Consider a duopoly with two firms, Firm 1 and Firm 2, who simultaneously choose their own quantity of output. For simplicity, each firm has only two output quantities to choose from: Low and High. The payoff of each firm is its profit. The interaction between the two firms is modeled in the following strategic-form game:
Firm 1
Low High
Firm 2
Low 5,7 0,8
High 0,8 3,3
a) Find the Nash equilibrium (there is only one).
b) Find the cooperative outcome. Explain why this is not strategically stable.
Now, suppose that instead of competing with each other only once, the two firms share the market in the way modeled above indefinitely. After each period, both firms can tell what occurred in the game in the previous period: Each one discounts the future using a discount factor denoted by δ and γ, for Firm 1 and Firm 2, respectively. Moreover, suppose that both firms use the grim-trigger strategy of cooperating as long as the other firm has cooperated, and playing the Nash equilibrium strategy forever if it has not.
c) What is the smallest value of δ that makes cooperation in every period an equilibrium choice for Firm 1? You must prove your answer.
d) What is the smallest value of γ that makes cooperation in every period an equilibrium choice for Firm 2? You must prove your answer.
e) Explain any difference in your answer to parts c) and d).
f) An employee at the Federal Trade Commission, Ignatius Quale - commonly known as IQ - comes up with a complicated new policy that would lower the profit for each firm when both produce a high quantity of output from 3 to 2. Since it makes the market more competitive, IQ argues, his policy must be good for social welfare. Is he right or is he wrong? (You don't need to answer this question)
