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CHAPTER 27
OLIGOPOLY
27.1 Choosing a Strategy
Oligopoly: There are a number of competitors
in the market, but not so many as to regard
each of them as having a negligible effect on
price.
 Duopoly: Two firms only.

27.1 Choosing a Strategy
Sequential game: Players take actions in a
given order.
 Quantity leader: The firm who chooses its
quantity first.
 Quantity follower: The firm who chooses its
quantity following the leader.

27.2 Quantity Leadership

The follower chooses its output, given the
output of the leader.
max p( y1  y2 ) y2  c2 ( y2 )
y2
F.O.C. MR2  p( y1  y2 )  p( y1  y2 ) y2  MC2
 Reaction curve: the optimal y2 as a function of
y1.

y2  f 2 ( y1 )
27.2 Quantity Leadership
Linear demand: p( y )  a  by
 Zero marginal cost for simplicity.
 The profit function of firm 2:
 2 ( y1 , y2 )  [a  b( y1  y2 )] y2
 Isoprofit: the locus of (y1, y2) that yield a
constant level of profit.
ay2  by1 y2  by22   2


Reaction curve:
a  by1
y2 
2b
27.2 Quantity Leadership



Higher profits for
inner isoprofits.
The reaction curve
attains the
maximal profit for
a given y1.
The reaction curve
passes though the
top of the
isoprofits.
27.2 Quantity Leadership

The leader chooses y1 in recognition of the
follower’s reactions.
max p( y1  f 2 ( y1 )) y1  c1 ( y1 )
y1


Profit function:
1 ( y1 , y2 )  p( y1  y2 ) y1  ay1  by12  by1 y2
F.O.C.:
a
b 2
 y1  y1
2
2
a
*
y1 
2b
27.2 Quantity Leadership


The leader
knows that the
system will settle
on firm 2’s
reaction curve.
The leader
choose the
lowest isoprofit
on firm 2’s
reaction curve.
27.3 Price Leadership
The leader has set a price p;
 The follower take this price as given and set its
output S(p):

max py2  c2 ( y2 )
y2
The demand faced by the leader is the residual
demand R(p)=D(p)-S(p);
 The leader’s problem:

max( p  c) R( p)
p
27.3 Price Leadership
27.3 Price Leadership
Follower’s cost function: c2(y)=y2/2
 Follower’s supply curve: y2=p
 Linear market demand: D(p)=a-bp
 Residual demand: R(p)=a-(b+1)p
 The leader’s problem:

1
max
(a  y1 ) y1  cy1
y1 b  1

The leader’s output:
a  c (b  1)
y 
2
*
1
27.5 Simultaneous Quantity Setting
Each firm forms a belief about the other firm’s
output;
 Each firm chooses its profit-maximizing
output according to this belief;
 Nash equilibrium: the belief is consistent with
the outcome and no firm has incentives for
further adjustment.

27.5 Simultaneous Quantity Setting
Firm 1 expects that firm 2 produces y2e units;
 Firm 2 expects that firm 1 produces y1e units;
 Firm 1’s problem:

max p( y1  y ) y1  c( y1 )
y1
e
2
Firm 1’s output: y1=f1(y2e);
 Firm 2’s problem:

max p( y1e  y2 ) y2  c( y2 )
y2

Firm 2’s output: y2=f2(y1e).
27.5 Simultaneous Quantity Setting
The equilibrium satisfies
y1*=f1(y2*) and y2*=f2(y1*).
 The equilibrium is given by the intersection of
the two reaction curves.

27.6 An Example of Cournot
Equilibrium
Linear market demand: p(y)=a-by;
 Zero marginal cost;
a  by2e
a  by1e
y1 
y2 
 The reaction curves:
2b
2b
 Equilibrium conditions: y1=y1e, y2=y2e

a  by2
y1 
2b

a  by1
y2 
2b
Equilibrium outputs:
a
y y 
3b
*
1
*
2
27.7 Adjustment to Equilibrium
27.8 Many Firms in Cournot
equilibrium
Suppose that there are n firms;
 Total industry output: Y=y1+…+yn;
 Firm i’s F.O.C.:

p
p(Y ) 
yi  MC ( yi )
Y
 p Y yi 
p(Y ) 1 
 MC ( yi )

 Y p(Y ) Y 
27.8 Many Firms in Cournot
equilibrium

Firm i’s market share: si=yi/Y

si 
p(Y ) 1 
  MC ( yi )
  (Y ) 

With a large number of firms, each firm’s market
share is negligible, and the Cournot equilibrium is
effectively the same as pure competition.
27.9 Simultaneous Price Setting
Firms set their prices and let the market
determine the quantity sold;
 Assuming constant marginal cost c;
 Prices can never be lower than c;
 If ph>c, the low bidder can always increase its
profits by charging a slightly lower price;
 Both firms charging p=c is the unique
equilibrium.

27.10 Collusion
Cartel: Firms set prices and outputs so as to
maximize total industry profits.
 Cartel’s problem:

max p( y1  y2 )[ y1  y2 ]  c1 ( y1 )  c2 ( y2 )
y1 , y2

F.O.C.:
*
*
*
*
*

p( y  y )  p ( y1  y2 )( y1  y2 )  MC1 ( y1 )
*
1
*
2
p( y1*  y2* )  p( y1*  y2* )( y1*  y2* )  MC2 ( y2* )
27.10 Collusion

Marginal profits of firm 1:
*
*
*
*

MR1 ( y )  p( y  y )  p ( y1  y2 ) y1  MC1 ( y1 )
*
1
*
1
*
2
  p( y1*  y2* ) y2*  0
Each firm has incentives to deviate from the
cartel solution.
 Cartel is unstable in the short run.
 Punishment mechanism is needed to maintain
the cartel in the long run.

27.10 Collusion
27.11 Punishment Strategies

Punishment strategy
 Coorperation
stage: produce the Cartel quantity
until a deviation is observed;
 Punishment stage: produce the Cournot quantity
for ever.

The firm being punished will choose the
Cournot quantity in response.
27.11 Punishment Strategies

Present value of cartel behavior:
m
m
m
m 

  m 
2
1 r

r
Present value of cheating:
c
c
c
d 

  d 
2
1 r

(1  r )
(1  r )
r
Cheating won’t happen if
m
c
m 
 d 
r
m c
r
d m
r