Question

Question 2: 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 and B?

Answer 1

2.1 Profit of A = TRa - TCa = Qa*Pa - LACa*(Qa) = (72 – 4Pa + 4 Pb)Pa - 2(72 – 4Pa + 4 Pb)
= 72Pa – 4Pa2 + 4PbPa - 144 + 8Pa - 8Pb = 80Pa – 4Pa2 + 4PbPa - 144 - 8Pb

So, d(Profit of A)/dPa = 80 - 2(4Pa) + 4Pb = 0
So, 8Pa = 80 + 4Pb
So, Pa = (80/8) + 4(Pb/8)
So, Pa = 10 + 0.5Pb

2.2 Profit of B = TRb - TCb = Qb*Pb - LACb*(Qb) = (100 – 3Pb + 4Pa)Pb - 6.67(100 – 3Pb + 4Pa)
= 100Pb – 3Pb2 + 4PaPb - 667 + 20.01Pb - 26.68Pa = 120.01Pb – 3Pb2 + 4PaPb - 667 - 26.68Pa

So, d(Profit of B)/dPb = 120.01 - 2(3Pb) + 4Pa = 0
So, 6Pb = 120.01 + 4Pa
So, Pb = (120.01/6) + 4(Pa/6)
So, Pb = 20 + 0.67Pa

2.3 Substituting Pa into Pb, we get,
Pb = 20 + 0.67(10 + 0.5Pb) = 20 + 6.7 + 0.335Pb
So, Pb - 0.335Pb = 0.665Pb = 26.7
So, Pb = 26.7/0.665
So, Pb = 40.15

Pa = 10 + 0.5Pb = 10 + 0.5(40.15) = 10 + 20.075
So, Pa = 30.08