UNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Homework 6
Solutions
1) Because Q can be produced with either L or K alone
, we know that the production
technology for Q is that of perfect
substitutes
.
Given that each L produces 4 units of Q
and each K produces (1/2) unit of Q, the firm’s production technology can be described
as: Q = (4)L + (1/2)K.
The firm’s profitmaximizing problem is therefore:
subject to:
Q = (4)L + (1/2)K
There are now three possible cases:
Recall that we can produce Q using L or K alone.
If it is not profitable to produce Q with either L or K,
then we should not produce at all:
CASE 1:
If
(
29
0
P
(4)
P
L
Q
<

⋅
and
(
29
0
P
)
2
/
1
(
P
K
Q
<

⋅
, then L*=0, K*=0, Q*=0
On the other hand, if it is profitable to produce Q with either L or K, then we want to produce using the
input that is most profitable:
CASE 2: If
(
29
0
P
)
4
(
P
L
Q

⋅
and
(
29
(
29
K
Q
L
Q
P
)
2
/
1
(
P
P
)
4
(
P

⋅

⋅
, then
∞
=
L*
, K*=0,
∞
=
Q*
CASE 3: If
(
29
0
P
)
2
/
1
(
P
K
Q

⋅
and
(
29
(
29
L
Q
K
Q
P
)
4
(
P
P
)
2
/
1
(
P

⋅

⋅
, then
0
L*
=
,
∞
=
K*
,
∞
=
Q*
In this problem, since
(
29
$30
P
)
4
(
P
L
Q
=

⋅
and
(
29
$1
P
)
2
/
1
(
P
K
Q
=

⋅
, we have CASE 2;
so, the answer is
∞
=
L*
, K
*
=0,
∞
=
Q*
.
1
(
29
K
P
L
P
Q
P
ofit
Pr
K
L
Q
,
,
max
⋅
+
⋅

⋅
=
Q
K
L
(
29
(
29
K
P
L
P

K
)
2
/
1
(
L
(4)
P
ofit
Pr
K
L
Q
,
max
⋅
+
⋅
⋅
+
⋅
⋅
=
K
L
(
29
(
29
K
P

)
2
/
1
(
P
L
P
(4)
P
ofit
Pr
K
Q
L
Q
,
max
⋅
⋅
+
⋅

⋅
=
K
L