4 market outcome is e cient third degree price

price discrimination (Figure 14.4),
market outcome is e¢ cient!
Third-degree 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
) .
Second-degree 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 price-setting/price-competition 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 price-setting 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 quantity-setting 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 quantity-setting game
To ±nd a Nash equilibrium
(
q
±
1
, q
±
2
)
,
set q
1
= BR
1
( q
2
) and q
2