1. Theory of Industrial Organisation
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Theory of Industrial Organisation
Oligopoly

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Introduction
We now consider markets with more than two firms – moving from duopoly to more general oligopoly.
We still assume that the firms produce homogeneous products.
We start by extending the Bertrand and Cournot models to more firms, before considering free entry later.

3. Bertrand oligopoly This is effectively the same as Bertrand duopoly:
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Bertrand oligopoly
This is effectively the same as Bertrand duopoly:
If more than one firm has the equal lowest MC, they share the market (so n firms with the lowest cost each have a market share of 1/n) selling at P = MC.
If one firm has lower MC than all others, it will be the only firm in the market, selling at a price just below the next most efficient firm’s MC.

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Cournot oligopoly
With n firms, the maximisation problem faced by firm 1 is:
max 𝑞 1 Π 1 𝑞 1 , 𝑞 2 ,…, 𝑞 𝑛 = 𝑃 𝑗=1 𝑛 𝑞 𝑗 𝑞 1 − 𝐶 1 𝑞 1

5. In the linear demand, constant MC case, this becomes:
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In the linear demand, constant MC case, this becomes:
max 𝑞 1 Π 1 𝑞 1 , 𝑞 2 ,…, 𝑞 𝑛 = 𝑣− 𝑗=1 𝑛 𝑞 𝑗 − 𝑐 1 𝑞 1 −𝑓
F.O.C.:
𝑣− 𝑗=1 𝑛 𝑞 𝑗 − 𝑐 1 − 𝑞 1 =0 ⇒𝑣−2 𝑞 1 − 𝑗=2 𝑛 𝑞 𝑗 − 𝑐 1 =0
⟹ 𝑞 1 = 1 2 𝑣− 𝑐 1 − 𝑗=2 𝑛 𝑞 𝑗

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This means that firm 1’s best-reply output now depends on the joint output of all the other n – 1 firms in the market.
More generally, the best-reply for firm i is given by: 𝑞 𝑖 = 1 2 𝑣− 𝑐 𝑖 − 𝑗=1,𝑗≠𝑖 𝑛 𝑞 𝑗
= 1 2 𝑣− 𝑐 1 − 𝑄− 𝑞 𝑖
where 𝑄= 𝑗=1 𝑛 𝑞 𝑗 is the sum of output of all n firms in the market.

7. Summing over the n firms’ best-replies gives:
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Summing over the n firms’ best-replies gives:
𝑄= 𝑗=1 𝑛 𝑞 𝑗 = 𝑗=1 𝑛 𝑣− 𝑐 𝑗 − 𝑄− 𝑞 𝑗
= 1 2 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 −𝑛𝑄+ 𝑗=1 𝑛 𝑞 𝑗
= 1 2 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 − 𝑛−1 𝑄
2𝑄=𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 − 𝑛−1 𝑄

8. 2𝑄=𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 − 𝑛−1 𝑄 𝑛+1 𝑄=𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑄= 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1
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2𝑄=𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 − 𝑛−1 𝑄 𝑛+1 𝑄=𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑄= 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1

9. where 𝑐 is the average MC for a firm in the industry.
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We can substitute this output into the expression for qi above to give:
𝑞 𝑖 = 1 2 𝑣− 𝑐 𝑖 − 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1 − 𝑞 𝑖
2𝑞 𝑖 =𝑣− 𝑐 𝑖 − 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1 + 𝑞 𝑖
𝑞 𝑖 = 𝑛+1 𝑣− 𝑐 𝑖 − 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1
= 𝑣− 𝑛+1 𝑐 𝑖 +𝑛 𝑐 𝑛+1 = 𝑣− 𝑐 𝑖 +𝑛 𝑐 − 𝑐 𝑖 𝑛+1
where 𝑐 is the average MC for a firm in the industry.

10. Price is found by substituting Q into the inverse demand equation:
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Price is found by substituting Q into the inverse demand equation:
𝑃=𝑣−𝑄=𝑣− 𝑛𝑣− 𝑗=1 𝑛 𝑐 𝑗 𝑛+1
= 𝑣+ 𝑗=1 𝑛 𝑐 𝑗 𝑛+1

11. What does this tell us?
Each firm’s output is now lower, the more firms there are in the market.
However the lower the cost of an individual firm, the more it will produce.
The more firms there are in the market, the higher the total output and the lower the price.

12. What does this tell us?
In the symmetric case, with ci = c for all firms, 𝑞 𝑖 = 𝑣−𝑐 𝑛+1 ;𝑄= 𝑛 𝑣−𝑐 𝑛+1 ;𝑃= 𝑣+𝑛𝑐 𝑛+1
For n = 1, these give us monopoly values; for large n, they approach perfect competition.

13. Q-i = Q – qi gives combined output for all firms other than i. Q-i
Q-i QC
Each firm sells the same output, so we find the point on i’s BR curve where the combined output of the rest of the market is the same as i’s for duopoly 
Q-i = qi
BRi
qi(2) qm qi

14. With 3 firms we find the point on i’s BR curve where the combined output of the rest of the market is twice as high as i’s, and so on…
Q-i
Q-i = 3qi
Q-i = 2qi QC
Q-i = qi
As n increases, firm i’s output falls towards zero, but total market output rises towards the competitive level.
BRi
qi(4)
qi(2) qm qi

15. Oligopoly and market power
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Oligopoly and market power
From above, it should be clear that with enough firms, Cournot will start to approach perfect competition. But how quickly does this happen?
Cabral analyses efficiency losses from oligopoly.

16. Oligopoly and market power
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Oligopoly and market power
On an inefficiency scale from zero (perfect competition) to 100% (monopoly), he shows that a 7-firm symmetric Cournot model scores 6%, while 15 firms score 1.5%.
This suggests that when firms are symmetric and set outputs, we don’t need many firms for perfect competition to be a reasonable approximation.

17. Oligopoly and market power
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Oligopoly and market power
However, if firms are not symmetric, using number of firms alone can be misleading – one large firm and a number of small firms is closer to monopoly than perfect competition. 
Two common measures of market power and concentration are the Lerner index and the Herfindahl index.

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Lerner index
The Lerner index is calculated as the weighted average of firms’ price-cost margins, weighted by market shares (si for firm i):
𝐿= 𝑖=1 𝑛 𝑠 𝑖 𝑝− 𝑐 𝑖 𝑝
This gives a good estimate of market power in an industry, but is generally hard to measure because it requires cost data for each firm.

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Herfindahl index
The Herfindahl index is easier to calculate as we only need data on market shares:
𝐻= 𝑖=1 𝑛 𝑠 𝑖 2

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Free entry
What will happen in Cournot if new firms are free to enter the market?
If firms are symmetric and face no fixed costs, we will reach perfect competition!
Any firm will sell a (small) positive output 𝑞 𝑖 = 𝑣−𝑐 𝑛+1 at a price above MC, 𝑃= 𝑣+𝑛𝑐 𝑛+1 .
So entry is always profitable.
As n → ∞, qi → 0 and P → c, that is we approach perfect competition.

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Free entry
This tells us that without entry barriers, oligopoly is only sustainable in the long run if firms face fixed costs.
With fixed costs, entry will continue until there are n firms in the market, such that firm n earns Π ≥ 0, but firm n + 1 would earn Π < 0 if it entered.

22. Is entry always good for welfare?
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Is entry always good for welfare?
Generally, we might think of more firms as reducing the welfare loss from monopoly.
In the absence of fixed costs, this is certainly true under Cournot oligopoly – each additional firm would increase the sum of profits and consumer surplus.
However, once we allow for fixed costs, this might no longer be true – it is possible that the fixed cost might make entry that is profitable to the firm inefficient in terms of social welfare.

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Areas a and b are the welfare gain from firm n + 1 entering the market in the absence of fixed costs. Pn
Pn+1 a b MC Qn
Qn+1 Q

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Areas a and d are the rise in consumer surplus, of which d is transferred from producers. Pn
Pn+1 d a b MC Qn
Qn+1 Q

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Areas b and c are the profits of the new entrant, of which c is ‘stolen’ from existing firms. Pn
Pn+1 d a c b MC
(Qn+1 – qn+1) Qn
Qn+1 Q

26. Now consider what happens with a positive fixed cost.
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Now consider what happens with a positive fixed cost.
The new firm will enter as long as b+c is bigger than its fixed cost. Pn
Pn+1 d a c b MC
(Qn+1 – qn+1) Qn
Qn+1 Q

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But if the fixed cost is bigger than a+b, but smaller than b+c, profitable entry will reduce total welfare. Pn
Pn+1 d a c b MC
(Qn+1 – qn+1) Qn
Qn+1 Q