One solution is
v
= (2
,

1)
T
. To find an eigenvector corresponding to
λ
= 5, we
solve
4

2

2
1
u
=
0
0
.
One solution is
u
= (1
,
2)
T
.
c
) Although
u
·
v
= 0, which shows the eigenvectors are orthogonal, they do not have
unit norm. We divide by their norms to get orthonormal eigenvectors:
2
√
5

1
√
5
,
1
√
5
2
√
5
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 12, 2002
Page 2
3. Robinson Crusoe has utility function
u
(
x, y
) =
x
2
+
y
2
. Crusoe has a production possi
bilities set given by
{
(
x, y
) :
x
≥
0
, y
≥
0
, x
+
y
2
≤
4
}
.
a
) Is the production possibility set a closed set? Is it a bounded set?
b
) Show that Crusoe’s problem of maximizing utility over his production set has a
solution.
Answer:
a
) The set is closed.
There are many ways to see this, one is to realize that the
functions
f
(
x, y
) =
x
,
g
(
x, y
) =
y
, and
h
(
x, y
) =
x
+
y
2
are all continuous. Then
f

1
([0
,
∞
)),
g

1
([0
,
∞
)) and
h

1
((
∞
,
4]) are all closed sets (as the inverse image
of closed intervals).
The production possibility set is closed because it is the
intersection of 3 closed sets.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 STAFF
 Economics, Topology, Eigenvalue, eigenvector and eigenspace, Compact space, limit point

Click to edit the document details