An Analysis of Profit and Consumer Surlus Imlications of Resale Qiong Wang 1, Jon M. Peha, Marvin A. Sirbu 3 Abstract When a monooly carrier rovides multile services(voice, data, video) through a single network, its rofit-maximizing ricing olicy usually induces resale. In many cases, resale can benefit consumers by roviding them with a cheaer alternatives to a highriced service. Nevertheless, in this aer, we demonstrate that there are situations in which the carrier can use resale to imrove its rofits at consumers exense. We also find that even in cases where resale costs the carrier rofit, total consumer surlus does not necessarily always increase. In fact, resale always results in higher consumer surlus in for some users and lower consumer surlus for others. Those findings suggest a regulator should exercise caution in defining the olicy for resale. 1. Introduction With telecommunication deregulation in recent years, many new ventures have started to challenge dominant carriers who reviously monoolized telecommunications markets. Given disadvantages in customer base and brand recognition, it is usually difficult for start-us to engage in direct facility-based cometition with dominant carriers. Therefore, many new comanies choose to comete with incumbent carrier in an indirect way, such as via resale of the incumbent s services. Even for entrants who are building new networks, resale is a good way to start u servicing customers before all facilities are in lace([bear98]). To understand how resale can be rofitable in telecommunications markets, consider a dominant carrier who, enabled by acket-switching technology, offers two services at rices 1 and through a single network. Assume further that demand for service is less elastic than demand for service 1. To maximize rofit, the carrier should set a higher rice er unit of resource for services with inelastic demand ([WANG97]). Therefore, 1 should be smaller than _ even if calls of service 1 consume exactly half as much caacity as service. This rice difference can be exloited by a reseller who buys two service 1 calls at rice 1, then multilexes and resells them as one service call at a rice between 1 and. The business is rofitable as long as the multilexing cost is lower than - 1. Regulatory olicy has a strong influence on the availability of resale. The carrier has an incentive to encourage resale when it imroves rofit, in which case the regulator can choose to allow or forbid resale. When resale reduces rofit, the carrier can drive resellers out of the market by denying service to them or discriminating against them 1 Member of Technical Staff, Bell Laboratories, Lucent Technology. Address: 600 Mountain Avenue, Room D-457, Murray Hill, NJ 07974 Phone: 908/58-3798. email: chiwang@lucent.com Associate Professor, Deartment of Engineering and Public Policy, Carnegie Mellon University. Address: Deartment of Electrical and Comuter Engineering, Carnegie Mellon University, Pittsburgh, PA 1513. Phone: 41/68-716, email: eha@ece.cmu.edu 3 Professor, Deartment of Engineering and Public Policy, Carnegie Mellon University. Address: Deartment of Engineering and Public Policy, Carnegie Mellon University, Pittsburgh, PA 1513. Phone: 41/68-3436, email: sirbu@andrew.cmu.edu
through high rices. In this case, the regulator can hel resellers to survive by requiring the carrier to give equal access to both end users and resellers. Determining the aroriate resale olicy is a roblem facing not only regulators in the US and other industrialized countries, but also those in develoing countries. For examle, in China, the wireline communications network is monoolized by the Ministry of Information Industry, whose rice for international telehone calls is three to six times higher than US rices. In January 1999, a charge was brought against a rivate oerator for reselling the ministry s Internet service as international telehone calls. Though the accused were acquitted, the debate goes on as whether his oeration should be allowed 4. Similar situations have arisen in Haiti where, Teleco, the government owned monooly, accused ACN, the nation s largest ISP, of reselling international telehone service, and took away two thirds of ACN s access to local telehone lines([peha99]). Intuitively, one would exect that resale benefits consumers by offering a cheaer alternative to users of high-riced services([bear98], [KASE97]). Given resellers take away customers, one might also think that resale will cause the carrier s rofit to dro, so the carrier will block it. Therefore, to rotect consumer benefits, the regulator should always favor resale, and enforce it by reventing the carrier from charging different rices to resellers from end users. However, as we will show in this aer, while the intuition is true in many cases, there are situations in which resale can hel the carrier and hurt consumers. Consequently, it is the carrier who should encourage and the regulator who should resist resale. Furthermore, even in cases when resale reduces the carrier s rofit, overall consumer benefits, as measured by total consumer surlus, do not always increase. Those results suggest that there is no simle answer to whether the regulator should oen the resale market and force the carrier to give equal access to resellers. As indicated by our analysis, to make the right decision, the regulator should consider some seemingly unrelated factors such as the extent to which multilexing costs and consumer willingness to ay are correlated. The aer is organized as follows: the mathematical formulation of the roblem is given in Section. Analysis and major results are resented in Section 3, and the aer is summarized in Section 4. 4 Source: News reort of Guangzhou Daily, 01/4/99. htt://www.asia1.com.sg/gzbao
. Problem Formulation and Assumtions: We consider an incumbent carrier who offers two constant bit rate services (indexed as services 1 and ) with the same average call duration. Assume the data rate of service is k times the rate of service 1. We assume in the absence of resale, the carrier chooses 1,, rices for services 1 and, and C T, network caacity, to maximize the following rofit function: T [ ( 1 β ) ( 1 ) 1 1λ1 β λ + ] dt F( CT ) (.1) 0 r1 where λ 1, λ are call arrival rates, which are functions of 1,, resectively. T is the lanning eriod. F(C T ) is the caacity cost allocated to that eriod. β 1 and β are blocking rates of services 1 and, which are functions of call arrival rates and caacity. Suose demand for service 1 is more elastic than that for service ; so to maximize (.1), the carrier will over-charge for service in the sense that >k 1. In the resence of resale, resellers will buy service 1 calls at 1, multilex/demultilex and resell them as service calls. Denote the mux/demux cost as h, which is assumed to be a random variable distributed over users. Let φ(h w) be the conditional Probability Density Function (PDF) of the mux/demux cost, and ψ(w) be the PDF of willingness to ay, then γ(h,w)= φ(h w)ψ(w) is the PDF of the joint distribution of consumer willingness to ay and mux/demux cost. Assume the resale market is cometitive, so resellers charge users at marginal cost, i.e. k 1 +h. We divide users of service into four segments: segment 1 are users whose willingness to ay is below both the resellers and carrier s rices; segment are users whose willingness to ay is below the carrier s rice but above the resellers rice; segment 3 are users whose willingness to ay is above both rices, and for whom the mux/demux costs are lower than the rice difference; and segment 4 are users whose willingness to ay is also above both rices, but whose mux/demux costs are higher than the rice difference. Assume users always make their decisions so as to maximize their utility, which is their willingness to ay minus the rice they ay. In the resence of resale, segments and 3 users will buy service from resellers, segment 4 users will buy the service from the carrier, and segment 1 users will not buy service from either of them. Define λ i (i=,3,4) as call arrival rates from segment i users, soλ = λ + λ + λ 3 4. Denote maximum mux/demux cost er user as H. Figure 1(a-c) shows the division of those segments in different situations, and the formula following each figure gives call arrival rates from each grou in that situation: If ( k ): 1 Figure 1 (a) Segmentation of Users
H h segment 1 segment 4 If ( k1 < k1 + H) max λ =0, λ 3 =0, and λ = max 4 λ max γ ( w, h) dhdw 0 h Figure 1 (b) Segmentation of Users H w H segment 1 segment 4 -k 1 h=w-k 1 segment segment 3 k 1 = w k max 1 max 1 λ λmax γ ( w, h) dhdw, λ λ max γ ( w, h) dhdw, k1 0 max and λ λmax γ ( w, h) dhdw = 4 k1 H = k 3 0 w If ( k1 + H ) H h segment 1 Figure 1 (c) Segmentation of Users h=w-k 1 segment segment 3 k 1 max w = k + H w k 1 1 λ λmax γ( w, h) dhdw, λ = 1 λ γ k 0 3 max ( w, h) dhdw, and λ 0 4 =0 1 k + H H The carrier collects 1 er minute from segments and 3 users of service and er minute from segment 4 users. Including both revenue from service 1 and caacity investment cost, the rofit function for the carrier can be written as:
T { ( 1 β ) ( ) 1 1λ1 1 β + [ ( ) ]} ( ) 0 1 λ + λ + λ dt F CT r1 where β 1 and β are blocking rates of services 1 and, the values of which we can calculate based on steady-state queuing system analysis, using λ 1,λ, and C T as inuts ([OZEK90]). 3. Analysis 3 4 (.) In the following, we discuss the imact of resale on the dominant carrier and consumers in 3.1 and 3. resectively. We demonstrate that resale may or may not reduce the carrier s rofit, so the carrier may not always act against it. Furthermore, resale can cause consumer surlus to increase or decrease, deending on the situation. Therefore, whether the regulator should romote resale is a question without unique answer. Our analysis is based on the model and assumtions develoed in section. We assume the demand function - i.e. call arrival rate as a function of rice - takes the following form: i αi λi = λi max[ 1 ( ) ] i=1, (3.1) i max In that formulation, λ imax is the maximum call arrival rate, which equals call arrival rate when rice is zero. imax is the maximum consumer willingness to ay, which equals the lowest rice at which there will be no call arrival. α i is a arameter that characterize how fast demand falls as rice increases. In all cases, we assume the carrier offers two constant bit rate services of 64kbs and 18kbs, resectively. The average call duration is 10 minutes for both services. 3.1 Imact of Resale on the Carrier Given resellers cometing for customers with the monooly carrier, it is not difficult to imagine that resale can reduce the carrier s rofit. However, in this section, we will first demonstrate that there can be situations where resale causes the carrier s rofit to increase, and exlain why. We will then discuss under what situations, resale always reduces the carrier s rofit. Consider the following examle: the dominant carrier serves two grous of users, indexed by a and b. Both grous have the same demand function for service 1 but different ( g) demand functions for service. Let λ j reresent the demand of grou g for service j. We assume: ( g) 1 04. λ 1 = 0*[ 1 ( ) ] g=a,b 06. and ( a) 05. ( b) 05. λ = 10*[ 1 ( ) ] and λ 15. = 10 *[ 1 ( ) ] We also assume that the two grous have different mux/demux costs, h. For grou a, h=0, while for grou b, h=.5. In essence, grou a consists of users with low willingness to ay and low mux/demux cost, while grou b consists of users with high willingness to ay and high mux/demux cost.
As formulated in Section, we assume the carrier maximizes rofit as defined in (.1) in the absence of resale, and the rofit function defined in (.) in the resence of resale. Based on solving strategy secified in WANG98, we obtain otimal rices, revenue, rofit, and consumer surluses. Those results are comared in Table 1. Table 1 Comarisons of Price, Revenues, Profit, and Consumer Surlus rice service 1 service / 1 resale 0.43 1.17.71 no resale 0.399 1.04.61 revenue investment rofit service 1 service resale 901 780 83 849 no resale 351 1318 83 838 consumer surlus service 1 service service total (grou 1) (grou ) resale 58 07 63 58 no resale 85 106 360 551 Since resellers buy service 1 from the carrier and resell it as service to end users, it can be exected that in the resence of resale, the carrier s revenue from service 1 increases, and that from service decreases, just as shown in Table 1. What is interesting in this examle is that when resale occurs, the carrier s otimal rice of service also increases: it is otimal for a carrier to charge an even higher rice for service when it faces cometitions from resellers than in the monooly situation. Furthermore, it is also shown in the table that under such a ricing strategy, the carrier s rofit increases as a result of resale. Under our assumtions, the carrier is able to increase rofits via rice discrimination as we exlain below. A comany can usually achieve a higher revenue by dividing markets into different segments, and charging a different rice to each segment. Sometimes, the rice discrimination can be racticed through a third arty if the comany can t do it itself. For examle, fashion designer s may rovide goods labeled with a store brand, which hels to divide buyers into a segment that is willing to ay a high rice for brand-names, and a segment that wants cheaer store brands. Those manufacturers can then charge a higher rice to the higher willingness to ay segment, while still retain the lower willingness-toay segment through the store brand. Similarly, a carrier can ractice rice discrimination through resellers. In the above examle, grou a users have lower demand, and grou b users have higher demand. As a result, the carrier can choose to rovide service only to grou b and leave grou a to resellers. This segmentation gives the service rovider the leverage to increase the rice of service since they can still indirectly sell the service to the low willingness to ay users through resellers at a lower rice ( 1 instead of ). Like other forms of rice discrimination, the strategy works only when the carrier is caable of retaining higher willingness to ay users. Therefore, it is crucial in the above examle that users of grou b have higher mux/demux costs, and would thus ay a higher total rice by switching to resellers. In fact, such a ositive correlation between consumer willingness to ay and mux/demux costs is a necessary condition for the carrier to benefit
from resale. In the following roosition, we show that when they are uncorrelated, the carrier can not benefit from resale(see aendix for roof). Proosition 1 Let F(C) be the incumbent carrier s investment as a function of C, the amount of caacity invested. Let G(x,y) be the minimum caacity required to maintain given blocking rates for two services when call arrival rates are x and y, and let φ(h w) be the conditional PDF of mux/demux cost. Let ( 1a, a ) and ( 1, ) be the carrier s rofitmaximizing rices in the absence and resence of resale, resectively. Π a ( 1a, a ) and Π ( 1, ) are corresonding rofits: 1). F 0 for any C, C ). Gxy (, + y) Gxy (, ) + Gx (, y) for all x, y, and y>0, 3). φ(h w)=φ(h), then Π, ) Π (, ). ( 1 a 1a a In summary, we demonstrated in this section that resale may not always cause the carrier s rofit to decrease, and derived a necessary condition if the carrier is to imrove rofit from resale. The discussion indicates two ublic olicy making scenarios: in the case where resale benefits the carrier, the regulator s olicy choice is whether or not to ban resale. In the case resale cause the carrier s rofit to decrease, the regulator s olicy choice then becomes whether or not to enforce resale by requiring the carrier to charge the same rice to resellers and end users for the same service. In either scenario, regulator s choice deends uon the imact on consumer benefits, to which now turn our attention. 3. Imact of Resale on Consumers Assume that the regulator s objective is to enhance consumer benefits, and will thus use consumer surlus to evaluate the imact of resale. In this section, we comare total consumer surlus with and without resale, and discuss the olicy imlications for the regulator. In section 3.1, we demonstrated in our examle that when there is a ositive correlation between consumer willingness to ay and mux/demux costs, resale can be used by the carrier to imrove its rofit through rice discrimination. While rice discrimination is not necessarily always harmful to consumers, in our examle, total consumer surlus does decrease as a result of resale. The examle shows there can be cases in which resale allows the carrier to rofit at consumers exense. We now consider other scenarios consistent with Proosition 1, in which resale always reduces the carrier s rofit because we assume there is no ositive correlation between consumer willingness to ay the mux/demux cost. Our analysis below demonstrates that even in these situations, resale may or may not benefit consumers. Consider the following examle: assume the mux/demux cost is indeendent of consumer willingness to ay, and is uniformly distributed in the interval [0,H], where H=0.1. Assume the demand functions for services 1 and are the same as in Equation 3.1. Let λ 1max = λ max =6, α 1 =0.4, α =1.0, max =1.5, and vary 1max from 0.5 to 0.75. The difference in rofit with and without resale is comared in Figure, the difference in total consumer surlus is comared in Figure 3, and differences in rices of services 1 and are comared in Figure 4. As indicated by Proosition 1, given there is no ositive correlation between mux/demux cost and consumer willingness to ay, the carrier s rofit is always lower in
the resence of resale (See Figure ). Furthermore, unlike the revious case in which the carrier can increase rice because of the discrimination effect, the carrier s rice of service is always lower with resale. What is interesting is that desite dros in both rofit and rice, resale does not necessarily always result in higher total consumer surlus. As shown in Figure 3. As shown in Figure 3, when 1max =0.75, total consumer surlus is actually lower in the resence of resale. This henomena can be exlained as follows: Figure Comarison of Profit 650 60 no resale 590 resale 560 0.50 0.55 0.60 0.65 0.70 0.75 1max Figure 3 Comarison of Total Consumer Surlus 360 345 resale 330 315 no resale 300 1max 0.50 0.55 0.60 0.65 0.70 0.75 Figure 4 (a) Comarison of Price of Service 1
0.45 1 0.4 0.35 resale 0.3 0.5 no resale 0. 0.50 0.55 0.60 0.65 0.70 0.75 1max Figure 4 (b) Comarison of Price of Service 0.85 0.8 no resale 0.75 resale 0.7 1max 0.50 0.55 0.60 0.65 0.70 0.75 To comete with resellers, the carrier has to reduce the discreancy in the rice er unit of caacity between the two services, which can be accomlished either by increasing the rice of service 1 or decreasing the rice of service. Given that the original rices are otimized for the situation without resale, increasing rice 1 and reducing rice will cause the carrier s revenues from services 1 and to fall, resectively. From the carrier s ersective, changes in the two rices should be balanced to minimize the decrease in total rofit. From the consumers ersective, increasing rice 1 results in a smaller consumer surlus from service 1 and decreasing rice leads to greater consumer surlus from service. Therefore, whether or not resale can cause total consumer surlus to increase deends on how much rice 1 is increased versus how much rice is decreased. For the demand function assumed by Equation 3.1, a smaller value for 1max imlies a greater elasticity of demand of service 1. This means the same ercentage increase of rice for service 1 leads a larger ercentage decrease in demand, and thus revenue. Therefore, the carrier s otimal resonse to resale is to decrease rice more when 1max is smaller, and increase rice 1 more when 1max becomes larger. As 1max kees increasing, the carrier relies more on increasing rice 1 to eliminate the difference of rice er unit of caacity between the two services. Eventually, the decrease of consumer surlus due to resale of service 1 outweighs the increase of consumer surlus of service, and total consumer surlus starts to decrease. The analysis has the following two imlications for the regulator. First, the fact that the rice of service 1 is always higher with resale (see Figure 4.4a) imlies there is a tradeoff for the regulator, i.e. oening u the resale market will inevitably hurt some users
even if it hels to imrove overall consumer welfare. As a result, if a regulator considers that the welfare of some users are more imortant than others, s/he may still choose not to enforce resale. For examle, if the regulator s first riority is to kee down household hone bills, then s/he will not authorize reselling of residential POTS lines as business lines, even if doing so would increase the combined welfare of residential and business customers. More imortantly, even if the regulator is indifferent to distributional effects and choose to maximize total consumer surlus, resale may still be unattractive. This haens in two situations. First, when there is a ositive correlation between consumer willingness to ay and mux/demux costs, resale can be used by the carrier as a means of rice discrimination, which may lower total consumer surlus. When this haens, the carrier will embrace resale, but the regulator should forbid it. Second, even if there is no such ositive correlation and resale always hurts the carrier s rofit, the carrier may still increase the rice of service bought by resellers to such an extent that total consumer surlus decreases. In both cases, the regulator may wish to eschew resale as a means to constrain ricing by a monoolist. 4. Summary To maximize rofit, a monooly carrier always charges a higher rice for a service than the cost of roviding it. In comarison with a cometitive market where rice equals marginal cost, the monooly carrier earns monooly rents while consumers enjoy a smaller surlus. The difference between rice and cost is most significant in service with inelastic demand, and the carrier extracts more rents from users of those services. To rotect consumer interests, regulatory rules have been develoed to constrain monooly ricing. Requiring the carrier to allow resale is one of those aroaches. If the carrier is rohibited by regulation from denying service to resellers or charging resellers a different rice from ordinary users, it can be rofitable to buy one service with elastic demand and resell it as a service with inelastic demand at a lower rice. Therefore, one would exect that resale can give cnsumers a cheaer alternative to a reviously highlyrice service, thereby increasing consumer surlus. Our analysis in this aer demonstrates while the exectation is true in many cases, there are also situations under which consumer benefits can t be imroved by romoting resale. Furthermore, there even can be cases where for the benefit of consumers, a regulator should ban rather than require resale. We resent a numerical examle that demonstrates that when there is a ositive correlation between consumer willingness to ay and mux/demux cost, the carrier may rofit from resale. In this case, instead of lowering rice to comete with resellers, the carrier can charge a higher rice to high willingness-to-ay users who can t switch to resellers because of the high mux/demux cost. Because of resale, the carrier is not unished for losing customers by raising rices, since it can still sell to low willingness-toay users through resellers. As a result, in our examle, resale causes the carrier s rice and rofit to increase and total consumer surlus to decrease. Therefore, the carrier will encourage resale even without being required by regulation, and the regulator should ban resale to maximize consumer surlus. We rove that resale always causes the carrier s rofit to decrease if there is no correlation between a reseller s mux/demux costs and consumers willingness to ay for the service being resold. In those cases, the carrier has no incentive to allow resale, so the regulator has to mandate it. Whether the regulator should do so deends on circumstances. Resale can benefit users of the service with inelastic demand because: 1) users who buy the
service from resellers get a lower rice; ) some users whose willingness to ay is below the carrier s rice can get the se4rvice form resellers; and 3) the carrier may reduce the rice to retain customers, thus benefiting those who continue use its service. However, resale also causes the rice of the service with the more elastic demand to increase and hurts the original users. In some cases, the decrease in consumer surlus from that service exceeds the increase in consumer surlus from other services, so total consumer surlus decreases as a result of resale. Even in cases when resale leads to an increase in total consumer surlus, the regulator may hesitate to mandate it in consideration of the differential imact on consumers of the different services. Resale may imrove total consumer welfare only by increasing the welfare of one grou more than it decreases the welfare of another. In summary, while requiring the carrier to allow resale has been viewed as a way to enhance cometition and imrove consumer welfare, our research identifies some ossible itfalls in this aroach. In deciding whether or not to require resale, the regulator should consider the conflict of interests among different grous of users, and be aware that under certain situations, resale can reduce instead of increase consumer welfare. This work could be extended in several different directions. It would be interesting to consider the imact of resale on other otential forms of cometition. For examle, one might want to examine whether mandating resale would enhance or retard facility based cometition. It would also be interesting to examine the dynamics of resale growth and ricing resonse over time. The result should be interesting for olicy making with resect to the growing market for Internet Telehony, where acket data service is resold as voice service. Reference: [BEAR98] [KASE97] [OZEK90] [PEHA99] [WANG97] [WANG98] T. R. Beard, D. L. Kaserman, and J. W. Mayo(1998), The Role of Resale Entry in Promoting Local Exchange Cometition, Telecommunications Policy, Vol., No. 4/5,. 315-36. D. L. Kaserman and J. W. Mayo(1997), An Efficient Avoided Cost Pricing Rule for Resale of Local Exchange Telehone Service, Journal of Regulatory Economics, 11,. 91-107. Queuing Theory and Alications, Ozebici eds.(1990), Hemishere Publishing Cororation. Peha, J. M., "Lessons from Haiti's Internet Develoment," to aear in Communications of the ACM, June 1999. Q. Wang, J. W. Peha, and M. A. Sirbu(1997), Otimal Pricing for Integrated-Services Networks, Internet Economics, L. W. McKnight and J. P. Bailey eds., The MIT Press,. 353-378. Q. Wang (1998), Pricing of Integrated-Services Networks, Ph.D. Thesis, Carnegie Mellon University.
Aendix - Proof of Proosition 1: Denoting the demand function of service i as λ i (), i=1,, we start by considering the carrier s rofit in four scenarios: 1) Suose in the absence of resale, ia β ia are otimal rices and blocking rates for service i (i=1,), C a is the otimal caacity, then: ( 1 β1 a) 1aλ1( 1a) ( 1 βa) aλ( a) Π a = + FC ( a ) (A-1) r1. Suose in the resence of resale, i and β i are the otimal rices and blocking rates for services i (i=1,). k a 1a Define θ = φ() hdh (A-) 0 Define λ r as the call arrival rate for resellers services. The call arrival rate of service for the carrier is (1-θ)λ ( ). Define C as the minimum caacity needed to kee call blocking rates below β 1 and β when the call arrival rates of services 1 and are λ 1 ( 1 ) andλ r + (1-θ)λ ( ). The carrier s maximum rofit in the resence of resale is: ( 1 β1) 1λ1( 1) ( 1 β)[ k1λr + ( 1 θ) λ( )] Π = + FC ( ) (A-3) r1 3). Suose in the absence of resale, the carrier charges 1 and, and kees blocking rates below β 1, β. The minimum caacity needed is C g. The carrier s rofit is: ( 1 β1) 1λ1 ( 1) ( 1 β) λ ( ) Π g = + FC ( g ) (A-4) r1 4). Suose in the resence of resale, the carrier charges 1 and k 1, and kees blocking rates below β 1,β. The minimum caacity needed is C y. The carrier s rofit is: ( 1 β1) 1λ1( 1) ( 1 β) k1λ( k1) Π Y = + FC ( y ). (A-5) r1 We now rove the theorem ( Πa Π ) by showing if Π a < Π, then Π < Π g, which contradicts the assumtion that Π is the maximum rofit in the resence of resale. If Π a < Π, then Π g Π because in the absence of resale, Π a is the otimal solution and Π g is just one feasible solution. By equations (A-3) and (A-4): ( 1 β ) Π Π a = [ k1λr θλ ( )] FC ( ) + FC ( g ) 0 (A-6) Define λ g =λ r -θλ ( ). From Figures 1(b), λ g >0. From (A-6) ( 1 β ) [ k1lg θ ( k1) λ ( )] F( C) F( Cg) (A-7) Define λ b =λ (k 1 ) -λ ( ) - λ g. From Figure 1(b) λ b θ 1 (A-8) λ θ g Multily λ 1b at both sides of (A-7) and aly equation (A-8), λ1a ( 1 β ) λ b [ k1λg ( 1 θ )( k1 ) λ ( )] [ F ( C ) F ( C g )] (A-9) r λ g
C, C g, and C y are caacities needed to accommodate λ 1 ( 1 ) and λ ( ), λ 1 ( 1 ) and λ r =λ ( )+λ g, λ 1 ( 1 ) and λ (k 1 )=λ ( )+λ g + λ b, and kee blocking rates at β 1,β. Since Gxy (, + y) Gxy (, ) + Gx (, y) for all x, y, y, F 0 C λ b [ FC ( ) FC ( g)] FC ( y) FC ( g) (A-10) λ g Combine equations (A-9) and (A-10): ( 1 β ) [ k1λb ( 1 θ )( k1 ) λ ( )] FC ( y ) FC ( g ) (A-11) From Equations (A-3), (A-5), and (A-11) ( 1 β ) ΠY Π = [ k1λ ( 1 θ)( k1) λ( )] FC ( y) + FC ( g) 0 which contradicts the assumtion Π is the maximum rofit in the resence of resale. Therefore, Π Π a b,