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

Yohanes E. Riyanto 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

Yohanes E. Riyanto 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 – bP with a = A/B and b= 1/B

Yohanes E. Riyanto 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 p 1   if firm 2 sets a price greater than p 1 she will sell nothing   if firm 2 sets a price less than p 1 she gets the whole market   if firm 2 sets a price of exactly p 1 consumers are indifferent between the two firms: the market is shared, presumably 50:50 So we have the derived demand for firm 2   q 2 = 0 if p 2 > p 1   q 2 = (a – bp 2 )/2 if p 2 = p 1   q 2 = a – bp 2 if p 2 < p 1

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 5 Bertrand Competition This can be illustrated as follows: Demand is discontinuous p2p2 q2q2 p1p1 aa - bp 1 (a - bp 1 )/2 There is a jump at p 2 = p 1 The discontinuity in demand carries over to profit

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

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 7 Bertrand Competition With p 1 > (a + bc)/2b, Firm 2’s profit looks like this: Firm 2’s Price Firm 2’s Profit c(a+bc)/2b p1p1 p 2 < p 1 p 2 = p 1 p 2 > p 1 What price should firm 2 set? The monopoly price What if firm 1 prices at (a + c)/2b? So firm 2 should just undercut p 1 a bit and get almost all the monopoly profit Firm 2 will only earn a positive profit by cutting its price to (a + bc)/2b or less At p2 = p1 firm 2 gets half of the monopoly profit Π 2 (p 1,, p 2 ) = (p 2 - c)(a - bp 2 )

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 8 Bertrand Competition Now suppose that firm 1 sets a price less than (a + bc)/2b Firm 2’s Price Firm 2’s Profit c (a+bc)/2b p1p1 p 2 < p 1 p 2 = p 1 p 2 > p 1 Firm 2’s profit looks like this: What price should firm 2 set now? As long as p 1 > c, Firm 2 should aim just to undercut firm 1 What if firm 1 prices at c? Then firm 2 should also price at c. Cutting price below cost gains the whole market but loses money on every customer Of course, firm 1 will then undercut firm 2 and so on

Yohanes E. Riyanto 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 p 1 > (a + bc)/2b   p* 2 = p 1 - “something small (  )” if c < p 1 < (a + bc)/2b   p* 2 = c if p 1 < c We have a symmetric best response for firm 1   p* 1 = (a + bc)/2b if p 2 > (a + bc)/2b   p* 1 = p 2 - “something small (  )”if c < p 2 < (a + bc)/2b   p* 1 = c if p 2 < c

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 10 Bertrand Competition These best response functions look like this p2p2 p1p1 c c R1R1 R2R2 The best response function for firm 1 The best response function for firm 2 The equilibrium is with both firms pricing at c The Bertrand equilibrium has both firms charging marginal cost (a + bc)/2b

Yohanes E. Riyanto 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?

Yohanes E. Riyanto 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 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) 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

Yohanes E. Riyanto 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

Yohanes E. Riyanto 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 – 60P or P = – Q/60 in inverse form.   marginal revenue is then MR = – Q/30

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

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 17 Capacity Constraints Residual demand and MR: Price Quantity Demand 1,400 $83.33 $60 $36.66 $10MC MR 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 Q R = 1,400 Firm 2 is at capacity and does not want to reduce price Same logic applies to Firm 1 so P = $60 is a Nash equilibrium for this game.

Yohanes E. Riyanto 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

Yohanes E. Riyanto 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.

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

Yohanes E. Riyanto 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

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

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 23 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’). Sequential Move Game

Yohanes E. Riyanto 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) terrorist bomb -1,-1 1,1 not bomb terrorist bomb -1,-1 2,0 not bomb

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 25 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. Subgame Perfect Equilibrium , 8 7, 9 1, 2 2, , 4 0, 5 4, 0 8, 3 1 1

Yohanes E. Riyanto 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

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

Yohanes E. Riyanto 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(q 1 + q 2 ) Marginal cost for for each firm is c Firm 1 is the market leader and chooses q 1 In doing so it can anticipate firm 2’s actions. So consider firm 2. Demand for firm 2 is: PP = (A – Bq 1 ) – Bq 2 Marginal revenue therefore is: MMR 2 = (A - Bq 1 ) – 2Bq 2

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 29 Stackelberg Competition MR 2 = (A - Bq 1 ) – 2Bq 2 MC = c Equate marginal revenue with marginal cost  q* 2 = (A - c)/2B - q 1 /2 q2q2 q1q1 R2R2 (A – c)/2B (A – c)/B This is firm 2’s best response function Firm 1 knows that this is how firm 2 will react to firm 1’s output choice Firm 1 knows that this is how firm 2 will react to firm 1’s output choice So firm 1 can anticipate firm 2’s reaction So firm 1 can anticipate firm 2’s reaction Demand for firm 1 is: P = (A - Bq 2 ) – Bq 1 But firm 1 knows what q 2 is going to be P = (A - Bq* 2 ) – Bq 1 P = (A - (A-c)/2) – Bq 1 /2  P = (A + c)/2 – Bq 1 /2 Marginal revenue for firm 1 is: MR 1 = (A + c)/2 - Bq 1 (A + c)/2 – Bq 1 = c Solve this equation for output q 1  q* 1 = (A – c)/2B (A – c)/2B  q* 2 = (A – c)4B (A – c)/4B S Equate marginal revenue with marginal cost 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

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 30 Firm 1’s best response function is “like” firm 2’s Stackelberg Competition Aggregate output is 3(A-c)/4B So the equilibrium price is (A+3c)/4 q2q2 q1q1 R2R2 (A-c)/2B (A-c)/ B Compare this with the Cournot equilibrium Compare this with the Cournot equilibrium (A-c)/2B Firm 1’s profit is (A-c) 2 /8B Firm 2’s profit is (A-c) 2 /16B (A-c)/B R1R1 S C We know that the Cournot equilibrium is: q C 1 = q C 2 = (A-c)/3B (A-c)/3B The Cournot price is (A+c)/3 Profit to each firm is (A-c) 2 /9B Leadership benefits the leader firm 1 but harms the follower firm 2 Leadership benefits consumers but reduces aggregate profits (A-c)/4B

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 31 OutputPriceProfitConsumer Surplus FirmIndustryFirmIndustry Monopoly Cournot Duopoly Stackelberg Duopoly  Leader  Follower Competitive Market (P=MC) Market demand function: P = 1 – 0.001Q and MC=0.28 (linear demand and constant MC) A Comparison of Oligopoly Equilibria

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 32 Bertrand & Competitive Solution Cournot Stackelberg Monopoly A Comparison of Oligopoly Equilibria Profit Possibility Frontier

Yohanes E. Riyanto 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

Yohanes E. Riyanto 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

Yohanes E. Riyanto 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 Time Period t=1 firm 1 chooses advertising level ( a ) in order to enhance demand t=2 firm 1 and 2 compete in a Cournot fashion (choosing quantity Level) Firm 2 observes the choice of a of firm 1

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 36   (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. Application: Advertising & Competition

Yohanes E. Riyanto EC 3322 (Industrial Organization I) 37   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 Application: Advertising & Competition