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Chapter 5
Imperfect competition
Outline.

An introduction to the theory of games

Some oligopoly models

Monopolistic competition
Collusion between firms


Collusion is difficult to achieve on a
competitive market. Might seem less
difficult to achieve among oligopolists,
that is in industries served by only a small
number of firms.

However, collusion usually appears extremely
difficult to sustain.

Because it is the interest of each firm
individually is usually not in the interest of all
firms taken as a whole.
Similar to the prisoner’s dilemma
Prisoner’s dilemma

2 prisoners



If one of the 2 confesses, the confessor will be freed
while the other one will spend 20 years in jail.
If both confess, they both get an intermediate sentence
(say 5 years).
Payoff matrix:
Table 1
Prisoner Y
Confess
Confess
5 years for each
Prisoner X
Remains silent

20 years for X
0 years for Y
Dominant strategy = confess
Remains silent
0 years for X
20 years for Y
1 year for each
The sustainability of collusion


Consider 2 firms that are the sole providers of
mineral water on a given market.

Market demand curve is given by: P = 20 – Q

MC = 0
Collusion: each firm will offer half of the
monopoly output and sell it at the monopoly
price.

Monopoly quantity: 20 – 2Q = 0  Q = 10 and P = 10.

If both firms abide by this agreement:


Each will sell 5 units of output
Profit: P = 50 – 0 = 50.
The sustainability of collusion (ctd 1)

Each firm actually has two options: it can
abide by the agreement or defect.
Assume that defection means cutting the
price from 10 to 9.

If one firm abides by the agreement and the
other one defects. What happens?



The defector will capture the entire market because
of its lower price. He will sell Q = 20 – 9 = 11 and
make a profit of 99.
The other firm sells nothing and makes zero profit.
If both firms defect, they will end up splitting
the 11 units of output sold at a price of 9 and
will make an economic profit of 49.5
The sustainability of collusion (ctd 2)

Payoff matrix:
Table 2
Firm 1
Firm 2

Cooperate
(P = 10)
Defect
(P = 9)
Cooperate
(P = 10)
P1 = 50
P2 = 50
P1 = 0
P2 = 99
Defect
(P = 9)
P1 = 99
P2 = 0
P1 = 49.5
P2 = 49.5
Dominant strategy = defect

As this example is set up firms do not do much worse
when they defect that when they cooperate.

However, if firms find it in their interest to defect once,
they are likely to defect again.
Advertising


Oligopolists compete on prices but also by
advertising. When a firm advertises its
product, its demand increases for 2
reasons.

First people who never used that type of
product before learn about it, which leads
some of them to buy it.

Second other people who already consumed a
different brand of the product will switch brand
because of advertising.
US cigarette industry: brand-switching
effect of advertisement is very strong.
Advertising (ctd)

Payoff matrix:
Table 3
Don't advertise
Firm 2
Advertise
 Dominant
Firm 1
Don't advertise
P1 = 500
P2 = 500
P1 = 0
P2 = 750
strategy = advertise
Advertise
P1 = 750
P2 = 0
P1 = 250
P2 = 250
Nash equilibrium

In many games, not every player has a
dominant strategy.
Table 4
Don't advertise
Firm 2
Advertise
Firm 1
Don't advertise
P1 = 500
P2 = 400
P1 = 200
P2 = 0
Advertise
P1 = 750
P2 = 100
P1 = 300
P2 = 200

The dominant strategy for Firm 1 is to advertise.
But Firm 2 has no dominant strategy

is the Nash equilibrium of the game

Nash equilibrium: definition

A Nash equilibrium is a combination of
strategies such that each player's strategy
is the best he can choose given the
strategy chosen by the other player.

At a Nash equilibrium, neither player has any
incentive to deviate from his current strategy.

In a prisoner's dilemma, the equilibrium is a
Nash equilibrium.

But a Nash equilibrium does not require both players
to have a dominant strategy.
The maximin strategy



In the previous example, we have assumed that
Firm 2 believes that Firm 1 will act rationally.

However, Firm 2 may not be sure that Firm 1 will act
rationally.

When Firm 2 has no dominant strategy and is not sure
of what Firm 1 will do, what should it do?
If firm 2 is extremely cautious, it may choose the
maximin strategy: it will choose the strategy that
maximises the lowest possible value of its own
payoff.
In this situation, the maximin strategy is not to advertise.
Repeated play in prisoner's dilemma


Strategy to prevent defection: tit-for-tat

How it works: the first time you interact with somebody,
you cooperate. In each subsequent interaction you do
what the person did in the previous interaction.

Robert Axelrod (The Evolution of Cooperation, 1984):
tit-for-tat performs very well against a large number of
alternative strategies
Conditions for tit-for-tat to be successful:


There must be a rather stable set of players each of
whom can remember what the others have done in
previous interactions.
Players must allocate a sufficiently high value to the
future.
Tit-for-tat

These conditions are often met in human
populations. Many people interact repeatedly and
keep track of what others did in the past.
Examples
 World War I
 Business world

Additional condition: there is not a known fixed
number of future interactions.

Does tit-for-tat generate widespread collusion?
By no means: cartels tend to be highly unstable.


Problem of selective punishment
Risk of entry
Sequential games

In many games, one player moves first and the
other one can choose his strategy with full
knowledge of the first player's choice

Example: USA versus Soviet Union during much
of the Cold War

Given the assumed payoff, USA may threaten to
retaliate, but if the payoffs are as displayed above, this
is not credible.

In order to be sure that there won't be any risk of
nuclear war, the USA should install a "doomsday"
machine
USA versus Soviet Union
Useless investments: Sears Tower

Consider the example of the Sears Tower in
Chicago (the highest building).

A company X considers whether to build a higher
building. Its concern is that Sears may react by building
an even higher building.
Sears Tower: strategic entry deterrence

Before Sears had originally built its tower: option
of building a platform at the top on which it could
subsequently build an addition that would make
the building taller.

Building the platform costs 10 units but reduces
the cost of making the building taller by 20 units.

This platform is an example of strategic entry
deterrence
Sears Tower (ctd)
Outline.

Some oligopoly models

Monopolistic competition
The Cournot model

The central assumption of the model is that each
firm treats the amount produced by the other
firm as a fixed quantity that does not depend on
its own production decisions.

Suppose the market demand curve for mineral
water is given by:
P  a  b(Q1  Q2 )
and suppose MC = 0

The demand curve for firm 1's water is:
P  (a  bQ2 )  bQ1
The Cournot duopoly (ctd 1)
The Cournot model (ctd2)
Firm 1's demand curve is the portion of the
original demand curve that lies to the right of this
vertical axis.
 So, it is sometimes called the residual demand
curve.


The rule for firm 1's profit maximisation is: MR =
MC = 0.

Marginal revenue has twice the slope as demand so
that:
MR1  (a  bQ2 )  2bQ1
The reaction functions

The optimal output level is given by:
(a  bQ 2 )
MR 1  0  Q 
2b
*
1

This is firm 1’s reaction function: Q1*  R 1 (Q 2 )

Firm 2’s reaction function is given by:
(a  bQ1 )
Q  R 2 (Q1 ) 
2b
*
2
The reaction functions (ctd)
The Nash equilibrium of the Cournot
model

The intersect of both reaction functions is the
Nash equilibrium of the Cournot model:
(a  bQ1 ) (a  bQ 2 )

 Q1  Q 2
2b
2b
(a  bQ1 )
a
 Q1 
 Q1  Q 2 
2b
3b
How profitable are Cournot duopolists?

Since their combined output is 2a/3b, the market
price will be:
a
P  a  b ( Q1  Q 2 ) 
3

At this price, each will have a total revenue equal
to the economic profit given by:
2
a
TR  Pr ofit 
9b
The Bertrand model

Each firm chooses its price on the assumption
that its rival's price remains fixed.

Suppose that the market demand and cost
conditions are the same as in the Cournot
0
example. Suppose firm 1 charges an initial price P1

Then firm 2 faces essentially 3 choices:




It can charge more than firm 1 and sell nothing.
It can charge the same as firm 1 in which case both
firms will split the market demand at that price.
It can sell at a marginally lower price than firm 1 and
capture the entire market.
This last option is always the most profitable.
The Bertrand model (ctd)

As in the Cournot model the situations of the
duopolists are completely symmetric in the
Bertrand model.

So, the strategy of selling at a marginally lower price will
be chosen by both firms.

In this case, there is no stable equilibrium: the pricecutting process will go on until the price reaches the
marginal cost =0.

In this case, both duopolists will share the market
equally.
The Stackelberg model

What would a firm do if knowing that its rival is a
naïve Cournot duopolist?


This firm would choose its own output level by taking
into account the effect of that choice upon the output
level of its rival.
Returning to the Cournot model, assume that
firm 1 knows that firm 2 will treat firm 1's output
as given.

Firm 2's reaction function is:
Q*2  R 2 (Q1 ) 

(a  bQ1 )
2b
Knowing this, firm 1 can substitute R2(Q1) for Q2 in the
equation for the market demand curve.
The Stackelberg model (ctd 1)
 The demand curve addressed to firm 1
a  bQ1  a  bQ1

P  a  b[Q1  R 2 (Q1 )]  a  b Q1 

2b 
2

The Stackelberg model (ctd 2)
 Firm 1 = Stackelberg leader
Firm 2 = Stackelberg follower
Comparison of outcomes

A monopoly with the same demand and cost
curves as the Cournot duopolist would have
produced:
P  a  b(Q1  Q2 )  a  bQ
MR  a  2bQ  0  Q 

and
P
a
2
The Cournot duopolists



a
2b
P = a/3
Q = 2a/3b.
The Bertrand duopolists



P = MC = 0
Q = a/b, so that each of them produce a/2b.
This is similar to the perfect competition situation.
Comparison of outcomes (ctd 1)

The Stackelberg model:

P = a/4
Q = 3a/4b

P1 = a2/8b and P2 = a2/16b

 In the Stackelberg model, the leader fares better than
the follower.
Table 5
Model
Shared monopoly
Cournot
Stackelberg
Bertrand
Perfect competition
Industry output Q
Qm=a/2b
(4/3)Qm
(3/2)Qm
2Qm
2Qm
Market price P
Pm=a/2
(2/3)Pm
(1/2)Pm
0
0
Industry profit P
Pm=a2/4b
(8/9)Pm
(3/4)Pm
0
0
Comparison of outcomes (ctd 2)
Contestable markets

William Baumol, John Panzer and Robert Willig,
Contestable Markets and the Theory of Industry
Structure, 1982.



The idea is that monopolies sometimes behave just like
perfectly competitive firms.
This will happen when entry and exit are perfectly free.
Costless entry: there are no sunk costs
associated with entry and exit

When sunk costs are high, new firms will not enter the
market even if the incumbent is making high profits

When sunk costs are almost zero, new competitors will
enter the market with the idea that they will pull out if
post-entry business proves non profitable.
Contestable markets (ctd)

The contestable market theory



Cost conditions will determine how many firms
will end up serving the market.
But there is no clear relationship between the
actual number of competitors in a market and
the extent to which prices and quantities are
similar to what we would see under perfect
competition.
Critics: there are substantial sunk costs involved
in all activities
Competition under increasing returns to scale

Suppose that there exists a duopoly in an
industry where there are increasing returns to
scale.



2 firms started at an early stage of development
Should we expect that one firm will drive the other one
out of the market?
2 solutions


Merge: problem of antitrust laws.
Price war: none of the 2 firms has any interest in doing
that.

without a threat of entry, a live-and-let-live strategy is
very likely to be adopted
Competition under increasing returns to scale
(ctd)

Suppose now that a firm has a monopoly
position and that potential entrants face
substantial sunk costs.
 Potential
entrants may be reluctant to enter
the industry and face a potentially ruinous
price war with the incumbent

Last solution


Buyers may be willing to approach a potential
entrant.
Local authorities usually do that
Outline.

Monopolistic competition
Monopolistic competition

Monopolistic competition occurs when:



Many firms serve a market with free entry and
exit
But in which the product of each firm is not a
perfect substitute to the product of the other
firms on the market.
The degree of substitutability between
products determines how closely the
industry resembles perfect competition
The Chamberlin model


Developed in the 1930s by Edward Chamberlin and
Joan Robinson.

Assumption: there exists a clearly defined market
composed of many firms producing products that are
close but imperfect substitutes for one another.

So, each firm faces a downward-sloping demand curve
but behaves as if its price and quantity decisions should
not affect the behaviour of the other firms in the industry
Firms are perfectly symmetric so if it makes sense
for a firm to alter its price in one direction, it will
make sense for the other firms to do the same
The individual firm’s demand curves

Each firm faces 2 demand curves
Chamberlinian equilibrium in the short
run
Chamberlinian equilibrium in the long
run
Perfect competition versus Chamberlinian
monopolistic competition

Perfect competition generates allocative efficiency
whereas monopolistic competition does not.
 The Chamberlin model is more realistic than the
perfect competition model on, at least, one point.



Perfect competition: the price is equal to the marginal
cost  firms are indifferent to the opportunity of filling a
new order.
Monopolistic competition: the price is higher than the
marginal cost  firms will be very keen on filling an
additional order.
In both market structures, long-run profits are 0.
Criticisms of the Chamberlin model


The model considers a group of products which
are different in some unspecified way but that are
likely to appeal to any given buyer.

George Stigler: it is impossible to draw operational
boundaries between groups of products in this way.

The Chamberlinian industry group quickly expands to
contain all possible consumption goods in the economy.

Complicates the perfect competition model without
altering its most important predictions.
Does not depart sufficiently from the perfect
competition model

Assumption that each firm has an equal chance to
attract any of the buyers in an industry: not always true
The spatial interpretation of monopolistic competition

One concrete way of thinking about the lack of
substitutability is distance.
 The seminal paper in this literature has been published by
Harold Hotelling in the Economic Journal in 1929

Consider a small island with a big lake in the
middle. Business activities are necessarily located
at the periphery of the island. Restaurants: meals
are produced under increasing returns to scale.

Circumference of the island is 1 mile. Initially: 4
restaurants, evenly spaced.
Min. distance = 0 and Max distance = 1/8 miles.

L customers scattered uniformly around the circle and

the cost of travel is t€ per mile.
The initial location of restaurants

Total cost curve of the restaurant is:
TC = F + MQ
 ATC = TC/Q = F/Q + M
The average cost of a meal with 4
restaurants


If TC = 50 + 5Q where Q is the number of meals served
each day.

If L = 100 and there are 4 restaurants, each restaurant will
serve 100/4 = 25 persons a day.

So, the total cost is TC = 50 + (5x25) = 175€ per day.
Average total cost is TC/25 = 7€ per meal.

Clearly this is higher than if there are only 2 restaurants: each
serve 50 meals per day with AC = [50+(5x50)]/50 = 6€.
What is the average cost of transportation if there are 4
restaurants?

In this case the farthest someone can live from a restaurant is
1/8 miles so that he round trip is 1/4 miles.

If the travel cost is 24€ per mile the total travel cost will be
6€.
The average cost of a meal with 4
restaurants(ctd)
 Since people are uniformly scattered around the
loop, straightforward calculation (!) show that the
average round trip is 1/8 miles


Trick: this is the average between maximum distance
(=1/4) and minimum distance which is 0

So, the average transportation cost is 3€.
The overall average cost per meal is 7€ + 3€ =
10€
The optimal number of locations

Results from a trade-off between the fixed cost of
opening new locations and the savings from
lower transportation costs

What is the best number of outlets to have?

If we increase the number of restaurants from 4 to 5,
what happens to the average cost?
 Each restaurant serves 20 meals per day, with an
ATC = [50+(5x20)]/20 = 7.5€

The distance between 2 restaurants is now 1/5 miles.

So the maximum round-trip distance is 1/5 miles.
The minimum being 0, the average round-trip distance is
1/10 miles.

So, the average transportation cost is 24€ x 1/10 = 2,4€.

The optimal number of locations
(ctd 1)


The overall average cost is therefore: 7.5 + 2.4 = 9.9€.
So, adding a fifth restaurant reduces the average cost of
the meal by 0.1€.

Adding a 6th restaurant, the overall AC goes up

 The optimal number of restaurants is 5.
What is the best number of outlets to have?

If we increase the number of restaurants from 4 to 5,
what happens to the average cost?

Each restaurant serves 20 meals per day, with an
ATC = [50+(5x20)]/20 = 7.5€
The optimal number of locations
(ctd 2)


The distance between 2 restaurants is now 1/5 miles. So
the maximum round-trip distance is 1/5 miles.

The minimum being 0, the average round-trip distance
is 1/10 miles.

So, the average transportation cost is 24€ x 1/10 =
2,4€.
The overall average cost is therefore:

AC = 7.5 + 2.4 = 9.9€.

So, adding a fifth restaurant reduces the average cost of
the meal by 0.1€
Generalisation


In order to generalise this result, let us assume
that there are N outlets around the loop.

The distance between adjacent outlets is 1/N and the
maximum one-way trip length is half of that: 1/2N.

If people are uniformly distributed around the loop, the
average one-way trip length is 1/4N and the average
round-trip distance is 1/2N.
The average transportation cost is :
C trans

tL

2N
The total cost of meals is:
C meals  LM  NF
Generalisation (ctd 1)
Generalisation (ctd 2)

The first order condition is:
 (C trans  C meals )
0
N


tL
tL
*

F

0

N

2F
2N 2
Applying this to our example yields:
24 100
N 
 4.9  5
2  50
*
An example

Why are there many so many fewer small food
stores than 30 years ago (and so much larger
supermarkets)?

Food stores face strongly increasing returns to scale.

Transportation costs have decreased
The analogy to product characteristics

Consider the various airline flights between two
cities on a given day. People have different
preferences for travelling at different times of the
day.
The analogy to product characteristics
(ctd)

Virtually, any consumer product can be
interpreted in the context of the spatial model.

On the automobile market, there exists a very
large variety of options.




Of course, it would be much cheaper if there were only a
couple of models.
But people are willing to pay a little extra for variety as
they are willing to pay some more for a more
conveniently located shop.
Car manufacturers are said to "locate" on the "productspace". Their aim is to make sure that few buyers are
left without a choice that lies "close" to the car that best
suits them.
Similar interpretations apply to cameras,
vacations, bicycles, etc
Paying for variety

Wastefully high levels of product variety?

In our model we have assumed that all
customers face the same transportation costs.



This is clearly not the case in reality.
Demand for variety increases sharply with income: it is
a luxury.
So, firms usually set prices in a different way for
their different products.


Typically, they will price their basic products very close
to the marginal cost
And the more fancy products several times the marginal
cost.
The Hotelling model

2 hot-dog vendors who can choose where to
settle along a beach.

Suppose the beach is 1 mile long and is bounded by
natural obstacles.

Suppose the vendors charge the same price and
customers are evenly distributed along the beach. They
buy one hot-dog from the nearest vendor.

Where should vendors position themselves?
The Hotelling model (ctd)


A and B are the locations that minimise average
travel distance for all customers.

Yet, they are not optimal from the perspective of the
vendors.

The only stable outcome is for each to locate at C. Each
gets half of the market as before, but now the average
one-way distance is ¼ of miles, i.e. twice as much as
before.
Having both vendors at the centre of the beach is
optimal for vendors but not for customers.

In this case, the "invisible hand" does not guide resource
allocation so as to produce the greatest good for all.
U.S. political parties
Liberal
A
Democrats
A'
C
B'
B
Republicans
Conservative