© 2012  Steven Tschantz
Math 256  Assignment 3
Duopoly
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1/24/12
Problem
Two firms sell similar products in competition, setting prices to attract consumers. Consumers don't view these
products as exactly equivalent, but if the price of one product goes up, some who would buy that product instead
switch to the other. The firms do not cooperate in setting prices, they set price to do the best for themselves. What
prices do they set? What would happen to prices if the firms merge?
Model
Economists describe this as Bertrand competition between firms selling differentiated products. The products are
(partial) substitutes. The assumption is that firms set prices independently, without cooperation or coordination, each
in order to maximize their own profits.
The competition can be thought of as a game. Each player makes a choice of strategy (price) and the result for each
player is the payoff they get (the profit they make). Players make their choices simultaneously without communicating
or negotiating. Each player will choose a strategy to maximize its payoff knowing that the other player will choose its
strategy to maximize its payoff. If player
A
were to choose strategy
a
1
say, and player
B
knows this, then player
B
might want strategy
b
2
to maximize its payoff. But if player
A
knows
B
will choose strategy
b
2
, then
A
may want
a
3
as
its strategy. Then
B
might prefer strategy
b
4
, and back and forth endlessly. What we want is a strategy
a
for
A
for
which the best
B
can do is a strategy
b
and such that the best
A
can do in response to
b
is the original
a
. A pair of
strategies
H
a
,
b
L
satisfying this condition is called a Nash equilibrium of the game and is thought of as a solution for the
game.
What we assume then is that each firm takes the competition's price as a given, looks at the demand for its own product
as a function of its own price, and sets the price that maximize its own profits. As before, to determine a firm's profit
maximizing price we need to know the demand it faces and its costs. The cost of each firm can reasonably be assumed
to be a function of the quantity it produces and sells. However, the demand for each firm's product is a function of the
prices of both products.
Let the firms, and their corresponding products, be denoted by 1 and 2, with prices
p
1
and
p
2
and quantities
q
1
and
q
2
.
Suppose firms face costs
C
1
H
q
1
L
and
C
2
H
q
2
L
. Assume demands are specified by functions
q
1
=
q
1
H
p
1
,
p
2
L
and
q
2
=
q
2
H
p
1
,
p
2
L
. Then the profit for firm 1 is
P
1
H
p
1
,
p
2
L
=
p
1
q
1
H
p
1
,
p
2
L

C
1
H
q
1
H
p
1
,
p
2
LL
and similarly for the profit of 2,
P
2
H
p
1
,
p
2
L
.
When firm 1 sets price, it takes
p
2
as given. It's maximum profit should be at a critical point of
P
1
taken as a function
of
p
1
. The first order condition on firm 1's maximum profit is then
0
=
¶P
1
¶
p
1
=
q
1
H
p
1
,
p
2
L
+
H
p
1

mc
1
L
¶
q
1
H
p
1
,
p
2
L
¶
p
1
(1)
where mc
1
=
C
1
'
H
q
1
H
p
1
,
p
2
LL
is the marginal cost for product 1. If
q
1
doesn't depend on
p
2
this reduces to finding the
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optimal price for a monopolist. At the same time, firm 2 takes
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 Spring '11
 Schantz
 Math, Supply And Demand, p1, P2i

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