ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
Chapter 9 Basic Oligopoly Models
Cournot Model
A.
B.
C.

Given a linear (inverse) demand function: 𝑃 = 𝑎 − 𝑏(𝑄1 + 𝑄2 )
Cost functions: 𝐶1 (𝑄1 ) = 𝑐1 𝑄1 and 𝐶2 (𝑄2 ) = 𝑐2 𝑄2 ,
So Marginal cost functions: 𝑀𝐶1 (𝑄1 ) = 𝑐1 and 𝑀𝐶2 (𝑄2 ) = 𝑐2
Derive their reaction functions:
 Derive Cournot Equilibrium: a situation in which neither firm has an incentive to change its
output given the other firm’s output
EX1: Suppose the inverse demand function in a Cournot duopoly is given by
𝑃 = 10 − (𝑄1 + 𝑄2 ) and their marginal costs are 2.
a. What are the reaction functions for the two firms?
b. What are the Cournot equilibrium outputs?
c. What is the equilibrium price?
d. What is the equilibrium profit for each firm?
Cournot Oligopoly: Collusion
EX2: Suppose the inverse demand function in a Cournot duopoly is given by
𝑃 = 10 − (𝑄1 + 𝑄2 ) and their marginal costs are 2. Suppose they decide to collude, then
a. What are the equilibrium outputs of each firm?
b. What is the equilibrium price?
c. What is the equilibrium profits for each firm?
d. Compared to the profit in example1, which profit is higher?
ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
Stackelberg Oligopoly
Given a linear (inverse) demand function 𝑃 = 𝑎 − 𝑏(𝑄1 + 𝑄2 ) and marginal cost functions are
𝑀𝐶1 (𝑄1 ) = 𝑐1 and 𝑀𝐶2 (𝑄2 ) = 𝑐2 . Then, The follower sets output according to the reaction
function 𝑄2 = 𝑟2 (𝑄1 ) =
𝑎−𝑐2
2𝑏
1
2
− 𝑄1 and The leader’s output is 𝑄1 =
𝑎+𝑐2 −2𝑐1
2𝑏
EX3: Suppose the inverse demand function for two firms in a homogeneous-product, Stackelberg
oligopoly is given by 𝑃 = 50 − (𝑄1 + 𝑄2 ) and their marginal costs are: 𝑀𝐶1 (𝑄1 ) = 𝑀𝐶2 (𝑄2 ) = 2
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?
Betrand Oligopoly
EX4: Consider a Bertrand oligopoly consisting of four firms that produce an identical product at a
marginal cost of $100.The inverse market demand for this product is P = 500 - 2Q.
a. Determine the equilibrium market price.
b. Determine the equilibrium level of output in the market.
c. Determine the profits of each firm.
ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
EX5: The table below compares and contrasts the output levels and profits for the Cournot,
Stackelberg, Bertrand and Collusion models. Fill in the table assuming that there are two firms in
the market, the market demand is given by Q=150-1/10 P, each firm has a marginal cost of $20, an
average variable cost
of 20, and fixed costs of zero.
Firm One's
Output
Firm Two's
Output
Total
Output
Market Firm One's
Price
Profit
Firm Two's
Profit
a. Cournot
b.Stackelberg (leader)
(follower)
c. Bertrand
d. Collusion
Answer:
a. Cournot
P=1500 -10(q1+q2)
For Firm one we have:
MR1=1500-20q1 -10q2
MC=20
MR1 =MC => 1500-20q1 -10q2=20
Solving for q1 we get the Firm One’s best response function: q1=74- ½ q2 (1)
Similarly, for Firm 2 we have that:
R2=1500q2-10q22 -10q1q2
MR2=1500-20q2 -10q1
MC=20
MR2=MC => 1500-20q2 -10q1=20
Solving for q2 we get the Firm Two’s best response function: q2=74- ½ q1 (2)
The equilibrium is the intersection of the two best response function, substituting (2)
into (1) we get:
q1=74- ½ (74- ½ q1)
Solving for q1 we get q1=49.333.
Substituting this into (2) we get q2=49.333 and Q= q1+q2=98.66
Substituting Q into the inverse demand equation we get P=1500 -10(98.66) =>
P=513.33
With these values, the profits for both firms are $24,338
ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
b. Stackelberg
From the previous question Firm Two’s best response function is given by: q2=74- ½ q1
Substituting this into the inverse demand function, we get:
P=1500 -10q1-10(74- ½ q1) which simplifies to: P=760-5q1
Thus, we have that the marginal revenue of the leader is:
M R1=760-10 q1
MC1=20
From MR1= MC1 => 760-10q1=20 => q1=74
Substituting this into the best response function of the follower, we get:
q2=37 and Q= q1+q2=111
Substituting Q into the inverse demand equation we get P=1500 -10(111) =>
P=390 With these values, the profits for leader firm are $27,380 and the profits
for the follower firm are $13,690.
c. Bertrand
Firm set P=MC => P=20 => 1500-10Q=20 =>Q=148
The firms split the total output in half => q1=q2=74 and both firms earn zero
economic profits.
d. Collusion
When firms collude, they act as a multi-plant monopoly. Because both firms have the
same marginal cost, we can solve the problem for the regular monopolist and split
the total output in half.
P=1500 -10Q
For the monopolist we have:
MR=1500-20Q
MC=20
MR =MC => 1500-20Q=20
Solving for Q we get the monopolist output: Q=74 => q1=q2=37
Substituting Q into the inverse demand equation we get P=1500 -10(74) => P=760
The profits for both firms are $27,380
4
ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
EX6. Consider a market consisting of two firms where the inverse demand curve is given by P =
500 - 2Q1 - 2Q2. Each firm has a marginal cost of $50. Based on this information, we can
conclude that equilibrium price in the different oligopoly models will follow which of the
following orderings?
A. PBertrand < PStackelberg < PCournot < PCollusion
B. PStackelberg < PCollusion < PCournot < PBertrand
C. PCollusion < PCournot < PStackelberg < PBertrand
D. PBertrand < PCournot < PStackelberg < PCollusion
EX7. Which of the following are quantity-setting oligopoly models?
A. Stackelberg.
B. Cournot.
C. Bertrand.
D. Stackelberg and Cournot.
EX8. With linear demand and constant marginal cost, a Stackelberg leader's profits are
___________ the follower.
A. less than
B. equal to
C. greater than
D. either less than or greater than
EX9. If firms compete in a Cournot fashion, then each firm views the:
A. output of rivals as given.
B. prices of rivals as given.
C. profits of rivals as given.
D. All of the statements associated with this question are correct.
EX10. The Bertrand model of oligopoly reveals that:
A. capacity constraints are not important in determining market performance.
B. perfectly competitive prices can arise in markets with only a few firms.
C. changes in marginal cost do not affect prices.
D. All of the statements associated with this question are true.
EX11. A new firm enters a market which is initially serviced by a Bertrand duopoly charging a
price of $20. What will the new price be should the three firms coexist after the entry?
A. $25
B. $20
C. $15
D. None of the answers is correct.
5
ECO 3320-001
Chapter 9
Instructor: Lanlan Chu
Summary of Four Market Structures:
# of firms
Perfect Competition
Monopolistic Competition
Oligopoly
Monopoly
Very many
many
few
One
idential or
differentiated
unique
(no close
substitutes)
Product
identical
differentiated (but highly
substituable)
Entry/Exit
Easy
Easy
difficult
Nearly impossible
Profit max rule
MR=MC(or P=MC)
MR=MC
MR=MC
MR=MC
Short run
profit
π<, > or =0
π<, >, or =0
π<, > or =0
π<, > or =0
Long run profit
π=0
π=0
π> or =0
π> or =0
Ppc<Pmc<Po<Pm
Price(P)
Qpc>Qmc>Qo>Qm
Quantity(Q)
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