# 1838 Investments - PowerPoint

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```					Course outline I
   Introduction
   Game theory
   Price setting
– monopoly
– oligopoly
Homogeneous
   Quantity setting
goods
– monopoly
– oligopoly

   Process innovation
1
Quantity and cost competition
 Bertrand versus Cournot
 Simultaneous quantity competition (Cournot)
 Sequential quantity competition (Stackelberg)
 Quantity Cartel
 Concentration and competition

2
Price or quantity competition?

Cournot (1838)
x1        1
x2
2

Bertrand (1883)
p1       1
p2       2
3
Capacity + Bertrand = Cournot
   Bertrand (1883) criticized Cournot’s model (1838) on the
grounds that firms compete by setting prices and not by
setting quantities.
   Kreps and Scheinkman (1983) defended Cournot’s
model. They developed a two-stage game with capacities
k1         p1           1
k2         p2           2
and proved that capacities in a Nash equilibrium are
determined by Cournot’s model.
4
Cournot versus Stackelberg

   Cournot duopoly (simultaneous quantity
competition)
x1       1
x2       2
   Stackelberg duopoly (sequential quantity
competition)
1
x1     x2
2

5
Homogeneous duopoly
(linear case)
 Two firms (i=1,2) produce a homogenous
good.
 Outputs: x1 and x2, X= x1+x2
 Marginal costs: c1 and c2
 Inverse demand function:
pX  a bX  a bx1  x2 
   Profit function of firm 1:
1 x1, x2   p X x1  c1x1  a  bx1  x2   c1 x1
6
Cournot-Nash equilibrium
 Profit functions: 1 ( x1 , x2 ),  2 ( x1 , x2 )
 Reaction functions:
x ( x2 )  arg max x1 1 ( x1 , x2 )
R
1

x ( x1 )  arg max x2  2 ( x1 , x2 )
R
2
C        C
   Nash equilibrium: ( x , x )    1        2
x1R ( x2 )  x1
C      C

x2 ( x1 )  x2
R    C      C

7
Computing the Cournot equilibrium
(accommodation)
   Profit function of firm 1
1 ( x1 , x2 )  p( X ) x1  c1x1  (a  b( x1  x2 )  c1 ) x1
   Reaction function of firm 1
a  c1 x2                                a  c2 x1
x ( x2 ) 
R
1                          analogous : x ( x1 ) 
R
2            
2b     2                                  2b    2
   Nash equilibrium
 C     a  2c1  c2 C       a  2c2  c1 
 x1                 , x2               
            3b                   3b      
( a  c1  c2 )
p 
C

3
( a  2c1  c2 ) 2                      (a  2c2  c1 ) 2
1 
C
analogous :  2 
C

9b                                     9b
8
Depicting the Cournot equilibrium
x2

x1R ( x2 )

M
Cournot-Nash
x   2
C          equilibrium
C
x2
R
x2 ( x1 )

x1C   x1M                   x1
9
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
and c2 = 2.

20           23
S. : x 
C
1       and   x 
C
2
3           3

10
Common interests

 c 1, c 2
Obtaining government subsidies and
negotiating with labor unions.

   a , b 
(e.g. CMA).

11
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?
dp 2
S. : a) p  1.9 b)   
dt 3                          12
   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

13
Direct approach, analytically
   1 c1 , c2   1 c1 , c2 , x1 c1 , c2 , x2 c1 , c2 
C                             C              C

   Direct approach (reduction of your own
marginal costs):
 1  1  1 x1  1 x2
C               C       C
                     0
c1   c1   x1 c1 x2 c1

        
              
 
0       0             0  0
 

0
direct                    strategic
effect                    effect
14
Direct approach, graphically
x2        x1R ( x2 )

equilibria: increase in
production of firm 1

R
x2 ( x1 )
x1
15
Exercise (direct approach)
   Who has a higher incentive to reduce own
costs, a monopolist or a firm in Cournot-
Duopoly?

16
Indirect approach, analytically
    c1 , c2   1 c1 , c2 , x c1 , c2 , x c1 , c2 
C
1
C
1
C
2

   Indirect approach (raising rival’s cost):
d1 C
1      1 dx2 1 dx1
C             C
                                   0
c2      x2 dc2       x1 dc2
dc2
  
=0           <0 <0           =0
 
 
0
direct          strategic
effect          effect
17
Indirect approach, graphically
x2   equilibria: increase
in production of firm 1

x1R ( x2 )

R
x2 ( x1 )

x1
18
Reaction curve in the linear case
x2
 a  c2 x1           a  c2
            if x1 
M                        x2 ( x1 )   2b
R
2             b
x2                                    0
              otherwise

R
x2 ( x1 )

x1L
x1
a  c2
Note: x 
L
1         alone leads to a price of c2.
b
19
x2

x1R ( x2 )

C            firm 1
R                       as a monopolist
M     x2 ( x1 )
x2
M
x2  0                    L   M
x1
x x
1   1
20
   Entry is blockaded for each firm:
c1  a and c2  a
   Entry is blockaded for firm 2:
c1  a and
a  c2 a  c1
x x
L
1
M
1        i.e.           
b     2b
1
 c2  p (c1 )  (a  c1 )
M

2
21

c2                        no
firm 1 as a         supply
a
monopolist

1
a
2
firm 2 as a
duopoly       monopolist
1
2
a       a            c1
22
Cournot – Executive summary
 A duopoly will occur only, if entry is
 Firms have common and competing interests
with respect to demand and cost functions.
 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“.

23
Stackelberg equilibrium
   Profit functions 1 ( x1 , x2 ),  2 ( x1 , x2 )

   Follower’s reaction function (2nd stage)
x2 ( x1 )  arg max x2  2 ( x1 , x2 )
R

   Leader’s optimal quantity (1st stage)
x1S  arg max x1 1 x1 , x2 ( x1 ) 
R

   Nash equilibrium: ( x1S , x2R )

24
Finding the profit-maximizing point
on the follower’s reaction curve
x2      Accommodation

M
R
x   2
deterrence
R
x2 ( x1 )

x1           x 
L   a  c2         x1
1
b
25
Computing the Stackelberg
equilibrium (accommodation)
   Reaction function of firm 2:
a  c2 x1
x ( x1 ) 
R
2               
2b    2
   Profit function of firm 1:
            a  c2 x1     
1 ( x1 , x ( x1 ))   a  b x1 
R
2                                 c1  x1

              2b   2      
   Nash equilibrium
 S a  2c1  c2 R 
 x1           , x2 
        2b          
a  2c1  3c2                  a  2c1  c2
with x2 
S
and     p 
S
4b                             4
26
Depicting the Stackelberg
outcome (both firms produce)
x2

x1R ( x2 )
quantities in a
Stackelberg equilibrium
M
x2
C
C
x   2
x   S
2
S      R
x2 ( x1 )

x1C    S
x
1
x1
27
Exercise (equilibria)
   Which is an equilibrium in the Stackelberg
model?
xS
1
R
 
, x2 x1S ,
xS
1   ,x   R
2   ,
xC
1   ,x   C
2   ?
   Are there any additional Nash equilibria ?

28
Cournot versus Stackelberg
 Profit function of firm 1
1 ( x1 , x2 )  p( X ) x1  C1 ( x1 )
 First order condition for firm 1
dR1               dp dX                dp  dx1 dx2 
R
 p( X )  x1         p( X )  x1            
dx1               dX dx1               dX  dx1 dx1 
R
dp      dp dx2 !
 p( X )  x1     x1         MC1 ( x1 )
dX      dX dx1
direct effect     follower or strategic effect,
Cournot: 0, Stackelberg: >0
29
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.
S. : x1S  10, x2 
R

   Possible or not:    ?
C
1
S
1

30
Exercise (three firms)
Three firms compete in a homogenous good market
with X(p)=100-p. The costs are zero. At stage 1, firm
1 sets its quantity; at stage 2, firms 2 and 3
simultaneously decide on their quantities.
Calculate the price on the market!
50
S. : p 
3

31
p
for firm 2
a        p( x1 )
c2
p1M                      M

c1

x1L     x1M        L            x1
x
1
32
Reaction functions in the case of
x2

x1R ( x2 )
R
x2 ( x1 )

L          M
x1
x 1          x
1

33
Profit function of firm 1 in the
case of blockaded entry of firm 2
1
1
M

1  x1 ,0

1 x1 , x2  x1 
R

x1L           x1M                 x1
34
Deterring firm 2’s entry
p

a
p( x1 )
p1M
c2

L
1
c1

M
x1   x1L
x1
35
d1 x1 , x2  x1 
R
Deterrence pays,                    dx1
0
L
x1

1

1
M

1  x1 ,0

1 x1 , x2  x1 
R

x1M x1L                           x1
36
d1 x1 , x2  x1 
R
Deterrence does not pay                             dx1
0
L
x1

1
1
M

1  x1 ,0

1 x1 , x2  x1 
R

x1M    x1S   x1L                  x1
37
   Entry is blockaded for each firm:
c1  a and c2  a
c   2    p M ( c1 ) or   x1L  x1M  and   c1  a
a  c1
 c2                 and    c1  a
2

38
   Deterred entry (firm 2):
a  c1
–   Entry is not blockaded if    c2          p1M
2
–   Deterrence pays if
d1 x1 , x2  x1 
R
1   1           1   3
0                          bx1  a  c2  c1   a  c2  c1
L

dx1             L
x1
2   2           2   2
1   2
 c2  a  c1
3   3
Deterrence if           1    2          a  c1
a  c1  c2 
3    3            2
39
c2
no supply
firm 1 as a
monopolist
1
a
2
1
firm 2 as a
a
3        duopoly           monopolist

a
1
2
a               c1
40
Exercise (entry and deterrence)
Suppose a monopolist faces a demand of the
form p(X)=4-0.25X. The firm’s unit costs are 2.
a) Find the profit-maximizing quantity and price.
Is entry blockaded for a potential entrant
(firm 2) with unit costs of 3.5?
b) How about unit costs of c2=1?
c) Find firm 1’s limit output level for c2=1.
Should the incumbent deter entry?
c) 1  0.5  12  1 , accomodation
S                 L

41
Deterrence and sunk costs I
We now introduce quasifix costs of 3:
p(X)=4-0.25X

3  2 x1 , x1  0
C1  x1   
0,         x1  0
Follower' s cost function
3  x2 , x2  0
C2  x2   
0,       x2  0
42
Deterrence and sunk costs II
?               ?
x  4,
M
1          p  4   3
M

 1 4,0  1  0   1 0,0  x1  4
M
p   M
4   3
Comparison p x   2 is not sufficient
M
1
c

x2  x1   6  x1  x2 x1   4
R             1              R     M

2
 2 4,4   4  0,25  4  4   4  3  1  4   1  0
43
Deterrence and sunk costs III
   c) Should firm 1 deter?
x1Lq  limit quantity w ith quasifixed costs, x1Lq  x1L (why?)
0   2 x1Lq , x2 x1Lq   p  X   x2 x1Lq   C2 x2 x1Lq 
R                       R                R

 4  1  x1Lq  x2 x1Lq  x2 x1Lq   3  x2 x1Lq 
4
R             R                 R

 4  1  x1Lq  6  1 x1Lq  6  1 x1Lq   3  6  1 x1Lq 
4               2               2                    2

 2  8 x1Lq  6  1 x1Lq   3  6  1 x1Lq 
5   1
2                   2
Lq 2
 15  x   3
4
Lq
1     x 5
4
Lq
1        1
16   x
1       3  6  1 x1Lq
2
Lq 2
6 x  3
2
Lq
1        1
16   x
1

x1Lq1  12  4 3,                                  x1Lq : x1Lq 2  12  4 3    12  x 
L
1

44
Deterrence and sunk costs IV
 1 x1Lq ,0  4  1 12  4 3  0  12  4 3  0  3  24  8 3 
4

 0,71    1 x1S , x2 x1S  (see exercise " entry and deterrence " )
1    S         R

2
 deterrence pays

45
Deterrence and sunk costs V

x2

6

x2  x1 
R

x1S   x1M x1Lq               12   x1
46
Deterrence and sunk costs VI
x1S

7,2

x1M  4

x1S  2

0,5 1,44        3       4                   9          CF

47
xd       1
s
xf       2

   Two firms, one domestic (d), the other foreign (f),
compete on a market in a third country.
   The domestic government subsidizes its firm’s
exports using a unit subsidy s.
   The subsidy grants the domestic firm an advantage
that is higher than the subsidy itself (Brander /
Spencer (1981, 1983)).
48
 In the setting just described, assume c : c1  c2
and p(X)=a-bX.
 Since the two firms sell to a third country, the
rent of the consumers is without relevance and
domestic welfare given by
W s    C c  s, c   sxd c  s, c 
d
C

   Which subsidy s maximizes domestic welfare?
ac
S. : s 
*

4                                     49
interpretation
    Direct effect of subsidy for domestic welfare is zero.
    Strategic effect:  d x C
f
        0
xf                                     x f       s
<0         <0
R ,s
x   d
Cournot-Nash-
s           equilibria
x   R                                                  xd c  s, c   xd c, c  !!
C                S
d

xM                                                          (firm d Stackelberg
xR
f

M                     xd
x                                                          50
 The recommendation depends on whether
there is price or quantity competition.
 „One can always do better than free trade, but
the optimal tariffs or subsidies seem to be
small, the potential gains tiny, and there is
plenty of room for policy errors that may lead
to eventual losses rather than gains.“

Trade Policy and Market Structure; Helpman/Krugman, S. 186
51
Stackelberg – Executive Summary
   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.
 Costs of entry (even in the form of identical
quasifix costs) make the follower’s
deterrence easier.
 Strategic trade policy may conceivably pay.

52
Example: The OPEC Cartel
   The best known cartel is the OPEC, which was
formed in 1960 by Saudi Arabia, Venezuela,
Kuwait, Iraq and Iran. Each member nation must
agree to an individual output quota, except for
Saudi Arabia, which adjusts its production as
necessary to maintained high prices.
   In 1982, OPEC set an overall output limit of 18
million barrels per day (before 31 million).
   Production quota at 28 million barrels per day
effective July 1, 2005.
53
The quantity cartel
   The firms seek to maximize joint profits
1 ( x1 , x2 )   2 ( x1 , x2 )
 p( X )( x1  x2 )  C1 ( x1 )  C2 ( x2 )

   Optimization conditions
 ( 1   2 )                         dp              !
 p ( X )  ( x1  x2 )     MC1 ( x1 )  0
x1                                dX
 ( 1   2 )                         dp              !
 p ( X )  ( x1  x2 )     MC2 ( x2 )  0
x2                                dX
   Compare monopoly with two factories.
54
The cartel agreement
   The optimization condition is given by
d1               dp              !     dp
 p( X )  x1     MC1 ( x1 )  x2    0
dx1               dX                    dX
 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.

55
Cartel quantities
x2
quantities in a
symmetric cartel
x1R ( x2 )

M
x2
C
C
x2                         S
1 M
x2  x2
S                              R
x2 ( x1 )
2           K
1
x x x x
M C      S    M                 x1
2   1 1      1    1

56
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?

Intermediate Microeconomics; Hal R. Varian

57
Cartel – Executive Summary
 If all firms keep the cartel agreement, they
can increase their profits compared to
Cournot competition.
 Nevertheless cartels are unstable from a static
point of view.
 However, cartel agreements may be stable
from the point of view of repeated games.

58
The outcomes of our models
price
a

pM             monopoly (M) and cartel (K)
pC               Cournot (C)
pS                  Stackelberg (S)
perfect competition (PC)
p PC = c

X   M   X   C   X   S   X   PC
quantity
59
Antitrust laws and enforcement,
Germany
   laws
–   Gesetz gegen unlauteren Wettbewerb (1896)
–   Gesetz gegen Wettbewerbsbeschränkungen
(GWB), (1957)
   enforcement
–   Bundeskartellamt

60
Ck concentration ratio
xi
Setup: n firms, si         and    s1  s2    sk    sn
X
k
k
Definition: Ck   si , for n identical firms : Ck  ,k  n
i 1                               n
monopoly:     n  k 1  k 1   n

perfect comp.: lim k  0 for identical firms
n
n

Exercise: Calculate C2 for
 2 firms with equal market shares,

 3 firms with shares of 0.1, 0.1 and 0.8 or

 3 firms with shares of 0.2, 0.6 and 0.2 ?

61
GWB, §19 (3)
 One firm is called „market dominating“ if
C1>1/3.
 A group of firms is called „market
dominating“ if
Ck  1 / 2, k  3
or
Ck  2 / 3, k  5.

62
The Herfindahl (Hirschman) index
monopoly : H  1
   Definition:                                                    1
2                      n identical firms : H 
n
 xi    n                                            n
H       si     2
perfect competitio n
i 1  X   i 1                 (n  ) : H  0

   Exercise: Calculate H for
   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 ?

63
n firms in Cournot competition
 Total industry output: X  x1  x2  ...  xn
 Firm i’s profit function:
 i ( x1 ,..., xn )  p( x1  ...  xn ) xi  Ci  xi 
   Firm i’s marginal revenue:
dp           dp d  x1    xn           dp
MR ( xi )  p  xi      p  xi                       p  xi    1
dxi          dX        dxi                 dX
 xi X dp                       
 p 1              p  1  si     
    X p dX                    
     X ,p   

64
Lerner index of market power
   First order condition:
dp dX !
MR( xi )  p( X )  xi         MC ( xi )
dX dxi
   Lerner index for one firm:
           
p p  1  si    
p  MCi !                
           X,p     si
p             p            X,p
   Lerner index for the industry:
n
p  MCi ! n    si  H
 si p   si   
i 1           i 1 X,p X,p

65
Exercise (Replication)
In a homogenous good market there are m identical costumers
and n identical firms. Every costumer demands the quantity 1-p at
price p. The cost function of firm j is given by C j x j   0,5x 2 .
j

a) Calculate the inverse market demand function!
b) Calculate the reaction function of firm j and the total market
output X C  x1C  x2    xn and pC in the symmetric Cournot-
C    C

equilibrium! Hint: Use X  j  x1  ...  x j 1  x j 1  ..  xn
c) Now the number of firms and costumers is multiplied by .
Calculate again pC and MCj! Prove that for    the gap
between price and marginal costs converges to zero!
S. : a) p  X   1  m
X

m                  n
b) X C  n           , pC  1              Theorie der Industrieökonomik; Bester
66
m 1 n            m 1 n

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