Question

Managers at Firm A and Firm B must make pricing decisions simultaneously. The following demand and long-run cost conditions are common knowledge to the managers:

Qa = 72 – 4Pa + 4 Pb                LACa = LMCa = 2

Qb = 100 – 3Pb + 4 Pa              LACb = LMC b = 6.67

2.1. Derive the best response curve for firm A

2.2. Derive the best response curve for firm B.

2.3. What will be the prices charged by firms A an B?

Answer 1

Profit is maximized when marginal profit is zero

a) For firm A, we have profit = TR - TC = QaPa - ACaQa

= (72 – 4Pa + 4 Pb)Pa - 2(72 – 4Pa + 4 Pb)

= 72Pa - 4Pa^2 + 4PaPb - 144 + 8Pa - 8Pb

= 80Pa - 4Pa^2 + 4PaPb - 8Pb + 144

Marginal profit = 0

80 - 8Pa + 4Pb = 0

2Pa - Pb = 20

Pa = 10 + 0.5Pb..............This is the best response curve for firm A

b) For firm B we have profit = QbPb - ACbQb

= (100 – 3Pb + 4 Pa)Pb - 6.67(100 – 3Pb + 4 Pa)

= 100Pb - 3Pb^2 + 4PaPb - 66.7 + 20Pb - 26.67Pa

= 120Pb - 3Pb^2 + 4PaPb -26.67Pa - 66.7

Marginal profit = 0

120 - 6Pb + 4Pa = 0

Pb = 20 + 0.66Pa...........This is the  best response curve for firm B.

Solve them

Pb = 20 + 0.66*(10 + 0.5Pb)

Pb = 20 + 6.6 + 0.33Pb

Pb = $40 (approx) and Pa = 10 + 0.5*40 = $30.