Part III. Sources of market power Chapter 5. Product differentiation Slides Industrial Organization: Markets and Strategies Paul Belleflamme and Martin Peitz, 2d Edition © Cambridge University Press 2015 Introduction to Part III • Where does the market power come from? • Consequence of firms’ conduct • Marketing mix: Price - Product - Promotion • Product  Closer look at different types of product differentiation  Chapter 5 • Promotion  Advertising strategies Chapter 6 • Price  Discrimination (Part IV) • Consumer inertia (skipped)  Chapter 7 • Search costs, switching costs, behavioural issues (statusqua bias) etc… © Cambridge University Press 2015 2 Chapter 5 - Objectives Chapter 5. Learning objectives • Understand that product differentiation involves two conflicting forces: it relaxes price competition, but it may reduce the demand that the firm faces (niche market). • Distinguish between horizontal (location) and vertical (quality) product differentiation. • Reconsider the question of entry into product market. • Discuss some basic approaches to estimate differentiated product markets. © Cambridge University Press 2015 3 Chapter 5 - On product differentiation • Horizontal product differentiation • Each product would be preferred by some consumers, depending on their tastes. • Vertical product differentiation • Everybody would prefer one over the other product. • More formally: if, at equal prices, • consumers do not agree on which product is the preferred one  products are horizontally differentiated; • all consumers prefer one over the other product  products are vertically differentiated. © Cambridge University Press 2015 4 Chapter 5 - Horizontal differentiation Location choice with fixed price • Suppose constant price (e.g., regulated price): 𝑝ҧ > 𝑐 • Product positioning: Two firms choose where to locate their product in “linear city”: 𝑙1 , 𝑙2 ∈ 0,1 . • Consumers are as in the earlier Hotelling model: • uniformly distributed on ; location represents the ideal point in product space; linear transportation cost. • Need to buy one unit from one of the firms. • 𝑢𝑥 (𝑖, 𝑝𝑖 ) is the utility of buying from firm i at the price 𝑝𝑖 for the consumer located at the point 𝑥: 𝑢𝑥 𝑖, 𝑝𝑖 = 𝑟 − 𝜏 𝑥 − 𝑙𝑖 − 𝑝𝑖 © Cambridge University Press 2015 5 Chapter 5 - Horizontal differentiation Location choice with fixed price • Since 𝑝1 = 𝑝2, each consumer selects the nearest firm. • If 𝑙𝑖 < 𝑙𝑗 , there is a unique indifferent consumer: 𝑥ො =   𝑄𝑖 = 𝑥ො = 𝑙1 + 𝑙2 2 𝑙1 + 𝑙2 2 𝑄𝑗 = 1 − 𝑥ො = 1 − (demand for firm 𝑖) 𝑙1 + 𝑙2 2 (demand for firm 𝑗) • If 𝑙1 = 𝑙2 , firms share the market equally: 𝑄1 = 𝑄2 = 12 • Firm i’s problem: Given 𝑙𝑗 , max 𝜋𝑖 (𝑙1 , 𝑙2 ) = (𝑝ҧ − 𝑐)𝑄𝑖 (𝑙1 , 𝑙2 ) 𝑙𝑖 ∈[0,1] © Cambridge University Press 2015 6 Chapter 5 - Horizontal differentiation Location choice with fixed price • It follows that: 𝜋𝑖 𝑙1 , 𝑙2 (𝑝ҧ − 𝑐)(𝑙1 + 𝑙2 )/2 (𝑝ҧ − 𝑐)/2 =൞ 𝑝ҧ − 𝑐 1 − (𝑙1 + 𝑙2 )/2 if 𝑙𝑖 < 𝑙𝑗 , if 𝑙𝑖 = 𝑙𝑗 , if 𝑙𝑖 > 𝑙𝑗 . Note: • For 𝑙𝑖 < 𝑙𝑗 , 𝜋𝑖 is increasing with 𝑙𝑖 . Getting closer to firm 𝑗 brings more customers from the right side without losing anybody from the left side. • Similarly, when 𝑙𝑖 > 𝑙𝑗 , firm 𝑖 has an incentive to get closer to the other firm by moving leftward.  There is no Nash equilibrium with 𝑙1 ≠ 𝑙2 . © Cambridge University Press 2015 7 Chapter 5 - Horizontal differentiation Location choice with fixed price • 𝑙1 = 𝑙2 = 1/2 is an equilibrium because 1 𝜋𝑖 𝑙𝑖 , 2 • = 𝜋𝑖 1 1 , 2 2 ∀𝑙𝑖 ∈ 0,1 . 𝑙1 = 𝑙2 ≠ 1/2 is not an equilibrium. • • ≤ ҧ 𝑝−𝑐 2 For example, if 𝑙1 = 𝑙2 < 1/2, both firms get half of the market. Any firm 𝑖 can move slightly rightward, to 𝑙𝑖 + 𝜀, so as to increase its market share to ≅ 1 − 𝑙𝑖 > 1/2. CONCLUSION: In the unique equilibrium, both firms select the midpoint; there is no product differentiation. © Cambridge University Press 2015 8 Chapter 5 - Horizontal differentiation Location choice with fixed price • Lesson: If duopolists were not able to entertain distinct prices, they would offer the same product. This is because differentiation reduces the demand for a given product by effectively targeting a smaller niche in the market. © Cambridge University Press 2015 9 Chapter 5 - Horizontal differentiation Socially Efficient Locations: • Minimize total distance: 𝑙1 min න (𝑙1 − 𝑥)𝑑𝑥 + න 0 𝑙1 +𝑙2 2 𝑙1 1 (𝑥 − 𝑙1 )𝑑𝑥 𝑙2 + න (𝑥 − 𝑙2 )𝑑𝑥 + න 𝑙1 +𝑙2 2 𝑙2 s.t. • Solution: 𝑙1 , 𝑙2 ∈ 0,1 𝑙1 = 1/4 and 2 (𝑙2 − 𝑥)𝑑𝑥 and 𝑙1 ≤ 𝑙2 . 𝑙2 = 3/4  Insufficient differentiation in equilibrium. © Cambridge University Press 2015 10 Chapter 5 - Horizontal differentiation Hotelling model (full version) • Firms choose location and price. • 2 stage model 1. Location choice (long term decision) 2. Price choice (short term decision) • • We already studied (in Chapter 3) the price stage with extreme locations (i.e., 0 and 1). We will “repeat” (not really) the analysis for any pair of locations under two different scenarios: • • Linear transportation costs Quadratic transportation costs © Cambridge University Press 2015 11 Chapter 5 - Horizontal differentiation Linear Hotelling model • As before, consumers are distributed on [0,1] with the utility function: 𝑢𝑥 𝑖, 𝑝𝑖 = 𝑟 − 𝜏 𝑥 − 𝑙𝑖 − 𝑝𝑖 • If both products are identical, firms share the market equally. Otherwise, there is at most one indifferent consumer. • Firms: • Choose first 𝑙𝑖 in [0,1] and then 𝑝𝑖 • Move simultaneously in both stages. • Look for subgame perfect equilibria (backward induction). • First step: Fix 𝑙1 and 𝑙2. Analyse the ensuing price stage. © Cambridge University Press 2015 12 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) • Price stage: • • If 𝑙1 = 𝑙2 , we are back to Bertrand: The cheaper firm gets the whole market.  𝑝1 = 𝑝2 = 𝑐 If 𝑙1 < 𝑙2 , depending on prices, either there is an indifferent consumer between the two locations, or all consumers select the same firm: 𝑟 − 𝜏(𝑥ො − 𝑙1 ) − 𝑝1 = 𝑟 − 𝜏 𝑙2 − 𝑥ො − 𝑝2  So, 𝑥ො = 𝑙1 +𝑙2 2 − 𝑝1 −𝑝2 2𝜏 = 𝑙1 +𝑙2 2 + 𝑝2 −𝑝1 2𝜏 𝑥ො ≥ 𝑙1  𝑝1 − 𝑝2 ≤ 𝜏(𝑙2 − 𝑙1 ) 𝑥ො ≤ 𝑙2  𝑝1 − 𝑝2 ≥ −𝜏(𝑙2 − 𝑙1 ) © Cambridge University Press 2015 13 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) • If price difference is “not too large” relative to locations, meaning that 𝑝1 − 𝑝2 ≤ 𝜏 𝑙2 − 𝑙1 , then there is an indifferent consumer given by: 𝑥ො = • 𝑙1 +𝑙2 2 − 𝑝1 −𝑝2 2𝜏 Otherwise, i.e., if 𝑝1 − 𝑝2 > 𝜏 𝑙2 − 𝑙1 , the cheaper firm gets the whole market. (Note the role of linear costs here.) © Cambridge University Press 2015 14 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) Total cost of buying from 2 Total cost of buying from 1 © Cambridge University Press 2015 15 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) • Profit of firm 1 (assuming 𝑙1 < 𝑙2 , given 𝑝2 ):  0  l1 l2 ( p  c)  1 ( p1 p2 ;l1 ,l2 )   1 2   ( p1  c)   p2  p1 2  if p1  p2   (l2  l1 ), if p1  p2   (l2  l1 ), if p1  p2   (l2  l1 ). • Note: 𝑝1 = 𝑝2 − 𝜏 𝑙2 − 𝑙1  𝜋1 = (𝑝1 − 𝑐)𝑙2 < (𝑝1 − 𝑐) (unless 𝑙2 = 1)  Discontinuity at 𝑝1 = 𝑝2 − 𝜏 𝑙2 − 𝑙1 Intuition: Firm 2 has zero demand if it can’t attract the consumer located at 𝑙2 . Otherwise, its demand is at least 1 − 𝑙2 , because any 𝑥 ≥ 𝑙2 buys from 2. So, 𝑄2 falls from 1 − 𝑙2 to 0, suddenly, at a certain level of 𝑝1 . © Cambridge University Press 2015 16 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) A sharp fall in 𝜋1 Local max. Won’t be a global max when the arc is narrow, and closer to the point of discontinuity, on the left. Price equilibrium may fail to exist Happens when locations selected in stage 1 are too close, so that the arc is narrow. © Cambridge University Press 2015 17 Chapter 5 - Horizontal differentiation Linear Hotelling model (cont’d) • • Price equilibrium fails to exist for some pairs of location  no subgame perfect equilibrium Recall that if firms don’t expect a price difference in stage 2, they would select the same location in stage 1 to maximize their demand. But when the locations are truly close, there is no price equilibrium. So, firms may indeed want to move towards a zone where price equilibrium does not exist. • Instability in competition • Lesson: Although product differentiation relaxes price competition, firms may have an incentive to offer better substitutes to generate more demand, which may lead to instability in competition. © Cambridge University Press 2015 18 Chapter 5 - Horizontal differentiation Quadratic Hotelling model • Transport costs increase quadratically: 𝑢𝑥 𝑖, 𝑝𝑖 = 𝑟 − 𝜏 𝑥 − 𝑙𝑖 2 − 𝑝𝑖 • Suppose 𝑙1 < 𝑙2. Since 𝑥  𝜏 𝑥 − 𝑙2 2 is strictly convex, even if 𝑥 = 𝑙2 selects firm 1, consumers further on the right may select firm 2 because the additional distance to firm 1 will be costlier for them. • Formally, for 𝑙1 < 𝑙2 and 𝑥 ≥ 𝑙2, the added cost of traveling to the distant firm 𝑙1 increases with 𝑥: • 𝑑 𝑑𝑥 𝑥 − 𝑙1 2 − 𝑥 − 𝑙2 2 = 2(𝑙2 − 𝑙1 ) > 0.  No discontinuity in demand, in the price stage. © Cambridge University Press 2015 19 Chapter 5 - Horizontal differentiation Quadratic Hotelling model (cont’d) • Price Stage: • Indifferent consumer (assuming 𝑙1 < 𝑙2): 𝑟 − 𝜏 𝑥 − 𝑙1 𝜏 𝑥 − 𝑙2 2 2 − 𝑝1 = 𝑟 − 𝜏 𝑥 − 𝑙2 − 𝑥 − 𝑙1 2  𝑥ො = + − 𝑝2 = 𝑝1 − 𝑝2 𝜏 2𝑥 − (𝑙1 + 𝑙2 ) ∗ 𝑙1 − 𝑙2 𝑙1 +𝑙2 2 2 = 𝑝1 − 𝑝2    𝑝2 −𝑝1 2𝜏 𝑙2 −𝑙1 • So, 𝑥ො  with 𝑝2 − 𝑝1, given 𝑙1 < 𝑙2 • If 𝑥ො > 1, we must set 𝑄2 ≡ 0, and similarly for 𝑥ො < 0. • Nevermind: Equilibrium prices will be such that 𝑥ො ∈ 0,1 . © Cambridge University Press 2015 20 Quadratic Hotelling model (cont’d) • Price Stage (cont’d): • Assuming 𝑙1 < 𝑙2, the firms’ problem are: max(𝑝1 − 𝑐)𝑥ො 𝑝1 , 𝑝2 and max 𝑝2 − 𝑐 1 − 𝑥ො 𝑝1 , 𝑝2 𝑝1 ≥0  • 𝑝2 ≥0 𝑝1∗ =𝑐 𝑝2∗ =𝑐 2𝜏 + 3 2𝜏 + 3 𝑙2 − 𝑙1 𝑙2 − 𝑙1 𝑙1 +𝑙2 1+ 2 𝑙1 +𝑙2 2− 2 Note: Holding constant the midpoint are increasing with 𝑙2 − 𝑙1 . (1) 𝑙1 +𝑙2 , 2 both prices • Illustrates how product differentiation helps relax the price competition. • Note: Eqn (1) is also valid with 𝑙1 = 𝑙2 (Bertrand) © Cambridge University Press 2015 21 Chapter 5 - Horizontal differentiation Quadratic Hotelling model (cont’d) • Location Stage: • Remember, however, closer locations increase the demand with fixed prices. At the location stage, firms need to take into account both effects simultaneously. • Subsitute eqn (1) into profits, and the definition of 𝑥ො  ̂ 1  181  (l2  l1 )(2  l1  l2 )2 ̂ 2   (l2  l1 )(4  l1  l2 ) 1 18 2  ̂ 1 / l1  0 for all l1 [0,l2 ) ̂ 2 / l2  0 for all l2 (l1 ,1] • Subgame perfect equilibrium: firms locate at the extreme points  “maximum differentiation” • The dominant force here is to relax price competition. © Cambridge University Press 2015 22 Chapter 5 - Horizontal differentiation Quadratic Hotelling model (cont’d) • But this is just a particular example. Different results obtain: • if we remove the boundaries, 0 and 1. (Optimal locations will be −1/4 and 5/4.) • if we select a non-uniform distribution for the consumers. • if we select a different function for cost of traveling. © Cambridge University Press 2015 23 Chapter 5 - Horizontal differentiation Quadratic Hotelling model (cont’d) • General Conclusions: • 2 forces at play • Competition effect  differentiate to enjoy market power  drives competitors apart • Market size effect  meet consumers preferences  brings competitors together • Balance (equilibrium)depends on distribution of consumers, shape of transportation costs function and feasible product range • Lesson: With endogenous product differentiation, the degree of differentiation is determined by balancing • the competition effect (drives firm to  differentiation) • the market size effect (drives firm to  differentiation). © Cambridge University Press 2015 24 Chapter 5 - Vertical differentiation Vertical product differentiation • All consumers agree that one product is preferable to another, i.e., has a higher quality. • Consumers: • Preference parameter for quality: 𝜃 ∈ 𝜃, 𝜃 ⊆ ℝ+ • larger   consumer more sensitive to quality changes • Each consumer chooses 1 unit of 1 of the products • Distributed uniformly on 𝜃, 𝜃 , total mass is 𝑀 = 𝜃 − 𝜃 • 𝑠𝑖 ∈ 𝑠, 𝑠 ⊆ ℝ+ stands for the quality of product 𝑖 © Cambridge University Press 2015 25 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Utility of consumer  from one unit of product 𝑖: 𝑢𝜃 𝑠𝑖 , 𝑝𝑖 = 𝑟 + 𝜃𝑠𝑖 − 𝑝𝑖 Key feature: If 𝑠2 > 𝑠1 , then 𝑢𝜃 𝑠2 , 𝑝 − 𝑢𝜃 𝑠1 , 𝑝 = 𝜃(𝑠2 − 𝑠1 ) is  in 𝜃  Higher 𝜃 = stronger sensitivity to quality differences • Firms: Duopolists • Stage 1: Choose quality: s1, s2 • Stage 2: Choose price: p1, p2 • Simultaneous move in both stages • Constant marginal cost, c  © Cambridge University Press 2015 26 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Price stage: Suppose s1  s2 • The indifferent consumer 𝜃෠ is given by (if it exists): ෠ 1 − 𝑝1 = 𝑟 + 𝜃𝑠 ෠ 2 − 𝑝2 𝑟 + 𝜃𝑠 𝑝2 −𝑝1 ෠  𝜃= ; 𝑠2 −𝑠1 𝜃 < 𝜃መ prefers the lower quality s1 𝜃 > 𝜃መ prefers the higher quality s2 Compare with the endogenous sunk-costs/quality augmented Cournot: • In that model price-quality ratio is the same for every firm; consumers are indifferent between all products. • Here, a given consumer prefers one or the other good, depending on their sensitivity to quality. (Consumers are more heterogeneous.) © Cambridge University Press 2015 27 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Price stage (cont’d) • An indifferent consumer truly exists iff: 𝜃 ≤ 𝜃መ ≤ 𝜃  𝜃 𝑠2 − 𝑠1 ≤ 𝑝2 − 𝑝1 ≤ 𝜃 𝑠2 − 𝑠1 Hence (assuming 𝑠2 > 𝑠1 ): 𝜋1 = 𝑝1 0 𝑝2 − 𝑝1 −𝜃 𝑠2 − 𝑠1 𝑝1 𝜃 − 𝜃 if 𝑝1 > 𝑝2 − 𝜃 𝑠2 − 𝑠1 , if 𝜃 𝑠2 − 𝑠1 ≤ 𝑝2 − 𝑝1 ≤ 𝜃 𝑠2 − 𝑠1 , if 𝑝1 < 𝑝2 − 𝜃 𝑠2 − 𝑠1 . • No discontinuity because 𝑝1 = 𝑝2 − 𝜃 𝑠2 − 𝑠1  𝜋1 = 𝑝1 𝜃 − 𝜃 • Moreover, 𝜋1 is increasing in 𝑝1 up to this point. © Cambridge University Press 2015 28 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Price stage (cont’d) 𝑝1 (𝑝2 ) • The quadratic part 𝑝1 𝑝2 −𝑝1 𝑠2 −𝑠1 −𝜃 is maximized at 1 2 𝑝1 (𝑝2 ) = [𝑝2 − 𝜃(𝑠2 − 𝑠1 )] • This is the best response if it is positive. Otherwise, 𝜋1 = 0 because firm 1 has no demand for any 𝑝1 ≥ 0, and a best response is 𝑝1 = 0. © Cambridge University Press 2015 29 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Price stage (cont’d). Similarly: 𝜋2 = 𝑝 2 𝑝2 𝜃 − 𝜃 𝑝2 − 𝑝1 𝜃− 𝑠2 − 𝑠1 0 if 𝑝1 > 𝑝2 − 𝜃 𝑠2 − 𝑠1 , if 𝜃 𝑠2 − 𝑠1 ≤ 𝑝2 − 𝑝1 ≤ 𝜃 𝑠2 − 𝑠1 , if 𝑝1 < 𝑝2 − 𝜃 𝑠2 − 𝑠1 . 1 2  𝑝2 (𝑝1 ) = [𝜃 𝑠2 − 𝑠1 + 𝑝1 ] (best response) • Equilibrium: 𝑝1∗ = 𝑝2∗ = 1 3 1 3 𝜃 − 2𝜃 𝑠2 − 𝑠1 2𝜃 − 𝜃 𝑠2 − 𝑠1 (assuming 𝜃 > 2𝜃) (2)  Even the price of the low-quality firm increases with the quality difference! © Cambridge University Press 2015 30 Vertical product differentiation (cont’d) • Price stage (cont’d) Note: Suppose 𝜃 ≤ 2𝜃. Set 𝑝2 ≡ 𝑝2 (0) = 𝜃 𝑠2 − 𝑠1 /2. 𝑝2 −𝑝1 ෠ 𝜃= ≤  𝑠2 −𝑠1 • 𝑝2 𝑠2 −𝑠1 = 1 𝜃 2 ≤𝜃 That is, firm 1 cannot get a positive demand even if it sets 𝑝1 = 0. This makes 𝑝1∗ = 0 a best response. • Then, 𝑝2∗ ≡ 𝑝2 (0) is a best response too.  Conclusion: If 𝜃 ≤ 2𝜃 and 𝑠1 ≠ 𝑠2 , the high quality firm gets the whole market, the other firm shuts down in the price stage. • Henceforth, assume 𝜃 > 2𝜃, so that both firms remain active in stage 2. © Cambridge University Press 2015 31 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Quality stage • Substitute 𝑝1∗ and 𝑝2∗ from eqn (2) into stage 1 profit function: 𝜋1 𝑠1 , 𝑠2 = 𝜋2 𝑠1 , 𝑠2 = • • 1 9 1 9 𝜃 − 2𝜃 2𝜃 − 𝜃 2 2 𝑠2 − 𝑠1 𝑠2 − 𝑠1 Both profits  in the quality difference 𝑠2 − 𝑠1 .  equilibrium quality choices: 𝑠1∗ = 𝑠 and 𝑠2∗ = 𝑠 • The converse is also possible by symmetry in this simultaneous move game. • Note: Sequential quality selection would imply a first mover advantage, because of strategic substitutability. The first mover would select 𝑠 and get a higher profit as in 𝜋2 above. © Cambridge University Press 2015 32 Chapter 5 - Vertical differentiation Vertical product differentiation (cont’d) • Note that the marginal cost of quality is assumed to be 0 here. If we were to take into account the cost of producing a high quality product, optimal quality choices may not be so extreme. The general conclusion is the following: • Lesson: In markets in which products can be vertically differentiated, the firms offer different qualities in equilibrium so as to relax price competition. © Cambridge University Press 2015 33 Chapter 5 - Vertical differentiation Case. VLJ industry: “Battle of bathrooms” • Very Light Jets • 4 to 8 passengers, city-to-city, 60 to 90-minute trips You are not going to have women on a plane unless it has a lavatory. Jim Burns, Founder of Magnum Air Vertical differentiation Ed Iacobucci, CEO of DayJet Corp. VS Adam Aircraft A700 Bigger, more expensive Has a lavatory Having a bathroom on board is not an issue for short trips. Eclipse 500 Less expensive No lavatory © Cambridge University Press 2015 34 Chapter 5 - Vertical differentiation Vertical differentiation: Entry problem revisited • • Recall Chapter 4: Quality augmented Cournot may bound the number of firms in oligopolistic markets. (Requires costly quality choice.) The present model predicts a limited number of firms even for costless quality choice and arbitrarily small entry costs. • • The presence of a small entry cost creates a small economies of scale, which turns out to be sufficient to limit the number of active firms. We may even have a natural monopoly. But the equilibrium number of firms goes to ∞ as the mass of consumers 𝑀 = 𝜃 − 𝜃  ∞. © Cambridge University Press 2015 35 Chapter 5 - Vertical differentiation Vertical differentiation: Entry problem revisited • Formally, recall that if 𝜃 ≤ 2𝜃 the low quality firm shuts down. • No other firm will have an incentive to enter.  Natural monopoly. • More generally, it can be shown that, for arbitrarily small entry costs, the equilibrium number of active firms is the smallest integer 𝑛 such that 𝜃 ≤ 2𝑛 𝜃 • See the book for the details. • Note: In contrast to the earlier model, equilibrium number of firms “slowly”  ∞ as the mass of consumers, 𝜃 − 𝜃, goes to ∞. © Cambridge University Press 2015 36 Chapter 5 - Empirical analysis Probabilistic choice • • Discrete choice problem: Choose one among few options. Empirical analysis of discrete choice problems are based on the so called “random” or “probabilistic” choice models. • Random component is meant to capture consumer heterogeneity in tastes or quality sensitivity etc. • There can also be unpredictable variations in the behaviour of a given consumer. © Cambridge University Press 2015 37 Chapter 5 - Empirical analysis Probabilistic choice & horizontal differentiation • • Suppose the utility of a product 𝑖 is a random variable: 𝑣𝑖 ≡ 𝑣ҧ𝑖 + 𝜀𝑖 𝑣ҧ𝑖 ≡ 𝑢(𝑟, 𝑝𝑖 ) is the (mean) utility, including the effect of price • The observable part of utility that we can estimate. • 𝜀𝑖 is the random part: Exogenous. • Think of it as a random taste parameter: For example, this can be the distance between a particular product location 𝑖 and a randomly chosen consumer. • Assumption: The expected value of 𝜀𝑖 is 0 •  E(𝑣𝑖 ) = 𝑣ҧ𝑖 . So, 𝑣ҧ𝑖 is the mean or expected utility from product 𝑖. In the “linear city” this is the utility of the consumer 𝑥 = 1/2 from the product 𝑖 © Cambridge University Press 2015 38 Chapter 5 - Empirical analysis Probabilistic choice & horizontal diff. (cont’d) • • • Let 𝑒𝑖 denote the realization of 𝜀𝑗 − 𝜀𝑖 . Our randomly chosen consumer selects the product 𝑖 over 𝑗 iff 𝑣𝑖 > 𝑣𝑗 , iff 𝑣ҧ𝑖 − 𝑣𝑗ҧ > 𝑒𝑖 Thus, with two products, and assuming continuous distributions, the choice probability of 𝒊 is: Pr(𝑒𝑖 ≤ 𝑣ҧ𝑖 − 𝑣𝑗ҧ ) ≡ 𝐹𝑖 (𝑣ҧ𝑖 − 𝑣𝑗ҧ ), where 𝐹𝑖 is the distribution function of 𝜀𝑗 − 𝜀𝑖 . • Typically 𝜀1 and 𝜀2 are assumed to be i.i.d. with a well behaved distribution (e.g., logistic distribution). •  Particular functional form for 𝐹𝑖 (𝑣ҧ𝑖 − 𝑣𝑗ҧ ) © Cambridge University Press 2015 (optional: see the book) 39 Chapter 5 - Empirical analysis Probabilistic choice & horizontal diff. (cont’d) • • Let 𝛼𝑖 denote the market share of product 𝑖. Our first demand equation is: 𝛼𝑖 = 𝐹𝑖 (𝑣ҧ𝑖 − 𝑣𝑗ҧ ) • • LHS is observable, RHS is an exogenously given function of the variables 𝑣ҧ1 and 𝑣ҧ2 . Second demand equation decomposes 𝑣ҧ𝑖 : 𝑣ҧ𝑖 = 𝛽𝑥𝑖 −𝛾𝑝𝑖 +𝜉𝑖 (D2) • • • • (D1) 𝑥𝑖 is the vector of observed product characteristic (location, level of sugar or alcohol etc.) 𝛾 measures the effect of price 𝜉𝑖 is an error term, that will be left unexplained Use (D1) and (D2) to estimate (𝛽, 𝛾) and thereby 𝑣ҧ1 , 𝑣ҧ2 © Cambridge University Press 2015 40 Chapter 5 - Empirical analysis Probabilistic choice & product diff.: Final remarks • • First order conditions of the firms will also depend on the demand function/market share, which will give one more equation that depends on 𝛾. If there are 𝑛 products, choice probability of 𝑖 will be: Pr 𝑣𝑖 = max 𝑣1 , … , 𝑣𝑛 • • = Pr 𝑣ҧ𝑖 − 𝑣𝑗ҧ ≥ 𝜀𝑗 − 𝜀𝑖 ∀𝑗 ≠ 𝑖 This can be computed as a function of 𝑣ҧ1 , … , 𝑣𝑛ҧ given the joint distribution of 𝜀1 , … , 𝜀𝑛 . In case of vertical differentiation, we need an additional random variable 𝜃𝑘 that represents the quality-sensitivity of consumer 𝑘. (The main methodological ideas are similar.) © Cambridge University Press 2015 41 Chapter 5 - Review questions Review questions • What makes firms locate close to each other in the product space? And what does it make them differentiate themselves from their competitors? • Explain the main difference between horizontal and vertical product differentiation. • Determine if the following statements are true or false. Explain your answer. • In horizontal product differentiation, firms always select most extreme positions. • In a model of vertical product differentiation with sequential moves, the firm that selects the quality first is advantageous. • The number of firms in an industry with constant marginal costs necessarily converges to infinity as the entry cost goes to zero. © Cambridge University Press 2015 42