Econ 210A
Problem Set 6 Answer Key
Jehle and Reny Exercise
2.19 Using the de°nition of risk aversion given in the text, prove that an individual is risk
averse over gambles involving nonnegative wealth levels if and only if her VNM utility
funciton is strictly concave on
R
+
.
Suppose that an individual is risk averse, which implies that
u
(
E
(
g
))
> u
(
g
)
:
u
(
E
(
g
))
> u
(
g
)
)
u
(°
p
i
w
i
)
>
°
p
i
u
(
w
i
)
°
u
(
p
1
w
1
+
:::
+
p
n
w
n
)
> p
1
u
(
w
1
) +
:::
+
p
n
u
(
w
n
)
This holds for all
p
0
s
that are between 0 and 1 and sum to one and
u
(
p
1
w
1
+
:::
+
p
n
w
n
)
°
u
(
w
t
)
and
p
1
u
(
w
1
) +
:::
+
p
n
u
(
w
n
)
°
T
1
u
(
w
1
) +
:::
+
T
n
u
(
w
n
)
Hence,
u
(
w
t
)
> T
1
u
(
w
1
) +
:::
+
T
n
u
(
w
n
)
which implies
u
is strictly concave.
,
Suppose that a consumer has a concave utility function over all wealth levels that
are positive. This implies that
u
(
E
(
g
)) =
u
(°
n
i
p
i
w
i
)
>
°
n
i
p
i
w
i
=
u
(
g
)
. Therefore the
consumer is risk averse.
2.23 Consider the quadratic VNM utility function
U
(
w
) =
a
+
bw
+
cw
2
.
a)
What restrictions if any must be placed on parameters
a; b
and
c
for this function
to display risk aversion?
Risk aversion is characterized by the utility function when
U
0
(
w
)
>
0
and
U
00
(
w
)
<
0
.
U
0
(
w
) =
b
+ 2
cw >
0
U
00
(
w
) = 2
c <
0
)
c <
0
b)
Over what domain of wealth can a quadratic VNM utility function be de°ned?
In order for the utility function to be de°ned,
U
0
(
w
)
>
0
)
b
+ 2
cw >
0
or
w <
±
b
2
c
c)
Given the gamble
g
= ((1
=
2) (
w
+
h
)
;
(1
=
2) (
w
±
h
))
show that
CE < E
(
g
)
and
P >
0
.
U
(
CE
) =
1
2
U
(
w
+
h
) +
1
2
U
(
w
±
h
)
)
a
+
bCE
+
cCE
2
=
1
2
°
a
+
b
(
w
+
h
) +
c
(
w
+
h
)
2
±
+
1
2
°
a
+
b
(
w
±
h
) +
c
(
w
±
h
)
2
±
)
u
(
CE
) =
a
+
b
(
CE
) +
c
(
CE
)
2
=
a
+
bw
+
cw
2
+
ch
2
> u
(
w
) =
u
(
E
(
g
))
Because utility is strictly increasing,
CE < E
(
g
)
.
P
=
E
(
g
)
±
CE
is then greater
than 0.
d)
Show that this function, satisfying the restrictions in part (a), cannot represent
preferences that display decreasing absolute risk aversion.
Risk aversion
R
a
(
w
) =
±
u
00
(
w
)
u
0
(
w
)
=
±
c
b
+2
cw
dR
a
(
w
)
dw
=
4
c
2
(
b
+2
cw
)
2
>
0
, which implies that
R
a
(
w
)
is increasing in wealth and cannot
represent decreasing absolute risk aversion.
1
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2.24 Let
u
(
w
) =
±
(
b
±
w
)
c
. What restrictrions on
w; b;
and
c
are required to ensure that
u
(
w
)
is strictly increasing and strictly concave? Show that under those restrictions,
u
(
w
)
displays increasing absolute risk aversion.
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 Spring '10
 fallahi
 Utility, Dracula, absolute risk aversion, Randy Variable

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