Question

Consider a market with 2 identical firms (a and b). The market demand is P = 14 - Q

where Q = Qa + Qb. For both firms AC=MC= 2.

A. Solve for the Cournot-Nash reaction functions of each firm.

B. Solve for the Cournot- Nash equilibrium. Solve for Q, Qa, Qb, Price, and each firms profit.

C. Compare the Cournot-Nash equilibrium with perfect competition, and monopoly (you can refer to your results from question 2, if you’ve already done it)

Answer 1

Market demand is P = 14 - Qa - Qb.

a) For firm 1, reaction function is arrived at MR = MC. This gives 14 - 2Qa - Qb = 2 and so the reaction function of firm A is Qa = 6 - 0.5Qb. Since MC is same for both, the reaction function of firm B is Qb = 6 - 0.5Qa.

b) Solve the reaction functions to get Qa = 6 - 0.5*(6 - 0.5Qa) or Qa = 3 + 0.25Qa. This gives Qa = Qb = 4 units and Q = 8 units. Price P = 14 - 8 = $6 per unit, and profit for each firm is Pr1 = Pr2 = (6 - 2)*4 = $16

c) In perfect competition, both firm charges P = MC = $2 and so they produce 14 - 2 = 12 units in total and Qa = Qb = 6 units. Profits are zero. Monopoly has MR = MC or 14 - 2Q = 2 which gives Q = 6 units and Qa = Qb = 3 units. Price is 14 - 6 = $8 per unit and Profits = (8 - 2)*6 = $36 for the market and $18 for each firm.