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October 25, 2010
1. Short questions (31 points)
(a) (15 points; 3 points each) For each of the following vNM utility functions
u
(
x
) (with
x >
0), do they represent risk averse preferences, risk neutral preferences, risk loving
preferences, or neither?
(i)
u
(
x
) = 24
x
2
+ 9
x
{
Note that
u
0
(
x
) = 48
x
+ 9, and
u
00
(
x
) = 48
>
0. Thus, this individual is risk
loving.
(ii)
u
(
x
) = exp(7
x
+ 18)
{
Note that
u
0
(
x
) = 7 exp(7
x
+ 18), and
u
00
(
x
) = 49 exp(7
x
+ 18)
>
0. Thus, this
individual is risk loving.
(iii)
u
(
x
) = log(
x
) +
p
x
{
Note that
u
0
(
x
) =
1
x
+
1
2
p
x
, and
u
00
(
x
) =
1
x
2
1
4
x
p
x
<
0. Thus, this individual is
risk averse.
(iv)
u
(
x
) = 2
x
+ 22
{
Note that
u
0
(
x
) = 2, and
u
00
(
x
) = 0. Thus, this individual is risk neutral.
(v)
u
(
x
) =
x
+ log(
x
)
{
Note that
u
0
(
x
) = 1 +
1
x
, and
u
00
(
x
) =
1
x
2
<
0. Thus, this individual is risk
averse.
(b) (4 points) There are two goods (
a
and
b
). Rob’s utility function is
u
R
(
a; b
) =
a
+
b
.
Tom’s utility function is
u
T
(
a; b
) = 3
a
+ 3
b
. Jerry’s utility function is
u
J
(
a; b
) =
10
a
+ 10
b
. There are 10 units of each good. Describe the set of
all
allocations. (You can use words or math, but it has to be precise).
±
(c) (4 points) Consider the same setting as in part (b) above, but suppose now that Jerry’s
utility is
u
J
(
a; b
) =
10
a
10
b
. Describe the set of
all
can use words or math, but it has to be precise).
±
Jerry doesn’t like either
a
and
b
nothing, and will split the goods between Rob and Tom. Any allocation that allocates
(d) (4 points) Consider the same setting as in part (b) above, but suppose now that Jerry’s
utility is
u
J
(
a; b
) = 10
a
10
b
. Describe the set of
all
can use words or math, but it has to be precise).
1
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This note was uploaded on 03/19/2011 for the course ECON 51 taught by Professor Tendall,m during the Winter '07 term at Stanford.
 Winter '07
 Tendall,M
 Utility

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