BUS 525: Managerial Economics

Lecture 9

Basic Oligopoly Models

OverviewOverviewI. Conditions for Oligopoly?II. Role of Strategic InterdependenceIII. Profit Maximization in Four Oligopoly Settings

– Sweezy (Kinked-Demand) Model– Cournot Model– Stackelberg Model – Bertrand Model

IV. Contestable Markets

9-2

Oligopoly EnvironmentOligopoly EnvironmentA market structure there are only a fewFirms, each of which is large relative the totalindustry• Relatively few firms, usually less than 10.

– Duopoly - two firms– Triopoly - three firms

• The products firms offer can be either differentiated or homogeneous.

• Firms’ decisions impact one another.• Many different strategic variables are modeled:

– No single oligopoly model.

9-3

Role of Strategic InteractionRole of Strategic Interaction• Your actions affect the profits of

your rivals.• Your rivals’ actions affect your

profits.• How will rivals respond to your

actions?

9-4

An ExampleAn Example• You and another firm sell

differentiated products.• How does the quantity demanded for

your product change when you change your price?

9-5

P

Q

D2 (Rival holds itsprice constant)

P0

PL

D1 (Rival matches your price change)

PH

Q0 QL2 QL1QH1 QH2

9-6

B

P

Q

D1

P0

Q0

D2 (Rival matches your price change)

(Rival holds itsprice constant)

D2

Demand if Rivals Match Price Reductions but not Price Increases

9-7

Note that demand is more inelastic when rivals match a price change than when they do notReason: For a given price reduction, a firm will sell more if rivals do not cut their prices D2 than it will if they lower their prices D1

Key InsightKey Insight• The effect of a price reduction on the

quantity demanded of your product depends upon whether your rivals respond by cutting their prices too!

• The effect of a price increase on the quantity demanded of your product depends upon whether your rivals respond by raising their prices too!

• Strategic interdependence: You aren’t in complete control of your own destiny!

9-8

Sweezy (Kinked-Demand) Sweezy (Kinked-Demand) Model EnvironmentModel Environment

• Few firms in the market serving many consumers.

• Firms produce differentiated products.• Barriers to entry.• Each firm believes rivals will match (or

follow) price reductions, but won’t match (or follow) price increases.

• Key feature of Sweezy Model– Price-Rigidity.

9-9

Sweezy Demand and Sweezy Demand and Marginal RevenueMarginal Revenue

P

Q

P0

Q0

D1(Rival holds itsprice constant)

MR1

D2 (Rival matches your price change)

MR2

DS: Sweezy Demand

MRS: Sweezy MR

9-10

Sweezy Profit-Maximizing Sweezy Profit-Maximizing DecisionDecision

P

Q

P0

Q0

DS: Sweezy DemandMRS

MC1MC2

MC3

D2 (Rival matches your price change)

D1 (Rival holds price constant)

9-11

A

C

E

Sweezy Oligopoly SummarySweezy Oligopoly Summary

• Firms believe rivals match price cuts, but not price increases.

• Firms operating in a Sweezy oligopoly maximize profit by producing where

MRS = MC.– The kinked-shaped marginal revenue curve

implies that there exists a range over which changes in MC will not impact the profit-maximizing level of output.

– Therefore, the firm may have no incentive to change price provided that marginal cost remains in a given range.

9-12

Cournot Model Cournot Model EnvironmentEnvironment

• A few firms produce goods that are either perfect substitutes (homogeneous) or imperfect substitutes (differentiated).

• Firms’ control variable is output in contrast to price.

• Each firm believes their rivals will hold output constant if it changes its own output (The output of rivals is viewed as given or “fixed”).

• Barriers to entry exist.

9-13

Inverse Demand in a Cournot Inverse Demand in a Cournot DuopolyDuopoly

• Market demand in a homogeneous-product Cournot duopoly is

• Thus, each firm’s marginal revenue depends on the output produced by the other firm. More formally,

212 2bQbQaMR

121 2bQbQaMR

21 QQbaP

9-14

Best-Response FunctionBest-Response Function• Since a firm’s marginal revenue in a

homogeneous Cournot oligopoly depends on both its output and its rivals, each firm needs a way to “respond” to rival’s output decisions.

• Firm 1’s best-response (or reaction) function is a schedule summarizing the amount of Q1 firm 1 should produce in order to maximize its profits for each quantity of Q2 produced by firm 2.

• Since the products are substitutes, an increase in firm 2’s output leads to a decrease in the profit-maximizing amount of firm 1’s product.

9-15

Best-Response Function for a Best-Response Function for a Cournot DuopolyCournot Duopoly

• To find a firm’s best-response function, equate its marginal revenue to marginal cost and solve for its output as a function of its rival’s output.

• Firm 1’s best-response function is (c1 is firm 1’s MC)

• Firm 2’s best-response function is (c2 is firm 2’s MC)

21

211 21

2Q

bcaQrQ

12

122 21

2Q

bcaQrQ

9-16

Graph of Firm 1’s Best-Graph of Firm 1’s Best-Response FunctionResponse Function

Q2

Q1

(Firm 1’s Reaction Function)

Q1M

Q2

Q1

r1

(a-c1)/b Q1 = r1(Q2) = (a-c1)/2b - 0.5Q2

9-17

Cournot EquilibriumCournot Equilibrium• Situation where each firm produces

the output that maximizes its profits, given the the output of rival firms.

• No firm can gain by unilaterally changing its own output to improve its profit.– A point where the two firm’s best-

response functions intersect.

9-18

Graph of Cournot EquilibriumGraph of Cournot Equilibrium

Q2*

Q1*

Q2

Q1

Q1M

r1

r2

Q2M

Cournot Equilibrium

(a-c1)/b

(a-c2)/b

9-19

AB

CE

Summary of Cournot Summary of Cournot EquilibriumEquilibrium

• The output Q1* maximizes firm 1’s

profits, given that firm 2 produces Q2*.

• The output Q2* maximizes firm 2’s

profits, given that firm 1 produces Q1*.

• Neither firm has an incentive to change its output, given the output of the rival.

• Beliefs are consistent: – In equilibrium, each firm “thinks” rivals will

stick to their current output – and they do!

9-20

The Isoprofit CurveThe Isoprofit Curve

Firm 1’s Isoprofit CurveFirm 1’s Isoprofit Curve• The combinations of outputs of the two firms

that yield firm 1 the same level of profit

Q1Q1M

r1

0 = $100

1 = $200

Increasing Profits for

Firm 1D

Q2

A

B C

9-22

2 = $300E

Another Look at Cournot Another Look at Cournot DecisionsDecisions

Q2

Q1Q1M

r1

Q2*

Q1*

Firm 1’s best response to Q2*

1 = $200 2 = $300

9-23

0 = $100A B CD

QA QB QD

Another Look at Cournot Another Look at Cournot EquilibriumEquilibrium

Q2

Q1Q1M

r1

Q2*

Q1*

Firm 1’s Profits

Firm 2’s Profits

r2

Q2M Cournot Equilibrium

9-24

Collusion Incentives in Collusion Incentives in Cournot OligopolyCournot Oligopoly

Q2

Q1

r1

Q2M

Q1M

r2

Cournot2

Cournot1

9-25

Impact of Rising Costs on the Impact of Rising Costs on the Cournot EquilibriumCournot Equilibrium

Q2

Q1

r1**

r2

r1*

Q1*

Q2*

Q2**

Q1**

Cournot equilibrium prior to firm 1’s marginal cost increase

Cournot equilibrium after firm 1’s marginal cost increase

9-26

Price Leadership ModelPrice Leadership Model• In a price leadership model, one dominant firm takes

reactions of all other firms into account in its output and pricing decisions

• Competitive fringe: A group of firm that act as a price taker in a market dominated by a price leader

• A dominant firms demand curve is the residual demand curve that shows what it can sell after accounting for sales by other firms

• Other firms accept whatever price is set by the dominant firm and produce an output where P=MC

• Note that P>MC for dominant firm, total industry output is less than competitive output

Dominant Firm Dominant Firm ModelModel

Fig : Equilibrium in the Dominant Firm Model

Stackelberg Model Stackelberg Model EnvironmentEnvironment

• Few firms serving many consumers.• Firms produce differentiated or

homogeneous products.• Barriers to entry.• Firm one is the leader.

– The leader commits to an output before all other firms.

• Remaining firms are followers.– They choose their outputs so as to

maximize profits, given the leader’s output.

9-29

The Algebra of the Stackelberg The Algebra of the Stackelberg ModelModel

• Since the follower reacts to the leader’s output, the follower’s output is determined by its reaction function

• The Stackelberg leader uses this reaction function to determine its profit maximizing output level, which simplifies to

12

122 5.02

QbcaQrQ

bccaQ

22 12

1

9-30

Stackelberg SummaryStackelberg Summary• Stackelberg model illustrates how

commitment can enhance profits in strategic environments.

• Leader produces more than the Cournot equilibrium output.– Larger market share, higher profits.– First-mover advantage.

• Follower produces less than the Cournot equilibrium output.– Smaller market share, lower profits.

9-31

Bertrand Model Bertrand Model EnvironmentEnvironment

• Few firms that sell to many consumers.• Firms produce identical products at

constant marginal cost.• Each firm independently sets its price in

order to maximize profits (price is each firms’ control variable).

• Barriers to entry exist.• Consumers enjoy

– Perfect information. – Zero transaction costs.

9-32

Bertrand EquilibriumBertrand Equilibrium• Firms set P1 = P2 = MC! Why?• Suppose MC < P1 < P2.• Firm 1 earns (P1 - MC) on each unit sold,

while firm 2 earns nothing.• Firm 2 has an incentive to slightly undercut

firm 1’s price to capture the entire market.• Firm 1 then has an incentive to undercut firm

2’s price. This undercutting continues...• Equilibrium: Each firm charges P1 = P2 = MC.

9-33

Contestable MarketsContestable Markets• Key Assumptions

– Producers have access to same technology.– Consumers respond quickly to price changes.– Existing firms cannot respond quickly to deter entry

by lowering price.– Absence of sunk costs.

• Key Implications– Threat of entry disciplines firms already in the

market.– Incumbents have no market power, even if there is

only a single incumbent (a monopolist).

9-34

ConclusionConclusion• Different oligopoly scenarios give rise to

different optimal strategies and different outcomes.

• Your optimal price and output depends on …– Beliefs about the reactions of rivals.– Your choice variable (P or Q) and the nature

of the product market (differentiated or homogeneous products).

– Your ability to credibly commit prior to your rivals.

9-35

..Basic modelBasic model

Q2 - 120Q P.Q have we

120

120

costs, zero Assuming

caseMonopoly

QP

PQ

3600 PQ

60 P

thereforeand,

60 Qor

0 2Q - 120 dQd

0 /dQdπ settingby found bemay profits Maximum

Duopoly case

0 2Q - Q - dQdQQ - 120

dQdπ

0 Q - dQdQQ - 2Q - 120

dQdπ

2Q - QQ - 120Q P.Q π

QQ2 Q 120Q P.Q π

Q (Q - 120 P

212

11

2

2

21

211

1

1

221222

211111

2121 Q Q Q , )

2Q - 120 Q

2Q - 120 Q

0 2Q - Q - 120 dQd

0 Q - 2Q - 120 dQdπ

0 dQdQ

dQdQ

SolutionCournot

12

21

21

2

2

211

1

2

1

1

2

]profit)oly 3600(monop3200profit [Cournot 1600 PQ 1600 PQ

40 80 - 120 )Q (Q - 120 P

402Q - 120 Q

40 Q

Q 120 Q4

]Q of valuengsubstituti[by 2)/2Q - (120- 120

2Q- 120 Q

22

11

21

12

1

11

212

1

process in the erodedseriously been hasprofit s2' Firm reaction. s2' firm of knowledge its usingby profit its increase toablebeen has 1 Firm 900 PQ

1800 PQ 30 90 - 120 )Q (Q - 120 P

30 Q60 Q

32 - 80 Q

0 Q Q21 2Q - 120

dd

21 -

dQdQ

suppose function,reaction rivals its discovered has firm Onesolution gStackelber

22

11

21

2

1

21

2111

1

1

2

Q

Q

...

2Q - 120 Q: R 2

11

120

60

40

060 120

40

Equilibrium

Q = 120 - P

120

60

40

60 80 120

Price

Q per period

0

60

60Q

0 120

120

P

P

Q

Zero cost monopoly

Cournot solution Stackelberg solutionMR

Q = 120 - P

2Q - 120 QR 1

2 2 :

Monopoly

Cournot

Stackelberg