Incentives for the Expansion of Network Capacity In a “Peering” Free Access Settlement Roberto Roson*
Abstract The absence of an access fee may hinder the development of capacity in interfaced networks, because the benefits of a larger capacity partly accrue to external consumers. This paper shows, however, that the implied result of under-investment in network capacity may be reversed in a number of circumstances. Different cases are discussed by means of a model, which adopts fairly general assumptions about some key functions. This allows accommodating for a variety of alternative specifications found in the literature.
Keywords: Peering, Internet, Network Capacity, Investment Incentives, Congestion.
*
Dipartimento di Scienze Economiche, Università Ca’Foscari di Venezia, Cannaregio 873, Venice, Italy. E-mail:
[email protected]. 1
1.Introduction Many telecommunications networks are operated on the basis of “peering” or “bill and keep” arrangements. This means that all traffic destined to the customers of a network provider is accepted without imposing any access fee, whereas all outgoing traffic is routed freely on external, interfaced networks. The most notable example is the Internet, where peering schemes regulate the exchange of data packets between backbone operators of similar size. Analogous settlements are used in other telecommunications or transportation networks1. Recently, some papers have addressed the relationship between peering schemes and the incentives for investment in network capacity (e.g., bandwidth, routers speed, etc.). In particular, Little and Wright [6] show that if regulation forbids settlement payments between firms, there will be under-investment in capacity and under-pricing of usage2. This is because the absence of an access fee allows outside users to free-ride on the improved quality (reduced congestion) of accessed networks. To overcome these problems, firms that are net providers of infrastructure should be allowed to charge firms that are net users. The first-best outcome is achieved if settlement fees are set at the incremental cost of capital provided. This and other models found in the literature are typically stated as two (or three) stage games: capacity levels are chosen first, and conventional market competition follows (assuming reciprocal free access). In this paper, I propose a model, which retains this type of approach, but adopts fairly general assumptions about some key functions, like utility and cost functions, with the aim of highlighting the critical factors affecting the amount of capacity investment.
1
DeGraba [3] argues that a Bill and Keep scheme may lead to efficient interconnection pricing. On this issue, see Wright [10] and DeGraba [4]. 2 Related papers are Crèmer, Rey and Tirole [2], and Foros and Hansen [5], where two networks are connected by an interface, whose capacity is determined by the minimum of two capacity levels, independently chosen by the operators (for a critical comparison of these two papers, see Roson [7]). 2
Instead of modelling capacity constraints as upper bounds on traffic volumes, I adopt here a more flexible specification in terms of congestion functions, which include the traditional “inverted-L” shaped cost function as a special case. Congestion is assumed to be increasing in traffic volume and decreasing in capacity levels, but the exact relationship between congestion costs and the capacity level chosen by individual operators is left unspecified. This allows accommodating for a variety of alternative specifications found in the literature. For example, if only total capacity matters (like in Little and Wright (ibid.)) the individual capacities enter additively in the congestion function (in other words, they are perfect substitutes). If, on the contrary, only the minimum capacity level matters (like in Crèmer, Rey and Tirole (ibid.)), individual capacities enter the congestion function as perfect complements. The adoption of continuous, differentiable, and convex congestion functions is quite standard in transportation economics (e.g., Small [8]) but not very common in network and industrial economics. However, their utilization here provides the double advantage of improving the mathematical tractability of the models (discontinuities are avoided), while improving the model realism at the same time. For instance, it not generally true that Internet users cannot access the network when the capacity is saturated (however saturation may be defined). Typically, data packets are “dropped” and sent at a later time. From a user point of view, all this translates into longer delays in sending and receiving information3. In my model, the utility of a representative consumer is supposed to be influenced by two distinct elements: “on-net” and “off-net” traffic flows. The former involves the use of a network infrastructure owned by a single “home” operator, whereas the latter involves the sharing of some external infrastructure. Again, the functional form for the utility is not specified so, for example, it is possible that only off-net traffic matters. These two elements are considered separately, because it is possible that an investment in capacity affects the two networks asymmetrically: an higher capacity affects directly the
3
In a different setting, the precise instant of access to a network may have a random component. The congestion function could, in this case, be interpreted as the probability of “collision” with a request for access made by another user. 3
on-net flows, whereas off-net flows are affected only through the interaction between capacity of the source network and capacity of the destination network. Even if external consumers may benefit from investments in the home network, the impact of quality improvements may be stronger within the network in which the capacity has been expanded. This means that network operators may compete in quality through capacity investment (as in Valletti and Cambini [9]). Since outside firms and consumers are affected by the investment choices, the market equilibrium is characterized by externalities and consequently it is not, in general, socially efficient. This is not the point of the paper, though. The paper is aimed at highlighting the role played by some factors in determining the sign of the externality (a negative externality implies over-investment, a positive externality implies underinvestment), as well as its magnitude. The work is organized as follows. Basic assumptions of the model are set out in the next section. In the subsequent sections, four alternative model versions are presented and discussed, mainly differing in terms of pricing and consumers’ mobility hypotheses. As usual, a final section summarizes the main findings and draws some concluding remarks.
2.Basic Assumptions There are two groups of consumers of telecommunication services, initially connected to two networks, called 1 and 2. The number of consumers in each group is normalized to one. All consumers have the same characteristics, and derive utility from the consumption of on-net and off-net calls, in the following way:
Assumption A1. A representative consumer connected to network i derives utility from consumption of on-net calls qi and off net calls qic, through a sub-utility function U(qi,qic), having the following properties4:
4
In the following, superscripts will be used to indicate derivatives. 4
U = U qi ≥ 0 qi U = U qic ≥ 0 qic
2
U qi,qi ≤0 2 =U qi 2
U qic,qic ≤0 2 =U qic
[1]
2
U = U qi,qic = 0 i ∈ {1,2 } qi qic
Assumption A2. The supply of telecommunications services is carried out at zero cost, with the exception of congestion costs, which are different for the two types of flows: Ci = Ci(ki ,qi ,q1c ,q2c ) Cc = Cc (k1,k2 ,q1,q2 ,q1c ,q2c ) C− C− = C−k − ≤ 0 = C−q− ≥ 0 k− q−
[2]
Assumption A3. Consumers in each group i are assumed to maximize: Vi = Ui (qi,qic ) − piqi − picqic − Ci (.)qi − Cc (.)qic
[3]
while taking the congestion costs as independent from the individually chosen quantity levels. This implies the following f.o.c.: Uiqi = pi + Ci (.) Uiqic = pic + Cc (.)
[4]
On this type of setting, we now analyse some model variants.
3.Two-parts Tariffs, Captive Consumers Consider first a situation in which every consumer is connected to one network, but switching to the alternative network is impossible. There is perfect information, and each provider applies a two-part tariff (a fixed fee and a price per consumed unit). A standard result, which I recall here without demonstration, states that a profitmaximizing supplier will charge a marginal cost price (in this case, zero5) and a fixed fee, which fully expropriates the representative consumer of her surplus. Since social surplus coincides with total profits, the choice of capacity investment (independently done by the two providers) will be efficient in the absence of pecuniary 5
Alternatively: the service provider bears the congestion, and charges the marginal congestion cost. 5
externalities, that is, if an increase in a network capacity does not affect the other operator’s profits. If, for example, expanding the capacity in a network creates positive effects for the external consumers (e.g., benefiting from reduced congestion in off-net calls), the other network operator may be able to increase the fixed fee. This positive spillover would not be internalised by the investment-making operator, who may then select a network capacity level lower than the socially optimal one (e.g., when marginal investment costs are strictly increasing). To verify if this is the case, it is important to make a preliminary observation.
Proposition P1. The volume of on-net and off-net calls changes in opposite ways, when capacity is increased in the external network. As a consequence, the (gross) utility of the representative consumer may either increase or decrease (or stay constant), depending on how the two types of services are weighted in the utility function. (Proof: see the appendix)
For example, if k2 is expanded, we could observe a reduction of congestion costs for offnet calls, raising the inter-network traffic. If k1 is left unchanged, this creates an extra pressure on network one, raising congestion costs for intra-network calls. It is not difficult to build specific examples in which this induced congestion effect reduces the utility of consumers linked to network one6. In general, we cannot rule out the possibility that an increase of capacity k2 reduces q1c flows. This may happen when q2 increases so much that congestion costs for the internetwork calls are higher, despite the larger capacity. However, even in this case, q1c and q1 would move in opposite directions. On the whole, the effects of an expansion of k2 on the different traffic flows can be obtained through differentiation of first order conditions, which can be expressed in matrix form as:
6
This is the case, for instance, when the utility is linear logarithmic in consumption of the two services, and congestion functions are expressed as the inverse of the difference between total capacity and total traffic. 6
sq1 c cq1 c q1 c 0
c1q1c c1q2c sq1c c cq2c ccq1c sq2c c2q1c c 2q2c
0 q1k 2 0 c qc 2 q1ck2 −cck 2 ⋅ = c qc 2 qk2c2 −cck 2 sq2 qk2 2 −c2k 2
[5]
where sqx = - (Uqx,qx-cxqx) > 0 (because of second order conditions associated to [4]). Here, the right hand side vector expresses the direct effect of a capacity expansion on congestion costs. The matrix on the left hand side, instead, accounts for the systemic effect of changes in the traffic flows.
4.Linear Pricing, Captive Consumers The situation in which the network providers use a simple price per unit of traffic, instead of a two-parts tariff, can be considered as a simple variant of the case discussed above. Two key observations are necessary here. First, linear pricing fails in fully expropriating the consumer surplus. This means that, generally speaking, a network operator cannot fully internalise, in terms of profits, all the benefits of an investment in capacity. As a consequence, the level of investment in network capacity will tend to be lower than the socially optimal one. Second, even if linear pricing is an imperfect device to extract the consumer surplus, it remains true that more profits can be achieved if the consumer surplus increases. To see this, consider the f.o.c. in [4] and suppose that an investment in capacity has reduced the congestion cost C(.) for one type of service (on or off-net). To restore the equality, we must observe an increase in the price, and/or an increase in the quantity (reducing the marginal utility). In both instances, profits increase. Combining these two elements, we can conclude that investments in capacity are associated with two types of externalities. One positive externality affects the consumers, and a second externality affects the operator in the interfaced network. The latter may either reinforce or weaken the first effect. The conditions determining the direction of the secondary externality are the same as those discussed in the previous section.
7
5.Two-parts Tariffs, Endogenous Market Shares Suppose that network services are horizontally differentiated and the consumers may change provider, on the basis of utility differentials and preference parameters. If consumers are homogeneously distributed in terms of absolute preferences, we can adopt a standard Hotelling-type competition framework, and compute the market shares in the following way (ruling out the possibility of a corner solution with zero customers):
Assumption A4. Let the market size i
for the operator i be determined by7:
= 1+ (Vi − V j ) > 0 i, j ∈ {1,2 }
where
[6]
is a parameter, measuring the intensity of competition (degree of service
differentiation).
In this setting, network congestion may change because more or less customers are connected to a network (like in Baake and Mitusch [1]), even if the traffic generated by each single consumer does not change. Congestion costs may then be expressed as: Ci = Ci(ki , qi , q1c ,(2 − )q2c ) Cc = Cc (k1,k 2 , q1 , q1c ,(2 − )q2c ,(2 − )q2 ) where
=
and
j
=[ +
j
]
=
[7]
To keep the model tractable, as well as to isolate
this effect from the one discussed in the previous sections, the following hypotheses are introduced:
Assumption A5. For small variations in capacity levels, individual traffic demand does not change: qi q q q = i = ic = ic = 0 i, j ∈ {1,2 } ki kj ki kj
[8]
7
So far, the number of consumers connected to each network (a constant) was normalized to one. Here, this number is variable, but it is still one if the utility differential is zero. Consumers choose their provider on the basis of the sub-utility function V and of some other subjective preference parameters, related to the identity of the provider but independent from the price/quantity characteristics of each network. 8
Assumption A6. The impact of additional traffic on congestion costs does not depend on the traffic type: Ci C Ci = i = = Cit qi qic q jc
[9]
Cc C C C = c = c = c = Cct i, j ∈ {1,2} qi qj qic q jc
Considering again two-part tariffs, implying the expropriation of consumer surplus, the incentives for the expansion of network capacity continue to be influenced by the cross effect of a capacity increase on profits, which can be expressed here as: Πi = kj
Π i = iVi ⇒
(V
kj
i
− V jkj )Vi + iVi kj
[10]
To evaluate [10], the effects of some additional capacity on the consumer surpluses must be computed first. These can be obtained by solving the following system8:
(
)) −q (C + C ( (V − V )(q + q − q − q ))) V = −q (C ( (V − Vj )(q + q − q ))) − −q (C + C ( (V − V )(q + q − q − q ))) V jkj = −q j C kjj + C tj jc
kj c
t c
kj
i
i
ic
kj c
t i
( (V
kj j
kj j
i
kj
i
t c
kj
i
− Vi kj )(q j + q jc − qic ) −
kj
j
jc
ic
kj i
i
ic
jc
kj j
i
ic
jc
i
[11]
j
The solution of [11] is rather messy. To easy its interpretation, let us consider a situation in which: q1=q2=q and q1c=q2c=qc. In this case, the solution of [11] simplifies to: C kjj (q + Cit q3 ) + Cckjqc Ψ >0 Ψ C kjC t q3 + Cckjqc Ψ Vikj = − j i >0 Ψ V jkj = −
[12]
8
Basic utility functions U(.) do not appear in [11] because of assumption A5. However, even without A5, they would not appear because of the envelope theorem. 9
where Ψ =1+ C1tq 2 + C2t q2 > 0. Subtracting these two equations, we can see that the variation in the utility differential is strictly positive:
V jkj − Vikj = −
C kjj q >0 Ψ
[13]
The meaning of [13] is rather straightforward. When capacity in a network is expanded, this network attracts consumers. In this way, some additional congestion reduces the initial benefits of a larger capacity, but at the same time few customers remain in the other networks, which also experience less congestion. The numerator accounts for the direct effect of a capacity expansion on intra-net calls, which matters only for those consumers attached to the improved network. The denominator accounts instead for a smoothing effect, due to the shifting of customers. Indeed, this term is increasing in the parameter
: a higher degree of mobility in the
consumers implies a lower differential advantage, obtained by the capacity increase. This result does not depend on the hypothesis q1=q2=q and q1c=q2c=qc, which is a sufficient but not a necessary condition. To see this, consider the position of the marginal consumer, who is indifferent in terms of network membership. Her utility, when connected to network i, must be the same as her utility, when connected to network j9: Ui − Ci(.)qi − Cc (k j ,.)qic = U j − C j (k j ,.)q j − Cc (k j ,.)q jc
[14]
From this situation of balanced utility, suppose that capacity in network j is augmented. Both left and right hand sides of [14] would then change, possibly breaking the equality. There is here a direct effect due to the larger capacity on congestion costs, and an indirect
9
The parameters U in [14] comprise quantity-dependent utilities U and subjective preference parameters. Both terms are constant, as they depend neither on market shares, nor on capacity levels. 10
effect, due to the induced changes in the market shares. The indirect effect is of second order, however, and does not affect the sign of the variation in utility differentials. In other words, to make sure that the marginal consumer switches to network j, when capacity in this network is increased, the following requirement must be met: Assumption A6: Let −C kjj q j > −Cckj (qic − q jc ) i, j ∈ {1,2} .
This condition is, of course, satisfied when q1=q2=q and q1c=q2c=qc, since the right hand side would be zero. It would also be satisfied if the switching consumers do not significantly change their demand for off-net calls, after having changed their provider. Now we can state the following:
Proposition P2. Under assumption A6, an increase in capacity for a network benefits the consumers directly connected to this network, as well as the external consumers, connected to the interfaced networks. However, the welfare gain is larger for the consumers belonging to the network in which the capacity expansion takes place.
Coming back to equation [10], notice first that an increase in the capacity of an external network affects the profits in two opposite ways: some customers are lost (attracted by the outside network), but the remaining ones enjoy a higher potential surplus (which is actually fully extracted by the home network provider). Therefore, as in the first model, the sign of the externality cannot be determined a priori. One could conjecture that a lower degree of service differentiation, or a higher degree of consumers mobility, boosts the “business stealing” mechanism and makes the sign of [10] negative (remember that this would bring about over-investment in capacity). This is only partly true, though, as can be seen by plugging [12] into [10]: kj t 2 kj Π i C j q(1− Ci q ) − Cc qc Ψ = kj Ψ
[15]
11
Here, the denominator is positive. The numerator is given by the algebraic sum of two terms. The second one is positive, but the first one, which is weighted by
, has an
ambiguous sign. However, for a sufficiently large Cit, even this element is positive. This case, emerging when congestion is high for internal calls within network i, is characterized by positive cross-effects on profits (and, consequently, under-investment in capacity, despite the fact that firms use capacity investments to compete in quality). This happens when customers of network i experience a significant reduction in the on-net congestion, due to the shrinking of their network group size. This positive externality adds to the one associated with lower congestion costs in the inter-network calls (last term in the numerator of [15]).
6.Variable Market Shares and Network Externalities Network externalities arise when belonging to a network becomes more valuable as network membership increases10. Roughly speaking, network externalities work like a sort of “reverse congestion” (although congestion refers to traffic flows, whereas network externalities refer to the network size). Network externalities can be easily accommodated in the framework illustrated above in which, because of assumption A5 (constant individual traffic demand), the distinction between traffic volumes and network size is irrelevant. This could be done in many different ways. For example, by replacing assumption A3 with:
Assumption A3’. Consumers in each group i are assumed to maximize: Vi = Ui (qi,qic ) − piqi − picqic − Ci (.)qi − Cc (.)qic + ( iqi +
q )
j ic
[3’]
where we have added two extra terms, making the utility function V increasing in the market shares (networks size) and in the parameter , weighting the relative impact of network externalities. 10
When networks are interfaced, the utility of a representative customer may be defined as an increasing function of the degree of accessibility to other customers (including those belonging to external networks). 12
Repeating the same steps as before, the following results can be obtained:
V =− kj j
Vi = − kj
C kjj (q + Cit q3 − q C kjj (Cit q3 − q
(q − qc )) + Cckjqc (Ψ − 2 Ψ −2 (q − qc )
(q − qc )) + Cckjqc (Ψ − 2 Ψ − 2 (q − qc )
(q − qc ))
(q − qc ))
C kjj q V − Vi = − Ψ −2 (q − qc ) kj j
kj
[12’]
[13’]
The only difference between [13] and [13’] is a smaller denominator in [13’] if q>qc. This occurs because total market size is fixed ( +
j
2). When a customer is attracted in
a network with larger capacity, the other members of this network experience a utility gain if the membership in their home network is valued more than the membership in their outside network. This is the case in [3’] when the on-net individual traffic volume (q) is larger than the off-net one (qc). Since network externalities operate here like a reverse congestion effect, utility differentials are widened, counteracting the smoothing effect of conventional congestion. In other words, network externalities reinforce the negative spillovers generated by the capacity-driven competition, whereas congestion effects tend to work in the opposite direction. Therefore, if network externalities are relatively important in a market, free access may bring about over-investment in network capacity.
7.Conclusion When evaluating the pros and cons of an access regime, the implications for the development of network capacity should be carefully evaluated. From this point of view, reciprocal free access settlements, like “peering” in the Internet or “bill and keep” in telephony, seem to bring about under-investment in capacity, because investment costs cannot be recovered through access fees. This paper shows, however, that this result may not hold in a number of circumstances, characterized by the differentiated impact of a capacity expansion for on-net and off-net flows. Whereas intra-network flows are directly affected by the increase in capacity, the
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influence of congestion on inter-network flows depends on both capacity levels of the source and destination networks. Outside users, having access to a capacity-improved network, typically experience a reduction in the cost of off-net calls, but at the same time congestion may increase in their home network, raising the cost of on-net calls. In addition, as it is typical in price regulation regimes, firms may compete in quality, in this case through capacity investments. Even if a firm may loose customers, when a competitor network expands its capacity, the remaining customers may benefit from reduced congestion, if the reduction in the number of users reduces the pressure on the home network infrastructure. When investment in capacity produces positive spillovers for outside firms and customers, the investing firm does not internalise all the benefits, and the investment level tend to be lower than the socially optimal one. The opposite case emerges when the externalities are negative. This paper has highlighted some important factors, determining the direction of the externality effects. How the users ponder the various types of traffic flows is clearly important, especially when demanded quantities change in opposite ways. Furthermore, competition in quality (“business stealing”) and network externalities generate negative spillovers (= over-investment), but direct and indirect reductions of congestion costs work in the reverse direction (= under-investment). All these factors may be more or less important in the various networks in which peering access schemes are applied. As a consequence, regulatory policies should consider the implications of peering for capacity development, on the basis of the specific characteristics of each network.
14
Appendix Proof of Proposition P1 Consider the f.o.c. associated with quantity q1: U1q1 − C1 (k1,q1,q1c ,q2c ) = 0
[A1]
Differentiate with respect to k2, and explicit q1k2: q1k2 =
C1q1c q1ck2 + C1q2c qk2c2 U1q1,q1 − C1q1
[A2]
The denominator is negative if [A1] identifies a maximum. On the numerator, cost derivatives are positive, by assumption. On the other hand, derivatives of off-net traffic k2 quantities must have the same sign ( sign[q1ck2 ] = sign[q2c ]), because these are determined
on the basis of the same level of congestion cost, Cc, and marginal utilities are monotonically non-decreasing. Therefore: sign[q1k 2 ] = −sign[q1ck2 ]. Q.E.D.
Acknowledgements Carlo Cambini and Tommaso Valletti provided useful comments on an earlier draft of this paper. The usual disclaimer applies.
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References 1. Baake, P., Mitusch, K., Competition with Congestible Networks (mimeo, Humboldt and Technical University, Berlin, 2002). 2. Crèmer, J., Rey, P., Tirole, J., Connectivity in the Commercial Internet, Journal of Industrial Economics 48 (2000) 433–472. 3. DeGraba, P., Bill and Keep at the Central Office as the Efficient Interconnection Regime (OPP Working Paper #33, Federal Communications Commission, 2000). 4. DeGraba, P., Bill and Keep as the Efficient Interconnection Regime?: A Reply, Review of Network Economics 1(2002) 60-64. 5. Foros, Ø, Hansen, J., Competition and compatibility among Internet Service Providers, Information Economics and Policy 13 (2001) 411–425. 6. Little, I., Wright, J., Peering and Settlement in the Internet: an Economic Analysis, Journal of Regulatory Economics 18 (2000) 151-173. 7. Roson, R., Two Papers on Internet Connectivity and Quality, Review of Network Economics 1(2002) 74-79. 8. Small, K.A, Urban Transportation Economics (Harwood Academic Publishers, 1992). 9. Valletti, T. e Cambini, C., Investment and Network Competition (mimeo, Imperial College, London and Politecnico di Torino, 2002). 10. Wright, J., Bill and Keep as the Efficient Interconnection Regime?, Review of Network Economics 1(2002) 53-59.
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