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Two identical firms compete as a Cournot duopoly. The inverse market demand they face is P...

Two identical firms compete as a Cournot duopoly. The inverse market demand they face is P = 128 - 4Q. The cost function for each firm is C(Q) = 8Q. The price charged in this market will be

a. $32.

b. $48.

c. $12.

d. $56.

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Answer #1

b. $48.

(Q = Q1 + Q2 where Q1 is firm 1's output and Q2 ia firm 2's output.
So, P = 128 - 4(Q1+Q2) = 128 - 4Q1 - 4Q2
MC = dC/dQ = 8

Each firm maximizes profit where MR = MC.

Firm 1: TR1 = P*Q1 = (128 - 4Q1 - 4Q2)Q1 = 128Q1 - 4Q12 - 4Q2Q1
So, MR1 = d(TR1)/dQ1 = 128 - 2(4Q1) - 4Q2 = 128 - 8Q1 - 4Q2
So, MR1 = MC gives,
128 - 8Q1 - 4Q2 = 8
So, 8Q1 = 128 - 4Q2 - 8
So, Q1 = (120/8) - (4Q2/8)
So, Q1 = 15 - 0.5Q2

Using symmetry, Q2 = 15 - 0.5Q1 = 15 - 0.5(15 - 0.5Q2) = 15 - 7.5 + 0.25Q2
So, Q2 - 0.25Q2 = 0.75Q2 = 7.5
So, Q2 = 7.5/.75 = 10
So, Q1 = 10

P = 128 - 4(Q1+Q2) = 128 - 4(10+10) = 128 - 80 = 48)

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