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Unformatted text preview: 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 p1 and p2 and quantities q1 and q2 .
Suppose firms face costs C1 q1 and C2 q2 . Assume demands are specified by functions q1 q1 p1 , p2 and
q2 q2 p1 , p2 . Then the profit for firm 1 is
p1 q1 p1 , p2 C1 q1 p1 , p2
1 p1 , p2
and similarly for the profit of 2, 2 p1 , p2 .
When firm 1 sets price, it takes p2 as given. It's maximum profit should be at a critical point of
of p1 . The first order condition on firm 1's maximum profit is then
0 1 p1 q1 p1 , p2 p1 mc1 q1 p1 , p2
p1 1 taken as a function (1) '
where mc1 C1 q1 p1 , p2 is the marginal cost for product 1. If q1 doesn't depend on p2 this reduces to finding the
optimal price for a monopolist. At the same time, firm 2 takes p1 as given and maximizes 2 with respect to p2 .
Similarly then, 0 2 p2 q2 p1 , p2 p2 mc2 q2 p1 , p2
p2 (2) '
with mc2 C2 q2 p1 , p2 the marginal cost of product 2. These two equations can be solved simultaneously for the
prices p1 and p2 such that 1 can do no better than set price at p1 when 2 sets its price at p2 , and at the same time 2 can 2 When firm 1 sets price, it takes p2 as given. It's maximum profit should be at a critical point of
of p1 . The first order condition on firm 1's maximum profit is then m256hw03soln.nb
1 0 q1 p1 , p2 p1 p1 mc1 q1 p1 , p2
p1 1 taken as a function (1) '
where mc1 C1 q1 p1 , p2 is the marginal cost for product 1. If q1 doesn't depend on p2 this reduces to finding the
optimal price for a monopolist. At the same time, firm 2 takes p1 as given and maximizes 2 with respect to p2 .
Similarly then,
2 0 q2 p1 , p2 p2 p2 mc2 q2 p1 , p2
p2 (2) '
with mc2 C2 q2 p1 , p2 the marginal cost of product 2. These two equations can be solved simultaneously for the
prices p1 and p2 such that 1 can do no better than set price at p1 when 2 sets its price at p2 , and at the same time 2 can
do no better than set price p2 when 1 sets its price at p1 . Since neither firm wants to change price...
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This note was uploaded on 07/12/2013 for the course MATH 256 taught by Professor Schantz during the Spring '11 term at Vanderbilt.
 Spring '11
 Schantz
 Math

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