Oligopoly Pricing
EC 202 – Lecture IV
Francesco Nava
London School of Economics
January 2011
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EC 202 – Lecture IV
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Summary
The models of competition presented in MT explored the consequences on
prices and trade of two extreme assumptions:
Perfect Competition
Monopoly
[Many sellers supplying many buyers]
[One seller supplying many buyers]
Today intermediate assumption is discussed:
Oligopoly
[Few sellers supplying many buyers]
Two models of competition among oligopolists are presented:
Quantity Competition
[aka Cournot Competition]
Price Competition
[aka Bertrand Competition]
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EC 202 – Lecture IV
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A Duopoly
Consider the following economy:
There are two …rms N = f1, 2g
Each …rm i 2 N produces output qi with a cost function
ci (qi ) mapping quantities to costs
Aggregate output in this economy is q = q1 + q2
Both …rms face an aggregate inverse demand for output
p (q ) mapping aggregate output to prices
The payo¤ of each …rm i 2 N is its pro…ts:
ui ( q1 , q2 ) = p ( q ) qi
ci ( q i )
Pro…ts depend on the output decisions of both
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Cournot Competition: Duopoly
Competition proceeds as follows:
All …rms simultaneously select their output to maximize pro…ts
Each …rm takes as given the output of its competitors
Firms account for the e¤ects of their output decision on prices
In particular the decision problem of player i 2 N is to:
max ui (qi , qj ) = max p (qi + qj )qi
qi
qi
ci ( q i )
[Historically this is the …rst known example of Nash Equilibrium – 1838]
[Example: Visa vs Mastercard]
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Cournot Competition: Equilibrium
If standard conditions on primitives of the problem hold:
a pure strategy Nash equilibrium exists
the PNE is characterized by the FOC
If so, the problem of any producer i 2 N satis…es:
∂ui (qi , qj )
∂p (qi + qj )
= p ( qi + qj ) +
qi
∂qi
∂qi
|
{z
}
Marginal Revenue
∂ci (qi )
∂q
| {zi }
0 if qi = 0
= 0 if qi > 0
Marginal Cost
Marginal revenue accounts for the distortion in prices
Prices decrease if inverse demand is downward-sloping
FOC de…nes the best response (aka reaction function) of player i:
qi = bi ( qj )
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Cournot Example
Consider the following economy:
p (q ) = 2
q
c1 (q1 ) = q12 and c2 (q2 ) = 3q22
Firm i’s problem is to choose production qi given choice of the other qj :
max(2
qi
qi
qj ) qi
ci ( q i )
The best reply map of each …rm is determined by FOC:
2
2q1
q2
2
2q2
q1
2q1 = 0 ) q1 = b1 (q2 ) = (2
6q2 = 0 ) q2 = b2 (q1 ) = (2
Cournot Equilibrium outputs are:
Perfect competition outputs are larger:
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q2 )/4
q1 )/8
q1 = 14/31 and q2 = 6/31
q1 = 3/5 and q2 = 1/5
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Collusion and Cartels
Suppose that the producers collude by forming a cartel
A cartel maximizes the joint pro…ts of the two …rms:
max p (q )q
q 1 ,q 2
c 1 ( q1 )
c 2 ( q2 )
First order optimality of this problem requires for any i 2 N:
∂p (q )
p (q ) +
q
∂q
|
{z
}
Marginal Revenue
∂ci (qi )
∂q
| {zi }
0 if qi = 0
= 0 if qi > 0
Marginal Cost
Aggregate pro…ts are higher in the cartel
Players account for e¤ects of their output choice on others
But the pro…ts of each individual do not necessarily increase
Nava (LSE)
EC 202 – Lecture IV
Jan 2011
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Collusion Example
Consider the previous duopoly, but suppose that a cartel is in place
If so, FOC for the cartel production satisfy:
2
2q 1
2q 2
2
2q 2
2q 1
2q 1 = 0 ) q 1 = (1
6q 2 = 0 ) q 2 = (1
q 1 )/4
q 1 = 3/7 and q 2 = 1/7
Cartel outputs are:
q1 = 14/31 and q2 = 6/31
Cournot outputs are larger:
u 1 = 3/7 and u 2 = 1/7
Cartel pro…ts are:
Cournot pro…ts are:
q 2 )/2
u1 = 392/961 and u2 = 144/961
Total pro…ts are larger with a cartel in place, but not all …rms may bene…t
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Defection from a Cartel
Suppose that the …rm j produces the cartel output q j
If so, …rm i may bene…t by producing more than the cartel output since:
bi ( q j ) > q i
In this scenario sustaining a cartel may be hard without output monitoring
In the example this was the case as …rms preferred to increase output:
b1 (1/7) = 13/28 > 3/7
b2 (3/7) = 11/56 > 1/7
If so the problem of sustaining the cartel becomes a Prisoner’s dilemma
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Bertrand Competition: Duopoly
Competition proceeds as follows:
All …rms simultaneously quote a price to maximize pro…ts
Each …rm takes as given the price quoted by its competitors
Firms account for the e¤ects of their pricing decision on sales
Consider an economy with:
Two producers with constant marginal costs c
Aggregate demand for output given by q (p ) = (b0
Given the prices demand of output from …rm i
8
< q (pi ) if
q (pi )/2 if
qi ( pi , pj ) =
:
0
if
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p )/b
2 N is:
pi < pj
pi = pj
pi > pj
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Bertrand Competition: Monopoly
If only one …rm operated in such market it would choose the price to:
max u (p ) = max q (p )(p
p
p
c)
Thus a pro…t maximizing monopolist would sell goods at a price:
p = (b0 + c )/2
With two producers the problem of each …rm becomes:
max ui (pi , pj ) = max qi (pi , pj )(pi
pi
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pi
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c)
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Bertrand Competition: Best Responses
In the Bertrand model, i’s optimal pricing is aimed at “maximizing sales”
In particular …rm i would set prices as follows (for ε small):
If pj > p, set pi = p and capture all the market at the monopoly price
If p
pj > c, set pi = pj
If c
pj , set pi = c as there are no bene…ts by pricing below MC
ε, undercut j and capture all the market
Such logic requires the best response of each player to satisfy:
8
if
pj > p
< p
p
ε if p pj > c
pi = bi ( pj ) =
: j
c
if
c pj
Thus in the unique NE both …rms set p1 = p2 = c and perfect
competition emerges with just 2 …rms!
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Cournot vs Bertrand Competition
Bertrand model predicts that duopoly is enough to push down prices to
marginal cost (as in perfect competition)
Cournot model instead predicts that few producers do not su¢ ce to
eliminate markups (prices above marginal cost)
In both models there are incentives to form a cartel and to charge the
monopoly price
Neither model is intrinsically better
Accuracy of either model depends on the fundamentals of the economy:
Bertrand works better when capacity is easy to adjust
Cournot works better when capacity is hard to adjust
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