Suppose that two identical firms produce widgets and that they are the only firms in the market. their costs are given by c1 = 60q1 and c2 = 60q2, where q1 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 = q1 + 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 1’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?

Consider the following calculations

Explanation:

a. π1 = P Q1 − C1 = (300 − Q1 − Q2 )Q1 − 60Q1 = 300Q1 − Q1^2 − Q1 Q2 − 60Q1

π2 = P Q2 − C2 = (300 − Q1 − Q2 )Q2 − 60Q2 = 300Q2 − Q1 Q2 − Q2^2-60Q2

Take the FOCs:

∂π/(∂Q1)= 300 − 2Q1 − Q2 = 0 ⇒ Q1 = 120 − 0.5Q2

∂π/(∂Q2)= 300 − Q1 − 2Q2 = 0 ⇒ Q2 = 120 − 0.5Q1

Q1 = 120 − 0.5[120 − 0.5Q1 ] = 60 − 0.25Q1 ⇒ Q1 = 80

Similarly find Q2 = 80 such that π1 = π2 = 6, 400.

b. The two firms act as a monopolist, where each firm 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 tofind that Q = 120 and Q1 = Q2 = 60, respectively. Therefore:

π1 = π2 = 180 × 60 − 60 × 60 = 7, 200.

c. It would be higher because they could make more money.

d. Firm 2 knows that Q1 = 60 and given the reaction function derived in part (a) firm 2 sets Q2 = 120 − 0.5 × 60 = 90. Overall, QT = 150 and P = 300 − 150 = 150. Hence:

π1 = 150 × 60 − 60 × 60 = 5, 400

π2 = 150 × 90 − 60 × 90 = 8, 100.

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