Price-Output Determination in
Oligopolistic Market Structures
We have good models of priceoutput determination for the
structural cases of pure
competition and pure monopoly.
Oligopoly is more problematic,
and a wide range of outcomes is
possible.
Cournot
1
Model
•Illustrates the principle of mutual interdependence
among sellers in tightly concentrated markets--even
where such interdependence is unrecognized by
sellers.
•Illustrates that social welfare can be improved by
the entry of new sellers--even if post-entry structure
is oligopolistic.
1 Augustin
Cournot. Research Into the Mathematical Principles of
the Theory of Wealth, 1838
Assumptions
1. Two sellers
2. MC = $40
3. Homogeneous product
4. Q is the “decision variable”
5. Maximizing behavior
Let the inverse demand function be given by:
P = 100 – Q
[1]
The revenue function (R) is given by:
R = P • Q = (100 – Q)Q = 100Q – Q2
[2]
Thus the marginal revenue (MR) function is given by:
MR = dR/dQ = 100 – 2Q
[3]
Let q1 denote the output of seller 1 and q2 is the output of
seller 2. Now rewrite equation [1]
P = 100 – q1 – q2
[4]
The profit () functions of sellers 1 and 2 are given by:
1 = (100 – q1 – q2)q1 – 40q1
[5]
2 = (100 – q1 – q2)q2 – 40q2
[6]
Mutual interdependence is revealed by the profit
equations. The profits of seller 1 depend on the
output of seller 2—and vice versa
Monopoly case
Let q2 = 0 units so that Q = q1—that is, seller 1 is a monopolist.
Seller 1 should set its quantity supplied at the level
corresponding to the equality of MR and MC.
Let MR – MC = 0
100 – 2Q – 40 = 0
2Q = 60  Q = QM = 30 units
Thus
PM = 100 – QM = $70
Substituting into equation [5], we find that:
 = $900
Finding equilibrium
Question: Suppose that seller 1 expects that seller 2
will supply 10 units. How many units should seller 1
supply based on this expectation?
By equation [4], we can say:
P = 100 – q1 – 10 = 90 – q1
[7]
The the revenue function of seller 1 is given by:
R = P • q1 = (90 – q1)q1 = 90q1 – q12
[8]
Thus:
MR = dR/dq1 = 90 – 2q1
[9]
Subtracting MC from MR
90 – 2q1 – 40 = 0
[10]
2q1 = 50  q1 = 25 units
[11]
Thus the profit maximizing output for seller 1, given that
q2 = 10 units, is 25 units.
We repeat these calculations
for every possible value of q2
and we find that the
-maximizing output for seller
1 can be obtained from the
following equation:
q1 = 30 - .5q2
[12]
Best reply function
Equation [12] is a best reply function (BRF) for seller 1. It
can be used to compute the -maximizing output for seller
1 for any output selected by seller 2.
60
30 - .5q2
30
10
0
15
25 30
Output of seller 1
In similar fashion, we derive a best reply function for
seller 2. It is given by:
q2 = 30 - .5q1
[13]
q2
30
0
q2 = 30 - .5q1
60
q1
So we have a system with 2 equations and 2 unknowns
(q1 and q2) :
The solutions are:
q1 = 30 – .5q2
q1 = 20 units
q2 = 30 – .5q1
q2 = 20 units
q2
60
Seller 1’s BRF
30
20
0
Equilibrium
20
30
Equilibrium is established
when both sellers are on
their best reply function
Seller 2’s BRF
60
q1
Cournot duopoly solution
QCOURNOT = 40 Units (20 units each)
PCOURNOT = $60
1 = 2 = $400
Note that:
PCOMPETITIVE = $40
QCOMPETITIVE = 60 Units
Therefore
PCOMPETITIVE < PCOURNOT < PMONOPOLY
Implications of the model
The Cournot model predicts that,
holding elasticity of demand constant,
price-cost margins are inversely related to
the number of sellers in the market
This principle is expressed by the following
equation
( P  MC ) 1

P
n
[14]
Where  is elasticity of demand and n is the number of
sellers. So as n  , the price-margin approaches zero—
as in the purely competitive case.