Download Topic 1: Introduction: Markets vs. Firms

Lecture 6
Oligopoly
1
Introduction

A monopoly does not have to worry about how rivals will react to its action
simply because there are no rivals.

A competitive firm potentiall faces many rivals, but the firm and its rivals
are price takers  also no need to worry about rivals’ actions.

An oligopolist operating in a market with few competitors needs to
anticipate rivals’ actions/ strategies (e.g. prices, outputs, advertising, etc), as
these actions are going to affect its profit.

The oligopolist needs to choose an appropriate response to those actions
 similarly, rivals also need to anticipate the firm’s response and act
accordingly  interactive setting.

Game theory is an appropriate tool to analyze strategic actions in such an
interactive setting  important assumption: firms (or firms’ managers) are
rational decision makers.
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Introduction …
A ‘game’ consists of:




A set of players (e.g. 2 firms (duopoly))
A set of feasible strategies (e.g. prices, quantities, etc) for all players
A set of payoffs (e.g. profits) for each player from all combinations of
strategies chosen by players.

Equilibrium concept  first formalized by John Nash  no player (firm)
wants to unilaterally change its chosen strategy given that no other player
(firm) change its strategy.

Equilibrium may not be ‘nice’  players (firms) can do better if they can
cooperate, but cooperation may be difficult to enforced (not credible) or
illegal.

Finding an equilibrium:  one way is by elimination of all (strictly)
dominated strategies, i.e. strategies that will never be chosen by players 
the elimination process should lead us to the dominant strategy.
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Oligopoly Models



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There are three dominant oligopoly models

Cournot

Bertrand

Stackelberg
They are distinguished by

the decision variable that firms choose

the timing of the underlying game
We will start first with Cournot Model.
The Cournot Model

Consider the case of duopoly (2 competing firms) and there are no entry..

Firms produce homogenous (identical) product with the market demand
for the product:
P  A  BQ  A  B  q1  q2 
q1  quantity of firm 1
q2  quantity of firm 2

Marginal cost for each firm is constant at c per unit of output. Assume that
A>c.

To get the demand curve for one of the firms we treat the output of the
other firm as constant. So for firm 2, demand is
P   A  Bq1   Bq2

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It can be depicted graphically as follows.
The Cournot Model
P = (A - Bq1) - Bq2
The profit-maximizing
choice of output by firm
2 depends upon the
output of firm 1
Marginal revenue for
firm 2 is
TR2
MR2 =
= (A - Bq1) - 2Bq2
q2
MR2 = MC
A - Bq1 - 2Bq2 = c
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If the output of
firm 1 is increased
the demand curve
for firm 2 moves
to the left
$
A - Bq1
A - Bq’1
Demand
c
MC
MR2
q*2
 q*2 = (A - c)/2B - q1/2
Quantity
The Cournot Model

We have
q*2 
 A  c   q1
2B
2
 this is the best response function for firm 2 (reaction function for
firm 2).

It gives firm 2’s profit-maximizing choice of output for any choice of
output by firm 1.

In a similar fashion, we can also derive the reaction function for firm 1.
q1* 

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 A  c   q2
2B
2
Cournot-Nash equilibrium requires that both firms be on their
reaction functions.
The Cournot Model
q2
(A-c)/B
Firm 1’s reaction function
The Cournot-Nash
equilibrium is at
the intersection
of the reaction
functions
(A-c)/2B
qC
C
2
Firm 2’s reaction function
qC1 (A-c)/2B
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(A-c)/B
q1
The Cournot Model
q*1 = (A - c)/2B - q*2/2
q2
q*2 = (A - c)/2B - q*1/2
(A-c)/B
Firm 1’s reaction function
 3q*2/4 = (A - c)/4B
 q*2 = (A - c)/3B
(A-c)/2B
(A-c)/3B
 q*2 = (A - c)/2B - (A - c)/4B
+ q*2/4
C
Firm 2’s reaction function
(A-c)/2B
(A-c)/3B
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(A-c)/B
q1
 q*1 = (A - c)/3B
The Cournot Model

In equilibrium each firm produces
q1*c  q2*c 

2 A  c
3B
Demand is P=A-BQ, thus price equals to
2 A  c
A  2c
*
P  A

3B
Total output is therefore
Q* 

 A  c
3

3
Profits of firms 1 and 2 are respectively
1*   *2   P*  c  q1*c   P*  c  q2*c
 A  c
1*   *2 
2
9B

A monopoly will produce
max 1M   P  c  q1   A  Bq1  c  q1
q
1
1M 
10
 A  c
4B
2

A  c

q 
M
1
2B
The Cournot Model

Competition between firms leads them to overproduce. The total output
produced is higher than in the monopoly case. The duopoly price is lower
than the monopoly price.
2 A  c
 A  c
 q1M 
3B
2B
A  2c
Ac
P* 
 P m  A  Bq1 
because A  c
3
2
Q* 
The overproduction is essentially due to the inability of firms to
credibly commit to cooperate
 they are in a prisoner’s dilemma kind of situation

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The Cournot Model (Many Firms)

Suppose there are N identical firms producing identical products.

Total output:
Q  q1  q2  q3  ...  qN
Demand is:




P  A  BQ  A  B  q1  q2  q3  ...  qN 
Consider firm 1, its demand can be expressed as:
P  A  BQ  A  B  q2  q3  ...  qN   Bq1
Use a simplifying notation:
Q1  q2  q3  ...  qN
So demand for firm 1 is:
P   A  BQ1   Bq1
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The Cournot Model (Many Firms)
P = (A - BQ-1) - Bq1
The profit-maximizing
choice of output by firm
1 depends upon the
output of the other firms
Marginal revenue for
firm 1 is
If the output of
the other firms
is increased
the demand curve
for firm 1 moves
to the left
$
A - BQ-1
A - BQ’-1
Demand
c
MC
MR1
MR1 = (A - BQ-1) - 2Bq1
MR1 = MC
q*1
A - BQ-1 - 2Bq1 = c  q*1 = (A - c)/2B - Q-1/2
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Quantity
The Cournot Model (Many Firms)
q*1 = (A - c)/2B - Q-1/2
 Q*-1 = (N - 1)q*1
 q*1 = (A - c)/2B - (N - 1)q*1/2
 (1 + (N - 1)/2)q*1 = (A - c)/2B
 q*1(N + 1)/2 = (A - c)/2B
 q*1 = (A - c)/(N + 1)B
Q* 1  A  c 

N
B  N  12
 A  Nc   c
 Q* = N(A - c)/(N + 1)B
lim
N   N  1
 P* = A - BQ* = (A + Nc)/(N + 1)
Profit of firm 1 is Π*1 = (P* - c)q*1 = (A - c)2/(N + 1)2B
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Cournot-Nash Equilibrium: Different Costs

Marginal costs of firm 1 are c1 and of firm 2 are c2.

Demand is P = A - BQ = A - B(q1 + q2)

We have marginal revenue for firm 1 as before.

MR1 = (A - Bq2) - 2Bq1

Equate to marginal cost: (A - Bq2) - 2Bq1 = c1
 q*1 = (A - c1)/2B - q2/2
 q*2 = (A - c2)/2B - q1/2
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Cournot-Nash Equilibrium: Different Costs
q*1 = (A - c1)/2B - q*2/2
q2
(A-c1)/B
q*2 = (A - c2)/2B - q*1/2
R1
 q*2 = (A - c2)/2B - (A - c1)/4B
+ q*2/4
 3q*2/4 = (A - 2c2 + c1)/4B
 q*2 = (A - 2c2 + c1)/3B
(A-c2)/2B
R2
C
(A-c1)/2B
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 q*1 = (A - 2c1 + c2)/3B
(A-c2)/B
q1
Cournot-Nash Equilibrium: Different Costs

In equilibrium the firms produce:
q1C 
 A  2c1  c2  and q C   A  2c2  c1 
2
3B
Q*  q1C  q2C 



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 2 A  c1  c2 
3B
3B
The demand is P=A-BQ, thus the eq. price is:
 2 A  c1  c2  A  c1  c2
P*  A  

3
3


Profits are:
2
2
A  2c1  c2 
A  2c2  c1 


*
*
1 
and  2 
9B
9B
Equilibrium output is less than the competitive level.
Concentration and Profitability

Consider the case of N firms with different marginal costs.

We can use the N-firms analysis with modification.

Recall that the demand for firm 1 is P   A  BQ1   Bq1


So then the demand for firm 1 is : P   A  BQi   Bqi , so the MR
can be derived as MR  A  BQ i  2 Bqi
Equate MR=MC  and denote the equilibrium solution by *.
A  BQ*i  2 Bq*i  ci

A  BQ*i  Bq*i  Bq*i  ci
A  B  Q*i  q*i   Bq*i  ci  0
P
P*  Bqi*  ci  0
P*  Bqi*  ci
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Concentration and Profitability
P* - ci = Bq*i
Divide by P* and multiply the right-hand side by Q*/Q*
P* - ci
P*
=
BQ* q*i
P* Q*
But BQ*/P* = 1/ and q*i/Q* = si
so: P* - ci = si

P*
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The price-cost margin
for each firm is
determined by its
market share and
demand elasticity