Strategic investments under competition for access provision

Telecommunications Policy xxx (2017) 1–18

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Strategic investments under competition for access provision☆ Noriaki Matsushima a, Keizo Mizuno b, * a b

Institute of Social and Economic Research, Osaka University, Japan School of Business Administration, Kwansei Gakuin University, Japan

A R T I C L E I N F O

A B S T R A C T

JEL classification: L43 L51 L96

We examine competition for access provision when symmetric vertically integrated firms invest in infrastructure upgrades. Spillovers through access have two effects (a wholesale-profit effect and a retail-production effect) on infrastructure investment made by vertically integrated firms. When the vertically integrated firms freely set access charges, due to the dominance of the wholesale-profit effect, quality differentials endogenously occur between these firms (asymmetric equilibria). When access charges are regulated, symmetric equilibria occur with multiple equilibrium investments due to the retail-production effect. Because competition for access provision induces a strong incentive for infrastructure investment, it also achieves a higher social welfare than does access regulation.

Keywords: Access provision Infrastructure upgrades Spillovers

1. Introduction In this paper, we examine wholesale competition in industries where vertically integrated firms (or facility-based firms) and unintegrated downstream firms (or service-based firms) coexist. Wholesale competition in industries with two-tier structures abounds in the business world. A typical example is found in the telecommunications industry. In broadband markets, facility-based firms, such as regional telephone companies or cable TV companies, have their own infrastructures to provide Internet services to customers, while service-based firms, such as independent Internet service providers, need to borrow infrastructure to offer the services to their customers. In the retail market (i.e., the Internet market), there exists competition that includes not only facility-based firms but also service-based firms. In mobile communications markets, mobile virtual network operators (MVNO, service-based firms) need to purchase wholesale mobile services from mobile network operators (MNO, facility-based firms) to offer mobile services to their end users. All these firms, including MVNOs and MNOs, compete with each other in the retail market.1 In addition, further investments for 5G network deployment are also required in mobile markets. There are two approaches in the existing studies that examine two-tier competition with several vertically integrated firms (facilitybased firms) and unintegrated downstream firms (service-based firms). The first approach focuses on vertically integrated firms' incentive to grant access to their infrastructures to unintegrated firms. Ordover and Shaffer (2007) find that unintegrated firms are likely to obtain access when inputs are homogeneous (i.e., their products are not horizontally differentiated). H€ offler and Schmidt (2008) ask ☆ We would like to thank three anonymous referees for their valuable comments and suggestions. We also thank Patrick Rey, Wilfried Sand-Zantman, Yosuke Yasuda, and the seminar participants at Osaka University, University of Hyogo, the Contract Theory Workshop, the 5th Workshop on the Economics of ICTs (Porto), and EARIE (Milan) for beneficial discussions. Financial support of JSPS KAKENHI #JP15H03349 and #24530285 is gratefully acknowledged. * Corresponding author. School of Business Administration, Kwansei Gakuin University, 1-1-155 Uegahara, Nishinomiya, Hyogo, 662-8501, Japan. E-mail address: [email protected] (K. Mizuno). 1 The licensing of intellectual property provides another example of competition within two-tier structures. In licensing markets, firms that own their IP-protected technologies can be considered vertically integrated firms (facility-based firms). They decide whether to license their technologies to potential rival firms, and if they license their technologies, they compete with potential rivals (service-based firms) in product markets.

http://dx.doi.org/10.1016/j.telpol.2017.09.005 Received 17 March 2017; Received in revised form 22 August 2017; Accepted 7 September 2017 Available online xxxx 0308-5961/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Matsushima, N., & Mizuno, K., Strategic investments under competition for access provision, Telecommunications Policy (2017), http://dx.doi.org/10.1016/j.telpol.2017.09.005

N. Matsushima, K. Mizuno

Telecommunications Policy xxx (2017) 1–18

whether granting access to unintegrated downstream firms always enhances social welfare. They find that if final products supplied in a downstream market are horizontally product-differentiated, resale (i.e., granting access to unintegrated downstream firms) can be harmful to consumers. Brito and Pereira (2009, 2010) examine the endogenous determination of horizontal product differentiation and show the possibility of high retail prices and the realization of asymmetric product differentiation in the sense that both an entrant and an access provider prefer a closer substitute of products between them to a substitute of products between the entrant and other vertically integrated firms. The second approach examines the (non-)necessity of regulatory intervention under the wholesale competition among vertically integrated firms. Bourreau, Hombert, Pouyet, and Schutz (2011) find that competition between vertically integrated firms can induce the input to be priced above marginal costs even when the firms offer a homogeneous input. They discuss the impacts of several regulatory tools such as a wholesale price cap and vertical separation. Bourreau, Cambini, and Hoernig (2015) compare the effect of uniform access pricing on investment in rolling out infrastructure with that of deregulating access price in a geographical setting. This paper proposes a different model that relates to the second approach by shedding light on the following three aspects. First, in our model, two vertically integrated firms have an opportunity to invest in the upgrades of inputs, called infrastructure upgrades. Second, through the infrastructure upgrades, the quality of services supplied downstream is endogenously determined, which means that the relative magnitude of vertical product differentiation among the services provided by all firms is endogenously determined. Third, we introduce various degrees of spillovers generated from infrastructure upgrades through access, and the benefits of the upgrades to an unintegrated downstream firm depends on the degree of spillovers through access.2 In an open access environment, the degree of spillovers through access is interpreted as the relative inferiority or superiority of an unintegrated firm's retail production technology compared with that of a vertically integrated firm that it accesses. Featuring these three aspects gives new insight on wholesale competition for access provision. In fact, when vertically integrated firms compete for access provision in the wholesale market and have an opportunity to invest in infrastructure upgrades, the likelihood that a vertically integrated firm is accessed by an unintegrated downstream firm depends on the investment level of its rival vertically integrated firm. This is because an unintegrated downstream firm wants to access a firm that generates the greatest benefits generated from infrastructure upgrades among vertically integrated firms. Then, because of the existence of access profits, the profit obtained by a vertically integrated firm can change discontinuously according to the investment level of its rival. This in turn implies that the investment of the vertically integrated firm cannot change smoothly according to the investment level of its rival (i.e., the non-smoothness of a vertically integrated firm's reaction function). We show that the non-smoothness of a vertically integrated firm's reaction function emerges from the two opposite effects generated from spillovers of infrastructure upgrades when the firm is accessed by an unintegrated downstream firm. The first effect is the wholesaleprofit effect, and the second is the retail-production effect. The wholesale-profit effect means that one vertically integrated firm can obtain access profit in the wholesale market by giving an unintegrated downstream firm a more favorable term than its rival on the benefit of spillovers. This wholesale-profit effect has a positive impact on a vertically integrated firm's incentive to invest in infrastructure upgrades. The retail-production effect means that an unintegrated downstream firm can obtain the benefit of enhancing its retail production without investment, which has a negative impact on the incentive to invest in infrastructure upgrades. The relative magnitudes of these two effects determine whether a vertically integrated firm is willing to invest more when it is accessed than when it is not accessed. In particular, in the free competition regime in which access charges are not regulated, each of the vertically integrated firms has a stronger incentive to invest in infrastructure upgrades when it is accessed than when it is not accessed: the wholesale-profit effect dominates the retail-production effect when a vertically integrated firm is accessed by an unintegrated downstream firm. In this case, an analytical result similar to that in a preemption game is derived. That is, asymmetric equilibria occur, where a vertically integrated firm that is accessed by an unintegrated downstream firm invests more than does a vertically integrated firm that is not accessed. This means that vertical quality differentials occur in equilibrium between vertically integrated firms that have the same technologies ex ante. In contrast, in the access regulation regime with regulatory non-commitment, an access charge is set equal to the access cost (a costbased access charge) by a regulator. In that case, the wholesale-profit effect does not appear, whereas the retail-production effect remains. Then, a vertically integrated firm has a stronger incentive to invest in infrastructure upgrades when it is not accessed by an unintegrated downstream firm than when it is accessed. In that case, each vertically integrated firm is unwilling to invest in infrastructure upgrades or to be accessed by an unintegrated downstream firm. As a result, symmetric investments emerge in equilibrium in the access regulation regime. Furthermore, the difference in the relative magnitudes of the wholesale-profit effect and the retail-production effect in the two regimes generates the difference in total investment levels in equilibrium. In fact, comparing the total investment between the two regimes, we show that the free competition regime achieves a larger total investment in infrastructure upgrades than the access regulation regime does. This result in turn indicates a higher social welfare in the free competition regime than in the access regulation regime. Therefore, building a simple but solid model, we provide a theoretical justification for the policy recommendation that when there is effective competition in the wholesale broadband market, access regulation should be removed (see Ofcom (2013)). Furthermore, based on a sample of OECD countries, Bouckaert, van Dijk, and Verboven (2010) show that inter-platform competition contributes to broadband penetration.3 Our analytical results are also consistent with their empirical evidence. Similarly, in the mobile communications market, the wholesale access regulation should not be introduced for the deployment of 5G networks.

2 Although Bourreau et al. (2015) address an integrated firm's investment incentive, they implicitly assume that an unintegrated downstream firm can obtain the same level of benefits generated from infrastructure upgrades as an integrated firm does. 3 See also Williamson, Lewin, and Wood (2016).

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The effect of spillovers generated from infrastructure upgrades in broadband markets is examined by Foros (2004) and Kotakorpi (2006).4 These studies show that access regulation can be harmful to consumers through small investments depending on the degree of spillover. Although the logic behind their results is similar to our logic, they do not examine the effects of wholesale competition between vertically integrated firms. Our paper extends their analysis by indicating that because of the existence of access profits, a vertically integrated firm's reaction function does not smoothly change according to the investment level of its rival. We then show that the wholesale-profit effect dominates the retail-production effect in the free competition regime, which in turn induces a higher investment incentive than in the access regulation regime. Section 2 presents the framework of the model. Section 3 gives a preliminary analysis of the case without firms' investment opportunities, which is useful as a benchmark. Section 4 derives the equilibrium in the free competition regime. Section 5 gives the equilibrium in the access regulation regime and compares it with the equilibrium in the free competition regime from the welfare and policy perspectives. Section 6 concludes the paper. All proofs are relegated to the Appendix. 2. The model We examine a simple game of competition for access provision. We call it a free competition regime because there is no government intervention in the game. There are two vertically related sectors in a market: an upstream sector and a downstream sector. The two sectors are required to supply services to consumers in a market. There are three firms; firm 1, firm 2, and firm S. Firms 1 and 2, called facility-based firms, are incumbents, and each has its own infrastructure upstream and a production facility downstream (i.e., they are vertically integrated firms). On the contrary, firm S, called a service-based firm, is a potential entrant that has only a production facility downstream (i.e., it is an unintegrated downstream firm). To serve consumers, firm S needs to access the infrastructure owned by a facility-based firm k (k ¼ 1 or 2) by paying an access charge ak set by firm k. We assume the same production technology between the two facility-based firms. One unit of input (i.e., the output produced upstream) produces one unit of output downstream. The (constant) marginal access cost that a facility-based firm incurs for firm S's access is the same as its marginal production cost upstream, and for analytical simplicity, they are assumed to be zero. In addition, we assume that the production cost downstream for each firm is zero. Each of facility-based firms 1 and 2 has an opportunity for investment to upgrade its own infrastructure. Investment to upgrade infrastructure has a demand-enhancing effect because it improves the quality of services sold in the downstream sector. A typical example would be an investment in broadband technology such as fiber to the home (FTTH) that upgrades the speed or quality of information searches, which would thereby enhance consumer demand. Following Katz and Shapiro (1985) and Foros (2004), we employ a linear inverse demand system with vertically differentiated services by supposing that heterogeneous consumers with a unit demand for a service are uniformly distributed. Then, the inverse demand function for service j (j ¼ 1, 2, S) is given by

pj ¼ vj  Q;

ðj ¼ 1; 2; SÞ

where vj represents the quality of service j, and Q ≡ q1 þ q2 þ qS . Here, vjs (j ¼ 1, 2, S) are given by, respectively,

v1 ¼ α þ x1 ;

v2 ¼ α þ x2 ;

vS ¼ α þ sxk

where α is a positive constant that represents consumers' willingness-to-pay for a service supplied using an old infrastructure, xk (k ¼ 1, 2) is the investment level of firm k to update its infrastructure, and s is the degree of spillover of infrastructure upgrades. Note that the quality of services sold in the downstream sector is endogenously determined by facility-based firms' investments in infrastructure upgrades upstream. For analytical simplicity, we assume the same investment technology between firms 1 and 2, which is represented by Iðxk Þ ¼ ðγx2k Þ∕2, where γ (>0) is an investment cost parameter of firms 1 and 2. The degree of spillover s reflects the relative inferiority or superiority of firm S's retail production technology compared with that of the facility-based firm that firm S accesses. In particular, the case in which s <(>)1 means that firm s's retail technology is inferior (superior) to those of firms 1 and 2. The timing of the game is as follows. In the first stage, the two facility-based firms, firm 1 and firm 2, simultaneously invest in infrastructure upgrades. In the second stage, observing the levels of investments x1 and x2, firm 1 and firm 2 simultaneously set the access charges a1 and a2 independently. In the third stage, firm S decides whether to enter the market by accessing one of the facilitybased firms.5 For analytical simplicity, we assume that firm S's entry sunk cost is zero. In the fourth stage, all active firms compete downstream in a Cournot fashion. Cournot competition in the downstream sector is suitable for the broadband market because of the capacity constraints that Internet service providers face.6 After analyzing equilibrium in the free competition regime, we examine the access regulation regime for comparison with the performance of the free competition regime. The access regulation regime is defined as the regime in which a benevolent regulator instead

4 The effect of spillovers is mainly examined in the R&D literature, such as d’Aspremont and Jacquemin (1988) and Suzumura (1992). See Cambini and Jiang (2009) for a survey regarding access regulation and investment in broadband markets. 5 We assume that when the access conditions offered by firm 1 and firm 2 are indifferent from firm S's point of view, the probability of accessing each of the facilitybased firms is 0.5. 6 See Faulhaber and Hogendorn (2000) for this point.

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of the facility-based firms determines the access charge. That is, in the second stage of the model, a regulator sets an access charge to maximize social welfare by taking as given the level of investments the two facility-based firms make in the first stage. This regime is prevalent in telecommunications worldwide, especially in the Internet and broadband markets. We should note here that the limited commitment ability of a regulator is assumed in our model. This regulatory non-commitment is justified by the fact that the economic life of investment in infrastructure upgrades is longer than the period of a regulatory contract in real policy arenas. To preserve analytical tractability, we make the following Assumptions. Assumptions. (i) ak  0 (k ¼ 1, 2), (ii) γ  2, (iii) 9∕11  s < 2. Assumption (i) is set for practical reasons.7 In fact, it is rare for access charges to be set below the marginal access cost in the real policy arena. Assumption (ii) guarantees the strict concavity of an incumbent's profit function with respect to investment, and it restricts our attention to the range of xk  α, as ensured in the following analysis. For the purposes of this paper, we make Assumption (iii) to avoid unnecessary complexity in the analysis. In particular, this assumption excludes the possibility of the multiplicity of equilibria in the free competition regime.8 It also allows us to focus on the regulator's setting of a cost-based access charge in the access regulation regime, as shown in Section 5.9 Assumption (iii) also means that firm S's retail technology is not sufficiently different from those of firms 1 and 2. However, we should note that even under Assumption (iii), we still allow both the inferiority and superiority of firm S's retail technology. 3. Preliminary analysis In this section, we first characterize the equilibria in the third and fourth stages of the model in Section 2. Then, as a benchmark, we examine the case in which two facility-based firms do not have an opportunity to invest in infrastructure upgrades. In that case, the equilibrium in the free competition regime is compared with that in the access regulation regime. 3.1. Equilibria in the third and fourth stages The profit function of a facility-based firm k that firm S accesses and that of a facility-based firm l that firm S does not access are represented by

π k ¼ pk qk þ ak qS  Iðxk Þ;

π l ¼ pl ql  Iðxl Þ;

k; l ¼ 1; 2; and k≠l;

respectively. The profit function of firm S if it enters the market is represented by

π S ¼ ðpS  ak ÞqS ; and π S ¼ 0 if firm S does not enter the market. When firm S enters the market in the third stage, a triopoly appears in the fourth stage. Solving the maximization problems of the firms, we obtain the equilibrium quantities as follows:

qk ðak ; xk ; xl Þ ¼

α þ ak þ ð3  sÞxk  xl ; 4

(1)

ql ðak ; xk ; xl Þ ¼

α þ ak  ð1 þ sÞxk þ 3xl ; 4

(2)

qS ðak ; xk ; xl Þ ¼

α  3ak  ð1  3sÞxk  xl ; 4

(3)

Qðak ; xk ; xl Þ ¼

3α  ak þ ð1 þ sÞxk þ xl : 4

(4)

π k ðak ; xk ; xl Þ ¼ ðqk ðak ; xk ; xl ÞÞ2 þ ak qS ðak ; xk ; xl Þ  Iðxk Þ;

(5)

where qk(.) (ql(.)) is the equilibrium quantity of firm k that firm S accesses (does not access) and Q(.) is the total equilibrium quantity. Note that when the degree of spillover s is larger than 1∕3, the investment of firm k increases the quantity of firm S. Using (1) to (4), the equilibrium profits in the subgame of the fourth stage are given by

2

π l ðak ; xk ; xl Þ ¼ ðql ðak ; xk ; xl ÞÞ  Iðxl Þ;

(6)

7

This assumption also appears in Foros (2004). When s < 7∕9, two types of equilibria emerge even in the subgame of the stage where two facility-based firms set their access charges. See the proof of Lemma 1 for details. The multiplicity of equilibria in the free competition regime is also reported by Bourreau et al. (2011). 9 Similarly, Foros (2004) restricts the range of spillovers to focus on a cost-based access regulation. 8

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2

π S ðak ; xk ; xl Þ ¼ ðqS ðak ; xk ; xl ÞÞ :

(7)

If firm S does not enter the market, a duopoly appears in the fourth stage. The equilibrium quantities supplied by firms 1 and 2 and the total equilibrium quantity are given by

qe1 ðx1 ; x2 Þ ¼

α þ 2x1  x2 ; 3

~ ðx1 ; x2 Þ ¼ Q

2α þ x1 þ x2 : 3

q~2 ðx1 ; x2 Þ ¼

α þ 2x2  x1 ; 3

(8)

(9)

The equilibrium profits in the subgame in which firm S does not enter the market are given by

π~1 ðx1 ; x2 Þ ¼

 2 q~1 ðx1 ; x2 Þ Iðx1 Þ;

 π~2 ðx1 ; x2 Þ ¼

2 q~2 ðx1 ; x2 Þ

Iðx2 Þ

In the third stage, firm S determines whether to enter the market by accessing the infrastructure of firm k (k ¼ 1 or 2) with the payment of ak. The necessary and sufficient conditions for firm S to enter the market and to access firm k are given by qS ðak ; xk ; xl Þ  0 and qS ðak ; xk ; xl Þ  qS ðal ; xk ; xl Þ, which we can rewrite as follows:



ak

ak ≡

ak  sxk



1 ðα  ð1  3sÞxk  xl Þ; 3

(10)

al  sxl :

(11)

(10) means that the access charge set by firm k must be small so that firm S can obtain a positive profit in the market. (11) implies that firm k needs to offer a more profitable condition to firm S than firm l does. Note that the profitability condition for firm S includes not only the access charge but also the spillovers generated from infrastructure upgrades. Social welfare is defined as the sum of consumer surplus (CS) and firms' profits in a given market structure. Social welfare in a triopoly, SWT, is represented by

SW T ¼ CS þ π 1 þ π 2 þ π S ; where CS ¼ (Q(ak; x1, x2))2∕2. Social welfare in a duopoly, SWD, is defined in a similar manner. 3.2. A benchmark: the equilibrium without investment opportunities Before examining the equilibrium in the free competition regime, we provide as a benchmark the results when the two facility-based firms do not have an opportunity to invest in infrastructure upgrades: that is, we impose x1 ¼ x2 ¼ 0. We examine the stage of each facility-based firm's decision on access charges. Consider firm 1's decision, for example. When a2 > a2 ≡ ð1∕3Þα, firm S does not access firm 2's infrastructure. Hence, firm 1's best response is to allow firm S to access firm 1's ~1 ¼ a ~M ~M infrastructure by setting a 1 ¼ ð3∕11Þα where a 1 is the profit-maximizing access charge when allowing entry. In contrast, when a2  a2 ≡ ð1∕3Þα, there is a likelihood that firm S accesses firm 2's infrastructure. Then, firm 1's best response is

~a1

 ¼

~aM 1 ¼ ð3∕11Þα a2  ε

i  for a2 2 a~M a2 ; 2 ;   for a2 2 0; a~M 2 ;

where ε is an infinitesimal number. We have firm 2's best response in a similar manner. Therefore, under Assumption (i), i.e., ak  0 10 no (k ¼ 1,2), the equilibrium access charges are ano 1 ¼ a2 ¼ 0, which is consistent with the result of Bourreau et al. (2011). To evaluate the performance of the free competition regime in this benchmark, we next examine the equilibrium in the access regulation regime. As mentioned in Section 1, the access regulation regime is defined as the regime in which a benevolent regulator, whose objective is to maximize social welfare, determines the access charge instead of the facility-based firms. Without loss of generality, we assume that firm S accesses firm 1 0 s infrastructure if it wants to enter the market under a regulated access charge. Comparing the social welfare in a triopoly SW T with an optimal access charge ano and that in a duopoly SW D, we verify that the regulator sets ano ¼ 0 and chooses a triopoly under Assumption (i).11 Summarizing the above results, we obtain the following Proposition.

10 Bourreau et al. (2011) show that when the downstream competition is fierce, such as Bertrand competition with homogenous goods, there may exist the softening effect, which makes the access charge higher than the access cost (i.e., ano > 0) if an upstream supplier (a facility-based firm) can enhance its upstream profits by i behaving softly in the downstream sector. However, when Cournot competition (quantity competition with simultaneous moves) prevails in the downstream sector, the upstream supplier can scarcely impact its upstream profit through its downstream behavior, so that the softening effect does not work under Cournot competition. 11 This result comes from the facts that SW T is strictly concave in a and SW T ¼ SW D at a ≡ ð1∕3Þα.

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Proposition 1. When there is no opportunity to invest in infrastructure upgrades, the free competition regime achieves the same outcome as that in the access regulation regime. In the model of Foros (2004), where only one incumbent exists with one potential entrant, the free competition regime induces foreclosure when there is no opportunity to invest in infrastructure upgrades. Then, the free competition regime is welfare-inferior to the access regulation regime. 12 By contrast, in our model, two incumbents exist in the market, and the wholesale competition between the two incumbents gives birth to granting access to a potential entrant, thus contributing to welfare enhancement. Then, a cost-based access charge is realized in the free competition regime, which achieves the same level of social welfare as that in the access regulation regime. 4. Equilibria in the free competition regime 4.1. Equilibrium access charges Let us turn to the model with investment opportunities. We examine the second stage in which two facility-based firms, firms 1 and 2, set their own access charges a1 and a2 independently. The access charge set by each of the two facility-based firms depends not only on the rival's access charge but also on the levels of the investments in infrastructure upgrades x1 and x2 determined in the first stage. The equilibrium access charges in the second stage are characterized in the following Lemma. Lemma 1. follows13:

Under Assumptions (i) to (iii), the equilibrium access charges a1 and a2 in the second stage of the game are characterized as

ak ðxk ; xl Þ ¼ al ðxk ; xl Þ ¼ 0 if xk ¼ xl ; ak ðxk ; xl Þ ¼ sðxk  xl Þ and al ðxk ; xl Þ ¼ 0 if xk > xl ; k; l ¼ 1; 2; and k≠l: In the latter case, firm k is accessed by firm S with probability 1. Proof. See Appendix. ■ The result of Lemma 1 is very intuitive. ”Competition for access provision” occurs in the upstream sector, similar to the case in which the facility-based firms do not have investment opportunities. In particular, when the investment levels are the same between firms 1 and 2, each of them has an incentive to reduce its access charge to attract firm S. As a result, access charge is equal to access cost in equilibrium, and each of the incumbents expects to be accessed by firm S with probability 0.5. Again, this result comes from the absence of the softening effect due to Cournot competition in the downstream sector. However, when the investment of one of the facility-based firms is larger than that of the other firm, that firm can set an access charge that is higher than the access cost and it is accessed by firm S with probability 1. Then, the firm with larger investment can obtain a positive access profit. Note here that the level of the access charge depends on the degree of spillover s. In particular, as s increases, the access charge increases, when taking x1 and x2 as given. This fact is explained as follows. As s increases, differences in the investment levels are more relevant in terms of the service-based firm’s profit. In other words, a high s makes the two facility-based firms’ networks more vertically differentiated in the service-based firm’s perspective. Hence, the service-based firm is willing to pay a higher access charge to access the best facility-based firm. Concerning Lemma 1, it is essential for us to clarify the meaning of the restriction made by Assumptions (ii) and (iii). If γ<2 and s<9∕11 (especially, s<7∕9), a facility-based firm i(¼ 1,2) has an incentive to exclude firm S from the market by making a large investment in infrastructure upgrades and by setting a high access charge, i.e., ai > ai ≡ ðα ð1 3sÞxi  xj Þ∕3. This suggests the possibility of foreclosure equilibrium. However, the foreclosure equilibrium occurs only when the rival facility-based firm j also sets a high access charge, aj > aj ≡ ðα ð1 3sÞxj  xi Þ∕3. In other words, even for γ<2 and s<9∕11, a facility-based firm i sets the competitive access charge defined in Lemma 1 as long as the rival incumbent sets aj  aj . Hence, when γ<2 and s<9∕11, we may have multiple equilibria (i.e., access provision equilibrium and foreclosure equilibrium) in the second stage of the game. In this sense, competition for access provision is a robust phenomenon in our model. 4.2. Strategic investments and access provision In the first stage, firms 1 and 2 invest in infrastructure upgrades. The following Proposition states the characterization of the equilibria in the free competition regime. Proposition 2. In the free competition regime, there exist asymmetric access provision equilibria (AAPE) in which aAAPE ¼ sðxAAPE  xAAPE Þ k k l AAPE AAPE ¼ 0 (k, l ¼ 1,2, and k≠l) and the equilibrium investments are x ¼ Φðs; γÞ∕Θðs; γÞ and x ¼ Ψðs; γÞ∕Θðs; γÞ, where and aAAPE l k l

 Θðs; γÞ ≡ ð8γ ð9  4sÞÞ 8γ  ð3  sÞ2  3ð2s þ 1Þð1  sÞð3  sÞ; Φðs; γÞ ≡ 2α 4ð3 þ 2sÞγ  ð3  sÞ 6 þ 3s  4s2 ; Ψðs; γÞ ≡ 2αð3  sÞð4γ  ð6  sÞÞ:

12

See Lemma 1 of Foros (2004). In this Lemma, the expression of ak ¼ sðxk  xl Þ (k, l ¼ 1, 2, and k≠l) ignores the small reduction of ε. That is, the equilibrium access charge is precisely written as ak ¼ sðxk  xl Þ ε. 13

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*2 *1 Fig. 1. Asymmetric Access Provision Equilibrium (AAPE). Note: Firm 1's (firm 2's, respectively) reaction function is the solid parts of x*1 1 ðx2 Þ. and x 1 ðx2 Þ (x 2 ðx1 Þ and x*2 2 ðx1 Þ, respectively).

and xAAPE > xAAPE . Then, firm k is accessed by firm S with probability 1. k l Proof. See Appendix. ■ According to Proposition 2, asymmetric access provision equilibria (AAPE) occur in the free competition regime.14 Fig. 1 illustrates the two facility-based firms’ reaction functions associated with AAPE. In Fig. 1, firm 1's (firm 2's, respectively) reaction function consists of 2 1 2 ^j the solid parts of x1 1 ðx2 Þ and x1 ðx2 Þ (x2 ðx1 Þ and x2 ðx1 Þ, respectively). We note that each firm’s reaction function is discontinuous at x (j ¼ 1, 2). The discontinuity of the reaction function is explained by the difference in the marginal benefit of investment in infrastructure upgrades between the case in which a facility-based firm is accessed by a service-based firm and the case in which it is not accessed. We explain this in the following. See firm 1's reaction function, for example. Consider the case in which firm 2's investment x2 is small. In this case, firm 1 has a strong incentive to win a competition for access provision by investing more than firm 2. In fact, if firm 1 invests more than firm 2, it can obtain not only a larger retail profit but also a wholesale profit (or access profit) by setting an access charge higher than the access cost (see Lemma 1). The latter is called the wholesale-profit effect, which induces a strong incentive to invest in infrastructure upgrades. Of course, when firm 1 is accessed by firm S, the investment in infrastructure upgrades increases not only firm 1's own retail production but also firm S's production. This is the so-called retail-production effect, which means that the investment automatically gives a spillover benefit (i.e., positive externality) to a service-based firm. Then, the retail-production effect weakens an incentive to invest in infrastructure upgrades. However, because the access charge is set higher than the access cost, an increase in firm S ’s production increases firm 1's access profit. In that sense, the wholesale-profit effect dominates the retail-production effect in the free competition regime. Hence, when x2 is small, firm 1 wants to win a competition for access provision by investing more than firm 2. This is the part of x2 1 ðx2 Þ of firm 1's reaction function. In contrast, when firm 2's investment x2 is large, firm 1 has no incentive to invest more than firm 2 to win a competition for access provision. If firm 1 invests more than firm 2, firm 1's retail profit is expected to be small because of a small difference of service quality, whereas the investment cost is sufficiently large. In addition, when x2 is large, firm 1's access profit is small because the access profit is monotonically decreasing in x2.15 Hence, when x2 is large, firm 1 chooses x1 1 ðx2 Þ and it does not have an incentive to win a competition for access provision.

14 15

Because we restrict our attention to pure strategy equilibria, there is no symmetric equilibrium in the free competition regime. Substituting (3) and the result of Lemma 1 into the access profit a1qS, we can verify that it is monotonically decreasing in x2 for s 2 ½9∕11; 2. 7

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2 Then, there is a threshold of x^ 2 at which firm 1's reaction function jumps from x1 1 ðx2 Þ to x1 ðx2 Þ (i.e., the non-smoothness of a facility2 based firm’s reaction function). Note that x1 ðx Þ stands farther left than x ðx Þ. This means that firm 1 has more incentive to invest in 2 2 1 1 infrastructure upgrades when it is accessed by firm S than when it is not accessed. We should note that the non-smoothness of a facilitybased firm’s reaction function is generated by spillovers of infrastructure upgrades when it is accessed by a service-based firm. In fact, if s ¼ 0, we find that each of the reaction functions is smoothly continuous in the rival’s investment (see (13) and (15) in the Appendix). There are two AAPEs in Fig. 1. In each AAPE, one of the facility-based firms is accessed by firm S, and that firm can set the access charge higher than the access cost so that it obtains a positive access profit. The origins of this positive access profit are twofold; one is the presumption that the right to determine the access charge is delegated to the facility-based firms, and the other is the existence of spillovers through access. When a facility-based firm is accessed by firm S, an increase in firm S’s production increases a wholesale profit obtained by a facility-based firm because of the wholesale effect. In addition, an increase in firm S’s production decreases the production of a rival facility-based firm. Thus, in our model, a service-based firm can be interpreted as an affiliated firm of a facility-based firm. Because the investment in infrastructure upgrades is a strategic substitute, the facility-based firm that is accessed by firm S achieves more investment than the other (i.e., xAAPE > xAAPE ) as a result of competition for access provision. k l An interesting thing is that in the equilibria of the free competition regime, the heterogeneity in service quality (i.e., vertical product differentiation) endogenously occurs between two facility-based firms even though both the production and the investment technologies are the same ex ante between the two firms. This is a critical property generated from spillovers of infrastructure upgrades in the ¼ xAAPE , so that there is no heterogeneity in wholesale competition for access provision. We again verify that if s ¼ 0, we have xAAPE k l 16 service quality between two facility-based firms when there is no spillover. The result of the asymmetric access provision equilibria is reminiscent of an R&D race. In fact, the free competition regime is similar to a preemption game in an open access environment (see Hori and Mizuno (2006) and Vareda and Hoernig (2010)). They show that the preemption effect emerges even when access seekers have an opportunity to follow their leaders. As in the preemption equilibria, the driving force for the asymmetric access provision equilibria is the access profit. Hence, an access charge set by a facility-based firm or a regulator plays a critical role in motivating investments for infrastructure upgrades.

5. Comparison with the access regulation regime 5.1. Access regulation equilibria In the free competition regime, there is competition for access provision among the facility-based firms. As shown in Proposition 2, competition for access provision induces voluntary access provision with a large investment in infrastructure upgrades. Thus, the natural question arises of whether we really need government intervention when there is competition for access provision in industries with two-tier structures. To answer this question, we examine a government intervention regime in this subsection. Then, in the next subsection, we evaluate the performance of the free competition regime by comparing it with the government intervention regime. As in Section 3.2, we restrict our attention to the access regulation regime as a government intervention regime. In the access regulation regime, a benevolent regulator determines the access charge instead of a facility-based firm in the second stage. Here, we assume that the access regulation is symmetric in the sense that the regulated access charge is applied to each of the two facility-based firms if it is accessed by firm S.17 The equilibria in the third and fourth stages are the same as they are in the free competition regime. In the second stage, the regulator sets an access charge to maximize social welfare by taking the facility-based firms' investments x1 and x2 as given. The optimal access charge set by the regulator is characterized in the following Lemma. Lemma 2. The optimal access charge set by the regulator, a , is characterized by

a ðxk ; xl Þ ¼

8 > > <0

if

> > : α þ ð5  11sÞxk þ 5xl

if

1 xl  ðα  ð5  11sÞxk Þ; 5 1 1 ðα  ð5  11sÞxk Þ < xl  ðα  ð4  9sÞxk Þ; 5 4

where xk>xl (k, l ¼ 1,2, and k≠l). Otherwise, a > ak ≡ 13 ðα ð1 3sÞxk  xl Þ. Proof. See the Appendix. ■ According to Lemma 2, the regulator may set three types of access charges, depending on the levels of the facility-based firms’ investments. However, we ensure that only the cost-based access charge, i.e., a ðx1 ; x2 Þ ¼ 0, is realized under Assumptions (i) to (iii), as shown in the following lemma. Lemma 3.

Under Assumptions (i) to (iii), the optimal access charge set by the regulator is a cost-based access charge (i.e., a ðx1 ; x2 Þ ¼ 0),

16 One may expect that as s increases, xAAPE increases and xAAPE decreases. For s 2 ½9∕11; 2, we can ensure that ∂xAAPE ∕∂s < 0, whereas the sign of ∂xAAPE ∕∂s is k l l k ambiguous. 17 In reality, asymmetric access regulation, also known as ”dominant regulation”, has been prevalent in the telecommunications industries of many countries. However, because the bypassed market share gradually expands as a result of technological progress or technological convergence, some countries now try to adopt symmetric access regulation. For example, Belgium has set out to adopt symmetric access regulation.

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Fig. 2. Access Regulation Equilibrium (ARE). Notes: Firm 1's (firm 2's, respectively) reaction function is the solid parts of x*AR1 ðx2 Þ and x*AR2 ðx2 Þ (x*AR1 ðx1 Þ and 1 1 2     ARA ARA x*AR2 (x2 ¼ x1 for x1 2 xARB , respectively). ðx1 Þ, respectively) and x1 ¼ x2 for x2 2 xARB 2 2 ; x2 1 ; x1

regardless of the levels of x1 and x2. Proof. See the Appendix. ■ Then, the following Proposition characterizes the equilibria in the access regulation regime. Proposition 3.

xARB ¼

In the access regulation equilibria (ARE), aARE ð¼ aARE ¼ aARE Þ ¼ 0 and xARE ¼ xARE 2 ½xARB ; 1 2 1 2

xARA  where

ð3  sÞα 3α and xARA ¼ : 8γ  ð3  sÞð2  sÞ 8γ  3ð2  sÞ

Then, each of the facility-based firms is accessed by firm S with probability 0.5, irrespective of the degree of spillover and the investment cost. Proof. See the Appendix. ■ Fig. 2 shows the facility-based firms’ reaction functions in the access regulation regime. Contrary to Fig. 1, we see that each firm’s reaction function is continuous but kinked at xARA and xARB (i ¼ 1,2). That is, we again observe the non-smoothness of a facility-based i i firm’s reaction function. In particular, firm 1's (firm 2 ’s) reaction function stands more leftward (downward) when x1>(<)x2 than it does when x1<(>)x2. This means that each of the facility-based firms has less incentive to invest when it is accessed by firm S than it does when it is not accessed. The reason is explained as follows. Consider the case in which a facility-based firm is accessed by a service-based firm. As in the free competition regime, the investment in infrastructure upgrades increases not only a facility-based firm’s own production but also a service-based firm’s production because of the retail-production effect. However, contrary to the free competition regime, the facility-based firm cannot obtain an increase in wholesale profit because the access charge is set equal to the access cost by the regulator. That is, there is no wholesale-profit effect. Hence, when a facility-based firm is accessed by a service-based firm, its incentive to invest in infrastructure upgrades, which is motivated by an increase in retail profit through an increase in its own production, is weakened by the retail-production effect. On the contrary, when it is not accessed, there is no retail-production effect. Therefore, each of the facility-based firms has less incentive to invest when it is accessed by firm S than it does when it is not accessed. In Fig. 2, we find the access regulation equilibria (ARE). As a result of the non-smoothness of the facility-based firms’ reaction functions, there are multiple equilibrium investments in ARE. Note that in the range of the equilibrium investments, infrastructure investment is a strategic complement rather than a strategic substitute. This is explained as follows. Consider firm 1's incentive for investment. Then, suppose that firm 2's investment is very large. In that case, firm S accesses firm 2's infrastructure, so that it gains spillovers from firm 2 (i.e., the retail-production effect). If firm 2 reduces its investment, firm 2's production and firm S's production both decrease. Then, firm 1 has a strong incentive to invest in infrastructure upgrades. This is the case of strategic substitutes. In contrast, 9

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suppose that firm 2's investment is not large, but firm S still accesses firm 2's infrastructure. In that case, if firm 2 further reduces its investment, there is a high possibility that firm S would like to move its access from firm 2 to firm 1 when taking firm 1's investment as given. If firm 1 recognizes this possibility of firm S’s access to its own infrastructure, it cares about the retail-production effect generated by firm S’s access. Hence, firm 1 also has an incentive to reduce its investment. This is the case of strategic complementarity. In fact, strategic complementarity occurs in ARE. From the above discussion, we verify that spillovers of infrastructure upgrades generate the multiple equilibria with strategic complementarity regarding infrastructure investment. In fact, if s ¼ 0, the reaction function of a facility-based firm does not have a kinked portion, and we ensure a unique equilibrium investment level (see (21) and (23)). Furthermore, when s ¼ 0, this equilibrium investment is exactly the same as that in the AAPE; xARE ¼ xARE ¼ 3α∕ð2ð4γ 3ÞÞ ¼ xAAPE ¼ xAAPE . That is, when a service-based 1 2 1 2 firm does not have any kind of new retail technology that fits for infrastructure upgrades, the free regulation regime achieves the same outcome as that in the access regulation regime in our model. 5.2. Welfare comparison and policy implications We now compare the equilibria in the free competition regime and the equilibria in the access regulation regime from a welfare perspective. First, we compare the investment level between the two regimes. For that comparison, we adopt the equilibrium with the largest total investment among the multiple equilibria of the access regulation regime. We can then ensure that a total investment in infrastructure upgrades is larger in the free competition regime than in the access regulation regime, as shown in the following Proposition. Proposition 4. Under Assumptions (i) to (iii), the total investment in infrastructure upgrades is larger in the free competition regime than it is in the access regulation regime. Proof. See Appendix. ■ From the result of Proposition 4, we expect that social welfare in the free competition regime is also higher than that in the access regulation regime. This is because investments in infrastructure upgrades enhance demand by improving service quality, which may benefit not only consumers but also firms. Needless to say, however, there is also a possibility of overinvestment in the free competition regime from a welfare viewpoint because the private benefit for the investing firm is obtained at the expense of its rival’s profit. Irrespective of this concern of overinvestment, in our model with Assumptions (i) to (iii), we find that social welfare in the free competition regime is higher than that in the access regulation regime. See Figs. 3-1 and 3-2. Figs. 3-1 and 3-2 show the welfare comparison between the AAPE in the free competition regime and the ARE in the access regulation regime with the setting of α ¼ 50.18 Figs. 3-1 and 3-2 illustrate social welfare (SW  ), consumer surplus (CS ), the profit of a firm that is accessed by firm S (firm k’s profit), and the profit of a firm that is not accessed (firm l’s profit), as well as firm S’s profit for the case in which the investment cost is low (γ ¼ 3) and for the case in which the investment cost is high (γ ¼ 15), respectively. We figure out the following four properties. First, social welfare is higher in the free competition regime than it is in the access regulation regime, irrespective of the investment cost and the degree of spillovers. This result verifies the expectation stated above. Moreover, as the degree of spillover increases, the social welfare in each of the two regimes increases. This result ensures that the improvement of a service-based firm’s retail technology contributes to welfare enhancement. Second, from Figs. 3-1 and 3-2, we find that the main factor that drives the welfare superiority of the free competition regime is the larger profit obtained by a facility-based firm that is accessed by a service-based firm in the free competition regime than that obtained by that firm in the access regulation regime. Interestingly, as the degree of spillover increases, the profit of a facility-based firm that is accessed by a service-based firm increases in the free competition regime, whereas it decreases in the access regulation regime. This implies that in the free competition regime, a facility-based firm that is accessed by a service-based firm can appropriate the benefit of the improvement of the service-based firm’s retail technology in terms of access profits by investing in infrastructure upgrades. However, in the access regulation regime, the benefit of the improvement of the retail technology contributes to the service-based firm’s profit because of the cost-based access charge set by the regulator. This point is re-ensured by the change in firm S’s profit in the figures. As the degree of spillover increases, firm S’s profit (almost) decreases in the free competition regime, whereas it increases in the access regulation regime. Third, consumer surplus in the free competition regime is always lower than it is in the access regulation regime, except for the case in which the investment cost is high and the degree of spillover is small (see the case for s<0.94 in Fig. 3-2). Again, this result implies that the benefit of the larger total investment in the free competition regime than in the access regulation regime is appropriated by a facilitybased firm that is accessed by a service-based firm through a high access charge and the associated high retail price. Lastly, the profit of a facility-based firm that is not accessed by a service-based firm is higher in the access regulation regime than it is in the free competition regime. Furthermore, as the degree of spillover increases, it decreases in each of the two regimes. Recognizing these four properties from the comparison between the free competition regime and the access regulation regime, we obtain the following policy implications. In our model, we presume Cournot competition that implies capacity constraints or limited capacities. The situation of limited capacities corresponds to limited spectrums in mobile communications markets. Then, the analytical result derived in this paper insists that the free competition regime with multiple facility-based firms can induce more incentives to

18

The qualitative features of the welfare comparison do not change when the value of α changes. 10

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Fig. 3-1. Welfare Comparison between AAPE and ARE: Case for (The upper 5 figures)γ ¼ 3: Notes: (i) α ¼ 50. (ii) SW * : Social welfare. CS* : Consumer surplus. (iii) “Firm k” is the firm that is accessed by firm S, whereas “firm l” is the firm that is not accessed. (k, l ¼ 1, 2 and k≠l). (iv) In the figures, “AAPE” (“ARE”) is the line associated with the free competition regime (the access regulation regime).

invest for demand-enhancing innovation than the access regulation regime. In this respect, wholesale access regulation should not be introduced into a mobile communications market that requires innovation for the deployment of 5G networks.19 Furthermore, as mentioned in Section 1, cost-based access regulation should be abandoned also for the penetration of ultra-broadband infrastructures. However, we need to understand that consumer welfare may decrease through an increase in access charges, even though high-quality services are provided under 5G networks or ultra-broadband infrastructures.

6. Concluding remarks In this paper, we have examined wholesale competition with an access provision when vertically integrated firms have an

19

See Alexiadis and Shortall (2016) and Frias and Martínez (2017) for a discussion about regulatory intervention in a 5G network environment. 11

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Fig. 3-2. Welfare Comparison between AAPE and ARE: Case for γ ¼ 15: Notes: (i) α ¼ 50. (ii) SW * : Social welfare. CS* : Consumer surplus. (iii) “Firm k” is the firm that is accessed by firm S, whereas “firm l” is the firm that is not accessed. (k, l ¼ 1, 2 and k≠l) (iv) In the figures, “AAPE” (“ARE”) is the line associated with the free competition regime (the access regulation regime).

opportunity to invest in infrastructure upgrades. Spillovers of infrastructure upgrades through access have two opposite effects: the wholesale-profit effect and the retail-production effect. The difference in the relative magnitudes of these effects generates different characteristics of the equilibria between the free competition regime and the access regulation regime. In the free competition regime in which access charges are not regulated, each vertically integrated firm has a stronger incentive to invest in infrastructure upgrades when it is accessed than it does when it is not accessed (the non-smoothness of a vertically integrated firm's reaction function). This is because the wholesale-profit effect dominates the retail-production effect when the firm is accessed by an unintegrated downstream firm. In this case, asymmetric equilibria occur in which a vertically integrated firm that is accessed by an unintegrated downstream firm invests more than a vertically integrated firm that is not accessed. This means that vertical quality differentials occur in equilibrium between vertically integrated firms that have the same technologies ex ante. In contrast, in the access regulation regime with regulatory non-commitment, an access charge is set equal to the access cost (a costbased access charge) by a regulator. In that case, the wholesale-profit effect does not appear, whereas the retail-production effect remains. Then, a vertically integrated firm has a stronger incentive to invest in infrastructure upgrades when it is not accessed by an unintegrated downstream firm than it does when it is accessed (i.e., non-smoothness of a vertically integrated firm's reaction function). 12

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In that case, each vertically integrated firm is unwilling to invest in infrastructure upgrades and to allow access to an unintegrated downstream firm. As a result, symmetric investments emerge in equilibrium in the access regulation regime. Furthermore, the difference in the relative magnitudes of the wholesale-profit effect and the retail-production effect in the two regimes generates the difference of total investment levels in equilibrium. In fact, comparing the total investment between the two regimes, we find that the free competition regime achieves a larger total investment for infrastructure upgrades than the access regulation regime. Irrespective of the possibility of overinvestment in the free-competition regime, we ensure that social welfare is higher in the free competition regime than in the access regulation regime. However, the access competition regime is always more beneficial to consumers than the free competition regime, except for the case in which the investment cost is high and the degree of spillover is small. Finally, we mention the effect of fierceness of retail competition on firms' incentive for strategic investment. In our analysis, we have presumed Cournot competition without horizontal product differentiation. In that case, a softening effect mentioned by Bourreau et al. (2011) does not work as in the analysis of Section 3.2. That is, access charges are set to be equal to access costs in the free competition regime when there is no opportunity for strategic investment. Then, as mentioned in our analysis, introducing the opportunity for strategic investment makes access charges higher than access costs in the free competition regime, which in turn induces a larger investment than in the access regulation regime. On the contrary, when retail competition becomes fiercer as under price competition with homogeneous goods, access charges set by vertically integrated firms may become higher than access costs due to the softening effect even when there is no opportunity for strategic investment. We then need to clarify how the opportunity for strategic investment affects the softening effect on access charges generated by the fierceness of retail competition, which is an issue that remains for future research. Appendix Proof of Lemma 1. First, we characterize the access pricing strategy of each of the two incumbents. In the following, we focus on firm 1's strategy. Firm 2's strategy is characterized in a similar way. Under a pair of investments, (x1, x2), set in the first stage, firm 1 determines the optimal access charge, a1 , taking a2 as given. There are two cases. (i) When a2 > a2 ≡ ðα ð1 3sÞx2  x1 Þ∕3, firm S does not access firm 2's infrastructure to enter the market. Then, if firm 1 offers a1 > a1 ≡ ðα ð1 3sÞx1  x2 Þ∕3, firm S does not enter the market. Thus, firm 1's profit is π~1 ðx1 ; x2 Þ. In contrast, if firm 1 offers a1  a1 , firm S enters the market by accessing firm 1's infrastructure, and firm 1's profit becomes π 1(a1;x1, x2). Maximizing π 1(a1;x1, x2) with respect to a1 gives

aM 1 ¼

1 ð3α þ ð1 þ 5sÞx1  3x2 Þ: 11

(12)

We should note that because π 1(a1;x1, x2) is continuous in a1, we have π 1 ða1 ; x1 ; x2 Þ ¼ π~1 ðx1 ; x2 Þ at a1 ¼ a1 . Moreover, because π 1(a1;x1, x2) is concave in a1, we have

 π 1 aM π 1 ðx1 ; x2 Þ if and only if aM 1 ; x1 ; x2  ð < Þ~ 1  ð > Þa1 : Furthermore, we have

aM 1  ð > Þa1 if and only if α  ð7  9sÞx1  x2  ð < Þ0:  That is, if α ð7 9sÞx1  x2  0, a1 ¼ aM 1 . Otherwise, a1 2 ½a1 ; ∞Þ. However, under Assumptions (ii) and (iii) (i.e., γ  2 and s 2 ½9∕ 11; 2), the only relevant case is that α ð7 9sÞx1  x2  0. Therefore, when a2 > a2 , a1 ¼ aM 1 .

(ii) When a2  a2 , firm S has an incentive to access firm 2's infrastructure to enter the market. In this case, firm 1's access pricing strategy depends on whether the profit with firm S's access is greater than the profit without it. In particular, we have

 a1 ¼ min aM if π 1 ða1 ; x1 ; x2 Þ > π 1 ða2 ; x1 ; x2 Þ; 1 ; a2  sðx2  x1 Þ a1 2 ða2  sðx2  x1 Þ; ∞ Þ if π 1 ða1 ; x1 ; x2 Þ < π 1 ða2 ; x1 ; x2 Þ In addition, a1 remains unchanged if π 1(a1;x1, x2) ¼ π 1(a2;x1, x2). In the characterization of a1 above, we should note that aM 1 can be smaller than a2  s(x2  x1) if a2  a2 > a 2 ≡ ð3α ð6s 1Þx1 þ ð11s 3Þx2 Þ∕11. That is, firm 1 can set its ”monopoly” access charge aM 1 if a2 is in the range of ða2 ; a2 . Then, we can ensure that π 1(a1;x1, x2)>π 1(a2;x1, x2) as long as a1 < a1 and a2  a2 . The reason is as follows: First, we ensure that π 1 ða1 ; x1 ; x2 Þ > π 1 ða2 ; x1 ; x2 Þ at a1 ¼ a2  sðx2  x1 Þ for a2  a2 . Then, if firm 1 raises the access charge from a1 for a small amount, i.e., a1 þ ε, its profit becomes π 1(a2;x1, x2), which is smaller than π 1 ða1 ; x1 ; x2 Þ. Hence, a1 ¼ a2  sðx2  x1 Þ for any a2  a 2 . Similarly, this argument holds when a1 ¼ aM 1 for a2 2 ða2 ; a2 . That is, in the case of (ii), two incumbents face price competition for

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access provision in the upstream sector. Summarizing the discussions of (i) and (ii), firm 1's access pricing strategy is stated as follows: When a2 > a 2 , a1 ¼ aM 1 , while when a2  a 2 , a1 ¼ a2  sðx2  x1 Þ.20 Firm 2's access-pricing strategy is the same as that of firm 1 by changing the identification number of the firm. Given each firm's access pricing strategy, we can derive Nash equilibria in the second stage. In fact, we ensure that when x1 ¼ x2, the

 Nash equilibrium access charges are fa1 ; a2 g ¼ f0; 0g. When x1>x2, fa1 ; a2 g ¼ fsðx1  x2 Þ; 0g. When x1x2, and x1 ¼ x2. First, we examine the case in which x1
Max x1

  2 π 11 ðx1 ; x2 Þ ¼ q1 a2 ; x1 ; x2  Iðx1 Þ:

Using (2), we derive the following reaction function.

x1 1 ðx2 Þ ¼

ð3  sÞðα  x2 Þ : 8γ  ð3  sÞ2

(13)

The associated profit when x2 is taken as given is 2  γðα  x2 Þ π 11 x1 : 1 ; x2 ¼  2 8γ  ð3  sÞ2

(14)

Second, we examine the case in which x1>x2. In this case, a1 ¼ sðx1  x2 Þ and a2 ¼ 0, and firm 1 is accessed by firm S. Thus, firm 1's problem is formulated as follows:

Max x1

  2  π 21 ðx1 ; x2 Þ ¼ q1 a1 ; x1 ; x2 þ a1 qS a1 ; x1 ; x2  Iðx1 Þ:

Using (1) and (3), we derive the following reaction function.

x2 1 ðx2 Þ ¼

ð3 þ 2sÞα  3ð2s þ 1Þð1  sÞx2 : 8γ  ð9  4sÞ

(15)

The associated profit when x2 is taken as given is

 π 21 x2 1 ; x2 ¼

Ωðx2 Þ 4ð8γ  ð9  4sÞÞ

2;

(16)

where

   Ωðx2 Þ ≡ ðð4γ þ 5sÞα  ð4ð1 þ sÞγ þ sð2  7sÞÞx2 Þ2 þ 2s ð3 þ 2sÞα  8γ  6s2  7s þ 6 x2    2  ð4γ  ð6  sÞÞα  4ð1  3sÞγ  3s  14s þ 6 x2  2γðð3 þ 2sÞα  3ð2s þ 1Þð1  sÞx2 Þ2 : Finally, when x1 ¼ x2, a2 ¼ a1 ¼ 0, and each of firms 1 and 2 is accessed by firm S with probability 0.5. Then, the associated profit when x2 is taken as given is:

π 31 ðx2 ; x2 Þ ¼

1 γ 2 2 ðα þ ð2  sÞx2 Þ  ðx2 Þ : 16 2

(17)

Then, firm 1's reaction function is derived by choosing the investment that corresponds with the highest profit among (14), (16), and (17) when x2 is taken as given. We explain the procedure for the comparison among (14), (16), and (17). We denote the intersection of (15) and x1 ¼ x2 by A and the associated investment by xA ð¼ xA1 ¼ xA2 Þ. Similarly, we denote the intersection of (13) and x1 ¼ x2 by B and the associated investment by xB ð¼ xB1 ¼ xB2 Þ. Using (15) and (13), we ensure that

xA ¼

ð3 þ 2sÞα ð3  sÞα ; and xA > xB : ; xB ¼ 8γ  ð6s2  7s þ 6Þ 8γ  ð3  sÞð2  sÞ

20 For a2  a 2 , the expression of a1 ¼ a2  sðx2  x1 Þ ignores the small reduction of ε. That is, firm 1's access pricing strategy is precisely written as a1 ¼ a2  sðx2  x1 Þ ε.

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2 2 A 2 2 A 3 A A 2 A We compare π 11 ðx1 1 ; x2 Þ and π 1 ðx 1 ; x2 Þ at x 2 . First, we note that π 1 ðx 1 ; x 2 Þ ¼ π 1 ðx 1 ; x 2 Þ because x 1 ¼ x 1 . On the contrary, because 1 1 1 1 1 A A 1 1 A 3 A A 2 2 x1 ≡ argmaxπ ðx ; x Þ, π ðx ; x Þ  π ðx ; x Þ for any x . Now, because x ≠x at x , we obtain that π ðx 1 2 2 1 2 1 1 1 1 1 1 1 1 2 1 1 ; x 2 Þ > π 1 ðx 1 ; x 2 Þ ¼ π 1 ðx 1 ; A A 1 x2 Þ. Hence, at x2 , firm 1's optimal strategy is x1 . 2 2 B 1 1 B 3 B B 1 B Similarly, we compare π 11 ðx1 1 ; x2 Þ and π 1 ðx1 ; x2 Þ at x2 . Note that π 1 ðx1 ; x2 Þ ¼ π 1 ðx1 ; x2 Þ because x1 ¼ x1 . On the contrary, because 2 2 2 2 2 B B 2 2 B 3 B B 1 1 x2 ≡ argmaxπ ðx ; x Þ, π ðx ; x Þ  π ðx ; x Þ for any x . Now, because x ≠x at x , we obtain that π ðx 2 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 ; x2 Þ > π 1 ðx1 ; x2 Þ ¼ π 1 ðx1 ; xB2 Þ. Hence, at xB2 , firm 1's optimal strategy is x2 . 1   2 2 ^ 2 because π 11 ðx1 These results indicate that there exists x^ 2 2 x2B ; x2A such that π 11 x11 ; x^ 2 ¼ π 21 x12 ; x 1 ; x2 Þ and π 1 ðx1 ; x2 Þ are monotone in x2. Therefore, firm 1's reaction function, (R1), is derived as follows:

8 ð3  sÞðα  x2 Þ > > > < 8γ  ð3  sÞ2 for x2  ^x 2 ; x1 ðx2 Þ ¼ > ð3 þ 2sÞα  3ð2s þ 1Þð1  sÞx2 > > : for x2  ^x 2 : 8γ  ð9  4sÞ

ðR1Þ

Firm 2's reaction function, (R2), is similarly derived by replacing the identification of firms. That is,

8 ð3  sÞðα  x1 Þ > > for x1  ^x 1 ; > < 8γ  ð3  sÞ2  x2 ðx1 Þ ¼ > > ð3 þ 2sÞα  3ð2s þ 1Þð1  sÞx1 > : for x1  ^x 1 : 8γ  ð9  4sÞ

ðR2Þ

These reaction functions of firm 1 and firm 2 are drawn in Fig. 1. Then, there are two equilibria in the figure; E AAPE1 and E AAPE2 . E AAPE1 (EAAPE2 ) represents the asymmetric access provision equilibrium (AAPE) in which firm 1 (firm 2) is accessed by firm S with probability 1. Simple calculations show that the investments in equilibrium are

xAAPE1 ¼ 1

Φðs; γÞ Ψðs; γÞ and xAAPE1 at E AAPE1 ; ¼ 2 Θðs; γÞ Θðs; γÞ

¼ xAAPE2 1

Ψðs; γÞ Φðs; γÞ and xAAPE2 at E AAPE2 ; ¼ 2 Θðs; γÞ Θðs; γÞ

where

 Θðs; γÞ ≡ ð8γ ð9  4sÞÞ 8γ  ð3  sÞ2  3ð2s þ 1Þð1  sÞð3  sÞ; Φðs; γÞ ≡ 2α 4ð3 þ 2sÞγ  ð3  sÞ 6 þ 3s  4s2 ; Ψðs; γÞ ≡ 2αð3  sÞð4γ  ð6  sÞÞ: All the results are summarized as a Proposition. ■ Proof of Lemma 2. The regulator's problem consists of two subproblems: The first subproblem is to find an optimal access charge a if she allows firm S's entry (i.e., in a triopoly). The second subproblem is to determine whether to allow firm S's entry by comparing the social welfare in a triopoly SW T with a and that in a duopoly SW D. Note that both SW T and SW D are continuous in a (in particular, SW T ¼ SW D at a ≡ ð1 3sÞxk  xl Þ). Hence, we only need to check if a >< a ≡ 13 ðα ð1 3sÞxk  xl Þ to solve the second subproblem. Assuming that x1>x2 so that firm S accesses firm 1's infrastructure when entering the market, we can formulate the regulator's problem to find an optimal access charge a in a triopoly as follows:

1 3 ðα

Max a

¼

SW T ¼ CS þ π 1 þ π 2 þ π S

ðQða; x1 ; x2 Þ Þ2 þ ðq1 ða; x1 ; x2 Þ Þ2 þ aqS ða; x1 ; x2 Þ  Iðx1 Þ þ ðq2 ða; x1 ; x2 Þ Þ2  Iðx2 Þ þ ðqS ða; x1 ; x2 Þ Þ2 2

s:t:

0aa≡

1 ðα  ð1  3sÞxk  xl Þ 3

~ as follows. Ignoring the constraints of the problem, we have an interior solution a

~a ¼ α þ ð5  11sÞx1 þ 5x2 : ~  0, a ¼ 0. If 0 < a ~  a, a ¼ a ~ . If a ~ > a, a 2 ½a; ∞Þ. The Then, the optimal access charge a is characterized as follows: If a   ~ ~  0 and 0 < a  a are rewritten as conditions that a

~a  0

⇔

1 x2  ðα  ð5  11sÞx1 Þ; 0 < ~a  a 5

⇔

15

1 1 ðα  ð5  11sÞx1 Þ < x2  ðα  ð4  9sÞx1 Þ; 5 4

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which is the statement of the Lemma. ■ Proof of Lemma 3. According to Lemma 2, the optimal access charge a set by the regulator depends on the levels of the two facility-based firms' investments. Hence, we examine the range of the investments that can be realized under Assumptions (ii) and (iii). For s 2 ½9∕11; 2Þ, we ensure that x1 and x2 that satisfy (α  (5  11s)x1)∕5x2 or (α  (5  11s)x2)∕5x1 cannot be within the range of x1  α and x2  α. In fact, under Assumption (ii) (i.e., γ  2), neither of the facility-based firms has an incentive to invest more than α even in a duopoly. Thus, the relevant case is only the one where ðα ð5 11sÞxk Þ∕5  xl and xk > xl (k,l ¼ 1,2, and k≠l). Therefore, a ¼ 0 for s 2 ½9∕11; 2. ■ Proof of Proposition 3.  As shown in Lemma 3, the regulator sets the cost-based access charge, i.e., a ð¼ a 1 ¼ a2 Þ ¼ 0 in the second stage. We then examine the investment game between firms 1 and 2 in the first stage under a cost-based access regulation. With a ¼ 0, the conditions under which firm S accesses firm 1 are

x1  x2 and α  ð1  3sÞx1  x2  0:

(18)

Similarly, the conditions under which firm S accesses firm 2 are

x2  x1 and α  ð1  3sÞx2  x1  0:

(19)

The conditions under which firm S does not enter the market are

α  ð1  3sÞx1  x2 < 0 and α  ð1  3sÞx2  x1 < 0:

(20)

We can easily verify that under Assumption (iii) (i.e., s 2 ½9∕11; 2Þ), there do not exist x1  0 and x2  0 that satisfy (20). Hence, the foreclosure cannot occur in the access regulation regime. We examine firm 1's strategy for investment. There are three cases; x1x2, and x1 ¼ x2. When x1
Max x1

 AR   2 π AR1  Iðx1 Þ: 1 ðx1 ; x2 Þ ¼ q1 a2 ; x1 ; x2

Using (2), we derive the following reaction function.

x1AR1 ðx2 Þ ¼

3ðα  ð1 þ sÞx2 Þ : 8γ  9

(21)

The associated profit when x2 is taken as given is

 AR1 γðα  ð1 þ sÞx2 Þ2 : π AR1 x1 ðx2 Þ; x2 ¼ 1 2ð8γ  9Þ

(22)

When x1>x2, firm 1 is accessed by firm S. Thus, firm 1's problem is formulated as follows:

Max x1

 AR   2 π AR2  Iðx1 Þ: 1 ðx1 ; x2 Þ ¼ q1 a1 ; x1 ; x2

Using (1), we derive the following reaction function.

x1AR2 ðx2 Þ ¼

ð3  sÞðα  x2 Þ : 8γ  ð3  sÞ2

(23)

The associated profit when x2 is taken as given is

 AR2 γðα  x2 Þ2 π AR2 x1 ðx2 Þ; x2 ¼  ; 1 2 8γ  ð3  sÞ2

(24)

Finally, when x1 ¼ x2, firm 1 is accessed by firm S with probability 0.5. Then, the associated profit when x2 is taken as given is

π AR3 1 ðx2 ; x2 Þ ¼

1 γ ðα þ ð2  sÞx2 Þ2  ðx2 Þ2 : 16 2

(25)

Firm 1's reaction function is derived by choosing the investment level that corresponds with the highest profit among (22), (24), and (25) when x2 is taken as given. ðx2 Þ and x1 ¼ x2 by A and the associated investment by xARA ð¼ xARA ¼ xARA We denote the intersection of xAR1 1 1 2 Þ. Similarly, we AR2 ARB ARB ARB denote the intersection of x1 ðx2 Þ and x1 ¼ x2 by B and the associated investment by x ð¼ x1 ¼ x2 Þ. Using (21) and (23), we ensure that

16

N. Matsushima, K. Mizuno

xARA ¼

Telecommunications Policy xxx (2017) 1–18

ð3  sÞα 3α ; xARB ¼ ; and xARA > xARB : 8γ  3ð2  sÞ 8γ  ð3  sÞð2  sÞ

We also ensure that the slope of xAR2 ðx2 Þ is steeper than that of xAR1 ðx2 Þ in (x1, x2) plane. Then, it is apparent that for x2  xARA 1 1 2 , AR1 AR2 ARB firm 1's optimal strategy is x1 ðx2 Þ because firm 1 cannot take x1 ðx2 Þ for x2  xARA 2 . By the same reasoning, for x2  x 2 , firm 1's ðx2 Þ or x1AR2 ðx2 Þ, so it sets x1 ¼ x2. optimal strategy is x1AR2 ðx2 Þ. For x2 2 ðxARB ; xARA Þ, firm 1 can take neither xAR1 1 In sum, firm 1's reaction function, (ARR1), is derived as follows:

ðARR1Þ

8 > 3ðα  ð1 þ sÞx2 Þ 3α > > > for x2  ; > > 8γ  9 8γ  3ð2  sÞ > > > > < ð3  sÞα 3α < x2 < ; xAR ðx2 Þ ¼ x2 for 1 8γ  ð3  sÞð2  sÞ 8γ  3ð2  sÞ > > > > > > ð3  sÞα > ð3  sÞðα  x2 Þ > > : for x2  > : 8γ  ð3  sÞ2 8γ  ð3  sÞð2  sÞ

Firm 2's reaction function, (ARR2), is similarly derived by replacing the identification of firms. That is

ðARR2Þ

8 > 3ðα  ð1 þ sÞx1 Þ 3α > > > for x1  ; > > 8γ  9 8γ  3ð2  sÞ > > > > < ð3  sÞα 3α < x1 < ; xAR ðx1 Þ ¼ x1 for 2 8γ  ð3  sÞð2  sÞ 8γ  3ð2  sÞ > > > > > > ð3  sÞðα  x1 Þ ð3  sÞα > > > : for x1  > : 8γ  ð3  sÞ2 8γ  ð3  sÞð2  sÞ

These reaction functions of firm 1 and firm 2 are drawn in Fig. 2. From the figure, we ensure that there are multiple equilibria with ¼ aARE Þ ¼ 0, and each of the x1ARE ¼ x2ARE 2 ½xARB ; xARA  in the access regulation regime. In each of the equilibria, aARE ð¼ aARE 1 2 facility-based firms is accessed by firm S with probability 0.5, irrespective of the degree of spillover and the investment cost. ■ Proof of Proposition 4. Let us denote the total investment in the free competition regime by TX AAPE and that in the access regulation regime by TX ARE . Because there are multiple equilibrium investments in the access regulation regime, as mentioned in Proposition 3, we take the largest investments among them for the comparison with those in the free competition regime. Then, we have

TX AAPE ¼

4α ð2ð6 þ sÞγ  ð3  sÞð2s þ 3Þð2  sÞÞ; Θðs; γÞ

TX ARE ¼

6α ; 8γ  3ð2  sÞ

where Θ(s,γ) is defined in Proposition 2. Taking the difference between TX AAPE and TX ARE gives

8sα χðγ; sÞ; where Θðs; γÞð8γ  3ð2  sÞÞ  χðγ; sÞ ≡ 8γ 2  8s2  37s þ 36 γ þ 3ð3  sÞð4  sÞð1  sÞ: ΔTX ≡ TX AAPE  TX ARE ¼

From now on, we show the claim that χ(γ; s) > 0 for γ  2 and s 2 ½9∕11; 2Þ (i.e., under Assumptions (ii) and (iii)). Because χ(γ; s) is a quadratic function of γ, it would take the minimum at γ  , where

γ ¼

1  2 8s  37s þ 36 ; 16

if there is no constraint on the range of γ. For s 2 ½9∕11; 2Þ, we have γ  2 ð 0:375; 0:69. Hence, χ(γ; s) takes the minimum at γ ¼ 2 under the constraint of γ  2. Then, it is sufficient to show that χ(γ; s) > 0 at γ ¼ 2 to ensure the claim. Substituting γ ¼ 2 into χ(γ; s) gives

χð2; sÞ ¼ 3s3 þ 8s2 þ 17s  4 ≡ ZðsÞ: Here, Z(s) has a minimum at s ¼ 9∕11 under the constraint of s 2 ½9∕11; 2Þ. In fact, we have Zð9∕11Þ ¼ 11; 714∕1; 331ð≃8:8Þ > 0. Therefore, we ensure the claim that χ(γ; s) > 0 for γ  2 and s 2 ½9∕11; 2Þ. That is, TX AAPE > TX ARE for γ  2 and s 2 ½9∕11; 2Þ. ■ References Alexiadis, P., & Shortall, T. (2016). The advent of 5G: Should technological evolution lead to regulatory revolution? CPI Antitrust Chronicle November, 3. Available at: SSRN https://ssrn.com/abstract¼2876484. 17

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