Price-Output Determination in Oligopolistic Market Structures

1
Price-Output Determination in Oligopolistic
Market Structures
We have good models of price-output determination
for the structural cases of pure competition and
pure monopoly. Oligopoly is more problematic, and
a wide range of outcomes is possible.
2
Cournot Model1

• 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
3
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
4
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
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Mutual interdependence is revealed by the profit
equations. The profits of seller 1 depend on the
output of seller 2and vice versa
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Monopoly case
Let q2 0 units so that Q q1that 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
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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
7
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
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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
Output of seller 2
30
10
30
25
0
15
Output of seller 1
9
In similar fashion, we derive a best reply
function for seller 2. It is given by q2 30 -
.5q1
13
q2
30
q2 30 - .5q1
0
60
q1
10
So we have a system with 2 equations and 2
unknowns (q1 and q2) q1 30 .5q2q2 30
.5q1
The solutions are q1 20 units q2 20 units
q2
Equilibrium is established when both sellers are
on their best reply function
Seller 1s BRF
60
30
Equilibrium
Seller 2s BRF
20
0
20
30
q1
60
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Cournot duopoly solution
QCOURNOT 40 Units (20 units each)PCOURNOT
60?1 ?2 400
Note that PCOMPETITIVE 40QCOMPETITIVE 60
Units
Therefore PCOMPETITIVE lt PCOURNOT lt PMONOPOLY
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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
14
Where ? is elasticity of demand and n is the
number of sellers. So as n ? ?, the price-margin
approaches zeroas in the purely competitive case.