Next generation access networks and Smart Grid services - an applied model Preliminary paper

Transcription

1 Next generation aess networks and Smart Grid servies - an applied model Preliminary paper François-Noël Bernal yz Grenoble Applied Eonomis Laboratory January 30, 07 Abstrat Smart grid spreading requires a spei and integrated ommuniation infrastruture at the residential level. Distribution Network Operators, as private rms, are expeted to provide new servies to this market. They ould provide it through two ommuniation networks, whih ould be perfet substitutes or omplements, with respet to the smart grid servie onsidered. We fous here on residential load shedding servie. With regard to spei networks onditions, DNOs should use BPL network and private rm the optial ber network. We aim to determine the impats of a di erential on ost e ieny on the industrial organization in a mixed duopoly. Thus, relying on a standard prie model on the Hotelling line, we examine the relationship between the industrial organization and the degree of ost di erentiation in a mixed duopoly in whih a welfaremaximizing publi enterprise ompetes against a pro t-maximizing private rm. We then onsider Ramsey-Boiteux priing to ful ll balaned pro t onditions for the regulated rm. We show that several industrial ompetition strutures are implied by this riterion. Also, ost e ieny gap between the two rms lead to di erent situations in terms of welfare and in terms of surplus distribution, providing insights of best situations, depending ost onditions. Keywords : load shedding, mixed duopoly, Hotelling JEL : L3 Grenoble Alpes University, GAEL, F Grenoble, Frane y mailto:franois-noel.bernal@univ-grenoble-alpes.fr z

2 Introdution While at the level of the eletriity generation, strong evolutions and innovations are denoted, the remark is more quali ed in an empirial way on industries of distribution and transmission (Erlinghagen & Markard [0]). The development of the Smart Grid onept is on this level a key framework. Atually, it ontains the most important potential tehnologial and operational innovations on the downstream setor of the distribution, by inreasing the exibility of the overall eletriity demand. And this, in so far as it potentially makes possible to serve new needs: management of onsumption, aquisition of information, themselves vetor of behaviors knowledge. But it also allows to supplement and diversify the tari o ers, besides being a vetor of integration for the new pro- les (ompare to integration in the network of renewable energies, the eletri vehiles) (Clastres [0]). The spei deployment of the intelligent meters, with strong objetivediretives for the European ountries for example (80% of smart meters by 00), must be aordingly onsidered. If the omplete property and features are empirially di erent aording to the studied ountries, it remains interesting to see that on the level of the transmission of information, between the plae whih is the household and the one allowing the treatment upstream, this arhiteture neither is harmonized nor lear-ut. In fat, two types of means are used as infrastruture-key of the feedbak. Struturally speaking, all the servies onsidered (ounting, load shedding) ould be transmitted by the two types. In pratie, it seems to exlusively tend to have a servie of ounting on a network, while the servie of load shedding management is mainly treated by another network. Nevertheless, knowing the neessary ost of the improvement of the networks to serve these more demanding needs, it appears interesting to see what the literature brings and let develop on the behavior of these agents of potentially di erent nature. Namely, how the substituability of the network to provide the servie impat the industrial organization is the onern. Espeially as substitutability is stressed by potential e ieny gaps between providers, due to their networks. Initially, the empirial developments onerning the supply of servies stamped smart grid within the households and and of the methods of ommuniations expeted and used will be treated. Then the teahing of the literature sine it is onsidered that two substitutable servies an provide the onsumer will be looked at. This will lead to present a model taking into aount theses servies and help determine the e ets of xed osts on rms pro tability. Smart grid and teleommuniation networks Initially, let s onsider the smart grid, that is to say the way the market downstream is arranged between the eletrial network and the ommuniation networks. A potential substitutability of the ommuniation network will be then onsider, and in a seond time, the development of a model trying to formalize

3 this situation. It an be useful rst to note that the development of Smart Grid is arried out in a planned way, with horizons of over in the long term. For example, the amount of investment expeted until 00, for European Union only, is about 56 Be (Pike Researh, 0). With some amount oming diretly from subsidies, like, in this ase, more or less than 00 Me from the European Energy Programme for Reovery (EEPR). Although this is onsidering the whole Smart Grid vision and do not spei ally apply to the downstream market, it helps to see at whih extent smar grid are expeted. These plans of quanti ed deployments leave nevertheless ertain latitude in their deployment. And in partiular, onerning the distribution of the roles between the ators, whih is a major onern. Objetives for wih the need is inreased by an asymmetry of innovation upstream-downstream The setor of eletriity is reognized like having made emerge, and having integrated, many innovations on the upstream side of the hain of value. So onerning the eletriity generation, transmission and distribution setor. In these various ases, important e orts were already arried out, drawn by industrial nature and the eonomies from osts. On the other hand, and although the potential is theoretially important, the integration of innovations on the level downstream remained weaker. Beyond the assoiated ost, various points ould justify this irrefutable fat. On the one hand, a faility on the level of the establishment of the innovation: tehnial improvements installed during the replaements, low resistane to the innovation from the level where those are operated. In addition, a ertain onnetion with the produtive e etiveness and the maintenane of balane an be the soure of greater e orts. Lastly, a ertain limitation due to the di ulties during a time to transmit in a reliable and regular way the onsumer information to the supplier. In fat, the supply of servies known as intelligent requires either of important means of ommuniation, or of important means of treatment, or both. For example, to redue a onsumption to a given time : it is neessary to take note of the need for reduing the onsumption, to see of whih aggregate potential of redution the soiety lays out (whih is tantamount to knowing, for the same moment, the various di use potential omposing this aggregate), then ating to liquidate this potential. And we an note here that beyond the infrastruture of ommuniation, to improve the exibility of the eletriity demand implies to treat with a stronger unertainty. Thus, to have a onsequent ritial mass to overpass this unertainty. Servies of energy e ieny and households Conretely, the intelligene of the eletrial network needs to be seen as a mean of operating this lever whih is the exibility of the request. From the industries, but typially from the households. This exibility being treated here via servies of energy e ienies whih are dynami priing and obliteration. Indeed, there servies an be realized by larifying the signal prie, through a dynami priing. The 3

4 idea is then to better show the real tensions on the market in order to bring loser the behaviors to household onsumption to those optimal ones in suh situations. To better show the pries whih allow, while leaving to the onsumer the hoie to alloate voluntarily its onsumption, to restore the signal-prie. But beyond that, one an also foresee other servies: for example a supply of obliteration. That onsists of the provision of a monetary ounterpart related to a relative loss of the deisional power on the operation of one or more eletri omponents. For example by adjusting downwards the general thermostat, or by di ering the exeution from equipment household applianes. These various approahes require all a onsiderable development of the interfaes of ommuniation as well as methods of use. And this, although various studies relating to the installation of suh interfaes re et at the same time their impat but also the derease of their use [soure]. This development, beyond ergonomis, must also take into aount the issues relating to the transit of information (reliability -lateny- and seurity onerns). Next Generation Aess Networks(NGAN) and households Aordingly, it an be useful to reall that a great diversity of supports of ommuniation is used within the value hain. Indeed, various means of monitoring and transfer of information are mobilized, and this by various types of networks (power line ommuniation, Ethernet lines, mobile networks... ). Nevertheless, to provide the servies Smart Grid to the market downstream, the networks with broad band are onsidered. Beause they are the only ones being able to guarantee the viability of suh servies, respeting the various onstraints of reliability and volume. To deploy the intelligent network, it is to develop bidiretional interfaes of ommuniation along the eletriity generation, going through transmission assets to distribution setor. Having as an underlying neessity, a network support able to allow the transit of the noted information. For the ase of the downstream setor of the supply, there is a diversi ation, as well on the level of the property as of the use of this network. Beause there are not only one and single network, but several ommuniation networks used either exlusively, or jointly. In this ase, there are two tehnologies of networks able to allow the feedbak to distributors: the network by optial ber on the one hand, and the network by power line ommuniation (PLC) on the other hand. The rst, the optial ber, depends on the installations of the networks suppliers, but takes advantage from positive spillovers from its inreasing use for other needs in the onsumers, who are moreover austomed to its utilization and its funtion. Spillover insofar as the realization of this servie is only one faet of the various possibilities made possible by the optial ber (VODHD, television via Internet box, et). The seond, the PLC, on the other hand is already used by the operators of the energy for the metering servies. However, it is not neessarily pereived like a true ommuniation network for the households. By the way, the types of rms ating on these networks are di erent: in a ase there is a rm either publi, or privatized following the unbundling. And beyond, 4

5 these tehnologies of networks have di erent maturities. Nevertheless, in these two situations, the rm using the network remains potentially subjeted to onstraints of strong regulations. Contrary to the ase of optial ber, network where the servies of obliteration are also arried out by private rms. From the point of view of the property of the ommuniation network, this di erent nature tends to justify potentially di erent development strategies. Indeed, in the ase of the optial ber, this servie being not the prinipal use, it should not ondition signi antly its deployment. Whereas in the ase of the deployment of the broadband over power lines (BPL), this network supporting almost only this use, it is thus the main motor. And although initially, logis of savings in osts would tend to provide the same geographial zones given a weaker marginal ost of installation (ase of the very onentrated urban areas), the extension of suh networks at stronger marginal osts ould not be made. It is true that the zones thikly populated ould be a priority for the establishment of BPL network, onsidering that it may be possible to maximize the aggregates relative to the invested osts. Nevertheless, areas requiring a partiularly important exibility request may have a more sattered population ; and thus a ost-e etiveness arbitration integrating more parameters. Talking about BPL, a reent ase have to be takled here. National broadband plan other the US led to a private rm promoting BPL. Willingly to bypass ommon networks suh as DSL ones, they enounter several issues linked to the reent nature of their tehnology. Even if they failed to reah their objetive and at the end at being pro table, this failure should not be understood at a dead-end. However, it is true that it enhanes the fat that BPL is not as known as OF. And, as many servies belonging to the smart grid area, new management strategies have to be found, either in energy inumbent business models (Shomali & Pinkse, 05) or ompetitors. Here, BPL as a network owned by the energy inumbent does not fae the same goals, neither the same ompetitors. The aim is not to by-pass the optial ber network in the internet furniture, but to provide some restrited aess required by their energy-linked servies. ICT rms and eletriity utilities are ollaborating, as it an be see in the various smart grid experiment. But they may beome ompetitors on the provision of ommuniation support, as they already turned in many smart grid projets, where they overpass their usual role (as shown in Erlinghagen and Markard, 0). 3 Literature Then there is the ase where ators of the eletri setor (whih is regulated in Frane) would be in ompetition with ators of the teleommuniations setor (ativity where the regulation is less strong onerning the supply of the servie). Mainly, an extreme situation shows up : a rm is harged of a publi servie mission a eting its maximization program on the one hand, and some private rm on the other hand. Whih is a "lassi" mixed oligopoly. However, on this topi, eonomi literature makes it possible to have an out- 5

6 line a little more general. Mixed oligopolies an be observed in many setors, through all the various types of eonomies (developed, developing, in transition). The trend of establishment of mixed industries, following the programs of privatization of the publi monopolies at the end of the 0th entury, is one of the most fundamental reasons. Finally, various situations where rms, held totally of partially by publi authorities, are in ompetition with private rms exist. The study of these mixed oligopolies beame onsequently very ative sine the end of the 80 s, in partiular after the researh of Fraja & Delbono (989). Various works then posed the stakes, allowing the analysis of mixed oligopolies at various levels and from various models : those in the line of Cremer and al [99], relating to the e ets of spae ompetition and the hoie of loalization of a mixed oligopoly, but also others, questioning the degree of privatization (the nationalization degree of a rm and its impat). The oexistene of di erent type of rms generated various analyses on the potential reasons of a variation of the quality when serving a demand. An assumption, generally found in the literature of mixed oligopolies, is that a di erential of quality exists between what is provided by the publi rm and what is provided by the private one (Ishibashi & Kaneto [008]), with a better quality for the private ompany. It an be supposed, as Benassi and al. [06], that the di erential of quality is onseutive with a willingness to furnish a greater aessibility to the servie. Less quality would re et this di erential on the pries, downwards, and would then allow serving a bigger market share. It should be noted that if various studies show this di erential of quality on various setors (Ros [999] for teleommuniations, Mizutani & Uranishi [003] for the postal servie), others point out that empirially, for a given setor, the result an be di erent from one ountry to another (Sanjo[009], Cremer & Maldonado [03]), and thus tend to moderate this observation. Nevertheless, this di erential ould also be explained by a better management of the osts (or rather a di erent struture of osts) and thus would re et a better e etiveness for a homogeneous good. Many works also treat existene of pure strategy equilibrium, as d Aspremont, Gabszewiz and Thisse [979] showed that there is no pure strategy equilibrium where transport ost are not quadrati. This existene issue has sine been investigated in many way, suh as introduing heterogeneity of prodution osts, or onsidering mixed strategies (Matsumara & Matsushima, 008). Overall, the models of loalization in Hotelling are usually used in the industrial eonomy to treat ases of imperfet ompetitions between various produers. The analysis made here is based on a Hotelling model of linear ity. The hoie here will not be made on the strategy of loalization. Whih is a major drawbak when omparing the result obtained by the literature. Indeed, mosf of them who introdue ost di erentiation put their matter on how this impat the hoie of loation, as in Nilssen & Sorgard [00]. They develop loation hoie models with asymmetri distane osts, with pries xed exogenously. As Weijde, Verhoef and Van Den Berg showed [05], there seems to be no horizontal di erentiation models whih inlude a prie-sensitive demand in a ontext of asymmetri osts. 6

7 Many papers in the teleommuniation eonomis literature study network interonnetion. Main onerns are enhaning ompetition between networks, and the matter on whether substitutability implies ollusive equilibrium (as in Shniewindt [00]). Indeed, given that interonnetion between ompetitive networks are realised, a major onern is how ollusive agreements impat interonnetion harges, and when do they arise. But in our ase, even if there is a ompetition between two networks, they are not as similar as teleommuniation ones. As an example, "on-net" and "o -net" all rules do not apply, due to the spei nature of Internet. That s why, even if some argument ould be applied, main results of Armstrong [998] nor Baumol [999] ould not t. This is why a dediated applied model is required, to give the proper insights. 4 Competition between rms serving energy ef- ieny servies - an applied model Basially, even if ICT rms and eletriity utilities are ollaborating in the various smart grid experiments, they may beome ompetitors on the provision of ommuniation support. This has already been the ase in many smart grid projets, where they overpass their usual role. Indeed, when looking at the role of ICT rms in the setoral transformation towards a smart eletriity grid from 005 to 0, Erlinghagen and Markard [0] had shown that, despite energy inumbents domination, ICT rms presene is rising. But, ontrary to Geels et al (008) results, energy inumbents are not passive but pro-ative (mainly through mergers and aquisitions). Willingly to provide some insights on ompetition between networks, a standard linear model of Hotelling [99] and of Aspremont et al. [979] is onsidered, mainly in the manner of Kitahara & Matsumara [03]. We onsider a linear ity of length, on whih the onsumers are uniformly distributed. Let i = fi; Eg be two rms produing a homogeneous servie. Where I represents the publi rm, here in fat the DSO, and E the onurrent private rm. Their ost funtions are suh as : I = q + d et E = q + d., and not normalized to zero like in Kitahara & Matsumara [0]. This gives an aount of two things: rstly, the important xed ost related to the establishment of the network that allow providing the servie (d and d). And seondly, the variable ost binding the ost of this network to the mass of demand served (q and q). In fat, this onsideration tends to give an aount of the struture of assoiated osts to the establishment of suh network arhiteture. Conerning the fator, it allows onsidering the e etiveness of the rm E. On the one hand, this makes it possible to test the impat of a better e etiveness of its produtive proess. On the other hand, that indiretly allows onsidering the same subjaent as the one leading at a di erential of quality. "on-net" is a term used when the all or message originates and terminates within the same operator. In other words, the same network is used. "O -net" applies when at least two networks are involved. 7

8 Indeed, this one induing a better quality at the same ost for the private rm, it lets suppose a lower ost for a qualitatively similar good. The onsideration of not standardized osts makes the analysis di erent from the various quoted models, insofar as in their mixed oligopolies, the funtion objetive of the publi rm is a maximization of the total surplus, and thus a minimization of transport osts. Let x i [0; ] be the loalization of the rm i. A onsumer loated at x i [0; ] undergoes a quadrati ost of transport t(x i x) when he buys the servie of the rm i. The onsumers have then the following indiret utility funtion, suh as v is su iently broad in order to let all the onsumers buy a p servie (V (p i ) = v ) As far as eah onsumer onsumes only one unit of the servie of energy e ieny onsidered, the total demand is then inelasti. To take an inelasti demand thus implies having V (p i ) = v p i with v a su iently large onstant, for i = fi; Eg: There are then the following utilities: V (p i ) (x i x) the withdrawn utility if the purhase is made from rm i = fi; Eg Let x n be the indi erent onsumer, of whih the withdrawn utility for the purhase from I is the same as one from E, heking the following equality: V i (x i x n ) = V i (x i x n ) V i V i+x i x i (x i x i) Whih gives x n = For simpli ation reasons, let suppose x I = 0 and x E = the loalization, respetively, of rm I and rm E. Then the utility funtions are U I = V (p I ) (0 x n ) and U E = V (p E ) ( x n ) Consequently: x n = p E p I + : Let onsider the following demand funtions, positive and limited to : D i = ( + p i p i ) The onsumer s surplus is alulated as follows: R x n R SC = V (pi )x n + V (pe )( x n ) (x 0 I x) dx (x x n E x) dx Whih leads to SC = + 6p I + 6p E v 3p I + 6p Ip E 3p E Whene the expression of Welfare, as the sum R of the surplus : W = I + E +V (p I )x n +V (p E )( x n x ) n (0 x) dx R ( x) dx x n 0 Wih leads to W = 6 + d v + 6p I + 6p E + 3p I + 3p E 6p I p E d Baseline senario 5. Timing We onsider a duopoly, with simultaneous pro t maximisation I and E (respetively through p I and p E ). This is the ase when two rms ompetes 8

9 with two substituable networks. Hene, the servie they o er is the same to the onsumer, but osts inurred to serve it ould be di erent : one is served through broadband over powerline tehnology, when the other one is served through optial ber. The rst ould be, as an example, the inumbent servie of load shedding, when the seond ome within the ompeteny of another ompetitor from Internet-ommuniation setor. Pro t funtions are, as lassially expressed : I = (p I )D I d, I = (p I )( (p E p I + )) d E = (p E )D E d, E = (p E )( ( + p I p E )) d 5. Resolution p 0 At the equilibrium, the set of prie solution is I = p 0 E = Whih means that prie order between p 0 I and p0 E is, as expeted, determined by the value. If the rival is more e ient, meaning <, then the rival an harge, with the same marginal revenue, p 0 E < p0 I, and thus attrats a larger demand. Funtions demand follow : DI 0 = and DE 0 = 6 (+3 ) Proposition Swithing ost ould be understand as a measure of relative aptive demand variability. Thus, it allows a rm to harge a higher prie, despite its ost disadvantage. The intuition is the following : as expeted, at the equilibrium, the prie harged by the inumbent rise with it s own ost of furniture but also with its rival. And It is also rising with the opportunity ost : the higher the ost to hange the servie provider, the higher the relative "aptive" demand. Indeed, the opportunity ost allows him to be more interesting for some onsumer even when its ompetitor is more e ient in its providing ost. Thus, given this level of "aptive" demand, the provider an harge and higher prie when is high. We an also see that the absolute ost di erentiation threshold = is relevant here : when is below, the rival is more e ient than the inumbent, as their ost struture are the same. Its prie is, all things being equal, lower than the inumbent s one, with the same margin. It ould be noted that positive demand funtions imply a range of the ef- ieny riterion between ( 3) and ( + 3) : Given that we onsider a potentially more e ient rival, this riterion should be always positive. Thus, we onsider 3. This assertion ould be illustrated by the teleommuniation networks setor. Indeed, onsidering the ase of moving from a servie provider, the opportunity ost implied ould be higher than the marginal ost of furniture arried by the provider. Inidentally, maintaining a high opportunity ost is a key to lower the elastiity of one s demand, lowering the saving for a onsumer to swith to a rival servie. 9

10 Proof. D 0 I > 0, > 3 DE 0 > 0, < +3 Equilibrium values of the baseline senario : SC 0 = v W 0 = d 36v d 0 I = d 0 E = d As expeted, pro t funtion of the rival E, E (), is monotonous and dereasing on the subset onsidered. Pro ts 0 E are thus positive for an e ieny < 0 E, and negative when is above this threshold. When we ompare this 0 E threshold with the absolute e ieny threshold =, we obtain a relation linking xed and swithing osts. Indeed, depending on whether the xed osts level is below, equal or above half of swithing osts, 0 E is above, equal or below. Thus, three ases emerge. Proposition When xed osts are below half of the opportunity ost, there a range of e etiveness where both rms over all their osts and draw positive bene ts. Below or above this range of e ieny, rm i make positive pro ts whereas rm i limits its loss. When the inomer is more e etive than the inumbent, it provides a large part of the demand, thanks to a lower prie. The less e etive it is, the more its prie is high, whereas the prie of the inumbent inreases to a lesser extent with the e etiveness. Consequently, it beomes advantageous for the inumbent to serve the demand sine the limit of e etiveness I is reahed. Then, both pro ts from the both rms are positive. The pro ts of the rm E dereasing in, meanwhile the ones of the inumbent are inreasing in, knowing that more the e etiveness of the inomer inreases, more the prie proposed is high ompared to the one of the inumbent. When the ompetitor is su iently less e etive than the inumbent, meaning for an e etiveness > E, the prie is not high enough to allow a positive pro t. Nevertheless, it an over the variable osts. Thus, E stays on the market limiting the loss. The prie of the inumbent is then, even if high, lower than the one of the inomer: allowing the DSO to serve the remaining demand. This ombined e et of high pries and volume of demand served allowing the DSO to get high levels of pro ts. Thus, given the relation between the xed osts and the swithing osts (from one supplier to another), the simple relative ine ieny of the inomer ompared to the inumbent is not enough to determine the positivity of its pro t, as I < < E : Note here that when xed osts are equal to half of the opportunity ost, rm i make positive pro ts when rm i only over its variable osts. This ase is similar as the Hotelling model without xed osts. The inomer only realizing pro ts when it is more e ient than the inumbent. 0

11 Fig. : Pro t funtions for eah e ieny riterion, given xed osts below half of the swithing osts Remark 3 When xed osts are above half of the opportunity ost, either only one rm make positive pro ts or both of them ould just limit the loss. The importane of the xed osts ompared to the swithing ost implies a stronger in uene of the struture of osts on the pro ts of the two rms. In fat, when the inomer is very e etive, it serves the market at a prie p E. When this prie inreases, and as the diminution of the demand is linked to this inrease, a limit E omes, from whih its inomes do not exeed anymore its osts. Consequently, sine it an over its variable osts, the inomer remains in the market but have negative pro ts. However, this limit does not imply positive pro ts for the inumbent. Indeed, the level of prie of the inumbent allows him to serve an inreasing demand, but does not make enough revenue to over both its variable and xed osts. Consequently, only for a ritial ine ieny I > of its ompetitor, the inumbent makes positive pro ts. The more the ine ieny of its ompetitor is, starting from this limit, the more the prie tari ed at its inreasing demand enables him to inrease its pro ts. But, in so far there is 0 E < < 0 I, this situation implies a delta of e ieny [ 0 E ; 0 I ], where the pro ts of both rms are negative.

12 Fig. : Pro t funtions for eah e ieny riterion, given xed osts above half of the swithing osts Needless to say that this last ase, allowing an e ieny gap of negative profits, provides strong negative inentives. In the presene of network investments, these inentives have to be handled. Indeed, in a dynami way of thinking, information asymmetries ould avoid entrane of ompetitors, who ould fear this e ieny gap. Thus, enhaning ompetition ould require regulation rules in order to fae investment dilemma. 5.3 First senario : introduing a welfare onstraint on the publi objetive demand funtion As faing xed osts inertitude and willingly to enhane ompetition on the overall downstream market, authorities ould set up a welfare maximization rule on the behalf of the inumbent. Introduing this welfare-onstraint stresses the importane of xed osts in this model. Indeed, di erent ases are enountered aording to its level. Proposition 4 Introduing a welfare onstraint enhane the absolute e ieny threshold ( = ) importane. Both pries are higher when the ompetitor is more e ient, and are lower otherwise. p Proof. At the equilibrium, the new set of prie solution is I = + p E = + : D Aordingly, demand funtions are I = ( + ) DE = ( + ) Note here that as in the baseline senario, swithing ost allows rms to keep a larger demand, even when it is less e ient than its rival. Indeed, when is below, the derivative funtion D I of is positive. Thus, when rises, the demand faed by the inumbent rises. As its base demand is weaker

13 than E s, a strong swithing ost let the inumbent preserve a bigger part of "aptive" demand. In other words, onsumers fae a lower inentive to hange their supplier, even if the other one is more e ient, beause the swithing ost to assume is high. In the same way, then is above, the derivative funtion of D I is negative. The demand faed by the inumbent is larger than E s, but a high level of swithing ost implies a lower demand. D I variation is negative as E is keeping a larger demand than in the ase where only its e ieny matters. Proof. DI = ( + ), d(d I ) d = ( ) and DE = ( + ), d(d E ) d = ( ) Requiring positive demands imply a within the subset [ ; + ]. We make the hypothesis that. Inumbent demand is then always positive, when E s is onstrained by a minimum e ieny riterion < +. Comparing to the baseline senario, we should note that here p pries are negative when the rival is way ine ient. Due to the requirements on our model, this ine ieny level is beyond the subset of onsidered. Thus, in the subset, pries are always positive. Proposition 5 Introduing a mixed duopoly model implies a potential shift of the e ieny threshold ;E ompared to the absolute e ieny threshold =. It depends on the xed and swithing osts levels. Indeed, for every d <, then ;E > : As in the baseline senario, when d =, ;E = and for every d > then ;E < : Proof. As in the baseline senario, rival s pro t funtion is monotonous and dereasing in its ine ieny. Its positive for every < p ;E, with ;E = + d d + d + d + in the subset [ ; + ]: Negative above. ;E >, p + d d + d + d + >, d < In regards to the baseline senario, it ould be interesting to note that when E is more e ient ( < ), this rm serves a lower demand but at a higher prie, whih lets it release higher levels of pro ts. Indeed, here, the prie e et overrides the negative volume e et. Nevertheless, when sine E is less e ient, the provided demand is weaker due to a more aggressive prie poliy on behalf of the inumbent (whom provides then a more important demand DI > D0 I ). This least demand, ombined with a prie p E < p I, involves then a level of pro t weaker than if there is no regulation. Proof. We have DE 0 < D E, 6 ( + 3 ) < ( + ), < and p 0 E < p E, < +, < Also, E > 0 E, < : Indeed E > 0 E, 4 ( ) ( + 3 ) > 0 3

14 Whih is true when < or > ( + 3): We ould easily show that + ( + 3) > :Then E > 0 E imply <. In ontrast with the fat that new rival pro t funtion is quite similar to the baseline one, the new objetive funtion of the inumbent drives di erent ases. Proposition 6 Introduing a welfare onstraint on the behalf of the inumbent fores us to onsider various trade-o s between onsumer and rms surpluses, when xed ost level is su iently low ompared to the opportunity ost (Fig. 4). The intuition is the following : given a xed ost level below half of the opportunity ost, when the rival is very ost-e ient, it rst drives both negative onsumer surplus and negative pro ts for the inumbent. Indeed, a too high prie p I prevents the DSO from being ompetitive ompared to p E = +. But this level of prie p E, too high, generates an unwilling from the onsumer to buy the servie, when their raw utility is not large enough ompared to and. In this ase, although the potential pro ts of E are very high, there is thus no market, as the onsumers do not see enough added values to the servie. This being, in the ase of the supply of Smart Grid servies as load shedding or smart metering, the mentioned servies are, in pratie, usually already provided. The matter is by whom and how. As this market exists (or begin to exist), the raw utility is quite higher than the ritial level : and thus, when the rm E is very e etive, it provides a demand. Thus, this ase ould be remote from our analysis. However, the onsumer s surplus is relatively weak when a very e ient E serves the demand, the welfare being then mainly driven by the pro t of the rm. But, when the inomer is more e etive than the inumbent, but not enough to stop the inumbent from having positive pro ts (p I dereasing with ), then the inumbent provides a part of the demand through its prie. Both rms release positive pro ts. And, when the rival is ine ient ompared to the inumbent, ( > ), this senario provides new insights as inumbent s pro ts are dereasing even though its rival make positive pro ts. Sine the inomer is less e etive than the inumbent, meaning >, pro ts of the inumbent derease. Indeed, even if dereasing, the pro ts of E are still positive. Sine p I follows dereasing, the demand provided by I inreases. The volume e et allows the inumbent to release positive pro ts. As p I is dereasing, the marginal revenue olleted by inumbent on its demand is lower, whih is higher. This inreases the level of onsumer s surplus. Note that there is an e ieny threshold ;E where rm E do not make positive pro ts anymore. But despite the fat that E is not pro table, I pro ts still dereasing. Indeed, for a still lower e ieny, and onsidering d,,, the rm E is too ine ient to over its variable osts and its xed ones d. However, it 4

15 ontinues to serve a demand for a prie p I = +, in order to minimize the loss by overing its variable osts. The inumbent still releases positive pro ts even if they are dereasing with. Indeed, a largest part of the demand is provided but at a weaker prie. The surplus is then higher, and it ompensates for the falls of the levels of pro ts on the olletive welfare. There is a last ine ieny threshold ;I where both rms make negative pro ts. The intuition is the following : when the rm E is very ine etive, negative prie e et overrides the volume e et for the inumbent. Atually, the high level of provided demand do not su iently ompensate the evolution downwards of the prie p I, whih leads to a total revenue for the inumbent not overing any more all the osts. Indeed, the high level of prie used by E leads, in order to maximize the welfare and thus here the surplus, for I to serve the demand at a very low prie. In this model, the inumbent provides nevertheless always a demand, in order to over its variable osts. Fig. 4. Welfare maximizing rule on the inumbent when d < However, when xed osts are above half of the swithing osts, there is no situation where the inumbent make positive pro ts. Indeed, the welfare maximizing rule enfored to the inumbent make him very sensitive to xed osts. Full analysis is the same as explained in the ase where xed osts are below, but their importane prevent the revenues to be higher than the osts. 5

16 Fig. 5. Welfare maximizing rule on the inumbent when d > This senario pushes to a limit of positivity of the pro ts I () later than if the two ompanies maximize their pro ts. Indeed, 0 ;I < ;I. This an be explained by a level of prie p 0 I < p I as long as the ompetitor is more e ient than the inumbent ( < ). Consequently, the lower level of prie lets the inumbent ath a more important demand (DI 0 > D I, when < ) in this ase. This volume e et ompensates the negative prie e et, whih lets him more quikly release positive pro ts. This is true as long as the xed osts are lower than the swithing osts, but not exeeding the lower threshold. Indeed, in the ase where the xed osts are almost negligible (d < < ), the inumbent releases a positive pro t even if the ompetitor is more e etive. This is explained by a derease of the osts ompared to the previous situation, derease of the osts whih, for the same level of prie p I and demand D I, makes pro table the ativity in an antiipated way. Proof. S0 ;I < ;I, 0d 5 < 3p ( d) As d < then 0d 5 < 0. So S0 ;I < ;I is true as long as d <. And it ould easily show that p 0 I < p I, < and D0 I < D I, > 5.3. Comments and onlusions These results are very onsistent with the literature of homogenous ompetition without produt di erentiation. Also, it should be kept in mind that many reasons justify to onsider a mixed duopoly here. Indeed, as ommuniation networks require huge investments, rms strategies are important. Espeially as they fae di erent development strategies : in the ase of the optial ber, this servie being not the prinipal use, it should not determine signi antly its deployment. Whereas in the ase of the deployment of the broadband over power lines (BPL), this network supporting almost only this use, it is thus the main motor. And although initially, logis of savings in osts would help providing the same geographial zones given a weaker marginal ost of installation 6

17 (ase of onentrated urban areas), the extension of suh networks at stronger marginal osts ould not be made. It is true that the zones thikly populated ould be a priority for the establishment of BPL network, onsidering that it may be possible to maximize the aggregates relative to the invested osts. Nevertheless, areas requiring a partiularly important exibility demand may have a more sattered population ; and thus a lower ost-e etiveness ratio. Although investments are not onsidered here, it remains that it ould be easier to enfore suh priorities on a publi rm. 5.4 Seond senario : introduing a positive onstraint on the behalf of the publi rm Likely, these situations where negative bene ts our on the inumbent behalf, ould not be tolerated. Given the welfare maximizing rule, introduing an inumbent s positive onstraint ould be the way to enhane ompetition. Both ompetitors and regulating authorities are aware of these mehanisms, and used to them. It ould be argued that, nowadays, few rms are really stated owned. Nevertheless, it does not mean that no rules are applied to bind these rms in those behaviors. That s why we don t take into aount the diversity of state owned level of rms in this set up. Even if, as Kumar & Saha[008] showed, the degree of privatization of a rm a ets its output strategies (as it a ets the internal trade-o between pro t maximization and onsumer surplus maximization). Assuming spei rules enfored to spei agents, asymmetri regulation literature seems to provide the adequate set up. Main onstraints here are the enhaning of the ompetition and a welfare maximizing approah whih has to allow a pro table situation for the agent enfored. Asymmetri regulation is more likely to have the means to promote ompetition through its enforing inumbent prie. In this set up, the inumbent faes a welfare maximization through p I with a positive onstraint on its global revenue. On its side, the rival still maximize its own pro t as previously. Proposition 7 Welfare maximization approah through the prie of the ompetitor I with a positive onstraint on its pro ts have a mixed e et on pries strategies. It does not hange I 0 s, but it fores a prie harged by the ompetitor E higher than in the private duopoly ase, regardless of its e ieny. Proof. At the equilibrium, only one set of solutions ts the model. It gives the following 8 set : < p I = p E : = :Where p I = p0 I. Prie strategy p E gives always an = output higher than in the baseline senario, regardless of its e ieny. Indeed, p E > p0 E, > 3. For more details : the inumbent program is Max W o/ I 0 (). 7

18 FOC give us, for the inumbent program : dl dp I = 0 I 0, ( p I + p E + + p I + p E ) = 0 ( d ( p I) ( p I + p E )) = 0 Equilibria 8 are found with the both programs : < ( p I + p E + + p I + p E ) = 0 ( d : ( p I) ( p I + p E )) = 0 p E = ( + + p I) K&T Disussion The ase where the onstraint on the DSO is released is the previous one, where no onstraint was holding on DSO pro t. pi = + p E p p E = ( + + p, I = + I) p E = + The ase interesting here is the one where the onstraint is maintained : the only equilibria whih allows non null pro t funtions (and non-normalized ) is the8following : < p I = p I : = : = Positive demand funtion imply a range of suh as [ 3 ; 9+ ]: We onsider > 3. DI > 0, > 3 DE > 0, < +9 Note here that if +9 positive imply > 3: Thus, is always positive, 3 the demand faing the inumbent is always positive, whereas rival one s have an e ieny threshold < +9. Proposition 8 Considering the opportunity ost lower than a ertain level a variable ost, the "new" prie harged by E is lower than in the previous senario when E is very ost-e ient. When it is not ost-e ient enough, the prie harged is higher. Indeed, as p E > p 6 E yields > 3 5 : This e ieny threshold ould not be above one, as long as a needed assumption of this senario is that > 3: But as long as the opportunity ost is below a ertain level of variable ost ( < ), this threshold is positive and should be onsidered. This expeted result ould be explained by the high level of prie harged in the previous senario, where the prie is not linked with its e ieny. Proposition 9 When <, any rise in the opportunity ost improve the level of the demand the inumbent is faing. When <, then the derivate funtion d(d I ) d is positive. Given the fat that the demand the inumbent is faing is lower than the one its rival faes, as a result of its ost-e ieny, any high opportunity ost allow him to keep a larger "aptive" demand. Indeed, as in the baseline senario, onsumers have a lower inentive to swith their provider as the ost linked to this swith ounterbalanes the savings. 8

19 Proof. DI = (3 + ) and d(d I ) d = ( ) DE = d( ( + 9 ) and (+9 )) d = ( ) When the e ieny riterion is positive but below, derivate funtion of the demand D E is negative, thus the higher the opportunity ost the lower the demand faing the ompetitor E. And DI > D E, > +3. Proposition 0 Introduing positive onstraint has modi ed the threshold where demands meet, translating the threshold e ieny from to +3 :. Indeed, the pries harged by the rms are suh as p I is higher than p E for every < +3. Also, in the baseline senario, the negative threshold 0 ;E was determined by the level of xed osts as follows : for every d <, then 0 ;E was below. Here, this model translate the negative threshold to d = 9 8 : Proposition As expeted, and as in the private duopoly model, inumbent pro t funtion is rising in, but xed osts level impats its positivity. The intuition is the following : as the prie harge by the inumbent is quite high, even when faing a low demand it overs its osts. When xed osts are higher, given the prie welfare-maximization strategy, it stresses the pressure to maintain a pro table situation. (3 ) When xed osts are below 8, then the relatively high prie of the inumbent meet a relatively low demand. Nevertheless, it allows him to over all his osts, due to the low level of xed osts. Nonetheless the ompetitor e ieny, his osts are always overed. But a lower e ieny of the ompetitor allows him to prie stronger. Whih gives him a more pro table situation. When xed osts are above 8, then the relatively high prie of the inumbent does not allow him to meet a large enough demand, as the previous ase. But given the high osts, he always faes negative bene ts. Last ase is when the xed osts are between these two levels. Then, the logi is similar. Linearity providing a ase where the prie harged by the inumbent does not allow him to have positive bene ts. It ould be interesting to note that in the previous regulated ase, the absolute e ieny threshold ( = ) was also the demand threshold : when the e ieny was above one, then the demand faed by the ompetitor was greater than the inumbent one. Whereas here, this e ieny threshold has been translated to = +3. This level of e ieny was also the range upper limit in the baseline senario. Regulation thus expand the range of e ieny were both rm fae positive demands. Proof. Inumbent pro t funtion I is rising and monotonous in in the onsidered ranged of. Nevertheless, its sign depends on the level of xed (3 ) osts. If these xed osts are below a d < then the pro t funtion is positive for every (note here that 8 (3 ) 8 < in our setup - where > (3 ) 3). If 8 < d < 8 then it is negative for every < ;I, with ;I = 9

20 3 + 3 p p d, positive above. Lastly, if xed osts are above 8 then the inumbent pro t is negative, although inreasing. Pro t funtion is I = d : Given its stritly positive disriminant ( = 7 d), two real roots have to be onsidered. ;I = ( p p +6) 7 d and ;I = ( +6)+ (7 d) ;with d > 0 8d; ; R + : Here only ;I is positive, thus relevant. (3 ) ;I > 0, d > 8 (3 ) Note that 8 I;extremum < d < 8 does exist. is negative and is reahed when = 3 monotonous on the range of onsidered. The order between ;I and depends on d: if d > then ;I >.. Therefore, I is Proposition Obviously, the rst mixed duopoly senario (S) is welfare improving, regardless the e ieny of the ompetitor. Proof. Let s remain the di erent welfare levels obtained : W 0 = d 36v d W = d v d W = d 44v d W > W 0 leads to 9 ( ) > 0 whih is always veri ed. Proposition 3 Introduing the positive onstraint enhane the role of the opportunity ost as a key mover. Indeed, a high level imply a welfare under S below the welfare obtained under S. But when the opportunity ost level is very low, levels of welfare are higher in the rst senario S than in the seond, regardless the e ieny. Nevertheless, when this opportunity ost rises it allows an e ieny threshold. When the ompetitor is very ost-e ient, welfare under the last senario ould be higher. When this advantage dereases, the welfare beomes lower. Proof. W > W yields < 3 + p 6( 69+4) 3, onsidering < When it is not the ase, the welfare under the rst senario is always higher. Let s denote W = 3 + p 6( 69+4) 3 : It ould be noted that W <. W is positive, if and only if > 3 : Reminding that the main assumption assuring positive demands is > 3 : - if we have 3 < < 3, thus W < 0, whih means that W is lower than W, regardless the rival rm s e ieny. 0

21 3 - if < < 4 69, there s a range of suh as [0; W ] where W is higher than W. Then, lower, for the range [ W ; + ]: Proposition 4 Opportunity ost level also determine the trades on produers surplus. When the opportunity ost dereases, above its threshold, the range of e ieny where pro ts under the seond senario exeed pro ts under the rst one widen. And when it dereases below its threshold, the range of e ieny shrinks. Indeed, if we onsider the sum of produers surplus, it ould be shown that pro ts under the seond senario are higher than under the rst one when rival s e ieny level is higher than a ertain threshold. The intuition of the widening is the following : even when the rival is very ost e ient (situation that enhane pro ts under the rst senario), the deline of the opportunity ost enhanes pro ts under the seond senario. Proof. Let s note i the sum of pro ts for senario i (i = ; ). > leads to > 6 + p d( ( 6+ p 6+7)) d = p p whih is positive for every > 9p 3 p Comments and onlusions Overall, the positive onstraint reated a trade-o senario between the simple duopoly ase and the welfare maximizing senario (if we do not onsider a ompetitor whih has a very high ost advantage). Note here that a major drawbak of this model is the range of xed osts onsidered. Relating to our topi, meaning the ase of investment linked to next generation aess networks implementation, the level of whih relation between xed osts and swithing osts (over, below half of swithing osts) is faed in reality is di ult to apprehend. Also, a model with aess fee ould probably give new insights and t better with what s probably going to happen. Aording to the instrument hosen by regulating authorities (inumbent prie, aess fee), it may have di erent e ets on welfare distribution. Indeed, given that internet networks ould not be explained as usual teleommuniation networks, results ould be slightly di erent than what the teleommuniation literature provides. 6 Main onlusion Mixed markets whih are haraterized by rms having di erent objetive funtions have been widely interpreted in the literature as empirial mixed markets have to be explained in many setors. Nevertheless, it seems that there are no applied models fousing on mixed markets on the downstream eletriity setor