PartialRegulationinVerticallyDifferentiated. Industries

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1 PartialRegulationinVerticallyDifferentiated Industries Angela S. Bergantino Etienne Billette de Villeeur Annalisa Vinella Preliinary and Incoplete Abstract In this paper we provide a theoretical foundation for price-and-quality cap regulation of the recently liberalized utilities. We odel a partially regulated oligopoly where vertically differentiated services, such as high-speed and low-speed transportation services, are provided by a regulated incubent and a unregulated entrant copeting in price and quality. The odel ay equally well represent copetition within or across industries. We establish that weights in the cap need to depend also on the arket served by the entrant, despite the latter is not directly concerned by the partial regulation. This calls for the possibility for regulators to use inforation about the whole industries, rather than the sole incubents. Keywords: Price-and-Quality Cap; Partial Regulation; Vertical Differentiation J.E.L. Classification Nubers: L11, L13, L51, L9 This paper has been presented at the XVIIIth eeting of SIEP, Università di Pavia and at the 5th Conference on Applied Infrastructure Research, Technische Universität Berlin. We would like to thank seinar participants for their coents and in particular Carlo iorio and Ingo Vogelsang. All reaining errors are ours. University of Bari, Departent of Econoics, Via C. Rosalba, 53, Bari (Italy). E-ail: a.bergantino@dse.uniba.it Corresponding Author; University of Toulouse, IDEI and GREMAQ, Manufacture des Tabacs, Aile J.-J. Laffont, 21 Allée de Brienne, Toulouse (rance). E-ail: etienne.devilleeur@univ-tlse1.fr University of Toulouse, GREMAQ, Manufacture des Tabacs, Aile J.-J. Laffont, 21 Allée de Brienne, Toulouse (rance); University of Bari, Departent of Econoics, Via C. Rosalba, 53, Bari (Italy); European University Institute, RSCAS, lorence School of Regulation, Via delle ontanelle, 19, I-50016, San Doenico di iesole (Italy). 1

2 "Price cap regulation rates as one of the success stories of applied econoic theory. (...) it strikes a very good coproise between the theoretically rigorous foundation of the theory of optial regulation for ultiproduct firs (...) and the practitioner s requireent of the siple, easy-to-understand, easy-to-apply rule." (De raja and Iozzi [16], p.1) 1 Introduction In a seinal piece of work, Laffont and Tirole [21] show that capping the prices of a ultiproduct onopolist by eans of a constraint in which any such price is attributed a weight equal (or proportional) to the correctly forecasted optial quantities induces the regulated fir to charge the Rasey prices, so that the second-best optiu for a onopolistic arket is entailed. This result "restates" the one tracing back to Brennan [10] that the price cap eerges as the solution to a proble of welfare-constrained profit axiization, output quantities being the appropriate weights to be attributed to the allowable deviations fro the socially desirable prices. The ones previously entioned are definitely not the sole research studies about the socalled ideal price cap. Indeed, nuerous other authors have concentrated on the subject. Soe such papers generically look at industries providing services of general interest. Aong others, one ay recall De raja and Iozzi [16] and [15] as well as Iozzi, Poritz and Valentini [20]. As for the works which, instead, focus on particular sectors, one ay reeber Billette de Villeeur, Creer, Roy and Toledano [8], who investigate the price cap regulation in the postal sector. irs that are copelled to eet regulatory obligations often provide services of socially suboptial quality. Such a phenoenon has been deonstrated on a theoretical ground as well as observed in several real-world situations. In consideration of this, regulators tend to exert special effort for controlling the quality supplied by the regulated operators. The concern for quality is particularly strong when utilities are subject to price cap regulation, due to the incentive the latter provides to cost cutting, which ay translate into reductions in the level of offered quality. In one of the contributions previously listed, 2

3 De raja and Iozzi [15] recall that, as fro Rovizzi and Thopson (1992), a noticeable reduction in quality was registered in British Teleco s services iediately after privatization, as soon as the copany was subject to price cap regulation without specific quality provisions. Then the Authors address the issue by laying down a theoretical foundation of the price-and-quality regulation, which eerges by integrating the price cap with the quality diension. As the latter is introduced to obtain a global price-and-quality cap, it is found that appropriate price weights are still the optial quantities, whereas opportune quality weights are given by the consuer arginal surplus evaluated at the optial prices and qualities 1. Observe that the overall econoic foundation of the ideal price(-and-quality) cap, to which all the contributions previously entioned belong, refers to onopolistic industries. This circustance is easily explained. After the second World War, alost all governents chose to structure the provision of services of general interest as State controlled-andowned vertically integrated onopolies. Over tie, this organizational odel has lost oentu, due to the dissatisfactory perforance experienced in ters of investent, efficiency, incentives and the growing budgetary probles. In the recent decades, network industries, such as telecounications, water, energy and transportation, have either undergone a ajor process of privatization 2 or becoe coercial (hence, profit-oriented rather then welfare-concerned) players to be regulated, ownership being (at least forally) separated fro regulatory tasks. Having in ind these real-world scenarios, econoists have devoted close attention to onopoly regulation in general, and to price cap onopoly regulation in particular, with and without quality adjustents. However, in a plurality of cases, the segents of the network industries that are not concerned by subadditive technologies have also been opened up to copetition. As a consequence of the liberalization process, partially regulated oligopolies arise, where the 1 The list of papers about price cap regulation we ention is far fro exhaustive. There also exists a full class of works which focus on the practical application of the price cap regulation, either in a static or in a dynaic version. Soe such papers are Vogelsang and insinger [28], Littlechild [19], orean [17] and Billette de Villeeur [7]. However, we do not strictly hinge on the literature background about ipleentation, to which those studies belong. or the sae reason, Brennan [10] and De raja and Iozzi [15] are solely referred to as for the theoretical foundation of the price cap and of the price-and-quality cap respectively, not as for the ipleentation schees they propose. More generally, see Vogelsang [27] for a survey of the literature about incentive regulation in general and its ipleentation over the last two decades. See Arstrong and Sappington [2] as well. 2 See Martiort [23] for a discussion about the costs and benefits associated with privatization in an incentive theory perspective. 3

4 incubent firs (naely, the forer onopolists) are subject to regulatory obligations, whereas the entrants are allowed to operate uncontrolled, though strategic. Such policies are eant, on one side, to proote access and (soe) copetition and, on the other side, to guarantee the collectivity with a reliable supply of the concerned services at affordable prices, by aking the incubents less arket powerful but still financially viable. Biglaiser and Ma [6] provide the exaple of AT&T, which operates as a regulated doinant fir in the long-distance telecounications arket, where it copetes with the unregulated MCI and Spring 3. A crucial question arises. Can we expect the by now established onopoly regulation to be valid also for iperfectly copetitive partially regulated sectors? Clearly, there is no econoic reason for this to be apriorithe case. This is confired by the conclusions achieved by Biglaiser and Ma [6] who, investigating the optial regulation of a doinant fir playing the arket gae against an unregulated copetitor, show that the regulatory prograe is sensitive to the presence of a second operator endowed with arket power. New constraints and trade-offs add up in the regulatory process, beyond those faced in the onopoly regulation, and the result is ultiately a third-best one. This finding, which eerges fro a Bayesian context of optial regulation under asyetric inforation, conveys a lesson that usefully applies to price cap regulation as well. Provided that the latter lacks a norative basis with reference to the new relevant situations, the necessity arises to revise the underlying theory, so that the ipleentation practice can then be updated accordingly. Notably, this need is envisaged already in Brennan [10]. Indeed, on one side, his results are specified to easily extend to scenarios where a copetitive fringe is present, the relevant output quantities being just those of the regulated fir, rather than total arket quantities. On the other side, conclusions are recognized to be hardly applicable to situations where copeting firs are not price takers 4. The ultiate goal of the present paper is to provide a theoretical foundation for the price-and-quality cap regulation in oligopolistic environents, where a regulated doinant 3 See also Hel and Jenkinson [18], who report that regies of partial regulation apply to railways and truck freight transport in Argentina and in the USA, as well as to the natural gas and oil sectors in Gerany, inland and Hong Kong. Other fors of partial regulation are those concerning forer onopolies currently engaged both in regulated and unregulated activities in different arket segents. 4 Check footnote 6 at page 144 and footnote 10 at page 145 in Brennan [10]. 4

5 fir (a Stackelberg leader) copetes in price and quality with one (or ore) unregulated follower(s) not behaving as price taker(s). In particular, we focus on a arket where vertically differentiated services are supplied to a population of consuers exhibiting heterogeneous valuations for quality. This is eant to account for the iportance quality takes in the provision of basically all services of general interest. The interpretation of our odel is twofold. irst of all, it represents copetition between asyetrically regulated operators within soe given industry. To fix ideas, one ay think about a regulated doinant train operator copeting with one or ore unregulated rivals, whose services display different qualities, such as high-speed and lowspeed train services. Secondly, our odel stylizes copetition between regulated and unregulated industries. Insisting on transportation, one ay consider the inter-odal copetition between regulated train operators and deregulated air carriers, whose services differ as for speed, frequency, scheduling reliability, cofort. Albeit transportation is particularly well placed to illustrate the flexibility of our odel, it is noteworthy that the latter does apply to the ajority of the utilities. Indeed, as further exaples of relevant quality diensions, one ay recall the continuity and constancy of the supply of Internet services as well as the reaction lags in electricity generation and provision 5. Within the context so far described, we first characterize the (constrained) optial partial regulatory policy. This preliinary step is necessary because, as we entioned, due to the presence of the rivals, such a policy is no longer the well-known second-best onopoly one. In particular, Rasey pricing no longer yields (constrained) efficiency. Therefore, the second-best optiu for a onopoly does not constitute the objective of the regulatory echanis. Relevant target is, instead, the policy which arises whenever a welfare-axiizing fir is in the position of a Stackelberg leader with respect to one (or ore) profit-axiizing rival(s), facing the requireent that its profits be non-negative. In this perspective, our work is close to the ixed oligopoly odels, naely Rees [25] and Bos [9], in which the public fir is taken to be a first over vis-à-vis the private operator(s) 6. 5 Crapes and Moreaux [11] stress that this quality aspect of the energy provision introduces a heterogeneity diension in the generated electricity, which is otherwise an hoogeneous product. 6 In particular, addressing the issue of which pricing policy a public fir should pursue if it faces constraints (naely, the requireent of operating at zero profits), Bos [9] stresses that the fir should 5

6 At later stage, we establish that the optial policy can be decentralized to the incubent by eans of an ideal price-and-quality cap, the truly practical ipleentation issue being, instead, left for further research. Our ajor finding is that, despite partial regulation does not directly concern the follower(s), decentralization of the optial policy requires that weights in the price-and-quality cap depend not only on the portion of arket that is served by the incubent, but also on the one that is covered by the unregulated copetitor(s). As a result, regulatory bodies of the incubents in liberalized industries should be allowed to use the available knowledge (if any) and/or to extract inforation (otherwise) about the overall industry, rather than being restricted to solely access and rely upon the (available) inforation about the targeted operators. The paper is organized as follows. Section 2 presents the general theoretical fraework. In particular, it details over consuer preferences and behaviour and producer technologies and profit functions. Section 3 illustrates the ipact of the incubent s actions on the entrant s decision and characterizes the optial partial regulatory policy in a Stackelberg oligopoly. Section 4 focuses on the decentralization of the target policy by eans of an appropriate price-and-quality cap. The latter is further investigated as for the special case of unit deand, which returns especially useful and intuitive insights. Section 5 concludes. 2 The Model We consider an industry where two firs provide vertically differentiated products. One fir, denoted I for incubent, is subject to regulation. The second fir, denoted, is a newly entered operator. The two providers play a Stackelberg gae, the incubent in the role of the leader and the entrant in that of the follower. Strategic variables are prices (p I and p ) and qualities (q I and q ). Given the latter, the goods that are provided by the two firs turn out to be perfect substitutes. The population of consuers is heterogeneous in that each individual can be identified by eans of a paraeter θ. The latter expresses the personal valuation for the quality of the purchased product. Consuers do not differ in other diensions. The characteristic opt for a odified Rasey-pricing rule. In their turn, Beato and Mas Colell [4] show that the solution to a hoogeneous-product Stackelberg gae, where the public fir oves first, corresponds to average-cost pricing for the public fir. Indeed, since the latter selects its output before the private copetitor, it chooses so that, at equilibriu, it obtains zero profits. See Nett [24] for a survey about the hoogeneousgood ixed oligopoly literature. 6

7 θ is distributed on the copact interval θ, θ, with θ > 0, density f (θ) and cuulative distribution function (θ). Given her quality valuation, the θ consuer patronizing fir i {I,} faces the so-called generalized price ep i (θ) p i θq i, which is equal to the unit onetary price (p i ) net of the individual benefit associatedwiththeproductquality (θq i ). The θ consuer prefers purchasing the good fro fir i, rather than fro fir j, whenever she bears a saller generalized price (ep i < ep j ) by doing so. The arginal consuer, who is indifferent between the two operators, is characterized by the paraeter value θ p i p j q i q j, (1) where q i >q j and p i p j without loss of generality. Individuals whose θ exceeds θ patronize fir i, whereas individuals whose θ is saller than θ select operator j. 2.1 Consuer Surplus and Deands The net surplus the θ client obtains fro the consuption of x i units of the good purchased fro fir i is given by v θ (x i )u (x i ) ep i (θ) x i, i {I,}, (2) where ep i is the generalized cost and u (x i ) the gross utility derived fro consuing x i once quality effects has been taken into account and. The optial quantity of good i for the client under scrutiny is pinned down by axiizing (2) with respect to x i, which yields u x i ep i, i {I,}, (3) showing that the optial quantity x i (p i,q i ; θ) is, indeed, a function of the sole generalized price of the consued coodity. Relying upon (3), it is possible to establish the relationship between the ipact on the consued quantity of arginal changes in price and quality. To see that, observe first that (3) holds for any p i and any q i. Differentiating both sides with respect to p i and to 7

8 q i and cobining both equation, we obtain x i / q i x i / p i θ, i {I,}. (4) This evidences that, for the θ consuer deand to reain unchanged, as the price p i is increased by one unit, the quality q i should be augented by an aount equal to her arginal valuation for the quality, naely θ. Aggregate deands are then iediately obtained by suing over the relevant ranges of θ 0 s. They are given by X j (p i,q i,p j,q j ) θ R θ x j (p j,q j ; θ) f (θ) dθ (5a) and θr X i (p i,q i,p j,q j ) x i (p i,q i ; θ) f (θ) dθ θ (5b) for fir j and i respectively. The properties they display are rather standard. We hereafter briefly recall the for any i, j {I,} : ³ 1. fir i 0 s deand decreases in its own price p Xi i p i < 0 ; ³ 2. fir i 0 s deand increases in its own quality q Xi i q i > 0 ; ³ 3. fir i 0 s deand increases in the rival price p Xi j p j > 0 ; ³ 4. fir i 0 s deand decreases in the rival quality q Xi j q j < 0. Moreover, it is straightforward to obtain the aggregate consuer surplus as a function of prices and qualities. At this ai, it is enough to plug the individual deand pinned down by (3) into the surplus function (2) and then su over the relevant ranges of θ, which ultiately returns V (p i,q i,p j,q j ) θ R θ v (p j,q j ; θ) f (θ) dθ + θ R θ v (p i,q i ; θ) f (θ) dθ. 2.2 Technologies and Profits ir i 0 s cost function is given by C i (X i,q i ),i {I,}, which is increasing both in ³ ³ the production level Ci X i > 0 andintheoffered quality Ci q i > 0. 8

9 We assue that C i (, + ) +. This equality eans that providing very high quality is infinitely costly for the operator, hence such a quality is never actually provided on the arket. We further suppose that firs never offer zero quality. Taken together, these hypotheses ensure that q i (0, q), i {I,}, q identifying a finite quality level. inally, defining fir i 0 s revenues as R i p i X i, the profit function is written π i (p i,q i,p j,q j )R i C i (X i,q i ), i {I,}. (6) 3 The (Constrained) Optial Partial Regulatory Policy The present section is devoted to the characterization of the (constrained) optial partial regulatory policy. By the latter we ean the policy which would aterialize in a ixed duopoly where a welfare-axiizing (public) fir were to play the arket gae as a Stackelberg leader vis-à-vis aprofit-axiizing (private) copetitor, under a nonnegative profit constraint. As the follower s reaction to the incubent s policy is taken into account, the analysis is to be perfored backward. Therefore, we start by investigating the entrant s behaviour and we subsequently look for the leader s price-and-quality bundle. 3.1 The Price-and-Quality Policy of ir ir behaves as a follower vis-à-vis the incubent. Therefore, it takes fir I 0 sprice and quality as given and optiizes its own accordingly. The first-order condition for a axiu of (6) with respect to price p gives the standard forula: p C / X p 1 ε Xp, (7) wherewehavedefined ε Xp (p /X )( X / p ) the(absolutevalueofthe)deand elasticity to price. It is straightforward to recognize in (7) the onopoly inverse elasticity rule. This suggests that fir acts as a onopolist vis-à-vis the residual deand. urtherore, the first-order condition with respect to quality q is given by p X q C q + C X X q. (8) 9

10 As (8) reveals, q is chosen so that the arginal revenue of quality (the left-hand side) equals the arginal cost of quality (the right-hand side), as expressed by the su of quality direct ipact on cost (first ter) and indirect ipact (second ter). Equation (8) can be rearranged and cobined with the first-order condition with respect to p in order to obtain X / q X / p 1 X C q. (9) Clearly, the ratio on the left-hand side of (9) can be interpreted as arginal value of quality for We denote p,q the price-and-quality pair fir selects to satisfy the above conditions. This choice is anticipated while deterining the leader s policy. 3.2 The Partial Regulatory Policy We next characterize the incubent s policy, taking the follower s behaviour (as illustrated in the previous Section) into account. or sake of shortness, let us denote p and q the vectors of overall arket prices and qualities respectively. The (constrained) optial partial regulatory policy is pinned down by axiizing the unweighed social welfare function W (p, q) V (p, q)+π I (p, q)+π (p, q) (10) with respect to p I and q I, under the constraint that the incubent akes non-negative profits (π I 0). Let λ the Lagrange ultiplier associated with fir I 0 s budget constraint. The firstorder condition for a constrained axiu of (10) with respect to price p I is given by µ µ dπ I 1 dv dπ. (11) 1+λ The left-hand side of (11) is the total derivative of fir I 0 sprofits with respect to p I, which also internalizes the indirect ipact a variation in the own price induces on profits π I through the rival s price p and quality q ; it writes as dπ I π I p I + π I p dp + π I q dq. 10

11 Siilarily the ipact on consuer surplus of a change in the price p I can be decoposed as: dv V p I + V p dp + V q dq. (12) By contrast, since ( π / p )0and ( π / q )0, the derivative of fir 0 sprofits with respect to p I siplifies to dπ π µ p C p I X X µ X / p I X / p X p I, (13) wherewehaveusedthefirst-order condition streaing fro fir 0 sprofit-axiization. Assue that the regulated fir provides higher quality services so that fir I serves the consuers whose θ belongs to the interval θ, θ and fir those with θ [θ,θ ). Clearly, fro Roy s identity, we have V θr x I (p I,q I ; θ) f (θ) dθ X I, p I θ (14) V θr x (p,q ; θ) f (θ) dθ X, p θ (15) V θr x (p,q ; θ) θf (θ) dθ q θ X. (16) θ The assuption that a regulated incubent provides higher quality, hence it sells its products to the custoers exhibiting a relatively larger valuation for quality, and receives a price that is sufficiently large to reunerate such a quality provision, sees to be rather reasonable and will be aintained all along the analysis. Nevertheless, it is noteworthy that, utatis utandis, the investigation would siilarly be perfored in the event that the unregulated follower served the arket segent θ, θ. Plugging (13) and (12) into (11) ultiately yields µ µ dπ I 1 dp X I + X 1+λ dp e dq X / p I θ X X. (17) I X / p Turning next to the second relevant diension, the first-order condition for a con- 11

12 strained axiu of (10) with respect to quality q I is given by dπ I µ µ 1 dv + π. (18) 1+λ q I A siilar analysis yields to a decopostion of consuer surplus variation that writes dv e θ I X I X dp + e θ X dq, (19) where e θ I is the average valuation of quality by fir I 0 s clients, as weighted by their relative deand: θr e x I (p I,q I ; θ) θi θf (θ) dθ. θ X I Again, relying upon the first-order conditions fro fir profit-axiization, we can write π q I X µ X / q I X / p. (20) Replacing (19) and (20) into (18), we ultiately obtain dπ I µ µ 1 e X / q I dp θi X I + X X 1+λ X / p + e dq θ X. (21) In definitive, the optial partial regulatory policy under the incubent s budget constraint is given by the price-and-quality cobination p I,qI which siultaneously satisfies the pair of conditions (17) and (21). 4 Decentralization through a Price-and-Quality Cap In this section, we propose a echanis that allows to decentralize the allocation identified above. The incubent is left free to choose both prices and qualities provided that a price-and-quality cap constraint is satisfied. orally, this eans that the incubent solves the prograe Max π I p I X I C I (X I,q I ) {p I,q I }. (22) s.t. αp I βq I P + γp δq 12

13 Let μ the ultiplier associated with the price-and-quality constraint in (22). The first-order condition for a constrained axiu of π I with respect to p I and q I are given by dπ I μ α γ dp + δ dq, (23) dπ I μ β + γ dp δ dq. (24) Given α, β > 0, the regulatory constraint is tightened by an increase in the price p I, relaxedbyanincreasequalityq I. These effects are itigated or enhanced through their ipact on fir s decisions that are explicitly considered in the price-and-quality cap constraint. or the iposition of the price-and-quality cap to ultiately ipleent the partial regulatory policy p I,qI as previously characterized, it is sufficient to have: μ 1 1+λ (25a) together with α X I X β e θ I XI µ X / p I X / p + X µ X X, (25b) / q I / p, (25c) and with γ X δ e θ X, (25d) where the presence of the superscript indicates that ters are evaluated at the partial regulation prices and qualities. Notice that the derivatives X / p I and X / q I in the right-hand sides of the previous expressions both express arginal variations, as fir 0 s infra-arginal consuption units are not concerned by changes in the incubent s price and quality. More precisely, we have X p I x f θ dθ (26a) 13

14 as well as X q I x f θ dθ, (26b) where x easures the consuption of the arginal client, the one characterized by θ p I qi p q at those prices and qualities, the relevant density being f θ. Plugging (26a) and (26b) into (25b) and (25c) ultiately returns ore interpretable expressions for the regulatory price and quality weights, that is, α X I X " θ R θ x f θ dθ x p f (θ) dθ + x f θ θ p # (27a) and β e θ I XI X " θ R θ x f θ dθ x p f (θ) dθ + x f θ θ p # (27b) respectively. Let us closely investigate each of the weights we have found. According to (27a), the appropriate price weight α is given by the difference between two ters. The first ter is the incubent s deand evaluated at p, q, naely XI. The second ter consists in the deand faced by fir E evaluated at p, q, naely X, as ultiplied by a ratio which displays the arginal variation that is induced in deand X by an increase in the incubent s price at the nuerator and the (absolute value of the) overall (arginal and infra-arginal) variation that is caused in the sae deand by an increase in the copetitor s price at the denoinator. The (cross) effect of price p I on the follower s deand is less iportant than the (own) effect of price p. Therefore, the ratio under scrutiny is saller than unity. This involves that the incubent s deand is attributed a larger relevance than the rival s in the coposition of the price weight. In definitive, α is obtained by subtracting a properly calibrated portion of the entrant s deand, at the desirable prices and qualities, fro the doinant operator s. urtherore, (27b) suggests that also the quality weight β equals the difference between two ters. As already entioned, the first ter, naely e θ I XI, is an aggregate 14

15 easure of the quality appreciation of fir I 0 s consuers. The second ter is given by the copetitor s deand X as ultiplied by a ratio between the arginal variation that is induced in deand X by an increase in the incubent s quality and, again, the (absolute value of the) overall (arginal and infra-arginal) variation that is caused in the sae deand by an increase in the copetitor s price. Observe that it is harder to draw further considerations about the ultiate coposition of β, than it is about α, because a direct coparison between nuerator and denoinator of the ratio just described cannot be perfored in the general scenario under scrutiny. Nevertheless, a clear essage eerges fro (27a) and (27b), naely that, for either weight to be properly deterined, the relevant arket conditions about the incubent and the follower are to be taken into account. Therefore, while under regulated onopoly relevant such conditions are the overall arket conditions, under partially regulated oligopoly relevant conditions are the ones pertaining to each active player and to the arket as a whole. In the next Section, we investigate a specific case that returns particularly intuitive insights for soe interesting real-world environents. 4.1 The Unit Deand Case We hereafter focus on the case where each custoer has to allocate only one unit of consuption to her preferred operator 7. This kind of consuption decision fits the transportation sector especially well. To see this, consider that, in soe given day, an individual who needs to ake a travel and faces two operators, chooses the one fro which she will purchase her ticket. As an alternative way to view the situation, consider the case of an individual to who two transportation odes are available, who selects one such ode to perfor a single travel. In the unit deand case, neither changes in prices nor in quality ipact infra-arginal 7 As the reader should recall, for each individual, the preferred operator is the one which ensures the lower generalized cost, given her personal valuation for quality. 15

16 consuer decisions. Thus, we have X f θ dθ, p I X f θ dθ. q I with, under ild conditions (See appendix), dθ θ dθ. It follows that X / p I / p X X X / q I / p 1, θ which leads to the easy to interpret equations: π X, p I (28) π θ X. q I (29) Replacing (28) into (27a) and (29) into (27b) respectively yields α U X I X (30a) and β U e θ I XI θ X (30b) A very intuitive result is ebodied in (30a). The appropriate weight to be attributed to the incubent s price is siply the difference between the incubent s deand XI and the rival s deand X, which are evaluated at the optial partial regulation prices and qualities p, q. Interestingly enough, in the current scenario, equal relevance is recognized to the operators deands. Hence, the fotllower s deand needs be entirely subtracted fro the doinant fir s. 16

17 In turn, (30b) suggests that the proper weight to be assigned to the incubent s quality ³ equals the difference between the aggregate valuation of such a quality eθ I XI and a easure of the aggregate valuation of fir 0 squality θ X. In particular, in this aggregate easure, the arginal valuation θ is used to weight all the consuption units X. Recall that θ is the upper bound of the range of θ 0 sservedbythe follower. Therefore, while the volue of the incubent s services is weighed with the overall interval of relevant θ 0 s, the one of the rival s services is wholly weighed with the highest possible value of θ. This reveals that the valuation of quality q, which is relevant to the coposition of β U, "overstates" the one in the population of fir 0 sclients,whereas not so is as for the valuation of the regulated quality. Suing up, the generalised price-and-quality cap writes X I X pi ³ eθ I XI ³ θ X q I P p e θ q X. (31) This aounts to say that the incubent ust choose p I and q I as to have an average generalised price ep I that verifies ³ ep I XI p I e θ h pi I q I XI P + θ ³ q I p e θ i q X. Note that, by definition of the arginal consuer, p I θ q I p θ q, so that ep I X I ³ P θ e θ q X The later eans that there are no distortions in the incentives to be given to the regulated fir that follow fro its strategic interaction with the unregulated one. Indeed, ultiately, what atters is ep I, the average generalised price of incubent product or services. However, the regulatory pressure ought to be odulated according to the unregulated fir activity. More precisely, the bigger the arket of the unregulated (low quality) fir, the higher should be the regulatory pressure on the (high quality) fir. Siilarily, the regulatory pressure should increase with the difference θ e θ. Both effects are to due to 17

18 thefactthatalotistobegainedfroaslight decrease in the generalised price ep I since a good aount of consuers would shift to the high quality product. If the quality of the follower product q increases, it should also increase the copetitive pressure for the regulated fir. The quality offered by the unregulated fir is indeed an indicator of how uch the regulator should forder fro the regulated fir in ters of quality. 5 Concluding Rearks In this paper, we have provided a theoretical foundation for the price-and-quality cap regulation of the recently liberalized utilities. or this purpose, we have stylized a partially regulated oligopoly where vertically differentiated services, such as high-speed and lowspeed train services, are provided by a regulated incubent (the Stackelberg leader) and a strategic unregulated provider (the follower) copeting in price and quality. We have as well proposed an alternative interpretation of this odel in ters of copetition between regulated and unregulated industries, naely regulated train operators and deregulated air carriers. Within the context described above, we have first characterized the (constrained) optial partial regulatory policy as the target policy for the industry regulation. In the environent under scrutiny, this is the policy which arises whenever a welfare-axiizing fir is in the position of a Stackelberg leader with respect to one (or ore) profit-axiizing rival(s) and is copelled to the requireent that its profits be non-negative. We have subsequently addressed the issue of properly decentralizing the target policy. We have established that the appropriate weights to be inserted in the incubent s priceand-quality cap depend both on the "optial" deand of the regulated incubent and on that faced by the follower, despite the latter is not directly concerned by the partial regulation. In particular, in the special case of unit deand, under ild conditions, the price weight is siply the difference between the leader s and the follower s deand at the optial prices and qualities. On the other hand, the quality weight is given by the difference between the aggregate valuation of the regulated quality and a easure of the aggregate valuation of the rival quality that overstates the real valuation in the population of the follower s clients. In order to better highlight the core contribution of our investigation, we find it useful 18

19 to copare our results about the appropriate price weight with soe points raised in Brennan [10]. The latter [10] explains that, when the regulated fir faces a price-taking copetitive fringe, as long as fringe profits are included in the social welfare, the relevant output quantity in the price cap is just that of the regulated fir. If, instead, fringe profits are not considered to be a social welfare coponent, then the relevant output quantity is the total arket supply, precisely as in a onopoly. Our analysis shows that things are different in the presence of an unregulated strategic follower. In particular, for the price cap to be desirably set, the incubent s optial deand has to be partially or totally diinished by the copetitor s optial deand. The intuition behind this finding is that, since rival profits contribute to social welfare just as the surplus accruing to any other econoic agent and since the rival operator is not a price-taker, the weight to be attributed to the incubent s price in the cap needs to ensure that the follower be not crowded out and that, at the sae tie, the distortion induced by the arket power it exerts be iniized. Siilar intuition underlies the proper coposition of the quality weight, but this part of the cap is not coparable to any outcoe in Brennan [10], where attention is devoted to a pure price cap and the quality issue reains unaddressed. inally, hinging on the results of our analysis, an iportant conclusion can be drawn as for the regulatory process of the sectors under scrutiny. Regulatory bodies ipleenting the "partial version" of the price-and-quality cap should be allowed to use inforation about both the concerned industry as a whole and each of its active players. Indeed, contrary to what loose intuition ay suggest, relevant inforation for the regulatory process is not just the one about the doinant fir,eveninaworldwherethisfir is the sole regulated agent. References [1] Arstrong, M., S. Cowan and J. Vickers (1994), Regulatory Refor. Econoic Analysis and British Experience, The MIT Press, Cabridge [2] Arstrong, M., and D. E. M. Sappington, "Recent Developents in the Theory of Regulation", in Arstrong, M., and R. Porter (ed.), Handbook of Industrial Organization, Vol.III,Elsevier,North-Holland 19

20 [3] Arstrong, M., and D. E. M. Sappington (2005), "Regulation, Copetition and Liberalization", ieo [4] Beato, P., and A. Mas-Colell (1984), "The Marginal Cost Pricing Rule as a Regulation Mechanis in Mixed Markets", in Marchand, M., P. Pestieau and H. Tulkens (eds.), The Perforance of Public Enterprises, North-Holland, Asterda, pp [5] Bergantino, A. S., E. Billette de Villeeur and A. Vinella (2006), "A Model of Partial Regulation in the Maritie erry Industry", Southern Europe Research in Econoic Studies (S.E.R.I.E.S.), Quaderni del Dipartiento di Scienze Econoiche, Working Paper No.10, University of Bari [6] Biglaiser, G., and C. A. Ma (1995), "Regulating a doinant fir: unknown deand and industry structure", RAND Journal of Econoics, 26(1): 1-19 [7] Billette de Villeeur, E. (2004), "Regulation in the Air: Price-and-requency Caps", Transportation Research Part E, 40: [8] Billette de Villeeur, E., H. Creer, B. Roy and J. Toledano (2003), "Optial Pricing and Price-Cap Regulation in the Postal Sector", Journal of Regulatory Econoics, 24 (1): [9] Bos, D. (1989), Public Enterprise Econoics, North-Holland, Asterda, 2nd revised edition [10] Brennan, T. (1989), "Regulating by Capping Prices", Journal of Regulatory Econoics, 1(2), [11] Crapes, C., and M. Moreaux (2001), "Water Resource and Power Generation", International Journal of Industrial Organization, 19, [12] Creer, H., M. De Rycke and A. Griaud (1997), "Service Quality, Copetition, and Regulatory Policies in the Postal Sector", Kluwer Acadeic Publishers [13] Creer, H., J.-P. lorens, A. Griaud, S. Marcy, B. Roy and J. Toledano (2001), "Entry and Copetition in the Postal Market: oundations for the Construction of Entry Scenarios", Journal of Regulatory Econoics, 19(2),

21 [14] De raja, G., and E. Iossa (1998), "Price Caps and Output loors: A Coparison of Siple Regulatory Rules", TheEconoicJournal, 108: [15] De raja, G., and A. Iozzi (2004), "Bigger and Better: A Dynaic Regulatory Mechanis for Optiu Quality", CE, Discussion Paper No [16] De raja, G., and A. Iozzi (2000), "Short-Ter and Long-Ter Effects of Price Cap Regulation", Discussion Papers in Econoics, No.2000/61, University of York [17] orean, R.D. (1995), "Pricing incentives under price-cap regulation", Inforation Econoics and Policy, 7: [18] Hel, D., and T. Jenkinson (1998), Ed., Copetition in Regulated Industries, Oxford University Press, Oxford [19] Littlechild, S. C. (1983), Regulation of British Telecounications Profitability, HMSO, London [20] Iozzi, A., J. Poritz and E. Valentini (2002), "Social Preferences and Price Cap Regulation", Journal of Public Econoic Theory, 4 (1), [21] Laffont, J.-J., and J. Tirole (1996), "Creating Copetition through Interconnection: Theory and Practice", Journal of Regulatory Econoics, 10: [22] Laffont, J.-J., and J. Tirole (1993), A Theory of Incentives in Procureent and Regulation, The MIT Press, Cabridge [23] Martiort, D. (2005), "An Agency Perspective on the Costs and Benefits of Privatization", Journal of Regulatory Econoics, forthcoing [24] Nett, L. (1993), "Mixed Oligopoly with Hoogeneous Goods", Annales de l Econoie Publique, [25] Rees, R. (1984), Public Enterprise Econoics, Weidenfeld and Nicolson, 2nd Edition [26] Tirole, J. (1988), The Theory of Industrial Organization, TheMITPress, Cabridge [27] Vogelsang, I. (2002), "Incentive Regulation and Copetition in Public Utility Markets: A 20-Year Perspective", Journal of Regulatory Econoics, 22(1),

22 [28] Vogelsang, I., and G. insinger (1979), "Regulatory Adjustent Process for Optial Pricing by Multiproduct irs", Bell Journal of Econoics, 10, A Appendix Let us consider the unit deand case and suppose that θ p I p q I q. ir I serves the θ 0 s belonging to the interval θ, θ, fir the ones within the interval [θ,θ ). In this scenario, the equalities and X p X q f (θ ) θ p f (θ ) q I q f (θ ) θ q θ f (θ ) q I q are true. Therefore, we can write X q f (θ ) θ q θ f (θ ) q I q θ X p θ f (θ ) θ p or, equivalently, θ q θ θ p. (32) We hereafter check whether and, if so, under which conditions, one as well has dθ θ dθ. (33) or this purpose, we first copute the total derivative of θ with respect to p I and obtain dθ θ p I 1 + θ p + θ q p p I q p I q I q 1 q I q 1 q I q p p I + µ 1 p p I + θ q p I θ q q I q p I. (34a) 22

23 We next calculate the total derivative of θ with respect to q I, which is given by dθ θ + θ p q I p q I θ q I q 1 q I q + θ q q q I 1 p + θ q q I q q I q I q q I µ θ p q I + θ q q I. (34b) The first-order condition for a axiu of π with respect to p is given by µ p C f (θ ) (θ ). X q I q Moreover, the first-order condition for a axiu of π with respect to q is written µ p C f (θ ) C 1. X q I q q θ Cobining the two conditions above, it follows that θ (θ ) C q. (35) Differentiating both sides of (35) with respect to p I yields dθ (θ )+θ f (θ ) dθ 2 C q 2 dq + 2 C q X µ X dp + X dq. p q Recalling that we have together with X f (θ ) p q I q X q θ f (θ ) q I q, we can ultiately write dθ 2 C q 2 ³ dq + 2 C f(θ ) dq q X q I q θ (θ )+θ f (θ ) dp. (36) 23

24 Letusnextdifferentiate both sides of (35) with respect to q I, which returns dθ (θ )+θ f (θ ) dθ 2 C q 2 dq + 2 C q X µ X p dp + X q dq. This is equivalent to dθ [ (θ )+θ f (θ )] 2 C dq q 2 + f (θ ) q I q 2 C q X µ dq θ dp, which finally yields dθ 2 C q 2 dq + f(θ ) q I q ³ 2 C dq q X θ (θ )+θ f (θ ) dp. (37) Relying upon (34a) and (34b), we can write µ [ (θ )+θ f (θ )] 1 dp 2 C dq (q I q ) q 2 + dq + θ 2 C f (θ ) q X q I q µ dq θ dp together with µ [ (θ )+θ f (θ )] θ dp 2 C dq (q I q ) q 2 + f (θ ) 2 C q I q q X dq + θ µ θ dq dp. Hence, we obtain [ (θ )+θ f (θ )] 2 C (θ )+θ f (θ ) f (θ ) q X µ dp dq θ +(q I q ) 2 C dq q 2 as well as θ [ (θ )+θ f (θ )] 2 µ C dp (θ )+θ f (θ ) f (θ ) q X dq θ +(q I q ) 2 C dq q 2. 24

25 It follows that dp θ dq (θ )+θ f (θ ) (q I q ) 2 C q 2 (θ )+θ f (θ ) f (θ ) 2 C q X dq and dp θ dq θ [ (θ )+θ f (θ )] (q I q ) 2 C q 2 (θ )+θ f (θ ) f (θ ) dq. 2 C q X If C ( ) is linear in q (i.e., ifitis 2 C / q 2 0), then one has and (q I q ) dθ 1 dp dq + θ (θ )+θ f (θ ) 1 (θ )+θ f (θ ) f (θ ) 2 C q X (θ )+θ f (θ ) f (θ ) (θ )+θ f (θ ) f (θ ) f (θ ) 2 C q X (θ )+θ f (θ ) f (θ ) 2 C q X (θ ) θ f (θ ) 2 C q X 2 C q X (q I q ) dθ θ dp + θ dq # (θ )+θ f (θ ) θ "1 (θ )+θ f (θ ) f (θ ) 2 C q X θ (θ )+θ f (θ ) f (θ ) 2 C q X (θ ) θ f (θ ) (θ )+θ f (θ ) f (θ ) θ f (θ ) 2 C q X (θ )+θ f (θ ) f (θ ). 2 C q X 2 C q X It follows that dθ θ dθ. 25

26 Letusnextlookatthecasewhere 2 C / q In particular, fro (35) we have 2 C q 2 [θ (θ )] q θ (θ ) θ (θ )+θ q θ q θ [ (θ )+θ f (θ )] q θ [ (θ )+θ f (θ )]. q I q Recalling that X (θ ), fro (35) we also deduce that 2 C [θ (θ )] q X X (θ ) [θ (θ )] θ. Using these results, we can write dp θ dq q I q [ (θ )+θ f (θ )] dq (θ θ )+θ f (θ ) (q I q ) (θ )+θ f (θ ) θ f (θ ) (θ )+θ f (θ ) θ [ (θ )+θ f (θ )] dq (θ ) (θ )+θ f (θ ) (θ ) µ 1 θ dq as well as dp dq θ θ [ (θ )+θ f (θ )] θ [ (θ )+θ f (θ )] (θ )+θ f (θ ) θ f (θ ) µ (θ )+θ f (θ ) θ 1+ dq. (θ ) dq It follows that (q I q ) dθ 1 dp dq + θ 1 (θ )+θ f (θ ) (θ ) θ (θ ) µ dq 1 θ ½ [ (θ )+θ f (θ )] dq f (θ ) ¾ 26

27 and so that dθ ½ θ [ (θ )+θ f (θ )] dq ¾ f (θ ). (q I q ) (θ ) It also follows that (q I q ) dθ θ dp dq + θ (θ )+θ f (θ ) θ + θ θ (θ ) µ 1+ dq (θ ) ½ [ (θ )+θ f (θ )] dq + θ f (θ ) ¾ and so that dθ ½ θ [ (θ )+θ f (θ )] dq ¾ + θ f (θ ). (q I q ) (θ ) Therefore, in the general case, for the condition dθ θ dθ to hold, it ust be the case that [ (θ )+θ f (θ )] dq ½ + θ f (θ ) θ [ (θ )+θ f (θ )] dq ¾ f (θ ) θ [ (θ )+θ f (θ )] dq + θ f (θ ), that is, we need to have [ (θ )+θ f (θ )] dq θ [ (θ )+θ f (θ )] dq or, equivalently, dq dq I dq θ. 27