1. Basic Oligopoly Models
Lecture 9
Basic Oligopoly Models
9/18/2018
Agyapomaa Gyeke-Dako(PhD) .

2. Agyapomaa Gyeke-Dako(PhD)
Lecture Outline
Conditions for Oligopoly
Role of beliefs and strategic interaction
Profit maximization in four oligopoly settings
Sweezy oligopoly
Cournot oligopoly
Stackelberg oligopoly
Bertrand oligopoly
Comparing oligopoly models
Contestable markets
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

3. Agyapomaa Gyeke-Dako(PhD)
Introduction
Lecture 7 examined profit-maximizing behavior in perfectly competitive, monopoly, and monopolistically competitive markets. One distinguishing feature is the absence of strategic interaction among the firms (in monopoly markets, strategic interaction is irrelevant since only one firms exists).
This lecture focuses on how managers select the optimal price and quantity in the following oligopoly market environments:
Sweezy
Cournot
Stackelberg
Bertrand
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

4. Agyapomaa Gyeke-Dako(PhD)
Key Conditions
Oligopoly market structures are characterized by only a few firms, each of which is large relative to the total industry.
Typical number of firms is between 2 and 10 controlling about 75% of the total industrial output.
Products can be identical or differentiated.
An oligopoly market composed of two firms is called a duopoly.
Oligopoly settings tend to be the most difficult to manage since managers must consider the likely impact of his or her decisions on the decisions of other firms in the market.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

5. Conditions for Sweezy Oligopoly
There are few firms in the market serving many consumers.
The firms produce differentiated products.
Each firm believes its rivals will cut their prices in response to a price reduction but will not raise their prices in response to a price increase. This leads to price rigidities in oligopolistic markets
Barriers to entry exist.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

6. Conditions for Sweezy Oligopoly
Unlike a monopoly or a monopolistically competitive firm that has a degree of market power along the length of a single demand curve, oligopolist faces a demand curve which is kinked.
Demand curve for the product comprises two demand curves, D1 and D2.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

7. Agyapomaa Gyeke-Dako(PhD)
Sweezy Oligopoly
Price
Sweezy Demand
MC0 A B
𝑃 0
MC1 C
Demand1
(rival holds price constant) E
MR1
Demand2
(rival matches price change) MR F
𝑄 0
Output
MR2
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

8. Agyapomaa Gyeke-Dako(PhD)
Sweezy Oligopoly
Illustrates why prices in oligopolistic industries do not change more infrequently
Because of the discontinuity of the marginal revenue curve, marginal cost can fluctuate between a certain range without a change in output or price
Implication
Adoption of more efficient production technologies which results in lower cost may not lead to a reduction in the price of a product
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

9. Conditions for Cournot Oligopoly
First Formulated by Augustine Cournot (1897)
There are few firms in the market serving many consumers.
The firms produce either differentiated or homogeneous products.
Each firm believes rivals will hold their output constant if it changes its output.
Barriers to entry exist.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

10. Cournot Oligopoly: Reaction Functions
Consider a Cournot duopoly. Each firm makes an output decision under the belief that is rival will hold its output constant when the other changes its output level.
Implication: Each firm’s marginal revenue is impacted by the other firms output decision.
The relationship between each firm’s profit-maximizing output level is called a best-response or reaction function.
Assume that firm A is the first to enter the industry
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

11. Firm and Industry Output
Iteration QA QB
QA + QB 1
1/2Q 2
1/4Q
3/4Q 3
3/8Q
5/8Q 4
11/16Q 5 .. i
1/3Q
2/3Q
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

12. Cournot Oligopoly: Reaction Functions Formula
Given a linear (inverse) demand function
𝑃=𝑎−𝑏 𝑄 1 + 𝑄 2
and cost functions, 𝐶 1 𝑄 1 = 𝑐 1 𝑄 1 and 𝐶 2 𝑄 2 = 𝑐 2 𝑄 2 the reactions functions are:
𝑄 1 = 𝑟 1 𝑄 2 = 𝑎−𝑐 1 2𝑏 − 1 2 𝑄 2
𝑄 2 = 𝑟 2 𝑄 1 = 𝑎−𝑐 2 2𝑏 − 1 2 𝑄 1
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

13. Cournot Reaction Functions
Quantity2
Firm 1’s Reaction Function
𝑄 1 = 𝑟 1 𝑄 2
𝑄 2 𝑀𝑜𝑛𝑜𝑝𝑜𝑙𝑦
Cournot equilibrium
𝑄 2 𝐶𝑜𝑢𝑟𝑛𝑜𝑡 C
Firm 2’s Reaction Function
𝑄 2 = 𝑟 2 𝑄 1 D A B
𝑄 1 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝑄 1 𝑀𝑜𝑛𝑜𝑝𝑜𝑙𝑦
Quantity1
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

14. Cournot Oligopoly: Equilibrium
A situation in which neither firm has an incentive to change its output given the other firm’s output.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

15. Cournot Oligopoly In Action: Problem
Suppose the inverse demand function in a Cournot duopoly is given by
𝑃=10− 𝑄 1 + 𝑄 2
and their costs are zero.
What are the reaction functions for the two firms?
What are the Cournot equilibrium outputs?
What is the equilibrium price?
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

16. Cournot Oligopoly In Action: Answer
The reaction functions are: 𝑄 1 = 𝑟 1 𝑄 2 = 10 2 − 1 2 𝑄 2 and 𝑄 2 = 𝑟 2 𝑄 1 = 10 2 − 1 2 𝑄 1 .
Equilibrium output is found as:
𝑄 1 =5− − 1 2 𝑄 1 ⟹ 𝑄 1 = Since the firms are symmetric, 𝑄 2 =
Total industry output is 𝑄= 𝑄 1 + 𝑄 2 = = So, the equilibrium price is: 𝑃=10− 20 3 =
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

17. Cournot Oligopoly In Action
𝑃=200−2 𝑄 1 + 𝑄 2
TC1 = 4Q1
TC2 = 4Q1
Calculate each firm’s reaction function
Calculate the equilibrium price, profit maximising output and profits for each firm.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

18. Firm 1’s Best Response to Firm 2’s Output
Quantity2
𝑟 1 (Firm 1’s reaction function) A B C
𝑄 2 ∗
𝜋 1 𝐴
𝜋 1 𝐵
𝜋 1 𝐶
Firm 1’s profit increases as isoprofit
curves move toward 𝑄 1 𝑀
𝑄 1 𝐴
𝑄 1 𝐵
𝑄 1 𝐶
𝑄 1 𝑀
Quantity1
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

19. Firm 2’s Reaction Function and Isoprofit Curves
Quantity2
Monopoly point for firm 2
𝑄 2 𝑀
Firm 2’s profit increases as isoprofit
curves move toward 𝑄 2 𝑀
𝜋 3 𝐶
𝑟 2 (Firm 2’s reaction function)
𝜋 2 𝐵
𝜋 1 𝐴
Quantity1
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

20. Agyapomaa Gyeke-Dako(PhD)
Cournot Equilibrium
Quantity2
𝜋 2 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝑄 2 𝑀
Cournot Equilibrium
𝑄 2 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝜋 1 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝑄 1 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝑄 1 𝑀
Quantity1
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

21. Cournot Oligopoly: Collusion
Markets with only a few dominant firms can coordinate to restrict output to their benefit at the expense of consumers.
Restricted output leads to higher market prices.
Such acts by firms is known as collusion.
Collusion, however, is prone to cheating behavior.
Since both parties are aware of these incentives, reaching collusive agreements is often very difficult.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

22. Conditions for Stackelberg Oligopoly
There are few firms serving many consumers.
Firms produce either differentiated or homogeneous products.
A single firm (the leader) chooses an output before all other firms choose their outputs.
Leader believes that follower will take its output as given. Leader will then incorporate known reaction of follower into his production decision.
All other firms (the followers) take as given the output of the leader and choose outputs that maximize profits given the leader’s output.
Barriers to entry exist.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

23. Stackelberg Equilibrium
Quantity Follower
𝜋 2 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝑟 𝐿𝑒𝑎𝑑𝑒𝑟
𝜋 2 𝑆𝑡𝑎𝑐𝑘𝑒𝑙𝑏𝑒𝑟𝑔 𝐹𝑜𝑙𝑙𝑜𝑤𝑒𝑟
𝑟 𝐹𝑜𝑙𝑙𝑜𝑤𝑒𝑟
𝑄 2 𝑀
𝑄 2 𝑆𝑡𝑎𝑐𝑘𝑒𝑙𝑏𝑒𝑟𝑔 𝐹𝑜𝑙𝑙𝑜𝑤𝑒𝑟
𝜋 1 𝐶𝑜𝑢𝑟𝑛𝑜𝑡
𝜋 1 𝑆𝑡𝑎𝑐𝑘𝑒𝑙𝑏𝑒𝑟𝑔 𝐿𝑒𝑎𝑑𝑒𝑟
𝑄 1 𝑀
𝑄 1 𝑆𝑡𝑎𝑐𝑘𝑒𝑙𝑏𝑒𝑟𝑔 𝐿𝑒𝑎𝑑𝑒𝑟
Quantity Leader
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

24. Stackelberg Oligopoly: Equilibrium Output Formulae
Given a linear (inverse) demand function
𝑃=𝑎−𝑏 𝑄 1 + 𝑄 2
and cost functions 𝐶 1 𝑄 1 = 𝑐 1 𝑄 1 and 𝐶 2 𝑄 2 = 𝑐 2 𝑄 2 .
The follower sets output according to the reaction function
𝑄 2 = 𝑟 2 𝑄 1 = 𝑎− 𝑐 2 2𝑏 − 1 2 𝑄 1
The leader’s output is
𝑄 1 = 𝑎+ 𝑐 2 −2 𝑐 1 2𝑏
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

25. Stackelberg Oligopoly In Action: Problem
Suppose the inverse demand function for two firms in a homogeneous-product, Stackelberg oligopoly is given by
𝑃=50− 𝑄 1 + 𝑄 2
and their costs are zero. Firm 1 is the leader, and firm 2 is the follower.
What is firm 2’s reaction function?
What is firm 1’s output?
What is firm 2’s output?
What is the market price?
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

26. Stackelberg Oligopoly In Action: Answer
Find the follower’s reaction function and incorporate that into profit decision of leader.26
The follower’s reaction function is: 𝑄 2 = 𝑟 2 𝑄 1 =24− 1 2 𝑄 1 .
The leader’s output is: 𝑄 1 = 50+2−4 2 =24.
The follower’s output is: 𝑄 2 =24− =12.
The market price is: 𝑃=50− =$14.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

27. Conditions for Bertrand Oligopoly
There are few firms in the market serving many consumers.
Firms produce identical products at a constant marginal cost.
Firms engage in price competition and react optimally to prices charged by competitors.
Consumers have perfect information and there are no transaction costs.
Barriers to entry exist.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

28. Bertrand Oligopoly: Equilibrium
The conditions for a Bertrand oligopoly imply that firms in this market will undercut one another to capture the entire market leaving the rivals with no profit. All consumers will purchase at the low-price firm.
This “price war” would come to an end when the price each firm charged equaled marginal cost.
In equilibrium, 𝑃 1 = 𝑃 2 =𝑀𝐶.
Socially efficient level of output.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

29. Comparing Oligopoly Outcomes
Consider the following inverse market demand function:
𝑃=1,000− 𝑄 1 + 𝑄 2
and the cost function for each firm in this market is identical, and given by
𝐶 𝑖 𝑄 𝑖 =4 𝑄 𝑖
Under these condition, the different oligopoly outputs, prices and profits are examined.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

30. Comparing Oligopoly: Cournot Outcome
The Cournot oligopoly reaction functions are
𝑄 1 =498− 1 2
𝑄 2 =498− 1 2
These reaction functions can be solved for the equilibrium output. These quantities can be used to compute price and profit.
𝑄 1 = 𝑄 2 =…….
𝑃=$336
𝜋 1 = 𝜋 2 =$110,224
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

31. Comparing Oligopoly: Stackelberg Outcome
Comparing Oligopoly Models
Comparing Oligopoly: Stackelberg Outcome
The Stackelberg leader’s output is
𝑄 𝑙𝑒𝑎𝑑𝑒𝑟 = 1,000+4−2×4 2×1 =498
𝑄 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑟 =498− 1 2 ×498=249
The market price is: 𝑃=1,000−498−249=$253
𝜋 𝑙𝑒𝑎𝑑𝑒𝑟 =$124,002
𝜋 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑟 =$62,001
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

32. Comparing Oligopoly: Bertrand Outcome
Since 𝑃=𝑀𝐶, 𝑃=$4.
Total output is found by: $4=1,000−𝑄
Solving yields: 𝑄=996
Given symmetric firms, each firm gets half the market, or 498 units.
𝜋 1 = 𝜋 2 =$0
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

33. Comparing Oligopoly: Collusion Outcome
Since the output associated with collusion is the same as monopoly output, the inverse market demand function implies that monopoly marginal revenue function is:
𝑀𝑅=1,000−2𝑄
Setting marginal revenue equal to marginal cost yields:
1,000−2𝑄=4
Solving this: 𝑄=498 units. Each firm will produce half of these units.
Price is: 𝑃=1,000−498=$502
Each firm earns profits of $124,002.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

34. Agyapomaa Gyeke-Dako(PhD)
Key Conditions
Contestable markets involve strategic interaction among existing firms and potential entrants into a market.
A market is contestable if:
All producers have access to the same technology.
Consumers respond quickly to price changes.
Existing firms cannot respond quickly to entry by lowering price.
There are no sunk costs.
If these conditions hold, incumbent firms have no market power over consumers.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)

35. Agyapomaa Gyeke-Dako(PhD)
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.
9/18/2018
Agyapomaa Gyeke-Dako(PhD)