UNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Monopoly (single seller) and Monopsony (single buyer)
Example Problems
Monopoly (single seller) Example Problem
Assumptions:
•
A single producer is producing product Q using two inputs, L and K
•
The market for Q is a monopoly (the producer has market power in the market for Q; thus, P
Q
is a choice
variable for the producer).
Suppose a market research study finds that consumers’ demand function for
product Q is given by:
Q = 4P
Q
2
.
(Note that the demand function can also be written as P
Q
= 2Q
1/2
.)
•
The markets for L and K are perfectly competitive (the producer does NOT have market power in the
markets for L and K; thus, P
L
and P
K
are given constants for the producer)
•
In this example, suppose that the producer uses an imperfect substitutes technology to produce product Q
from inputs L and K.
Suppose that the producer’s production function is:
Q = 10L
1/2
K
1/4
.
Set up the optimization problem:
subject to:
4
1
2
1
K
L
10
Q
⋅
⋅
=
, technology
Q = 4P
Q
2
,
consumer demand
Replace TR and TC with their definitions, and rewrite consumer demand as P
Q
= 2Q
1/2
.
subject to:
4
1
2
1
K
L
10
Q
⋅
⋅
=
technology
P
Q
= 2Q
1/2
consumer demand (rewritten)
Substitute consumer demand for P
Q
in the profit function (notice that P
Q
then disappears from the list of
choice variables).
subject to:
4
1
2
1
K
L
10
Q
⋅
⋅
=
technology
Simplify the profit function by combining the factors that contain Q (that is, add the exponents on the Q’s):
subject to:
4
1
2
1
K
L
10
Q
⋅
⋅
=
technology
1
(
29
K
P
L
P
Q
P
ofit
Pr
K
L
Q
Q
P
,
Q
,
K
,
L
max
⋅
+
⋅

⋅
=
TC
TR
ofit
Pr
max
Q
P
,
Q
,
K
,
L

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

⋅
=

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

=
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UNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
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 Fall '08
 Dumas
 Microeconomics, Pk, Monopoly, Supply And Demand, Department of Economics and Finance, Dr. Chris Dumas, max Pr ofit

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