The pricing of critical applications in the Internet Jacques Crémer CNRS, IDEI and GREMAQ Université de Toulouse 1 Cyril Hariton Université de Toulouse 1 July 1999 Abstract A number of authors have recently proposed techniques for pricing the access to Internet resources in case of congestion. However, these approaches do not take into account the fact that some applications necessitate guaranteed capacity over a relatively long period of time. This paper discusses some elements of the theory of a mechanism that would accommodate such applications. We begin by reviewing both current practice and theory. We then build in…nite horizon stationary models with asymmetry of information, which we …rst use to show the limits of smart markets (McKie-Mason and Varian). Finally, in a very simpli…ed model, we compute the optimal mechanism, and in a speci…c example, we show that the optimal mechanism favors the high type long term user.

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The strikingly rapid development of the Internet in recent years has created a large number of new policy problems, and for economists is also raising interesting analytical issues: the types of externalities are di¤erent on the Internet and in other network industries, new products are being designed for distribution on the Internet, old products are being redesigned, the ‡ow of information between …rms and their clients is improved. On a more ‘day to day’ plane, the pricing of the network requires new thinking, and in this paper, we would like to contribute to this task. We will argue that the literature has neglected the dynamic aspect of the pricing problem, and more speci…cally the needs of what we call ‘critical applications’, that is applications that require high quality connections for an extended and continuous time. We will propose a new methodology to explore this issue and present examples of its use. As the story is often, and we believe correctly, told, the Internet grew out of a collaborative e¤ort by academics. The e¢cient use of the resource was enforced through social pressure. Since the Internet has been privatized, new modes of allocation of resources must be found. In section 1.3, we review the existing arrangements. Small users, those who connect through a dial-up connection are mostly subject to a ‡at fee, independent of their use of the network. Larger users typically pay an amount that is roughly proportional to their peak consumption. None of these methods allow for congestion based pricing or for di¤erentiation between high priority and low priority usage. However, as we argue in section 1.2, there are a number of new applications which are important for the future of the network and require guaranteed high quality connections over an extended period of time. The formal models that we build are an attempt to understand the type of instruments that will be necessary to provide this guarantee. We consider a transmission facility, which we call a ‘pipe’, and assume that at any period new potential customers appear who want to use this pipe, some of them for one period, others for several periods. The aim of a pricing policy is to establish under which conditions a potential customer should be given the use of the pipe for one or several periods, and under which conditions he or she should receive a …rm guarantee of uninterrupted use. In our …rst model, presented in section 3.2, a new customer appears every period and every customer wants to use the pipe for two consecutive periods. The policy issue in this framework is to know whether any guarantee should be given to a customer who has already purchased one unit of the resource. We will use this model to study the use of the ‘smart markets’ proposed by McKie-Mason and Varian. We will show that in a dynamic framework, they loose one of their main theoretical advantages, the fact that they induce truthful revelation of the willingness to pay of customers. 2

In our second model, in section 3.3, two consumers appear in each period, a short run consumer who would like to use the resource for one period and a long run consumer who would like to use it for two consecutive periods. We assume that any long run consumer who is given the pipe for one period is guaranteed a second period (for instance because of the presence of large setup costs). In this framework, we are able to compute the optimal selling mechanism, which enables us to study the way in which customers who ask for long run commitment of the resource should be treated. The paper is organized as follows. In section 1.1, we explain brie‡y the basic technical features of the Internet. In section 1.2, more details are given to clarify the reasons why current protocols are not adequate for allocating resources to high priority users, a need required by new applications. Current pricing methods are explained in section 1.3. Section 2 surveys brie‡y the existing state of the research on Internet pricing, and we present our formal analysis in section 3. Finally, section 4 is a conclusion that brie‡y surveys other important issues in the economics of the Internet.

1

The Internet

1.1

Traveling from users to backbones

The main function of the Internet is to connect computer networks — it is a ‘network of networks’. One way to understand its functioning is to trace the ‡ow of information from a user A to another user B. First, user A connects its computer, through a standard telephone call (dial-up access), TV cables or a permanent open communication line (dedicated access), to a computer which belongs to an Internet Service Provider (ISP1 , hereafter called ISP A) which is in charge of arranging its delivery. This computer, a point of presence (POP) of ISP A, is the gateway through which A has access to the Internet. If user B is also a client of ISP A, and is linked to the same POP, then the message is stored in the computer of the POP until B connects to the ISP. Otherwise, ISP A transmits the message to the computer of ISP B, of which B is a customer. ISP A leases telephone lines from the POP where A is connected to Internet eXchanges (IXs) where it can exchange tra¢c with other ISPs2 . If there 1

Some Internet Access Providers (IAPs) only provide the basic access to the Internet, without services. For the ease of the description, both IAP ans ISP will be called ISPs. 2 Actually data is exchanged both at ‘NAP’s and IXs. Basically NAPs have public rules of acceptance while IXs are only ruled by private contracts. In the rest of this paper, the

3

exists an IX to which ISP A and ISP B are both connected, the message will be directly transmitted to ISP B. Otherwise, a path that the message must follow in order to reach ISP B is computed. Indeed, IXs are themselves connected to each other by ISPs or other companies (such as telephone companies or resalers of telephone services) in a basically hierarchical pattern. This interconnection structure is repeated at each level of the Internet. Small networks or individual users are connected to ISPs that are themselves connected to each other through one or more IXs. At the top level of the hierarchy lie the backbones, that provide overall connectivity and high transmission capacity, over long distances if needed. Backbones are usually designed in rings, with redundant parts, in order to provide security and di¤erent paths for moving data from one point to another3 . Other backbones are just pipes, linking di¤erent geographic areas (transoceanic cables e.g.). As all ISPs must be connected at least to one IX, this must also be the case for ISP B. Therefore, there must be at least one route leading from ISP A to ISP B. This route is found according to information on the computer of the …rst IX to which ISP A sends the message: availability of networks and of connections between them, peering agreements between ISPs, routing protocol implemented. A …rst intermediary (a transit ISP) is handed the message and takes it to a second IX. There, the route is recomputed in order to include some more information, and the message is given to another ISP, and so on until ISP B gets the message. Then, it is stored while waiting for the connection of user B (in case of dial-up connection), or arrives directly in B’s computer.

1.2 1.2.1

Internet applications Internet protocols

All these exchanges of data among di¤erent networks, using di¤erent physical supports for transportation (copper lines, …ber cables, wireless communications), are possible thanks to the interoperability between these networks. This is achieved with some standards of exchange, called protocols, such as the Internet protocol (IP) for the structure of data or the transfer control protocol (TCP) for the transportation of data. There are other levels where term IX will be used to refers to both NAPs and IXs. 3 Maps of networks are often available on line: PsiNet [http://www.psi.net/network/euconnectivitymaps.html], Tcg CerfNet [http://www.cerf.net/about/Bbone-map/Bbone-large.gif], or GTE [http://www.bbn.com/products/maps].

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standardization also operates, but ‘TCP/IP’ has practically become synonymous with Internet. Protocols operate in the following way. Before a message is sent, it is cut in small pieces (packets) which are sent independently on the Internet. In principle, packets could all follow di¤erent paths and after they have been collected at computer B, it is the job of that computer to put them back in the original order and to reconstitute the original message. This ‡exibility allows each individual piece of data to follow the most e¢cient path to …nd its addressee, given the level of congestion in the network. In reality, the necessity to implement pricing schemes reduces the ‡exibility of routing. Indeed, given the extremely decentralized nature of the Internet, there is no centralized body that can implement rules that ensure that every entity on the path of a packet receives payment for the service it has rendered. Under these conditions, ISPs have favored partners through which they send data. In particular, in most cases, they are direct or indirect clients of only one backbone. (See section 2). When packets arrive at a router, they are processed in order of arrival (…rst-in …rst-out, FIFO). The Internet protocol allows the user to set a priority for his or her message, but routing algorithms ignore them: there is no discrimination among packets at each router level. If the capacity of the router is reached, it queues the packets and the senders of data are ordered to slow down the rate of transmission of data. If, despite this slowing down, the queue over‡ows, packets are lost. As a consequence, Internet protocols authorize neither di¤erentiation between services/applications, nor service guarantee (delivery time e.g.). Other techniques, such as asynchronous transfer mode (ATM)4 , can give quality of service guarantees, and bene…t from a comparative advantage on that score. 1.2.2

Internet applications

The most important current applications of the Internet are electronic mail (e-mail) to send messages and documents between users, …le transfer protocol (FTP) to download documents from a server, or browsers and the hypertext transport protocol (HTTP) to navigate the World Wide Web (WWW). More demanding applications are developing or being planned, such as telephony, video-conferencing, video on demand or interactive gaming. These applications exhibit di¤erent sensitivity to congestion. For some, 4

ATM is well developed by now. It tries to reconciliate the very adaptable, but also unpredictable, TCP/IP to the circuit switching technique used by the usual vocal telecommunication …rms. Please refer to a documentation written by Sprint for an introduction to ATM [http://www.sprintbiz.com/news/downloads/atm.pdf].

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such as electronic mail, the cost of a few minutes delay are usually very small. For others, they are more substantial but do not threaten the use of the network. For instance, the fact that web pages that take ‘too long’ to load, although costly in time and frustration, do not prevent people from sur…ng in search of information or amusement. However, some applications are just worthless in the presence of delays, specially if these delays are not predictable. Many of applications that should yield the most spectacular bene…ts for the widespread use of the Internet technology fall into this class. For instance, it has been suggested that classes taught by specially competent professors could be broadcast over the network to other institutions. This is only feasible if there is reasonable assurance that su¢cient bandwidth will be available for the planned duration of the class. It has also been suggested that improvements in telecommunications technology will enable doctors in remote areas5 to consult with specialists in major research hospitals. Clearly, if a surgeon in a rural hospital is being adviced during an operation by a specialist some hundreds of miles away, a very strong guarantee that he will be able to follow the whole operation is necessary. In a more prosaic range, Internet telephony and video-conferencing will only be able to reach their full potential if one can guarantee high enough quality for long periods of time. In all these applications the Internet is competing with connection oriented networks, and there are some doubts over its capacity to guarantee a good quality connection. As usual, the issues are both technical and economic, and the economics are quite interesting. Most of the discussion of the pricing of the Internet has been conducted in a static framework: given a certain demand for bandwidth, what kind of tari…cation can be used to ensure that social welfare is maximized? The critical applications that we have discussed brie‡y create a new problem: how should we determine whether it is worthwhile reserving some resources over a period of time for a speci…c user?

1.3

Current pricing of Internet usage

As in the traditional telecommunication sector, users of the Internet di¤er in their demand for volume of communication and for quality of service. Some customers only need to send e-mails and download small documents. Some …rms use the Internet for a large proportion of their communications with their customers and subcontractors: huge databases are available on line, 5 On 24 February 1999, a surgeon realized an operation with the help of a specialist through the …rst version of the new Internet (Abilene).

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live audio and video conferences are given through the Internet. Some other …rms also use the Internet to sell their products and services. All these customers do not have the same needs. Light users require only dial-up connections, with small bandwidth, whereas heavy users require permanent connections, with high bandwidth, certi…ed delivery times, secured transactions and high reliability of the systems. But all value, at di¤erent levels speed, quality and security of data transfers. 1.3.1

Light users

Many ISPs all over the world provide Internet connectivity to individuals and very small …rms. There are more than 4000 ISPs with highly di¤ering characteristics (size, users pro…les, owned infrastructure, geographical locations of POPs) in the OECD countries (OECD, 1998a, p. 14). The tari¤s are, for the most part, of three types: unlimited (monthly) access, a …xed number of connection hours plus a usage based fee for extra hours, and a full-usage based (constant) fee. There is often, for the two last types of subscription, some free connection hours given for new customers. Moreover, set-up fees are sometimes required by providers, mainly in the US (Uunet, GTE, Sprint, for instance). In terms of number of ISPs, the industry is mainly composed by national or regional players and by a few international ones, mainly American …rms with agreements with national ISPs all over the world. International ISPs, such as American OnLine (AOL), o¤er more or less the same packages in all countries. Some of AOL’s o¤ers are summarized in table 1. Unlimited access is also proposed by AOL in some of these countries: Canada6 for $13.95 (US$9:3) and France for FF95 (US$16:41). In France, the unlimited package corresponds to 5:16 (2 + 3:16) hours of connection at the Essentiel package prices. The analysis of AOL o¤ers re‡ects the development of the Internet in the di¤erent countries7 . These Internet access packages do not contain any 6

There is a condition for this package: the user has to get access to the Interet through another ISP than AOL, but has an unlimited use of the services of AOL (e-mail, personal page e.g.). This is shaped for individual having access to the Internet from work, shool or University. 7 Germany and France o¤ers seem to assume that most of the clientele will have little use of the Internet: the unlimited package becomes pro…table for customers who are connected slightly more than 6 hours per month. On the other hand, Canada has clearly two types of o¤ers: discovery of the Internet (Light and Basic) with few hours, and ‘heavy’ individual use (Value) with a 50 hours package. O¤ers in the US re‡ect the fact there are many other users: packages of 50, 100 or 150 hours are often available. These are usually a variation on unlimited access modi…ed so that consumers do not stay connected twenty four hours a days, which they would be

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Australia - Extra Australia - Now Canada - Value Canada - Basic Canada - Light France - Essentiel Germany - Standard Germany - Card UK US

Prices per hour (local – US$) Hours In package Extra hours 15 2.00 – 1.27 4.00 – 2.53 3 3.92 – 2.48 4.00 – 2.53 50 0.54 – 0.36 1.95 – 1.29 5 2.79 – 1.85 3.95 – 2.62 3 2.32 – 1.54 3.95 – 2.62 2 17.50 – 3.02 19.00 – 3.28 3 3.30 – 1.91 4.95 – 2.86 2 4.95 – 2.86 2.00 – 1.16 3 1.65 – 2.71 2.35 – 3.86 NA NA NA

Source: National websites of AOL. Table 1: AOL Internet access packages. In the price columns, the …rst number is in local currency, the second in US dollars. provision for congestion, priority or quality guarantees. The characteristics, or the reputation, of each ISP is often the only information that the consumer has ex ante about the quality of the service. A recent trend is observed in the ISP industry. Some …rms8 o¤er free unlimited Internet access (but the customer must pay the local telecommunication bill), with their revenues generated by the fact that customers see advertisements when reading their e-mail, and/or by taking advantage of the di¤erence between interconnection charges and costs, through agreements with telecommunication operators. 1.3.2

Heavy users

Heavy users generally require a permanent connection to the Internet with high bandwidth. The services provided are often di¤erentiated by bandwidth: ISPs typically provide leased lines with 56 Kbps bandwidth, T1 lines (up to 1.54 Kbps), double T1 lines, T3 lines (up to 45 Mbps) and OC3 lines (from 60 to 155 Mbps). All these lines require additional equipments in order to deal with the incoming and outcoming ‡ows of data. Comparison are not easy because services are heterogenous, but it is interesting to compare the tempted to do with a ‡at rate for local telephony. 8 For example, in France, FNAC [http://www.fnac.net] or WorldOnLine [http://www.worldonline.fr] and, in U.K., Virgin [http://www.virgin.net] or BTClick [http://www.btclick.net].

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Company Geonet Exodus @Home Agis Cais Internet Savvis C&W Usa Sprint AT&T Concentric Ibm Uunet

Setup fee Monthly fee Monthly cost Setup/Monthly 0 1,395 1,395 0 2,499 1,199 1,407 2.08 1,500 1,400 1,525 1. 07 2,000 1,500 1,666 1.33 1,500 1,750 1,875 0.86 2,000 1,800 1,967 1.11 1,000 1,900 1,983 0.53 1,000 2,061 2,144 0.49 1,000 2,100 2,183 0.48 3,000 2,095 2,345 1.43 3,000 3,000 3,250 1 5,000 3,000 3,417 1.67

Source: Boardwatch (1998, p. 32-256). Table 2: Pricing of Internet access T1 packages. The monthly cost is obtained by dividing the total annual cost, that is the setup fee plus 12 times the monthly fee, by 12. The last column is the ratio of the setup fee to the monthly fee. o¤ers for the basic US small business Internet access: a full T1 leased line. Table 2 presents the o¤ers of some operators. Not all users require the full use of a leased line. Some operators catter to this clientele by o¤ering two kinds of services. The …rst one is the possibility to tier the bandwidth. The customer knows that its current tra¢c is one third of a T1 line, but believes that he may rapidly need a full T1. Therefore, it wants to keep the possibility to upgrade its connection with as small a change as possible. With a tiered T1, the customer is allocated a full T1 but buys only one third of the total bandwidth and has the right to upgrade its connection simply by changing one option of its contract, without any physical change in its installation. The second possibility is to buy a burstable T1. The user is also given a full T1, but is charged according to some computation of its actual average use or peak use. For instance, Uunet o¤ers for a tiered and a burstable T3 access line in the US are composed of a $6,000 one-time startup charge and the usage fees listed in table 3. Heavy users are often highly concerned by the risk of failure and some global ISPs guarantee availability of the network, as well as some upper limit for the latency between points of the network. For example, MCIWorldcom guarantees availability 100 % of the time (unavailability is credited to cus9

Port Speed (tiered) Sustained Usage (burst) 0 - 3 Mbps 3 - 6 Mbps 6 - 7.5 Mbps 7.5 - 9 Mbps 9 - 10.5 Mbps 10.5 - 12 Mbps 12 - 13.5 Mbps 13.5 - 15 Mbps 15 - 16.5 Mbps 16.5 - 18 Mbps 18 - 19.5 Mbps 19.5 - 21 Mbps 21 - 24 Mbps 24 - 27 Mbps 27 - 30 Mbps 30 - 33 Mbps 33 - 36 Mbps 36 - 39 Mbps 39 - 45 Mbps Sources:

Monthly price tiered burst $6,000 $8,000 $12,000 $14,000 $10,000 $17,000 $19,000 $12,000 $22,000 $26,000 $14,000 $29,000 $32,000 $17,000 $37,000 $43,000 $19,000 $48,000 $22,000 $26,000 $29,000 $32,000 $37,000 $43,000 $54,000 $55,000

Uunet US website: [http://www.us.uu.net/products/pricing.cgi/t3], Boardwatch (1998).

Table 3: Uunet tiered and burstable T3. The left column indicates, for the tiered T3, the authorized port speed. For the burstable T3, it indicates the 95th percentile of line usage. Each price column should be read the following way: each service (left column) is sold at a price that is written either on the same line in the corresponding price column or, if no price is written, to the next higher price in the same column. Examples: the monthly price for a tiered T3 with an authorized port speed less than 12 Kpbs, is $12,000; the monthly price for a burstable T3 with an average use between 21 Kbps and 45 Kbps is $55,000.

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tomers), and its latency guarantee is 85 ms between some American backbone routers and 120 ms for its transatlantic pipe (failure also implies credits for customers). Measurement of performance has therefore become a vital tool for networks in order to certify their quality9 . Despite these sophisticated measurement possibilities, consumers cannot choose among di¤erent classes of services for speci…c transfers of data. Furthermore, ISPs can only guarantee quality on their own network, and as a consequence of the structure of the Internet, it is di¢cult to guarantee quality of service from end to end.

2

Internet pricing in the economics literature

Much of the existing work on the pricing of Internet usage can best be seen as attempts by economists to convince engineers that the use of prices could yield a more e¢cient allocation of resources. The Internet grew from a collaborative e¤ort among universities and during the …rst years of its explosive growth many of the …rst participants felt uneasy with its commercialization. The economists …rst tried to convince them that a large system could not rely on reputational e¤ects to maintain discipline in the use of the resources. They also argued that the proper objective function for the system was some measure of the satisfaction of users rather than technical criteria given exante. Finally, they tried to show that properly computed short run prices could give information about the value of capacity, and provide useful indication for investment planning. These facts are by now well accepted, but it is not clear whether this is due to this pedagogical e¤ort: the privatization of the Internet in any case ensured that an economic logic would take over. All this discussion used well established economic ideas, in particular that users should be charged to re‡ect the negative externalities that they impose on others. Hence, if the capacity of the network is constrained, a user should be charged for the fact that the packets he transmits increase congestion and therefore decrease the utility of other users. The welfare theorems of general equilibrium theory then show that some type of ‘demand equals supply’ condition will yield optimality. The Internet raises speci…c economic issues once one turns to the much more complex issue of implementation of these pricing schemes. First, as we have seen above, packets are routed dynamically and in principle di¤erent packets originating, let us say, from server A and destined to browser B, containing information about the same webpage, can travel through di¤erent routes. Assume that A is responsible for the communication charge. In 9

See Boardwatch (1998, p. 20-25) and the website of Keynotes, a …rm that specializes in measurements [http://www.keynote.com].

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principle, A should pay for the marginal cost of congestion on every router and every link along the route. Clearly, this raises formidable problems, both technical and economic. Among the technical problems, the accounting problem of measuring the ‡ow, potentially packet by packet, is overwhelming. Among the economic problems, A would be made responsible for costs over which it has very little control as they depend on the ‡ow of packets on the network, which is determined by the routers. The hierarchical nature of the Internet, which seems to contradict somewhat the image of an ‘amorphous mesh’ where the routing of packets is extremely ‡exible, may be a response to this di¢culty. The second pricing problem stems from the fact that the shadow prices of capacity vary over time, and it is possible that in some parts of the network where capacity is specially limited and demand specially bursty, these shadow prices may vary very fast. This makes it very di¢cult to organize a pricing scheme that tracks the shadow prices …nely (although the loss of welfare associated with a second best pricing is a very di¢cult empirical question). The di¤erent models used to discuss these questions provide a nice sampling of the techniques that economists have developed to study pricing. The modern theory of mechanism design has been used by McKie-Mason and Varian in their proposal for ‘smart markets’ (McKie-Mason-Varian, 1995, 1996 and 1997, and McKie-Mason, 1997), which are basically an implementation of Vickrey auctions in order to price access to the network. As is by now well understood, these auctions have the bene…t that they allocate resources e¢ciently and induce participants to reveal their true willingness to pay. In the implementation that McKie-Mason and Varian propose, each packet would carry a maximum price that the sender is willing to pay in order to send the message. In case of congestion, the network would accept to forward the packets who carry a willingness to pay superior to a threshold computed in such a way that the total number of packets that are transmitted is equal to the capacity. In a number of papers, Gupta, Stahl and Whinston (e.g. 1997) have applied general equilibrium theory to solve the same problem. At every point of time the network is monitored in order to determine whether packets are slowed down by congestion. The prices charged for the nodes at which congestion is severe are increased, whereas the prices for nodes at which congestion is less severe are decreased. Each user of the network is informed dynamically of the prices and can decide whether or not to send packets accordingly. In theory, convergence of a tâtonnement process to equilibrium prices is guaranteed only under stringent conditions (see Hahn, 1982). In this case the problem is even more complex as the tâtonnement process is pursuing a moving target. However, Gupta, Stahl and Whinston have simu12

lated their proposed algorithm and shown that, under the conditions of their experiments, it tracks the equilibrium prices rather well. Clark (1995) has focussed on the description of simple priority schemes. His basic proposal gives users the opportunity to buy ‘priority ‡ags’, which can be attached to specially important packets. Packets are ‡agged according to an expected capacity requirement. For example, the user pays for the guarantee to be able to send a certain amount of bits within a certain period. Expected capacity pricing scheme has the main advantages of being predictable for the user, easy to implement for the provider (no tracking of data) and, moreover, captures directly the fact that marginal cost of sending is non-zero only during congestion (because the user does not pay according to her usage in non-congested period). Expected capacity is also used by providers for capacity provisioning, as well as interconnection negotiation. These proposals, and all the other proposals for pricing the Internet that we have seen, are ‘spot market’ proposals. This creates some di¢culties if, as we have argued before, there are some applications for which availability of bandwidth over a long period is indispensable.

3

Modeling congestion

The aim of this section is to introduce a dynamic model of pricing congestion. As does much of the literature, we abstract from the fact that the Internet has no supervising authority that can enforce a pricing scheme, but we take seriously the dynamics of the utilization of the resource and the asymmetry of information between the seller and the buyers.

3.1

A general model for resource reservation

In the sequel, we explore two very simple models, but it might be worthwhile explaining informally the general model of which they are a special case. We consider the network as a pipe along which information can ‡ow, and that at any point of time can accommodate a given …nite number of users. At each period arrive a number of consumers who desire to use it. Each of these consumers has a desired length of use as well as a willingness to pay, neither of which is known by the monopolist who owns the pipe. Because the capacity of the pipe is limited, the monopolist will organize a mechanism to allocate access to the consumers. As a result of this mechanism, a consumer will owe a certain amount to the monopolist, and maybe allocated a right to use the pipe. Typically, this right will be unconditional for the …rst period, but will be conditional on the realized demand for subsequent periods. For instance, 13

a consumer who arrived in a low demand period t, and has indicated a small willingness to pay for the pipe for two periods will be told: ‘we give you the right to use the pipe in period t, but you will be entitled to use it in period t + 1 only if the expressed demand of new consumers arriving in that period is not too high’. The mechanism set in place in each period will be a function of the state10 of the system, that is of the number of consumers carried over from previous periods, and of the promises that they have been made. Assuming that the monopolist can commit itself at the beginning of times, it will announce a series of mechanisms, that is of functions of the state of the system and of announcements of the consumers into rights of use. The revelation mechanism holds in this case and one can compute the optimal mechanism. Notice that there are two ways in which the consumer can lie about his or her type: misrepresenting her willingness to pay and misrepresenting the length of time for which she would like to use the resource. To this basic frame can be added a large number of complications, among them: ² the intensity of demand could also vary among consumers, some of them wanting to send a greater amount of information per period; ² consumers might be willing to wait for the use of the pipe if there is congestion and the price at their preferred period is too high (if they predict congestion, they might also be willing to use the pipe at an earlier period). Some of these issues have been tackled in the literature on road congestion (see Arnott et al. (1999) for a recent contribution and an extensive bibliography). However, the question of the length of use does not arise in these models. In the next two sections, we study some very special stationary examples of this general model, in order to provide insights on specially important aspects of the economics of congestion pricing. Both models assume that the resource can only serve one consumer at the same time. In section 3.2, we study a model in which one new customer arrives at each period, and wants to use the resource for two periods. In this framework, we study the consequences of implementing the ‘smart market’ mechanism which has been proposed by McKie-Mason and Varian. We show that it does not induce 10

If demand is not independent from period to period, the state would include all the information which is relevant for predicting it in future periods. We neglect this aspect, implicitly assuming that demand is stationary.

14

revelation of the true willingness to pay of the consumers, and present some counterintuitive results on the pro…ts that result from the presence of di¤erent types of consumers. Even in this simple model, non-standard functional equations appear, that indicate that the general problem will be quite hard to solve. In section 3.3, we study a simple implementation of the general model, in which there are two new consumers in each period, one of whom wants the use of the resource for one period, and one who wants it for two periods. We compute the optimal stationary mechanism in this case, and derive some economic consequences.

3.2 3.2.1

Smart market and multi-periods resource reservation A story

Consider for instance the case of a young man who wants to deliver through an Internet video a poem to the girl of his dreams. The recitation of the poem will last for half an hour (he is very old fashioned), and he is willing to pay $1800 in order to deliver it. Because the last verse, where he asks for her hand, is the most important, nothing less than the full 30 minutes has any value for him, and he is not willing to pay more than $0 even for 29 minutes and …fty seconds. The video stream is divided into packets, each of them carrying one second of the poem, so that he is willing to pay on average $1 per packet. Assume that a smart market is implemented, and that he decides to bid $1 per packet. After 29 minutes, he realizes that there is only one minute left, and this minute is worth $1800 to him. If he follows the same strategy of bidding the expected valuation per packet, he should attach to every packet a bid of $30. It is easy to see that such a strategy could easily lead him to pay more than his valuation of the delivery of the poem, and could make him regret not having bought a plane ticket. Of course, his optimal strategy would be to begin by bidding less that $1 and to increase his bid progressively; in any case, the last packet will have a $1800 attached to it (assuming away any liquidity constraint). Of course economic theoreticians do not care about the vows of young lovers, but we should be distressed by the fact that in this dynamic framework, the strategies of the users become much more complex than the simple dominant strategy of bidding one’s own valuation.

15

3.2.2

The strategy of the customers

There is a in…nite number of agents t = 1; 2; : : : ; +1. Agent t is willing to pay vt for using the pipe during periods t and t + 1, if she can use it during both periods. She assigns no value to the use of the pipe during a single period. The vt s are independent of each other and uniformly distributed on [0; 1]. We assume that the use of the pipe is decided through a ‘smart market’ mechanism. If, in period t, two agents are present (they would be t and t¡1), they each bid for the right to use the pipe in the next period. The pipe can be used by the agent that bids the highest at the price o¤ered by the other agent (the second highest bidder). If, in period t, only one agent is present (because agent t ¡ 2 won the auction in period t ¡ 1), she obtains the use of the pipe for the next period for a price of 0. We assume that the discount factor of the monopolist and of all the consumers is equal to ±. Consumer t is willing to pay vt , evaluated at period t, for the use of the resource for the two periods t and t + 1. If she wins the auction in period t, at the beginning of period t + 1 she will be willing to pay vt =±, and given the rules of the auction, bidding this amount will be a dominant strategy. It is more di¢cult to compute the amount that agent t should be willing to bid in period t. Given the rules of the auction, it will be a dominant strategy to bid the maximum amount that she is willing to pay. We look for a stationary solution, and call b(v) the function that indicates the maximum amount that an agent of type v is willing to pay in order to have the use of the resource for the …rst period where she could use it. We have h hv ii b(v) = ±Ev+1 max ¡ b(v+1 ); 0 (1) ± = Ev+1 [max [v ¡ ±b(v+1 ); 0]] : (2) To understand equation (1), note …rst that if the agent of type v in period t obtains the use of the pipe, in period t + 1, she will bid v=±. If v=± is greater than b(vt+1 ); the bid of her competitor in period t+1, she will obtain a surplus of v=± ¡ b(vt+1 ) discounted at t + 1, otherwise she will loose the auction and obtain a surplus of 0. The right hand side of equation (1) is the value of this surplus discounted to the beginning of period t.11 Following the strategy of Holt (1980) we look for a function b that is strictly increasing and continuous, and therefore such that the inverse function b¡1 11

Note that, given the rules of the smart market mechanism, the participation constraint of the agents is automatically met.

16

exists. From (2), we have Z v+1 =1 b(v) = [v ¡ ±b(v+1 )] dv+1 if v ¸ ±b(1)

(3)

v+1 =0

and

b(v) = If v

Z

v+1 =b¡1 (v=±) v+1 =0

[v ¡ ±b(v+1 )] dv+1 if v

±b(1):

(4)

±b(1), di¤erentiating both sides of (4) with respect to v, we obtain

@b¡1 (v=±) £ (v ¡ ±b(v+1 ))jv+1 =b¡1 (v=±) @v Z v+1 =b¡1 (v=±) @ [v ¡ ±b(v+1 )] dv+1 + @v v+1 =0 Z v+1 =b¡1 (v=±) @b¡1 (v=±) £0+ 1 £ dv+1 = @v v+1 =0 v = b¡1 ( ): (5) ± Note the unusual form of this functional equation. We try to …nd a solution12 of (5) of the type b(v) = ¯v ® . This yields b0 (v) = ¯®v®¡1 , b¡1 (v) = ¯ ¡1=® v 1=® and b¡1 (v=±) = ¯ ¡1=® ± ¡1=® v1=® . Hence, we must have b0 (v) =

1

1

¯® = ¯ ¡ ® ± ¡ ®

(6)

and ®¡1=

1 . ®

(7)

Equation (7) implies ®2 ¡ ® ¡ 1 = 0

and therefore (as we know that b(v) is increasing, and therefore that ® > 0) 1p 1 ®= 5 + ' 1: 618, 2 2 the golden number!13 From (6) and (7) we obtain Ãp ! p5¡1 µ ¶ ®1 2 1 5¡1 ¡ 12 ¡ 12 ¯=± ® =± ® ' 0: 743 £ ± ¡0: 382 : ® 2 12 13

By doing so, we are only …nding one equilibrium of the game; there could exist others. For another apparition of the golden number in economic theory, see Crémer (1994).

17

Writing ° = (1=®)1=® , we obtain 1

b(v) = °± ¡ ®2 v ® ' 0: 743 £ ± ¡0: 382 £ v 1: 618 for v

±b(1). R1 In order to compute b(v) for v ¸ ±b(1), let us write I = 0 b(v)dv. We have b(v) = v ¡ ±I, which implies (8)

b(1) = 1 ¡ ±I: Because the function b is continuous at ±b(1) we have 1

±b(1) ¡ ±I = °± ¡ ®2 +® [b (1)]® .

(9)

Furthermore I=

Z

±b(1)

°±

¡

1 ®2

®

v dv +

This implies

1

±b(1)

0

=

Z

(v ¡ ±I) dv

1 1 °± ¡ ®2 +®+1 [b (1)]1+® 1+® ¤ 1£ + 1 ¡ ± 2 [b (1)]2 ¡ ±I [1 ¡ ±b (1)] : 2

(10)

¸ 1 ¡ b(1) 1 ¡ 12 +®+1 = °± ® [b (1)]1+® ± 1+® µ ¶¸ ± 1 + 1 [b (1)]2 + [1 + ±] [b (1)] ¡ ; ¡ ± 2 2 and therefore14 µ ¶¸ 1+± 2+± 2 0= ± ¡ [b (1)]2 1+® 2

¸ ¸ ® ± + 1+±+± [b (1)] ¡ 1 + : 1+® 2 2

14

Modify equation (10) with equation (8) in order to get an equation with b(1). Isolate 1+® the term in [b (1)] and replace it by its value extracted from equation (9) multiplied by b(1). The resulting equation is a polynome of order 2 in b(1).

18

which has two solutions, of which only one is smaller than 1:

b(1) =

¡

3+

where

q¡ p ¢¡ 2 ¡ p ¢ p ¢¢ p ¢ ¡ 5 2± + 1 + 5 ± + 1 + 5 ¡ 2 3 + 5 A p ¡ ¢ 4± 2 3 + 5 + ±

³ ³ ³ ³ p ´ p ´ p ´ p ´ A = 2±4 ¡ 2 1 + 5 ± 3 ¡ 1 + 5 ± 2 + 2 7 + 3 5 ± + 7 + 3 5 : The bidding function for ± = 0:9 is represented on …gure ??.

3.2.3

The pro…t of the seller

The bidding function is quite complicated and makes it impossible to compute a closed form solution for the pro…t of the seller. On the other hand, it is possible to derive some interesting comparative statics that are summarized in proposition 1. Let us call ¦(v) the expected pro…t of the seller over the horizon t; t+1; : : : discounted at the beginning of period t if, in period t, customer t ¡ 1 of type v is bidding against customer t for the right to use the pipe for a second period. Let us also call ¦0 the expected pro…t of the seller over the same horizon in the case where no one is bidding against customer t. It is clear that we have ¦(0) = ¦0 . For v ¸ b(1), we have Z v+1 =1 ¦(v) = b(v+1 ) dv+1 + ±¦ (0) (11) v+1 =0

= I + ±¦(0):

Indeed, customer t ¡ 1 will win the auction with probability 1, and she will pay b(vt+1 ). The …rst term on the right hand side of (11) is equal to the pro…t in period t; the second term is equal to ±¦0 , that is the pro…t discounted to the beginning of period t + 1. Notice that over this range we have ¦0 (v) = 0. If v b(1), we have Z v+1 =b¡1 (v) Z v+1 =1 ¦(v) = [b(v+1 ) + ±¦ (0)] dv+1 + [v + ±¦ (v+1 )] dv+1 . v+1 =b¡1 (v)

v+1 =0

In particular ¦(0) =

Z

v+1 =1

±¦ (v+1 ) dv+1 0

19

(12)

The …rst term of the right hand side of (12) is similar to the right hand side of (11) and corresponds to the states of nature where t wins the auction. If t + 1 wins the auction — when her type is greater than b¡1 (v) — she pays v and the corresponding pro…t is expressed by the second term on the right hand side of (12). This implies the following proposition. Proposition 1 The function ¦ is not monotone in v. Proof. We have ¦0 (v) =

¡ ¢¤ £ ¤ @b¡1 (v) £ ± ¦ (0) ¡ ¦ b¡1 (v) + 1 ¡ b¡1 (v) @v

and therefore ¦0 (0) = 1 and

¯ @b¡1 (v) ¯¯ ¦ (b (1) ) = £ ± [¦ (0) ¡ ¦ (1)] : @v ¯v=b(1)¡ 0

¡

If ¦ were monotone, it would have to be increasing. We would then have ¦ (0) < ¦ (1), and therefore ¦0 (b (1)¡ ) would be negative as @b¡1 =@v is increasing, which establishes the contradiction. Notice that the pro…t at time 1 of the seller, conditional on the type v1 of the period 1 customer, is ±¦(v1 ). Proposition 1 implies that the seller could prefer having a customer with a smaller willingness to pay! The economic intuition behind this is quite clear. The type of the agent does not a¤ect the revenue in the …rst period. If customer 1 wins the second period auction, the revenue will be zero in the third period. Hence, too high a type for customer 1, which basically guarantees that she wins the second period auction is not desirable. We believe that the same type of paradoxes would arise if there were several new customers at each period. The same type of di¢culties would arise with every single one of the pricing schemes that have been proposed in section 2. Faced with this problem, we have two research strategies. The …rst is to try to study the behavior of the pricing proposals which have been made in a dynamic framework; this can been done both in theoretical models and in simulations. The other strategy is more ‘fundamentalist’. It attempts to identify the optimal mechanism in a framework where we explicitly model the fact that some users have long term demands. This is the approach which is followed in the following section, with a simpli…ed model of multi-periods resource reservation.

20

3.3

A two types case of the general model

We have not (yet!) been able to solve for the optimal strategy in the model of section 3.1. In order to tackle the problem of the computation of the optimal mechanism, we introduce another speci…c model, with discrete types. This will also enable us to study the consequences of the presence of customers with demands of di¤erent lengths. There are two types of customers or buyers. Short term buyers want to use the pipe for one period, while long term buyers want to use it for two consecutive periods. More precisely, we assume that at the beginning of each period one long term and one short term customer appear. If the bandwidth for the next period is already committed, they ‘disappear’; if the pipe is not committed for the next period, they participate in a mechanism organized by the seller. Hence, contrary to what happens in the previous model, once a user has begun using the pipe she will never be asked to relinquish the use. This would be appropriate if there are large setup costs to begin to use the pipe, which are lost if it has to be relinquished. The seller does not know the willingness to pay for bandwidth of the two buyers, and each of the buyers does not know the willingness to pay of the other. We will assume that the short term buyer has a willingness to pay equal to v1` ¸ 0 with probability ¼1` and a willingness to pay equal to v1h > v1` with probability ¼1h = 1 ¡ ¼1` (the subscript ‘1’ indicates that she wants one period of bandwidth). We will also assume that the long term buyer has a willingness to pay equal to v2` with probability ¼2` and a willingness to pay equal to v2h > v2` with probability ¼2h = 1 ¡ ¼2` (the subscript ‘2’ indicates that she wants bandwidth for two periods). We will say that we are in, for instance, state of nature ‘`h’ to indicate that at the beginning of a period where the pipe is available, buyer 1 ‘is of type `’, that is her willingness to pay is equal to v1` , and buyer 2 ‘is of type h’, that is her willingness to pay is equal to v2h . The probability ¼1` ¼2h of state of nature `h will also be written ¼ `h (we are therefore assuming that the types of the agents are independent). We restrict ourselves to revelation mechanisms in which the seller asks each buyer to reveal her type. If buyer 1 announces that she is of type j and buyer 2 announces that she is of type k, the buyer allocates the bandwidth jk to buyer 1 with probability pjk 1 ¡ pjk 1 and to buyer 2 with probability p2 1 , j k and buyer 1 pays t1 while buyer 2 pays t2 (there is no bene…t to making the payment of a buyer depend on the announced type of the other). h jh We will call pj1 = ¼2` pj` 1 + ¼2 p1 the probability that buyer 1 obtains the bandwidth if she is of type j and pj2 = ¼1` p`2 + ¼1h phj 2 the probability that buyer 2 obtains the bandwidth if she is of type j. 21

3.3.1

The problem of the seller

Agent 1 will announce her true type if and only if the incentive compatibility conditions ½ IC h1 : ph1 v1h ¡ th1 ¸ p`1 v1h ¡ t`1 ; IC `1 : p`1 v1` ¡ t`1 ¸ ph1 v1` ¡ th1 ; are satis…ed. She will accept to participate in the mechanism if and only if the individual rationality constraints ½ IRh1 : ph1 v1h ¡ th1 ¸ 0; IR`1 : p`1 v1` ¡ t`1 ¸ 0; are satis…ed. As usual in this type of problems constraint (IRh1 ) is satis…ed if the other three constraints are satis…ed, and we will assume that constraint (IC `1 ) is not binding at the optimum and check ex-post that it is indeed satis…ed by the solution that we …nd. Then, constraints (IC h1 ) and (IR`1 ) are binding and we have p`1 v1` ¡ t`1 = 0 =) t`1 = p`1 v1` and ph1 v1h ¡ th1 = p`1 v1h ¡ t`1 =) th1 = ph1 v1h ¡ p`1 (v1h ¡ v1` ): Similarly t`2 = p`2 v2` and th2 = ph2 v2h ¡ p`2 (v2h ¡ v2` ): In order to write the objective function of the seller, it will be convenient jk to add a few other notations. We will call qm = ¼ jk pjk m , the probability that the state of nature is jk and that buyer m obtains the bandwidth. For instance, q1`` = ¼ `` p`` 1 is the probability that the state of nature is (`; `) and that agent 1 obtains the bandwidth. Then q1j = q1j` + q1jh is the probability that agent 1 is of type j and obtains the bandwidth. Similarly, q2k = q2`k + q2hk is the probability that agent 2 ` h is of type k and obtains the bandwidth. Finally let qm = qm + qm be the probability that buyer m obtains the bandwidth. 22

Let ¦ be the value of the stream of pro…ts of the seller discounted to the beginning of a period where the connection is free, before the buyer has been chosen. We have ¦ = ¼1` t`1 + ¼1h th1 + ¼2` t`2 + ¼2h th2 + q1 ±¦ + q2 ± 2 ¦ + (1 ¡ q1 ¡ q2 )±¦:

(13)

The …rst four terms in equation (13) represent the payment that the seller will receive in the current period. With probability q1 the bandwidth will be sold to a short term buyer, and it will be possible to sell it at the beginning of the next period, which has a value ±¦. With probability q2 the bandwidth will be sold to a long term buyer, and it will be possible to sell it in two periods, which has a value ± 2 ¦. Finally, with probability 1 ¡ q1 ¡ q2 , the bandwidth will not be sold and will be replaced on the market at the end of next period. This yields ¦= Using the notation

¼1` t`1 + ¼1h th1 + ¼2` t`2 + ¼2h th2 : (1 ¡ ±)(1 + ±q2 )

m1 =

v1` ¡ ¼1h v1h ¼1h h ` = v ¡ (v ¡ v1` ) < v1` < v1h 1 ¼1` ¼1` 1

m2 =

¼2h h v2` ¡ ¼2h v2h ` = v ¡ (v ¡ v2` ) < v2` < v2h; 2 ¼2` ¼2` 2

and

easy substitutions enable us to show that the problem of the seller is to maximize (1 ¡ ±)¦ =

q1`` m1 + q1`h m1 + q1h` v1h + q1hh v1h + q2`` m2 + q2h` m2 + q2`h v2h + q2hh v2h 1 + ±q2`` + ±q2`h + ±q2h` + ±q2hh (14)

subject to the feasibility constraints 8 `` C : q1`` + q2`` ¼ `` ; > > < `h C : q1`h + q2`h ¼ `h ; C h` : q1h` + q2h` ¼ h` ; > > : hh C : q1hh + q2hh ¼ hh ;

and the non-negativity constraints ½ C1 : q1`` ¸ 0; q1`h ¸ 0; q1h` ¸ 0; q1hh ¸ 0; C2 : q2`` ¸ 0; q2`h ¸ 0; q2h` ¸ 0; q2hh ¸ 0: 23

3.3.2

Optimal solutions

The following lemma shows that we can restrict ourselves to considering monotone solutions of the problem. All proofs can be found in the appendix. Lemma 1 In any optimal solution, the probability that a buyer obtains the object is increasing in its own valuation: p`` 1

`h ph` 1 ; p1

`` phh 1 ; p2

h` p`h 2 ; p2

phh 2 ;

and decreasing in the valuation of the other: `h h` hh `` h` `h hh p`` 1 ¸ p1 if m1 6= 0; p1 ¸ p1 ; p2 ¸ p2 ; p2 ¸ p2 :

If m1 = 0, there exists an optimal solution to the problem that satis…es `h p`` 1 ¸ p1 (other inequalities remaining satis…ed by any optimal solution). This lemma ensures that constraints (IC `1 ) and (IC `2 ) are satis…ed by the solution that we identify. A priori, it would seem that solving this problem whose objective function is non linear would force us to consider interior solutions with some of the pjk m s strictly between 0 and 1. It turns out that this is not the case, as the following lemma shows. Lemma 2 The problem of the seller has an optimal solution such that for jk all m 2 f1; 2g, and all j and k in fh; `g the probability qm is either equal jk to 0 or to ¼ . Lemma 2 looks a priori as a natural extension of standard auction theory. The seller organizes an auction, and the fact that we are in a dynamic system implies that it attaches positive value to not selling the object. This should not change the well known fact that there exists an optimal non stochastic mechanism. Note however that in this case, the ‘cost of production’ of the object depends on the identity of the buyer — indeed, foregone revenues are not the same if the use of the pipe is sold to the short run or to the long run customer. Lemma 2 implies that pjk m is equal to either 0 or 1: there exists a solution such that conditional on the announcement of the buyers the mechanism is not random. Lemmas 1 and 2 allow us to identify the solutions of the problem.

24

Proposition 2 There are nine possible solutions, which are represented by the following table p`` 1 p`h 1 ph` 1 phh 1 p`` 2 p`h 2 ph` 2 phh 2

s0 0 0 1 1 0 0 0 0

s1 1 1 1 1 0 0 0 0

s2 0 0 0 0 1 1 1 1

s3 0 0 1 0 0 1 0 1

s4 1 0 1 0 0 1 0 1

s5 0 0 1 0 1 1 0 1

s6 0 0 1 1 0 1 0 0

s7 1 0 1 1 0 1 0 0

s8 0 0 1 1 1 1 0 0

where each of columns corresponds to a solution, whose name is indicated in h` h` the top row. For instance in solution s4 , ph` and 1 = 1 and therefore q1 = ¼ `` `` in solution s7 , p2 = q2 = 0. There exits numerical examples showing that, depending on the value of the parameters ¼ ij and vji , each of these nine strategies is indeed optimal solution15 . On the other hand, the conditions under which each type of solution is optimal are quite complex and not really interpretable at that level of generality. We therefore move to a speci…c set of parameters. 3.3.3

An example

In order to get a better handle on the consequences of the dynamic aspects of the problem, we consider the following family of examples: ½ ` ¼1 = ¼1h = ¼2` = ¼2h = 1=2; v1` = 1; v1h = v; v2` = (1 + ±); v2h = v(1 + ±): The examples are built in such a way that long run and short run buyers of low (resp. high) types have the same per period willingness to pay (remember that v2` and v2h represent the amount that agent 2 would be willing to pay immediately for the use of the pipe in this period and the next; therefore v2` =(1 + ±) and v2h =(1 + ±) represent the amount that she would be willing to pay per period, assuming, as before, that she has the same discount rate than the seller). The payo¤s associated with the di¤erent solutions are represented in the following table: 15

These examples are available from the authors upon request.

25

s0 s1 s2 s3 s4 v 1:5v + v± 1 + v + v± 1 1 2 2+± 2+± s5 s6 s7 s8 1 + v + ± + v±=2 3v + v± 2 + 2v + v± 1 + v + ± 2 + 1:5± 4+± 4+± 2+± It is easy to verify that for all relevant values of ± 2 (0; 1) and v > 1, s4 dominates s1 , s2 , s5 , s7 and s8 and that s3 dominates s0 and s6 . The only possible solutions are therefore s3 and s4 , and it is immediate that the optimal solution is s3 if v > 2 and s4 if v < 2. Whereas ex-ante per period valuation of the long term and the short term consumers are equal, the optimal mechanism favors the high type long term user (for large v, the optimal solution has the long run user obtaining the use of the pipe with a higher probability).

4

Conclusion

In this paper, we have presented a new stationary approach to the pricing of capacity on networks that permits the computation of the optimal mechanism, and have argued that it could throw light on the tari…cation of the Internet. Many problems are left open, in particular problems of implementation and problems of queuing, which are left for future research. But it is clear that the economic analysis is needed for the Internet in other …elds than its usage pricing. Among these …elds are: ² the applicability of telecommunication regulation to ISPs with a special interest for Internet telephony16 ; ² market power and abuse of dominant position17 specially for the de…nition of standards18 (Internet governance) and interconnectivity19 (peering agreements); ² the standardization of software and hardware user interfaces; 16

² the reform of the domain name system20 ;

See Werbach (1997) for a general discussion of the regulatory challenges. See OECD (1998c) which tries to estimate the market power of US backbones. 18 See Coates (1995) for some concerns about standardization. 19 For more on the subject, see Bailey (1995), OECD (1998a) and Crémer, Rey and Tirole (1999) for an economic perspective, and Husson (1999) for technical details. 20 A good description of the system can be found in OECD (1997a). See also Coates (1998). 17

26

² intellectual property rights de…nition and enforcement.

5

References

Arnott Richard, de Palma André, Lindsey Robin (1999): ‘Information and time-of-usage decisions in the bottleneck model with stochastic capacity and demand’, European Economic Review, 43(3):525-548, March 1999 Bailey Joseph P. (1995): ‘Economics and Internet interconnection agreements’, MIT Workshop on Internet Economics, March 199521 Boardwatch (1998): Boardwatch Magazine’s Directory of Internet Service Providers, 3(2), Winter 1998-Spring 1999 Clark David D. (1995): ‘A model for cost allocation and pricing in the Internet’, MIT Workshop on Internet Economics, March 199522 Coates Kevin (1998): ‘Competing for the Internet’, European Commission, DG IV-C-123 Crémer Jacques (1994): ‘The value of large extensive form games’, Games and Economic Behavior, 7(3):309-317, November 1994 Crémer Jacques, Rey Patrick, Tirole Jean (1999): ‘Market power and interconnection in the Internet’, forthcoming. Esbin Barbara (1998): ‘Internet cable overview: De…ning the future in term of the past’, Federal Communications Commission, O¢ce of Plans and Policy working paper series n: 30, August 199824 European Commission (1998): ‘Status of voice communication on Internet under Community law and, in particular, under Directive 90/338/EEC’, O¢cial Journal n. C6, p. 4, 10/01/9825 Gupta Alok, Stahl Dale O., Whinston Andrew B. (1997): ‘A stochastic equilibrium model of internet pricing’, Journal of Economic Dynamics and Control, 21:697-722, 1997 Hahn F. (1982): ‘Stability’, in K. Arrow and M. Intriligators (eds), “Handbook of mathematical economics”, vol. II, ch. 16, North-Holland, Amsterdam Holt Charles A. (1980): ‘Competitive bidding for contracts under alternative auction procedure’, Journal of Political Economy, 88(3):433-435, June 1980 21

[http://www.press.umich.edu/jep/works/BailEconAg.html] [http://www.press.umich.edu/jep/works/ClarkModel.html] 23 [http://europa.eu.int/comm/dg04/speech/eight/en/sp98006.htm] 24 [http://www.fcc.gov/Bureaus/OPP/working_papers/oppwp30.pdf] 25 [http://europa.eu.int/comm/dg04/lawliber/en/voice.htm] 22

27

Huson Geo¤ (1999): ‘Interconnection, peering and settlements’, January 199926 OECD (1998c): ‘Internet infrastructure indicators’, DSTI/ICCP/TISP(98)7/ FINAL, October 199827 OECD (1998b): ‘Internet voice telephony developments’, DSTI/ICCP/ TISP(97)3/FINAL, April 199828 OECD (1998a): ‘Internet tra¢c exchange: Developments and policy’, DSTI/ICPP/TISP(98)1/FINAL, March 199829 OECD (1997b): ‘Webcasting and convergence: Policy implications’, OCDE/ GD(97)221, September 199730 OECD (1997a): ‘Internet domain names: Allocation policies’, OCDE/ GD(97)207, 199731 McKie-Mason Je¤rey K. (1997): ‘A smart market for resource reservation in a multiple quality of service information network’, mimeo32 McKie-Mason Je¤rey K. and Varian Hal R. (1997): ‘Economic FAQs about the Internet’, in L. W. McKnight and J. P. Bailey (eds), “Internet Economics” (pp. 27-62), Cambridge, MA: The MIT Press McKie-Mason Je¤rey K. and Varian Hal R. (1996): ‘Some economics of the Internet’, in B. W. Sichel and D. L. Alexander (eds), “Networks, infrastructure, and the new task for regulation” (pp. 107-136), Ann Arbor: The University of Michigan Press McKie-Mason Je¤rey K. and Varian Hal R. (1995): ‘Pricing the Internet’, in B. Kahin and J. Keller (eds), “Public access to the Internet” (pp. 269-314), Cambridge, MA: The MIT Press33 Werbach Kevin (1997): ‘Digital Tornado: The Internet and telecommunications policy’, Federal Communications Commission, O¢ce of Plans and Policy working paper series n: 29, March 199734

26

[http://www.telstra.net/peerdocs/peer.html] [http://www.oecd.org/dsti/sti/it/cm/prod/tisp98-7e.pdf] 28 [http://www.oecd.org/dsti/sti/it/cm/prod/tisp97-3e.pdf] 29 [http://www.oecd.org/dsti/sti/it/cm/prod/traffic.pdf] 30 [http://www.oecd.org/dsti/sti/it/cm/prod/e_97-221.pdf] 31 [http://www.oecd.org/dsti/sti/it/cm/prod/e_97-207.pdf] 32 [http://www-personal.umich.edu/~jmm/papers/reservev3.pdf] 33 [http://www-personal.umich.edu/~jmm/papers/Pricing_the_Internet.pdf] 34 [http://www.fcc.gov/Bureaus/OPP/working_papers/oppwp29.pdf] 27

28

Appendix In this appendix, we present the proofs of lemmas 1 and 2, and of proposition 2.

Preliminary results Let …rst go through the following lemmas. Lemma A.1 If m1 < 0, any optimal solution of the problem satis…es q1`` = q1`h = 0: Proof. The objective function ¦ de…ned by (14) is linear in these two variables, with a coe¢cient of the sign of m1 . Lemma A.2 If m2 0, any optimal solution of the problem satis…es q1h` = ¼ h` and q2`` = q2h` = 0. Proof. As m2 0, there is no gain to increase q2`` and q2h` which are set to zero. Then, because the coe¢cient of q1h` is strictly positive, its optimal value makes the constraint binding, and q1h` = ¼ h` . Lemma A.3 The constraints (C h` ) and (C hh ) are binding at any optimum. Furthermore, if m1 > 0 the constraints (C `` ) and (C `h) are also binding. Proof. First, given q2hh , the objective function (14) is linear in q1hh with positive coe¢cient. Therefore, the constraint (C hh ) is binding at any optimum. h` Second, if m2 > 0, the same argument can be used with qm , m = 1; 2. h` h` h` If m2 0,¡ then by lemma A.2 q = 0 and q = ¼ . In all cases, the 2 1 ¢ h` constraint C is binding at any optimum. Third, if m1 > 0, given the values of q2`` and q2`h, the objective function (14) is linear in q1`` and q1`h with positive coe¢cients. Therefore, the constraints (C `` ) and (C `h ) are binding at any optimum.

Proof of lemma 1 We now turn to the proof of lemma 1 that the probability that a buyer obtains bandwidth is increasing in its own valuation and decreasing in the valuation of the other agent.

29

Lemma A.4 Let D > 0, E > 0 and B > C. Then any solution (x¤ ; y ¤ ) of the problem max

0 x kx 0 y ky

A + Bx + Cy D + Ex + Ey

satis…es either y¤

x¤ = kx

or 0 = y¤

x¤ :

Proof. We cannot have y ¤ > 0 and x¤ < kx : there would exist an " such that (x¤ + "; y ¤ ¡ ") would still be feasible and yield a strictly greater value for the objective function as the denominator would be unchanged and the numerator increased. Lemma A.5 If m1 < 0 and m2 < 0, then any optimal solution satis…es the monotonicity conditions of lemma 1. Proof. If m1 < 0 and m2 < 0, lemmas A.1 and A.2 yield q1`` = q1`h = q2`` = q2h` = 0, which implies `h `` h` p`` 1 = p1 = p2 = p2 = 0,

This implies that the monotonicity conditions p`` 1

`h ph` 1 ; p1

`` phh 1 ; p2

h` p`h 2 ; p2

phh 2 ;

and `h `` h` p`` 1 ¸ p1 ; p2 ¸ p2

hh `h hh are satis…ed. We must still prove ph` 1 ¸ p1 and p2 ¸ p2 . h` h` hh From lemma A.2, we have q1 = ¼ , which implies ph` 1 = 1 ¸ p1 . Because the constraint (C hh ) is binding, by lemma A.3, and q1`h = 0, substituting q1hh = ¼ hh ¡ q2hh and …xing the value of all the other variables allow the problem of the seller to be written under the form of the following problem in q2`h and q2hh

max

0 q2`h ¼ `h 0 q2hh ¼ hh

A + q2`h v2h + q2hh (v2h ¡ v1h ) ; C + ±q2`h + ±q2hh

and lemma A.4 implies that either 30

² q2`h = ¼ `h, which in turn implies phh 2

p`h 2 = 1,

² or 0 = q2hh , which in turn implies 0 = phh 2

p`h 2 .

Lemma A.6 If m1 > 0 and m2 ¸ 0, then any optimal solution satis…es the monotonicity conditions of lemma 1. Proof. By lemma A.3, all the C ij constraints are binding at the optimum. Susbstituting away the q1ij s the objective function of the seller can be written under the form A + (m2 ¡ m1 )q2`` + (m2 ¡ v1h )q2h` + (v2h ¡ m1 )q2`h + (v2h ¡ v1h )q2hh ; 1 + ±q2`` + ±q2h` + ±q2`h + ±q2hh which must be maximized under the constraints 0

q2ij

¼ ij .

We have m2 ¡ v1h < m2 ¡ m1 < v2h ¡ m1 and m2 ¡ v1h < v2h ¡ v1h < v2h ¡ m1 : ij By lemma A.4, this implies that ph` 2 > 0 implies p2 = 1 for all (i; j) 6= (h; l). The inequalities involving ph` 2 are therefore proved. Similarly if any `h pij > 0, we have p = 1, and the inequalities involving p`h 2 2 2 are proved. This proves all the monotonocity conditions for agent 2. To prove the monotonicity conditions for agent 1, notice …rst that, be`h cause all the constraints are binding, if p`h 1 > 0, we have p2 < 1 and therefore ij ij all the other p2 = 0, which implies that all the other p1 = 1, and the two ij h` inequalities involving p`h 1 are proved. Similarly, p1 < 1 implies p1 = 0 for the other (i; j) and the result is proved.

Lemma A.7 If m1 > 0 and m2 0, then any optimal solution satis…es the monotonicity conditions of lemma 1. Proof. By lemma A.3, all the constraints C `j are binding at the optimum h` and by lemma A.2, q2`` = q2h` = 0. This implies p`` 2 = p2 = 0 and also h` p`` 1 = p1 = 1. All the monotonicity conditions are thus proved except for 31

`h hh hh p`h phh is binding at the 1 1 and p2 ¸ p2 . Noticing that the constraint C optimum we can write the objective function under the form

A + (v2h ¡ m1 )q2`h + (v2h ¡ v1h )q2hh ; C + ±q2`h + ±q2hh and the same reasoning than in lemma A.6 concludes the proof. Lemma A.8 If m1 < 0 and m2 ¸ 0, then any optimal solution satis…es the monotonicity conditions of lemma 1. `h Proof. By lemma A.1, q1`` = q1`h = p`` 1 = p1 = 0 and the two monotonic`h ity conditions involving p`` 1 and p1 are satis…ed. Noticing by lemma A.3 that ¡ hj ¢ the C constraints must be binding the objective function can be written under the form

A + m2 q2`` + v2hq2`h + (m2 ¡ v1h )q2h` + (v2h ¡ v1h)q2hh 1 + ±q2`` + ±q2`h + ±q2h` + ±q2hh subject to the constraints 0 q2ij ¼ ij . The monotonicity conditions on pij 2 follow immediately. There is one condition left to prove: phh ph` 1 1 . To prove it note that hh hh hh p1 > 0 implies q2 < ¼ , and by lemma A.4 this implies q2h` = ph` 2 = 0, and h` therefore p1 = 1. If m1 = 0, it is impossible to guarantee monotonicity. To see why assume m2 0. Then, by lemma A.2, at the optimum q2`` = 0 and q1`` can take any value between 0 and ¼ `` . If v2h is small enough q2`h will be equal to 0 at the optimum, and there will be some solutions that contradict the monotonicity condition p`h p`` 1 1 : On the other hand, we can show the following lemma: Lemma A.9 If m1 = 0, there exists an optimal solution that satis…es the monotonicity conditions of lemma 1. Proof. If m1 = 0, the seller can set q1`` = q1`h = 0 without loss of income. This implies that all the monotonocity conditions for agent 1, except hh ph` 1 ¸ p1 , hold. Depending on the value of m2 , the rest of the proof ends as hh in the preceding lemmas for the condition ph` 1 ¸ p1 and the monotonicity conditions of agent 2.

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Proof of lemma 2 To prove lemma 2, we will use the following two lemmas. Lemma A.10 If C > 0 and D ¸ 0, the problem max f(y) =

y2[0;k]

A + By C + Dy

has a solution in which y is either equal to 0 or to k. Proof. Assume that there exists a solution y ¤ 2 (0; k). We have f 0 (y ¤ ) =

BC ¡ AD = 0: (C + Dy ¤ )2

Therefore f 0 (y) is equal to 0 for all feasible y and the result is proved. Lemma A.11 Assume that D > 0 and E ¸ 0. Then the problem max f (x; y) =

x+y k x¸0 y¸0

A + Bx + Cy D + Ey

(15)

has an optimal solution where both x and y are equal to either 0 or k. Proof. If B 0, there is an optimal solution in which x = 0, and the lemma is a direct consequence of lemma A.10. If B > 0, at any optimal solution the constraint x + y k must be binding and the problem can be rewritten under the form A + Bk + (C ¡ B)y ; y2[0;k] D + Ey max

and the result is proved by lemma A.10. Proof of lemma 2. Assume, say, that there exists an optimal solution such that q1`` and q2`` are both strictly positive. Given the values of the other qhij , the problem of …nding the optimal q1`` and q2`` is of the type of problem (15), and therefore there exists another optimal solution such that q1`` and q2`` are both equal either to 0 or to ¼ `` , with the other qhij unchanged. Reitering the reasoning over all pairs fq1ij ; q2ij g such that at least one of the pair is not equal to either 0 or ¼ ij , the result is proved.

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Proof of proposition 2 Finally, we turn to the proof of proposition 2. Proof of proposition 2. To see that we need only consider these solutions, notice …rst that if q2`h = 0, we have q2`` = q2h` = q2hh = 0 by lemma 1. By lemma A.3 we have q1hh = ¼ hh and q1h` = ¼ h` . Because the derivatives of the objective function with respect to q1`` and q1`h are equal to m1 , either we h` `` h` will have p`` 1 = p1 = 0 if m1 < 0 or p1 = p1 = 1 if m1 > 0. This justi…es solutions s0 and s1 . If q1h` = 0, we have q1`` = q1`h = q1hh = 0 by lemma 1. By lemma A.3 we have q2hh = ¼ hh and q2h` = ¼ h` . Because the derivatives of the objective function with respect to q2`h and q2hh are equal and strictly positive, we can …nd a solution such that q2`h = ¼ `h , and similarly we can …nd a solution such that q2`` = ¼`` . This justi…es solution s2 . All the other relevant solutions satisfy q1h` = ¼ h` and q2`h = ¼ `h . By (C h` ) and (C `h ), they therefore satisfy q2h` = q1`h = 0: Because constraint (C hh ) is binding, one of q1hh and q2hh must be equal to 0 and the other one to ¼ hh . This characterizes the solution s3 to s8 , by choosing the possible values of q1`` and q2`` .

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