the cournot model

Lectures in Microeconomics-Charles W. Upton

The Cournot ModelA’s Output

B’s Output

A’s Reaction Function

B’s Reaction Function

45

45

90

90

The Cournot Model

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

D

P

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

D

P

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

P

Q

qb

Da

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

P

Q

qb

Da

An important assumption, the heart of the Cournot model.

The Cournot Model

Solving A’s problem   

DDa

The Cournot Model

Solving A’s problem   

DDa

MCMR

The Cournot Model

Solving A’s problem   

DDa

MCMR

qa*

p*

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 isP = 50 – (1/2)Q

The Cournot Model

A Reaction Function

• The industry demand function Q = 100 – 2p.

• The inverse demand function isP = 50 – (1/2)Q

• A’s demand function is thenP = 50 –(1/2)(qA+qB)

The Cournot Model

A Reaction Function

A’s demand function is thenP = 50 –(1/2)(qA +qB)

• The firm’s profits are = PqA – 5qA

The Cournot Model

A Reaction Function

A’s demand function is thenP = 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

baaa qqqq21

2145 2

The Cournot Model

A Reaction Function

baa

baaa

qqdqd

qqqq

2145

21

2145 2

The Cournot Model

A Reaction Function

ba

baa

qq

qqdqd

2145

02145

The Cournot Model

Symmetry

•There is a similar reaction function for B

qB = 45 – (1/2)qA

qA = 45 – (1/2)qB

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 = 22.5qA = (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 = 22.5qA = (4/3)22.5

qA = 30qB = 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 FunctionA’s Output

B’s Output

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

If B produces the competitive

output, A produces nothing.

If B produces

nothing, A acts like a monopoly45

90

900

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

A’s Reaction Function

45

90

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

A’s Reaction Function

If A produces the competitive output, B produces nothing.

If A produces nothing, B acts like

a monopoly.45

45

90

90

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

A’s Reaction Function

B’s Reaction Function

45

45

90

90

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

If A and B are off their reaction functions, they react and change output. Here B expands, A contracts.

90

45

9045

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

If A is here, B wants to

be here

The Cournot Model

Graphing the Reaction FunctionA’s Output

B’s Output

If B is here, A wants to

be here

The Cournot Model

EquilibriumA’s Output

B’s Output

A’s Reaction Function

B’s Reaction Function

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