The Multiproduct Monopolist Under Vertical Differentiation: an Inductive Approach

Lambertini, Luca Working Paper The Multiproduct Monopolist Under Vertical Differentiation: an Inductive Approach Quaderni - Working Paper DSE, No. 226 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (1995) : The Multiproduct Monopolist Under Vertical Differentiation: an Inductive Approach, Quaderni - Working Paper DSE, No. 226, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5087 This Version is available at: https://hdl.handle.net/10419/159069 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ THE MULTIPRODUCT MONOPOLIST UNDER VERTICAL DIFFERENTIATION: AN INDUCTIVE APPROACH Luca Lambertini Keywords: monopoly, quality distortion, output restriction JEL classification: L12 Abstract The behaviour of a multiproduct profit seeking monopolist is evaluated vis à vis that of a social planner, in a model where there is a continuum of consumers characterized by different marginal willingnesstopay for quality.When themarket iscompletely covered,the monopolist undersupplies all qualities as long as their number is finite. When quality becomes continuous, the richest consumer is provided with the socially optimal quality. Under the alternative assumption of partial market coverage, the monopolist supplies the same qualities as the social planner, restricting though totaloutput. Finally, it turnsout that, for a given number of varieties, under partial market coverage the monopolist can make at least as good as under full market coverage, so that she prefers to distort quantity rather than quality. Acknowledgements I would like to thank Guido Candela and Martin Slater for helpful comments and suggestions. the usual disclaimer applies. 1. Introduction Theissueofevaluatingthebehaviourofaprofitseekingmonopolistunderverticalproduct differentiation, as compared to the social optimum, has been the focus of several influential papers. Spence (1975, 1976) and Sheshinski (1976) have established that, although the monopolist tends to restrict output for a given quality, she introduces a bias in the provision of quality for a given output level, since in selecting quality the private monopolist takes into account the willingnessto pay of the marginal consumer,while a social planner would take into account that of the average consumer. Thus, the monopolist ends up undersupplying quality for a given output level if the average consumer’s valuation for quality is higher than the marginal consumer’s, and viceversa. The monopolist may offer several qualities of the same good in order to extract more consumer surplus. This is the subject of the contributions due to Mussa and Rosen (1978), Itoh (1983), Maskin and Riley (1984), and Besanko et al. (1987). A conclusion common to all these authors is that the monopolist resorts to an enlargement of the quality spectrum, as compared tothesocialoptimum,asascreeningdevicewhichenableshertodiscriminateamongconsumers. In particular, it has been shown that with either (i) two types of consumers (rich-poor) and two qualities (high-low) or (ii) with a continuum of consumers and qualities, the richest consumers, i.e., those with the highest valuation for quality, are provided with the socially optimal quality, while the poorest ones buy a suboptimal quality (for an illustrative argument, see Tirole, 1988, p.150). The above contributions, though, leave largely unanswered the following questions: If there exixts a continuum of consumers characterized by different incomes and thus different marginalwillingnesstopayforquality,whatkindofdistortionshallthemultiproductmonopolist prefer to introduce in the market? Will she exploit exclusively the quality distortion or alternatively a restriction in output, or rather a mix of both? In order to provide an answer, I shall adopt here a model which respects the general assumptions made in Mussa and Rosen (1978) as for technology and consumer tastes. 1 Considering a continuum of consumers characterized by different valuation for quality, under the assumption of full market coverage I show that, as long as the the quality spectrum of the privatemonopolistisdiscrete,i.e.,thenumberofvarietiesbeingprovidedisfinite,themonopolist undersupplies all qualities. Furthermore, the extent to which she distorts quality is inversely related to the marginal willingness to pay, both for a given number of varieties and as their number increases, so that one can rather paradoxically conclude that rich consumers would favourably consider product proliferation, while the opposite holds for poor consumers. Then, it is shown that if the monopolist only partially serves the market, she offers the same qualities that would be supplied by a social planner aiming at the maximization of social welfare. In such a situation, though, the monopolist produces only half the output of the social planner, both overall and for each variety. Finally, it appears that the monopolist is at least as welloffby restrictingoutput than sheis byservingall the market,so thatshe shouldbe expected to exert her monopoly power so as to exclude some individuals from consumption rather than providing them with a range of suboptimal qualities. 2. The setting Consider a monopolistic market for a vertically differentiated good where the firm supplying the good may be thought of as being alternatively run by a social planner taking care of social surplus or by a profit seeking monopolist. I shall compare the behaviour of these two agents assuming they provide the same number of varieties, n. Consumers are uniformly distributed with unit density over the interval so that the total number of individuals is normalised to 1. Parameter represents consumer’s marginal willingness to pay forquality, and itmay be interpreted asthe reciprocal ofthe marginal utilityof nominal income, or money (Tirole, 1988, pp.96-7). Each consumer buys one unit of the variety of the product maximizing the net surplus he obtains, provided that the latter is non-negative: [θ,θ], θ=θ+1, θ>0, θ 2 where qiis the quality of variety iand piis the price charged by the monopolist for that quality. Production involves variable costs only: where xiis the quantity produced and tis a positive parameter. Hence, if marginal cost pricing is considered, it is possible to define the range of consumers’ preferred qualities as the interval (see Cremer and Thisse, 1994). 2.1. Full market coverage Under the full market coverage assumption, condition (1) holds as an equality for the poorestconsumer, whileitholdsasastrictinequalityforanyotherindividual.Themainfeatures ofthebehaviourofthesocialplannerandtheprivatemonopolistasfarastheprovisionofproduct quality is concerned are summarized in the following PROPOSITION 1. As long as the number of varieties is finite, the profit seeking monopolist strictlyundersuppliesallqualitiesascomparedtothesocialoptimum.Asthenumberofvarieties tends to infinity, the highest quality she offers tends to coincide with the socially optimal one, while the difference between her lowest quality and the social planner’s lowest quality is increasing in the number of varieties and in the limit it is equal in size to the whole range of consumers’ preferred qualities. PROOF. In order show what is stated in the above Proposition, I shall proceed by induction. Hence, assume first that a single quality is supplied. The objective of the social planner is the U=θqi−pi≥0,i=1,2...n,(1 C=tqi2xi,i=1,2...n,(2 [θ/2t,θ/2t] 3 maximization of social welfare, defined as the sum of profit and consumer surplus: It can be quickly verified that the price level is irrelevant as for the problem described in (3), since it can only redistribute surplus from the consumers to the producer or viceversa, without modifying the overall level of welfare.1Thus, from the first order condition of (3) w.r.t q,I obtain i.e.,the social plannerlocates her product exactlyin themidpoint of thespectrum ofconsumers’ preferred qualities, as it also happens in spatial models àlaHotelling (see Bonanno, 1987, and Lambertini, 1995). Consider now the problem faced by a profit-maximizing monopolist, whose objective function is: max qsw =⌠ ⌡ θ θ(θq−tq2)dθ. (3 qsp =θ−1 4t,(4 πm=p−tq2.(5 1. Obviously, the same cannot be expected to hold under partial market coverage, since in such a case a price change brings about a change in quantity as well. 4 Thederivativew.r.t.priceisalwayspositive.Consequently,themonopolistwillsetthemaximum price consistent with the assumption of full market coverage, i.e., the price at which the poorest consumergives up all his surplusto purchasethe good, Accordingly, the profit function can be rewritten as follows: whichisconcaveandsingle-peaked,withthemaximumat thatisthequalitypreferred by the poorest consumer. This allows to conclude that when a single variety is available, the monopolist undersupplies quality as compared to the social optimum, and the "distance" amounts to one half of the interval of consumers’ preferred qualities. Assume now that two varieties are being supplied, qH>qL>0, produced at costs i=H, L. Market demands are given by: The objectivefunction ofthe profit-seeking monopolist andthe social planner are, respectively: and p= θq. πm= θq−tq2,(6 qm=θ/2t, qsp −qm=1/4t Ci=tqi2xi, xH=θ−(pH−pL) qH−qL;xL=(pH−pL) qH−qL−(θ−1). (7 πm=(pH−tqH 2)xH+(pL−tqL 2)xL;(8 sw =⌠ ⌡ θ h(θqL−tqL 2)dθ+⌠ ⌡ h θ(θqH−tqH 2)dθ. (9 5 Let’s take into account the problem of the private monopolist. Provided that pLis such that the poorest consumer is indifferent between purchasing or not, the price of the high quality good can be obtained from the first order condition (FOCs) derived from (8). Thus, equilibrium qualities are univocally determined by the FOCs of (8) w.r.t. qHand qL, with and Solving then the social planner’s problem yields and It immediately appears that the monopolist undersupplies both qualities, with and i.e., the "distance" between varieties in the high quality segment of the market amounts to one quarter of the range of consumers’ preferred qualities, while it amounts to three quarters in the low quality segment. Furthermore, the degree of differentiation adopted by the profit-seeking monopolist is twice as wide as that adopted by the social planner, due to the the monopolist’s attempt to extract as much consumer surplus as possible by enhancing differentiation beyond the socially preferable level. This, coupled with thefactthatthemonopolistundersuppliesqualitysincebyassumptionshecannotrestrictoutput, yields the result observed here. The same problem can be easily reformulated in the case of three varieties, where the private monopolist supplies and while the social planner produces and Subscript M indicates the intermediate quality. As compared to the previous case, enlarging the number of varieties leads to (i) an increase in the degree of differentiation under both market regimes; (ii) an increase (decrease) in the differentiation degree between the low (high) qualities supplied in the two market regimes, with the monopolist always undersupplying all qualities respect to the social optimum. Thus, I am now able to extend the analysis to the setting where nvarieties are offered, or, in other terms, where the distance between contiguous varieties tends to zero so that quality becomes a continuous variable, with qHand qLindicating now the highest and lowest qualities being supplied, respectively. In such a case, qH m=(2θ−1)/4t qL m=(2θ−3)/4t.qH sp =(4θ−1)/8t qL sp =(4θ−3)/8t. qH sp −qH m=1/8tq L sp −qL m=3/8t, qL m=(3θ−5)/6t,qM m=(θ−1)/2tq H m=(3θ−1)/6t, qL sp =(6θ−5)/12t,qM sp =(2θ−1)/4tq H sp =(6θ−1)/12t. 6 Thus, when nvarieties are available, the differentiation degrees under the two market regimes are, respectively: with and The result in (13) implies that as the number of varieties tends to infinity, the social planner provides each individual with his own most preferred quality, so that under social planning the degreeof differentiationcoincidesinthelimitwith therangeofconsumers’ preferencesinterms of quality. Asfortheextenttowhichtheprofitmaximizingmonopolistundesuppliesquality,observe that with qH m=nθ−1 2nt ;qL m=nθ−2n+1 2nt ;(10 qH sp =2nθ−1 4nt ;qL sp =2nθ−2n+1 4nt .(11 qH m−qL m=∆qm=n−1 nt ;qH sp −qL sp =∆qsp =n−1 2nt ,(12 ∆qm/∆qsp =2 lim n→∞∆qm=1 t,lim n→∞∆qsp =1 2t.(13 qH sp −qH m=∆qH=1 4nt =   1 2n       1 2t   ;qL sp −qL m=∆qL=2n−1 4nt =   2n−1 2n       1 2t   ,(14 7 Appendix Assume ten varieties are being supplied by the profit maximizing monopolist. From expression (10), it can be checked that must be greater than 19/10 for the lowest quality, to be positive. Compare then the total profits accruing to the monopolist in the two alternative settings, i.e., full and partial market coverage, as defined by expressions (27) and (28), respectively. In the case under analysis , the difference between the two amounts to: which simplifies to It is immediate to verify that the above difference is non-negative for all and it is nil in correspondence of Besides, straightforward calculations are needed to show thatthe criticalvalue of forwhichthe monopolistis indifferent betweenpartial andfull market coverage (which we have seen to correspond to 3 for n=1 and 21/10 for n=10) is decreasing in the number of varieties, so that as the latter becomes arbitrarily large, the profit seeking monopolist strictly prefers not to serve the entire market. qL m,θ πpc m−πfc m=55θ3 1323t−100θ3−200θ2+133 400t,(a1 πpc m−πfc m=(10θ−21)2(220θ−399) 529200t.(a2 θ∈]19/10,∞[, θ=21/10. θ 14 References Bonanno, G. (1987), Location Choice, Product Proliferation and Entry Deterrence, Review of Economic Studies, 54, pp.37-46. Besanko, D., S. Donnenfeld and L. White (1987), Monopoly and Quality Distortion: Effects and Remedies, Quarterly Journal of Economics, 102, pp.743-68. Cremer, H., and J.-F. Thisse (1994), Commodity Taxation in a Differentiated Oligopoly, International Economic Review, 35, pp.613-33. Itoh, M. (1983), Monopoly, Product Differentiation and Economic Welfare, Journal of Economic Theory, 31, pp.88-104. Lambertini, L. (1995), A Note on Spatial Monopoly, Rivista Internazionale di Scienze Economiche e Commerciali, forthcoming. Maskin, E., and J. Riley (1984), Monopoly with Incomplete Information, Rand Journal of Economics, 15, pp.171-96. Mussa, M., and S. Rosen (1978), Monopolyand Product Quality, Journal of Economic Theory, 18, pp.301-17. Sheshinski, E. (1976), Price, Quality and Quantity Regulation in Monopoly, Econometrica, 43, pp.127-37. Spence,M.(1975),Monopoly,QualityandRegulation,BellJournalofEconomics,6,pp.417-29. Spence, M. (1976), Product Differentiation and Welfare, American Economic Review, 66, pp.407-14. Tirole, J. (1988), The Theory of Industrial Organization, Cambridge, MA, MIT Press. 15