Question

a) A duopolist faces a market demand curve given by
P = 56-2Q . Each firm can produce output at a constant MC
of $20 per unit. find the equilibrium quantity for the market.

b) A duopolist faces a market demand curve given by
P = 56-2Q . Each firm can produce output at a constant MC of $20 per unit. Firm one is the leading firm and makes the first decision. find the equilibrium quantity for the market.

Answer 1

a) For both firms demand is P = 56-2Q .

There is a constant MC of $20 per unit.

Use Cournot model.

We have demand function P = 56 - 2Q1 - 2Q2.

This gives MR1 = 56 - 4Q1 - 2Q2 and MR2 = 56 - 2Q1 - 4Q2

Use MR = MC

56 - 4Q1 - 2Q2 = 20 and 56 - 2Q1 - 4Q2 = 20

Q1 = 9 - 0.5Q2 and Q2 = 9 - 0.5Q1

Q1 = 9 - 0.5*(9 - 0.5Q1)

Q1 = 4.5 + 0.25Q1

Q1 = 6 units and Q2 = 6 units

Market equilibrium quantity is 12 units

b) Firm one is the leading firm and makes the first decision.

Derivation of firm 2’s reaction function

Total revenue of firm 2 = P*(q2) = (56 – 2(q1 + q2))q2 = 56q2 – 2q2² – 2q1q2

Marginal revenue = 56 – 4q2 – 2q1

Marginal cost = 20

Solve for the reaction function

56 – 4q2 – 2q1 = 20

This gives q2 = 9 - 0.5q1

Incorporate this in the reaction function of firm 1

Total revenue for firm 1 = P*(q1) = (56 – 2(q1 + q2))q1

TR = 56q1 - 2q1^2 - 2q1q2

= 56q1 - 2q1^2 - 2q1*(9 - 0.5q1)

= 56q1 - 2q1^2 - 18q1 + q1^2

= 38q1 - q1^2

MR = MC

38 - 2q1 = 20

q1 = 9 and so q2 = 4.5

Market quantity is 13.5 units