production and operations management -...

Option Contracts and CapacityManagement—Enabling Price Discrimination

under Demand Uncertainty

Fang Fang • Andrew WhinstonHigh Tech Management Department, College of Business Administration, California State University

at San Marcos, 333 S. Twin Oaks Valley, San Marcos, California 92096, USADepartment of IROM, McCombs School of Business, The University of Texas at Austin,

1 University Station, Austin, Texas 78703, [email protected] • [email protected]

We explore using an option contract as a price discrimination tool under demand uncertainty. In ourcapacity game model, a monopolistic supplier has to build capacity before observing the uncertain

demand. The demand is generated by two potential customers, who privately know their own types.The types could be either high or low, differing in willingness to pay for each unit of demand. Todiscriminate between the customer types, the supplier designs option contracts so that only the hightype will buy options in advance. The high type will do so because the options can hedge their risk ofdemand loss when capacity is tight. The supplier profits in three ways. First, the high type customers payhigher marginal prices on average. Second, the high type customers’ demand is satisfied as a firstpriority, guaranteeing allocation efficiency. Third, the supplier can observe the number of options beingpurchased and so determine customer types, improving capacity investment efficiency. We compare ourresults to those of classical second degree price discrimination. We show that our proposed frameworkguarantees the same level of supplier profit even when the supplier cannot discriminate between thecustomers by bundling products.

Key words: option contracts; capacity management; price discrimination; demand uncertainty; monopolyrevenue management

Submissions and Acceptance: Received June 2005; revision received November 2005; accepted December2005.

1. IntroductionReal options contracts are used in supply chain man-agement to protect risk-averse partners from potentialuncertainties, such as demand and material costchanges (see e.g., Huchzermeier and Cohen 1996). Inan incomplete contract setup, options can also im-prove contracting efficiency by solving the hold-upproblem (see e.g., Noldeke and Schmidt 1995). In thispaper, we explore using options as a price discrimi-nation tool. In the classic economics literature, pricediscrimination relies on the supplier’s ability to deter-mine customers’ different levels of willingness to payand to charge different prices. When customer typesare not observable, the supplier can offer a menu ofbundles for the customers to choose from. This prac-

tice is known as second degree price discrimination(Tirole, 1988).

When customer demand is stochastic, bundling isan inefficient discrimination tool. This is because thecustomers’ desired quantities vary according to theirrealized demand. In addition, when the capacity mustbe determined before the demand is fully revealed, thesupplier’s optimal capacity decision is critical to rev-enue management. Moreover, the supplier alwaysseeks ways to better utilize tight capacity.

In this paper, we propose using a new form ofoption contract to improve the supplier’s revenue. Insituations when capacity may be tight, customers havean incentive to hedge the risk of demand loss. Theincentive is higher for those with higher willingness to

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pay. To meet customers’ hedging demands, the sup-plier could provide option contracts where customershave the right but not the obligation to exercise. Exer-cising an option guarantees the availability of one unitof demand. If the option is priced so that only custom-ers with higher willingness to pay will buy it, thenthose customers will self-select to pay more than theircounterparts and so ensure execution of their demand.

To demonstrate the effectiveness of option con-tracts, we present a game theoretic model where onemonopolistic supplier or service provider (he) facestwo customers each with uncertain future demand.Each customer (she) has private information abouttheir willingness to pay for one unit of demand. Thesupplier has to build capacity before the customers’demand is realized. Since capacity may be insufficientfor the highest level of demand realization, customerssuffer from potential demand losses. When demandexceeds capacity, the supplier can only serve the de-mand randomly.

Option contracts can be adopted in the followingmanner. The supplier opens the option market to thecustomers before the capacity investment and uncer-tain demand realization. At that time, customers canpurchase options with unit price po. After observingcustomers’ option purchase decisions, the supplier in-vests in capacity. The customers then find out theiractual demand, observe the supplier’s capacity anddecide whether to exercise their options. Each cus-tomer pays a strike price, pe, for each unit optionexercised. The amount of demand protected by theoptions (referred to as the “option demand” in thecontext) will be satisfied as the first priority. The re-maining demand will be satisfied at a unit price p ifthere is leftover capacity.

Our proposed framework improves the supplier’srevenue in three ways. First, customers with a higherwillingness to pay will pay a higher unit price whendemand is tight, increasing the supplier’s overall rev-enue. Second, the customers’ option demands are sat-isfied as a first priority. The remaining demand will besatisfied only when there is extra capacity. Third, cus-tomers’ options purchases reveal their types. Thisknowledge allows the supplier to more efficiently ad-just capacity levels, better accommodating the poten-tial demand. The last two effects also improve thesupplier’s decision efficiency, leading to an enhancedoverall social welfare.

In order to successfully induce high type customersinto purchasing options, the supplier needs to con-vince the customers that capacity could be insufficient.If a supplier is able to change capacity after the optionsare purchased, however, it may undermine the cus-tomers’ initial incentives to purchase them. This isbecause customers know the supplier will want toguarantee enough capacity to meet the demand, so as

to maximize the revenue by serving as much customerdemand as possible. Based on knowing their owntypes and the supplier’s capacity cost, a rational cus-tomer can always conjecture the capacity level that thesupplier will invest in contingent on their counter-part’s type. They can therefore calculate the overallbenefit of purchasing the options based on these “ra-tionally expected” capacity levels. The supplier cannotmislead the customers. However, the capacity invest-ment levels could be different if the supplier adoptsdifferent option contracts (e.g., different po and pe).Hence, the option contract design is critical in ourframework.

The supplier can decide not to build sufficient ca-pacity to guarantee the execution of the entire optiondemand. The decision depends on what commitmentthe supplier makes in the option contracts when hefails to meet the option demand. If the supplier doesnot compensate unsatisfied customers, the customerswill have less incentive to purchase the options. Thisdisappoints the high type customers and reduces theirvaluation of the options. If the supplier promises toohigh a compensation, he has to always ensure to over-invest in capacity. This may be inefficient especiallywhen the capacity cost is high. In this paper, wesuggest the supplier offer an option buy-back price asa compensation mechanism which leaves the hightype customers indifferent about exercising their op-tions or selling them back to the supplier. We willshow that this buy-back price scheme reduces the hightype customers’ strategic decision of exercising theoptions and induces the supplier’s efficient capacityinvestment which maximizes ex ante social welfare.

Our option framework can improve revenue man-agement in many industries wherever demand uncer-tainty and information asymmetry exist. One potentialapplication is in network traffic management. Sincebusiness communication relies heavily upon emailsand video conferences, network congestion can causesevere economic losses. Companies are willing to paya premium more than what the regular users pay toensure important business emails being deliveredpromptly.

Another potential application of the option frame-work is in the ticket sale business. Many people wantto go to a concert or a game, but face the risk of notbeing able to attend ex post. They do not want to paythe full price for the tickets in advance because they donot want to waste money if they are unable to attend.However, if they wait too long, the tickets could besold out. The option contract is a good solution forthese people. Similar applications can be implementedin airline ticket management, hospital facility manage-ment, and the hotel reservations business. In hotelreservation business, customers who care more aboutgetting the hotel room can make reservations before

Fang and Whinston: Option Contracts and Capacity Management126 Production and Operations Management 16(1), pp. 125–137, © 2007 Production and Operations Management Society

arrival. It bears resemblance to the option framework.However, in practice, reservation is often free (i.e.,option price pe is zero). This paper provides an appro-priate analytical framework to determine the optionprice.

In the option framework, the supplier employs pricediscrimination by extending the customers’ decisionproblem into an intertemporal one. That is, the cus-tomers have to decide whether and how many optionsthey should purchase to hedge the future risk of de-mand loss before their demand realization and thesupplier’s capacity investment. To make the decision,they have to figure out the capacity level the suppliermay invest in and the possibility they will use theoptions. When they evaluate the options, rational cus-tomers also take into account the fact that their optionpurchase reveals their types to the supplier and affectsthe capacity level.

An alternative pricing scheme is the spot pricingscheme, which suggests that the price should be dy-namically adjusted according to congestion levels.Gupta et al. (1996 and 1999) suggests using priorityclasses with different spot prices to efficiently allocatenetwork resource. The idea is that the customers withhigher delay costs can choose to send their demand tothe network with lower expected delay time. Afeche(2004) suggests to add strategic delay in the queue tofurther discrimination customers and maximize therevenue of the supplier. However, under the spotpricing mechanism, customers decide whether to ex-ecute the demand when observing the spot price. Thesupplier cannot discover the customers’ type distribu-tion before adjusting the capacity.

Maskin and Riley (1984) have proposed a mecha-nism for price discrimination without demand uncer-tainty. In their mechanism, the monopolist decides theallocation rules and pricing mechanism according toeach customer’s reported willingness to pay. The cus-tomers then decide what to report. In a truth-tellingmechanism, every customer finds it optimal to reporttheir true private information. Deshpande andSchwartz (2002) extend the mechanism into a con-strained capacity setup. In this mechanism, non-linearpricing rules are adopted to guarantee incentive com-patibility. The fact that demand is uncertain in oursetup changes the favorite allocations for the twotypes of customers. In this sense, Maskin and Riley’smechanism cannot be directly applied to solve ourproblem. Boyaci and Ray (2006) discussed the impactof capacity costs on product differentiation in a prod-uct delivery model.

The literature on demand uncertainty and con-strained capacity is prodigious. As an example, Birge(2000) used option pricing theory to quantify the risksassociated with capacity planning under uncertain-ty.Ozer and Wei (2004) look at managing demand

uncertainty of non-perishable product with a capaci-tated inventory model. Sodhi (2004) compares manag-ing demand risks under a deterministic model to thatunder a stochastic model. Iyer et al. (2003) proposed ademand postponement model to handle demandsurges for perishable products. They justified thevalue of demand postponement, in which the suppliercan choose to satisfy only a fraction of the aggregatedemand when new information is incorporated. How-ever, they only examined demand surge on an aggre-gate level. In addition, demand will be postponedrandomly under demand surge. If the customers havedifferent postponement preferences, there is potentialfor designing a revelation mechanism to find custom-ers with lower delay costs. This would save the feespaid by suppliers to compensate for the postponeddemand. Our framework complements the postpone-ment mechanism by differentiating among customers’delay costs.

To illustrate the implementation of option contracts,we present a two-period game-theoretic model. Themodel has one monopolistic supplier and two poten-tial customers with private valuation of the service.Figure 1 shows the time line of events.

In the first period—the contracting period—the mo-nopolistic supplier announces the option price, po,and the strike price, pe, to maximize his expectedoverall profit. Then each customer decides the number ofoptions to purchase, Oi, according to their types.

Before the beginning of the second period—theconsumption period—the supplier observes the cus-tomers’ options purchases and decides on an opti-mal capacity investment, K. Afterwards, the cus-tomers demands, Di , are realized. Each customerdecides how many options they are going to exer-cise (Di

o) based on the observation of the aggregatedemand and the capacity level. The supplier satis-fies the option demand as a first priority. If Do � K,some of the options cannot be satisfied and thesupplier will compensate the customers. If Do � K,the extra capacity will be used to satisfy the remain-ing demand.

The rest of the paper is organized as follows.Section 2 presents the model and analyzes the equi-librium strategies of the supplier and two custom-ers. We compare the model outcome to two bench-mark cases. In one of the benchmarks, the suppliercannot distinguish the customer types at all. In theother, the supplier can determine each customer’stype and can charge different prices to each of them.In Section 3, we discuss the implications and exten-sions of our model. Section 4 concludes the paper.

2. The ModelA monopolistic supplier sells to two risk neutral cus-tomers (i � 1, 2). Each customer can be one of two

Fang and Whinston: Option Contracts and Capacity ManagementProduction and Operations Management 16(1), pp. 125–137, © 2007 Production and Operations Management Society 127

unknown types. A “high” type customer enjoyshigher marginal utility from the good than a “low”type customer. Specifically, customer i receives totalutility ui � vtiDi

e � mi if she is of type ti � {l, h}. vh � vl

represents the marginal value for each type, Die is

customer i’s demand being satisfied by the supplier,and mi is the total monetary transfer customer i pays tothe supplier.

Each customer knows her own type ti but does notknow for certain the other’s type. The supplier cannotobserve the customers’ types either. The common be-lief is that ti � h with probability � � (0, 1) and ti � lwith probability 1 � �. The realizations of t1 and t2 areindependent.

Each customer demand, Di, is uncertain and couldbe either DH (with probability �) or DL. The realizationof Di is independent of customers’ type realization, ti.We denote � � (D1, D2) as the demand vector and D� D1 � D2 as the aggregate demand. In addition, weassume that DH � 2DL. After Di is realized, eachcustomer decides how much of the demand should besubmitted to the supplier, which we denote as Di

s. Inthis model, we restrict Di

s � Di, implying that thecustomer cannot submit a demand higher than therealization of their actual demand. This restrictionmakes sense when the demand can be verified ex post.Taking the example of network traffic management,customer sends out files of certain sizes. A customercan increase the demand by extending the size of thefile. However, she cannot benefit from doing so. Fail-ure to impose this restriction introduces customers’strategic behavior when submitting their demand. Ca-chon and Lariviere (1999) analyzed this kind of stra-tegic behavior under different allocation rules. Theyshow that customers’ order inflation could be an equi-librium strategy and the supplier is worse-off due tothe concern of such strategic behavior. However, thisis not the focus of our paper.

To serve the customer demand, the supplier has to

invest a certain level of capacity K before the demand� is realized. The marginal cost of capacity investmentis c0. Observing the customers’ submitted demand �s

�� (D1

s , D2s), the supplier decides how much demand to

satisfy for each customer, �e �� (D1

e , D2e), to maximize

his expected revenue. The total amount of satisfieddemand, De �

� D1e � D2

e is constrained by the supplier’scapacity K. In addition, we assume that c0 � min{(2�� 2)vl, �2vh} so that the capacity cost is high enoughthat the supplier will never choose to simply invest infull capacity. We also assume that c0 � (2� � �2)vh.This condition guarantees that the high type custom-ers’ willingness to pay is high enough so that theefficient capacity level is higher if both customers arehigh type than that if at least one customer is low type.c0 � (2� � �2)vh is imposed to guarantee that thechance that a customer is of high type is not bigenough for the supplier to only serve the high typecustomers. Assuming the supplier is risk neutral andthere is no cost for executing the customers’ demand,the potential supplier profit � is calculated as

� � m1 � m2 � c0 K.

We propose that the supplier can use option con-tracts to manage congestion (i.e., in the event that D� K). When capacity is insufficient to meet theaggregate demand, customers can choose to exercisetheir options, guaranteeing that their demand beexecuted. One unit of the option contract guaranteesthe customer one unit of satisfied demand regard-less of congestion. To implement price discrimina-tion, the options are priced such that only high typecustomers will buy the options because they suffermore from demand loss than low types. By usingthe options, high type customers can avoid demandloss but may pay a higher price. In addition, lowtype customers’ demand will be executed at a lowerpriority, increasing their potential demand loss. Thesupplier benefits from using the option contracts

Figure 1 The time line of the base model.


since he can, in effect, charge the high types a higherfee and is able to adjust capacity after observingcustomers’ actual types. From a social optimal per-spective, the allocation efficiency is always guaran-teed since those demands with higher marginalvalue always receive first priority execution.

We use two benchmark models for comparison withthe options framework. In the first benchmark model,the supplier does not have the ability to distinguishcustomers’ types. He invests in capacity before thedemand uncertainty is resolved and can only charge alinear price, p, for each unit of executed demand. Thesecond benchmark model is a full information case, inwhich the supplier can observe the customers’ typesbefore the capacity investment. The supplier chargesdifferent prices ph and pl to the high types and lowtypes, respectively. In this case, allocation efficiency isguaranteed since the supplier always satisfies demandfrom high type customers first and uses the remainingcapacity (if there is any) to serve the low type custom-ers. In addition, the supplier can set the prices thatleave no surplus to both types of customers and thecapacity investment will be efficient. We will examineboth the supplier’s expected profit E� and the effi-ciency W—defined as the sum of the supplier’s ex-pected profit and the expected utilities of both custom-ers—in these two benchmark models. We thencompare them to our options framework.

2.1. Benchmark I—No Discrimination CaseIn this section, we examine the case where the suppliercan only charge a single linear price p to the customersregardless of their types. From the definition of De, wehave that De � min{K, Ds}. Each customer is chargedmi � pDi

e and the supplier’s profit is �(p, K) � pDe

� c0K. In this case, we have

Dis� p � � Di if p � vti

0 otherwise.

By charging a price higher than vl, the supplier onlyserves the high type customers. The supplier can enjoya higher marginal profit from each unit of demand,but the supplier loses business if a customer is of lowtype. Proposition 2.1 states that the supplier shouldserve both types of customers by charging a price p� vl when c0 � (2� � �2)vh. It also determines theoptimal capacity decision KND, the supplier’s expectedprofit E�ND, and the overall efficiency WND �

� E(�ND

� u1ND � u2

ND). The superscript ND represents the nodiscrimination case.

Proposition 2.1. Assume the supplier can only chargea unit price, p. The optimal price pND � vl and the capacityKND � 2DL. Both types of customers will submit theirdemand and the supplier’s profit is

E�ND � �vl � c0 2DL.

The overall efficiency is

WND � ��vh � �1 � �vl � c0 2DL.

Note. See appendix for all the proofs.Proposition 2.1 provides a benchmark outcome

where the supplier must charge all customers a singleunit price after the capacity is invested. In this case,the capacity level is barely enough for the minimallevel of aggregate demand, that is, K* � 2DL � inf{D}.The low type customers are left no surplus and thehigh type customers can make strictly positive surpluswhich is proportional to the difference in the marginalutility of these two types, that is vh � vl.

2.2. Benchmark II—Full Information CaseWe use superscript FI to indicate the “full informa-tion” case. If the supplier can distinguish the types ofthe customers before the capacity is invested andcharge different prices according to types, the optimalprices will be pFI(t � h) � vh and pFI(t � l ) � vl. Thesupplier hence leaves no surplus to both types ofcustomers. When D � K, the supplier will satisfy thehigh type customers’ demand first since he can get ahigher margin.

The supplier’s optimal capacity decision KFI is de-termined based on the customers’ types (t1, t2). Due tosymmetry, KFI(h, l ) � KFI(l, h). Lemma 2.2 specifies thisoptimal capacity level.

Lemma 2.2. In the full information benchmark case, thecapacity investment decision is made contingent on theactual types of the two customers.

KFI�t1 , t2 � � DH � DL if t1 � t2 � h2DL otherwise.

Since the supplier charges prices which leave bothtypes of customers zero surplus, the optimal capacitylevel maximizes both the supplier’s expected profitE�FI and the overall efficiency WFI. Hence, KFI repre-sents the efficient capacity level.

Proposition 2.3. When the supplier can observe cus-tomer types before making a capacity investment andcharges different prices accordingly, the expected supplierprofit equals the overall efficiency. That is, E�FI � WFI.Comparing with E�ND and WND yields

� E�FI � E�ND � �1 � �2 � �3

WFI � WND � �1 � �3

where:

(1) �1 � 2�(1 � �)�(DH � DL)(vh � vl),(2) �2 � (vh � vl)�2DL and(3) �3 � �2((2� � �2)vh � c0)(DH � DL).

It can be shown that �1 , �2 , and �3 are all strictlypositive. The results that E�FI � E�ND and WFI

� WND always hold. The supplier’s profit gain comes


in three parts. �1 represents the profit gain by priori-tizing the customers’ demand. �2 represents the profitgain from charging different prices according to types.�3 , which is proportional to (DH � DL) � KFI(h, h)� KND, represents the profit gain from the ability tochange capacity after actual realization of the cus-tomer types. The efficiency gain comes in two partsbecause the supplier’s ability to charge the high typecustomers a higher price only affects the monetary trans-fer among parties; it does not change overall efficiency.

2.3. Option-Capacity GameIn this section, we discuss the framework in which thesupplier offers buyers an option contract to hedgetheir risk of demand loss. The customers choose thenumber of options to buy before the capacity is built.The supplier will then observe the number of optionspurchased by each customer and decide how muchcapacity he should build to meet the demand. Afterthe capacity is built and the demand is realized, cus-tomers observe the supplier’s capacity and the aggre-gate demand and decide how many options to exer-cise, if they have any. The number of options acustomer chooses to exercise is called the option de-mand and is denoted as Di

o. We assume that the cus-tomer cannot exercise more options than the actualdemand, that is Di

o � Di.To successfully discriminate the customers, the sup-

plier must set the option contract parameters (po, pe) insuch a way that only high type customers will buy them.The supplier also sets a unit price p � vl for regulardemand. If the aggregate option demand, Do �

� D1o � D2

o,exceeds the capacity K, the supplier needs to buy backsome of the options. The option buy back price has to behigh enough so that the high type customers are willingto sell it back. Meanwhile, it can not be too high to makethe customers want to sell all the options back instead ofexecuting them. Thus, the buy-back price, pb, shouldequal vh � pe, the marginal benefit the high type custom-ers get from the demand being executed.

According to the time line in Figure 1, the strategicinteractions among the supplier and the two custom-ers can be described in a three-stage game. In the firststage, the supplier announces the option contract pa-rameters, po and pe. The customers simultaneouslyde-cide the number of options, Oi, to buy based on theirown types. We are interested in the equilibrium caseswhere O*i (t � l ) � 0 and O*i (t � h) � 0. With ourassumption of the customer demand, the high typecustomers have actually two choices: whether to buyOi � DL for a minimal hedge or to buy Oi � DH for amaximal hedge.

The supplier observes (t1, t2) through the sale of theoptions and decides the capacity, K(t1, t2), to maximizehis expected payoff in the following period, E�(K).

After the demand is realized, the customers simulta-

neously decide how much demand to submit to thesupplier, denoted as Di

s, and how much of Dis is submit-

ted as option demand, Dio. When the regular price p � vl,

Dis � Di hold for both types of customers. The supplier

gathers the total demand (Do, Ds) and decides how toallocate the constrained capacity with the priority of theoption demand. The total charge to a customer will be

mi � po Oi � pe Dio � vl�Di

e � Dio� � vh�Di

o � Die�,

where the last term is the customer’s expected compen-sation if the option demand is not executed. For the lowtype customers, since they won’t buy any option inour equilibrium, we can simplify the total charge asmi(t � l ) � vlDi

e. The customer’s overall utility:

ui � � (vh � pe)Dio � (vh � vl)(Di

e � Dio)� � poOi if t � h

0 if t � l

In the following subsections, we use backward in-duction to resolve the three-stage game.

2.3.1. The Consumption Period. In this period,each customer observes their demand, which couldbe either DH or DL. After observing the aggregatedemand, D, and the capacity, K, the two customers si-multaneously decide how many options to exercise.

Denote � �� (O1, O2). There are six configurations that

need to be discussed: � � (DH, DH), (DH, 0), (DL, 0), (DL,DL), (0, 0) and (DH, DL). Due to symmetry, we do notneed to analyze the cases of (DL, DH), (0, DH) and (0, DL).On the equilibrium path, only the high type customerswill buy options. Therefore, supplier will infer that bothcustomers are of high type when � equals (DH, DH), (DL,DL), or (DH, DL). Configurations (DH, 0) and (DL, 0)indicate that only one customer is high type, and con-figuration (0, 0) indicates both customers are of low type.

For the low type customers, it is straightforwardthat Di

o � 0 since they don’t have any options. For thehigh types, Di

o is decided based on the following max-imization problem:

maxDi

o�minOi ,Di�

�vh � peDio � �vh � vl�Di � Di

o�

s.t. � � min�1,(K � D�i

o � Dio)�

D � D�io � Di

o �where D�i

o is the option demand submitted by theother customer. D�i

o � 0 if that other customer is of lowtype. � indicates the probability that regular demand issatisfied. � � 1 when D � K and � � 0 when Do � K.

Lemma 2.4. Denoting �o as the solution set of the abovemaximization problem, we have �o � {0, min{Di, Oi}}.

Lemma 2.4 states that a high type customer willeither exercise all the available options to execute thedemand or not exercise the options at all. When pe

increases, the customer pays more to exercise the op-tions and hence is increasingly reluctant to use them.


Tables 1 through 6 show the solutions of Dio as

functions of different realizations of �, �, pe and K.The results in Tables 1 through 6 show that the

amount of options a high type customer will exercisedecreases as the option strike price pe increases. More-over, we can see that when pe vh � (vh � vl)(K/2DH),no option will be exercised in any configuration of � anddemand realization �. This is because that the optionsare too expensive to exercise. Therefore, the options haveno value. In the following discussion, we focus on thecases where the strike price pe � vh � (vh � vl)(K/DH).

2.3.2. Capacity Investment Game. In the abovesection, we analyze both customers’ equilibrium de-cisions on Di

o. This decision is contingent on thenumber of options both customers have purchased�, the realized demand �, the option strike price pe,and the capacity K. In this section, we analyze thesupplier’s optimal capacity decision K*, which ismade before the actual demand is realized. Theoptimal capacity decision K* is hence based on thesupplier’s observation of option purchase by thecustomers � and the strike price pe, while expectingthe possible future demand realization � and thecustomers’ reaction to Di

o, i � 1, 2.

Once the customers purchase the options, the sup-plier will only look at the future revenue excluding therevenue from option purchase, po(O1 � O2), to decidethe optimal capacity. When K* increases, the chance ofcongestion decreases. The supplier’s revenue comesmore from serving the regular demand. When pe � vl,it is more profitable for the supplier to increase capac-ity. However, if pe � vl, the supplier tends to inducethe customers into exercising their options by restrict-ing the capacity K. This may create problems when K� Do because the supplier fails to satisfy the optiondemand as they promised in the option contract. Hemust compensate customers whose option demandscannot be satisfied.

Customers are more willing to exercise their optionsas pe decreases. Foreseeing this, the supplier tends toincrease the capacity level to guarantee all the exer-cised options can be satisfied. This will minimize theexpected compensation. Hence, we predict capacity K*decreases as pe increases.

Proposition 2.5. In a subgame perfect equilibrium, thesupplier’s optimal capacity decision is as follows:

(1) When � � (DH, DH), the optimal capacity

K* � �DH � DL if vh � (vh � vl)

DL

DH pe � vh

vh � pe

vh � vl � 2DH if vh � (vh � vl)DH � DL

2DH pe � vh � (vh � vl)DL

DH

2DL if pe � vh � (vh � vl)DH � DL

2DH

(2) When � � (DH, DL), the optimal capacity

K* � � DH � DL if vh � (vh � vl)2DL

DH � DL pe � vh

vh � pe

vh � vl � (DH � DL) if vl pe � vh � (vh � vl)2DL

DH � DL .

2DL if pe � vl

(3) When � � {(DL, DL), (DH, 0), (DL, 0), (0, 0)}, the optimal capacity K* � 2DL for all pe � vh.

Table 1 Option Demand When � � (DH, DH)

� � (DH, DH) � � (DH, DH) (DH, D L) (D L, DH) (D L, D L)

vh � (vh � vL) min {K/2DH, 1} pe � vl D1o � 0 D1

o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0

vh � (vh � vl) min {K/(DH � DL), 1} pe � vh � (vh � vl) min {K/2DH, 1} D1o � DH D1

o � 0 D1o � 0 D1

o � 0D2

o � DH D2o � 0 D2

o � 0 D2o � 0

vh � (vh � vl) min {K/2DL, 1} pe � vh � (vh � vl) min {K/(DH � DL), 1} D1o � DH D1

o � DH D1o � DL D1

o � 0D2


o � DH D2o � 0

pe � vh � (vh � vl) min {K/2DL, 1} D1o � DH D1


o � DL

D2o � DH D2

o � DL D2o � DH D2

o � DL


Figure 2 summarizes the optimal capacity level K*as a function of pe in all six configurations of �. Thecapacity is at least the minimal level of the aggregatedemand 2DL. It is greater than 2DL only when bothcustomers have purchased the options and at least one

Table 2 Option Demand When � � (DH, D L)

� � (DH, D L) � � (DH, DH) (DH, D L) (D L, DH) (D L, D L)

vh � (vh � vl) min {K/2DH, 1} pe � vh D1o � 0 D1

o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0


o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0

vh � (vh � vl) min {K/2DL,1} pe � vh � (vh � vl) min {K/(DH � DL), 1} D1o � DH D1


o � 0D2

o � 0 D2o � DL D2

o � 0 D2o � 0

vl pe � vh � (vh � vl) min {K/2DL, 1} D1o � DH D1


o � DL

D2o � 0 D2

o � DL D2o � 0 D2

o � 0pe � vl D1

o � DH D1o � DH D1

o � DL D1o � DL

D2o � DL D2

o � DL D2o � DL D2

o � DL

Table 3 Option Demand When � � (DH, 0)

� � (DH, 0) � � (DH, DH) (DH, DL) (DL, DH) (DL, DL)

vh � (vh � vl) min {K/2DH, 1} pe � vh D1o � 0 D1

o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0


o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0

vl pe � vh � (vh � vl) min {K/(DH � DL), 1} D1o � DH D1


o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0

pe � vl D1o � DH D1


o � DL

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0

Table 4 Option Demand When � � (DL, DL)

� � (DL, DL) � � (DH, DH) (DH, DL) (DL, DH) (DL, DL)

vh � (vh � vl) min {(K � 2(DH � DL))/(DH � 2(DH � DL)), 1} pe � vh D1o � 0 D1

o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0

vh � (vh � vl) min {K/(DH � DL), 1} pe � vh � (vh � vl) min{(K � 2(DH � DL))/(DH � 2(DH � DL)), 1}

D1o � DL D1

o � 0 D1o � 0 D1

o � 0D2

o � DL D2o � 0 D2

o � 0 D2o � 0

vl pe � vh � (vh � vl) min {K/(DH � DL), 1} D1o � DL D1


o � 0D2


o � DL D2o � 0

pe � vl D1o � DL D1


o � DL

D2o � DL D2


o � DL

Table 5 Option Demand When � � (DL, 0)

� � (DL, 0) � � (DH, DH) (DH, DL) (DL, DH) (DL, DL)

vh � (vh � vl) min {( 1

2K-DL � DH/(2DH-DL), 1} pe � vh D1

o � 0 D1o � 0 D1

o � 0 D1o � 0

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0vh � (vh � vl) min {K/(DH � DL), 1} pe � vh � (vh � vl) min

{ (12

K-DL � DH)/(2DH(2DH � DL)), 1}D1

o � DL D1o � 0 D1

o � 0 D1o � 0

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0vh � (vh � vl) min {(KDL � DHDH � DLDL)/((DH � DL)DH), 1}

pe � vh � (vh � vl) min {K/(DH � DL), 1}D1

o � DL D1o � 0 D1

o � DL D1o � 0

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0vl pe � vh � (vh � vl) min {(KDL � DHDH � DLDL)/(DH � DL)DH, 1} D1


o � DL D1o � DL

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0pe � vl D1


o � DL D1o � DL

D2o � 0 D2

o � 0 D2o � 0 D2

o � 0

Table 6 Optimal Demand When � � (0, 0)

� � (0, 0) � � (DH, DH) (DH, DL) (DL, DH) (DL, DL)

For all pe � vl D1o � 0 D1

o � 0 D1o � 0 D1

o � 0D2

o � 0 D2o � 0 D2

o � 0 D2o � 0


of them bought DH options. In all other cases, we haveO1 � O2 � 2DL and hence the total number of optionsdemanded, which is smaller than the total number ofoptions purchased, must be smaller than 2DL. As aresult, the supplier will remain unconcerned about thecompensation even when he sets the capacity K � 2DL.

However, when both have purchased options and atleast one bought DH, the supplier has to worry aboutthe situation that both customers exercise all theiroptions and the capacity may be insufficient for theoption demand. If he stays with the capacity level K� 2DL, the probability of providing compensation willbe as high as 2� � �2 when pe is low enough. So ourassumption that (2� � �2)vh � c0 suggests that thesupplier should increase the capacity from 2DL toavoid the situation. The assumption that �2vh � c0suggests that the supplier cannot be better off byincreasing the capacity from DH � DL to avoid thepossible compensation when � � (DH, DH). The opti-mal capacity is between 2DL and DH � DL.

2.3.3. Optimal Option Prices. Given the optimaldecisions of K* and (D1

o, D2o), we can now analyze the

first stage of the option-capacity game to derive theoptimal option contract (po, pe) and the supplier’s ex-pected profit by using the option contracts. All thedecisions analyzed in Sections 2.3.1 and 2.3.2 are con-tingent on both the option strike price pe and thecustomer’s option purchase �. In addition, the cus-tomers decide � based on their own types (t1, t2) and

how the supplier prices the options. The fundamentalquestion to be resolved is: what is the optimal po andpe that induces the customer to purchase the rightnumber of options and maximizes supplier profit?

In this period, the supplier announces the optionprices (po, pe). Then the customers, who do not knoweach other’s type, simultaneously submit their optionpurchase demand Oi to the supplier. The option ispriced such that the low type customers will not feelprofitable purchasing the options but the high typecustomers will.

The customer’s incentive of buying options is dif-ferent when the other customer’s option purchase de-cision, O�i , varies. If we denote the expected value ofan option as fo(Oi, O�i), fo can be calculated as follows:

fo �Oi , O�i �1Oi

Euih�Oi , O�i � Eui

h�0, O�i �

where Euih(Oi, O�i) represents the expected utility a

high type agent gets after purchasing Oi units of op-tions while the other customer has purchased O�i

units of options.

Lemma 2.6. fo(Oi, DH) fo(Oi, DL) fo(Oi, 0).

Lemma 2.6 states the fact that the option is morevaluable to a high type customer if the other customerbuys more options. This is so because the two custom-ers will compete for the limited capacity resource in

Figure 2 Optimal capacity levels.


the consumption period. Buying more options pro-tects them in the competition. Consequently, if thesupplier can induce one customer to buy DH units ofoptions, the other customer would want to pay morefor the options if she is a high type. This implies thatthe supplier prefers the equilibrium where the hightype customers choose the maximal hedging strategy,as shown in Proposition 2.7.

Proposition 2.7. The supplier maximizes his expectedprofit when p*e � vh � (vh � vl)((DH � DL)/2DH) and p*o� �fo(D

H, DH) � (1 � �)fo(DH, 0). In such an equilibrium,

a high type customer will choose a maximal hedging strat-egy Oi � DH. The expected supplier profit

E�* � �vl � c0 2DL � 2��vh � vl�DH � DL

� �2��2� � �2vh � c0 �DH � DL

and the equilibrium capacity investment

K* � � DH � DL if O1 � O2 � DH

2DL otherwise.

The supplier’s capacity investment K* is the same asthe one in the full information benchmark case (seeSection 2.2) given the belief that O*i (t � h) � DH andO*i (t � l ) � 0. By using option contracts, the suppliercan make a contingent capacity investment based onthe customer’s option purchase decision �, which re-veals the customer types (t1, t2) in equilibrium. Thisflexibility improves the capacity investment decision.The option compensation pb � vh � pe is important inachieving the efficient capacity level K* � KFI. Underthe options framework, the supplier’s incentive to in-crease the capacity when both customers are of hightype is to satisfy as many options as possible whilereducing the compensation. That is, the marginal ben-efit the supplier gets is pe � pb � vh when K � Do. Thisincentive is well-aligned with the full informationbenchmark case where the incentive of increasing ca-pacity is to increase the ability of serving high typedemand, which yields a marginal profit pFI(t � h) � vh.

We can rewrite the supplier’s expected profit as:

E�* � E�ND � �1 � �3 � �4

where �1 and �3 are defined in Proposition 2.3. �1represents the gain from prioritizing the high typecustomer’s demand over the low type one’s. �3 refersto the gain achieved when the supplier increases ca-pacity after observing two high type customers. �4 �

�

2��2(vh � vl)(DH � DL) � 0. �(vh � vl)(DH � DL) is theexpected loss a high type customer suffers if shedoesn’t buy options but competing for capacity withanother high type customer who has options. �4 is 2�2

times this expected loss representing the supplier’sexpected gain from the competition between two hightype customers.

Comparing E�* to E�FI, we can characterize sup-plier’s profit loss when he cannot distinguish the hightype customer and extract full surplus from them asfollows:

E�FI � E�* � �2 � �4 � 2��Euih�0, DH

� �1 � � Euih�0, 0�

{�Euih(0, DH) � (1 � �)Eui

h(0, 0)} is the reserve utility ahigh type customer has when she does not buy op-tions. This reserve utility is also known as the Infor-mation Rent in the price discrimination literature, rep-resenting the cost the supplier pays to induce a hightype customer to reveal her type. The expected infor-mation rent the supplier pays is exactly �2 � �4 ,which increases as probability � increases.

Proposition 2.8. W* � WFI.

The capacity investment is efficient in equilibriumand high type customers will always be served as thefirst priority. Hence, it is not surprising that the optioncontracts can achieve the same efficiency level as thefull information benchmark case. Proposition 2.8 jus-tifies the optimality of our proposed option frame-work.

3. Discussion and Extensions3.1. Using Options for Price DiscriminationIn the literature of price discrimination under unob-servable types, the supplier’s ability to employ a non-linear pricing scheme (e.g. quantity discount and bun-dling) is critical to his profit. In this paper, we showthat with properly priced options, the supplier canachieve the same profit level with a simple linearpricing function using an option framework. Ourframework also works when the customer’s demandis uncertain.

In our framework, a customer faces two purchasedecisions. She chooses the number of options to pur-chase before the demand realization and the amountof demand (including the numbers of regular demandand option demand) to request afterwards. In bothdecisions, she has full flexibility to choose the quanti-ties. A linear pricing scheme is applied in all thepurchases. The customers prefer such flexibility whenthey suffer from demand uncertainty.

To discriminate the customers, we pay attention tothe case where only high type customers will buy theoptions. A high type customer’s total charge in equi-librium can be divided into two parts. A fixed pay-ment poD

H is charged ex ante regardless of her actualdemand realization to guarantee the priority of herdemand execution. In addition, she pays a contingentpayment based on her actual demand realization. Ad-justing the option price po and pe, the supplier essen-


tially changes the ratio between the ex ante and expost payments to affect the high type customers hedg-ing incentive and exploit their willingness to pay forthe demand.

The option framework helps the supplier to conductprice discrimination when the customer’s demand isuncertain. When there is no demand uncertainty (i.e.,DH � DL), the capacity will always be enough for theaggregate demand under our assumption. Hence, theoption has no value and the supplier can not discrim-inate among the customers.

Our assumption of the supplier’s marginal capacitycost is critical to derive the result. The discriminationframework is built based on the high type customer’sconcern of potential demand loss. If the capacity costis too small, the customers can infer that the supplierwill always build enough capacity for the demand andwon’t pay money ex ante to hedge the potential de-mand loss. The discrimination is not implementable inthis case. If the capacity is too large, the supplier willfind it profitable to charge a high regular price (e.g. vh)to exploit the high type customers’ surplus only.

3.2. Multiple Agents and Multiple TypesIn this paper, we use a parsimonious model with onesupplier and two customers to illustrate price discrim-ination. The model can be extended to a case wheremultiple customers with two possible types. In such amodel, we could still design the option contracts sothat only those high type customers will buy them.The major difference would be a more complicatedaggregate demand pattern. In a symmetric equilib-rium where all high type customers adopt the samestrategy, the supplier can separate the customers intotwo groups and treat them as two representativeagents. The same analysis can then be applied to fig-ure out the optimal option contract. The efficiencylevel of the second degree price discrimination canstill be achieved.

The model gets further complicated with multipletype customers. The supplier could design multipleoption contracts with different combinations of strikeprices pe and compensation pb for each type. The cus-tomers could then self-select the options and decidewhen to exercise them. The demand should then beprioritized according to the supplier’s marginal pun-ishment of not fulfilling the demand (i.e., pb � pe). Asa result, different demand associated with differentoption contracts are categorized into different prioritylevels. Allocation efficiency can be achieved if thecustomers with higher willingness to pay will alwaysbuy the option contracts with higher priority (charac-terized by pb � pe). The challenge is how to price theoptions to induce the customers to purchase the rightoptions to reveal their types.

3.3. Spot Exchange Options MarketIn our setup, customers purchase options to hedgetheir demand risk. After their demand is realized, theycan decide how many of their options to exercise. Inthis setup, a customer cannot buy additional optionsfrom other customer ex post. This raises a question:what if they can exchange their options after observ-ing their demand?

On the one hand, allowing to exchange optionsleads to more efficient option utilization. If a customeris able to sell her extra options after the demand isrealized, she is more willing to buy options ex ante.Thus, the ex post exchange encourages the optionpurchase ex ante. On the other hand, the customers’incentive for a maximal hedge decreases with the pos-sibility of option exchange. This is because she mayfind someone who will sell options to her if her de-mand is high. Between this conflicting incentives, it isnot clear which incentive is stronger in general.

However, if we assume that customer types maychange ex post, then the existence of an option ex-change market helps in some cases. A detailed analy-sis of when a spot exchange market helps increase amonopolist’s profit can be found in Geng, Wu, andWhinston (2007). Moreover, ex post exchange reducesthe customer’s ex post risk of purchasing options.Hence, the existence of an ex post exchange marketalways improves the option purchase incentive if thecustomers are risk-averse.

4. ConclusionIn this paper, we propose an option framework thatallows a monopolistic supplier to conduct price dis-crimination among customers, thereby maximizinghis expected revenue under demand uncertainty and acapacity constraint. Our analysis shows that optioncontracts can benefit the supplier because the hightype customers will pay more for hedging potentialdemand loss. The supplier gains the additional benefitof being able to adjust capacity according to his ob-servation of customer types.

We also analyze the strategic interactions among thesupplier and customers. We show that, in equilibrium,the efficient capacity level can be induced by setting acompensation price which leaves the high type cus-tomers indifferent about whether to exercise their op-tions or to ask for compensation. Overall efficiency isguaranteed and the supplier and the high type cus-tomers share the efficiency gain from the efficientcapacity investment.

Our proposed structure replicates the classical pricediscrimination outcome where the low type customersdo not gain surplus and the high type customers enjoyan information rent. Our proposed structure can easilybe adopted in situations where the supplier is not


allowed to sell a bundled product with fixed quantityand situations where the actual demand and capacityis not contractible. Our framework has significant rev-enue management implications for various industrialapplications such as network capacity management,airline ticket reservation, and telephone and electricityproviders.

AcknowledgmentThis work was funded in part by NSF grant No. IIS-0219825.

Appendix

Proof of Proposition 2.1Proof. If p � vl, both customers submit all their demand

to the supplier,

Dis � Di � � DH with pr � �

DL with pr � 1 � �.

The supplier’s expected profit

E� � p��2 min K, 2DH� � 2��1 � � min K, DH � DL

� �1 � �2 min K, 2DL� � c0K.

Maximizing the profit under the condition p � vl, we havep* � vl and K* � 2DL. The supplier’s expected profit is E�(p� vl) � (vl � c0)2DL. Customer i’s expected utility is ui � (vi

� vl)DL.If p � (vl, vh], only high type customers will submit the

demand. Dis � Di when ti � h and Di

s � 0 for ti � l.When vl � p � vh, the supplier’s expected profit

E�� p � �2p��2 min K, 2DH� � 2��1 � �� min K, DH � DL�

� �1 � �2 min K, 2DL� � 2��1 � �p�� min K, DH�

� �1 � � min K, DL� � c0K

Maximizing the expected profit and applying the assump-tion (2� � �2)vh � c0, we have p* � vh and K* � 0. Thereby,E�*(p � vh) � 0 and it is never profitable to serve high typecustomers only.

In summary, we conclude that the supplier’s best strategyis always to set pND � vl to serve both types of customers.The optimal capacity will be KND � 2DL. The expected profitE�ND � (vl � c0)2DL and the overall efficiency:

WND � E�ND � �EuiND�ti � h � �1 � � Eui

ND�ti � l

� ��vh � �1 � �vl � c0 2DL.�

Proof of Lemma 2.2Proof. The optimal capacity KFI is made contingent on the

customer types � � (t1, t2). When � � (h, h), the supplier’sexpected profit

E�FI�K, � � vh��2 min K, 2DH� � 2��1 � � min K, DH

� DL� � �1 � �2 min K, 2DL� � c0K

which is maximized when K(h, h) � DH � DL from theassumption �vh � c0 � (2� � �2)vh.

When � � (l, l ), similarly, we can have KFI(l, l ) � 2DL,since (2� � �2)vl � c0 � vl.

When � � (h, l ) or (l, h) and K � DH, the supplier’sexpected profit

E�FI�K, �h, l � �2�vhDH � vl min K � DH, DH� � ��1

� ��vhDH � vl min K � DH, DL� � ��1 � ��vhDL � vl

min K � DL, DH� � �1 � �2�vhDL � vl min K � DL, DL�

� c0K

which is maximized when K* � 2DL. It can also be shownthat K � DH cannot be optimal. Therefore, KFI(h, l ) � KFI(l,h) � 2DL. �

Proof of Proposition 2.3Proof. For the supplier, the probability that both custom-

ers are high types is �2. The expected profit

E��h, h � �vl � c0 2DL � �vh � vl2DL

� ��2� � �2vh � c0 �DH � DL.

With probability 2�(1 � �), one customer is of high typeand the other is of low type. The expected profit E�(h, l )� (vl � c0)2DL � (vh � vl)(�DH � (1 � �)DL). With proba-bility (1 � �)2, both customers are of low type. The expectedprofit E�(l, l ) � (vl � c0)2DL. Since ui(t � h) � ui(t � l ) � 0,

WFI � E�FI � �2E��h, h � 2��1 � � E��h, l

� �1 � �2E��l, l � �vl � c0 2DL � �vh � vl2��1 � �

� �DH � DL � �vh � vl�2DL � �2��2� � �2vh � c0

� �DH � DL

compared to E�ND � (vl � c0)2DL and WND � (�vh � (1� �)vl � c0)2DL, we can easily conclude that

� E�FI � E�ND � �1 � �2 � �3

WFI � WND � �1 � �3

�

Sketch Proof of Lemma 2.4Proof. The proof is straightforward since we can show

that the second order condition of the above objective func-tion is non-negative. �

Sketch Proof of Proposition 2.5Proof. Applying the outcomes from Table 1–6, we can

derive the optimal capacity directly. �

Sketch Proof of Lemma 2.6Proof. From the result of(1) When pe VI � (VI � VII)(DL/DH): K* � 2DL for all

configurations of �. No options will be exercised forrealized �. Hence, the option has no value, it isstraightforward to conclude that p*o � 0;

(2) When VI � (VI � VII)(DH/(2DH � DL)) � pe � VI

� (VI � VII)(DL/DH): if the agent has bought Oi � DL,she will never exercise the options no matter what theother type of customer is. Hence, she would not payfor any positive po to purchase the option;

(3) When VI � (VI � VII)(2DL/(DH � DL)) � pe � VI � (VI


� VII)(DH/(2DH � DL)): if the agent has bought Oi

� DL, she will only exercise it when � � (DH, DH) andD�i

o � DH. Hence if the other agent is of type I, she isalways better off by purchasing O�i � DH than O�i

� DL. However, given O�i � DH, agent i will neverexercise her options and the value of the options is 0.If the other agent is of type II, the agent’s expectedutility from the options are(a) Oi � DH: ui(D

H) � �2(VI � pe)DH � (1 � �2)DL

� poDH;

(b) Oi � DL: ui(DL) � �2((VI � pe)D

L � (VI � VII)(DLDH/(2DH � DL))) � (1 � �2)DL � poD

L; and(c) Oi � 0: ui(0) � (VI � VII)DL

We can show that (ui(DH) � ui(0) � poD

H)/(ui(DH) � ui(0)

� poDL) � (DH/DL), meaning that the agent is better off by

purchasing Oi � DH.Following the same way, we can show that DL is not the

optimal choice when pe � VII. �

Sketch Proof of Proposition 2.7We need to discuss the profit according to the different pe

segments:(1) pe VI � (VI � VII)(DL/DH): no option is exercised, po

� 0 and E� � (VII � c0)2DL;(2) VI � (VI � VII)((DH � DL)/2DH) � pe � VI � (VI

� VII)(DL/DH). K*(DH, DH) � ((VI � pe)/(VI � VII))2DH.

E� � E� � 2�po DH

� �2�((2� � �2)VII � c0)VI � pe

VI � VII 2DH � (1 � �2)VII2DL�� 2��1 � ��2�pe � VIIDH � �VII � c02DL

� �1 � �2�VII � c02DL � 2��EuiI�DH � Eui

I�0

which we can obtain dE�/dpe � 0.(3) VII � pe � VI � (VI � VII)((DH � DL)/2DH): K*(DH,

DH) � DH � DL, K*(DH, 0) � K*(0, 0) � 2DL. One can get

E� � �VII � c0 2DL � 2��VI � VII�DH � DL

� �2��2� � �2VI � c0 �DH � DL.

Here, dE�/dpe � 0.

Sketch Proof of Proposition 2.8Proof. The proof is straightforward from the proof

of Proposition 2.7. �

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