### 18. Oligopoly

```18. Oligopoly
Varian, Chapter 27
Two firms, two issues
• Concentrate on duopoly – easy notation
• Two issues:
1. What are firms’ choices?
– Choose a quantity/quality of output; or
– Choose a price
2. What is the timing of firms’ actions?
– Simultaneous decisions; or
– Sequential decisions
Four interactions
Timing
Simultaneous Sequential
Strategy Prices
Quantities
Bertrand
Stackelberg-p
Cournot
Stackelberg-q
We’ll do these two
Costs and profits
• Two firms, 1 and 2
• Single good, outputs y1 and y2
• Cost for firm i is
c(yi)
Market price depends
on total output, but not
• Inverse demand function is
on which firm makes it
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)
• 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)
Revenue
Costs
• Firm 2’s output satisfies
p(y1+y2) + p’(y1+y2) y2 = c’(y2)
Marginal revenue
Marginal cost
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
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)
Graphical treatment of linear case
y2
Iso-profit lines for firm 2
Profit
increasing
Firm 2’s reaction function
y2 = f2(y1) = (A – By1)/2B
y1
• 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)
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)
Stackelberg equilibrium
y2
Firm 2’s reaction function
y2 = f2(y1) = (A – By1)/2B
Stackelberg equilibrium
Profit
increasing
Iso-profit lines for firm 1
y1
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?
Pareto efficiency
y2
Is the Stackelberg equilibrium
Pareto efficient from the
perspective of the two firms?
Stackelberg equilibrium
2’s Profit
increasing
Room for a
Pareto improvement
1’s Profit
increasing
y1
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
Firm 2’s reaction
the best it can:
function
y2 = f2(y1)
• Whatever 2’s output, y2 , firm 1 must do
the best it can:
Firm 1’s reaction
y1 = f1(y2)
function
Cournot equilibrium
y2
y1 = f1(y2)
Cournot equilibrium
2’s Profit
increasing
y2 = f2(y1)
1’s Profit
increasing
y1
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)
Pareto efficiency
y2
Is the Cournot equilibrium
y1 = f1(y2) Pareto efficient from the
perspective of the two firms?
Still room for a
Pareto improvement
Cournot equilibrium
y2 = f2(y1)
y1
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)
Comparing output levels
y2
y1 = f1(y2)
YP
YS
YC
45o
Pareto efficient from
firms’ and consumers’
perspective
YM
y2 = f2(y1)
Pareto efficient from
firms’ perspective
y1
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
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
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