Product differentiation What is vertical product differentiation?
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4820–5 Product different. II Geir B. Asheim Introduction Demand Structure Product differentiation Vertical differentiation 4820–5 Equilibrium prices Equilibrium quality Equilibrium entry Geir B. Asheim Department of Economics, University of Oslo ECON4820 Spring 2010 Last modified: 2010.02.15 What is vertical product differentiation? 4820–5 Product different. II Geir B. Asheim Introduction Outline Ex: Cars of different quality and in different price segments Products differ in some characteristic (called quality ) in which all consumers agree what is best. Demand Structure Equilibrium prices Equilibrium quality Equilibrium entry For equal prices, all prefer the variant with higher quality. The optimal choice is the same for everyone. For different prices, consumer make different choice (e.g. due to differences in taste or income). They differ in their willingness to pay for increased quality. Key reference: Shaked & Sutton, Relaxing price competition through product differentiation, Rev Econ Stud (1982) 3–13 Outline 4820–5 Product different. II Demand facing two vertically differentiated firms Geir B. Asheim Introduction Structure of the game Outline Demand Structure Equilibrium prices when two vertically differentiated firm compete in prices Equilibrium prices Equilibrium quality Vertical differentiation in equilibrium when two firms have entered Equilibrium entry How many firms will enter in equilibrium? Demand facing two vertically differentiated firms 4820–5 Product different. II s — quality θ — measure ⎧ ⎪ ⎨θs − p U = ⎪ ⎩ 0 Geir B. Asheim Introduction Demand Structure Equilibrium prices Equilibrium quality Equilibrium entry of a consumer’s taste for quality if a consumer with taste θ buys quality s at price p if no purchase F (θ) — cumulative distribution function of consumer type Let s1 < s2 and write Δs = s2 − s1 . If p2 s2 < p1 s1 , then: D2 (p1 , p2 ) = 1 − F D1 (p1 , p2 ) = 0 If p2 s2 > p1 s1 , then: p2 s2 −p1 D2 (p1 , p2 ) = 1 − F p2Δs −p1 D1 (p1 , p2 ) = F p2Δs − F ps11 Structure of the game 4820–5 Product different. II Geir B. Asheim Introduction Demand (1) Firms choose whether to enter (2) Entered firms choose quality (3) Given entry and quality, firms choose prices (4) Demand determines quantities (unit cost = c) Structure Equilibrium prices Equilibrium quality Equilibrium entry Assumption θ is uniformly distributed on [θ, θ̄], where θ̄ = θ + 1. Assumption (Ensures that, in equilibrium, θΔs ≤ p2 − p1 ) θ̄ ≥ 2θ Assumption c+ θ̄−2θ 3 Δs ≤ θs1 (Ensures that, in equil., p1 ≤ θs1 ) Equilibrium prices when two vertically differentiated firm compete in prices 4820–5 Product different. II Geir B. Asheim Introduction Let Δ̄ ≡ θ̄Δs & Δ ≡ θΔs. If Δ̄ ≥ p2 − p1 ≥ Δ & θs1 ≥ p1 , then −p1 −p1 D2 (p1 , p2 ) = 1 − p2Δs − θ = θ̄ − p2Δs = Δ̄−(pΔs2 −p1 ) Δ̄−p2 maxp2 (p2 − c)D2 (p1 , p2 ) = maxp2 (p2 − c) p1 +Δs 0= Demand Structure Equilibrium prices Equilibrium quality Equilibrium entry D1 (p1 , p2 ) = p1 +c+Δ̄−2p2 Δs p2 −p1 − θ − Δs ⇒ 0= p2 = 12 (p1 + c + Δ̄) p2 −p1 Δs −θ = maxp1 (p1 − c)D1 (p1 , p2 ) = maxp1 (p1 − 0= p2 +c−Δ−2p1 Δs ⇒ (p2 −p1 )−Δ Δs p2 −Δ−p1 c) Δs p1 = 12 (p2 + c − Δ) Prices are strategic complements. Nash equilibrium prices: p2c = c + p1c = c + 2Δ̄−Δ 3 Δ̄−2Δ 3 =c+ =c+ 2θ̄−θ 3 Δs θ̄−2θ 3 Δs ∈ [c, θs1 ] by the assumptions we have made. Vertical differentiation in equilibrium when two firms have entered 4820–5 Product different. II Geir B. Asheim Introduction Demand Structure Equilibrium prices Equilibrium quality Equilibrium entry Assume that it is possible to choose qualities s1 & s2 in [s, s̄], where c + θ̄−2θ 3 (s̄ − s) ≤ θ s. Assume w.l.o.g. that s1 ≤ s2 . By plugging in equilibrium prices in the profit expressions (where we use that p2 − p1 = θ̄+θ 3 Δs): 2θ̄−θ 2 θ̄+θ Δs θ̄ − Δs Π2 (s1 , s2 ) = 2θ̄−θ = 3 3 3 θ̄+θ θ̄−2θ 2 Π1 (s1 , s2 ) = θ̄−2θ Δs − θ Δs = 3 3 3 If s1 = s2 , then Π2 (s1 , s2 ) = Π1 (s1 , s2 ) = 0. If s1 > s2 , then Π2 (s1 , s2 ) is increasing in s2 and Π1 (s1 , s2 ) is decreasing in s1 . Therefore, there are 2 Nash equilibria in pure strategies (2 equilibria as the firms can be labeled in 2 different ways): The one firm chooses s̄; the other firm chooses s. Maximal differentiation to relax price competition. The firm with higher quality (= s̄) earns higher profit. With sequential entry, the first entrant will choose s̄. How many firms will enter in equilibrium? 4820–5 Product different. II Geir B. Asheim Introduction What happens if three or more firms enter? They all set quality = s̄ and price = c and earn zero profit. What happens if only one firm enters? Profitable for another firm to enter. Demand Structure Equilibrium prices Equilibrium quality Equilibrium entry Conclusion: In any pure strategy subgame-perfect equilibrium, two (and only two) firms enter and they maximize their vertical product differentiation. In this model of vertical differentiation, the number of firms is determined by consumer heterogeneity (θ̄ ≥ 2θ). Note that the model assumes that it does not cost more to produce the high quality variant than the low quality variant.
