Assumptions
• Two firms A, and B
produce widgets
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
The Cournot Model
Assumptions
• Two firms A, and B
produce widgets
• The industry demand
function is D
Assumptions
P
• Two firms A, and B
produce widgets
• The industry demand
function is D
• Firm A produces qA;
firm B produces qB
D
Q
The Cournot Model
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
D
Q
The Cournot Model
Assumptions
P
The Cournot Model
P
qb
• Two firms A, and B
produce widgets
• The industry demand
function is D
• Firm A produces qA;
firm B produces qB
D
Da
Q
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
1
Solving A’s problem
Solving A’s problem
Da
Da
D
D
MR
The Cournot Model
The Cournot Model
Solving A’s problem
Symmetry
• Just as Firm A is choosing qA to maximize
profits, so too is Firm B choosing qB to
maximize profits.
Da
D
p*
MR
MC
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.
• If B changes its output, A will react by
changing its output.
The Cournot Model
The Cournot Model
A Reaction Function
• We do the mathematical approach first and
then the graphical approach.
The Cournot Model
2
A Reaction Function
• The industry demand function
Q = 100 – 2p.
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
The Cournot Model
A Reaction Function
A’s demand function is then
P = 50 –(1/2)(qA +qB)
• The firm’s profits are
π = PqA – 5qA
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
The Cournot Model
A Reaction Function
π = [50 –(1/2)(qA + qB)]qA – 5qA
π = [50 –(1/2)(qA +qB)]qA – 5qA
The Cournot Model
The Cournot Model
3
A Reaction Function
A Reaction Function
π = [50 –(1/2)(qA + qB)]qA – 5qA
π = [50 –(1/2)(qA + qB)]qA – 5qA
π = 50 qA–(1/2) qA 2– (1/2)qBqA –
5qA
π = 50 qA–(1/2) qA 2– (1/2)qBqA – 5qA
The Cournot Model
π = 45qA –(1/2)qA2 – (1/2)qBqA
The Cournot Model
A Reaction Function
A Reaction Function
1
1
π = 45qa − qa2 − qa qb
2
2
π = 45qa − qa2 − qa qb
The Cournot Model
A Reaction Function
1
2
1
2
dπ
1
= 45 − qa − qb
dqa
2
The Cournot Model
Symmetry
qA = 45 – (1/2)qB
dπ
1
= 45 − qa − qb = 0
dqa
2
1
qa = 45 − qb
2
The Cournot Model
• There is a similar reaction
function for B
qB = 45 – (1/2)qA
The Cournot Model
4
Solving for A’s Output
qA = 45 – (1/2)qB
qB = 45 – (1/2)qA
Solving for A’s Output
qA = 45 – (1/2)[45 – (1/2)qA]
qA = 22.5 + (1/4)qA
qA = 45 – (1/2)[45 – (1/2)qA]
The Cournot Model
The Cournot Model
Solving for A’s Output
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 = 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
The Cournot Model
A Graphical Approach
qA = 45 – (1/2)qB
• We want to use the reaction function to
come to a graphical solution,
qA = 30
qB = 30
The Cournot Model
The Cournot Model
5
A Graphical Approach
A Graphical Approach
qA = 45 – (1/2)qB
• When B produces nothing A should
react by producing the monopoly
output (45).
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
The Cournot Model
A Graphical Approach
Graphing the Reaction Function
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.
A’s
Output
• Similar rules apply for B’s reactions.
B’s
Output
The Cournot Model
The Cournot Model
Graphing the Reaction Function
A’s
90
Output
45
0
If B
produces
nothing, A
acts like a
monopoly
The Cournot Model
If B produces
the competitive
output, A
produces
nothing.
90
Graphing the Reaction Function
A’s
Output
A’s
Reaction
Function
45
B’s
Output
The Cournot Model
90
B’s
Output
6
Graphing the Reaction Function
A’s
90
Output
45
A’s 90
Output
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
90
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
A’s
Reaction
Function
45
90
The Cournot Model
B’s
Output
Graphing the Reaction Function
A’s
Output
B’s
Output
B’s
Output
The Cournot Model
Graphing the Reaction Function
A’s
Output
B’s
Reaction
Function
45
B’s
Output
Graphing the Reaction Function
A’s
90
Output
Graphing the Reaction Function
Graphing the Reaction Function
A’s
Output
If A is here,
B wants to
be here
If B is here,
A wants to
be here
B’s
Output
The Cournot Model
B’s
Output
The Cournot Model
7
Equilibrium
A’s
Output
The Basic Steps
B’s
Reaction
Function
• Plot the reaction functions
A’s
Reaction
Function
– If B produces nothing, A behaves like a
monopoly
– If B produces competitive output, A produces
nothing
• Solve for their intersection
B’s
Output
The Cournot Model
The Cournot Model
End
©2003 Charles
W. Upton
The Cournot Model
8