EC2066 MICROECONOMICS

TOPIC 9: OLIGOPOLY AND STRATEGIC BEHAVIOUR

Learning Outcomes

By the end of this chapter, and having completed the Essential reading and activities, you should
be able to:
 Define the conditions for various Oligopoly model.
 Understand the application of game theory into various game theoretic Oligopoly models.
 Define the market equilibrium price and quantity under various Oligopoly models.
 Describe the society welfare under various Oligopoly models.

Essential Reading
 Morgan, Katz and Rosen (MKZ) chapter 15: Oligopoly and strategic behaviour
 Perloff (P) chapter 14: Oligopoly
 Pindyck and Rubenfeld (P&R) chapter 12: Monopolistic competition and oligopoly;
chapter 13: Game Theory and Competitive Strategy

1. Game Theory and the Difference Between Oligopoly, Monopolistic, Monopoly

And Perfect Competition
 Some theoretical characteristic of Oligopoly, Monopoly and Perfect Competition.

Characteristics Oligopoly Monopoly Perfect competition

Number of sellers A few sellers Only one Many sellers
Price setting power Yes Yes No, firms are price takers
Number of buyers Many Many Many
Strategic behaviour Yes No No

Characteristics Monopolistic Oligopoly

Number of firms A few A few
Price setting power Yes Yes
Strategic behaviour No Yes
Product differentiation Some differentiation Homogeneous good

 The main difference between Oligopoly and other markets lie in strategic behaviour. In
monopolistic competition, there are a few firms in the market but the firms do not
consider their opponents’ strategies.

 In Oligopoly competition, there are also a few firms and but they consider their rivals’
strategies prior to their decision making. For example, how much to produce, how much
is the price of their good, which firm moves first, is it a one-time game or repeated game
etc.?

Copyright: Singapore Institute of Management Page |1

EC2066 MICROECONOMICS

Various Oligopoly models

Oligopoly

Repeated
Non-Cooperative Cooperative
games

Sequential moves Simultaneous moves Collusion

Quantity competition Quantity competition

Stackelberg model Cournot model
Price competition
Bertrand model

 We will cover the following models of oligopoly:

Simultaneous competition Sequential competition

Price competition Bertrand
Quantity competition Cournot Stackelberg

 Important definitions

 Price competition: Firms compete with each other by choosing the price for their
product.

 Quantity competition: Firms compete by choosing their unit of output.

 Bertrand model: Firms produce the same good and choose the price simultaneously.

 Cournot model: Firms produce the same good and choose their output
simultaneously.

 Stackelberg model: Firms produce the same good, but there is sequential quantity
competition among firms, where one of the firm chooses its output first and the rest
firms will learn the leader’s production decision and decide on their productions.

Copyright: Singapore Institute of Management Page |2

EC2066 MICROECONOMICS

2. Cournot Model: Foundations

 Under Cournot model, firms compete with each by choosing their output individually and
simultaneously, with reference to their competitor’s reaction.

 Here, we will examine a one-period quantity competition model where each firm has to
guess the other firm’s output choice, and then the firm will produce its own profit-
maximising output based on the guess.

 Note that firms will guess each other’s output level but they don’t know the exact output
level! Assuming there are only 2 firms in the Cournot model and, for simplicity, suppose
both Firm 1 and 2 have similar cost functions. A description of the model is shown
below:

 Both firms face the same market.

 The buyers have no bargaining power at all and will accept whatever price set by
both firms, as long as the price is lower than their reservation price.
 The market has high barriers to entry so that no other firms can enter the market.
 There is symmetric information such that firm 1 and 2 know about each other’s cost
and production functions, thus they know how each other will react if they choose a
particular output.
 Both firms decide on only their output but do not decide on price. The price of the
good will depend on the market equilibrium where the market supply intersects the
market demand (aggregate output of firm A and B).

 Algebraic example

Assuming an inverse market demand, , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost,

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,

1) Firm 1’s profit function,

2) For profit-maximisation, Max

3) Obtain First Order Condition (FOC) with respect to

, 2 0

Copyright: Singapore Institute of Management Page |3

EC2066 MICROECONOMICS

4) Obtain the reaction function of firm 1,

5) By symmetry, due to similar cost function, the reaction function of Firm 2,

6) Solve the simultaneous equations,

7) Market quantity,

8) Market price,

9) Firm 1 and 2 profit,

CS
E

PS DW
c MC

D Q

Copyright: Singapore Institute of Management Page |4

EC2066 MICROECONOMICS

Q2

R2

R1
Q1

 The Cournot-Nash equilibrium E is simply the pair of outputs produced by both firms
where the two reaction function curves intersect. At that point, both firms are producing
a profit-maximising level of output given the output choice of their rival.

 Proof: Firm 2’s Reaction Function:

1. Firm 2’s profit function,

2. For profit-maximisation, Max

3. Obtain First Order Condition (FOC) with respect to ,

2 0

4. Obtain the reaction function of Firm 2,

 As one can see, if Firm 1 and 2 have similar cost functions and are facing the same
market demand, one only has to work out one of the firm’s reaction functions like that of
Firm 1’s reaction function. Firm 2’s reaction function is the just the mirror of Firm 1’s
reaction function.

Copyright: Singapore Institute of Management Page |5

EC2066 MICROECONOMICS

3. Numerical Example for Cournot

Assuming inverse market demand, 10 ,

Firm 1 and 2 cost function,

Firm 1 and 2 marginal cost, 1

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,

1. Firm 1’s profit function, 10 1

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to , 9 2 0

4. Obtain the reaction function of Firm 1,

5. By symmetry, due to similar cost functions, Firm 2’s reaction function is,

6. Solve the simultaneous equations, 3

7. Market quantity, 6

8. Market price, 10 6 4

9. Firm 1 and 2’s profit, 4 1 3 9

Copyright: Singapore Institute of Management Page |6

EC2066 MICROECONOMICS

10

CS
E
4

PS DW
1 MC

D Q
6 10

Q2

R2

4.5

E
3

R1
Q1
3 4.5

 The Cournot-Nash equilibrium is E, where Firm 1 and 2 each produces 3 units of output.

The consumer surplus, 10 4 6 18

The industrial profit / producer surplus, 4 1 6 18

Copyright: Singapore Institute of Management Page |7

EC2066 MICROECONOMICS

The deadweight loss, 9 6 4 1 4.5

 Comparison with monopoly and perfect competition

10

Monopoly
5.5
4 Cournot

Perfect Competition
1 MC

D Q
4.5 6 10

Monopoly Cournot Perfect Competition

Consumer Surplus 10.125 18 40.5
Producer Surplus 20.25 18 0
Deadweight Loss 10.125 4.5 0

4. Cournot Model: A Shift in the Demand Curve

Assuming a new inverse market demand, 40 ; ∑

Firm 1 and 2 cost function,

Firm 1 and 2 marginal cost, 1

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,
1. Firm 1’s profit function, 40 1

2. For profit-maximisation, Max 39

Copyright: Singapore Institute of Management Page |8

EC2066 MICROECONOMICS

3. Obtain First Order Condition (FOC) with respect to , 39 2 0

4. Obtain the reaction function of Firm 1,

5. By symmetry, due to similar cost functions, the reaction function of Firm 2 ,

6. Solve the simultaneous equations, 13

7. Market quantity, 26

8. Market price, 40 26 14

9. Firm 1 and 2’s profit, 14 1 13 169

4
0

CS
E
1
4
1
0
PS
DW
1 MC

D D’ Q
1 2 4
0 6 0

 A shift in the demand curve will increase the firms’ output and profit, market output and
price.

5. Cournot Model: A Change in the Cost Function

Assuming an inverse market demand, 10 and ∑

Firm 1 and 2 new cost functions, 4

Firm 1 and 2 new marginal cost, 4

Copyright: Singapore Institute of Management Page |9

EC2066 MICROECONOMICS

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,

1. Firm 1’s profit function, 10 4

2. For profit-maximisation, Max 6

3. Obtain First Order Condition (FOC) with respect to , 6 2 0

4. Obtain the reaction function of Firm 1,

5. By symmetry, due to similar cost functions, the reaction function of Firm 2,

6. Solve the simultaneous equations, 2

7. Market quantity, 4

8. Market price, 10 4 6

9. Firm 1 and 2 profit’s, 6 4 2 4

10

CS
E
6

PS DW
4 MC’

1 MC
D
Q
4 10

Copyright: Singapore Institute of Management P a g e | 10

EC2066 MICROECONOMICS

Q2

R2

E1
3
2 E2

R1
Q1
2 3

 An increase in the cost function will decrease the firms’ output and profit, market output
and price. The reaction function curves of the firms will shift inward too.

6. Cournot model: Asymmetric Cost Function

 Example
Assuming an inverse market demand, 10 , ∑

Firm 1’s cost function,

Firm 1’s marginal cost, 1
Firm 2’s cost function, 4
Firm 2’s marginal cost, 4

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,
1. Firm 1’s profit function, 10 1

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) Q,

Copyright: Singapore Institute of Management P a g e | 11

EC2066 MICROECONOMICS

9 2 0

4. Obtain the reaction function of Firm 1,

For Firm 2,
1. Firm 2’s profit function, 10 4

2. For profit-maximisation,Max 6

3. Obtain First Order Condition (FOC) with respect to (wrt) ,

6 2 0

4. Obtain the reaction function of Firm 2,

After obtaining the reaction function for both Firm 1 and 2,

Solve the simultaneous equations, 4, 1

Market quantity, 5
Market price, 10 5 5
Firm 1’s profit, 5 1 4 16
Firm 2’s profit, 5 4 1 1

10

CS
E
5

PS DW
1 MC

D Q
5 10

Copyright: Singapore Institute of Management P a g e | 12

EC2066 MICROECONOMICS

Q2

R2

E
1

R1
Q1
4 4.5

 Due to asymmetric cost, the output and profit of Firm 1 and 2 will be different.

7. Cournot Special Case: N Firms in Cournot

Assuming inverse market demand, and

⋯ ∑ , ∈ 1,2 …

Firm i's cost function,

Firm i's marginal cost,

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,
1. Firm 1's profit function
Becomes ∑
Becomes ∑

2. For profit-maximisation, Max ∑

3. Obtain First Order Condition (FOC) with respect to ,

Copyright: Singapore Institute of Management P a g e | 13

EC2066 MICROECONOMICS

2 0

∑
4. Obtain the reaction function of firm 1,

5. By symmetry, due to similar cost functions, firm i’s reaction function is:

1
2

6. Solve the simultaneous equations,

7. Market quantity, ∑

8. Market price,

9. Firm 1's profit,

 You can plug in the number of firms to check the validity of this equation. Try pluggin in
N = 2 and obtain the same equations from the earlier example. In the exam, there is a
possibility that the examiner will ask you to obtain the number of firms when the
equilibrium profit is zero. You can solve it by simply setting 0, and find
out the value of N.

8. The impact of fixed cost on firms in Cournot model

Assuming inverse market demand, 10 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,
1. Firm 1’s profit function, 9

2. For profit maximization,Max 9

Copyright: Singapore Institute of Management P a g e | 14

EC2066 MICROECONOMICS

3. Obtain First Order Condition (FOC) with respect to , 9 2 0

4. Obtain the reaction function of firm 1,

5. By symmetry, due to similar cost function, the reaction function of firm 2,

6. Solve the simultaneous equations, 3

7. Market quantity, 6

8. Market price, 10 6 4

9. Firm 1 and 2 profit, 4 1 3 9

 Comparing to the earlier example that has a cost function without fixed cost, the
presence of fixed cost does not affect the firm’s reaction function and output decisions;
the market equilibrium quantity and price. However, it affects the firm’s profit. Here, if
the fixed cost, F > 9, Firm 1 will make losses and will exit the market.

9. The Impact of a Production Cap on Firms in a Cournot Model

 Example
Assuming inverse market demand, 10 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

However, the government restricts Firm 1’s production up to only 2 units.

The steps for solving the Cournot-Nash equilibrium are:

For Firm 1,

1. Firm 1’s profit function, 9

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to , 9 2 0

4. Obtain the reaction function of Firm 1,

Copyright: Singapore Institute of Management P a g e | 15

EC2066 MICROECONOMICS

5. By symmetry, due to similar cost functions, the reaction function of Firm 2,

6. Solve the simultaneous equations, 3

7. However, 3 is not feasible because Firm 1 can produce up to 2 units only.

Hence, Firm 1’s output 2 and then substitute 2 into Firm 2’s reaction
function to solve for Firm 2’s output, for which Firm 2’s output 3.5

8. Market quantity, 5.5

9. Market price, 10 5.5 4.5

10. Firm 1 and 2’s profit, 7, 12.25

Q2

R2
Production cap

4.5
3.5 E
3

R1
Q1
2 3 4.5

10. Stackelberg Model: Foundations

 Under the Stackelberg model, firms compete with each other by choosing their output
individually and sequentially with reference to their competitor’s reaction.

 Here, we will examine a one-period (2-stage) quantity competition where one firm (also
known as leader) has the first-mover advantage to choose its output first, and then
another firm (the follower) will produce its own profit-maximising output with reference
to the leader’s output.

Copyright: Singapore Institute of Management P a g e | 16

EC2066 MICROECONOMICS

 Note that when the follower makes its decision, it already knows the leader’s output
decision! Assuming there are only 2 firms in the Stackelberg model and, for simplicity,
suppose both Firm 1 and 2 have similar cost functions, a description of the model is
shown below:

 Firm 1 moves first and decides its output.

 After seeing Firm 1’s output, Firm 2 moves and decides its output.
 Both firms seek to maximise their profit.
 Both face the same market.
 The buyers have no bargaining power at all and will accept whatever price set by
both firms, as long as the price is lower than their reservation price.
 The market has high barrier to entry so that no other firm can enter the market.
 There is symmetric information such that Firm 1 and 2 know about each other’s
cost and production functions, thus they know how each other will react if they
choose a particular output.
 Both firms decide only their output but do not decide on price. The price of the good
will depend on the market equilibrium where the market supply intersects market
demand (aggregate output of firm A and B).

 Example

Assuming the inverse market demand is , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost,

 Assuming Firm 1 moves first and chooses its output. Subsequently, Firm 2 chooses its
profit-maximising output with reference to Firm 1’s output. Since Firm 1 has the first
mover advantage, it will maximise its profit by including Firm 2’s reaction function into its
own profit function.

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:
1. Firm 2’s profit function,

2. For profit-maximisation, Max

3. Obtain First Order Condition (FOC) with respect to ,

2 0

4. Obtain the reaction function of Firm 1,

Copyright: Singapore Institute of Management P a g e | 17

EC2066 MICROECONOMICS

Now consider the leader, Firm 1:

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,

2 2

6. For profit-maximisation, Max

7. Obtain First Order Condition (FOC) with respect to (wrt) , 0

8. The output of Firm 1 and 2,

Under symmetric cost, follower output is always

4 half of the leader’s output.

9. Market quantity,

10. Market price,

11. Firm 1 and 2’s profit,

16

Copyright: Singapore Institute of Management P a g e | 18

EC2066 MICROECONOMICS

CS
E

PS DW
c MC

D Q

Q2

R2

Cournot

Stackelberg

R1
Q1

 The Stackelberg equilibrium S, is simply the pair of outputs of both firms where Firm 1
produces its output first after guessing the output decision of Firm 2. At that point, both
firms are producing a profit-maximising level of output given the output choice of their
rival.

Copyright: Singapore Institute of Management P a g e | 19

EC2066 MICROECONOMICS

11. Numerical Example for Stackelberg

Assuming an inverse market demand, 10 , ∑
Firm 1 and 2 cost functions,
Firm 1 and 2 marginal cost, 1

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2 profit function, 9

2. For profit maximization, Max 9

3. Obtain First Order Condition (FOC) with respect to , 9 2 0

4. Obtain the reaction function of Firm 1,

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,
9
9
2
4.5
2

6. For profit-maximisation, Max 4.5

7. Obtain First Order Condition (FOC) with respect to (wrt) , 4.5 0

8. The output of Firm 1 and 2,

4.5
Under symmetric cost, follower output is always
2.25 half of the leader’s output.

9. Market quantity, 6.75

10. Market price, 10 6.75 3.25

11. Firm 1 and 2’s profit,

10.125
5.0625

Copyright: Singapore Institute of Management P a g e | 20

EC2066 MICROECONOMICS

10

CS
E
3.25

PS DW
1 MC

D Q
6.75 10

Q2

R2

Cournot
3
Stackelberg
2.25
R1
Q1
3 4.5

 The Stackelberg equilibrium is S, where Firm 1 produces 4.5 units of output and Firm 2
produces 2.25 units of output.
The consumer surplus, 10 3.25 6.75 22.78
The industrial profit / producer surplus, 3.25 1 6.75 15.19
The deadweight loss, 3.25 1 9 6.75 2.53

 The following graph shows a simple comparison among Cournot, monopoly and perfect
competition:

Copyright: Singapore Institute of Management P a g e | 21

EC2066 MICROECONOMICS

10

Monopoly
5.5
4 Cournot

Stackelberg
3.25
Perfect Competition
1 MC

D Q
4.5 6 6.75 9 10

Perfect
Monopoly Cournot Stackelberg
Competition
Consumer
10.125 18 22.78 40.5
Surplus
Producer Surplus 20.25 18 15.19 0
Deadweight Loss 10.125 6 2.53 0

12. Stackelberg Model: A Shift in the Demand Curve

 Example
Assuming an inverse market demand, 40 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2’s profit function, 39

2. For profit-maximisation, Max 39

3. Obtain First Order Condition (FOC) with respect to , 39 2 0

Copyright: Singapore Institute of Management P a g e | 22

EC2066 MICROECONOMICS

4. Obtain the reaction function of Firm 1,

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,
39
39
2
19.5
2

6. For profit-maximisation, Max 19.5

7. Obtain First Order Condition (FOC) with respect to (wrt) , 19.5 0

8. The output of Firm 1 and 2,

19.5

Under symmetric cost, follower output is always

9.75 half of the leader’s output.

9. Market quantity, 29.25

10. Market price, 40 29.25 10.75

11. Firm 1 and 2’s profit,

190.125
95.0625

Copyright: Singapore Institute of Management P a g e | 23

EC2066 MICROECONOMICS

40

CS
E
10.75

10
PS
DW
1 MC

D D Q
10 29.25 40

 A shift in the demand curve will increase the firms’ output and profit, market output and
price.

13. Stackelberg Model: A Change in the Cost Function

 Example
Assuming an inverse market demand, 10 and ∑

Firm 1 and 2 new cost functions, 4

Firm 1 and 2 new marginal cost, 4

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2’s profit function, 6

2. For profit-maximisation, Max 6

3. Obtain First Order Condition (FOC) with respect to , 6 2 0

4. Obtain the reaction function of Firm 2,

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,

Copyright: Singapore Institute of Management P a g e | 24

EC2066 MICROECONOMICS

6
6
2
3
2

6. For profit-maximisation, Max 3

7. Obtain First Order Condition (FOC) with respect to (wrt) , 3 0

8. The output of Firm 1 and 2,

Under symmetric cost, follower output is always

1.5 half of the leader’s output.

9. Market quantity, 4.5

10. Market price, 10 4.5 5.5

11. Firm 1 and 2 profit,

4.5
2.25

10

CS
E
5.5

PS DW
4 MC’

1 MC
D Q
4.5 10

Copyright: Singapore Institute of Management P a g e | 25

EC2066 MICROECONOMICS

Q2

R2

Stackelberg
2.25
R1
1.5
Q1
3 4.5

 An increase in the cost function will decrease the firms’ output and profit, market output
and price. The reaction function curve of firms will shift inward too.

14. Stackelberg Model: Asymmetric Cost Function

 Example
Assuming an inverse market demand, 10 and ∑

Firm 1’s cost function,

Firm 1’s marginal cost, 1
Firm 2’s cost function, 2
Firm 2’s marginal cost, 2

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2’s profit function, 10 2

2. For profit-maximisation, Max 6

3. Obtain First Order Condition (FOC) with respect to , 8 2 0

4. Obtain the reaction function of Firm 2,

Copyright: Singapore Institute of Management P a g e | 26

EC2066 MICROECONOMICS

For Firm 1,

5. Firm 1’s profit function,

10 10 1

6. Substitute the Firm 2’s reaction function into Firm 1’s profit function,
8
9 5
2 2

7. For profit-maximisation, Max 5

8. Obtain First Order Condition (FOC) with respect to (wrt) , 5 0

9. The output of Firm 1 and 2,

5

1.5 Under asymmetric cost, follower output is not

necessary half of the leader’s output.

10. Market quantity, 6.5

11. Market price, 10 6.5 3.5

12. Firm 1 and 2’s profit,

12.5
2.25

Copyright: Singapore Institute of Management P a g e | 27

EC2066 MICROECONOMICS

10

CS
E
3.5

PS DW

1 MC
D Q
6.5 10

Q2

R2

Stackelberg
1.5
R1
Q1
5

 Due to asymmetric cost will, the output and profit of firm 1 and 2 will be different.

Copyright: Singapore Institute of Management P a g e | 28

EC2066 MICROECONOMICS

15. Special Case: The Role of Fixed Cost

 Example
Assuming inverse market demand, 10 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2’s profit function, 9

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) ,

9 2 0

4. Obtain the reaction function of Firm 1,

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,
9
9
2
4.5
2

6. For profit-maximisation, Max 4.5

7. Obtain First Order Condition (FOC) with respect to (wrt) , 4.5 0

8. The output of Firm 1 and 2,

4.5

Under symmetric cost, follower output is

2.25 always half of the leader’s output.

9. Market quantity, 6.75

10. Market price, 10 6.75 3.25

11. Firm 1 and 2’s profit,

Copyright: Singapore Institute of Management P a g e | 29

EC2066 MICROECONOMICS

10.125
5.0625

 In a similar fashion to the Cournot model, the presence of fixed cost does not affect the
firms’ reaction function, the market equilibrium quantity and price. However, it will affect
the firms’ profit and thus the firms’ decision to stay or exit from the market.

 For example, if 5.0625, which implies Firm 2 is making losses, then Firm 2 will exit
the market and Firm 1 becomes the monopolist in the market. This will be discussed
further during a later part - Entry Deterrence and Accommodation.

16. Optional Material: Multi-Stage Stackelberg

 Assuming there are three firms in the market: Firm 1, 2 and 3. They compete in the
Stackelberg model where Firm 1 moves first, Firm 2 moves second and Firm 3 moves
last. Therefore, by backward induction, Firm 2 will consider Firm 3’s reaction function
when it decides its output, and Firm 1 will consider Firm 2’s reaction function when it
decides its output. Assuming all firms are profit-maximising firms.
Assuming inverse market demand, 10 , ∑

Firm 1, 2 and 3 cost functions,

Firm 1, 2 and 3 marginal cost, 1

The steps for solving the Stackelberg equilibrium are:

For Firm 3,
1. Firm 3’s profit function,
10 1

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to (wrt)

9 2 0

4. Obtain the reaction function of Firm 3,

9
2

For Firm 2,

5. Firm 2’s profit function,

10 1

Copyright: Singapore Institute of Management P a g e | 30

EC2066 MICROECONOMICS

6. Substitute the Firm 3’s reaction function into Firm 2’s profit function,
9 1 1
9 4.5
2 2 2

7. For profit-maximisation, Max 4.5

8. Obtain First Order Condition (FOC) with respect to (wrt)

1
4.5 0
2

9. Obtain the reaction function of Firm 2,

9
2

For Firm 1,

10. Firm 1’s profit function,

10 1

11. Substitute the reaction functions of Firms 2 and 3 into Firm 1’s profit function,
9 q 9 q q
π 9 q q
2 2
9 q 9 q
π 9 q
2 4
9 q
π q
4 4

12. For profit-maximisation, Max

13. Obtain First Order Condition (FOC) with respect to (wrt) , 0

14. The output of Firm 1, 2 and 3, , ,

15. Market quantity, Q q q q

16. Market price, P 10

Copyright: Singapore Institute of Management P a g e | 31

EC2066 MICROECONOMICS

Firm 1, 2 and 3 profit,

81
16
81
32
81
64

17. Optional Material: One Leader and N Followers in the Cournot Model
 This is a more complicated case that combines Stackelberg and Cournot models. For
simplification and easier understanding, we will work with real numbers. Assuming there
are 3 firms: Firm 1, 2 and 3 in a market. Firm 1 has first-mover advantage and decides
its output first; subsequently Firm 2 and Firm 3 observes Firm 1’s output and produce
their output simultaneously. Note that Firm 2 and 3 are engaged in Cournot
competition.

 Assuming all firms are profit-maximising firms.

Assuming inverse market demand, 10 , ∑

Firm 1, 2 and 3 cost functions,

Firm 1, 2 and 3 marginal cost, 1

First, we adopt the backward induction strategy and solve the Cournot equilibrium.

1. Firm 2’s profit function,

10 1

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to ,

9 2 0

4. Obtain the reaction function of Firm 2,

5. By symmetry, due to similar cost functions, the reaction function of Firm 3,

6. Solve the simultaneous equations, with similar reaction functions,

9 9
2 3

Copyright: Singapore Institute of Management P a g e | 32

EC2066 MICROECONOMICS

 Now, after obtaining the output of Firm 2 and 3, we can substitute it into Firm 1’s profit
function and work out the output of Firm 1 in the Stackelberg model.

For Firm 1,
7. Firm 1’s profit function,
10 1

8. Substitute the reaction function of Firm 2 and 3 into Firm 1’s profit function,
9 9
9 3
3 3 3

9. For profit-maximisation, Max 3

10. Obtain First Order Condition (FOC) with respect to , 3 0

11. The output of Firm 1, 2 and 3,

9 9 9
4.5, 1.5, 1.5
2 6 6

12. Market quantity, 7.5

13. Market price, 10 7.5 2.5

14. Firm 1, 2 and 3’s profit,

6.75
2.25
2.25

18. Bertrand Model

 The Bertrand model is an oligopoly model whereby firms compete in price
simultaneously.

 Assuming firms are producing a homogenous good and there is symmetric information,
under simultaneous price competition, consumers will buy the good from the firm that
offers the lowest price.

 Further assuming firms do not have capacity constraints and have production that
exhibits constant returns-to-scale, logically, firms will have the incentive to offer a price
that is lower than what other firms are selling in order to capture the entire market
demand.

Copyright: Singapore Institute of Management P a g e | 33

EC2066 MICROECONOMICS

 However, the lowest price that the firm can offer is its marginal cost. Consequently, all
firms will set its price such that . If all firms have the same cost function, all
firms will sell their product at the same price and share the market equally. If one firm
has lower MC than others, then this firm will set a price equal to the second lowest MC
minus a small amount    .

Assuming inverse market demand, 10 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

At the Bertrand equilibrium, 1 and 9

Both firms will share the market equally 4.5
Firm 1 and 2’s profit, 0

 This is the equilibrium because there is no incentive for firms to increase or decrease
the price. If Firm 1 increases its price such as , it will lose its market share and
Firm 2 will capture the entire market demand. If Firm 1 decreases its price such that
, it will capture the entire market but it will also make losses because
.

 However, the Bertrand model does not imply perfect competition, although perfect
competition can be considered as one form of the Bertrand model! Consider the
following model:
Assuming inverse market demand, 10 , ∑

Firm 1’s cost function, 2

Firm 2’s cost function,
Firm 1’s marginal cost, 2
Firm 2’s marginal cost, 1

At the Bertrand equilibrium is, 2 8

Firm 2 will capture the entire market demand, 8
Firm 1’s profit, 0
Firm 2’s profit, 8

 Since Firm 1 has 2, Firm 2 could capture the entire market by pricing its output
at 2, such as 1.99. At the Bertrand equilibrium, Firm 2 makes
positive profit which violates the perfect competition model.

 Meanwhile, this is the Bertrand equilibrium because there is no incentive for Firm 2 to
change its price. If Firm 2 increase the price, 2, Firm 1 will capture the entire
market demand at a lower price. If Firm 2 decreases the price to 2, its profit will
fall.

Copyright: Singapore Institute of Management P a g e | 34

EC2066 MICROECONOMICS

 Besides, Firm 2 will be the monopoly in the market because Firm 1 is unable to compete
with Firm 2 in terms of price and will leave the industry.

 This example also shows that it is not necessary that a monopolist will always price its
goods at . With only one firm in the market, there are many ways for the
monopolist to set the price for its output. The following ways have been covered:

 Pricing method of finding the optimum quantity at

 Perfect price discrimination
 Two-part tariff
 3rd degree price discrimination if there is a multi-market
 Limit pricing Not in your syllabus, for general knowledge only
 Predatory pricing

19. Collusion
 All three models we learnt earlier, namely Cournot, Stackelberg and Bertrand are non-
cooperative and one-period games.

 Instead of competing with each other in the form of quantity and price, firms might
choose to cooperate and form a monopoly like a cartel. In order to form a monopoly in
the market, the member firms must collectively decide:

 What will be the industry output?

 What will be the individual firm’s output?
 What will be the consequent market price?

 Under cooperation, both firms form a cartel and act like a monopolist. Assuming the
collusion is such that firms:

 Play a one-period game.

 Contains only Firm 1 and Firm 2 that produce a homogeneous good.
 If one of the firms cheat, the cheating firm will maximise its profit by assuming that
the other firm will cooperate and produce the output when both firms are co-
operating.
 If both cheat, this will gravitate back to simultaneous quantity competition under the
Cournot model.

 Amidst the interaction between Firm 1 and 2, there are four possible scenarios in terms
of payoff for firms.

Firm 2
Cooperate Cheat
Cooperate Cooperate, Cooperate Cooperate, Cheat
Firm 1
Cheat Cheat, Cooperate Cheat, Cheat

Copyright: Singapore Institute of Management P a g e | 35

EC2066 MICROECONOMICS

Assuming an inverse market demand, 10 , ∑

Firm 1 and 2 cost functions,

Firm 1 and 2 marginal cost, 1

 The steps to obtain the collusion-monopolist equilibrium are:

1. Write down the monopoly profit function, 9

2. For profit-maximisation, max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) Q, 9 2 0

4. Obtain the value of market quantity, 4.5

5. Obtain the value of market price, 10 4.5 5.5

6. The value of monopoly profit, 20.25

7. Assuming equal share, quantity produced by each firm, 2.25

8. The profit earned by each firm, 10.125

 At the monopoly equilibrium when both cooperate:

Consumer surplus, 10 5.5 4.5 10.125

Producer surplus, 5.5 1 4.5 20.25
Deadweight loss, 5.5 1 9 4.5 10.125

 The outcome when both cooperate is shown in the following graph:

Copyright: Singapore Institute of Management P a g e | 36

EC2066 MICROECONOMICS

10

E
5.5

1 MC
D Q
4.5 9 10

 The steps to obtain the equilibrium when one firm cheats and another firm cooperates.

Assuming Firm 1 cheats and Firm 2 cooperates,

1. Firm 2 cooperates and produces, 2.25

2. Firm 1’s profit function, 9 6.75

3. For profit-maximisation, max 6.75

4. Obtain First Order Condition (FOC) with respect to (wrt) ,

6.75 2 0

5. Firm 1’s output, 3.375

6. Market output, 5.625

7. Market price, 10 5.625 4.375

8. Firm 1’s profit, 11.39

9. Firm 2’s profit, 7.59

Copyright: Singapore Institute of Management P a g e | 37

EC2066 MICROECONOMICS

 The outcome when firm 2 cooperates while firm 1 takes advantage of firm 2’s
cooperative behaviour is shown in the following graph:

Q2

R2

Firm 1 cheats while Firm 2 is cooperating

Collusion
2.25

R1
Q1
3.375

 The following graph shows the market outcome:

10

E
4.375

1 MC
D Q
5.625 9 10

 The steps for solving the Cournot equilibrium are:

1. Firm 1’s profit function, 10 1

Copyright: Singapore Institute of Management P a g e | 38

EC2066 MICROECONOMICS

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) ,

9 2 0

4. Obtain the reaction function of Firm 1,

5. By symmetry, due to similar cost functions, the reaction function of Firm 2,

6. Solve the simultaneous equations, 3

7. Market quantity, 6

8. Market price, 10 6 4

9. Firm 1 and 2’s profit, 4 1 3 9

 The following graphs show the outcome when both firms refuse to cooperate.

Q2

R2

4.5

Cournot
3
One firm cheating
2.25
Collusion
R1
Q1
3 3.375

Copyright: Singapore Institute of Management P a g e | 39

EC2066 MICROECONOMICS

10

E
4

1 MC
D Q
6 9 10

 The payoff chart will look like this:

Firm 2
Cooperate Cheat
Cooperate 10.125, 10.125 7.59, 11.39
Firm 1
Cheat 11.39, 7.59 9, 9

 Obviously, the dominant strategy for both firms is to cheat and the Nash equilibrium is
(9, 9). Notice that it is a prisoner’s dilemma game where both firms could have earned
more profit if they cooperated, but they will cheat because it is their dominant strategy.

 Even if it is not a one-period game, as long as it is finite game, it will still yield the same
outcome in that both firms will cheat by the logic of backward induction.

 Assuming it is a three-period game, the dominant strategy for both firms in the last game
is to cheat. Since both firms realise that the other player will cheat in the last game,
they will cheat in the second last game, and then the second last game too. In the end,
if this is a finite game, both players will cheat in every period.

 In short, if the competition is repeated over a finite number of periods, firms will play
according to the Nash equilibrium of the one-period game in each period and there is no
room for collusion.

 However, if the game is played for infinite periods, then there is a possibility that firms
might collude instead of engaging in competition whether in terms of price or quantity.

Copyright: Singapore Institute of Management P a g e | 40

EC2066 MICROECONOMICS

 One of a strategies to elicit collusion is the trigger strategy. Specifically, the strategy
stipulates that players will cooperate so long as the other players also cooperate. They
will switch to compete for the remaining periods if one of the players are found out to be
cheating.

 Cheating gives highest level of profit. This is followed by the level of profit if both firms
cooperate. When both firms compete, the level of profit is the lowest. The trigger
strategy thus induces cooperation as it does not pay to cheat and earn a one-off
attractively high profit and then the lowest amount of profits for all subsequent periods.

20. Entry Deterrence and Accommodation

 Assuming:

 There are two periods: Period 1 and 2.

 There are two firms: Firm 1 and 2 engaged in quantity competition.
 Firm 1 is the incumbent firm and has first-mover advantage such that it decides its
output in Period 1.
 Firm 2 is the entrant firm and will choose whether to enter or not in Period 1. If it
enters, it will decide its output in Period 2 after observing the output of Firm 1 in
Period 1.
 If Firm 2 decides not to enter or stay out of the industry, its profit will be zero.

Assuming an inverse market demand, 10 and ∑

Firm 1’s cost function,

Firm 2’s cost function,

Assuming Firm 2’s fixed cost, 4

Firm 1’s marginal cost, 1

Firm 2’s marginal cost, 1

 Knowing that Firm 2 could enter the market, Firm 1 could deter Firm 2’s entry if it pays.
If not, Firm 1 will accommodate the entry. The structure of interaction is shown in the
following decision tree.

Copyright: Singapore Institute of Management P a g e | 41

EC2066 MICROECONOMICS

Payoff
Firm 1 Firm 2

Deter 10 0
entry
Incumbent
Firm 1
Enter
Sub‐game
Entrant Accomodate
entry 10.125 1.0625 Perfect Nash
Firm 2 Equilibrium

No Enter Monopoly
Incumbent 20.25 0
Firm 1

 First scenario: firm 2 stays out and firm 1 becomes the monopoly in the market.

1. Write down Firm 1’s profit function, 9

2. For profit-maximisation, max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) Q, 9 2 0

4. Obtain the value of market quantity, 4.5

5. Obtain the value of market price, 10 4.5 5.5

6. Monopoly profit, 20.25

 Second scenario: Firm 2 enters and Firm 1 accommodates its entry.

 Since Firm 1 decides its output in Period 1 and Firm 2 decides its output in Period 2,
Firm 1 has first-mover advantage, it would be a Stackelberg equilibrium.

Since the interaction is a sequential game with 2 players, the steps for solving the
Stackelberg equilibrium is based on the concept of backward induction. We will start
with the follower, Firm 2:

1. Firm 2’s profit function, 9

Copyright: Singapore Institute of Management P a g e | 42

EC2066 MICROECONOMICS

2. For profit-maximisation, Max 9

3. Obtain First Order Condition (FOC) with respect to (wrt) q2,

9 2 0

4. The reaction function of firm 2,

5. Substitute Firm 2’s reaction function into Firm 1’s profit function,
9
9 4.5
2 2

6. For profit-maximisation, Max 4.5

7. Obtain First Order Condition (FOC) with respect to (wrt) , 4.5 0

8. The output of Firm 1 and 2, 4.5, 2.25

9. Market quantity, 6.75

10. Market price, 10 6.75 3.25

11. Firm 1 and 2’s profit,

10.125

5.0625

 Scenario 3: Firm 2 enters and Firm 1 deters its entry.

 First, we need to think: under what kind of situation will Firm 2 not enter the market?
Answer: Firm 2 will not enter the market if it makes losses or zero profit.

The steps are:

1. Firm 2 profit function, 9

2. The reaction function of firm 2,

3. Substitute the reaction function of Firm 2 into its profit function, Firm 2’s profit
function becomes,
9 9
9
2 2

Copyright: Singapore Institute of Management P a g e | 43

EC2066 MICROECONOMICS

9 9
9
2 2

9
2
4. Set Firm 2’s profit to zero to find out the value of q1 that will cause Firm 2 to earn
zero profit. When Firm 2’s profit equals zero:
9
0
2
5. Firm 1’s output,
9 2√

6. Given F = 4, then, Firm 1’s output, 9 2√ 5

7. As then 2 if 5.

Thus, if 2 and 5, Firm 2 will not enter.

Market output, 7
Market price, 10 3
Firm 1’s profit, 3 1 5 10
Firm 2’s profit, 9 9 2 5 2 4 0

 Notice that Firm 1’s profit is higher if it accommodates than if it deters Firm 2. The sub-
game perfect equilibrium is then (10.125, 1.0625) that can be obtained through the
backward-induction method.

 Specifically, if Firm 2 enters, Firm 1 will choose to accommodate entry as profit from
accommodating > profit from deterring, i.e., 10.125 > 10. Secondly, Firm 2 knows that
Firm 1 will accommodate its entry, it will decide to enter as entering gives it higher profit
than staying out, i.e., 1.0625 > 0

 What if F = 9?

Firm 1 output, 9 2√ → 3

 It implies Firm 1 has to produce at least 3 units of output to deter Firm 2’s entry.
Therefore, even when Firm 1 acts as a monopolist and produces 4.5 units of output,
Firm 2 will not enter as 4.5 > 3. This would imply firm 1 will produce the monopolist
output of 4.5.

Copyright: Singapore Institute of Management P a g e | 44

EC2066 MICROECONOMICS

Firm 1’s output, 4.5

Firm 2’s output, 2.25

Market output, 6.75

Market price, 10 3.25

Firm 1’s profit, 3.25 1 4.5 10.125

Firm 2’s profit, 9 9 9 6.75 2.25 9 3.9375

 Hence, due to high fixed cost F, even if Firm 1 act as a monopolist and produces
monopoly output at 4.5 units of output, Firm 2 is unable to enter.

Copyright: Singapore Institute of Management P a g e | 45