Internet Peering, Capacity and Pricing

Transcription

1 Internet Peering, Capacity and Pricing Haim Mendelson Shiri Shneorson June, 2003 Abstract We analyze peering on the Internet with consumer delay costs, finding that they have a substantial effect on market structure. When network operators make capacity decisions, both the first- and second-best solutions result in a natural monopoly due to economies of pooling. The oligopoly equilibrium is symmetric, but in general only a limited number of networks participate in the market, reflecting a minimum operating scale. We characterize the optimal access charge and its dependence on market parameters. When both the consumer and Website markets compete a-la Cournot, the socially optimal and the operator-preferred access charges are both equal to the access charge that maximizes network traffic. Graduate School of Business, Stanford University, Stanford, CA , haim@stanford.edu Graduate School of Business, Stanford University, Stanford, CA , shiri@stanford.edu 1

2 1 Introduction As its name suggests, the Internet is a network of interconnected networks. Since April 1995, when the National Science Foundation ended its direct financial involvement in the Internet, each of the networks composing it has been owned by an independent operator. To achieve inter-network connectivity, operators make agreements that enable address advertising and data exchange across these different networks. Many of these agreements are peering agreements, whereby network operators agree to exchange data that is originated on one network and destined to the other. 1 Historically, peering agreements were Bill and Keep contracts involving no transfer of money, but as peering became commonplace and the Internet went mainstream, companies started implementing settlement-based peering agreements that include charges for imbalanced traffic across network boundaries. Similar to the case of voice networks, these schemes involve a settlement known as an access charge paid by one network operator to the other in return for the carriage of traffic. As of 2003, the Internet transport market is unregulated, unlike the heavy regulatory environment that characterizes traditional telecommunication carriers. 2 And yet, as the commercial backbone market is evolving, industry observers and policy makers are questioning whether the government can, or should, maintain a hands-off approach to Internet network operations (cf. [9]). This issue is brought up by pressures (primarily from less-developed countries) to change the current structure of International Charging Arrangements for Internet Services (ICAIS) (cf. [1]). 1 Another type of agreement is a transit agreement, where one network operator agrees to carry traffic originated by the other and terminated on any other network (including third-party networks) in exchange for payment. 2 In implementing the 1996 Telecommunications Act, the FCC determined that basic services (referred to as telecommunications services ) and enhanced services (referred to as information services ) were separate and distinct categories of service. Hence, telecommunications services are regulated as common carrier services whereas information services are not. For more details, see, e.g., [18]. 2

3 Laffont, Rey and Tirole [13] [14] analyzed access charges, interconnection and market structure for the telephone network, and their analysis forms the foundation for both the theory (cf. [2], [7], [8]) and application of access charges to telecommunication networks. Crémer, Rey and Tirole [3] studied the Internet backbone market and analyzed strategies that would be used by a dominant backbone, focusing on the quality of interconnections. Laffont, Marcus, Rey and Tirole [12]-[11] (LMRT in short) extended the analysis of telecommunication networks to peering agreements between Internet backbone operators. A key difference between traditional telecommunications agreements and Internet peering is the asymmetry between consumers and Websites (content providers). Taking this difference into account, LMRT determined how each operator should price downloads and consumer access when their costs are proportional to the traffic they carry. They also derived the socially optimal access charge set by a regulator and showed that when backbones have market power in the Website market, they prefer an access charge lower than the socially optimal one. LMRT showed that networks that engage in peering follow the off-net-cost pricing principle, whereby prices are set as if all traffic is entirely off-net (i.e., moves across network boundaries). They analyzed the first-best prices and identified the importance of network externalities between consumers and Websites. They also showed how the access charge determines the allocation of costs between Websites and consumers and how it affects the level of traffic. Throughout their analysis, all network operators participate in the market. Another key difference between telephone and Internet services is the importance of delay in the latter, as the delay incurred by consumers at the application layer is an important determinant of their willingness to pay for service. The key role of this delay has been recognized since the early days of the ARPANET, when the speed at which messages could be delivered over the network was used as a primary measure of network performance (cf. [10]). The effect of congestion and delay on the demand for Internet services is important, indeed: If a consumer needs to wait for too long for a news Website to download, the next 3

4 day she might prefer to buy a newspaper. 3 From its early days, one of the stumbling blocks in Internet adoption was the World Wide Wait phenomenon, which is now being replaced by poor user experience for rich content. Clearly, consumers willingness to pay for services is a function of the quality of service they receive, with delay being a key attribute. Little and Wright [15] modeled congestion using a rationing model. Assuming that utility is independent of congestion as long as demand does not exceed total capacity, whereas any demand beyond the capacity constraint receives zero utility, they showed that if regulation forbade settlement payments between two competing networks, there would be under-investment in capacity and under-pricing of usage, both of which lead to excessive congestion. To overcome this problem, they suggested that networks with excess capacity should be allowed to charge firms that have insufficient capacity, leading to a more efficient allocation. In this paper we explicitly introduce consumer delay costs and study their effects on pricing, market structure, capacity investment and access charges. In our model, network operators price their services considering the effect of delay cost on demand. In setting capacity, they strategically balance the cost of capacity against the cost of delay. The question that arises, then, is what happens to the classical results when the effects of congestion are incorporated into the analysis. Does congestion totally alter the outcome? Does it have a noticeable effect on market structure? Does it only require an adjustment of parameters, reverting back to classical conditions? And, does incorporating congestion into the analysis make a qualitative difference? To answer these questions, we combine a queueing model with the LMRT framework, assuming that consumers create random download events from each Website. Delay has a negative effect on consumers utility, which we explicitly introduce into our analysis. Each network is characterized by its capacity, 4 which is set by a profit-maximizing network 3 The popularity of Netflix, a service where consumers order DVDs on the Web for delivery via the U.S. Postal Service rather than over the Internet is a manifestation of this phenomenon. 4 Capacity refers to the effective throughput of the network. This throughput is typically determined by 4

5 operator. We first analyze the case of fixed-capacity networks. We find that when networks have fixed and equal capacities, both the first-best solution and the oligopoly equilibrium are full-participation symmetric solutions. We show how our results can be interpreted as extensions of LMRT s, although the delay sensitivity leads to higher consumer prices and smaller market sizes. We also show that the first-best solution is not attainable when the regulator s only instrument is the access charge. Delay sensitivity implies a difference between usage and capacity. When we allow network operators to make capacity decisions, the first-best solution calls for a single network. This natural monopoly result is due to the economies of pooling that characterize queueing phenomena. Under the first-best solution, the network operator loses money due to positive network externalities between Websites and consumers. Further, when consumers are delay sensitive, network operators have to invest in excess capacity, which the first-best price does not compensate them for. The Ramsey prices in our model include an equal premium paid by consumers and Websites to recover the investment in excess capacity, as well as the more common premium (inversely proportional to the demand elasticities) needed to recover the losses due to network externalities. We find that the oligopoly equilibrium is symmetric, but in general only a limited number of operators will enter the market, reflecting a minimum operating scale. We find the equilibrium number of networks an oligopoly market can support and study its determinants and the associated access charge. In addition to the capacity and delay cost parameters, the access charge depends on the utility functions, marginal costs, and on the variance of download sizes. We find that the operator-preferred access charge is larger than the socially optimal access charge, and that both are larger than the access charge that maximizes network traffic. However, when both the consumer and Website markets are characterized by Cournot competition, all three access charges are the same. resources such as link bandwidth, buffers, computing resources etc. 5

6 Our results suggest that delay-sensitivity has important market structure effects, which we analyze. This happens because capacity decisions are separable from usage decisions, and the economies of pooling that characterize queuing phenomena affect pricing, scale and market structure. In what follows, we present our model in Section 2 and analyze the fixed-capacity problem in Section 3. Section 4 studies the case where network operators also make capacity decisions. Section 5 compares alternative ways of setting the access charge, and Section 6 considers Cournot competition in both the consumer and Website markets. Section 7 offers our concluding remarks. 2 The Model We model n network operators who provide access and transport services to consumers and Websites. The expected number of accesses made by a consumer is proportional to the number of available Websites. 5 Specifically, we assume that each consumer-website pair generates a random flow of downloads; without loss of generality, the expected number of downloads per pair is unity, which makes our results directly comparable to LMRT. Consumers and the Websites they access may be served by different networks. When a download crosses two different networks, we call the network that hosts the Website originating network, and the network that hosts the consumer terminating network. 6 Other traffic, such as consumer to consumer and consumer to business, is neglected, as in LMRT. The marginal cost of a download is c o for the originating network and c t for the terminating network. Hence, the total marginal cost of traffic is c = c o + c t per download. There are q consumers and q Websites. The gross utility of the marginal consumer from joining the network when no delays are present is determined by the downward-sloping (inverse) demand curve V (q, q) q, where V (q, q), the total gross utility of q consumers when 5 This assumption is relaxed in Section These terms are commonly used in the telephony context. 6

7 they can download content from q Websites, is strictly concave in q and differentiable three times. The dependence on q means that a consumer s utility is an increasing function of the number of downloads she has access to. The downward-sloping demand curve assumption is equivalent to assuming that for a given q, consumer valuations are i.i.d. (cf. [4]). We model consumers as atomistic users: a consumer does not consider her individual decisions as having an effect on either price or market congestion. This specification is common in congestion control models (cf. [4], [16], [17]). We also assume that 2 2 V (q, q) q 2 < ξ 3 V (q, q) q 3 for all 0 ξ q. 7 The utility derived by the marginal Website from being on the network is linear in the number of consumers and follows the strictly decreasing (inverse) demand curve U(q, q) q = qu( q). To make the market viable, we assume that 1 q V (q, q) q + 1 q U(q, q) q > c for some positive q and q. Networks generate primary revenues by charging their customers by the download. Network operator i charges consumers p i per download and Websites p i per download. An additional access charge a is transferred between the originating and terminating networks for every download. This fee is also known as Termination-Charge, since in traditional telephony networks, where receiving a call is free, the originating operator pays the terminating operator to access and use its network. In the Internet setting, both networks bill their respective customers, hence it is not a-priori clear which network should pay the access charge. Thus, we do not restrict the sign of a: if a > 0, the originating network pays the terminating network, as in telephony networks; if a < 0, the terminating network pays the originating one. Figure 1 illustrates the model s traffic and payment transfers in a market with two network operators. Our model focuses on the effects of network delays in the consumer market. Consumers are delay-sensitive, with a delay of one time unit resulting in v units of consumer disutility. The expected delay is a function of the number of consumers on the network, the network s capacity, the number of Websites and the distribution of download sizes. We assume that all 7 This assumption is made for technical reasons that will become clear later. 7

8 Figure 1: An illustration of the model for a market with two network operators, 1 and 2 with capacities µ 1 and µ 2, respectively. The figure shows all possible payments and traffic flows in the market for either on-net or off-net traffic. download sizes are i.i.d., having a general distribution with finite variance. We denote the expected delay experienced by a consumer on network i as W i. To derive the consumer s net utility from a download, we subtract from the consumer s gross utility both her network charge p i and her delay cost. Thus, the marginal consumer s (expected) net utility on network i is 1 q V (q, q) q p i v W i. Consumers join the network that gives them the highest net utility, and they continue to join as long as their net utility is positive. In equilibrium, the number of consumers on network i, q i, satisfies 1 V (q, q) = p i + v W i, (1) q q where q = n i=1 q i. Equation (1) has the usual marginal value equals marginal cost interpretation. Here, a consumer s marginal cost consists of two components: the price she pays the network operator, p i, and her expected delay cost, v W i. The expected delay W i in turn depends on the level of activity on network i and on the network s capacity. Other things equal, consumers are willing to pay more for faster service. From the point of view of a network operator with a given capacity, equation (1) is the consumer market s effective demand relationship. 8

9 Our analysis applies to all queueing regimes in which the expected delay function W i is twice differentiable, convex and increasing in the traffic volume. The expected traffic on network i is q i q downloads. Its capacity µ i is the number of average downloads that an otherwise empty network can carry per unit of time. We denote by L i (q i q,µ i ) the expected number of downloads on the network and assume that L i = f ( ) qi q µ i for an increasing function f. This means that the expected number of downloads on the network is independent of time units, e.g., measuring both traffic and capacity in downloads per second yields the same expected number as measuring both in downloads per minute. 8 By Little s law (cf. [6], Section 2.4), L i = q i q W i, that is, the expected number of downloads on network i is equal to the traffic times the expected delay per download. It follows that the average delay on network i satisfies W i = 1 q i q f ( qi q µ i ). In several places, we gain further insight by considering the special case where the download sizes are exponentially distributed, giving rise to networks that are modeled as M/M/1 queues. Network operators compete in both the consumer and Website markets. We model competition among network operators in the consumer market using the Cournot model. 9 To simplify matters, we assume there is an additional network that serves only Websites and its price is set at the marginal cost, p = c o + a. In competition, this induces pricing at p = c o + a for all Websites, consistent with LMRT s off-net-cost pricing principle. In Section 6, we consider Cournot competition in the Website market as well. Performing the analysis for general demand curves requires complex conditions on the mixed derivatives of the utility functions without gaining commensurate insights. Thus, while we perform the analysis for general demand curves, explicit results will focus on the 8 Needless to say, this assumption is satisfied by all common queueing models. 9 We use Cournot competition since for most reasonable parameter values, a Bertrand equilibrium does not exist. This is the case, for example, for linear or constant elasticity demand curves. 9

10 special case of linear demand curves, with Consumer gross utility per download: 1 V (q, q) = α βq, (2) q q and Website utility per download: 1 U(q, q) = α q q β q. (3) 3 Equilibrium and Access Charge Consider n networks with capacities µ i (i = 1,2,...,n), and assume that the access charge is fixed at a. Since u( q) is strictly decreasing and all Websites pay p = c o + a, the access charge a uniquely determines the number of Websites at q = u 1 (c o +a). When the demand curve is linear, q = 1 β( α a c o ). The number of consumers q i on network i must satisfy (1) subject to the constraint 0 q i q < µ i, i.e., the total traffic on each network must be non- negative and strictly less than its capacity. 10 The capacity constraint is the common stability condition of queueing theory (cf. [6], Section 5.1), which requires that the utilization of each network, q i q µ i, be less than unity. 11 Since in our model traffic is strictly less than capacity, we need to explicitly distinguish between the two. In this Section we consider the scenario where the networks have fixed, predetermined capacities. To facilitate exposition and comparisons, we present our analysis for the case of symmetric networks with the same capacity µ. We first consider the benchmark of welfare maximization. 10 If no feasible solution exists, we set q i = If this condition is not satisfied, the system becomes unstable and the expected delay cost becomes infinite. 10

11 3.1 Welfare Maximization Consider a centralized market where a social planner sets the prices charged by the networks to both consumers and Websites so as to maximize expected social welfare, defined as the sum of consumers and Websites gross surplus net of the networks expected operating costs and the consumers expected delay costs. Since each consumer on network i experiences an average delay of W i per download, the problem is to maximize: n SW(q 1,...,q n, q) = V (q, q) + U(q, q) q i q vw i c q q, (4) subject to the constraints that the traffic volume on each network is non-negative and less than its capacity, and that consumer demand satisfies (1). The solution to (4) is given by the following proposition. i=1 Proposition 1 The first-best solution is a symmetric solution with n networks in the market. The first-best prices satisfy: p fb = 1 q V (q, q) q vw i = c 1 q U(q, q) q + qv n W i q i (5) and p fb = 1 q U(q, q) q = c ( ) 1 V (q, q) vw i + qv W i q q q. (6) Proof : The first order conditions with respect to q i imply that for all i and j, vw i + vq i W i q i = vw j + vq j W j q j. (7) Since all networks have the same capacity, the expressions on both sides of (7) are identical monotone increasing functions of their respective quantity arguments, implying that the q i are the same for all i = 1,2,...,n. Now let ˆq n and ˆ q n be the efficient consumer and Website 11

12 market sizes in a market with n networks. Then, ) SW (q 1 = ˆqn 1 n 1,...,q n 1 = ˆqn 1 n 1,q n = 0, ˆ q n 1 ) SW (q 1 = ˆqn 1 n,...,q n = ˆqn 1 n, ˆ q n 1 SW (q 1 = ˆqnn,...,q n = ˆqnn ), ˆ q n, where the first equality holds due to symmetry and the fact that W i is convex and increasing in q i q, and the second follows from (4). Thus, border solutions in which less than n networks participate in the market are less efficient than the interior solution. It follows that the firstbest solution is the interior stationary point, and is symmetric in the consumer market. Now, (5) follows from (1) and the optimality conditions of (4), and (6) follows from the optimality conditions of (4) and the fact that the marginal Website cost p must equal its marginal utility. In LMRT, the first-best prices for the consumer and Website markets are: p fb LMRT = c 1 q U(q, q), (8) q and p fb LMRT = c 1 q V (q, q), (9) q where U(q, q) q = u( q) and V (q, q) q = v(q). That is, each market segment (consumers or Websites) is charged a price equal to marginal cost minus a discount that reflects the positive externality to the other segment (for example, v(q) is the additional gross consumer surplus generated by an additional Website). In our model, the first-best prices reflect (i) the marginal cost, (ii) a cross-market network externality, and (iii) a delay externality. We discuss each of the two externalities in turn. Cross-Market Network Externality. An extra Website generates an additional gross utility of 1 q V (q, q) q to the average consumer. On the other hand, the consumer s access to the additional Website increases the expected delay cost by v W i, resulting in a net additional 12

13 utility of 1 V (q, q) q q vw i. This net utility reflects the net cross-market externality. 12 Delay Externality. An additional consumer on network i increases the number of downloads, increasing the delay of all consumers on the network. The delay increase per additional consumer on network i, W i q i, affects all q i consumers on the network. The total cost this additional delay inflicts on all other consumers is the premium charged to the consumer that increased the network load. Although the Websites do not experience a delay, each additional Website leads to more consumer downloads and congestion. An extra Website implies an additional delay W i q to all consumers when visiting the existing q Websites. The total cost this additional delay creates is the premium charged the entering Website. We next study the outcomes of an oligopoly network operator market. 3.2 Oligopoly For a given (exogenous) access charge a, all network operators charge Websites the same price, p = c o +a. Thus, the distribution of Websites across networks is indeterminate. We assume that network i serves a fraction θ i of the Websites (0 θ i 1). 13 We assume quantity-setting (Cournot) competition in the consumer market, i.e., each network operator determines how many consumers to acquire given the number of consumers acquired by its competitors. Given its opponents strategies, network operator i solves, max q i q i q(p i c t ) + q i q(1 θ i )a + q qθ i ( p i c o ) qθ i (q q i )a (10) Subject to: 0 q i q < µ and (1). Simplifying 14 equation (10), the problem becomes: 12 This discussion assumes that 1 q max q i q i q(p i (c t a)), (11) V (q, q) V (q, q) vw q i 0. If say information overload leads to a negative, q which is possible, Websites should be charged a premium to deter excess entry. 13 The results of this section are independent of the θ i. 14 The simplification is achieved by adding and subtracting θ i qq ia to expression (10), and using the fact that p = c o + a. 13

14 subject to the same constraints. Note that for any fixed symmetric µ, the consumer delay W i is only a function of the network traffic q i q and can therefore be denoted W(q i q). For any network i, the equilibrium may result in a positive number of consumers or in no consumer participation. In our linear model, this depends on the parameter S 0 = α vw(q i q = 0) (c t a), which is the marginal surplus of a consumer entering an empty market. Theorem 1 There is a unique equilibrium for the linear demand model (2) (3). If S 0 0, all quantities are zero. If S 0 > 0, the equilibrium is symmetric with positive quantities. The equilibrium consumer market size q satisfies: α n + 1 βq vw(q q n n ) vq W( q q n ) (c t a) = 0. (12) n q Proof : Using the optimality conditions, an n-firm equilibrium must satisfy the following system of equations: 15 α n j=1 βq j βq i vw i (c t a) vq i W i q i = 0 for some i, (13) βq i + vw i + vq i W i q i = βq j + vw j + vq j W j q j for all i,j, (14) and 0 q j < µ j for j = 1,...,n. Since the left and right hand sides of equation (14) are monotone increasing in their respective arguments, any equilibrium must be symmetric. For an n-firm equilibrium to exist, equation (13) must have a solution q > 0 so that q i = q n for i = 1,...,n, which implies (12). 15 The optimality conditions imply that an equilibrium must satisfy the system α nx j=1 βq j βq i vw i (c t a) vq i W i q i = 0 for all i = 1,..., n, where 0 q i < µ i for all i. Equations (13) (14) are equivalent. 14

15 Our assumption that 2 2 V (q, q) < ξ 3 V (q, q) q 2 q 3 for all 0 ξ q implies that (11) is concave and twice differentiable in q, hence the LHS of (13) is continuous and decreasing in q i. It follows that a solution exists if and only if the LHS of (13) is positive at q = 0, which is equivalent to S 0 > 0. To complete the proof, it remains to show that any equilibrium with l firms where l < n is not feasible. Any l-firm equilibrium must satisfy (13) (14) with n = l. It follows that no solution exists when S 0 0. When S 0 > 0, it suffices to show that another network will respond by entering the consumer market. Denote the total market size under the l-firm equilibrium by q l. If the best response is q br > 0, it must satisfy: α 2βq br βq l vw br (c t a) vq br W br q br = 0. (15) Expression (15) is decreasing in q br and therefore the best response is positive if and only if: βq l < α vw(q br q = 0) (c t a). By (13), the market size q l must satisfy: βq l = l (α vw(ql q ) vql W( ql q l ) l + 1 l l q l (c t a)) < l l + 1 (α vw(q q = 0) (c t a)) < α vw(q q = 0) (c t a). The last inequality holds since S 0 > 0 by assumption. The above implies that q br > 0 exists, and therefore no l-firm equilibrium, for l < n, is feasible. In LMRT, when 1 q V (q=0, q) q p > 0, the market equilibrium is always a symmetric duopoly, prices are set at marginal cost and networks make zero profits. In our setting, the unique n-firm equilibrium is symmetric and firms participate whenever S 0 = 1 q V (q=0, q) q vw(q q = 0) p(q = 0) > 0. Thus, the market is viable if it is worthwhile for consumers to join, and it will then have a symmetric equilibrium. In equilibrium, network i prices a 15

16 download at p i = c t a q i 2 V (q, q) q q 2 + q i v W i (16) q i in the consumer market. In the linear consumer demand case (2), (16) is reduced to p i = c t a + βq i + q i v W i q i. (17) Network i s expected profit is q i q(βq i + q i v W i q i ), which is positive for positive q i. It follows that network operators charge consumers a premium for the delay externalities they inflict on the network, and the externality fees contribute to the networks profit directly. The other premium charged, βq i, is the result of Cournot competition. Thus, in our setting, equilibrium prices are higher than in LMRT, reflecting in part the negative delay externalities consumers exert on the network. The higher prices in turn lead to positive operator profits and to smaller consumer markets. 3.3 Optimal Access Charge As seen in Section 3.2, the decisions made by profit-maximizing network operators are affected by the access charge a. In this Section we consider how a regulator might set the access charge. Ideally, the regulator would want to set an access charge that induces the first-best solution. This, however, turns out to be impossible in general. Combining the first-best prices p fb and p fb ((5) (6)) with the equilibrium pricing strategy (16), the optimal access charge should satisfy on the one hand: and on the other hand: a(p) = 1 q a( p) = c t 1 q U(q, q) q V (q, q) q c o q 2 V (q, q) n q q 2, + vw i + qv W i q. In general, a(p) a( p) and the first-best solution is not attainable We can have a(p) = a( p) if one of the market segments (the consumer or Website market) has a fixed predetermined size and inelastic demand, as in LMRT. 16

17 We next derive the access charge a that maximizes expected welfare subject to the network operators equilibrium strategies in the linear demand setting. Given that the equilibrium is symmetric (Theorem 1), the regulator s problem is: max q, q 0, q q n <µ q q ( ) α β β q + α 2 2 q vw i c (18) subject to c o + a = α β q (19) α n + 1 n βq vw i qv n W i q i (c t a) = 0. (20) Proposition 2 Consider the model with linear demand curves. The optimal access charge satisfies a = α β q c o, where (q, q ) solve the system of equations: α + α n + 1 n βq β q vw i qv n W i q i c = 0, (21) and ( q + q q ) ( ) ( α β β q + α q 2 2 q vw β i c = q q 2 + β ) q 2 q + v W i q + v W i q q q (22) for q q q = a q a n+1 n = (β + v W i q i ) + qv n β + v W i q + qv n 2 W i q i q 2 W i q i q. (23) Proof : Combining constraints (19) and (20), we get (21). The Implicit Function Theorem and (21) imply (23). Now, using the optimality conditions 17 we obtain the system of equations (21) (22) as a necessary condition for the optimality of (q, q ). If the optimality conditions have multiple solutions, the maximizing point (q, q ) is determined by a straightforward welfare comparison. Plugging q into constraint (19), we find a. 17 The maximal welfare that can be achieved at a border solution is 0. Therefore, for viable markets, (18) has an interior maximizer that satisfies the first order conditions. 17

18 Proposition 2 provides a method to calculate the optimal access charge a regulator should set. Since W i is convex and increasing in the traffic volume, (23) implies that q q < 0, that is, with more consumers we must have less Websites. This is a result of the fact that other things equal, increasing a implies a price increase in the Website market and a price decrease in the consumer market. We further discuss the implications of the optimal access charge in Section 4.4, where we consider capacity decisions as well. 4 Capacity Decisions We now relax the constraint of fixed capacities and assume that network operators simultaneously choose both the capacity levels of their networks and the amount of traffic flowing through them. We assume a constant average cost g per unit of capacity and solve for the first-best, oligopoly and optimal access charge. 4.1 Welfare Maximization When the regulator can set both the traffic volumes and capacities to maximize overall surplus, she solves max q i, q,µ i 0,q i q<µ i i=1,...,n SW = V (q, q) n i=1 q i qvw i + U(q, q) cq q g n i=1 µ i. (24) The first order optimality condition with respect to µ i is q i qv W i µ i g = 0. (25) Focusing on the case of exponential download sizes, equation (25) implies that the optimal capacity is given by µ i = q i q + vq i q g. (26) That is, the optimal capacity is the sum of the actual traffic q i q and excess capacity vqi q g, which increases as the square root of the traffic. The average delay experienced by a 18

19 consumer, g vq i q, decreases as the square root of the traffic. Thus, queuing effects are characterized by economies of pooling: networks with more consumers can offer a superior service to their customers if they set their capacity optimally. As shown below, these pooling effects imply that the market behaves as a natural monopoly. Proposition 3 The first-best solution for n network operators and exponential download sizes has a single network operator in the market. The first-best prices satisfy p fb = c + g 1 q U(q, q), (27) q and p fb = c + g ( 1 V (q, q) q q ) vg, (28) q q where q and q are the first-best market sizes. Proof : Assume by contradiction that the first-best solution is at an interior point. Similar to (7), the first order conditions with respect to the consumer market sizes imply vw i + q i v W i q i = vw j + q j v W j q j for all i,j. Combining these equalities with (25) and using the fact that W i = 1 q i q f(q i q µ i ) for an increasing f, it follows that q i = q j and µ i = µ j for all i,j = 1,...,n. Thus, at this interior solution, denoted as (q 1 = ˆq n,...,q n = ˆq n, q), the expected welfare is: SW(q 1 = ˆq n,...,q n = ˆq n, q) = However, expected welfare at the border solution V (ˆq, q) + U(ˆq, q) (c + g)ˆq q 2 nˆq qvg. ( ) q 1 = ˆq n 1,...,q n 1 = ˆq n 1,q n = 0, q is: SW(q 1 = ˆq n 1,...,q n 1 = ˆq n 1,q n = 0, q) = V (ˆq, q) + U(ˆq, q) (c + g)ˆq q 2 (n 1)ˆq qvg. Clearly, SW(q 1 = ˆq n 1,...,q n 1 = ˆq n 1,q n = 0, q) SW(q 1 = ˆq n,...,q n = ˆq n, q), with equality only when ˆq = 0, which leads to a contradiction. The efficient solution must 19

20 therefore be a border solution. By a similar argument, a border solution that uses only n 2 networks is more efficient than a border solution that uses n 1 networks. Applying this argument iteratively, we conclude that the first-best solution has a single network. Now, equations (27) (28) directly follow from the first order conditions, equation (1) and the Website pricing p = c o + a. Proposition 3 says that social welfare is maximized by having only one network operator (a natural monopoly) that serves all consumers, due to economies of pooling in capacity investments. As might be expected, the first-best network operator loses money. We further discuss the nature of this loss and the second-best (Ramsey) prices in Section To compare the first-best pricing results of proposition 3 to LMRT, we interpret c t as the cost of capacity g. Under the exponential assumption, the expressions for the first-best consumer prices ((27) for our model and (8) for LMRT) are similar, although the actual prices are obviously different. Our results imply that optimal capacity is set so that the negative delay externality is equal to the marginal capacity cost. The delay externality is not eliminated, but capacity is expanded to the point where the consumer pricing equations exactly coincide with those of a deterministic model. As for the Website prices, our expression for p fb, (28), adds the delay due to an additional Website visit to the LMRT expression (9), since the externality term in our model corresponds to net utility. The other terms parallel those in LMRT, again taking advantage of the equality of the negative delay externality and the marginal capacity cost, g. This equality, however, does not hold when the download size distribution is not exponential, as we illustrate below Effects of Download Size Distribution The equality of the delay externality at optimal capacity to the cost of capacity is not obvious. In fact, it does not hold when the download size distribution is not exponential. If X is a random variable representing the download size and µ is capacity, then X µ is the time required for a download (excluding delays). To examine the effect of the download 20

21 15.05 (a) Delay Externality at First Best Solution Var(X) (b) Cross Market Externality at First Best Solution 10 capacity cost g Delay Externality 10.2 LMRT Queueing Model Var(X) (c) First Best Consumer Price LMRT Queueing Model Var(X) Figure 2: Effects of changing the variance of download size. Panel (a) shows the first-best consumer delay externality, Panel (b) the first-best cross-market network externality, and Panel (c) the first-best consumer price, all as functions of the download-size variance Var(X) for a market with n = 2 networks, α = 25, α = 20, β = 0.2, β = 0.5, c o =0, g = 15 and v =350. In all cases, we compare our results to LMRT. size distribution on the first-best prices, we set its expected value at 1 (EX=1) and vary its variance Var(X) while keeping all other parameters constant; for the exponential distribution, Var(X) = EX = 1. Figure 2 shows how the delay externality, cross-market network externality discount and first-best consumer price vary as functions of the download-size variance. Figure 2(a) shows that the first-best delay externality q i v W i q i is equal to the marginal capacity cost (which is what we found in (27)) only for exponentially-distributed downloads (Var(X) = 1). As the variance increases so Var(X)> 1, the delay externality increases so it is larger than the marginal capacity cost, g. Similarly, as the variance decreases so Var(X)< 1, the delay externality decreases and its value drops below g. However, the delay externality need not be increasing in Var(X), and one can construct examples where 21

22 the opposite holds. 18 Figure 2(b) shows the first-best price discount due to the crossmarket externality in our model (as a function of the download size variance) and in LMRT (where the market sizes are invariant to the download size variance). For our model, the externality increases (the discount increases in absolute value) with the variance since a variance increase leads to a decrease in both the consumer and Website market sizes, which imply that a higher externality is enjoyed by each Website in the market from an additional consumer. Figure 2(c) shows the first-best consumer prices, which in this case are given by the vertical summation of the delay externality (Fig. 2(a)) and the cross-market externality discount (Fig. 2(b)). Even when downloads are exponential (Var(X) = 1), so the delay externality coincides with the marginal capacity cost, the first-best prices of the two models differ. The difference is driven by the different first-best market sizes that lead to different cross-market network externalities. When Var(X) = 0, the download size is deterministic in both models. Yet, in our model price is different from that in LMRT. Even for deterministic download sizes, the stochastic timing of downloads and the sensitivity to delay result in a delay externality that leads to different market volumes and prices Ramsey Prices As discussed earlier, a network operator charging the first-best prices loses money. To see this explicitly, note that since the Websites utility is linear in the number of consumers, 1 U(q, q) q q = 1 q t= q t=0 u(t)dt. That is, the consumer price is adjusted downwards to reflect the external effects of an additional download on the average Website. Since the net utility of every Website on the network (except the marginal one) is positive, the net utility of the 1 t= q average Website is positive: q t=0 u(t) p > 0. Substituting this inequality into (27), we 18 This is because the delay externality is deceasing in the excess capacity and increasing in both the traffic volume and the variance, while excess capacity is increasing and traffic is decreasing in the variance. As a result, the delay externality may either increase or decrease with the variance. 22

23 find that p fb + p fb = c + g ( 1 q U(q, q) q p fb ) < c + g. Thus, even when the excess capacity investment is ignored, the total revenues collected do not cover the costs of operating the network. Taking into account the need to invest in excess capacity implies even larger losses. To address this issue, we solve for the Ramsey (second-best) prices. We assume that the consumer utility is linear in the number of Websites, so V (q, q) = qv (q). Proposition 4 The second-best solution for n network operators and exponential download sizes has a single network in the market. The second-best prices satisfy p sb p fb = λ(p sb + p sb c g) λqv (q), (29) and p sb p fb = λ(p sb + p sb c g) λ qu ( q), (30) for a constant λ > 0, where q and q are the second-best market sizes. Proof : We solve problem (24) subject to the constraints q i q ( ) ( ) 1 V (q, q) 1 U(q, q) vw i c t + q qθ i c o gµ i 0 for all i. (31) q q q q The first order condition with respect to µ i is proportional to (25) and therefore implies that the optimal capacity expression (26) is still valid. We now show that a single network operator will enter the market under the second-best solution. Assume by contradiction that an n- firm interior solution ((q 1, q 1 ),...,(q n, q n )) maximizes welfare subject to (31) where q i = qθ i. Then, in particular, SW ((q 1, q 1 ),...,(q n, q n )) SW ((q 1, q 1 ),...,(q n 1 + q n, q n 1 + q n )) (note that the n 1 networks solution satisfies (31)). This implies that q n 1 + q n qn 1 + q n, which leads to a contradiction. Thus, the n-network solution cannot be the second-best solution. Repeating this argument iteratively, we conclude that the second-best solution has a single network. 23

24 Using the Lagrange multiplier λ and the first-best prices (27) (28), the optimality condition with respect to the consumer market size reduces to (29). Similarly, the optimality condition with respect to the Website market size reduces to (30). Proposition 4 shows that economies of pooling lead to a single-firm second-best solution. The second-best price premia over the first-best prices recover both the losses due to crossmarket network externality and the costs of excess capacity. As usual, the price premia needed to recover the cross-market externality, λqv (q) for the consumer and λ qu ( q) for the Website market, are inversely proportional to the demand elasticities. However, the Ramsey prices incorporate the term, λ(p sb + p sb c g), which can be rewritten as λ(p fb + p fb c g) + λ(p fb + p fb p sb p sb ). The first term, λ(p fb + p fb c g), is proportional to the unrecovered cost of excess capacity in a first-best market with no network externalities. The second, λ(p fb + p fb p sb p sb ), accounts for the different market sizes and prices. This structure appears in both the consumer and Website prices despite the asymmetry in the two segments delay sensitivities. This happens because the expected traffic volume q i q is symmetric in q i and q, implying that a change in either has a similar effect on congestion and capacity investment. 4.2 Oligopoly Next consider the problem of competing network operators given a fixed access charge a. Given the values of q j,µ j for all j i and Website prices c o + a, network operator i solves: max q i,µ i 0,q i < µ ĩ q q i q ( ) 1 V (q, q) vw i (c t a) gµ i. (32) q q 24

25 The optimal strategy must satisfy the following system of equations for all i: q i qv W i µ i g = 0 1 V (q, q) + q i 2 V (q, q) q q q q 2 vw i q i v W i (c t a) = 0. q i Assuming exponential download sizes, the optimal capacity level is given by (26). Thus, pooling effects also come into play in the oligopoly setting: a network with more consumers has an advantage, as it can set capacity to achieve both higher utilization and lower delays. In the linear demand model, any n-firm equilibrium must satisfy the system: α β βq i + n j=1 vg q j q = c t a + g + βq i for some i, (33) i q vg q i q = βq j + vg q j q for all j i, (34) and q j 0 for j = 1,...,n. Moreover, for any l n, an equilibrium with l networks must satisfy the following. Lemma 1 Consider a market with n network operators, access charge a, linear demand curves and exponentially distributed downloads. Then, q l, the total consumer market size in an l-firm equilibrium, is the unique solution to the system: α βq l lvg q l q = c t a + g + βql, (35) l and ( q l l ) vg > β q. (36) The equilibrium is symmetric, and the consumer prices are given by p i = c t a + g + βql l for all i = 1,...,l. Proof : Equations (33) (34) are necessary condition for an n-firm equilibrium. The solution maximizes network i s profits if the expected profit function (32) is concave and 25

26 network i s profit is positive (otherwise network i would prefer q i = 0). These two requirements imply: q 3 2 i > 1 vg 4β q and q 3 2 i > 1 vg β q, (37) respectively, where the binding constraint is the positive profit constraint. Thus, any l-firm equilibrium must satisfy the system (33), (34) and (37) using n = l. For market volumes that satisfy (37), both sides of (34) are identical monotone increasing functions. Therefore, a symmetric l-firm equilibrium is the only l-firm equilibrium, and it is fully characterized by (35) (36). Since the LHS of (35) is continuous and decreasing and the RHS is continuous and increasing in ql l, there is a unique solution which is symmetric. Now, equations (1) and (35) imply that network i s equilibrium price is p i = c t a+g+ βql l. By Lemma 1, a network operator can be profitable in the consumer market only when its scale exceeds the minimum scale, ( 1 β vg q )2 3. Below this minimum scale, the network operator is better off staying out of the consumer market. The minimum market size condition is equivalent to requiring q i q(βq i ) > g(µ i q i q), where βq i is the optimal price premium over the direct cost c t a + g implied by the optimal capacity. Condition (36) therefore means that the total premium collected must be greater than the cost of excess capacity. The minimum scale is increasing with the consumer delay cost and marginal capacity cost, since both lead to larger capacity investments. It is decreasing in β (since a higher β increases the price premium) and in the number of Websites q (which contributes to the scale). The minimum scale also implies that regardless of the number of competing network operators, only a limited number will actually enter the market: Excess capacity investment serves as an entry barrier. Lemma 1 provides only necessary conditions for an l-firm equilibrium. The following Theorem provides existence conditions and shows how the equilibrium market structure changes with market parameters. 26

27 Theorem 2 Consider a market with n network operators, access charge a, linear demand curves and exponentially distributed downloads. Let S l = α (c t a + g) (l + 2)( βvg q )1 3 for any l n. Then, there exists a symmetric l-firm equilibrium if S l > 0 and βq l > S 1. If l = n, it suffices that S l > 0. Proof : Rearranging terms in (33) we get: ( )2 If we set q i = 1 3 β vg q α β l j=1 vg q j q (c t a + g) βq i = 0. (38) i q for all i, the LHS of (38) is exactly S l. Since (38) is decreasing and continuous in q i, and since any equilibrium is symmetric (Lemma 1), S l > 0 is a sufficient condition for the existence of a solution to the system (35) (36). To ensure that this is an equilibrium when l < n, we require that the best response of the l + 1 st network to this strategy is to stay out of the consumer market. Therefore for q br the best response consumer size, the best response function must satisfy: vg α β(q l + q br ) q (c t a + g) βq br q 2 br q q br = 1 3 < 0, β vg q which reduces to the inequality βq l > S 1. Theorem 2 introduces the parameter S l, which is the left hand side of (38) at the symmetric break-even point q i = ( 1 β vg q )2 3 for all i. S l > 0 implies that the break-even consumer will have positive surplus from joining the l th network, given that the market has networks at their break-even points. Theorem 2 suggests that an l-firm equilibrium in the consumer market is feasible when it is worthwhile for the break-even consumer to join the l th network while allowing the networks to earn positive profits. On the other hand, the condition βq l > S 1 can be written ( as V )2 (ql +q l+1, q) q vw l+1 < p l+1 for q l+1 = 1 3 β vg q, that is, the l + 1 st network cannot make positive profits and will choose to stay out of the consumer market. Note that for a given market there may be multiple equilibria with different numbers of networks in the market. 27

28 We next compare the market size in our model to that in LMRT, where we interpret c t, the cost of transmission on the terminating network, to be the cost of capacity g. Corollary 1 In equilibrium, the size of the consumer market is lower, and the consumer price is higher than is LMRT. While the higher consumer price in our model is due to the different assumptions, the smaller equilibrium consumer market size is a result of consumers delay sensitivity. In addition, our model implies a market structure that is quite different from LMRT s. A key reason for the difference is the pooling effect that characterizes queueing phenomena, coupled with consumers sensitivity to delay. In particular, these results hinge on consumers delay sensitivity being positive. If v = 0, network operators have no incentive to invest in excess capacity, and then, as in the fixed-capacity setting (Theorem 1), the unique market equilibrium is an n-firm equilibrium. Further, the number of network operators that enter the market is sensitive to market parameters: more network operators will enter more lucrative markets, namely markets where either the consumer utilities are high (high α); or the capacity cost g is low; or consumers delay sensitivity v is low; or there is a high positive access charge a; or a combination of the above. We illustrate these results for the case of two network operators. 4.3 Two-Operator Example Consider the changes in the structure of a market with n = 2 network operators as we change the marginal capacity cost g. For each value of g, we compare our solution to that of LMRT. 19 Figure 3 illustrates the change in equilibrium market structure implied by Theorem 2. For high values of g (approximately g > 22.9), the consumer market is not viable (denoted in the Figure by No Entry ). As g decreases, the market structure changes to a monopoly (denoted M ). Then, in the region g [17.9,20.6], two network 19 In the LMRT model, we interpret c t, the cost of transmission on the terminating network, as the cost of capacity, and set it equal to g. 28