9781118041581

Quantity Competition 365

The duopoly equilibrium lies between the pure-monopoly and purely com-

petitive outcomes. The latter outcome occurs at a quantity sufficiently large

that price is driven down to average cost, PcAC $6, so that industry profit

is zero. According to the demand curve, the requisite total quantity is Qc 24

thousand units. In contrast, a monopolist—either a single firm or the two firms

acting as a cartel—would limit total output (Q) to maximize industry profit:

Setting marginal revenue (with respect to totaloutput) equal to marginal cost

implies 30 2Q 6. The result is Qm12 thousand units and Pm$18 thou-

sand. Total industry profit is $144,000. In sum, the duopoly equilibrium has a

lower price, a larger total output, and a lower total profit than the pure-monop-

oly outcome.

The analysis behind the quantity equilibrium can be applied to any num-

ber of firms; it is not limited to the duopoly case. Suppose n firms serve the

market and the market-clearing price is given by

Then firm 1’s marginal revenue is MR [30 (Q 2 .. .Qn)] 2Q 1.

Setting MR equal to the firm’s $6 MC yields

[9.4]

Analogous expressions hold for each of the other firms. The equilibrium is found

by simultaneously solving n equations in n unknowns. In fact, the easiest method

of solution is to recognize that the equilibrium must be symmetric. Because all

firms have identical costs and face the same demand, all will produce the same

output. Denoting each firm’s output by Q*, we can rewrite Equation 9.4 as

implying the solution

[9.5]

Notice that in the duopoly case (n 2), each firm’s equilibrium output is 8

thousand, the same result we found earlier. As the number of firms increases,

each firm’s profit-maximizing output falls (becomes a smaller part of the mar-

ket). What is the impact on totaloutput? Total output is

QnQ*24n/(n1)

Q*24/ 3 n 14.

Q* 12 .5(n1)Q*,

Q 1  12 .5(Q 2 .. .Qn).

P 30 (Q 1 Q 2 ... Qn).

(30Q)Q6Q.

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