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 Bertrand Stackelberg-p

Quantities 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)

• 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 dependson total output, but noton which firm makes it

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

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

y1

y2

Iso-profit lines for firm 2

Firm 2’s reaction functiony2 = f2(y1) = (A – By1)/2B

Profitincreasing

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)

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

y1

y2

Iso-profit lines for firm 1

Firm 2’s reaction functiony2 = f2(y1) = (A – By1)/2B

Profitincreasing

Stackelberg equilibrium

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 outputYP = A/B

Why?

Pareto efficiency

y1

y2

Stackelberg equilibrium

2’s Profitincreasing

1’s Profitincreasing

Room for aPareto improvement

Is the Stackelberg equilibriumPareto efficient from theperspective of the two firms?

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 reactionfunction

Firm 1’s reactionfunction

Cournot equilibrium

y1

y2

Cournot equilibrium

2’s Profitincreasing

1’s Profitincreasing

y2 = f2(y1)

y1 = f1(y2)

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

y1

y2

Cournot equilibrium

y2 = f2(y1)

y1 = f1(y2)

Still room for aPareto improvement

Is the Cournot equilibriumPareto efficient from theperspective of the two firms?

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)

(2A)/(3B) (3A)/(4B)A/B

Comparing output levels

y1

y2

45o

y2 = f2(y1)

y1 = f1(y2)

YP

YM

YC

YS

Pareto efficient fromfirms’ perspective

Pareto efficient fromfirms’ and consumers’perspective

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