in parentheses in Equation 9.1, depends on the competitor’s output quantity.
Increases in Q 2 cause a parallel downward shift in demand; a decrease in Q 2 has
the opposite effect. Given a prediction about Q 2 , firm 1 can apply marginal
analysis to maximize profit in the usual way. The firm’s revenue is R 1 (30
Q 2 Q 1 )Q 1 (30 Q 2 )Q 1 Q 12. Marginal revenue, in turn, isSetting marginal revenue equal to the $6 marginal cost, we find that 30 Q 2
2Q 1 6,or [9.2]Firm 1’s profit-maximizing output depends on its competitor’s quantity. An
increase in Q 2 reduces firm 1’s (net) demand, its marginal revenue, and its
optimal output. For example, if firm 1 anticipates Q 2 6, its optimal output
is 9; if it expects Q 2 10, its optimal output falls to 7. In other words,
Equation 9.2 sets a schedule of optimal quantities in response to different
competitive outputs. For this reason, it is often referred to as the optimal reac-
tion function.A similar profit maximization for firm 2 produces the analogous
reaction function:[9.3]
Now we are ready to derive the quantity and price outcomes for the duop-
oly. The derivation rests on the notion of equilibrium.^7 Here is the definition:
In equilibrium, each firm makes a profit-maximizing decision, anticipating
profit-maximizing decisions by all competitors.
Before we discuss this definition further, let’s determine the equilibrium
quantities in the current example. To qualify as an equilibrium, the firms’
quantities must be profit-maximizing against each other; that is, they must
satisfy both Equations 9.2 and 9.3. Solving these equations simultaneously,
we find Q 1 Q 2 8 thousand. (Check this.) These are the unique equilib-
rium quantities. Since the firms face the same demand and have the same
costs, they produce the same optimal outputs. These outputs imply the mar-
ket price, P 30 16 $14. Each firm’s profit is $64,000, and total profit
is $128,000.Q 2 12 .5Q 1.
Q 1 12 .5Q 2.
MR R 1 / Q 1 (30Q 2 )2Q 1
364 Chapter 9 Oligopoly(^7) This concept frequently is called a Cournot equilibrium or a Nash equilibrium, after John Nash,
who demonstrated its general properties.
Suppose the duopoly example is as described earlier except that the second firm’s aver-
age cost is $9 per unit. Find the firms’ equilibrium quantities.
CHECK
STATION 1
c09Oligopoly.qxd 9/29/11 1:32 PM Page 364