On Optimal Service Differentiation in Congested Network Markets Mao Zou1,2, Richard T. B. Ma3, Xin Wang1,2, Yinlong Xu1,2 1 School of Computer Science and Technology, University of Science and Technology of China 2 AnHui Province Key Laboratory of High Performance Computing (USTC) 3 School of Computing, National University of Singapore raffaele@mail.ustc.edu.cn, tbma@comp.nus.edu.sg, yixinxa@mail.ustc.edu.cn, ylxu@ustc.edu.cn Abstract—As Internet applications have become more diverse decision-making process. In deciding the optimal strategy, in recent years, users having heavy demand for online video an ISP needs to answer two questions: what are the service services are more willing to pay higher prices for better services qualities to offer and what are the associated prices for them. 7 than light users that mainly use e-mails and instant messages. For the former question, even the range of the optimal set 1 This encourages the Internet Service Providers (ISPs) to explore 0 service differentiations so as to optimize their profits and al- of service qualities is hard to determine; for the latter one, 2 location of network resources. Much prior work has focused it needs to find the optimal quality-price mapping for the on the viability of network service differentiation by comparing chosen service qualities. The problem is actually a dynamic n with the case of a single-class service. However, the optimal a optimization problem in a functional space and the domain service differentiation for an ISP subject to resource constraints J of the price mapping is to be determined. As the realized has remained unsolved. In this work, we establish an optimal 5 control framework to derive the analytical solution to an ISP’s quality in a service class depends on the allocated network 1 optimal service differentiation, i.e. the optimal service qualities capacity and the aggregate user demand, which is determined andassociatedprices.Byanalyzingthestructuresofthesolution, bytheavailableservicequalitiesandthecorrespondingprices, ] we reveal how an ISP should adjust the service qualities and I anothermajorchallengeistomodeltheISP’scapacityalloca- N prices in order to meet varying capacity constraints and users’ tionplansubjecttonetworkcapacityconstraintsanddetermine characteristics. We also obtain the conditions under which ISPs s. have strong incentives to implement service differentiation and the realized quality in terms of network congestion. In other c whether regulators should encourage such practices. words, ISPs need to allocate limited network capacity to each [ service class so as to guarantee the promised service quality. 1 I. INTRODUCTION Inviewofthecomplexityoftheoptimalservicedifferentia- v Recent years have witnessed increasing commercialization tion,thecommonpracticeofpriorworkistoadoptsimplifying 6 of the Internet and considerably rising diversities of Inter- assumptions. For instance, Jain et al. [8] assumed the ISP 7 net applications and users. Users having heavy demand for only provides two service classes. Shakkottai et al. [20] 0 quality-sensitive services such as online video services are overlookedthecongestionexternalitiesandconsideredaloose 4 0 often willing to pay higher prices for better service qualities, upper bound (under the first degree price discrimination) of . while light users tend to have low willingness-to-pay for the ISP’s optimal profits instead. Chau et al. [4] considered 1 0 elementary services such as e-mails and instant messages. a finite number of service classes, analyzed the viability of 7 Such heterogeneity in the users’ characteristics has rendered differentiated pricing, but did not solve the optimal strategy. 1 the traditional practice of uniform prices, regardless of the In this work, we solve an ISP’s optimal service differen- v: quality of service, inefficient in both generating profits and tiation in generic settings by explicitly modelling the ISP’s i allocating network capacity. In light of this, there has been capacityallocationplansubjecttonetworkcapacityconstraints X tremendous interest in designing pricing strategies to provide andincorporatingthedualapproachinmicroeconomics[14]to r a differentiated services. Many schemes [17–19] have been transformtheproblemintoatractableoptimalcontrolproblem. designed, e.g., Paris Metro Pricing (PMP) [17] proposed by As for the network capacity constraints, we consider two Andrew Odlyzko. In such a scheme, network is partitioned typical scenarios: a) the fixed capacity scenario, where an into several logically separated channels, each of which uses ISP has limited network capacity, and b) the variable capacity a fraction of the entire network capacity and a different price, scenario, where an ISP can invest to expand the capacity of and the channels with higher prices attract fewer users but its network infrastructure. Our findings include the following. provide better services. • We formulate a profit maximization problem subject to An important theoretical question naturally arises: what is capacity constraints under an optimal control framework the optimal service differentiation for an ISP subject to its and derive the analytical optimal service differentiation capacity constraints so as to maximize its profit? Despite the of an ISP, i.e., the qualities and prices of service classes. vast literature on network service differentiation, this question • We find that the optimal differentiation strategy offers has remained unanswered. Prior work [4, 8, 10, 20] mostly service classes with sufficiently high qualities and prices focusedontheviabilityofPMP-typeofdifferentiationanddid such that users with low value will not join the services, notinvestigatetheoptimalservicedifferentiation.Totacklethe even under abundant capacity. We characterize an ISP’s optimal service differentiation, we delve a bit into the ISP’s market share by users’ virtual valuation function [16]. • WeshowthatwhenanISPexpandsitscapacity,itshould studies investigated the competition among service providers. increase market share by offering more premium service For example, Gibbens et al. [6] analyzed the competition classes and reducing prices. When there are more high- between two ISP’s and show that in any equilibrium compet- end users, it is optimal for the ISP to offer a narrower itive outcome, both ISPs offer a single-class service. Ma [10] range of premium service classes with higher prices. studied usage-based pricing scheme and competition among • We show that if network capacity is scarce, an ISP has oligopolistic service provides. These studies focused on the strongincentivestoimplementservicedifferentiationand viability of PMP and its variants by comparing with a single- regulators should encourage such practices in order to class service. Our work, however, focuses on the theoretical improve the total user surplus; otherwise, it is optimal problem of solving the ISP’s optimal service differentiation for the ISP to provide a single-class service. subjecttocapacityconstraints.Weestablishanoptimalcontrol To the best of our knowledge, our work is the first to frameworktoderivetheanalyticalsolutiontotheISP’soptimal analytically derive and characterize an ISP’s optimal service service differentiation and characterize its properties. differentiation subject to network capacity constraints. We III. MODEL believethatourfindingswillhelpISPstobetterpricenetwork servicesandguideregulatorstodesignregulations.Therestof A. ISP and User Model the paper is organized as follows. Section II discusses related We consider an ISP and a continuum of users with hetero- work. Section III presents the model and formulate the ISP’s geneous characteristics. Suppose the ISP offers a set N of profit-maximizing problem as a nested optimization problem. service classes and adopts flat-rate1 pricing [1] for all service In Section IV, we derive the analytical solution of the optimal classes. For each service class i ∈ N, we denote p as the i service differentiation and study the structure of the solution. price and q as the congestion level, a measure of service i We reveal the dynamics of the solution under varying market quality. Correspondingly, we denote p (cid:44) {p : i ∈ N} and i environments in Section V and conclude in Section VI. q(cid:44){q :i∈N} as the price and congestion vectors. i Once the menu of services (p,q) is decided, users will II. RELATEDWORK make choices according to their preferences over the bundles Much theoretical economics literature has considered prod- {(p ,q ):i∈N}. We characterize each user by her intrinsic uctdifferentiation.MussaandRosen[15]analyzedtheoptimal i i value of the network service θ ∈Θ=[0,1]. We denote f(θ) product mix for a monopolist offering several qualities of andF(θ)astheprobabilitydensityandcumulativedistribution a product to consumers with different preferences for qual- functionsofusers’value,andassumethatf(θ)iscontinuously ity. Maskin and Riley [11] characterized the optimal selling differentiable and strictly positive over Θ. When a user with strategy of a monopolist, and Champsaur and Rochet [2] value θ chooses a service class (p,q), we define her utility as studied the product differentiation for two competing firms. Both adopt a common assumption that the quality of the u(p,q;θ)=θv(q)−p, (1) product can be unilaterally decided by the seller. However, which can be interpreted as follows. these models are inapplicable to network services because the quality of services are influenced by both the user demand • v(·)isasatisfactiondiscountfunctionwhichcapturesthe and the network capacity. Chander and Leruth [3] studied the negative effect of congestion on users’ intrinsic values. best strategy of a monopolist in the presence of congestion • θv(q) is the user’s achieved value over the service under effects. Nevertheless, they did not provide the solution of the a congestion level q, and thus u(p,q;θ) is the user’s net optimal strategy and their congestion model does not capture surplus, i.e., the achieved value θv(q) minus the price p. the scarcity of capacity in network markets. • Without loss of generality, we normalize the range of Various pricing schemes for providing differentiated net- congestion level q to the interval [0,1] and assume that work services has been proposed [17–19]. In particular, Paris v(·) : [0,1] → R+ is continuously differentiable and Metro Pricing (PMP) [17] has attracted considerable attention strictly decreasing, satisfying v(0) = 1 and v(1) = 0. for its simplicity. Since PMP does not guarantee the quality This models that the user’s value is at maximum without of service for users, which depends on the spontaneous user congestion, i.e., q = 0, and decreases to zero when the demand, its viability and effectiveness need to be established. network is heavily congested, i.e., q →1. Prior studies [4, 6, 8, 10, 20] addressed the viability of PMP. Weassumethatusersareindividuallyrational,i.e.,anyuser Shakkottai et al. [20] characterized the types of environments with value θ chooses either a) to join the service class k that in which simple flat-rate pricing is efficient in extracting yields the highest nonnegative utility2: profits. Lee et al. [9] extended the model of [20] to explicitly k =argmax u(p ,q ;θ) (2) considerthecongestionexternalitiesandindicatethatwhenthe j j j∈N network is congested, a simple flat-rate pricing is inefficient in extracting profits. Jain et al. [8] analyzed the ISP’s profit- 1Our analysis can be extended to handle usage-based pricing schemes. maximizingproblemwhentheISPisrestrictedtoprovidertwo InterestedreadersarereferredtotheAppendixfordetails. service classes. Chau et al. [4] provided sufficient condition 2Technically, we should write k ∈ argmaxj∈N u(pj,qj;θ). By using theequalsign,weassumewhentherearemorethanoneserviceclassesthat of congestion externalities for the viability of PMP. Several maximizetheuser’sutility,theuserwillchoosetheonebyherowncriteria. or b) to opt out of the ISP’s services if joining any service function w(q) models the capacity required to maintain the class induces a negative utility, i.e., u(p ,q ;θ) < 0,∀i ∈ N. congestion level q given that the service class has a unit user i i To unify the above two cases, we assume the existence of a demand. Under Assumption 1, the implied capacity function dummy service class 0 which satisfies p = 0 and q = 1, satisfies C((cid:15)d,q) = (cid:15)C(d,q), indicating that users will not 0 0 and therefore, any user can always choose service class 0 to perceive any difference in terms of congestion after service gain zero utility. We denote the set of user types that choose partitioning or multiplexing. This form is a generalization of the service class i as Θ (p,q). Notice that if the ISP offers C(d,q) = d/q (to see this, let w(q) = 1/q), which has been i two service classes i and j such that p > p and q > q , considered in much prior work [4, 6, 8, 10]. i j i j no user would choose class i because it is inferior to class j WemodeltheISP’scapacityallocationplanbyavectorc(cid:44) in terms of both price and congestion. Therefore, without loss {c : ∀i ∈ N}, where c is the capacity allocated to service i i of generality, we can sort the indexes of non-dummy service classi.Toguaranteethecongestionlevelofeachserviceclass, classes in an ascending order of congestion level as follows: the ISP’s capacity allocation plan c must satisfy p1 >p2 >...>p|N| and q1 0, θˆdepends on both the user distribution F(·) values, the ISP needs to sacrifice part of the low-end market and the fixed capacity CM. Given any F(·), θˆdecreases share and dedicate its capacity to premium service classes for in CM with the asymptotic limit limCM→C¯ =θ0. high-end users so as to maximize its profits. Corollary3statesthattheoptimalservicedifferentiationoffers V. DYNAMICSOFOPTIMALSERVICEDIFFERENTIATION the service classes with sufficiently low levels of congestion and high prices such that the users with low value will not In this section, we study the dynamics of an ISP’s optimal use any service, even when the network capacity is abundant. servicedifferentiationandthecorrespondinguserbehavior.We The intuition behind Corollary 3 is that by offering premium firstchoosetheparametersofthesystem.Inparticular,weuse services with high quality and high prices, the gains from the the exponential form e−q to construct the user’s satisfaction high-end users outweigh the loss of low-end market share. discount on congestion level. As we normalize the domain of Next,weturntothestructureoftheoptimalchoicesofusers congestion q to be [0,1], we adopt a normalized form v(q)= with values θ above θˆ. Letting θ¯(cid:44)min{G−1(µh(1)),1}, we e−q−e−1 which satisfies v(0) = 1 and v(1) = 0. We further 1−e−1 rewrite their choices q∗(θ) as follows. model the implied capacity by the quadratic convex function w(q)=(1−q)2,underwhichthemarginaldemandofcapacity  (cid:18)G(θ)(cid:19) h−1 , ∀θ ∈[θˆ,θ¯]; is increasing as the required service quality becomes better, q∗(θ)= µ i.e., as q decreases. Under these settings, the virtual capacity 0, ∀θ ∈[θ¯,1]. function satisfies h(0)<+∞ and h(1)=0. Overtheinterval [θˆ,θ¯],auser oftypeθ joinstheservice class A. Impacts of Network Capacity Constraint q∗(θ) such that G(θ) = µh(q∗(θ)), i.e., the user’s virtual We study the dynamics of the ISP’s optimal service differ- valuation G(θ) equals the cost of maintaining service class entiation with respect to network capacity constraint, i.e., the q∗(θ), which is the shadow price multiplied by the virtual maximum capacity C in the fixed capacity scenario and the M capacity. In other words, under the ISP’s optimal service cost function S(·) in the variable capacity scenario. differentiation,eachuseroftypeθwillchoosetheserviceclass with highest quality and price that it can afford. Moreover, U[0,1] U[0,1] 0.8 1 bhwoihguehnnedrehdt(h0ab)ny1) or lower (α<1) values. Corollary 6: UnderafixeduserdistributionF(·),anISP’s Fig 3 plots the ISP’s optimal pricing curves (left) and the optimal profit J∗ increases with its maximum network capac- corresponding user choice curves (right) under various user ity C over (0,C¯) and have the following asymptotic limit: distributions and fixed maximum capacity C = 0.1. We M M observethatwhenuservaluationshiftstowardsthehigherend, lim J∗ =θ [1−F(θ )]. (19) 0 0 1) the ISP’s optimal pricing curve and the corresponding user CM→C¯ choice curve both shift upwards, 2) the ISP offers a narrower Corollary 6 states that if an ISP’s capacity is not abundant, range of low congestion services and fewer users choose the i.e., C < C¯, it can gain more profits under the optimal M network service, 3) the bunching effect vanishes when α is service differentiation when its maximum capacity expands. large, and 4) the ISP’s profit rises. The second observation Next we study the variable capacity scenario. We focus on is a demonstration of Corollary 3, i.e., when α increases, the the impacts of cost function S(·) on the optimal consumption marginal user value θîncreases. The third observation is due W∗(1)ofcapacityandtheprofitsJ.Tothisend,wechoosea to the fact that the threshold C¯ (cid:44) [1−F(θ )]w(0) increases 0 family of cost functions parameterized by a single parameter with the parameter α. t : S(c) (cid:44) 0,∀c ∈ [0,0.1] and S(c) (cid:44) t(c − 0.1)2,∀c ∈ Summaryofimplications:Whentherearemorehigh-value (0.1,+∞). Our settings model the scenario where the ISP’s users in the market, the ISP’s optimal service differentiation existing capacity equals C = 0.1 and only need to pay for will focus on these high-value users by offering a narrower costsoftheadditionalcapacity.Ingeneral,alargertmeasures range of premium service classes and charging higher prices, higher costs for expanding capacity. thus extracting more profits from these high-value users. Figure 2 plots the optimal consumption W∗(1) of capacity (on the left) and the optimal profits J∗ (on the right) versus C. Comparison with the Optimal Single-Class Service theparametertvaryingalongthex-axis.Weobservethatboth We evaluate whether an ISP has the incentive to implement W∗(1) and J∗ decrease with t, which indicates that when the service differentiation so as to obtain higher profits. To this cost of expanding capacity is cheaper, it is optimal for the end, we use the optimal pricing strategy of a single-class ISP to purchase more capacity so as to gain more profits. In service8 (p∗,q∗) as our benchmark and compare the ISP’s particular, when t→+∞, the ISP will not invest in any extra profits and the total user surplus under our optimal service capacity and the optimal consumption level is W∗(1)=0.1. differentiation with that under the benchmark. We conduct Summary of implications: In the fixed capacity scenario, simulations under the various user distributions. Since the when the ISP increases network capacity, its optimal service general trends are almost the same, we adopt the uniform differentiation scheme will a) enlarge the range of service distribution U[0,1] as the representative setting. congestionsbyofferingmorepremiumservices,andb)charge Figure4plotstheISP’sprofitsJ∗ (ontheleft)andthetotal each service class a lower price. In this scenario, the ISP is user surplus s (cid:44) (cid:82)1V(θ)dθ (on the right) versus the maxi- 0 able to allocate more capacity for the premium services and mum capacity C under the two strategies. We observe that M 7We also observe that the optimal prices satisfy p2(q) < p1(q) for all 8The optimal single-class pricing strategy (p∗,q∗) can be solved by q∈Q1∩Q2\{1}.However,thisobservationhasnotbeenrigorouslyproved. restricting|N|=1intheoptimizationproblem(7). 0.25 0.1 low value will not use the services, even when the network OptimalDi, OptimalDi, BestSingleClass 0.08 BestSingleClass capacity is abundant. When the ISP expands its capacity, its 0.2 0.06 market share will be increased by offering more premium $J s service classes and reducing prices. Furthermore, when there 0.04 0.15 exist more high-value users, the ISP should focus on them 0.02 by offering a narrower range of premium service classes and 0.1 0 charging higher prices. Our results also show that the ISP has 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 CM CM strongincentivestoimplementservicedifferentiationwhenits Fig.4. BenefitsofServiceDifferentiation capacityisscarceandsuggestthatregulatorsshouldencourage CM=0.11 CM=0.22 such practices as they increase the total user surplus. 0.2 OptimalDi, OptimalDi, 0.15 BestSingleClass 0.2 BestSingleClass ACKNOWLEDGMENT 0.15 s 0.1 s This work is supported by National Science Foundation of 0.1 China (61379038), the Huawei Innovation Research Program 0.05 0.05 (HIRP) and Ministry of Education of Singapore AcRF grant 0 0 R-252-000-572-112. 0.4 0.6 0.8 1 0.4 0.6 0.8 1 3 3 Fig.5. Surplusofeachuser APPENDIX A. Technical Proofs when the capacity is scarce, the ISP has stronger incentives to differentiate services so as to gain more profits. However, Proof of Lemma 1: If 1 ≥ θ1 ≥ ... ≥ θ|N|, we define asthecapacitykeepsgrowing,theincreaseinprofitsbecomes |N| intervals as follows: marginal. This observation is consistent with Corollary 6 and I (cid:44)[θ ,1], ∀i∈N. Theorem1,whichstatethatwhenC approachesC¯,theISP’s i i M profit converges to the upper bound which is reached without Note that for i=1,2,...,|N|−1, thecapacityconstraints.Figure1alsoillustratesthis:thecurve p −p with CM = 0.3345 shows that when capacity expands, the θ ∈Ii ⇐⇒ θ ≥θi = v(q )i−vi(+q1 ) i i+1 majority of users are bunched up in the best service class ⇐⇒ θv(q )−p ≥θv(q )−p q = 0 and the profit gained by service classes q ∈ (0,1] is i i i+1 i+1 small compared to that of the service class q =0. ⇐⇒ θ prefers class i to class i+1 The trend of total user surplus s is similar, except for that and for i = |N|, θ ∈ I ⇐⇒ θ subscribes to the ISP. i the increase is also marginal when capacity is scarce. To see Therefore, we have this more clearly, we plot the user surplus s versus the user type θ when CM =0.11 (on the left) and CM =0.22 (on the θ ∈[θi,θi−1] ⇐⇒ θ ∈IN ∩...∩Ii∩I¯i−1∩...∩I¯1, right) in Figure 5. We observe that under the ISP’s optimal which means θ prefers class i to i+1, i+1 to i+2, ..., N service differentiation, more users subscribe to the services. to opting-out, and i to i−1, ..., 2 to 1. Hence we have However, when the capacity is insufficient, mid-tier uers are sacrificed to subsidize the high-end and low-end users. Θ (p,q)=[θ ,θ ], i=1,2,...,|N|. i i i−1 Summary of implications: The ISP has strong incentives to implement service differentiation if its network capacity is Proof of Lemma 2: Suppose q(·) is not nonincreasing in scarce and under this scenario, regulators should encourage θ, then there exist two user types θ and θ such that θ >θ such practices because more users will have access to the 1 2 1 2 and q(θ ) > q(θ ). Let q = q(θ ), q = q(θ ), p = p(q ) services and the total user surplus is higher. 1 2 1 1 2 2 1 1 and p = p(q ). Since the pricing function p(·) is strictly 2 2 VI. CONCLUSION decreasing,wehavep

θ . 1 2 Proof of Lemma 3: Since the utility function (1) has capacity C¯ (cid:44) [1 − F(θ )]w(0), the corresponding capacity 0 the strict and smooth single crossing differences properties consumption level. [13], we can apply the constraint simplification theorem (see If W(1)−C = 0, the slackness condition gives µ > 0. M Theorem 4.3 in [13]) to derive this lemma. Then q(·) is the point-wise maxima of (cid:101) Proof of Lemma 4: Since u(p,q;θ = 0) = −p ≤ 0 and u(0,1;θ = 0) = 0 (choosing the dummy service class), we Ψ(θ,q)=G(θ)·v(q)−µ·w(q). have V∗(0)=0. When θ ∈ [0,θ ), G(θ) < 0 and Ψ(θ,q) is decreasing in q. 0 Proof of Lemma 5: Define the set of users that choose Therefore q(·) = 1 for all θ ∈ [0,θ ). When θ > θ , we (cid:101) 0 0 service class q as calculate the partial derivative with respect to q to find the Φ(q)(cid:44){θ ∈Θ:q(θ)=q}. maxima. ∂Ψ(θ,q) We perform summation in a user-wise fashion instead of the =G(θ)v(cid:48)(q)−µ·w(cid:48)(q)=−µ·v(cid:48)(q)[h(q)−G(θ)/µ] ∂q class-wise one: (cid:90) (cid:0)(cid:90) f(θ)dθ(cid:1)w(q)dq =(cid:90) (cid:0)(cid:90) f(θ)w(q)dθ(cid:1)dq Since −µ·v(cid:48)(q) > 0, the sign of ∂Ψ∂(θq,q) is determined by h(q)−G(θ)/µ. Then under Assumption 3, we have Q Φ(q) Q Φ(q) (cid:90) (cid:90) (cid:40) = (cid:0) f(θ)w(q(θ))dθ(cid:1)dq 1, θ ∈[0,θˆ) q(·)= Q Φ(q) (cid:101) h−1(G(θ)/µ), θ ∈[θˆ,1]; (cid:90) 1 = w(q(θ))f(θ)dθ where the marginal user is θˆ=G−1(µ·h(1)). 0 By far we have completed the first step, i.e. to solve the candidate optimal control q(·) for the relaxed optimal control Proof of Lemma 6: The envelop theorem [12] together (cid:101) problem.Nextweprovethatq(·)isindeedtheoptimalcontrol with Lemma 3 have established the one-one correspondence (cid:101) of the relaxed optimal control problem. This is guaranteed by between the any service differentiation strategy and the cor- theArrowsufficiencytheorem[5]duetothattheHamiltonian responding user choice function q(·) together with an indirect is concave in V. As for the third step, it is easy to show that utilitylevelV(0).FromLemma4wehavesetV(0)=0.From (13) we obtain W(cid:48)(θ) = w(q(θ))f(θ) and the total capacity q(cid:101)(·) is nonincreasing under Assumption 3. Hence we have q∗(·)=q(·) and the proof of Theorem 1 is completed. consumption is W(1) by Lemma 5. The first order derivative (cid:101) of V(·) is given by the envelop theorem. The ISP’s profits is (cid:32) (cid:33) Proof of Theorem 2: For the variable capacity scenario, (cid:90) (cid:90) (cid:90) 1 p(q) f(θ)dθ dq = p(q(θ))f(θ)dθ the transversality conditions become Q Φ(q) 0  dλ ∂H =(cid:90) 1[θv(q(θ))−V(θ)]f(θ)dθ  dθ1 =−∂V =f(θ) dλ ∂H 0 2 =− =0 ThenPLroeomfmofaT6heisoraemsu1m:mIanriSzeactitoionnoIfV-thCe,wabeohvaevreeisnutlrtosd.uced λd1θ(1)=0∂,Wλ2(1)=−S(cid:48)(W(cid:102)(1)). thethreestepsoftacklingtheoptimalcontrolproblem(14).In Therefore, the costate variable associated with W(·) is a particular, we have proved that the candidate optimal control q(·) for the relaxed optimal control problem is the point-wise constantλ2(θ)≡−S(cid:48)(W(cid:102)(1)).InparallelwithTheorem1,the (cid:101) marginal cost S(cid:48)(W(cid:102)(1)) at the optimal capacity consumption maxima of (cid:18) (cid:19) level W(cid:102)(1) here plays the same role as the shadow price µ in 1−F(θ) Ψ(θ,q)(cid:44) θ− v(q)−µ·w(q). the fixed capacity scenario. Assuming S(·) is nondecreasing f(θ) and convex, the three steps in the proof of Theorem 1 can be The shadow price µ satisfies the slackness condition: readily applied in this scenario and we omit the details here. µ≥0, µ·[W(1)−C ]=0 M Proof of Corollary 1: Note that the indirect utility level If W(1)−C <0, then we have µ=0, q(·) is the point- isV∗(0)=0,thenCorollary1isstraightforwardfromLemma M (cid:101) wise maxima of 3. Proof of Corollary 2: By Theorem 1, If C

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