Oligopoly

Oligopoly

Two firms, two issues Concentrate on duopoly – easy notation Two issues: 1. What are firms’ strategies? –

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 on which firm makes it

• Inverse market demand function is p(y1+y2) • If outputs are y1 and y2, profits are 1(y1,y2)

= p(y1+y2) y1 - c(y1)

2(y1,y2)

= p(y1+y2) y2 - c(y2)

π π

Quantity leadership: Stackelberg • Firm 1 goes first; firm 2 follows • The follower’s problem: Given y1 = y*1 , choose y2 to max π2(y*1,y2) = [p(y*1+y2) y2] - c(y2) Revenue

Costs

• Firm 2’s output satisfies p(y*1+y2) + p’(y*1+y2) y2 = c’(y2) Marginalrevenue

Marginalcost

Firm 2’s reaction function • Thus 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 π2

(y1,y2) = (A – B(y1+y2)2 ) y2 = (A - By1) y2 - B y2

• Firm 2’s best choice of output given y1 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

The leader’s problem • Firm 1 anticipates firm 2’s reaction to its output choice • Firm 1 wants to choose y1 to max π1(y1,y2) = [p(y1+y2) y1] - c(y1) but it knows that y2 = f2(y1) so the problem becomes max [p(y1+ f2(y1)) y1] - c(y1)

Again Linear demand, zero costs π

1

= [A-B(y1+f2(y1)] y1

Firm 1 knows that f2(y1) = (A – By1)/2B So π = [A- By – B {(A – By )/2B }]y 1 1 1 1 = (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

y1

1’s Profit increasing

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’s reaction function is y2 = f2(y1) = (A – By1)/2B • 1’s 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 Pareto efficient from the perspective of the two firms? Still room for a Pareto improvement Cournot equilibrium

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)

Externalities in competition • Firms produce too much when they compete from the perspective of maximizing their joint profits

• 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

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