Oligopoly
Economics 302 - Microeconomic Theory II: Strategic Behavior
Shih En Lu
Simon Fraser University
(with thanks to Anke Kessler)
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Oligopoly
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Topics
1
Quick review of game theory basics
2
Cournot competition
3
Bertrand competition
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Most Important Things to Learn
1
Solidify understanding of normal-form games: de…nitions, …nding NE,
relation between ISD and NE
2
Find best-response functions in Cournot game
3
Determine the Cournot outcome in an oligopolistic market
4
Determine the Bertrand outcome(s) in an oligopolistic market
5
Be able to explain how to rule out other outcomes in the Bertrand
game
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Some Review Questions
What’s the di¤erence between a strategy, an outcome and a Nash
equilibrium?
Does it ever make sense to compare di¤erent players’payo¤s?
When you’re doing ISD, do comparisons between player 2’s payo¤s
help determine whether you should delete a row?
Why can a NE, whether or not it is in pure strategies, only involve
strategies that survive ISD?
If a game is dominance solvable, what can you say about its NE?
If you need to …nd the NE of a game that’s bigger than 2x2, what
could you do …rst to reduce your workload?
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Oligopoly
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Oligopoly
Armed with game theory, let’s return to the study of market power.
What happens when the number of sellers is small, but greater
than one?
Intuitively, we expect the …rms to have some market power, but not
as much as a monopoly, so we expect an intermediate outcome.
First: Assume …rms simultaneously pick quantities (Cournot model)
Then: Assume …rms simultaneously pick prices (Bertrand model)
Later this semester: Firms make decisions sequentially
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Cournot Competition: Basic Model
Let’s start with the simplest case: two …rms with the same constant
marginal cost c produce a homogeneous good with linear market
demand P = a bQ.
We look for a Nash equilibrium.
Actions: Firm 1 picks q1
actions!
0, …rm 2 picks q2
0. In…nitely many
Let’s …x …rm 2’s quantity q2 , and …gure out …rm 1’s best response.
Firm 1 maximizes:
(P
c ) q1 = ( a
=
b ( q1 + q2 )
bq12
+ (a
bq2
c ) q1
c ) q1
(This is strictly concave in q1 for any q2 . Is …rm 1’s best response
ever a mixed strategy?)
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Cournot Competition: Basic Model (II)
First-order condition:
2bq1 + a
a
= 0
bq2 c
q2
c
2b
2
= q1
Similarly, …rm 2’s best response to …rm 1 picking q1 is:
a
c
2b
q1
= q2
2
In a NE, …rms best respond to each other, so both these
best-response functions (or reaction functions) must hold.
Thus we solve the system of equations, which gives:
q1 = q2 =
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Oligopoly
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c
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Cournot Competition: Basic Model (III)
The total market quantity is thus
Q = q1 + q2 = 23 a b c > 12 a b c = qm .
The price is therefore lower under a Cournot duopoly than under a
monopoly.
Deadweight loss is also lower, but this conclusion changes if …xed
costs are high enough.
You can check that the total pro…t is
monopoly pro…t of
2
1 (a c )
.
4
b
2
2 (a c )
,
9
b
which is less than the
The Cournot duopoly fails to maximize pro…t: …rms are not
internalizing the negative e¤ect of their production on the other …rm’s
price.
Firms could increase pro…t by each producing 21 qm (cooperating), but
each would have an incentive to produce more (defect).
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Oligopoly
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Exercise
Same setup (constant MC = c, demand P = a
…rms.
bQ), but now n
Solve for the Cournot quantities.
What happens as n ! ∞?
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Oligopoly
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Additional Topics (Time Permitting)
Graphical representation
Merging …rms with increasing marginal cost (you will see this on the
problem set)
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Cournot Summary
Firms simultaneously pick quantity.
Look for Nash equilibria: where the best response functions intersect.
With linear demand (and typically), a …rm wants to produce less
when others produce more (quantities are strategic substitutes).
Total quantity and price are between monopoly and perfect
competition cases.
Sellers fail to maximize joint pro…t.
Outcome converges to perfect competition outcome when the number
of …rms ! ∞.
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Bertrand Competition
The Cournot model delivered sensible results. But do we observe
…rms setting quantity or price?
Bertrand model: …rms simultaneously choose price.
Let Qd (.) be the demand curve.
Firm with lowest price pL sells Qd (pL ) units, while …rm(s) with higher
price(s) sell nothing. (Homogeneous good)
The following is often, but not always, assumed: if n …rms share the
lowest price pL , then each sells n1 Qd (pL ) units.
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Bertrand Competition: Simplest Case
Two …rms with the same constant marginal cost c (and no …xed
cost) face downward-sloping demand that crosses c at a positive
quantity.
We look for pure-strategy Nash equilibria.
(There sometimes exist NE that are not in pure strategies. They are
hard to …nd, and we will ignore them.)
For simplicity, we assume that any real number can be a price (not
just multiples of 0.01).
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Bertrand Competition: Simplest Case (II)
In a NE, can a …rm pick a price below c?
In a NE, can both …rms pick prices strictly above c?
In a NE, can one …rm choose c and the other …rm choose a price
strictly above c?
What case is left? Is that a NE?
Graphical representation
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Bertrand Competition: Discussion
Does the NE seem plausible?
Some explanations:
1
2
3
Product di¤erentiation
Capacity constraints
Collusion (later this semester)
Even though in the examined case, the Bertrand model seems easier
to solve than the Cournot model, it gets very messy very fast when
some simplifying assumptions are omitted! See problem set for
examples.
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Oligopoly
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