oligopoly 寡头垄断. a monopoly is an industry consisting a single firm. a duopoly is an...
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Chapter Twenty-Seven
Oligopoly寡头垄断
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.
The Four Types of Market Structure
Copyright © 2004 South-Western
• Tap water• Cable TV
Monopoly
• Novels• Movies
MonopolisticCompetition
• Tennis balls• Crude oil
Oligopoly
Number of Firms?
Perfect
• Wheat• Milk
Competition
Type of Products?
Identicalproducts
Differentiatedproducts
Onefirm
Fewfirms
Manyfirms
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 (数量竞争) 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).
Augustin Cournot
(1838) created a model that is
based on relatively simple
assumption: ignore the
interdependency with rivals.
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 2
215( ) . 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
So, given y2, firm 1’s profit-maximizingoutput level solves
y
y y y1
1 2 160 2 2 0 .
I.e. firm 1’s best response to y2 isy R y y1 1 2 215 1
4 ( ) .
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
So, given y1, firm 2’s profit-maximizingoutput level solves
y
y y y2
1 2 260 2 15 2 0 .
I.e. firm 2’s best response to y1 isy R y y
2 2 1145
4 ( ) .
Quantity Competition; An Example
y2
y1
Firm 2’s “reaction curve”y R y y
2 2 1145
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 (古诺均衡) if ).( *
12*2 yRy )( *
21*1 yRy and
Quantity Competition; An Example
y R y y1 1 2 215 14
* * *( ) y R y y2 2 1
1454
* **
( ) . and
Substitute for y2* to get
y y y11
115 14
454
13**
*
Hence y245 13
48* .
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 y1 1 2 215 1
4 ( ) .
y R y y2 2 1
1454
( ) .
45/4
45
Quantity Competition; An Example
y2
y1
Firm 2’s “reaction curve”
48
60
Firm 1’s “reaction curve”y R y y1 1 2 215 1
4 ( ) .
8
13
Cournot-Nash equilibrium y y1 2 13 8* *, , .
y R y y2 2 1
1454
( ) .
Quantity Competition
1 1 2 1 2 1 1 1( ; ) ( ) ( )y y p y y y c y
11
1 2 11 2
11 1 0
yp y y y p y y
yc y ( ) ( ) ( ) .
Generally, given firm 2’s chosen outputlevel 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
22
1 2 21 2
22 2 0
yp y y y p y y
yc y ( ) ( ) ( ) .
Similarly, given firm 1’s chosen outputlevel 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 equilibriumy1* = R1(y2*) and y2* = R2(y1*)y2
*
y R y2 2 1 ( ).
y1*
Case: Cournot Solution:假设两家欧洲电子公司西门子 ( 厂商 S) 和汤姆森 -CSF( 厂商 T) 共同持有一项用于机场雷达系统的零件的专利权 .
.50.2995
25.199570000
25.05700001000
15.05110000
25.0570000
1000
2
2
2
2
STS
S
SSTSS
SSSTSSSS
TTT
SSS
TS
QQQ
QQQQ
QQQQQTCPQ
:。,
QQTC
QQTC
:。P,
QQP:
于西门子公司的总利润等最大化各自都谋求自己利润的假设两家厂商独立行动
零件的总成本函数为两家厂商制造和销售此为市场销售价格的销售量式中和分别为两家厂商
此零件的需求函数为(13-1)
(13-3)
(13-2)
.17.3536$;00.22695$
,/47.413*P.21.314Q,32.272Q
,
995Q30.2Q995QQ50.2
,
.Q30.2Q995Q
Q15.1QQQ995110000
Q15.0Q5110000QQQ1000
TCPQ:,
TS
TS
TS
TS
TST
T
2TSTTT
2TTTTST
TTT
单位美元单位单位
量水平为解得两家厂商的最优产
联立两公式
于汤姆森公司的总利润等同样
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 profitincreases as y2 decreases.
y2
y1
Increasing profitfor 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 thatmaximizes firm 1’s profit?
y2’
y2
y1
Iso-Profit Curves for Firm 1Q: Firm 2 chooses y2 = y2’.Where along the line y2 = y2’ is the output level thatmaximizes firm 1’s profit? A: The point attaining thehighest 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 1Q: Firm 2 chooses y2 = y2’.Where along the line y2 = y2’ is the output level thatmaximizes firm 1’s profit? A: The point attaining thehighest 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(反应曲线) passes through the “tops”of firm 1’s iso-profitcurves.
Iso-Profit Curves for Firm 1
y2
y1
Iso-Profit Curves for Firm 2
Increasing profitfor firm 2.
y2
y1
Iso-Profit Curves for Firm 2
Firm 2’s reaction curvepasses through the “tops”of firm 2’s iso-profitcurves.
y2 = R2(y1)
If there are N-firms in the market
Question: How to find Cournot Nash Equilibrium
solution? What if N is a number big enough?
Collusion (串谋) Q: Are the Cournot-Nash equilibrium
profits the largest that the firms can earn in total?
Collusiony2
y1y1*
y2*
Are there other output levelpairs (y1,y2) that givehigher profits to both firms?
(y1*,y2*) is the Cournot-Nashequilibrium.
Collusiony2
y1y1*
y2*
Are there other output levelpairs (y1,y2) that givehigher profits to both firms?
(y1*,y2*) is the Cournot-Nashequilibrium.
Collusiony2
y1y1*
y2*
Are there other output levelpairs (y1,y2) that givehigher profits to both firms?
(y1*,y2*) is the Cournot-Nashequilibrium.
Collusiony2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nashequilibrium.
Higher 2
Higher 1
Collusiony2
y1y1*
y2*
Higher 2
Higher 1y2’
y1’
Collusiony2
y1y1*
y2*y2’
y1’
Higher 2
Higher 1
Collusiony2
y1y1*
y2*y2’
y1’
Higher 2
Higher 1
(y1’,y2’) earnshigher profits forboth firms thandoes (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.
Collusiony2
y1y1*
y2*y2’
y1’
Higher 2
Higher 1
(y1’,y2’) earnshigher profits forboth firms thandoes (y1*,y2*).
Collusiony2
y1y1*
y2*y2’
y1’
Higher 2
Higher 1
(y1’,y2’) earnshigher profits forboth firms thandoes (y1*,y2*).
(y1”,y2”) earns stillhigher profits forboth firms.
y2”
y1”
Collusiony2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profitwhile leaving firm 2’s profit at the Cournot-Nash equilibrium level.
~ ~
Collusiony2
y1y1*
y2*
y2~
y1~
(y1,y2) maximizes firm 1’s profitwhile leaving firm 2’s profit at the Cournot-Nash equilibrium level.
~ ~
y2
_
y2
_
(y1,y2) maximizes firm2’s profit while leaving firm 1’s profit at the Cournot-Nash equilibrium level.
_ _
Collusiony2
y1y1*
y2*
y2~
y1~
y2
_
y2
_
The path of output pairs thatmaximize one firm’s profit while giving the other firm at least its CN equilibrium profit.
Collusiony2
y1y1*
y2*
y2~
y1~
y2
_
y2
_
The path of output pairs thatmaximize 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.
Collusiony2
y1y1*
y2*
y2m
y1m
(y1m,y2
m) denotesthe output levelsthat maximize thecartel’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 y1
m units, is it profit-maximizing for firm 2 to continue to produce y2
m units?
Collusion Firm 2’s profit-maximizing response
to y1 = y1m is y2 = R2(y1
m).
Collusiony2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm1 choosing y1 = y1
m.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(y1
m) > y2m.
Firm 2’s profit increases if it cheats on firm 1 by increasing its output level from y2
m to R2(y1m).
Similarly, firm 1’s profit increases if it cheats on firm 2 by increasing its output level from y1
m to R1(y2m).
Collusiony2
y1
y2m
y1m
y2 = R2(y1m) is firm 2’s
best response to firm1 choosing y1 = y1
m.
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.
Collusion versus Competition? Sometimes collusion succeeds Sometimes forces of competition win out
over collective action When will collusion tend to succeed?
There are six factors that influence successful collusion as follows:
Factors Affecting Likelihood of Successful Collusion
1. Number and Size Distribution of Sellers. Collusion is more successful with few firms or if there exists a dominant firm.
2. Product Heterogeneity. Collusion is more successful with products that are standardized or homogeneous
3. Cost Structures. Collusion is more successful when the costs are similar for all of the firms in the oligopoly.
4. Size and Frequency of Orders. Collusion is more successful with small, frequent orders.
5. Secrecy and Retaliation. Collusion is more successful when it is difficult to give secret price concessions.
6. Percentage of External Orders. Collusion is more successful when percentage of orders outside of the cartel is small.
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 (决策顺序) What if firm 1 chooses its output level
first and then firm 2 responds to this choice?
Firm 1 is then a leader (领导者) . Firm 2 is a follower (追随者) .
The competition is a sequential game in which the output levels are the strategic variables.
The Order of Play Such games are Stackelberg (斯塔克尔伯格) 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 then chooses y1 to maximize its profit level.
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) = y1
2 and c2(y2) = 15y2 + y2
2. Firm 2 is the follower. Its reaction
function isy R y y
2 2 1145
4
( ) .
Stackelberg Games; An Example
1 1 1 2 1 1 12
11
1 12
1 12
60
60 454
1954
74
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 12
11
1 12
1 12
60
60 454
1954
74
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
472
13 91 1 y ys .
Stackelberg Games; An Example
Q: What is firm 2’s response to theleader’s choice ys
1 13 9 ?
Stackelberg Games; An Example
Q: What is firm 2’s response to theleader’s choice
A:
ys1 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 theleader’s choice
A:
ys1 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 itsC-N output and the follower produces lessthan its C-N output. This is true generally.
Stackelberg Gamesy2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nashequilibrium.Higher 2
Higher 1
Stackelberg Gamesy2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nashequilibrium.
Higher 1
Follower’sreaction curve
Stackelberg Gamesy2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nashequilibrium. (y1
S,y2S) is the
Stackelberg equilibrium.
Higher 1
y1S
Follower’sreaction curve
y2S
Stackelberg Gamesy2
y1y1*
y2*
(y1*,y2*) is the Cournot-Nashequilibrium. (y1
S,y2S) is the
Stackelberg equilibrium.
y1S
Follower’sreaction curve
y2S
Price Competition (价格竞争) 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.
Case : 方便面的价格战 据尼尔森最新统计:康师傅,华龙与统一在市场占有率上依次排列,形成三分天下格局。 为什么华龙集团能够从一家小型乡镇民营企业发展起来? 低价面铺市场,中档面创效益,高档面树形象。
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 (价格领导) .
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(剩余需求曲线)
Hence the leader’s profit function is
L p D p Y pf( ) ( ) ( ).
(p))Y(D(p)c(p))Yp(D(p)(p) fLfL
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*).
(p))Y(D(p)c(p))Yp(D(p)(p) fLfL
Demand: D(p)=a-bp Cost:
Leader: c1(y1)=cy1 Follower: c2(y2)= y2
2/2 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
£
QO
Sall other firms
Dmarket
Dleader
PL
MRleader
QL QF QT
f t
Determination of price and output
l
Dominant firm price leadership
Dleader
MCleader
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