1. 18. Oligopoly
Varian, Chapter 27

2. Two firms, two issues Concentrate on duopoly – easy notation
What are firms’ choices?
Choose a quantity/quality of output; or
Choose a price
What is the timing of firms’ actions?
Simultaneous decisions; or
Sequential decisions

3. Four interactions Timing Simultaneous Sequential Strategy Prices
Bertrand
Stackelberg-p
Quantities
Cournot
Stackelberg-q
We’ll do these two

4. Costs and profits Two firms, 1 and 2 Single good, outputs y1 and y2
Cost for firm i is
c(yi)
Inverse demand function is
p(y1+y2)
If outputs are y1 and y2, profits are
p1(y1,y2) = p(y1+y2) y1 - c(y1)
p2(y1,y2) = p(y1+y2) y2 - c(y2)
Market price depends
on total output, but not
on which firm makes it

5. Quantity leadership: Stackelberg
Firm 1 goes first; firm 2 follows
The follower’s problem: Given y1, choose y2 to
max p2(y1,y2) = [p(y1+y2) y2] - c(y2)
Firm 2’s output satisfies
p(y1+y2) + p’(y1+y2) y2 = c’(y2)
Revenue
Costs
Marginal revenue
Marginal cost

6. Firm 2’s reaction function
Firm 2’s profit-maximizing output depends on firm 1’s choice
That is,
y2 = f2(y1)
for some function f2(.)
f2(.) is called firm 2’s reaction function

7. Example: linear demand and zero costs
Suppose the inverse demand function is
p(y1+y2) = A – B(y1+y2)
Firm 2’s profit is
p2(y1,y2) = (A – B(y1+y2) ) y2
= (A - By1) y2 - B y22
Firm 2’s best choice of output is
y2 = (A – By1)/2B = f2(y1)

8. Graphical treatment of linear casey2
Iso-profit lines for firm 2
Profit
increasing
Firm 2’s reaction function
y2 = f2(y1) = (A – By1)/2B y1

9. The leader’s problem
Firm 1 anticipates firm 2’s reaction to its output choice
So it chooses y1 to
max p1(y1,y2) = [p(y1+y2) y1] - c(y1) or
max [p(y1+ f2(y1)) y1] - c(y1)

10. Linear demand, zero costs
We know
f2(y1) = (A – By1)/2B So
p1 = (A-B(y1+f2(y1)) y1
= {A-By1 – B [(A – By1)/2B ]} y1
= (A/2) y1 - (B/2) y12
Best choice of y1:
y1 = A/(2B)

11. Stackelberg equilibriumy2
Firm 2’s reaction function
y2 = f2(y1) = (A – By1)/2B
Stackelberg equilibrium
Profit
increasing
Iso-profit lines for firm 1 y1

12. y2 = f2(y1) = (A – By1)/(2B) = A/(4B)
Stackelberg outcome
Firm outputs
y1 = A/(2B)
y2 = f2(y1) = (A – By1)/(2B) = A/(4B)
Total industry output
YS = y1 + y2 = (3A)/(4B)
Pareto efficient output
YP = A/B
Why?

13. Pareto efficiency Is the Stackelberg equilibrium
Pareto efficient from the
perspective of the two firms?
Stackelberg equilibrium
2’s Profit
increasing
Room for a
Pareto improvement y1
1’s Profit
increasing

14. Cournot competition Now both firms choose output simultaneously
We assume their choices constitute a Nash equilibrium
Whatever 1’s output, y1 , firm 2 must do the best it can:
y2 = f2(y1)
Whatever 2’s output, y2 , firm 1 must do the best it can:
y1 = f1(y2)
Firm 2’s reaction
function
Firm 1’s reaction
function

15. Cournot equilibrium y2 y1 = f1(y2) y2 = f2(y1) y1 Cournot equilibrium
2’s Profit
increasing
y2 = f2(y1) y1
1’s Profit
increasing

16. Linear demand, zero costs
2’ reaction function is
y2 = f2(y1) = (A – By1)/2B
1’ reaction function is
y1 = f1(y2) = (A – By2)/2B
Solve these two equations for y1 and y2 :
y1 = y2 = A/3B
Industry output
YC = y1+y2 = (2A)/(3B)

17. Pareto efficiency Is the Cournot equilibrium Pareto efficient from the
perspective of the two firms?
y1 = f1(y2)
Still room for a
Pareto improvement
Cournot equilibrium
y2 = f2(y1) y1

18. Maximizing joint profits
Suppose the firms cooperatively choose outputs, y1 and y2
When costs are zero, they choose aggregate output Y = y1 + y2 like a single monopolist:
YM = A/(2B)
Note that
YM < YC < YS < YP
A/(2B)
A/B
(2A)/(3B)
(3A)/(4B)

19. Comparing output levelsy2
y1 = f1(y2)
45o YP YS
Pareto efficient from
firms’ and consumers’
perspective YC YM
y2 = f2(y1)
Pareto efficient from
firms’ perspective y1

20. Externalities in competition
Firms produce too much when they compete
Where does the inefficiency come from?
Each firm ignores the effect on the other’s profit when it expands output
i.e., there is a negative externality
Compared to monopoly, oligopoly pushes result closer to perfectly competitive outcome

21. Sustaining a cartel Beat-any-price clauses
It sounds very competitive
….but maybe each firm is using consumers to check that other firms are not “cheating”
VERs – voluntary export restraints in Japan
US negotiated with Japan for Japanese firms to reduce sales in US
Benefited US car makers
…..but not US car consumers