Part II. Market power
Chapter 3. Static imperfect competition
MATH SLIDES
Slides
Industrial Organization: Markets and Strategies
Paul Belleflamme and Martin Peitz
© Cambridge University Press 2009
Chapter 3M - Salop model
Salop model
• Setting
• Firms equidistantly
located on circle with
circumference 1
• Consumers uniformly
distributed on circle
• They buy at most one unit,
from firm with lowest
‘generalized price’
• Unit transportation cost, 
i 1
n
xi,i 1
r   (xi,i 1  ni )  pi  r   ( i 1
n  xi,i 1 )  pi 1
 xi,i 1
2i  1 pi 1  pi


2n
2
© Cambridge University Press 2009
i
n
xi,i 1
Firm i’s
demand
2
i 1
n
Chapter 3M - Salop model
Salop model (cont’d)
• Focus on symmetric equilibrium
• Firm i’s problem:
 1 p  pi 
max pi ( pi  c)Q( pi , p)  ( pi  c)  

n
 
• FOC: 1 / n  ( p  2 p  c) /   0
• Setting pi p yields: p  c   / n
i
*
• n   closer substitutes on the circle
 competitive pressure   p* 
• If n, then p*c (perfect competition)
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Chapter 3M - Asymmetric Hotelling
Asymmetric competition with differentiated
products
• Same setting as Hotelling model
• Only difference: product 1 is of superior quality
• Consumer’s indirect utility:

r1   x  p1

 r2   (1  x)  p2
if buy 1
if buy 2
with r1  r2
• Assume: r2 r1  product 2 more attractive for
some consumers
• Indifferent consumer
1 (r1  r2 )  ( p1  p2 )
x̂  
 Q1 ( p1 , p2 )
2
2
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4
Chapter 3M - Asymmetric Hotelling
Asymmetric competition with differentiated
products (cont’d)
• Firm 1 chooses p1 to maximize (p1c)Q1(p1,p2)
• Similarly for firm 2.
• Solving for the two FOCs:
 p*  c    1 (r  r )
 1
2
3 1
 *
1
 p2  c    3 (r1  r2 )
1 r1  r2
*
*
Q1 ( p1 , p2 )  
2
6
• High-quality firm sets a higher price and sells more.
© Cambridge University Press 2009
5
Chapter 3M - Asymmetric Hotelling
Asymmetric competition with differentiated
products (cont’d)
• Welfare maximization sell at marginal cost
1 r1  r2
1 r1  r2
*
*
Q1 (c, c)  
 Q1 ( p1 , p2 )  
2
2
2
6
• Firm 1’s equilibrium demand is too low from a social
point of view.
• Same analysis if r1 r2 r, but c1 < c2
• Lesson: Under imperfect competition, the firm
with higher quality or lower marginal cost sells
too few units from a welfare perspective.
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Chapter 3M - Asymmetric Hotelling
Cournot pricing formula
• F.O.C. of profit maximization for Cournot firm
P(q)qi  P(q)  Ci(qi )  0 
P(q)  Ci(qi )   P(q)qi 
P(q)  Ci(qi )  P(q)q qi 1

 i
P(q)
P(q) q 
• Suppose constant marginal costs: Ci(qi)  ciqi
n

(
p

 i 1 i ci )q
p  ci  i


   i   ( p  ci ) i q  
n
pq
2
p

i 1
i 1

  i 1  i

n
p   i 1 i ci
n

p
n


n
2

i
i 1


IH

 Lerner index (weighted by
market shares) is proportional
to Herfindahl index
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Chapter 3M - Kreps-Scheinkman
Capacity-then-price model
• Setting
• Stage 1: firms set capacities
qi and incur cost of
capacity, c
• Stage 2: firms set prices pi; cost of production is  up
to capacity (and infinite beyond capacity); demand is
Q(p)  a  p.
• Subgame-perfect equilibrium: firms know that
capacity choices may affect equilibrium prices
• Efficient rationing
• Upper bound on capacity at stage 1
cqi  maxq (a  q)q  a2 / 4  qi  a2 / (4c)
© Cambridge University Press 2009
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Chapter 3M - Kreps-Scheinkman
Capacity-then-price model (cont’d)
• Claim: if c  a  (4/3)c, then both firms
set the
*
p

p

p
 a  q1  q2
market-clearing price: 1
2
• Proof
• Let p1 p* and show that 2’s best-response is p2
p*.
• p2 p* doesn’t pay: same quantity (because firm 2
sells all its capacity) sold at lower price
• p2 p* could pay as firm 1 is capacity constrained...
For this, revenues should be increasing at p* ...
• Firm 2’s revenues:

 p (a  p2  q1 ) if a  p2  q1 ,
p2Q( p2 )   2
0
else


© Cambridge University Press 2009
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Chapter 3M - Kreps-Scheinkman
Capacity-then-price model (cont’d)
• Proof (cont’d)
• Max reached at p  (a  q ) / 2
• Revenues are decreasing at p* if
2
1
p*  p2  a  q1  q2  (a  q1 ) / 2  a  q1  2q2
but q1 , q2  a 2 / (4c)  q1  2q2  (3 / 4)(a 2 / c)
and a  (4 / 3)c  a  (3 / 4)(a 2 / c)
• Hence, not profitable to set p2 p*. QED
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