The Pure Theory of Oligopoly
A list of strategies ( such as "confess--confess",
or "high--high ") is said to be a Nash
Equilibrium if the strategy selected by each
player yields the highest payoff after taking into account the strategies
selected by all other players. That is, we have Nash
Equilibrium if every player is making their
best play based on the strategies selected by the other players.
So, for example, in the
advertising
rivalry game, the
"high--high" payoff cell fulfills our criteria for Nash Equilibrium.
The Cournot Model
Augustin Cournot. Research Into
the Mathematical Principles of the Theory of Wealth, 1838
The Cournot 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.
Assumptions
Two
sellers
MC
= $40
Homogeneous
product
Q
is the "decision variable"
Maximizing
behavior
Let the inverse demand function
be given by:
P = 100 - Q [1]
The total revenue (TR) function is
given by:
TR = PQ = (100 - Q)Q = 100Q - Q2
[2]
Thus marginal revenue (MR) is given
by:
MR = dTR/dQ = 100 - 2Q [3]
(power rule)
Let q1
denote the output of seller 1 and q2
is the output of seller 2. [1] can be rewritten
as:
P = 100 - q1
- q2 [4]
Thus the profits of sellers 1 and
2 are given
by:
P 1
= (100 - q1 - q2)q1 - 40q1 [5]
P2 = (100 - q1 - q2)q2
- 40q2 [6]
b
Mutual
interdependence is revealed by the profit equations--i.e., profits of seller
1 depend on the quantity -supplied of seller 2, and the reverse is also
true.
The monopoly case
Let q2 = 0 units
so that Q = q1 --i.e., seller 1
is a monopolist. Seller 1 should set quantity-supplied at the level corresponding
to the equality of marginal revenue and marginal cost. That is, let MR
- MC = 0
100 - 2Q - 40 = 0
2Q = 60 \
Q = QM = 30 units
Thus, the monopoly price is given
by:
PM = 100 - QM
= $70
Click here to see the graph
Substituting back into [5], we find
that
P = $900
Finding Nash Equilibrium
Seller 1 expects seller 2 to supply
10 units. What is the profit-maximizing quantity-supplied for seller 1
based on this expectation?
By equation [4], we can say:
P = 100 - q1 - 10
= 90 - q1 [7]
Thus TR of seller 1 is given by:
TR = Pq1 = (90
- q1) q1 = 90 q1 - q12
[8]
Thus:
MR = 90 - 2 q1 [9]
Subtracting MC from MR
90 - 2 q1 - 40
= 0 [10]
Therefore
2 q1 = 50 \
q1 = 25 units [11]
Thus, the profit maximizing output
for seller 1, given that q2 = 10 units, is 25 units.
Repeating this process for all possible
values of q2, we find that the profit maximizing quantity supplied
of seller 1 can be obtained from the following equation:
q1 = 30 - .5q2
[12]
[12] is the best
reply function (BRP) of seller 1. See
graph
.In similar fashion, we can derive a best
reply function (BRP) for seller 2:
q2 = 30 - .5q1 [13]
Nash Equilibrium is established
when both sellers are on their best reply functions. See graph.
Thus:
QCournot = 40 units
(20 each)
PCournot = $60
P 1 = P
2 = $400
Note that:
PC = $40
QC = 60 units
Therefore
PC < PCournot
< PM
The policy implications:
The Cournot model predicts that, holding elasticity of demand (h
) constant, price-cost margins are inversely related to the number of sellers
in the market. This principle is expressed by equation [14]
(P - MC)/P = 1/(h
n) [14]
where n is the number of
sellers in the market. Thus as n moves to infinity, the price-cost margin
approaches zero-as in competition.
