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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
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