Econ 110 / Pol Sci 135 Suggested Exercises
Set 2 Solutions
Fall 2009
Page 1 of 10
1. (a)
Jack’s expected utility is 0.5*
√
250,000 + 0.5*
√
90,000 = 400.
(b)
Since Jack’s utility is the square root of his income, to achieve a utility of 400 Jack would
need a certain income of 400
2
= $160,000.
(c) The four possible luck outcome pairs for Jack and Janet are:
(good, good)
(good, bad)
(bad, good)
(bad, bad)
Each of these outcomes happens with probability (0.5)(0.5) = 0.25.
(d)
Under the agreement, the expected utility for each participant is 0.25*
√
250,000 +
0.25*
√
170,000 + 0.25*
√
170,000 + 0.25*
√
90,000
≈
406.1552.
(e)
For either Jack or Jill to achieve a utility of 406.1552, he or she would need a certain income
of 406.1552
2
= $164,962.11.
(f)
The eight possible luck outcome triplets for Jack, Janet, and Chrissy are:
(good, good, good)
(good, good, bad)
(good, bad, good)
(bad, good, good)
(good, bad, bad)
(bad, good, bad)
(bad, bad, good)
(bad, bad, bad)
Each of these outcomes occurs with probability (0.5)(0.5)(0.5) = 0.125.
(g)
Under the agreement, the expected utility for each participant is 0.125*
√
250,000 +
3*0.125*
√
196,666.667 +3*0.125*
√
143,333.333 + 0.125*
√
90,000
≈
408.2744.
(h)
For Jack, Jill, or Chrissy to achieve a utility of 408.2744, he or she (or she) would need a
certain income of 408.2744
2
= $166,687.99.
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Econ 110 / Pol Sci 135 Suggested Exercises
Set 2 Solutions
Fall 2009
Page 2 of 10
2.
The table of certainty equivalents is as follows:
(a)
In order to plot the utility function, we need to show the relationship between a given
input, the dollar value, which we can put on the x-axis, and the utility of that dollar
value, which we put on the y-axis.
Note that because the utility of $1000 is 10 and
the probabilities above denote probabilities of winning $1000, we have, for example
0.8*
u
(1000)=
u
(512).
u
(1000)=10, so 8=
u
(512).
We can graph this system:
0
1
2
3
4
5
6
7
8
9
10
0
100
200
300
400
500
600
700
800
900
1000
Dollar Payment
Utility of Dollar Payment
Econ 110 / Pol Sci 135 Suggested Exercises
Set 2 Solutions
Fall 2009
Page 3 of 10
Note that the utility relationship is the familiar cube-root utility that we have seen in
lecture.
So, the utility function is:
0
1
2
3
4
5
6
7
8
9
10
0
45
90
135
180
225
270
315
360
405
450
495
540
585
630
675
720
765
810
855
900
945
990
Dollar Payment
Utility of Dollar Payment
(b)
The expected utility of the lottery is
( )
[
]
)
u(outcome2
*
2}
Pr{outcome
)
u(outcome1
*
1}
Pr{outcome
+
=
L
U
E
.
In this case, the expected utility is
( )
[
]
( )
(
)
2.0127
2.1544
4
3
1.5874
4
1
10
4
3
4
4
1
10
4
3
4
4
1
3
1
3
1
=
+
=
+
=
+
=
u
u
L
U
E
The certainty equivalent of the lottery is the dollar value that will yield a utility equal
to the utility of the lottery.
So, u(Certainty Equivalent) = 2.0127.
So,
0127
.
2
3
1
=
CE
Which implies
1533
.
8
0127
.
2
3
=
=
CE
.
(c)
Using the reasoning from part (b),
( )
[
]
( )
( )
( )
(
)
3
1
3
1
3
1
10
24
7
6
4
1
2
3
1
0
8
1
10
24
7
6
4
1
2
3
1
0
8
1
+
+
+
=
+
+
=
u
u
u
u
L
U
E
Solving,
( )
[
]
5477
.
1
6824
.
0
4453
.
0
4200
.
0
2144
.
2
24
7
8171
.
1
4
1
2599
.
1
3
1
0
=
+
+
=
+
+
+
=
L
U
E
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Econ 110 / Pol Sci 135 Suggested Exercises
Set 2 Solutions
Fall 2009
Page 4 of 10
3.
(a) There is no pure-strategy Nash equilibrium here, hence the search for an equilibrium in
mixed strategies. Row’s p-mix (probability p on Up) must keep Column indifferent and so
must satisfy 16p + 20(1 – p) = 6p + 40(1 – p); this yields p = 2/3 = 0.67 and (1 – p) = 0.33.
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- Fall '09
- Zenou
- Game Theory, Utility, suggested exercises
-
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