Chapter 13a - Oligopoly
Goals:
1. Cournot: compete on quantity simultaneously.
2. Bertrand: compete on price simultaneously.
3. Stackelberg: compete on quantity in a sequential
setting
4. Hotelling (differentiated products)
Brief Introduction of Game Theory

Five elements of a game:
◦
◦
◦
◦
The players
The timing of the game.
The list of possible strategies for each player.
The payoffs associated with each combination
of strategies.
◦ The decision rule.
Cournot Model of Quantity
Competition

Setting:
◦ Homogeneous product market with 2 firms
◦ Firm sets quantity q1, q2 respectively.
 Total market output: q=q1+q2
 Linear cost functions: Ci(qi)=ciqi where I = 1, 2.
◦ Market price given by P(q)=a−bq
Cournot Model of Quantity
Competition
◦ The players: Firm 1 and Firm 2
◦ The timing of the game: Simultaneous
◦ The list of possible strategies for each player:
All possible choices of quantity q1 and q2.
◦ The payoffs associated with each combination
of strategies: profits
◦ The decision rule: maximize profit.
Cournot Model of Quantity
Competition

Solve the model:
◦ Firm 1’s problem:
 Max 1= (a – bq)q1 – cq1
 Firm 1’s best-response function (reaction function)
 q1 = (a – bq2 – c)/2b
◦ Firm 2’s problem:
 Max 2= (a – bq)q2 – cq2
 Firm 2’s best-response function (reaction function)
 q2 = (a – bq1 – c)/2b
◦ Nash Equilbrium:
 q1 = q2 = a/3b and P = a/3
Cournot Model of Quantity
Competition
Cournot Model of Quantity
Competition

Exercise:
◦ A market demand curve for a pair of
duopolists is given as: P = 36 – 3Q where Q =
Q1 + Q2. Each duopolist has a constant
marginal cost equal to 18 (fixed cost is zero).
Fill the below table.
Model
Cournot
Q1
Q2
Q1+Q
2
P
1
2
1+ 2
Bertrand Model of Price
Competition

Setting:
◦ Homogeneous product market with 2 firms
◦ Firm sets prices P1, P2 respectively and have
unlimited capacity.
◦ Market demand given by P(q)=a−bq
◦ Linear cost functions: Ci(qi)=ciqi where i = 1, 2.
 C1 = C2
Bertrand Model of Price
Competition.
◦ The players: Firm 1 and Firm 2
◦ The timing of the game: Simultaneous
◦ The list of possible strategies for each player:
All possible choices of quantity P1 and P2.
◦ The payoffs associated with each combination
of strategies: profits
◦ The decision rule: maximize profit.
Bertrand Model of Price
Competition

Firm’s problem:
◦ Firm faces the following demand schedule:
 Q = a – bP1
if P1 < P2
 Q = ½(a – bP) if P1 = P2 = P
 Q=0
if P1 >P2
◦ Nash Equilibrium:
 With symmetric cost functions: P1 = P2 = MC and two
firms slit the market demand equally.
 With asymmetric cost functions:
 c1 < c2 then P2 = c2 and P1 = P2 whole market.

and firm 1 captures the
Bertrand’s Paradox: Only 2 firms but
achieve the perfectly competitive market
outcome.
Bertrand Model of Price Competion
Cournot Model of Quantity
Competition

Exercise:
◦ A market demand curve for a pair of
duopolists is given as: P = 36 – 3Q where Q =
Q1 + Q2. Each duopolist has a constant
marginal cost equal to 18 (fixed cost is zero).
Fill the below table.
Model
Q1
Q2
Q1+Q
2
P
Cournot
2
2
4
24
Bertrand
1
2
12
12
1+ 2
24
Stackelberg Sequential Quantity
Competition

Setting:
◦ Homogeneous product market with 2 firms:
one leader and one follower
◦ Leader sets quantityq1, then follower sets
quantity q2.
◦ Market demand given by P(q)=a−bq
◦ Linear cost functions: Ci(qi)=ciqi where I = 1, 2.
Cournot Model of Quantity
Competition
◦ The players: Firm 1 and Firm 2
◦ The timing of the game: Sequential where firm
1 moves first and firm 2 moves later.
◦ The list of possible strategies for each player:
All possible choices of quantity q1 and q2.
◦ The payoffs associated with each combination
of strategies: profits
◦ The decision rule: maximize profit.
Stackelberg Sequential Quantity
Competition

Solving the model: backward induction.
◦ Follower’s Problem:
 Max 2 = (a – bq)q2 – cq2
 Where q = q1 + q2
 Best-response function for firm 1
 q2 = (a – bq1 – c)/2b
◦ Leader’s Problem:
 Max 2 = (a – bq)q1 – cq1
 Where q = q1 + (a – bq1 – c)/2b
 Best-response function for firm 1
 q1 = (a – c)/2b and q2 = (a – c)/4b
Stackelberg Sequential Quantity
Competition.

Exercise:
◦ A market demand curve for a pair of
duopolists is given as: P = 36 – 3Q where Q =
Q1 + Q2. Each duopolist has a constant
marginal cost equal to 18 (fixed cost is zero).
Fill the below table.
1
2
24
12
12
24
18
0
0
0
Model
Q1
Q2
Q1+Q2 P
Cournot
2
2
4
Bertrand
3
3
6
Stackelberg
1+ 2
Stackelberg Sequential Quantity
Competition

First mover advantage: Leader earns
higher profit than follower.
◦ In the price competition however, there is a
second mover advantage as the follower can
always undercut leader’s price.
A Comparison across models.
1
2
24
12
12
24
6
18
0
0
0
1.5
4.5
22.5
13.5
6.75
20.25
1.5
3
27
13.5
13.5
27
Model
Q1
Q2
Q1+Q2 P
Cournot
2
2
4
Bertrand
3
3
Stackelberg
3
Shared
Monopoly
1.5
1+ 2
Duopoly

Exercise:
◦ Firm A and B face a market demand
 P = 24 – Q.
◦ They both have 0 fixed cost and MCA=6 and MCB=0.
 If they behave as Cournot duopolist, derive the best response
function for the 2 firms. Compute equilibrium market price,
quantities and profits for firm A and B.
 Suppose now they behave as Bertrand duopolist, compute the
market price, outputs and profit for each firms.
 Still under Bertrand, if Firm B could bribe firm A to shut down
his production, what is the max. firm B would be willing to
pay? What is the min amount firm A would accept.
Hotelling’s Model

Setting:
◦ Heterogeneous products market with 2 firms. In this case, it is the
distance to the store.
◦ Firm sets prices P1, P2 respectively and have unlimited capacity.
◦ Linear cost functions: Ci(qi)=ciqi where i = 1, 2.
 C1 = C2
◦ Consumer has a cost of travelling equal to a.
  ax+p1=cost to the xth consumer from buying from firm 1.
  a(1-x) +p2 = cost to the xth consumer from buying from firm 2.
 In equilibrium, the xth consumer must be indifferent between buying from
either firm.
Hotelling’s Model

Firm 1’s Problem:
◦ Max 1 = (P1 – c)*x
 Where x is the demand for firm 1 and (1-x) is the
demand for firm 2.
 In equilibrium the xth consumer must be
indifferent between buying from firm 1 or firm 2.
  ax+P1 =a(1-x)+P2 => x*=
a P 2 P1
2a
 Substitute the value of x* into firm 1’s objective
function:
MAX 1
a P 2 P1
P1 C1
2a
Hotelling’s Model
 Firm 1’s best response function (reaction function):
 P1 = ½(p2 + c2 + a)

Firm 2’s Problem:
◦ Max 2 = (P2 – c2)
1
a P 2 P1
2a
 Firm 2’s best response function (reaction function):
 P2 = ½(p1 + c2 + a)
◦ Equilibrium prices when c1 = c2 = c:
 P1 = P2 = P = c + a
 Higher degree of production differentiation
increases prices.