Regulating a Multiproduct Monopolist with Unknown Demand: Cross-Subsidization and Countervailing Incentives

Transcription

1 Regulating a Multirodut Monoolist with Unknown emand: Cross-Subsidization and Countervailing Inentives by IÑAKI AGUIRRE AN ARANTZA BEITIA This aer studies the regulation of a multirodut monoolist that has rivate information about demand onditions. In artiular we onsider the regulation of a two-rodut monoolist with interdeendent demands when it has better information onerning the demand of one rodut than the regulator and ubli funds are ostly. e show that the otimal regulation oliy in this ase ruially deends on whether goods are substitutes or omlements. Cross-subsidization arises with demand omlementarities making it likely that ountervailing inentives haraterize the otimal ontrat. JEL: 8 L 5 Introdution It seems natural that regulated firms have better information about their oerating environment than regulators do: Beause of its suerior resoures its ongoing management of rodution and its frequent diret ontat with ustomers a regulated firm will often be better informed that the regulator about both its oerating tehnology and onsumer demand ARMSTRONG AN SAPPINGTON [7. 564]. Most regulated firms suly a range of roduts. For instane railroads offer freight and assenger servies eletri utilities generate ower at different times of the day or seasons of the year for both residential and industrial onsumers telehone omanies serve residential onsumers small and big businesses and so on see for examle CRE AN KLEINORFER [986] and LAFFONT AN TIROLE [993]. Hene the multirodut framework seems the most aroriate senario in whih to analyze regulatory oliies. e are grateful to three anonymous referees Ignaio Palaios-Huerta and artiiants in the XX Jornadas de Eonomía Industrial Granada 4 and the XXIX Simosio de Análisis Eonómio Pamlona 4. Finanial suort from the Ministerio de Cienia y Tenología and FEER SEJ and from the eartamento de Eduaión Universidades e Investigaión del Gobierno Vaso IT-3-7 is gratefully aknowledged. hile we onsider a multirodut monoolist we do not mean that the firm neessarily sells different roduts. Our model also fits for instane ontexts where a single-rodut firm sells the same good in different markets but demands are interdeendent. In this ontext it is imortant to understand whether the demands in the two markets are omlements or substitutes see LAYSON [998] and AACHI [5] for interesting examles many of them haraterized by onsumtion externalities. JITE 64/4 Art.-Nr. 4

2 In artiular in this aer we are interested in studying the regulation of a multirodut monoolist that has rivate information about demand onditions and whose regulator is allowed to make monetary transfers to the regulated firm. More reisely we onsider the regulation of a two-rodut monoolist with interdeendent demands that has better information onerning the demand for one rodut than the regulator. The assumtion of a single soure of asymmetri information for examle an idiosynrati shok affeting solely a market seems aroriate for industries suh as teleommuniations where either there is a new market or a new rodut so that the regulator annot redit its demand from ast history or where beause of tehnologial rogress or new information that affets onsumers valuation of the good demand is stohasti and the evaluation of demand requires ostly information that is heaer to obtain for the regulator see for examle IOSSA [999]. In these and similar ases it is likely the ase that the regulated firm will be better informed than the regulator about onsumer demand. e shall show that the otimal regulation oliy deends ruially on whether goods are substitutes or omlements. Although examles with demand substitutes arise naturally demand omlementarities aear in arguably rather ommon irumstanes. For instane: i network externalities and swithing ost: FARRELL AN KLEMPERER [7]; ii bandwagon effets: ROHLFS [7] onsiders two tyes: a network externalities whereby existing subsribers benefit from being able to ommuniate with a larger user set and b omlementary bandwagon effets whereby urhasers of the base rodut e.g. hardware benefit from the greater availability of ometitively sulied omlementary roduts e.g. software as the user set exands iii diret demand omlementary; iv a single rodut sold in different markets see AACHI [5]. ith demand substitutes the firm s inentive is always to understate the market demand whih is the one unknown to the regulator; and in order to redue informational rents the regulator sets ries for both roduts below those that would take lae in the full-information ase. ith demand omlements however the haraterization of the otimal regulatory oliy is more omlex and deends on whether rodut is subsidized sold below marginal ost under full information. In this ase the firm s inentive is always to overstate the market demand and in order to redue informational rents the regulator inreases the rie of rodut and redues the rie of rodut relative to the fullinformation ase. hen rodut is not subsidized under full information however the firm s inentive is always to understate the market demand and to redue informational rents the regulator redues the rie of rodut and inreases the rie of rodut. Finally we analyze the ossibility that ountervailing inentives in the sense of LEIS AN SAPPINGTON [989] may arise with demand omlements; in those ases the firm may be temted either to overstate or to understate its rivate information deending uon the realization of the demand arameter. This aer is organized as follows. Setion briefly reviews the literature on the regulation of monooly under asymmetri information. Setion 3 develos the

3 3 basi model. Setion 4 resents the full-information ase as a benhmark. In setion 5 we analyze the design of the otimal regulatory oliy under rivate information on demand onditions. Finally setion 6 offers onluding remarks. Related Literature For the ase of a single rodut monooly there is an extensive literature fousing on the design of otimal regulatory mehanisms under asymmetri information about: i ertain ost arameters see for examle BARON AN MYERSON [98] LAFFONT AN TIROLE [986] and BARON [989]; ii market demand RIORAN [984] LEIS AN SAPPINGTON [988a] and AGUIRRE AN BEITIA [4]; and iii both osts and demand onditions LEIS AN SAPPINGTON [988b] and ARMSTRONG [999]. The rie of the rodut is established by the regulator who is allowed to make monetary transfers to the regulated firm. The aliation of the rinial agent method leads to wellknown results: asymmetry of information auses in general a loss of effiieny due to the neessity of limiting monooly informational rents. The literature on the regulation of a multirodut monooly has mainly foused on the ase of asymmetri information onerning osts SAPPINGTON [983] LAFFONT AN TIROLE [99a] [99b] ANA [993] and ARMSTRONG AN ROCHET [999]. esite its imortane however the analysis of the design of regulatory oliies when the regulated multirodut firm is better informed about demand than the regulator has been mostly negleted in the literature. Notable exetions are IOSSA [999] and ARMSTRONG AN VICKERS []. IOSSA [999] addresses the issue of how to organize with a multirodut monooly or with a differentiated duooly a two-rodut industry with interdeendent demands when the regulator has worse information than the firm. As in her setting we assume that there exists a single soure of asymmetri information: the multirodut firm observes rivately the realization of an idiosynrati shok affeting the demand of only one of its roduts. However she assumes linear demands whereas we allow for a more general demand onfiguration; and she analyzes the effets of asymmetri information on the otimal industry struture whereas we fous on the otimal regulation of the multirodut firm. Further one more imortant differene rests in the treatment of omlementary goods in that she does not allow riing one rodut below marginal ost. This riing oliy ross-subsidization however is relevant theoretially and emirially to multirodut firms whether they are regulated or not. In AGUIRRE AN BEITIA [4] we analyze the otimal regulation oliy when a single-rodut firm has better information onerning the market demand than the regulator. This aer reresents a generalization that fits better the multirodut nature of regulated firms in the real world where the interdeendene between the demands of different roduts lays an imortant role.

4 4 The ossibility of a multirodut monoolist deiding to rie one good below marginal ost when roduts are demand omlements has been analyzed for examle by TIROLE [988] and more reently by AVIS AN MURPHY []. These authors show not only that a multirodut firm may be interested in riing below marginal ost but also that the rofit-maximizing outome an involve a zero or negative rie for one good. 3 On the other hand CRE AN KLEINORFER [986] analyze the roblem of multirodut Ramsey riing with interdeendent demands and disuss in deth the ase in whih one rodut is subsidized sold below marginal ost beause of the benefiial effets that it may have on the sales of the other rodut. They also inlude very interesting aliations to the teleommuniations industry. In fat some authors like SRINAGESH [984] onsider that ross-subsidization is so ommon in teleommuniations that it might be labeled a stylized fat. 4 Further muh literature has doumented the resene of ross-subsidization in the water gas and eletriity utility industries see for examle SAKINS AN REI [7] for the water industry in transortation and in hositals HARRIS [979]. ARMSTRONG AN VICKERS [] disuss the regulation of a multirodut monoolist when the firm has rivate information about ost or demand onditions. Contrary to the standard literature on the otimal regulation under asymmetri information they do not ermit the use of lum-sum transfer. They fous on the question of how muh riing disretion should be granted to a regulated multirodut firm. Although we onsider a standard model of regulation under adverse seletion that allows transfers we adot the framework they roosed in their Examle 4 by onsidering that the soure of asymmetri information is an additive shok rivately observed by the firm. 3 The Model e onsider the following regulatory environment. The market demands for the roduts of the firm are given by and where i is the unit rie of rodut i i and the arameter atures the firm s rivate information onerning the market demand of rodut. e assume that the i greater the rie of the good the lower the demand / i < that the own effet is greater than the ross effet i i / > / i j j i i j 3 Their analysis shows that although several fators lay a role in the design riing and distribution of Mirosoft s Internet Exlorer a key fator for this tehnology being inluded in indows at no searate harge is the omlementary nature of the demand for indows and the demand for eb use. 4 ROHLFS [979] shows that substantial ross-subsidization ours between loal servie whih is ried aroximately 5% below marginal ost and long distane whih is ried at two or three times marginal ost.

5 5 and that the higher the realization of the greater the quantity of outut rodut onsumers demand at any nonnegative rie / >. Moreover we assume that / /. Note that any setting in whih higher realizations of orresond to arallel outward shifts in demand onstitutes a setting where this roerty is satisfied: that is we might onsider the demand of rodut as d g with g >. e may see then the onsumer surlus as v v g where v is a known onsumer surlus funtion and is some high referene rie and v / v / g where d v /. That is we onsider an additive shok affeting market similar to that in Examle 4 in ARMSTRONG AN VICKERS []. 5 The regulator s unertainty about the arameter is reresented by a robability distribution F with assoiated density funtion f stritly ositive on the suort [ ]. The funtion is ommon knowledge but the realization of is observed only by the firm. The ost known of roduing q and q is given by Cq q q q F where and are the onstant marginal osts and F is the fixed ost. The regulator is endowed with the ower to set unit ries and for the firm s oututs and to seify a transfer t from onsumers to the firm. To simlify the analysis we assume that the onsumer demand deends only on ries and. 6 The regulator an observe and therefore enfore the regulated ries. The quantities sold at the regulated ries are assumed to be too ostly for the regulator to monitor diretly see for examle LEIS AN SAPPINGTON [988a] for a disussion of this assumtion. However the regulator an still be sure that the firm fulfills its mandate to serve all demand at the regulated ries. The regulator need only invite onsumers to reort any inident in whih they were either refused servie at the established ries or harged higher ries and enalize the firm for suh rationing. e onsider a generalized soial welfare funtion that inororates both distributional onsiderations through a oeffiient α [ ] affeting firm s 5 LEIS AN SAPPINGTON [988a] [988b] ARMSTRONG [999] IOSSA [999] and AGUIRRE AN BEITIA [4] also onsider additive shoks. 6 The total fixed harge may be thought of as aortioned among onsumers in suh a manner that no onsumer is exluded from urhasing the good. The analysis an thus be onduted in terms of the aggregate fixed harges t aid to the firms by onsumers.

6 6 rofits and ostly ubli funds: raising and transferring $ through ubli hannels osts soiety $ : t v α t t where v is the onsumer surlus and the rofit of the firm is given by t F t. It is standard in the literature to introdue distributional onsiderations through a oeffiient α [ ]. ARMSTRONG AN SAPPINGTON [7. 56f.] stated: The regulator s referene for onsumer surlus over rent indiated by α < reflets a greater onern with the welfare of onsumers than the welfare of shareholders. This might be due to differenes in their average inome or beause the regulator ares about the welfare of loal onstituents and many shareholders reside in another jurisditions. 7 The arameter is usually alled the shadow ost of ubli funds. Transfers between a firm and either onsumers or the state may involve administrative osts tax distortions or ineffiienies that must be taken into aount in the design of the regulatory mehanism. See for examle LAFFONT AN TIROLE [993]. 4 The Full-Information Case: A Benhmark Consider the benhmark ase in whih the regulator knows all omonents of demand funtions. The roblem of the regulator under full information is given by max t t subjet to t. Solving for t and substituting t in the roblem is seen to be equivalent to max subjet to. That is max v F 3 α subjet to. The first-order onditions are i 4 i i [ ] i 7 CAILLAU et al. [988] resent a omlete disussion on welfare funtions in the regulated ontext.

7 7 i i i i 5 6 α. e an interret the onditions 4 and 5 as follows: A marginal inrease in i redues onsumer surlus in i while the rofit of the firm inreases in i i i i / i j j j / i. Given that the firm s rofit is zero at the otimum the regulator an redue the transfer by the equivalent of that amount whih imlies a saving in soial ost of [ i i i i / i j j j / i ]. At the otimal rie the marginal soial ost of a hange in rie equals the marginal soial benefit. e an rewrite these onditions as 7 j i i i i ji 8 i ii Ri ii where ii is the rie elastiity of market i s demand ji is the ross elastiity ii i / i i / i ji j / i i / j and R i i i i j j i. The ondition 8 an be exressed as 9 i i i ii where ij ji ii jj ii ii R j ji R i j j i. i jj The suerelastiities ii were first derived by BOITEUX [956]. e obtain that the otimal ries are given by the Ramsey formula: the Lerner index or rie marginal-ost ratio of eah good is inversely roortional to its suerelastiity of demand. In artiular the Lerner index is / times the reiroal of the suerelastiity of demand where / is known as the Ramsey number. hen is zero taxation is not distortive and the otimal rie of eah good is therefore equal to its marginal ost. hen the shadow ost of ubli funds beomes very high ries tend to the monooly ries.

8 8 hen roduts and are demand substitutes ji > i j j i the suerelastiities are always ositive ii > i given that the own effets are greater than the ross effets ii jj > ij ji. The otimal ries under full information are therefore higher than the marginal osts: i i > i. On the other hand when roduts are demand omlements ij < i j j i the suerelastiity of one and only one rodut may be negative and therefore its rie might be lower than its marginal ost. 8 In that ase one rodut would be subsidized sold below its marginal ost beause of the ositive effets it has on sales of the other rodut. However both suerelastiities annot be simultaneously negative. 9 The full-information oliy onsists of ries i i given by and transfer ayments t suh that firms obtain zero rofits. It is easy to hek that the otimal rie of rodut is an inreasing funtion of the onsumer demand: d > d given that / >. On the other hand the effet of on the otimal rie of rodut is given by d d d d. d d d d Therefore the sign of d / d deends ruially on the sign of Note that when > then d / d >. In onsequene in resonse to a hange in it would be otimal to move both ries in the same diretion: d / d > and d / d >. In that ase we say that ries are soial omlements. On the other hand when < then d / d < ; hene in resonse to a hange in it would be otimal to move the two ries in oosite diretions: d / d > and d / d <. In that ase we say that ries are soial substitutes. Note that if i / i j then ries are soial omlements substitutes when roduts are demand substitutes omlements. 8 Note that the suerelastiity of rodut i is negative when Ri jj R j ji < that is j j i j when / i > / j. 9 Suose that the suerelastiity of rodut j were also negative. Then we would i i j i have that / j > / i. But these two onditions would then imly that i j i j / i / j > / j / i whih ontradits the fat that own effets are greater than ross effets. Of ourse when roduts are demand-indeendent or the regulated monoolist sells the same good in two erfetly searated markets and therefore

9 9 5 Regulation under Private Information on emand e now study the regulation model when the firm has rivate information about the demand of rodut. The arameter is ontinuously distributed on the suort Θ [ ] aording to the umulative distribution funtion F and stritly ositive density f. e assume that F satisfies the monotone-hazardrate ondition; that is the ratios f / f and F / f are nondereasing funtions of. The single-rossing roerty whih states that the greater the demand the more systematially willing a firm is to forgo transfer ayments to obtain a higher unit rie for rodut holds if the firm s marginal rate of substitution MRS of rodut rie for transfer ayment grows with. Given the firm s rofit defined by we have MRS /. t t Sine we have assumed that / then MRS t >. On the other hand the firm s MRS of rodut rie for transfer ayment does not hange with. To haraterize the otimal regulatory oliy under rivate information we first determine the lass of feasible oliies and then selet the otimal oliy from within that lass that is we adot the aroah of BARON AN MYERSON [98] and GUESNERIE AN LAFFONT [984]. At the first stage we restrit the analysis to diret revelation mehanisms by the revelation rinile. A diret revelation mehanism is omosed of transfer funtions and assoiated rie levels given by t { } [ ]. Therefore we may be restrited to regulatory oliies that require the firm to reort its rivate information arameter truthfully that is inentive-omatible regulatory oliies to determine the lass of feasible oliies. The regulator d / d. Then the analysis would be similar to the regulation of a single monooly under unknown demand see for examle AGUIRRE AN BEITIA [4]. These roerties require the density funtion not to inrease too raidly and are satisfied by frequently used distribution funtions for examle the uniform normal and exonential funtions. The revelation rinile was established by MYERSON [979] and ASGUPTA HAMMON AN MASKIN [979].

10 maximizes the exeted soial welfare subjet to the following inentive omatibility and individual rationality onstraints: Inentive omatibility onstraints IC: the firm reorts truthfully if the rofit it exets to obtain by announing its tye is at least as great as the exeted rofit from any other reort. That is IC Θ where and. [ ] [ ] F t Individual rationality onstraints IR: the regulator annot fore the firm to artiiate if it exets negative rofits. That is IR Θ. The regulator s roblem an be written as 3 t t max f d subjet to IR and IC. The following lemma haraterizes the lass of oliies that satisfies IC. LEMMA Neessary and suffiient onditions for IC are: i d [ ]. d ii The regulated rie is a nondereasing funtion of. The otimal regulatory oliy deends on whether the ries of the two roduts are soial omlements or soial substitutes whih are onets in general that are inversely related to the notions of demand substitutes and demand omlements resetively.

11 5. Soial Comlements hen ries are omlements from a soial welfare oint of view S i j > 3 otimal ries under omlete information are always higher than marginal osts. The next lemma restrits the otimal regulatory oliy. LEMMA Given that > [ ] the otimal regulatory rie of rodut under rivate information must over marginal ost. That is [ ]. Therefore Lemma imlies that for any inentive-omatible oliy informational rents must inrease with the demand arameter that is d / d > [ ]. As a onsequene the individual rationality onstraint needs to be satisfied only at and we an rewrite the firm s rofits of tye as d 4. The regulation roblem an thus be rewritten as 5 max f d subjet to a d b d d d. By introduing ondition a into the objetive funtion and by taking into aount that ondition b is binding at the otimum we an rewrite the soial welfare in state as 3 Reall that if i / ries are soial omlements when roduts are demand substitutes. i j

12 6 [ ]. α f F F v The regulator s maximization roblem under inomlete information is given by 7 d f max subjet to and d. First we solve the roblem without taking into aount onstraints and d. Then we analyze the imliations of these restritions on the otimal regulatory oliy. The first-order onditions are 8 [ ] F f α 9. The following roosition states the main result of this subsetion. PROPOSITION The otimal regulatory rie oliy under rivate information PI and PI is given by PI PI where and solve 8 and 9 that is f F R α R and is suh that. By solving and jointly we get f F α

13 3 α R F 3. R f By omaring the omlete-information solution and the asymmetri information riing oliy PI PI in Figure we see that for all [ we have > PI and > PI. 4 Figure Regulatory Poliy with Soial Comlements under Full and under Asymmetri Information PI PI hen the regulator hooses the otimal rie levels under rivate information she maximizes the exeted soial welfare subjet to the inentive omatibility onstraint. In order to redue the informational rents of the monooly she will make this onstraint as small as ossible while still being omatible with the maximization of the soial surlus. To redue this rent the regulator must redue for all [. Given that > and she ahieves this objetive by dereasing. Moreover given that ries are omlements from a soial welfare oint of view it is otimal to move these variables in the same diretion and as a onsequene is redued. Figure also shows how the two rodut ries hange with under both full information and asymmetri information. e next disuss the effet of distributional onsiderations measured through α and the ost of ubli funds on the otimal regulatory oliy under rivate information. It must be stressed that if the ost of ubli funds is zero then marginal-ost riing would be otimal under both omlete and inomlete information indeendently of distributional onsiderations. Therefore the results of LEIS AN SAPPINGTON [988a] may be generalized to a multirodut firm 4 For the sake of simliity we reresent in the figures the ase of linear funtions. This would be the ase for examle when the distribution funtion is uniform and / g.

14 4 setting. 5 Under ostly ubli funds Ramsey riing haraterizes the otimal oliy under full information whih imlies that informational rents neessarily aear in order to guarantee inentive omatibility. 5. Soial Substitutes hen ries are substitutes from a soial welfare oint of view the otimal ries under omlete information may be smaller than marginal osts deending on the value of. e shall distinguish three ases: 5.. emand Comlements when Produt is Not Cross-Subsidized The first ase ours when rodut is not subsidized by rodut in the sense that under full information the otimal rie of rodut is above marginal ost for any realization of ; that is > [ ]. Given that Lemma alies the haraterization of the otimal regulatory ries under rivate information is similar to the ase of soial omlements. By omaring the omlete-information solution and the PI PI asymmetri information riing oliy in Figure we see that PI PI for all [ we have > and <. Figure Regulatory Poliy with Soial Substitutes under Full Information and Asymmetri Information PI PI hen the regulator hooses the otimal rie levels under rivate information she maximizes the exeted soial welfare taking into aount the inentive omatibility onstraint and in order to redue informational rents the regulator 5 If there is no ost of ubli funds then the full-information oliy whih oinides with the first-best oliy is imlementable and rivate information is inonsequential for regulation.

15 5 must redue for all [. Given that > and she again ahieves this objetive by dereasing. Moreover given that ries are substitutes from a soial welfare oint of view it is otimal to move these variables in oosite diretions and as a onsequene is inreased. 5.. emand Comlements hen Produt Is Cross-Subsidized The seond ase ours when rodut is subsidized by rodut in the sense that under full information the otimal rie for rodut is below its marginal ost for any realization of. That is < [ ]. The next lemma restrits the otimal regulatory oliy. LEMMA 3 Given that < [ ] the otimal regulatory rie of rodut under rivate information must be lower than or equal to marginal ost. That is [ ]. Therefore Lemma 3 imlies that for any inentive-omatible oliy informational rents must not inrease with the demand arameter: d / d [ ]. The individual rationality onstraint needs to be satisfied only at and we an rewrite the firm s rofits of tye as d 4. The regulation roblem may be written as 5 max f d subjet to a d b d d d. By introduing a into the objetive funtion and by taking into aount that b is binding at the otimum we rewrite the soial welfare in state as

16 6 6 [ ]. α f F F v The regulator s roblem under inomlete information is therefore given by 7 d f max subjet to and d. The first-order onditions [without taking into aount onstraints and d] are then 8 F f α 9. The next roosition states the otimal regulatory rie oliy. PROPOSITION The regulatory oliy under rivate information PI and PI is PI PI where and solve 8 and 9 that is 3 f F R α 3 R and is suh that. By solving 3 and 3 jointly we get 3 f F α

17 7 α R F 33. R f Figure 3 Regulatory Poliy with Soial Substitutes and Cross-Subsidization PI PI Figure 3 illustrates the omarison between regulatory oliies under both full PI information and rivate information: for all ] we have < PI and > In order to redue informational rents the regulator redues the differene with reset to the omlete-information ase for all in ]. Given that this differene is negative the regulator inreases and sine ries are soial substitutes this imlies a derease in Countervailing Inentives The third ase ours when rodut subsidizes rodut only for low realizations of ; that is [ ]. As we shall show this ase is haraterized by the existene of ountervailing inentives. Let be the value of the demand arameter suh that. Given that d / d > then < for all [ and > for all ]. The next lemma restrits the otimal regulatory oliy. LEMMA 4 The otimal regulatory rie of rodut under rivate information may be above or below than marginal ost deending on. In artiular [ ] and [ ]. Lemma 4 imlies that / d and d / d for all [ ]. Countervailing inentives are then unavoidable see for examle LEIS AN SAPPINGTON [989] MAGGI AN d for all [ ]

18 8 RORIGUEZ-CLARE [995] JULLIEN [] or ARMSTRONG AN SAPPINGTON [7]. The individual rationality onstraint needs to be satisfied only at and we an rewrite the firm s rofits of tye as 34 ] [ d 35 ]. [ d The regulation roblem an be written as 36 d f max subjet to a ] [ d a ] [ d b d d d ] [ d ]. [ By introduing a and a into the objetive funtion and by taking into aount that b is binding at the otimum we an rewrite the soial welfare in state as 37 [ ] ]. [ ] [ α α f F f F F v

19 9 Thus the regulator s roblem is 38 max f d subjet to d and d. The next roosition haraterizes the otimal regulatory rie oliy. PROPOSITION 3 The otimal regulatory rie oliy under rivate information PI and PI is given by PI PI where and are defined by 3 3 and 9 resetively; is suh that and is suh that. Figure 4 Countervailing Inentives PI PI Figure 4 illustrates the relationshi between otimal ries under the two informational regimes full information and asymmetri information as a funtion of the tye. Note that there are ountervailing inentives under the otimal regulatory oliy: for low realizations of the firm s inentive to overstate will dominate its inentive to understate while for higher realizations the dominant inentive will be to understate. The otimal regulatory oliy will resribe rodut s ries rodut s ries below above marginal ost for low realizations of and rodut s ries rodut s ries above below marginal ost for high realizations. For intermediate

20 realizations the ountervailing inentives omel the regulator to set a rie for eah rodut that does not vary with the realized demand of rodut. As a onsequene no riing authority is delegated to the firm and the regulator relies on her rior beliefs to establish a single regulated rie for eah rodut. The tye of ountervailing inentives in our model is similar to that in LEIS AN SAPPINGTON [989]. Informational rents are dereasing for low tyes low realizations of equal to zero for an intermediate interval of tyes intermediate realizations of and inreasing for high tyes high realizations of. 6 6 Conluding Remarks In this aer we have studied the otimal regulation oliy of a multirodut monoolist when the regulated firm has better information about demand onditions than the regulator. e find that the otimal regulation oliy deends ruially on whether goods are substitutes or omlements. ith demand substitutes the firm s inentive is always to understate the market demand; to redue informational rents the regulator sets ries for both roduts below those under full information. ith demand omlements the haraterization of the otimal regulatory oliy is more omlex and deends on whether rodut is subsidized under full information. In that ase the firm s inentive is always to overstate the market demand and in order to redue informational rents the regulator inreases the rie of rodut and redues the rie of rodut. hen rodut is not subsidized under full information the firm s inentive is always to understate the market demand; and to redue informational rents the regulator redues the rie of rodut and inreases the rie of rodut. Finally we analyze the ossibility that ountervailing inentives may arise with demand omlements. In those ases the firm may be temted either to overstate or to understate its rivate information deending uon the realization of the arameter of demand. Our results may be generalized in several diretions. The assumtion of onstant marginal ost ould be relaxed in order to allow inreasing and searable marginal ost without altering our main results. The analysis might also be extended to onsider the regulation of a monoolist selling n substitute roduts the ase with omlements would be very omlex under rivate information onerning one market. 7 One further ossible extension that we leave for further researh is to onsider the otimal regulation of a multirodut 6 MAGGI AN RORIGUEZ-CLARE [995] identify other tye of otimal ontrol where informational rents are bell-shaed with both extreme tyes earning no rents. 7 The ase of multidimensional rivate information affeting the demand of different roduts would involve onsiderable tehnial diffiulties. This situation an be analyzed only to the best of our knowledge in very simle settings where either there is erfet orrelation between the rivate information arameters in this setting our results ould be generalized to the ase of demand substitutes or the tye sae is disrete rather than ontinuous.

21 firm with rivate information onerning the degree of substitutability between its roduts. A. Proof of Lemma Aendix The roof of Lemma is standard see for examle BARON AN MYERSON [98] and GUESNERIE AN LAFFONT [984]. The rofit of the firm of tye when it reorts is given by [ ] [ ] F t. If truth-telling is a loal otimal resonse for the monoolist then the following two onditions are satisfied: the first-order ondition and. By differentiating the rofit funtion with reset to we obtain that d d. d d If we evaluate this exression at from the first-order ondition we have d d d whih orresonds to art i in Lemma. By differentiating the first-order ondition with reset to we have. Note that if the seond-order ondition is satisfied then [ d / d ]. Given that then d / d is equivalent to the suffiient loal order ondition. It is easy to show that d / d also imlies global otimality. If truthtelling is a global otimum for tye we have that or equivalently Θ x dx. By using the first-order ondition we have

22 [ x x x ] dx or Therefore / d imlies x y dxdy. x [ / ] d d and given that x when > and x when < a loal otimum is always global. Q.E.. A. Proof of Lemma Assume that is a solution to the regulatory roblem under rivate information suh that < for [ ]. Note that < imlies that < for [ ] given art ii of Lemma. At [ ] the sloe of required to indue truth-telling given Lemma art i is d d < whih seifies how informational rents must hange with the rodut demand arameter. Individual rationality IR and d / d < imly that any tye [ ] would obtain a stritly ositive information rent. Consider a rie oliy suh that for [ ] for ]. Note that when moving from to total surlus inreases sine from a soial welfare oint of view the rie of rodut must be greater than its marginal ost beause of the ost of ubli funds and informational rents also derease. This argument therefore means that is not a solution to the regulatory roblem. Q.E.. A.3 Proof of Lemma 3 Assume that is a solution to the regulatory roblem under rivate information suh that > for [ ]. Note that > imlies that > for [ ] given art ii of Lemma. At [ ] the sloe of required to indue truth-telling given Lemma art i is d d >

23 3 whih seifies how informational rents must hange with the rodut demand arameter. Individual rationality IR and d / d > imly that any tye would obtain a stritly ositive information rent. Consider a rie [ ] oliy suh that for [ for [ ]. Note that when moving from to total surlus inreases sine from a soial welfare oint of view the rie of rodut must be lower than its marginal ost beause of the ost of ubli funds and informational rents also derease. This argument therefore ontradits as a solution to the regulatory roblem. Q.E.. A.3 Proof of Lemma 4 The roof is straightforward taking into aount the roofs of Lemma and Lemma 3. The tye is the value of the demand arameter suh that and < for all [ and < for all ]. Lemma alies when > and Lemma 3 when <. Therefore from Lemma 3 it follows that [ ] and from Lemma that [ ]. Q.E.. A.4 Proof of Proosition Firstly if we do not take onstraint d into aount then it is straightforward to obtain the rie oliy from the onditions 9 and. If we inlude onstraint d then the otimal regulatory oliy for rodut may be written as PI max { }. enote by the tye suh that the onditions 9 and are satisfied but the rie of rodut is equal to marginal ost that is. By evaluating 9 and at we obtain < [ > ] [ ] beause > and. Given that ries are soial omlements > and that seond-order onditions must be satisfied the above inequalities imly that for any ] there is a soial benefit a saving of ubli funds from inreasing the rie of rodut beyond its marginal ost and

24 4 from inreasing the rie of rodut over whih imlies PI and PI for all ]. However for tyes in the range [ an inrease in beyond its marginal ost along with the orresonding inrease in generates a loss of soial welfare. Given Lemma must be equal to and onsequently must be at the otimal level when the rie of rodut equals its marginal ost that is. Therefore when < the otimal rie oliy is PI and PI. Q.E.. A.5 Proof of Proosition and Proof of Proosition 3 The roofs are almost idential to the roof of Proosition and hene omitted. Referenes AACHI T. [5] Third-egree Prie isrimination Consumtion Externalities and Soial elfare Eonomia AGUIRRE I. AN A. BEITIA [4] Regulating a Monoolist with Unknown emand: Costly Publi Funds and the Value of Private Information Journal of Publi Eonomi Theory ARMSTRONG M. [999] Otimal Regulation with Unknown emand and Cost Funtions Journal of Eonomi Theory AN R. PORTER eds. [7] Handbook of Industrial Organization Vol. 3 North- Holland: Amsterdam. AN J.-C. ROCHET [999] Multi-imensional Sreening: A User s Guide Euroean Eonomi Review AN. E. M. SAPPINGTON [7] Reent eveloments in the Theory of Regulation in: ARMSTRONG AN PORTER eds. [7]. AN J. VICKERS [] Multi-Produt Prie Regulation under Asymmetri Information Journal of Industrial Eonomis XLVIII BARON. [989] esign of Regulatory Mehanisms and Institutions in: R. Shmalensee and R. illig eds. Handbook of Industrial Organization Vol. North-Holland: Amsterdam. AN R. MYERSON [98] Regulating a Monoolist with Unknown Costs Eonometria BOITEUX M. [956] Sur la gestion des monooles ublis astreints à l equilibre budgétaire Eonometria 4 4. CAILLAU B. R. GUESNERIE P. REY AN J. TIROLE [988] Government Intervention in Prodution and Inentives Theory: A Review of Reent Contributions RAN Journal of Eonomis 9 6. CRE M. A. AN P. R. KLEINORFER [986] The Eonomis of Publi Utility Regulation Mamillan: London. ANA J.. [993] The Organization and Soe of Agents: Regulating Multi-Produt Industries Journal of Eonomi Theory ASGUPTA P. P. HAMMON AN E. MASKIN [979] The Imlementation of Soial Choies Rules: Some General Results on Inentive Comatibility Review of Eonomi Studies

25 5 AVIS S. J. AN K. M. MURPHY [] A Cometitive Persetive on Internet Exlorer Amerian Eonomi Review: Paers and Proeedings FARRELL J. AN P. KLEMPERER [7] Coordination and Lok-In: Cometition with Swithing Costs and Networks Effets in: ARMSTRONG AN PORTER eds. [7]. GUESNERIE P. AN J.-J. LAFFONT [984] A Comlete Solution to a Class of Prinial Agent Problems with an Aliation to the Control of a Self-Managed Firm Journal of Publi Eonomis HARRIS J. E. [979] Priing Rules for Hositals Bell Journal of Eonomis IOSSA E. [999] Informative Externalities and Priing in Regulated Multi-Produt Industries Journal of Industrial Eonomis XLVII JULLIEN B. [] Partiiation Constraints in Adverse Seletion Models Journal of Eonomi Theory LAFFONT J.-J. AN J. TIROLE [986] Using Cost Observation to Regulate Firms Journal of Politial Eonomy AN [99a] The Regulation of Multi-Produt Firms Part I: Theory Journal of Publi Eonomis AN [99b] The Regulation of Multi-Produt Firms Part II: Aliations to Cometitive Environments and Poliy Analysis Journal of Publi Eonomis AN [993] A Theory of Inentives in Prourement and Regulation MIT Press: Cambridge MA. LAYSON S. K. [998] Third-egree Prie isrimination with Interdeendent emands Journal of Industrial Eonomis LEIS T. R. AN. E. M. SAPPINGTON [988a] Regulating a Monoolist with Unknown emand Amerian Eonomi Review AN [988b] Regulating a Monoolist with Unknown emand and Cost Funtions RAN Journal of Eonomis AN [989] Countervailing Inentives in Ageny Problems Journal of Eonomi Theory MAGGI G. AN A. RORIGUEZ-CLARE [995] On Countervailing Inentives Journal of Eonomi Theory MYERSON R. [979] Inentive-Comatibility and the Bargaining Problem Eonometria RIORAN M. [984] On elegating Prie Authority to a Regulated Firm RAN Journal of Eonomis ROHLFS J. [979] Eonomially Effiient Bell-System Priing Bell Laboratory isussion Paer No. 38. [7] Bandwagon Effets in Teleommuniations. 8 7 in: S. K. Majumdar I. Vogelsang and M. E. Cave eds. Handbook of Teleommuniations Eonomis Vol. North-Holland: Amsterdam. SAPPINGTON. E. M. [983] Otimal Regulation of a Multirodut Monooly with Unknown Tehnologial Caabilities The Bell Journal of Eonomis SAKINS J.. AN S. REI [7] The Measurement and Regulation of Cross Subsidy: The Case of Sottish ater Industry Utilities Poliy A mathematial sulement is available at htt://gsbwww.uhiago.edu/fa/ steven.davis/researh.

26 6 SRINAGESH P. [984] Mixed Linear Nonlinear Priing with Bundling Journal of Regulatory Eonomis TIROLE J. [988] The Theory of Industrial Organization MIT Press: Cambridge MA. Iñaki Aguirre Arantza Beitia eartamento de Fundamentos del Análisis Eonómio I University of the Basque Country Avda. Lehendakari Aguirre Bilbao Sain inaki.aguirre@ehu.es arantza.beitia@ehu.es