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A Duopoly Example.
Consider an industry with two firms. Firms are identical and produce an
homogenous product. Firms have to select outputs (capacity) in order to maximize
profits. Each firm knows its own total cost of production, the total cost of production of
the competitor and the industry demand.
We analyze two different scenarios:
(i) one–shot scenario, i.e., the life of the industry lasts one period
(ii) repeated scenario, i.e., the life of the industry lasts several periods.
The following data are known by both firms and describe the industry
situation:
1) p = 140 – (Q1+Q2) (industry demand)
2) TC1 = 20Q1 (total cost of firm 1),
3) TC2 = 20Q2 (total cost of firm 2).
Observe that the industry price, equation 1, depends on the output of both firms. This
feature has two implications: a) since the profits of each firm depend on the price, they
depend on the choice of the competitor (strategic interaction), b) in order to establish
profit maximizing decisions, each firm has to guess what the competitor will do.
1. One shot case.
We analyze and compare two different situations. In the first, firms compete
strategically. In order to maximize their profits, they guess and take into account what the
competitor does (Cournot – Nash). In the second, firms collude and coordinate their
actions by forming a Cartel.
1.1. Cournot–Nash Competition.
We know that firms with market power maximize profits by selecting a level of
output at which their marginal revenue is equal to their marginal costs. However, contrary
to what we have seen up to now, in a situation of strategic interaction, the marginal
revenue of a firm depends on the competitors choices.
We divide the analyses in three steps. In the first, we compute the marginal revenue
curves. In the second, the profit maximizing outputs. In the third, equilibrium price and
profits.
Step 1: Compute the marginal revenues for both firms.
Start with firm 1. The total revenues of firm 1, R1, are:
2
R1= pQ1 = (140–(Q1+Q2))Q1 = 140 Q1 – Q12 – Q2Q1.
The marginal revenue of firm 1, MR1, is just the derivative of the total revenues with
respect to Q1. Hence
5) MR1 = 140– 2Q1 – Q2 (Marginal revenue of firm 1).
Similarly, the total revenues of firm 2, R2 are:
R2= pQ2 = (140–(Q1+Q2))Q2 = 140 Q1–Q2 2–Q2Q1.
Hence, the marginal revenue for firm 2, MR2, is:
6) MR2 = 140– 2Q2 – Q1, (Marginal revenue of firm 2)
Step 2: Compute the profit maximizing outputs for both firms.
To start with observe that equations 2) and 3) imply that MC1 =MC2 = 20.
Start with firm 1. Profit maximization for both firms entails selecting an output at which
the marginal revenue equates the marginal cost. Hence for firm 1, MR1 = MC1 implies by
equation 4):
140– 2Q1 – Q2 = 20 (Marginal revenue of firm 1)
