Lecture 6 Oligopoly 1

2 Introduction A monopoly does not have to worry about how rivals will react to its action simply because there are no rivals. A competitive firm potentiall faces many rivals, but the firm and its rivals are price takers  also no need to worry about rivals’ actions. An oligopolist operating in a market with few competitors needs to anticipate rivals’ actions/ strategies (e.g. prices, outputs, advertising, etc), as these actions are going to affect its profit. The oligopolist needs to choose an appropriate response to those actions  similarly, rivals also need to anticipate the firm’s response and act accordingly  interactive setting. Game theory is an appropriate tool to analyze strategic actions in such an interactive setting  important assumption: firms (or firms’ managers) are rational decision makers.

3 Introduction … A ‘game’ consists of:   A set of players (e.g. 2 firms (duopoly))   A set of feasible strategies (e.g. prices, quantities, etc) for all players   A set of payoffs (e.g. profits) for each player from all combinations of strategies chosen by players. Equilibrium concept  first formalized by John Nash  no player (firm) wants to unilaterally change its chosen strategy given that no other player (firm) change its strategy. Equilibrium may not be ‘nice’  players (firms) can do better if they can cooperate, but cooperation may be difficult to enforced (not credible) or illegal. Finding an equilibrium:  one way is by elimination of all (strictly) dominated strategies, i.e. strategies that will never be chosen by players  the elimination process should lead us to the dominant strategy.

4 Oligopoly Models There are three dominant oligopoly models   Cournot   Bertrand   Stackelberg They are distinguished by   the decision variable that firms choose   the timing of the underlying game We will start first with Cournot Model.

5 The Cournot Model Consider the case of duopoly (2 competing firms) and there are no entry.. Firms produce homogenous (identical) product with the market demand for the product: Marginal cost for each firm is constant at c per unit of output. Assume that A>c. To get the demand curve for one of the firms we treat the output of the other firm as constant. So for firm 2, demand is It can be depicted graphically as follows.

6 The Cournot Model P = (A - Bq 1 ) - Bq 2 $ Quantity A - Bq 1 If the output of firm 1 is increased the demand curve for firm 2 moves to the left A - Bq’ 1 The profit-maximizing choice of output by firm 2 depends upon the output of firm 1 Demand Marginal revenue for firm 2 is MR 2 = = (A - Bq 1 ) - 2Bq 2 MR 2 MR 2 = MC A - Bq 1 - 2Bq 2 = c  q* 2 = (A - c)/2B - q 1 /2 c MC q* 2

7 The Cournot Model We have  this is the best response function for firm 2 (reaction function for firm 2). It gives firm 2’s profit-maximizing choice of output for any choice of output by firm 1. In a similar fashion, we can also derive the reaction function for firm 1. Cournot-Nash equilibrium requires that both firms be on their reaction functions.

8 q2q2 q1q1 (A-c)/B (A-c)/2B Firm 1’s reaction function (A-c)/2B (A-c)/B Firm 2’s reaction function The Cournot-Nash equilibrium is at the intersection of the reaction functions C qC1qC1 qC2qC2 The Cournot Model

9 q2q2 q1q1 (A-c)/B (A-c)/2B Firm 1’s reaction function (A-c)/2B (A-c)/B Firm 2’s reaction function C q* 1 = (A - c)/2B - q* 2 /2 q* 2 = (A - c)/2B - q* 1 /2  q* 2 = (A - c)/2B - (A - c)/4B + q* 2 /4  3q* 2 /4 = (A - c)/4B  q* 2 = (A - c)/3B (A-c)/3B  q* 1 = (A - c)/3B The Cournot Model

10 The Cournot Model In equilibrium each firm produces Total output is therefore Demand is P=A-BQ, thus price equals to Profits of firms 1 and 2 are respectively A monopoly will produce

11 The Cournot Model Competition between firms leads them to overproduce. The total output produced is higher than in the monopoly case. The duopoly price is lower than the monopoly price. The overproduction is essentially due to the inability of firms to credibly commit to cooperate  they are in a prisoner’s dilemma kind of situation

12 The Cournot Model (Many Firms) Suppose there are N identical firms producing identical products. Total output: Demand is: Consider firm 1, its demand can be expressed as: Use a simplifying notation: So demand for firm 1 is:

13 P = (A - BQ -1 ) - Bq 1 $ Quantity A - BQ -1 If the output of the other firms is increased the demand curve for firm 1 moves to the left A - BQ’ -1 The profit-maximizing choice of output by firm 1 depends upon the output of the other firms Demand Marginal revenue for firm 1 is MR 1 = (A - BQ -1 ) - 2Bq 1 MR 1 MR 1 = MC A - BQ Bq 1 = c  q* 1 = (A - c)/2B - Q -1 /2 cMC q* 1 The Cournot Model (Many Firms)

14 q* 1 = (A - c)/2B - Q -1 /2  Q* -1 = (N - 1)q* 1  q* 1 = (A - c)/2B - (N - 1)q* 1 /2  (1 + (N - 1)/2)q* 1 = (A - c)/2B  q* 1 (N + 1)/2 = (A - c)/2B  q* 1 = (A - c)/(N + 1)B  Q* = N(A - c)/(N + 1)B  P* = A - BQ* = (A + Nc)/(N + 1) Profit of firm 1 is Π* 1 = (P* - c)q* 1 = (A - c) 2 /(N + 1) 2 B The Cournot Model (Many Firms)

15 Cournot-Nash Equilibrium: Different Costs Marginal costs of firm 1 are c 1 and of firm 2 are c 2. Demand is P = A - BQ = A - B(q 1 + q 2 ) We have marginal revenue for firm 1 as before. MR 1 = (A - Bq 2 ) - 2Bq 1 Equate to marginal cost: (A - Bq 2 ) - 2Bq 1 = c 1  q* 1 = (A - c 1 )/2B - q 2 /2  q* 2 = (A - c 2 )/2B - q 1 /2

16 Cournot-Nash Equilibrium: Different Costs q2q2 q1q1 (A-c 1 )/B (A-c 1 )/2B R1R1 (A-c 2 )/2B (A-c 2 )/B R2R2 C q* 1 = (A - c 1 )/2B - q* 2 /2 q* 2 = (A - c 2 )/2B - q* 1 /2  q* 2 = (A - c 2 )/2B - (A - c 1 )/4B + q* 2 /4  3q* 2 /4 = (A - 2c 2 + c 1 )/4B  q* 2 = (A - 2c 2 + c 1 )/3B  q* 1 = (A - 2c 1 + c 2 )/3B

17 Cournot-Nash Equilibrium: Different Costs In equilibrium the firms produce: The demand is P=A-BQ, thus the eq. price is: Profits are: Equilibrium output is less than the competitive level.

18 Concentration and Profitability Consider the case of N firms with different marginal costs. We can use the N-firms analysis with modification. Recall that the demand for firm 1 is So then the demand for firm 1 is :, so the MR can be derived as Equate MR=MC  and denote the equilibrium solution by *.

19 Concentration and Profitability P* - c i = Bq* i Divide by P* and multiply the right-hand side by Q*/Q* P* - c i P* = BQ* P* q* i Q* But BQ*/P* = 1/  and q* i /Q* = s i so: P* - c i P* = sisi  The price-cost margin for each firm is determined by its market share and demand elasticity