Going beyond duopoly: Connectivity breakdowns under receiving party pays

Information Economics and Policy 36 (2016) 1–9

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Going beyond duopoly: Connectivity breakdowns under receiving party paysR Steffen Hoernig∗ Nova School of Business and Economics, Universidade Nova de Lisboa, Campus de Campolide, 1099-032 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Received 8 July 2015 Revised 16 May 2016 Accepted 6 June 2016 Available online 7 June 2016 JEL Classification: L13 L51

a b s t r a c t We show that the prediction of strategic connectivity breakdowns under a receiving-party-pays system and discrimination between on- and off-net prices does not hold up once more than two mobile networks are considered. Indeed, if there are at least three competing networks and enough utility is obtained from receiving calls, no strategic connectivity breakdowns occur. Private negotiations over access charges then achieve the efficient outcome. Bill & keep (zero access charges) and free outgoing and incoming calls are efficient if and only marginal costs of calls are zero. © 2016 Elsevier B.V. All rights reserved.

Keywords: Mobile network competition Receiving party pays Connectivity breakdown Termination rates

1. Introduction The regulation of mobile termination rates (MTRs or access charges, the fees that mobile networks receive from their competitors to terminate calls) in the European Union has come a long way over the last decade, moving away from the paradigm of full network cost recovery towards an approach based on recovering only the incremental cost of termination. The advent of next-generation networks and IP interconnection have made the analysis of zero access charges (bill & keep) on mobile networks more urgent. Indeed, in 2009 the communications regulator in the United Kingdom, Ofcom, held a consultation about the future regulation of MTRs, explicitly mentioning bill & keep as one of the options to consider.1 While charging very low MTRs is standard practice in the US, in Europe there has been some anxiety about the effects of MTR reductions on the mobile telephony market, in particular because the US has an RPP (receiving party pays) retail model. Contrary to

R This paper is a renamed and extended version of Hoernig (2012). I would like to thank Carlo Cambini and Tommaso Valletti for valuable comments. ∗ Corresponding author. Fax +351-213870933. E-mail address: [email protected], [email protected] 1 See European Commission (2009) on incremental cost and Tera (2010) report for the European Commission on Bill & Keep. The Ofcom consultation and responses are available at http://stakeholders.ofcom.org.uk/consultations/mobilecallterm (as of 27/09/2012).

http://dx.doi.org/10.1016/j.infoecopol.2016.06.003 0167-6245/© 2016 Elsevier B.V. All rights reserved.

the CPP (calling party pays) model common in Europe, under RPP subscribers pay both for making and receiving calls. While opponents of RPP claim that consumers should not pay for calls they receive if they already pay for making calls, they overlook that, in a nutshell, paying for reception tends to go together with paying less for making calls. More worrying are theoretical results in the academic literature which indicate that with a high likelihood RPP will lead to ”connectivity breakdowns” due to strategic considerations (Jeon et al. (2004), JLT). These breakdowns are predicted to occur exactly in the most realistic case considered, i.e. under price discrimination between on- and off-net calls (within or between networks, respectively) and unregulated receiver prices. The analysis has been performed for duopoly, and each firm has a strategic incentive to reduce the surplus of its rival’s subscribers by shutting off crossnetwork traffic through prohibitively high call or receiving prices. That this kind of pricing behavior has not been observed in reality may well be due to regulatory pressure not included in the original JLT modeling framework, but it seems important to check whether the theoretical prediction is robust in the first place. In this paper we demonstrate that it is not robust to a change in the number of firms: going beyond duopoly leads to a major change in predictions. Results and intuitions. Below we show that strategic connectivity breakdowns are not robust to an increase in the number of networks, as there is a significant range of fundamental parameter

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values for which no strategic breakdown occurs if and only if the number of networks is larger than two. This range only depends on the relation between the strength of the ”call externality”, i.e. the utility that call receivers obtain, and the number of networks. More precisely, if the parameter β measures the strength of the call externality on a scale from 0 (no call externality) through 1 (the receiver obtains the same utility as the caller), with n > 2 symmetric networks the range of β where no strategic connectivity breakdowns occur is given by

1 < β < n − 1. n−1

(1)

Clearly, in duopoly (n = 2) the lower and upper bounds are equal and no value of β satisfies this condition. Already with three networks no strategic breakdown occurs for 1/2 < β < 2, and the interval of call externalities without breakdown increases with the number of competing networks. The intuition behind the effect of the number of networks is the following. Strategic breakdowns are caused by networks setting either a very high call or receiver price for cross-network traffic. First consider an equilibrium where the breakdown is caused by a high price for making off-net calls, while at the corresponding receiving price subscribers would accept to receive calls. This calling price is the result of very high ”strategic marginal cost”, i.e. the combined value of the marginal cost of network usage and the externalities bestowed on rival networks’ customers, including the call externality. The key to understanding the pricing incentives in this case is the size of the externality as compared to the marginal utility of the caller: If the strategic externality becomes too large, the strategic marginal cost outweighs the corresponding benefits, and the network maximizes its profits by causing a connectivity breakdown. Yet, as the number of networks grows, the total externality received by any specific rival network becomes smaller, and thus the externality component of strategic marginal cost declines. Once strategic marginal cost falls below a certain level (we show below that this happens when β < n − 1) the call price will be set low enough and no strategic breakdown occurs. In a similar fashion, consider now a strategic breakdown caused by a very large receiving price, while the off-net call price is finite. In this case the strategic marginal cost of receiving calls, which includes the externality imposed on the rival networks’ customers by accepting an incoming call, is very high. Similar to the previous case, as the number of networks increases, the externality on each single rival network becomes smaller, and below a certain level the strategic marginal cost and receiver price take on low enough values. The exact condition for this to happen is β > 1/(n − 1 ). As a second step we study the jointly profit-maximizing and socially optimal levels of access charges. We find that the latter coincide and are at a level below termination cost but generally different from zero. This implies on the one hand that under RPP the regulation of access charges is no longer necessary, but on the other that firms would not agree on bill & keep unless the marginal costs of origination and terminating calls are effectively zero. Indeed, in the latter case the market would settle on both bill & keep and free incoming and outgoing calls (”bucket pricing”). Finally, in an extension section, (i) we show that larger networks have higher incentives for provoking connectivity breakdowns, (ii) consider the efficiency of bill & keep, and (iii) investigate non-negative reception charges. Previous literature. JLT consider competition between two mobile networks under call externalities and payments for receiving calls. They consider combinations of uniform pricing (the same price is charged for on-net and off-net calls) vs. price discrimination, and regulated vs. unregulated receiver charges. With the latter and uniform pricing , JLT find that call and reception charges are set at off-

net cost (thus at a finite level), and that the socially optimal volume of calls can be achieved by setting the mobile access charges below termination cost. On the other hand, with discrimination between on-net and off-net calls, which is the more realistic case that we also consider here, connection tends to break down in equilibrium, regardless of the strength of the call externality: There are always equilibria where for strategic reasons networks choose to set either call or receiver charges so high that no off-net calls will occur. Lopez (2011) confirms the result of JLT in a setting with noise in both caller and receiver utility. On the other hand, Kim and Lim (2001) assume that the originating network sets the price for receiving calls and show that in this (unrealistic) case no breakdown occurs. We show that the duopoly model does not lead to a robust prediction of market outcomes, as with more than two networks a new class of equilibria without breakdowns appears for feasible values of the call externality parameter. Hurkens and López (2014) independently derive results similar to those of the present paper, under the assumption of ”passive beliefs” where consumers’ expectations about network sizes do not adjust to firms’ out-ofequilibrium tariff offers. Cambini and Valletti (2008) show that the possibility that call receivers phone back reduces the probability of breakdowns in duopoly while not eliminating them unless networks can jointly set access charges. They also show that if callers end calls first then networks will jointly choose the efficient access charge. Our paper obtains the same result for the more general case where either callers or receivers end calls, and for more than two firms.2 Hermalin and Katz (2011) assume that networks commit to subscriber numbers before setting retail prices, which decouples call pricing decisions from competition for subscribers. As a consequence, strategic considerations are reflected only in setting the fixed part of tariffs, and no connectivity breakdowns occur in their model. Some authors, such as Littlechild (2006) and Dewenter and Kruse (2011), interpret RPP as conflating payment for incoming calls and the imposition of bill & keep, resulting in the idea that the receivers of calls pay termination charges. The latter would then be subject to consumer choice and competition, which might help to keep them low. Here we interpret RPP differently: The receiver price is a missing price under CPP, while it is charged under RPP; simultaneously, the access charge is a wholesale price that is either chosen by networks or set by a regulator. Indeed, since for call externality values in the range (1) no strategic connectivity breakdown occurs, we can meaningfully study the negotiated and efficient levels of the access charge. DeGraba (2003) determines socially optimal call and receiving prices and access charges, but does not check whether these could actually be implemented in market equilibrium. We show that if the call externality value is in the correct range then the market equilibrium at the efficient access charge implies that both call and receiver prices are set efficiently. Our work shows that it is necessary to consider both the number of networks and the strength of the call externality to come to this conclusion. Littlechild (2006) and Harbord and Pagnozzi (2010) present stylized facts and policy arguments concerning the RPP versus CPP regimes, while Dewenter and Kruse (2011) present an econometric analysis of mobile penetration. Overall, their conclusions are that CPP and RPP lead to similar mobile penetration, while usage tends to be higher under RPP. There is no mentioning of breakdowns

2 It can be shown that combining a larger number of networks with the possibility of calling back further increases the range of equilibria. A referee also pointed out that one should take further JLT’s suggestion to consider cooperative calling behavior, and explore the consequences of different specifications.

S. Hoernig / Information Economics and Policy 36 (2016) 1–9

having ever happened, which underlines the need to have theory models that predict finite (or even zero) call prices. 2. Model setup Our model setup is a generalization to many networks of JLT’s case of price discrimination between on- and off-net calls and unregulated receiver charges (Section 5 in their paper). It is precisely in this case that connectivity breakdown occurs. Our basic model setup is also significantly more general, in that we have a adopted a rather general formulation subscription demand. The actual derivations of the equilibrium conditions are identical to those in JLT. Therefore, in the main text we only present the model setup and the discussion of the equilibrium conditions; all derivations are relegated to the Appendix. We assume that there are n ≥ 2 symmetric mobile networks i = 1, . . . , n which compete in multi-part tariffs of the form  pi , ri , pˆ i , rˆi , Fi , where pi and ri are the per-minute calling and reception charges for on-net calls, pˆ i and rˆi those for off-net calls, and Fi is a monthly fixed fee. Networks’ marginal on-net cost of a call is c > 0, the cost of termination of an off-net call is c0 > 0, and networks charge each other the access charge a per incoming call minute. Thus the marginal cost of an off-net call is c + m, where m = a − c0 is the termination margin. There is also a monthly fixed cost f per customer. Market shares are given by the following demand system. If consumers obtain surplus wi from subscribing to networks i = 1, . . . , n, the market share of network i is

αi = A(wi − w1 , . . . , wi − wi−1 , wi − wi+1 , . . . , wi − wn ).

(2)

Here A : Rn−1 → R is differentiable, strictly increasing at 0, and n symmetric in its arguments, with 0 ≤ α i ≤ 1, i=1 αi = 1 and A(0, . . . , 0 ) = 1/n.3 From making a call of length q, a consumer obtains utility u(q), where u(0 ) = 0, u > 0 and u  < 0. For call price p, the corresponding call demand q(p) is defined implicitly by u (q ) = p. Receiving a call of length q yields utility u˜ (q ) = β u(q ) + ε q. Here β ≥ 0 indicates the strength of the call externality, and ε is a random ”noise” term with zero expected value.4 For each caller and receiver pair, the socially optimal call volume q would be given by u (q ) + u˜ (q ) = c, i.e. where expected marginal benefits are equal to marginal cost. Since the latter depends on ε , in the presence of noise the social optimum cannot be achieved. For vanishing noise, the condition for optimal call volume becomes u (q ) = c/(1 + β ). This optimal call volume can be implemented at call and receiving prices p∗ = c/(1 + β ) and r ∗ = β c/(1 + β ), since both callers and receivers then end calls simultaneously at the optimal quantity. Define the access charge level

a∗ = c0 −

βc . 1+β

(3)

We will see below that a∗ implements the efficient outcome at equilibrium call prices, as discussed in DeGraba (2003) and Cambini and Valletti (2008) in different settings. Denote the expected length of a call by D(p, r), and the corresponding utilities of making or receiving a call as U(p, r) and

3

This demand specification encapsulates both the generalized Hotelling model of   Hoernig (2014) and the logit model αi = exp(wi )/ nj=1 exp(w j ) = 1/ nj=1 exp(w j − wi ). Thus we allow for a much more general discrete choice setup than JLT. 4 The existence of randomness in receiver utility implies that both callers and receivers may determine the length of calls. If call length was determined by only callers or receivers then in equilibrium either the calling or receiving price would be indeterminate. In studying equilibrium conditions, JLT then let the noise converge to zero.

3

U˜ ( p, r ), respectively. In order  to reduce notational complexity, denote Dii = D( pi , ri ), Di j = D pˆ i , rˆ j , and similar for utilities. We can now express the expected surplus of a subscriber to network i, assuming a uniform calling pattern, as

wi =

  αi Uii + U˜ii − ( pi + ri )Dii    + α j Ui j + U˜ ji − pˆ i Di j − rˆi D ji − Fi . j=i

The first term contains the utility and payments for making and receiving on-net calls, while the second term refers to off-net calls to and from networks j = i. Correspondingly, network i’s profits are

πi = αi {Fi − f + αi ( pi + ri − c )Dii       + α j pˆ i − c − m Di j + rˆi + m D ji }.

(4)

j=i

The first line contains the profits from fixed fees and on-net calls, and the second line those from incoming and outgoing off-net calls. 3. Equilibrium outcomes In this model, there are two types of equilibria. First, there are equilibria where both off-net calling and reception charges are infinite. These can be interpreted as arising due to a coordination failure: If the reception charge on the other networks is infinite, then the call demand will still be zero even for a finite calling charge, and vice versa. In the following we will concentrate on the second type of equilibrium, where at least one of these prices is finite. Connectivity breakdown then occurs exclusively due to strategic motives, when networks choke off either outgoing or incoming calls. In the Appendix we lay out the derivation of the conditions describing the equilibrium call prices. These derivations are rather complex and follow JLT very closely, apart from having a larger number of firms, and are therefore not included in the main text. Rather, we explain the effect of the number of networks on the equilibrium conditions, so as to highlight the economic intuition behind the absence of connectivity breakdown without the accompanying mathematical complexity. As one might expect, the number of networks has no effect on how on-net calls are priced: We obtain the usual result that on-net prices are set at the efficient levels p = p∗ and r = r ∗ . This follows from the known logic under multi-part tariffs that usage prices are set to maximize total surplus. Now we turn to off-net call and receiving prices. First consider equilibrium outcomes where callers end the call first. In this case it is the price of making off-net calls, pˆ, that determines call length. Networks set this price by equating a customer’s marginal utility from making a call to its marginal costs. The relevant marginal costs, as has been pointed out by JLT, contain strategic elements, over and above the purely technical cost elements. For this reason, they have coined the term ”strategic marginal costs”, which in the present case contain three elements:5

u (q( pˆ )) = c + m +

 1   β u (q( pˆ )) − rˆ . n−1

(5)

Here u and β u are the caller’s and receiver’s marginal utilities, respectively, and c + m is (technical) off-net cost. The last term on the right is the strategic term: It describes the expected benefit of a customer of the (n − 1 ) competing networks from this outgoing call. If he receives this call then the direct utility is β u , from 5

This condition is obtained from equation (9) in the Appendix.

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which he deducts the receiver charge rˆ, and finally the probability of receiving this charge is 1/(n − 1 ). This term thus describes a utility that accrues to customers of competing networks, which the first network needs to match through lower fixed fees if it wants to retain the marginal customer. This cost exceeds own consumers’ benefit u if and only if β > n − 1, and thus the network will want to impede off-net calls by setting pˆ to infinity. On the other hand, for β < n − 1 there is a well-defined interior optimum at a finite off-net price. Clearly, the condition β < n − 1 becomes less strict as the number of networks is larger, therefore there is more room for equilibrium without connectivity breakdown. Similarly, if receivers end the call first, then the network maximizes profits by equating the marginal utility of receivers to its strategic marginal cost of receiving calls,6

β u (q(rˆ/β )) = −m +

 1   u (q(rˆ/β )) − pˆ . n−1

(6)

The direct technical ”cost” −m is given by the profits from terminating a call, while the strategic term consists of a rival network consumer’s utility from making the call minus the call price, adjusted for the probability of being on a specific rival network. As above, the network must compensate its consumers for this additional utility that they could obtain by switching to a competitor. This strategic cost is larger than the network’s own receiver utility if β < 1/(n − 1 ), and the network will set an infinite receiver price to cut off incoming calls. On the other hand, for β > 1/(n − 1 ), receiver utility is larger and the network maximizes profits by setting a finite receiving price. Again, connectivity breakdown becomes less likely for a larger number of networks. Finally, note that in the case n = 2 considered by JLT the first condition becomes β < 1, while the second condition is β > 1. Thus duopoly is just the knife-edge case where connectivity breakdown always occurs. Now we can state our first result. Proposition 1. Let ε be regularly distributed, and n > 2 networks compete in multi-part tariffs with on/off-net price discrimination. As ε 1 vanishes, for n−1 < β < n − 1 there is no strategic connectivity breakdown in symmetric equilibrium.7 Depending on the access charge level, we obtain the following equilibrium prices: n−1 c+mn 1. for a > a∗ , callers end the call first, with pˆ = pˆc ≡ ( n−1) −β > p∗

and rˆ = rˆc ≡ −m < r ∗ ; 2. for a = a∗ , callers and receivers end the call simultaneously, with pˆ = p∗ and rˆ = r ∗ ; 3. for a < a∗ , receivers end the call first, with pˆ = pˆr ≡ c + m < p∗ β (c+nm ) and rˆ = rˆr ≡ − (n−1 > r∗ . )β −1 The configurations of equilibria with n = 2 and n > 2 are depicted in Figs. 1 and 2, respectively. Fig. 1 represents the result of JLT. With two networks, for all combinations of call externality values β and access charge levels a there are equilibria with strategic connectivity breakdown (areas indicated by horizontal lines). For β < 1 the receiving price is infinite, while for β > 1 the call price is infinite. Simultaneously, equilibrium candidates exist where no breakdown occurs; these lie strictly above (below) the dashed line to the left (right) of β = 1, and are indicated by diagonal lines.8 With more than two networks, this structure of breakdowns carries over to either very small β , i.e. β < 1/(n − 1 ), or very large β , i.e. β > n − 1, while for intermediate values β a whole new range of equilibria opens up. In this range, for a > a∗ callers end

the call first, while for a < a∗ receivers end the call first. Crucially, no strategic connectivity breakdown occurs, and for all values of β in this range a market equilibrium has finite prices, including when the access charge is set efficiently at a = a∗ . Moreover, this region includes realistic values for the call externality at the prevailing number of networks in most countries. For example, with three or four networks, there is no connectivity breakdown for 1/2 < β and 1/3 < β , respectively (Evidently, if one follows the common assumption that β ≤ 1 then only the lower bound is relevant in practice). An additional significant piece of good news is that with more than two networks the efficient call volumes can be achieved by setting the access charge equal to a∗ , without having to fear strategic connectivity breakdowns. Indeed, a look at Fig. 1 shows that the same is not true in duopoly: The line indicating a = a∗ only passes through areas where breakdown is unavoidable. This implies that while in duopoly efficient call volumes can only be achieved if access and receiver charges are regulated, with more networks it is enough to set the access charge at the right level and let the market choose equilibrium retail prices. As a further point, we consider how call and receiver prices change as a function of the number of networks: 1 Corollary 1. Let n−1 < β < n − 1. For all n > 2, rˆc and pˆr are equal to off-net cost. As n increases, pˆc and rˆr converge from above to offnet cost.

Proof. Follows from the expressions in Proposition 1.



This Corollary implies that the relevant charges, i.e. pˆc when callers end the call first and rˆr when receivers do so, are higher than they would be under uniform pricing, where even with many networks charges continue to be equal to off-net cost as in JLT.9 In other words, if there is no connectivity breakdown, for strategic reasons fewer off-net calls will be made with discrimination between on-net and off-net calls, just as in the case without receiver charges. As the number of networks increases, though, more calls will be made off-net and therefore there are fewer incentives to distort off-net call prices upwards (the externality term in (5) disappears), and the call pricing structure approaches that under uniform pricing. Finally, we consider which access charge maximizes networks’ joint profits, i.e. which is the access charge that networks would choose if they were to negotiate between themselves. To our knowledge, only Cambini and Valletti (2008) have previously considered this issue under RPP with discriminatory pricing, since all other authors found that only equilibria with connectivity breakdowns existed, including JLT.10 They found that if there is a positive probability that call receivers make return calls then equilibria without breakdown exist, and that if callers end calls first then a network duopoly would agree on the efficient access charge. We show that this finding carries over to the case of multiple firms in our setting (The proof is relegated to the Appendix): 1 Proposition 2. Let n−1 < β < n − 1. For all n > 2, the jointly profitmaximizing access charge is equal to the efficient level, i.e. a = a∗ .

The intuition behind this result is that an access charge above or below the efficient the efficient level creates network externalities that make operators compete more fiercely, and thus low-

6

See (10) in the Appendix. Similar to JLT, these equilibria exist if either m is sufficiently close to zero or if networks are sufficiently differentiated. 8 JLT do not formally prove that these are indeed equilibria; they only show that all other candidates involve connectivity breakdown. The dashed line is given by πirˆ (rˆr ) = 0 (πipˆ ( pˆc ) = 0) to the left (right) of β = 1, as defined in the Appendix. 7

9

The proof is straightforward and therefore omitted. Hurkens and López (2014) independently derived profit-maximizing access charges in their model and found the same result as in our paper. 10

S. Hoernig / Information Economics and Policy 36 (2016) 1–9

5

Fig. 1. Equilibria in duopoly.

Fig. 2. Equilibria with at least three networks.

ers profits. Take a higher access charge, a > a∗ : The sum of offn−1 c+ 1+β m net call and receiver prices, pˆc + rˆc = ( n)−1(−β ) , increases with a higher access charge a, while on-net prices remain constant. This larger wedge between on-net and off-net prices strengthens tariff-mediated network effects, and as Gans and King (2001) have shown this makes networks compete harder. They can thus benefit from jointly agreeing on a lower access charge a ≥ a∗ . Interestingly, the opposite logic holds for an access charge below a∗ : Here the n−2 β −1 c−(1+β )m sum of off-net prices, pˆr + rˆr = (( )(n−1 ))β −1 , increases with a lower access charge a. Thus network effects become stronger if a is chosen below a∗ , and networks can make higher profits it they agree on an access charge exactly equal to a∗ . This outcome implies that if mobile networks were to adopt RPP, regulatory determination of termination charges would no longer be necessary. It would be enough to impose that access charges must be reciprocal and then let networks negotiate. The outcome would be efficient, with both call and receiver charges

set at their socially optimal levels. This outcome would also be obtained if networks were prohibited to discriminate between on-net and off-net calls, but in the present case even this regulatory imposition would not be needed. Contrary to what some authors have claimed (e.g. Dewenter and Kruse, 2011; Littlechild, 2006), networks would not agree on bill & keep (zero access charges), though, as the efficient access charge a∗ is (for almost all parameter combinations) different from zero – unless one whishes to argue that network costs are indeed identically zero. 4. Additional issues Asymmetric networks.. Here we give a quick stab at the question of how strategic connectivity breakdown depends on networks’ relative sizes. For simplicity, we continue to assume that networks j = i are symmetric, thus derivatives (7) and (8) in the Appendix still apply to network i even if α i is different from 1/n in equilibrium.

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For vanishing noise, the derivatives defining the off-net call and receiver prices that influence call duration, i.e. pˆci and rˆir , become

βαi αi pˆ i + rˆ − c − m q ( pˆ i ), 1 − αi 1 − αi    rˆ ∂πi αi rˆi αi ∼ pˆ + 1 − rˆ + m q . 1 − αi β ∂ rˆi (1 − αi )β i

∂πipˆ ∼ ∂ pˆ i



cannot set negative reception charges then if a > c0 the symmetric equilibrium involves off-net receiving price r˜ = 0, the off-net call price m p˜ = 1−βc+ if β < n − 1, and p˜ = ∞ if β > n − 1. /(n−1 )

1−

Note that call the externality terms depend on the relative size of network i as measured by αi /(1 − αi ): these terms gain in importance as α i increases. Reframing the conditions for the existence of local maxima in terms of market share α i (i.e. the sign of the derivative must change from positive to negative at the solution), the condition for a finite call price becomes αi < 1+1β , β while the condition for a finite receiver price becomes αi < 1+ β αi (or β > 1−α ). The latter condition is stricter in the more relevant i

case β < 1. We find in this indicative example that larger networks have a stronger incentive to cause strategic connectivity breakdowns, by setting a high off-net receiver price. Put differently, for any given number of networks, the risk of connectivity breakdown increases with the relative asymmetry between networks, much as in the case without receiver charges, and the exact condition for no conαi nectivity breakdown is β > maxi 1−α . i

Optimality of bill & keep. An unavoidable question is whether and when bill & keep (a = 0) can achieve the social optimum. This question has been hotly discussed in Europe under the CPP system, and now we pose it assuming RPP. First of all, even at the risk of repeating ourselves, we would like to stress that this question could not have been meaningfully posed in the duopoly case. With multiple networks, though, fine-tuning of the access charge becomes possible. 1 Corollary 2. Let n−1 < β < n − 1. If c > 0 then bill & keep is efficient iff β = c0 /(c − c0 ). If c = c0 = 0 then bill & keep is efficient for all β ≥ 0, and equilibrium retail prices are pˆ = rˆ = 0.

Proof. Follows from the definition of (3) and a∗ = 0.



This condition for optimality of bill & keep has been proven before by DeGraba (2003, p. 213), but without considering whether an equilibrium without breakdown exists at all. We add to this condition the certainty that for a large range of values of β no strategic connectivity breakdown occurs. On the other hand, it may be that the marginal cost of both origination and termination are effectively zero, and positive cost values only arise due to the accounting practice of attributing common costs. This argument is only bound to get stronger with the routing of traffic over cheaper IP-based networks. In other words, if marginal costs are indeed zero, under bill & keep the market would move to “pure bucket pricing” , where consumers pay a subscription fee and then make and receive calls for free. Non-negative reception charges. As several authors have pointed out (e.g. Cambini and Valletti (2008); Lopez (2011), networks may not find it possible to set negative reception charges (payments to consumers for receiving calls), since the latter may invite arbitrage or opportunistic behavior by clients.11 In this case the restriction rˆ ≥ 0 is binding whenever the access charge is high enough. Corollary 3. In the (unconstrained) symmetric equilibrium, networks will choose positive reception charges if and only if a < c0 . If networks 11 For example, in Portugal and Italy, where such tariffs existed for a while, users were using office phones to ”charge” their own mobile accounts.

Proof. In the unconstrained equilibrium, rˆ < 0 only occurs in the case a > a∗ , where rˆ = −m. Thus rˆ < 0 if and only if m > 0. The expression for p˜ follows from the first-order condition for the offnet price in Proposition (1) with r˜ = 0.  Assuming CPP from the outset, JLT and Hoernig (2014) derive the same pricing formula for n = 2 and arbitrary n ≥ 2, respectively.

5. Conclusions In this paper we have shown that the stark prediction in Jeon et al. (2004) of a strategic connectivity breakdown under RPP (receiving party pays, i.e. subscribers also pay for receiving calls) and discrimination between on- and off-net prices does not hold up once more than two networks are considered in the model. Indeed, for feasible values of the call externality, connectivity breakdowns for strategic reasons do not arise in symmetric equilibrium. Intuitively, in the presence of multiple rivals it becomes essential that off-net calls, both incoming and outgoing, are priced reasonably, while strategic externalities and their associated costs lose importance. Since we have shown when call prices obtain finite values, one can now ask policy-relevant questions. Our main take-away from a policy perspective is that an access charge value below cost is socially optimal in the presence of RPP, and that under RPP networks would voluntarily choose this value if they were able to negotiate a reciprocal access charge. It also follows that bill & keep is exactly socially optimal if marginal costs are zero, and that the associated RPP call pricing model includes free calls made and received. This lends further support to regulatory policies that induce access charges at or close to zero, at least in a transition phase.

Appendix Model setup and proof of Proposition 1: Proof. Here we reproduce the technical assumptions and computations of JLT, adapted to the many-firm case, for readers who want to refresh their memory of the latter’s treatment. Let G be the distribution function of ε and g its density, on a support [ε , ε¯ ] that is large enough so either callers or receivers end calls. From u˜ (q ) = β u(q ) + ε q it follows that, with a reception price r, utilitymaximizing receiver demand is determined by β u (q ) + ε = r. Both callers and receivers can end the call, thus for each caller and receiver pair the length of a call is given by q(max { p, (r − ε )/β} ). Since for high (small) values of ε the caller (receiver) hangs up first, the expected length of a call is given by

D( p, r ) = [1 − G(r − β p)]q( p) +



r−β p

ε

q

r − ε β

g( ε ) d ε .

In the first term on the right callers hang up first ( p > (r − ε )/β or ε > r − β p), and thus a call of length q(p), while in the second term receivers hang up first ( p < (r − ε )/β ) with uncertain call length q( (r − ε )/β ). The corresponding expected utilities for

S. Hoernig / Information Economics and Policy 36 (2016) 1–9

making and receiving calls are

U ( p, r ) = [1 − G(r − β p)]u(q( p)) + U˜ ( p, r ) =

ε¯ r−β p



+

r−β p

ε

  r − ε g( ε ) d ε , β

u q

[β u(q( p)) + ε q( p)]g(ε )dε

r−β p

ε



  r − ε  r − ε  βu q + εq g( ε ) d ε . β β

JLT consider the equilibrium conditions for ”vanishing noise”, i.e. for a sequence of distributions Gn which converges to zero in probability (but whose support remains sufficiently large that both callers or receivers sometimes end the call). They also assume that this sequence is regular in the following sense: Denoting by En the expected value according to Gn , for ε− < 0 < ε+ we have 12

lim En [ε|ε ≤ ε− ] = ε− , lim En [ε|ε ≥ ε+ ] = ε+ .

n→∞

n→∞

We will now determine the call and reception charges that arise in a symmetric equilibrium for n ≥ 2. Assume that   all networks j = i choose the same tariff p, r, pˆ, rˆ, F , resulting in identical market shares α j = (1 − αi )/(n − 1 ), which we will hold constant together with α i while determining call prices. For these symmetric tariffs we have w j = w for all j = i and some expected surplus w. Thus we can state network i’s market share as

αi = A(wi − w, . . . , wi − w ) = A¯ (wi − w ), with the strictly increasing function A¯ . Solving the latter condition for the variable Fi and substituting the result into (4) leads to the following profits of network i:

  πi = αi2 Uii + U˜ii − cDii   +αi (1 − αi ) Uik − (c + m )Dik + U˜ki + mDki   −αi2 Uki − pˆDki + U˜ik − rˆDik + const.

The first line contains the surplus and cost from making and receiving on-net calls, while the second line contains those for offnet calls. The third line indicates the externalities on customers of rival networks, direct ones via the utilities Uki and U˜ik , and pecuniary ones via the payments pˆDki and rˆDik . Finally, there is a term that does not depend on network i’s call prices. The expressions corresponding to on-net calls do not depend on the number of networks. Rather, network i will maximize Uii + U˜ii − cDii as in the duopoly case, which leads to the efficient choices pi = p∗ and ri = r ∗ . This result arises because network i fully internalizes the externalities on callers and receivers. For off-net calls, denote the partial profits related to call prices and receiving prices, respectively, by

   πipˆ ( pˆ i ; αi , rˆ) = αi (1 − αi )(Uik − (c + m )Dik ) + αi rˆDik − U˜ik ,      πirˆ (rˆi ; αi , pˆ ) = αi (1 − αi ) U˜ki + mDki + αi pˆDki − Uki .

While these expressions are ostensibly identical to those found in the duopoly case, allowing for multiple networks will make all the difference. Note that the relative weights on direct effects and those related to externalities on rival networks are (1 − αi ) and α i , respectively. With a larger number of networks, the balance shifts away from externalities and towards direct effects. Since infinite calling or receiving prices choke off demand we pˆ pˆ have πi (∞ ) = πirˆ (∞ ) = 0, so that in equilibrium both πi and πirˆ must be non-negative. The first derivatives with respect to pi and

12 These assumptions imply that analogous conditions hold for any continuous and bounded function of ε (my thanks to Iliyan Georgiev for this observation).

7

ri are

     ∂πipˆ = αi 1 − F rˆ − β pˆ i × {(1 − αi ) pˆ i − c − m ∂ pˆ i     − αi β pˆ i + E ε|ε ≥ rˆ − β pˆ i − rˆ }q ( pˆ i ),

(7)

    F rˆi − β pˆ ∂πirˆ = αi × E[{(1 − αi ) rˆi + m β ∂ rˆi       rˆi − ε rˆ − ε  +αi pˆ − u q } q i ε ≤ rˆi − β pˆ]. β β (8) As noise vanishes, and assuming symmetric market shares αi = 1/n pˆ from now on, we can restate ∂ πi /∂ pˆ i , omitting positive leading factors, as

  pˆ i − c − m − pˆ i − c − m −

or

  1−

β

1 n−1 1 n−1

   β pˆ i − rˆ q ( pˆ i )   β pˆ i + rˆ − β pˆ i − rˆ q ( pˆ i )





1 pˆ i + n−1 rˆ − c − m q ( pˆ i )   pˆ i − c − m q ( pˆ i ) n−1

if if

if if

rˆ ≤ β pˆ i , rˆ ≥ β pˆ i

rˆ ≤ β pˆ i . rˆ ≥ β pˆ i

(9)

This derivative is continuous, even at β pˆ i = rˆ, and for β < n − 1 pˆ changes sign (from positive to negative) at most once. Thus πi is quasiconcave and any local maximizer is global and must imply non-negative profits. If β = n − 1 then the derivative does not depend on pˆ i and indicates a maximum if and only if rˆ = (n − 1 )(c + m ). On the other hand, for β > n − 1 profits πipˆ are not quasi-concave, and the previous candidate becomes a local minimum. Indeed, following the same steps JLT (p. 109) in the case of duopoly, it can be shown that independently of the level of the access charge there is an equilibrium involving connectivity breakdown with pˆ = ∞.13 Similarly, as noise vanishes we find that ∂ πirˆ /∂ rˆi becomes

  ˆ  r1i + m +



  

pˆ − pˆ q pˆ     1 pˆ + 1 − (n−1 )β rˆi + m q βrˆi n−1 1 n−1

if if

rˆi ≤ β pˆ . rˆi ≥ β pˆ

(10)

These profits are quasiconcave with a non-negative value at the maximizer if β > 1/(n − 1 ). If β = 1/(n − 1 ) then there is a local maximum if pˆ = (n − 1 )m. For β < 1/(n − 1 ) it can be shown that independently of the access charge level there is an equilibrium where breakdown is caused by rˆ = ∞.   Assuming rˆ ≤ β pˆ, the symmetric equilibrium candidate pˆc , rˆc is given by the conditions



1−

β n−1



pˆc +

1 c rˆ − c − m = 0, rˆc + m = 0. n−1

n−1 c+nm The solution is rˆc = −m and pˆc = ( n−1) −β . We have rˆc ≤ β pˆc if

βc ∗ and only if m ≥ − 1+ β , or a ≥ a .

In a similar manner,   assuming rˆ ≥ β pˆ the symmetric equilibrium candidate is pˆr , rˆr with

1 pˆ − c − m = 0, pˆr + n−1



r

1 1− (n − 1 )β



rˆr + m = 0,

c+mn with solution pˆr = c + m, rˆr = −β (n−1 . We have rˆr ≥ β pˆr if and )β −1

βc ∗ only if m ≤ − 1+ β or a ≤ a .



Proof of Proposition 2: 13 We do not discuss this further here, as our focus is on equilibria without breakdown.

8

S. Hoernig / Information Economics and Policy 36 (2016) 1–9

Proof. This proof is performed in various steps. It uses techniques from Hoernig (2014) but is completely self-contained. 1. First we determine an expression for equilibrium profits using the first-order condition for fixed fees. From (4), the first-order condition for profit-maximizing fixed fees is

0=

∂πi ∂αi πi = ∂ Fi ∂ Fi αi   ∂α j   ∂αi + αi 1 + pˆ i − c − m Di j ( pi + ri − c )Dii + ∂ Fi ∂ Fi j=i     + rˆi + m D ji

.

This condition contains equilibrium profits π i , which can be solved out (Solving for the equilibrium fixed fee and plugging it back into profits leads to the same solution but duplicates the work):



πi = −α

2 i

∂α

1

+ ( pi + ri − c )Dii + ∂αi ∂ Fi





+ rˆi + m D ji

 

 ∂Fj i j=i

∂αi ∂ Fi





pˆ i − c − m Di j

h

dα −I dF

+



⇐⇒

dα −1 = − ( I − Qw h ) Qw , dF

where Qw is the Jacobian of Q, and I is the n × n identity matrix.14 Since due to (2) changing all wj , j = 1, . . . , n, by the same amount does not affect market shares, we have n ∂ Qi j=1 ∂ w = 0 for all i. In a symmetric equilibrium this implies



n(1 + β )Uii 1 − + n−1 (n − 1 )σ

n(1 + β )Ui j − n−1



 p+r n−1





+ c Dii



pˆ + rˆ + c Di j . n−1

The first line contains the usual Hotelling-type profit due to market power, plus the profits related to on-net calls. The terms of interest for the following are those related to off-net calls, which are on the second line. 4. In order to see the effect of the access margin m on profits we need to consider separately the cases where callers end calls first and where receivers end calls first. βc Case 1: β pˆ ≥ rˆ, i.e. callers end calls first (m ≥ m∗ = − 1+ β ): In

n−1 c+mn this case we have Di j = q( pˆ ), Ui j = u(q( pˆ )), pˆ = ( n−1) −β and rˆ =

−m. The derivative of profits with respect to the access margin m becomes



  d πi 1 1+β = 2 n(1 + β ) pˆ + pˆ − pˆ − m − (n − 1 )c dm nηˆ n

.

and w, F the n × 1 -vectors of wi and Fi , we can write w = hα − F . Stating market shares individually as αi = Qi (w ) and as vectors α = Q (w ) = Q (hα − F ) for a function Q : Rn → Rn , we obtain the market share derivatives



πi

1 = 2 n

×

2. [This point can be skipped by those who are unfamiliar with matrix algebra] We now need to determine the derivatives of market shares ∂ α i /∂ Fj , which is a non-trivial exercise due the price discrimination between on-net and offnet calls (see Hoernig, 2014). In order to compress notation, write consumers’ benefits of subscribing to network i as wi = n ˜ j=1 hi j α j − Fi , where hii = Uii + Uii − ( pi + ri )Dii and hi j = Ui j − pˆ i Di j + U˜ ji − rˆi D ji for j = i. Letting h be the n × n-matrix of hij ,

dα = Qw ∗ dF

one finally obtains the equilibrium profits



q ( pˆ ) n n−1n−1−β

with the demand elasticity ηˆ = − pˆq ( pˆ )/q( pˆ ). At the lower bound this derivative becomes



dπi  dm 

=

m=m∗

1 c q ( pˆ ) . n2 ηˆ (n − 1 )(n − 1 − β )

If no breakdown occurs, i.e. β < n − 1, the latter is negative. There is a unique solution to the first-order condition dπi /dm = 0, at

m = −(n − 1 )c

1 + β + (n + 1 )nηβ ˆ







n n2 − 1 ηˆ + 1 (1 + β )

.

The latter lies below m∗ if and only if β < n − 1. Thus on this (right-hand) branch of profits there is a unique global maximum at m = m∗ if β < n − 1. βc Case 2: β pˆ ≤ rˆ, i.e. receivers end calls first (m ≤ m∗ = − 1+ β ):



 

In this case we have Di j = q βrˆ , Ui j = u q βrˆ

, pˆ = c + m and

β (c+nm ) rˆ = − (n−1 . The derivative of profits with respect to the access )β −1

margin m now becomes



n 1 + β ) βrˆ + ( 

1+β rˆ nηˆ β

  rˆ 

∂ Qi ∂ Qi ∂ wi = −(n − 1 ) ∂ w j for all i and j = i. Furthermore, we have

1 d πi =− 2 dm n

σ ≡−

with ηˆ = −(rˆ/β )q (rˆ/β )/q(rˆ/β ), which at the upper bound m = m∗ simplifies to

j

 ∂ Qi ∂ A(x, 0, . . . , 0 )  =  > 0, ∂wj ∂x x=0

i.e. σ is the partial derivative of A at equal surplus on all networks (and as such is a constant). In symmetric equilibrium all hii are identical, and so are all hij , for j = i. Using these simplifications in dα /dF, we find

dαi dαi σ (n − 1 )σ  ,  . =− = dFi dF 1 − σ n hii − hi j 1 − σ n hii − hi j j 3. After substituting the market share derivatives into profits, and applying symmetry via D ji = Di j , pˆ i = pˆ, rˆi = rˆ and αi = 1/n, 14 There is a unique solution for the market share equation α = Q (hα − F ) if the eigenvalues of (I − Qw h ) are positive, which corresponds to the stability condition of Hoernig (2014).



dπi  dm 

m=m∗

− c + m + rˆ − (n − 1 )c

q β n , n − 1 (n − 1 )β − 1

 

q βrˆ 1 c =− 2 . n ηˆ (n − 1 )[(n − 1 )β − 1]

If no breakdown occurs, i.e. β > 1/(n − 1 ), the latter is positive. The unique solution to dπi /dm = 0 is

    m = −c  , n n2 − 1 ηˆ + 1 (1 + β ) 1 + β + n2 − 1 nηβ ˆ

which lies above m∗ if and only if β > 1/(n − 1 ). Thus on the second (left-hand) branch of profits there is a unique global maximum at m = m∗ if β > 1/(n − 1 ). Thus we can conclude that the profits are indeed maximized at m = m∗ . This concludes the proof. 

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