Suppose a market exists where there are only two firms. Assume that the firms (called A and B) are symmetric, which means that they have identical costs. Assume also that they produce a homogeneous (identical) product. The market demand and firm A and B's costs are:

Market Demand:

P = 100 - qA - qB

(P = market price, q = output)

There are two possibilities: (1) they maximize their own profits, or (2) they maximize their collective (joint) profits. We'll consider both.

1. If each firm maximizes its own profits.

This is saying that the firms are engaged in (typical) competitive behavior. To maximize their own profits, the firms must produce where their own marginal revenue is equal to their marginal cost. Each firm's MR and MC are:

Firm A's MR	MRA = 100 - 2qA - qB
Firm B's MR	MRB = 100 - qA - 2qB
Firm A's MC and AC	MCA = ACA = 20
Firm B's MC and AC	MCB = ACB = 20

A. Find equilibrium output for each firm.

	Firm A	Firm B
1. Set MR = MC	100 - 2qA - qB = 20	100 - qA - 2qB = 20
2. Solve for (own) q	qA = 40 - 0.5qB	qB = 40 - 0.5qA
3. Plug qA equation into qB equation		qB = 40 - 0.5(40 - 0.5qB)
4. Solve for qB*		qB* = 80/3
5. Plug qB* into qA equation	qA = 40 - 0.5(80/3)
6. Solve for qA*	qA* = 80/3

B. Find the equilibrium market price.

1. Plug qA* and qB* into the market demand equation	P = 100 - (80/3) - (80/3)
2. Solve for P*	P* = $140/3

C. Find each firm's equilibrium profits.

	Firm A	Firm B
1. Find (P - AC)q* for each firm	((140/3) - 20)(80/3)	((140/3) - 20)(80/3)
2. Calculate profits	pA* = $711.11	pB* = $711.11

Summary: the firms each produce 80/3 units, charging a common price of $140/3 and making $711.11 in profits.

2. If the two firms maximize their collective profits.

This is says that the firms are colluding and may have formed a cartel. It is as though the firms merged into a single firm (forming a monopoly). The cartel sets the (market) marginal revenue equal to their marginal cost. The cartel's MR and MC are:

Cartel's MR	MRC = 100 - 2Q
Cartel's MC and AC	MCC = ACC = 20

A. Find equilibrium output and price for the cartel.

	Firm A
1. Set MR = MC	100 - 2Q = 20
2. Solve for Q*	Q* = 40
5. Plug Q* into market demand	P = 100 - (40)
6. Solve for P*	P* = $60

B. Find each firm's equilibrium profits.

	Firm A	Firm B
1. Find each firm's output share	qA* = 40/2 = 20	qB* = 40/2 = 20
1. Find (P - AC)q* for each firm	(60 - 20)20	(60 - 20)20
2. Calculate profits	pA* = $800	pB* = $800

Summary: the firms each produce 20 units, charge a common price of $40 and make $800 in profits. With a lack of competition, we see prices rising and output falling.

If the firms operate independently, maximizing their own profits, they make profits of $711.11. However, if they collude, then they can make profits of $800. Direct collusion is illegal in the United States, but even if it were not there are reasons why many cartels would not remain intact for long.

3. Collusion and the incentive to cheat

When firms form an agreement to fix prices (i.e. collude) there is always some incentive to cheat on the agreement. To understand why, consider the following.

Suppose firm A decides to increase their output slightly, to 80/3 units (i.e. the amount they would produce if maximizing their own profits). This would have no effect on their own AC, but would change the market price. Firm A would produce 80/3 units while B would continue producing their share of the cartel output, so the price would be: P = 100 - (80/3) - 20 = 160/3.

Profits for firm A become: p_A* = ((160/3) - 20)(80/3) = $888.89

While firm B's profits are: p_B* = ((160/3) - 20)(20) = $666.67

Firm A obviously benefits, at firm B's expense, from cheating on the collusive agreement. Of course, once firm B realizes that firm A is doing this, then firm B should increase their output too. The result is that the cartel falls apart and both firms end up producing their original (competitive) output levels. This problem illustrates a classic conflict between maximizing one's own welfare - in this case, profits - vs. that of the group.