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OLIGOPOLY THEORY
1. Cournot Oligopoly
Here we are considering a generalized version of a simple example of the
linear Cournot oligopoly in the market for some homogeneous good. We will
suppose that there are N identical …rms, that entry by additional …rms is
e¤ectively blocked, and that each …rm has identical constant marginal costs.
C (qi ) = cqi ;
c
0;
and
i = 1; 2; :::; N:
Firms sell output on a common market, so market price depends on the
total quantity sold by all …rms in the industry. Let the inverse market
demand be the linear form,
p=a
b
N
P
qi
i=1
where a > 0; b > 0; and we will require that a > c. We can write the
pro…t for form i as
i
(q1; q2 ; :::qN ) =
a
b
N
P
qj
j=1
!
qi
cqi
We seek a vector of outputs (q1 ; q2 ; :::; qN ) such that each …rm’s output is
pro…t maximizing given the best output choices choosen by the other …rms.
Such a vector of outputs is called a Cournot-Nash equilibrium.
If (q1 ; q2 ; :::; qN ) is a Cournot-Nash equilibrium, qi must maximize i (q1; q2 ; :::qN )
when qj = qj for all j 6= i: Consequently, the derivative of i (q1; q2 ; :::qN )
with respect to qj must be zero when qj = qj for all j = 1; 2; :::; N: Thus,
multiplying qi to simplify the expression yields
aqi
bqi (q1 + q2 + :::qi
1
+ qi + qi+1 + :::qN
bqi2
i (q1; q2 ; :::qN ) = aqi
bqi
N
P
j6=i
1
qj
1
+ qN )
cqi
cqi
The marginal pro…t is
@
=a
@qi
@
@qi
=a
2bqi
N
P
qj
c
j6=i
bqi
bqi
b
N
P
qj
c
j6=i
@
=a
@qi
@2
=
@qi2
b
bqi
b
N
P
qj
c
j=1
2b < 0 so the following optimal solution yields the maximum value.
We set the …rst order condition to equal zero to …nd the maximized pro…t.
a
bqi
b
N
P
qj
c=0
j=1
bqi = a
c
b
N
P
qj
j=1
Note that the right hand side is independent of which …rm i we are considering. We conclude that all …rms must produce the same amount of output
in equilibrium. By letting q denote this common equilibrium output so that
q1 = q2 = ::: = qN = q ; the previous equation can be written as
bq = a
c
bN q
bq (N + 1) = a
c
Each …rm’s output q =
a c
b (N + 1)
Industry’s output Q =
N (a c)
b (N + 1)
Market Price P = a
b
N (a c)
=a
b (N + 1)
Each …rm’s pro…t
2
i
=
N (a c)
<a
(N + 1)
(a c)2
b (N + 1)2
Equilibrium in this Cournot oligopoly has some interesting features. We
can calculate the deviation of price from marginal cost,
P
a c
>0
N +1
c=
and observe that equilibrium price will typically exceed the marginal cost
of each identical …rm. When N = 1, this coincides with the monopoly
market and the deviation of price from marginal cost is greatest. At the other
a c
extreme, when the number of …rms, N ! 1;we see that lim
= 0:
N !1 N + 1
This tells us that price will approach marginal cost as the number of …rms
becomes large. This suggests that prefect competition can be viewed as a
limiting case of imperfect competition when the number of …rms becomes
in…nity.
2. Stackelberg Oligopoly
We are considering the linear Stackelberg duopoly before moving to the
case of N …rms since this is a little bit trickier than Cournot with N identical
…rms. Assume that the market inverse demand function is P = a bQ; where
Q = q1 + q2 : We assume that both …rm have constant marginal cost c < a;
produce identical products, and that …rm 1 is the Stackelberg leader. Firm
2 will chooses q2 given q1 :
max (a
bQ) q2
q2
max (a
bq1
q2
max aq2
q2
cq2
bq2 ) q2
cq2
bq22
cq2
bq1 q2
The …rst order condtion is
@
=a
@q2
@2
=
@q22
a
c
bq1
2bq2
2b < 0 ! maximum
c
q2 =
bq1
a
2bq2 = 0
c bq1
:
2b
3
This is called …rm 2’s reaction function. Firm 2 thinks that …rm 1 will
choose its output as its best strategy so …rm 2 chooses its output given what
…rm 2 thinks of …rm 1’s best strategy. In the Stackelberg duopoly; however,
…rm 1 is a leader so it can choose its output based on what …rm 2 thinks it
a c bq1
so
will produce. That is, …rm 1 thinks …rm 2 will choose q2 =
2b
we insert this expression into …rm 1’s maximization problem.
max (a
bQ) q1
cq1
c
bq1
bq2 ) q1
b
q1
max (a
q1
max
a
c
bq1
max
a
c
bq1
q1
q1
max
a
q1
a
c bq1
2b
q1
a
c bq1
2
q1
c bq1
2
q1
The …rst order condition yields
q1 =
a
c
2b
a c
;
Note that the competitive output, where P = MC = c; is qc =
b
a c
: Hence, the
and the monopoly output where MR = MC, is qm =
2b
Stackelberg leader produces exactly as the monopolist and half of competitive
output. We can then compute the quantity produced by the follower as
q2 =
a
c bq1
a c
=
2b
2b
q1
a c
=
2
2b
a
c
4b
=
a
c
4b
=
qc
:
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Therefore, the equilibrium quantity in the Stackerlberg duopoly is that
a c
a c
q1 =
and q2 =
:
2b
4b
Now, we are examining the generalized case where there are N identical
…rms. We assume that each …rm chooses output sequentially, in which …rm
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1 is the …rst leader, …rm 2 follows …rm 1, until …rm N follows …rm N -1. How
much does the leader produce?
qc
qc
Again, the leader will produce , …rm 2 will produce ; :::; and …rm N
2
4
qc
will produce N : Thus, if we rank each …rm by the order they are choosing
2
qc
the output, we have that any …rm j will produce exactly an amount of j for
2
j = 1; 2; :::; N: As N ! 1; we see that the total industry output equals
qc
qc qc
+ +:::+ +::: . This approaches qc so pro…t will becomes zero and price
2 4
N
will be reduced to equal marginal cost. How could this happen? If we work
backwards, …rm j’s optimal output cuts pro…t of each of its predecessors in
half. Therefore, …rm N-1 acts as residual monopolist, as do all predecessors.
3. Bertrand Oligopoly
We are considering the price competition instead of quantity competition
described previously. First, we will examine the case of Bertrand duopoly
before generalizing the idea to the case of N competitors.
There is one misleading point. Some authors summarize the di¤erences between Cournot Oligopoly and Bertrand Oligopoly by referring to
the Cournot and Bertrand equilibria. Such usage refers to the di¤erence
between the equilibrium behavior in these models, not to a di¤erence in
the equilibrium concept used in the games. In both games, the equilibrium
concept used is the Nash equilibrium.
We …rst translate the problem into a normal form game. There are again
two players. This time, however, the strategies available to each …rm are the
di¤erent prices it might charge, rather than the di¤erent quantities it might
produce.
Hence, each …rm’s strategy space can be written as
Si = [0; 1); i = 1; 2
The basic concepts of Bertrand duopoly is that …rms choose prices pi
simultaneously with homogeneous product. Both …rms have identical constant marginal costs equal to c. Suppose that there is no …xed cost from
entering the industry. The market demand is given by the general function
D (p). It is not hard to see that the pro…t will be categorized as follows.
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P
MC1
MC1+MC2
Inverse Demand
Q
i
=
8
<
0
D (pi ) (pi c) =2
:
D (pi ) (pi c)
if pi > pj
if pi = pj
if pi < pj
The highest possible price that can be set is the monopoly price, pm ;
which can be solved from di¤erentiating D (pi ) (pi c) with respect to pi :
Typically, …rm i has no best response for c < pi
pm since i wants to
undercut marginally, i.e., pi = pj " for any " in…nitestimally more than
zero (i.e., " = 10 1;000;000 ): If …rm j reasons the same way, it will set price
0
pj = pi " = pj 2": This will continues until pi = pj = c: This is
often called a Bertrand Paradox; even with only two …rms, the price is
similar as that of perfect competition. As an exercise, you are to show that
this is a Nash Equilibrium, and show that it is the only pure strategy Nash
Equilibrium.
Bertrand Model of Di¤erentiated Products
Although this topic is to be studied in the product di¤erentiation chapter,
for the sake of writing the handout it is more appropriate to include some
variations of Bertrand model here. We consider the case of di¤erentiated
products. If …rms 1 and 2, assuming they have the same constant marginal
costs, choose prices p1; and p2 ; respectively, the quantities that consumers
demand from each …rm are
q1 = a
p1 + bp2
q2 = a
p2 + bp1
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We can see whether both goods are substitutes or complements from
the sign of b. These are unrealistic demand functions because demand for
each …rm’s product is positive even when it charges an arbitrarily high price,
provided that another …rm also charges high enough price. The pro…ts for
both …rms are
1
(p1 ; p2 ) = (p1
c) q1 (p1 ; p2 ) = (p1
c) (a
p1 + bp2 )
2
(p1 ; p2 ) = (p2
c) q2 (p1 ; p2 ) = (p2
c) (a
p2 + bp1 )
The objective function is quadratic.
midpoint of its roots.
A quadratic is maximized at the
Here, the roots for each function is, c; and a
responses for each …rm are
p1 =
a + bp2 + c
2
p2 =
a + bp1 + c
2
bpj ; j = 1; 2: The best
Solving these pair of equations yields the Nash Equilibrium:
p1 = p2 =
a+c
:
2 b
What can we conclude from the case of di¤erentiated products? We see
that the introduction of product di¤erentiation results in price charged at
higher level than the marginal cost, provided b < 2:
The increased product di¤erentiation can result in higher prices does not
imply that increased product di¤erentiation lowers social welfare, because in
return for higher prices consumers receive increased variaety of products.
Other variations of Bertrand Oligopoly
Before moving to the N …rms case, it is interesting to consider other
variations in the Bertrand model. Suppose that …rm 1 has higher marginal
cost than that of …rm 2, c1 > c2 : There are two cases. The boring case is
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that c1 > pm2 = arg max D (p) (p
c2 ) : So, at …rm 2’s monopoly price, …rm
p
1 is just not pro…table and is out of the market. The interesting case is that
when c1 < pm2 : Mathematically, there is no equilibrium exists since there
is no best response. Intuitively, if p1 = c1 ; then p2 = c1 " for any " > 0:
Therefore, there are many Nash Equilibria such that p1 = p and p2 = p "
for any c2 < p
c1 : Although these are all Nash Equilibria, the strategy
p1 = c1 weakly dominates all p1 < c1 for player 1.
There is another interesting case if a …rm faces an increasing marginal
cost, as it should be due to the law of diminishing returns. Bertand is much
tougher problem now since a lower priced …rm may choose to serve only a
part of quantity demanded at its price.
From the …gure in page 6, the rationing rule used by the lower priced
…rm can a¤ect the sales and pro…ts of the other …rm. For example, if …rm 1
sells only to the most eager buyers (i.e. those who have highest reservation
price), then …rm 2 will sell nothing.
Finally, we consider the case of N …rms in Bertrand oligopoly. Assume
for simplicity that they have identical constant marginal cost c. Now, there
are in…nitely many Nash Equilibria. All …rms i will set prices pi
c and
at least 2 …rms i will set price pi = c: It is left to the reader why this is so.
(be aware that this might be on the midterm exam.)
4. Price Leadership Model
This is probably one of the models that I see little bene…t from it. Those
of you who will pursue a graduate degree may plausibly not see this in your
graduate study.
Suppose that a market consists of one dominant …rm that controls a large
percentage of total industry output and a signi…cant number of relatively
small "fringe" …rms. Suppose the total industry inverse demand is given by
p = a bQ = a bqd bqf ; where qd is the quantity demanded from the
dominant …rm and qf is the quantity demanded from the small fringe …rms.
We assume that the fringe’s total inverse supply curve is given by p = c+dqf ,
and the dominant …rm’s marginal cost curve is given by M Cd = c+eqd , where
d > e; and a > c: To obtain the dominant …rm’s residual demand curve,
we need to subtract the fringe supply curve from the total industry demand
curve at every price greater than c.
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dqf = p
c ! qf =
p
c
d
;
Q=
a
p
b
The residual demand, written as qd = R (p) is calculated from
qd = R (p) = Q
qf =
a
b
p
p
b
p c
ad + bc
+ =
d d
bd
(b + d)
ad + bc
=
bd
bd
Inverse Residual Demand : p =
Marginal Revenue : MR =
ad + bc
b+d
ad + bc
b+d
p
(b + d)
bd
qd
bd
qd
b+d
2bd
qd
b+d
The dominant …rm equate MR to equal MC,
ad + bc
b+d
2bd
qd = c + eqd
b+d
ad + bc
b+d
c = eqd +
2bd
qd
b+d
ad cd
e (b + d) + 2bd
=
qd
b+d
b+d
qd =
ad cd
e (b + d) + 2bd
The dominant …rm sets price from its inverse residual demand
p=
=
ad + bc
b+d
bd
ad cd
b + d e (b + d) + 2bd
1
ad + bc
b+d
abd2 + bcd2
e (b + d) + 2bd
=
1
abde + ad2 e + 2abd2 + b2 ce + bcde + 2b2 cd
b+d
e (b + d) + 2bd
=
abde + ad2 e + abd2 + b2 ce + bcde + 2b2 cd + bcd2
1
b+d
e (b + d) + 2bd
9
abd2 + bcd2
Now, I understand why there is no textbook explaining the dominant price
leadership model in the parameterized form because of this tremendously
intimidating,but little intuitive expression. The …nal step is to …nd the
quantity produced by the small fringe …rms. That is, they are behaving like
competitive price takers so they equate this (ugly) price to their supply (or
MC).
abde + ad2 e + abd2 + b2 ce + bcde + 2b2 cd + bcd2
1
= c + dqf
b+d
e (b + d) + 2bd
1
abde + ad2 e + abd2 + b2 ce + bcde + 2b2 cd + bcd2
qf =
bd + d2
e (b + d) + 2bd
c
d
As I expected, little insight would be gained from working the price leadership model in this general form. Perhaps you may better understand this
model by consulting the Waldman and Jensen’s textbook.
5. Collusive Oligopoly
In this models we are exploring how collusive equilibrium could form, and
why each oligopolist has an incentive to move away from the equilibrium. It
the …rms collude, then they will act like a single monopolist to set prices and
outputs so as to maximize total industry pro…ts, not individual pro…t. Such
collusion is called cartel.
Consider the simple duopoly where each …rm acts to collude to set prices
and quantities. the total industry pro…ts could be written as
total
= max p (Q) Q
q1 ;q2
c (Q) = p (q1 + q2 ) [q1 + q2 ]
c1 (q1 )
The …rst order conditions are
@
@q1
p (Q ) + p0 (Q ) [Q ]
c01 (q1 ) = 0
@
@q2
p (Q ) + p0 (Q ) [Q ]
c02 (q2 ) = 0
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c2 (q2 )
p (q1 + q2 ) + p0 (Q ) [q1 + q2 ] = M C1 (q1 )
p (q1 + q2 ) + p0 (Q ) [q1 + q2 ] = M C2 (q2 )
) M C1 (q1 ) = M C2 (q2 )
Two marginal costs are equal in equilibrium. If one …rm has a cost lower
than that of another, it will necessarily produce more output in equilibrium
in the cartel solution.
The problem of forming a cartel is that there is always a temptatiuon to
cheat. Suppose that two …rms are producing at the maximizing quantities q1
and q2 : Firm 1 considers producing a little bit more output, by the amount
dq1 : That is, …rm 1 is considering maximizing its own pro…t.
1
@ 1
@q1
= max p (Q ) q1
q1
p (Q ) + p0 (Q ) q1
c1 (q1 )
M C1 (q1 ) ; but
p (Q ) + p0 (Q ) q1 + p0 (Q ) q2 = M C1 (q1 )
Therefore,
@ 1
= p (Q ) + p0 (Q ) q1
@q1
M C1 (q1 ) =
p0 (Q ) q2 > 0:
The last inequality follows since p0 (Q ) is negative due to negatively inverse demand slope.
Hence, if …rm 1 believes that …rm 2 will keep its output …xed, then it
will believe that it can increase pro…ts by increasing its own production
@ 1
since
> 0: Firm 2 also believes similarly. Thus, each …rm will be
@q1
tempted to increase its own pro…ts by expanding its own output, and then
the equilibrium will go to Counot Nash Equilibrium.
Consider the case of linear collusive equilibrium. Suppose that the inverse demand function is p (Q) = a bQ = a b (q1 + q2 ) : The marginal
costs for both …rms are c1 q1 and c2 q2 : The total cartel pro…t is
11
= [a b (q1 + q2 )] (q1 + q2 ) c1 q1 c2 q2 =
a (q1 + q2 ) b (q1 + q2 )2 c1 q1 c2 q2
@
=a
@q1
@
=a
@q2
2b (q1 + q2 )
2b (q1 + q2 )
c 1 = 0 ! q1 + q2 =
c 2 = 0 ! q1 + q2 =
a
c1
2b
a
c2
c1 = c2
2b
If marginal costs are constant, both of them must be equal in the equilibrium. As you can see, the two equations are not independent so that we
can only determine the total output. Nonetheless, the division of output
between the two …rm does not matter.
We have seen that a cartel is fundamentally unstable in that it is always
in the best interest of each …rms to increase their production. Then, how
could a collusive equilibrium be sustained? We will explore this matter
in the Grim Trigger Strategy from the study of dynamic game of complete
information.
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