EC 3322 Semester I – 2008/2009

1. Topic 6:Static Games Bertrand (Price) Competition EC 3322 Semester I – 2008/2009 EC 3322 (Industrial Organization I)

2. Introduction • In a wide variety of markets firms compete in prices • Internet access • Restaurants • Consultants • Financial services • In monopoly, setting price or quantity first makes no difference • But, in oligopoly the strategic variable matters a great deal  price competition is much more aggressive than quantity competition EC 3322 (Industrial Organization I)

3. Bertrand Competition • In the Cournot model price is set by some market clearing mechanism • An alternative approach is to assume that firms compete in prices  it leads to dramatically different results • Take a simple example • two firms producing (or selling) an identical product (mineral water or fruits) • firms choose the prices at which they sell their products • each firm has constant marginal cost of c • inverse demand is P = A – B.Q • direct demand is Q = a – bPwith a = A/B and b= 1/B EC 3322 (Industrial Organization I)

4. Bertrand Competition • We need the derived demand for each firm  demand conditional upon the price charged by the other firm • Take firm 2. Assume that firm 1 has set a price of p1 • if firm 2 sets a price greater than p1 she will sell nothing • if firm 2 sets a price less than p1 she gets the whole market • if firm 2 sets a price of exactly p1 consumers are indifferent between the two firms: the market is shared, presumably 50:50 • So we have the derived demand for firm 2 • q2 = 0 if p2 > p1 • q2 = (a – bp2)/2 if p2 = p1 • q2 = a – bp2 if p2 < p1 EC 3322 (Industrial Organization I)

5. Bertrand Competition p2 There is a jump at p2 = p1 • This can be illustrated as follows: • Demand is discontinuous p1 • The discontinuity in demand carries over to profit a - bp1 a q2 (a - bp1)/2 EC 3322 (Industrial Organization I)

6. Bertrand Competition Firm 2’s profit is: Π2(p1,, p2) = 0 if p2 > p1 Π2(p1,, p2) = (p2 - c)(a - bp2) if p2 < p1 For whatever reason! Π2(p1,, p2) = (p2 - c)(a - bp2)/2 if p2 = p1 Clearly this depends on p1. Suppose first that firm 1 sets a “very high” price: greater than the monopoly price of pM = (a +bc)/2b EC 3322 (Industrial Organization I)

7. Bertrand Competition What price should firm 2 set? So firm 2 should just undercut p1 a bit and get almost all the monopoly profit With p1 > (a + bc)/2b, Firm 2’s profit looks like this: At p2 = p1 firm 2 gets half of the monopoly profit What if firm 1 prices at (a + c)/2b? The monopoly price Firm 2’s Profit Firm 2 will only earn a positive profit by cutting its price to (a + bc)/2b or less p2 < p1 Π2(p1,, p2) = (p2 - c)(a - bp2) p2 = p1 p2 > p1 p1 c (a+bc)/2b Firm 2’s Price EC 3322 (Industrial Organization I)

8. Bertrand Competition Now suppose that firm 1 sets a price less than (a + bc)/2b Firm 2’s profit looks like this: What price should firm 2 set now? As long as p1 > c, Firm 2 should aim just to undercut firm 1 Firm 2’s Profit Of course, firm 1 will then undercut firm 2 and so on Then firm 2 should also price at c. Cutting price below costgains the whole market but loses money on every customer p2 < p1 What if firm 1 prices at c? p2 = p1 p2 > p1 p1 (a+bc)/2b c Firm 2’s Price EC 3322 (Industrial Organization I)

9. Bertrand Competition • We now have Firm 2’s best response to any price set by firm 1: • p*2 = (a + bc)/2b if p1 > (a + bc)/2b • p*2 = p1 - “something small ()” if c < p1< (a + bc)/2b • p*2 = c if p1< c • We have a symmetric best response for firm 1 • p*1 = (a + bc)/2b if p2 > (a + bc)/2b • p*1 = p2 - “something small ()” if c < p2< (a + bc)/2b • p*1 = c if p2< c EC 3322 (Industrial Organization I)

10. Bertrand Competition The best response function for firm 1 The best response function for firm 2 These best response functions look like this p2 R1 The Bertrand equilibrium has both firms charging marginal cost R2 (a + bc)/2b The equilibrium is with both firms pricing at c c p1 c (a + bc)/2b EC 3322 (Industrial Organization I)

11. Bertrand Equilibrium • The Bertrand model shows that competition in prices gives very different result from competition in quantities. • Since many firms seem to set prices (and not quantities) this is a challenge to the Cournot approach • But the result is “not nice”  there are only 2 firms and yet firms charge p=MC  Bertrand Paradox. • Two extensions can be considered • So far, firms set prices  quantities adjust  what if we have capacity constraints? • What happen if products are differentiated? EC 3322 (Industrial Organization I)

12. Bertrand Equilibrium Diaper Wars The Kimberly-Clark Corporation, a leading diaper manufacturer, attempted to improve profits during the economic downturn of the Summer of 2002. The company decreased the number of diapers in each pack in order to increase the price per diaper by 5% for its Huggies brand. Kimberly-Clark’s chief executive officer, Thomas J. Falk, expected Procter & Gamble (P&G), the second largest producer of diapers, to respond with a similar price increase for its Pampers brand. P&G had followed Kimberly-Clark’s price moves in the past. Kimberly-Clark and P&G had cooperated previously to increase the profits of both firms. Cooperation is often the profit maximizing response in repeated games. However, P&G did not respond with a cooperative price increase in this instance. P&G increased promotional expenses to encourage retailers to cut prices on larger Pampers packs or to put up special displays. P&G also marked its Pampers packs with “Compare,” to highlight the price difference between brands. P&G deviated from its past cooperative strategy with Kimberly-Clark in an attempt to increase its market share. Given the poor conditions of the market, P&G executives believed that this one-time, non-cooperative response would maximize profits, and that any future punishment from Kimberly-Clark would not offset the gains from improving its market position. The conditions of the market determined the level of cooperation P&G employed. Source: Ellison, Sarah, “In Lean Times, Big Companies Make a Grab for Market Share,” Wall Street Journal, September 5, 2003. EC 3322 (Industrial Organization I)

13. Capacity Constraints • For the p = c equilibrium to arise, both firms need enough capacity to fill all demand at p = c • But when p = c they each get only half the market • So, at the p = c equilibrium, there is huge excess capacity • So capacity constraints may affect the equilibrium • Consider an example • daily demand for product A Q = 6,000 – 60P • Suppose there are two firms: Firm 1 with daily capacity 1,000 and Firm 2 with daily capacity 1,400, both are fixed • marginal cost for both is $10 EC 3322 (Industrial Organization I)

14. Capacity Constraints • Is a price P = c = $10 an equilibrium? • total demand is then 5,400, well in excess of capacity • Suppose both firms set P = $10: both then have demand of 2,700 • Consider Firm 1: • Normally, raising price loses some demand • but where can they go? Firm 2 is already above capacity • so some buyers will not switch from Firm 1 at the higher price • but then Firm 1 can price above MC and make profit on the buyers who remain • so P = $10 cannot be an equilibrium EC 3322 (Industrial Organization I)

15. Capacity Constraints • Assume that at any price where demand is greater than capacity there is efficient rationing. • Buyers with the highest willingness to pay are served first. • Then we can derive residual demand. • Assume P = $60 • total demand = 2,400 = total capacity • so Firm 1 gets 1,000 units • residual demand to Firm 2 with efficient rationing is Q = 5000 – 60Por P = 83.33 – Q/60 in inverse form. • marginal revenue is then MR = 83.33 – Q/30 EC 3322 (Industrial Organization I)

16. Capacity Constraints (Efficient-Rationing Rule) Efficient rationing rule: consumers with highest willingness to pay are served first. Price 100 1000 units residual demand 83.33 60 1400 units (firm 2) 1000 units (firm 1) Quantity 2400 units EC 3322 (Industrial Organization I)

17. Capacity Constraints • Residual demand and MR: Price • Suppose that Firm 2 sets P = $60. Does it want to change? $83.33 Demand • since MR > MC Firm 2 does not want to raise price and lose buyers $60 MR $36.66 • since QR = 1,400 Firm 2 is at capacity and does not want to reduce price $10 MC 1,400 Quantity • Same logic applies to Firm 1 so P = $60 is a Nash equilibrium for this game. EC 3322 (Industrial Organization I)

18. Capacity Constraints • Logic is quite general • firms are unlikely to choose sufficient capacity to serve the whole market when price equals marginal cost • since they get only a fraction in equilibrium • so capacity of each firm is less than needed to serve the whole market • but then there is no incentive to cut price to marginal cost • So we avoid the Bertrand Paradox when firms are capacity constrained EC 3322 (Industrial Organization I)

19. Product Differentiation • Original analysis also assumes that firms offer homogeneous products • Creates incentives for firms to differentiate their products • to generate consumer loyalty • do not lose all demand when they price above their rivals • keep the “most loyal” • We will discuss this when we cover the product differentiation topic. EC 3322 (Industrial Organization I)

20. Topic 7:Sequential Move Games Stackelberg Competition EC 3322 Semester I – 2008/2009 EC 3322 (Industrial Organization I)

21. Introduction • In a wide variety of markets firms compete sequentially • One firm makes a move. • new product • advertising • Second firms sees this move and responds. • These are dynamic games. • May create a first-mover advantage • or may give a second-mover advantage • May also allow early mover to preempt the market • Can generate very different equilibria from simultaneous move games EC 3322 (Industrial Organization I)

22. Pilot Y X -1,-1 bomb BB’ -1 , -1 -1 , -1 terrorist fly to X pilot not bomb 1,1 BN’ 0 , 2 -1 , -1 Terrorist -1,-1 bomb fly to the original destination Y -1 , -1 1 , 1 NB’ 2, 0 not bomb 0 , 2 1 , 1 Pilot-Terrorist Game Sequential Move Game NN’ Nash-Equilibria: (X,NB’); (Y,BN’); (Y,NN’) EC 3322 (Industrial Organization I)

23. Sequential Move Game • Thus, there are multiple pure strategy NE. This greatly reduce our ability to generate predictions from the game. • We need another solution concept that can narrow down the set of NE outcomes into a smaller set of outcomes. • We need to eliminate NE that involves non-credible threat (unreasonable). • From the example: terrorist’ strategy that involves bomb threatis not credible, because once his information set is reached he will never carried out the threat. • Thus, we need to be able to eliminate(X, NB’) and (Y, BN’). EC 3322 (Industrial Organization I)

24. Sequential Move Game • Refinement:  Subgame Perfect Nash Equilibrium: A Strategy profile is said to be a subgame perfect Nash equilibrium if it specifies a Nash Equilibrium in every subgame of the original game. • For the entire game, the NE are (X,NB’); (Y,BN’); (Y,NN’) • For the two subgames: • Hence, (X,NB’); (Y,BN’); are not SPE. The terrorist will always choose Not Bomb (NN) -1,-1 bomb -1,-1 bomb terrorist terrorist not bomb 1,1 not bomb 2,0 EC 3322 (Industrial Organization I)

25. Subgame Perfect Equilibrium • To get the SPE  “backward induction” method (‘look ahead reason back’) • Analyze a game from back to front (from information sets at the end of the tree to information sets at the beginning). At each information set, one eliminates strategies that are dominated, given the terminal nodes that can be reached. 3, 8 1 2 7, 9 1 1, 2 1 2, 1 10, 4 1 2 0, 5 1 4, 0 8, 3 EC 3322 (Industrial Organization I)

26. Stackelberg Competition • Let’s interpret first in terms of Cournot • Firms choose outputs sequentially • Leader sets output first, and the choice is observed by the follower. • Follower then sets output upon observing the leader’s choice. • The firm moving first has a leadership advantage • It can anticipate the follower’s actions • can therefore manipulate the follower • For this to work the leader must be able to commit to its choice of output • Strategic commitment has value EC 3322 (Industrial Organization I)

27. Stackelberg Competition t=1 t=2 Time Period firm 1 choosing its optimal quantity (q*1) to maximize its profit. firm 2 observes the optimal quantity choice of of firm 1 (q*1)and sets its optimal quantity (q*2(q1)) EC 3322 (Industrial Organization I)

28. Stackelberg Competition • Assume that there are two firms with identical products • As in our earlier Cournot example, let demand be: • P = A – B.Q = A – B(q1 + q2) • Marginal cost for for each firm is c • Firm 1 is the market leader and chooses q1 • In doing so it can anticipate firm 2’s actions. So consider firm 2. Demand for firm 2 is: • P = (A – Bq1) – Bq2 • Marginal revenue therefore is: • MR2 = (A - Bq1) – 2Bq2 EC 3322 (Industrial Organization I)

29. Stackelberg Competition Equate marginal revenue with marginal cost This is firm 2’s best response function MR2 = (A - Bq1) – 2Bq2 q2 Firm 1 knows that this is how firm 2 will react to firm 1’s output choice MC = c But firm 1 knows what q2 is going to be  q*2 = (A - c)/2B - q1/2 So firm 1 can anticipate firm 2’s reaction Demand for firm 1 is: From earlier example we know that this is the monopoly output. This is an important result. The Stackelberg leader chooses the same output as a monopolist would. But firm 2 is not excluded from the market P = (A - Bq2) – Bq1 (A – c)/2B P = (A - Bq*2) – Bq1 P = (A - (A-c)/2) – Bq1/2 Equate marginal revenue with marginal cost S (A – c)/4B P = (A + c)/2 – Bq1/2 R2 Solve this equation for output q1 Marginal revenue for firm 1 is: q1 MR1 = (A + c)/2 - Bq1 (A – c)/B (A – c)/2B (A + c)/2 – Bq1 = c  q*1 = (A – c)/2B  q*2 = (A – c)4B EC 3322 (Industrial Organization I)

30. Stackelberg Competition Leadership benefits the leader firm 1 but harms the follower firm 2 Aggregate output is 3(A-c)/4B Leadership benefits consumers but reduces aggregate profits q2 Firm 1’s best response function is “like” firm 2’s So the equilibrium price is (A+3c)/4 (A-c)/B R1 Compare this with the Cournot equilibrium Firm 1’s profit is (A-c)2/8B Firm 2’s profit is (A-c)2/16B We know that the Cournot equilibrium is: (A-c)/2B C (A-c)/3B qC1 = qC2 = (A-c)/3B S (A-c)/4B The Cournot price is (A+c)/3 R2 Profit to each firm is (A-c)2/9B q1 (A-c)/3B (A-c)/ B (A-c)/2B EC 3322 (Industrial Organization I)

31. A Comparison of Oligopoly Equilibria EC 3322 (Industrial Organization I)

32. A Comparison of Oligopoly Equilibria Profit Possibility Frontier Cournot 57.6 Stackelberg 32.4 Monopoly Bertrand & Competitive Solution 57.6 64.8 129.6 EC 3322 (Industrial Organization I)

33. Stackelberg and Commitment • It is crucial that the leader can commit to its output choice • without such commitment firm 2 should ignore any stated intent by firm 1 to produce (A – c)/2B units • the only equilibrium would be the Cournot equilibrium • So how to commit? • prior reputation • investment in additional capacity • place the stated output on the market • Given such a commitment, the timing of decisions matters • But is moving first always better than following? • Consider price competition EC 3322 (Industrial Organization I)

34. Stackelberg and Price Competition • With price competition matters are different • first-mover does not have an advantage • suppose products are identical • suppose first-mover commits to a price greater than marginal cost • the second-mover will undercut this price and take the market • so the only equilibrium is P = MC • identical to simultaneous game EC 3322 (Industrial Organization I)

35. Application: Advertising & Competition • The game (firm 1 and 2) • The market demand faced by the two firms • Firms produce at zero costs, but firm 1 incurs advertising costs of t=1 t=2 Time Period firm 1 chooses advertising level (a) in order to enhance demand firm 1 and 2 compete in a Cournot fashion (choosing quantity Level) Firm 2 observes the choice of a of firm 1 EC 3322 (Industrial Organization I)

36. Application: Advertising & Competition • (Start from t=2): Solve the Cournot best response function of the two firms at the end of the game (t=2), taking the advertising level determined in t=1 as given. • Derive the f.o.c. w.r.t. q1 and solve for q1, we get the best response function. • Similarly derive the best response fu. For firm 2. EC 3322 (Industrial Organization I)

37. Application: Advertising & Competition • The Cournot Nash equilibrium can be obtained. • The equilibrium price is then, • Hence, firm 1’s profit function as a function of a is, • (Now at t=1): Firm 1 chooses its advertisement level (a) to maximize its profit. • The SPE strategy profile is EC 3322 (Industrial Organization I)