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Eco 403: Industrial Organization Economics, Fall 2012

Dr. Abdel-Hameed H. Nawar

Oligopoly

Isoprofit Curve

An isoprofit Curve of a firm, �����, ���, is defined to be the

set of all possible combinations of each firm’s output levels

that give that firm the same level of profit.

.. ���, ���: ����, ����� − ����� = ���, � = 1,2�

How is an isoprofit curve shaped?

Property 1. An isoprofit curve is concave and reaches a

maximum on the firm’s reaction (best response) curve.

Recall that for an �-variable function, the hyperplane that is

tangent to the function at a particular point lies above the

function if it is concave and lies below the function if it is

convex. For a single variable function, i.e. where � = 1,

At point E, firm 1 maximizes its profit given firm 2’s

Note that ����� �same output level �Since at point �, firm 2’s output level

� � �at point �then ����� � ����

Similarly, ����� �2

At point E, firm 1 maximizes its profit given firm 2’s

� ����� since at � and �, firm 2 has the

�� = ��∗, because ����, ����� −, firm 2’s output level is greater, �

and thus � at point � � � at point

��.

� � ����� and ����� � �����

At point E, firm 1 maximizes its profit given firm 2’s ��∗.

, firm 2 has the

− ���

� at point

at point � ,

Property 2. A lower isoprofit curve ha

profit.

Property 3: Two

each other.

Proof.

Consider points � ,

definition, ����� =

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ower isoprofit curve has a greater level of

isoprofit curves of a firm cannot

, � and � . Since ����� � ��� � = ����� = �����. Contradiction.

a greater level of

of a firm cannot cross

���� . By

. Contradiction. ■

Property 4: An isoprofit curve

reaction curve more than once.

Proof.

Consider points �

However, for any given

■

One Period Model of Collusion

If firm 1 and firm 2 decide to collude to improve their

profit, then this is possible by producing lower output level.

What is the total output level that maximizes the joint

profits?

In collusion, Firm 1 and Firm 2 act as a monopolist and

will produce the monopolist output level jointly.

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isoprofit curve of a firm cannot cross its

reaction curve more than once.

and �. By Property 2, ����� �given ��∗, ����� � �����. Contradiction.

One Period Model of Collusion

If firm 1 and firm 2 decide to collude to improve their

profit, then this is possible by producing lower output level.

total output level that maximizes the joint

irm 1 and Firm 2 act as a monopolist and

uce the monopolist output level jointly.

of a firm cannot cross its

� � � �����. Contradiction.

If firm 1 and firm 2 decide to collude to improve their

profit, then this is possible by producing lower output level.

total output level that maximizes the joint

irm 1 and Firm 2 act as a monopolist and

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Firm 1’s and Firm 2’s isoprofit functions are largest.

�� !" = �� !" = ���#

�� !" = �� !" = ����# � �� $ = �� $

Note that:

�� !" � �� $

Is unilaterally deviation gainful? As we will see for firm 1,

��% � �� !" � �� $ � �� %

This is symmetric game:

• Deviation profit when the rival colludes: ��% = �� %

• Collusion profit when the rival deviates �� % = ��%

Firm 1

Firm 2

Collude Deviate

Collude �� !" , �� !" �� % , �� %

Deviate ��% , ��% �� $ , �� $

��% firm 1’s optimal output when firm 2 produces �� !"

Clearly, collusion cannot be sustained.

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Collusion is not sustainable in one period model since

cheater cannot be punished.

Repeated Games

• Firms in some industries do not play one-shot game but

rather strategically interact repeatedly. This may affect

the equilibrium behavior. In particular, it opens

possibilities for collusion in an industry.

• Repeated game is a game that is played over and over

again. Repetition could be finite or infinite times and

time is discrete.

• Due to the time-value of money, a 1 dollar earned during

the first period is worth more than a dollar in later

repetitions. Players must discount future payoffs when

they make current decisions.

An infinitely repeated game of Collusion

Present Value

& = interest rate per period

Period 1 Period 2 … Period T

� ��1 + &� … ��1 + &�()� ��1 + &�()�

��1 + &�()� … �

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* = ��+, , 0 � * � 1 is a discount factor, which measures

the time value of the money.

The value today of a future payoff is called the present

value (PV), which is an increasing function of *.

• If * approaches 0, the firm does not care about the

future (impatient),

• If * is close to 1, the firm really cares about the future

(patient).

Trigger Strategy

Collusion could be achieved in an infinitely repeated game

by using the following set of strategies:

1. Each firm produces the collusion level of output each

period as long as its rival does the same.

2. If any firm produces a different level of output, then

beginning from the next period and forever its rivals

will punish the firm by playing the CN equilibrium.

Collusion is sustainable if the

• PV(collusion) > PV(deviation once & then punished

forever); or

• PV(punishment) > PV(gains from collusion)

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Let PV(collusion) be denoted by ./01234 and PV(deviation

once & then punished forever) be denoted by ./.5436

./01234 = � !" + *� !" + *�� !"+. . .= � !"81 + * + *� +⋯:= � !" ; 1

1 − *<

./.5436 = �% + *� $ + *�� $+. ..= �% + *� $81 + * + *� +⋯:= �% + � $ ; *

1 − *<

Collusion is sustainable if

� !" = ��)>? � �% + � $ = >

�)>?

Thus � !" � �1 − *��% + *� $� !" � �% − *�% + *� $� !" � �% − *��% − � $�

Thus

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*��% − � $� � ��% − � !"�

* � ��% − � !"���% − � $�

• If firms care less about the future than the present,

then deviation may be attractive.

• If firms care about future more, then collusion is

sustainable.

• Collusion is sustainable if the firm is sufficiently

patient (i.e. * is close to 1).

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Note:

PV(gains from deviation)

=�% − � !"

PV(punishment)

= *��% − � $� + *���% − � $� + ⋯

= >�)> ��% − � $�

We can obtain

PV(punishment) > PV(gains from deviation)

which gives the same condition for sustainability:

* � ��% − � !"���% − � $�

Numerical Example

Inverse demand function: @��� = 175 − 2��� + ��� Total cost: ������ = 7�� ,� = 1,2

Monopoly Case

maxFGH �175 − 2��� − 7�

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FOC

175 − 4� − 7 = 0

�# = 42

@# = 175 − 2�#= 175 − 2�42�= 91

�# = @#�# − ���#�= 91�42� − 7�42�= 3528

In case of collusion we assume that each firm makes 1 2M of

�#.

� !" = ���# = 21.

� !" = ��� !" = 1769

Duopoly Case

Firm 1’s profit

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maxOPGH �� = Q175 − 2��� + ���R�� − 7��= 168�� − 2��� − 2����

FOC

S��S�� = 168 − 4�� − 2�� = 0

Firm1’s best response curve:

�� = 42 − �� ��

By symmetry,

�� = 42 − �� ��

Solving for �� and ��,

.. �� = 42 − �� =42 − �

� ��?

.. �� $ = 28

By symmetry, �� $ = 28

@ $ = 175 − 2��� $ + �� $�= 175 − 2�28 + 28�= 63

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�� $ = @ $�� $ − ���� $�= 63�28� − 7�28�= 1568

By symmetry,�� $ = 1568

Homework Exercise

• Draw firm 1's best response curve and the isoprofit curve

corresponding to its profit in the Cournot-Nash

equilibrium.

• Draw firm 2's best response curve and the isoprofit curve

corresponding to its profit in the Cournot-Nash

equilibrium.

Hint: for a given level of profit �T�, �T� = UV − WQ�� + �XRY�� − ���, hence�X = =Z)[\ ? − �� − = ]̂_

\O_?.

Collusion

If firm 1 thinks that form 2 will play the collusive level of

output, firm 1 has an incentive to deviate taking firm 2’s

output level as given and then maximizing its profit:

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Using the reaction function �� = 42 − �� ��,

��% = 42 − �� �� !"

= 42 − �� �21�= 31.5

@% = 175 − 2���% + �� !"�= 175 − 2�31.5 + 21�= 70

��% = @% ��% − ����% �= 70 × 31.5 − 7�31.5�= 1983.5

��% = @% ��% − ����% �= 70 × 21 − 7�21�= 1323

The above results can be summarizes in as follows:

Firm 1

Firm 2

Collude Deviate

Collude 1764,1764 1323, 1984.5

Deviate 1984.5, 1323 1568,1568 *

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Clearly, (Deviate, Deviate) is the NE in a one period game.

In an infinitely repeated game, collusion outcome

(Collude, Collude) is sustainable if:

./01234 � ./.5436

and hence if

* � Q]ab)]bcdR�]ab)]be�

* � ��fgh.i)�jkh���fgh.i)�ikg� = 0.5291

If the discount factor * � 0.5291 , then collusion is

sustainable.

CNE, Monopoly and Perfect Competition

The l-firm linear symmetric CNE

For � = 1,2,… �, the following hold

nopq values as function of l

�� $ = Z)[�r+��\ decreasing

� $ = = rr+�? =Z)[\ ? Increasing

@ $ = � + =Z)[r+�? Decreasing

�� $ = �Z)[�s�r+��s\

�t = \� ur

�Z)[��r+�� v

�

t = r�r+����r+��s

�Z)[\

What are some of the insights?

As the number of firms in the industry goes up, each

individual firm produces less, but total output goes up, the

market price decreases and the total surplus increases.

Optimal Entry in a Cournot

We assume:

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Decreasing

v Increasing

� [�s Increasing

What are some of the insights?

As the number of firms in the industry goes up, each

individual firm produces less, but total output goes up, the

market price decreases and the total surplus increases.

Optimal Entry in a Cournot Nash World

As the number of firms in the industry goes up, each

individual firm produces less, but total output goes up, the

market price decreases and the total surplus increases.

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Stage 1: Foreseeing a Cournot game, firms decide whether

or not to enter at a cost of w.

Stage 2: the number of entrants (say �) is observed and the

firms strategically choose Cournot Nash Equilibrium:

��� $, �� $ , … , �r $�

Backward Induction: in a multiple stage game, we solve

the game backwards from the last stage.

Stage 2: �, the number of entrants

�� $ = �Z)[�s�r+��s\

Stage 2: �ntry will occur until profit equals entry cost:

�� $�� $� = w

where � $ is the equilibrium number of firms

�V − ����� $ + 1��W = w

�� $ + 1�� = �Z)[�s\x

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� $ = y�Z)[�s\x − 1

The social planner’s decision problem:

maxr t��� = r�r+����r+��s�Z)[�s

\ − �w

FOC

z({�r�zr = �Z)[�s

�\ = �r+��r+��s − r�r+��

�r+��|? − w = 0

�∗ = y�Z)[�s\x

| − 1� � $

• In the linear symmetric Cournot game , the market

provides too much entry.

• In general there are two extremes associated with

entry.

1. Surplus appropriability: an entrant ignores an

increase in consumer surplus due to entry

2. Business stealing: an entrant ignores declines of

the profits of existing firms caused by entry

(negative externality)