }1) + M in{y1 , ..., yS }.
By Lemma 4 and the properties of W , this function has to be continuous and
nondecreasing. However, a preference structure can satisfy CRRA if and only if its
certainty equivalent satis…es positive linear homogeneity. (This is obvious from the
part of the proof of Result 1 that is omitted.) Hence, g must satisfy positive linear
homogeneity and Lemma 4 establishes necessity. Sufficiency follows exactly as in
the proof of Result 1. One particularly informative example of a monotone increasing, risk-averse preference function displaying both CARA and CRRA is given by W (y) = E [y] [y],
3 We
are indebted to Zvi Safra for this reference.
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 9
where 0 < < 1/S and is the standard deviation operator. It is trivially obvious
that this form can be expressed in a form that satis…es Lemma 3. The combination
of both CARA and CRRA will also be observed for any risk averse member of the
family of rank-dependent expected utility functions (Quiggin, 1982) having linear
utility. Such functions characterize the ‘dual model’ examined by Yaari (1987).
More generally, any member of the class of functions de…ned by the mean and
the Hardy-Littlewood-Polya generalized mean of the absolute value of deviations
from the mean, of which the mean-standard error representation is a special case,
will exhibit both CRRA and CARA:
r
E (y) + (k ak kyk
E (y)k )1/r .
Taking this expression as the certainty equivalent then allows us to interpret (k ak kyk
as minus the risk premium. Neither this case nor the special case given by the meanstandard deviation speci…cation above is consistent with expected-utility theory.
Even more general preference structures can be speci…ed which are consistent
with both CARA and CRRA. For example, specifying the certainty equivalent as
the sum of the mean and a generalized Leontief function of the absolute values of
the deviation from the mean also yields both forms of constant risk aversion:
E (y) + +1/2 k m akm kyk
1
E (y)k 2 kym
1
E (y)k 2
where akm = amk . Here the generalized Leontief function can be taken as minus
the risk premium.
Note that the linear homogeneity of the certainty equivalent implies that strictly
risk-averse preferences with constant absolute and relative risk aversion must display …rst-order risk aversion in the sense of Segal and Spivak (1990). Hence, as
shown by Safra and Segal (personal communication), the only preference functionals that are smooth, in the sense of Fréchet differentiability, and display constant
absolute and relative risk aversion are expected-value functionals.
More generally we derive:
Result 4: Assume the preference functional is Fréchet differentiable. Strictly riskaverse preferences satisfying CARA cannot display constant or increasing relative
risk aversion. Strictly risk-averse preferences satisfying CRRA cannot display constant or decreasing absolute risk aversion.
Proof : Consider the risk premium for variables of the form ty + k 1. For a …xed
risky prospect y and k , this is a function R(t) = r (ty + k 1) corresponding to a
multiplicative increase in risk. For preferences satisfying CRRA, and 0 t 1 we
have
tR(1) = tr (y + k 1) = r (ty + tk 1) (=, ) r (ty + k 1) = R(t)
whence
R(t)
t
if preferences satisfy DARA (CARA, IARA) with R(1) > 0 by risk aversion.
Risk-aversion requires R0 (t) 0. Since R takes its minimum value of 0 for t = 0,
smoothness, in the sense of Fréchet differentiability, requires that R0 (t)|0 = 0.
Recognizing that R(t)|0 = 0 and applying l’Hôpital’s rule yields when preferences
R(1) (=, )
r
E (y)k )1/r
10
JOHN QUIGGIN AND ROBERT G. CHAMBERS
satisfy both CRRA and CARA that R0 (t)|0 = R(1) and when preferences satisfy
both CRRA and IARA that R0 (t)|0 R(1) contradicting smoothness.
For preferences satisfying CARA, and 0 t 1
tR(1) = tr (y + k 1) = tr (y + (k/t)1) (=, ) r (ty + k 1) = R(t)
if preferences satisfy IRRA (CRRA, DRRA). Proceeding as above establishes that
CARA is inconsistent with CRRA and DRRA for smooth preferences.
The proof of Result 4 relies only on smoothness. However, the additive separability of expected utility provides a separate argument to show that, in the expected
utility case, only risk-neutral preferences display both CRRA and CARA. If the
i-th element of the gradient Wi (y) is equal to u0 (yi ) for some utility function u,
Corollary 2.1 shows that CRRA and CARA can hold jointly in a neighborhood of
y if and only if u is linear. Applied to the rank-dependent model, this argument
yields:
Corollary 4.1: In the rank-dependent expected-utility model, linearity of the
utility function is necessary and sufficient for CRRA and CARA to hold jointly.
2.5. Equivalent and compensating bene…ts as measures of risk. Consider
any move from y to y0 . For such moves, Luenberger (1996) shows that the bene…t
function yields a measure of compensating bene…t B (W (y), y0 ) and a measure of
equivalent bene…t B (W (y0 ), y). As noted above, when y0 = E [y]1, so that the
move involves the elimination of risk, the risk premium is a compensating bene…t
measure.
The distance function yields analogous measures of compensating and equivalent
relative bene…ts. For a move from y to y0 , we de…ne the compensating relative
bene…t as the scaling factor for the new prospect that leaves the decision-maker
indifferent to the original prospect, that is, D(y0 , W (y)). Similarly, the equivalent
relative bene…t is the inverse of the scaling factor for the original prospect that
leaves the decision-maker indifferent to the new prospect, that is, D(y, W (y0 )) 1 .
In these terms, the relative risk premium is just the compensating relative bene…t
when y0 = E [y]1.
Thus, both traditional measures of risk aversion can be thought of as a form
of compensating bene…t. However, the absolute and relative equivalent bene…t
concepts are equally legitimate measures of aversion to risk and, as noted earlier,
can be used to de…ne conditional absolute and relative risk premiums.
We have the following extension of a result developed in the context of welfare
measurement by Chambers and Färe (1997):
Result 5: The compensating bene…t always equals the equivalent bene…t if and
only if preferences display CARA. The compensating relative bene…t always equals
the equivalent relative bene…t if and only if preferences display CRRA.
Proof : The relative measures always equal one another only if
D(y0 , W (y)) =
1
D(y, W (y0 ))
for all y, y0 . Because this must apply for all y, y0 , we can …x y0 at an arbitrary
reference y0 to obtain
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 11
0
D(y , W (y)) =
1
0
D(y, W (y ))
0
where W (y ) is now a number. This establishes that W (y) is a monotonic transformation of a positively linearly homogeneous function by Lemma 2a and 2b.
Now apply Result 1. Sufficiency is obvious. The proof is parallel for the absolute
measures.
Corollary 5.1: Equality between the compensating bene…t and equivalent bene…t
and between the relative compensating bene…t and the relative equivalent bene…t
holds if and only if W satis…es the conditions of Result 3.
2.6. Generalized Schur-concavity. The notion of Schur-concavity is naturally
related to risk-aversion in the case where the function W is symmetric, so that
all elements of are equal. This idea has been extended to the case where is
an interval with Lebesgue measure (Chew and Mao, 1995). In this section, we
begin with some preliminaries to deal with the case where is …nite and there
exists a unique with elements that are not necessarily equal. Thus, probabilistic
sophistication is assumed and there is no loss of generality in referring to as a
probability distribution.
Given E [y] = E [y0 ] and the probability distribution , we say that y0 is a
mean-preserving spread of y (denoted notationally as y M P S y0 ) if for all t0
Z
t0
Z
0
F (t; y , )dt t0
F (t; y, )dt
where
F (t; y0 , )
=
=
Pr{y t}
X
s
y s t
.
The relation y M P S y0 is also stated as y0 -majorizes y (Marshall and Olkin,
1979) and is equivalent to the statement that y is less risky than y0 in the sense
of Rothschild and Stiglitz (1970). If the requirement that E [y] = E [y0 ] is replaced
by E [y] E [y0 ] , y second-order stochastically dominates y0 (Hadar and Russell,
1969).
Following Chambers and Quiggin (1997), for a given probability distribution ,
we de…ne a preference function W to be generalized Schur-concave for if W :
<S → < satis…es:
y M P S y0 ⇒ W (y) W (y0 ).
Three comments: First, generalized Schur convexity is given by the requirement
that y M P S y0 ⇒ W (y) W (y0 ). Second, both generalized Schur concavity and
generalized Schur convexity are conditional on the probability measure . Since
y and y0 are equivalent in the sense that y M P S y0 and y0 M P S y whenever
Fy = Fy0 , generalized Schur-concave or generalized Schur convex preferences are
probabilistically sophisticated. And third, for differentiable preference structures,
Chambers and Quiggin (1997) have shown:
12
JOHN QUIGGIN AND ROBERT G. CHAMBERS
Lemma 6: If a function H : <S → < is differentiable and Schur-concave:
Hi (y) Hj (y)
(yi yj ) 0.
i
j
From this lemma, we can conclude immediately that for a differentiable generalized
Schur-concave preference structure:
Wi (y) Wj (y)
(yi yj ) 0.
i
j
In this context, the key result of Hadar and Russell (1969), Hanoch and Levy
(1969), and Rothschild and Stiglitz (1970) is that if u is a concave function, then, under expected utility, W (y) = u(y) is generalized Schur concave for . Moreover,
an obvious consequence of Lemma 6 is the well-known consequence of concavity of
u:
(u0 (yi )
u0 (yj )) (yi
yj ) 0.
Thus, for expected utility, generalized Schur-concavity is equivalent to risk-aversion
in the sense de…ned above, namely that the risk premium is non-negative for all y.
That this is not true for general preference functions was …rst observed by Machina
(1984).
In the present context the importance of generalized Schur-concavity arises from
the fact that the distance function and the bene…t function inherit the preference
function’s generalized Schur concavity. We have the following extension of results
originally due to Blackorby and Donaldson (1978, 1980) in the context of inequality
measurement.
Result 6: If W is generalized Schur concave, then for all w
(i) D(y, w ) is generalized Schur concave. y M P S y0 ⇒ D(y, w ) D(y0 , w ),
and
(ii) B (y, w ) is generalized Schur concave, y M P S y0 ⇒ B (y, w ) B (y0 , w )
Proof of Result 6: (i) Suppose to the contrary that y y0 , but D(y, w ) <
D(y0 , w ). Because multiplication by a positive scalar preserves the partial order
M P S , y / D(y0 , w ) M P S y0 / D(y0 , w ). The generalized Schur concavity of W
implies W (y / D(y0 , w )) W (y0 / D(y0 , w)) = w , where the last equality follows
by Lemma 2.c. Because we have presumed that D(y, w) < D(y0 , w ), we now have
a contradiction to the de…nition of the distance function. This establishes (i). (ii)
is established by an exactly parallel argument.
An immediate consequence of the generalized Schur concavity of the preference
function is:
Corollary 6.1: If W is generalized Schur concave, the certainty equivalent is also
generalized Schur concave.
Moreover, it follows immediately from the de…nition of the risk premiums and
Lemmas 4, 1, and 2 that
Corollary 6.2: If W is generalized Schur concave, the absolute and relative risk
premiums are generalized Schur convex.
We close this section by remarking that combining Lemma 6 with Result 6 and
its corollaries allows one to establish that the bene…t and distance functions, the
certainty equivalent, and the absolute and relative risk premiums all satisfy similar
derivative properties.
RISK PREMIUMS AND BENEFIT MEASURES FOR GENERALIZED-EXPECTED-UTILITY THEORIES 13
3. Concluding comments
In the choice between expected-utility theory and more general models, the availability of tractable functional forms for expected utility, and the usefulness of CARA
and CRRA have led many analysts to use expected-utility models even in situations
where there is empirical evidence to suggest that decision-makers do not obey the
expected-utility axioms. In this paper, it has been shown that risk premiums can
be derived for a very wide class of models in a way that permits the characterization
of constant, increasing and decreasing absolute and relative risk aversion. Further,
the use of the bene…t and distance functions shows that a wide range of ‡exible
functional forms familiar from modern production theory can be used to characterize preferences under uncertainty. Because these functional forms are not additively
separable, they are not consistent with the expected-utility hypothesis. Concepts of
risk aversion are shown to apply regardless of whether preferences satisfy expected
utility and even regardless of whether decision-makers have well-de…ned subjective
probabilities in the Savage-Machina-Schmeidler sense.
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