Partial Regulation in Vertically Differentiated Industries

Transcription

1 Partial Regulation in Vertically Differentiated Industries Angela S. Bergantino Etienne Billette de Villeeur Annalisa Vinella Abstract In this paper we provide theoretical foundations for a price-and-quality cap regulation of recently liberalized utilities. We odel a partially regulated oligopoly where vertically differentiated services are provided by a regulated incubent and an 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 regulation. This calls for the possibility that regulators use inforation about the whole industries, rather than on the sole incubent. In the unit deand case, however, regulation can be conducted as if the regulated fir were still a onopoly. 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 XVIII SIEP Meeting, Università di Pavia, at the 5th Conference on Applied Infrastructure Research, Technische Universität Berlin, at the 4th Annual Conference on Railroad Industry Structure, Copetition and Investent, Universidad Carlos III de Madrid, and at the 47th SIE Meeting, Università di Verona. We would like to thank seinar participants for their coents and, in particular, Carlo iorio and Ingo Vogelsang. All reaining errors are ours. The paper was copleted while the second author was visiting Université de Montreal whose hospitality is gratefully acknowledged. The third author also whish to thank the lorence School of Regulation, at the EUI that she visited in quality of Jean Monnet ellow. University of Bari, (Departent of Econoics), Italy. Corresponding author: Toulouse School of Econoics, (IDEI and GREMAQ), rance. E-ail: etienne.devilleeur@univ-tlse1.fr University of Bari (Departent of Econoics) Italy and Toulouse School of Econoics (GRE- MAQ), rance. 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 [17], p.1) 1 Introduction In a seinal work, Laffont and Tirole [23] 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 quantity leads to Rasey pricing. The second-best optiu under onopoly thus entails. This result "restates" Brennan [11] s finding that price cap eerges as the solution to a proble of welfare-constrained profit axiization, in which quantities are appropriate weights to be attached to allowable deviations fro socially desirable prices. Those aforeentioned are definitely not the sole papers about the so-called ideal price cap. Indeed, nuerous other authors have worked on the subject. Soe generically look at industries providing services of general interest. One ay recall, for instance, De raja and Iozzi [16] and [17] as well as Iozzi, Poritz and Valentini [22]. Others focus on particular sectors, such as Billette de Villeeur, Creer, Roy and Toledano [8], who propose a price-cap schee that fits postal sector specificities. Under price-cap regulation, firs have incentives to cut costs, which likely translates into quality under-provision. Rovizzi and Thopson [28] point out that noticeable quality reduction was registered in British Teleco s services as soon as the privatized copany went subject to price-cap regulation. In consideration of this drawback, De raja and Iozzi [16] lay down theoretical foundations for joint regulation of onopoly price and quality. Integrating quality diensions into standard price cap, they find that appropriate price weights are still the optial quantities, whereas quality weights should equal consuer arginal surplus evaluated at the optial prices and qualities 1. Siilar results eerge in Billette de Villeeur [7] as for the airline industry. The contributions recalled above all refer to onopoly regulation. This circustance is easily explained. After the second World War, the provision of services of general interest was typically concentrated in State controlled-and-owned vertically integrated onopolies. Over tie, this organizational odel has lost oentu. 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 price cap regulation, either in a static or in a dynaic version. Soe such papers are Vogelsang and insinger [30], Littlechild [21] and orean [19]. However, we do not strictly hinge on the literature background about ipleentation, to which those studies belong. or the sae reason, Brennan [11] and De raja and Iozzi [16] are solely referred to as for the theoretical foundation of price cap and price-andquality cap respectively, not as for the ipleentation schees they propose. More generally, see Vogelsang [29] 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

3 Network industries, such as telecounications, water, energy and transportation, have undergone a ajor privatization process. Thus statutory onopolists have often becoe regulated coercial players 2. With such real-world scenarios in ind, econoists have long focused on onopoly regulation, with and without quality adjustents. However, at ore recent stage, the segents of network industries that are not concerned by sub-additive technologies have been opened up to copetition. Partially regulated oligopolies thus arise, where incubents are subject to regulatory obligations, while entrants are allowed to operate uncontrolled. On one side, such policies are eant to proote access and (soe) copetition. On the other side, they ai at guaranteeing reliable service supply at affordable prices. This calls for aking incubents less arket powerful, though still financially viable 3. A crucial issue arises. Consolidated onopoly regulation does not need to suit iperfectly copetitive partially regulated sectors. Biglaiser and Ma [6] show that regulatory progras targeted to sole incubents are yet sensitive to the presence of copetitors endowed with arket power. New constraints and trade-offs addupin the regulatory process, leading to third-best outcoes. Observe in particular that price-and-quality-cap regulation lacks any norative basis with reference to these (newly relevant) situations. As for pure price cap, Brennan [11] proves that, if an incubent copetes with a passive fringe whose profits are taken to contribute to social welfare, then relevant quantities are the optial ones of the sole regulated fir. Yet, he acknowledges, when copetitors are not price takers, a different recipe is required, which he does not identify though 4. The ultiate goal of our paper is to provide theoretical foundations to price-andquality cap regulation of oligopolies where a regulated doinant fir (a Stackelberg leader) copetes in price and quality with one (or ore) strategic follower(s). To capture the iportance of quality provision in nearly all utilities, we focus on a arket where vertically differentiated services are supplied to consuers exhibiting heterogeneous quality valuations. The interpretation of our odel is twofold. irstly, 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- 2 See Martiort [25] for a discussion about the costs and benefits that are associated with privatization in an incentive theory perspective. 3 Biglaiser and Ma [6] provide the exaple of AT&T copeting as a regulated doinant fir with the unregulated MCI and Spring in the long-distance telecounications arket. See also Hel and Jenkinson [20], 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 eerge when forer onopolies are now engaged in both regulated and unregulated activities in different arket segents. 4 See footnote 6 at page 144 and footnote 10 at page 145 in Brennan [11]. The latter also points out that, if fringe profits were not to be included in social welfare, the optial price-cap would not differ fro the onopoly case (i.e. price weights in the price-cap forula are proportional to optial quantities). By contrast, we will consider that unregulated fir s profits do contribute to social welfare. 3

4 speed and low-speed services). Secondly, our odel stylizes copetition between regulated and unregulated industries. Insisting on transportation, one ay consider inter-odal copetition between regulated train operators and deregulated air carriers, whose services differ as for speed, frequency, scheduling reliability or cofort. Albeit transportation is particularly well placed to illustrate the flexibility of our odel, the latter does apply to ost utilities. Indeed, as further exaples of relevant quality diensions, one ay recall continuity and constancy of Internet service supply as well as reaction lags in electricity generation and provision 5. We begin by characterizing the optial (partial) regulatory policy. In the presence of strategic rivals, such a policy is no ore the second-best onopoly policy. In particular, Rasey pricing no longer yields (constrained) efficiency. Relevant benchark for the regulator is now the policy that arises when a welfare-axiizing fir acts as a Stackelberg leader with respect to one (or ore) profit-axiizing rival(s), facing the requireent that profits be non-negative. In this perspective, our work is reiniscent of ixed oligopoly odels where the public fir is taken to be a first over vis-à-vis the private operator(s) 6. At a later stage, we deonstrate how the optial policy can be decentralized by eans of a price-and-quality cap targeted to the sole doinant fir. Decentralization requires that weights in the cap depend not only on the arket served by the incubent, but also on the arket covered by the unregulated copetitor(s). It follows that regulatory bodies of liberalized industries should not be restricted to access and use inforation about the sole regulated firs. They should rather be allowed to extract and utilize inforation about the overall industry. However, in the particular case of unit deand, the optial policy can yet be decentralized by eans of a standard price-and-quality cap, which uniquely refers to the regulated fir. This ay coe out as an interesting result for those sectors (e.g., transportation) where relevant arkets are difficult to define. The reainder of the paper is organized as follows. Section 2 presents the fraework. Section 3 illustrates the ipact of incubent s actions on entrant s decisions and characterizes the optial (partial) regulatory policy in a Stackelberg oligopoly. Section 4 evidences how the policy target can be decentralized by eans of an appropriate price-and-quality cap. The latter is further investigated for the unit deand case, which returns especially useful and intuitive insights. Section 5 concludes. 5 Crapes and Moreaux [12] stress that this quality aspect of energy provision introduces a heterogeneity diension in 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), Bös [10] stresses that the fir should opt for a odified Rasey-pricing rule. Beato and Mas Colell [4] show in turn 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 [26] for a survey about the hoogeneous-good ixed oligopoly literature. 4

5 2 The Model Consider an industry where two firs provide vertically differentiated products. The two providers play a Stackelberg gae. One fir, denoted L (for leader), is subject to regulation. The other, denoted (for follower), is not. Strategic variables are prices (p L and p ) and qualities (q L and q ). They are both assued to be observable and verifiable. Although this assuption excludes any environents, it appears yet to be a natural and realistic hypothesis in a variety of contexts. Transportation, where length of trip or frequency of services ay play the role of quality, provides a good exaple of such a context. We will often refer to it all along the paper. Except for quality differences, the goods provided by the two firs turn out to be perfect substitutes. Consuers differ only in their valuation for quality, as represented by eans of a paraeter θ. More precisely, we adopt a quasi-linear fraework. The net surplus that a consuer of characteristic θ derives fro the consuption of x units of quality q bought at unit price p writes v θ (x, p, q) u (x) (p θq) x. (1) The paraeter θ is distributed over the interval θ, θ accordingtoacontinuous density function f (θ). The associated cuulative distribution function is denoted (θ). Given her quality valuation, a θ-consuer patronizing fir i {L, } faces the so-called generalized price ep (θ) p θq, which is equal to the unit price p net of the benefits θq, associated with product quality. A θ-consuer prefers to purchase the good fro fir i rather than fro fir j, whenever she bears a saller generalized price by doing so, i.e. whenever ep i (θ) < ep j (θ). Observe that, by construction, no consuer finds it profitable to patronize both firs. 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, (2) where without any loss of generality p i p j and q i >q j. Individuals whose θ exceeds θ patronize fir i, whereas individuals whose θ is saller than θ patronize fir j. 2.1 Consuer Valuation of Quality, Deand and Surplus The consuption of a θ-consuer is pinned down by axiizing (1) with respect to x, which yields u ep (θ), (3) x wherewehaveep (θ) argin{ep L (θ), ep (θ)}. This says that individual consuption x i (p i,q i ; θ) appears to be a function of the sole generalized price of the consued 5

6 coodity ep i (θ). Sinceep (θ) decreases with θ and ep i (θ )ep j (θ ), the ranking of consuers in ters of quality valuation conveys when individual consuptions x θ are considered. orally x θ1 x θ2 whenever θ 1 θ 2. Relying upon (3), it is possible to establish the relationship between the ipacts on consuption of arginal changes in price and quality. To see this, observe first that (3) holds for any p i and q i. Differentiating both sides with respect to p i and to q i and cobining the two equations, we obtain ( x i / q i ) ( x i / p i ) θ, i L,. (4) This evidences that, for θ-consuer deand to reain unchanged as price p i is increased by one unit, quality q i should be augented by an aount equal to individual arginal valuation for quality, naely θ. It also follows that a consuer with a strictly higher quality valuation that patronizes the sae fir would consider an increase in (p i,q i ) that leaves a θ-consuer indifferent as strictly beneficial. Conversely a consuer with a strictly lower valuation would find it detriental. Opposite appreciations would arise if a decrease in (p i,q i ) that leaves a θ-consuer indifferent were considered. Aggregate deands are iediately obtained by suing over the relevant ranges of θ 0 s. They are given by X j (p, q) θ R x j (p j,q j ; θ) f (θ) dθ X i (p, q) θ R θ θ x i (p i,q i ; θ) f (θ) dθ (5a) (5b) where θ (p i p j ) / (q i q j ) while p and q stand for the vector of prices and quality respectively. The properties they display are rather standard. We hereafter briefly recall the. or any i, j {L, } : 1. ( X i / p i ) < 0:fir i 0 s deand decreases in its own price p i ; 2. ( X i / q i ) > 0:fir i 0 s deand increases in its own quality q i ; 3. ( X i / p j ) > 0:fir i 0 sdeandincreasesintherivalpricep j ; 4. ( X i / q j ) < 0:fir i 0 s deand decreases in the rival quality q j. It is also straightforward to obtain aggregate consuer surplus as a function of prices and qualities. To this ai, plug individual deands pinned down by (3) into the surplus function (1) and su over the relevant ranges of θ 0 s. This ultiately returns V (p, q) θ R R v θ (x θ,p j,q j ) f (θ) dθ + θ v θ (x θ,p i,q i ) f (θ) dθ. (6) θ θ 6

7 2.2 Technologies and Profits Let C i (X i,q i ) be fir i 0 s cost function. This function is assued to be continuous and increasing in both production level and quality. orally ( C i / X i ) > 0 and ( C i / q i ) > 0. We further assue that C i (, + ) +. This says that high quality products are so costly to iprove that perfect products (q i + ) are never actually provided on arkets. We finally assue that firs never find it profitable to decrease the quality of their products down to zero. Taken together, these hypotheses ensure an interior solution to the definition of quality. orally, there exists a finite q such that 0 <q i < q, for i L,. inally, defining fir i 0 s revenues as R i p i X i, the profit function is written π i (p, q) R i C i (X i,q i ), i L,. (7) 3 Optial Partial Regulation This section is devoted to the characterization of optial (partial) regulatory policy. By the latter we ean the policy which would aterialize in a ixed duopoly where a welfare-axiizing (public) firweretoplaythearketgaeasastackelberg leader vis-à-vis aprofit-axiizing (private) copetitor, under a non-negative profit constraint. As follower s reaction to incubent s policy is taken into account, analysis is perfored backward. We therefore start by investigating entrant s behaviour and look subsequently for leader s price-and-quality bundle. 3.1 Price-and-Quality Policy of ir ir behaves as a follower vis-à-vis the incubent. Therefore, it takes fir L 0 s price and quality as given and optiizes accordingly. The first-order condition that derives fro the axiization of (7) with respect to price p givesrisetothestandardforula p ( C / X ) p 1 ε, (8) where ε (p /X )( X / p ) is (the absolute value of) the deand elasticity with respect to price. Equation (8) is nothing but the Lerner forula, evidencing that fir acts as a onopolist vis-à-vis residual deand. The first-order condition that derives fro the axiization of (7) with respect to quality q is given by p X q C q + C X X q. (9) As revealed by equation (9), qualityq is chosen so that arginal benefits fro 7

8 quality iproveents (the left-hand side) equate their arginal costs (the right-hand side), as expressed by the su of direct costs of quality (the first ter) and indirect costs (as reflected by the second ter), that follow fro deand increases resulting fro quality increases. The latter equation can be rearranged and cobined with the first-order condition with respect to p to obtain X / q 1 C. (10) X / p X q By analogy with equation (4), the ratio on the left-hand side of (10) can be interpreted as the aggregate arginal valuation of quality by fir 0 s clients. The right-hand side of (10) is nothing but the average cost of a arginal increase in quality for this sae fir. Although the fir is profit axiizing, no distortion is introduced by the fir s choices in ters of quality. Given consuer valuation, a further quality iproveent would not appear to be worth its costs Optial Regulatory Policy We now characterize the socially optial policy. Reind that fir L, the regulated fir is assued to act as a Stackelberg leader vis-à-vis fir, the (profitaxiizing) follower. Optial policy is thus pinned down by axiizing the unweighted social welfare function W (p, q) V (p, q)+π L (p, q)+π (p, q) (11) with respect to p L and q L, while p and q are assued to obey the rules (8) and (9) identified above. The set of policy considered is also restricted to allow fir L to break-even. orally, the pair (p L,q L ) should be such that π L (p, q) 0. Let λ be the Lagrange ultiplier associated with fir L 0 s budget constraint. The first-order condition that follows fro the constrained axiization of (11) with respect to price p L is given by dπ L µ 1 1+λ µ dv + dπ. (12) The left-hand side of (12) is the total derivative of fir L 0 sprofits with respect to p L. In other words, it also accounts for the indirect effects on π L of a variation in p L. The latter indeed result in induced adjustents of rival s price and quality, p and q. orally, we have: dπ L π L p L + π L p dp + π L q dq. 7 This does not ean that consuer and fir s objectives are perfectly aligned, even when attention is restricted to quality issues. In fact, were the price be lower, the deand would be larger. As a result, the average cost of quality would be lower, calling for a strict iproveent in ters of quality. 8

9 Siilarly the ipact on consuer surplus of a change in price p L can be decoposed as dv V + V dp + V dq. (13) p L p q By contrast, since ( π / p )0and ( π / q )0, thederivativeoffir 0 s profits with respect to p L siplifies to dπ π p L µ p C X X µ X / p L X / p X p L. (14) Assue that the regulated fir provides higher quality services than fir. In other words, assue that fir L serves consuers with quality valuation in θ, θ and fir consuers with quality valuation in [θ,θ ).Thisisanaturalassuption in a variety of contexts. In particular, it happens to be often the case when fir L is the incubent of a previously fully regulated industry while fir is a frail newcoer, that entered arket after liberalization. Observe however that, utatis utandis, the analysis would be identical if the unregulated follower would have served the high arket segent θ, θ. The assuption is thus to be considered as an explanatory device introduced to avoid redundancy. It does not put any restriction upon the analysis. ro Roy s identity, we have where e θ L and e θ as defined by V R θ x L (p L,q L ; θ) f (θ) dθ X L, p L θ (15) V θ R x (p,q ; θ) f (θ) dθ X, p θ (16) V R θ x L (p L,q L ; θ) θf (θ) dθ q θ L X L, L θ (17) V θ R x (p,q ; θ) θf (θ) dθ q θ X, (18) θ R e θl θ x L (p L,q L ; θ) θf (θ) dθ, θ X L e θ θ R x (p,q ; θ) θf (θ) dθ, θ X denote the average valuation of quality by each fir clients, as weighed by their relative deand. Plugging (15), (16) and (18) into (13) and then (14) and (13) into (12), we 9

10 ultiately obtain dπ L µ 1 1+λ µ X / p L dp X L X + X X / p dp e dq θ X. (19) L Turning now to the second relevant diension, the first-order condition for a constrained axiu of (11) with respect to quality q L is given by µ µ dπ L 1 dv + π. (20) 1+λ q L A siilar analysis yields a decoposition of consuer surplus variation that writes dv e θ L X L X dp + e θ X dq. (21) Again, relying upon the first-order conditions of fir 0 sprofit-axiization, we can write µ π X / q L X. (22) q L X / p Replacing (21) and (22) into (20), we ultiately obtain µ µ dπ L 1 e X / q L dp θl X L + X X + 1+λ X / p dq e dq θ X. (23) L To su up, the optial partial regulatory policy under the incubent s budget constraint is given by the price-and-quality cobination (p L,q L ) which siultaneously satisfies the pair of conditions (19) and (23). Observe that both equations contain ters that reflect strategic interactions across firs (like (dp / ) and (dp / )) and cross-price effects (like X / p L ). Soe ay thus consider the definition of optial policy as a purely theoretical exercise, with no practical value. If optial policy does not find an explicit expression, exact ipleentation is indeed likely to be beyond reach. This akes ore striking the results we present hereafter. 4 Decentralization through an Extended Price Cap In this Section, we propose an extended version of the price cap schee and evidence that it allows to decentralize the optial allocation identified above. Assue that the incubent is left free to choose both price and quality provided that a price-and-quality cap is satisfied. orally, let ( Max π L p L X L C L (X L,q L ), {p L,q L } (24) s.t. αp L βq L P + γp δq, be the regulated fir s progra and denote μ the ultiplier associated with the price-and-quality constraint. The first-order conditions for a constrained axiu 10

11 of π L with respect to p L and q L are respectively given by dπ L μ α γ dp + δ dq, (25) dp L dπ L μ β + γ dp δ dq. (26) With α, β > 0, the regulatory constraint is tightened by an increase in price p L, relaxedbyanincreaseinqualityq L. These effects are itigated or enhanced through their ipact on fir 0 s decisions, which are explicitly considered in the extended price cap. or the proposed extended price cap to ultiately ipleent the partial regulatory policy (p L,q L ) previously characterized, it is sufficient (i) to set coefficients (α, β) to α X PR L X PR β e θ PR L XL PR (ii) to set coefficients (γ,δ) to µ X PR X PR + X PR µ X PR / p L / p, (27a) / q L / p, (27b) X PR γ X PR δ e θ PR X PR, (27c) and (iii) to decrease P enough as to wash out fir L 0 sprofits so that 1 μ 1+λ PR. (28a) The presence of the superscript PR indicates that exact values are those obtained at the Partial Regulatory optiu. Observe that both derivatives X PR/ p L and X PR / q L in the right-hand sides of (27a) and (27b) reflect only arginal variations, as fir 0 s infra-arginal custoers are not concerned by changes in fir L s price and quality. More precisely, we have X PR x PR f θ PR p L X PR θ PR, p L θ PR q L (29a) X PR p L, (29b) x PR f θ PR θ PR q L where x PR f θ PR easures consuption by arginal clients, i.e. consuers characterized by quality valuation θ PR ppr L ql PR ppr qpr. 11

12 Plugging (29a) and (29b) into (27a) and (27b) ultiately returns ore interpretable expressions for the regulatory price and quality weights where ν θr θ α X PR L νx PR, (30a) β e θ PR L XL PR θ PR νx PR, (30b) x PR f θ PR θ PR p L xpr p f (θ) dθ + x PR f θ PR θ PR p. (31) The coefficient ν as defined by (31) displays, at the nuerator, the arginal variation in X PR inducedbyanincreaseinp L ; at the denoinator, the (absolute value of the) overall (arginal and infra-arginal) variation in the sae quantity X PR as induced by an increase in p. The (cross) effect of price p L on the follower s deand is likely to be saller than the (own) effect of price p. Therefore, the ratio under scrutiny can be assued to be saller than one. According to (30a), the appropriate price weight in the cap is given by the difference between two ters. The firstteristheregulatedfir s quantity evaluated at p PR, q PR, naely X PR L. The second ter consists in fir 0 s quantity evaluated at p PR, q PR, naely X PR, as ultiplied by the coefficient ν. Since ν 1, fir L 0 s output is given a larger relevance than fir 0 s output in the coposition of the price weight. In other words, α is obtained by subtracting fro the regulated fir s quantity XL PR (the standard weight in cap forulae) a fraction of its (unregulated) copetitor s one, νx PR, where the downsizing of X PR reflects the difference between cross and own-price effects. Siilarly, the quality weight β defined by (30b) is also given by the difference between two ters. The first ter, e θ PR X PR, is an aggregate easure of the quality appreciation by fir L 0 s consuers. The second ter is linked to the appreciation of quality by fir 0 s consuers. However, since the sole arginal clients of fir are concerned by changes in q L, the quality appreciation refers to θ PR and not to e θ. Interestingly enough, the arginal quality valuation is to be ultiplied by νx PR (the exact sae part of α that refers to fir ), i.e. the coefficient is to be calculated by using the whole deand for fir 0 sproducts,x PR (which is eventually observable), and not the consuption by fir 0 sarginalclients(whichisnot). A clear essage eerges fro (30a) and (30b). irst,properregulationoffir L cannot spare reference to its copetitor, fir. Second, the larger the arket share of the unregulated fir, the lower the required regulatory pressure on the regulated fir. In fact, if X PR denotes the arket size at p PR, q PR, the optial weight (30a) attached to price p in the price-cap forula rewrites α X PR (1 + ν) X PR That is to say, copetition has a bigger ipact on arkets than what appears when considering the sole arket share of unregulated firs X PR. Third, the agnitude of this copetitive effect depends on the coefficient ν, as defined by (31), which reflects firs differentiation. ourth, the higher θ PR, the lower the rewards for. 12

13 regulated fir product quality. We now turn to the unit deand case that yields particularly siple results, i.e. an alost ready-for-use schee to regulators. 4.1 The Unit Deand Case We hereafter focus on the case where each custoer allocates a single unit of consuption to her preferred operator 8. Despite the restrictions attached to this case, it does fit nuerous real world situations where regulation is actually perfored. Consider, for instance, (passenger) transportation arkets. Deand is naturally odelled as a discrete choice across available alternatives. The unit deand assuption corresponds to adopting trips as consuption units and considering that alternatives are attached to the various ways to reach the envisioned (single) destination. Note also that a quality attribute like travel tie is both observable and verifiable. The latter ay thus be used to regulate, say, the operator attached to one of two copeting odes. In the unit deand case, neither changes in prices nor in quality ipact infraarginal consuer decisions. Thus changes affect arginal consuers only. 9 Clearly, any lost consuer is a gain for the copetitor. It follows that X p f (θ ) θ p f (θ ) θ p L X p L hence which yields ν XPR X PR / p L / p 1, α U X PR L X PR, (32a) β U e θ PR L XL PR θ PR X PR. (32b) Suing up, the generalized price-and-quality cap writes as ³ X PR L X PR pl eθ PR L XL PR θ PR X PR q L P ³p e θ PR q X PR. (33) This says that the regulated fir is copelled to choose p L and q L so that its average generalized price ep L verifies ep L XL PR ³p L e h θ PR pl ³ L q L XL PR P + θ PR q L p e i θ PR q X PR. 8 Recall that, for each individual, the preferred operator is the one which ensures the lower generalized price, given her personal valuation for quality. 9 This does not ean however that regulation can be drawn according to the sole characteristics of arginal consuers. See appendix. 13

14 Recall that, by definition of the arginal consuer, p L θ PR q L p θ PR q. Thus, the regulatory constraint can finally be rewritten ³ ep L XL PR P θ PR e θ PR q X PR. (34) The latter forula deserves a few coents. irst, strategic interactions across firs do not yield any change in the fundaental structure of the incentives to be given to the fir. Indeed, ultiately, what atters is ep L, the average generalized price associated to the regulated fir services. Second, this average generalized price ep L p L e θ PR L q L involves an estiate of the arginal value of quality. This estiate e θ PR L is nothing but the average quality valuation by fir L s consuers. Third, at optiu, the cap P is adjusted by the regulator so as to wash out fir L 0 s profits, according to (28a). This does ean that the other ters on the right-hand side of (34) can be safely reoved. 10 As a result, in the unit deand case, optial regulation follow fro iposing a standard quality adjusted price-cap: p L e θ PR L q L p. (35) The proposed regulatory schee hinges upon a single paraeter, exogenously set by the regulator. If quality is to be taken into account by the regulator, this paraeter is the siplest inforation one can think of. This is indeed an average arginal valuation of quality by consuers of the regulated fir, upon which the regulator ay legitiately and ore easily collect data. De raja and Iozzi [16] evidence a difficulty attached to the convergence of their quality adjusted price-caps. or this reason, they introduced an additional constraint that further liits the regulated fir choices. Observe however that this difficulty streas directly fro their abitious approach. Indeed, in their regulatory schee, quality valuation is endogenously deterined by coputing at each step consuer arginal surplus ( V/ q). By contrast, and along the actual practice, we assue that social valuation of quality is a policy attribute chosen and ade public by the regulator. Of course, with our echanis, a bias in quality valuation e θ PR L would ultiately result in a sub-optial allocation. However, the schee is ore transparent, hence ore likely to get public support. Moreover, with a fixed coefficient for quality valuation it is less prone to anipulations. Interestingly, although the regulatory policy accounts for its ipacts on the whole industry, the optial allocation can be characterised by the eans of an extended price-cap that looks at the sole regulated fir.thisaycoeouttobeofipor- tance when handling with industries where relevant arkets are difficult to define. 10 This holds true at optiu only. In ters of incentives (i.e. in order to guarantee the convergence of the adjustent process), this second ter ay appear to be useful. In any case, observe that the price p of the copetitor does not play any role in the schee; by contrast, a higher quality q and a larger supply X appear to call for a greater regulatory pressure. 14

15 5 Concluding Rearks In this paper, we provide theoretical foundations to a price-and-quality cap regulation of recently liberalized industries. or this purpose, we stylize a partially regulated oligopoly where vertically differentiated services are provided by a regulated incubent (the Stackelberg leader) and a strategic unregulated operator (the follower), copeting in price and quality. The odel ay equally well represent copetition across asyetrically regulated industries. Within the context just described, we first characterize the optial (partial) regulatory policy. In the environent under scrutiny, this is the policy that arises when a welfare-axiizing fir acts as a Stackelberg leader vis-à-vis one (or ore) profit-axiizing rival(s), under the constraint firs ust break-even. We subsequently propose a regulatory schee able to decentralize properly this policy target. The latter is part of the broad class of extended price(-and-quality) caps. Appropriate weights depend on (optial) quantities provided by both the regulated incubent and the follower, despite the latter is not directly concerned by regulation. In the particular case of unit deand, optial policy is however characterised by a standard price-and-quality cap, that refers to the sole regulated fir. There are essentially two insights to be drawn fro our analysis. irst, in a partially regulated industry, there is need for the regulatory agency to be able to hinge upon inforation on the whole industry; inforation about the sole regulated fir does not appear to allow an efficient regulation, in general. Second, in the particular case where consuers would typically consue only one unit of the patronized good, the optial policy is however characterised by the sole regulated fir settings; regulators can thus legitiately focus on one fir and regulate the latter as if it were a onopolist. The present contribution also presents several liits; it is far fro exhausting the research questions attached to the topic. In particular, it would be of iportance to consider the issue of convergence in the line of the recent contribution by Earle, Schedders and Tatur [18]. This is left however for further research. References [1] Arstrong, M., S. Cowan and J. Vickers (1994), Regulatory Refor. Econoic Analysis and British Experience, The MIT Press, Cabridge [2] Arstrong,M.,andD.E.M.Sappington,"RecentDevelopentsintheTheory of Regulation", in Arstrong, M., and R. Porter (ed.), Handbook of Industrial Organization, Vol.III, Elsevier, North-Holland [3] Arstrong, M., and D. E. M. Sappington (2006), "Regulation, Copetition and Liberalization", Journal of Econoic Literature, 44(2), [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 15

16 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] Billette de Villeeur, E., and A. Vinella (2007), "Regolaentazione Parziale di Quantità", Mieo [10] Bös, D. (1989), Public Enterprise Econoics, North-Holland, Asterda, 2nd revised edition [11] Brennan, T. (1989), "Regulating by Capping Prices", Journal of Regulatory Econoics, 1(2), [12] Crapes, C., and M. Moreaux (2001), "Water Resource and Power Generation", International Journal of Industrial Organization, 19, [13] Creer, H., M. De Rycke and A. Griaud (1997), "Service Quality, Copetition, and Regulatory Policies in the Postal Sector", Kluwer Acadeic Publishers [14]Creer,H.,J.-P.lorens,A.Griaud,S.Marcy,B.RoyandJ.Toledano (2001), "Entry and Copetition in the Postal Market: oundations for the Construction of Entry Scenarios", Journal of Regulatory Econoics, 19(2), [15] De raja, G., and E. Iossa (1998), "Price Caps and Output loors: A Coparison of Siple Regulatory Rules", TheEconoicJournal, 108: [16] De raja, G., and A. Iozzi (2004), "Bigger and Better: A Dynaic Regulatory Mechanis for Optiu Quality", CEPR, Discussion Paper No [17] 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 16

17 [18] Earle, R. L., Schedders, K. and T. Tatur (2007), "On Price Caps Under Uncertainty", Review of Econoic Studies, 74: [19] orean, R.D. (1995), "Pricing incentives under price-cap regulation", Inforation Econoics and Policy, 7: [20] Hel, D., and T. Jenkinson (1998), Ed., Copetition in Regulated Industries, Oxford University Press, Oxford [21] Littlechild,S.C.(1983),Regulation of British Telecounications Profitability, HMSO,London [22] Iozzi, A., J. Poritz and E. Valentini (2002), "Social Preferences and Price Cap Regulation", Journal of Public Econoic Theory, 4(1), [23] Laffont, J.-J., and J. Tirole (1996), "Creating Copetition through Interconnection: Theory and Practice", Journal of Regulatory Econoics, 10: [24] Laffont, J.-J., and J. Tirole (1993), A Theory of Incentives in Procureent and Regulation, The MIT Press, Cabridge [25] Martiort, D. (2006), "An Agency Perspective on the Costs and Benefits of Privatization", Journal of Regulatory Econoics, 30(1), 5-44 [26] Nett, L. (1993), "Mixed Oligopoly with Hoogeneous Goods", Annales de l Econoie Publique, [27] Rees, R. (1984), Public Enterprise Econoics, Weidenfeld and Nicolson, 2nd Edition [28] Rovizzi, L. and D. Thopson (1992), "The Regulation of Product Quality in the Public Utilities and the Citizen s Charter", iscal Studies 13 (3), [29] Vogelsang, I. (2002), "Incentive Regulation and Copetition in Public Utility Markets: A 20-Year Perspective", Journal of Regulatory Econoics, 22(1), 5-27 [30] Vogelsang, I., and G. insinger (1979), "Regulatory Adjustent Process for Optial Pricing by Multiproduct irs", Bell Journal of Econoics, 10,

18 Appendix Consider the unit deand case and assue that fir L produces the high quality good while fir produces the low quality one. This says that fir L serves consuers with a arginal valuation of quality θ in θ, θ, while fir serves consuers with arginal variation in [θ,θ ). Reind that the threshold valuation θ is given by the ratio θ (p I p ) / (q I q ). By definition, in the unit deand case price and quality changes have only an ipact on the arginal consuer. As a result, changes in p and q have the following arginal ipact on deand X : It follows that that is to say, X f (θ ) θ p p f (θ ), q I q (36) X f (θ ) θ q q f (θ ) θ. q I q (37) X q θ q θ X p, θ θ p. (38) We hereafter check whether and, if so, under which conditions, a siilar relationship holds true when changes in p L and q L are considered. Observe that in the considered fraework, changes ade by fir L (the leader) have an ipact on fir (the follower) choices. This explains why the ipact of changes in p L and q L on the arginal consuer are not straightforward. A standard decoposition leads to: dθ θ p L + θ p p q L q dθ θ + θ p q L p 1 1 q L q + θ q p L q p µ L 1 p q + θ p L p L + θ q q L q q µ L θ p q + θ q L q L, (39a). (39b) If, as assued, fir profit axiization gives rise to an interior solution, p and q are deterined by the syste of first-order conditions µ p C f (θ ) (θ ), X q L q µ p C f (θ ) 1 C. X q L q θ q 18

19 This yields θ (θ ) C. (40) q Differentiating both sides of (40) with respect to p L and q L yields the following pair of equalities: dθ (θ )+θ f (θ ) dθ µ 2 C dq + 2 C X dp + X dq, p q 2 dθ (θ )+θ f (θ ) dθ 2 C q 2 dθ 2 C q 2 dq + 2 C f(θ ) q X q L q q X 2 C q X q µ X p dp + X q Making use of both (36) and (37), they can be rewritten respectively as ³ dq dp dq θ dθ 2 C q 2 (θ )+θ f (θ ) dq 2 C q X f(θ ) ³ dp q L q (θ )+θ f (θ ) dq., (41) dq θ. (42) Reind that both left-hand sides were defined in (39a) and (39b). This allows to rewrite both equations as a function of the sole follower reactions to leader decision changes, i.e. (dp / ), (dq / ), (dp / ) and (dq / ). More precisely, we obtain: dp dq θ (θ )+θ f (θ ) (q L q ) 2 C q 2 dq, (43) (θ )+θ f (θ ) f (θ ) 2 C q X dp dq θ θ [ (θ )+θ f (θ )] (q L q ) 2 C q 2 dq. (44) (θ )+θ f (θ ) f (θ ) 2 C q X If C ( ) is linear in q so that ( 2 C / q 2 ) 0, it follows iediately that µ µ dp dq dp dq θ θ θ. As a result, (41) and (42) yields to dθ θ dθ, (45) an equality that exactly irrors the relationship (38) obtained for the follower. 19

20 More generally, we know fro (40) that 2 C q 2 [θ (θ )] q θ [ (θ )+θ f (θ )]. q L q ro the unit-deand assuption, we have X (θ ) so that (40) also yields 2 C q X X [θ (θ )] θ. By plugging this last two results into (41) and (42), one obtains: µ µ dθ θ dq f (θ ) dp dq θ, (46) q L q (θ )+θ f (θ ) dp µ µ L dθ θ dq f (θ ) dp dq θ, (47) q L q (θ )+θ f (θ ) while substituting these sae results into (43) and (44) yields µ dp dq θ 1+ θ µ f (θ ) dq 1 θ, (48) (θ ) dp µ L dp dq θ θ 1+ θ µ f (θ ) 1+ dq. (49) (θ ) Cobining (46) with (48) and (47) with (49) we obtain: µµ dθ θ 1+ θ f (θ ) dq f (θ ), q L q (θ ) (θ ) µµ dθ θ 1+ θ f (θ ) dq + θ f (θ ). q L q (θ ) (θ ) Thus: µ dθ dθ + θ θ q L q µ 1+ θ f (θ ) (θ ) µ dq dq + θ, (50) that is (dθ / ) θ (dθ / ) if and only if θ (dq / ) (dq / ). In words, even in the unit deand case, an increase of one unit in q L is not equivalent to a decrease of θ units in p L, in general. Unless costs can be assued to be linear in quality, the ipact of price and quality changes are not related to each other through arginal consuers quality valuation θ. 20