Question

Use the information below to answer the folloiwng question: Inverse demand function:

P = 300 – 0.5Q

Marginal revenue: MR = 300 – Q

Total Cost function: C = 4000 + 90Q

Marginal Cost: MC = 90

The equilibrium P and Q under Duopoly are: P = 160, Q= 280 with each firm's output = 110

Select one: True False

Answer 1

False.

P = 160, Q = 280 with each firm's output = 140 (= 280/2).

P = 300 - 0.5Q1 - 0.5Q2 [since Q = Q1 + Q2]

For firm 1,

TR1 = P x Q1 = 300Q1 - 0.5Q12 - 0.5Q1Q2

MR1 = TR1/Q1 = 300 - Q1 - 0.5Q2

Setting MR1 = MC,

300 - Q1 - 0.5Q2 = 90

Q1 + 0.5Q2 = 210..........(1) [reaction function, firm 1]

For firm 2,

TR2 = P x Q2 = 300Q2 - 0.5Q1Q2 - 0.5Q22

MR2 = TR2/Q2 = 300 - 0.5Q1 - Q2

Setting MR2 = MC,

300 - 0.5Q1 - Q2 = 90

0.5Q1 + Q2 = 210..........(2) [reaction function, firm 2]

Equilibrium is achieved by solving (1) and (2). Multiplying (2) by 2,

Q1 + 2Q2 = 420............(3)

Q1 + 0.5Q2 = 210............(1)

(3) - (1) yields:

1.5Q2 = 210

Q2 = 140

Q1 = 420 - 2Q2 [from (3)] = 420 - 2 x 140 = 420 - 280 = 140

Q = 140 + 140 = 280

P = 300 - 0.5 x 280 = 300 - 140 = 160