Would you like to enter first with a low-quality good?

Lambertini, Luca; Tedeschi, Piero Working Paper Would you like to enter first with a low-quality good? Quaderni - Working Paper DSE, No. 494 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Tedeschi, Piero (2003) : Would you like to enter first with a low-quality good?, Quaderni - Working Paper DSE, No. 494, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4800 This Version is available at: https://hdl.handle.net/10419/159335 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Luca Lambertini§and Piero Tedeschi# §Dipartimento di Scienze Economiche Universit`adiBologna Strada Maggiore 45, 40125 Bologna, Italy fax: 0039-051-2092664; [email protected] # Dipartimento di Statistica Universit`a di Milano-Bicocca Dipartimento di Statistica Via Bicocca degli Arcimboldi, 8, 20126 Milano, Italy fax: 0039-02-6473312; piero.tedesc[email protected] December 3, 2003 Abstract Using a two-period duopoly model with vertical differentiation, we show that there exists a unique subgame perfect equilibrium where the first entrant supplies a lower quality and gains higher profits than the second entrant. We also prove that this entry sequence is also socially efficient. JEL Classification: C73, D43, L13 Keywords: entry, vertical differentiation 1Introduction According to the established wisdom concerning vertically differentiated markets, earlier entrants appropriate the high-quality niches, while later entrants fill the remaining lower part of the quality spectrum (Gabszewicz and Thisse, 1979, 1980; Shaked and Sutton, 1982, 1983; Donnenfeld and Weber, 1992; Lehmann, 1997). This is due to two basic assumptions according to which the distribution of cumsumers’ willingness to pay is uniform and the game unravels in a single period. Here, we want to relax the second assumption, by adopting a simple two-period setup, with sequential entry.1Using a model whose original formulation is in Cremer and Thisse (1991), we show that profit incentives drive firms toward a unique subgame perfect equilibrium where the first entrant supplies a lower quality and gains higher profits as compared to the second entrant. Moreover, we also prove that this entry sequence is also socially efficient, in that it entails a higher average quality level than the alternative one. The remainder of the note is structured as follows. The setup is laid out in section 2. The entry process and the welfare performance are investigated in section 3. 1.1 The model Themarketexistsovertwoperiods,t∈{0,1}. In each period, a population of consumer of unit size is uniformly distributed over the interval £θ, θ¤,with θ=θ−1.Parameter θ∈£θ, θ¤measures a consumer’s marginal willingness 1In the above mentioned literature, entry is euristically considered in one-shot games, without allowing any explicit role for calendar time. One exception is Dutta et al. (1995) where, however, a high-quality advantage obtains. 1 to pay for quality, and the net surplus from consumption is: U=θqi−pi≥0(1) where piand qiare the price and quality of the product supplied by firm i. We assume that (1) holds for all consumers in both periods, so that the market is always fully covered. On the supply side, any firm imust bear total cost Ci=cq2 ixiper period, where xiis the market demand for her product and cis a positive parameter. Accordingly, firm i’s profit function is πi=( pi−cq2 i)xiin each period. In the remainder, we will consider the following game. Each firm irreversibly sets quality at the time of entry. At t=0 ,the firm 1 enters and remains a monopolist in that period. At t=1 ,firm 2 enters and the market becomes a duopoly. Hence, the problem of the first entrant (the leader) consists in choosing whether to offer a lowor high-quality good, correctly anticipating the optimal behaviour of the second entrant (the follower). That is, the stage describing quality choices is going to be solved `alaStackelberg. Once both qualities are set, Bertrand-Nash competition takes place. The objective of the leader (firm 1) is: max pM,p1,q1 Π1≡πM 1+δπD 1=pM−cq2 1+δ¡p1−cq2 1¢x1(2) where πM 1=pM−cq2 1are monopoly profits at t=0 ,π D 1=( p1−cq2 1)x1 are duopoly profits at t= 1, the latter being discounted by the factor δ≡ 1/(1 + ρ),with δ∈[0,1] for all ρ∈[0,∞).Theobjectiveofthefollower (firm 2) consists in maximising duopoly profits πD 2=( p2−cq2 2)x2w.r.t. p2 and q2. The two firms will supply qualities qH≥qL>0 at duopoly prices pH≥ pL,and either q1=qL;q2=qHor the opposite. In either case, at t=1 ,the 2 consumer indexed by bθ=( pH−pL)/(qH−qL)willbeindifferent between the two goods, so that we may define duopoly demands as follows: xH=θ−pH−pL qH−qL ;xL=pH−pL qH−qL −¡θ−1¢.(3) 1.2 Optimal pricing behaviour The optimal monopoly pricing at t= 0 can be quickly characterised, for any given q1.Under full coverage, firm 1 sets the price driving to zero the net surplus of the poorest consumer located at θ=θ−1,i.e., pM=¡θ−1¢q1. Hence, monopoly profits are πM 1=¡θ−1¢q1−cq2 1. The Nash game in prices is well known; therefore we omit a detailed exposition (see Cremer and Thisse, 1991, and Lambertini, 1996, inter alia). Equilibrium prices are: pH=(qH−qL)(θ+1)+2 cq2 H+cq2 L 3;pL=(qH−qL)(2 −θ)+2 cq2 L+cq2 H 3(4) so that profit functions simplify as follows: πH=(qH−qL)£θ+1−c(qH+qL)¤2 9;πL=(qH−qL)£2−θ+c(qH+qL)¤2 9. (5) Hence, we have two alternative scenarios. The first, where q1=qLand q2=qH,is labelled as low-quality leadership; the second, where q1=qHand q2=qL,is labelled as high-quality leadership. 3 2Entry 2.1 Low-quality leadership In this case, πD 1=πLand πD 2=πH.The leader’s problem consists in:2 max qL Π1L=¡θ−1¢qL−cq2 L+(qH−qL)£2−θ+c(qH+qL)¤2 9(1+ρ) s.t. :∂πH ∂qH =0⇔q∗ H=θ+1+cqL 3c(6) Plugging q∗ Hinto Π1Land solving the first order condition (FOC) ∂Π1L/∂qL= 0 w.r.t. qL,we obtain: qL=16θ−81ρ−113 ±3p9ρ(81ρ+ 194) + 1081 32c(7) with lim ρ→∞ q− L=−∞ ; lim ρ→∞ q+ L=θ−1 2c the latter being the single-period optimal monopoly quality (see Lambertini, 1997). Therefore, Stackelberg equilibrium qualities are: ql L=16θ−81ρ−113 ±3p9ρ(81ρ+ 194) + 1081 32c qf H=16θ−27 (1 + ρ)+p9ρ(81ρ+ 194) + 1081 32c (8) where superscripts land fstand for leader and follower, respectively. The associated equilibrium profits are: Π1L=©1152θ¡θ−2¢(1 + ρ)−ρ¡19683ρ2+ 70713ρ+ 84969¢+ −33427 + Ψ3ª/[4608c(1 + ρ)] 2There exists another solution to ∂πH/∂qH=0,i.e., qH=¡θ+1−cqL¢/c. However, this can be excluded on the basis of concavity conditions. 4 πH=ρ(19683ρ2+ 82377ρ+ 115641) + 54739 −[1657 + 9ρ(81ρ+242)]Ψ 2304c(9) where Ψ≡p9ρ(81ρ+ 194) + 1081.Resorting to numerical simulations, it can be ascertained that Π1L>π Hin the admissible range of parameters.3 Equilibrium market shares in the duopoly phase are: xl L=Ψ−27ρ−19 24 ;xf H=27ρ+43−Ψ 24 ;xl L>x f Halways. (10) Finally, bθ∈¡θ, θ¢always. 2.2 High-quality leadership Now πD 1=πHand πD 2=πL.The leader’s problem consists in:4 max qH Π1H=¡θ−1¢qH−cq2 H+(qH−qL)£θ+1−c(qH+qL)¤2 9(1+ρ) s.t. :∂πL ∂qL =0⇔q∗ L=θ−2+cqH 3c(11) Adopting the same procedure as in the previous case, we can find the optimal qualities: qf L=16θ+27 ρ+11−p9ρ(81ρ+ 226) + 1369 32c ql H=16θ+81 ρ+97−3p9ρ(81ρ+ 226) + 1369 32c (12) with limρ→∞ qH=¡θ−1¢/(2c).Equilibrium profits are: Π1H=©1152θ¡θ−2¢(1 + ρ)−ρ¡19683ρ2+ 82377ρ+ 111753¢+ −48547 + Ψ3ª/[4608c(1 + ρ)] 3The ranges are ρ∈[0,∞),i.e., δ∈[0,1] ,and θ≥7/2.The latter condition ensures that full coverage obtains in duopoly. If so, then it also holds under monopoly, where θ≥3wouldsuffice (see Lambertini, 1996, 1997). 4There exists another solution to ∂πL/∂qL=0,i.e., qL=¡θ−2−cqH¢/c. Again, this can be excluded on the basis of second order conditions. 5 πL=ρ(19683ρ2+ 88209ρ+ 130761) + 64027 −(9ρ+13)(81 ρ+ 133) Φ 2304c(13) where Φ≡p9ρ(81ρ+ 226) + 1369.Again, numerical simulations show that Π1H>π Lin the admissible range of parameters.5Equilibrium market shares at t=1are: xl L=27ρ+43−Φ 24 ;xf H=Ψ−27ρ−19 24 ;xl L<x f Halways. (14) Again, bθ∈¡θ, θ¢always. 2.3 The subgame perfect equilibrium and welfare assessment In order to complete the characterisation of the subgame perfect equilibrium, consider the following inequalities: Π1L>Π1Hand πH>π Lfor all ρ∈[0,∞).(15) This holds for any admissible value of θ. Accordingly, we may state: Proposition 1 The first entrant prefers to supply a low-quality good, while the second entrant prefers to supply a high-quality good. Therefore, the subgame perfect equilibrium is unique and involves q1=ql L;q2=qf H. Now we pass on to examine the welfare performance of the market in the two cases. In general, the definition of the discounted social welfare over the two periods is: SW ≡πM 1+πD 1+πD 2 1+ρ+CS (16) 5In this case, the result is intuitive, in that high-quality supply combines with a period of monopoly power, hence the first entrant’s profits are necessarily higher than the second entrant’s. 6