Chapter Twenty-Seven
Oligopoly
寡头垄断
Structure
Non-collusive moves
– Simultaneous moves
Quantity competition –Cournot model
Price competition – Bertrand model
– Sequential moves
Quantity leadership – Stakelberg
model
Price leadership
Collusion
Oligopoly
A monopoly is an industry consisting
a single firm.
A duopoly is an industry consisting of
two firms.
An oligopoly is an industry consisting
of a few firms,Particularly,each firm’s
own price or output decisions affect its
competitors’ profits.
Oligopoly
How do we analyze markets in which
the supplying industry is
oligopolistic?
Consider the duopolistic case of two
firms supplying the same product.
Quantity Competition
Assume that firms compete by
choosing output levels.
If firm 1 produces y1 units and firm 2
produces y2 units then total quantity
supplied is y1 + y2,The market price
will be p(y1+ y2).
The firms’ total cost functions are
c1(y1) and c2(y2).
Quantity Competition
Suppose firm 1 takes firm 2’s output
level choice y2 as given,Then firm 1
sees its profit function as
Given y2,what output level y1
maximizes firm 1’s profit?
1 1 2 1 2 1 1 1( ; ) ( ) ( ).y y p y y y c y
Quantity Competition; An Example
Suppose that the market inverse
demand function is
and that the firms’ total cost
functions are
p y yT T( )60
c y y1 1 12( )?c y y y2 2 2 2215( ),and
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
So,given y2,firm 1’s profit-maximizing
output level solves?
y
y y y
1
1 2 160 2 2 0,
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
So,given y2,firm 1’s profit-maximizing
output level solves?
y
y y y
1
1 2 160 2 2 0,
I.e,firm 1’s best response to y2 is
y R y y1 1 2 215 14( ),
Quantity Competition; An Example
y2
y1
60
15
Firm 1’s,reaction curve”
y R y y1 1 2 215 14( ),
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
So,given y1,firm 2’s profit-maximizing
output level solves?
y y y y2 1 2 260 2 15 2 0,
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
So,given y1,firm 2’s profit-maximizing
output level solves?
y y y y2 1 2 260 2 15 2 0,
I.e,firm 1’s best response to y2 is
y R y y2 2 1 145 4( ),
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”y R y y
2 2 1 1
45
4
( ),
45/4
45
Quantity Competition; An Example
An equilibrium is when each firm’s
output level is a best response to the
other firm’s output level,for then
neither wants to deviate from its
output level.
A pair of output levels (y1*,y2*) is a
Cournot-Nash equilibrium (古诺 -纳什均衡 ) if
y R y2 2 1* *( ).?y R y1 1 2* *( )? and
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y
1 115
1
4
45
4
* *
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Hence y 2
45 13
4 8
*,
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Hence y 2
45 13
4 8
*,
So the Cournot-Nash equilibrium is
(,) (,).* *y y1 2 13 8?
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”
60
15
Firm 1’s,reaction curve”y R y y
1 1 2 215
1
4( ),
y R y y2 2 1 145 4( ),
45/4
45
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”
48
60
Firm 1’s,reaction curve”y R y y
1 1 2 215
1
4( ),
8
13
Cournot-Nash equilibriumy y
1 2 13 8* *,,.?
y R y y2 2 1 145 4( ),
Quantity Competition
1 1 2 1 2 1 1 1( ; ) ( ) ( )y y p y y y c y
1
1
1 2 1 1 2
1
1 1 0y p y y y
p y y
y c y
( ) ( ) ( ),
Generally,given firm 2’s chosen output
level y2,firm 1’s profit function is
and the profit-maximizing value of y1 solves
The solution,y1 = R1(y2),is firm 1’s Cournot-
Nash reaction to y2.
Quantity Competition
2 2 1 1 2 2 2 2( ; ) ( ) ( )y y p y y y c y
2
2
1 2 2 1 2
2
2 2 0y p y y y
p y y
y c y
( ) ( ) ( ),
Similarly,given firm 1’s chosen output
level y1,firm 2’s profit function is
and the profit-maximizing value of y2 solves
The solution,y2 = R2(y1),is firm 2’s Cournot-
Nash reaction to y1.
Quantity Competition
y2
y1
Firm 1’s,reaction curve”
Firm 1’s,reaction curve” y R y1 1 2? ( ).
Cournot-Nash equilibrium
y1* = R1(y2*) and y2* = R2(y1*)y2*
y R y2 2 1? ( ).
y1*
Iso-Profit Curves
For firm 1,an iso-profit curve
contains all the output pairs (y1,y2)
giving firm 1 the same profit level?1.
What do iso-profit curves look like?
y2
y1
Iso-Profit Curves for Firm 1
With y1 fixed,firm 1’s profit
increases as y2 decreases.
y2
y1
Increasing profit
for firm 1.
Iso-Profit Curves for Firm 1
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
y2’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1.y2’
y1’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1,y1’ is firm 1’s
best response to y2 = y2’.
y2’
y1’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1,y1’ is firm 1’s
best response to y2 = y2’.
y2’
R1(y2’)
y2
y1
y2’
R1(y2’)
y2”
R1(y2”)
Iso-Profit Curves for Firm 1
y2
y1
y2’
y2”
R1(y2”)
R1(y2’)
Firm 1’s reaction curve
passes through the,tops”
of firm 1’s iso-profit
curves.
Iso-Profit Curves for Firm 1
y2
y1
Iso-Profit Curves for Firm 2
Increasing profit
for firm 2.
y2
y1
Iso-Profit Curves for Firm 2
Firm 2’s reaction curve
passes through the,tops”
of firm 2’s iso-profit
curves.
y2 = R2(y1)
Collusion （ 共谋）
Q,Are the Cournot-Nash equilibrium
profits the largest that the firms can
earn in total?
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?2
Higher?1
Collusiony
2
y1y1*
y2*
Higher?2
Higher?1y2’
y1’
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
Collusion
So there are profit incentives for both
firms to,cooperate” by lowering
their output levels.
This is collusion.
Firms that collude are said to have
formed a cartel （ 卡特尔）,
If firms form a cartel,how should
they do it?
Collusion
Suppose the two firms want to
maximize their total profit and divide
it between them,Their goal is to
choose cooperatively output levels
y1 and y2 that maximize
m y y p y y y y c y c y(,) ( )( ) ( ) ( ).1 2 1 2 1 2 1 1 2 2
Collusion
The firms cannot do worse by
colluding since they can cooperatively
choose their Cournot-Nash
equilibrium output levels and so earn
their Cournot-Nash equilibrium
profits,So collusion must provide
profits at least as large as their
Cournot-Nash equilibrium profits.
Allocate Output in a Cartel
FOC of the maximization problem,
Which implies that MC1(y1)=MC2(y2)
m y y p y y y y c y c y(,) ( )( ) ( ) ( ).1 2 1 2 1 2 1 1 2 2
1 2 1 2 1 1( ) ( ) ( ),
dpp y y y y M C y
dY
1 2 1 2 2 2( ) ( ) ( ),
dpp y y y y M C y
dY
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
(y1”,y2”) earns still
higher profits for
both firms.
y2”
y1”
Collusiony
2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profit
while leaving firm 2’s profit at
the Cournot-Nash equilibrium
level.
~ ~
Collusiony
2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profit
while leaving firm 2’s profit at
the Cournot-Nash equilibrium
level.
~ ~
y2_
y2_
(y1,y2) maximizes firm
2’s profit while leaving
firm 1’s profit at the
Cournot-Nash
equilibrium level.
_ _
Collusiony
2
y1y1*
y2*
y2~
y1~
y2_
y2_
The path of output pairs that
maximize one firm’s profit
while giving the other firm at
least its CN equilibrium
profit.
Collusiony
2
y1y1*
y2*
y2~
y1~
y2_
y2_
The path of output pairs that
maximize one firm’s profit
while giving the other firm at
least its CN equilibrium
profit,One of
these output pairs
must maximize the
cartel’s joint profit.
Collusiony
2
y1y1*
y2*
y2m
y1m
(y1m,y2m) denotes
the output levels
that maximize the
cartel’s total profit.
Collusion
Is such a cartel stable?
Does one firm have an incentive to
cheat on the other?
I.e,if firm 1 continues to produce y1m
units,is it profit-maximizing for firm
2 to continue to produce y2m units?
Cheating in a Cartel
At collusive output levels,firm 1’s
profit is:
What happens if raising own output?
From Cartel’s FOC we know that
1 1 2 1 2 1 1 1( *,* ) ( * * ) * ( * ),y y p y y y c y
1
1 2 1 1 1
1
( * * ) * ( * ),d dpp y y y M C y
d y d Y
1 2 1 1 1 2( * * ) * ( * ) * 0,
d p d pp y y y M C y y
d Y d Y
Collusion
Firm 2’s profit-maximizing response
to y1 = y1m is y2 = R2(y1m).
Collusiony
2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm
1 choosing y1 = y1m.R2(y1m)
y1 = R1(y2),firm 1’s reaction curve
y2 = R2(y1),firm 2’s
reaction curve
Collusion
Firm 2’s profit-maximizing response
to y1 = y1m is y2 = R2(y1m) > y2m.
Firm 2’s profit increases if it cheats
on firm 1 by increasing its output
level from y2m to R2(y1m).
Collusion
Similarly,firm 1’s profit increases if it
cheats on firm 2 by increasing its
output level from y1m to R1(y2m).
Collusiony
2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm
1 choosing y1 = y1m.
R1(y2m)
y1 = R1(y2),firm 1’s reaction curve
y2 = R2(y1),firm 2’s
reaction curve
Collusion
So a profit-seeking cartel in which
firms cooperatively set their output
levels is fundamentally unstable.
E.g,OPEC’s broken agreements.
The Order of Play
So far it has been assumed that firms
choose their output levels
simultaneously.
The competition between the firms is
then a simultaneous play game in
which the output levels are the
strategic variables.
The Order of Play
What if firm 1 chooses its output level
first and then firm 2 responds to this
choice?
Firm 1 is then a leader,Firm 2 is a
follower.
The competition is a sequential game （
有序游戏） in which the output levels are
the strategic variables.
The Order of Play
Such games are von Stackelberg
games.
Is it better to be the leader?
Or is it better to be the follower?
Stackelberg Games
Q,What is the best response that
follower firm 2 can make to the
choice y1 already made by the leader,
firm 1?
A,Choose y2 = R2(y1).
Firm 1 knows this and so perfectly
anticipates firm 2’s reaction to any y1
chosen by firm 1.
Stackelberg Games
This makes the leader’s profit
function
The leader chooses y1 to maximize its
profit.
Q,Will the leader make a profit at
least as large as its Cournot-Nash
equilibrium profit?
1 1 1 2 1 1 1 1s y p y R y y c y( ) ( ( )) ( ).
Stackelberg Games
A,Yes,The leader could choose its
Cournot-Nash output level,knowing
that the follower would then also
choose its C-N output level,The
leader’s profit would then be its C-N
profit,But the leader does not have
to do this,so its profit must be at
least as large as its C-N profit.
Stackelberg Games; An Example
The market inverse demand function
is p = 60 - yT,The firms’ cost
functions are c1(y1) = y12 and c2(y2) =
15y2 + y22.
Firm 2 is the follower,Its reaction
function isy R y y2 2 1 145
4
( ),
Stackelberg Games; An Example
1 1 1 2 1 1 1
2
1
1
1 1
2
1 1
2
60
60
45
4
195
4
7
4
s
y y R y y y
y
y
y y
y y
( ) ( ( ))
( )
.
The leader’s profit function is therefore
Stackelberg Games; An Example
1 1 1 2 1 1 1
2
1
1
1 1
2
1 1
2
60
60
45
4
195
4
7
4
s
y y R y y y
y
y
y y
y y
( ) ( ( ))
( )
.
The leader’s profit function is therefore
For a profit-maximum,195
4
7
2 13 91 1y y
s,
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice y s1 13 9
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice
A:
y s1 13 9
y R ys s2 2 1 45 13 94 7 8( ),
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice
A:
y s1 13 9
y R ys s2 2 1 45 13 94 7 8( ),
The C-N output levels are (y1*,y2*) = (13,8)
so the leader produces more than its
C-N output and the follower produces less
than its C-N output,This is true generally.
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?2
Higher?1
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?1
Follower’s
reaction curve
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium,(y1S,y2S) is the
Stackelberg equilibrium.
Higher?1
y1S
Follower’s
reaction curve
y2S
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium,(y1S,y2S) is the
Stackelberg equilibrium.
y1S
Follower’s
reaction curve
y2S
Price Competition
What if firms compete using only
price-setting strategies,instead of
using only quantity-setting
strategies?
Games in which firms use only price
strategies and play simultaneously
are Bertrand games.
Bertrand Games
Each firm’s marginal production cost
is constant at c.
All firms simultaneously set their
prices.
Q,Is there a Nash equilibrium?
A,Yes,Exactly one,All firms set
their prices equal to the marginal
cost c,Why?
Bertrand Games
Suppose one firm sets its price
higher than another firm’s price.
Then the higher-priced firm would
have no customers.
Hence,at an equilibrium,all firms
must set the same price.
Bertrand Games
Suppose the common price set by all
firm is higher than marginal cost c.
Then one firm can just slightly lower
its price and sell to all the buyers,
thereby increasing its profit.
The only common price which
prevents undercutting is c,Hence
this is the only Nash equilibrium.
Sequential Price Games
What if,instead of simultaneous play
in pricing strategies,one firm
decides its price ahead of the others.
This is a sequential game in pricing
strategies called a price-leadership
game.
The firm which sets its price ahead of
the other firms is the price-leader.
Sequential Price Games
Think of one large firm (the leader)
and many competitive small firms
(the followers).
The small firms are price-takers and
so their collective supply reaction to
a market price p is their aggregate
supply function Yf(p).
Sequential Price Games
The market demand function is D(p).
So the leader knows that if it sets a
price p the quantity demanded from
it will be the residual demand
Hence the leader’s profit function is
L p D p Y pf( ) ( ) ( ).
L f L f( p ) p ( D ( p ) Y ( p ) ) c ( D ( p ) Y ( p ) ),
Sequential Price Games
The leader’s profit function is
so the leader chooses the price level
p* for which profit is maximized.
The followers collectively supply
Yf(p*) units and the leader supplies
the residual quantity D(p*) - Yf(p*).
L f L Fp p D p Y p c D p Y p( ) ( ( ) ( )) ( ( ) ( ))
Demand,D(p)=a-bp
Cost,
–Leader,c1(y1)=cy1
–Follower,c2(y2)= y22/2
Follower’s supply,p= y2
Residual demand for leader:
–R(p)=D(p)-S2(p)=a-bp-p=a-(b+1)p
Solve for a monopoly problem
Sequential Price Games - Example
Sequential Price Games
Oligopoly
寡头垄断
Structure
Non-collusive moves
– Simultaneous moves
Quantity competition –Cournot model
Price competition – Bertrand model
– Sequential moves
Quantity leadership – Stakelberg
model
Price leadership
Collusion
Oligopoly
A monopoly is an industry consisting
a single firm.
A duopoly is an industry consisting of
two firms.
An oligopoly is an industry consisting
of a few firms,Particularly,each firm’s
own price or output decisions affect its
competitors’ profits.
Oligopoly
How do we analyze markets in which
the supplying industry is
oligopolistic?
Consider the duopolistic case of two
firms supplying the same product.
Quantity Competition
Assume that firms compete by
choosing output levels.
If firm 1 produces y1 units and firm 2
produces y2 units then total quantity
supplied is y1 + y2,The market price
will be p(y1+ y2).
The firms’ total cost functions are
c1(y1) and c2(y2).
Quantity Competition
Suppose firm 1 takes firm 2’s output
level choice y2 as given,Then firm 1
sees its profit function as
Given y2,what output level y1
maximizes firm 1’s profit?
1 1 2 1 2 1 1 1( ; ) ( ) ( ).y y p y y y c y
Quantity Competition; An Example
Suppose that the market inverse
demand function is
and that the firms’ total cost
functions are
p y yT T( )60
c y y1 1 12( )?c y y y2 2 2 2215( ),and
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
So,given y2,firm 1’s profit-maximizing
output level solves?
y
y y y
1
1 2 160 2 2 0,
Quantity Competition; An Example
( ; ) ( ),y y y y y y1 2 1 2 1 1260
Then,for given y2,firm 1’s profit function is
So,given y2,firm 1’s profit-maximizing
output level solves?
y
y y y
1
1 2 160 2 2 0,
I.e,firm 1’s best response to y2 is
y R y y1 1 2 215 14( ),
Quantity Competition; An Example
y2
y1
60
15
Firm 1’s,reaction curve”
y R y y1 1 2 215 14( ),
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
So,given y1,firm 2’s profit-maximizing
output level solves?
y y y y2 1 2 260 2 15 2 0,
Quantity Competition; An Example
( ; ) ( ),y y y y y y y2 1 1 2 2 2 2260 15
Similarly,given y1,firm 2’s profit function is
So,given y1,firm 2’s profit-maximizing
output level solves?
y y y y2 1 2 260 2 15 2 0,
I.e,firm 1’s best response to y2 is
y R y y2 2 1 145 4( ),
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”y R y y
2 2 1 1
45
4
( ),
45/4
45
Quantity Competition; An Example
An equilibrium is when each firm’s
output level is a best response to the
other firm’s output level,for then
neither wants to deviate from its
output level.
A pair of output levels (y1*,y2*) is a
Cournot-Nash equilibrium (古诺 -纳什均衡 ) if
y R y2 2 1* *( ).?y R y1 1 2* *( )? and
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y
1 115
1
4
45
4
* *
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Hence y 2
45 13
4 8
*,
Quantity Competition; An Example
y R y y1 1 2 215 14* * *( )y R y y2 2 1 145 4* *
*
( ),and
Substitute for y2* to gety y y
1 1 115
1
4
45
4
13*
* *
Hence y 2
45 13
4 8
*,
So the Cournot-Nash equilibrium is
(,) (,).* *y y1 2 13 8?
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”
60
15
Firm 1’s,reaction curve”y R y y
1 1 2 215
1
4( ),
y R y y2 2 1 145 4( ),
45/4
45
Quantity Competition; An Example
y2
y1
Firm 2’s,reaction curve”
48
60
Firm 1’s,reaction curve”y R y y
1 1 2 215
1
4( ),
8
13
Cournot-Nash equilibriumy y
1 2 13 8* *,,.?
y R y y2 2 1 145 4( ),
Quantity Competition
1 1 2 1 2 1 1 1( ; ) ( ) ( )y y p y y y c y
1
1
1 2 1 1 2
1
1 1 0y p y y y
p y y
y c y
( ) ( ) ( ),
Generally,given firm 2’s chosen output
level y2,firm 1’s profit function is
and the profit-maximizing value of y1 solves
The solution,y1 = R1(y2),is firm 1’s Cournot-
Nash reaction to y2.
Quantity Competition
2 2 1 1 2 2 2 2( ; ) ( ) ( )y y p y y y c y
2
2
1 2 2 1 2
2
2 2 0y p y y y
p y y
y c y
( ) ( ) ( ),
Similarly,given firm 1’s chosen output
level y1,firm 2’s profit function is
and the profit-maximizing value of y2 solves
The solution,y2 = R2(y1),is firm 2’s Cournot-
Nash reaction to y1.
Quantity Competition
y2
y1
Firm 1’s,reaction curve”
Firm 1’s,reaction curve” y R y1 1 2? ( ).
Cournot-Nash equilibrium
y1* = R1(y2*) and y2* = R2(y1*)y2*
y R y2 2 1? ( ).
y1*
Iso-Profit Curves
For firm 1,an iso-profit curve
contains all the output pairs (y1,y2)
giving firm 1 the same profit level?1.
What do iso-profit curves look like?
y2
y1
Iso-Profit Curves for Firm 1
With y1 fixed,firm 1’s profit
increases as y2 decreases.
y2
y1
Increasing profit
for firm 1.
Iso-Profit Curves for Firm 1
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
y2’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1.y2’
y1’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1,y1’ is firm 1’s
best response to y2 = y2’.
y2’
y1’
y2
y1
Iso-Profit Curves for Firm 1
Q,Firm 2 chooses y2 = y2’.
Where along the line y2 = y2’
is the output level that
maximizes firm 1’s profit?
A,The point attaining the
highest iso-profit curve for
firm 1,y1’ is firm 1’s
best response to y2 = y2’.
y2’
R1(y2’)
y2
y1
y2’
R1(y2’)
y2”
R1(y2”)
Iso-Profit Curves for Firm 1
y2
y1
y2’
y2”
R1(y2”)
R1(y2’)
Firm 1’s reaction curve
passes through the,tops”
of firm 1’s iso-profit
curves.
Iso-Profit Curves for Firm 1
y2
y1
Iso-Profit Curves for Firm 2
Increasing profit
for firm 2.
y2
y1
Iso-Profit Curves for Firm 2
Firm 2’s reaction curve
passes through the,tops”
of firm 2’s iso-profit
curves.
y2 = R2(y1)
Collusion （ 共谋）
Q,Are the Cournot-Nash equilibrium
profits the largest that the firms can
earn in total?
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
Are there other output level
pairs (y1,y2) that give
higher profits to both firms?
(y1*,y2*) is the Cournot-Nash
equilibrium.
Collusiony
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?2
Higher?1
Collusiony
2
y1y1*
y2*
Higher?2
Higher?1y2’
y1’
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
Collusion
So there are profit incentives for both
firms to,cooperate” by lowering
their output levels.
This is collusion.
Firms that collude are said to have
formed a cartel （ 卡特尔）,
If firms form a cartel,how should
they do it?
Collusion
Suppose the two firms want to
maximize their total profit and divide
it between them,Their goal is to
choose cooperatively output levels
y1 and y2 that maximize
m y y p y y y y c y c y(,) ( )( ) ( ) ( ).1 2 1 2 1 2 1 1 2 2
Collusion
The firms cannot do worse by
colluding since they can cooperatively
choose their Cournot-Nash
equilibrium output levels and so earn
their Cournot-Nash equilibrium
profits,So collusion must provide
profits at least as large as their
Cournot-Nash equilibrium profits.
Allocate Output in a Cartel
FOC of the maximization problem,
Which implies that MC1(y1)=MC2(y2)
m y y p y y y y c y c y(,) ( )( ) ( ) ( ).1 2 1 2 1 2 1 1 2 2
1 2 1 2 1 1( ) ( ) ( ),
dpp y y y y M C y
dY
1 2 1 2 2 2( ) ( ) ( ),
dpp y y y y M C y
dY
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
Collusiony
2
y1y1*
y2*
y2’
y1’
Higher?2
Higher?1
(y1’,y2’) earns
higher profits for
both firms than
does (y1*,y2*).
(y1”,y2”) earns still
higher profits for
both firms.
y2”
y1”
Collusiony
2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profit
while leaving firm 2’s profit at
the Cournot-Nash equilibrium
level.
~ ~
Collusiony
2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profit
while leaving firm 2’s profit at
the Cournot-Nash equilibrium
level.
~ ~
y2_
y2_
(y1,y2) maximizes firm
2’s profit while leaving
firm 1’s profit at the
Cournot-Nash
equilibrium level.
_ _
Collusiony
2
y1y1*
y2*
y2~
y1~
y2_
y2_
The path of output pairs that
maximize one firm’s profit
while giving the other firm at
least its CN equilibrium
profit.
Collusiony
2
y1y1*
y2*
y2~
y1~
y2_
y2_
The path of output pairs that
maximize one firm’s profit
while giving the other firm at
least its CN equilibrium
profit,One of
these output pairs
must maximize the
cartel’s joint profit.
Collusiony
2
y1y1*
y2*
y2m
y1m
(y1m,y2m) denotes
the output levels
that maximize the
cartel’s total profit.
Collusion
Is such a cartel stable?
Does one firm have an incentive to
cheat on the other?
I.e,if firm 1 continues to produce y1m
units,is it profit-maximizing for firm
2 to continue to produce y2m units?
Cheating in a Cartel
At collusive output levels,firm 1’s
profit is:
What happens if raising own output?
From Cartel’s FOC we know that
1 1 2 1 2 1 1 1( *,* ) ( * * ) * ( * ),y y p y y y c y
1
1 2 1 1 1
1
( * * ) * ( * ),d dpp y y y M C y
d y d Y
1 2 1 1 1 2( * * ) * ( * ) * 0,
d p d pp y y y M C y y
d Y d Y
Collusion
Firm 2’s profit-maximizing response
to y1 = y1m is y2 = R2(y1m).
Collusiony
2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm
1 choosing y1 = y1m.R2(y1m)
y1 = R1(y2),firm 1’s reaction curve
y2 = R2(y1),firm 2’s
reaction curve
Collusion
Firm 2’s profit-maximizing response
to y1 = y1m is y2 = R2(y1m) > y2m.
Firm 2’s profit increases if it cheats
on firm 1 by increasing its output
level from y2m to R2(y1m).
Collusion
Similarly,firm 1’s profit increases if it
cheats on firm 2 by increasing its
output level from y1m to R1(y2m).
Collusiony
2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm
1 choosing y1 = y1m.
R1(y2m)
y1 = R1(y2),firm 1’s reaction curve
y2 = R2(y1),firm 2’s
reaction curve
Collusion
So a profit-seeking cartel in which
firms cooperatively set their output
levels is fundamentally unstable.
E.g,OPEC’s broken agreements.
The Order of Play
So far it has been assumed that firms
choose their output levels
simultaneously.
The competition between the firms is
then a simultaneous play game in
which the output levels are the
strategic variables.
The Order of Play
What if firm 1 chooses its output level
first and then firm 2 responds to this
choice?
Firm 1 is then a leader,Firm 2 is a
follower.
The competition is a sequential game （
有序游戏） in which the output levels are
the strategic variables.
The Order of Play
Such games are von Stackelberg
games.
Is it better to be the leader?
Or is it better to be the follower?
Stackelberg Games
Q,What is the best response that
follower firm 2 can make to the
choice y1 already made by the leader,
firm 1?
A,Choose y2 = R2(y1).
Firm 1 knows this and so perfectly
anticipates firm 2’s reaction to any y1
chosen by firm 1.
Stackelberg Games
This makes the leader’s profit
function
The leader chooses y1 to maximize its
profit.
Q,Will the leader make a profit at
least as large as its Cournot-Nash
equilibrium profit?
1 1 1 2 1 1 1 1s y p y R y y c y( ) ( ( )) ( ).
Stackelberg Games
A,Yes,The leader could choose its
Cournot-Nash output level,knowing
that the follower would then also
choose its C-N output level,The
leader’s profit would then be its C-N
profit,But the leader does not have
to do this,so its profit must be at
least as large as its C-N profit.
Stackelberg Games; An Example
The market inverse demand function
is p = 60 - yT,The firms’ cost
functions are c1(y1) = y12 and c2(y2) =
15y2 + y22.
Firm 2 is the follower,Its reaction
function isy R y y2 2 1 145
4
( ),
Stackelberg Games; An Example
1 1 1 2 1 1 1
2
1
1
1 1
2
1 1
2
60
60
45
4
195
4
7
4
s
y y R y y y
y
y
y y
y y
( ) ( ( ))
( )
.
The leader’s profit function is therefore
Stackelberg Games; An Example
1 1 1 2 1 1 1
2
1
1
1 1
2
1 1
2
60
60
45
4
195
4
7
4
s
y y R y y y
y
y
y y
y y
( ) ( ( ))
( )
.
The leader’s profit function is therefore
For a profit-maximum,195
4
7
2 13 91 1y y
s,
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice y s1 13 9
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice
A:
y s1 13 9
y R ys s2 2 1 45 13 94 7 8( ),
Stackelberg Games; An Example
Q,What is firm 2’s response to the
leader’s choice
A:
y s1 13 9
y R ys s2 2 1 45 13 94 7 8( ),
The C-N output levels are (y1*,y2*) = (13,8)
so the leader produces more than its
C-N output and the follower produces less
than its C-N output,This is true generally.
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?2
Higher?1
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium.
Higher?1
Follower’s
reaction curve
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium,(y1S,y2S) is the
Stackelberg equilibrium.
Higher?1
y1S
Follower’s
reaction curve
y2S
Stackelberg Gamesy
2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nash
equilibrium,(y1S,y2S) is the
Stackelberg equilibrium.
y1S
Follower’s
reaction curve
y2S
Price Competition
What if firms compete using only
price-setting strategies,instead of
using only quantity-setting
strategies?
Games in which firms use only price
strategies and play simultaneously
are Bertrand games.
Bertrand Games
Each firm’s marginal production cost
is constant at c.
All firms simultaneously set their
prices.
Q,Is there a Nash equilibrium?
A,Yes,Exactly one,All firms set
their prices equal to the marginal
cost c,Why?
Bertrand Games
Suppose one firm sets its price
higher than another firm’s price.
Then the higher-priced firm would
have no customers.
Hence,at an equilibrium,all firms
must set the same price.
Bertrand Games
Suppose the common price set by all
firm is higher than marginal cost c.
Then one firm can just slightly lower
its price and sell to all the buyers,
thereby increasing its profit.
The only common price which
prevents undercutting is c,Hence
this is the only Nash equilibrium.
Sequential Price Games
What if,instead of simultaneous play
in pricing strategies,one firm
decides its price ahead of the others.
This is a sequential game in pricing
strategies called a price-leadership
game.
The firm which sets its price ahead of
the other firms is the price-leader.
Sequential Price Games
Think of one large firm (the leader)
and many competitive small firms
(the followers).
The small firms are price-takers and
so their collective supply reaction to
a market price p is their aggregate
supply function Yf(p).
Sequential Price Games
The market demand function is D(p).
So the leader knows that if it sets a
price p the quantity demanded from
it will be the residual demand
Hence the leader’s profit function is
L p D p Y pf( ) ( ) ( ).
L f L f( p ) p ( D ( p ) Y ( p ) ) c ( D ( p ) Y ( p ) ),
Sequential Price Games
The leader’s profit function is
so the leader chooses the price level
p* for which profit is maximized.
The followers collectively supply
Yf(p*) units and the leader supplies
the residual quantity D(p*) - Yf(p*).
L f L Fp p D p Y p c D p Y p( ) ( ( ) ( )) ( ( ) ( ))
Demand,D(p)=a-bp
Cost,
–Leader,c1(y1)=cy1
–Follower,c2(y2)= y22/2
Follower’s supply,p= y2
Residual demand for leader:
–R(p)=D(p)-S2(p)=a-bp-p=a-(b+1)p
Solve for a monopoly problem
Sequential Price Games - Example
Sequential Price Games
