The Cournot Model
A’s 90
Output
B’s
Reaction
Function
A’s
Reaction
Function
45
45
90
B’s
Output
Lectures in Microeconomics-Charles W. Upton
Assumptions
• Two firms A, and B
produce widgets
The Cournot Model
Assumptions
• Two firms A, and B
produce widgets
• The industry demand
function is D
P
D
The Cournot Model
Q
Assumptions
• Two firms A, and B
produce widgets
• The industry demand
function is D
• Firm A produces qA;
firm B produces qB
P
The Cournot Model
D
Q
Assumptions
• Two firms A, and B
produce widgets
• The industry demand
function is D
• Firm A produces qA;
firm B produces qB
• Firm A takes its
demand function as D
-qB
P
The Cournot Model
qb
D
Da
Q
Assumptions
• Two firms A, and B
produce widgets
• The industry demand
function is D
• Firm A produces qA;
firm B produces qB
P
An important
assumption, the heart
of the Cournot model.
D
qb
• Firm A takes its
demand function
as D -qB
The Cournot Model
Da
Q
Solving A’s problem
Da
D
The Cournot Model
Solving A’s problem
Da
D
MR
MC
The Cournot Model
Solving A’s problem
Da
D
p*
MR
MC
qa*The Cournot Model
Symmetry
• Just as Firm A is choosing qA to maximize
profits, so too is Firm B choosing qB to
maximize profits.
The Cournot Model
Symmetry
• Just as Firm A is choosing qA to maximize
profits, so too is Firm B choosing qB to
maximize profits.
• If B changes its output, A will react by
changing its output.
The Cournot Model
A Reaction Function
• We do the mathematical approach first and
then the graphical approach.
The Cournot Model
A Reaction Function
• The industry demand function
Q = 100 – 2p.
The Cournot Model
A Reaction Function
• The industry demand function
Q = 100 – 2p.
• The inverse demand function is
P = 50 – (1/2)Q
The Cournot Model
A Reaction Function
• The industry demand function
Q = 100 – 2p.
• The inverse demand function is
P = 50 – (1/2)Q
• A’s demand function is then
P = 50 –(1/2)(qA+qB)
The Cournot Model
A Reaction Function
A’s demand function is then
P = 50 –(1/2)(qA +qB)
• The firm’s profits are
 = PqA – 5qA
The Cournot Model
A Reaction Function
A’s demand function is then
P = 50 –(1/2)(qA +qB)
• The firm’s profits are
 = [50 –(1/2)(qA +qB)]qA – 5qA
The Cournot Model
A Reaction Function
 = [50 –(1/2)(qA + qB)]qA – 5qA
The Cournot Model
A Reaction Function
 = [50 –(1/2)(qA + qB)]qA – 5qA
 = 50 qA–(1/2) qA 2– (1/2)qBqA –
5qA
The Cournot Model
A Reaction Function
 = [50 –(1/2)(qA + qB)]qA – 5qA
 = 50 qA–(1/2) qA 2– (1/2)qBqA – 5qA
 = 45qA –(1/2)qA2 – (1/2)qBqA
The Cournot Model
A Reaction Function
1 2 1
  45qa  qa  qa qb
2
2
The Cournot Model
A Reaction Function
1 2 1
  45qa  qa  qa qb
2
2
d
1
 45  qa  qb
dqa
2
The Cournot Model
A Reaction Function
d
1
 45  qa  qb  0
dqa
2
1
qa  45  qb
2
The Cournot Model
Symmetry
qA = 45 – (1/2)qB
• There is a similar reaction
function for B
qB = 45 – (1/2)qA
The Cournot Model
Solving for A’s Output
qA = 45 – (1/2)qB
qB = 45 – (1/2)qA
qA = 45 – (1/2)[45 – (1/2)qA]
The Cournot Model
Solving for A’s Output
qA = 45 – (1/2)[45 – (1/2)qA]
qA = 22.5 + (1/4)qA
The Cournot Model
Solving for A’s Output
qA = 45 – (1/2)[45 – (1/2)qA]
qA = 22.5 + (1/4)qA
(3/4)qA = 22.5
The Cournot Model
Solving for A’s Output
qA = 45 – (1/2)[45 – (1/2)qA]
qA = 22.5 + (1/4)qA
(3/4)qA = 22.5
qA = (4/3)22.5
The Cournot Model
Solving for A’s Output
qA = 45 – (1/2)[45 – (1/2)qA]
qA = 22.5 + (1/4)qA
(3/4)qA = 22.5
qA = (4/3)22.5
qA = 30
qB = 30
The Cournot Model
A Graphical Approach
qA = 45 – (1/2)qB
• We want to use the reaction function to
come to a graphical solution,
The Cournot Model
A Graphical Approach
qA = 45 – (1/2)qB
• When B produces nothing A should
react by producing the monopoly
output (45).
The Cournot Model
A Graphical Approach
qA = 45 – (1/2)qB
• When B produces nothing A should react by
producing the monopoly output (45).
• When B produces the output of the
competitive industry (90), A should
react by producing nothing.
The Cournot Model
A Graphical Approach
qA = 45 – (1/2)qB
• When B produces nothing A should react by
producing the monopoly output (45).
• When B produces the output of the
competitive industry (90), A should react by
producing nothing.
• Similar rules apply for B’s reactions.
The Cournot Model
Graphing the Reaction Function
A’s
Output
B’s
Output
The Cournot Model
Graphing the Reaction Function
A’s
90
Output
45
If B
produces
nothing, A
acts like a
monopoly
0
The Cournot Model
If B produces
the competitive
output, A
produces
nothing.
90
B’s
Output
Graphing the Reaction Function
A’s
Output
A’s
Reaction
Function
45
The Cournot Model
90
B’s
Output
Graphing the Reaction Function
A’s
90
Output
45
If A produces the
competitive output,
B produces nothing.
A’s
Reaction
Function
If A produces
nothing, B acts like
a monopoly.
45
The Cournot Model
B’s
Output
90
Graphing the Reaction Function
A’s 90
Output
B’s
Reaction
Function
A’s
Reaction
Function
45
45
90
The Cournot Model
B’s
Output
Graphing the Reaction Function
A’s
90
Output
45
If A and B are off their
reaction functions, they
react and change output.
Here B expands, A
contracts.
45
The Cournot Model
90
B’s
Output
Graphing the Reaction Function
A’s
Output
B’s
Output
The Cournot Model
Graphing the Reaction Function
A’s
Output
If A is here,
B wants to
be here
B’s
Output
The Cournot Model
Graphing the Reaction Function
A’s
Output
If B is here,
A wants to
be here
B’s
Output
The Cournot Model
Equilibrium
A’s
Output
B’s
Reaction
Function
A’s
Reaction
Function
B’s
Output
The Cournot Model
The Basic Steps
• Plot the reaction functions
– If B produces nothing, A behaves like a
monopoly
– If B produces competitive output, A produces
nothing
• Solve for their intersection
The Cournot Model
End
©2003 Charles
W. Upton
The Cournot Model