The Dominant firm model
Both the small firms and the dominant firm have constant marginal cost of c. The small firms
have a total capacity of K units. The market demand is QM(P). The dominant firm sets its
price, the small firms take that price as the market price (they behave as price takers) and they
assume that they can sell as much as they want subject to their capacity constraint. We are
interested in finding the profit maximizing price for the dominant firm.
The profits of the dominant firm is
π = [QM(P) − K]·P − cQM(P)
Differentiate π with respect to price set equal to 0:
dQ M
dQ M
P + QM − K − c
dP
dP
= 0. Modify the equation as follows
dQ M
( P − c ) = − (QM − K )
dP
Divide and multiply the left side by P, divide both sides by QM:
(Q − K )
dQ M P ⎛ P − c ⎞
⎟=−
M ⎜
dP Q ⎝ P ⎠
QM
M
εP
Rewrite this as
M
1 (Q − K )
P−c
=−
εP
P
QM
Note that
QM − K
is the dominant firm’s market share.
QM
Applications
Example 1
The dominant firm has a constant marginal cost of $2, the fringe firms have a constant
marginal cost of $5, and they have a total capacity of K = 4 units. The market demand is D(P)
= 16 − P. The dominant firm sets its price, the fringe firms take that price as given and sell as
much as they want subject to the capacity constraint. For simplicity, we assume that when the
dominant firm and the fringe firms set the same price all consumers go to the fringe firms
first.
Compute the profit maximizing price for the dominant firm.
Hint : the dominant firm can either set its price slightly below the MC of the fringe firms to
drive them off completely, or can set a higher price that leaves part of the market to the fringe
firms.
Suggested answer
If the dominant firm sets a price higher than 5, its residual demand will be (16 − P − 4).
Its profit function will be
(16 − P − 4)P − 2(16 − P − 4)
Differentiate with respect to P set = 0
12 − 2P + 2 = 0
P* = 7.
Profits = (7−2)5 = 25.
Profits with P = 4.99999
(4.9999999 − 2)11 = 33.
The dominant firm is better off with setting price low enough to kill off the smaller firms.
Example 2
There are 6 fringe firms with cost function c(q) = 2q +q2. Their marginal cost is mc(q) = 2 +
2q. The dominant firm’s cost function is Cd(q) = 2q. Its marginal cost is MCd(q) = 2. The
inverse market demand is p(Q) = 100 − Q.
a. Compute the profit maximizing price the dominant firm will charge taking into account the
effect of the fringe firms.
b. What is the market share of the fringe?
Suggested answer
a. Given the inverse demand P = 100 − Q, the marginal revenue is MR = 100 − 2Q. Set
MR = MC ⇒ 100 − 2Q = 2. The monopoly quantity and price are QM = 49, and PM = 51.
b. With the smaller firms in the marker, the monopolist must share the market. The
monopolist’s residual demand is given by QR(P) = QD(P) − So(P), where QD(P) is the market
demand, and So(P) is the total supply of the six small firms, provided that the price exceeds 2.
(CAN YOU SEE WHY P > 2 is needed for the smaller firms to operate?)
We now compute the supply of the smaller firms. Since they behave as price takers, they
maximize profits by choosing an output level at which their marginal cost is equal to market
price. MC = P ⇒ 2 +2qf = P ⇒ qf = (P − 2)/2, if P > 2, qf = 0 if P ≤ 0. With six firms we
have Qf = 6qf = 3P − 6.
QR(P) = 100 − P − (3P − 6) = 106 − 4P if P > 2, QR(P) = 100 − P if P ≤ 2.
c. The inverse residual demand is P = 106/4 − QR/4. The marginal revenue is
MRR = 106/4 − QR/2. The monopolist sets qM so that MR = MC. We have
106/4 − QR/2 = 2, so we have qM = 49, PM = 14.25. Each small firm produces 6.125, so in
total the 6 firms produce 36.75. Their market share is 36.75/(36.27+49) = 43%. The average
cost of the small firm is 2 + q, so at q = 6.125 we have AC = 8.125. Each small firm makes a
profit of (P−AC)×q = (14.25 − 8.125)×6.125 = 37.51.
More on the Dominant Firm Model
The dominant firm model is used to explain the behavior in industries where there is one
large firm who acts as a price searcher followed by small competitive firms that are price
takers. The price searching firm sets the market price first, followed by the small firms
choosing outputs in response to the market price.
The dominant firm is a firm typically with lower costs than other firms and therefore has
a large market share. It acts as a price searcher.
The dominant firm could also be a group of colluding firms (see collusion issues from
before)
The competitive fringe or fringe firms are those with higher costs and therefore each
one has a small market share. They act as price takers. The fringe follows the dominant
firm (timing) by taking its price.
Examples: Microsoft, IBM, Kodak, Boeing, General Electric.
Reasons for a dominant firm:
1) lower costs – efficiency/technology, first-mover advantage from learning
2) Superior product or reputation (advertising)
3) Collusion by a group
Results: A dominant firm may or may not price so low that the fringe is forced out of the
market. Second, the presence of a competitive fringe will force the dominant firm to
typically price below the otherwise monopoly level.
Assumptions
1) One firm has lower production costs than the other firms and is therefore
larger (total output).
2) All other firms are price takers.
3) The number of fringe firms is fixed.
4) The dominant firm knows the market demand and can predict the behavior of
the fringe in response to the price it chooses.
The market demand is given by D(p). The
fringe firms choose output according to their
combined supply S(p). The graph shows the
market less the dominant firm. Anticipating the
response of the fringe, the dominant firm has a
residual demand according to the remaining
quantity of demand at different prices. We let
R(p) denote the residual demand for the
dominant firm : R(p) = D(p)-S(p)
Q
Q
Q
For example, if the dominant firm sets a price p , the fringe firms will supply and output equal
Q which equals the quantity demanded and there will be no residual output for the dominant
firm to sell to. Alternatively, if the dominant firm sets a price p or lower, the competitive
fringe will supply zero output and the residual demand will equal the market demand. We can
plot the residual demand faced by the dominant firm as follows:
After we find the residual demand, the
dominant firm behaves like a monopolist
on that demand by equating the marginal
revenue of the residual demand to its
marginal cost. The price is set according
to the residual demand curve.
The dominant firm is able to set price above
marginal cost and thus make a profit.
Similarly, the fringe may also earn profit
because there is no entry (by assumption).
However, the fringe firms are choosing their
output where p = MCf. Alternatively, the
dominant firm may prefer to set a price
below p , in which case the competitive
fringe is forced out of the market. In this
case, the dominant firm will be a monopoly.
This will occur if the dominant firm’s costs
are sufficiently lower than the fringe firms’.
The graph shows three different MC curves for the dominant firm.
If the cost curve is MC1, will the dominant firm let the fringe firms have a market share?
If the cost curve is MC3, will the dominant firm let the fringe firms have a market share?
More complicated : What happens if the cost curve is MC2?