Price Competition Chapter 10 Price Competition 1 Introduction

Price Competition Chapter 10: Price Competition 1

Introduction • In a wide variety of markets firms compete in prices – – Internet access Restaurants Consultants Financial services • With monopoly setting price or quantity first makes no difference • In oligopoly it matters a great deal – nature of price competition is much more aggressive the quantity competition Chapter 10: Price Competition 2

Price Competition: Bertrand • In the Cournot model price is set by some market clearing mechanism • An alternative approach is to assume that firms compete in prices: this is the approach taken by Bertrand • Leads to dramatically different results • Take a simple example – – – two firms producing an identical product (spring water? ) 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 – b. P with a = A/B and b= 1/B Chapter 10: Price Competition 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 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 – q 2 = (a – bp 2)/2 – q 2 = a – bp 2 if p 2 > p 1 if p 2 = p 1 if p 2 < p 1 Chapter 10: Price Competition 4

Bertrand competition 2 • This can be illustrated as follows: • Demand is discontinuous • The discontinuity in demand carries over to profit p 2 There is a jump at p 2 = p 1 a - bp 1 (a - bp 1)/2 Chapter 10: Price Competition a q 2 5

Bertrand competition 3 Firm 2’s profit is: p 2(p 1, , p 2) = 0 if p 2 > p 1 p 2(p 1, , p 2) = (p 2 - c)(a - bp 2) if p 2 < p 1 p 2(p 1, , p 2) = (p 2 - c)(a - bp 2)/2 if p 2 = p 1 Clearly this depends on p 1. For whatever reason! Suppose first that firm 1 sets a “very high” price: greater than the monopoly price of p. M = (a +c)/2 b Chapter 10: Price Competition 6

What price should firm 2 set? Bertrand competition 4 With p 1 Firm > (a 2+will c)/2 b, Firm only earn a 2’s profit looks like this: positive by cutting its Firm profit 2’s Profit At p 2 = p 1 The monopoly price to (a + c)/2 b or less firm 2 gets half of the price So firm 2 should just monopoly profit undercut p 1 a bit and p 2 < p 1 get almost all the What if firm 1 monopoly profit prices at (a + c)/2 b? p 2 = p 1 p 2 > p 1 c (a+c)/2 b Chapter 10: Price Competition p 1 Firm 2’s Price 7

Bertrand competition 5 Now suppose that firm 1 sets a price less than (a + c)/2 b Firm 2’s profit looks like this: What price As long as p 1 > c, Firm 2’s Profit Of course, firm 1 Firm 2 should firmaim 2 just to undercut will then undercut set now? firm 1 firm 2 and so on p 2 < p 1 Then firm 2 should also price at c. Cutting price below cost gains the whole market but loses What if firm money on 1 every customer prices at c? p 2 = p 1 p 2 > p 1 c p 1 (a+c)/2 b Chapter 10: Price Competition Firm 2’s Price 8

Bertrand competition 6 • We now have Firm 2’s best response to any price set by firm 1: – p*2 = (a + c)/2 b – p*2 = p 1 - “something small” – p*2 = c if p 1 > (a + c)/2 b if c < p 1 < (a + c)/2 b if p 1 < c • We have a symmetric best response for firm 1 – p*1 = (a + c)/2 b – p*1 = p 2 - “something small” – p*1 = c if p 2 > (a + c)/2 b if c < p 2 < (a + c)/2 b if p 2 < c Chapter 10: Price Competition 9

Bertrand competition 7 These best The best response function for response firm functions 1 p 2 look like R 1 The best response thisfunction for firm 2 R 2 (a + c)/2 b The Bertrand The equilibrium has isboth with both firms charging firms pricing at marginal cost c c c (a + c)/2 b p 1 Chapter 10: Price Competition 10

Bertrand Equilibrium: modifications • The Bertrand model makes clear that competition in prices is very different from competition in quantities • Since many firms seem to set prices (and not quantities) this is a challenge to the Cournot approach • But the extreme version of the difference seems somewhat forced • Two extensions can be considered – impact of capacity constraints – product differentiation Chapter 10: Price Competition 11

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 skiing on Mount Norman Q = 6, 000 – 60 P – Q is number of lift tickets and P is price of a lift ticket – two resorts: Pepall with daily capacity 1, 000 and Richards with daily capacity 1, 400, both fixed – marginal cost of lift services for both is $10 Chapter 10: Price Competition 12

The Example • Is a price P = c = $10 an equilibrium? – total demand is then 5, 400, well in excess of capacity • Suppose both resorts set P = $10: both then have demand of 2, 700 • Consider Pepall: – – raising price loses some demand but where can they go? Richards is already above capacity so some skiers will not switch from Pepall at the higher price but then Pepall is pricing above MC and making profit on the skiers who remain – so P = $10 cannot be an equilibrium Chapter 10: Price Competition 13

The example 2 • Assume that at any price where demand at a resort is greater than capacity there is efficient rationing – serves skiers with the highest willingness to pay • Then can derive residual demand • Assume P = $60 – total demand = 2, 400 = total capacity – so Pepall gets 1, 000 skiers – residual demand to Richards with efficient rationing is Q = 5000 – 60 P or P = 83. 33 – Q/60 in inverse form – marginal revenue is then MR = 83. 33 – Q/30 Chapter 10: Price Competition 14

The example 3 • Residual demand MR: • Suppose that Richards sets P = $60. Does it want to change? – since MR > MC Richards does not want to raise price and lose skiers – since QR = 1, 400 Richards is at capacity and does not want to reduce price Price $83. 33 Demand $60 MR $36. 66 $10 MC 1, 400 Quantity • Same logic applies to Pepall so P = $60 is a Nash equilibrium for this game. Chapter 10: Price Competition 15

Capacity constraints again • 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 the efficiency property of Bertrand equilibrium breaks down when firms are capacity constrained Chapter 10: Price Competition 16

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” Chapter 10: Price Competition 17

An example of product differentiation Coke and Pepsi are similar but not identical. As a result, the lower priced product does not win the entire market. Econometric estimation gives: QC = 63. 42 - 3. 98 PC + 2. 25 PP MCC = $4. 96 QP = 49. 52 - 5. 48 PP + 1. 40 PC MCP = $3. 96 There at least two methods for solving for PC and PP Chapter 10: Price Competition 18

Bertrand product differentiation Method 1: Calculus Profit of Coke: p. C = (PC - 4. 96)(63. 42 - 3. 98 PC + 2. 25 PP) Profit of Pepsi: p. P = (PP - 3. 96)(49. 52 - 5. 48 PP + 1. 40 PC) Differentiate with respect to PC and PP respectively Method 2: MR = MC Reorganize the demand functions PC = (15. 93 + 0. 57 PP) - 0. 25 QC PP = (9. 04 + 0. 26 PC) - 0. 18 QP Calculate marginal revenue, equate to marginal cost, solve for QC and QP and substitute in the demand functions Chapter 10: Price Competition 19

Bertrand product differentiation 2 Both methods give the best response functions: PC = 10. 44 + 0. 2826 PP PP PP = 6. 49 + 0. 1277 PC These can be solved for the equilibrium prices as indicated The Note. Bertrand that these equilibrium are upwardis atsloping their intersection $8. 11 RC B The equilibrium prices $6. 49 are each greater than marginal cost $10. 44 Chapter 10: Price Competition RP PC $12. 72 20

Bertrand competition and the spatial model • An alternative approach: spatial model of Hotelling – – a Main Street over which consumers are distributed supplied by two shops located at opposite ends of the street but now the shops are competitors each consumer buys exactly one unit of the good provided that its full price is less than V – a consumer buys from the shop offering the lower full price – consumers incur transport costs of t per unit distance in travelling to a shop • Recall the broader interpretation • What prices will the two shops charge? Chapter 10: Price Competition 21

marks themodel location of the Bertrand thexmspatial Price marginal buyer—one who Assume that shop 1 sets between. Price What if shop 1 raises is indifferent price shop 2 either sets firm’s good its pprice? 1 andbuying price p 2 p’ 1 p 2 p 1 x’m Shop 1 xm All consumers to the And all consumers x moves to the left of xm buy from m to the right buy. Shop from 2 left: some consumers shop 1 shop 2 switch to shop 2 Chapter 10: Price Competition 22

Bertrand the spatial model 2 p 1 + txm = p 2 + t(1 - xm) 2 txm = p 2 - p 1 + t xm(p 1, p 2) = (p 2 - p 1 + t)/2 t How is xm determined? This is the fraction There are N consumers in total of consumers who 1 So demand to firm 1 is D = N(p 2 - p 1 +buy t)/2 t from firm 1 Price p 2 p 1 xm Shop 1 Shop 2 Chapter 10: Price Competition 23

Bertrand equilibrium Profit to firm 1 is p 1 = (p 1 - c)D 1 = N(p 1 - c)(p 2 - p 1 + t)/2 t This is the best p 1 = N(p 2 p 1 - p 12 + tp 1 + cp 1 -response cp 2 -ct)/2 t function Solve this Differentiate with respect to p 1 for firm 1 for p 1 N p 1/ p 1 = (p 2 - 2 p 1 + t + c) = 0 2 t p*1 = (p 2 + t + c)/2 This is the best response function for firmit 2 has firm 2? By symmetry, What about similar best response function. a p*2 = (p 1 + t + c)/2 Chapter 10: Price Competition 24

Bertrand equilibrium 2 p*1 = (p 2 + t + c)/2 p 2 R 1 p*2 = (p 1 + t + c)/2 2 p*2 = p 1 + t + c R 2 = p 2/2 + 3(t + c)/2 c + t p*2 = t + c (c + t)/2 p*1 = t + c Profit per unit to each firm is t (c + t)/2 c+t p 1 Aggregate profit to each firm is Nt/2 Chapter 10: Price Competition 25

Bertrand competition 3 • Two final points on this analysis • t is a measure of transport costs – it is also a measure of the value consumers place on getting their most preferred variety – when t is large competition is softened • and profit is increased – when t is small competition is tougher • and profit is decreased • Locations have been taken as fixed – suppose product design can be set by the firms • balance “business stealing” temptation to be close • against “competition softening” desire to be separate Chapter 10: Price Competition 26

Strategic complements and substitutes • Best response functions are very different with Cournot and Bertrand – they have opposite slopes – reflects very different forms of competition – firms react differently e. g. to an increase in costs q 2 Firm 1 Cournot Firm 2 q 1 p 2 Firm 1 Firm 2 Bertrand p 1 Chapter 10: Price Competition 27

Strategic complements and substitutes q 2 – suppose firm 2’s costs increase – this causes Firm 2’s Cournot best response function to fall Firm 1 • at any output for firm 1 firm 2 passive now wants to produce less – firm 1’s output increases and response by firm 1 firm 2’s falls – Firm 2’s Bertrand best response function rises aggressive response by firm 1 Cournot Firm 2 q 1 p 2 • at any price for firm 1 firm 2 now wants to raise its price – firm 1’s price increases as does firm 2’s Chapter 10: Price Competition Firm 1 Firm 2 Bertrand p 1 28

Strategic complements and substitutes 2 • When best response functions are upward sloping (e. g. Bertrand) we have strategic complements – passive action induces passive response • When best response functions are downward sloping (e. g. Cournot) we have strategic substitutes – passive actions induces aggressive response • Difficult to determine strategic choice variable: price or quantity – output in advance of sale – probably quantity – production schedules easily changed and intense competition for customers – probably price Chapter 10: Price Competition 29

Empirical Application: Brand Competition and Consumer Preferences • As noted earlier, products can be differentiated horizontally or vertically • In many respects, which type of differentiation prevails reflects underlying consumer preferences • Are the meaningful differences between consumers about what makes for quality and not about what quality is worth (Horizontal Differentiation); Or • Are the meaningful differences between consumers not about what constitutes good quality but about how much extra quality should be valued (Vertical Differentiation) Chapter 10: Price Competition 30

Brand Competition & Consumer Preferences 2 • Consider the study of the retail gasoline market in southern California by Hastings (2004) • Gasoline is heavily branded. Established brands like Chevron and Exxon-Mobil have contain special, trademarked additives that are not found in discount brands, e. g. Race. Trak. • In June 1997, the established brand Arco gained control of 260 stations in Southern California formerly operated by the discount independent, Thrifty • By September of 1997, the acquired stations were converted to Arco stations. What effect did this have on branded gasoline prices? Chapter 10: Price Competition 31

Brand Competition & Consumer Preferences 3 • If consumers regard Thrifty as substantially different in quality from the additive brands, then losing the Thrifty stations would not hurt competition much while the entry of 260 established Arco stations would mean a real increase in competition for branded gasoline and those prices should fall. • If consumers do not see any real quality differences worth paying for but simply valued the Thrifty stations for providing a low-cost alternative, then establish brand prices should rise after the acquisition. • So, behavior of gasoline prices before and after the acquisition tells us something about preferences. Chapter 10: Price Competition 32

Brand Competition & Consumer Preferences 4 • Tracking differences in price behavior over time is tricky though • Hastings (2004) proceeds by looking at gas stations that competed with Thrifty’s before the acquisition (were within 1 mile of a Thrifty) and ones that do not. She asks if there is any difference in the response of the prices at these two types of stations to the conversion of the Thrifty stations • Presumably, prices for both types were different after the acquisition than they were before it. The question is, is there a difference between the two groups in these before-and-after differences? For this reason, this approach is called a difference-indifferences model. Chapter 10: Price Competition 33

Brand Competition & Consumer Preferences 5 Hastings observes prices for each station in Feb, June, Sept. and December of 1997, i. e. , before and after the conversion. She runs a regression explaining station i’s price in each of the four time periods, t pit = Constant + i+ 1 Xit + 2 Zit + 3 Ti+ eit i is an intercept term different for each that controls for differences between each station unrelated to time Xit is 1 if station i competes with a Thrifty at time t and 0 otherwise. Zit is 1 if station i competes with a station directly owned by a major brand but 0 if it is a franchise. Ti is a sequence of time dummies equal reflecting each of the four periods. This variable controls for the pure effect of time on the prices at all stations. Chapter 10: Price Competition 34

Brand Competition & Consumer Preferences 6 The issue is the value of the estimated coefficient 1 Ignore the contractual variable Zit for the moment and consider two stations: firm 1 that competed with a Thrifty before the conversion and firm 2 that did not. In the pre-conversion periods, Xit is positive for firm 1 but zero for firm 2. Over time, each firm will change its price because of common factors that affect them over time. However, firm 1 will also change is price because for the final two observations, Xit is zero. Before Firm 1: Firm 2: After αi + β 1 αi + time effects αj αj + time effects Chapter 10: Price Competition Difference - β 1 + time effects 35

Brand Competition & Consumer Preferences 6 Thus, the estimated coefficient 1 captures the difference in movement over time between firm 1 and firm 2. Hastings (2004) estimates 1 to be about -0. 05. That is, firms that competed with a Thrifty saw their prices rise by about 5 cents more over time than did other firms Before the conversion, prices at stations that competed against Thrifty’s were about 2 to 3 cents below those that did not. After the removal of the Thrifty’s, however, they had prices about 2 to 3 cents higher than those that did not. Conversion of the Thrifty’s to Arco stations did not intensify competition among the big brands. Instead, it removed a lost cost alternative. Chapter 10: Price Competition 36