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EC 3322 Semester I – 2008/2009

Topic 5 : Static Games Cournot Competition. EC 3322 Semester I – 2008/2009. Introduction. A monopoly does not have to worry about how rivals will react to its action simply because there are no rivals .

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EC 3322 Semester I – 2008/2009

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  1. Topic 5:Static Games Cournot Competition EC 3322 Semester I – 2008/2009 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. 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?”…. 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. EC 3322 (Industrial Organization I)

  5. 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. 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. EC 3322 (Industrial Organization I)

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

  9. Player 1 H L In millions HH’ 1 , 1 2 , 0 H 1,1 2 H 1 , 1 ½ , ½ HL’ 1 L 0,2 2 , 0 LH’ 0 , 2 H’ 2,0 L 2 0 , 2 ½ , ½ LL’ L’ ½, ½ extensive form - sequential move normal form - sequential move Introduction … • 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. Normal Form Player 2 EC 3322 (Industrial Organization I)

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

  11. 1-a,0 1-a,0 exit exit 1 1 2 2 stay in stay in a-a2, ¼ - a/2 a-a2, ¼ - a/2 Introduction … • When players can choose infinite number of actions, instead of only 2 actions  e.g. quantities, advertising expenditures, prices, etc. a a sequential move simultaneous move 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. EC 3322 (Industrial Organization I)

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

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

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

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

  19. Nash Equilibrium (PH, PH) is a Nash equilibrium. If both are pricing high then neither wants to change There are two Nash equilibria to this version of the game There is no simple way to choose between these equilibria (PL, PL) is a Nash equilibrium. If both are pricing low then neither wants to change (PH, PL) cannot be a Nash equilibrium. If Qantas prices low then SIA should also price low The Pay-Off Matrix Custom and familiarity might lead both to price high “Regret” might cause both to price low Qantas (PL, PH) cannot be a Nash equilibrium. If Qantas prices high then SIA should also price high PH = $500 PL = $220 ($9000, $9000) ($0, $3600) PH = $500 ($9000,$9000) ($0, $3600) SIA ($3600, $0) ($1800, $1800) PL = $220 ($3600, $0) ($1800, $1800) 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 2 Confess Don’t Confess (6,6) (1,10) Confess criminal 1 (10,1) (3,3) Don’t confess EC 3322 (Industrial Organization I)

  21. Nash Equilibrium • 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. Firm B H L (100, 100) (25, 140) H Firm A (140, 25) (80, 80) L 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 si* must be firm i’s best response to firm j’s strategy, sj* , and conversely sj*must be firm j’s best response to strategy si*. EC 3322 (Industrial Organization I)

  23. Mathematical Presentation of Nash Eq. • 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 si* 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. 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. 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. EC 3322 (Industrial Organization I)

  26. The Cournot Model If the output of firm 1 is increased the demand curve for firm 2 moves to the left $ P = (A - Bq1) - Bq2 The profit-maximizing choice of output by firm 2 depends upon the output of firm 1 A - Bq1 A - Bq’1 Marginal revenue for firm 2 is Solve this for output q2 Demand c MC MR2 = = (A - Bq1) - 2Bq2 MR2 MR2 = MC q*2 Quantity A - Bq1 - 2Bq2 = c  q*2 = (A - c)/2B - q1/2 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. EC 3322 (Industrial Organization I)

  28. The Cournot Model q2 If firm 2 produces (A-c)/B then firm 1 will choose to produce no output The reaction function for firm 1 is q*1 = (A-c)/2B - q2/2 The Cournot-Nash equilibrium is at the intersection of the reaction functions (A-c)/B Firm 1’s reaction function If firm 2 produces nothing then firm 1 will produce the monopoly output (A-c)/2B The reaction function for firm 2 is q*2 = (A-c)/2B - q1/2 (A-c)/2B C qC2 Firm 2’s reaction function q1 (A-c)/2B (A-c)/B qC1 EC 3322 (Industrial Organization I)

  29. The Cournot Model q*1 = (A - c)/2B - q*2/2 q2 q*2 = (A - c)/2B - q*1/2 (A-c)/B  q*2 = (A - c)/2B - (A - c)/4B + q*2/4 Firm 1’s reaction function  3q*2/4 = (A - c)/4B (A-c)/2B  q*2 = (A - c)/3B C (A-c)/3B  q*1 = (A - c)/3B Firm 2’s reaction function q1 (A-c)/2B (A-c)/B (A-c)/3B 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 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. 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 EC 3322 (Industrial Organization I)

  33. The Cournot Model (Many Firms) If the output of the other firms is increased the demand curve for firm 1 moves to the left P = (A - BQ-1) - Bq1 $ The profit-maximizing choice of output by firm 1 depends upon the output of the other firms A - BQ-1 A - BQ’-1 Marginal revenue for firm 1 is Solve this for output q1 Demand c MC MR1 = (A - BQ-1) - 2Bq1 MR1 MR1 = MC q*1 Quantity A - BQ-1 - 2Bq1 = c  q*1 = (A - c)/2B - Q-1/2 EC 3322 (Industrial Organization I)

  34. The Cournot Model (Many Firms) q*1 = (A - c)/2B - Q-1/2 As the number of firms increases output of each firm falls 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 As the number of firms increases aggregate output increases  q*1 = (A - c)/2B - (N - 1)q*1/2 As the number of firms increases price tends to marginal cost As the number of firms increases profit of each firm falls  (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)2B EC 3322 (Industrial Organization I)

  35. Cournot-Nash Equilibrium: Different Costs • Marginal costs of firm 1 are c1 and of firm 2 are c2. • Demand is P = A - BQ = A - B(q1 + q2) • We have marginal revenue for firm 1 as before. • MR1 = (A - Bq2) - 2Bq1 • Equate to marginal cost: (A - Bq2) - 2Bq1 = c1 Solve this for output q1 A symmetric result holds for output of firm 2  q*1 = (A - c1)/2B - q2/2  q*2 = (A - c2)/2B - q1/2 EC 3322 (Industrial Organization I)

  36. Cournot-Nash Equilibrium: Different Costs q*1 = (A - c1)/2B - q*2/2 q2 The equilibrium output of firm 2 increases and of firm 1 falls If the marginal cost of firm 2 falls its reaction curve shifts to the right q*2 = (A - c2)/2B - q*1/2 What happens to this equilibrium when costs change? (A-c1)/B R1  q*2 = (A - c2)/2B - (A - c1)/4B + q*2/4  3q*2/4 = (A - 2c2 + c1)/4B (A-c2)/2B  q*2 = (A - 2c2 + c1)/3B C R2  q*1 = (A - 2c1 + c2)/3B q1 (A-c1)/2B (A-c2)/B 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. 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* EC 3322 (Industrial Organization I)

  39. Concentration and Profitability The price-cost margin for each firm is determined by its market share and demand elasticity P* - ci = Bq*i Divide by P* and multiply the right-hand side by Q*/Q* P* - ci BQ* q*i = Average price-cost margin is determined by industry concentration P* P* Q* But BQ*/P* = 1/ and q*i/Q* = si P* - ci si so: =  P* Extending this we have P* - c H = P*  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 Heavy Advertising Low Price Low Price (56, 45) (60, 35) No Pure Strategy Eq. McDonalds Heavy Advertising (58, 50) (60, 40) EC 3322 (Industrial Organization I)

  41. Final Remarks • 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. 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 Heavy Advertising Low Price Low Price (56, 45) (60, 35) McDonalds Heavy Advertising (58, 50) (60, 40) EC 3322 (Industrial Organization I)

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

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