Two firms, 1 and 2, sell an identical product and engage in Bertrand competition
ID: 1107175 • Letter: T
Question
Two firms, 1 and 2, sell an identical product and engage in Bertrand competition where each firm simultaneously chooses a single price. Total demand for the product
is q(p) = 60 p. Buyers purchase from whichever firm has the lower price. However, assume that buyers like the owner of firm 1 more, so if prices are identical then all purchase from firm 1. Each firm has a cost function given by C(q) = 10q. However, for some strange reason, firms can only set integer prices.
(a) Derive each firm’s best response function.
(b) Solve for all Bertrand equilibria.
Explanation / Answer
The Bertrand model examines price competition among firms that produce differentiated products but highly substitutable products.
C = 10q
MC = 10
Total demand: q = 60 – p
q = 60 – (p1 + p2) or (60 – p1 – p2).
Firm 1’s total revenue function = p1*q
TR1 = p1*(60-p1-p2) = 60p1 – p1^2 – p1p2
MR1 = 60 – 2p1 – p2
60 – 2p1 – p2 = 10
p2 = 50 – 2p1 (Reaction function of firm 2).
Repeating the steps to find reaction function of p1:
TR2 = p2*(60 – p1 – p2)
TR2 = 60p2 – p1p2 – p2^2
MR2 = 60 – p1 – 2p2
60 – p1 – 2p2 = 10
p1 = 50 – 2p2 (Reaction function of firm 1).
Substituting firm 2’s reaction function into firm 1’s reaction function:
p1 = 50 – 2(50 – 2p1)
p1 = 50 – 100 + 4p1
p1 = $50/3 or $17
p2 = 50 – 2(50/3)
P = 50 – 100/3
p2 = $50/3 or $17
q = 60 – (17 + 17)
q = 24 units.
(a) Best Response function of firm 1: p1 = 50 – 2p2.
Best Response function of firm 2: p2 = 50 – 2p1.
(b) Bertrand’s Eqilibiuria:
p1 = $27. p2 = $27. q = 24.
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