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
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View Full DocumentUNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
2)
Because L and K must be used in fixedproportions
in the production of Q, we know that the
production technology for Q is that of perfect complements
, or the “Leontief
” technology
.
Given that the
firm can produce 1 unit of Q for every 8 units of L used in production, and 4 units of K must be used with
each unit of L, the firm’s production technology can be described as:
⋅
=
⋅
=
L
4
K
L
8
1
Q
or
⋅
⋅
=
K
32
1
L,
8
1
min
Q
The firm’s profitmaximization problem is therefore:
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 Fall '08
 Dumas
 Economics, Microeconomics, Pk, Optimization, Supply And Demand, Dr. Chris Dumas, max Pr ofit

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