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Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C, = 60Ql and C2 = 60Q2″ where Ql is the output of Firm 1 and Q2 the output of Firm 2. Price is determined by the following demand curve:
P =300- Q
where Q = Ql + Q2
a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.
b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm”s profit.
c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm L”s profit differ from that found in part (b) above?
d. Returning to the duopoly of part (b),suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm”s profits?
a. p1 = P Q1 – C1 = (300 – Q1 – Q2 )Q1 – 60Q1 = 300Q1 – Q1^2 – Q1 Q2 – 60Q1 p2 = P Q2 – C2 = (300 – Q1 – Q2 )Q2 – 60Q2 = 300Q2 – Q1 Q2 – Q2^2-60Q2 Take the FOCs: ?p/(?Q1)= 300 – 2Q1 – Q2 = 0 ? Q1 = 120 – 0.5Q2 ?p/(?Q2)= 300 – Q1 – 2Q2 = 0 ? Q2 = 120 – 0.5Q1 Q1 = 120 – 0.5[120 – 0.5Q1 ] = 60 – 0.25Q1 ? Q1 = 80 Similarly ?nd Q2 = 80 such that p1 = p2 = 6, 400. b. The two ?rms act as a monopolist, where each ?rm produces an equal share of total output. Demand is given by P = 300 – Q, M R = 300 – 2Q, and M C = 60. Set M C = M R to?nd that Q = 120 and Q1 = Q2 = 60, respectively. Therefore: p1 = p2 = 180 60 – 60 60 = 7, 200. c.f Firm 1 were the only firm, it would produce where marginal…

nue is equal to marginal cost, as found in part (b). In this case Firm 1 would produce the entire 120 units of output and earn a profit of $14,400. d. Firm 2 knows that Q1 = 60 and given the reaction function derived in part (a) ?rm 2 sets Q2 = 120 – 0.5 60 = 90. Overall, QT = 150 and P = 300 – 150 = 150. Hence: p1 = 150 60 – 60 60 = 5, 400 p2 = 150 90 – 60 90 = 8, 100. Firm 2 increases its profits at the expense of Firm 1 by cheating on the agreement.

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