1. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 1 EC 3322 Semester I – 2008/2009 Topic 5: Static Games Cournot Competition

2. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 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. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 3 Introduction … Consider the following story (taken from Dixit and Skeath (1999), Games of Strategy)   …”There were two friends taking Chemistry at Duke. Both had done pretty well on all of the quizzes, the labs, and the midterm, so that going to the final they had a solid A. They were so confident that the weekend before the final exam they decided to go to a party at the University of Virginia. The party was so good that they overslept all day Sunday, and got back too late to study for the Chemistry final that was scheduled for Monday morning. Rather than take the final unprepared, they went to the professor with a sob story. They said they had gone to the University of Virginia and had planned to come back in good time to study for the final but had had a flat tire on the way back. Because they did not have a spare, they had spent most of the night looking for help. Now they were too tired, so could they please have a make-up final the next day?   The two studied all of Monday evening and came well prepared on Tuesday morning. The professor placed them in separate rooms and handed the test to each. Each of them wrote a good answer, and greatly relieved, but …   when they turned to the last page. It had just one question, worth 90 points. It was: “Which tire?”….

4. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 4 Introduction … Why are Professors So Mean (taken from Dixit and Skeath (1999), Games of Strategy)   Many professors have an inflexible rule not to accept late submission of problem sets of term papers. Students think the professors must be really hard heartened to behave this way.   However, the true strategic reason is often exactly the opposite. Most professors are kindhearted, and would like to give their students every reasonable break and accept any reasonable excuse. The trouble lies in judging what is reasonable. It is hard to distinguish between similar excuses and almost impossible to verify the truth. The professor knows that on each occasion he will end up by giving the student the benefit of the doubt. But the professor also knows that this is a slippery slope, As the students come to know that the professor is a soft touch, they will procrastinate more and produce even flimsier excuses. Deadline will cease to mean anything.

5. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 5

6. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 6 Introduction … Non-cooperative game theory vs. cooperative game theory. The former refers to a setting in which each individual firm (a player) behave non- cooperatively towards others (rivals players). The latter refers to a setting in which a group of firms cooperate by forming a coalition. We focus on non-cooperative game theory. Different in timing of actions: simultaneous vs. sequential move games. Different in the nature of information: complete vs. incomplete information. Oligopoly theory  no single unified theory, unlike theory of monopoly and theory of perfect competition  theoretical predictions depend on the game theoretical tools chosen. Need a concept of equilibrium  to characterize the chosen optimal strategies.

7. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 7 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.

8. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 8 Introduction … Ways of representing a Game:   Extensive Form Representation (Game Tree)   Normal Form Representation Extensive Form 1 H L 2 H L 2 H L In $ millions 1,1 0,2 2,0 ½, ½ 1 H L 2 H L 2 H’ L’ In $ millions 1,1 0,2 2,0 ½, ½ sequential move simultaneous move

9. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 9 Introduction … Normal Form Definition: A strategy is a complete contingent plan (a full specification of a player’s behavior at each of his/ her decision points) for a player in the game. Player 2 normal form - sequential move HL HH’ HL’ LH’ LL’ Player 1 1, 12, 0 1, 1 ½, ½ 0, 2 2, 0 0, 2 ½, ½ extensive form - sequential move 1 H L 2 H L 2 H’ L’ In millions 1,1 0,2 2,0 ½, ½

10. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 10 Introduction … 1 H L 2 H L 2 H L In millions 1,1 0,2 2,0 ½, ½ L H L H Player 2 Player 1 1/2, 1/2 0, 2 2, 0 1, 1 extensive form - simultaneous movenormal form - simultaneous move

11. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 11 Introduction … When players can choose infinite number of actions, instead of only 2 actions  e.g. quantities, advertising expenditures, prices, etc. 1 2 1-a,0 a-a 2, ¼ - a/2 stay in exit 1 2 1-a,0 a-a 2, ¼ - a/2 exit stay in a a sequential movesimultaneous move

12. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 12 Example … Two airline companies, e.g. SIA and Qantas offering a daily flight from Singapore to Sydney. Assume that they already have set a price for the flight, but the departure time is still undecided  the departure time is the strategy choice in this game. 70% of consumers prefer evening departure while 30% prefer morning departure. If both airlines choose the same departure time, they share the market share equally. Payoffs to the airlines are determined by the market share obtained. Both airlines choose the departure time simultaneously  we can represent the payoffs in a matrix firm.

13. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 13 Example … The Pay-Off Matrix Qantas SIA Morning Evening (15, 15) The left-hand number is the pay-off to SIA (30, 70) (70, 30)(35, 35) What is the equilibrium for this game? The right-hand number is the pay-off to Qantas

14. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 14 Example … The Pay-Off Matrix Qantas SIA Morning Evening (15, 15) If Qantas chooses a morning departure, SIA will choose evening (30, 70) (70, 30)(35, 35) If Qantas chooses an evening departure, SIA will also choose evening The morning departure is a dominated strategy for SIA Both airlines choose an evening departure (35, 35) The morning departure is also a dominated strategy for Qantas

15. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 15 Example … Suppose now that SIA has a frequent flyer program. Thus, when both airlines choose the same departure times, SIA will obtain 60% of market share. This will change the payoff matrix.

16. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 16 Example … The Pay-Off Matrix Qantas SIA Morning Evening (18, 12)(30, 70) (70, 30)(42, 28) However, a morning departure is still a dominated strategy for SIA If SIA chooses a morning departure, Qantas will choose evening But if SIA chooses an evening departure, Qantas will choose morning Qantas has no dominated strategy Qantas knows this and so chooses a morning departure (70, 30)

17. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 17 Example … What if there are no dominated strategies? We need to use the Nash Equilibrium concept. To show this  consider a modified version of our airlines game  instead of choosing departure times, firms choose prices  For simplicity, consider only two possible price levels. Settings:   There are 60 consumers with a reservation price of $500 for the flight, and another 120 consumers with the lower reservation price of $220.   Price discrimination is not possible (perhaps for regulatory reasons or because the airlines don’t know the passenger types).   Costs are $200 per passenger no matter when the plane leaves.   airlines must choose between a price of $500 and a price of $220   If equal prices are charged the passengers are evenly shared. The low price airline gets all passengers.

18. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 18 Example The Pay-Off Matrix Qantas SIA P H = $500 ($9000,$9000)($0, $3600) ($3600, $0)($1800, $1800) P H = $500 P L = $220 If both price high then both get 30 passengers. Profit per passenger is $300 If SIA prices high and Qantas low then Qantas gets all 180 passengers. Profit per passenger is $20 If SIA prices low and Qantas high then SIA gets all 180 passengers. Profit per passenger is $20 If both price low they each get 90 passengers. Profit per passenger is $20

19. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 19 Nash Equilibrium The Pay-Off Matrix Qantas SIA P H = $500 ($9000,$9000)($0, $3600) ($3600, $0)($1800, $1800) P H = $500 P L = $220 (P H, P L ) cannot be a Nash equilibrium. If Qantas prices low then SIA should also price low ($0, $3600) (P L, P H ) cannot be a Nash equilibrium. If Qantas prices high then SIA should also price high ($3600, $0) (P H, P H ) is a Nash equilibrium. If both are pricing high then neither wants to change ($9000, $9000) (P L, P L ) is a Nash equilibrium. If both are pricing low then neither wants to change ($1800, $1800) There are two Nash equilibria to this version of the game There is no simple way to choose between these equilibria Custom and familiarity might lead both to price high “Regret” might cause both to price low

20. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 20 Nash Equilibrium Another very common game  prisoner’s dilemma game  illustrates that the resulting NE outcome may be ‘inefficient’. So the only Nash equilibrium for this game is (C,C), even though (D,D) gives both 1 and 2 better jail terms. The only Nash equilibrium is inefficient. criminal 1 (6,6)(1,10) (10,1) (3,3) Confess Don’t confess ConfessDon’t Confess criminal 2

21. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 21 Nash Equilibrium Firm A (100, 100)(25, 140) (140, 25) (80, 80) H L HL Firm B Consider the following price game between Firm A and Firm B Had the firms been able to cooperate, they would have been able to obtain higher payoffs.

22. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 22 Mathematical Presentation of Nash Eq. Suppose that there are 2 firms, 1 and 2  it can be generalized to n firms. The profit of each firm is denoted by with is the set of all feasible strategies from which i can choose. Thus, are the pair of strategies chosen by players i and j from the set of feasible strategies. Then, a pair of strategies is a Nash equilibrium if, for each firm i: Thus, for a strategy combination to be a Nash eq., the strategy s i * must be firm i’s best response to firm j’s strategy, s j *, and conversely s j * must be firm j’s best response to strategy s i *.

23. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 23 Example:   is firm i’s profit and are firms i and j’s quantities (outputs). If the profit function is continuous, concave and differentiable, we can solve for the optimal strategy s i * by solving the first-order condition for the max. problem:   Similarly firm j will also choose its strategy optimally:   Finally the pair of Nash eq. outputs can be obtained by solving the system of equation (1) and (2) simultaneously. To guarantee that are the maximands we have to check for the second order condition for profit maximization. Mathematical Presentation of Nash Eq.

24. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 24 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.

25. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 25 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.

26. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 26 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 Solve this for output q 2  q* 2 = (A - c)/2B - q 1 /2 c MC q* 2

27. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 27 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.

28. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 28 q2q2 q1q1 The reaction function for firm 1 is q* 1 = (A-c)/2B - q 2 /2 The reaction function for firm 1 is q* 1 = (A-c)/2B - q 2 /2 (A-c)/B (A-c)/2B Firm 1’s reaction function The reaction function for firm 2 is q* 2 = (A-c)/2B - q 1 /2 The reaction function for firm 2 is q* 2 = (A-c)/2B - q 1 /2 (A-c)/2B (A-c)/B If firm 2 produces nothing then firm 1 will produce the monopoly output (A-c)/2B If firm 2 produces (A-c)/B then firm 1 will choose to produce no output Firm 2’s reaction function The Cournot-Nash equilibrium is at the intersection of the reaction functions C qC1qC1 qC2qC2 The Cournot Model

29. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 29 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

30. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 30 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

31. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 31 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. It can be verified that, the duopoly output is still lower than the competitive output  where P=MC. 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  more on this when we discuss collusion.

32. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 32 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: This denotes output of every firm other than firm 1

33. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 33 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 -1 - 2Bq 1 = c Solve this for output q 1  q* 1 = (A - c)/2B - Q -1 /2 cMC q* 1 The Cournot Model (Many Firms)

34. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 34 q* 1 = (A - c)/2B - Q -1 /2 How do we solve this for q* 1 ? The firms are identical. So in equilibrium they will have identical outputs  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) As the number of firms increases output of each firm falls As the number of firms increases aggregate output increases As the number of firms increases price tends to marginal cost Profit of firm 1 is Π* 1 = (P* - c)q* 1 = (A - c) 2 /(N + 1) 2 B As the number of firms increases profit of each firm falls The Cournot Model (Many Firms)

35. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 35 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 Solve this for output q 1  q* 1 = (A - c 1 )/2B - q 2 /2 A symmetric result holds for output of firm 2  q* 2 = (A - c 2 )/2B - q 1 /2

36. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 36 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 What happens to this equilibrium when costs change? If the marginal cost of firm 2 falls its reaction curve shifts to the right The equilibrium output of firm 2 increases and of firm 1 falls

37. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 37 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. Output is produced inefficiently  the low cost firm should produce all the output.

38. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 38 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 *. But Q* -i + q* i = Q* and A - BQ* = P*

39. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 39 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 Extending this we have P* - c P* = H  Average price-cost margin is determined by industry concentration

40. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 40 Final Remarks So far we consider only “pure” strategy equilibria  a player picks the strategy with certainty (prob.=1), e.g. choosing ‘kick the ball to the middle’ in a soccer penalty shootout.. “Mixed” strategies  the player uses a probabilistic mixture of the available strategies, e.g. left, middle, right  thus randomize the strategies  sometimes aims the left, middle or right. Burger King Low Price Heavy Advertising Low Price Heavy Advertising McDonalds (60, 35) (56, 45) (58, 50) (60, 40) No Pure Strategy Eq.

41. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 41 Suppose Burger King believes that McDonald will play strategy L with prob and H with prob.. When BK plays L, its expected payoff is: If BK plays H, its expected payoff is: BK will be indifferent between L and H iff: Thus, when McDonald plays the optimal mixed strategy eq. with the above prob. distribution then BK will be indifferent between playing L or H. Final Remarks

42. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 42 Final Remarks Similarly, when BK plays its optimal mixed strategy eq. then McDonald will be indifferent between playing L or H. Burger King Low Price Heavy Advertising Low Price Heavy Advertising McDonalds (60, 35) (56, 45) (58, 50) (60, 40)

43. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 43 Next … (Bertrand Price Competition)