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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 
    Advertising by the agricultural industry
    (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
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

                                                   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
Blockaded entry, graphically
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
Blockaded entry
   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
Blockaded entry (overview)

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
  blockaded for other firms.
 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
      x2                      Blockade or
                     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
Blockaded entry
     p
                         blockaded entry
                         for firm 2
a        p( x1 )
c2
p1M                      M

c1


           x1L     x1M        L            x1
                             x
                             1
                                                32
Reaction functions in the case of
blockaded entry
 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
Blockaded and deterred entry I
   Entry is blockaded for each firm:
    c1  a and c2  a
   Blockaded entry (firm 2):
    c   2    p M ( c1 ) or   x1L  x1M  and   c1  a
           a  c1
     c2                 and    c1  a
             2




                                                          38
Blockaded and deterred entry II
   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
Blockade and deterrence
c2
                                 no supply
a       blockade
      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?
S. : a) blockaded entry, b) entry is not blockaded
     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

Leader' s cost function
            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
   b) Entry blockaded ?
        ?               ?
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
  Entry not blockaded
                                                                 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
     Accommodation   Deterrence       Accommodation       Blockade

                                                                      47
Strategic trade policy
                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
Exercise (Strategic trade policy)
 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
Solution (Strategic trade policy) -
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
                                                            leader)
                                      xR
                                       f

                                 M                     xd
                             x                                                          50
Strategic trade policy - problems
 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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