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Topic 6: Static Games Bertrand (Price) Competition EC 3322 Semester I – 2008/2009 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 1 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 2 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 – bP with a = A/B and b= 1/B Yohanes E. Riyanto EC 3322 (Industrial Organization I) 3 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 4 Bertrand Competition This can be illustrated as follows: Demand is discontinuous The discontinuity in demand carries over to profit p2 There is a jump at p2 = p1 p1 a - bp1 (a - bp1)/2 Yohanes E. Riyanto EC 3322 (Industrial Organization I) a q2 5 Bertrand Competition Firm 2’s profit is: Π2(p1,, p2) = 0 if p2 > p1 Π2(p1,, p2) = (p2 - c)(a - bp2) if p2 < p1 Π2(p1,, p2) = (p2 - c)(a - bp2)/2 if p2 = pFor 1 whatever Clearly this depends on p1. reason! Suppose first that firm 1 sets a “very high” price: greater than the monopoly price of pM = (a +bc)/2b Ac 2B A c aB c a bc p m A BQ m 2 2 2b MR MC A 2 BQ c Q m Yohanes E. Riyanto EC 3322 (Industrial Organization I) 6 Bertrand Competition So firm 2 should just What price should firm 2 set? With p1 > (aundercut + bc)/2b, Firm 2’s profit looks like this: p a bit and At p2 = p1 all the Firmget 2’salmost Profit firm 2 ifgets half What firm 1 of the monopoly profit profit pricesmonopoly at (a + c)/2b? 1 The monopoly price p2 < p1 Π2(p1,, p2) = (p2 - c)(a - bp2) Firm 2 will only earn a positive profit by cutting its price to (a + bc)/2b or less p2 = p 1 p 2 > p1 c Yohanes E. Riyanto (a+bc)/2b EC 3322 (Industrial Organization I) p1 Firm 2’s Price 7 Bertrand Competition Now suppose that firm 1 sets a price less than (a + bc)/2b Firm 2’s profit looks like this: What price As long as p1 > c, Firm 2’s Profit Of course, firm 1 Firm 2 should should firmaim 2 just will then undercut to undercut set now?firm 1 firm 2 and so on p2 < p1 Then firm 2 should also price at c. Cutting price below cost gains the whole market but loses What if firm money on1every customer prices at c? p2 = p 1 p 2 > p1 c Yohanes E. Riyanto p1 (a+bc)/2b EC 3322 (Industrial Organization I) Firm 2’s Price 8 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 9 Bertrand Competition The best response function for The best response These best response functions look like this firm 1 function for p2 firm 2 R1 R2 (a + bc)/2b The Bertrand The equilibrium equilibrium has isboth with both firms charging firms pricing at marginal cost c c p1 c Yohanes E. Riyanto (a + bc)/2b EC 3322 (Industrial Organization I) 10 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? Yohanes E. Riyanto EC 3322 (Industrial Organization I) 11 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. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 12 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 13 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 14 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 – 60P or P = 83.33 – Q/60 in inverse form. marginal revenue is then MR = 83.33 – Q/30 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 15 Capacity Constraints (Efficient-Rationing Rule) Efficient rationing rule: consumers with highest willingness to pay are served first. Price 100 83.33 residual demand 1000 units 60 1400 units (firm 2) 1000 units (firm 1) 2400 units Yohanes E. Riyanto EC 3322 (Industrial Organization I) Quantity 16 Capacity Constraints Residual demand and MR: Price Suppose that Firm 2 sets P = $60. Does it want to change? since MR > MC Firm 2 does not want to raise price and lose buyers since QR = 1,400 Firm 2 is at capacity and does not want to reduce price $83.33 Demand $60 MR $36.66 $10 MC 1,400 Quantity Same logic applies to Firm 1 so P = $60 is a Nash equilibrium for this game. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 17 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 18 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. Yohanes E. Riyanto EC 3322 (Industrial Organization I) 19 Topic 7: Sequential Move Games Stackelberg Competition EC 3322 Semester I – 2008/2009 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 20 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 21 Sequential Move Game X terrorist fly to X pilot fly to the original destinati on Y bomb -1,-1 not bomb 1,1 bomb -1,-1 not bomb Pilot-Terrorist Game 2, 0 Pilot Y BB’ -1 , -1 -1 , -1 BN’ -1 , -1 0,2 NB’ 1,1 -1 , -1 NN’ 1,1 0,2 Terrorist Nash-Equilibria: (X,NB’); (Y,BN’); (Y,NN’) Yohanes E. Riyanto EC 3322 (Industrial Organization I) 22 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 threat is 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’). Yohanes E. Riyanto EC 3322 (Industrial Organization I) 23 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: terrorist bomb not bomb -1,-1 terrorist 1,1 bomb not bomb -1,-1 2,0 Hence, (X,NB’); (Y,BN’); are not SPE. The terrorist will always choose Not Bomb (NN) Yohanes E. Riyanto EC 3322 (Industrial Organization I) 24 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. 1 3, 8 1 7, 9 1, 2 1 2, 1 10, 4 2 1 2 0, 5 1 4, 0 8, 3 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 25 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 26 Stackelberg Competition t=1 firm 1 choosing its optimal quantity (q*1) to maximize its profit. Yohanes E. Riyanto t=2 Time Period 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) 27 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 28 Stackelberg Competition Equate This is marginal firm 2’s revenue with marginal cost MR2 = (A - Bq1) – 2Bq2 best response q2 function MC = c But firm 1 knows Firm 1 knows that this is how firm 2 q*2 = (A - c)/2B what - q1/2q2 is going to be will reactSotofirm firm11’s can Demand for firm 1 is: output choicefirm 2’s anticipate P = (A - Bq2) – Bq1 (A – c)/2B From earlier example we know reaction P = (A - Bq*2) – Bqthat this is the monopoly output. This is an 1 important result. The Stackelberg leader P = (A - (A-c)/2) – Bq 1/2 S Equate marginal revenue (A – c)/4B chooses the output as a monopolist would. P = (A + c)/2 – Bq1/2 with same marginal cost R2 But 2 isequation not excluded from the market Marginal revenue for firm 1firm is: this Solve q1 MR1 = (A + c)/2 - Bq1 for output q1 (A – c)/B (A – c)/2B (A + c)/2 – Bq1 = c q*1 = (A – c)/2B Yohanes E. Riyanto q*2 = (A – c)4B EC 3322 (Industrial Organization I) 29 Stackelberg Competition Aggregate output is 3(A-c)/4B So the equilibrium price is (A+3c)/4 Leadership benefits Firm 1’s best the leader firm 1response but harms Leadership the follower benefits function is “like” consumers firm but Compare this with firm2 2’s reduces aggregate the Cournot profits equilibrium q2 (A-c)/B Firm 1’s profit is (A-c)2/8B R1 Firm 2’s profit is (A-c)2/16B We know that the Cournot equilibrium is: qC1 = qC2 = (A-c)/3B (A-c)/2B C (A-c)/3B The Cournot price is (A+c)/3 S (A-c)/4B R2 Profit to each firm is (A-c)2/9B (A-c)/3B (A-c)/2B Yohanes E. Riyanto EC 3322 (Industrial Organization I) (A-c)/ B q1 30 A Comparison of Oligopoly Equilibria Output Price Firm Industry Monopoly 360 360 Cournot Duopoly 240 Stackelberg Duopoly Leader Follower Competitive Market (P=MC) Profit Consumer Surplus Firm Industry 64 129.6 129.6 64.8 480 52 57.6 115.2 115.2 540 46 97.2 145.8 0 259.2 360 180 64.8 32.4 720 28 0 Market demand function: P = 1 – 0.001Q and MC=0.28 (linear demand and constant MC) Yohanes E. Riyanto EC 3322 (Industrial Organization I) 31 A Comparison of Oligopoly Equilibria 2 57.6 32.4 Cournot Stackelberg Monopoly Bertrand & Competitive Solution Yohanes E. Riyanto 57.6 64.8 EC 3322 (Industrial Organization I) 129.6 1 32 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 33 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 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 34 Application: Advertising & Competition The game (firm 1 and 2) 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 The market demand faced by the two firms p a q1 q2 Firms produce at zero costs, but firm 1 incurs advertising costs of Yohanes E. Riyanto EC 3322 (Industrial Organization I) 2a 3 81 35 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. 2a 3 1 a q1 q2 q1 81 Derive the f.o.c. w.r.t. q1 and solve for q1, we get the best response function. q*1 f q2 a q2 2 Similarly derive the best response fu. For firm 2. 2 a q1 q2 q2 a q1 q 2 f q1 * 2 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 36 Application: Advertising & Competition * * The Cournot Nash equilibrium can be obtained. q 1 q 2 The equilibrium price is then, p a a 3 a a a 3 3 3 Hence, firm 1’s profit function as a function of a is, 2a 3 1 a q1 q2 q1 81 2 2a 3 a 1 81 3 (Now at t=1): Firm 1 chooses its advertisement level (a) to maximize its profit. 1 2a 6a 2 f .o.c. 0 a* 3 a 9 81 The SPE strategy profile is a a a* 3 q1a q2 a 3 3 Yohanes E. Riyanto EC 3322 (Industrial Organization I) 37