The result follows.
Exercise 6.27 (exercise:
mean-variance).
Suppose a consumer’s utility for a
CDF is
U
(
F
) =
μ
F
−
α
Var(
F
), where
Var(
F
) =
integraldisplay
(
x
−
μ
F
)
2
dF
is the variance of
F
.
1. Prove that if
α > α
′
then the preference corresponding to
α
is more risk averse
than that corresponding to
α
′
.

2. Prove that
U
violates expected utility.

Section 6: vNM Representation Theorem and Risk
6-23
Exercise 6.31 (12.12).
Prove the following proposition: Suppose
followsorequal
is represented
by a twice continuously differentiable vNM index
v
. Then
followsorequal
exhibits constant ab-
solute risk aversion if and only if there exists
a >
0 and
b
∈
R
such that either
v
(
x
) =
ax
+
b
for all
x
or
v
(
x
) =
ae
−
λx
+
b
for all
x
. This is sometimes called the
CARA utility index for wealth.