price discrimination (Figure 14.4),
market outcome is e¢ cient!
Thirddegree price discrimination:
price depends on the identity but not on quantity.
Separate the market into di/erent segments.
Example 14.5:
p
1
=
24
²
q
1
and
p
2
=
12
²
0
.
5
q
2
,
MC
=
6
.
max
[(
24
²
q
1
)
q
1
+ (
12
²
0
.
5
q
2
)
q
2
]
²
6
(
q
1
+
q
2
)
.
Seconddegree price discrimination:
price depends on quantity but not on the identity.
Q. Wen
(UW)
Econ 400
University of Washington
33 / 52
Chapter 15 Imperfect Competition
Many ±rms, compete in either price or quantity,
simultaneously or sequentially.
Bertrand model
: simultaneous pricesetting/pricecompetition game.
Two ±rms, 1 and 2, each has a constant MC
c
>
0
.
Products of the two ±rms are
homogenous
/identical.
Firm
i
chooses price
p
i
³
0, for
i
=
1 and 2.
Market demand function:
q
=
a
²
bp
.
Because of
homogenous products
, ±rm 1°s pro±t is not continuous:
π
1
(
p
1
,
p
2
) =
8
>
<
>
:
(
a
²
bp
1
) (
p
1
²
c
)
if
p
1
<
p
2
,
1
2
(
a
²
bp
1
) (
p
1
²
c
)
if
p
1
=
p
2
,
0
if
p
1
>
p
2
.
Q. Wen
(UW)
Econ 400
University of Washington
34 / 52
Chapter 15 Imperfect Competition
Bertrand model
: simultaneous pricesetting game.
Firm 2°s pro±t function is not continuous as well:
π
2
(
p
1
,
p
2
) =
8
>
<
>
:
(
a
²
bp
2
) (
p
2
²
c
)
if
p
2
<
p
1
,
1
2
(
a
²
bp
2
) (
p
2
²
c
)
if
p
2
=
p
1
,
0
if
p
2
>
p
1
.
Because discontinuous payo/ functions,
a ±rm does not always have a
best response
.
However, if
p
2
=
c
, then any
p
1
³
c
is a best response of ±rm 1.
There is a Nash equilibrium where
p
1
=
p
2
=
c
,
which is the same as the competitive equilibrium outcome.
Problem 15.4:
Bertrand model with product
di/erentiation
.
Demand for ±rm 1°s product is
q
1
=
1
²
p
1
+
bp
2
with
b
<
2,
For simplicity, assume production cost is 0.
Q. Wen
(UW)
Econ 400
University of Washington
35 / 52
Chapter 15 Imperfect Competition
Example 15.5
Hotelling°s Beach Model
Horizontal product di/erentiation
Consumers are distributed uniformly on
[
0
,
2
]
.
Two ±rms, ±rm
A
is located at 0 and ±rm
B
is located at 2.
A consumer°s transportation cost is
1
4
°
(
distance)
2
.
Consumer
x
is indi/erence to buy from ±rm
A
or ±rm
B
if
p
A
+
1
4
x
2

{z
}
cost to buy from
A
=
p
B
+
1
4
(
2
²
x
)
2

{z
}
cost to buy from
B
See Figure 15.5
Solve
x
=
1
+
p
B
²
p
A
.
Q. Wen
(UW)
Econ 400
University of Washington
36 / 52
Chapter 15 Imperfect Competition
Example 15.5
Hotelling°s Beach Model
Demand function for ±rm
A
:
q
A
(
p
A
,
p
B
) =
x
=
1
²
p
A
+
p
B
.
Demand function for ±rm
B
:
q
B
(
p
A
,
p
B
) =
2
²
x
=
1
+
p
A
²
p
B
.
If two ±rms chooses prices simultaneously (Bertrand competition)
max
p
A
(
1
²
p
A
+
p
B
)
p
A
max
p
B
(
1
+
p
A
²
p
B
)
p
B
BR
A
(
p
B
) =
1
+
p
B
2
BR
B
(
p
A
) =
1
+
p
A
2
Nash equilibrium:
p
±
A
=
p
±
B
=
1
.
Firms°pro±t:
π
±
A
=
π
±
B
= (
1
²
1
+
1
)
°
1
=
1
.
Q. Wen
(UW)
Econ 400
University of Washington
37 / 52
Chapter 15 Imperfect Competition
Cournot model:
simultaneous quantitysetting game
There are 2 ±rms, each has a constant MC
c
³
0.
Denote ±rm
i
°s quantity output by
q
i
³
0.
Let the
inverse
market demand function be
p
=
a
²
bQ
=
a
²
b
(
q
1
+
q
2
)
.
Firm
i
°s pro±t/payo/ function is
π
i
(
q
i
,
q
j
) = [
a
²
b
(
q
1
+
q
2
)]
q
i
²
cq
i
,
which is continuous.
To ±nd ±rm
i
°s best response function,
max
q
i
[
a
²
b
(
q
1
+
q
2
)]
q
i
²
cq
i
.
Find ±rm
i
°s best response function:
BR
i
(
q
j
) =
a
²
c
2
b
²
1
2
q
j
.
Q. Wen
(UW)
Econ 400
University of Washington
38 / 52
Chapter 15 Imperfect Competition
Cournot model:
simultaneous quantitysetting game
To ±nd a Nash equilibrium
(
q
±
1
,
q
±
2
)
,
set
q
1
=
BR
1
(
q
2
)
and
q
2
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 Fall '08
 Ellis,G
 Microeconomics, Game Theory, Supply And Demand, general equilibrium, Wen