problem code : 220609MICRO

[vishruti]

Consider two firms. The demand faced by the two firms is Q1 = a – 2P1 + P2 and Q2 = a + P1 - 2P2. Assume zero marginal cost. Find the cournot equilibrium price, quantity and profit for both firms. What is the relationship between the commodities produced by two firms?

[Smriti]

q1=q2=2a/3
p1=p2=a/3
profit for both firms=2a^2/9
the two goods are substitutes

[vishruti]

The goods are substitutes because dQ1/dP2 = 1 > 0 & dQ2/dP1 = 1 > 0

The answer given by smriti is correct for the bertrand market model but INCORRECT for the cournot model and the questions asks cournot equilibrium price , quantity and profit. Please note that we maximize profit wrt quatity in cournot model and NOT wrt price . Firms choose the level of output in cournot market model and prices are set in bertrand model

The answer is Q1 = Q2 = 3a/5 , P1 = P2 = 2a/5 , profit for both firms is PQ = 6a^2/25 because the question assumes 0 marginal cost

NOTE: In order to find the cournot equilibrium , it is required to re-calulate demand function for both firms such that P = f ( Q ). This can be done by substituting P2 from firm2's demand function into the demand function of firm1. This gives P1 = 1/3 [ 3a - Q2 - 2Q1 ]. Similary find demand function of firm 2 and this is reduced to profit maximization problem wrt Q