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Question 3. Duopolies Dilemma

Consider two firms, 1 and 2, which produce in a duopoly market.

Firm 1 and firm 2 simultaneously choose their respective quantities, q1and q2.

Given these outputs, the market price is given by p = 6q1 q2.

There are zero costs of production.

1. Write the profit functions for firms 1 and 2. Find the optimal quantity of output q1 for firm 1 as a function of q2 , and the optimal quantity of output q2 for firm 2 as a function of q1 (i.e. find each firms best response function).

2. Define the notion of Nash equilibrium in pure strategies for this game. Find the (Cournot or Nash) equilibrium quantities of this duopoly market. Having done this, calculate the equilibrium payoff for each firm.

3. Now find the monopoly output for the market. Assuming that each firm produces half of this output, calculate their profits. This is the joint-profit-maximizing output. If the firms could collude, this would be an obvious pair of outputs for them to pick.

4. How do your answers in 2 and 3 compare? If firm 2 produces half of the monopoly output you find in 3, does firm 1 have incentives to deviate? Calculate the output that maximizes firm 1s profit when firm 2 produces half of the monopoly output? Therefore, are the outputs in 3 going to constitute an equilibrium? 1

5. People often say that repeated interactions facilitate cooperation. Now suppose that the two firms produce and sell their output yearly, for a total of 3 years. Each firm tries to maximize the sum of the profits it earns in the 3 years. Does there exist an equilibrium in which each firm produces half the monopoly output in the first year? Briefly explain your answer. What if the two firms produce and sell yearly, for a total of 100 years?

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