intercept of the two bestanswer curves.
In this point, neither of the companies has an
incentive to change strategy unilaterally. At
x
*
B
the optimum supply quantity of A is
x
*
A
and vice versa.
If we place the reaction function (72’) in the reaction function (71)
′
we get:
x
A
=
90
2

90

x
A
4
(73)
=
90 +
x
A
4
(74)
This we get the equilibrium supply quantity of Company A
x
A
= 30
. Placing this in the
reaction function of Company B then produces the supply quantity
x
B
= 30
. The identical
supply quantities
x
*
A
=
x
*
B
are due to the symmetrical assumptions made above.
Price duopoly with differentiated goods
We again observe a market with two companies, A and B (symmetrical duopoly). Now,
however, the suppliers produce differentiated goods, which are substitutes. Accordingly,
187
the demand function for the goods of Company A (B) is
x
A
= 100

p
A
+
p
B
(
x
B
=
100

p
B
+
p
A
) and marginal costs are constant at 10. This time the strategic decision
variable is the price.
The profit functions with differentiated goods are:
P
i
=
(
p
i

10)
x
i
(75)
=
(
p
i

10)(100

p
i
+
p
j
)
i, j
=
A, B
i
̸
=
j
with first order conditions:
∂P
i
∂p
i
=
100

p
i
+
p
j

(
p
i

10)
(76)
=
100

2
p
i
+
p
j
+ 10
!
= 0
From the FOCs the reaction functions follow:
p
i
=
100 +
p
j
+ 10
2
(77)
and with that
p
A
=
100 +
p
B
+ 10
2
(78)
p
B
=
100 +
p
A
+ 10
2
(79)
The reaction function (79) placed in (78) yields the equilibrium price:
p
A
=
100 +
(100+
p
A
+10)
2
+ 10
2
=
330
3
= 110
(80)
Due to the symmetry properties the result
p
A
=
p
B
can be derived directly.
188
p
*
A
p
*
B

6
p
B
p
A
.
.
p
A
(
p
B
)
p
B
(
p
A
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.................
..............
...
.............
...
.............
...
..............
..
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
.
.
.
.
...
Graphical illustration:
Again the Nash equilibrium is given at the intercept of the two reaction curves. Neither
of the two duopolists has an incentive to move away from the price combination
(
p
*
A
, p
*
B
)
.
From A’s perspective the price
p
*
A
is optimal given B’s optimal price (
p
*
B
).
From B’s
perspective the price
p
*
B
is optimal given A’s optimal price (
p
*
A
).
Price duopoly with homogenous goods
Again there is a simultaneous determination of the prices of both companies.
The so
called Bertrand duopoly delivers important knowledge that can often be very helpful as an
approximation of the actual market situation. With homogenous goods, the company with
the higher price does not receive any demand, even if the prices differ only marginally.
Proposition:
In the Nash equilibrium both suppliers set the same price, which corresponds
to the marginal costs
c
:
p
A
=
p
B
=
c
.
Intuition:
With a market price that lies above the marginal costs, it is always worthwhile to underbid,
so that the entire demand can be exploited. Ultimately, the consumer does not care about
who sells him the homogeneous good. Only at
p
A
=
p
B
=
c
does no company have any
incentive to change strategy.
189
More detailed evidence:
At
e
p
A
>
e
p
B
> c
there are absolutely no sales and zero profit for Company A:
x
A
= 0
and
G
A
= 0
. In undercutting the price of Company B by a very small amount
ϵ
(
p
A
=
e
p
B

ϵ >
c
) Company A can achieve a positive profit. For this reason, the original price
e
p
A
cannot
have been optimal.
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