Answer:
In part (a) we found that
A
is invertible. It follows that the system of
equations has exactly one solution. For the record, the solution is (4
/
3
,

2
,
3)
t
.
3. Suppose a firm’s production function is
Q
=
K
1
/
3
L
2
/
3
and that
K
= 8000 and
L
= 64.
a
) How much can the firm produce?
b
) What are the marginal products of capital (
K
) and labor (
L
)?
c
) Suppose that the available capital falls by 2 units, while labor increases by 5
units. Without plugging the new numbers for
K
and
L
into the production func
tion, compute approximately how much the firm can now produce.
4. Let
f
(
x, y, z
) =
x
2
+ 3
y
+
z
3

5.
a
) Find an (
x
0
, y
0
, z
0
) satisfying
f
(
x
0
, y
0
, z
0
) = 0.
Answer:
The point (1
,
1
,
1) works.
b
) Can
x
be expressed as a function
g
(
y, z
) in some neighborhood of (
x
0
, y
0
, z
0
)?
Answer:
Since
∂
f/
∂
x
= 2
x
,
∂
f/
∂
x
= 2 at (
x
0
, y
0
, z
0
). The Implicit Function The
orem yields such a function
g
. Alternatively, note that
g
(
y, z
) = (5

3
y

z
3
)
1
/
2
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MATHEMATICAL ECONOMICS FINAL, DECEMBER 11, 2003
Page 2
works.
c
) Compute
dg
.
Answer:
By the Implicit Function Theorem,
dg
= (

1
/
2
x
)(
∂
f/
∂
y,
∂
f/
∂
z
) =

(3
/
2)(1
,
3
z
2
). At (1
,
1
,
1), this has the value (

3
/
2
,

3
/
2).
5. Consider the problem of maximizing
x
2
+
y
2
subject to the constraints
x
≥
1,
y
≥
2,
and
x
+ 2
y
≤
10.
a
) Does this problem have a solution?
b
) Is constraint qualification satisfied?
c
) Assuming there is a solution, find it. Make sure to check the secondorder con
ditions.
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 Economics, Linear Algebra, Invertible matrix, linearly independent columns, linearly independent rows

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