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Cournot versus Stackelberg
Cournot duopoly (simultaneous quantity competition)
Stackelberg duopoly (sequential quantity competition)
x2x1
1
2
x1
x2 2
1
Homogeneous duopoly(linear case) Two firms (i=1,2) produce a homogenous
goods. Outputs: x1 and x2, X= x1+x2
Marginal costs: MC1 and MC2
Inverse demand function:
Profit function of firm 1:
21 xxbabXaXp
1121111211 , xcxxbaxcxXpxx
Cournot-Nash equilibrium
Profit functions: Reaction functions:
Nash equilibrium: ),( 21CC xx
),(),,( 212211 xxxx
),(maxarg)(
),(maxarg)(
21212
21121
2
1
xxxx
xxxx
xR
xR
CCR
CCR
xxx
xxx
212
121
)(
)(
Computing the Cournot equilibrium(accommodation) Profit function of firm 1
Reaction function of firm 1
Nash equilibrium
1121111211 ))(()(),( xcxxbaxcxXpxx
22)( 21
21
x
b
caxxR
bcca
xx
ccap
bcca
xb
ccax
C
C
CC
9)2(
),(
3)(
32
,3
2
221
211
21
122
211
Depicting the Cournot equilibrium
x 1
Cournot-NashequilibriumC
)( 21 xxR
)( 12 xxR
x 2
Mx2
Cx2
Cx1Mx1
Exercise (Cournot)
Find the equilibrium in a Cournot competition. Suppose that the demand function is given by p(X) = 24 - X and the costs per unit by c1 = 3, c2 = 2.
Common interests
c1, c2 obtaining government subsidies and negotiating with labor unions
a , b advertising by the chemistry industrie
Exercise (taxes in a duopoly)
Two firms in a duopoly offer petrol. The demand function is given by p(X)=5-0.5X.Unit costs are c1=0.2 and c2=0.5.
a) Find the Cournot equilibrium and calculate the price.
b) Now suppose that the government imposes a quantity tax t (eco tax). Who ends up paying it?
Two approaches to cost leadership
Direct approach (reduction of own marginal costs)- change of ratio between fixed and variable
costs- investments in research and development
(R&D) Indirect approach (“raising rivals’ costs”)
- sabotage- minimum wages, enviromental legislation
Direct approach, analytically
Direct approach (reduction of your own marginal costs):
<0 <0 >0 =0direct strategiceffect effect
0
1
1
1
1
1
2
2
1
1
1
1
1
dcdx
xdcdx
xcdcd CCC
12111111 ,, cxcxcc CCC
Direct approach, graphically
x 2
x 1
equilibria: increase in production of firm 1
marginal cost
reduction of firm 1)( 21 xxR
)( 12 xxR
Indirect approach, analytically
Indirect approach (raising rival’s cost):
=0 <0 <0 =0 direct strategic effect effect
02
1
1
1
2
2
2
1
2
1
2
1
dcdx
xdcdx
xcdcd CCC
22212121 ,, cxcxcc CCC
Indirect approach, graphically
incr
ease
of m
argi
nal
costs
of f
irm 2
x 2
x 1
equilibria: increase in production of firm 1
)( 21 xxR
)( 12 xxR
Reaction curve in the linear case
x 1
)( 12 xxR
x 2
Mx2
Lx1
Note: alone leads to a price of .b
caxL 2
1
2c
otherwise
b
caxif
x
b
caxxR
022)(
21
12
12
Blockaded entry, graphically
x 2
x 1
C
M
firm 1as a monopolist
)( 21 xxR
)( 12 xxR
Mx1
Mx2
02 x
Blockaded entry
Entry is blockaded for each firm:
Entry is blockaded for firm 2:
acandac 21
)(21
)(
32
0
112
122
cacpc
bcca
x
M
C
andac 1
Blockaded entry (overview)
c 2
c 1
duopoly
no supplyfirm 1 as a
monopolist
firm 2 as a monopolist
a2
1
a2
1
a
a
Exercise (Strategic trade policy)
Two firms, one domestic (d), the other foreign (f), engage in Cournot competition on a market in a third country. Assume
The domestic government subsidizes its firm’s exports using a unit subsidy s.
Which subsidy s maximizes domestic welfare
?scsxscsW Cd
Cd
.: 21 ccc
The Herfindahl index
The Herfindahl index measures the concentration in an industry.
Exercise:Which market is the most concentrated:
n
ii
n
i
i sX
xH
1
22
1
2 firms with equal market shares,
3 firms with shares of 0.8, 0.1 and 0.1 or
3 firms with shares of 0.6, 0.2 and 0.2 ?
0.50.660.44
n firms in Cournot competition
Total industry output:
Firm i’s profit function:
Firm i’s marginal revenue:
nxxxX ...21
iiin
ini
xcxxxba
xXpxx
))...((
)(),...,(
1
1
pX
iiii
sp
dXdp
pX
Xx
pdXdp
xpxMR,
11)(
Lerner index of monopoly power
First order condition:
Lerner index for one firm:
Lerner index for the industry:
)()()(!
ii
ii xMCdx
dX
dX
dpxXpxMR
pX
ipX
i
i s
p
spp
p
MCp
,
,
1
pX
n
i pX
ii
n
i
ii
Hss
p
MCps
,1 ,1
Stackelberg equilibrium
Profit function
Follower’s reaction function
Leader’s optimal quantity
Nash equilibrium:
),(),,( 212211 xxxx
),(maxarg)( 21212 2xxxx x
R
)(,maxarg 12111 1xxxx R
xS
),( 21RS xx
Finding the profit-maximizing point on the follower's reaction curve
Accommodation
x 1
)( 12 xxR
x 2
Mx2
Lx1
Blockade or deterrence
Computing the Stackelberg equilibrium (accommodation) Reaction function of firm 2:
Profit function of firm 1:
Nash equilibrium
with and
22)( 12
12
x
b
caxxR
1112
11211 22))(,( xc
x
b
caxbaxxx R
b
ccax
xb
ccax
S
RS
4
32
,2
2
212
221
1
4
2 21 ccapS
Depicting the Stackelberg outcome (both firms produce)x 2
x 1
quantities in a Stackelberg equilibrium
CS
)( 21 xxR
)( 12 xxR
Mx2
Cx2
Sx1Cx1
Sx2
Exercise (Equilibria)
Which is an equilibrium in the Stackelberg model?
Are there any additional Nash equilibria ?
?,
,,
,,
21
21
121
CC
RS
SRS
xx
xx
xxx
Cournot versus Stackelberg II
Profit function of firm 1
First order condition for firm 1
direct effect follower effect Cournot: 0
)()(),( 111211 xCxXpxx
)()(
)()(
11
!
1
211
1
2
1
11
11
1
1
xMCdx
dx
dX
dpx
dX
dpxXp
dx
dx
dx
dx
dX
dpxXp
dx
dX
dX
dpxXp
dx
dR
R
R
Exercise (Stackelberg)
Find the equilibrium in a Stackelberg competition. Suppose that the demand function is given by p(X) = 24 - X and the costs per unit by c1 = 3, c2 = 2.
Possible or not? : ? SC11
Blockaded entry
p
x 1x L1
blockaded entry for firm 2
)( 1xp
MS xx 11
Mp1
2c
1c
a
Lx1
Reaction functions in the case of blockaded entry
Lx11x
2x
MS xx 11
)( 12 xxR
)( 21 xxR
Deterring firm 2’s entry
p
x 1
1 L
)( 1xpa
Mp1
2c
1c
Mx1LS xx 11
Reaction functions in the case of deterred entry
1x
2x
Mx1
)( 12 xxR
)( 21 xxR
LS xx 11
Blockaded and deterred entry
Blockaded entry (firm 2):
Deterred entry (firm 2):
acand
cac
acandxxorcpc MLM
11
2
11112
2
)(
2,
32 1
21
2
111
cac
cac
xxx MLS
Blockade and deterrence
firm 1 as a monopolist
c 2
c 1
duopoly
no supplyblockade
firm 2 as a monopolist
a
a2
1
a3
1
a2
1 a
deterrence
Exercise I (entry and deterrence)
1) Suppose a monopolist faces a demand of the form p(X)=10-0.5X.The firm’s unit costs are 2.
a) Find the profit-maximizing quantity and price. Is entry blockaded for a potential entrant with unit costs of 8?
b) Assume now that the potential entrant’s unit costs decrease to 4. Is entry still blockaded?
c) Find the limit output level and price. Should the incumbent deter?
Exercise II (entry and deterrence)
2) In addition, the firms are supposed to face quasi-fixed costs of 1.Hence, the cost functions are given by
a) What is the monopolist’s profit-maximizing quantity and price?
b) Is entry blockaded for the potential entrant?
c) Find the incumbent’s profit-maximizing output level. (Hint: Compare the profits.)
00
041)(
00
021)(
2
2222
1
1111
xif
xifxxc
xif
xifxxc
The quantity cartel
The firms seek to maximize joint profits
Optimization conditions
0)()()()(
0)()()()(
!
22212
21
!
11211
21
xMCdX
dpxxXp
x
xMCdX
dpxxXp
x
)()())((
),(),(
221121
212211
xCxCxxXp
xxxx
Cartel quantities
x 2
x 1
CS
CAMx12
1
quantities in a symmetric cartel
)( 21 xxR
)( 12 xxR
Cx1MS xx 11
SM xx 222
1
Cx2
Mx2
The cartel agreement
The optimization condition is given by
Each firm will be tempted to increase its profits by unilaterally expanding its output.
In order to maintain a cartel, the firms need a way to detect and punish cheating, otherwise the temptation to cheat may break the cartel.
0)()( 2
!
111 dX
dpxxMC
dX
dpxXp
Summary
The cartel is unstable if the firms meet only once. Thus, the cartel agreement is not an equilibrium.
Supposing that the number of repetitions is unknown, the cartel agreement can be an equilibrium outcome.
Exercise (cartel quantities)
Consider a cartel in which each firm has identical and constant marginal costs. If the cartel maximizes total industry profits, what does this imply about the division of output between the firms?
Solution:Nothing. Since all firms have the same marginal cost, it doesn’t matter which of them produces the output.
Hal R. Varian, Intermediate Microeconomics
The outcomes of our models
quantity
pricea
monopoly (M) and cartel (CA)Cournot (C)
Stackelberg (S)perfect competition (PC)
p M
p C
p S
p PC = c
X M X C X S X PC
Cournot – Executive summary I A duopoly can only be expected if entry is blockaded
for other firms. Otherwise industry profits would attract potential competitors. An entry can be blockaded by cost structures or by certain laws or conditions (legal/administrative barriers).
All competing firms have the incentive to lower costs in common. Firms‘ profit and the industry profit increase. Thus, activities in associations which aim at improving cost structures are worthwhile (collective agreements, government technology promotion,...)
...43 ccpC
Cournot – Executive summary II
All competing firms have the incentive to increase market demand by appropriate marketing activities. Firms‘ profit and the industry profit increase. Thus, activities in associations which aim at increasing market demand are worthwhile (cooperative advertising campaigns,...)
The attempt to become a cost leader before a simultaneous competition is worthwhile. The firm with the lower marginal cost or unit cost respectively faces higher turnovers and profits. Cost leadership even leads to monopolization.
Cournot – Executive summary III
There are two approaches to cost leadership. The direct approach is to lower your own marginal cost. The indirect approach is known as “raising rivals‘ costs“.
Stackelberg – Executive Summary I
Time leadership is worthwhile: in a Stackelberg equilibrium the leader realizes a profit that is higher than the follower‘s and his own in a Cournot equilibrium. Even with a slight cost disadvantage, the leader‘s profit might be higher than the follower‘s.
If a firm is both Stackelberg leader and cost leader, it can consider deterring the follower‘s entry by offering the limit quantity as a strategic entry barrier.
Stackelberg – Executive Summary II Entry costs make the follower‘s deterrence
easier. The amount of its costs at which entry is deterred increases with the entry costs.
Since the Stackelberg leader offers a higher quantity than the follower, he might enlarge his cost advantage by learning curve effects.
Cartel – Executive Summary I
If all firms keep the cartel agreement, then they can increase their profits compared to the Cournot competition.
Nevertheless cartels are unstable. Cheating even makes higher profits possible. When all firms break up the cartel agreement, their profits decrease. Moreover, they might face a worse position than in the Cournot competition.
Cartel – Executive Summary II
Cartels will not continue to exist when potential competitors‘ entries, who are attracted by the cartel profit, are not blockaded. Therefore, the firms have to be able and willing to restrict or deter entry. If there are administrative barriers, investments in maintenance of these barriers are necessary. If the barriers are more structural, then the firms have to try to keep their cost and technology advantage (cooperation in R&D).