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# profit level associated with the Cournot quantity

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Suppose the market demand curve in an industry is characterized by P=1-Q, where P is the market price and Q is the total quantity supplied to the market. Suppose there are three firms in this industry. All have a zero marginal cost of production.

1. In a one-shot interaction, what is the Cournot quantity for each firm?
2. What is the profit level associated with the Cournot quantity for each firm?

Now consider an infinitely repeated version of the above game. Suppose that the firms agree to act collectively as a monopolist. All firms have the same discount factor .

3. How much will each individual firm produce under the cartel arrangement?
4. How much profit will each individual firm make under the cartel arrangement?
5. Suppose a firm decided to cheat on the cartel arrangement. How much would it produce i.e. what is the best possible deviation it could make? How much profit would it get from making this deviation?
6. How big does the discount factor have to be in order to support the cartel arrangement through the threat of playing Cournot forever as punishment for deviation?
7. Compare this with the two firm case. Is the delta that supports the two-firm case higher or lower than the three-firm case i.e. is it easier to support a cartel with two firms or with three firms? Explain.

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#### Solution Preview

Please see the attached file.

Suppose the market demand curve in an industry is characterized by P=1-Q, where P is the market price and Q is the total quantity supplied to the market. Suppose there are three firms in this industry. All have a zero marginal cost of production.

1. In a one-shot interaction, what is the Cournot quantity for each firm?
2. What is the profit level associated with the Cournot quantity for each firm?

Now consider an infinitely repeated version of the above game. Suppose that the firms agree to act collectively as a monopolist. All firms have the same discount factor .

3. How much will each individual firm produce under the cartel arrangement?
4. How much profit will each individual firm make under the cartel arrangement?
5. Suppose a firm decided to cheat on the cartel arrangement. How much would it produce i.e. what is the best possible deviation it could make? How much profit would it get from making this deviation?
6. How big does the discount factor have to be in order to support the cartel arrangement through the threat of playing Cournot forever as punishment for deviation?
7. Compare this with the two firm case. Is the delta that supports the two-firm case higher or lower than the three-firm case i.e. is it easier to support a cartel with two firms or with three firms? Explain.

Question 1
Let's call Q1, Q2 and Q3 to the quantities produced by each of the firms. We get that the ...

#### Solution Summary

The profit level associated with the Cournot quantity is determined.

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## Cournot equilibrium outputs

HOW MUCH ?

1. Suppose two rival firms face the following industry demand curve, and respective cost curves:

P = 500 - q1 - q2 (Market demand)

TC1 = 50q1 and TC2 = 50q2 (Respective total cost curves for firm 1 and 2).

MC1 = 50 and MC2 = 50 (Respective marginal cost curves for firm 1 and 2)

a.) Find the Cournot equilibrium outputs for each firm.
b.) Find the industry price that is associated with these output levels.
c.) Find the profit level for each firm and for the industry.

2.) Suppose firm-1 is the first-mover and considers firm-2's reaction when determining its output.

a.) Find the new set of equilibrium outputs for these two rivals.
b.) Find the industry price that is associated with these output levels.
c.) Find the new profit level for each firm and for the industry.

3.) Suppose these firms choose to compete

a.) Find the new set of equilibrium outputs for these two rivals.
b.) Find the industry price that is associated with these output levels.
c.) Find the new profit level for each firm and for the industry.

4.) Suppose these firms are able to successfully collude

a.) Find the new set of equilibrium outputs for these two rivals.
b.) Find the industry price that is associated with these output levels.
c.) Find the new profit level for each firm and for the industry.
d.) What are the implications of the differing firm profit levels derived from using these three different models? For instance, why would firm-1 receive higher profit when it behaves as a first-mover as opposed to behaving in the standard Cournot manner?

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