This preview shows pages 1–3. Sign up to view the full content.
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
(
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
(
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 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
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
 Spring '08
 DANNICATAMBAY
 Microeconomics, Utility

Click to edit the document details