Chapter 13a - Oligopoly

Transcription

1 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)

2 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.

3 Cournot Model of Quantity 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

4 Cournot Model of Quantity 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.

5 Cournot Model of Quantity 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

6 Cournot Model of Quantity

7 Cournot Model of Quantity 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

8 Bertrand Model of Price 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

9 Bertrand Model of Price. 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.

10 Bertrand Model of Price 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.

11 Bertrand Model of Price Competion

12 Cournot Model of Quantity 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 Bertrand

13 Stackelberg Sequential Quantity 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.

14 Cournot Model of Quantity 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.

15 Stackelberg Sequential Quantity 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

16 Stackelberg Sequential Quantity. 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 Cournot Bertrand Stackelberg

17 Stackelberg Sequential Quantity 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.

18 A Comparison across models. Model Q1 Q2 Q1+Q2 P Cournot Bertrand Stackelberg Shared Monopoly

19 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.

20 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.

21 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 P2 P1 2a Substitute the value of x* into firm 1 s objective function: MAX 1 P1 C1 a P2 P1 2a

22 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 a P2 P1 2a Higher degree of production differentiation increases prices. 1