The Price Leadership or Dominant Firm Model
I think this model is easiest to learn diagrammatically,
and then mathematically. Here is the graph and then an
explanation of what is happening:
Price
MCCF - Sum of marginal
costs of competitive
fringe
Total
Demand
P*
MCDF - Marginal
Cost of
Dominant Firm
DDF
Q*CF Q*DF
MRDF
Quantity
Notice first the total market demand curve for the
industry as a whole. Then notice the marginal cost curve
for the competitive fringe of firms. This is a model in
which there is one firm which is dominant and then a
fringe of small firms who are so small that they behave
like perfectly competitive firms – they take the price
that is give by the dominant firm (and then set P = MC to
profit maximize).
The basic story in this model is that the dominant firm
leaves room for the competitive fringe (and therefore
profit maximizes according to the “residual” demand
curve. Since the fringe of firms behaves like perfect
competitors, the sum of their marginal cost curves is
essentially their supply curve. It represents the amount
that these firms together will want to supply at any
possible price.
Therefore, the residual demand curve is total demand
minus this supply by the competitive fringe. This is
exactly what the curve labeled DDF represents.
Our story is that the dominant firm profit maximizes
using this residual demand curve. That means setting MR
= MC for this demand curve. This is exactly where Q*DF
comes from (it is the quantity at which MR is just equal
to MC for the dominant firm. The dominant firm will
charge the profit-maximizing price, which is P*.
Once P* is established by the dominant firm, the
competitive fringe (who are price takers) will just take
this price and set P* = MC. This gives us the profitmaximizing quantity Q*CF for the competitive fringe.
We can take an algebraic example. Assume that the
overall industry demand curve is P = 100 – Q and that the
sum of the marginal costs of the competitive fringe is P =
10 + 4Q. The marginal cost of the dominant firm is
constant at MC = 18.
The price at which the total demand and the competitive
fringe marginal cost curve intersect will give us the
vertical intercept of the residual demand curve.
Therefore:
100 – Q = 10 + 4Q or 5Q = 90 or Q = 18 and P = $82.
Therefore, the vertical intercept is $82.
The residual demand curve will join with the industry
demand curve exactly at the price at which the quantity
supplied by the competitive fringe = 0. Since the
equation of the competitive fringe’s MC curve is P = 10 +
4Q, the competitive fringe will supply nothing when P =
$10. The quantity demanded according to the industry
demand curve is 10 = 100 – Q or Q = 90 at a price of $10.
We now have two points on the dominant firm’s residual
demand curve. It starts at P = $82 and Q = 0 and it joins
the industry demand curve at P = $10 and Q = 90. Since
the demand curve is linear between these two points, we
can calculate the slope to be (82 – 10)/(90 – 0) = 72/90 =
4/5 or 0.8. Therefore, the equation of the (top part of
the) dominant firm demand curve is P = 82 – 0.8Q
Therefore, the dominant firm’s MR curve is MR = 82 –
1.6Q. Since the MC curve of the dominant firm is MC =
18, we have 82 – 1.6Q = 18 or Q*DF = 40. Substitute this
into the equation for the dominant firm demand curve to
get the price the dominant firm will charge: P* = 82 –
0.8(40) = $50.
At a price of $50, the competitive fringe will supply 50 =
10 + 4Q, or Q*CF = 10.