Econ 300  Solutions to Problem Set 1  Julien Bengui
10.1.8
First we write the firm’s profit function using the factor and output price information as
Π(
K, L
)
=
4
*
9
L
1
/
3
K
1
/
3

12
L

6
K
=
36
L
1
/
3
K
1
/
3

12
L

6
K.
To find the optimal levels for each input, we write the first order conditions:
Π
K
=
12
L
1
/
3
K

2
/
3

6 = 0
,
Π
L
=
12
L

2
/
3
K
1
/
3

12 = 0
.
Solving the second equation for
K
yields
K
=
L
2
. This can be substituted into the first
equation:
2
L
1
/
3
(
L
2
)

2
/
3
= 1
,
which can be simplified to 2
L

1
= 1, or
L
= 2.
Plugging this back into
K
=
L
2
yields
K
= 4. Thus, the stationary point of the profit function is (
K, L
) = (4
,
2). To check that
this stationary point is a maximum, we compute the second order derivatives of the profit
function and evaluate them at the stationary point:
Π
KK
=

8
L
1
/
3
K

5
/
3
=

8
*
2
1
/
3
4

5
/
3
=

8
*
2
1
/
3
2

10
/
3
=

8
*
2

9
/
3
=

8
*
2

3
=

1
<
0
,
Π
LL
=

8
L

5
/
3
K
1
/
3
=

8
*
2

5
/
3
4
1
/
3
=

8
*
2

5
/
3
2
2
/
3
=

8
*
2

3
/
3
=

8
*
2

1
=

4
<
0
.
We also calculate the cross partial derivative
Π
KL
= 4
L

2
/
3
K

2
/
3
= 4
*
2

2
/
3
4

2
/
3
= 4
*
2

2
/
3
2

4
/
3
= 4
*
2

6
/
3
= 4
*
2

2
= 4
/
4 = 1
,
from which we see that Π
KK
Π
LL
= 4
>
1 = (Π
KL
)
2
. We can thus conclude that the point
(
K
*
, L
*
) = (4
,
2) is a maximum.
10.1.10
(a)
With linear total costs, the profit function of AWL is given by
Π(
Q
T
, Q
M
)
=
(
α

βQ
T
)
Q
T
+ (
γ

θQ
M
)
Q
M

Ψ

c
(
Q
M
+
Q
T
)
=

βQ
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 Spring '08
 cramton
 Derivative, Optimization, Fermat's theorem, Stationary point

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