Question 2 (20 marks) Consider a homogeneous product duopoly in which the (inver

Question

Question 2 (20 marks)

Consider a homogeneous product duopoly in which the (inverse) market demand is given by p = 30 – (q1 + q2): q1, q2 are the outputs produced by firms 1 and 2 respectively. Each firm has constant marginal and average cost equal to zero.

(a) Find the Bertrand equilibrium prices, quantities and profits for firms 1 and 2. Is this a Nash equilibrium? (6 marks)

(b) Find the Cournot equilibrium prices, quantities and profits for firms 1 and 2. Is this a Nash equilibrium? (7 marks)

(c) Explain, in the context of either Bertrand or Cournot oligopolists, why they may want to collude. Why is such a collusive agreement unlikely to be sustained? (7 marks)

Explanation / Answer

Note that we have a homogeneous product duopoly with market demand is given by p = 30 – (q1 + q2) and a constant marginal and average cost = 0. Here P = 30 - Q or Q = 30 - P can be the demand function form

(a) Under homogeneous product duopoly with same market demand and zero marginal cost, Bertrand equilibrium has maximized revenue for both duopolist

TR1 = 30P1 - P1^2 - P1P2.............. MR1 = 30 - 2P1 - P2 = 0

TR2 = 30P2 - P1P2 - P2^2...............MR2 = 30 - 2P2 - P1 = 0

Solve them to get P1 = P2. Since MR1 = 0 at equilibrium, we have 30 - 2P1 - P1 = 0 or P1 = P2 = Common price = 10. Q1 = 10 units each Profit = 100 each.

There is an incentive to deviate because a lower price would attract all consumers and that firm will all the market. Hence it is not a Nash equilibrium

(b) Cournot equilibrium prices, quantities and profits for firms 1 and 2 are:

q1* = q2* = q = (30 - 0)/3 = 10 units each and price = $10. Profits = $100.

(c) Explain, in the context of either Bertrand or Cournot oligopolists, they may want to collude as this will provide them greater market power and profit.

MR = 0

30 - 2Q = 0

Q* = 15 and price = 15. Profits are 15*15 = 225 and separately profits are 112.50 to each firm. Hence higher profits encourage them to develop a monopoly. But this is not sustainly as there is an incentive to cheat and produce slightly more at the same price and earn even higher profit.

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