May 26, 2007
Dominant Firm and Competitive Fringe
This model joins together some aspects of monopoly and perfect competition.
The “competitive fringe” of firms behave competitively – they are price takers.
The dominant firm, however, profit-maximizes by setting MR = MC and behaves
like a monopolist, with one important difference. The dominant firm realizes
that the competitive fringe of firms exists and takes into account the amount they
will produce. In other words, the dominant firm profit-maximizes using the
residual demand rather than the whole market demand curve.
Here is an example (discussed in class, but there is more detail here).
There is an industry with a dominant firm and 100 firms that make up the
competitive fringe. The 100 firms have identical cost curves. The industry
demand curve is given by P = 1000 - .005Q. The total cost curve of each of the
competitive fringe firms is TC = 600q + q2.
The first thing to do is to find the residual demand curve facing the dominant
firm. In order to do that, we need to know how much the competitive fringe of
firms will produce (their supply curve) and subtract this from the whole market
demand curve.
MC of each firm in the competitive fringe is MC = 600 + 2q. These firms produce
where P = MC, so P = 600 + 2q is the supply curve for each of these firms. Since
there are 100 identical firms, q = .01Q. Substituting, we can conclude that P = 600
+ 2(.01Q), or P = 600 + .02Q is the supply curve for all of these 100 firms when
added together.
We now need to subtract this competitive fringe supply from the market demand
curve to get the residual demand curve faced by the Dominant Firm. There are 2
critical points – (1) the price at which the competitive fringe will be producing
enough to serve the entire market, leaving no residual demand for the Dominant
Firm, and (2) the price below which the competitive fringe is unwilling to supply
any output to the market (because it is does not cover their costs).
Point (1) can be found as the intersection of the market demand curve and the
competitive fringe supply curve, or where 1000 - .005Q = 600 + .02Q. This will
happen where Q = 16,000 and where P = $920.
Point (2) can be found as the point on the market demand curve where the
competitive fringe supply crosses the horizontal axis (Q = 0), which is at P =
$600. Substituting this into the market demand curve we have 600 = 1000 .005Q, or Q = 80,000.
A line joining the point ($920, 0) to the point ($600, 80000) will form the main
part of the residual demand curve of the Dominant Firm. Because the slope of
this line (rise over the run) will be (920 – 600)/80,000 = .004, the equation of this
part of the residual demand curve of the Dominant Firm will be P = 920 - .004Q.
The second part of the residual demand curve applies to quantities greater than
80,000 units of output. Beyond this output (and therefore for an equilibrium
price less than $600), there is no output from the competitive fringe, so the
Dominant Firm now faces the entire market demand curve. The equation of the
entire residual demand curve would therefore be:
P = 920 - .004Q for Q <= 80,000, and P = 1000 - .005Q for Q > 80,000.
The Dominant Firm will profit-maximize by setting MR = MC, with MR coming
from the residual demand curve. Because the residual demand curve is twopart, the MR curve will also be two-part. The Dominant Firm’s MR curve will be
MR = 920 - .008Q for Q<= 80,000 and MR = 1000 - .01Q for Q > 80,000.
There are two possibilities for the behaviour of the Dominant Firm. If the
Dominant Firm’s MC curve is low enough to cut the second part (the bottom
part) of the MR curve, the Dominant Firm will sell at a low enough price (and a
high enough quantity) that there will be no competitive fringe left in the
industry. If the Dominant Firm’s MC curve is higher than this, it will cut the top
part of the MR curve and the Dominant Firm will profit-maximize by sharing the
market with the competitive fringe of firms.
In this case, we assume that the Dominant Firm’s MC curve is MC = AC =
500 (so the Dominant Firm has a cost advantage over the competitive fringe).
Therefore, MR = MC where 920 - .008Q = 500 or QDF* = 52,500 and P* = 920 .004(52,500) = $710. The profit of the Dominant Firm will be TR – TC = (710 x
52,500) – (500 x 52,500) = $11,025,000.
The competitive fringe is a group of price-taking firms. They therefore take the
price established by the Dominant Firm and produce the profit-maximizing
quantity at that price. Therefore, 710 = 600 + .02Q, so that QCF* = 5,500. Since
there are 100 firms in the competitive fringe, each must be producing 55 units of
output. Profit for each of the competitive fringe of firms is (710 x 55) – [(600 x 55)
+ (55 x 55)] = $3,025.
We can ask several questions about this solution, and the answers help us to
understand the model:
(1) What is the Deadweight Loss associated with this equilibrium solution?
First, we must determine the “competitive” result. The competitive result
comes where MC = P, in other words, where MC crosses the market
demand curve. The Dominant Firm’s MC curve is the lowest, so we
should use this for the calculations. Imagining that the Dominant Firm
supplied the whole market at a price of $500, this would mean the sale of
1000 - .005Q = 500 or 100,000 units. The Dominant Firm/Competitive
Fringe equilibrium involved a price of $710 and the production of 52,500 +
5,500 = 58,000 units. The Deadweight Loss is therefore [(710 – 500) x
(100,000 – 58,000)]/2 = $4,410,000.
(2) Why does the Dominant Firm choose this alternative, rather than
charging a price just low enough to drive the competitive fringe out of
the market? The highest price at which the competitive fringe could not
compete would be $600. At this price the competitive fringe would
produce nothing. If the Dominant Firm charged this price, it could sell
1000 - .005Q = 600 or Q = 80,000 units. The profit for the Dominant Firm
would be TR – TC = (600 x 80,000) – (500 x 80,000) = $8,000,000. This is a
lower profit than the Dominant Firm can earn by sharing the market at a
price of $710, so sharing the market is a better alternative.
(3) How is the Dominant Firm/Competitive Fringe equilibrium different
from the monopoly model? If there were no competitive fringe, the
Dominant Firm would set MR = MC, so 1000 - .01Q = 500, or Q* = 50,000.
The monopoly price would be 1000 - .005(50,000) = $750. The equilibrium
profit would be (750 x 50,000) – (500 x 50,000) = $12,500,000. Here we can
see that the competitive fringe (part of the“structure” of the industry) acts
to reduce the ability of the Dominant Firm to raise price and earn profits.
You can check to see that the Deadweight Loss is lower under the
Dominant Firm/Competitive Fringe situation than it would be under
monopoly.
(4) What would happen in the Dominant Firm model if the Dominant Firm
had much lower costs, say MC = AC = $100? In this case, the MC curve
would intersect MR in its lower part, so we would have MR = MC: 1000 .01Q = 100, or QDF* = 90,000 and P* = $450. The profit of the Dominant
Firm would be (450 x 90,000) – (100 x 90,000) = 40,500 – 9,000,000 =
$31,500,000. This equilibrium is now the same as the Monopoly model
(because there is only one firm left in the industry).