Advanced Microeconomics
Price and quantity competition
Harald Wiese
University of Leipzig
Harald Wiese (University of Leipzig)
Advanced Microeconomics
1 / 92
Part C. Games and industrial organization
1
Games in strategic form
2
Price and quantity competition
3
Games in extensive form
4
Repeated games
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Price and quantity competition
overview
1
Monopoly: Pricing policy
2
Price competition
3
Monopoly: Quantity policy
4
Quantity competition
Harald Wiese (University of Leipzig)
Advanced Microeconomics
3 / 92
De…nitions
Monopoly: one …rm sells
Monopsony: one …rm buys
p
Π
X
Π
…rst: pricing policy for a monopolist
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The linear model
demand properties
Demand function
X (p ) = d
d, e
0, p
ep
d
e
Problem
Find
the saturation quantity,
the prohibitive price and
the price elasticity of demand
of the above demand curve!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
5 / 92
The linear model
demand properties
Solution
X
saturation quantity
X (0) = d
ε X ,p = 0
d
X (p)
prohibitive price de
(solve X (p ) = 0 for
the price)
ε X ,p = 1
ε X ,p = ∞
price elasticity of
demand
εX ,p
=
d
2e
dX p
dp X
= ( e)
d
e
p
p
d
Harald Wiese (University of Leipzig)
ep
Advanced Microeconomics
6 / 92
The linear model
pro…t
De…nition
X is the demand function.
Π (p ) : =
| {z }
pro…t
R (p )
| {z }
revenue
= pX (p )
C (p )
| {z }
cost
C [X (p )]
– monopoly’s pro…t in terms of price p.
Π (p ) = p (d
c, d, e
0, p
ep )
d
e
c ((d
ep )) ,
– pro…t in linear model.
Note the dependencies: Price 7! Quantity 7! Cost
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The linear model
decision situation
De…nition
A tuple
(X , C )
is the monopolist’s decision situation with price setting;
X – demand curve
C – function
pro…t-maximizing price de…ned by
p R (X , C ) := arg max Π(p )
p 2R
p M := p R (X , C ) – monopoly price.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The linear model
decision situation: graph I
X,R
d
R
X
p?
p Rmax
d
e
p
Problem
Find the economic meaning of the question mark!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
9 / 92
The linear model
decision situation: graph I
Solution
No meaning!
X,R
d
R
X
p?
p Rmax
d
e
p
Units:
Prices:
monetary units
quantity units
Revenue = price
quantity:
monetary units
quantity units
quantity units
= monetary units
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Advanced Microeconomics
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The linear model
decision situation: graph II
C, R
cd
C
R
p?
p?
p?
p?
p
Problem
Find the economic meaning of the question marks!
Harald Wiese (University of Leipzig)
Advanced Microeconomics
11 / 92
The linear model
decision situation: graph II
Solution
C, R
cd
C
R
p Π =0
Harald Wiese (University of Leipzig)
p Rmax
p Π max
p X =C = R= Π =0
Advanced Microeconomics
p
12 / 92
Marginal revenue and elasticity
di¤erentiating with respect to price
Marginal revenue with respect to price:
dR (p )
d [pX (p )]
dX
=
= X +p
dp
dp
dp
Amoroso-Robinson equation:
dR (p )
=
dp
... > 0
... = 0
for
for
X (p ) [jεX ,p j
1]
εX ,p < 1
εX ,p = 1
Problem
Comment: A …rm can increase pro…t if it produces at a point where
demand is inelastic, i.e., where 0 > εX ,p > 1 holds.
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Advanced Microeconomics
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Maximizing revenue
X ,R
d
ε X , p = −1
R
X
p Rmax =
R (p ) = p (d
d
p R max =
2e
Harald Wiese (University of Leipzig)
d
2e
ep ) = pd
Advanced Microeconomics
d
e
p
ep 2
14 / 92
Marginal cost
w.r.t. price and w.r.t. quantity
dC
dX
dC
dp
: marginal cost (with respect to quantity)
: marginal cost with respect to price
dC dX
dC
=
< 0.
dp
dX
dp
|{z} |{z}
>0
Harald Wiese (University of Leipzig)
<0
Advanced Microeconomics
15 / 92
Pro…t maximization
FOC:
dR ! dC
=
dp
dp
equivalent to “price-cost margin” rule (shown later):
dC
dX
p
p
!
=
1
jεX ,p j
.
Problem
ce
Con…rm: For linear demand p M = d +
2e . What price maximizes revenue?
M
How does p change if c changes?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
16 / 92
Price di¤erentiation
First-degree price di¤erentiation — > monopoly quantity policy
Third-degree price di¤erentiation
Problem
Two demand functions:
X1 (p1 ) = 100
p1
X2 (p2 ) = 100
2p2
c = 20.
a) Price di¤erentiation
b) No price di¤erentiation
Hint 1: Find prohibitve prices in each submarket in order to
sum demand
Hint 2: You arrive at two solutions. Compare pro…ts.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Price di¤erentiation
solution
Third degree: Solve two isolated pro…t-maximization problems; obtain
p1M
p2M
= 60,
= 35.
The prohibitive prices are 100 and 50. The aggregate demand is
8
p > 100
< 0,
100 p, 50 < p 100
X (p ) =
:
200 3p, 0 p 50.
Two local solutions: p =43 31 and p =60. Comparison of pro…ts:
maximum at
1
p M = 43 .
3
Harald Wiese (University of Leipzig)
Advanced Microeconomics
18 / 92
Price and quantity competition
overview
1
Monopoly: Pricing policy
2
Price competition
3
Monopoly: Quantity policy
4
Quantity competition
Harald Wiese (University of Leipzig)
Advanced Microeconomics
19 / 92
Price versus quantity competition
Cournot 1838 :
Bertrand 1883 :
x1
Π1
x2
Π2
p1
Π1
p2
Π2
Bertrand criticizes Cournot, but Kreps/Scheinkman 1983:
simultaneous capacity competition
+ simultaneous price competition (Bertrand competition)
= Cournot results
Harald Wiese (University of Leipzig)
Advanced Microeconomics
20 / 92
Simultaneous versus sequential competition
Cournot 1838 :
Stackelberg 1934:
x1
Π1
x2
Π2
x1
x2
Π1
Π2
…rst: simultaneous pricing game = Bertrand model
Harald Wiese (University of Leipzig)
Advanced Microeconomics
21 / 92
The game
demand and costs
Assumptions:
homogeneous product
x1
X ( p1 )
consumers buy best
linear demand
Demand for …rm 1:
8
< d ep1 , p1 < p2
d ep 1
x1 (p1 , p2 ) =
,
p1 = p2
: 2
0,
p1 > p2
Unit cost c1 :
Π1 (p1 , p2 ) = (p1
Harald Wiese (University of Leipzig)
1
2
X ( p2 )
p2
p1
c1 )x1 (p1 , p2 )
Advanced Microeconomics
22 / 92
The pricing game
De…nition
Γ = N, (Si )i 2N , (Πi )i 2N ,
– pricing game (Bertrand game)
with
N – set of …rms
Si := 0, de – set of prices
Πi : S ! R – …rm i’s pro…t function
Equilibria: ‘Bertrand equilibria’or ‘Bertrand-Nash equilibria’
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Advanced Microeconomics
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Accomodation and Bertrand paradox
How is the incumbent’s position toward entry
Bain 1956:
Accomodated entry
Blockaded entry
Deterred entry
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Advanced Microeconomics
24 / 92
Accomodation and Bertrand paradox
Bertrand paradox
Assumption: c := c1 = c2 <
d
e
Highly pro…table undercutting
) Nash-equilibrium candidate: p1B , p2B = (c, c )
Lemma
Only one equilibrium p1B , p2B = (c, c ) .
x1B
ΠB1
1
d ec
= x2B = X (c ) =
2
2
= ΠB2 = 0
Problem
Assume two …rms with identical unit costs of 10. The strategy sets are
S1 = S2 = f1, 2, ..., g . Determine both Bertrand equilibria.
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Advanced Microeconomics
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Economic genius: Joseph Bertrand
Joseph Louis François Bertrand (1822
– 1900) was a French mathematician
and pedagogue.
In 1883, he developed the
price-competition model while
criticising the Cournot model of
quantity competition.
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Advanced Microeconomics
26 / 92
Accomodation and Bertrand paradox
Escaping the Bertrand paradox
Theory of repeated games — > chapter after next
Di¤erent average costs — > this chapter
Price cartel — > agreement to charge monopoly prices
Products not homogeneous, but di¤erentiated — > next chapter
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Advanced Microeconomics
27 / 92
Blockaded entry and deterred entry
market entry blockaded for both …rms (case 1)
Now c1 < c2
c1
d
e , c2
d
e
Market entry blockaded for both …rms
Problem
Which price tuples (p1 , p2 ) are equilibria?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Blockaded entry and deterred entry
market entry of …rm 2 blockaded (case 2)
c1 <
d
e
and c2 < p1M
Market entry of …rm 2 blockaded
Take p2 := c2 in the …gure
Π1
p1M ≤ p 2
c1
Harald Wiese (University of Leipzig)
p1M
p2
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p1
29 / 92
Blockaded entry and deterred entry
market entry of …rm 2 blockaded (case 2)
Equilibrium:
p1B , p2B
Π1
=
p1M , c2
=
d
c1
+ , c2
2e
2
p1M ≤ p 2
c1
x1B
ΠB1
= (d
= (d
ec1 ) /2,
p1M
p2
p1
x2B = 0
ec1 )2 /(4e ),
ΠB2 = 0
Problem
Can you …nd other equilibria?
All strategy combinations p1M , p2 ful…lling p2 > p1M are also equilibria.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
30 / 92
Blockaded entry and deterred entry
market entry of …rm 2 deterred (case 3)
c1 < de and c2 p1M .
Market entry of …rm 2 deterred
Take p2 := c2 in the …gure
…rm 1 prevents entry by setting limit price
p1L (c2 ) := c2
Π1
c1 < p2 < p1M
c1
Harald Wiese (University of Leipzig)
ε.
p2
p1M
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p1
31 / 92
Blockaded entry and deterred entry
market entry of …rm 2 deterred (case 3)
One Bertrand-Nash equilibrium is
p1B , p2B
= p1L (c2 ), c2 = (c2
x1B
d
ΠB1
( c2
ec2 ,
x2B = 0,
c1 ) ( d
Harald Wiese (University of Leipzig)
ε, c2 )
ec2 ) ,
ΠB2 = 0.
Advanced Microeconomics
Π1
c1 < p2 < p1M
c1
p2
p1M
p1
32 / 92
Blockaded entry and deterred entry
summary I
c2
no supply
(case 1)
d
e
blockade (case 2)
monopoly 1
d
2e
deterrence
(case 3)
monopoly 2
deterrence
(case 5)
d
2e
Harald Wiese (University of Leipzig)
duopoly,
Bertrand-Paradox
(case 4)
blockade (case 6)
d
e
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c1
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Blockaded entry and deterred entry
summary II
1. no supply,
2. Entry of …rm 2 blockaded
3. Entry of …rm 2 deterred
4. Bertrand-Paradox
5. Entry of …rm 1 deterred
6. Entry of …rm 1 blockaded
Harald Wiese (University of Leipzig)
d
c1
and
e
d
c2
e
0 c1 < de and
p1M = d +2eec1 < c2
0 c1 < de and
d +ec1
c1 < c2
= p1M
2e
c1 = c2 =: c and
0 c < de
0 c2 < de and
d +ec2
c2 < c1
= p2M
2e
0 c2 < de and
p2M = d +2eec2 < c1
Advanced Microeconomics
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Price and quantity competition
overview
1
Monopoly: Pricing policy
2
Price competition
3
Monopoly: Quantity policy
4
Quantity competition
Harald Wiese (University of Leipzig)
Advanced Microeconomics
35 / 92
The linear model
preliminaries
Problem
Assume linear inverse demand p (X ) = a
bX , a, b > 0. Determine
1
the slope of the inverse linear demand function,
2
the slope of its marginal-revenue curve,
3
saturation quantity and
4
prohibitive price.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The linear model
preliminaries
Solution
1
2
p
The slope of the inverse
demand curve is
dp/dX = b
Revenue: R (X )
= p (X ) X = aX
MR: dR (X ) /dX
= a 2bX .
Slope: 2b
1
b
bX 2
p (X )
2b
1
MR
a
2b
3
Saturation quantity: a/b
4
a is the prohibitive price.
Harald Wiese (University of Leipzig)
a
Advanced Microeconomics
a
b
X
37 / 92
The linear model
de…nition pro…t function
De…nition
X
0; p inverse demand function.
Π (X ) : = R (X )
| {z }
| {z }
pro…t
revenue
C (X ) = p (X ) X
| {z }
C (X )
cost
– monopoly’s pro…t in terms of quantity
Linear model:
Π (X ) = (a
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bX ) X
cX ,
Advanced Microeconomics
X
a
,
b
38 / 92
The linear model
de…nition decision situation
De…nition
A tuple
(p, C ) ,
– monopolist’s decision situation with quantity setting where
p – inverse demand function
C – cost function
Quantity setting monopolist’s problem: Find
X R (p, C ) := arg max Π(X )
X 2R
– pro…t maximizing quantity
Notation:
X M := X R (p, C ) – monopoly quantity
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Advanced Microeconomics
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Marginal revenue
... and elasticity ... and price
Marginal revenue and elasticity
MR
dp
dX
1
= p 1+
εX ,p
= p+X
=p 1
1
> 0 for
jεX ,p j
dp
dX
Marginal revenue equals price: MR = p + X
dp
dX
=p
= 0 horizontal (inverse) demand: MR = p + X
…rst “small” unit, X = 0: MR = p + X
=0
dp
dX
— > see chapter on production theory
First-degree price di¤erentiation MR = p + X
=p=
=0
jεX ,p j > 1.
dp
dX
=0
=p
R (X )
X
dp
dX
— > see below
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Advanced Microeconomics
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Monopoly pro…t
average versus marginal de…nition
Pro…t at X̄ :
p
Π (X̄ )
a
= p (X̄ )X̄ C (X̄ )
= [p (X̄ ) AC (X̄ )] X̄
= pro…t (average de…nition)
=
ZX̄
p(X )
G
E
F
c
[MR (X )
D
MR
H
C
B
p (X )
AC
MC
A
MC (X )] dX
X
X
0
F (perhaps)
= pro…t (marginal de…nition)
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Advanced Microeconomics
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Pro…t maximization
…rst order condition
FOC (w.r.t. X ):
!
MC = MR.
Problem
Find X M for p (X ) = 24
X and constant unit cost c = 2!
(a haon déag)
Problem
Find X M for p (X ) =
Harald Wiese (University of Leipzig)
1
X
and constant unit cost c!
Advanced Microeconomics
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Pro…t maximization
linear model
p
p
ε X ,p = ∞
MC = AC
c
a
pM
ε X ,p = 1
D
ε X ,p = 0
M
E
A
p(X )
p (X )
MR
c
a
MR
F
B
XM
X PC
X
M
M
= X (c, a, b ) =
Harald Wiese (University of Leipzig)
X
X
(
1 (a c )
2 b ,
0,
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c a
c>a
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Pro…t maximization
comparative statics
X M (a, b, c )
p M (a, b, c )
ΠM (a, b, c )
=
=
=
1 (a c )
2 b ,
where
1
2 (a + c ),
2
1 (a c )
,
4
b
where
where
∂X M
∂c
< 0;
∂p M
∂c
> 0;
∂ΠM
∂c
< 0;
∂X M
∂a
∂p M
> 0;
∂a
> 0;
∂ΠM
∂a
> 0;
∂X M
∂b
∂p M
∂b
∂ΠM
∂b
< 0,
= 0,
< 0.
Problem
Consider ΠM (c ) =
2
1 (a c )
4
b
Harald Wiese (University of Leipzig)
and calculate
d ΠM
dc
Advanced Microeconomics
! Hint: Use the chain rule!
44 / 92
Pro…t maximization
the e¤ect of unit cost on pro…t I
Solution
d ΠM
dc
d
=
=
=
Harald Wiese (University of Leipzig)
2
1 (a c )
4
b
dc
1
2(a c ) ( 1)
4b
a c
2b
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Pro…t maximization
the e¤ect of unit cost on pro…t III
Reduced-form pro…t function ΠM (c ) = Π c, X M (c ) .
Forming the derivative with respect to c yields
d ΠM (c )
=
dc
∂Π
∂c
|{z}
∂Π
∂X X =X M
| {z
}
=0
+
<0
direct e¤ect
<0
FOC for pro…t maximization
|
dX M
.
| dc
{z }
{z
=0
higher marginal cost,
indirect e¤ect
lower output
}
Envelope theorem — > manuscript chapter “Comparative statics and
duality theory”
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Pro…t maximization
price and quantity
p
II
I
a
pM =
a+c
2
M
p (X )
MR
AC = MC
c
Π
(a − c)2
XM =
4b
a −c
2b
a−c
b
X
(a − c )2
III
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4b
IV
Advanced
Π Microeconomics
47 / 92
Pro…t maximization
exercise
Consider a monopolist with
the inverse demand function p (X ) = 26
the cost function C (X ) =
X3
14X 2
2X and
+ 47X + 13
Find the pro…t-maximizing price!
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Advanced Microeconomics
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Alternative expressions for pro…t maximization
1
!
MC = MR = p 1
1
!
p=
p
1
1
jεX ,p j
MC ! p
=
p
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jεX ,p j
jεX ,p j
MC
jεX ,p j 1
h
i
p 1 jε 1 j
1
X ,p
=
p
jεX ,p j
MC =
Advanced Microeconomics
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Alternative expressions for pro…t maximization
Lerner index
De…nition
In a monopoly:
p
MC
p
is the Lerner index of market power
perfect competititon: p = MC
Note:
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p
1
MC !
=
p
jεX ,p j
Advanced Microeconomics
50 / 92
Alternative expressions for pro…t maximization
Lerner index: monopoly power versus monopoly pro…t
p
pM
Cournot point
AC
p (X )
MR
XM
p > MC but AC X M
Harald Wiese (University of Leipzig)
MC
X
=
C XM
XM
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= pM
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First-degree price di¤erentiation
bachelor-level derivation
Every consumer pays his willingness to pay
MR = p + X
=0
dp
=p
dX
Price decrease following a quantity increase concerns
the marginal consumer,
not the inframarginal consumers.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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First-degree price di¤erentiation
formal analysis
Objective function
Marshallian willingness to pay
=
Z X
p (q ) dq
cost
C (X )
0
Di¤erentiating w.r.t. X :
!
p (X ) =
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dC
dX
Advanced Microeconomics
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First-degree price di¤erentiation
graph
p
a
pM
ε X ,p = ∞
ε X ,p = 1
D
p (X )
MR
c
A
ε X ,p = 0
M
E
B
XM
F
X PC
X
Pro…t for non-discriminating (Cournot) monopolist: ABME
Pro…t for discriminating monopolist: AFD
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Advanced Microeconomics
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Third-degree price di¤erentiation (two markets, one
factory)
optimality condition
Pro…t
Π (x1 , x2 ) = p1 (x1 ) x1 + p2 (x2 ) x2
C (x1 + x2 ) ,
FOCs
∂Π (x1 , x2 )
∂x1
∂Π (x1 , x2 )
∂x2
!
= MR1 (x1 )
MC (x1 + x2 ) = 0,
= MR2 (x2 )
MC (x1 + x2 ) = 0.
!
!
MR1 (x1 ) = MR2 (x2 )
Assume, to the contrary, MR1 < MR2 ...
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Advanced Microeconomics
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Third-degree price di¤erentiation (two markets, one
factory)
graph
p
market 2
market 1
p1*
p
p2
x2
*
2
x 2*
p1
x1*
MR2
x1
MR1
total output
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Advanced Microeconomics
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Third-degree price di¤erentiation (two markets, one
factory)
elasticities
MR1 (x1 ) = MR2 (x2 ) :
p1M 1
1
!
= p2M 1
j ε1 j
1
j ε2 j
jε1 j > jε2 j ) p1M < p2M .
Harald Wiese (University of Leipzig)
Advanced Microeconomics
57 / 92
Third-degree price di¤erentiation
exercise
Problem
A monopolist sells his product in two markets:
p1 (x1 ) = 100 x1 , p2 (x2 ) = 80 x2 .
1
Assume price di¤erentiation of the third degree and the cost function
given by C (X ) = X 2 . Determine the pro…t-maximizing quantities
and the pro…t.
2
Repeat the …rst part of the exercise with the cost function
C (X ) = 10X .
3
Assume, now, that price di¤erentiation is not possible any more.
Using the cost function C (X ) = 10X , …nd the pro…t-maximizing
output and price. Hint: You need to distinguish quantities below and
above 20.
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Advanced Microeconomics
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Third-degree price di¤erentiation
exercise: solution
Solution
1
The …rm’s pro…t function is
Π (x1 , x2 ) = p1 (x1 ) x1 + p2 (x2 ) x2
= (100
x1 ) x1 + (80
C (x1 + x2 )
x2 ) x2
(x1 + x2 )2 .
Partial di¤erentiations yield x1M = 20 and x2M = 10;
ΠM (20, 10) = 1400.
2
We …nd: x1M = 45 and x2M = 35; ΠM = 3250.
3
Aggregate inverse demand
p (X ) =
100
90
X,
1
2X,
X < 20
X
20.
At X M = 80, the monopolist’s pro…t is 3200 < 3250.
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One market, two factories
optimality condition
Pro…t
Π (x1 , x2 ) = p (x1 + x2 ) (x1 + x2 )
C1 (x1 )
C2 (x2 ) .
FOCS
∂Π (x1 , x2 )
∂x1
∂Π (x1 , x2 )
∂x2
!
= MR (x1 + x2 )
MC1 (x1 ) = 0,
= MR (x1 + x2 )
MC2 (x2 ) = 0.
!
!
MC1 = MC2
Assume MC1 < MC2 ...
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One market, two factories
graph
p
factory 2
factory 1
MC2
x2
MC1
x 2*
x1*
x1
total output
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Welfare-theoretic analysis of monopoly
introduction
Normative economics
Concepts
Marshallian consumers’rent
Producers’rent
Taxes
Monetary evaluation
The government is often assumed to maximize welfare
benevolent dictatorship
support maximization (chances of reelection) by bene…tting
consumers,
producers,
bene…ciaries of publicly provided goods and
tax payers.
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Welfare-theoretic analysis of monopoly
perfect competition as benchmark
Price taking & pro…t-maximizing
) p = MC
Marginal consumer’s willingness
to pay
=
marginal …rm’s loss
compensation
p
supply
CR
R
PR
Consumers’
+
producers’rents maximal
demand
X
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Advanced Microeconomics
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Welfare-theoretic analysis of monopoly
Cournot monopoly
Note:
X M < X PC
p
MC
pM
p = MC
p PC
MR = MC
MR
XM
X PC
p (X )
X
Problem
No price di¤erentiation, marginal-cost curve MC = 2X and inverse
demand p (X ) = 12 2X . Determine the welfare loss! Hint: Sketch and
apply the triangle rule!
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Advanced Microeconomics
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Welfare-theoretic analysis of monopoly
Cournot monopoly
Problem
No price di¤erentiation, marginal-cost curve MC = 2X and inverse
demand p (X ) = 12 2X . Determine the welfare loss!
p
Solution
The welfare loss is equal
to
12
MC
(8
4) (3
2
2)
8
= 2.
p (X )
4
MR
2
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3
6
X
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Welfare-theoretic analysis of monopoly
Cournot monopoly
Loss due to
CR (X̄ ) =
Z X̄
p (X ) dX
p (X̄ ) X̄
0
dCR (X̄ )
d X̄
=
d
R X̄
0
= p (X̄ )
Harald Wiese (University of Leipzig)
p (X ) dX
d X̄
d [p (X̄ ) X̄ ]
d X̄
dp
p (X̄ ) +
X̄ =
d X̄
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dp
X̄ > 0.
d X̄
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Cournot monopoly
Benevolent monopoly
max [p (X̄ )X̄
FOC:
p (X̄ ) +
dp
X̄
d X̄
C (X̄ )] + CR (X̄ )
dC
d X̄
or
!
p (X̄ ) =
Harald Wiese (University of Leipzig)
dp
!
X̄ = 0
d X̄
dC
d X̄
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Price and quantity competition
overview
1
Monopoly: Pricing policy
2
Price competition
3
Monopoly: Quantity policy
4
Quantity competition
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Advanced Microeconomics
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Quantity competition
price versus quantity competition
Cournot 1838, Bertrand 1883
Quantity or price variation
Capacity
simultaneous capacity construction
+ Bertrand competition
= Cournot results
x1
Π1
x2
Π2
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Economic genius:
Antoine Augustin Cournot
Antoine Augustin Cournot
(1801-1877) was a French philosopher,
mathematician, and economist.
In 1838, Cournot presents monopoly
theory and oligopoly theory for
quantity setting in his famous
“Recherches sur les principes
mathématiques de la théorie des
richesses”.
De…nes the Nash equilibrium for the
special case of quantity competition
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Advanced Microeconomics
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Quantity competition
the Cournot game
De…nition
Cournot game (simultaneous quantity competition)
Γ = N, (Si )i 2N , (Πi )i 2N
N – set of …rms
Si := [0, ∞) – set of quantities
Πi : S ! R – i’s pro…t function X
i
:=
∑ i 6 = j = 1 xj
n
Πi (xi , X i ) = p (xi + X i ) xi
C (xi )
Equilibria: ‘Cournot equilibria’or ‘Cournot-Nash equilibria’
Recall: x1C , x2C is de…ned by x1C = x1R x2C and x2C = x2R x1C
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Quantity competition
Equilibrium
Linear case
∂Π1 (x1, x2 )
= MR1 (x1 )
∂x1
MC1 (x1 ) = a
2bx1
bx2
!
c1 = 0
Quantities are strategic substitutes:
x1R (x2 ) =
a
c1
2b
= x1M
1
x2
2
1
x2 .
2
Solve the two reaction functions in the two unknowns x1 and x2
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Quantity competition
Equilibrium
x1C =
1
3b
(a
2c1 + c2 ) , x2C =
1
3b
(a
2c2 + c1 )
x2
x1R (x 2 )
Cournot-Nash
equilibrium
x 2M
C
x
C
2
x 2R (x1 )
x1C
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x1M
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x1
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Quantity competition
Equilibrium
X C = x1C + x2C =
pC =
ΠC1
=
ΠC2
=
1
(2a
3b
c1
c2 )
1
( a + c1 + c2 )
3
1
(a
9b
1
(a
9b
2c1 + c2 )2
2c2 + c1 )2
ΠC = ΠC1 + ΠC2 < ΠM
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Quantity competition
Iterative rationalizability
Reaction function:
x2R (x1 ) =
a c2
2b
x1
2,
0,
x1 < a bc2
otherwise
x2
x 2M
x 2R (x1 )
x1L
x1L :=
Harald Wiese (University of Leipzig)
a
x1
c2
b
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Quantity competition
Iterative rationalizability
x2
x 2M
x 2R (x1 )
x1L
x1
For …rm 2, any quantity between 0 and x2M is rationalizable:
h
i h
i
I1 := x2R x1L , x2R (0) = 0, x2M
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Quantity competition
Iterative rationalizability
h
i
I1 : = 0, x2M ,
x2
x1R (x2 )
I3
I3
1 a c1 M
, x1
4 b
1 a c2 3 a c2
: =
,
4 b
8 b
I2 : =
x2R (x1 )
I1
x1
I2
Convergence towards the Cournot equilibrium
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Cartel treaty between two duopolists
Cartel pro…t
Π1,2 (x1 , x2 ) = Π1 (x1 , x2 ) + Π2 (x1 , x2 )
= p (x1 + x2 ) (x1 + x2 )
C1 (x1 )
C2 (x2 ) .
with …rst-order conditions
∂Π1,2
∂x1
∂Π1,2
∂x2
dp
(x1 + x2 )
dX
dp
= p+
(x1 + x2 )
dX
= p+
dC1 !
= 0 and
dx1
dC2 !
=0
dx2
Equal marginal cost (as in “one market, two factories”)
2
Negative externality ∂Π
∂x1 =
care of in the cartel treaty
Harald Wiese (University of Leipzig)
dp
dX x2
< 0 in the Cournot model is taken
Advanced Microeconomics
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Quantity competition
Comparative statics and cost competition
Common interests with respect to
demand (parameters a and b): common advertising campaign
cost (parameter c): lobby for governmental subsidies or take a
common stance against union demands
Problem
Two …rms sell gasoline with unit costs c1 = 0.2 and c2 = 0.5, respectively.
The inverse demand function is p (X ) = 5 0.5X .
1
Determine the Cournot equilibrium and the resulting market price.
2
The government charges a quantity tax t on gasoline. How does the
tax a¤ect the price payable by consumers?
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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Quantity competition
Comparative statics and cost competition
Problem
Two …rms sell gasoline with unit costs c1 = 0.2 and c2 = 0.5, respectively.
The inverse demand function is p (X ) = 5 0.5X .
1
Determine the Cournot equilibrium and the resulting market price.
2
The government charges a quantity tax t on gasoline. How does the
tax a¤ect the price payable by consumers?
1
2
x1C = 3.4, x2C = 2.8 and p C = 1.9
2
p C = 1.9 + 23 t. Di¤erentiationg w. r. t. t : dp
dt = 3 , i.e., a tax
increase by one Euro leads to a price increase by 23 Euros.
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Quantity competition
Comparative statics and cost competition (envelope theorem)
Reducing own cost
cost saving
R&D
ΠC1 (c1 , c2 ) = Π1 c1 , c2 , x1C (c1 , c2 ) , x2C (c1 , c2 ) .
∂ΠC1
=
∂c1
∂Π1
∂c1
|{z}
<0
direct e¤ect
+
∂Π1 ∂x1C
+
∂x1 ∂c1
|{z}
=0
∂Π1 ∂x2C
∂x2 ∂c1
|{z}
|{z}
<0 >0
| {z }
<0
< 0.
strategic e¤ect
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Quantity competition
Comparative statics and cost competition (graphical analysis)
x2
x1R (x 2 )
c1 is reduced
Cournot-Nash
equilibrium
x 2R (x1 )
x1
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Quantity competition
Comparative statics and cost competition
Increasing rival’s cost
sabotage
level playing …eld with respect to pay, environment, ...
ΠC1 (c1 , c2 ) = Π1 c1 , c2 , x1C (c1 , c2 ) , x2C (c1 , c2 ) .
∂ΠC1
=
∂c2
∂Π1
∂c2
|{z}
=0
direct e¤ect
+
∂Π1 ∂x1C
+
∂x1 ∂c2
|{z}
=0
∂Π1 ∂x2C
∂x2 ∂c2
|{z}
|{z}
<0 <0
| {z }
>0
> 0.
strategic e¤ect
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Quantity competition
Comparative statics and cost competition
x2
x1R (x 2 )
Cournot-Nash
equilibrium
c2 increases
x 2R (x1 )
x1
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Quantity competition
Replicating the Cournot model
m identical consumers, n identical …rms
demand: 1
p for i = 1, ..., m
= m (1 p )
m X
p (X ) =
=1
m
X
X
m
for j = 1, ..., n : C (xj ) = 12 xj2
j’s pro…t
Πj (X ) = p (X ) xj C (xj )
xj + ∑i 6=j xj
1 2
=
1
xj
x
m
2 j
xj + X j
1 2
=
1
xj
x
m
2 j
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Quantity competition
Replicating the Cournot model
x j +X
m
Πj (X ) = 1
j
xj
1 2
2 xj
xjR (X j ) =
with X
j
= (n
1) xj :
xj =
m
xjC =
X C = nxjC =
m X j
m+2
nm
m +1 +n
and p (X ) =
pC = 1
Harald Wiese (University of Leipzig)
(n 1) xj
m+2
m
m+1+n
m X
m
:
n
.
m+1+n
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Quantity competition
Replicating the Cournot model
Consider λn …rms and λm consumers
Price - marginal cost (= equilibrium quantity) equals
p C (λ)
MCj (λ) =
=
=
λn
λm + 1 + λn
1
λm + 1 + λn
1
!0
λ (m + n ) + 1 λ ! ∞
1
λm
λm + 1 + λn
so that we obtain the price-takership result known from perfect
competition.
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Advanced Microeconomics
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Quantity competition
Blockaded entry and deterred entry
Assume c1 < c2
Market entry blockaded for both if
c1
a
and
c2
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a
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Quantity competition
Blockaded entry and deterred entry
Assume c1 < c2
x2
Market entry
blockaded for …rm 2
if
c2
p M ( c1 ) =
x1R (x2 )
a + c1
.
2
or
x1L
x1M
x
M
2
x 2R (x1 )
C
monopoly
firm 1
M
x1L x1M
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x1
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Quantity competition
Blockaded entry and deterred entry
Summary
c2
no supply
a
monopoly 1
a
2
duopoly
monopoly 2
a
2
Market entry blockaded for …rm 2 if c2
Harald Wiese (University of Leipzig)
a
c1
p M (c1 ) = 21 a + 12 c1
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Further exercises
Problem 1
Consider a monopolist with cost function C (X ) = cX , c > 0, and
demand function X (p ) = ap ε , ε < 1.
1
Find the price elasticity of demand and the marginal revenue with
respect to price!
2
Express the monopoly price p M as a function of ε!
3
Find and interpret
dp M
d jεj
!
Problem 2
Assume simultaneous price competition and two …rms where …rm 2 has
capacity constraint cap2 such that
1
2X
(c ) < cap2 < X (c ) .
Is (c, c ) an equilibrium?
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Further exercises
Problem 3
Three …rms operate on a market. The consumers are uniformly distributed
on the unit interval, [0, 1]. The …rms i = 1, 2, 3 simultaneously choose
their respective location li 2 [0, 1]. Each consumer buys one unit from the
…rm which is closest to her position; if more than one …rm is closest to her
position, she splits her demand evenly among them. Each …rm tries to
maximize its demand. Determine the Nash equilibria in this game!
Problem 4
Assume a Cournot monopoly. Analyze the welfare e¤ects of a unit tax and
a pro…t tax.
Consider the welfare e¤ects of a unit tax in the Cournot oligopoly with
n > 1 …rms, linear demand, and constant average cost. Restrict attention
to symmetric Nash equilibria! What happens for n ! ∞?
Problem 5
Assume a Cournot monopoly. Analyze the quantity e¤ects of a price cap.
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