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Game Theory
Solutions to Exercises:
Models of Duopoly
JanJaap Oosterwijk
Fall 2007
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View Full Document3 Models of Duopoly
3.5.1
(a)
Suppose in the Cournot model that the ﬁrms have diﬀerent production costs. Let
c
1
and
c
2
be the costs of production per unit for ﬁrms 1 and 2 respectively, where both
c
1
and
c
2
are assumed less than
a/
2
. Find the Cournot equilibrium.
Solution:
The price function stays the same. The only thing that changes is the payoﬀ function for
each of the two players:
±
u
1
(
q
1
,q
2
) =
q
1
P
(
q
1
+
q
2
)

c
1
q
1
=
q
1
(
a

q
1

q
2
)
+

c
1
q
1
u
2
(
q
1
,q
2
) =
q
2
P
(
q
1
+
q
2
)

c
2
q
2
=
q
2
(
a

q
1

q
2
)
+

c
2
q
2
.
Setting the partial derivates to zero,
(
∂
∂q
1
u
1
(
q
1
,q
2
) =
a

2
q
1

q
2

c
1
= 0
∂
∂q
2
u
2
(
q
1
,q
2
) =
a

q
1

2
q
2

c
2
= 0
,
gives
±
q
*
2
=
a

2
q
*
1

c
1
q
*
1
=
a

2
q
*
2

c
2
,
i.e.
q
*
1
=
a

2(
a

2
q
*
1

c
1
)

c
2
=

a
+ 4
q
*
1
+ 2
c
1

c
2
. Therefore,
q
*
1
=
a
+
c
2

2
c
1
3
=
(
a

c
1
) + (
c
2

c
1
)
3
,
and because of symmetry,
q
*
2
=
a
+
c
1

2
c
2
3
=
(
a

c
2
) + (
c
1

c
2
)
3
.
The payoﬀ that Player I receives from this Cournot equilibrium is
u
1
(
q
*
1
,q
*
2
) =
q
*
1
(
a

q
*
1

q
*
2
)
+

c
1
q
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