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Demand in a market is given by P = 20 Q. The cost function is C = Q2 and marginal cost of production is 2Q.
(a) What are the monopoly output and profits?
(b) Suppose that there are two firms in the market. If they were to agree to maximize industry profits, how much would each produce? What would the profits of each firm and industry profits be?
(c) Why are monopoly profits and collusive profits not the same?
(d) Is the collusive agreement in (b) a Nash equilibrium?
Solution: a)Profit= PQ-cost =Q[20-Q] Q^2 =20Q- Q^2 Q^2 =20Q -2Q^2 Differentiating both sides with respect to Q: Profit/ Q = 20 4Q The FOC for the firm gives us: Profit/ Q=0 20 4Q=0 Or, 4Q=20 Or, Q=5 Therefore, Monopoly Output, Q=5 Price,P= 20-5 =15 Profit = 5[20-5] 5^2 =5*15- 25 =75-25 =50 Profit= PQ-cost =Q[20-Q] Q^2 =20Q- Q^2 Q^2 =20Q -2Q^2 Differentiating both sides with respect to Q: Profit/ Q = 20 4Q The FOC for the firm gives us: Profit/ Q=0 20 4Q=0 Or, 4Q=20 Or, Q=5 Therefore, Monopoly Output, Q=5 Price,P= 20-5 =15 Profit = 5[20-5] 5^2 =5*15- 25 =75-25 =50 b) Profit for firm 1: Profit= PQ-cost =q1[20-(q1+q2)] (q1^2) =20 q1 q1^2 q1 q2 -q1^2 Partially differentiating both sides with respect to q1: Profit/ q1 =20 2q1 q2 -2q1 The FOC for firm 1 gives us: Profit/ q1=0 q1 = 1/4[20- q2]..(1) Again, For firm 2 : Profit= PQ-cost =q2[20-(q1+q2)] (q2^2) =20 q2 q2^2 q1 q2 -q2^2 Partially differentiating both sides with respect to q1: Profit/ q2 =20 2q2 q1 -2q2 The FOC for firm 1 gives us: Profit/ q2=0 q2 = 1/4[20- q1]..(2) Solving the above equations, we get, q1= q2 = 5 and industry output = 5*2 = 10 Therefore,…

fit of each firm: Profit= PQ-cost =5[20-10] (5^2) =5*10-25 =50-25 =25 Profit of the industry = 25*2 =50 c)Monopoly profits and collusive profits are not the same . This is because, in case of monopoly a single firm controls the prices and it is this firm which has control over the price and hence can earn a higher profit. On the other hand, when two firms collude, there are two firms competing for the market share and hence profit too gets divided between the two firms. Firms collude to restrict competition or to fix prices so as to earn monopoly profits. d)The collusion agreement in (b) is a Nash equilibrium In a Nash equilibrium, each firm is doing the best it can given what its competitors are doing and thus ,in a Nash equilibrium, none of the players have an incentive to deviate from their position. In the above collusion agreement too, each firm is doing the best it can given ehat is competitors are doing. Hence it is Nash equilibrium.

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