# Course outline I Monopoly _quantity setting_ Decision problem by ghkgkyyt

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```									Stratégies des entreprises

Course outline I
Introduction
Game theory
Price setting
–   monopoly
–   oligopoly
Homogeneous
Quantity setting
goods
–   monopoly
–   oligopoly
Process innovation
1

Monopoly (quantity setting)
Inverse demand function
Revenues, costs, profits
Profit maximizing quantity
–   Basic model
–   Price discrimination
–   Several factories
Double marginalization
Welfare analysis
Executive summary
2

Decision problem

X                        Π
3

Harald Wiese                                                   1
Stratégies des entreprises

Inverse demand function
p

demand
p           function

inverse demand
p( X )                                 function

X
X ( p)               X
4

Revenue, costs and profit
Revenue:           R( X ) = p ( X ) ⋅ X
Costs:             C(X )
Profit:            Π ( X ) = R( X ) − C ( X )

5

Marginal revenue with respect to
quantity
dR                   dp( X )
= MR X = p( X ) +         X
dX                    dX
When a firm increases the quantity by one unit, revenue
goes up by p (the price of the last unit),
but goes down by dp/dX X (the quantity increase diminishes
the price and this price decrease is applied to all units)
⎛         X    dp ( X ) ⎞
Amoroso-Robinson MR X = p ( X ) ⎜ 1 +                         ⎟
relation:                       ⎝     p( X )           dX ⎠
⎛    1      ⎞      ⎛    1      ⎞
= p ⋅⎜ 1+        ⎟ = p ⋅⎜ 1−        ⎟
⎜ ε         ⎟      ⎜ ε         ⎟
⎝    X ,p   ⎠      ⎝    X ,p   ⎠
6

Harald Wiese                                                                             2
Stratégies des entreprises

Marginal revenue = price?
dR            dp ( X )
= p( X ) +          X
dX             dX

dp ( X )
1.                =0 ⇒      i.e. horizontal demand curve, perfect
dX                competition

2. X=0               ⇒      sale of first unit
⇒      first degree price discrimination

7

Linear demand curve in a
monopoly
Demand:                            p( X ) = a − bX

Revenue:                           R ( X ) = aX − bX 2

Marginal revenue:                  MR = a − 2bX

8

Exercise (Depicting the linear
demand curve p( X ) = a − bX )
Slope of demand curve: ....
Slope of marginal revenue curve: ....
The ............................             has the same
vertical intercept, ..., as the demand curve.
Economically,
–   the vertical intercept is .................,
–   the horizontal intercept is ................. .
9

Harald Wiese                                                                           3
Stratégies des entreprises

Depicting demand and marginal
revenue
p
a

1
b
2b

1                   p( X )
MR
X
a/(2b)                   a/b
10

First order condition
dΠ ( X )              dp ( X ) dC ( X ) !
= p( X ) + X         −        =0
dX                    dX       dX

MR X                 MC X
Notation:                     MR := MRX
MC := MC X

11

First order condition, alternative
formulations
!         ⎛    1                 ⎞
MC = MR = p ⋅⎜ 1−                   ⎟
⎜ ε                    ⎟
⎝    X ,p              ⎠
!
1                  ε X ,p
p=                     MC =               MC
1−
1           ε X ,p − 1
ε X ,p

p − MC ! 1
=                       (price-cost margin)
p     ε X ,p
12

Harald Wiese                                                                        4
Stratégies des entreprises

Depicting the Cournot monopoly
p

MC
Cournot
M
p                    point

p( X )

MR
X
XM
13

Exercise (Quantity)
Consider a monopolist facing the inverse demand
function p(X)=24-X. Assume that the average and
marginal costs are given by AC=2.
Find the profit-maximizing quantity!
S. : X M = 11

14

Profit in a monopoly
Marginal point of view:                  Average point of view:
p                                            p

p(X)
M
pM                                    p
MC
MC

p(X)
AC
MR
X                                   X
XM                                        XM   MR
15

Harald Wiese                                                                              5
Stratégies des entreprises

Exercise (monopoly)
Consider a monopoly facing the inverse demand
function p(X)=40-X2. Assume that the cost function
is given by C(X)=13X+19.
Find the profit-maximizing price and calculate the
profit.
S. : Π M = 35

16

Price discrimination
First degree price discrimination:
Every consumer pays a different price which is equal to his
or her willingness to pay.

Second degree price discrimination:
Prices differ according to the quantity demanded and sold
(quantity rebate).

Third degree price discrimination:
Consumer groups (students, children, ...) are treated
differently.

17

Monopolistic price
discrimination (two markets)
p

p1
p2
x1 (p1 )                                           x 2 (p 2 )
MR 1                MR 2
MC

*
x1                 x1        total output          x*
2                x2
18

Harald Wiese                                                                             6
Stratégies des entreprises

Inverse elasticities rule for third
degree price discrimination
Supplying a good X to two markets results in the inverse demand
functions p1(x1) and p2(x2).
Profit function:                    Π ( x1 ,x2 )= p1 ( x1 )⋅x1 + p2 ( x2 )⋅x2 −C ( x1 + x2 )
4 4 4 3
1 2 3 1 24
R1 (x1 )      R2 (x2 )
∂ Π ( x1 ,x2 )                                  !
First order conditions:                            =MR1 ( x1 )− MC ( x1 + x2 )=0
∂ x1

=
∂ Π (x1 ,x2 )                                   !
= MR2 ( x2 )− MC ( x1 + x2 )=0
∂x 2
Equating the marginal revenues (using the Amoroso-Robinson relation) leads to:
⎛        1 ⎞!              ⎛        1 ⎞
⎜ ε ( x ) ⎟= p2 ( x 2 )⋅⎜1− ε ( x ) ⎟
p1 ( x1 )⋅⎜1−            ⎟           ⎜             ⎟
⎝      1 1 ⎠               ⎝      2    2 ⎠
ε 1 ( x1 ) < ε 2 ( x2 ) ⇒ p1 ( x1 ) > p2 ( x2 )
19

Exercise (two markets or one)
A monopoly sells in two markets:
p1(x1)=100-x1 and p2(x2)=80-x2.
a) Calculate the profit-maximizing quantities and the profit
at these quantities, if the cost function is given by C(X)=X2.
b) Calculate the profit-maximizing quantities and the profit
at these quantities, if the cost function is given by
C(X)=10X.
c) What happens if price discrimination between the two
markets is not possible anymore? Consider C(X)=10X.
Hint: Differentiate between quantities below and above 20.
S. : a) Π M = 1400, b) Π M = 3250, c) Π M = 3200                                              20

Solution III (one market)

p

100

90
80

50

MR                             p( X )

10                                                MC
20          50          80       100
X

21

Harald Wiese                                                                                                       7
Stratégies des entreprises

One market, two factories
Profit function:
Π ( x1 , x2 ) = p ( x1 , x2 )( x1 + x2 ) − C1 ( x1 ) − C2 ( x2 )
First order conditions:
∂Π ( x1 , x2 )                               !
= MR ( x1 , x2 ) − MC1 ( x1 ) = 0
∂x1
=

∂Π ( x1 , x2 )                                !
= MR ( x1 , x2 ) − MC 2 ( x2 ) = 0
∂x2
!
⇒              MC1 ( x1 ) = MC 2 ( x2 )
22

One market, two factories II
factory 2                           factory 1
p
MC 2
MC1

x2                     *                       *                   x1
x2                      x1

total output
23

Double marginalization - idea
Retailer, not producer sells to consumers.
Assumptions:
–   Zero costs for retailing.
–   Producer decides on a quantity and charges a price
ppro to the retailer.
–   ppro is the retailer‘s marginal cost.
–   The retailer‘s MR=MC condition defines the
producer‘s demand function.

24

Harald Wiese                                                                                 8
Stratégies des entreprises

Double marginalization - linear case
p(x)=a-bX
MCpro=ACpro=c
Second stage: First order condition for the retailer:
!
MC = p pro = a − 2bX = MR
First stage: First order condition for the producer:
!
MC pro = c = a − 4bX = MR pro
a−c                         a+c
X =                    ⇒ p pro* =
4b                           2
3a + c
p* =
4
25

Double marginalization - depicting
the solution
p                                            Producer decides
a                                            on quantity and
retailer   announces ppro*
to retailer.
p * = 3a4+ c                               producer
Retailer decides
on quantity (fore-
p pro* = a 2 c
+                                           seen by producer).

MR pro            MR = p pro ( X )
c

a −c              a                      a
X
4b               2b                     b
26

Double marginalization - exercise
p(X)=110-X
c=10

a) Calculate the price the consumers have to pay!
b) What price would they pay if the producer sold
directly to the consumers?

S. : a) p M = 85
b) p M = 60

27

Harald Wiese                                                                                9
Stratégies des entreprises

Welfare Analysis
Evaluation of economic policy measures
Welfare = consumer surplus (CS)
+ producer surplus (PS)
+ taxes - subsidies
CS = willingness to pay - price
PS = revenue - variable costs
= profit + fixed costs

28

CS, PS - graphically
p

supply (=MC)
CS

ï       Welfare is
PS
demand             maximized at
the equilibrium.

0
X

29

The deadweight loss of a monopoly
p    Without price discrimination a monopoly

MC

pM
Perfect
p∗                                       p = MC      competition
MR = MC
p( X )
MR
XM        X∗
X
30

Harald Wiese                                                                                  10
Stratégies des entreprises

Consider a monopoly where the demand is
given by p(X)=-2X+12. Suppose that the
marginal costs are given by MC=2X.
–       without price discrimination,
–       with perfect price discrimination.
S. :          DLwopd = 2
DLwpd = 0
31

Exercise (price cap in a monopoly)
p
How does a price cap influence the
demand and the marginal revenue curves?

MC
pM
p cap

p( X )

MR
X   M                                   X
32

Right or wrong? Why?
p

MC
cap
p
pM
p( X )

MR
XM                            X

33

Harald Wiese                                                                                   11
Stratégies des entreprises

Price cap and welfare
p

MC
pM
p   cap
=p   M ,cap

p( X )

X
X M X M ,cap    MR
34

Taxes on profits
p
C( X )
R( X )

Π(X)

MR
pM
MC
Π(X)(1−t)
p( X )

XM
X
35

Taxes on profits and welfare
Quantity / price unchanged
CS is constant
PS decrease; in the same extent net public
revenues increase

36

Harald Wiese                                                                                         12
Stratégies des entreprises

to quantity tax
p
consumer’s surplus: A+B+C→A
producer’s surplus: T+E+F →E+B
additional       net public revenue: 0 →T
A
pT                        C
p
B                                                        MC + t
MC
E    F

T                                           p( X )
MR
X
XT X
37

Exercise (quantity taxes)
A monopolist is facing a demand curve given by p(X)=a-X.
The monopoly’s unit production cost is given by c>0.
Now, suppose that the government imposes a specific tax
of t dollars per unit sold.
a) Show that this tax would raise the price paid by
consumers by less than t.
curve is given by p(X)=-ln(X)+5.
c) If the demand curve is given by p(X)=X-1/2, what is the
influence on price?
1                                                                        Industrial Organization ; Oz Shy
S. : a) , b) 1, c) 2
2                                                                                                     38

Illustrating the solutions
p
a)                                                   b) p

pM(c+t)
pM(c+t)
pM(c)                                   c+t                                                c+t

p( X )               p (c)
M                              p( X )
MR                    c
c

X
MR
X
39

Harald Wiese                                                                                                                      13
Stratégies des entreprises

Lerner index of monopoly power
First order condition:
!
dp    ⎛      1 ⎞
MC ( X ) = MR ( X ) = p( X ) + X      = p⎜1 −        ⎟
dX    ⎜    ε X ,p ⎟
⎝           ⎠
Lerner index:
⎛    1        ⎞
p − p ⎜1 −          ⎟
p − MC         ⎜ ε           ⎟
!
=       ⎝    X,p      ⎠= 1
p             p             εX ,p

40

Monopoly profits
and monopoly power
p
AC             pM = AC ⇒ ΠM = 0

Cournot
pM                  point

MC

p( X )
MR
X
XM
41

Executive summary
A profit-maximizing monopolist always sets the
quantity in the elastic region of the demand curve.
Monopolistic power: price will be set above the
marginal cost by a profit maximizer.
If the demand curve is tangent to the average cost
curve, the profit-maximizing price is set above
marginal cost and equal to average cost.
⇒ monopolistic power and zero profits
Monopolistic quantities without price
discrimination (!) lead to a welfare loss.
A quantity tax leads to a welfare loss, a tax on
profits does not.                                     42

Harald Wiese                                                               14

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