The Comparative Statics of Changes in Risk for Most
Decision Makers
Rachel J. Huang
National Taiwan University of Science and Technology
Pai-Ta Shih
Larry Y. Tzeng
National Taiwan University
National Taiwan University
July 10, 2012
Abstract
Gollier (1995) has found the necessary and su¢ cient condition under a change in risk
for all risk-averse individuals to have unambiguous comparative statics. By extending
Gollier (1995), this paper provides the necessary and su¢ cient condition for unambiguous
comparative statics in a standard portfolio problem for all non-pathological concave
preferences as characterized by Leshno and Levy (2002). We further generalize our results
to the cases with non-linear payo¤ functions. Conditions for higher-order preferences are
also analyzed.
JEL classi…cation: D80; D81
Keywords: Stochastic Dominance; Almost Stochastic Dominance; Central Riskiness
1
1
Introduction
The e¤ect of changes in risk on optimal decisions has been well studied for decades since
Rothschild and Stiglitz’s (1971) pioneering work. They found that a risk-averse individual
may not decrease her demand for a risky asset when there is an increase in risk in the return
on the risky asset de…ned by second-order stochastic dominance.1 In the literature, one
way to …nd the unambiguous comparative statics is to provide the restrictions on stochastic
dominance such that the restrictive stochastic dominance could generate the desired comparative statics.2 The proposed restrictive stochastic dominance orders include, for example,
monotone likelihood ratio dominance (Landsberger and Meilijson, 1990), monotone probability ratio order (Eeckhoudt and Gollier, 1995) and a strong increase in risk (Eeckhoudt
and Hansen, 1980; Meyer and Ormiston, 1985).
Instead of restricting the existing stochastic orders, Gollier (1995) solved the problem by
providing a correct stochastic order. He found the necessary and su¢ cient condition under
a change in risk for all risk-averse individuals to have unambiguous comparative statics. In
a static portfolio choice problem, Gollier (1995) concluded that all risk-averse investors will
reduce their investment in a risky asset if and only if the distribution of the risky asset
becomes centrally riskier (CR), i.e., the location-weighted probability mass function under
the new cumulative distribution function (CDF) is uniformly less than that under the initial
CDF times some scalars.
Although Gollier (1995) has provided a correct stochastic order to …nd the unambiguous
comparative statics, the condition found by Gollier (1995) may not be easy to meet and
could limit its application. For example, assume that an investor is considering investing in
a risk-free asset and a risky asset where the risk-free rate of return is zero and the return
on the risky asset is with a 0:01 probability of obtaining
10% and a 0:99 probability of
obtaining 1; 000%. Now, let the distribution of the risky asset return shift and become one
with a 0:01 probability of obtaining
9:99% and a 0:99 probability of obtaining 1%. It is not
surprising that, in practice, most of the risk-averse investors would reduce their investment
1
See Levy (1992 and 1998) for a detailed survey regarding stochastic dominance.
The other line of research tries to identify the constraints on the individual’s preferences given that
the change in the risk is characterized by second-order stochastic dominance as in some papers (see Dreze
and Modigliani, 1972; Diamond and Stiglitz, 1974; Dionne and Eeckhoudt, 1987; and Briys, Dionne, and
Eeckhoudt, 1989).
2
2
u′
100
0
0.1
0.05
wealth
10
Figure 1: The marginal utility of u(z) = 1
e
100z .
in the risky asset while facing this change in risk. However, according to Gollier (1995), one
cannot have an unambiguous prediction.3 This example is not a special case. It is easy to
construct many cases where most investors would decrease their investment in risky assets
but the changes in risk do not satisfy CR.
The major reason for this CR problem is that, rather than for most risk-averse individuals, CR seeks to apply to all individuals which include some extreme preferences. In order
to include these decision makers with extreme preferences, the condition for unambiguous
comparative statics becomes strict and limited. For example, in the above case, the individuals with utility function u(z) = 1
e
100z
would increase their investment in risky assets
from 0:91% to 20:87% while facing such a change in risk. This type of utility function leads
to a di¤erent decision from that of most decision makers mainly because the preferences are
extreme. When u(z) = 1
e
100z ,
the marginal utility of large wealth is almost zero, but
that of very small wealth is quite large as shown in Figure 1. Since such types of preferences
are extreme and cannot represent most decision makers, it is straightforward to exclude them
while considering the comparative statics.
3
See Appendix A for the proof.
3
In this paper, we would like to extend Gollier (1995) by limiting the set of decision makers
by assuming that individuals exhibit preferences characterized by Leshno and Levy (2002).
Leshno and Levy (2002) shed light on indicating the inability of stochastic dominance and
mean-variance rules to distinguish two options which could be easily compared by most
decision makers. They further establish a new concept of stochastic orders, i.e., almost
stochastic dominance rules (ASD), for the decision makers with non-pathological preferences
to rank distributions.4 To de…ne “most” decision makers, they exclude some pathological
preferences, e.g., the marginal utility for some wealth level approaches zero or in…nity, which
is similar to the previously mentioned utility function. Since these preferences would rank
distributions in a way unlike most decision makers, they are economically irrelevant.
Our paper contributes to the literature by providing the unambiguous comparative statics
for most decision makers characterized by non-pathological preferences as in Leshno and Levy
(2002). Our paper, however, di¤ers from Leshno and Levy (2002) in that, for most decision
makers, we …nd the conditions for the unambiguous comparative statics while they provided
conditions to rank distributions. Since the set of the decision makers in this paper is smaller
than that in Gollier (1995), who considers all risk-averse decision makers, we could obtain
a weaker distribution condition than those in Gollier (1995). Thus, our condition could be
applied to more cases in the real world.
The remainder of this paper is organized as follows. Section 2 examines the standard
static portfolio problem and provides the main theorem which shows the necessary and
su¢ cient condition such that the decision makers with non-pathological concave preferences
would decrease their demand for risky assets. Section 3 extends our main theorem for all
individuals excluding the “pathological higher-order preferences.”Section 4 relaxes the linear
payo¤ assumption in Section 2 and provides a more general condition for the main theorem.
Section 5 concludes the paper.
4
Following Leshno and Levy (2002), researchers have further applied their theorem from several perspectives. For example, Levy, Leshno and Leibovitch (2010) construct experiments to characterize the sets of
non-pathological preferences. Bali, Ozgur Demirtas, Levy and Wolf (2009) provide empirical support for the
popular investment practices which are not supported by stochastic dominance rules. Huang, Shih and Tzeng
(2012) further provide a correct theorem regarding almost second-order stochastic dominance.
4
2
Portfolio Problem
Let us …rst discuss a standard static portfolio choice problem. Assume that the investor with
one dollar respectively chooses to invest
and 1
in the risky asset and the risk-free asset.
Let r denote the risk-free return and x
~ 2 [x; x] denote the return on the risky asset where
x
~ follows a probability density function f (x). Furthermore, let the expected return on the
risky asset be greater than r, i.e.,
Ef (x) > r,
(1)
so that a risk-averse investor would consider investing in the risky asset. Let u denote
the investor’s utility function with u0 > 0. The objective function of an expected utility
maximizer can be written as
max Eu =
Z
x
u ( x + (1
)r) f (x)dx.
(2)
x
The corresponding …rst-order condition (FOC) for the optimal
Z
x
(x
r) u0 ( x + (1
)r) f (x)dx = 0.
is
(3)
x
Let us assume that the second-order condition (SOC) holds, i.e.,5
Z
x
(x
r)2 u00 ( x + (1
)r) f (x)dx
0.
(4)
x
Thus, Equation (3) is the necessary and su¢ cient condition for the optimization problem.
Let
denote the optimal solution under f (x).
f
If the probability density function of x
~ shifts from f (x) to g(x), then the FOC will become
Z
x
(x
r) u0 (
gx
+ (1
g )r) g(x)dx
= 0;
(5)
x
The SOC holds if u00 < 0. So far, we do not assume the sign of u00 . The SOC could also hold if u00 > 0
at some but not all wealth levels.
5
5
where
g
Z
is the optimal solution under g(x). Thus,
x
(x
r) u0 (
fx
+ (1
f )r) g(x)dx
g
f
if and only if
0:
(6)
x
The necessary and su¢ cient condition can be restated as that 9 2 R such that
Z
x
(x
r) u0 (
fx
+ (1
f )r) [g(x)
f (x)] dx
0:
(7)
x
De…ne
(x
r) f (x) = tf (x) and (x
r) g(x) = tg (x).
Condition (7) can be rewritten as 9 2 R such that
Z
x
u0 (
fx
+ (1
f )r) [tg (x)
tf (x)] dx
0:
(8)
x
In the following, we will impose restrictions on the utility function for the individuals.
First, let us restrict the marginal utility. As proposed by Leshno and Levy (2002), de…ne
U1 ("1 ) as follows:
U1 ("1 ) =
u : u0 (z) > 0 and u0 (z)
inffu0 (z)g
1
"1
1 ; 8z ,
(9)
where "1 2 0; 12 . Equation (9) means that, for all utility in U1 ("1 ), the marginal utility
is positive and the ratio between the maximum and the minimum value of the marginal
utility is bounded by
1
"1
1. If "1 approaches 21 , then only the linear utility functions will
be in the set of U1 ("1 ). If "1 approaches zero, then the set U1 ("1 ) is the same as the set
of all decision makers with u0 (z) > 0. The utility function mentioned in the Introduction,
u(z) = 1
e
100z ,
does not belong to U1 ("1 ) when "1 is set to be greater than 3:73
10
44 .
Furthermore, let
1(
) = fx : tg (x)
tf (x) > 0g :
Thus, if x > r, then x is in the set of
(10)
1(
) if g(x)
6
f (x) > 0. If x < r, then x is in the
set of
1(
) if g(x)
ktg (x)
f (x) < 0. Denote
Z
tf (x)k =
x
jtg (x)
x
tf (x)j dx.
(11)
The following Proposition gives the necessary and su¢ cient conditions on distributions for
all DMs satisfying u 2 U1 ("1 ) to decrease their demand for risky assets.
Proposition 1 Given the SOC, we have
g
f
for all u 2 U1 ("1 ) if and only if 9 2 R
such that
Z
[tg (x)
1(
tf (x)] dx
"1 ktg (x)
)
tf (x)k .
(12)
Proof. Please see Appendix B.
If the decision maker is risk averse, then the SOC will always hold in this problem. Therefore, we can apply Proposition 1 to …nd the unambiguous comparative statics. Interestingly,
the decision maker does not have to be risk averse for all wealth to apply Proposition 1.
When the decision maker is locally risk averse, e.g., risk averse for small wealth but risk loving for large wealth (Machina 1982) or risk loving for small wealth but risk averse for large
wealth (Green and Jullien 1988), Proposition 1 can still provide an unambiguous comparative
statics as long as the SOC holds.
For the sake of illustration, consider r = 0 and f (x) as a uniform distribution in [ 2; 10].
Assume that g(x) also follows a uniform distribution in [ 1; 2]. Thus, we have
tf (x) =
8
>
<
tg (x) =
8
>
<
and
x
12
if
2
x
10
>
: 0 otherwise
x
3
if
1
x
2
>
: 0 otherwise
According to Levy, Leshno and Leibovitch’s (2010) experiments, the maximun critical value
of "1 is 5:9%. Thus, let us assume that "1 = 5%. As shown in Figure 2, we can …nd a
7
=4
tg(x),γtf(x)
γtf(x)
B
-2
-1
x
2
A
10
tg(x)
Figure 2: tg (x) and tf (x).
such that
1(
) = fx :
Area A =
Z
2
x
[tg (x)
1(
)
c(
1
)
1g ,
tf (x)] dx =
1
2
and
Area B =
Z
[
1 tf (x)
tg (x)] dx = 16.
It is easy to …nd that Condition (12) is satis…ed, i.e.,
R
1( 1)
[tg (x)
ktg (x)
1 tf (x)] dx
1 tf (x)k
=
1
2
1
2
+ 16
=
1
= 3:03%
33
5%.
In other words, individuals with "1 = 5% will decrease their investment in the risky asset
when the distribution of the risky assets shifts from f to g.
Now, let us further examine the optimal decision of the individuals with a constrained
8
concave utility function. De…ne U2 ("2 ) as follows:
u : u0 (z) > 0, u00 (z) < 0 and
U2 ("2 ) =
u00 (z)
inff u00 (z)g
1
"2
1 ; 8z . (13)
This set includes all risk-averse individuals with non-pathological concave preferences.
To examine the necessary and su¢ cient conditions on distributions for all decision makers
satisfying u 2 U2 ("2 ) to decrease their optimal decision, we can integrate Equation (8) by
parts and obtain
Z
x
u0 (
fx
fx
x
+ (1
x
0
= u (
Z
fu
00
+ (1
(
f )r) [tg (x)
f )r) [Tg (x)
fx
+ (1
tf (x)] dx
Tf (x)]
f )r) [Tg (x)
Tf (x)] dx,
(14)
x
where
Tf (x) =
Z
x
tf (s)ds
(15)
tg (s)ds.
(16)
x
and
Tg (x) =
Z
x
x
Furthermore, let
2(
) = fx : Tg (x)
Tf (x) > 0g ;
(17)
and
kTg (x)
Tf (x)k =
Z
x
x
jTg (x)
Tf (x)j dx.
(18)
Thus, we can de…ne Almost Central Riskiness (ACR) as follows:
De…nition 1 ACR. For 0 < "2 <
1
2,
f dominates g by "2 -Almost Central Riskiness if
9
9 2 R such that
Z
[Tg (x)
2(
Tf (x)] dx
)
and Tg (x)
"2 kTg (x)
Tf (x)k ;
Tf (x)
0. It is denoted by f ACR g.
Z
r) g(s)ds = Eg (x)
(19)
Note that
Tg (x) =
x
(s
r,
x
and Tf (x) = Ef (x)
Eg (x)
Ef (x)
r. Thus, the condition Tg (x)
Tf (x)
0 can be written as
r
,
r
(20)
since Ef (x) > r. Thus, Equation (20) provides a lower bound for .
The following Proposition shows the necessary and su¢ cient conditions on distributions
for all DMs satisfying u 2 U2 ("2 ) to decrease their optimal decision:
Proposition 2 For all u 2 U2 ("2 ),
g
f
if and only if f dominates g by "2 -Almost
Central Riskiness.
Proof. See Appendix C.
Proposition 2 indicates that all decision makers with non-pathological concave preferences
will decrease their investment in risky assets when the distribution of the risky assets shifts
from f to g if and only if f ACR g. Note that almost central dominance is a weaker
distribution condition than central dominance. Suppose that, as the condition of CR, there
exists a
such that Tg (x)
Tf (x)
0 for all x. We can easily …nd that
2(
) is an empty
set, and Condition (19) holds. In other words, ACR is a necessary condition for CR.
Furthermore, in the case where the shift in the distribution does not satisfy CR, we might
still be able to provide unambiguous comparative statics by the ACR rule. Let us use the
previous example to demonstrate it. Figure 3 shows Tg (x) and Tf (x). C, D and E denote
the areas where Tg (x)
Tf (x). To satisfy CR, we need to …nd a
E disappear. However, an increase in
Therefore, we cannot …nd a
such that Areas C, D and
will increase Areas C and D but decrease Area E.
such that Tg (x)
10
Tf (x)
0 for all x. In other words, g is not
Tg(x), Tf(x)
Tf(x)
F
Tg(x)
3/6
E
-2
-1
x
D 2
-1/6
C
10
-2/6
Figure 3: Tg (x) and Tf (x).
centrally riskier than f . On the other hand, our dominance order can give an unambiguous
comparative statics. We can vary
which satis…es Equation (20) to possibly change the
Areas C, D, E, F , such that the following condition holds:
R
2(
) [Tg (x)
kTg (x)
Tf (x)] dx
Tf (x)k
=
C +D+E
C +D+E+F
In other words, ACR allows Tg (x)
"2 .
Tf (x) > 0 for some x but the area that violates the
condition of CR cannot be too large.
3
Non-pathological Higher-order Preferences
In the previous section, we analyze the condition such that all decision makers with nonpathological concave preferences would decrease their demand for risky assets. The literature
has demonstrated that higher-order preferences are important for investment and other decisions.6 Tzeng (2001) extended Gollier’s (1995) model to all prudent individuals and found the
condition for an unambiguous comparative statics. However, the individuals with patholog6
For example, Eeckhoudt and Schlesinger (2008) demonstrated the necessary and su¢ cient
conditions on higher-order preferences for an Nth-degree change in risk to increase saving.
11
ically prudent preferences are included in his set of decision makers and thus his dominance
order su¤ers the same critiques as Gollier’s CR order. Thus, in this section, we will examine
the condition for all decision makers with non-pathological higher-order preferences.
Following Leshno and Levy (2002), Huang, Shih and Tzeng (2012) proposed a way to
exclude the pathological N th-order preferences.7 We adopt their setting and de…ne
n
UN = u : ( 1)n+1 u(n)
o
0, n = 1; 2; :::; N ,
UN ("N ) =
N +1 (N )
(21)
and
u 2 UN : ( 1)
u
n
o
N +1 (N )
inf ( 1)
u (x)
(x)
1
"N
1 ,8x . (22)
where u(n) denotes the nth derivative of the utility function u, and N > 2. Thus, an
individual with a utility function belonging to UN ("N ) is the one whose N th derivative is
bounded and the nth derivative of the utility function alters in sign from u0 > 0, n =
1; 2; :::; N . As noted by Huang, Shih and Tzeng (2012), the preferences with extreme values
of the N th derivative could be viewed as the “pathological N th-order preferences” and are
therefore excluded by Equation (22).
Let us further de…ne
(n)
Tf (x) =
Z
x
x
(n 1)
Tf
(s)ds
(23)
(1)
(n)
and Tf (x) = Tf (x). Similiarly, we can de…ne Tg (x). Furthermore, let
N(
n
) = x : Tg(N
1)
(x)
(N 1)
Tf
o
(x) > 0 .
(24)
Thus, the following Proposition is obtained.
Proposition 3 For all u 2 UN ("N ),
Z
N( )
h
Tg(N
1)
(x)
(N 1)
Tf
g
i
(x) dx
f
if and only if 9 2 R such that
"N Tg(N
7
1)
(x)
(N 1)
Tf
(x) ;
(25)
Huang, Shih and Tzeng (2012) provide the if and only if condition for most decision makers to rank
distributions, whereas we provided the condition for the unambiguous comparative statics.
12
(n)
(n)
and Tg (x)
Tf (x)
0, n = 1; 2; :::; N
1.
Proof. See Appendix D.
4
A Model with a General Payo¤ Function
Our …nding in the previous portfolio problem could be generalized to cases with a general
payo¤ function rather than a linear function.8 Assume that the payo¤ function of the DM
@z(x; )
@x
is z (x; ). For simplicity, assume that
= zx (x; ) > 0 as in Gollier (1995). Thus, the
objective function of an expected utility maximizer can be written as
max Eu =
Z
x
u(z(x; ))f (x)dx.
x
The corresponding FOC for the optimal
Z
x
z (x;
f )u
0
(z(x;
f ))f (x)dx
is
= 0;
(26)
x
where z denotes the derivative of z with respect to . Assume that z
(x;
f)
0 so that
Equation (26) is the necessary and su¢ cient condition for the optimization problem.
Let the pdf of x
~ shift from f (x) to g(x). Thus, the optimal decision under g(x), denoted
as
g,
will be no greater than
Z
x
u0 (z(x;
f ))) [tg (x;
f
f)
if and only if 9 2 R such that
tf (x;
f )] dx
0;
(27)
x
where z (x;
Tg (x;
f )g(x)
f) =
Z
= tg (x;
f)
and z (x;
f )f (x)
= tf (x;
f ).
Furthermore, de…ne
x
tg (s;
f )ds;
(28)
tf (s;
f )ds;
(29)
x
Tf (x;
f)
=
Z
x
x
1(
8
) = fx : tg (x;
f)
tf (x;
f)
> 0g ;
The general payo¤ function assumption is also adopted by Gollier (1995).
13
(30)
and
2(
) = fx : Tg (x;
f)
Tf (x;
f)
> 0g :
(31)
We can follow the previous Propositions and obtain the next one:
Proposition 4 We have
g
f
1. for all u 2 U1 ("1 ) if and only if 9 2 R such that
Z
[tg (x;
1(
f)
tf (x;
f )] dx
)
"1 ktg (x;
f)
tf (x;
f )k .
(32)
2. for all u 2 U2 ("2 ) if and only if 9 2 R such that
Z
zx (x;
2(
f ) [Tg (x;
f)
Tf (x;
f )] dx
)
and Tg (x;
"2 kzx (x;
f ) [Tg (x;
f)
Tf (x;
f )]k
(33)
f)
Tf (x;
f)
0.
Proof. The proof is similar to the proofs of Propositions 1 and 2 and thus is omitted.
5
Conclusion
Gollier (1995) has found the necessary and su¢ cient condition under a change in risk for all
risk-averse individuals to have unambiguous comparative statics. However, a small violation
of Gollier’s (1995) rule could make the comparative statics undetermined even in the case
where most decision makers would obviously decrease their optimal value of the decision
variable due to the increase in risk. Thus, in this paper, we have provided the necessary
and su¢ cient condition for the unambiguous comparative statics for most decision makers
who exhibit non-pathological preferences and are economically important. We have further
analyzed the conditions for higher-order preferences. In addition, we have generalized our
results to the cases with non-linear payo¤ functions.
14
References
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15
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16
Appendix A
Proof of the example in the Introduction
Let x
~ denote the risky return on the risky asset. The distribution of x
~ is
8
>
>
0 if x < 10%
>
>
<
F (x) =
1% if
10% x < 1000%
>
>
>
>
: 100% if x 1000%.
Assume that the distribution of x
~ shifts to G where
8
>
>
0 if x < 9:99%
>
>
<
G (x) =
1% if
9:99% x < 1%
>
>
>
>
: 100% if x 1%.
It can be shown that the location-weighted probability mass function under F and G are
and
8
>
>
0 if x < 10%
>
>
<
TF (x) =
0:1% if
10% x < 1000%
>
>
>
>
: 989:9% if x 1000%,
8
>
>
0 if x < 9:99%
>
>
<
TG (x) =
9:99% if
9:99%
>
>
>
>
: 0:8901% if x 1%.
x < 1%
Gollier (1995) had shown that all risk-averse investors will reduce their investment in the
risky asset if and only if there exists a real scalar
TG (x)
such that
TF (x) , 8x.
He further indicated that the above necessary and su¢ cient condition could be restated as
TG (x)
f xjTF (x)<0g TF (x)
inf
TG (x)
.
f xjTF (x)>0g TF (x)
sup
17
However, in this case, we have
TG (x)
0:8901%
TG (x)
0:8901%
=
<
sup
=
.
0:1%
989:9%
f xjTF (x)<0g TF (x)
f xjTF (x)>0g TF (x)
inf
Thus, we cannot apply Gollier’s (1995) condition for this case.
Appendix B
Proof of Proposition 1
“if” part: We would like to prove that if 9 2 R such that
Z
[tg (x)
1(
tf (x)] dx
)
"1 ktg (x)
tf (x)k ,
(A.1)
then
Z
x
u0 (
fx
+ (1
f )r) [tg (x)
tf (x)] dx
0;
(A.2)
x
for all u 2 U1 ("1 ). Let
c(
1
) denote the complement of
1(
) and
and
respectively
denote supfu0 (x)g and inffu0 (x)g. Thus, rewriting the above Equation yields
Z
Z
=
=
Z
+
Z
0
u (
1(
fx
[tg (x)
1(
tf (x)] dx +
)
[tg (x)
1(
Z
[tg (x)
1(
tf (x)] dx
)
[tg (x)
Z
Z
+
)
c(
1
)
Z
tf (x)] dx
u0 (
fx
+ (1
f )r) [tg (x)
tf (x)] dx
)
[tg (x)
tf (x)] dx
[tg (x)
tf (x)] dx
[tg (x)
1(
c(
1
tf (x)] dx
)
ktg (x)
tf (x)k :
(A.3)
. Thus, Equation (A.1) can be written as
tf (x)] dx
)
tf (x)] dx +
c(
1
)
[tg (x)
1(
tf (x)] dx +
)
+
1(
f )r) [tg (x)
)
Furthermore, let "1
Z
+ (1
Z
+
ktg (x)
tf (x)k ,
(A.4)
or,
+
Z
[tg (x)
1(
)
tf (x)] dx
ktg (x)
18
tf (x)k
0,
(A.5)
which ensures that Equation (A.3) is negative. Thus, we will have
Z
x
u0 (
fx
+ (1
f )r) [tg (x)
tf (x)] dx
0:
x
“only if” part: We would like to prove that if 8 such that
Z
[tg (x)
1(
tf (x)] dx > "1 ktg (x)
)
tf (x)k ,
(A.6)
tf (x)] dx > 0.
(A.7)
then there exists a u 2 U1 ("1 ) such that
Z
x
u0 (
fx
+ (1
f )r) [tg (x)
x
Without losing any generality, assume that
1(
) = [a; b],
c(
1
) = ]a; b[ and "1 =
+
. Thus,
we have 8 such that
Z
b
[tg (x)
tf (x)] dx >
a
+
ktg (x)
tf (x)k .
(A.8)
De…ne
8
>
>
x
if x x a
>
>
<
u(x) =
if a x b :
(x a) + a
>
>
>
>
:
(x b) + b + a if b x x
It is obvious that u 2 U1 ("1 ). We have
Z
x
u0 (
fx
+ (1
f )r) [tg (x)
x
=
Z
b
a
=
+
[tg (x)
Z
tf (x)] dx +
b
[tg (x)
Z
tf (x)] dx
c(
1
tf (x)] dx
a
=
+
Z
[tg (x)
)
Z
[tg (x)
tf (x)] dx
+
a
> 0.
x
x
b
tf (x)] dx
jtg (x)
ktg (x)
tf (x)j dx
tf (x)k
(A.9)
The last inequality holds from Equation (A.8).
19
Appendix C
Proof of Proposition 2
The proof is very similar to the proof of Proposition 1. Since for all u 2 U2 ("2 ),
g
f
if
and only if
u0 (
f x + (1
Tf (x)]+
f )r) [Tg (x)
f
Z
x
u00 (
fx
+ (1
f )r)
[Tg (x)
Tf (x)] dx
0.
x
(A.10)
Since u0 could be any positive number in U2 ("2 ), to make the above condition hold, we must
have
Tg (x)
Note that
Z
Tf (x)
f
x
0.
> 0. To make Equation (A.10) hold, we should ask
u00 (
fx
+ (1
f )r)
[Tg (x)
Tf (x)] dx
0.
x
Thus, we can follow similar steps to those in the proof of Proposition 1 to prove this Proposition.
Appendix D
Proof of Proposition 3
Integrating Equation (A.10) by parts yields
0
u (
fx
+ (1
f )r) [Tg (x)
Tf (x)] +
f
Z
x
u00 (
fx
+ (1
f )r)
x
= u0 (
fx
+(
2
f)
+ (1
Z x
f )r) [Tg (x)
u000 (
fx
+ (1
x
= :::
N
X1
=
(
n
f)
n=1
+(
N 1
f)
1
Z
h
x
( 1)n
xh
1
( 1)N
u(n) (
1
Tf (x)] + f u00 ( f x + (1
h
i
(2)
(2)
Tf (x) dx
f )r) Tg (x)
ih
)r)
Tg(n) (x)
f
f x + (1
u(N ) (
fx
+ (1
20
ih
f )r)
Tg(N
f )r)
h
Tg(2) (x)
i
(n)
Tf (x)
1)
(x)
(N 1)
Tf
[Tg (x)
i
(x) dx.
Tf (x)] dx
i
(2)
Tf (x)
(A.11)
Since, in UN ("N ), only u(N ) is bounded and the nth derivative of the utility function alters
in sign from u0 > 0, to make sure the above equation is non-positive, we have
(n)
Tg(n) (x)
Tf (x)
0; n = 1; 2; :::; N
1
(A.12)
and
Z
x
xh
( 1)N
1
u(N ) (
f x + (1
ih
)r)
Tg(N
f
1)
(x)
(N 1)
Tf
i
(x) dx
0:
(A.13)
To …nd the if and only if condition for Equation (A.13), we can follow the proof of Proposition
1 and …nd that the condition is: 9 2 R such that
Z
N(
)
h
Tg(N
1)
(x)
(N 1)
Tf
i
(x) dx
"N Tg(N
21
1)
(x)
(N 1)
Tf
(x) :
(A.14)