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Econ 210A
Problem Set 6 Answer Key
Jehle and Reny Exercise
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)
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
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
 Spring '10
 fallahi
 Utility

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