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).