Mathematical Economics Midterm #1, October 3, 2002
You have until 4:45 to complete this exam. Answer all ﬁve questions. Each question is worth
20 points, for a total of 100 points. Good luck!
1. Consider the vector
x
=
3
2
1
. Find the coordinates of
x
in the basis
B
=
{
b
1
,
b
2
,
b
3
}
=
1
1
0
,
0
1
1
,
0
0
1
.
2. Let
A
=
1 6

7 3
1 9

6 4
1 3

8 4
.
a
) What is the rank of
A
?
b
) Does the equation
A
x
=
b
have a solution
x
∈
R
4
for every
b
∈
R
3
?
c
) If
A
x
=
b
can be solved, how many solutions does it have?
3. The stationary distribution of the Markov employment model obeys
(
q

1)
x
+
py
= 0
(1

q
)
x

py
= 0
x
+
y
= 1
where 0
< p <
1 and 0
< q <
1. Does this system have a solution (
x,y
)
0
? If so, is there
a unique solution or multiple solutions? Finally, does the system have a nonnegative
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 Fall '08
 STAFF
 Economics, Macroeconomics, mathematical economics midterm

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