Chapter 13a - OligopolyGoals:

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 Q1 Q2 Q1+Q

2

P 1 2 1+ 2

Cournot

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 - and firm 1 captures the whole market.

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 1 2 1+ 2

Cournot 2 2 4 24 12 12 24

Bertrand

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.

Model Q1 Q2 Q1+Q2 P 1 2 1+ 2

Cournot 2 2 4 24 12 12 24

Bertrand 3 3 6 18 0 0 0

Stackelberg

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.

Model Q1 Q2 Q1+Q2 P 1 2 1+ 2

Cournot 2 2 4 24 12 12 24

Bertrand 3 3 6 18 0 0 0

Stackelberg 3 1.5 4.5 22.5 13.5 6.75 20.25

Shared

Monopoly

1.5 1.5 3 27 13.5 13.5 27

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*=

Substitute the value of x* into firm 1’s objective

function:

2 1

2

a P P

a

2 11 1 1

2

a P PMAX P C

a

Hotelling’s Model

Firm 1’s best response function (reaction function):

P1 = ½(p2 + c2 + a)

Firm 2’s Problem:

◦ Max 2 = (P2 – c2)

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.

2 11

2

a P P

a