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.
