Double-Exponential Utility Functions
David E. Bell
Mathematics of Operations Research, Vol. 11, No. 2. (May, 1986), pp. 351-361.
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Mon Feb 11 16:52:16 2008
MATHEMATICS OF OPERATIONS RESEARCH
Vol. l I, No. 2, May 1986
Printed in U.S.A.
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS*
DAVID E. BELL
Harvard University
Ordinal additivity between two attributes X and Y is a property that permits the utility
function u(x, y) to be represented as cp(o(x) + w(y)) for some functions cp, u and w where cp
may be thought of as a single attribute utility function over o + w . In applications cp is usually
taken to be either linear (the additive decomposition) or exponential (the multiplicative
decomposition). The utility function can be shown to be exactly one of these forms whenever
the two attributes are mutually utility independent. In this paper we introduce a relaxation of
utility independence and show that it too implies ordinal additivity. We identify the class of
utility functions cp that are consistent with the relaxed assumptions.
One member of this class, cp(z) = -exp(bexp(-cz)), which we call double-exponential,
seems particularly appealing, for when b and c are positive it is increasing, risk averse and
decreasingly risk averse for all z . The multiattribute decomposition corresponding to it
-exp(b exp(- c(a + w))) may be represented as u(x, y) = c exp(cu(x)w(y)) which we call the
exponential product.
1. Introduction. Let u(x, y) be a two-attribute utility function on a continuous
y*], then the conditional utility function u(x Iy)
interval space X x Y = [xO,x*]x
is defined by
In this paper we will restrict attention to the case where x and y are both scalar, for
generalizations to the vector case will normally be obvious. For simplicity, we will also
assume that preferences for X increase from x0 to x* and, for Y, increase from to
y*. Without loss we will presume u(xO, = 0 and u(x*, y*) = 1 and denote u(x*,
= a and u(xO,y*) = b. Though the letter u will be used for three functions u(x, y),
u(x ( y ) and u(y I x), no confusion should arise. The equation above may be rearranged
to illustrate the assessment problem:
U(X,y ) = bu(y 1 xO)+ [ a
+ (1 - a)u(y I x*) - bu(y I xO)]u(xly).
(1)
Without simplifying assumptions we need to assess two constants a and b, two
one-attribute utility functions u(y ( xO)and u(y I x*) as well as, in principle, an infinite
number of one-attribute utility functions, u(x 1y).
Keeney and Raiffa (1976) exploited the utility independence assumption that u(x 1 y)
= u(x IyO) for ally. If, in addition, u(y I x) = u(y 1x4, Keeney (1968) showed that
known as the multiplicative form. The special case in which b = 1 - a is known as the
additive form.
Kirkwood (1976) considered the case in which u(x y ) could be assumed always to
*Received June 4, 1984; revised April 11, 1985. A MS 1980 subject classification. Primary: 90A 10. IAOR 1973 subject classification. Main: Decision theory. Cross references: Decision. O R / MS Index 1978 subject classification. Primary: 857 Utility/preference/theory. Key wordr. Utility independence, ordinal additivity, utility theory. 0364-765X/86/ 1102/0351$01.25
Copyright 0 1986. The Institute of Management Sciences/Operations Research Society of America
352
DAVID E. BELL
be a member of the same parametric family, for example u(x 1 y) = - exp(- c(y)x). In
this case the assessment of u(x ly) is reduced to determining the one-dimensional
function c(y). Kirkwood described this kind of condition as Xparametrically dependent
on Y.
Bell (1979) approached the assessment problem by assuming an interpolation
condition:
If X is interpolation independent of Y and vice versa, Bell showed that
which requires the assessment of 3 constants and 4 one-attribute utility functions.
While interpolation independence has many advantages for the assessment of
complex multiattribute utility functions, it does have a disadvantage not shared by
utility independence or parametric dependence. If u(x 1
and u(x 1 y*) are both
exponential, one might wish that so too should be u(x 1 y) for other values of y. More
generally we might ask that local risk aversion properties shared by u(x lyO) and
u(x 1 y*) would be common in all u(x 1 y).
Pratt (1964) defined (local) risk aversion for a utility function u(x) by the function
r(x) = - u"(x)/u'(x), a quantity that is proportional to the risk premium for small
(local) gambles. Define the conditional risk aversion of X given Y, by the function
r(x 1 y) = - U"(XI y)/u'(x 1 y). We note the following properties:
(i) r(x I y) is independent of x and y if and only if X is utility independent of Y and
for all x and y, for some constant c,
u(x 1 y) = cx or - ce -'"
(ii) r(x 1 y) is independent of y if and only if X is utility independent of Y, and
(iii) r(x ( y ) is independent of x if and only if X is parametrically dependent on Y
with the exponential form.
These properties suggest that insight about multiattribute assessment procedures
may be gained by considering the problem as one of assessing the conditional risk
aversion functions. In this paper we exploit the interpolation idea but apply it to
r(x I y) rather than u(x 1 y).
2. Interpolating risk aversion. Suppose that, for some function a(y),
For the sake of a name, we will call this condition risk aversion interpolation independence with the abbreviation X RAII Y. This property includes utility independence as
a special case and, in some cases, it includes parametric dependence as a special case
and
also. For if u(x 1 y4 and u(x 1 y*) are exponential, so too is u(x 1 y); if u(x 1
u(x 1 y*) have constant proportional risk aversion, so too has u(x 1 y). However, (4)
does not preserve the logarithmic form. For example, if u(x 1
= log(x + a,) and
U(XIy *) = log(x + a,), then (4) does not produce the result u(x 1 y) = log(x a(y)).
Nor does it preserve the more general power function family of (x + alb. [These forms
do satisfy the interpolation equation
+
but we leave a study of that relationship for another time.]
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS
353
An assumption of X RAII Y together with Y RAII X we will denote by X MRAII
Y, or, equivalently Y MRAII X ( M standing for "mutually"). We note that X MRAII
Y is a strictly stronger condition than X RAII Y, for example u ( x , y ) = [ x + l / c ( y ) ]
exp(- x c ( y ) ) satisfies X RAII Y but not Y RAII X.
The use of r ( x 1 y ) implicitly requires that u ( x , y ) be differentiable at least twice. The
proofs that accompany the following results require u ( x , y ) to be differentiable at least
three times. Since the proofs are neither elegant nor insightful they have been gathered
in a final section of the paper.
LEMMA1. X is utility independent of Y and Y RA 11 X if and only if u ( x , y ) has an
additive or multiplicative decomposition.
Lemma 1 provides slightly weaker assumptions than those required by Keeney for
the multiplicative form.
2. If X MRAII Y then X and Y are ordinally additive.
LEMMA
Our main result will utilize a family of utility functions characterized by a property
of the risk aversion function as described by the following lemma.
LEMMA3. The risk aversion function r ( x ) = - u " ( x ) / u f ( x ) satisfes the condition
r'(x) = cr(x) + d for some constants c # 0 and d , if and only if
where b > 0, k is any constant, and 0 is f 1.
The indirect representation of u in this result may mask some of its more useful
special cases. For example if k is zero then u ( x ) = - exp(b exp(- ex)) a function
which for all x is increasing, concave and exhibits decreasing risk aversion so long as b
and c are each positive. We will refer to this form as the double exponential.
If k is a positive integer and z = becx then either
a family that might be useful for small values of k .
The case r'(x) = c2 leads to a u ( x )which is the cumulative of a normal distribution,
the case r ' ( x ) = - c2 to an integral of the form j:oexp(t2)dt. For expositional ease it is
useful to have a name for the family of utility functions characterized by the equation
r'(x) = cr(x) + d.
DEFINITION.A utility function u ( x ) with risk aversion function r ( x ) is one of the
extended double exponentials if and only if r f ( x )= c r ( x ) + d for some constants c and d
(that might be zero).
Note that another family of utility functions can be characterized by their relation) ~
the linear and
ship between r f ( x )and r ( x ) .The equation r f ( x )= - k r ( ~characterizes
exponential families ( k = O), the logarithmic utility function (k = l), and the power
family (0 # k f 1).
THEOREM.The following conditions are equivalent:
(i) X MRAII Y,
(ii) X and Y are ordinally additive and X RAZZ Y .
(iii) u ( x , y ) = +(u(x)+ w ( y ) )for some functions +, u and w, where
the extended double exponential family.
+ is a member of
354
DAVID E. BELL
3. Discussion. The assessment of utility functions is a task requiring great care.
Decision makers are often neither consistent with the axioms of von Neumann and
Morgenstern, nor do they always satisfy certain established desiderata such as, in the
case of the attribute money, being risk averse and decreasingly so. Although procedures have been devised to construct a single attribute utility function to be consistent
with directly assessed points and with declared local properties [Meyer and Pratt 1968,
Schlaifer 19711, a common route taken is to find a family of utility functions with the
desired local properties and then to select the available parameters in a way that best
approximates the direct assessments.
Multiattribute utility assessment is also best thought of as an exercise in the
approximation of functions [Fishburn 19771. The task is made more delicate by the
desire to preserve local properties of the true function in the approximation. Even
though the interpolation scheme (2) is quite sound as an approximation to the absolute
value of the conditional utility function it does not preserve desirable qualitative
properties of the generating functions.
The interpolation scheme
appears to permit a great deal of flexibility in approximating the utility function, yet,
as Lemma 2 showed, it and the equivalent relation for r(y I x) require X and Y to be
ordinally additive. The ordinally additive representation, u(x, y) = +(u(x) + w(y))
when associated, through +, with families of single attribute utility functions, provides
a range of multiattribute decompositions. If is linear we have the additive form, if
is exponential we have the multiplicative. If + is the double exponential, u(x) =
- exp(b exp(- cx)), then we have u(x, y) = - exp(b exp(- c ( u + w))). This may be
written equivalently as u(x, y) = exp(bv,(x)v,(y)) a representation for which the term
exponentialproduct seems appropriate.
+
+
4. Proofs.
+
PROOFOF LEMMA 1. If X UI Y then u(x, y) = f(y) g(y)h(x) for some functions
In this case r(y I x) = - (f "(y) g"(y)h (x))
f, g and h, where h(x) = u(x 1
/( f'(y) + g'(y)h(x)) so that Y RAII X only if
+
where I have used the scaling convention u(xOlyO)= 0, u(x* Iy0) = 1 to deduce
h(xO)= 0, h(x*) = I . Clearing the denominators in the equation we have
=
afl'f'(f'
+ g') + (1 - a)fl'f'f'+ (1 - a)gl'flf'
- f r ' f ' ( f ' + g') or
Now if f"g' - gUf' = 0 then Y UI X immediately since then r(y 1 xO)= r(y 1 x*). If
(ag' + f')h = f'(l - a ) then f' = kg' for some constant k so that f(y) = kg(y) + c.
Hence in this case, Y UI X also. Now we use Keeney (1968) to show that u is
multiplicative.
355
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS
PROOFOF LEMMA
2. The proof is rather lengthy. The reader who wishes merely to
be satisfied of the plausibility of the result may be able to do so by consideration of
equation (7) in light of Lemma 4 (later in this section).
Integrating (4) we have
U t (1 y~) = eS(~')u'(xy*)
1
Since u(x, y) = f ( y ) + g(y)u(xly) for some functions f, g we also have au/ax = g(y)
u'(x 1 y). Therefore we can say that, for arbitrary functions a , , a,, b, and b, and a
constant k , , we have
where b,(xO)= b2(y0)= 0 and b,(x*) = b2(y*) = 1. By symmetry,
where d,(xO)= d,(y") = 0 and d,(x*) = d,(y*) = 1. (In future we will abbreviate
d,(xO) as dp and d,(x*) as d: and so on.) To show X OA Y we must show that
log(au/dx) - log(au/ay) = v(x) + w(y) for some functions u and w (Luce and Tukey
1964), that is, we must show a , + a, + k,b,b, - c, - c, - k2dld2= V ( X ) ~ ( y for
)
some functions o and w. Equivalently, we must establish that k,b,b2 - k2d,d2
= o,(x) + w2(.y) for some functions u, and w,. Since by = dp = 0 we have w2(y) = 0
and similarly o,(x) = 0. Hence we must show that k,b,b, = k,d,d,. We will do this by
demonstrating, somewhat tediously, that this condition is implied by (5) and (6).
Differentiating (5) with respect t o y we have a2u/axay = (a; k,b,b;)(au/ax) and,
similarly, a2u/ayax = (c; + k2d;d2)(au/ay). If these second order derivatives are
continuous they are identical so that
+
+
(a;
+ k,b,b;)exp(a, + a, + k, b, b,)
= (c;
+ k,d;d,)exp(c, + c2 + k2dld2)
is a necessary and sufficient condition for X MRAII Y.
Substitutingy = y o in (7) we have (a;'
k,b,b;")exp(a, a,") = c;exp(c,
larly,
a; exp(ap + a,) = (ciO+ k2d;Od2)exp(c~ cZ).
+
+
(7)
+ c a . Simi-
+
Substituting y = y * in (7) we have
Similarly,
(a;
+ k,b;)exp(a: + a, + k,b,)
= (c;*
+ k2d;*d2)exp(c: + c, + k,d,).
From these four equations we may identify a;, c;, d; and b;:
a;
+ k2d;Od2)exp(c2- a2 +
= (ciO
C:
-
a:),
k , d ~ = ( a ~ * + k , b , b ~ * ) e x p ( a , + a , * + k , b , - c , - c : - k , d , ) - c a; nd
(9)
(10)
356
DAVID E. BELL Also, c;' = a;Oexp(af- cf
c;* = (a;'
+ k,b;)exp(a:
+ a: - c:),
-
c:
(12)
+ a: - c:),
+ a,* - cf - c:) - ciO,
= (a;* + k,b;*)exp(a: + a: + k , - k , - c: - c:) - ci*.
k2di0= a;* exp(a;
k,d;*
(I3)
( 14)
(I5)
Substituting in (7) using (8), (9), ( l o ) , ( 1 1 ) :
exp(al + a,
+ k,b,b,)[(l - b,)(ciO+ k2d;Od2)exp(c2- a, + e 0l - a 0, )
+ b,(c;* + k2d;*d2)exp(c:+ c, + k2d2- a: - a, - k l b 2 ) ]
=[(I -
d2)(ai0+ k,b,b;o)exp(a, - c , + 0: - c:)
+d,(a;*
+ k,b,b;*)exp(a, + a: + k l b l - c , - c:
-
k2d,)]exp(c,+ c 2 + k2d,d2).
Now using (14) and (15): e x p ( k , b l b 2 ) [ (l b,)(c;O(l - d2)+ d2a;* exp(ap + a: - cp - c:))exp(cp - a:) + b,((l - d2)c;*+ d2(a;* + k,b;*)exp(a:
+ a:
+ k , - k , - cy - c:))
+ k2d2- a: - k,b,)] = exp(k,dld2)[(l- d,)(a; + k,b,bf)exp(a: - c:) + d2(a;* + k,b,b;*)exp(a; - c: + k , b , - k 2 d l ) ] .
x eip(c:
Now use (12) and (13): e x p ( k ,b,b,) [ ( 1 - b , ) ( ( l - d2)a;0exp(a:- c:)
+ d2a;*exp(a: - c:)) + b , ( ( l - d2)(ap+ k ,bp)exp(a; - c: + k2d2- k,b,)
+ d2(a;* + k,b;*)exp(a:
= exp(k2d,d2)
[(1 -
-
c:
+ k , - k , + k2d2- k,b,))]
d2)(ai0+ k,bl bp)exp(a: - c;)
+ d2(a;* + k,b,b;*)exp(a: - c: + k l b , - k i d l ) ] .
(16)
Let
A'
= aiOexp(a; -
c:),
BO
A* = a;* exp(a: - c?) and
= k,biOexp(a; - c:), B* = k , b;* exp(a; - c:). W e may assume that not all o f these four constants are zero for otherwise we would
have a2u/axay = 0 which would imply that u ( x , y ) is additive. Temporarily, let us use
357
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS
the notation
+ b, [(A' + B0)(1 - d,) + (A* + ~ * ) d , e * l - ~ ~ ] e x ~ ( k-, dk,b,)
,
and
so that (16) may be written as
exp(k,b,b,)f(x, Y) = exp(k2d1d2) g(x, Y).
(17)
Differentiate this with respect to x:
and use (17) to deduce
klb;b2 - k2d;d2 = g f / g - f'/ f.
We have
+ ( ( A 0 + BO)(l- d 2 ) + ( A * + B*)d2eki-*2)(exp(k2d2- klb2))] and
x (B* + (A* + blB*)(kl - k2d;(x)/b;(x)))].
Note that
f(x*, y )
= [(A'
g(x*, y ) = [(A'
+ B ')(I
-
d,)
+ (A* + ~ * ) d , e ~ ~ - ~ ~ ] -e xk1b2)
~ ( k ~and
d~
+ BO)(l- d2)+ (A* + B*)d2ekl-*2]
Note that f(x*, y ) = g(x*, y)exp(k,d, - klb2).
In what follows we wish to assume that b;(x) i
0 Z d;(y). Note that b,(x)
= loguf(x lyO)- loguf(x ly*) so that b;(x) = r(xly*) - r(x lyO). It' b;(x) = 0 in a
subinterval I of [xO,x*]then X RAII Y implies X UI Y on this subinterval. By Lemma
y*]. Hence it remains to
1 we know that b,(x) = d,(x) and b,(y) = d2(y) on I x
prove that b, = dl and b, = d, on intervals in which b; # 0. The idea is to partition
[xO,x*]into intervals I , , . . . , I,, and
y*] into J l , . . . , Jkand prove the result on
each I, x 4 separately, then rely on the continuity of b,, d l , b, and d, to prove the
result for the entire space.
Substituting x = x0 in (18), and letting so = dio/b;', we obtain
358
DAVID E. B E L L
Similarly, substitution of x* in (18), with s* = d ; * / b , gives
(klb2 - k,s*d2)
-~
+
~ - d,)
( 1 d 2 e k l - k 2 ( ~+* ( A * + B * ) ( k , - k2s*))+ g ( x O ,y ) e x p ( k l b 2- k2d2)- g ( x * , y )
g(x*3 Y )
+
Let z = d 2 ( y )and w = k , b , - k2d2so that k , b , = w k2z. The function b, may be
regarded as a function of z because d ; ( y ) .z. 0. Also let l l ( z )= g ( x O y, ) , 12(z)= g ( x * ,
y), then (19) becomes
and (20) becomes
+ ( A * + B * ) ( k l - k2s*))+ l l e w- 1,.
12(w + k,z(l - s*)) = B O ( l - z ) +
Finally, we may write (21) and (22) as
I l w + 1,e-"'= q , ( z ) and
It will be important later to remember that 1, and 1, are linear in z and that q l and 9,
are quadratic in z. The proof will continue by consideration of two cases, the first is
when differentiating (23) and (24) and solving for w' produces new information about
w, the second is when it doesn't.
So, differentiating both (23) and (24) with respect to z :
+ Iiw' + I;e-"' - w'I,e-" = q', ,
I;W + l2w' - l i e w - I l w J e w= q ; .
I;w
(25)
(26)
Eliminating w' we have
(1, - l l e w ) ( q ', 1;w - l i e w )= ( q ; - 1;w
+ l ; e w ) ( l 1- 12e-"').
Now use (23) and (24) to clear this equation of e" and e - " :
= (1,
Collecting terms in w:
i
1;
11
1
+ 1,w - q , ) q; - 1;w + - (12w - q2) .
359
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS
Since the term in w2 cancels to zero we conclude that either this equation is an identity
or w is representable as the ratio of two polynomials in z. But now use (24) and
Lemma 4 (see below) to see that in this latter case w must be constant.
If (23) and (24) produce equivalent differential equations in w' and since w(0) = 0 is
known, we may conclude that (23) and (24) are equivalent. That is, any solution w to
(23) is a solution to (24) and vice versa.
We can show that there cannot be two distinct solutions to (23) and (24) for suppose
that w = a and w = b were both solutions for some particular choice of z. We would
have
I,a+12e-"=q,,
1 2 a + I , e a = q 2 , I , b + 1 2 e - b = q , and 1 2 b + l , e b = q 2
from which we may deduce that l;(e-" - eCb) = l:(ea - eb), a condition only possible if a = b or if 1, = I, = 0. This latter case is ruled out since d;(y) # 0.
Equation (23) has exactly one solution only if either /,I2 < 0 for all z or if q, = t21,
for some constant t,. For consider the equation ax + be-" = c where a, b and c are
constants and ab > 0. We may as well assume b > 0. This equation has two solutions
whenever c > a(1ogb - loga + I), one solution when c = a(1ogb - loga 1) and no
solution otherwise. By contrast, if ab < 0 this equation has at most one solution since
ax be-" is a strictly monotone function of x .
So the only way we could have exactly one solution to (23) and yet have Ill2> 0 is if
+
+
If Ill2> 0 holds for some z it must hold for an open interval in which case (27) must
hold for all z (by analytic continuation). By Lemma 4 1, = t,l, and q, = t211for some
constants t, and t,.
If /,I2 < 0 for all z then 1, = - t,l, for some constant t, > 0. Substitute for I, in (23)
and (24) and note that a root of l,(z) = 0 must also be a root of q, and q2. Hence, for
some linear functions 1, and 1, we may write (23) and (24) as
~ - t , e - ~ = land
~
- t3w -
e w= I,.
P I
(29)
Recall that (23) and (24) yield the same equation in w'. From (28) w'[l + t3e-"1 = 1;
and from (29) w'[t3 + e W ]= - 1; where 1; and 1; are constants. If 1; = 0 then let S be
the subset of the real line on which w' # 0. On S we must have 1 + t,eCw = 0 so that w
is constant on S. Since w is continuous, it is constant everywhere. Assuming, therefore,
that 1; # 0 we have
Differentiating this twice with respect to z shows that w must be constant. The case in
which q2 = t211is equivalent to that in which 1; = 0.
We have shown that (5) and ( 6 ) require that k,b2 - k2d2be constant, and therefore
zero. Since b: = d: = 1 we have also shown that k, = k, so that b, = d,. By symmetry,
b, = d l . Hence k,blb2= k2dld2,hence u is additive. This ends the proof of Lemma 2.
z
PROOFOF LEMMA3. Differentiating the integral gives u'(x) =
becx. Differentiating once more gives
+ czk+leg' where
=
Hence r(x) = - c(k + 1 + 8z), so that rt(x) = - c28z = c(r(x) + c(k + I)), from
which we conclude that d, the constant in the lemma statement, is equal to c2(k + 1).
360
DAVID E. BELL
+
+
If rt(x) = cr(x) + d (c # 0), then log(cr d)2 = 2cx logai where a, is a nonzero
constant. Hence cr + d = a, exp ex. Integrating once more we have - c log ~ ' ( x +) ~
2dx + k, = 2(ao/c)expcx which is equivalent to u'(x) = ~ e " " e"PCX
e ~ for constants a, b
and A so that
Substitute t = @becY
where 0 = + 1 is chosen so that @b> 0. We have dt = c@be"dy
= ct dy, and eaY = (t/@b)"icso that
Let k = a / c - 1 and recognize that A/c can be any sign and we have
*Lo
b exp cx k 0,
U(X)=
t e dt
where b > 0.
The following result, used repeatedly in the proof of Lemma 2, is undoubtedly
proved in some textbook but I have been unable to locate a reference.
LEMMA4. If p l , p2, p3 and p, are polynomials such that p , /p2 = exp(p3/p4) then
p I / p 2 and p,/p, are constants.
PROOF. We may assume that p , and p, have no factors in common otherwise these
could be cancelled. Similarly with p3 and p,. If p2 has a root, say a where p2(a) = 0
then p4(a) must be zero also. Let p2(x) = (X - a)kp,(x) for a polynomial p,, where
p5(a) # 0 and k is an integer. We know that
Now p,/p5 behaves "sensibly" at x = a but (x - a)kexp(p,/p,) does not. More
precisely,p,/p, is analytic at x = a but (x - a)k exp(p,/p,) has an essential singularity
at x = a. (See Apostol 1957.) Hence p2 can have no real roots. If p, has a root, say
p4(b) = 0, then p3(b) < 0 is implied since p, is not zero and, therefore, we must have
pl(b) = 0. Let p,(x) = (x - b)'p,(x) where p,(b) # 0. Then
Now p2/p6 is nicely behaved at x = b but (x - b)'exp(-p3/p4) goes to infinity as
x -t b. Hence p4 has no real roots. Now consider the equation p2/p1 = exp(-p3/p4).
We already know that p2 and p4 have no real roots and by a similar sequence of
arguments neither does p , .
But, in fact, there is no need to restrict our consideration to real roots. By analytic
continuation, if pl/p2 = exp(p3/p,) on the real line it must hold in the complex plane
also (Apostol, p. 519). The above arguments now apply for complex roots. (Think of
complex roots as factors of the form x2 k where k +O. Now consider behavior as
x2+ - k.) Hence p , and p, have no real or complex roots and so must be constants.
+
PROOFOF THEOREM:(i) -+(ii) by Lemma 2. To show (ii) + (iii) note that X RAII Y
is equivalent to r(x 1 y) = a(x) + b(x)a(y) so that r'(x I y) = c(x) + d(x)a(y) for some
functions a, b, c and d. Hence
36 1
DOUBLE-EXPONENTIAL UTILITY FUNCTIONS
+W)
say. Now a u / a x = u'+'(u
- v " / v ' - v'+"/+'. Denoting
+
and a2u/ax2= on+' ulu'+" so that r ( x 1 y ) =
by r ( v w ) we have
+
-+"/+'
so that
Substituting for r ( x I y ) and rl(x ( y ) in (30) and rearranging we get an expression of the
form
r f ( v+ W ) = h ( x ) + j ( x ) r ( v + w )
(32)
for some functions h and j . If w ( y ) is not constant, so w' # 0 then differentiating (32)
with respect to y we have r"(v + w ) = j(x)r'(v w). Either r'(v + w ) = 0 in which
case r'(v w ) is trivially linear in r ( v + w ) or j ( x ) is constant. If j ( x ) is constant then
so too is h ( x ) . Hence r'(v + w ) is linear in r ( v w ) which shows that is one of the
extended double-exponential family.
Now to show (iii) +(i). In this case we know r'(v w ) = cr(v w ) + d . If c = 0
then r ( v + w ) = d ( u + w). Substituting in (31) we see that r ( x 1 y ) = - v " / v '
dvl(v w ) which is of the form a ( x ) b ( x ) a ( y ) as required. Hence X RAII Y and,
by symmetry Y RAII X. If c # 0 then log(cr d)2 = 2c(v w ) + logai or cr d
= aoe c ( c + w ) . Substituting once more in (31) gives
+
+
+
+
+
+
+
+
+
+
+
+
which is also of the form a ( x ) + b ( x ) a ( y ) .
Acknowledgments. John Pratt drew my attention to the undesirable risk aversion
properties of interpolated conditional utility functions. Richard Meyer and Richard
Stone provided me a refresher course in complex analysis.
References
Apostol, Tom M. (1957). Mathematical Analysis, Addison-Wesley, Reading, Mass. Bell, David E. (1979). Consistent Assessment Procedures for Conditional Utility Functions. Oper. Res. 27 1054- 1066.
Fishburn, Peter C. (1977). Approximations of Two-Attribute Utility Functions. Math. Oper. Res. 2 30-44.
Keeney, Ralph L. (1968). Quasi-Separable Utility Functions. Naval Res. Logisf. Quart. 15 551-565.
Kirkwood, Craig W. (1976). Parametrically Dependent Preferences for Multiattributed Consequences. Oper.
Res. 24 92-103.
Luce, R. Duncan and Tukey, John W. (1964). Simultaneous Conjoint Measurement: A New Type of
Fundamental Measurement. J. Math. Psychology 1 1-27.
Meyer, Richard F. and Pratt, John W. (1968). The Consistent Assessment and Fairing of Preference
Functions. IEEE Trans. Systems Sci. Cybernet. SSC-4 270-278.
Pratt, John W. (1964). Risk Aversion in the Small and in the Large. Econornetrica 32 122-136.
Schlaifer, Robert 0 . (1971). Computer Programs for Elementary Decision Analysis. Division of Research,
Graduate School of Business Administration, Harvard University, Boston.
Thompson, Silvanius P. (1914). Calculus Made Easy. MacMillan, New York, Second Edition.
GRADUATE SCHOOL O F BUSINESS ADMINISTRATION, HARVARD UNIVERSITY, SOLDIERS FIELD, BOSTON, MASSACHUSETTS 02163