Equity Justi cations for Universal Service Obligations

Transcription

1 Equity Justi cations for Universal Service Obligations Jean-Christophe Pouou an Michel Rolan yz January 29, 206 Abstract Equity is often invoke as a possible justi cation for the imposition of universal service obligations (USOs). However, no previous analysis supports a formal link between equity an USO. In this paper, we escribe the extent to which the imposition of USOs in oligopolistic network inustries can meet the objectives of an inequality-averse regulator. We show that USOs can be use for equity purposes provie that the regulator is able to control the competitive structure of the inustry. We also show that the uniform pricing constraint, which is an obligation to o er the same price conitions to all consumers, is welfare-enhancing but rather surprisingly, it oes not necessarily improve equity. Keywors: equity, coverage an price constraints, regulation, universal service obligations. JEL: D63, L5, K23 LAMETA, Université e Montpellier, Espace Richter, av. Raymon Dugran, CS 79606, Montpellier ceex 2, France. jean-christophe.pouou@lameta.univ-montp.fr y Corresponing author: CREATE, Département économique, Université Laval, 025, avenue es Sciences- Humaines, Québec, QC, GV 0A6, Canaa. Michel.Rolan@ecn.ulaval.ca. z We thank participants at the 55 th Congress of the Société canaienne e science économique an at the 32 th Journées e microéconomie appliquée for helpful comments on an earlier version of this paper. All of the errors or omissions are our own.

2 Introuction Universal service obligations (USOs) are common in network inustries such as telecommunications, electricity istribution, or postal services, where their objective is to ensure the greatest access to a service that is eeme essential. However, the economic rationale behin this objective is not clear an it is still a matter of ebate. Cremer et al. [8] argue that the most compelling theoretical justi cation for USOs is their relative e ciency as a reistributive policy. This justi cation comes from the uniform pricing constraint which is generally inclue in USOs: uniform pricing is a reistribution instrument through price that has the potential to be optimal in a secon-best worl when policy makers o not have the necessary information to implement (potentially) more e cient policies like irect transfers. 2 Although convincing, this argument is base on a case where the regulator has full control of the price in a natural monopoly inustry. In general, the equity justi cation for USOs, which is often consiere to be self-evient by policy makers, is not groune on rm theoretical grouns in a context where there is competition. In this paper, we escribe the extent to which USOs in oligopolistic network inustries can meet the objectives of an inequality-averse regulator. We use a moel where ientical consumers are istribute on a continuum of markets, which i er in terms of their xe connection costs to the network. 3 Two rms can potentially enter each market. Following Cremer et al. [9] an Valletti et al. [2], we consier USOs as constraints impose on the activities of rms. Three such constraints are analyze: a coverage constraint (CC), which obliges one of the rms to serve a given segment of the markets, a license constraint (LC), which controls competition by etermining whether there are one or two rms in each market, an a uniform pricing (UP) constraint, which forces rms to o er the goo at the same price in all the markets that they serve. The CC can be consiere the basic component of a USO because it correspons to the iea of making the service available to the largest possible group of consumers. The UP constraint is often consiere complementary because, as argue by Cremer et al. [9], it is meaningless for high-cost consumers to formally have access to a service if the rms are able to price them out of the market. For Cremer et al. [8], it is also the prime source of the equity justi cation for the USO. The LC is also complementary to the CC because it protects the rm that is subject to the latter (the USO Two types of reistribution are highlighte: from low-cost to high-cost consumers an from high-income to low-income consumers. 2 Cremer et al. [8], p As mentione in Choné et al. [6], this is meant to represent the geographical component of USOs, i.e., the component that aresses the reistribution from low-cost to high-cost consumers, as oppose to the social component, which aresses the reistribution from high-income to low-income consumers. This geographical component is prevalent in network inustries. 2

3 provier) from unfair competition: it can either force entry into high-cost markets in orer to avoi cream-skimming, or restrict entry to ensure the pro tability of the USO provier. 4 The analysis of the LC has been relatively neglecte in previous stuies of USOs. Inee, to the best of our knowlege, it has only been analyze by Crew an Kleinorfer [0] an by Pouou an Rolan [20] in terms of e ciency, whereas it has never been consiere for equity purposes. In orer to highlight an eventual reistributive function for USOs, in our main analysis, we employ assumptions that facilitate its emergence. First, we assume that the regulator can make lumpsum transfers between rms, thereby maximizing welfare uner a single inustry pro t constraint, rather than uner multiple rms participation constraints. This allows for more reistribution by softening the constraints facing the regulator. Secon, we forbi any transfer to consumers, either originating from the rms pro t or from other external sources, such as government subsiies. This means that there is no alternative to USOs for reaching an equity target. We relax the assumption on lump-sum transfers between rms in Section 6, an we n that the qualitative results are not moi e by this relaxation, although the reistributive capacity of USOs are unsurprisingly attenuate. 5 In our analysis, we rst stuy USOs without the UP constraint. We show that USOs can be use by the regulator for equity purposes provie that it imposes a LC. This is because when only a CC is impose, coverage is etermine solely by the pro t constraint, an thus it becomes impossible to moify the incumbent s coverage accoring to the preferences of the regulator. As a result, although it is clear that the CC improves welfare compare with an unregulate market an that the improvement is mae in terms of both the e ciency an the equity criteria, the same solution is obtaine regarless of the inequality aversion implie by the welfare function. In other wors, a utilitarian regulator that only consiers e ciency sets the same coverage as an inequalityaverse regulator. By contrast, aing a LC to the CC allows the regulator to translate its aversion to inequality into policy. We show that when the regulator s aversion to inequality is greater, the coverage impose on the universal service provier (USP) is also greater an the allowe competition is more restricte. Thus, a formal link is establishe between equity an universal service, an the preferences of the regulator can then be reveale by its regulatory choices. However, we note that the attainment of any equity target is impaire because of the pro t constraint an the fact that i erent serve consumers may face i erent prices. 4 In the latter case, the LC is referre to as the Reserve Area. 5 Relaxing the assumption on the absence of transfers to consumers allows a multitue of transfer mechanisms to be consiere, so that their analysis is beyon the scope of the present stuy. Intuitively, the presence of alternative reistribution instruments shoul again reuce the role of USOs for equity purposes. 3

4 We then introuce the UP constraint. A priori, this shoul levy the price i erential obstacle to equality. In fact, we show that the aition of UP to the CC an the LC e ectively increases welfare, but surprisingly, this increase is allowe through the relaxation of the pro t constraint, rather than through the equalization of the i erent market prices. Furthermore, even though the increase in welfare can sometimes be explaine partially by a reuction of inequality, this is not necessarily the case. In some cases, the imposition of UP ecreases inequality when aversion to inequality is low, but it increases inequality when aversion to inequality is high, so UP is preferre by less equalitarian regulators. These cases occur when the e ciency gains of imposing UP are su ciently strong to justify a less equitable outcome. The outcome is that there is an e ciency rationale when imposing a UP constraint but in contrast to conventional wisom, no general claim can be mae on its impact on equity. Again, we note that the welfare improvement of UP is conitional on imposing a LC. Our main contribution is proviing a formal analysis that links the working of USOs to equity in a competitive environment. Most economic moels of USOs assume that their imposition is exogenous an they then procee to compare the e ciency of i erent implementations of a universal service. 6 To the best of our knowlege, only Cremer et al. [8] explicitly state the role of USOs as a secon-best reistributive evice with the help of a moel. However, they assume that the price is fully controlle by the regulator, while the UP constraint oes not apply to the price level. Moreover, their argument is base on a monopolistic inustry context, whereas USOs are now generally applie in a oligopolistic context where strategic interactions moify the impact of UP. One of our intermeiate results con rms that the UP constraint improves equity when there is no competition, even in the case where the regulator oes not control the price level. However, we show that this result oes not hol when we consier the strategic interactions of rms in an oligopolistic market an we escribe the extent to which the imposition of USOs can meet the objectives of an inequality-averse regulator in this more general context. In the following section, we escribe our moel of the network inustry, the regulator s preferences, an the USO constraints analyze. Section 3 presents the benchmark scenarios use to evaluate the performance of USOs. Section 4 analyzes the properties of USOs without UP, rst with the basic CC component an then with both the CC an the LC. Section 5 then introuces the UP constraint. In Section 6, we analyze the impact of restricting the transfers between rms. Section 7 presents a numerical example featuring a linear eman function. In the conclusion, we iscuss a possible extension of our moel. 6 For example, see Anton et al. [2], Bourguignon an Ferrano [4], Calzaa [5], Choné et al. [6], [7], Foros an Kin [], Gautier an Minuzo [2], Gautier an Paolini [3], Gautier an Wauthy [4], [5], an Jaag [6]. 4

5 2 Moel Two rms can potentially supply a homogeneous goo on a continuum of locations 2 [0; ], which are istribute in the territory accoring to the ensity f () = F 0 (), where F () is logconcave. At each location, there is a mass of ientical consumers so that F (0) = 0 an F () =. Consumers are represente by a twice i erentiable eman function D(p), where p is the price of the goo. The consumers surplus is then given by v(p) = R p D(x)x. The net consumer s surplus for an agent locate in is enote by V () = v(p ()) + t () if the goo is actually consume an by V () = " + t () if not, where t() is a lump-sum transfer, 7 an " is the utility associate with a (poor) substitute that is set arbitrarily close to zero. 8 For each rm, the xe cost of entering location is k, so the locations are ranke in increasing orer of cost. Firms also have the same marginal cost of prouction c, which is assume to be constant an normalize to zero (i.e., c = 0). The operating pro t obtaine by a rm serving location at price p is enote by r (p) = pd (p) with r (0) = 0, so that the pro t at location is (p; ) = r (p) at a given location, they compete à la Cournot. k. When both rms are present We let (p) D0 (p) D(p) p be the elasticity of eman, (p) D00 (p) D 0 (p) p is the elasticity of the slope of the eman function, P (Q) is the inverse eman function, (Q) P 00 (Q)Q P 0 (Q) is the elasticity of the slope of the inverse eman function, an p m arg max p r(p) is the revenue maximizing price. In orer to ensure the existence of the Cournot equilibrium, we assume that the elasticity of the slope of the eman function is greater than or equal to. Lemma If (p), 8p, then for all p p m, (i) 0 (p) 0 (ii) (D(p)). We also assume that the eman function is such that 0 (p) 0. 9 This means that eman becomes more concave as the price is reuce. As explaine by Aguirre et al. [], a price change has less impact on welfare when the eman function is more concave. 7 The regulator will not be able to make such transfers, but they are possible in our rst-best benchmark. 8 A strictly positive " allows for a well-e ne welfare function even when the regulator has a strong aversion to inequality (e.g., a Rawlsian regulator). 9 This conition hols for a large class of eman functions, such as strictly concave, linear, an exponential functions. 5

6 An inequality averse regulator oversees the inustry accoring to social preferences which can be represente by the following family of generalize utilitarian functions: Z 0 (V ())f() for 0 an 6= ; () where V () represents the net consumers surplus of an agent locate in an (v) ( )v. When = 0, is the stanar utilitarian function; when!, h we have the leximin case, an R when!, either from the right or the left, we have exp 0 i. ln(v ())f() 0 As shown by Moulin [8], this family is, up to a multiplicative constant, the only family of functions that satis es the following properties simultaneously: (i) inepenence of unconcerne agents, which states that if consumers at location obtain the same surplus uner two i erent allocations, then V () has no impact on the comparison of these two allocations in terms of social welfare; (ii) the Pigou-Dalton transfer principle, which states that social welfare shoul increase if the utility gap between two agents is reuce while the aggregate utility of both agents is maintaine; an (iii) inepenence of common scale, which states that a rescaling of pro les fv ()g oes not a ect the social orering of the pro les. Note that is concave an represents a coe cient of the inequality aversion as = v We also use the fact that is a weighte mean of orer, which is strictly ecreasing in. 2 However, when we solve a problem for a given, we use the increasing transformation ~ = ( ) in objective functions in orer to simplify the presentation of rst-orer conitions. The only instrument that the regulator has to han is the imposition of USOs. More precisely, the regulator can impose constraints on the locations that rms can serve an on the prices that they can charge. Thus, regulation comprises one or a combination of the following components. CC: An obligation for one of the rms (esignate as the incumbent in the following) to serve all locations 2 [0; m ]. This obligation can be partly or fully compensate by a lump-sum transfer from the other rm (esignate as the entrant ). This constraint comprises the backbone of USOs in practice. When m =, this correspons to the ubiquity constraint. LC: A manate given to the entrant to serve all locations 2 [0; ], an only these locations. By construction, m. This constraint can be impose either to protect the incumbent from competition in high-cost markets or to expan competition in the face of cream-skimming. practice, the rst case is referre to as the incumbent s reserve area (RA), whereas the secon case correspons more generally to conitions inclue in operating licenses. 0 See Moulin [8] an Mitronović [7]. This inclues the case where!. 2 See Mitronović [7]. The weighte mean of orer is a continuous sum equivalent to the CES function with as the elasticity of substitution. In 6

7 UP constraint: An obligation for the incumbent to charge the same price in all markets that it serves. Finally, the regulator has the capacity to make lump-sum transfers between rms, so that its choices are subject to an inustry pro t constraint rather than iniviual rm s participation constraints. In practice, transfers between rms can be mae by establishing a USO fun, which serves to compensate the USP for the cost of USOs an it is nance by the inustry consumers or proucers. Allowing such transfers to be lump sums provies the best chance for the USO to have a role as a reistributive instrument. Similarly, we consier that no transfers are possible to the consumers, either from rms or from external sources such as taxation or subsiization from government. Thus, we eliminate any substitute for USOs in terms of transfers, which further enhances the eventual reistributive role of the USOs. 3 Benchmarks To evaluate the performance of USOs in terms of e ciency an equity, we compare the results obtaine uner various combinations of USO constraints with the polar cases of the rst-best allocation, which represents the ultimate goal of the regulator, an the unregulate allocation, which e nes the basic inustry performance that the regulator tries to improve. 3. First-Best (FB) In the rst-best (FB) benchmark, we assume that the regulator has full control of the inustry an that it can make lump-sum positive or negative transfers to consumers. Thus, it maximizes the value of function (). This requires the satisfaction of prouctive, allocative, an consumer participation e ciency as well as full equity whenever > 0. Consiering each of these requirements in turn, it is straightforwar to characterize the solution. Prouctive e ciency: since the inustry is a natural monopoly, only one rm is exploite; in other wors, = 0. Allocative e ciency: the price is set at marginal cost, which in our case means that the price is 0. Participation e ciency: the rm s coverage is given by: v(0) m min k ; ; (2) 7

8 so the marginal coverage cost never excees its marginal value. Equity: if > 0, transfers are mae to equalize the surplus across locations while compensating the rm for its e cit, so that each consumer obtains: 3 F ( m) v(0) kh( m): 3.2 Unregulate Market (UM) In the unregulate market (UM) benchmark, we assume that the regulator cannot intervene in markets an that the two pro t-maximizing rms are involve in the following two-stage game: Stage : Firms choose their coverage simultaneously. Stage 2: Firms compete à la Cournot in markets 2 [0; ], while the incumbent is a monopolist in markets 2 [ ; m ]. The secon stage leas to the usual monopoly an uopoly outcomes. If we let q I () an q E () represent the incumbent s an entrant s output at location, respectively, then we have: 8 8 < D(p ) 2 if 0 < D(p ) 2 if 0 q I () = ; q : E () = (3) D(p m ) if < : m 0 if < m where p is the price such that (p ) = 2 ; since we assume a zero marginal cost. Therefore, not surprisingly, p m > p, so that r(p m ) > r(p ). The incumbent s an entrant s problems in the rst stage, respectively, are: We thus have: max m 2 F ( )r(p ) + (F ( m ) F ( ))r(p m ) kh( m ) max 2 F ( )r(p ) kh( ) ~ r (pm ) m = min k ~ = min r (p ) 2k ; ; 3 If we let t() represent the lump-sum transfer to location, then we have: (4) : (5) 8 < ( F ( m))v(0) kh( m) < 0 t () = : F ( m) v(0) kh( m) > 0 if m if > m 8

9 It is straightforwar to see that in an UM, the incumbent s coverage is lower an the entrant s coverage is greater than the FB incumbent s coverage an the FB entrant s coverage, respectively. In aition, given that prices are strictly positive, that the inustry is serve by more than one rm, an that consumers surpluses vary across markets, it follows that our UM benchmark fails to meet any of the four requirements for a FB outcome, an thus that the regulator s intervention is require a priori. Our interest is to analyze more speci cally the extent to which the equity requirement can or cannot be met when this intervention takes the form of a USO. 4 USOs without Pricing Constraints We now stuy the imposition of USOs an we compare i erent combinations of constraints in terms of welfare. We separate cases with or without the UP constraint because their analyses are i erent. In this section, we start with the simplest cases where no UP constraint is impose. Without UP, the inustry pro t constraint is: ( ; m ) F ( )r(p ) + (F ( m ) F ( ))r(p m ) k(h( ) + H( m )) 0: 4. CC The CC is the basic component of USOs an it can be use as the sole USO component in practice. This type of USO scheme is moele as a three-stage game: Stage : The regulator sets m in orer to maximize welfare, subject to the inustry pro t constraint. Stage 2: The entrant chooses in orer to maximize pro t. Stage 3: Firms choose their output simultaneously at each location. Stage 3 is the same as that in the UM benchmark, so the equilibrium output results are given by (3). Since the entrant chooses its coverage inepenently of m, stage 2 still leas to the UM uopolistic coverage given in (5), i.e., = ~. Then, the stage regulator s problem is: max m F ( ~ ) (v(p )) + (F ( m ) F ( ~ )) (v(p m )) + (( F ( m ))(") s.t. ( ~ ; m ) 0 (6) m : 9

10 Using the FOC to express m in terms of, we obtain: c r(pm ) m = min + (v(p m)) (") ; ; (7) k k which, base on a simple comparison with (4), shows that ~ m c m m. Thus, a USO with the CC improves the allocative e ciency compare with the UM. However, we note that since p, p m, an ~ are exogenous variables for the regulator, then the value of m is etermine solely by the pro t constraint. In other wors, the optimal coverage uner the CC is c m = minff m ( ~ ; m ) = 0g ; g an the same solution is obtaine whatever is the function. As a result, imposing the CC alone cannot be justi e base on equity consierations. 4.2 CC an LC The LC is an instrument that allows the regulator to calibrate the competition within covere markets. The bene t of increasing the entrant s coverage is the improvement in the allocative e ciency within a market because the gap between the price an the marginal cost is reuce. The cost is the eterioration of the prouctive e ciency because the capital cost is uplicate in what woul otherwise be a natural monopoly inustry; because of the pro t constraint, this introuces a trae-o between an m. Moreover, in terms of equity, the price reuction on the covere markets increases the surplus gap between serve an unserve consumers. USOs uner CC+LC is a two-stage game: Stage : The regulator sets m an in orer to maximize welfare, subject to the inustry s non-negative pro t constraint. Stage 2: Firms choose their output simultaneously at each location. Stage 2 is the same as that in the UM benchmark, so the equilibrium outputs are given by (3). Then, the stage regulator s problem is: max F ( ) (v(p )) + (F ( m ) F ( )) (v(p m )) + ( F ( m ))(") m; s.t. ( ; m ) 0 an m 0: (8) The coverages obtaine uner the CC are clearly feasible uner CC+LC, so joining the LC to the CC cannot ecrease welfare. However, a more precise analysis of the solution is neee to evaluate the role of USOs in terms of equity. Several cases must be consiere. Here, we focus on 0

11 the most interesting case of an interior solution for both m an (i.e. 0 < < m < ) because this highlights the trae-o implie at the margin. 4 If we let represent the Lagrange multiplier of the pro t constraint, then an interior solution is such that: = (v(p m)) (") k m r(p m ) = (v(p )) (v(p m )) > 0; (9) k (r(p ) r(p m )) i.e., the bene t-cost ratio of extening monopolistic markets an uopolistic markets must be the same at optimal coverages. Then, we note that the marginal pro t of monopolistic markets, r(p m ) k m, is negative, thereby showing that the market coverage is greater than that in an UM. However, the marginal pro t on uopolistic markets, r(p ) because we must have r(p ) k, can be positive as well as negative k < r(p m ): the regulator coul choose lower uopolistic coverage as well as greater uopolistic coverage rather than the UM outcome. It is important to note that the aition of the LC makes the optimal solution irectly epenent on the regulator s aversion to inequality, as shown in the following proposition. Proposition Uner CC+LC, m. 0 an 0, with strict inequalities when 0 < < m < Then, greater aversion to inequality leas to an enlargement of the reserve area measure by m : The fact that m an have opposite signs comes from the pro t constraint: as an increase in m reuces pro t, it must be compensate by reuce competition in orer to increase the revenue in markets that were alreay covere. Then, because the utility gap between uncovere an covere markets is greater than that between monopolistic an uopolistic markets, a greater aversion to inequality makes an increase in monopolistic markets more valuable, which compels the regulator to exten the monopolistic coverage. Furthermore, ecreasing uopolistic coverage ecreases inequality between serve an unserve consumers as the average price on covere markets is increase. In brief, imposing USOs can be use for equity purposes provie that a LC is impose. This promotes equity in two ways: (i) by improving the participation e ciency; an (ii) by reucing the allocative e ciency, so the utility gap between serve an unserve consumers is reuce. 4 The full erivations of the rst-orer conitions are given in the Appenix.

12 5 UP Constraint In contrast to the CC an the LC, the UP constraint is not an instrument that is optimally calibrate accoring to the problem s parameters, but instea it is a rule that applies inepenently of the values of these parameters. For this reason, it is not clear a priori whether this rule leas to an increase or a ecrease of social welfare. Thus, the regulator will have to compare social welfare inepenently with an without the rule in orer to etermine whether the rule shoul be implemente or not. 5. UP Impose Separately This game is the same as that in the UM benchmark, except that in the secon stage, the incumbent is constraine to choose a quantity at each location such that the resulting price is ientical at all of the covere locations. This creates interepenence among markets, so that the thir stage oes not lea to the usual monopoly an uopoly outcomes, an its analysis is more involve. 5 Consier the secon stage an let q I () an q E () be the incumbent s an entrant s outputs, respectively, at location. The entrant is not constraine an thus its problem correspons to the classic Cournot problem: Z max fq E ()g 0 (P (q I () + q E ())q E ())f() kh( ): (0) This yiels a reaction function q E (q I ()), which is ecreasing in q I () an is such that: (p()) = q E (q I ()) q I () + q E (q I ()) ; () where p() = P (q I () + q E (q I ())). This reaction function is inepenent of. The incumbent must choose quantities such that the price p() = p; 8. The inverse eman function P is monotone, which implies that q I ()+q E () is inepenent of for uopolistic markets. is also monotone, so () then implies that q E () = q E, 8, i.e., q E () is inepenent of. In turn, q I () = q I, 8, because of the UP an monotonicity of P. Finally, the same price must also prevail in the monopolistic markets, so the quantity Q that is o ere by the incumbent on monopolistic markets must also be Q = q I + q E ; m. Therefore, the incumbent s problem is: max q I F ( m ) P (q I + q E )q I + [F ( m ) F ( )] P (q I + q E )q E kh( m ): 5 This game was analyze extensively by Pouou an Rolan [20], so we only outline their results. 2

13 The FOC is: P (q I + q E ) F ( m ) q I (p) q I + q E q E [F ( m ) F ( )] = 0; (p) q I + q E where p = P (q I + q E ). By solving for ; we n that the incumbent s reaction function q I (q E ) satis es: where s F ( ) F ( m) (p) = sq E q I (q E ) + q E ; (2) represents the uopolistic markets share of the overall markets covere. From (2) an () with p () = p, at equilibrium we obtain: q I = sq E : (3) Hence, the equilibrium price is a function of s, which we enote by p(s). From (2) an (3), p(s) is such that: (p(s)) = + s : (4) Note that when s = 0, all markets are monopolistic, so the uniform price is the usual monopolistic price p m. When s =, all markets are uopolistic so the price is p. Uner our assumptions on the elasticity of the slope of the eman function, Pouou an Rolan [20] showe that p(s) is strictly ecreasing, as well as the following Lemma. Lemma 2 For a given (s; m ), UP; (i) p(s) sp + ( s)p m, i.e., the average price per serve consumer oes not increase with (ii) the aggregate pro t is at least as great with UP than that without UP. Rather surprisingly, the imposition of UP on given coverages ecreases the average price pai per serve consumer, while it simultaneously increases the inustry pro t. Part (i) comes from the concavity of the revenue function uner assumption (p) : at initial quantities that prevail without UP, the imposition of UP makes the inustry s revenue jump by the amount r(sp + ( s)p m ) (sr(p ) + ( s)r(p m )), an thus the rms increase their output, thereby resulting in a ecrease in the price. Part (ii) comes from the fact that no rm has any interest in reucing the price uner the certainty equivalent of sp + ( s)p m given the other rm s strategy because the initial quantity without UP woul then be a better an still feasible strategy. 6 The basic argument that UP is prima facie an equity-oriente policy is re ecte in the next Lemma, which shows that if we focus on the subset of covere markets, then welfare within covere 6 Note that each rm s revenue also increases because it is proportional to the inustry s revenue. 3

14 markets is greater with UP than that without it, provie that the regulator has a su ciently strong aversion to inequality. This is ue to the fact that UP equalizes prices within covere markets. For xe m, uncovere markets are not a ecte by UP, an thus UP then increases global welfare. Lemma 3 For a given (s; m ), there exists a level of over which social welfare is greater with UP than that without it. The reason for which UP oes not increase welfare for all is that the inirect utility function v is convex in p. Thus, it is possible, for example, that a utilitarian regulator prefers unequal prices because the aggregate consumers utility is then lower uner UP espite the fact that average price p(s) sp +( s)p m. However, because increasing amounts to the concavi cation of iniviual utilities v, then there is necessarily a over which the preferences of the regulator ominate those of the consumers. 7 where In the rst stage, the entrant chooses ; or equivalently s, given m an solves: s +s max s sf ( m ) r(p(s)) + s kh(f (sf ( m ))); is the proportion of total eman serve by the entrant by virtue of (3). Assuming that s is given, the incumbent solves the following problem simultaneously: max I = F ( m ) r(p(s)) m + s kh ( m ) ; (5) where +s is the proportion of the total eman serve by the incumbent. The rst-orer conition is then: r(p(s)) m = min k( + s) ; : (6) Comparing (6) an (4) shows that the incumbent s coverage will never be greater uner UP than that in an UM. This is because UP reuces the price that the incumbent receives, so the marginal bene t of coverage is reuce, an thus the market coverage is reuce. We obtain the following proposition as a result. Proposition 2 There necessarily exists a level of over which welfare uner UP is less than welfare in an UM. 7 The average price is lower uner UP, i.e., p(s) sp + ( s)p m, so it is still possible that UP is also preferre by a utilitarian regulator (an thus by all types of regulators). This is the case in our numerical example. 4

15 This is because UP increases welfare within the covere markets but the maximum marginal gain obtaine within markets cannot compensate for the welfare loss of those who lose service, given that coverage is alreay sub-optimal in an UM. Moreover, the loss of service increases inequity between serve an unserve consumers so this loss is more highly weighte by regulators with a stronger aversion to inequality. It also follows irectly that UP cannot act a substitute for CC+LC if a regulator has a su ciently strong aversion to inequality: the i erence in market coverage between UP an CC+LC is even greater than that between UP an an UM; moreover, this i erence increases with. Proposition 3 There necessarily exists a level of over which using UP instea of CC+LC ecreases social welfare. In brief, contrary to intuition, the UP constraint tens to increase inequality when use as the sole instrument because it makes the incumbent reuce market coverage. It is possible that a regulator with a low aversion to inequality prefers UP to a coverage control (i.e., CC+LC), but a regulator with a su ciently high aversion will prefer to rop out UP, enlarge the reserve area, an reuce the competitive area. 5.2 Inclusion of the CC (CC+UP) We now consier the three-stage game that is equivalent to the one in Section 4., except that UP is impose on the incumbent in the thir stage. Thus, this thir stage is the same as that in the case where UP is the only constraint. The entrant s problem in the secon stage is the same as that in UP. Since p(s) 2 p, markets are more pro table for the entrant when a uniform constraint is impose. As a result, the entrant expans its coverage. Lemma 4 For a given m, the entrant s coverage uner CC+UP is greater than or equal to the entrant s coverage uner CC. In the rst stage, the regulator chooses the incumbent s coverage in orer to maximize welfare: max m F ( m )(v(p(s( m )))) + ( F ( m ))(") s.t. r(p(s( m )))F ( m ) k[h F (s( m )F ( m )) + H( m )] 0; (7) where s( m ) is the reaction function of the entrant obtaine in the secon stage. No statement can be mae about whether the incumbent s coverage is greater than, less than, or equal to the 5

16 CC coverage. To unerstan why, we note that by virtue of Lemma 4, aing UP to CC will make the entrant increase its coverage at the initial monopolistic coverage. If the regulator reacts by changing m, then there are two countervailing impacts on pro t: (i) a irect e ect, which is negative because the marginal pro t of the incumbent for a given s is alreay negative uner CC; an (ii) an inirect impact from the reaction s( m ), which is positive because s 0 ( m ) is negative 8, while the incumbent s pro t ecreases with the entrant s share for a given m. Thus, the overall impact is ambiguous. As a result, the aition of UP to CC cannot ensure an increase of welfare, as shown in the following proposition. Proposition 4 If the incumbent s coverage uner CC+UP is less than the incumbent s coverage uner CC, there exists a over which welfare uner CC+UP is less than welfare uner CC. As a result, although it is not exclue to n cases in practice where aing UP to CC is a equity-oriente policy by increasing welfare for say a Rawlsian regulator, the claim that a USO with CC+UP is a reistributive policy is not supporte by theoretical analysis. Section 7 presents an example where the incumbent s coverage is lower uner CC+UP than that uner CC, so Proposition 4 applies an welfare is lower uner CC+UP than that uner CC for a su ciently high aversion to inequality. 5.3 Inclusion of the LC (CC+LC+UP) The reason for which the regulator cannot ensure that the higher pro t attributable to UP for a given coverage results in higher welfare is that it has no irect control on the price, provie that s is etermine by the entrant. To make its policy re ect its equity objectives, the regulator can then bypass this i culty by appening the LC constraint. Thus, uner UP, the regulator s problem in stage is: max m; m;s F ( m ) (v (p (s))) + ( F ( m ))(") s.t. r (p (s)) F ( m ) k[h( ) + H( m )] 0 (8) s = F ( ) F ( m ) : Proposition 2 state that for a given pair (s; m ), the pro t cannot ecrease after the imposition of UP. This means that a feasible solution uner CC+LC is feasible uner CC+LC+UP. Rather surprisingly, uner competition, UP proves useful through the relaxation of the pro t constraint, rather than through the elimination of price iscrimination. 8 See Pouou an Rolan [20]. 6

17 Proposition 5 Social welfare is never lower uner CC+LC+UP than that uner CC+LC. Thus, the imposition of UP leas to an increase in social welfare provie that the incumbent s an the entrant s coverages are controlle by the regulator. The necessity to control coverage is ue to the externalities of UP: the lower price obtaine in monopolistic markets leas the incumbent to reuce market coverage, while the impact on the entrant s coverage is ambiguous. The externality on the incumbent s coverage clearly reuces welfare an this can be countere by the coverage constraint. However, coverages are strategically linke uner UP, so the regulator has to control both rms coverages in orer to reach a target. 6 Relaxing the Assumption on Transfers between Firms In practice, the assumption of the possibility of transferring pro t between rms correspons to the existence of a USO fun establishe to compensate the USP. This fun can be nance by taxes on operators, which can be applie on revenues, pro ts, or quantities. 9 These per unit taxes bring some e ciency an transaction costs. Lump-sum payments can also be levie, as in the case of telecom licenses allocate through auctions, but they are generally not allowe to amount to the full pro t of the rms. In other wors, the instruments use in practice to transfer pro ts between rms are not e cient an cost-free as we assume here. 20 fact in our moel. Let We now explain how to consier this I ( ; m ) = 2 F ( )r(p ) + (F ( m ) F ( ))r(p m ) kh( m ) (9) E ( ) = 2 F ( )r(p ) kh( ); (20) be the incumbent s an entrant s pro ts, respectively. Assume that the parameter 2 [0; ] represents the ease of making transfers between rms. Instea of constraint (8), we use the following pair of constraints: I ( ; m ) + E ( ) 0 (2) E ( ) + I ( ; m ) 0: (22) 9 See OXERA [9]. 20 In fact, the early stuies of USOs in competitive inustries mostly analyze the relative e ciency of i erent funing instruments. For example, see Choné et al. [6], Choné et al. [7], an Anton et al. [2]. 7

18 When =, we reprouce the case consiere previously where lump-sum transfers are allowe between rms. 2 When = 0, we have the case epicte by the constraints (9) an (20) where no transfers are allowe. Intermeiate values of lea to ine cient per-unit transfers or restrictions on total lump-sum transfers. Note that whenever a transfer is mae with 2 (0; ); only one of the constraints can be bining, which is because there can be a positive net transfer only in one irection, i.e. either from the entrant to the incumbent or in the reverse irection. 22 Constraint (2) applies when a transfer is mae from the entrant to the incumbent, while constraint (22) applies for a transfer from the incumbent to the entrant. Firms pro ts o not appear in the objective function, so allowing to be less than harens the constraints, but it oes not moify the basic trae-o between an m. As a result, the analysis of this choice is not moi e in qualitative terms. Proposition 6 Propositions to 5 hol when constraints (2) an (22) are consiere instea of constraint (8). Of course, ecreasing the ease of transfers quantitatively lowers welfare. This is why the case we stuie put the USOs uner their best light as a reistributive instrument, speci cally to make the case that they can be such a reistributive instrument. In practice, this case must be mae base on comparisons with other available instruments. We return to this point in the conclusion. 7 Numerical Example: The Case of Linear Deman We now illustrate the results base on a linear eman example with a uniform istribution of locations. This example highlights two factors that have been ignore in previous stuies of universal services. First, the mechanism by which UP brings more welfare is the increase in the inustry s pro t for given market an uopolistic coverages. This softens the pro t constraint. Secon, free competition between rms actually changes the market an uopolistic coverages following the introuction of UP, so the regulator shoul control the entrant s coverage as well as the incumbent s coverage. Aing LC to CC+UP avois an overexpansion of the entrant because of new pro t opportunities. 2 The aggregate pro t constraint (8) is then uplicate. 22 If both constraints were bining, we woul have I = E = 2 I, which is contraictory for 2 (0; ) unless I = E = 0. If (2) is bining an I 6= 0, we have I = E an E( 2 ) > 0, which implies a transfer from the entrant to the incumbent. Similarly, if (22) is bining an E 6= 0, we have a transfer from the incumbent to the entrant. 8

19 Let D(p) = a p, with a > 0, an F () =. In this case, we have p m = 2 a, p = 3 a, (p) = p a p, p(s) = a 2 2+s an H() = 2. The only exogenous parameters of the moel are a, k, an. In orer to focus on the impact of inequality aversion, we x the values of a an k: We arbitrarily set a = k = With these parameters, the UM outcome is ~ = 2 9 an ~ m = 2 ; so that s = 4 9. If UP is impose along with coverages ( ~ ; ~ m ), the price uner UP woul be p( 4 9 ) = 9 = 0:82. This is less than the average price pai without UP, which is 4 9 p p m = = 0:85: Although Lemma 3 only ensures that UP increases welfare in the covere markets for a su ciently large, in this case, the social welfare in the covere markets is increase for all, incluing the utilitarian case of = 0. Welfare with UP is equal to v(p( 4 9 )) = 0:698, whereas without UP, it changes from 0:672 to v m = 0:5 as is varie from 0 to. This strict ominance of UP makes this example a clear instance of the reistributive property that is presume for UP in the literature. 24 However, as shown in proposition 2, this price reistributive argument oes not hol in a unregulate competitive environment if we consier the reaction of rms following the introuction of UP. In our case, if UP is impose without restriction on coverage, the coverage vector ( ; m ) changes from (0:22; 0:5) to (0:7; 0:30), thereby reversing the result obtaine uner the assumption of xe coverages: here, UP causes a welfare reuction inepenently of the egree of inequality aversion. As shown in proposition 3, it then follows that there is a loss of welfare when imposing UP instea of CC+LC. Therefore, this example provies stronger results than those implie by propositions 2 an 3, which allow higher welfare for UP with a su ciently low, but, in oing so, it highlights the fragility of the reistributive argument in favor of UP in a competitive context. Figures an 2 show the coverages for i erent combinations of constraints uner a USO. In this example, the incumbent s coverage is lower uner CC+UP than that uner CC, an thus proposition 4 applies. The level of aversion to inequality for which welfare becomes higher without UP is approximately equal to 0:59, as shown in Figure 3. Above this value, the fact that fewer consumers are serve uner UP ominates the fact that the serve consumers enjoy lower prices in the calculation of welfare. As a result, CC is preferre to CC+UP espite the higher allocative e ciency of CC+UP. This shows that in the USO programs that are generally stuie in the economic literature, where the instruments consiere are the CC an the UP constraint, UP is an 23 The choice of a = 2 correspons to a monopoly price of. Then, with k = 2, the FB coverage is m = v(0) k =, so that we have participation e ciency with ubiquity. 24 Recall that UP also increases the inustry pro tability. For ( ; m) = ( 2 ; ), the pro t is equal to 0:84 uner 9 2 UP an to 0:76 without UP. 9

20 instrument that is aime less at equity than at e ciency. In fact, aing UP to CC can never be justi e base on inequality aversion because the CC+UP coverage is inepenent of CC+LC CC+LC+UP CC 0.90 CC+UP Inequality Aversion Figure : Incumbent s Coverage ( m ) CC+UP CC 0. CC+LC CC+LC+UP Inequality Aversion Figure 2: Entrant s Coverage ( ) By contrast, the coverages are epenent on inequality aversion uner CC+LC an CC+LC+UP. In agreement with proposition, the incumbent s coverage increases an the entrant s coverage ecreases with the regulator s aversion to inequality uner CC+LC. The welfare ominance of CC+LC+UP, as state in proposition 5, is also illustrate in Figure 3. It is interesting to note that regarless of its aversion to inequality, the regulator uses the higher inustry pro tability uner the UP constraint to increase the entrant s coverage up to a point where the incumbent s coverage is reuce. In other wors, the UP is again use more for increase allocative e ciency within the serve markets than for equity purposes. 20

21 CC+UP CC+LC+UP CC+LC CC Inequality Aversion Figure 3: Social Welfare This last point is illustrate more clearly in Figure 4, which presents the Atkinson inequality inex for CC+LC an CC+LC+UP. 25 Given the assumption that UP can serve equity purposes, we obtain the surprising result that aing UP increases inequality for a su ciently high aversion to inequality. For low, i.e., for cases where the impact of unserve markets is relatively low in the evaluation of welfare, inequality is reuce uner CC+LC+UP because inequality is reuce within the covere markets. However, because the impact of unserve markets is increase with, the fact that m is lower uner CC+LC+UP than that uner CC+LC increases inequality. This shows that the welfare gain of CC+LC+UP is ue to increase e ciency, which is su ciently strong to justify a less equitable outcome CC+LC+UP This inex is given by 0.0 CC+LC Inequality Aversion Figure 4: Atkinson Inequality Inex R 0. The coverages are constant uner CC an CC+UP, so this inex oes V ()f() not convey more information than the welfare function of Figure 2 in these two cases: the ecrease in welfare with is completely ue to the aversion to inequality, so the Atkinson inices are simply mirror images of the welfare functions. Therefore, these inices are increasing with : 2

22 This example also clari es the role of the control of competition through the LC. Figures an 2 show that without the LC, the aition of UP to CC leas to an increase in the entrant s coverage at the expense of the incumbent s coverage. The supplementary rent create by UP for xe coverages is then an incentive for the entrant to expan its coverage. As a result, in our case, CC+UP provies the lowest welfare for a su ciently high egree of aversion to inequality. This role of a LC in avoiing the entrant s overexpansion has been largely ignore in previous stuies. In brief, the nee to appen a LC to USOs in orer to support the equity objectives of the regulator as well as the prevalence of the e ciency role of UP over its presume equity role are strongly highlighte in the simplest of cases possible that are the linear eman an the uniform istribution of locations. 8 Conclusion We showe that when the market price level cannot be regulate irectly an when no e cient transfer mechanism to consumers is available, the prima facie claim that USOs respon to equity consierations is not supporte unless the regulator controls the competitive entry as well as the market coverage. This shoul not be a surprise because controlling the market structure can be a substitute for price regulation, which is itself a substitute for irect transfers. However, this fact has been neglecte in the literature. More surprisingly, we also showe that although the aition of a UP constraint to a LC an a CC unambiguously increases welfare, it oes not necessarily improve equity. Thus, our moel con rms the argument of Cremer et al. [8] that USOs can be viewe as a policy aime to achieve reistribution, but in a moel which is more clearly relate to the actual implementation of USOs an for reasons that are relate to the capacity of UP for creating an inustry rent (an the capacity of the regulator to capture this rent) rather than merely for price equalization. In practice, the use of USOs by an inequality averse regulator shoul follow a cost-bene t analysis in a secon-best worl where a panoply of reistributive instruments can exist. An extension of our stuy shoul involve relaxation of the assumption of the impossibility of transfers to consumers an the integration of particular transfer mechanisms, such as pro t reistribution (as in the case of public enterprise) or irect governmental taxation. We may conjecture that the presence of substitute reistributive instruments shoul weaken the link establishe between equity an USOs. In particular, as a transfer targete to unserve consumers reuces the marginal bene t of market coverage while the marginal cost of capacity remains the same at any location, we can expect that the optimal market coverage shoul not increase after the introuction of a supplementary 22

23 transfer mechanism. However, given the increase complexity of the strategic interactions that this extension entails, this analysis is left for future research. Appenix Proof of Lemma Assume that (p), 8p. (i) We have: 0 (p) = = (p) p [D 00 (p)p + D 0 (p)] D(p) pd 0 (p) 2 D(p) 2 [ (p) + + (p)] 0: (A.) (ii) Since the elasticity of the slope of inverse eman is (Q) = P 00 (Q) Q=P 0 (Q), we have: (D(p)) = D00 (p)d(p) D 0 (p) 2 = D00 (p) D(p) D 0 (p) p pd 0 = (p) (p) (p) : Since (p m ) =, from (A.), (p), 8p p m an (p) (p). Thus, we have (D(p)) = (p) (p). First-Orer Conitions for Problem (8) Let L( m ; ; ) = F ( )(v(p )) + (F ( m ) F ( )) (v(p m )) + ( F ( m )) (") + ( ; m ) be the Lagrangian. Then, an the FOC are: L 0 m = f( m )f[ (v(p m )) (")] + (r(p m ) k m )g L 0 = f( )f[ (v(p )) (v(p m ))] + ([r(p ) r(p m )] k )g L 0 m m = 0 L 0 m ( m ) = 0 0 m (A.2) L 0 = 0 L 0 ( m ) = 0 0 m (A.3) L 0 0 L 0 = 0 0 (A.4) Then, many cases can occur. Case : = m = an = 0 This is the case where uopolistic competition can be implemente in all locations. This requires a su ciently low capital cost k: k k = r(p ) 2H() : 23

24 Perfect equity is reache inepenently of the regulator s aversion to equity because every consumer is covere at the same price. Case 2: < m = an = 0 This case is impossible because it woul require [ (v(p )) (v(p m ))] = 0 : a non-bining pro t constraint with less than full uopolistic coverage is impossible at optimum, as increasing uopolistic coverage is feasible, which woul then increase welfare. Case 3: < m = an > 0 From the pro t constraint, the optimal uopolistic coverage is such that: kh( ) r(p ) + F ( )(r(p m ) r(p )) = r(p m ) kh(); where the left-han sie is the cost of uopolistic coverage in terms of capital cost an lost revenue, an the right-han sie is the potential monopolistic rent. This conition simply states that the regulator issipates the monopolistic rent completely in orer to exten competition as far as possible. Note that is etermine solely in terms of parameters p an p m, an thus it is inepenent of the regulator s preference for equity. From the rst-orer conition on ; we have: = 3 [ (v(p )) (v(p m ))] k (r(p ) r(p m )) : From the conition on m, we n that this solution prevails when: (v(p m )) + (r(p m ) k) 0; so that this case arises when k k < k 3 (v(pm)) 3 + r(p m ). Case 4: 0 < < m < an > 0 From the FOC on m an for an interior solution with < m, it must be the case that: = 4 (v(p m)) (") k m r(p m ) = (v(p )) (v(p m )) > 0: (A.5) k (r(p ) r(p m )) This is the case analyze in the main text. Case 5: 0 < = m < an > 0 This case is impossible because the pro t constraint an the FOC on m with m = together lea to: = (v(p m)) (") r(p ) 2H( ) r(p m ) < 0; 24