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This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
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Promoting network competition by regulating termination charges☆
Sjaak Hurkens a,b, Doh-Shin Jeon c,⁎a Institute for Economic Analysis (CSIC), Barcelona GSE, Spainb Public–Private Sector Research Center, IESE Business School, University of Navarra, Spainc Toulouse School of Economics, France
a b s t r a c ta r t i c l e i n f o
Article history:Received 4 April 2011Received in revised form 8 June 2012Accepted 11 June 2012Available online 19 June 2012
This paper contributes to the policy debate about the optimal termination charge when penetration rates areexplicitly taken into account. Although lowering termination charges towards cost leads to more efficientusage, its impact on consumer surplus is ambiguous since it induces an increase in fixed subscription feethrough the so-called waterbed effect. We show that a reduction in termination charge below cost has twoopposing effects on consumer surplus: it softens competition but also helps to internalize network external-ities. Hence, it can decrease or increase penetration, depending on whether or not the first effect dominatesthe second. We show that the first effect dominates the second when networks are weak substitutes or thepenetration rate is high.
In most countries, wireless telecommunication services are provid-ed by competing interconnected networks. However, competitionseems to be weak since regulators in many countries, including thoseof the European Union, are concerned about high retail prices. A maincause for the weak competition lies in the fact that each network paysa termination charge when a call is terminated on a different network,
which increases the cost and price of an off-net call.1 While regulatingretail tariffs in this sector is out of the question, regulators can regulatetermination charge, and thereby influence retail prices, because themarket for termination is generally considered to be monopolistic.2
There is no general consensus among regulators as to what is the op-timal termination charge. While some regulators seriously consider thedesirability of a Bill and Keep regime with zero termination charges,the European Commission recommends regulators to set terminationcharges equal to the cost of termination (European Commission, 2009,par. 5.2.4). Other regulators, such as Ofcom in the UK, have argued thattermination charges should be slightly above the cost of termination(Ofcom, 2007). Ofcom argued that such an externality surcharge leadsto higher mobile penetration which is desirable because of the networkfeature of the telecom industry: The social benefit of an additional sub-scriber is higher than just the private benefit of this additional subscriber
International Journal of Industrial Organization 30 (2012) 541–552
☆ Previous versions of this paper were circulated under the titles “A Retail BenchmarkingApproach to Efficient Two-WayAccess Pricing: Termination-Based PriceDiscriminationwithElastic Subscription Demand” and “Mobile Termination and Mobile Penetration”. We aregrateful for the comments and suggestions of the Editor and two anonymous referees ofthis journal. We thank participants of the presentations at IIOC09 (Boston), InternationalConference on Infrastructure Economics and Development (Toulouse), Telecom Workshopat ENST (Paris), Workshop on IO: The Telecoms Industry (SP-SP, IESE), 7th ZEW Conference(Mannheim) and the seminars at IST (Lisbon) and Korea University.We thank Byung-CheolKim,Ángel López, RebeccaMayer, JaeNahm, Patrick Rey, and FranciscoRuiz-Aliseda for com-ments and helpful discussions. This research was partially funded by the Kauffman Founda-tion and the Net Institute (www.netinst.org) whose financial support is gratefullyacknowledged. We also thank the support from the Barcelona GSE research network andof the Generalitat de Catalunya. Hurkens gratefully acknowledges the financial supportfrom the Spanish Ministry of Science and Innovation under ECO2009-12695.⁎ Corresponding author. Tel.: +33 5 61 12 86 05; fax: +33 5 61 12 86 37.
1 At least this is the case within the European Union according to European Commis-sion (2009): “Furthermore, the absolute level of termination rates remains high in anumber of Member States, thus continuing to translate into high, albeit decreasing,prices for end-consumers (p.4).”
2 “Each network operator has a monopolistic position on the market for terminatingcalls on its own network. (European Commission, 2009, p.5)”
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because all existing subscribers will be able to communicate with thisnew subscriber.
Our paper aims to contribute to this policy debate by analyzing the ef-fect of the termination charge on consumers' subscription decisions,which has been ignored by the vast majority of the literature. We studythe socially and privately optimal termination chargeswhen subscriptiondemand is elastic and network operators compete with non-linear pricesand can apply termination-based price discrimination.3 Although inelas-tic subscription demand is a standard assumption in the literature ontwo-way access pricing,4 this assumption is often not satisfied even inde-veloped countries.5
The lack of consensus among regulators is understandable. The ef-fect of termination charges on consumer surplus, which determinessubscription demand, is not straightforward because of the existenceof the so-called waterbed effect: A lower termination charge leads to alower off-net call price but at the same time to a higher fixed subscrip-tion fee. We disentangle two opposing effects on consumer surplus of areduction in termination charge belowcost. First, it softens competition.This effect was already identified by Gans and King (2001) and Calzadaand Valletti (2008) in the context of inelastic subscription demand.6
Competition is softened because the lower off-net prices make it moreattractive for consumers to subscribe to the smaller network, which inturn makes it harder for a firm to gain market share by lowering itsfixed fee. Second, a reduction in termination charge helps the firms tointernalize network externalities by increasing the degree of intercon-nection through lower off-net prices. 7 Firms realize that raising one'sfixed fee reduces the total number of consumers subscribed to networksand thus hurts its own customers. This negative feedback on one's owncustomers increases with the degree of interconnection.
Softening competition tends to reduce penetration, while internaliz-ing the network externality tends to increase it. Depending on whetherthe first or the second effect is stronger, a reduction in terminationcharge can decrease or increase consumer surplus and thus penetration.Although the effect on total penetration is an equilibrium phenomenon,we are able to characterize which of the two effects is stronger by justconsidering how the number of subscribers of one firm changes whenthe rival firm unilaterally increases its fixed fee. If the number of sub-scribers of the former increases, we say that there is a net business steal-ing effect. In this case the competition-softening effect is stronger thanthe network externality effect and a positive externality surcharge isneeded in order to increase total penetration. If the number of sub-scribers of the first firm decreases, we say that there is a net network ex-ternality effect. In this case a reduction in termination charge willincrease penetration. We show that a net business stealing effect exists
when total penetration is high or when the networks are relativelyweak substitutes.
Since the UKmarket is maturewith high penetration rates, our find-ing supports Ofcom's position about the need for a positive network ex-ternality surcharge. At first sight, the European Commission seems toignore the value of further increasing penetration by insisting on termi-nation charges equal to cost. However, one could also argue that theEuropean Commission's recommendation is justified by their interestin harmonizing the internal market and setting termination rates inall member states according to the same (simple) rule. Namely, ourfinding suggests that in some countries termination charges above costare optimal, while in others it could be optimal to set them below cost.
Of course, firms are also interested in penetration rates. Higher pen-etration rates mean more subscribers to make a profit from, and moresurplus for consumers as they can call more people. This extra surplusmay be partially extracted by the firms. We show that exactly whenthere is a net business stealing effect, firms prefer to soften competition,which means they prefer lower termination charges.8 When there is anet network externality effect, firms prefer to internalize this external-ity. Again, this means they prefer termination charges below cost.
Summarizing, we find that both the regulator and the firms want todepart from cost-based termination charge, and hence want to distortcall volumes in order to affect penetration. Since the regulator hasonly one instrument, the termination rate, he will have to find the opti-mal trade-off between efficient call volumes (which requires termina-tion rate equal to cost) and efficient penetration rates (which mayrequire termination below or above cost).
Ideally, the regulator would like to implement the Ramsey outcome.This requires both marginal cost pricing (in order to have efficientusage) and the subscription fee equal to the fixed cost per consumer(in order to maximize the total number of subscribers). In the secondpart of this paper, we show that the Ramsey outcome can indeed beachieved under a retail benchmarking approach that determines termi-nation charges as a function of (average) retail prices, introduced byJeon and Hurkens (2008). This approach gives the regulator an addi-tional instrument and allows to expand penetration without distortingcall volumes.
More technically, our paper contributes to the literature on networkcompetition by identifying a sufficient condition for the uniqueness of(continuous) consumer expectations about each firm's network size,and thus for the uniqueness of equilibrium in the subgame played byconsumers for given prices. A careful analysis of how these expectationsare affected by changes in prices (triggered by a change in terminationcharge) is both important and challenging because in our model thereare two sources of network effects: the direct one based on elastic par-ticipation and the tariff-mediated one based on termination-based pricediscrimination. The existing literature is mainly silent about the condi-tion that guarantees the uniqueness of rational expectations and ratherassumes that expectations are unique in the relevant range of tariffschedules.
Subscription demand has been assumed to be inelastic by the vastmajority of the literature. There are two exceptions that are closely relat-ed to the current paper and that deserve some discussion. Armstrongand Wright (2009) consider elastic subscription demand and allow for
3 In addition, we assume that there are no call externalities, that is, consumers onlyderive utility from placing calls, not from receiving them. Clearly, consumers will thennot pay for receiving calls and thus a Calling-Party-Pays (CPP) regime will be in place.Only in few countries such as the U.S.A. and Hong Kong a Receiving-Party-Pays (RPP)regime is used. See Jeon et al. (2004) and Cambini and Valletti (2008) for the analysisof RPP with inelastic subscription demand.
4 This literature is about how termination charges affect competition among inter-connected networks and the pioneers are Armstrong (1998) and Laffont et al.(1998a, 1998b).
5 For instance, according to some statistics from International TelecommunicationUnion (ITU), (see http://www.itu.int/ITU-D/ict/statistics/ (accessed on June 6, 2012))the mobile penetration rate (defined as mobile cellular subscribers per 100 inhabi-tants) in 2010 is 95.39 for Japan, 89.86 for USA, 70.66 for Canada, 64.04 for Chinaand 61.42 for India. The average rate in Western Europe is 114 (in 2007) but this highrate is due to subscribers having multiple phone numbers.
6 Calzada and Valletti (2008) consider an oligopolistic Logit model with inelasticsubscription while Gans and King (2001) consider the duopolistic Hotelling modelwith inelastic subscription.
7 To some extent, this effect is similar to the result of Katz and Shapiro (1985) thatan increase in compatibility among competing networks increases the total numberof subscribers. However, in their paper, firms are engaged in Cournot competitionand the cost of achieving compatibility is a fixed cost and hence does not directly affectthe (perceived) marginal cost. In our model, firms compete with non-linear tariffs andinterconnection is mediated by the access charge that directly affects retail competi-tion through the perceived marginal cost.
8 This theoretical result, first established by Gans and King (2001), seems at oddswith observed practice where operators systematically and heavily have opposed cutsin termination rates. This may of course be due to the fact that networks also competein linear prices for (pre-pay) consumers. Recently, a small theoretically literature hasaddressed the issue in order to reconcile theory and practice in various ways: Jullienet al. (2009) add inelastic participation of “pure receivers” who get a constant utilityfrom receiving calls; Armstrong and Wright (2009) emphasize the role of fixed-to-mobile termination rates; Hoernig et al. (2011) relax the standard assumption of bal-anced calling patterns; Hurkens and López (2010) relax the assumption that con-sumers always perfectly predict the size of networks. All these papers can explain (tosome extent) that firms are better off with termination charges above cost, in line withreal world practice. Furthermore, more recently small operators in many countries infact do campaign in favor of a Bill and Keep regime.
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termination-based price discrimination, as we do, but restrict attentionto a parameter range where there is a net business stealing effect. Ourmain innovation is to identify the interplay between the two opposingeffects associated with a change in termination charge and to showthat the network externality effect may dominate the business stealingeffect.9 Dessein (2003) considers elastic subscription demand but with-out termination-based price discrimination and obtains results that arevery similar to ours. When subscription demand is inelastic, in the ab-sence of termination-based price discrimination, a change in terminationcharge has a zero first order effect both on profit (profit neutrality) andon consumer surplus. It is then difficult to predict and understand the ef-fect of the termination charge on profit and welfare when subscriptiondemand is made elastic. In contrast, in the case of termination-basedprice discrimination, Gans and King (2001) show that lower terminationrates lead to higher profits and lower consumer surplus when subscrip-tion demand is inelastic. Intuitively, then, one would expect that lowertermination rates also lead to lower consumer surplus (and thus lowerpenetration) when subscription demand becomes elastic. However,this intuition turns out to be ill-founded and the effect on penetrationof a reduction in termination charge turns out to depend on the relativestrength of the network externality effect over the competition-softening effect. Our main contribution with respect to Dessein (2003)is that we identify and disentangle these two opposing effects.10
The rest of the paper is organized as follows. In Section 2, we intro-duce the Logit model formulation of network competition with elasticsubscription demand,11 explain how rational consumer expectationsare formed and describe the Ramsey benchmark. In Section 3we charac-terize the unique symmetric equilibrium in case of a fixed per‐minutetermination charge close to cost. We analyze how profits, consumer sur-plus and total welfare are affected by changes in the termination charge.In order to illustrate better how the termination charge can help to inter-nalize the network externality, we here also consider a variation of themodelwithout a business stealing effect. Namely,we consider “two inter-connected islands” in which each island is occupied by a monopolist fac-ing an elastic subscription demand. There is no competition between thetwo monopolists since consumers cannot move from one island to theother.We show that termination charge below cost, by increasing the de-gree of interconnection through lower off-net prices, helps them to inter-nalize better the network externalities and thus expands marketpenetration. In Section 4we showhow the retail benchmarking approachimplements the Ramsey outcome. Section 5 concludes. Appendix A col-lects all the proofs that are not explained in the main text.
2. The model
Our model is standard and identical to that of Laffont et al.(1998b) except the elastic subscription demand for which we use
the Logit specifications. After presenting the model, we describe ra-tional expectations and the Ramsey outcome.
2.1. The Logit model
We consider competition between twomobile operators.12 Each firmi (i=1, 2) charges a fixed fee Fi and may discriminate between on-netand off-net calls. Firm i's marginal on-net price is written pi and its off-net price is written pi. The total mass of consumers is normalized to 1.Consumer's utility from making calls of length q is given by a concaveand increasing utility function u(q). Demand q(p) is defined by u′(q(p))=p. The marginal cost of a call equals c and the termination cost equalsc0(≤c). The reciprocal access price (equivalently, termination charge) isdenoted a. Therefore, the marginal cost of on-net calls is c while that ofoff-net calls is c ¼ cþ a−c0. Networks incur a fixed cost of f per subscrib-er. We define v(p)=u(q(p))−pq(p). Note that v′(p)=−q(p). We alsomake the standard assumption of a balanced calling pattern, whichmeans that the fraction of calls originating from a given subscriber of agiven network and completed on another given (including the same)network is equal to the fraction of subscribers to the terminatingnetwork.
The timing of the game is as follows: First, a reciprocal access price(= termination charge) a is chosen either by the firms or by the regula-tor. Second, each firm i (=1, 2) chooses simultaneously retail tariffsTi ¼ Fi; pi; pið Þ. Third, consumers form expectations about the numberof subscribers of each network i (βi) with β1≥0, β2≥0 and β1+β2≤1and make subscription decisions. We will write β0=1−(β1+β2) forthe number of consumers expected to remain unsubscribed.
We consider rational expectations equilibria (see the next subsec-tion for details). This implies that all consumers will have the sameexpectations. Given such expectations, utility from subscribing to net-work 1 equals
V1 ¼ β1v p1ð Þ þ β2v p1ð Þ−F1;
while subscribing to network 2 yields
V2 ¼ β2v p2ð Þ þ β1v p2ð Þ−F2:
Finally, not subscribing at all yields utility V0.Define U1=V1+με1, U2=V2+με2, and U0=V0+με0. The parame-
ter μ>0 reflects the degree of product differentiation in a Logit model.A high value of μ implies that most of the value is determined by a ran-dom draw so that competition between the firms is rather weak. Thenoise terms εk (for k=0, 1, 2) are random variables of zero mean andunit variance, identically and independently double exponentially dis-tributed. They reflect consumers' preference for one good over another.A consumer will subscribe to network 1 if and only if U1>U2 andU1>U0; he will subscribe to network 2 if and only if U2>U1 andU2>U0; hewill not subscribe to any network otherwise. The probabilityof subscribing to network i is denoted by αi (where α0 represents theprobability to remain unsubscribed). This probability is given by
αi ¼exp Vi=μ½ �
∑2k¼0 exp Vk=μ½ � : ð1Þ
Expectations are rational if αi=βi for all i. For given rational ex-pectations, the profit of network i is equal to
9 Armstrong and Wright (2009) consider an extension of the Hotelling model thatallows for elastic demand. Although they focus on the case in which the fixed to mobile(FTM) termination charge and the mobile to mobile (MTM) termination charge are thesame due to arbitrage, when they study the case in which the two can be separatelychosen, they find that firms prefer the MTM charge below cost (due to competition-softening effect à la Gans and King) while the regulator prefers the charge above cost.This unambiguous result is driven by their assumption on the expansion parameter λin their Eq. (18) which implies that there is a net business stealing effect.10 In Dessein (2003), there is no clear competition-softening effect. A further techni-cal difference between our paper and Dessein (2003) is that we are being more explicitabout when exactly there is a net business stealing effect in terms of the product differ-entiation parameter. In addition, Dessein (2003) assumes unique rational expectationswhile we provide a condition for unique (continuous) rational expectations to existand show that the constraint imposed by this condition is weak enough that eitherthere can be a net business stealing effect or a net network externality effect. Dessein(2003) reports three numerical simulations but all of them correspond to the casewhere there is a net business stealing effect. In Section 3 we will present two numericalresults, including one case where there is a net network externality effect.11 For an introduction of Logit models see Anderson and de Palma (1992) and Andersonet al. (1992).
12 We consider neither the fixed phone networks nor the calls between the fixedphone networks and the mobile phone networks.
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The first term reflects the retail profit made on subscribers whilethe second term reflects the wholesale profit from terminating in-coming calls.
Consumer surplus in the Logit model has been derived by Smalland Rosen (1981) as (up to a constant)
CS ¼ μ lnX2k¼0
exp Vk=μð Þ !
¼ V0−μ ln α0ð Þ; ð2Þ
where the right-hand side follows from Eq. (1). Clearly, consumersurplus is increasing in market penetration 1−α0.
2.2. Rational expectations
As stated before, for expectations to be rational, βi=αimust hold forall i. For any price schedules (T1, T2), such self-fulfilling expectationsexist as these are fixed points of the mapping α :Δ2→Δ2 whereα(β1, β2)=(α1, α2). We can show:
Lemma 1. For any price schedules (T1, T2), rational expectations (α1, α2)are uniquely defined as long as μ is sufficiently high.
Proof. See Appendix A. ■If μ is relatively low (that is, operators are highly substitutable) and
rational expectations are not uniquely defined, one can potentially con-struct many equilibria by having even the tiniest of deviations lead toexpectations (and thus subscriptions) that jump discontinuously, inthe direction that makes such deviations unprofitable. We find it rea-sonable to restrict attention to equilibria where rational expectationsare continuous functions of the tariffs. Uniqueness of continuous ratio-nal expectations requires a milder condition on μ.
In particular, let expectationβ be rationalwhen tariffs (T1, T2) are cho-
sen, whereTi ¼ Fi;pi; pið Þ for i=1, 2. LetT ′1 ¼ F ′1;p
′1; p
′1
� �be an alterna-
tive tariff with small deviations in usage prices and fixed fee such that
β1v p1ð Þ þ β2v p1ð Þ−F1 ¼ β1v p′1� �
þ β2v p′1
� �−F ′1:
That is, the alternative tariff yields exactly the same utility if expecta-tions are not changed. Then it is clear that expectation β is also rationalwhen tariffs (T′1, T2) are chosen. The restriction of continuous rational ex-pectations then implies that expectations must remain fixedwhen tariffsare changed locally such that utility for its subscribers remains constant,given these expectations. By repeatedly applying the same argumentone can see that continuous rational expectations will remain fixedwhenp1 is changed into c and p1 is changed into c, as long as F1 is changedinto F ′1 ¼ β1 v cð Þ−v p1ð Þð Þ þ β2 v cð Þ−v p1ð Þð Þ þ F1. The same can bedone with prices of network 2. In the next section we use this argumentto establish the perceivedmarginal cost pricing principle, which says thatit is optimal for each firm to set usage prices at perceived marginal cost.Now, given that firms use perceived marginal cost pricing, the unique-ness of rational expectations is guaranteed if termination charge is closeto termination cost and μ>v(c)/4, which we will assume.
Assumption 1.
μ > v cð Þ=4: ð3Þ
Lemma 2. If firms set usage prices equal to perceived marginal cost,rational expectations are unique if a≈c0 and Assumption 1 holds.
Proof. See Appendix A.It will be convenient for the subsequent analysis to establish the rela-
tion between fixed fee F and the number of subscribers per firm α in asymmetric equilibriumcandidate inwhichfirms price usage at perceived
marginal cost, F; c; cð Þ. From Eq. (1) and the assumption of rational ex-pectations one obtains immediately:
F ¼ α v cð Þ þ v cð Þð Þ−μ lnα
1−2α
� �−V0: ð4Þ
2.3. Benchmark: the Ramsey outcome
Consider now the Ramsey outcome defined as the outcome thatmax-imizes social welfare under the break-even constraint of each firm. First,because of symmetry,we can look for the outcomeamong symmetric tar-iffsT ¼ F;p; pð Þ. Second, givenT ¼ F; p; pð Þ,maximizing socialwelfare re-quires each firm to realize zero profit. Otherwise, the social planner canfurther decrease F, which increases social welfare because it increasesthe number of subscribers (and therefore creates positive network exter-nalities on the existing subscribers). Third, given the binding zero profitconstraint, maximizing consumer surplus Eq. (2) is equivalent to maxi-mizing V(=V1=V2), which requires marginal cost pricing p ¼ p ¼ cð Þand F= f. Therefore, the Ramsey outcome is characterized by
p ¼ p ¼ c; F ¼ f :
Note that in the case of inelastic subscription, the Ramsey out-come is characterized by p ¼ p ¼ c and F≥ f since the fixed fee doesnot affect social welfare. With elastic subscription, the social plannerwants to increase the number of subscribers as long as the break-evenconstraint is satisfied.13 Note that the social planner maximizing onlyconsumer surplus under the break-even constraint will choose exact-ly the same allocation as the Ramsey outcome since consumer surplusincreases with penetration from Eq. (2).
3. Competition with a fixed per‐minute termination charge
In this section, we analyze the case with a constant reciprocal ter-mination charge a.
3.1. Preliminary results
3.1.1. Marginal cost pricingLet R(p)=(p−c)q(p). Although the number of subscribers αi and
αj depend on V0 and tariff schedule T1, T2, we will omit arguments forexpositional simplicity. Profit can be written as follows
Πi ¼ αi αiR pið Þ þ αjR pið Þ þ Fi−fh i
þ αiαj a−c0ð Þ q pj
� �−q pið Þ
� �: ð5Þ
Firm imaximizes profits by setting Ti, holding Tj constant. Note that achange inmarginal price pi or pi while holding Fi fixedwill affect not onlythe number of i's subscribers but also that of j's subscribers. For example,a decrease in Fi will make network imore attractive and will thus attractsome subscribers of j andwill also attract some consumerswhoprevious-ly did not subscribe to any network. This in turn canmake it alsomore at-tractive to subscribe to network j relative to staying unsubscribed,because of the network effect. It will be convenient to apply a change ofvariables and let network i maximize profits by choosing pi, pi, and αi,holding pj, pj and Fj fixed. This can be done because of the assumptionof continuous rational expectations, as explained in the previous section.
13 In fact, the first best outcome requires the social planner to subsidize each firm becauseof network externalities. Namely, the social planner would like to set usage prices equal tomarginal cost and choose fixed fee F and per firm number of subscribers α as to maximizethe sum of consumer surplus and industry profit, V0−μ ln(1−2α)+2α(F− f),subject tothe rational expectations condition (4). It is easily verified that this requires Fb f.
544 S. Hurkens, D.-S. Jeon / International Journal of Industrial Organization 30 (2012) 541–552
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Note that from Eq. (1) one immediately deduces that
αi
1−αi¼ exp Vi=μ½ �
exp Vj=μh i
þ exp V0=μ½ �: ð6Þ
From Eq. (6), one has
Fi ¼ αiv pið Þ þ αjv pið Þ−μ lnαi
1−αiexp Vj=μh i
þ exp V0=μ½ �� �� �
:
Holding everything but pi and Fi fixed, one obtains
∂Fi=∂pi ¼ αiv′ pið Þ:
Similarly, holding everything but pi and Fi fixed, one obtains
∂Fi=∂pi ¼ αjv′ pið Þ:
Note that if pi is changedwhile keeping αi, pi, pj, pj and Fj fixed, thenalso αj will remain fixed. MaximizingΠi with respect to on-net price pi(keeping αi fixed) thus yields
0 ¼ ∂Πi
∂pi¼ α2
i R′ pið Þ þ v′ pið Þ� �
¼ α2i pi−cð Þq′ pið Þ:
Hence, pi=c. In words, on-net calls are priced at marginal cost.Maximizing Πi with respect to off-net price pi (keeping αi fixed)
yields
0 ¼ ∂Πi
∂pi¼ αiαj R′ pið Þ þ v′ pið Þ− a−c0ð Þq′ pið Þ
� �¼ αiαj pi−c−aþ c0ð Þq′ pið Þ:
Hence, pi ¼ cþ a−c0≡c. Inwords, off-net calls are priced at perceivedmarginal cost (i.e. the off-netmarginal cost).We thus obtain the standard“perceived” marginal cost pricing result under non-linear pricing as inLaffont et al. (1998b).
Summarizing, we have:
Proposition 1. Under the assumption of continuous rational expecta-tions, it is optimal for each firm to price on-net call at the marginal cost(c) and off-net call at the perceived off-net marginal cost (c+a−c0).
In the sequel wewill write v=v(c) and v ¼ v cð Þ to reduce notation.
3.1.2. Net business stealing vs. net network externalityFor the equilibrium analysis and for the comparative statics exercises
we will perform later, it will be necessary to know how the number ofsubscribers to one of the networks changes when fixed fees are varied.Note that an increase in the fixed fee of network 1, for given consumers'expectations, will decrease the number of subscribers to network 1 andwill increase the subscribers to network 2. However, a change in F1also affects rational expectations. In particular, consumers will realizethat network 1 will become smaller and this may make network 2 alsoless attractive. So an increase in the fixed fee of network 1 could poten-tially reduce the number of subscribers to network 2. The followinglemma describes exactly when this happens.
Lemma 3. Suppose networks use tariffs T1 ¼ F1; c; cð Þ and T2 ¼ F2; c; cð Þ.Then
∂α1
∂F1¼ − α1
dμ2 1−α1ð Þμ−α2 1−α1−α2ð Þvð Þ ð7Þ
and
∂α2
∂F1¼ −α1α2
dμ2 v 1−α1−α2ð Þ−μð Þ ð8Þ
where
dμ2 ¼ μ2 þ μ α21−α1
� �vþ α2
2−α2
� �vþ 2α1α2v
h iþ α1α2 1−α1−α2ð Þ v2−v2
� �:
If a is close to c0 and Assumption 1 holds, then d>0 and
∂α1
∂F1b 0;
∂α1
∂F1þ ∂α2
∂F1< 0
and
∂α2
∂F1> 0 if and only if ∇ ¼ μ−v 1−α1−α2ð Þ > 0:
Proof. See Appendix A.When one firm increases its fixed fee, it loses subscribers such that
total market penetration (α1+α2) decreases. However, whetherthe rival firm loses or gains subscribers depends on the sign of∇ ¼ μ−v cð Þα0. If ∇>0, the rival firm gets more subscribers while if∇b0, it gets less subscribers. In what follows, we will say that there isa net business stealing effect if∇>0 and there is a net network externalityeffect if∇b0. For instance, in the extreme case of inelastic and full sub-scription (i.e. α1+α2=1), we have∇=μ>0 since there is only a busi-ness stealing effect: all consumers who leave firm 1 subscribe to firm 2.In general, when subscription is elastic, there exists a network external-ity effect since an increase in the fixed fee of firm 1 reduces the totalnumber of subscribers (α1+α2), which reduces the utility from sub-scribing to any given network. If the network externality effect domi-nates the business stealing effect, the number of subscribers of firm 2decreases as firm 1 increases its fixed fee.14
3.1.3. Unique symmetric equilibriumUnder the perceived marginal cost pricing (p1=p2=c and
p1 ¼ p2 ¼ c), profits can be rewritten as
Πi ¼ αi αjR cð Þ þ Fi−fh i
:
14 More precisely, an increase in F1 induces some subscribers of 1 to switch to firm 2 andsome others to become unsubscribed. If consumers do not immediately adjust their expec-tations, the ratio of the numbers of these switchers is equal toβ2/β0. Once consumers realizethat the value of subscription is reduced because there are more unsubscribed consumers,some of the subscribers of firm 2 will become unsubscribed. So firm 2 gains some sub-scribers due to the business stealing effect, but also loses some subscribers due to the net-work externality effect. Clearly, if β0 is relatively large, a relatively large fraction ofconsumers leaving firm 1 will become unsubscribed, so that the network externality effectbecomes large. In order to seewhether thenet effect is positive or negative, note that, in theLogitmodel ln(β2)− ln(β0)=(V2−V0)/μ. This implies that (when all usage prices are c) β-2=β0exp[((1−β0)v−F2−V0)/μ].The derivative of the right-hand side with respect to β0
equals (1+β0(−v/μ))exp[((1−β0)v−F2−V0)/μ], which is positive if and only if∇=μ−(1−β1−β2)v>0.
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Thus
∂Πi
∂Fi¼ ∂αi
∂FiαjR cð Þ þ Fi−fh i
þ αi∂αj
∂FiR cð Þ þ 1
" #:
So the first order condition reads
0 ¼ ∂αi
∂FiFi−fð Þ þ αi þ R cð Þ αj
∂αi
∂Fiþ αi
∂αj
∂Fi
!:
Solving for a symmetric solution, and using the marginal effects onthe number of subscribers of networks 1 and 2 with respect to achange in the fixed fee of network 1 derived in Lemma 3, yields
F−f ¼−α−R cð Þ α2 −1þ2αð Þ
μ−α 1−2αð Þ vþvð Þ∂αi∂Fi
:
This can be manipulated to yield F=Fequil(α, a) where
Fequil α; að Þ :¼ f þ μ− 1−2αð Þα vþ v þ R cð Þð Þ1−αð Þμ−α 1−2αð Þv μ−α v−vð Þð Þ: ð9Þ
On the other hand, rational expectations, by means of Eq. (4), needto be satisfied. We define:
FRE α; að Þ :¼ α vþ vð Þ−V0−μ lnα
1−2α
h i: ð10Þ
The equilibrium number of subscribers per firm is thus found bysolving Fequil(α, a)=FRE(α, a). We will denote this solution by α(a).In particular, for a=c0 the solution is given by
f þ μμ−2α 1−2αð Þv
1−αð Þμ−α 1−2αð Þv� �
− 2αv−V0−μ lnα
1−2α
� �h i¼ 0:
It can be shown (using Assumption 1) that there is a unique solutionα∗=α(c0) to this equation. There will then also be a unique solution fora≠c0 for small enough |a−c0|. Moreover, FRE(α, c0)>Fequil(α, c0) if andonly if αbα∗. That is, the rational expectations curve cuts the equilibri-um curve from above. Fig. 1 illustrates these findings for a given accesscharge a.
Proposition 2. Under Assumption 1, for |a−c0| small enough, thereexists a unique symmetric equilibrium F;p; pð Þ. This solution is givenby p=c, p ¼ c and F=FRE(α(a), a).
We will be particularly interested in how profits, consumer surplusand social welfare vary with a. It turns out that analyzing these effects
is not straightforward since there are two opposing effects at work.First, firmsmaywant to use the termination charge to soften price com-petition to raise fixed fees and profits. This is the only force at work inGans and King (2001) where subscription demand is inelastic. Howev-er, in the Logitmodelwith elastic subscription demand there is a secondforce atwork, namely network externalities. Firmsmay have a commonincentive to increase market penetration as this increases the value ofsubscription to each customer. Note that the second forceworks againstthe first one since softening competitionwould cause a reduction in thenumber of subscribers. It is not obvious which of the two effects domi-nates. Moreover, in case the network externality effect dominates, it isalso not clear whether firms or the regulator would like to increase ordecrease the termination charge. Therefore, inwhat follows,we first an-alyze an extreme case of two interconnectedmonopoly networks facingelastic subscription demand; this case allows us to isolate the networkexternality effect. After that we return to the case of competing net-works facing elastic subscription demand.
3.2. Extreme case of two interconnected monopolists
There are two islands and firm i (=1, 2) operates only in island i.Each island has a population of size 1/2. Inhabitants of an island can-not (or simply do not want to) subscribe to the operator of the otherisland. Hence, the two firms do not compete for the same customers.However, the inhabitants care indirectly about the pricing policy ofthe monopolist on the other island since it affects subscription ratesand thus affects how many calls can be made to its subscribers.Note that this model can alternatively be seen as one of two countries,each with a monopolist telecommunication operator who offers na-tional (on-net) and international (off-net) calls.
As before, given retail tariffs, consumers form expectations over thenumber of subscribers of network 1 (β1) and network 2 (β2), withβi≥0and β1+β2≤1. Given such expectations, utility from subscribing tonetwork i (for inhabitants of island i) equals Vi where V1, V2, V0 are de-fined as before. Define U1=V1+με1, U2=V2+με2 , and U0=V0+με0.Consumers from island i=1, 2 subscribe (to network i) when Ui>U0
and remain unsubscribed otherwise. The number of subscribers on is-land i now equals
since inhabitants of island i can only choose between subscribing (tonetwork i) or remaining unsubscribed.15 Rational expectations imply~α i ¼ βi. In Appendix A we show how rational expectations are affectedby a marginal change in firm 1's fixed fee, at a symmetric equilibriumcandidate ~F ; c; c
� �:
Lemma 4. In the Logit model with two interconnected monopolists, at asymmetric equilibrium candidate ~F ; c; c
15 In this subsection on interconnected monopolists we use tildes to distinguish thesymbols from the case of interconnected duopoly, whenever they are different.
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Proof. See Appendix A. ■Hence an increase of the fixed fee of firm 1 definitively results in a
decrease of the number of subscribers of firm 2. This is the networkexternality effect.
Given p1=p2=c and p1 ¼ p2 ¼ c :¼ cþ a−c0, profits can be re-written as
Πi ¼ ~α i ~α jR cð Þ þ ~F i−fh i
:
So the first order condition reads
0 ¼ ∂~α i
∂~F i
~F i−f� �
þ ~α i þ R cð Þ ~α j∂~α i
∂~F i
þ ~α i∂~α j
∂~F i
!:
Solving for a symmetric solution, using Lemma 4, yields
~F−f ¼−~α−R cð Þ −~α 2 1−2~αð Þ
μ−~α 1−2~αð Þ vþvð Þ∂~α i
∂~F i
:
This can be manipulated to yield ~F ¼ ~F equil ~α ; að Þ where
~F equil ~α ; að Þ ¼ f þ μ− 1−2~αð Þ~α vþ v þ R cð Þð Þ1−2~αð Þ μ−~α 1−2~αð Þvð Þ μ−~α 1−2~αð Þ v−vð Þð Þ:
ð14Þ
It is readily verified that the right-hand side of this equation is de-creasing in a at a=c0:
∂~F equil ~α ; að Þ∂a ja¼c0
¼ −~αq cð Þ μ−2~α 1−2~αð Þvμ−~α 1−2~αð Þv :
To have rational expectations fulfilled in this two island model, weobtain from Eq. (11)
~FRE ¼ ~α vþ vð Þ−V0−μ ln2~α
1−2~α
� �: ð15Þ
Note that, at a=c0, the right-hand side of this equation is decreas-ing in a:
∂~FRE ~α ; að Þ∂a ja¼c0
¼ −~αq cð Þ:
Hence, amarginal increase of a above c0 makes the rational expecta-tions curve drop by more than the equilibrium condition curve. Thismeans that the number of subscribers and the equilibrium fixed feego down when a is increased above c0. Or, equivalently, a reduction in
the termination charge below c0 increases the number of subscribersand the equilibrium fixed fee. This is illustrated in Fig. 2 below.
Lemma 5. In the Logit model with two interconnected monopolists, areduction in the termination charge below c0 increases overall marketpenetration and equilibrium fixed fees.
So if firms want to increase market penetration, they want termina-tion charge below cost. The intuition is that firms realize that raisingone's fixed fee reduces the size of the other network and thus hurts itsown customers. However, they fail to internalize the fact that this alsohurts the other network, and therefore they set a too high fixed fee andobtain a too low number of subscribers. By having abc0, the value ofmaking off-net calls is higher. This has two consequences: First, there isthe waterbed effect, which means that firms can increase the fixed fee(from point A to point B in Fig. 2) and still maintain the penetrationrate constant at ~α�. Second, the higher value from off-net calls impliesthat subscribers of a given network care more about the size of theother network. An (additional) increase in the fixed fee of network 1will now thus reduce the size of the other network more than whena=c0, which in turn hurts 1's own consumers more than when a=c0.Hence, letting abc0 exacerbates the negative feedback of raising one'sfee on its own subscribers and firms therefore lower the fixed fee(from point B to point C in Fig. 2) and this increases market penetration.
The above Lemma suggests that firms would prefer an accesscharge below termination as this increases market penetration andequilibrium fixed fees. The following proposition makes this formal.
Proposition 3. In the Logit model with two interconnected monopolists,firms prefer access fee abc0. This also improves consumer surplus. Thetermination charge that maximizes consumer surplus is lower than theone that maximizes industry profit.
Proof. See Appendix A. ■It is useful to note that the two monopolists would like to have the
number of subscribers closer to the one that would be chosen by an in-tegrated monopolist operating in both islands. Given that the two mo-nopolists do not fully internalize positive network externalities, theyend up having too small a number of subscribers and hence want to in-crease the subscribers by choosing an access charge below the termina-tion cost. At the point where firms' profits are maximized, a furtherreduction in termination charge would further increase the number ofsubscribers, and thus consumer surplus and total welfare. If terminationcharges are restricted to be non-negative,16 it may be that firms andregulator would agree that Bill and Keep (that is, access charge equalto zero) is the optimal regime.
3.3. Interconnected duopoly
We now return to the case of competing interconnected duopolists.As explained before, the case of interconnected duopolists exhibits bothnetwork externalities and business stealing effects.We here analyze theeffect of a change in termination charge around c0 for profits, consumersurplus and social welfare.
We first analyze how an increase in a effects market penetration.Let us define h(α, a)=Fequil(α, a)−FRE(α, a).
h α; að Þ ¼ μ− 1−2αð Þα vþ v þ R cð Þð Þ1−αð Þμ−α 1−2αð Þv μ−α v−vð Þð Þ−α vþ vð Þ þ V0 þ f
þ μ lnα
1−2α
h i:
Fig. 2. A reduction in a below c0 leads to higher market penetration.
16 Negative termination charges would in fact be origination charges. These do notmake much sense if one assumes, as we do in the paper, that there is no value of receiv-ing calls.
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We have already established that there is a unique solution α(a) ofh(α, a)=0. Moreover, at the solution hα>0. Hence,
We conclude that for ∇∗=μ−(1−2α∗)v>0 an increase in aabove c0 will increase the equilibrium number of subscribers, whilefor∇∗b0 such an increase in a results in a decrease in the equilibriumnumber of subscribers.
Lemma 6. Let ∇∗=μ−(1−2α∗)v.
dα� að Þda
ja¼c0> 0 if and only if ∇�
> 0
and
dα� að Þda
ja¼c0b 0 if and only if ∇�
b 0:
If ∇∗b0, then ∂α2/∂F1b0: the network externality effect dominatesthe business stealing effect. Therefore, as in the case of two inter-connected monopolists facing elastic subscription demand, a decreasein a below c0 increases the equilibrium number of subscribers by induc-ing the firms to internalize better the network externality. On the con-trary, if ∇∗>0, then ∂α2/∂F1>0: the business stealing effect dominatesthe network externality effect. Then, as in the case of competing duopolyfacing inelastic subscription demand, a decrease in a below c0 decreasesthe equilibrium number of subscribers by softening competition. There-fore, one would expect that firms would prefer termination chargebelow cost for opposite reasons: in order to boost market penetrationfor ∇∗b0 and in order to reduce market penetration for ∇∗>0 . Wenow proceed to verify that indeed firms always prefer below cost termi-nation charges.
Let H(a, α) denote the profit a firm makes when it has α sub-scribers, access charge is a and its fixed fee is FRE(α, c0). That is
H a;αð Þ ¼ α αR cð Þ þ F−fð Þ¼ α α R cð Þ þ v cð Þ þ v cð Þð Þ−V0−μ ln α= 1−2αð Þ½ �−fð Þ:
We will be interested in knowing what happens with this profit ata=c0 when α is moved away from the corresponding equilibriumvalue α∗. Note that we know that per consumer profit at the equilib-rium equals F− f, which by Eq. (9) equals (at a=c0)
Therefore, if Assumption 1 is satisfied, the sign of this derivative isopposite to the sign of ∇∗=μ−(1−2α∗)v. Thus, if an increase of a
above c0 increases (decreases) market penetration, profits decrease(increase) with the number of subscribers along the rational expecta-tions curve, in a neighborhood around c0.
Next, we have to account for the fact that when a is varied, the ra-tional expectations curve, and thus the equilibrium, will change. Thepartial effect on profits (keeping α∗
fixed) equals
∂H∂a ¼ α�� �2 a−c0ð Þq′ cð Þ;
so that at a=c0 a marginal change in a does not affect profits directly.The extra profit made from off-net calls is just offset by the decreasein fixed fee. However, profits are affected indirectly by a change inmarket penetration.
dHda
¼ Ha þ Hα � α′ að Þ:
Since the sign of Hα is the opposite of the sign of α′(a) at a=c0, oneobserves that profits are always decreasing in a in a neighborhoodaround c0. Firms thus always prefer a termination charge below cost.
Fig. 3 illustrates the findings in the case of interconnected duopolistswhen the business stealing effect dominates. For a=c0 the equilibriumis located at point A. An increase in the termination charge shifts theequilibrium point to C. At A, equilibrium profit equals the sum of areasi, ii and iii. Moving along the rational expectation curve, as market pen-etration increases, from A to B, profit is reduced by area i (lower fixedfee) and increased by areas iv and v (higher penetration). Our analysishas shown that the net effect of this move is negative. Then, movingfrom B to C, we account for the fact that the equilibrium fixed fee goesdown when a increases, keeping penetration constant. Our analysishas shown that the loss in revenues from fixed fees (areas ii and iv) isexactly offset by the gain in profits from off-net calls. The overall effecton profit of a marginal increase in a above c0 is thus negative when thebusiness stealing effect dominates.
Proposition 4. Firms prefer a termination charge below cost (i.e. abc0).Howdoes social welfare changewhen the access charge is changed?
Social welfare is the sum of consumer surplus and industry profit. Theexpression for consumer surplus in the Logit model was reported inEq. (2) and can clearly be rewritten as
CS a;α að Þð Þ ¼ μ lnX2j¼0
exp Vj=μ� �0
@1A ¼ V0−μ ln 1−2α að Þð Þ:
Fig. 3. Interconnected duopolists when the business stealing effect dominates. An in-crease in a above c0 leads to higher market penetration.
548 S. Hurkens, D.-S. Jeon / International Journal of Industrial Organization 30 (2012) 541–552
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Hence, consumer surplus is not directly affected by the access charge,but only through the equilibrium number of subscribers. Clearly, consumersurplus is increasing in the number of subscribers:
∂CS∂α ¼ 2μ
1−2α> 0:
SW a;α að Þð Þ ¼ CS a;α að Þð Þ þ 2H a;α að Þð Þ:
We thus obtain, at a=c0,
dSWda
¼ α′ að ÞCSα þ 2Ha þ 2Hαα′ að Þ
¼ α′ c0ð Þ 2μ1−2α
þ 2−α μ− 1−2αð Þvð Þ μ−2α 1−2αð Þvð Þ
1−2αð Þ 1−αð Þμ−α 1−2αð Þvð Þ�
It can be established that the term in brackets is strictly positive whenμ>v/4. This means that social welfare increases as market penetrationincreases.
Proposition 5. Let α∗ denote the number of subscribers per firm in theequilibrium when a=c0. If μ>(1−2α∗)v, the number of subscribers,and thus social welfare, increases as a is increased above c0. If μb(1−2α∗)v, the number of subscribers, and thus social welfare, increases asa is decreased below c0: in this case, the socially desirable access chargeis lower than the one that maximizes industry profit.
Both the firms and the regulator want to divert from the terminationcharge equal to termination cost in order to affect the number of sub-scribers. The firms want to make the number of subscribers closer tothe number chosen by a monopolist owning both networks while theregulator alwayswants to increase the number.When there is a net busi-ness stealing effect (i.e.∇∗>0), there is a conflict of interest between thefirms and the regulator since the firms want to decrease the number ofsubscribers, which requires them to choose a below c0 in order to softencompetition. When there is a net network externality effect (i.e.∇∗b0),there is a congruence of interests between the firms and the regulator inthe sense that firms want to increase the number of subscribers, whichagain requires them to choose a below c0 in order to internalize network
externalities. However, the firms do not internalize the positive effectthat an increase in their network size has on consumer surplus andtherefore the socially preferred access charge is lower than the one pre-ferred by the firms.
To conclude this section, we illustrate the results obtained bymeansof numerical examples. We assume the same demand functions andcost parameters (in cents) as de Bijl and Peitz (2004) and López andRey (2009): q(p)=(20−p)/0.015, c0=0.5, c=2, f=1000, μ=3000.Note that v(c)=10800 so that v(c)/2>μ>v(c)/4.
In the first case (Fig. 4, left panel) we assume V0=5000. This meansthe outside option is relatively attractive and network effectswill be im-portant. In this case, for a=c0, the (symmetric) equilibrium hasF≈2372 and total market penetration 2α∗≈0.58. So ∇∗=μ−(1−2α∗)v(c)≈−1517b0 and the network effect dominates.
The graphs illustrate that (i) market penetration decreases withaccess charge, (ii) industry profits are maximized at a=0, (iii) socialwelfare is maximized at a=0 (Bill and Keep).
In the second case (Fig. 4, right panel) we assume V0=0. In thiscase, for a=c0, the (symmetric) equilibrium has F≈5859 and totalmarket penetration 2α∗≈0.86. So ∇∗=μ−(1−2α∗)v(c)≈1535>0and the business stealing effect dominates.
The graphs illustrate that (i) market penetration increases withaccess charge, (ii) industry profits are maximized at a=0bc0 (Billand Keep), (iii) social welfare is maximized at about a=5.4.
4. Retail benchmarking approach and the Ramsey outcome
In this section,we show that the retail benchmarking approach allowsto achieve the Ramsey outcome. Before showing this result, we explainthe approach introduced in Jeon and Hurkens (2008) in amodel withouttermination-based price discrimination and with inelastic subscription.
Jeon and Hurkens (2008) find that the following family of accesspricing rules parameterized by κ(b1) induces each firm to adopt themarginal cost pricing (i.e. pi=c):
ai ¼ c0 þ κFi þ piq pið Þ
q pið Þ −c�
; ð16Þ
0.5 1 1.5 2 2.5 3access charge
3040506070
market penetration
1 2 3 4 5 6access charge
7580859095
100market penetration
0.5 1 1.5 2 2.5 3access charge
720740760780800
industry profit
1 2 3 4 5 6access charge
300032503500375040004250
industry profit
0.5 1 1.5 2 2.5 3access charge
8000
8200
8400
social welfare
1 2 3 4 5 6access charge
102001030010400105001060010700social welfare
Fig. 4. Illustration of the effect of access charges when network effects (left panel) or business stealing effect (right panel) are important.
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where ai represents the access charge that firm i pays to each rivalfirm. κ=0 corresponds to the fixed access charge equal to the termi-nation cost. According to the rule, the mark-up of the access charge thatfirm i pays to each rival firm is equal to thefirm i's average pricemark-upmultiplied by κ. They find that the retail benchmarking rule intensifiesretail competition such that higher values of κ translate into lower equi-librium fixed fee. However, when all consumers subscribe to one of thetwo networks, the level of fixed fee does not affect social welfare andchoosing termination charge equal to termination cost is enough toachieve the Ramsey outcome.
Let πi denote network i's retail profit per customer gross of thefixed cost f when a=c0:
πi ≡ pi−cð Þq pið Þ þ Fi:
Then, Eq. (16) is equivalent to
ai−c0ð Þq pið Þ ¼ κπi: ð17Þ
We now show that there is a simple modification of our access pric-ing rule (17) that achieves the Ramsey outcome as a Nash equilibriumin our model with termination-based price discrimination and elasticsubscription. Our aim is not so much to promote this modified accesspricing rule but to illustrate the power of the retail benchmarking ap-proach with respect to the approach based on a fixed access charge.Note that the Ramsey outcome is achieved when the firms charge theprices equal to the costs (i.e. pi ¼ pi ¼ c, Fi= f for i=1, 2) and this out-come can never be achieved under the approach based on a fixed per-minute access charge.
Let αR be each network's number of subscribers in the Ramseyoutcome. In a Logit model with duopoly, we have
0 b αRb 1=2:
Define κ∗ by 1−κ∗αR=0; hence κ∗>2. Let πi denote network i'sretail profit per customer gross of the fixed cost when a=c0;
πi ≡αi pi−cð Þq pið Þ þ αj pi−cð Þq pið Þ þ Fi:
We modify Eq. (17) as follows:
ai−c0ð Þq pið Þ ¼ κ�max πi; ff g ð18Þ
In Eq. (18), the mark‐up of the access charge that firm i pays is mul-tiplied by the off-net call volume q pið Þ that a caller of firm i makes to areceiver of firm j. In addition, we choose κ=κ∗ and add the max opera-tor such thatfirm i cannot realize any further reduction of its access pay-ment by pricing below costs. In the absence of retail benchmarking, firmi has no incentive to choose retail prices that give him a retail profit percustomer below the fixed cost per customer. However, under our retailbenchmarking approach, firm i may have an incentive to choose verylow retail prices only to reduce its access payments such that its net ac-cess profit (equal to access revenues minus access payments) morethan covers its net retail loss. Adding the max operator in Eq. (18)makes such a deviation not profitable, as will be shown below.
We now introduce one additional assumption:
Assumption 2. An increase in Fi increases the number of subscribersto firm j.
Assumption 2 holds if there is a net business stealing effect and issatisfied if μ is large enough. Then, we have:
Proposition 6. Suppose that the regulator proposes the access pricingrule Eq. (18). Then, under Assumption 2, the Ramsey outcome can beimplemented as a Nash Equilibrium: in the equilibrium, firm i choosespi ¼ pi ¼ c, Fi= f for i=1, 2.
Proof. See Appendix A. ■The Ramsey outcome requires to expand market penetration by re-
ducing the fixed subscription price to the minimum compatible withthe break-even constraint without distorting call volumes. The accesspricing rule (18) allows to achieve this Ramsey outcome. First, underthe rule, each network has an incentive to maintain the marginal costpricing,17 which implies that Eq. (18) is equivalent to
ai−c0ð Þq cð Þ ¼ κ�max Fi; ff g: ð19Þ
Eq. (19) shows that each network i has an incentive to reduce Fi (up to f)in order to reduce the access payment that it makes to network j.
5. Conclusion
We studied how access pricing affects network competition whenconsumers' subscription demand is elastic and firms compete withnon-linear tariffs and can apply termination-based price discrimination.We first considered the standard approach based on a fixed and recip-rocal (per-minute) termination charge and found that both the firmsand the regulator want to depart from cost-based termination charge(and hence want to distort call volumes) in order to affect market pen-etration. In particular, two opposing effects (softening competition andinternalizing network externalities) are associated with a reduction intermination charge. The former decreases market penetration whilethe latter increases it. We find that firms always prefer having termina-tion charge below cost for either motive while the regulator prefers ter-mination charge below cost only if this boosts penetration.
After studying the standard approach, we investigated the retailbenchmarking approach.We found that the approach allows to expandmarket penetration without distorting call volumes such that the Ram-sey outcome can be achieved. The approach intensifies retail competi-tion since a firm can reduce its access payment to rival firm(s) byreducing its retail prices (in particular, the fixed subscription price).
Although our analysis has been restricted to two firms, in an oligop-olistic setting themain insight that a decrease in the termination chargehas two opposing effects holds true. Whether the privately and sociallyoptimal termination charges increase with the number of firms is notclear. Calzada and Valletti (2008) showed that with inelastic subscrip-tion demand, a termination charge below cost softens competitionalso in an oligopoly with n firms. They show that the higher is n, thehigher is the termination charge that maximizes industry profit. Thisis because with more firms a higher fraction of calls is off-net and thecompetition-softening effect of a given decrease in termination chargeis stronger. On the other hand, consider a model with n islands each oc-cupied by a monopolist network. A decrease in the fixed fee of one mo-nopolist that increases the number of its subscribers by 1%, will have asmaller impact on the subscribers of the other networks the more net-works there are (as it increases the number of people that can be calledby 1/n per cent). Hence, in order to compensate for this smaller effect, tobetter internalize the network externality effect, the termination chargeshould be lower when there are more networks.
Appendix A
Proof of Lemma 1. We prove that the fixed point of the mappingα :Δ2→Δ2 where α(β1, β2)=(α1, α2) is unique for μ large enough.This can be done by looking at the index of zeros of the mappingg(β)=α(β)−β. The Jacobian of g is
Dβg ¼1μ
β1 1−β1ð Þv p1ð Þ−β1β2v p2ð Þ½ �−11μ
β1 1−β1ð Þv p1ð Þ−β1β2v p2ð Þ½ �1μ
β2 1−β2ð Þv p2ð Þ−β1β2v p1ð Þ½ � 1μ
β2 1−β2ð Þv p2ð Þ−β1β2v p1ð Þ½ �−1
0BB@
1CCA:
17 See our working paper, Hurkens and Jeon (2009) for more details.
550 S. Hurkens, D.-S. Jeon / International Journal of Industrial Organization 30 (2012) 541–552
Author's personal copy
Let d=detDβg. In the case at hand, the index of afixed pointβ is equalto +1 if d>0 and equal to −1 if db0. The Poincaré-Hopf Theorem im-plies that the sum of indexes of all fixed points equals +1 (the Eulerindex of the simplex). It is clear that for large enough μ, d>0 so thatthen every fixed point has index +1. This then implies that there is aunique fixed point. Thus for large enough μ rational expectations areuniquely defined for all tariff schedules.
Proof of Lemma 2. The proof uses elements from the proof of the pre-vious Lemma in the special case of firms using perceived marginal costpricing. Suppose first that a=c0 and that firms set both on-net and off-net price equal to cost, but possibly F1≠F2 (and thus α1≠α2). Let d de-note the determinant of the Jacobian used in the previous Lemma. In thiscase
d ¼ μ− 1−α1−α2ð Þ α1 þ α2ð Þv cð Þμ
¼ μ−α0 1−α0ð Þv cð Þμ
> 0
where the inequality follows from Assumption 1. By continuity the de-terminant will be strictly positive also when a is close to c0 and firmsprice at perceived marginal cost. This means that expectations areuniquely defined.
Proof of lemma 3. We continue to use the notation of the proof ofLemma 1. In particular, g(β)=α(β)−β. Note that
DF1g ¼ −β1 1−β1ð Þ=μ
β1β2=μ
� :
This implies that an increase in the fixed fee of network 1, everythingelse equal, will decrease the number of subscribers to network 1 and willincrease the subscribers to network 2.However, a change in F1 also affectsrational expectations and the total effect on the number of subscribers bya change in fixed fee F1 is given by the implicit function theorem as
DF1β ¼ − Dβg
h i−1DF1
g:
The results follow immediately.
Proof of Lemma 4. Rational expectations are zeros of the mapping~g βð Þ ¼ ~α1−β1; ~α2−β2ð Þ. The Jacobian of ~g is
Dβ~g ¼1μ
β1 1−2β1ð Þv p1ð Þ½ �−11μ
β1 1−2β1ð Þv p1ð Þ½ �1μ
β2 1−2β2ð Þv p2ð Þ½ � 1μ
β2 1−2β2ð Þv p2ð Þ½ �−1
0BB@
1CCA:
Let ~d denote the determinant of this Jacobian.For large enough μ, we have ~d > 0 and therefore rational self-
fulfilling expectations are then unique. Relying again on continuous ra-tional expectations one obtains again that firms will always set variableprice equal to perceived marginal cost: pi=c and pi ¼ c. Note that
DF1~g ¼ −β1 1−2β1ð Þ=μ
0
� :
This implies that an increase in the fixed fee of network 1, everythingelse equal, will decrease the number of subscribers to network 1 and willkeep the number of subscribers to network 2 constant. The latter illus-trates the fact that there is no business stealing effect in thismodel. How-ever, a change in F1 does affect expectations and the total effect on thenumber of subscribers by a change in fixed fee F1 is given by the implicitfunction theorem as
DF1β F1ð Þ ¼ − Dβ~g
h i−1DF1
~g :
One thus verifies that
∂β1
∂F1¼ −1
~dμ2β1 1−2β1ð Þ μ−β2 1−2β2ð Þvð Þð
and
∂β2
∂F1¼ −1
~dμ2β1β2 1−2β1ð Þ 1−2β2ð Þv:
Thus, at a symmetric solutionwith rational expectations ~α1 ¼ ~α2 ¼ ~α
∂~α1
∂F1¼ −1
~dμ2~α 1−2~αð Þ μ−~α 1−2~αð Þvð Þ b 0ð ð20Þ
and
∂~α2
∂F1¼ −1
~dμ2~α2 1−2~αð Þ2v b 0: ð21Þ
Proof of Proposition 3. At the equilibrium (at a=c0) per consumerprofit equals
F−f ¼ μðμ−2α� 1−2α�ð Þv1−2α�ð Þ μ−α� 1−2α�ð ÞvÞ:ð
The effect on total profit H(c0, α)=α(F− f) with respect to achange in α is thus
∂H∂α c0;α
�� � ¼ H c0;α�� �=α� þ α� 2v cð Þ− μ
α� 1−2α�ð Þ� �
¼ α�v μ−2α� 1−2α�ð Þvð Þμ−α� 1−2α�ð Þv > 0
Hence, profits increase along the rational expectations curve, in aneighborhood around α∗.
Next, we have to account for the fact thatwhen a is varied, the ratio-nal expectations curve, and thus the equilibrium, will change. The par-tial effect on profits (keeping α∗
fixed) equals
∂H∂a ¼ α�� �2 a−c0ð Þq′ cð Þ;
so that at a=c0 a marginal change in a does not affect profits directly.The extra profit made from off-net calls is just offset by the decreasein the fixed fee. However, profits are affected indirectly by a change inmarket penetration.
Therefore, we have:
dHda
¼ Ha þ Hα � α′ að Þ b 0;
profits are decreasing in a in a neighborhood around c0. Firms thus in-deed prefer an access fee below cost. Suppose a∗∈(0, c0) maximizesconsumer surplus, that is, α′(a∗)=0. Then
dHda ja¼a�
¼ ∂H∂a ja¼a�
¼ α a�� �2 a�−c0
� �q′ cð Þ > 0:
Hence, the termination charge that maximizes firms' profits ishigher than the one that maximizes consumer surplus. ■
Proof of Proposition 6. Suppose that firm j uses Fj ¼ f ;pj ¼ pj ¼ c.Then, we distinguish two cases depending on whether αi>αR orαibαR.
551S. Hurkens, D.-S. Jeon / International Journal of Industrial Organization 30 (2012) 541–552
Author's personal copy
Case 1. when αi>αR. αi>αR implies that πib f. Then, firm i's profit is
Πi ¼ αi πi−f½ �−αiαjκ� f−f½ �
¼ αi πi−f½ � b 0:
Case 2. when αibαR. αibαR implies, from Assumption 2, 1bαjκ∗. Con-sider first πi≥ f. Then, firm i's profit becomes
Πi ¼ αi 1−αjκ�� �
πi−f½ �≤ 0:
Consider now πib f. Then, firm i's profit is
Πi ¼ αi πi−f½ �−αiαjκ� f−f½ �
¼ αi πi−f½ �≤ 0:
■
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