Partial equilibrium analysis:
Oligopoly
Lectures in Microeconomic Theory
Fall 2010, Part 18
07.07.2010
G.B. Asheim, ECON4230-35, #18
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Examples

Telenor Mobil, NetCom
Rimi, Rema, others
SAS, low price airlines

Characteristica
Competition on price or quantity
Simultaneous or sequential decisions
Homogeneous or differentiated products
One-time competition, or long-term relation
Competition from potential entrants
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Model
2
G.B. Asheim, ECON4230-35, #18
price
 2 ( y1, y2 )  p (Y ) y2
 a  b( y1  y2 )  y2
Inverse
demand fn.:
p (Y )  a  bY
y1
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y2
Y
G.B. Asheim, ECON4230-35, #18
quantity
3
1
No competition — Monopoly
P
 1 ( y1, y2 )   2 ( y1, y2 ) 
a  b( y1  y2 )  y1  a  b( y1  y2 )  y2
 a  bY Y
max a  bY Y
p (Y m )
Y
FOC : a  2bY  0
Y
Y m  a / 2b
1 ( y1m , y2m )   2 ( y1m , y2m )
 a  bY Y m  a 2 / 4b
Ym
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P (Y m )  a / 2

m

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G.B. Asheim, ECON4230-35, #18
Sequential quantity setting: Stackelberg
Firm 1 chooses quantity first (1 is the leader)
Firm 2 chooses quantity after having
observed firm 1’s quantity (2 is the follower)
p
Is it best to be a leader or a follower?
The follower
follower' s problem :
max a  b( y1  y2 )  y2
y2
FOC : a  by1  2by2  0
Y
y1
y2  f 2 ( y1 )  ( a  by1 ) / 2b
y2
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Firm 2' s best respo nse fn :
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G.B. Asheim, ECON4230-35, #18
Stackelberg (cont.)
Τhe leader' s problem :
y2
The follower’s
best response fn
p


1
max a  b( y1  a 2by
b ) y1
y1
 max 12 a  by1  y1
y1
Iso profit curve FOC : a / 2  by  0
1
for the leader
y1

i e y1s  a / 2b
i.e.,

y2s  a  by2s / 2b  a / 4b
Y
y1
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y2
s
  y2s  3a / 4b p (Y s )  a / 4
Y 1 ( y1s , y2s )  p (Y s ) y1s  a 2 / 8b
 2 ( y1s , y2s )  p (Y s ) y2s  a 2 / 16b
y1s
G.B. Asheim, ECON4230-35, #18
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2
Simultaneous quantity setting: Cournot comp.
Are there quantities for the two firms so that
no firm will regret its own quantity when told
of the quantity of the other firm?
y2
1' s best respo nse fn : y1  ( a  by2 ) / 2b
2' s best
b t respo nse fn
f : y2  ( a  by1 ) / 2b
y1c  y2c  a / 3b
Y    2a / 3b p (Y c )  a / 3
1 ( y1c , y2c )   2 ( y1c , y2c )  a 2 / 9b
c
y1c
y2c
y1
07.07.2010
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G.B. Asheim, ECON4230-35, #18
A numerical example
a  60 b  1
y1c  y2c  20
Y c  40 p(Y c )  20
 1 ( y , y2c )   2 ( y1c , y2c )  400
y2
c
1
y1s  30 y2s  15
Y  45 p (Y s )  15
 1 ( y1s , y2s )  450  2 ( y1s , y2s )  225
s
y1m  y2m  30
Y
y1
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m
 30 p (Y m )  30
 1 ( y1m , y2m ) 
 2 ( y1m , y2m )  900
G.B. Asheim, ECON4230-35, #18
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Cournot competition with production costs
1 ( y1 , y2 )  ( p (Y )  c1 ) y1  a  b( y1  y2 )  c1  y1
y2
 2 ( y1, y2 )  ( p (Y )  c2 ) y2  a  b( y1  y2 )  c2  y2
FOC : y1  a  by2  c1  / 2b
y2  a  by1  c2  / 2b
Firm 1’s best
response
p
fn
y1c  (a  c2  2c1 ) / 3b
y2c  (a  c1  2c2 ) / 3b
Firm 2’s best
response fn
What happens if firm 1’s
costs are reduced?
y1
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G.B. Asheim, ECON4230-35, #18
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3
Observations about Cournot competition

Best response curves have negative slope

Cournot’s ”reaction
story” is stabile.
Reduced c1 shifts 1’s
curve outwards.

yy22

Reduced c1 leads to
increased y1 and reduced y2.

Reduced c1 leads to a direct and indirect advantage
for 1 and an indirect disadvantage for 2.
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yy11
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G.B. Asheim, ECON4230-35, #18
General analysis of Cournot competition
p (Y ) where Y  y1  y2 ( Homogeneous products)
 i ( yi , y j )  p ( yi  y j ) yi  ci ( yi )
FOC :
 i
 p (Y )  p(Y ) yi  ci ( yi )  0
yi
p (Y )  ci ( yi )
y
Y yi
s
  p(Y ) i   p(Y )
 i
p (Y )
p (Y )
p (Y ) Y

yi
where si 
is i' s market share,
Y
1 p (Y )
and  
is the elasticity of demand.
p(Y ) Y
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G.B. Asheim, ECON4230-35, #18
Characterization of Cournot competition
marginal revenue


Each firm has market power (P(Y)  P(Y)  P(Y) yi)
Outcome lies between monopoly and perfect comp.

Difference between price and marginal cost is
reduced when the demand becomes more elastic.

Each firm’s difference between price and marginal
cost is proportional to its market share.

A firm’s market share depends of its efficiency.

Even less efficient firms ”survive” with positive
market share.
07.07.2010
G.B. Asheim, ECON4230-35, #18
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4
Cournot competition with many firms
Assume ci ( yi )  c for all firms and quantities.
p (Y )  c
1

 perfect competitio n when n  .
p (Y )
n
p u (Y )
W lf analysis
Welfare
l i off Cournot
C
t comp.
p
Firms act as if they maximize
1
n
 p(Y )Y  cY   nn1 u (Y )  cY  where
u (Y )  
n  1 : Monopol y, maximize profit.
n  " large" : Almost maximize welfare.
07.07.2010
G.B. Asheim, ECON4230-35, #18
Y
0
~
p (Y )
~
Y Y
~ ~
p (Y ) dY
13
Condition for decreasing best response fn.
Profit function :  i ( yi , y j )
 i ( f i ( y j ), y j )
Best response fn : f i ( y j ) defined by
0
yi
 2 i
 2 i
Differenti ate totally :
dyi 
dy j  0
yi y j
yi2
2
SOC for
 i
-max.
2
y y
dy
 i
 2 i
f i( y j )  i
  2i j  0 if
0&
0
2
dy j  i 0
yi y j
 i
yi
yi
yi2
f i( y j )  0 : Strat. substitutes f i( y j )  0 : Strat. complem.
07.07.2010
G.B. Asheim, ECON4230-35, #18
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Simultaneous price setting: Bertrand comp.
Inverse demandfn.: p (Y )  a  bY
No costs.
The firm with the lowest price gets all the demand.
(If both set the same price, then each gets half the demand.)
Are there prices such that no firm will regret its own price
when told of the price of the other firm?
p1  p2  MC  0 ?
Each firm will regret that it did not set a lower price.
p1  p2  MC  0 eller p2  p1  MC  0 ?
The lower price firm will regret not having set a higher price.
Hence: p1b  p2b  MC  0
07.07.2010
y1b  y2b  a / 2b
G.B. Asheim, ECON4230-35, #18
 1b   2b  0
15
5
Observation
Firms in oligopoly markets set prices, but the competition
does not seem to be as hard as under perfect competition.
Why not?
1. There are capacity constraints. The Cournot model
can be interpreted as competition in capacity
(Example: Competition between tour operators)
2. Long-term relation. If one firm sets a lower price now,
then competitors will set lower prices in future
periods. (Example: Meet-the-competition-clause)
3. The products are differentiated. Types of differentiation.
(Example: Advantage programs for loyal customers)
07.07.2010
G.B. Asheim, ECON4230-35, #18
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