Production | Oligopoly 1
Cournot Duopoly
Suppose there are two flrms in an industry,Their strategy spaces are quantities,Their payofis are proflts.
Industry demand is given by the inverse demand function,P(Q),where industry production is Q = q1 +q2,They
have identical cost functions c(qi),Proflts for each flrm are therefore given by:
…1 = P(q1 +q2)q1| {z }
Revenue
c(q1)|{z}
Cost
and …2 = P(q1 +q2)q2| {z }
Revenue
c(q2)|{z}
Cost
This is a game,The Nash equilibrium occurs when both flrms are optimising given the behaviour of the other.
Throughout the lecture consider the following linear demand example with constant marginal costs,So:
P(Q)= a? Q = a? q1? q2 and c(q)= cq
Production | Oligopoly 2
Proflts and Best Responses
Proflts for each flrm are maximised where marginal revenue is equal to marginal cost,Recall flrms are interested in
flnding an optimal level of their quantity for each level their opponent might choose,Suppose flrm 2 chooses q2:
maxq
1
…1 =maxq
1
(a? q1? q2)q1? cq1
q2 acts like a constant,For linear demand,marginal revenue falls at twice the rate of demand.
Firm 1 sets marginal revenue,a? q2?2q1,equal to marginal cost,c,Hence:
q1(q2)= a? c? q22 and q2(q1)= a? c? q12
These two equations give the best response functions for the two flrms,Often called reaction or best reply functions.
Plotting these yield the reaction or best reply curves.
Production | Oligopoly 3
Reaction Curves
Drawing the reaction curves for both flrms on the same graph yields the picture below.
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0
q2
q1
q?2
q?1
q1(q2)
q2(q1)
The curve q1(q2) yields the optimal level of q1 for any given q2,The curve q2(q1) yields the optimal level of q2 for
any given q1,These curves will cross,Why?
A point at which there is no profltable unilateral deviation is a Nash equilibrium,That is a point which is a best
response to a best response (and so on) | where the two curves cross,written (q?1;q?2).
Production | Oligopoly 4
Nash Equilibrium
What is the value of q?1 and q?2? They could be read ofi from the graph,Alternatively,solve the two equations.
Substituting the value of q2 into the equation for q1 yields:
q?1 = 12
‰
a? c? a? c? q
1
2
=) q?1 = a? c3
Symmetrically solving for q?2 gives q?2 =(a? c)=3.
With the same linear demand and constant marginal cost assumptions in place but with n flrms in the industry it is
(mathematically) simple to show that each of the n flrms will produce at:
q?i = a? cn+1
Production | Oligopoly 5
Equilibrium Proflts and Prices
How much do the flrms charge and how much proflt do they make? Price is simply read ofi from the demand curve.
Equilibrium industry supply is Q? = q?1 +q?2,Hence equilibrium price is:
P? = P(Q?)= a? q?1? q?2 = a+2c3
Proflt is given by revenue less cost,…?1 =(P c)q?1 =(a? c)2=9.
It is just as easy to calculate the price and proflt in the case of n flrms:
P? = an+1 + ncn+1 and …?i =
a? c
n+1
2
Notice that,as the number of flrms grows,price gets closer to marginal cost and proflts get close to zero,These are
the conditions in a perfectly competitive market,An oligopoly with many flrms is like perfect competition.
Production | Oligopoly 6
Collusion
How does the case of Cournot duopoly difier from monopoly? If the two flrms could collude they would act like a
monopoly to maximise total proflts,Recall a monopolist faces the entire demand curve and sets MR = MC.
With linear demand P = a? Q and constant marginal costs c,the optimality condition requires:
MR = MC =) a?2Qm = c =) Qm = a? c2
A monopolist produces less and hence prices higher at P m =(a+c)=2,Proflts are …m =(a? c)2=4.
If the two flrms could collude they would be able to split the proflts in two,each flrm getting (a? c)2=8 by
producing qm1 =(a? c)=4,This is bigger than their Cournot equilibrium proflt.
But they cannot,If one of the flrms produced (a? c)=4 the other would not choose to produce the same,The best
response function reveals that the flrm has a better response where if (for example) flrm 1 produced (a? c)=4:
q2(q1)= a? c? q12 = 12
‰
a? c? a? c4
= 38(a? c)
This will yield higher proflts,How can collusion be explained?
Production | Oligopoly 7
Stackelberg Leadership
Suppose that flrm 1 (the leader) is already in the market,Firm 2 (the follower) is about to enter.
This situation is known as Stackelberg leadership,It is a difierent game and has a difierent equilibrium.
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Firm 1 chooses q1 Firm 2 chooses q2 =) (…1;…2)
Firm 2 has a relatively simple problem to solve,Knowing that flrm 1 has already produced q1 flrm 2 simply chooses
their quantity to maximise their proflts,Using the best response function reveals the action flrm 2 will take.
Working backwards,What should flrm 1 do? Firm 1 now knows which quantity flrm 2 will choose and how it
depends upon their own choice,Therefore,including this information in their proflt function,flrm 1 optimises.
Production | Oligopoly 8
Proflts and Best Responses
Suppose flrm 1 chooses q1,Using the same example,flrm 2 will choose q2 to maximise …2 =(a? q1? q2)q2? cq2.
The reaction curve gives the solution to this problem,Hence,flrm 2 chooses:
qs2(q1)= a? c? q12
Firm 1 knows this and uses this information to when maximising their proflt,Firm 1 has proflt equal to:
…1 =
a? q1? a? c? q12
q1
| {z }
Revenue
cq1|{z}
Cost
=) …1 = 12 f(a? q1)q1? cq1g
This is exactly half the proflt a monopolist would receive,Hence flrm 1,setting MR = MC,operates just like a
monopolist | and produces the monopoly output qs1 =(a? c)=2 | and receives half the proflt (a? c)2=8.
Firm 2’s output is worked out from the best response function yielding,qs2 =(a? c)=4.
Production | Oligopoly 9
Equilibrium
The equilibrium is characterised by flrm one producing the monopoly output (a? c)=2 and flrm 2 playing a best
response (a? c)=4,Equilibrium price is therefore P s = a? q1? q2 =(a+3c)=4.
Finally,equilibrium proflts are …1 =(a? c)2=8 and …2 =(a? c)2=16.
Compare this with Cournot,The leader does better,The follower does worse,Price is lower,Total output is larger.
Compare this with collusion,The leader does as well,The follower does worse,Price is lower,Output is larger.
However,like Cournot,comparing this to perfect competition reveals both flrms do better than perfectly
competitive flrms | who get zero proflt,Prices are higher and output is lower.
Production | Oligopoly 10
Bertrand Duopoly
Suppose instead flrms choose prices | does this make a difierence?
Firm 1 and flrm 2 choose p1 and p2 simultaneously,The flrm that charges the lower price serves the entire market.
If both flrms charge the same price,they both serve half the market.
Proflts are payofis,Writing down proflts,suppose that market demand is Q(P) where P is market price (the lower
of the two prices,p1 and p2),Suppose again that marginal costs are constant,MC = c.
…1 =
8>
><
>>:
p1Q(p1)? cQ(p1) if p1 < p2
1
2 fp1Q(p1)? cQ(p1)g if p1 = p2
0 if p1 > p2
To illustrate the best reply functions consider what flrm 1’s optimal response is if flrm 2 sets a price p2,If p2 is
greater than marginal cost then flrm 1 will wish to undercut flrm 2 by a small amount.
If p2 is equal to marginal cost,flrm 1 would be willing to set any price greater than or equal to p2 | all such prices
will result in zero proflts,Firm 1 will never set a price below p2 if p2 is less than c since this results in losses,In this
last case,flrm 1 would set any price strictly greater than p2 and get zero proflts.
Production | Oligopoly 11
Reaction Curves and Nash Equilibrium
Drawing this argument in an informal way gives the below \best response functions".
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p2
p1
c
c
p1(p2)
p2(p1)
The only place where the two curves cross | the Nash equilibrium | is at (p1;p2)=(c;c).
The flrms price at marginal cost | the e–cient outcome,It makes no difierence how many flrms are in the market.
The basic idea is that the flrms will continue to undercut one another until they reach marginal cost,They will go
no lower as this would involve making a loss.
This is very difierent to Cournot,However,the Cournot equilibrium can be recovered with capacity constraints.
Production | Oligopoly 12
A Price Leadership Game
Consider the following price leadership game | note the similarity with Stackelberg leadership,Note the difierence
between this and Varian’s rather odd game,where proflts accrue to flrm 1 during the game.
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Firm 1 chooses p1 Firm 2 chooses p2 =) (…1;…2)
Again,working from the back,Firm 2 will choose to undercut flrm 1’s initial price in order to gain the whole market.
Firm 1,unable to make positive proflts,can choose any price above marginal cost,Either flrm 2 will undercut and
make positive proflts or,if flrm 1 chooses to price at marginal cost both flrms make zero proflt.
In particular (p1;p2)=(c;c) is still an equilibrium | another difierence between price and quantity competition.
Again,it seems collusion could result in higher proflts,But flrms are unable to collude successfully,Why?
Cournot Duopoly
Suppose there are two flrms in an industry,Their strategy spaces are quantities,Their payofis are proflts.
Industry demand is given by the inverse demand function,P(Q),where industry production is Q = q1 +q2,They
have identical cost functions c(qi),Proflts for each flrm are therefore given by:
…1 = P(q1 +q2)q1| {z }
Revenue
c(q1)|{z}
Cost
and …2 = P(q1 +q2)q2| {z }
Revenue
c(q2)|{z}
Cost
This is a game,The Nash equilibrium occurs when both flrms are optimising given the behaviour of the other.
Throughout the lecture consider the following linear demand example with constant marginal costs,So:
P(Q)= a? Q = a? q1? q2 and c(q)= cq
Production | Oligopoly 2
Proflts and Best Responses
Proflts for each flrm are maximised where marginal revenue is equal to marginal cost,Recall flrms are interested in
flnding an optimal level of their quantity for each level their opponent might choose,Suppose flrm 2 chooses q2:
maxq
1
…1 =maxq
1
(a? q1? q2)q1? cq1
q2 acts like a constant,For linear demand,marginal revenue falls at twice the rate of demand.
Firm 1 sets marginal revenue,a? q2?2q1,equal to marginal cost,c,Hence:
q1(q2)= a? c? q22 and q2(q1)= a? c? q12
These two equations give the best response functions for the two flrms,Often called reaction or best reply functions.
Plotting these yield the reaction or best reply curves.
Production | Oligopoly 3
Reaction Curves
Drawing the reaction curves for both flrms on the same graph yields the picture below.
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0
q2
q1
q?2
q?1
q1(q2)
q2(q1)
The curve q1(q2) yields the optimal level of q1 for any given q2,The curve q2(q1) yields the optimal level of q2 for
any given q1,These curves will cross,Why?
A point at which there is no profltable unilateral deviation is a Nash equilibrium,That is a point which is a best
response to a best response (and so on) | where the two curves cross,written (q?1;q?2).
Production | Oligopoly 4
Nash Equilibrium
What is the value of q?1 and q?2? They could be read ofi from the graph,Alternatively,solve the two equations.
Substituting the value of q2 into the equation for q1 yields:
q?1 = 12
‰
a? c? a? c? q
1
2
=) q?1 = a? c3
Symmetrically solving for q?2 gives q?2 =(a? c)=3.
With the same linear demand and constant marginal cost assumptions in place but with n flrms in the industry it is
(mathematically) simple to show that each of the n flrms will produce at:
q?i = a? cn+1
Production | Oligopoly 5
Equilibrium Proflts and Prices
How much do the flrms charge and how much proflt do they make? Price is simply read ofi from the demand curve.
Equilibrium industry supply is Q? = q?1 +q?2,Hence equilibrium price is:
P? = P(Q?)= a? q?1? q?2 = a+2c3
Proflt is given by revenue less cost,…?1 =(P c)q?1 =(a? c)2=9.
It is just as easy to calculate the price and proflt in the case of n flrms:
P? = an+1 + ncn+1 and …?i =
a? c
n+1
2
Notice that,as the number of flrms grows,price gets closer to marginal cost and proflts get close to zero,These are
the conditions in a perfectly competitive market,An oligopoly with many flrms is like perfect competition.
Production | Oligopoly 6
Collusion
How does the case of Cournot duopoly difier from monopoly? If the two flrms could collude they would act like a
monopoly to maximise total proflts,Recall a monopolist faces the entire demand curve and sets MR = MC.
With linear demand P = a? Q and constant marginal costs c,the optimality condition requires:
MR = MC =) a?2Qm = c =) Qm = a? c2
A monopolist produces less and hence prices higher at P m =(a+c)=2,Proflts are …m =(a? c)2=4.
If the two flrms could collude they would be able to split the proflts in two,each flrm getting (a? c)2=8 by
producing qm1 =(a? c)=4,This is bigger than their Cournot equilibrium proflt.
But they cannot,If one of the flrms produced (a? c)=4 the other would not choose to produce the same,The best
response function reveals that the flrm has a better response where if (for example) flrm 1 produced (a? c)=4:
q2(q1)= a? c? q12 = 12
‰
a? c? a? c4
= 38(a? c)
This will yield higher proflts,How can collusion be explained?
Production | Oligopoly 7
Stackelberg Leadership
Suppose that flrm 1 (the leader) is already in the market,Firm 2 (the follower) is about to enter.
This situation is known as Stackelberg leadership,It is a difierent game and has a difierent equilibrium.
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Firm 1 chooses q1 Firm 2 chooses q2 =) (…1;…2)
Firm 2 has a relatively simple problem to solve,Knowing that flrm 1 has already produced q1 flrm 2 simply chooses
their quantity to maximise their proflts,Using the best response function reveals the action flrm 2 will take.
Working backwards,What should flrm 1 do? Firm 1 now knows which quantity flrm 2 will choose and how it
depends upon their own choice,Therefore,including this information in their proflt function,flrm 1 optimises.
Production | Oligopoly 8
Proflts and Best Responses
Suppose flrm 1 chooses q1,Using the same example,flrm 2 will choose q2 to maximise …2 =(a? q1? q2)q2? cq2.
The reaction curve gives the solution to this problem,Hence,flrm 2 chooses:
qs2(q1)= a? c? q12
Firm 1 knows this and uses this information to when maximising their proflt,Firm 1 has proflt equal to:
…1 =
a? q1? a? c? q12
q1
| {z }
Revenue
cq1|{z}
Cost
=) …1 = 12 f(a? q1)q1? cq1g
This is exactly half the proflt a monopolist would receive,Hence flrm 1,setting MR = MC,operates just like a
monopolist | and produces the monopoly output qs1 =(a? c)=2 | and receives half the proflt (a? c)2=8.
Firm 2’s output is worked out from the best response function yielding,qs2 =(a? c)=4.
Production | Oligopoly 9
Equilibrium
The equilibrium is characterised by flrm one producing the monopoly output (a? c)=2 and flrm 2 playing a best
response (a? c)=4,Equilibrium price is therefore P s = a? q1? q2 =(a+3c)=4.
Finally,equilibrium proflts are …1 =(a? c)2=8 and …2 =(a? c)2=16.
Compare this with Cournot,The leader does better,The follower does worse,Price is lower,Total output is larger.
Compare this with collusion,The leader does as well,The follower does worse,Price is lower,Output is larger.
However,like Cournot,comparing this to perfect competition reveals both flrms do better than perfectly
competitive flrms | who get zero proflt,Prices are higher and output is lower.
Production | Oligopoly 10
Bertrand Duopoly
Suppose instead flrms choose prices | does this make a difierence?
Firm 1 and flrm 2 choose p1 and p2 simultaneously,The flrm that charges the lower price serves the entire market.
If both flrms charge the same price,they both serve half the market.
Proflts are payofis,Writing down proflts,suppose that market demand is Q(P) where P is market price (the lower
of the two prices,p1 and p2),Suppose again that marginal costs are constant,MC = c.
…1 =
8>
><
>>:
p1Q(p1)? cQ(p1) if p1 < p2
1
2 fp1Q(p1)? cQ(p1)g if p1 = p2
0 if p1 > p2
To illustrate the best reply functions consider what flrm 1’s optimal response is if flrm 2 sets a price p2,If p2 is
greater than marginal cost then flrm 1 will wish to undercut flrm 2 by a small amount.
If p2 is equal to marginal cost,flrm 1 would be willing to set any price greater than or equal to p2 | all such prices
will result in zero proflts,Firm 1 will never set a price below p2 if p2 is less than c since this results in losses,In this
last case,flrm 1 would set any price strictly greater than p2 and get zero proflts.
Production | Oligopoly 11
Reaction Curves and Nash Equilibrium
Drawing this argument in an informal way gives the below \best response functions".
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0
p2
p1
c
c
p1(p2)
p2(p1)
The only place where the two curves cross | the Nash equilibrium | is at (p1;p2)=(c;c).
The flrms price at marginal cost | the e–cient outcome,It makes no difierence how many flrms are in the market.
The basic idea is that the flrms will continue to undercut one another until they reach marginal cost,They will go
no lower as this would involve making a loss.
This is very difierent to Cournot,However,the Cournot equilibrium can be recovered with capacity constraints.
Production | Oligopoly 12
A Price Leadership Game
Consider the following price leadership game | note the similarity with Stackelberg leadership,Note the difierence
between this and Varian’s rather odd game,where proflts accrue to flrm 1 during the game.
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.
Firm 1 chooses p1 Firm 2 chooses p2 =) (…1;…2)
Again,working from the back,Firm 2 will choose to undercut flrm 1’s initial price in order to gain the whole market.
Firm 1,unable to make positive proflts,can choose any price above marginal cost,Either flrm 2 will undercut and
make positive proflts or,if flrm 1 chooses to price at marginal cost both flrms make zero proflt.
In particular (p1;p2)=(c;c) is still an equilibrium | another difierence between price and quantity competition.
Again,it seems collusion could result in higher proflts,But flrms are unable to collude successfully,Why?
