1

EC 500

Chapter 9Basic Oligopoly Models

2

Headline

Crude Oil Prices Fall, but Consumers in Some Areas See No Relief at the Pump

• The price of crude oil declined during the summer of 2004, from about $43 to $35 per barrel. As a result of declining crude oil prices, consumers in most locations enjoyed lower gasoline prices.

• Not all consumers reaped the benefits of lower crude oil prices, however. In a few isolated areas, consumers cried foul because gasoline retailers did not pass on the price reductions to those who pay at the pump. Consumer groups argued that this corroborated their claim that gasoline retailers in these areas were colluding in order to earn monopoly profits. For obvious reason, the gasoline retailers involved denied the allegations.

• Based on the evidence, do you think that gasoline stations in these areas were colluding in order to earn monopoly profits? Explain.

3

OverviewI. Conditions for Oligopoly?

II. Role of Strategic Interdependence

III. Profit Maximization in Four Oligopoly Settings– Sweezy (Kinked-Demand) Model– Cournot Model– Stackelberg Model – Bertrand Model

IV. Contestable Markets

4

Oligopoly Environment

• Relatively few firms, usually less than 10.– Duopoly - two firms– Triopoly - three firms

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

5

Role of Strategic Interaction

• Your actions affect the profits of your rivals.

• Your rivals’ actions affect your profits.

6

An Example

• How does the quantity demanded for your product change when you change your price?

• There are two cases. – Rivals will not match price changes.– Rivals will match price changes.

• Point: If rivals MATCH price changes, demand becomes more INELASTIC! (STEEPER Demand curve; D2 in the next slide).

7

P

Q

D1 (Rival holds itsprice constant)

P0

PL

D2 (Rival matches your price change)

PH

Q0 QL2 QL1QH1 QH2

8

Why more Inelastic (D2) if matched?

• Suppose that you DECREASE price (P0 PL).– If rivals do not match price decrease, you will be able

to sell MORE (QL1) than they match price (QL2) .– If they will match price decrease, you will sell less

than they do not match price. (QL2 < QL1)• Suppose that you INCREASE price (P0 PH).

– If rivals do not match price increase, you will sell LESS (QH1) than they match price (QH2).

(QH1 < QH2)

• Thus, if rival match price changes, demand curve will be D2.

9

• Four Cases (see next slide)– Rivals match price decrease but not price

increase. (CAD) ; Sweezy model (kinked demand function!)

– Rivals match price increase but not price decrease. (EAF)

– Rivals match both price decrease and price increase. (EAD)

– Rivals match neither price decrease or price increase. (CAF)

10

P

Q

D1

P0

Q0

(Rival matches your price change)

(Rival holds itsprice constant)D

Demand if Rivals Match Price Reductions but not Price Increases (CAD)

A

CE

F

D2

11

Key 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!

12

Sweezy (Kinked-Demand) Model

• 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.

13

Sweezy Demand and Marginal Revenue

P

Q

P0

Q0

D1(Rival holds itsprice constant)

MR1

D2 (Rival matches your price change)

MR2

DS: Sweezy Demand

MRS: Sweezy MR

14

Sweezy Profit-Maximizing Decision

P

Q

P0

Q0

DS: Sweezy DemandMRS

MC1

MC2

MC3

D2 (Rival matches your price change)

D1 (Rival holds price constant)

15

Sweezy 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.

16

Cournot Model

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

• Firms set output, as opposed 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.

17

Inverse Demand in a Cournot Duopoly

• 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

18

Best-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.

19

Best-Response Function for a Cournot 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 2

1

2Q

b

caQrQ

12

122 2

1

2Q

b

caQrQ

20

Graph of Firm 1’s Best-Response Function

Q2

Q1

(Firm 1’s Reaction Function)

Q1M

Q2

Q1

r1

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

21

Cournot 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.

22

Graph of Cournot Equilibrium

Q2*

Q1*

Q2

Q1

Q1M

r1

r2

Q2M

Cournot Equilibrium

(a-c1)/b

(a-c2)/b

23

Summary of Cournot Equilibrium

• 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!

24

Numerical Example

P = 280 – 2(Q1 + Q2)

C1 = 3Q1, C2 = 2Q2

(a) Find MR1 and MR2.Rev1 = PQ1 = [280 – 2(Q1 + Q2)] Q1

MR1 = 280 - 2Q2 - 4Q1

Rev2 = PQ2 = [280 – 2(Q1 + Q2)] Q2

MR1 = 280 - 2Q1 - 4Q2

(b) Using MC1 and MC2, find the reaction functions for each firm. MR1 = MC1 implies 280 - 2Q2 - 4Q1 = 3

4Q1 = 280 - 2Q2 – 3 or Q1 = 69.25 - .5Q2

Similarly, from MR2 = MC2, we have Q2 = 69.5 - .5Q1

25

(c) How much output will each firm produce in equilibrium? Solve simultaneously,

Q1 = 69.25 - .5Q2

Q2 = 69.5 - .5Q1

Q1* = 46 Q2* = 46.5 P* = 280 – 2(Q1* + Q2*) = 95

(d) What are the equilibrium profits for each firm?Profit1 = PQ1 – 3Q1 = 95*46 – 3*46 = $4,232Profit2 = PQ2 – 2Q2 = 95*46.5 – 2*46.5 = $4,324.5.

• Exercise: Q. 2, p. 345P = 100 – 2(Q1 + Q2)C1 = 12Q1, C2 = 20Q2

26

Firm 1’s Isoprofit Curve

• The combinations of outputs of the two firms that yield firm 1 the same level of profit

• Profit will be maximized for firm 1 if it can produce Q1

M. Thus, moving downward increases its profit.

Q1Q1

M

r1

1 = $100

1 = $200

Increasing Profits for

Firm 1D

Q2

A

BC

27

Another Look at Cournot Decisions

Q2

Q1Q1

M

r1

Q2*

Q1*

Firm 1’s best response to Q2*

1 = $100

1 = $200

28

Another Look at Cournot Equilibrium

Q2

Q1Q1

M

r1

Q2*

Q1*

Firm 1’s Profits

Firm 2’s Profits

r2

Q2M Cournot Equilibrium

C

29

• Point: – Cournot equilibrium occurs at the

intersection of the two firms’ reaction functions (point C).

• We can do further analysis using this framework; next slides.

30

Impact of Rising Costs on the Cournot 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

31

• Points– r1 shifts in! Why?

– New output for firm 1 will be Q1**. Why?

– What happened to Q1 and Q2 in the end?

• Exercise• What if MC2 has declined?

• Draw a diagram and explain what will happen to Q1 and Q2 in the end.

32

33

Collusion Incentives in Cournot Oligopoly

Q2

Q1

r1

Q2M

Q1M

r2

Cournot2

Cournot1

34

Collusion Incentives in Cournot Oligopoly

Moving to Point D will make both firms better off! Why?

35

However, collusion is not easy. Why?

If Firm 1 cheats (point G), its profit increases! Why?

36

Stackelberg ModelIn the Sweezy and Cournot models,• We assume that firms are symmetric and they are equal.

But, there may be a leader, and others will follow the leader.

In the Stackelberg model,• 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.

• Firms produce differentiated or homogeneous products.• Barriers to entry.

37

• The leader can choose the point on the follower’s reaction curve that corresponds to the highest level of profits.

• Then, find the point tangent to the follower’s reaction function; Point S (next slide) and produce Q1

S.

• That is, firm 1 chooses its output level using the reaction function of firm 2.

• The profit for firm 1 (leader) is higher at Point S and Point C. Why?

38

Stackelberg Equilibrium

Q1 leaderQ1M

r1

Q2C

Q1C

r2

Q2

Followers

Q1S

Q2S

Follower’s Profits Decline

Stackelberg Equilibrium

πLS

π1C

πFS

π2C

S

C

39

Stackelberg 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.

40

Example of Stackelberg Model P = 50 – (Q1 + Q2)C1 = 2Q1, C2 = 2Q2

(a) Find MR1 and MR2.Rev1 = PQ1 = [50 – (Q1 + Q2)]Q1

MR1 = 50 - 2Q1 – Q2 Rev2 = PQ2 = [50 – (Q1 + Q2)]Q2

MR1 = 50 - Q1 - 2Q2

(b) Using MC1 and MC2, find the reaction functions for each firm. MR1 = MC1 implies 50 - 2Q1 – Q2 = 2

Q1 = 24 - .5Q2

Similarly, from MR2 = MC2, we have Q2 = 24 - .5Q1

Note: so far, the above analysis is the same as that of Cournot Models.

41

(c) Find Firm1’s output of Stackelberg model.

Rev1 = PQ1 – C1 = [50-(Q1+Q2)]Q1 – 2Q1

Point: Here, we replace Q2 with the reaction function of firm 2,

Q2 = 24 - .5Q1. Why?

Rev1 = [50-(Q1+ 24 - .5Q1)]Q1 – 2Q1

dRev1/dQ1 = 0 gives

dRev1/dQ1 = 50 – 2Q1 – 24 + Q1 – 2 = 0 Q1* = 24 Q2* = 24 - .5(24) = 12

P* = 50 – Q1* – Q2* = 50 – 24 – 12 = 14

42

(d) Find the profit of each firm.

Profit1 = PQ1 – 2Q1 = 14*24 – 2*24 = 288

Profit2 = PQ2 – 2Q2 = 14*12 – 2*12 = 144

Exercise: Re-do the above with the following.

P = 280 – 2(Q1 + Q2)

C1 = 3Q1, C2 = 2Q2

43

Bertrand Model• Question: Does oligopoly power always imply firms will

make positive profits? – Not necessarily if there is a price war. Firms compete for price and

react optimally to competitor’s prices. – Bertrand Model explains that P = MC is possible.

• Conditions for Bertrand Model– 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.– Barriers to entry.– Consumers enjoy Perfect information and Zero transaction costs.

44

Bertrand 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.

45

• The required key assumptions are– Identical MCs– Identical products– Price competition– Perfect information of consumers– But, Bertrand Model can be useful for on-line sales

• Example P = 50 – (Q1 + Q2)

C1 = 2Q1, C2 = 2Q2

Since P = MC, P = $2. Then, Q1 + Q2 = 48.

46

Contestable Markets

• Question: Should we have price wars to guarantee zero profits (P = MC)? – Not necessarily, even there is only one single firm,

if markets are contestable. There are potential entry

• Key Assumptions– Producers have access to same technology.– Consumers respond quickly to price changes.– Existing firms cannot respond quickly to entry by

lowering price.– Absence of sunk costs.

47

• 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).

48

Example again (summary)

Find the equilibrium outputs, and profits of each firm for each of

(i) Cournot Model (ii) Stackelberg Model (iii) Bertrand Model (iv) Collusion Model.

(Ex1) P = 50 – (Q1 + Q2)

C1 = 2Q1, C2 = 2Q2

(Ex2) P = 50 – (Q1 + Q2)

C1 = 2Q1, C2 = 2Q2

49

Answering the Headline

• Although the price of oil fell, in a few areas there were no declines in the price of gasoline. The headline asks whether this is evidence of collusion by gasoline stations in those areas. To answer this question, notice that oil is an input in producing gasoline. A reduction in the price of oil leads to a reduction in the marginal cost of producing gasoline—say, from MC0 to MC1.

• If gasoline stations were colluding, a reduction in marginal cost would lead the firms to lower the price of gasoline. Also, it could lead to a greater collusive output. Thus, collusion cannot explain why some gasoline firms failed to lower their prices. It is possible that these gasoline producers are Sweezy oligopolists; see the diagram.

50

51

52

Conclusion• 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.

53

Homework

• Q. 2 (Cournot)

• Q. 4 (Stackelberg)

• Q. 5 (Bertrand)

• Q. 7 (p. 346; all models)