b
) To ﬁnd an eigenvector corresponding to
λ
= 10, we solve
±

1

2

2

4
²
v
=
±
0
0
²
.
One solution is
v
= (2
,

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

2

2
1
²
u
=
±
0
0
²
.
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
²
,
±
1
√
5
2
√
5
²
.
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View Full DocumentMATHEMATICAL 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
((
∞
,
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 Spring '08
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 Economics, Topology, Eigenvalue, eigenvector and eigenspace, Compact space, limit point

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