Economics 101
Solutions to Problem Set 4
Due Tuesday, February 22, 2011
Soojin Kim
1. The
certainty equivalent
of a lottery (gamble)
X
is the amount of
money for which the individual (with utility for money
u
and wealth
w
0
) is indifferent between the lottery
X
and the certain amount
c
; that
is,
Eu
(
w
0
+
X
) =
u
(
w
0
+
c
)
.
(The certainty equivalent may be negative.)
Suppose Edward has
utility function over money
u
(
w
) =
√
w
, and initial wealth
w
0
= $4.
Consider the gamble
X
given by
X
=
(
$12
,
with probabilty
1
3
,

$4
,
with probabilty
2
3
.
(a) What is the expected value of the gamble
X
?
Solution
The expected value of the gamble
X
is
E
(
X
) =
1
3
·
12 +
2
3
·
(

4) =
4
3
.
(b) What is Edward’s expected utility if he accepts the gamble
X
?
Solution
Since Edward has the initial wealth of $4, his ex
pected utility is
Eu
(
w
0
+
X
) =
1
3
u
(4 + 12) +
2
3
u
(4

4) =
4
3
.
(c) What is Edward’s certainty equivalent of the above gamble?
Solution
By definition, we want to find
c
such that
Eu
(
w
0
+
X
) =
4
3
=
√
4 +
c
=
u
(
w
0
+
c
)
is satisfied. Thus,
c
=

20
9
.
(d) Discuss the relationship between the expected value of the gam
ble, Edward’s certainty equivalent of the gamble, and his atti
tudes towards risk.
1
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Solution
Notice that expected value of the gamble is positive
and Edward’s utility from getting expected value of the gamble
is
u
(4 +
4
3
) =
q
16
3
≈
2
.
03 which is greater than the expected
utility from the gamble (
Eu
(
w
0
+
X
) =
4
3
). This is due to the
fact that Edward is risk averse (notice that his utility function
is concave). Moreover, his certainty equivalent of the gamble is

20
9
, which implies that Edward is indifferent between taking the
gamble and being taken $
20
9
away from him. (Also refer to Figure
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 Spring '08
 DANNICATAMBAY
 Microeconomics, Utility, Certainty Equivalent, Tina

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