old technology upgrades, innovation, and competition in vertically differentiated markets

Information Economics and Policy 29 (2014) 10–31

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Information Economics and Policy

journal homepage: www.elsevier .com/locate / iep

Old technology upgrades, innovation, and competitionin vertically differentiated markets

http://dx.doi.org/10.1016/j.infoecopol.2014.08.0010167-6245/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (M. Bourreau),

[email protected] (P. Lupi), [email protected] (F.M. Manenti).1 The views and opinions expressed herein are solely those of the author

and do not necessarily reflect those of the Autorità per le garanzie nellecomunicazioni.

2 Fiber networks provide a higher speed than the standard DSband technology. Vectoring is an engineering technique thattraditional copper lines to achieve speeds that are close to the thlimits, and therefore also close to the speed of first-generatnetworks. For vectoring to be effective, however, all copper linrelevant area have to be under the control of a single provideimplies that vectoring is not compatible with sub-loop unbuwholesale service mandated by several European National ReAgencies.

3 See Czernich et al. (2011) for empirical evidence that the difbroadband has a positive impact on growth.

Marc Bourreau a, Paolo Lupi b,1, Fabio M. Manenti c,⇑a Telecom ParisTech, Department of Economics and Social Sciences, and CREST-LEI, Paris, Franceb Direzione Analisi dei mercati, concorrenza e studi, Autorità per le Garanzie nelle Comunicazioni, Centro Direzionale, Isola B5, 80143 Napoli, Italyc Dipartimento di Scienze Economiche ed Aziendali ‘‘M. Fanno’’, Università di Padova, Padova, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 November 2013Received in revised form 18 June 2014Accepted 1 August 2014Available online 27 August 2014

JEL codes:L1L51L96

Keywords:AccessInvestmentVertical differentiationMulti-product firms

In this paper we study how the migration from an old to a new technology is affected bythe access price to the old technology, when it is set after investments have taken place.We show that both the incumbent and the regulator are willing to set a very high accessprice to accelerate consumers’ migration to the new technology. When the quality of theold technology is exogenous and the entrant dominates investment in the new technology,the old technology is completely switched off in equilibrium. On the other hand, when theincumbent dominates investment, the old technology persists. When the incumbent candecide on an endogenous upgrade of the old technology, the migration to the new technol-ogy is slowed down, and the entrant might be foreclosed.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction new technology, called ‘‘vectoring,’’ which will allow them

L broad-enables

eoretical

In network industries, infrastructure investments arenecessary to maintain or improve the quality of servicesprovided to consumers. Such investments can involveeither upgrades of the old generation network or thedeployment of new infrastructures, i.e., next generationnetworks. For example, in the European telecommunica-tions industry operators have started to invest in high-speed fiber next generation access networks to replacethe old legacy copper networks. At the same time, somehistorical telecom operators are planning to introduce a

to upgrade copper networks to provide higher speeds forInternet access.2

As network infrastructures are expected to be a strongcontributor to economic activity and growth,3 a fast transi-tion from old network technologies to new ones is a keychallenge for policy makers. For example, the EuropeanCommission has set up a ‘‘Digital Agenda 2020’’ with very

ion fiberes in ther, which

ndling, agulatory

fusion of

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M. Bourreau et al. / Information Economics and Policy 29 (2014) 10–31 11

ambitious objectives for the migration of consumers fromstandard broadband (based on the old generation coppernetwork) to very-high speed broadband (based on next gen-eration fiber networks).4 A relevant and important questionis then which type of regulatory intervention could acceler-ate the transition.

The migration from an old to a new network technologycan indeed be a slow process, due in particular to the largeinvestments necessary to deploy next generation net-works, and to the competition on the consumers’ sidebetween the two technologies, when they coexist in a tran-sitory phase. Access regulations, which oblige the owner ofthe legacy network to provide access to competitors at agiven price, can affect both investment incentives and thecompetition between the old and the new technologies,and hence shape the transition process.

In this paper we study how the terms of access to theold generation network affect the competition betweenthe old and the new generation networks in the retailmarket, and thus the migration from the old to the newtechnology. This question is of particular relevance inEuropean telecoms, where some of the alternativeoperators investing in fiber lease access to the legacycopper network of incumbent operators.5

Aside from access obligations, we also consider twoother forms of regulatory intervention: (i) switching offthe old generation network after the new network hasbeen deployed and (ii) allowing or forbidding an upgradeof the old generation network. Switching off the old net-work forces consumer migration to the new network, andthereby reinforces firms’ incentives to invest in nextgeneration networks. In Australia, for example, a publiccompany (NBN Co) is rolling out a national fiber infrastruc-ture and has started a countdown for the switch-off of theold copper network. The European Commission also sees aswitch-off of the copper network as a means of providingproper investment incentives to operators.6 An upgrade ofthe old generation network is likely to have one of twocontrasting effects on investment incentives. It could eitherslow down the transition because of the tougher competi-tion between the old and the new technologies after theupgrade, or alternatively, spur investment because operatorswish to escape the competition from the old technology. It istherefore not surprising that in most European countriesregulators are still wondering whether they shouldauthorize the vectoring technology.7

4 The European Commission’s objective is that half of European house-holds subscribe to a broadband offer above 100 Mbps by 2020.

5 Our approach is less relevant in other markets (such as the US market),where the alternative operators investing in fiber do not lease access to thelegacy copper network, but rather use cable or electricity infrastructures.

6 For example, European Commission Vice-President Neelie Kroesdeclared that ‘‘the gradual switch-off of copper could reduce the cost tosuch a degree that new fiber investments break even in under 10 years. Andthus align the interests of investors and long-term financing providers.’’(See: Investing in digital networks: a bridge to Europe’s future, 3 October2011, http://europa.eu/rapid/press-release_SPEECH-11-623_en.htm). How-ever, to date, in Europe, there is no example of copper switch-off.

7 The only two European regulators that have withdrawn the sub-loopunbundling obligation to allow the deployment of vectoring are those ofBelgium and Ireland, but other NRAs are also considering the possibility ofat least partially removing this obligation.

We then investigate the two following questions. First, ifa switch off of the old generation network is socially desir-able, can it be achieved by the market players without anyregulatory intervention, or is a formal switch-off by the reg-ulatory authority necessary? Second, should the regulatorallow the owner of the legacy network to upgrade it?

In the stream of literature which studies the interplaybetween regulation and investment in network industries(see Cambini and Jiang (2009) and Vogelsang (2013) forrecent surveys), the new technology always replaces theold one. Some papers analyze investment by an incumbentfirm only, which can upgrade its old network (e.g., seeForos (2004), Kotakorpi (2006), etc.). Others focus onentrants’ incentives to bypass the old network by investingin a new alternative infrastructure (e.g., see Bourreau andDogan (2005, 2006), Avenali et al. (2010), Klumpp and Su(2010), etc.). Finally, some authors analyze investmentraces where firms compete to deploy new infrastructures,which completely replace the old ones (e.g., Gans (2001),Hori and Mizuno (2006), Vareda and Hoernig (2010),etc.). In all these studies, consumers cannot choosebetween the old and the new technology, and as a resultthe migration issue is absent.

We depart from this standard set-up by building aframework where two firms, an incumbent which ownsthe legacy old generation network (OGN), and an entrantthat leases access to the OGN, can invest to roll out nextgeneration networks (NGN) in order to offer multiplevertically differentiated services based on the old and thenew technologies. The migration from the old technologyto the new ones is endogenous to consumers’ decisions.It depends on firms’ pricing decisions, their initialinvestment decisions, and the upgrade of the OGN.8

We start by analyzing a benchmark with a monopolyprovider of OGN and/or NGN services. We show that, giventhe demand and cost structure of our model, it is profit andwelfare maximizing to supply only NGN services. Hence,consistent with policy makers’ position in favor of fibernetworks, a full migration to NGN is socially desirable.

We proceed by analyzing investments in a duopolisticcontext, when the OGN cannot be upgraded. The incum-bent and an entrant initially decide on the quality of theirnext generation networks (NGNs). Once investments havebeen made, the access charge to the OGN is set by eitherthe incumbent or the regulator. Finally, firms competewith vertically-differentiated multiple products.

Therefore, we assume that the incumbent and the regu-lator cannot commit ex-ante to an access price to the OGN.Consequently, they cannot influence investment decisionsdirectly with the access price. However, they can influencethe consumers’ choices at the retail level, when both theOGN and the NGN services are available, which affectsindirectly investment incentives.

Our results are as follows (see also the summary Table 1below). The game has two equilibria: a leapfrogging

8 Our model captures the actual situation in several European countries,such as France, Italy, Denmark and Finland, among others, where there aretwo (or more) competing fiber (FTTH) networks, and where incumbentoperators are upgrading their copper infrastructures to VDSL2/vectoring inorder to provide FTTC services.

http://europa.eu/rapid/press-release_SPEECH-11-623_en.htm

Table 1Summary of results.

Equilibria

Persistence ofleadership

Leapfrogging

No upgrade Migration Complete PartialSwitch-off Not necessary Desirable

Upgrade Migration Partial PartialSwitch-off Desirable Desirable

9 See also Bourreau et al. (2014), where the authors discuss other policyinstruments that could spur the migration to NGNs.

10 Our paper is also related to the literature on multiproduct competitionwith vertical differentiation (see, for example, Gal-Or (1983), Champsaurand Rochet (1989), De Fraja (1996), Johnson and Myatt (2003, 2006) andChisholm and Norman (2012)). We study competition between multiprod-uct firms with vertical differentiation, when the two products correspondto two successive technological generations, and a regulator aims atinfluencing the migration from the old to the new technology. However,since we adopt a simple and specific setting, we do not aim to contribute tothis literature.

11 For example, in telecoms, the OGN is the copper infrastructure that isnecessary to provide broadband DSL services.

12 The quality levels for the OGN can be viewed as a legacy from a marketequilibrium where the NGN was absent. If only the OGN were present, thetwo firms would choose to differentiate their qualities as much as possible.Given that the incumbent controls the OGN, we can assume that it was thefirst mover for OGN services, and chose to produce the high quality. We alsosolved our model when the entrant has OGN services of higher quality thanthe incumbent, that is, dE > dI . Our results were unchanged.

13 Note that the two assumptions of zero marginal cost and of multipli-cative quality utility functions drive our model towards outcomes wherehaving only one active network of high quality is both profit and welfaremaximizing.

12 M. Bourreau et al. / Information Economics and Policy 29 (2014) 10–31

equilibrium, where the entrant is leader in NGNinvestments, and a persistence-of-leadership equilibrium,where the incumbent is the leader. In the persistence-of-leadership equilibrium, the migration to the newtechnology is complete: whether or not access to theOGN is regulated, the two firms offer only NGN services,with the entrant providing the highest-quality NGN. TheOGN is therefore switched off, without any formalregulatory intervention. In the leapfrogging equilibrium,migration is only partial: the incumbent offers both OGNand NGN services, while the entrant offers only NGNservices. This market outcome is inefficient: welfare wouldincrease if the migration to the NGNs were complete.Therefore, a formal switch-off of the OGN would bewelfare-enhancing.

When the incumbent can upgrade the quality of theOGN at the same time as firms decide on their NGN invest-ments, two additional equilibria emerge: a leapfroggingequilibrium with upgrade, and a persistence-of-leadershipequilibrium with upgrade. We find that in the leapfroggingequilibrium with upgrade, the incumbent decides not toinvest in an NGN, and upgrades its OGN, whereas theentrant offers only NGN services. Up to a certain level ofquality achieved with the upgrade, welfare is lower thanunder the leapfrogging equilibrium without upgrade. Inthe persistence-of-leadership equilibrium with upgrade,the incumbent offers both upgraded OGN services andNGN services, whereas the entrant offers only NGN ser-vices of low quality. Welfare is lower than under the per-sistence-of-leadership equilibrium without upgrade;therefore, forbidding the incumbent to upgrade its OGNwould be welfare-enhancing.

Very few papers analyze the migration from an old to anew technology in network industries. de Bjil and Peitz(2009) study the transition from the old telephony tech-nology (PSTN) to the new one (VoIP). In their setting, theincumbent can offer both technologies, while the entrantoffers only the new one. The authors moreover model mar-ket segmentation in a different way than we do, using atwo-dimensional horizontal differentiation demandmodel, rather than a vertical differentiation model. Britoet al. (2012) and Bourreau et al. (2012) explore the migra-tion from the OGN to the NGNs in the broadband market.Brito et al. (2012) study a setting where an incumbentcan invest in an NGN and provide access to this infrastruc-ture on a volontary basis, while access to the OGN is regu-lated. They then analyze the entrant’s technology choice.Bourreau et al. (2012) study the impact of access to theold generation network on an incumbent’s and an entrant’s

investment incentives.9 However, in both papers, each firmoffers a single service (using either the OGN or the NGN),whereas in our setting firms can offer both OGN and NGNservices. Moreover, in both papers the access price is setex ante and affects investments, whereas in our setting itis set ex post, after investments have taken place. Comparedto this earlier literature, the access price affects not theinvestments, but rather the migration to the NGN at theretail level (i.e., access pricing affects the consumers’ choicebetween OGN and NGN services at a given provider).10

The rest of the paper is organized as follows. In Section2 we present the baseline model. We study two simplebenchmark cases in Section 3, and then solve for the equi-librium in Section 4. We extend our setting to account forthe possibility of an endogenous OGN upgrade in Section 5.Finally, we conclude.

2. The model

2.1. Firms

There are two firms: an incumbent firm (I), which con-trols an old generation network (OGN),11 and a rival firm(E). At the beginning of the game, the two firms are compet-ing for OGN services, and firm E is leasing access at the per-unit price a from the incumbent’s OGN. The access price isset by either the incumbent or a regulator. We denote bydi > 0 the quality of the OGN’s service for firm i ¼ I; E.Because the incumbent controls the OGN, we assume thatit provides a higher quality of service on the OGN than doesthe rival firm, that is, dI > dE.12

Firm I and firm E may decide to invest in a next gener-ation network (NGN). In order to build an NGN of qualityli, firm i ¼ I; E has to incur the quadratic investment costcðliÞ ¼ l2

i =2. Note that in our setting the NGN does notreplace the OGN, that is, the two technologies may coexist.We also rule out access to the NGN.

Finally, we consider that firms compete in prices andnormalize the marginal costs to zero.13 We denote by di

and mi the price for the OGN’s service and the NGN’s service,


respectively, for firm i ¼ I; E. We denote by pi firm i’s profit,gross of investment costs, and by Pi its net profit, that is,Pi ¼ pi � cðliÞ.

2.2. Consumers

A consumer is characterized by her type h, which repre-sents her marginal willingness to pay for the quality of theservice. Consumer types are distributed on the interval½0; �h�, with density 1. To obtain analytical solutions, weneed to specify a specific value for �h ; we set �h ¼ 100.14 Aconsumer of type h buys at most one unit of service in orderto maximize her net utility, U ¼ hx� p, where x is the qual-ity of the chosen service, p the price, andx;pf g 2 fdI; dIg; dE; dEf g; lI;mI

� �; lE;mE� ��

.

2.3. Timing

The timing of the game is as follows:

– Stage 1: the incumbent and rival firms choose theirNGN’s qualities lI and lE, respectively.

– Stage 2: the access charge to the OGN, a, is seteither by the incumbent or by a welfare-maximiz-ing regulator.

– Stage 3: the firms compete in prices with multipleproducts, and profits are realized.

Since our focus is on how the access price to the OGNcan affect the migration to NGN services once NGN invest-ments have been realized (i.e., at the retail level), ratherthan on how access affects NGN investments, we assumethat the regulator cannot commit to an access charge priorto the firms’ investment decisions.15

Finally, we make the following assumption16:A1. di < 228:29, for i ¼ I; E.Assumption A1 ensures the existence of two equilibria,

one where the incumbent is leader in NGN investments,and one in which the rival firm is the leader.17

3. A benchmark: Monopoly OGN and NGN provider

As a benchmark, we consider the case where firm I is amonopolist, which offers two different qualities: the OGNquality, dI , and the NGN quality, lI . The game then has onlytwo stages: in a first stage, firm I decides on its NGN invest-ment, and in a second stage, it sets the price for its lowquality and high quality services. At the second stage, firmI’s profit, gross of investment cost, is

pI dI;mIð Þ ¼ dI hL � hSð Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}OGN

þmI�h� hL� �|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

NGN

;

14 Our results would be qualitatively similar with other values of �h.15 This is in line with Foros (2004) and Kotakorpi (2006), for example.

Brito et al. (2010) also consider the possibility that the regulator is unableto commit to an access price.

16 This assumption and the next ones (Assumptions A2 and A3 in Section5) as well as all the numerical results are obtained under the normalization�h ¼ 100.

17 If A1 does not hold, the equilibrium where the incumbent is leader inNGN investments may not exist.

where hS ¼ dI=dI and hL ¼ ðmI � dIÞ=ðlI � dIÞ. Firm I choosesthe prices dI and mI to maximize its profit pI . We find thatd�I ¼ 50dI and m�I ¼ 50lI . Replacing for the equilibriumprices into the demands for the low quality and the highquality services, we find that hL � hS ¼ 0 and �h� hL ¼ 50.Hence, the incumbent monopolist sets prices so that onlythe high quality is active in equilibrium. Movingbackwards to the investment stage, the incumbentchooses an NGN quality lI to maximize its profit,PI ¼ 2500lI � ðlIÞ

2=2. The optimal NGN investment is

l�I ¼ 2500.To sum up, if the incumbent does not face competition,

it invests in an NGN infrastructure, and shuts down itsOGN. The result, that the monopolist may find it optimalto provide only the high quality, is well-known in the liter-ature on vertical differentiation with multiple productsfirms. When consumers have multiplicative preferencesand marginal production costs are non-increasing in qual-ity, there is a strong tendency for firms to offer only a sin-gle product rather than multiple products (see, forexample, Johnson and Myatt (2003)).18

In our setting, if the regulator can set the retail pricesfor the OGN and NGN services of the monopoly provider,it also sets prices such that only the high NGN quality isactive. To understand this result, note that since marginalcosts are equal to zero, profits are just transfers from theconsumers to the firms. Total welfare is therefore equalto the total gross consumer utility from using the services.When the price of the low quality service increases, there isa reduction of the number of consumers of this service,whereas some consumers of the low-quality service switchto the high-quality service. The former effect reduces wel-fare, whereas the latter increases welfare. However, wefind that the latter effect is always stronger than theformer.

To conclude, both the firms and the regulator haveincentives to fully migrate to the superior NGN technology.In the next section, we study how an access obligation tothe OGN affect these incentives.

4. The equilibrium

In this section we solve for the equilibrium of the game,when the incumbent and the rival firm invest in NGN infra-structures, and can offer both OGN and NGN services in theretail market.

As we will detail below, under our assumptions, thereare two market equilibria, which correspond to the follow-ing two configurations19: (1) dE < dI < lI < lE, firm E is thehigh quality NGN provider (‘‘leapfrogging’’ case); (2)dE < dI < lE < lI , firm I is the high quality NGN (‘‘persis-tence of leadership’’ case). We start by considering the‘‘leapfrogging’’ case, and then consider the other case(‘‘persistence of leadership’’).

18 In line with these theoretical findings, if we introduce positive marginalcosts of producing high quality services, we find that it is both profit andwelfare maximizing to have both the high and the low quality networks inoperation. This extension is available upon request from the authors.

19 In Appendix A.11, we show that there are no other equilibria than theones discussed in this section.

Fig. 1. Market segmentation with leapfrogging.


4.1. ‘‘Leapfrogging’’: Firm E is the high quality NGN provider

4.1.1. Competition stageAt the competition stage (Stage 3), four different net-

work qualities are available to consumers, with the follow-ing ranking: dE < dI < lI < lE. Since consumers have a unitdemand, each consumer chooses at most one quality. Themarginal consumer of type hS who is indifferent aboutbuying the lowest (OGN) quality dE or not buying at all ischaracterized by hSdE � dE ¼ 0, and therefore, hS ¼ dE=dE.The consumer of type hL who is indifferent about buyingfirm E’s OGN service or firm I’s OGN service is given byhLdE � dE ¼ hLdI � dI , hence, hL ¼ ðdI � dEÞ=ðdI � dEÞ. Simi-larly, we can find the consumer of type hM who is indifferentabout firm I’s OGN service or firm I’s NGN service and theconsumer of type hH who is indifferent about firm I’s or firmE’s NGN services. Formally, we have hM ¼ ðmI � dIÞ=ðlI � dIÞand hH ¼ ðmE �mIÞ=ðlE � lIÞ.

The demands for firm E’s and firm I’s OGN services arethen

DOE ¼ hL � hS ¼

dI � dE

dI � dE� dE

dEand DO

I ¼ hM � hL

¼ mI � dI

lI � dI� dI � dE

dI � dE;

respectively, while the demands for the NGN services offirm E and firm I are

DNE ¼ �h� hH ¼ �h�mE �mI

lE � lIand DN

I ¼ hH � hM

¼ mE �mI

lE � lI�mI � dI

lI � dI;

respectively. Fig. 1 shows the different market segmentsthat emerge. Note that a firm’s demand for a given servicedecreases with the price it sets for this service, butincreases with the other prices.

The firms’ profits are as follows:

PI ¼ ðhH � hMÞmI þ ðhM � hLÞdI þ aðhL � hSÞ � cðlIÞ;

and

PE ¼ ð�h� hHÞmE þ ðhL � hSÞðdE � aÞ � cðlEÞ:

The two firms set their prices, fdE;mEg and fdI;mIg, tomaximize their profits. Solving for the system of first orderconditions,20 we obtain the equilibrium prices.21

20 The second order derivatives for the incumbent are @2pI=@d2I ¼ �2=

ðlI � dIÞ � 2= ðdI � dEÞ < 0 and @2pI=@m2I ¼ �2=ðlE � lIÞ � 2=ðlI � dIÞ < 0,

a n d t h e d e t e r m i n a n t o f t h e H e s s i a n m a t r i x i s4ðlE � dEÞ=ððlE � lIÞðdI � dEÞðlI � dIÞÞ > 0. Similarly, the second orderderivatives for the entrant are @2pE=@d2

E ¼ �2=ðdI � dEÞ � 2=dE < 0 and@2pE=@m2

E ¼ �2=ðlE � lIÞ < 0, and the determinant of the Hessian matrixis 4dI=ðdEðdI � dEÞðlI � lEÞÞ > 0. Therefore, the solutions to the system offirst order conditions correspond to a maximum.

21 We omit the (analytically complex) equilibrium prices to simplify theexposition.

The following general result gives the effect of the OGNaccess charge on prices, and holds for all the marketconfigurations that we consider in this paper.

Lemma 1. The equilibrium OGN and NGN prices increasewith the OGN access charge.

Proof. See Appendix A.1. h

A higher OGN access charge inflates the entrant’s per-ceived marginal cost for the OGN service, and hence, itsOGN price. When the incumbent has a service that com-petes directly with the entrant’s OGN service, a higheraccess charge also inflates firm I’s price for this service.This is because the incumbent has an opportunity costfrom a price decrease, that is, foregone wholesale revenues.Finally, since OGN and NGN prices are strategic comple-ments, an initial lift-up of the entrant’s OGN price (andpossibly of the incumbent direct rival’s service price)percolates to all OGN and NGN prices.

This result shows that a low OGN access price leads tolow NGN retail prices, which reduces NGN investmentincentives.

We now replace for the equilibrium prices into thedemand functions, and obtain the demand for OGNservices for the entrant and the incumbent, respectively,at the equilibrium of the subgame:

DOE ¼

2dI 100dEðlE � lIÞ � ð4lE � lIÞa� �

gdEand

DOI ¼

100dEðlE � lIÞ � ð4lE � lIÞag

;

where g ¼ 4lEðdI � dEÞ þ 4dIðlE � lIÞ þ 8dIðlE � dEÞ þ dE

ðlI � dIÞ > 0, as dE < dI < lI < lE. Similarly, the demandfor NGN services for the incumbent and the entrant are

DNI ¼

100lEðdI � dEÞ þ 300dIðlE � dEÞ þ 3adI

g;

and

DNE ¼

2 100lEðdI � dEÞ þ 300dIðlE � dEÞ þ 3adI� �

g;

respectively. The firms’ demands for OGN servicesdecrease with the access charge a, and there is a thresholdvalue for the access charge, �aL ¼ 100ðlE � lIÞdE=ð4lE � lIÞ,above which there is no demand for OGN services. In con-trast, the demands for NGN services increase with theaccess charge.

4.1.2. Access pricing stageMoving backwards, the access charge is set either by

the incumbent or by the regulator. Let us start with the for-mer case, where the incumbent sets the access charge tomaximize its profit.


4.1.3. Profit-maximizing access chargeAt the equilibrium of the competition stage, the incum-

bent’s profit is PI ¼ PI a; d�I ; d�E;m

�I ;m

�E

� �. From the envelope

theorem, we have @PI=@dIjdI¼d�I¼ 0 and @PI=@mIjmI¼m�I

¼ 0.Hence, using the chain rule, the effect of a higher accesscharge on firm I’s profit is given by

dPI

da¼ @PI

@a|{z}ðþÞ

þ @PI

@mE|ffl{zffl}ðþÞ

@m�E@a|ffl{zffl}ðþÞ

þ @PI

@dE|{z}ðþÞ

@d�E@a|{z}ðþÞ

: ð1Þ

The first term in (1) represents the direct effect of the accesscharge on profit, and it is always positive (see Appendix A.2where we study the sign of (1)); taking prices as given, ahigher access charge means higher wholesale profits forthe incumbent. The second and third terms represent twoindirect effects, which go through firm E’s prices for theNGN and OGN services. The first indirect effect is alwayspositive; a higher access charge inflates the entrant’s NGNprice, which leads both to higher (OGN and NGN) retailand (OGN) wholesale sales for the incumbent. The sign ofthe second indirect effect is a priori ambiguous, because ahigher OGN retail price from the entrant has conflictingeffects on the incumbent’s profit. On the one hand, it bene-fits its retail operations. On the other, it reduces theentrant’s demand for the OGN service, and therefore, itdecreases the incumbent’s wholesale revenues. However,the former effect is always stronger than the latter; theindirect effect which goes through the entrant’s OGN ser-vice is therefore positive. Since the direct and indirecteffects are positive, firm I’s profits increase with the accesscharge; hence, we can state the following Lemma.

Lemma 2. Assume that dE < dI < lI < lE (‘‘leapfrogging’’).Then, firm I sets an access charge which leads to a complete‘‘switch-off’’ of the OGN.

23 The indirect effect with respect to price dI (resp., mI) is strictly positiveif firm I’s OGN (resp., NGN) service is in direct competition with at least oneof firm E’s services. Otherwise, it is equal to zero. Since at least one of firmI’s services competes directly with firm E’s services, at least one of the twoindirect effects is strictly positive.

Proof. In Appendix A.2, we show that dPI=da > 0 for alla < �aL. Therefore, the incumbent sets a P �aL, and foreclosesthe entrant’s OGN operations. At the same time, sincea P �aL, the demand for the incumbent’s OGN services isequal to zero. Therefore, everything is as if the incumbent‘‘switched off’’ the OGN. h

Compared to the benchmark where the incumbent andthe entrant compete in OGN services only, the incumbenthas an additional incentive to foreclose the entrant’s OGNoperations. When the access charge is low, the entrantoffers OGN services at a low price. Due to the strategiccomplementarity between OGN services and NGN services,this drives down the equilibrium prices for NGN services.Therefore, the incumbent has an incentive to foreclosethe entrant’s OGN service to soften competition in NGNservices. This is why the entrant also benefits from a highaccess charge to the OGN, since its NGN service remainsactive.

This effect is reminiscent of the retail-level businessmigration effect in Bourreau et al. (2012), except that inBourreau et al. firms I and E each offer a single service,whereas in the present setting, each firm offers both OGNand NGN services. Moreover, in Bourreau et al. the accesscharge is set ex ante and affects NGN investments, whereas

here the access charge is set ex post, after investmentshave taken place; it affects not the investments but ratherthe migration to the NGN at the retail level.

Finally, at the level of the access charge that foreclosesthe entrant’s OGN service, the demand for the incumbent’sown OGN service is also equal to zero. Therefore, in the equi-librium of the access subgame, the OGN is no longer active.

4.1.4. Welfare-maximizing access chargeLet us now consider the case where the access charge is

set by a welfare-maximizing regulator. We denote by CSOi

(resp., CSNi ) the surplus of the consumers who consume

firm i’s OGN (resp., NGN) service.22 Total welfare is definedas the sum of consumers’ surplus and industry profits, thatis, W ¼ CSO

E þ CSOI þ CSN

E þ CSNI þPI þPE. We obtain the

following result.

Lemma 3. Assume that dE < dI < lI < lE (‘‘leapfrogging’’).Then, the regulator sets an access charge which leads to acomplete ‘‘switch-off’’ of the OGN.


In contrast to a standard one-way access situationwhere the regulator would set a low access price, the regu-lator sets a very high access charge to accelerate the migra-tion from the old to the new technology. To understand thepositive effect of a higher access charge on welfare, we dis-cuss below the effect on consumers and on the rival firm.

As retail prices increase with the access charge (fromLemma 1), consumers are always worse off with a higheraccess charge. The effect of a higher access charge on theentrant’s profit is given by

dPE

da¼ @PE

@a|ffl{zffl}ð�Þ

þ @PE

@dI|ffl{zffl}ðþÞ

@d�I@a|{z}ðþÞ

þ @PE

@mI|ffl{zffl}ðþÞ

@m�I@a|ffl{zffl}ðþÞ

: ð2Þ

The first term in (2) represents the direct effect of a higheraccess charge on the entrant’s profit. We have@PE=@a ¼ �DO

E , and therefore it is always negative. Takingprices as given, a higher access charge inflates the entrant’sperceived marginal cost, and thus lowers its profit. The sec-ond and third terms represent two indirect effects whichoperate via firm I’s prices. From Lemma 1, we have@d�I =@a > 0 and @m�I =@a > 0. We also have

@PE

@dI

ðM� ;D�Þ

¼ p�E � a� � @DO

E

@dIþm�E

@DNE

@dIP 0;

and

@PE

@mI

ðM� ;D�Þ

¼ p�E � a� � @DO

E

@mIþm�E

@DNE

@mIP 0:

Therefore, the two indirect effects are always positive.23 Inother words, a higher access charge increases the entrant’s

Fig. 2. Market segmentation with persistence of leadership.


profit, due to its softening effect on the incumbent’s retailprices.

Since we have a negative direct effect and two positiveindirect effects, the overall effect of an increase in theaccess charge on firm E’s profit is a priori ambiguous. Inour setting, however, we find that we always have24

dPE=da > 0.To sum up, an increase in the access price has two

opposite effects on social welfare. On the one hand, higherretail prices lead to a lower consumer surplus. On theother, both firms benefit from an increase in the accesscharge, as it softens competition for NGN services and itstimulates migration. We find that, overall, the positiveeffect of a larger access charge on producers’ surplus isgreater than the negative effect it has on consumers’ sur-plus. This result explains why the regulator finds it optimalto set a large access price to induce firms to switch off theirOGN operations (Lemma 3).25

Note that while the two firms benefit if they both aban-don the old technology (since this reduces cannibalizationwithin their product lines), each firm has no unilateralincentive to shut down its OGN service (otherwise it wouldset an infinite retail price for it, which would be tantamountto closing the service). However, the OGN switch-off can beachieved via the access price to the OGN. Since the incum-bent controls the access terms, which affect both theentrant’s OGN demand and its own OGN demand, it canshut down all OGN services, to the benefit of both firms.

4.1.5. Investment stageWe now determine the firms’ investment decisions.

Assume that dE < dI < lI < lE holds in equilibrium. Lem-mas 2 and 3 show that in both the regulated and unregu-lated cases, the access charge is set at a P �aL, whichimplies that firms I and E compete for NGN services only.We determine the equilibrium NGN investments in thisconfiguration, and show that it is an equilibrium of thegame. The following result summarizes our analysis.

Proposition 1. There is a ‘‘leapfrogging’’ equilibrium, suchthat dE < dI < l�I < l�E. In both the regulated and unregulatedcases, the firms compete in NGN services only, and the OGN isswitched off.

24 See Appendix A.5.25 Lemma 3 concerns the welfare-maximizing access price. Things may go

differently if the regulator had (i) other instruments beyond the accesscharge and/or (ii) a different objective function. For example, the regulatormay impose firms to offer services of a minimum quality in order to favoraccess and/or to protect low income users. In this case, if low income usersare those with low willingness-to-pay for access services, operators may berequired to keep offering OGN services at a regulated price. Clearly, thisregulatory intervention would favor consumers, but it would hurt opera-tors, thus distorting their incentives to invest in NGNs.


When the entrant is the leader in NGN investments, themigration from the OGN to the NGN takes place com-pletely, that is, there is full migration. The OGN is replacedby two competing NGN infrastructures, and switched off.The OGN switch-off is realized through a very high accessprice, set by either the incumbent or the regulator, whichallows the two competing firms to coordinate on shuttingdown their OGN operations. Therefore, we can state thefollowing corollary.

Corollary 1. In the ‘‘leapfrogging’’ equilibrium, there is noneed to implement a formal switch-off of the OGN.

In this equilibrium, the unregulated outcome coincideswith the social optimum; therefore, the regulator coulddecide to unregulate access to the OGN.

Finally, note that an exogenous upgrade of theincumbent’s OGN has no effect on NGN investments,since the equilibrium does not depend on the incum-bent’s OGN quality dI . Therefore, whether or not the reg-ulator should allow an exogenous upgrade of the OGN isirrelevant.

4.2. ‘‘Persistence of leadership’’: Firm I is the high qualityprovider

4.2.1. Competition stageWhen the incumbent is the leader in NGN investments,

four different network qualities are available to consumers,with the following ranking: dE < dI < lE < lI . Followingthe same procedure as in Section 4.1, the demands forthe OGN services of firm E and I can be written as

DOE ¼ hL � hS ¼

dI � dE

dI � dE� dE

dEand DO

I ¼ hM � hL

¼ mE � dI

lE � dI� dI � dE

dI � dE;

respectively, while the demands for the NGN services offirm I and firm E are

DNI ¼ �h� hH ¼ �h�mI �mE

lI � lEand DN

E ¼ hH � hM

¼ mI �mE

lI � lE�mE � dI

lE � dI;

respectively. Fig. 2 shows the market segmentation withpersistence of leadership.

Firm I’s and firm E’s profits are as follows:

PI ¼ mIDNI þ dID

OI þ aDO

E � cðlIÞ and

PE ¼ mEDNE þ ðdE � aÞDO

E � cðlEÞ:

The two firms set their prices to maximize their profits.Solving for the system of first order conditions, we obtain


the equilibrium prices.26 Replacing for the equilibriumprices into the demand functions, we find that the demandfor the entrant’s OGN service is decreasing in the accesscharge27 and equals zero if and only if a P �aP , where28

�aP ¼50dE dIðlI � lEÞ þ lEðlE � dIÞ

� �lI½dI � 4lE� þ lE½lE þ 2dI�

:

The demand for the incumbent’s OGN service at a ¼ �aP isequal to

DOI

a¼�aP

¼50lE lI � lE

� �4lIlE � lE

� �2 � lIdI � 2lEdI

;

which is strictly positive as lI > lE > dI > dE. Therefore, incontrast to the ‘‘leapfrogging’’ case, when there is ‘‘persis-tence of leadership’’ and the incumbent forecloses theentrant’s OGN operations, it does not shut down its ownOGN operations. The intuition is that the incumbent’s highquality NGN service is not in direct competition with OGNservices, and hence, the cannibalization of its NGN sales byits own OGN service is less severe than in the ‘‘leapfrog-ging’’ case.

4.2.2. Access pricingWe now move backwards and determine the optimal

access charge, which is set by either the incumbent orthe regulator.

4.2.2.1. Profit-maximizing access charge. The effect of ahigher access charge on firm I’s profit can be written as

dPI

da¼ @PI

@a|{z}ðþÞ

þ @PI

@dE|{z}ðþÞ

@d�E@a|{z}ðþÞ

þ @PI

@mE|ffl{zffl}ðþÞ


: ð3Þ

The first term in (3) is positive, as @PI=@a ¼ DOE P 0. The

second and third terms are the two indirect effects, andthey are both positive. Indeed, from Lemma 1, we have@d�E=@a > 0 and @m�E=@a > 0. We find moreover that

@PI

@dE¼ d�I � dI=dEð Þa

dI � dE:

Since d�I � dI=dEð Þ�aP ¼ 0, we have @PI=@dE > 0 for all a < �aP .Finally, we find that

@PI

@mE¼ m�I

lI � lEþ d�I

lE � dI> 0:

Therefore, we can state the following result.

26 We omit the equilibrium prices to simplify the exposition. The secondorder derivatives for the incumbent are @2PI=@d2

I ¼ �2= ðlE � dIÞ � 2=ðdI � dEÞ < 0; @2 PI=@m2

I ¼ �2=ðlI� lEÞ < 0, and the determinant of theHessian matrix is 4ðlE � dEÞ= ððlI � lEÞðlE �dIÞðdI � dEÞÞ > 0. The secondorder derivatives for the entrant are @2PE= @d2

E ¼ �2= ðdI � dEÞ�2=dE < 0; @2PE= @m2

E ¼ �2=ðlI � lEÞ � 2=ðlE � dIÞ < 0, and the determi-nant of the Hessian matrix is 4ðlI � dIÞ dI=ððlE � dIÞ ðlI � lEÞ ðdI � dEÞdEÞ > 0. Therefore, the solutions to the first order conditions correspond toa maximum.

27 See Appendix A.7.28 Note that there is no clear-cut ordering between �aL and �aP .

Lemma 4. Assume that dE < dI < lE < lI (‘‘persistence ofleadership’’). Then, firm I sets a sufficiently high access chargeso as to foreclose the entrant’s OGN service. The incumbent’sOGN service remains active in equilibrium.

Proof. From the analysis above, we have dPI=da > 0 for alla < �aP . Therefore, the incumbent sets a P �aP , and fore-closes the entrant’s OGN operations. However, as seenabove (See SubSection 4.2.1), at a ¼ �aP ; DO

I > 0, thusimplying that for a P �aP the demand for the incumbent’sOGN services is strictly positive. h

Because the competition from firm E’s OGN servicesdrives down the profitability of firm I’s operations, firm Ihas an incentive to foreclose firm E’s OGN service. How-ever, as noted above, firm I’s OGN service is not in directcompetition with its own NGN service. Therefore, firm Ihas no incentive to shut it down.

4.2.2.2. Welfare-maximizing access charge. We now con-sider the case where the access charge is set by a wel-fare-maximizing regulator. We have the following result.

Lemma 5. Assume that dE < dI < lE < lI (‘‘persistence ofleadership’’). Then, the regulator sets the access charge so asto foreclose the entrant’s OGN service. The incumbent’s OGNservice remains active in equilibrium.


We know that the incumbent always benefits from ahigher access charge and that, since the access chargeinflates retail prices (from Lemma 1), consumers arealways worse off with a higher access charge. The effectof a higher access charge on the entrant’s profit is given by

dPE

da¼ @PE

@a|ffl{zffl}ð�Þ

þ @PE

@dI|ffl{zffl}ðþÞ

@d�I@a|{z}ðþÞ

þ @PE

@mI|ffl{zffl}ðþÞ

@m�I@a|ffl{zffl}ðþÞ

: ð4Þ

The first term in (4) represents the direct effect of a highera on the entrant’s profit, which is negative. The second andthird terms represent the two indirect effects, which gothrough firm I’s prices. These terms are both positive, asfrom Lemma 1, @d�I =@a > 0 and @m�I =@a > 0, and

@PE

@dI¼ m�E

lE � dIþ d�E � a

dI � dE> 0 and

@PE

@mI¼ m�E

lI � lE> 0:

Though the effect of the access charge on the entrant’sprofit is a priori ambiguous, as in the ‘‘leapfrogging’’ case,we find that dPE=da > 0.29

All in all, the positive effect of the access charge onindustry profits dominates the negative effect on consumersurplus. The intuition is similar to the one that we pro-vided for the benchmark. When the access price increases,the price of the low quality services goes up, which reducesthe number of low-end consumers and makes some con-sumers of the low quality service switch to the high qual-ity. The latter effect increases welfare, whereas the formerdecreases welfare. However, the positive effect is alwaysthe stronger. The regulator therefore chooses to set a very

29 See Appendix A.9.

Fig. 3. Incumbent’s investment.


high access price to accelerate migration from the old tothe new technology.

4.2.3. Investment stageWe now determine the firms’ investment decisions at

the first stage of the game. Assume that dE < dI < lE < lI

holds in equilibrium. From Lemmas 4 and 5, we know thatin both the regulated and unregulated cases, the accesscharge is set at a P �aP . Therefore, when firms decide ontheir NGN investments, they anticipate competitionbetween a multiproduct incumbent (with OGN and NGNservices) and a monoproduct entrant (with NGN servicesonly). We determine the equilibrium NGN investments inthis configuration, and show that it is indeed an equilibriumof the game. The following result summarizes the analysis.

Proposition 2. There is a ‘‘persistence of leadership’’ equi-librium, such that dE < dI < l�E < l�I . In both the regulatedand unregulated cases, firm I offers OGN services of quality dI

and NGN services of quality l�I dIð Þ , whereas firm E offers onlyNGN services of quality l�E dIð Þ. We have l�I dIð Þdecreasing withdI , and l�E dIð Þincreasing with dI . The OGN is not switched off.

Fig. 4. Entrant’s investment.


When the incumbent is the leader in NGN investments,the migration from the OGN to the NGN is only partial;some consumers from the incumbent still use OGN ser-vices. However, Figs. 5 and 6 show that the incumbent’sprofit and total welfare decrease with dI . Therefore, itwould be both profit-enhancing for the incumbent andwelfare-enhancing to shut down firm I’s OGN. This canbe done if, ex-ante, the incumbent or the regulator cancommit to switching off the OGN after NGN investmentshave been made.30 We can therefore state the followingCorollary.

Corollary 2. When there is ‘‘persistence of leadership’’, aformal switch-off of the OGN is socially desirable.

In our setting, the access price to the OGN is set ex post,after investments have taken place. Proposition 2 showsthat in this case, the incumbent does not find it profitableto set an extremely high access price to switch off theOGN. However, as we argue with Corollary 2, if the incum-bent could set the access price ex ante, it would set it to avery high level to switch off the OGN, and it would coincidewith the social optimum. The difference is that in the lattercase, the incumbent can influence investments with theaccess price, whereas in the former case it cannot.

In contrast to the ‘‘leapfrogging’’ case, when there is‘‘persistence of leadership’’ in equilibrium, an exogenousupgrade of the OGN affects the equilibrium outcome, inan ambiguous way. On the one hand, it decreases theincumbent’s NGN investment, but on the other it increasesthe entrant’s investment (see Figs. 3 and 4). However, as anOGN upgrade hurts total welfare (see Fig. 6), when itexpects ‘‘persistence of leadership’’, the regulator should

30 Such a commitment could be made credible by removing technicalequipments from the copper network, or even the copper cablesthemselves.

not allow an exogenous upgrade of the OGN. We can thenstate the following result.

Corollary 3. In the ‘‘persistence of leadership’’ equilibrium,an exogenous upgrade of the incumbent’s OGN decreasessocial welfare.

Finally, in Propositions 1 and 2 we have shown that thegame has two ‘‘leapfrogging’’ and ‘‘persistence of leader-ship’’ equilibria. In Appendix A.11 we show that there areno other equilibria to this game.

5. The impact of an old technology upgrade

In our baseline model, we considered that the quality ofthe OGN service was fixed, and that therefore, the only

33 The second order derivatives for the incumbent are @2pI=2 2 2


investment decision for the incumbent was related to theNGN. If the OGN can be improved, the incumbent will facea trade-off between improving the old technology anddeveloping the new one. For example, in the telecommuni-cations industry, incumbent firms can invest in the so-called ‘‘vectoring’’ technology to upgrade the quality ofthe broadband services that use the old generation coppernetwork.31 Therefore, in a given area, an incumbent has todecide whether to upgrade its OGN with the vectoring tech-nology and/or to roll out an NGN (fiber) infrastructure. Inthis section, we extend our baseline model to allow theincumbent to upgrade the quality of its OGN service, andinvestigate how this possibility of upgrading affects thefirms’ incentives to invest in NGNs.

We assume that in the same period as the firms decideon their NGN investments (i.e., Stage 1), firm I can alsoincrease the quality of its OGN service, up to ~dI 2 dI; �d

� �.

The upper bound �d > dI is the maximum quality achievablevia the upgrade. We assume that the incumbent incurs nocost for this upgrade. We assume furthermore that theincumbent’s quality upgrade does not spill over to theentrant’s OGN service, and that the incumbent does notprovide access to its upgraded OGN, but only to the stan-dard OGN. Therefore, when the incumbent upgrades itsold generation network, dE remains unchanged.

We assume that Assumption A1 still holds, and intro-duce the following additional assumptions:

A2. �d 2 ½284:8;1250�.A3. dI > 92:35.Assumptions A1, A2 and A3 ensure that with an endog-

enous OGN upgrade, there is both a ‘‘leapfrogging’’ equilib-rium and a ‘‘persistence of leadership’’ equilibrium.Assumption A2 also implies that there is no equilibriumwhere the quality of the upgraded OGN is higher thanthe quality achievable via the best NGN technology.32 Notethat with our assumptions, the quality achievable throughthe OGN’s upgrade can be either lower or larger than thequality of the NGN, that is, we can have either ~dI > lI orthe reverse. For example, in telecoms, upgraded copperbroadband lines are capable of achieving speeds that areeither lower or higher than those available on a low-qualityNGN, depending on the length of the line.

Under our assumptions, at the competition stage themarket can be in different potential configurations. How-ever, we show in Appendix B.1 that some of these configu-rations either correspond to the outcomes already studiedin Section 4 or are not proper equilibrium outcomes. There-fore, in what follows, we focus on the ‘‘leapfrogging’’ withupgrade case where dE < lI <

~dI < lE, and the ‘‘persistenceof leadership’’ case with upgrade, where dE < lE <

~dI < lI .

5.1. Leapfrogging with OGN upgrade

In this configuration, two OGN qualities and two NGNqualities are available to consumers, with dE < lI <

31 The vectoring technology allows operators to provide Internet accessservices with a higher speed than traditional broadband DSL services.

32 We exclude this possibility because we are interested in situationswhere the new technology provides an improvement over the oldtechnology, even if the latter is upgraded.

~dI < lE. We proceed as in Section 4 to determine the mar-ginal consumers between the different available qualities,which yields the firms’ demands

DOE ¼

mI � dE

lI � dE� dE

dE; DN

I ¼dI �mI

~dI � lI

�mI � dE

lI � dE;

DOI ¼

mE � dI

lE � ~dI

� dI �mI

~dI � lI

; DNE ¼ �h�mE � dI

lE � ~dI

:

5.1.1. Competition stageThe firms’ profits, gross of investment costs, are

pI ¼ dIDOI þmID

NI þ aDO

E and pE ¼ ðdE � aÞDOE þmEDN

E :

Firms I and E choose their prices to maximize their prof-its. Solving for the system of four first order conditions,33

we find the equilibrium prices. Replacing for the equilibriumprices into the demands, we find:

DOE ¼

2lI 100dEðlE � ~dIÞ � ð4lE � ~dIÞah i

jdE;

DOI ¼

100lEðlI � dEÞ þ 300lIðlE � dEÞ þ 3alI

j;

DNE ¼

2 100lEðlI � dEÞ þ 300lIðlE � dEÞ þ 3alI

� �j

;

DNI ¼

100dEðlE � ~dIÞ � ð4lE � ~dIÞaj

;

where j ¼ 16lIlE � 4lI~dI � 8lIdE � 4lEdE þ dEð~dI � lIÞ >

0, as dE < lI <~dI < lE.

The entrant’s demand for OGN services and the incum-bent’s demand for NGN services decrease with the accesscharge a, and there is a threshold value for the accesscharge, �a0L ¼ 100ðlE � ~dIÞdE=ð4lE � ~dIÞ, above which thereis no demand for the two low quality services. On the otherhand, the demands for the entrant’s NGN service and theincumbent’s upgraded OGN service increase with theaccess charge.

5.1.2. Access pricingMoving backwards, the access charge is set either by

the incumbent or by the regulator.

5.1.2.1. Profit-maximizing access charge. With an analysissimilar to the one in Section 4, we obtain the followingresult.

Lemma 6. Assume that dE < lI <~dI < lE (‘‘leapfrogging

with upgrade’’). Then, firm I sets the access charge so as toforeclose the entrant’s OGN service, which also shuts down itsown NGN operations.

Proof. See Appendix B.2. h

@dI ¼ �2=ðlE � dIÞ � 2=ðdI � lIÞ < 0 a n d @ pI=@mI ¼ �2=ðdI � lIÞ�2=ðlI � dEÞ < 0, and the determinant of the Hessian matrix is4ðlE � dEÞ=ððlI � dEÞðdI � lIÞðlE � dIÞÞ > 0. Similarly, the second orderderivatives for the entrant are @2pE=@d2

E ¼ �2=ðlI � dEÞ � 2=dE < 0 and@2pE=@m2

E ¼ �2=ðlE � dIÞ < 0, and the determinant of the Hessian matrix is4lI=ðdEðlI � dEÞðlE � dIÞÞ > 0. Therefore, the solutions to the first orderconditions correspond to a maximum.

Fig. 5. Incumbent’s and entrant’s profit.


This result resembles Lemma 2 in the ‘‘leapfrogging’’case without upgrade, except that firm I here shuts downits NGN operations rather than its OGN operations. Firm Ihas an incentive to shut down its low quality service (i.e.,the NGN service), as it is in direct competition with its highquality service (i.e., the upgraded OGN service). It also hasan incentive to foreclose the entrant’s low-quality OGN ser-vice to soften price competition for high-quality services.

5.1.2.2. Welfare-maximizing access charge. We now con-sider the case where the access charge is set by the regula-tor.34 We have the following result.

Lemma 7. Assume that dE < lI <~dI < lE (‘‘leapfrogging

with upgrade’’). Then, the regulator sets the access charge soas to foreclose the entrant’s OGN service and the incumbent’sNGN service.


Like the incumbent, the regulator is willing to have onlythe two highest qualities active in equilibrium. It thereforesets an access charge which leads to a zero demand for thetwo low quality services. We also find that the entrant ben-efits from a higher OGN access charge, i.e., dPE=da > 0.35

5.1.3. NGN investments and OGN upgradeFinally, we solve for the investment decisions at stage 1

of the game. Assume that dE < lI <~dI < lE holds in equi-

librium. The firms anticipate that once investments havebeen realized, in the continuation game, there will be nodemand for firm E’s OGN service and firm I’s NGN service.Therefore, at stage 1, everything is as if firm I and firm Ewere deciding on ~dI and lE , respectively. We determinethe optimal investments in this case, and check that thiscandidate equilibrium is indeed a Nash equilibrium ofthe game. The following result summarizes the analysis.

Proposition 3. There is a ‘‘leapfrogging with upgrade’’

equilibrium, such that dE < l�I < ~d�I < l�E. In equilibrium, firmI upgrades its OGN at the maximum achievable quality

~d�I ¼ �d �

, but does not invest in an NGN l�I ¼ 0� �

, while firm

E invests in an NGN of quality l�E �d� �

. We have ~d�I and l�Eð�dÞincreasing with �d.


In the ‘‘leapfrogging with upgrade’’ equilibrium, the oldand new technologies coexist. Therefore, in contrast to theleapfrogging case without upgrade, the migration from theold to the new technology is only partial. This is becausethe incumbent finds it less expensive to upgrade its OGNthan to invest in an NGN.

An interesting question is how welfare compares in the‘‘leapfrogging’’ and ‘‘leapfrogging with upgrade’’ equilibria.That is, provided that the entrant is the leader in NGNinvestments, should the regulator allow or forbid OGNupgrade? We obtain the following result.

34 See Appendix B.3 for the expressions of consumer surplus and welfarein the leapfrogging and persistence of leadership cases with upgrade.

35 See Appendix B.5.

Proposition 4. If �d < 413:25, welfare is higher under the‘‘leapfrogging’’ equilibrium than under the ‘‘leapfrogging withupgrade’’ equilibrium. Otherwise, the reverse is true.

We prove this result graphically. Fig. 7 plots the welfarelevels with and without OGN upgrade as a function of thequality of the upgraded OGN, �d. Without upgrade, the OGNis switched off and the equilibrium is independent of theOGN quality, while with an OGN upgrade, welfare increaseswith �d. The Proposition then follows from the plot.

This result suggests that if the quality upgrade is lim-ited, the regulator should forbid an OGN upgrade, becauseit would result in a partial migration to the NGN and lowerquality levels. Note that this is true even though theupgrade is costless in our setting. Otherwise, if the maxi-mum quality achievable under the OGN quality is suffi-ciently high, the regulator should allow the upgrade.

5.2. Persistence of leadership with OGN upgrade

In this configuration, two OGN qualities and two NGNqualities are available to consumers, with dE < lE <~dI < lI . As in the previous case, we determine the marginalconsumers between the different options, and find thefirms’ demands,

DOE ¼

mE � dE

lE � dE� dE

dE; DN

E ¼dI �mE

~dI � lE

�mE � dE

lE � dE;

DOI ¼

mI � dI

lI � ~dI

� dI �mE

~dI � lE

; DNI ¼ �h�mI � dI

lI � ~dI

:

5.2.1. Competition stageThe firms’ profits are pI ¼ dID

OI þmID

NI þ aDO

E andpE ¼ ðdE � aÞDO

E þmEDNE . Solving for the four first order con-

ditions,36 we find the demands in the price equilibrium. The

36 As in the previous cases, we have verified that the Hessian matrix isnegative definite, which ensures that the system of first order conditionsidentifies a maximum.

Fig. 9. I deviation in the leapfrogging with upgrade case.

Fig. 7. Welfare with leapfrogging.

Fig. 6. Total welfare. Fig. 8. E deviation in the leapfrogging with upgrade case.


entrant’s demand for OGN services in this price equilibriumis DO

E ¼ �alE=ð2dE lE � dE� �

Þ. As DOE 6 0 for any a P 0, there

is no interior equilibrium to the competition stage subgamewhere the entrant’s demand for OGN services is strictly posi-tive. This is because, when the entrant prices its OGN service,it seeks to minimize the cannibalization with its own NGNservice. Since the entrant’s OGN service cannot be active inequilibrium, we compute the equilibrium when the entrantoffers only an NGN service, while the incumbent offers bothOGN and NGN services. In equilibrium, the demands for thefirms’ services are then:

DOE ¼ 0; DN

I ¼ 50;DOI ¼

50lE

4~dI � lE

; and DNE ¼

100dI

4~dI � lE

:

5.2.2. Access pricingSince equilibrium prices and demands at the competi-

tion stage do not depend on the access charge, the choiceof the access price is irrelevant.

5.2.3. NGN investments and OGN upgradeAssume that dE < lE <

~dI < lI holds in equilibrium. Atstage 1, firm I and firm E decide on ~dI;lI and lE,respectively, and they anticipate that once investmentshave been realized, in the continuation game there willbe no demand for firm E’s OGN services. We determine

Fig. 10. Welfare with persistence of leadership.


the firms’ optimal investment in this case, and check thatthe outcome corresponds to an equilibrium. We can thenstate the following result.

Proposition 5. There is a ‘‘persistence of leadership with

upgrade’’ equilibrium, such that dE < l�E < ~d�I < l�I . Firm Iupgrades its OGN at the maximum achievable quality

~d�I ¼ �d �

, and invests in an NGN of quality l�I ¼ 2500. Firm

E sells only NGN services of quality l�E �d� �

.


This result shows that when it is the leader in NGNinvestments and can upgrade its OGN, the incumbent fullyupgrades the OGN (i.e., to the maximum achievable qual-ity), and also invests in an NGN of very high quality.37

The entrant invests in an NGN, but becomes the providerwith the lowest quality of service.

This is in contrast with the ‘‘persistence of leadership’’equilibrium without upgrade, where the entrant was offer-ing the second highest quality. Therefore, the possibility ofupgrading allows the incumbent to restrict its competitorto the lowest market segment.

We now study whether the regulator should forbid theOGN upgrade, by comparing the welfare under the ‘‘persis-tence of leadership’’ equilibrium and the ‘‘persistence ofleadership with upgrade’’ equilibrium.

Proposition 6. The social welfare is always lower under the‘‘persistence of leadership with upgrade’’ equilibrium thanunder the ‘‘persistence of leadership’’ equilibrium.


37 The incumbent’s NGN investment level actually corresponds to themonopoly level.

This result suggests that an OGN upgrade leads to aninefficient market outcome. Forbidding the incumbent toupgrade its OGN would therefore be welfare-enhancing.

5.2.3.1. Risk of foreclosure. The fact that the entrant is rel-egated to the lowest segment of the market, with onlyone active variety, suggests that there might be a risk offoreclosure. Assume that there is a fixed cost for theentrant of building an NGN of strictly positive quality(e.g., due to duct trenching).38 Instead of upgrading itsOGN to the maximum quality, the incumbent could ratherset its OGN quality ~dI at a sufficiently low level, close tothe entrant’s NGN quality lE , to deter the entrant frominvesting in an NGN. Due to the intense competition fromthe incumbent’s OGN, the entrant might not generateenough profits to cover its fixed cost.

In such a case, the entrant would then decide not toinvest in an NGN, and to offer only OGN services. However,if the entrant offers only OGN services, the incumbent hasan incentive to set the access charge to the OGN at a pro-hibitive level to foreclose completely its competitor–as inour benchmark of Section 2.

To sum up, if the incumbent is allowed to upgrade itsOGN infrastructure and is the leader in NGN invest-ments, a risk of foreclosure might emerge, which callsfor a regulation of the access to the OGN. Note that thisis in contrast with the other cases, where the incumbentalways sets the OGN access charge at the socially opti-mal level.

5.3. Multiplicity of equilibria

As we have shown, our game has multiple equilibria,which is not surprising in a game with vertical differentia-tion. When the quality of the OGN is exogenous, there aretwo equilibria that we called the leapfrogging and persis-tence of leadership equilibria. In the former, the entrantsupplies the highest quality of NGN services, whereas inthe latter it is the incumbent. A leapfrogging with upgradeand a persistence of leadership equilibrium with upgradeemerge when the incumbent is allowed to upgrade itsOGN. In addition to these two equilibria, the persistenceof leadership outcome without upgrade, where firm I doesnot upgrade its OGN and leads in NGN investments, canalso be an equilibrium if the maximum achievable OGNquality is not too large.39

We do not aim here at refining the equilibrium conceptto select a single equilibrium. Nonetheless, we can see thatfirms have clear preferences towards the various equilib-ria. As a general rule, we find that the high quality NGNprovider obtains larger profits than the low quality NGNprovider. Therefore, firm E always prefers leapfrogging topersistence, and vice versa for firm I. Furthermore, whenthe OGN can be upgraded, firm I prefers to upgrade itsOGN rather than not to upgrade it, that is, it prefers thepersistence of leadership equilibrium with upgrade to thesame equilibrium without upgrade.

38 The same argument would hold if we assumed a minimal quality forthe NGN.

39 See Appendix B.1 for details.


6. Conclusion

In this paper we have analyzed the migration from an oldto a new technology at the retail level, when an incumbentand an entrant can use both technologies, and the entrantleases access to the incumbent’s old generation network.

When the quality of the old generation network is exog-enous, two equilibria emerge: a persistence-of-leadershipequilibrium, where the incumbent remains the leader withthe new technology, and a leapfrogging equilibrium, wherethe entrant becomes the new leader. In the leapfroggingequilibrium, the migration to the new technology is com-plete, and the old network is switched off, via a very highaccess price set either by the incumbent or the regulator.By contrast, in the persistence-of-leadership equilibrium,the old network remains active. As it is an inefficient out-come, a formal intervention from the regulator to switchoff the old network would be desirable. In this equilibrium,an exogenous upgrade of the old generation network isalso welfare-decreasing.

We then extended our setting to allow the incumbentto upgrade the old network, at the same time as the twofirms invest in the new networks. Allowing for upgradingyields two additional potential equilibria: a persistence-of-leadership equilibrium with upgrade, and a leapfroggingequilibrium with upgrade.

In the former equilibrium, the incumbent does notinvest in a next generation network, and rather upgradesthe old network. If the quality achievable with the upgradeis not too high, welfare is lower than in the leapfroggingequilibrium without upgrade. In such a case, forbiddingthe incumbent from upgrading its network would be wel-fare-enhancing. In the latter equilibrium, the incumbentupgrades its old network to the maximum achievable qual-ity, and also invests in a new network of very high quality.We showed that this market outcome leads to a lower wel-fare than the persistence-of-leadership equilibrium with-out upgrade. We also argued that because the entrant isrelegated to the lowest market segment, a risk of foreclo-sure might arise. This suggests that the regulator shouldnot allow the OGN upgrade.

From a policy perspective, our results suggest that theaccess price to the old network might not be enough toachieve efficient outcomes, for the migration from an oldtechnology to a new one. Additional instruments, such alegal switch-off of the old network or forbidding upgradesto the old network, might be necessary for the regulator toorient the market towards efficient outcomes.

Our analysis has some limitations though. In particular,we assumed that the demand for NGN services is certainand known by the firms. In reality, operators take theirinvestment decisions under still strong uncertainty aboutthe future demand for NGN services. Introducing this impor-tant issue in our framework may affect our conclusions. Weleave to future research the analysis of this relevant issue.

Acknowledgements

We thank Martin Peitz, Hinnerk Gnutzmann, Henri deBelsunce, Dominique Torre, and Stefano Schiavo for usefulremarks. We also thank seminar participants at the

University of Nice Sophia Antipolis and the audiences atthe FSR Communications & Media Markets Conference(Florence, 2013), at the 4th Workshop on the Economicsof ICT (Evora, 2013), and at the 2013 annual meetings ofthe European Association of Industrial Economists (Evora),of the Società Italiana degli Economisti (Bologna) and ofthe Association of Southern European Economic Theorists(Bilbao). We also thank the Editor and two anonymous ref-erees for their valuable comments and suggestions.

Appendix A

A.1. Proof of Lemma 1

Proof. We start by showing that the access charge has apositive direct effect on the entrant’s OGN price. Theentrant always offers the lowest quality OGN service, dE.Let us denote by r (resp., s) the quality (resp., price) of theclosest rival product, where r > dE. The entrant’s best-response for the OGN service, dBR

E , then satisfies the FOC

@pE

@dE¼ 0 ¼ s� dBR

E

r� dE� dBR

E

dE�

r dBRE � a

�r� dEð ÞdE

: ð5Þ

From the implicit function theorem, since the SOC holds,

we have dBRE ¼ dBR

E a; s;r; dEð Þ , and the sign of @dBRE =@a is

the same as the sign of the derivative of (5) with respectto a, that is, s=ððs� dEÞdEÞ, which is strictly positive. There-

fore, @dBRE =@a > 0.

Besides, the access price can have a direct effect on theprice of one of the incumbent’s services, because it entersthe incumbent’s profits via the wholesale revenuesaðhL � hSÞ. However, it is the case only if the incumbentcommercializes the service which is contiguous to firm E’sOGN service. Let us consider that it is the case, that is, firm Ioffers the second lowest quality, which we denote by rI fora price of sI . The FOC of profit maximization with respect tosI can be written as

@pI

@sI¼ a

rI � dEþ . . .½ � ¼ 0; ð6Þ

where the term into brackets is independent of a. Let usdefine by sBR

I the solution of the FOC. From the implicitfunction theorem, since the SOC holds, the sign of @sBR

I =@ais the same as the sign of the derivative of (6) with respectto a, that is, 1=ðrI � dEÞ, which is strictly positive.Therefore, @sBR

I =@a > 0.We now prove that the prices for the network services

are strategic complements. From (5), using the implicitfunction theorem, we have @dBR

E =@s > 0. For the otherprices, we begin by considering firm I’s services. Firm I’sprofit can be written as

pI ¼ dIDOI dI;~rO

I

� �þ aDO

E dE;~rOE

� �þmID

NI mI;~rN

I

� �;

where ~rsi represents the vector of prices for the services

which compete with firm i’s technology s ¼ O;N. We have

@pI

@dI¼ DO

I dI;~rOI

� �þ dI

@DOI

@dIþ a

@DOE

@dIþmI

@DNI

@dI:

From the implicit function theorem, the effect of a given

price p on dBRI is given by the sign of @2pI=@dI@p. Since


demands are linear, @Dsi =@ps0

j is a constant, where

i; j ¼ I; E; s; s0 ¼ O;N, and ps0j designates the price of firm

j’s technology s0 ¼ O;N. Therefore, we have

@2pI

@dI@p¼ @DO

I

@p:

Besides, remark that @Dsi @ps0

j

.> 0 if s – s0. That is, the

demand for a firm’s service increases in the prices of therival services (controlled by the same firm or the rival firm).It follows that @2pI=@dI@p > 0, which implies that dI and pare strategic complements. Using the same methodology,we prove that all prices are strategic complements.

To sum up, when the access charge a increases, thispushes up dBR

E and possibly dBRI , due to the direct effects.

Then, since at least one price is directly increasing, due tothe strategic complementarities, all prices increase. h

;

40 Due to its algebraic complexity, we omit WðaÞ here.

A.2. access charge and firm I’s profits

The first term in (1) represents the direct effect of ahigher access charge on the profits of firm I. This term isalways positive as @PI=@a ¼ DO

E P 0. The second and thirdterms represent two indirect effects, which go through firmE’s prices for the NGN and OGN services, respectively. FromLemma 1, we know that @m�E

�@a > 0 and that @d�E

�@a > 0.

Moreover we have

@PI

@mE

ðM� ;D�Þ

¼ d�I@DO

I

@mEþm�I

@DNI

@mEþ a

@DOE

@mE;

where M� ¼ m�I ;m�E

� �and D� ¼ d�I ; d

�E

� �. As a given service’s

demand is increasing in the rival services’ prices, we have@PI=@mEj M� ;D�ð Þ > 0. This proves that the first indirect effectis always positive. Finally, we have

@PI

@dE

ðM� ;D�Þ

¼ d�I@DO

I

@dE|fflfflffl{zfflfflffl}ðþÞ

þm�I@DN

I

@dE|fflfflfflffl{zfflfflfflffl}ðþÞ

þ a@DO

E

@dE|fflffl{zfflffl}ð�Þ

: ð7Þ

Replacing for equilibrium prices into (7), we find that

@PI

@dE

ðM� ;D�Þ

¼4dI 100dE lE � lI

� �� a 4lE � lI

� �� gdE

P 0;

which proves that the indirect effect which goes throughthe entrant OGN price is always positive. This proves thatdPI=da > 0 for all a < �aL

A.3. Consumer surplus and total welfare

Total welfare is defined as the sum of consumer surplusand firms’ profits, that is, W ¼ CSO

E þ CSOI þ CSN

E þ CSNI þ

PI þPE, where CSsi denotes the surplus of the consumers

of firm i’s service that relies on technology s ¼ O;N. Below,we provide the expressions for consumer surplus in the twopossible cases.

A.3.1. ‘‘Leapfrogging’’ case

CSOE ¼

Z hL

hS

ðxdE � dEÞdx ¼ dEdI � dIdEð Þ2

2dE dI � dEð Þ2;

CSOI ¼

Z hM

hL

ðxdI � dIÞdx

¼ dI

2dI �mIð Þ2

dI � lI

� �2 �dI � dEð Þ2

dI � dEð Þ2

!� dI �mI

dI � lI� dI � dE

dI � dE

�dI;

CSNI ¼

Z hH

hM

ðxlI �mIÞdx ¼ lI

2mI �mEð Þ2

lI � lE

� �2 �dI �mIð Þ2

dI � lI

� �2

!

� mI �mE

lI � lE� dI �mI

dI � lI

�mI;

CSNE ¼

Z �h

hH

ðxlE �mEÞdx

¼ lE

2�h2 � mI �mEð Þ2

lI � lE

� �2

!� �h�mI �mE

lI � lE

�mE:

A.3.2. ‘‘Persistence of leadership’’ case

CSOE ¼

Z hL

hS

ðxdE � dEÞdx ¼ dEdI � dIdEð Þ2

2dE dI � dEð Þ2;

CSOI ¼

Z hM

hL

ðxdI � dIÞdx

¼ dI

2dI �mEð Þ2

dI � lE

� �2 �dI � dEð Þ2

dI � dEð Þ2

!� dI �mE

dI � lE� dI � dE

dI � dE

�dI;

CSNE ¼

Z hH

hM

ðxlE�mEÞdx

¼lE

2mE�mIð Þ2

lE�lI

� �2 �dI�mEð Þ2

dI�lE

� �2

!� mE�mI

lE�lI�dI�mE

dI�lE

�mE

CSNI ¼

Z �h

hH

ðxlI �mIÞdx

¼ lI

2�h2 � mE �mIð Þ2

lE � lI

� �2

!� �h�mE �mI

lE � lI

�mI:


Proof. Replacing for the equilibrium prices into thewelfare function, we derive welfare as a function of theaccess charge, a. This function WðaÞ is a second degreepolynomial in a, which is therefore either strictly concaveor strictly convex.40 To begin with, assume that WðaÞ isstrictly concave. To prove that the regulator sets an accesscharge above �aL, that is the level which drives the demandfor OGN services down to zero, it is sufficient to check thatthe slope of the welfare function is strictly positive at a ¼ �aL.We have

dWðaÞda

a¼�aL

¼600lI lE � lI

� �dI

4lE � lI

� �g

> 0;


as dI > dE and lE > lI . Now, assume that WðaÞ is strictlyconvex. We always have dW=daja¼0 > 0. Indeed,

dWðaÞda

a¼0

¼100 lI�lE

� ��11lI dI�dEð Þ�9dIlI�16dIlE�20dElEþ45dEdI� �

g2 ;

which is always positive for dE < dI < lI < lE. Therefore,WðaÞ is strictly increasing over 0; �aL½ �, which proves thatthe regulator sets a P �aL. Therefore, in equilibrium, theregulator ‘‘switches off’’ the OGN. h

A.5. dPE=da > 0 (‘‘leapfrogging’’ case)

A sufficient condition for dPE=da > 0 to hold is

@PE

@aþ @PE

@mI

@mI

@a> 0:

Since the LHS of this inequality is linear in a, in order for theinequality to be satisfied, it suffices for the expression on theLHS to have a positive slope and a positive intercept. We findthat the intercept is equal to 200ðlE � lIÞdIð24dIðlE � dEÞþ

�3dIðlI � dIÞ þ lIðdI � dEÞ þ 8lEðdI � dEÞÞÞ=g2, while the slopeis 2dI 16dI l2

E � dElI

� �þ 7dElIðlE � dIÞ þ 4dIlIðlI � dEÞþ

��32

dIlEðlE �lIÞ þ 16l2EðdI � dEÞ þlIdEðlE �lIÞÞÞ= dEg2

� �. Both

expressions are positive for dE < dI < lI < lE.

A.6. Proof of Proposition 1

Proof. Firm I’s and firm E’s profits are PI ¼ ðhH � hLÞmI�l2

I =2 and PE ¼ ð100� hHÞmE � l2E=2, where hH ¼

ðmE �mIÞ=ðlE � lIÞ, and hL ¼ mI=lI . Replacing for theequilibrium prices, the firms’ profits can be written as

PIðlI;lEÞ ¼ 10;000lE � lI

� �lIlE

4lE � lI

� �2 �lI

2

2; and

PEðlI;lEÞ ¼ 40;000lE � lI

� �l2

E

4lE � lI

� �2 �lE

2

2:

Firms choose their quality levels to maximize theirprofits. Solving for the system of first order conditions,we find that l�I ¼ 482:3 and l�E ¼ 2533:1.41 The firms’ equi-librium profits are P�I ¼ 152741:2 and P�E ¼ 2443858:8.

In order to prove that this is a Nash equilibrium, weneed to verify that both firms do not have incentive tounilaterally deviate from l�I ;l�E

� �. The entrant does not

have incentive to deviate: it has only a possible deviation,that is, to choose lE < l�I , and it is simple to check that it isnot profitable to become the low quality operator. Considernow the incumbent. Given l�E and also given the quality ofits OGN, here exogenously given, the incumbent mightdeviate by setting lI > l�E. In this case, the ranking of theoffered services would be dI < l�E < lI . This corresponds tothe ‘‘persistence of leadership’’ scenario analyzed in Sec-tion 4.2. From Lemma 4, we know that the high qualityincumbent does not switch off the OGN at the retail level.Given dI and lE, firm I’s profits from deviation are:

41 The second order conditions are @2PI @l2I

�¼ �2=ðlE � lIÞ � 2=lI < 0

and @2PE @l2E

�¼ �2=ðlE � lIÞ < 0, which clearly identify a maximum.

Al4I þ Bl3

I þ Cl2I � DlI þ E

2 2dIlE � 4lIlE þ l2E þ dIlI

� �2 ;

where A ¼ ðdI � 4lEÞ2; B ¼ 2ðð2lE � 2500Þ dI þ ð10;000þ

lEÞlEÞðdI � 4lEÞ; C ¼ lE ðlE � 5000Þ4d2I þ 75;000dI þ l2

E

� ��lE þ ðdI þ 20;000Þ4l2

EÞ; D ¼ 5000dIl2Eð22lE þ 5dIÞ and E ¼

5000dIl3Eð10dI � lEÞ. It is possible to check that for

lE ¼ l�E, these profits are always negative. Therefore, theincumbent does not have incentive to unilaterally deviatefrom l�I ;l�E

� �. h

A.7. DOE is decreasing with a (‘‘persistence of leadership’’)

We find that

@DOE

@a¼

2 2dIlE þ dIlI � 4lIlE þ l2E

� �dI

HðdEÞdE;

where HðdEÞ ¼ 2dIlE � 8dIlI þ 9d2I

�þl2

E � 4lIlEÞdE � 8d2I

lE � 4dIl2E þ 16dIlIlE � 4d2

I lE. The numerator of @DOE=@a

is negative, as dE < dI < lE < lI . The sign of the denomina-tor is given by the sign of HðdEÞ, which is linear in dE. Theslope of HðdEÞ is equal to 2dIlE � 8dIlI þ 9d2

I þ l2E � 4lElI ,

which is negative (using that dE < dI < lE < lI). Further-more, HðdEÞjdE¼dI

¼ 3dIðdI � lEÞð3dI � 4lI þ lEÞ > 0 for lI >

lE > dI . This proves that HðdEÞ is positive and that DOE

decreases with a.


Proof. Replacing for the equilibrium prices into thewelfare function, we derive welfare as a function of theaccess charge, a. This function WðaÞ is a second degreepolynomial in a, which is therefore either strictly concaveor strictly convex.42 To begin with, assume that WðaÞ isstrictly concave. To prove that the regulator sets an accesscharge above �aP , it is sufficient to check that the slope of thewelfare function is strictly positive at a ¼ �aP . We have

dWðaÞda

a¼�aP

¼150dIðlI�lEÞðdI�lEÞðlIdI�6dIlEþl2Eþ4lIlEÞ

GðdEÞðlIdIþl2Eþ2dIlE�4lIlEÞ

;

where GðdEÞ ¼ l2E þ 9d2

I � 8lIdI þ 2dIlE � 4lIlE

� �dE � 4dI

l2E � 4lEd

2I � 8d2

I lE þ 16lEdIlE.As dE < dI < lE < lI , the numerator of dWðaÞ=daja¼�aP

isnegative. The denominator is negative if GðdEÞ > 0. GðdEÞ islinear and decreasing in dE, since G0ðdEÞ ¼lE þ 9dI � 8lIdI þ 2dIlE � 4lElI < 0. On top of that, wehave GðdEÞjdE¼dI

¼ dIðdI � lEÞð3dI þ lE � 4lIÞ > 0, whichproves that dWðaÞ=daja¼�aP

> 0.

Now, assume that WðaÞ is strictly convex. We alwayshave dW=daja¼0 > 0. Indeed,

dWðaÞda

a¼0¼ �400ðlI � lEÞðdI � lEÞdI JðlIÞ

ðGðdEÞÞ2;

where JðlIÞ ¼ ð8dIlE þ dI þ dElE � 10dIdEÞlI � l2EdE � 10d2

I

lE þ 9d2I dE þ dEdIlE þ dIl2

E . The denominator of dWðaÞ=daja¼0 is strictly positive. As dE < dI < lE < lI , the

42 Due to its algebraic complexity, we omit to present the formalexpression of WðaÞ.


numerator is positive if JðlIÞ > 0. Since J0ðlIÞ ¼ 8dIlEþd2

I þ dElE � 10dIdE > 0, and JðlIÞjlI¼lE¼ 9dIðlI � dIÞ

ðlI � dEÞ > 0, we have JðlIÞ > 0 for all lI P lE. This provesthat dWðaÞ=daja¼0 > 0. Therefore, in equilibrium, theregulator switches off the OGN of the entrant firm. h

43 Assumption A1 guarantees that the optimal deviation lDI ðdIÞ is lower

than dI .

A.9. dPE=da > 0 (‘‘persistence of leadership’’ case)

We find that PEðaÞ is a second degree polynomial in a,with

d2PEðaÞda2 ¼ K

2dEM2 ;

where KðdEÞ ¼ �16dIl4Eþ

��64d2

I þ 128dIlI

� �l3

E þ 80d2I lI�

�256dIl2

I þ 80d3I Þ l2

E þ �64d3I lI þ 272d2

I l2I � 144d4

I

� �lE�

160d3I l2

I þ 144d4I lIÞdE þ 16d2

I l4E þ 64d3

I � 128d2I lI

� �l3

E þ 64ðd4

I � 224d3I lI þ 256d2

I l2I Þl2

E þ 64d4I lI � 128d3

I l2I

� �lE þ 16

d4I l2

I , and M ¼ �4dElIlE þ 9dEd2I þ dE l2

E þ 2dEdIlE � 4d2I lI�

8d2I lE þ 16dIlIlE � 4dIl2

E � 8dEdIlI .The denominator of d2PEðaÞ=da2 is positive, hence the

sign of the second order derivative is the same as the signof KðdEÞ. KðdEÞ is a linear function in dE, and it takes positivevalues at the extremes (namely, at dE ¼ 0 and dE ¼ dI).Hence, we have KðdEÞ > 0 for any admissible value of thequality parameters. This implies that PEðaÞ is a convexfunction. To prove that PEðaÞ increases with a, it is there-fore enough to show that its derivative is positive ata ¼ 0. We find that

dPEðaÞda

a¼0¼ 400ðlI � lEÞðlE � dIÞdI

M2 LðdEÞ;

where LðdEÞ ¼ 9d2I � 10dIlI þ dIlE þ lIlE � l2

E

� �dE þ d2

I lI�10d2

I lE þ 8dIlIlE þ dIl2E . LðdEÞ is linear in dE and at the

extremes it takes positive values. Hence, LðdEÞ is positive,and dPEðaÞ=daja¼0 > 0. This proves that dPE=da > 0 for all a.

A.10. Proof of Proposition 2

Proof. Firm I’s and firm E’s profits are PI ¼ ð100� hHÞmI þ ðhM � hLÞdI � l2

I =2 and PE ¼ ðhH � hMÞmE � l2E=2,

where hH ¼ ðmI �mEÞ=ðlI � lEÞ; hM ¼ ðmE � dIÞ=ðlE � dIÞ,and hL ¼ dI=dI . Replacing for the equilibrium prices, thefirms’ profits can be written as

PIðlI;lEÞ ¼ 2500dIlI � 4lIlE þ 3dIlE

� �2 lI � lE

� �2dIlE � 4lIlE þ l2

E þ dIlI

� �2

�dI dI � lE

� �lI � lE

� �2lE

2dIlE � 4lIlE þ l2E þ dIlI

� �2

!� lI

2

2; ð8Þ

and

PEðlI;lEÞ¼10;000lE

2 dI�lE

� �dI�lI

� �lI�lE

� �2dIlE�4lIlEþl2

EþdIlI

� �2 �lE

2

2: ð9Þ

Given dI , firms choose their NGN investments to maxi-mize profits. The system of first order conditions cannotbe solved analytically. Therefore, we revert to numericalsimulations to obtain the optimal functions l�I ðdIÞ and

l�EðdIÞ shown in Figs. 3 and 4. We use these equilibriumlevels of investment in NGN quality to derive theequilibrium profits and the social welfare, for a given dI

(see Figs. 5 and 6).In order to prove that this is a Nash equilibrium, we

need to verify that both firms do not have incentive tounilaterally deviate from ðl�I ðdIÞ;l�EðdIÞÞ. The entrant maydeviate by choosing lE > l�I ðdIÞ, that is, by playing a‘‘leapfrogging’’ game. In this case, the entrant’s profitfunction, given lI ¼ l�I ðdIÞ, is

40;000 lE � l�I ðdIÞ� �

lE

4lE � l�I ðdIÞ� �2 � l2

E

2:

It is possible to check that for any admissible value of dI

and lE, this expression is always negative. Hence, theentrant does not have incentive to unilaterally deviatefrom the candidate equilibrium.

Let us now consider the incumbent. Its possible devi-ation is to offer an NGN of a quality lower than l�EðdIÞ. Thegame would then become a ‘‘leapfrogging’’ game, wherethe incumbent switches off its OGN (see Lemma 2). Giventhat lE ¼ l�EðdIÞ, the profit function of the incumbent inthis case is

10;000 l�EðdIÞ � lI

� �lIl�EðdIÞ

4l�EðdIÞ � lI

� �2 � l2I

2:

The first order condition can be solved only through anumerical simulation. We obtain the optimal deviationlD

I ðdIÞ43 and the profits enjoyed by the incumbent fromdeviating. We then check that these profits are alwayslower than the profits at the candidate equilibriumP�I ðdIÞ. We can therefore conclude that the incumbent doesnot have incentive to unilaterally deviate froml�I ðdIÞ;l�EðdIÞ� �

. h

A.11. Alternative equilibria

We verify that there are no other equilibria than theones highlighted in Propositions 1 and 2. We consider thefollowing alternative market configurations: (1) dE < lI <

dI < lE; (2) dE < lE < dI < lI; (3) dE < lE < lI < dI; (4)dE < lI < lE < dI . Note that configuration (1) is identicalto configuration (L1) in the second part of the paper, config-uration (2) to (L2) and so on. The main difference is thathere the quality of the incumbent’s OGN is exogenous anddoes not exceed 228.29 (Assumption A1).

Configuration (1) : dE < lI < dI < lE.This scenario is similar to scenario (L1). From SubSec-

tion 5.1, we know that when dE < lI < dI < lE, at theequilibrium of the access price subgame, firm I doesnot offer NGN services, while firm E does not supply OGNservices. Firm I and firm E’s first-period profits are thereforeidentical to expressions (10) and (11) in Appendix B.6, withdI instead of ~dI . At the candidate equilibrium, firm I does notinvest in NGN, while l�EðdIÞ is given in expression (12);firms’ profits are the dotted lines in Figs. 8 and 9 (note: boththese pictures must be considered for the relevant part,


where dI < 228:29). We find that configuration (1)cannot be an equilibrium, as firm I finds it alwaysoptimal to deviate and to invest in an NGN of a qualitylarger than dI .

Configuration (2) : dE < lE < dI < lI .This scenario is similar to scenario (L2). From SubSec-

tion 5.2, we know that when dE < lE < dI < lI , at the equi-librium of the access price subgame, firm E does not supplyOGN services. Firm I’s optimal NGN investment is thenl�I ¼ 2500. We find that configuration (2) cannot be anequilibrium: firm E finds it optimal to deviate and to investin an NGN of a quality larger than dI . By deviating, firm Eprefers to play the game where dI < lE < 2500 that is stra-tegically identical to the one studied in Section 4.2. Firm E’sprofit in this game is similar to expression (9); a simple dif-ferentiation is enough to prove that firm E obtains higherprofits in case of deviation than at the candidateequilibrium.

Configurations (3) and (4) : dE < lE < lI < dI anddE < lI < lE < dI .

These scenarios are similar to scenarios (L3) and (L4) inSection 5 (see Appendix B.1). Following similar argumentsas above, we find that these two configurations cannot beNash equilibria since, under Assumption A1, both firmsmay profitably deviate by investing in NGN services of aquality higher than dI .

Finally, it is useful to stress that market configurationswhere li < dE cannot be Nash equilibria since investingin NGN services of a quality higher than dE would alwaysrepresent a profitable deviation.

Appendix B

B.1. Potential market configurations with endogenousupgrade

Under our assumptions and according to the firms’ NGNinvestments and the incumbent’s OGN quality upgrade, atthe competition stage, the market can be in one of thefollowing six potential configurations44:

44

li

(S1): dE < ~dI < lI < lE,

For the same reasons as in Section 4< dE . See Appendix A.11.

(S2): dE < ~dI < lE < lI ,
(L1): dE < lI <
~dI < lE,
(L2): dE < lE <~dI < lI ,
(L3): dE < lE < lI <~dI ,
(L4): dE < lI < lE <
~dI .

Cases (S1) and (S2) correspond to situations wherethe OGN is not upgraded in equilibrium (i.e., ~dI ¼ dI), orthe incumbent chooses a very small quality improvementfor its OGN service (i.e., ~dI < min lI;lE

� �). These two

cases are equivalent to the ‘‘leapfrogging’’ and ‘‘persis-tence of leadership’’ cases, respectively, that we studiedin Section 4. We show below that under AssumptionA2, only the equilibrium found in (S2) also persists whenfirm I is allowed to upgrade the OGN provided that �d, themaximum quality achievable with the OGN, is not toolarge.

Cases (L1) to (L4) correspond to situations where thequality of the upgraded OGN service is higher than the

, there is no equilibrium where

quality of at least some of the NGN services. However, aswe show below, (L3) and (L4) cannot be equilibrium out-comes. In both cases, firm I sets ~d�I ¼ �d: given AssumptionA2, firm E then finds profitable to deviate by setting anNGN quality larger than ~d�I .

B.1.1. Configurations (S1) and (S2) with endogenous upgradeLet us start with configuration (S1), characterized by

the following ranking dE < ~dI < lI < lE. The candidateequilibrium is l�I ¼ 482:3 and l�E ¼ 2533:1. In order to ver-ify if this is also a Nash equilibrium when firm I is allowedto upgrade its OGN, we need to consider the firms’ profitsin case of unilateral deviations.

Given l�E ¼ 2533:1, firm I may deviate as follows: iÞ byinvesting in a high quality NGN that is by playing a type(S2) game, ~dI < l�E < lI; iiÞ by upgrading the OGN, namelyplaying a type (L2) or (L3) game, that is l�E < ~dI < lI orl�E < lI <

~dI respectively, and iiiÞ by playing a type (L1)game, lI <

~dI < l�E.We find that deviation (i) is never profitable for firm I

due to the high investment cost; deviations (ii) areimpossible provided that �d � 1250. As deviation (iii) is con-cerned, we know that firm I uses prices to shut down itsNGN services (they are contiguous to OGN); hence thisdeviation collapses to ~dI < l�E. We find that this deviationis profitable for �d > 258:4; since, �d 2 ½284:8;1250�(Assumption A2), it follows that firm I finds always optimalto deviate and (S1) cannot be an equilibrium configurationwith upgrade.

Let us now move to configuration (S2), withdE < ~dI < lE < lI . We know that in this market configura-tion, firm I invests in NGN and it does not switch off theOGN. Firm I’s first period profit function is similar toexpression (8), with ~dI instead of dI . From the derivativeof this profit function with respect to ~dI , we immediatelysee that as dE < ~dI < lE < lI; @PIð~dI;lI;lEÞ=@~dI < 0, hencefirm I sets ~dI ¼ dI , that is, it does not upgrade its OGN.

Therefore the candidate equilibrium in the (S2) gamewith upgrade is exactly the same as without upgrade;hence, using Proposition 2, we know that firm I offersOGN services of quality dI and NGN services of qualityl�I dIð Þ , whereas firm E offers only NGN services of qualityl�E dIð Þ, with l�I dIð Þ and l�E dIð Þ as those given in Figs. 3and 4.

We check if this is an equilibrium, starting from firm I.Its possible deviations are: (i) to become the low qualityprovider, that is, to offer very low quality NGN servicesby playing either game (S1), with dI < lI < l�E dIð Þ or thegame with configuration lI < dI < l�E dIð Þ; (ii) to upgradethe OGN by playing a type (L2) game, withl�E dIð Þ < ~dI < lI; (iii) to play a type (L3) game, withl�E dIð Þ < lI <

~dI or, finally, (iv) to play a type (L4) gameby upgrading the OGN and by reducing the NGN invest-ment at the same time, lI < l�E dIð Þ < ~dI .

Deviations (i) are not profitable; as always, it is notdesirable to become the low quality provider. On top ofthat, they are not admissible for dI < 228:29. Considernow deviation (ii). In this case, given l�EðdIÞ, firm I upgradesits OGN to ~dI 6

�d. Firm I first period profits in case of devi-ation are:


PIðdI; ~dI;lIÞ ¼ 25004lI

~dI � 3~dIl�EðdIÞ � lIl�EðdIÞ �

4~dI � l�EðdIÞ �

þ 10;000l�EðdIÞ ~dIl�EðdIÞ

�~dI

4~dI � l�EðdIÞ �2 �

l2I

2

From the first order conditions, we find that firm I deviatesby upgrading the OGN at the maximum level, ~dI ¼ �d and byinvesting lI ¼ 2500. Plugging these values into the aboveprofit function we obtain the profits from deviation, as afunction of dI and �dI , firm I’s initial OGN quality, and theOGN maximum quality. We find that deviation (ii) is prof-itable for �d > �dLðdIÞ, where �dLðdIÞ is the function that implic-itly equates the profits that firm I obtains at the candidateequilibrium with those at deviation, formally �dLðdIÞ ¼0:058ðdIÞ2þ7:61dIþ1264:21

16:64þ0:02dI. �dLðdIÞ is increasing in dI and as

dI < 228:2, a sufficient condition for firm I to deviate is�d > 785:66.

Finally, it is possible to show that as �d 6 1250,deviations (iii) and (iv) are never profitable.

B.1.2. Configurations (L3) and (L4) are not Nash equilibriaThe quality ranking in configuration (L3) is dE < lI <

lE <~dI . As in the other cases, we find that firm I (or the

regulator) considers it optimal to set the access chargesuch that firm E does not supply OGN services. As a result,the market configuration collapses to lI < lE <

~dI . Follow-ing the by now standard procedure, we can determinesecond period prices and then, back to the investmentstage, the first period profit functions:

PIð~dI;lI;lEÞ ¼ 2500ðlI

~dI � 4~dIlE þ 3lIlEÞ2ð~dI � lEÞ

ð2lIlE þ lI~dI � 4~dIlE þ l2

EÞ2

� 2500lIð~dI � lEÞ

2ðlI � lEÞlE

2lIlE þ lI~dI � 4~dIlE þ l2

E

�2 �lI

2

2

PEð~dI;lI;lEÞ ¼10;000l2

EðlE � lIÞð~dI � lIÞð~dI � lEÞ

2lIlE þ lI~dI � 4~dIlE þ l2

E

�2 � l2E

2

From firm I’s first order condition, we find that @PI=@lI < 0and @PI=@~dI > 0; hence l�I ¼ 0 and ~dI ¼ �d. Plugging thesevalues into firm E’s first order condition, we find the opti-mal investment for firm E, l�Eð�dÞ. Once the investment lev-els have been derived, we compute firms profits at thecandidate equilibrium.

We find that this is not a Nash equilibrium. In fact, as�d ¼ 1250, firm E finds it optimal to deviate by increasingits NGN investment. More specifically, firm E optimallydeviates by investing lE >

�d, that is by playing game (L1).Let us now consider configuration (L4). In this case,

dE < lE < lI <~dI . As always, the model reduces to

lE < lI <~dI as firm I (or the regulator) sets the access

charge to shut down firm E’s OGN operations. Solving thesecond period stage and then back to the investment stage,firms’ profit functions are:

PIð~dI;lI;lEÞ ¼ 2500ð4lI

~dI � 3lElI � lE~dIÞ

4lI � lEþ 10;000

� ðlI � lEÞlElI

ð4lI � lEÞ2 �

l2I

2

PEðlI;lEÞ ¼ 10;000ðlI � lEÞlElI

ð4lI � lEÞ2 �

l2E

2

where PE does not depend on ~dI since firm E’s NGN ser-vices are not contiguous to firm I’s upgraded OGN. FirmI’s first order condition with respect to ~dI is@PI=@~dI ¼ 2500 > 0; hence, ~d�I ¼ �d. Solving the system offirst order conditions, we find that firm I and firm E NGNinvestment levels are, respectively, l�I ¼ 280:12 andlE ¼ 136:4. We also find in this case that this is not a Nashequilibrium; as for scenario (L3), firm E finds it optimal todeviate by investing lE >

~d�I .

B.2. Proof of Lemma 6

Using the envelope theorem, the effect of a higheraccess charge on firm I’s profit is given by

dPI

da¼ @PI

@a|{z}ðþÞ

þ @PI

@dE|{z}ðþÞ

@d�E@a|{z}ðþÞ

þ @PI

@mE|ffl{zffl}ðþÞ


:

The first term represents the direct effect of a higher accesscharge on profit, which is positive. The second and thirdterms are the two indirect effects. From Lemma 1, we have@d�E=@a > 0 and @m�E=@a > 0. Additionally, we find that

@PI

@dE

ðM� ;D�Þ

¼ 4lI

dE

100dE lE � ~dI

�� ð4lE � ~dIÞa

j

is positive for a 6 �a0L. Finally, we find that@PI=@mEjðM� ;D�Þ ¼ d�I =ðlE � ~dIÞ > 0. Therefore, the two indi-rect effects are positive. It follows that dPI=da > 0 for alla < �a0L. Therefore, the incumbent sets a P �a0L, and foreclosesthe entrant’s OGN operations. And, for a P �a0L, the demandfor the incumbent’s NGN service is equal to 0.

B.3. Consumer surplus and welfare with upgrade

B.3.1. Leapfrogging with upgrade

CSOE ¼

Z hL

hS

ðxdE � dEÞdx ¼dElI � dEmI� �2

2dE lI � dE� �2 ;

CSOI ¼

Z hH

hM

ðx~dI � dIÞdx ¼~dI

2dI �mEð Þ2

~dI � lE

�2 �dI �mIð Þ2

~dI � lI

�2

0B@

1CA

� dI �mE

~dI � lE

� dI �mI

~dI � lI

!dI;

CSNE ¼

Z �h

hH

ðxlE�mEÞdx¼lE

2�h2� dI�mEð Þ2

~dI�lE

�2

0B@

1CA� �h�dI�mE

~dI�lE

!mE;


CSNI ¼

Z hM

hL

ðxlI �mIÞdx ¼ lI

2mI � dIð Þ2

lI � ~dI

�2 �mI � dEð Þ2

lI � dE� �2

0B@

1CA

� dI �mI

~dI � lI

� dE �mI

dE � lI

!mI:

B.3.2. ‘‘Persistence of leadership with upgrade’’

CSOE ¼

Z hL

hS

ðxdE � dEÞdx ¼dElE � dEmE� �2

2dE lE � dE� �2 ;

CSIO ¼

Z hH

hM

ðx~dI � dIÞdx ¼~dI

2dI �mIð Þ2

~dI � lI

�2 �dI �mEð Þ2

~dI � lE

�2

0B@

1CA

� dI �mI

~dI � lI

� dI �mE

~dI � lE

!dI;

CSNI ¼

Z �h

hH

ðxlI �mIÞdx

¼ lI

2�h2 � dI �mIð Þ2

~dI � lI

�2

0B@

1CA� �h� dI �mI

~dI � lI

!mI;

CSNE ¼

Z hM

hL

ðxlE �mEÞdx ¼ lE

2mE � dIð Þ2

lE � ~dI

�2 �mE � dEð Þ2

lE � dE� �2

0B@

1CA

� dI �mE

~dI � lE

� dE �mE

dE � lE

!mE:

B.4. Proof of Lemma 7

The proof is similar to the one for Lemma 3. Replacingfor the equilibrium prices into the welfare function, wederive welfare as a function of the access charge. This func-tion WðaÞ is a second degree polynomial in a, which istherefore either strictly concave or strictly convex.45 Tobegin with, assume that WðaÞ is strictly concave. To provethat the regulator sets an access charge above �a0L, it is suffi-cient to check that the slope of the welfare function isstrictly positive at a ¼ �a0L. We have

dWðaÞda

a¼�a0L

¼600lI

~dI � lE

�~dI

~dI � 4lE

�j

;

where j ¼ 16lIlE � 4lI~dI � 8lIdE � 4lEdE þ dEð~dI � lIÞ. As

dE < lI <~dI < lE, the above expression is positive. Now,

assume that WðaÞ is strictly convex. We always havedW=daja¼0 > 0. Indeed,

dWðaÞda

a¼0¼

100 ~dI�lE

��11~dI lI�dE

� ��9lI

~dI�16lIlE�20dElEþ45dElI

h ij2 ;

45 Due to its algebraic complexity, we omit to present the formalexpression of WðaÞ.

which is always positive as dE < lI <~dI < lE. Therefore,

WðaÞ is strictly increasing over 0; �a0L� �

, which proves thatthe regulator sets a P �a0L. Therefore, in equilibrium, theregulator ‘‘switches off’’ the OGN.

B.5. dPE=da > 0 (‘‘leapfrogging’’ case with upgrade)

We proceed exactly as in Appendix A.5. A sufficient con-dition for dPE=da > 0 to hold is

@PE

@aþ @PE

@~dI

@~dI

@a> 0:

The LHS of this inequality is linear in a; in order for theinequality to be satisfied, it suffices for this function tohave a positive slope and a positive intercept. We find that

the intercept is equal to 200ðlE � ~dIÞlIð24lIðlE � dEÞþ

3lIð~dI � lIÞ þ ~dIðlI � dEÞ þ 8lEðlI � dEÞÞÞ=j2, while the

slope is 2lIð16lIðl2E � dE

~dIÞ þ 7dE~dIðlE � lIÞ þ 4lI

~dIð~dI�

dEÞ þ 32lIlEðlE � ~dIÞ þ :þ16l2

EðlI � dEÞ þ ~dIdEðlE � ~dIÞÞ�=ðdEj2Þ. Both expres-

sions are positive for dE < lI <~dI < lE.

B.6. Proof of Proposition 3 (‘‘leapfrogging’’ case with upgrade)

Assuming that dE < lI <~dI < lE holds in equilibrium, at

the access pricing stage, the access charge is set at such alevel that only the incumbent’s upgraded OGN and theentrant’s NGN are active. Replacing for the equilibriumprices into the profit functions, firm I’s and firm E’s first-stage profits can be written as

PIð~dI;lI;lEÞ ¼ 10;000lE � ~dI

�~dIlE

4lE � ~dI

�2 �lI

2

2; ð10Þ

PEð~dI;lI;lEÞ ¼ 40;000lE � ~dI

�l2

E

4lE � ~dI

�2 �lE

2

2: ð11Þ

A visual inspection of expression PIð~dI;lI;lEÞ revealsthat it is optimal for firm I not to invest in an NGN, thatis, l�I ¼ 0. Therefore, everything is as if firm I decided on~dI only and firm E on lE only. Note that the incumbent isconstrained by the maximum level of quality for theOGN, �d. The unconstrained solutions of the system of thefirst-order conditions @PI=@~dI ¼ 0 and @PE=@lE ¼ 0 are~dI ¼ 5000=3 and lE ¼ 8750=3. As �d < 5000=3 fromAssumption A2, in equilibrium the constraint ~dI 6

�d isbinding, and therefore, ~d�I ¼ �d. Firm E’s optimal NGNinvestment can then be derived from the first order condi-tion, evaluated at ~d�I ¼ �d,46

@PE

@lE

~dI¼�d

¼ 40;000l2E

4lE � �d� �2 �

80;000 �d� lE

� �lE

4lE � �d� �2

�320;000 �d� lE

� �l2

E

4lE � �d� �3 � lE ¼ 0:

46 The second order condition is @2PE=@ðlEÞ2j~dI¼�d ¼ �80;000�dð5lE þ �dÞ=

ð4lE � �dÞ4 � 1 < 0, which clearly identifies a maximum.

:

49 The second order derivatives for the incumbent are @2PI= @~d2I ¼

�80;000ðlEÞ2ð5~dIþ lEÞ= ð4~dI � lEÞ

4< 0 and @2PI=@l2

I ¼ �1 < 0, and the

determinant of the Hessian matrix is 80;000ðlEÞ2ð5~dI þ lEÞ= ð4~dI � lEÞ

4> 0.

2


Solving this expression for lE, we find that

l�E �d� �¼ H

6� 1250

H�d� 10;000

3

�þ 2500

3þ

�d4; ð12Þ

where47

H¼5

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3410�d2�30025�dþ109þ30�d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð5210623�10526�dþ243�d2Þ

q3

r

We find that l�E �d� �

> 2533 and dl�E �d� �

=d�d > 0: in orderto differentiate its services from those offered by theincumbent, firm E invests in a very high quality NGN andthe quality increases the higher the quality of firm I’supgraded OGN.

We now have to check that this configuration corre-sponds to a Nash equilibrium, namely that both the incum-bent and the entrant do not have incentives to deviateunilaterally from l�I ; ~d�I ;l�E

�. Consider the entrant first.

Firm E’s possible deviation is to invest in an NGN of qualitylower than ~d�I and eventually also to offer OGN services. Inline with our previous findings, we find that independentlyof the access charge, firm E is not willing to offer OGN ser-vices, since they would cannibalize its NGN services.Therefore, firm E’s optimal deviation is to offer NGN ser-vices only of a quality lower than ~d�I . We find that firmE’s first period profit function from this deviation is:

PdEðlEÞ ¼ 10;000

lE~d�I ~d�I � lE

�4~d�I � lE

�2 �ðlEÞ

2

2:

This is a concave function. Solving for the first order condi-tion we find firm E’s optimal NGN investment in case ofdeviation, and hence, its profits. Obviously, the profits thatfirm E obtains at the candidate equilibrium and those incase of deviation both depend on ~d�I ¼ �d. In Fig. 8, we plotthe two profit functions, and immediately see that for�d < 1250, firm E does not find it optimal to deviate.

Now, consider firm I’s deviations. Given l�E, firm I hastwo alternatives: either to invest in an NGN of a qualityhigher than l�E or to invest in an NGN of a quality lowerthan l�E. In the former case, the only possible service qual-

ity ranking is ~dI < l�E < lI .48 We already know that in this

case firm I does not switch off its OGN (Lemma 4) and thatits profit function is the same as in expression (8). We findthat, due to the high costs of NGN investments, the amountof profits firm I obtains from deviating are always negative.In case of a deviation characterized by an NGN service ofquality lower than l�E, the only possible ranking is~d�I < lI < l�E. From our previous analysis, we know that inthis case firm I would like to switch off its OGN, hence thebest deviation is to offer only NGN services of a lower qual-ity than l�E. Firm I’s first period profit function is then

PdI ðlIÞ ¼ 10;000lIl�Eðl�E � lIÞ= 4l�E � lI

� �2 �

� ðlIÞ2=2, This

function is concave in lI . Solving for the first order

47 A graphical representation of l�E �d� �

in the persistence of leadershipwith upgrade case is available upon request from the authors.

48 Note that due to Assumption A2, upgraded OGN services cannot be of aquality higher than l�E .

condition, we obtain firm I’s optimal NGN investment incase of deviation and the associated profits. As before, theprofits at the candidate equilibrium and those in case ofdeviation both depend on ~d�I ¼ �d. In Fig. 9 we plot the twoprofit functions, and immediately see that when �d is suffi-ciently large, firm I gets lower profits from deviation. For-mally, we find that for �d > 257:98 firm I does not deviate.This completes the proof.

B.7. Proof of Proposition 5 (‘‘persistence of leadership’’ casewith upgrade)

Assume that dE 6 lE 6~dI 6 lI holds in equilibrium.

Under this assumption, at the investment stage, firm I setsthe quality for its OGN and NGN services, while firm E setsthe quality of its NGN service. The three first-order deriva-tives of profit maximization are49

@PI

@lI¼ 2500� lI;

@PI

@~dI

¼2500ðlEÞ

2 20~dI þ lE

�4~dI � lE

�3 ;

and

@PE

@lE¼ðlEÞ

2ð6~dI�lEÞ2�4ð~dIÞ

2ð17;500�3lEþ16~dIÞlEþ40;000ð~dIÞ3

4~dI�lE

�3 :

Note that @PI=@~dI > 0 as dE 6 lE 6~dI 6 lI; hence ~d�I ¼ �d.

From @PI=@lI ¼ 0 it follows immediately that l�I ¼ 2500.Finally, we solve numerically the first order condition@PE=@lE ¼ 0 and we obtain the solution l�Eð�dÞ.

50

In order to check if this is a Nash equilibrium, we need todetermine firms’ profits with unilateral deviations. Considerfirm E first; the entrant has two possible deviations: (i) toinvest lE 2 ð�d;2500Þ and (ii) to invest lE > 2500. Deviation(ii) is never profitable due to the high cost of the NGN invest-ment that yields to negative profits. With deviation (i) firm Eplays a game similar to game (S2) (persistence of leadershipwith no upgrade). From Appendix A.10 we take the first per-iod profit function for the entrant. Plugging firm I’s invest-ments l�I ¼ 2500 and ~d�I ¼ �d into this function, we obtainfirm E profit function from deviation. Maximizing this func-tion we derive the optimal profits E can obtain by deviating.It is possible to show that for sufficiently high levels of �d (i.e.,�d > 284:8), firm E does not find it optimal to deviate.

Let us now consider firm I. Firm I’s first period profitfunction is:

P�I ðlI;lE;~dIÞ ¼

2500ðlIð4~dI � lEÞ � 3~dIlEÞ4~dI � lE

þ 10;000ð~dI � lEÞlE~dI

ð4~dI � lEÞ2 � ðlIÞ

2

2: ð13Þ

The second order condition for the entrant is @2PE= @l2E ¼ � 20; 000ð~dIÞ

ð8~dI þ 7lEÞ þ 256ð~dIÞ3 ð~dI � lEÞþ 16~dIðlEÞ

2ð6~dI� lEÞþ ðlEÞ4Þ=ð4~dI � lEÞ

4<

0. Therefore, the first order conditions correspond to a maximum.50 A graphical representation of the solution l�Eð�dÞ is available upon

request from the authors.


The candidate equilibrium is l�E ¼ l�Eð�dÞ, ~d�I ¼ �d andl�I ¼ 2500; replacing these investments levels in expres-sion (13) we obtain firm I’s level of profits at the candidateequilibrium.

Firm I’s deviation is to supply OGN services of a qualitylower than l�Eð�dÞ and, at the same time, to select a differentlevel of NGN investment. Therefore, as for firm E, deviationimplies playing a game of type (S2) with ~dI < l�E < lI .

51

Firm I’s first period profit function in case of deviation isgiven in expression (8). From our previous analysis, we knowthat these profits decrease with the quality of firm I’s OGNservices. Hence, the optimal deviation is not to upgrade theOGN at all and to set ~d�I ¼ dI . Using this fact and pluggingE’s NGN investments at the candidate equilibrium, l�Eð�dÞ, inexpression (8) and solving for the optimal lI in case of devi-ation, it is possible to show that a sufficient condition for firmI not to have an incentive to deviate is dI > 92:35.

B.8. Proof of Proposition 6

In Fig. 10 we plot the welfare levels with and withoutOGN upgrade as a function of the quality of the upgradedOGN, �d. Without an upgrade, firm I provides both OGNand NGN services and their quality depends on dI; there-fore, the welfare also depends on dI . In the figure, we plotthe welfare levels at the two extremes values, dI ¼ 92:35and dI ¼ 228:29; when dI 2 ð92:35;228:29Þ the welfaretakes intermediate values. With OGN upgrade, welfareincreases with �d. From the plot, we immediately see thatallowing for an upgrade never increases welfare.

References

Avenali, A., Matteucci, G., Reverberi, P., 2010. Dynamic access pricing andinvestment in alternative infrastructures. Int. J. Ind. Org. 28 (2),167–175.

Bourreau, M., Dogan, P., 2005. Unbundling the local loop. Eur. Econ. Rev.49, 173–199.

51 We do not consider deviation ~dI < lI < l�E: this would imply that firm Ibecomes the low quality provider and it cannot be profitable.

Bourreau, M., Dogan, P., 2006. Build-or-buy strategies in the local loop.Am. Econ. Rev. 96, 72–76.

Bourreau, M., Cambini, C., Dogan, P., 2012. Access pricing, competition,and incentives to migrate from ‘‘Old’’ to ‘‘New’’ technology. Int. J. Ind.Org. 30 (6), 713–723.

Bourreau, M., Cambini, C., Dogan, P., 2014. Access regulation and thetransition from copper to fiber networks in telecoms. J. Regul. Econ.45 (3), 233–258.

Brito, D., Pereira, P., Vareda, J., 2010. Can two-part tariffs promote efficientinvestment on next generation networks? Int. J. Ind. Org. 28, 323–333.

Brito, D., Pereira, P., Vareda, J., 2012. Incentives to invest and togive access to new technologies. Inform. Econ. Policy 24 (3–4), 197–211.

Cambini, C., Jiang, Y., 2009. Broadband investment and regulation. Aliterature review. Telecommun. Policy 33, 559–574.

Champsaur, P., Rochet, J.-C., 1989. Multiproducts duopolists.Econometrica 57 (3), 533–557.

Chisholm, D.C., Norman, G., 2012. Market access and competition inproduct lines. Int. J. Ind. Org. 30 (5), 429–435.

Czernich, N., Falck, O., Kretschmer, T., Woessmann, L., 2011. Broadbandinfrastructure and economic growth. Econ. J. 121, 505–532.

de Bjil, P., Peitz, M., 2009. Access regulation and the adoption of VoIP. J.Regul. Econ. 35, 111–134.

De Fraja, G., 1996. Product line competition in vertically differentiatedmarkets. Int. J. Ind. Org. 14 (3), 389–414.

Foros, O., 2004. Strategic investments with spillovers, vertical integrationand foreclosure in the broadband access market. Int. J. Ind. Org. 22 (1),1–24.

Gal-Or, 1983. Quality and quantity competition. Bell J. Econ. 14 (2), 590–600.

Gans, J., 2001. Regulating private infrastructure investment: optimalpricing for access to essential facilities. J. Regul. Econ. 20 (2), 167–189.

Hori, K., Mizuno, K., 2006. Access pricing and investment withstochastically growing demand. Int. J. Ind. Org. 24, 705–808.

Johnson, J.P., Myatt, D.P., 2003. Multiproduct quality competition:fighting brands and product line pruning. Am. Econ. Rev. 93 (3),748–774.

Johnson, J.P., Myatt, D.P., 2006. Multiproduct Cournot oligopoly. RAND J.Econ. 37 (3), 583–601.

Klumpp, T., Su, X., 2010. Open access and dynamic efficiency. Am. Econ. J.:Microecon. 2, 64–96.

Kotakorpi, K., 2006. Access price regulation, investment and entry intelecommunications. Int. J. Ind. Org. 24 (5), 1013–1020.

Vareda, J., Hoernig, S., 2010. Racing for investment under mandatoryaccess. BE J. Econ. Anal. Policy 10 (1) (Article 67).

Vogelsang, I., 2013. The Endgame of Telecommunications Policy? ASurvey. CESifo Working Paper Series No. 4545.

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old technology upgrades, innovation, and competition in vertically differentiated markets

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