Northwestern University
Marciano Siniscalchi
Fall 2009
Econ 3310
HOMEWORK 4
DUE MONDAY, NOVEMBER 16, 2009, IN CLASS.
PLUS ONE PRACTICE PROBLEM
1
Consider an economy with one riskfree asset, with return
r
f
= 1
.
01, and two risky returns
˜
r
1
,
˜
r
2
. The latter are jointly normally distributed, with means
μ
1
= 1
.
02 and
μ
2
= 1
.
05, standard
deviations
σ
1
= 0
.
02 and
σ
2
= 0
.
05, and covariance
σ
12
=

0
.
0003.
(a) Why is it OK to use the meanvariance approach in this setting?
(b) Construct the eﬃcient frontier. You need to ﬁnd an expression for the minimum amount
of
σ
required to achieve any target level
μ
of expected return; the expression will be of the form
“constant plus
μ
times another constant,” and you must specify what these constants are. If you
wish, you can also plot the eﬃcient frontier (using Excel, Matlab etc.: remember
σ
goes on the
horizontal axis); however, you
do
need to ﬁnd an explicit expression for the eﬃcient frontier.
2
Consider the following variant of the monopoly example we discussed in class. Demand is given by
Q
(
p,s
) =
e

sp
, the marginal cost is zero, and the possible values of
s
(i.e. the states) are
S
= [1
,
9].
Beliefs over
S
are uniform, as in the lecture notes. The information partition is
I
=
{
[1
,
5]
,
[5
,
9]
}
;
in other words, the ﬁrm learns whether
s
∈
[1
,
5] or
s
∈
[5
,
9].
Assume that the ﬁrm is riskneutral, as in the lecture notes. Calculate the optimal prices the
ﬁrm should set without any additional information, and conditional upon receiving each of the
signals [1
,
5] and [5
,
9]. Finally, compute the value of the information partition
I
.
3
A mine is oﬀered for sale at 2 million dollars. If the mine is rich in ore, it is worth 3 million
dollars; otherwise, it is only worth 1 million dollars. A test balloon can be lowered into a well;
if the balloon encounters a solid object, it breaks; otherwise the balloon is retrieved, and can be