MANAGEMENT SCIENCEArticles in Advance, pp. 1–17

http://pubsonline.informs.org/journal/mnsc/ ISSN 0025-1909 (print), ISSN 1526-5501 (online)

Cost Drivers of Versioning: Pricing and Product Line Strategiesfor Information GoodsRamnath K. Chellappa,a Amit Mehrab

aGoizueta Business School, Emory University, Atlanta, Georgia 30322; b Jindal School of Management, University of Texas at Dallas,Richardson, Texas 75080Contact: ramnath.chellappa@emory.edu (RKC); amit.mehra@utdallas.edu (AM)

Received: July 5, 2015Accepted: October 13, 2016Published Online in Articles in Advance:March 10, 2017

https://doi.org/10.1287/mnsc.2016.2698

Copyright: © 2017 INFORMS

Abstract. In this paper, we extend the understanding of versioning strategy of an infor-mation goods monopolist and provide new insights on when versioning is optimal. Todo so, we derive the optimal product line or versions of an information good and thecorresponding prices. By relaxing common assumptions on consumers’ usage costs, ver-sioning costs and capital research and development costs, we provide new insights as wellas reconcile extant findings on versioning. For a good with no-free-disposal (NFD), i.e.,one where consumers have usage costs, our results show that a monopolist’s marginalcost and consumers’ usage costs have the same impact on its versioning strategy, and thatthese factors are the sole reason for optimality of versioning of information goods. Byendogenizing the production of the highest-quality, we show that capital costs create adownward distortion of quality even for the highest types in the market even under fullinformation. Presence of separate versioning costs also lowers the qualities served to thehigh types and reduces the segment of consumers who are served with product versions.However, versioning costs do not affect market coverage or the price-quality menu itself.Further, when some of the consumer usage costs are absorbed by the firm (as in case ofcloud-based provisioning), it does not necessarily lead to market expansion.

History: Accepted by Chris Forman, information systems.

Keywords: versioning • information goods • mechanism design • pricing • free disposal • product line • second-degree pricediscrimination • NFD

1. IntroductionEarly research in economics has studied feature-differ-entiated product line and pricing decisions of phys-ical or industrial goods vendors, often called ver-tical differentiation or quality segmentation models(Mussa and Rosen 1978). Along these lines, in the lasttwo decades, quality-based differentiation for informa-tion goods or digital products has received significantattention where such product line strategies are calledversioning (Varian 1997). Usually, the idea behind sucha strategy is to create the highest version of a prod-uct and then create degraded versions or versions withless features or functionality by removing, disabling,or recombining functions from the flagship product(Wei and Nault 2014). Shapiro and Varian (2013) iden-tify many dimensions along which information goodsversioning is pursued including delay, access, features,functions, and quality and suggest that informationgoods vendors “design the high-end product first, thenremove features to make the low-end version” (p. 69).Although the context in which versioning is stud-

ied is wide and varied, in this body of literature onemajor theme that emerges is the interest in factorsthat influence/determine the versioning decision of afirm. These works include Bhargava and Choudhary

(2008), who study a monopoly that seeks to seg-ment the market by introducing additional lower-quality versions of its existing product; Chen andSeshadri (2007), who examine versioning when con-sumers have type-dependent reservation utilities; andWei and Nault (2014), who examine versioning in thepresence of group tastes among consumers. Many oth-ers (Chellappa and Shivendu 2005, Lahiri and Dey2013) have studied how presence of piracy impacts thedecision to do versioning. More recent works (Augustet al. 2014, Niculescu and Wu 2014) have investigatedsoftware versioning through feature-limited freemiummodels and software as a service model. While thesemodels provide a rich base from which to examineversioning, there are several other factors hitherto notexamined in literature that may potentially affect ver-sioning decisions. We identify four such factors anddescribe them in the following subsections. One of ourmajor research goals is to examine the role of each ofthese factors in the firm’s versioning decision.

1.1. Costs of Information Goods Productionand Consumption

The factors of interest to us can be grouped under“costs,” someofwhichmaybe incurredby theproducer

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while othersmay be incurred by the consumer in enjoy-ing the information good. While some elements of theformer have been examined in literature, the latter haslargely been ignored.1.1.1. Consumer’s Usage Costs (No Free Disposal).Many models in economics that employ utility func-tions generally assume a free disposal property (Mas-Colell 1992, Mas-Colell et al. 1995) implying that forthe same price consumers will weakly prefer more ofthe good. Utility functions in extant work on version-ing also embody this nonsatiation property, and thesefunctions are monotonic (often linear, or concave andincreasing) in quality or features. However, for manyinformation goods and services this assumption doesnot capture reality, e.g., Microsoft’s operating systemsand software are sometimes referred to as bloatware,as they are often packed with excessive features orfunctionalities. For these products, more is not neces-sarily better because software consumption is intrin-sically associated with memory usage; hence, at somepoint the diminishing return from additional featuresis overtaken by the increasing cost of using them. Suchexcess can be a particularly severe problem for mobileoperating systemswhere handsets and touchpads havelimited capacity and memory. While Microsoft’s Win-dows Mobile OS has always suffered from this criti-cism, more recently it has been reported that bloatwarehas crept into Google’s Android OS as well (Milian2010). Generally, this aspect of software consumptionhas been ignored with the exception of one work onsoftware bundling that recognizes this possibility, suchas when some consumers may find no value for add-ins and possibly even incur a penalty cost (Dewan andFreimer 2003).Recent research points out how utility from per-

sonalization services are also nonmonotonic (concave)in services due to the built-in disutility from pri-vacy costs (Chellappa and Shivendu 2010). Person-alization services are infeasible without sharing ofpersonal/preference information, which gives rise toprivacy concerns. Hence, consumers are known to onlyprefer a subset of the services offered, even if theymay be free. Indeed, the assumption of free disposalis increasingly being questioned in the case of infor-mation goods and services, “Unlike physical goods forwhich ‘free disposal’ is always an option and moreis, in general, always better, service delivery is intrin-sically participatory. Participation requires time com-mitment and physical effort on the part of consumers.Thus, there is no free disposal for service” (Essegaieret al. 2002, p. 151). However, there is little or scantresearch on mechanism design for goods with no-free-disposal (NFD) in both economics and IS research.Specifically, we do not know how these costs sufferedby the consumer will impact versioning and otherproduct line decisions.

1.1.2. Development Costs of Highest Quality Version.Extant research on information goods has ignoredthe impact of initial development costs on versioning;either it assumes that features or functionalities can bedeveloped costlessly (and therefore can create a prod-uct of infinite quality), or it has explicitly stated that“fixed costs of developing the highest quality are sunk,and the highest available quality is exogenously spec-ified” (Bhargava and Choudhary 2008, p. 1029). In theinformation goods context, one recent work (Jones andMendelson 2011) considers development costs, but thisdoes not impact the monopolist’s versioning strategy.In the model setup of one other work (Wei and Nault2013), the authors include a cost function for the pro-duction of the highest quality version, but they donot solve for the optimal menu, nor do they explic-itly derive the highest quality produced. Therefore,they are not able to comment on the optimality of ver-sioning as a strategy but they observe that “from thisanalysis it is clear that the quality of the high qual-ity version depends on the convexity of developmentcosts” (p. 500).

In this regard, there is also a work (Hahn 2000) inthe physical goods quality segmentation literature thatexamines the impact of initial fixed costs of developingthe highest quality good, although the firm also suffersa marginal cost of serving each consumer. Our workfollows this line of reasoning for information goodsand allows for the reconciliation of the impact of pro-duction cost (suffered by the firm) and usage cost (suf-fered by the consumer).

1.1.3. Versioning Costs. Since the cost of copying soft-ware or other digital goods is virtually zero, and asdegradation often just involves the disabling (or non-inclusion of) a subset of functions or features, priorresearch has generally examined versioning decisionstaking versioning costs to be zero (Chen and Seshadri2007). Sometimes there is simply no mention of thesecosts, and these are absorbed intomarginal costs. How-ever, versioning costs are distinct and different frommarginal costs of production and depend upon the sizeof the market segment served with type-customizedversions. This is particularly true when each versionneeds to be marketed/promoted separately and/orwhen they require a separate post-sales service invest-ments. We can frequently observe this for software thatis targeted toward different segments such as students,home use, small-business use, professional, enterprise,etc., all of which might require distinct sales channels.In other words, net versioning costs may not be all elec-tronic in nature, thus eschewing the common benefitsof digitization. In this work, we would like to under-stand whether versioning is optimal when costs of ver-sioning are considered and if so, the market segmentto whom type-customized versions should be served.

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1.1.4. Shifting of Usage Costs from Consumers toFirm Due to Cloud Hosting. Firms may reduce theusage costs incurred by their consumers by servingthem the software using cloud technology. This impliesthat the firms now have to incur an additional costof hosting and serving the software to the consumers.Due to this architecture, the capacity utilization ofthe machines at the consumer end is reduced, thusreducing their usage costs and increasing their utilityfrom consumption. Therefore, an interesting questionis whether such cloud technology would enable thefirm to serve the lower end of the market. Further, wewould like to know whether the firm can serve thehigher end of the market with higher quality.

1.2. Optimality of a Monopolist’s VersioningStrategy in Literature

As discussed earlier, literature suggests that the opti-mality of versioning strategy (as opposed to offeringa single version of the product) for a monopolist maydepend on a number of possible factors. These fac-tors are typically context specific. For example, inBhargava and Choudhary (2008) the presence of posi-tivemarginal costs that increasewith quality of the ver-sion and certain distributional properties are requiredfor versioning, whereas in Chen and Seshadri (2007)marginal costs are positive but independent of thequality of the version, yet versioning is found to beoptimal. So, a natural question to ask is whether thecause of versioning is fundamentally different as thecontext changes, or is there an underlying reason thatis common across different contexts where version-ing is observed. An answer to this question will helpdevelop a holistic and context independent under-standing of versioning. This is the second major goalof the research in our paper.In Section 2 we introduce a general model developed

along the lines of a standard monopolistic screeningmodel (Laffont and Martimort 2001), from which wenot only examine the impact of NFD and initial devel-opment costs but can also relate to extant results. Inthis section we first examine the full information caseas the results are not obvious; these results will serveas a benchmark for later comparison with informationasymmetry results. In Section 3 we analyze the mech-anism under information asymmetry where the firmdevelops a menu for self-selection by the consumers.We conclude with theoretical and managerial observa-tions in Section 4.

2. ModelThe standard vertical differentiationmodel of quality isthe inspiration for our base model formulation. Thereis a rich literature on vertical segmentation that exclu-sively uses the term quality differentiation or vertical

differentiation; however, in the information goods liter-ature, in addition to quality, terms such as versions, fea-tures, and functionality are also interchangeably used.Versioning itself is more formally defined by Varian(1997, p. 190), “[W]e will focus on a particular aspectof differential pricing known as quality discriminationor versioning. These terms describe situations in whichthe producer provides different qualities/versions ofa good which sell at different prices.” He further goeson to note that “If we think of quality as being ‘addi-tional features,’ an admittedly dangerous equivalence,this means that the producer should add features untilthe willingness to pay for an additional feature by thehigh end of the market just equals the cost of provid-ing that feature. Once the high end has been deter-mined, the producer then removes features to sell tothe lower segment of the market, recognizing that fea-ture it removes allows it to increase the price sold tothe high WTP consumers.” Similarly Wei and Nault(2013) use the term functionality in their model, e.g.,“degraded versions—versions with less functionality.”Generally, a product is considered to be an economicbundle of infinitesimally small features or functional-ities such that as more features/functions are added,the product becomes of higher quality.

Formally, our model consists of a principal—a digi-tal goods firm with a unique production cost structureand agents—consumers who face resource constraintsin consuming these goods. Let q: q ∈ �+ such thathigher q implies a good of higher quality. The firmmay costlessly damage its product of quality q̄ to anylower quality q ∈ [0, q̄] by “by removing, disabling orrecombining functions” (Wei and Nault 2013, p. 495).Along the lines of Mussa and Rosen (1978) and othersin the versioning literature (Varian 1997, Sundararajan2004, Bhargava and Choudhary 2008), consumers areindexedwith their marginal value for quality θ ∈ [

¯θ, θ̄]

which is distributed with density function f (θ) andcumulative density F(θ) that is continuously differen-tiable. Further, f (θ) is assumed to be single-peaked(unimodal) and is everywhere positive on its supportsuch that its hazard function h(θ) � f (θ)/F(θ), sat-isfies the monotone hazard rate property. The suffi-cient condition for this distribution requirement is sat-isfied by most parametric single-peak densities (Bag-noli and Bergstrom 2005), and the assumption is afairly standard one in models of price discrimination(Sundararajan 2004).

To consume the product, the consumers also incura resource cost. For example, this might be due tothe average memory consumed to run each softwarefeature, and we consider a market that is homoge-neous in its resource-cost coefficient given by a param-eter λ (λ > 0). Therefore, the utility for consuming a

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product with quality q priced at p (p ∈ �+) for a con-sumer with index θ is

U(θ, q , p)� θq − λq2 − p. (1)

Observe that the utility function is nonmonotonic inquality, i.e., up to a point (utility maximizing quality)the utility is increasing in quality, and then it decreases.In other words, there is no free disposal in quality;higher quality can actually make the consumer worseoff.1 We have adopted a specific functional form ofthe utility function which is standard in the extant lit-erature2 on vertical differentiation (Mussa and Rosen1978) and no-free disposal (Chellappa and Shivendu2005). For brevity, henceforth we refer to U(θ, q , p) asU(θ) only. Note that in our model the consumers donot face any budget constraints.Another aspect of our model is that the firm has to

decide on the highest quality it must produce alongwith any versioning and pricing decisions. To endo-genize this decision, we incorporate a fixed, quality-dependent cost of creating the highest quality. Weassume this cost to be convex in quality and givenby cq2. This convex cost function is commonly assumedfor information goods, since it is generally believedthat most cost-effective decisions are made first, andit becomes increasingly costly to improve qualityby adding features to information goods (Jones andMendelson 2011). Empirical research in software engi-neering also finds this cost to be convex, although thereare some differences in the degree of convexity (Boehmet al. 2000). This fixed cost is a one-time investmentin creating the highest quality, a kind of research anddevelopment (R&D) investment. Once this list of fea-tures is created, the firm can create versions of thegood. Here, we assume versioning costs to be zero (akacostless damaging), although we specifically relax thisassumption later to delineate the differential impactof versioning costs and marginal costs of production.Similarly, we also assume the marginal costs to benegligible at this time, i.e., firms can serve additionalconsumers for free once a version is created; how-ever, note that we also revisit this cost in the sectionwhere the firm adopts a cloud-based model of productprovisioning.A general understanding of versioning strategies is

provided by a mechanism design approach under in-formation asymmetry. While this is a common startingpoint for most literature on versioning, we also pro-vide a brief review of the full information case results(that is often left out because it is considered trivial).Full information results require a revisit in our paper,as there is a potential for full information strategies tobe different from extant models of versioning underfirst-degree discrimination because of the presence ofconsumer usage costs and endogenization of the max-imum quality decision.

2.1. Full Information Versioning Strategies(Welfare Maximizing Solution)

The timeline for the model is as follows: The vendor in-vests in research and development to create the highestquality product. Using this highest quality, he can cre-ate other reduced quality versions of the product andset their prices accordingly. It is costless for the vendorto create additional versions of reduced quality (zeroversioning costs), and he does not incur any additionalcosts in serving consumers (zero marginal cost). Todetermine this highest quality level, the vendor consid-ers his next stage decision of versions and correspond-ing prices and then employs backward induction.

In the full information case, the vendor knows eachconsumer’s type; hence, he will extract the maximumsurplus possible from each type. Note that the solutionto this problem is the same as a welfare-maximizingsolution but one where the vendor extracts all the sur-plus. Let q̂ be the highest quality level that is producedby the vendor and q(θ) be the quality offered to eachconsumer of type θ. Note that our utility function isnonmonotonic concave, i.e., each consumer has a sati-ation point at which he derives maximum benefit fromconsumption. This utility function is maximized at

q∗(θ)� arg maxq[θq − λq2]� θ

2λ ,

and the corresponding price to extract the full surpluscan be obtained from

U(θ)� θq(θ) − λq2(θ) − p(θ)� 0,

implying that the optimal price p∗(θ) � θ2/(4λ). It isvery simple to observe that even if the vendor had nocost of creating the highest quality, there is no pointin creating a quality greater than θ̄/(2λ), as this is thequality at which the highest type in the market derivesmaximum benefit from consumption.

However, when there is a cost associated with qual-ity production, we do not know if the vendor may evenbe able to supply this quality to the market. As q̂ isthe highest quality level, the consumer type for whomsurplus is maximized at this quality is θ̂ � 2λq̂, whereθ̂ ∈ [

¯θ, θ̄]. Since (∂/∂θ)(q∗(θ))> 0, this implies that con-

sumer types θ ∈ (θ̂, θ̄] will be served quality q̂ thatis less than their first-best (i.e., utility maximizing)quality. The corresponding price to extract full surplusfrom these consumers is p∗(θ)� θq̂−λq̂2; therefore, theobjective function of the vendor can be expressed as

Π� maxq̂

{∫ θ̂(q̂)

¯θ

θ2

4λ f (θ) dθ

+

∫ θ̄

θ̂(q̂)[θq̂ − λq̂2] f (θ) dθ− cq̂2

}. (2)

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Solving the maximization problem in (2) by Fubini’stheorem and pointwise maximization (details are inthe appendix), and using θ̂∗ and q̂∗ to represent optimalvalues, we have the following lemma.

Lemma 1. θ̂∗ is obtained by solving θ̄− θ̂�G(θ̄)−G(θ̂)+cθ̂/λ, where G(θ)�

∫ θ

θF(s) ds. The market is covered such

that the vendor provides q∗(θ) � θ/(2λ) ∀θ ∈ [¯θ, θ̂∗] and

q̂∗ � θ̂∗/(2λ) ∀θ ∈ (θ̂∗ , θ̄].All proofs are in the appendix.The properties of the above results make for inter-

esting analyses. If c � 0 the solution to the aboveEquation (2) is θ̂∗ � θ̄; i.e., all consumers will get theutility maximizing individualized version (q∗(θ)) andthe highest quality that will be produced is θ̄/(2λ).However, if c > 0, then θ̂∗ < θ̄ since G(θ) is an increasingsuperlinear function of θ; i.e., consumers with indexgreater than θ̂∗ are served with quality q̂, which isless than their utility maximizing quality. Therefore,the size of the consumer segment that is served withtype-customized versions is reduced in the latter casecompared to the situation where c � 0. In other words,as long as the vendor has some finite cost of creatingthe initial quality, he will not offer the first-best qual-ity to the highest types in the market even under fullinformation.Note that if both λ � 0 and c � 0, we do not get inte-

rior solutions. This should be fairly obvious in that ifthe consumers’ utility is strictly increasing in featuresand there is no cost to producing them, then an infi-nite quality/price would be the solution for all con-sumer types. Indeed, even when λ � 0 and for somepositive value of c, the solution will be to create thehighest possible quality and serve the same quality toall types but at different prices. While we use a stylizedquadratic utility function, using any other functionalform such as a higher-order polynomial does not dis-tract from the discussion. The key difference will bethat a higher-order polynomial represents lower sur-plus per consumer for each quality level; the surplusmaximizing quality and the break-even qualities willbe lower for a given type.Now, we examine the role of different cost factors

on versioning in Section 3, in the context of informa-tion asymmetry, i.e., when the firm does not know themarginal value for quality of an individual consumer.

3. Versioning Strategies UnderInformation Asymmetry

When the vendor cannot perfectly price discriminatebetween consumer types, it must develop a menu oftruth-revealing versions and prices such that the con-sumers self-select the version targeted at them. In thiscase, the vendor only knows of the distribution of thetypes but not the type of a particular consumer. Similar

to the full information case, the vendor has to decidethe highest quality that he will produce and the sub-sequent versions that he will create for the market.In determining his prices, he may have to pay infor-mation rent to high types so that they do not buya lower quality version. Suppose if the vendor cre-ates some maximum quality, qH , the correspondingprofit maximization problem for the firm, along withthe respective individual rationality (IR) and incentivecompatibility (IC) constraints is

max{q(θ), p(θ)}

∫ θ̄

¯θ

p(θ) f (θ) dθ− c[qH]2

s.t. U(θ) ≥ 0 ∀θ, (IR)U(θ) ≥Uθ(θ̃) ∀θ. (IC)

(3)

where Uθ(θ̃) represents the utility of the consumer oftype θ if she misrepresents her type as θ̃. The incentivecompatibility condition essentially states that a con-sumer prefers the price-quality meant for him becausethe utility from that pair is higher than provided byany other price-quality pair. Hence, it must be that

Uθ(θ)≥Uθ(θ̃)⇒ θq(θ)−λq2(θ)−p(θ)≥θq(θ̃)−λq2(θ̃)−p(θ̃) (4)

for any (θ, θ̃) ∈ [¯θ, θ̄]× [

¯θ, θ̄]. Similarly, for a consumer

of type θ̃, it must be true that declaring herself to beof type θ would result in lower utility for her, i.e., weneed that

Uθ̃(θ̃)≥Uθ̃(θ)⇒ θ̃q(θ̃)−λq2(θ̃)−p(θ̃)≥ θ̃q(θ)−λq2(θ)−p(θ). (5)

The formulation of the two IC constraints above en-sures that the consumer picks a version that is targetedto that consumer, i.e., there is no cannibalization in thedemand of a version. Equations (4) and (5) further leadus to understand that any optimal versioning menu,if one exists, needs to be nondecreasing in consumertypes (proof is in the appendix), i.e.,

q′(θ) ≥ 0. (6)

This gives us Lemma 2.

Lemma 2. The index of the lowest consumer type who isserved, θ∗L is a solution to θ−[(1−F(θ))/ f (θ)]� 0, and theindex of the lowest consumer type who gets served the highestquality, θ∗H is obtained by solving θ − [(1− F(θ))/ f (θ)] −2λqH � 0.

Using Lemma 2, we lay down the menu of qualitiesserved in the market.

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Proposition 1. The vendor serves the market such that

q∗(θ)�

0 for θ ∈ [

¯θ, θ∗L),

θ− [(1− F(θ))/ f (θ)]2λ for θ ∈ [θ∗L , θ∗H),

q(θ∗H) for θ ∈ [θ∗H , θ̄].

Lemma 2 and Proposition 1 are intermediate resultsin that we do not yet know what the optimal highestquality (q∗H � q(θ∗H))produced shouldbe. First, note thatthemonopolist delineates themarket into three distinctsegments where he does not serve a portion of the mar-ket given by type θ ∈ [

¯θ, θ∗L). While this is consistent

with extant segmentation models for physical goods(with marginal costs of production) where some lowtypes get left out of the market, this result is not thatintuitive for information goods; note that the monop-olist does not suffer any cost of serving the low typesin the market. In other words, he could have costlesslyserved this segment and potentially extracted a surplusequal to

∫ θ∗L

¯θ

p(θ) f (θ) dθ, and yet he finds it optimal notto do so. The economic rationale behind this decisionstems from the information rent that he has to pay tohigher typeswhenever a product of lower quality-priceis offered todeter thehigh-types frompurchasinga low-quality product. Themonopolist considers the trade-offbetween the revenue (as there are no costs) from theselow types and the net rent he has to pay to high typesif he serves the low types and decides not to serve thesegment given by θ ∈ [

¯θ, θ∗L) at all.

He also develops a nonlinear menu for a segmentgiven by θ ∈ [θ∗L , θ∗H) where each consumer gets a ver-sion q(θ) corresponding to his type. We can easily seethat this quality menu is decreasing in λ meaning thatwith increasing usage-related costs, each consumertype’s quality is lowered. For the consumer segmentdefined by θ ∈ [θ∗H , θ̄], the firm offers a single prod-uct. In extant segmentation models, the lowest type (

¯θ)

under asymmetry is either not served at all or receivesa lower quality than in the full information case. How-ever, the highest type (θ̄) should generally get the samequality as in the full information case. Therefore, tosolve for the complete quality-price schedule, we firstsolve for the maximum quality level the firm will pro-duce. Taking into account that the utility for the low-est served type (θ∗L) must be zero because the firmextracts full surplus from such a consumer (see theappendix formore details), we canwrite the expressionfor p(θ) as

p(θ)�[θ− λq(θ) − 1− F(θ)

f (θ)

]q(θ). (7)

Note that the first two terms of the price expressionabove represent the full surplus of the consumer (pricein the full information case). However, the third term is

negative and represents the information rent that mustbe paid by the firm because it does not know the type ofa given consumer. Substituting for p(θ) and incorporat-ing the appropriate limits for the integral, we can nowrewrite the objective function given in Equation (3) as

max{qH }

∫ θ∗H

θ∗L

[θ−λq(θ)−

[1−F(θ)

f (θ)

] ]q(θ) f (θ)dθ

+

∫ θ̄

θ∗H

[θ−λqH −

[1−F(θ)

f (θ)

] ]qH f (θ)dθ− cq2

H , (8)

where q(θ) � (θ − [(1 − F(θ))/ f (θ)])/(2λ). From Lem-ma 2, we see that θ∗H is expressed as a function of thehighest quality qH . Hence, to get a complete clarity onθ∗H , we would have to solve the firm’s optimizationproblem in (8) to obtain the equilibrium highest qual-ity q∗H . This result in presented in Lemma 3.

Lemma 3. The optimal highest quality, q∗H , produced by thevendor under asymmetric information is obtained by simul-taneously solving θ∗H−[(1−F(θ∗H))/ f (θ∗H)]−2λq∗H � 0 andθ∗H[1− F(θ∗H)]� 2q∗H[λ[1− F(θ∗H)]+ c].Lemma 3 provides us the lowest high-type con-

sumer who will receive the highest quality produced.We can now compare this highest quality that willbe developed under information asymmetry with thewelfare-maximizing or full information solution givenby Lemma 1. This comparison is given in Proposition 2,which also points to quality distortion for the hightypes.

Proposition 2. The highest quality under informationasymmetry is lower than the highest quality produced underfull information (q̂∗ > q∗H). Further, the optimal schedule ofquality under information asymmetry is also lower for everyconsumer.

Traditionally, under information asymmetry, themonopolist serves the same quality to the highest cus-tomer type as it would under the full information case.In other words, there is no distortion at the highestquality level under information asymmetry, and gen-erally it is the lower consumer types who are servedwith degraded quality (Srinagesh and Bradburd 1989).Even though the highest type receives the same qual-ity under both full and asymmetric information cases,the price she pays in the latter case is lower (due toinformation rent). In this regard, a recent unpublishedthesis on physical goods (Hahn 2000) with a standardutility function and a finite marginal cost of produc-tion also finds quality distortion for all consumer types.When it comes to information goods, the single ver-sion result of Jones and Mendelson (2011) also alludesto this reduction in quality. However, our finding ofquality distortion is much richer in the understandingit provides and is very general in its applicability. The

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Jones and Mendelson (2011) work finds that only onequality of information good is provided to the mar-ket and this quality is downward distorted; in otherwords, all served consumers get that same lower qual-ity. Our finding shows that quality distortion is only forthe high types, which is the interesting and counter-to-intuition result for information goods. We show thatwhile the low types are not served, the middle typesreceive personalized quality (not distorted) and thehigh types get a bunched solution that not only leadsto their getting a uniform quality but is also distorteddownward.Now we introduce Figure 1 to provide a visual com-

parison of quality menus under full and imperfectinformation. Note from Figure 1 that the versioningmenu is concave in consumer types under informationasymmetry, while it is linear in the full informationcase. Also note that the figure depicts only the casewhere θ̂∗ < θ∗H , although the converse is also possi-ble (see the proof of Proposition 2 for further discus-sion). We find that the monopolist divides the marketinto three segments, with the lowest segment not beingserved, the highest segment being served with a sin-gle product, and the mid segment being served withversions in accordance with their marginal value forquality.Further, the quality and price for consumers in the

mid segment increases with their type but at a decreas-ing rate. While the magnitude of the quality served inthe menu depends on the nature of the distributionof consumers, we find that the existence of this ver-sioning strategy is independent of the distribution ofconsumer types. Note that in some other extant mod-els of versioning, either explicit assumptions onmarketcoverage are introduced in themodel setup or only twoconsumer types are considered; in the latter, if version-ing is pursued, then market is always covered. In our

Figure 1. Market, Quality Menu, and Quality Distortion

First-best quality

Full informationmenu

Qualitydistortion

Information asymmetrymenu

q(�)

q*H

�

�*L �*

H �–�–

2��–

2��

�*

q*

q*(�) = 2�

� – (1 – F(�))/f (�)

case, as a result of deriving the full versioning-menufor a general distribution, we endogenize any marketcoverage decision.

Consider the case of Microsoft Office 2010. Microsoftoffers a fairly low-end option in a Starter edition fol-lowed by Home and Student, Home and Business,Standard, Professional, Professional Academic, andfinally a Professional Plus. Even though there are firmsthat seek and are willing to pay for more features, theyhave to make do with the highest version (ProfessionalPlus) that is offered. Evidence of this fact is commonlyseen in Microsoft’s technical forums3 where not onlyneeds for unmet features are expressed but solutions inthe form of either workarounds or third-party vendorsoffering such features are present. This is an exam-ple of a situation where there exist some high typeswhose optimal qualities are higher than what the firmprovides.

We can further understand market coverage thoughcomparative statics of the size of various market seg-ments derived in the model with respect to costs cand λ.

Proposition 3. The segment of consumers that are servedwith type-customized versions is decreasing in the develop-ment costs c but is increasing in the usage costs λ. Marketcoverage is independent of both these parameters.

Proposition 3 succinctly captures the differential im-pact of the initial development cost and the usage coston versioning providing key insights into market cov-erage and versioning as a strategy.4 Consider the com-parative statics of θ∗L with respect to the two costs;θ∗L refers to the lowest consumer type who would beoffered a version, and from Lemma 2 we can see thatthis boundary is independent of either cost parameters.In other words, under information asymmetry, marketcoverage when versioning is optimal is purely a func-tion of the distribution. The market is covered if andonly if

¯θ ≥ 1/ f (

¯θ). In the full information case, how-

ever, the market is always covered with the types lowerthan θ̂∗ receiving their first-best quality.Now consider the comparative statics of θ̂∗ and θ∗H ;

we can see that both these bounds are decreasing in cbut are increasing in λ. θ∗H refers to the highest con-sumer type who will be served a personalized version(second-best quality5) under information asymmetry,while θ̂∗ represents such consumer type (who gets herfirst-best quality) under full information. Not surpris-ingly, these bounds are decreasing in the developmentcost c as it plays a role in the production of the highestquality that corresponds to the personalized versionof θ∗H .However, it might be somewhat surprising to note

that these bounds are increasing in the consumer’susage cost λ, also implying that more people get theirsecond-best quality as usage costs increase. This can be

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explained as follows: The first implication of increasingusage costs is that the optimal quality served to eachconsumer is lowered. Therefore, for a given highestproduct quality, the interval of consumer typeswho getserved their second-best quality increases. However, asecond implication of increasing usage costs is that thesurplus of each consumer reduces. This implies thatthe market is less valuable for the firm, resulting in thecreation of a lowered highest-quality, and consequentlyfewer consumers get served their second-best quality.The result in Proposition 3 establishes that the impactof the first implication dominates, and so overall moreconsumers are served their second-best quality.

3.1. Impact of Versioning CostsThe difference between versioning and marginal costsis generally not captured in many extant models oninformation goods versioning. Versioning costs can in-deed be distinct and separate even for informationgoods that may have zero marginal costs of produc-tion. Often, these costs are due to specific marketingactivities and related investments incurred in manag-ing a segment of consumers for whom a particularversion is created. For example, in the context of soft-ware, Microsoft Office will have to incur packaging,marketing, and customer service costs that may bedistinct and different for the different versions it cre-ates. Often referred to as segment development costs(Dhebar 1990), even for a digital good, these mayinclude nondigital elements. This is not the cost ofserving each additional customer but rather this is theinvestment made when a separate version is created.The versioning cost is often ignored and is either

assumed away in information goods literature when“negligible marginal costs” form the core of the modelor is not differentiated from marginal costs. In fact,just as optimality of versioning is of interest, anotherelement of relevance is the monopolist’s decision withregard to the segment of consumer to whom productversions are to be offered. Many extant models con-sider only two consumer types that make it difficultto separate the decision on the optimality of version-ing from the decision on the consumer segment that isserved with the versions. This understanding becomesrelevant and important particularly when versioningcosts differ from marginal costs. Therefore, in this sec-tion we introduce a separate versioning cost, a one-time investment for each version that is created, andwerewrite the firm’s objective function in Equation (3) as

max{q(θ), p(θ)}

{∫ θ̄

θL

p(θ) f (θ) dθ−∫ θH

θL

k dθ− c[qH]2}, (9)

where k is the cost-coefficient for each version of theinformation good that is created. Borrowing from ourknowledge of what the quality-price schedule looks

like, we know that there are three possible regions:(a) a region θ ∈ [

¯θ, θL] where consumers may be

left unserved, (b) a region θ ∈ [θL , θH] where eachconsumer-type uses its own version, and (c) a regionθ ∈ [θH , θ̄]wherein all consumers use the same versionthat was created for customers of type θH which is thehighest quality created (qH � q(θH)). Similar to our ear-lier approach to deriving the truth-revealing menu, weneed to consider the consumers’ individual rational-ity and incentive compatibility constraints and derivea menu {q(θ), p(θ)} and then substitute it back andbackward induct to derive the cutoff points θ∗L and θ∗Hand then obtain the highest quality offered in the mar-ket, q∗H . The customers whose types lie between θ∗H andθ∗L are served with product versions, and the total ver-sioning cost is ∫θ

∗H

θ∗Lk dθ. Hence, we have Proposition 4.

Proposition 4. Versioning cost does not affect market cov-erage nor does it influence the quality schedule, however thiscost lowers the highest quality developed in the market andreduces the segment of consumer types who are served type-customized versions.

Proposition 4 tells us that even with the presence ofan additional cost of creating versions, the firm willcontinue to serve the same market as before. We cansee from the proof that θ∗L with versioning costs is thesame as what we have seen before in Lemma 3; sinceeveryone to the right of this consumer is served, themarket coverage is the same. In fact, it will also offer thesame quality menu in the market except that the seg-ment of consumers receiving their type-specific qualityreduces, i.e., θ∗H (no versioning costs) > θ∗H (with ver-sioning costs). Therefore, more high types now receivethe same version, also known as a bunched contract.

Note that our motivation for examining versioningcosts was to separate its influence from that ofmarginalcosts, and it is very evident from Proposition 4 thatthe impacts of these costs are indeed different. Forinformation goods, the relevance of this cost is directlyrelated to the density of the distribution—if there isonly one customer of each type, then essentially we canspeak of the marginal costs and versioning costs in thesame breath as in effect the cost of serving each con-sumer is the cost of the version itself. However, whenthere are multiple consumers for each type then eachconsumer type essentially forms a segment and thefirm may incur a marginal cost to serve each consumerin such a segment, whereas the versioning cost is a one-time cost of the developing the version for the wholesegment.

3.2. Cloud-Based Provisioning of NFD GoodsThe technological nature of usage costs to theconsumers is such that it can be partially transferredelsewhere such as through cloud-basedmodels of com-puting. While there are supply-side discussions about

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migrating to the cloud, such as trading off infrastruc-ture investments that are amortized over a period ver-sus incurring operational expenditure (CAPEX versusOPEX), there is little to no academic research on thespecific advantages of moving to the cloud for NFDgoods where opportunities for some cost transfer exist.Here we specifically wish to explore the implicationsof moving to software-as-a-service or cloud-type mod-els when there are intrinsic consumption-related coststhat could potentially be shared by the vendor.New technologies, such as those employed in cloud

computing, are developed on client-server principleswhere the amount of work done on the client end canbe reduced through virtualization and other technolo-gies. In other words, processing that would normallybe done on a client like a mobile device can possi-bly be accomplished on servers in a cloud implyingthat the burden of usage can be now partially borneby the firm that has invested in the cloud technol-ogy. Therefore, some disutility associated with mem-ory and/or processing power consumption is reducedfor the consumer, as it is now performed on thecloud. To capture this reduction in disutility, we intro-duce a parameter d ∈ [0, 1] such that provisioningthrough the cloud allows the consumer to enjoy util-ity of the form U(θ) � θq(θ) − λ(1 − d)q2(θ) − p(θ).A larger d implies that more and more processing isvirtualized.

However, this virtualization on the cloud is not cost-less, and cloud infrastructure providers charge vendorsfor this provisioning. While multiple types of tariffrelationships are possible, we consider a simple tariffstructure where the vendor pays the cloud infrastruc-ture provider on the basis of every feature-consumerserved. This implies that pricing is based on both thesize of the consumer segment as well as the qualityoffered to the consumers. We introduce a parameterδ(δ > 0) that captures any cloudification costs. Therefore,we can rewrite the objective function of the vendorconsidering cloud-based provisioning as

max{q(θ),p(θ)}

{∫ θ̄

¯θ

p(θ) f (θ)dθ−δ∫ θ̄

¯θ

q(θ) f (θ)dθ−c[qH]2}.

(10)Note that marginal cost suffered by the firm in this caseis not a traditional cost of production, but it specificallysuffers the cost itself to alleviate it for the consumers.Since a firmmight be able to better manage this aspect,it is not a one-to-one alleviation, i.e., the cost reduc-tion on the consumer side is not simply added onthe vendor side. Results from optimizing the objec-tive function given in Equation (10) is described inProposition 5.

Proposition 5. When the firm absorbs a portion of con-sumers’ usage costs (e.g., through cloud-based provisioning)

he will reduce market coverage leaving more low-type con-sumers unserved. The firm will improve the quality offered tohigh types if the cloud technology sufficiently lowers (relativeto cloudification costs) consumer specific usage costs.

We started with the premise that “cloudification”requires some transfer payment to a cloud vendorwhile it promises to reduce the computing burden onthe user end. Our results show that any such paymentwill automatically reducemarket coverage where somelow types will not be served under the cloud model.Therefore, our results suggest that we cannot alwaysassume that movement to a cloud-type technology willalways result in greater participation.

Not surprisingly, our results also suggest that anybenefit in the form of higher quality that can accrue tothe high types is possible only if the migration to thecloud results in substantial reduction in usage-relateddisutility. The cloudification costs themselveswill pushthe vendor to target more of the high types forcing himto serve individualized qualities to these consumers,i.e., θ∗H (noncloud) < θ∗H (with cloud model). However,whether or not the consumers who are served a menuwill get a higher quality under the cloud provision, i.e.,whether q(θ) under the cloud model will be greater orlower, is a function of the relative values of δ and d.There are some key theoretical understandings here,

primarily the finding that lowered costs from cloudi-fication are not passed on to the low-end consumers.In other words, the market coverage actually reduces—more low-types are left unserved (or lesser lower-endversions are offered), and the firm focuses on pro-viding more value to the high-types and extractingthat surplus. While this may be surprising at a firstglance, what we are essentially observing is the factthat monopoly considerations trump any cost benefitfrom cloud migration.

Indeed, there are other aspects to migrating to acloud, but in this model we are specifically interestedin the transfer of the usage burden to the cloud. Thegoal of this section is not to account for all possi-ble elements of cloud-based provisioning but to reallycompare versioning under product-provisioning withcloud-based provisioning. It is for this reason we haveconsidered the same market described in the ear-lier non-cloud-based offerings. A truly cloud-basedmodel will perhaps need to examine other elementsof usage, e.g., temporal changes in user demand. Asopposed to installed software products, it is possibleto enjoy cloud-provisioning on an on-demand basis,thus allowing for a very different pricing and tariffstructure. Even in current offerings, we observe thatpricing is offered at different levels of granularity, e.g.,Microsoft Office 365 does not allow for on-demandpricing, while Amazon’s AWS allows for more granu-lar options. Our current setup is more in line with theformer scenario.

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Having derived different quality schedules undersetups that include both extant and new factors affect-ing versioning, we have an opportunity to understandand reconcile differing observations on informationgoods versioning. So, in the following section, we fur-ther generalize our model with the sole intention ofseparating out the factor(s) that makes versioning opti-mal for a monopolist.

3.3. When Is Versioning Optimal?The industrial goods literature has shown that, whenconsumers differ in their preferences for quality, seg-mentation of consumers based on quality is an attrac-tive strategy. The benefit of this strategy comes fromthe fact that creating an individual, type-specific goodgenerates the highest consumer surplus, which canthen be extracted by the firm to maximize its prof-its. In the case of information goods, it would appearthat versioning should be even more attractive thanin industrial goods because of the possibility of cost-less degradation of such goods to create multipleversions. While this may be case, some works on infor-mation goods suggest that versioning is suboptimal,while some others suggest that versioning is optimal.As a result of our in-depth analyses of the variouscosts involved in information goods versioning, in thissection we are able to not only reconcile these appar-ently conflicting observations but also provide a holis-tic understanding of versioning.

We shall, therefore, begin by examining, from theinformation goods literature, the purported causesof versioning. In a seminal work, Bhargava andChoudhary (2008) observe about one of their proposi-tions that “the choice of versioning strategy dependson how the low- and high-type consumers vary in theirrelative valuations for low-and high-quality products”(p. 1031). Further, they also note that this proposition,“provides a precise quantifiable measure of the degreeof consumer heterogeneity needed to make versioningoptimal.” In other words, they suggest that the opti-mality of versioning depends on consumer heterogene-ity, i.e., distribution of their types. Similarly, Jones andMendelson (2011) claim that “for information goods,the losses from cannibalization always outweigh thebenefits of segmentation” (p. 166). They go on to sug-gest that the reason for this suboptimality of version-ing is due to specific distributional assumptions suchas, “consumers are distributed over an interval startingfrom zero, and we do not assume that the market iscompletely covered” (p. 166). In otherwords, this paperalso suggests that some property of the distribution isthe source of versioning.

Finally, a more recent paper has succinctly summa-rized the extant observations on information goodsversioning and pointed out that the prevalent viewis indeed that consumer type distributions and opti-mality of versioning are somehow related. August

et al. (2014) write about previous literature saying that(p. 493), “In this literature, a common point of concernis thatwhen consumers are heterogeneous in their tastefor quality and this taste parameter is uniformly dis-tributed, a software vendor will not find it optimal toversion its product.” To provide a holistic understand-ing of versioning that is fairly context independent, wedevelop the following result without assuming a func-tional form for the utility and a general distribution.

Proposition 6. For a monopolist, marginal costs of usage(from the consumers’ side) and marginal costs of production(from the supply side) have the same impact on versioningstrategies. For any standard utility function (monotonicallyincreasing concave or linear), such marginal costs are thesole reason for versioning. Further, optimality of versioningas a strategy is distribution independent.

Proposition 6 is critical to reconciliation of resultsfrom extant models discussed before, and this propo-sition is true under both the case of full informationand information asymmetry; we shall largely discussthe latter here since it is more general. This proposi-tion states that marginal usage costs (an example ofno-free-disposal) and production costs (generally dis-cussed in industrial goods literature) are duals in thattheir impact on all firm decisions including versioning,pricing, segmentation, and profit are the same evenif one appears to impact the consumer and the otheraffects the producer. This proposition also states thatthe presence of such a marginal cost (or any other fac-tor that has a similar role) is a necessary and sufficientcondition for versioning. Moreover, Proposition 6 findsthat whether versioning is optimal or not is indepen-dent of consumers’ type distribution bar common reg-ularity assumptions described earlier.

For the proof of Proposition 6,wefirst consider a gen-eral model without specifying a functional form for theutility function andwithout anydistributional assump-tions (bar standard regularity conditions) and proceedto develop the full solution space; we then impose spe-cific assumptions from extant models that correspondto various cost parameters to identify the source of ver-sioning in each of these models. We consider a generalutility function of the form u(θ, q) − p where the goalof the monopolistic firm is to design a price schedule{q(θ), p(θ)} to maximize its profit. Since the monopo-list does not knowwhich consumer is what type (infor-mation asymmetry), he has to design the menu orquality-price schedule to be truth-revealing, i.e., con-sumers must self-select into the intended quality-pricebased on their type. We show that to develop a menuwhere at least two different θ types get two differentqualities, it is necessary and sufficient that either con-sumers’ utility function itself is strictly concave, i.e.,uqq(θ, q)< 0 or/and if u(θ, q) is monotonically increas-ing concave or linear, the marginal cost c(q) suffered

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by the firm be strictly convex, i.e., cqq(q) > 0. In otherwords, the proof of the proposition shows the need fora marginal cost factor for the optimality of versioning.We shall now discuss this generalized result for

the specific utility function discussed in the paper.While there is no prior work that has examined no-free-disposal through usage costs and a nonmono-tonic utility function (case with positive λ (λ > 0)), wecan easily accommodate many extant models throughsuitable assumptions of our generalized model. Forexample, if we consider λ � 0 and maintain the devel-opment cost, we get the setup suggested by Jones andMendelson (2011)—strictly multiplicative monotonicutility with convex initial development cost and zeromarginal (and versioning) costs. Our results for thissetup will find that the monopolist will offer a sin-gle version (with quality distortion); versioning will besuboptimal, and these will be consistent with the afore-mentioned paper. However, our Proposition 6 showsthat this result is independent of the distributionalproperties or consumer heterogeneity, while Jones andMendelson (2011, p. 166) attribute the single productresult to distributional assumptions (quoted earlier inthis section). In other words, the aforementioned papersuggests that some property of the distribution is thesource of versioning; however, in reality the work hasno marginal cost and hence no versioning. Further, wecan also show that the quality distortion in this paperis not present for all served consumers but restrictedto the highest type consumers. A similar observa-tion can be made about the findings of Bhargava andChoudhary (2008) in that their versioning result is dueto the presence of a marginal cost and not due to dis-tributional aspects.On the other hand, consider a case where λ is pos-

itive with no initial development costs. We can easilysee that this leads to a versioning menu but withoutany distortion in quality for the high types; i.e., all con-sumers who are served receive their second-best qual-ity. Collectively, these observations tell us that whilethe capital cost is responsible for quality distortion forthe high types, the usage cost is responsible for theversioning decision.Now, consider a physical good equivalent with the

traditional multiplicative monotonic utility with con-vex initial development cost and a positive quality-dependentmarginal cost, andwhere the vendor suffersa marginal cost to serve each consumer. Proposition 6tells us that this will yield the same versioning strat-egy as the one for goods with no free disposal. It isinteresting to note that even though the vendor suffersthe cost in this case, he will offer the same menu, thusreconciling with a recent physical goods segmentationresult (Hahn 2000). Further examination of the surplusper consumer makes the economic intuition behindthis apparent; irrespective of who suffers the cost, the

vendor maximizes the net surplus per consumer, thenpays the corresponding rent (to ensure incentive com-patibility) and extracts the remainder through price.Therefore, it does not matter if the loss from consum-ing a good of a certain quality is through the no-free-disposal property of the consumer or from themarginal production costs to the vendor.

We can see that the only time a menu is derived asa strategy is when there is an interior solution to theterm inside the integrand in Equation (11).∫ θ̄

¯θ

[θq(θ) − λ[q(θ)]2 −

q(θ)(1− F(θ))f (θ)

]f (θ) dθ (11)

is concave in q(θ). Note that this is always true in thecase of goods with no free disposal, or when there isa convex marginal cost such as for physical goods. Inother words, we can categorically prove that version-ing is optimal only in the presence of usage costs ormarginal costs of production. Extant utility functions(increasing and multiplicative) considered in informa-tion goods (i.e., where marginal costs are zero) literaturecan never lead to optimality of versioning.

4. ConclusionsOur work builds upon a vast literature on physicalgoods quality segmentation and information goodsversioning to accommodate hitherto un-included fac-tors such as consumers’ usage costs and developmentcosts of the highest version. The motivation for ourwork is twofold: on the one hand we are interestedin realistically capturing the impact of different typesof costs such as no-free-disposal, versioning costs, andcost of quality development; while on the other hand,we wish to examine the theoretical underpinning ofversioning decisions.

Many information goods, in particular software,are consumed through mobile devices such as smart-phones and tablets. One cannot enjoy these informa-tion goods without them consuming resources suchas memory and processing power. Our work drawsattention to this fact and suggests that awareness ofthis consumption-related disutility is critical to fea-ture bundling. Hence, product managers need to beaware that more is not necessarily better and onecannot always expect higher featured product to bestrictly preferable to low quality products. However,our results also point to the fact that this disutility iscritical and useful as an instrument for consumer seg-mentation. This explains why we continue to see mostsoftware offered in multiple versions, while strictlyzero-marginal cost information goods vendors shouldreally find versioning suboptimal.

Further, as opposed to extant work on informationgoods that often considers versioning decisions sepa-rately from product development, we consider themsimultaneously. Results from this analysis tell us that

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firms need not produce the highest-possible qualityfor the highest-type consumer in the market. Such aresult is counter to the prevailing notion that qualitybe maximized and surplus be extracted through price.Further, our observation that highest types in the mar-ket should be served one quality while lower typesget their individualized quality is also an uncommonresult and stems from our ability to endogenize highestquality development.From a managerial viewpoint, differentiating ver-

sioning costs from marginal costs is an important con-tribution in that it clearly tells a product designerthat creating multiple versions where each versionmay have associated costs can actually be detrimen-tal to high types in the market. In other words, anincrease in versioning costs reduces the segment ofhigh types who get a custom feature-set. Finally, tech-nological advancements in cloud-based delivery area fairly attractive way of provisioning software; ourresults suggest that this may not necessarily be accom-panied by market expansion, and that the tariff struc-ture of the cloud vendor plays a large role. In fact, if thepayment is such that the firm pays the cloud vendor forevery feature-consumer served, then the market mayindeed contract. The final contribution of this work isthat it categorically identifies marginal cost (or usagecost) as the sole reason for optimality of versioning.

All our findings are fairly general even though weuse a specific form of the utility function for the con-sumer. The generalizability of our results is showcasedfrom the analysis conducted for Proposition 6, wherewe relaxed the assumption of the specific form of theutility function. We observed that the main insightsfrom our results that the lower end of the marketremains uncovered, the existence of the menu for themid-segment of the market and the top-quality distor-tion for the upper end of the market remains the same.

Appendix. Table of Notations and Proofs ofLemmas and Propositions

Table of Notationsθ ∈ [

¯θ, θ̄]θ∈�+ Market definition—Consumer types

distributed with pdf f (θ) and cdf F(θ)satisfying the monotone hazard rate property

θ̃ The misrepresented value of customer typewhen true customer type is θ

q ∈ �+ Qualityp ∈ �+ Price

λ (λ > 0) Usage cost parameterc Convex cost coefficient of creating the highest

qualityU(θ, q , p)� θq − λq2 − p Utility function{q∗(θ), p∗(θ)} Optimal menu or quality-price schedule

Full Information

θ̂∗ Index of the highest consumer type who will receive apersonalized quality

q̂∗ q̂∗ � q(θ̂∗)—highest quality produced in the market

Information Asymmetry

θ∗L Index of the lowest consumer type who will be served inthe market

θ∗H Index of the highest consumer type who will receive apersonalized quality

q∗H q∗H � q(θ∗H)—highest quality produced in the market

Proof of Lemma 1Using the analysis in Section 2.1 for the pricing and qualityoffered to the consumers, Equation (2) can be rewritten as∫ θ̂

¯θ

θ2

4λ f (θ) dθ+

∫ θ̄

θ̂

[θθ̂

2λ − λ[θ̂

2λ

]2]f (θ) dθ− c

[θ̂

2λ

]2

.

After integrating by parts and simplifying the above expres-sion becomes

14λ

[θ̂2F(θ̂)−

∫ θ̂

¯θ

2θF(θ)dθ]+θ̂

2λ

[θ̄− θ̂F(θ̂)−

∫ θ̄

θ̂

F(θ)dθ]

− θ̂2

4λ [1−F(θ̂)]− cθ̂2

4λ2 .

Expressing G(θ) �∫ θ

θF(s) ds and H(θ) �

∫ θ

θG(z) dz, the

above expression can be further simplified to

14λ [θ̂

2F(θ̂) − 2[θ̂G(θ̂) −G(¯θ)θ]+ 2[H(θ̂) −H(

¯θ)]]

+θ̂

2λ [θ̄− θ̂F(θ̂) −G(θ̄)+G(θ̂)] − θ̂2

4λ [1− F(θ̂)] − cθ̂2

4λ2 .

We represent the above expression by the symbol E, and con-sequently dE/dθ̂ � (1/(2λ))[[θ̄ − θ̂] − [G(θ̄) −G(θ̂)] − cθ̂/λ].The first-order condition can, therefore, be expressed asθ̄ − θ̂ � G(θ̄) − G(θ̂) + cθ̂/λ. Further, d2E/dθ̂2 � −1/(2λ) +F(θ̂)/(2λ) − c/(2λ2) < 0, since F[θ̂] ≤ 1. Thus, E is strictlyconcave in θ̂, and so an internal solution is possible. Also,note that at c � 0, the expression θ̄ − θ̂ � G(θ̄) −G(θ̂)+ cθ̂/λreduces to θ̄ − θ̂ � G(θ̄) −G(θ̂). It is easy to see that θ̂∗ � θ̄ isa solution to this equation. Further, no other value of θ canbe a solution to θ̂, since that can happen only when G(θ) isa linear function of θ. This would imply F(θ) �1, or that allprobability is a mass at one point and there is no distribu-tion of consumers. Since this is not the case, we conclude thatθ̂∗ � θ̄ is the unique solution to θ̄ − θ̂ � G(θ̄) − G(θ̂). Alsonote that dE/dθ̂ |c>0, θ̂�θ < dE/dθ̂ |c�0, θ̂�θ ∀θ. Thus, it must bethat θ̂∗ |c>0 < θ̂

∗ |c�0. Thus, the solution to θ̄− θ̂�G(θ̄)−G(θ̂)+cθ̂/λ is unique and internal.

Since θ̂∗ < θ̄, the implication is that consumers with θ > θ̂∗do not receive their most efficient quality. Further, all theseconsumers are served with the quality θ̂∗/(2λ). The remain-ing consumers with θ ∈ [

¯θ, θ̂∗] are served their most efficient

quality, since the firmmaximizes the consumers’ surplus andthen fully extracts that surplus to maximize its profits. �

Proof of Lemma 2 and Proposition 1The optimization problem for the vendor is given by

max{q(θ), p(θ)}

∫ θ̄

¯θ

p(θ) f (θ) dθ

s.t. U(θ) ≥ 0, (IR)U(θ) ≥Uθ(θ̃). (IC)

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Chellappa and Mehra: Versioning of Information GoodsManagement Science, Articles in Advance, pp. 1–17, ©2017 INFORMS 13

We first focus on the IC condition. Suppose the vendoroffers a quality/feature-price schedule {q(θ), p(θ)} for everytype θ. To make sure that the consumers self-select into buy-ing the appropriate version, it must be that each consumermaximizes her surplus by truthfully revealing her type θ.In other words, the consumers’ incentive compatibility con-straints (ICs) must be satisfied. We represent the utility of aconsumer of type θ who declares her type to be θ̃ as Uθ(θ̃).Hence, it must be that

Uθ(θ) ≥Uθ(θ̃)⇒ θq(θ) − λq2(θ) − p(θ) ≥ θq(θ̃) − λq2(θ̃) − p(θ̃) (A.1)

for any (θ, θ̃) ∈ [¯θ, θ̄] × [

¯θ, θ̄]. Similarly, for a consumer of

type θ̃, it must be true that declaring herself to be of type θwould result in lower utility for her. Corresponding to Equa-tion (A.1), we get

Uθ̃(θ̃) ≥Uθ̃(θ)⇒ θ̃q(θ̃) − λq2(θ̃) − p(θ̃) ≥ θ̃q(θ) − λq2(θ) − p(θ). (A.2)

Adding Equations (A.1) and (A.2), we get

[q(θ) − q(θ̃)][θ− θ̃] ≥ 0. (A.3)

Thus the incentive-compatibility constraint requires that theschedule of features q(θ) has to be nondecreasing, i.e.,

q′(θ) ≥ 0. (A.4)

Further, incentive compatibility also implies that truthfulrevelation of one’s type would result in utility maximiza-tion. Thus, for a consumer of type θ, it must be thatdUθ(θ̃)/dθ̃ |θ̃�θ � 0 because of the appropriate first-order con-ditions. This is simplified as

θq′(θ) − 2λq(θ)q′(θ) − p′(θ)� 0. (A.5)

For Equation (A.5) to be meaningful, the utility functionUθ(θ̃) must also satisfy the second-order condition, i.e.,d2Uθ(θ̃)/dθ̃ |θ̃�θ < 0. This requirement can be simplified to

θq′′(θ) − 2λ[q′2(θ)+ q(θ)q′′(θ)] − p′′(θ) < 0. (A.6)

Differentiating Equation (A.5) with respect to θ, we get

q′(θ)+ θq′′(θ) − 2λ[q′2(θ)+ q(θ)q′′(θ)] − p′′(θ)� 0. (A.7)

Substituting from Equation (A.7) in (A.6) we obtain q′(θ) ≥ 0.From Equation (A.4), we know that this condition is requiredfor truth revelation. Thus, the second-order conditions donot impose any further constraints. For local ICs to sat-isfy globally, we need that the crossing property or Spence-Mirrlees condition to be satisfied. Since, the cross-derivative(∂2U(q , p , θ)/∂q∂θ � ∂(θ − 2λq)/∂θ � 1) has a constant sign,the requisite conditions are met.

Next, we simplify the objective function utilizing the con-ditions imposed by the incentive compatibility (IC) constraintand expressed in Equation (A.5). Note that

U(θ)� θq(θ) − λq2(θ) − p(θ). (A.8)

Differentiating both sides of the above equation with respectto θ, we get

U′(θ)� q(θ)+ θq′(θ) − 2λq(θ)q′(θ) − p′(θ). (A.9)

Utilizing Equation (A.5), we can simplify Equation (A.9) to

U′(θ)� q(θ). (A.10)

Integrating Equation (A.10) between the limits¯θ and θ, we

get U(θ) − U(¯θ) � ∫θ

¯θ q(y) dy. Since the participation con-

straint of the lowest-type consumer must bind, we haveU(

¯θ)� 0. Hence, we have

U(θ)�∫ θ

¯θ

q(y) dy. (A.11)

Using Equations (A.8) and (A.11), we canwrite p(θ)�θq(θ)−λq2(θ) − ∫θ

¯θ q(y) dy. Thus, we can now rewrite the vendor’s

objective function to∫ θ̄

¯θ

[θq(θ) − λq2(θ)] f (θ) dθ−∫ θ̄

¯θ

[∫ θ

¯θ

q(y) dy]

f (θ) dθ.(A.12)

Using Fubini’s theorem we get∫ θ̄

¯θ

[∫ θ

¯θ

q(y) dy]

f (θ) dθ

�

[ [∫ θ

¯θ

q(y) dy]F(θ)

] θ̄¯θ

−∫ θ̄

¯θ

F(θ)q(θ) dθ.

Using the fact that F(θ̄)� 1 and F(¯θ)� 0, we can simplify the

right-hand side of the above equation to∫ θ̄

¯θ[1−F(θ)]q(θ) dθ.

Thus, we can further simplify the expression in (A.12) to∫ θ̄

¯θ

[θ− λq(θ) − 1− F(θ)

f (θ)

]q(θ) f (θ) dθ. (A.13)

At this point, we ignore the constraints and do an uncon-strained optimization. We later check that the constraints aresatisfied. By employing pointwise maximization, we need toonly maximize the integrand with respect to q(θ). This gives

q∗(θ)�θ− (1− F(θ))/ f (θ)

2λ (A.14)

We can now analyze the quality menu used to serve the mar-ket using Equation (A.14). Further, the quality being servedincreases with the consumer index until the highest possi-ble quality q̂ is reached (since (1 − F(θ))/ f (θ) is decreasingin θ). Hence, q∗(θ) is increasing in θ, which is exactly whatwe need to satisfy the constraint specified in Equation (A.4).

Note that the marginal consumer who is served gets aquality of 0. Let this consumer be indexed by θ∗L . Thenwe have

θ−[

1− F(θ)f (θ)

]� 0. (A.15)

The solution to the above equation, θ∗L , provides the indexof the lowest type of consumer who is served. Let the indexof the lowest consumer type who is served with full qualitybe θ∗H . This point is the solution to

qH �θ− (1− F(θ))/ f (θ)

2λ or

θ− 1− F(θ)f (θ) − 2λqH � 0.

(A.16)

Finally, note that θ∗H > θ∗L since the terms in the Equa-tions (A.15) and (A.16) are identical except for an additionalnegative constant term in Equation (A.16). Hence, versioningis optimal when consumers suffer from no free disposal. �

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Chellappa and Mehra: Versioning of Information Goods14 Management Science, Articles in Advance, pp. 1–17, ©2017 INFORMS

Proof of Lemma 3The first-order condition of Expression (8) with respect to qHyields ∫ θ̄

θ∗H

[θ− 2λq∗H −

[1− F(θ)

f (θ)

] ]f (θ) dθ � 2cq∗H

This can be simplified to

θ∗H[1− F(θ∗H)]� 2q∗H[λ[1− F(θ∗H)]+ c]. (A.17)

Substituting θ∗H in place of θ in Equation (A.16) and solv-ing it simultaneously with Equation (A.17), we obtain θ∗Hand q∗H . �

Proof of Proposition 2Proof that highest quality in full information is greater than high-est quality under information asymmetry: We need to provethat optimal highest quality in the full information case isgreater than the optimal highest quality in the incompleteinformation situation. We represent the objective function ofthe vendor under complete information (Expression (2)) as afunction of the highest quality q by OC(q). Hence, we have

dOC(q)dq

����q�q∗H

�

∫ θ̄

θ̂

[θ− 2λq∗H] f (θ) dθ− 2cq∗H

Substituting the value of 2cq∗H from Equation (A.17) in theabove equation, we get:

dOC(q)dq

����∗q�qH

�

∫ θ̄

θ̂

[θ−2λq∗H] f (θ)dθ

−∫ θ̄

θ∗H

[θ− 1−F(θ)

f (θ)

]f (θ)dθ+2λq∗H[1−F(θ∗H)].

Using the fact that q∗H � θ̂∗/(2λ), the above equation can beeasily simplified to

dOC(q)dq

����q�q∗H

�

∫ θ∗H

θ̂

[θ− θ̂] f (θ)dθ+∫ θ̄

θ∗H

[1−F(θ)]dθ.

(A.18)Note that θ̂∗(q∗H)� 2λq∗H and from Lemma 2, θ∗H(q∗H)� 2λq∗H +

(1− F(θ∗H(q∗H)))/ f (θ∗H(q∗H)). Clearly, it must be that θ∗H(q∗H) >θ̂∗(q∗H). This implies that the first term on the right-handside of Equation (A.18) must be positive. Also, the secondterm must be positive, since F(θ) < 1 for θ ≤ θ̄. Thus, wehave shown that dOC(q)/dq |q�q∗H

> 0. Further, we know thatdOC(q)/dq |q�q̂∗ � 0 and that OC(q) is a concave function in q.Thus, q̂∗ > q∗H . This completes the proof.

Proof that consumers suffer a quality distortion on the low sideunder incomplete information:

(A) Consider θ <min{θ̂∗ , θ∗H}From Proposition 1, the quality served under full informa-tion is q∗(θ) � θ/(2λ) and from Proposition 2, the qual-ity served under incomplete information is q∗(θ) � (θ −(1 − F(θ))/ f (θ))/(2λ). Since F(θ) < 1, it is obvious that thequality served under information asymmetry is lower.

(B) Consider θ >max{θ̂∗ , θ∗H}All such consumers are served with quality q̂∗ under full in-formation and q∗H under information asymmetry. We alreadyproved that q̂∗ > q∗H above. Hence, again, lower quality isserved under information asymmetry.

(C) Consider min{θ̂∗ , θ∗H} ≤ θ ≤max{θ̂∗ , θ∗H}Suppose θ̂∗ <θ∗H . So all consumers in this rangewill be servedquality q̂∗ under full information and a quality less than q∗Hunder information asymmetry since q∗(θ) is increasing (since(1− F(θ))/ f (θ) is decreasing in θ). Further, q̂∗ > q∗H . Hence, areduced quality is served under information asymmetry.

Suppose θ̂∗ > θ∗H . The consumer indexed by θ∗H will beserved quality q∗H under information asymmetry. Because oflogic similar to (A) above, this consumer must be served ahigher quality under full information. Further, as θ increases,the quality under information asymmetry remains at q∗H ,whereas the quality served under full information increases(since θ/(2λ) is increasing in θ). Hence, all consumers inthis range are served a reduced quality under informationasymmetry. �

Proof of Proposition 3For this proof, we need to first consider the comparativestatics of θ̂∗. So, differentiating the equation from Lemma 1,θ̄− θ̂∗ � G(θ̄) −G(θ̂∗)+ cθ̂∗/λ with respect to c, we get

− dθ̂∗

dc+

dG[θ̂∗]dθ̂∗

∗dθ̂∗

dc− θ̂

∗

λ− cλ

dθ̂∗

dc� 0.

This can be rewritten as (dθ̂∗/dc)[1 − F[θ̂∗] + c/λ] � −θ̂∗/λ.Since F[θ̂∗] ≤ 1, it must be that dθ̂∗/dc < 0.

Similarly, differentiating the equation in Lemma 1 withrespect to λ, we get (dθ̂∗/dλ)[1−F[θ̂∗]+ c/λ]� cθ̂∗/λ2. Fromthis, we can easily see that dθ̂∗/dλ > 0.

Comparative statics of θ∗L is fairly straightforward, as fromLemma 2 it is easy to see that θ∗L does not depend on either cor λ.

For the comparative statics of θ∗H , we know from Lemma 3that θ∗H is obtained by solving θ∗H − (1 − F(θ∗H))/ f (θ∗H) −2λq∗H � 0. Clearly, as λ increases, the solution to the aboveequation, i.e., θ∗H increases. Also, as c increases, q∗H reducesand hence θ∗H reduces. �

Proof of Proposition 4We solve for q∗H and q∗L exactly as in the case without version-ing costs:

θ∗H −1− F(θ∗H)

f (θ∗H)− 2λq∗H � 0

θ∗L −1− F(θ∗L)

f (θ∗L)� 0.

(A.19)

In the first stage, objective function of the vendor is given by∫ θ̄

¯θ

[θ−λq(θ)− 1−F(θ)

f (θ)

]q(θ) f (θ)dθ− k(θ∗H −θ∗L)− c(q∗H)2

where k > 0.The first-order condition of the above expression with

respect to qH yields∫ θ̄

θ∗H

[θ− 2λq∗H −

1− F(θ)f (θ)

]f (θ) dθ � k

d(θ∗H − θ∗L)dq∗H

+ 2cq∗H

Differentiating (A.19) w.r.t qH :

dθ∗Hdq∗H−− f 2(θ∗H) − f ′(θ∗H)(1− F(θ∗H))

f 2(θ∗H)·

dθ∗Hdq∗H

� 2λ.

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Chellappa and Mehra: Versioning of Information GoodsManagement Science, Articles in Advance, pp. 1–17, ©2017 INFORMS 15

Substituting dθ∗H/dq∗H from above into the first-order condi-tion, it can be written as follows:∫ θ̄

θ∗H

[θ− 2λq∗H −

1− F(θ)f (θ)

]f (θ) dθ

�kλ

(1+ ([1− F(θ∗H)] f ′(θ∗H))/(2 f 2(θ∗H)))+ 2cq∗H

This can be further simplified to

θ∗H[1− F(θ∗H)] �kλ

(1+ ([1− F(θ∗H)] f ′(θ∗H))/(2 f 2(θ∗H)))+ 2q∗H[λ(1− F(θ∗H))+ c]. (A.20)

Solving Equations (A.19) and (A.20) simultaneously, we get

θ∗H −(1− F(θ∗H))[c + λ(1− F(θ∗H))]

c f (θ∗H)

+2kλ2

c[2+ f ′(θ∗H)((1− F(θ∗H))/ f (θ∗H)) · (1/ f (θ∗H))]� 0

One can see that −(1 − F(θ∗H))[c + λ(1 − F(θ∗H))]/(c f (θ∗H)) isincreasing in θ∗H (as (1 − F(θ∗H))/ f (θ∗H) is decreasing). Also,as long as the monotone hazard rate property holds

d((1− F(θ∗H))/ f (θ∗H))dθ∗H

< 0

⇒− f 2(θ∗H) − f ′(θ∗H)(1− F(θ∗H))

f 2(θ∗H)< 0

⇒f ′(θ∗H)(1− F(θ∗H))

f 2(θ∗H)+ 1 > 0

⇒ 2kλ2

c[2+ f ′(θ∗H)((1− F(θ∗H))/ f 2(θ∗H))]> 0.

Therefore, the left-hand side is bigger compared to the casewhere k � 0, implying lower θ∗H to be the solution in the pres-ence of versioning costs. �

Proof of Proposition 5We can rewrite the maximization problem for the vendorwith cloud provisioning as follows:

max{q(θ),p(θ)}

{∫ θ̄

θ

p(θ) f (θ)dθ−δ∫ θ̄

θ

q(θ) f (θ)dθ−c[qH]2}

s.t. U(θ)≥0 (A.21)U(θ)≥U(θ̃) ∀ θ̃,θ

where δ > 0, and the second term represents the cost to thefirm to host the software as a service in the cloud. Note thatwe assume that the cost of hosting a version with more fea-tures is higher. Further, the cost of serving a given versionincreases as it serves more customers. Since the firm hoststhe software, the customers incur a reduced NFD disutilitywhich is represented by λ(1− d)q2(θ), where 0 < d < 1.

As in the proof for Proposition 2, we first focus on the ICcondition and simplify the objective function to∫ θ̄

θ

[θ− λ(1− d)q(θ) − 1− F(θ)

f (θ) − δ]q(θ) f (θ) dθ. (A.22)

An unconstrained optimization employing pointwise maxi-mization gives

q∗(θ)�θ− (1− F(θ))/ f (θ) − δ

2λ(1− d) (A.23)

We can now analyze the quality menu used to serve themarket using the above equation. Note that the marginal cus-tomer who is served gets a quality of 0. Let this customer beindexed by θ∗L . Then, the solution to the equation

θ− 1− F(θ)f (θ) − δ � 0 (A.24)

θ∗L , provides the index of the lowest type of customer who isserved. This θ∗L is greater compared to the case of softwarebeing sold as a product. Let the index of the lowest customertype who is served with full quality be given by θ∗H . Thispoint can be obtained by solving the following equation:

θ∗H −1− F(θ∗H)

f (θ∗H)− 2λ(1− d)q∗H − δ � 0. (A.25)

Note that θ∗H > θ∗L since the terms in the Equations (A.24) and

(A.25) are identical except for an additional negative constantterm in Equation (A.25). Hence, versioning is optimal in thissituation.

In the first stage, the objective function is∫ θ̄

θ∗H

[θ− 2λ(1− d)q∗H −

1− F(θ)f (θ) − δ

]q(θ) f (θ) dθ− c(q∗H)2

The first-order condition of the above equation with respectto q∗H yields∫ θ̄

θ∗H

[θ− 2λ(1− d)q∗H −

1− F(θ)f (θ) − δ

]f (θ) dθ � 2cq∗H

which can be written as

(θ∗H − δ)[1− F(θ∗H)]� 2q∗H[λ(1− d)(1− F(θ∗H))+ c]. (A.26)

Solving Equations (A.25) and (A.26), we get

δ � θ∗H −(1− F(θ∗H))[c + λ(1− F(θ∗H))(1− d)]

c f (θ∗H)

q∗H �(1− F(θ∗H))2

2c f (θ∗H)

(A.27)

Considering the earlier equation, we can see that the left-hand side is increasing in θ∗H (as (1−F(θ∗H))/ f (θ∗H) is decreas-ing). Also, both the left-hand side and the right-hand sideare bigger compared to the product case where d � δ � 0,implying when δ is sufficiently large or d is sufficiently small,optimal θ∗H is greater, and q∗H is lesser compared to the prod-uct case, and vice versa. �

Proof of Proposition 6We are able to show that Lemma 4 and Proposition 5 areapplicable to a more general utility function (and thereforeapplicable to our specific functional form aswell). So we shalluse new notation for this proof all along.

Let consumers of type θ enjoy a utility u(θ, q) − p whenchoosing some quality q and paying a monetary transfer p.

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Chellappa and Mehra: Versioning of Information Goods16 Management Science, Articles in Advance, pp. 1–17, ©2017 INFORMS

We assume that the standard sorting condition, uθq(θ, q) > 0holds for positive q with u(0, q) ≥ 0 and u(θ, 0)� 0.

Providing quality q is costly to the monopolistic firm witha marginal cost c(q)with c(0)� 0. The goal of the monopolis-tic firm is to design a price schedule {q(θ), p(θ)} to maximizeits profit. However, the monopolistic firm cannot arbitrarilychoose {q(θ), p(θ)}, as the consumer’s ability to choose qmust be respected (self-selection condition). Incentive com-patibility requires that q(θ) weakly increases in θ(q′(θ) ≥ 0)and the consumer’s marginal surplus is v′(θ)� uθ(θ, q(θ)) asgiven by the envelope theorem (with inequality constraints).The payoff per consumer can be written as the valuation perconsumer less the consumer surplus and the marginal cost

R(θ)� u(θ, q(θ)) − v(θ) − c(q(θ)). (A.28)

So the total expected profits for the firm is given by

max{qθ }

∫ θ̄

¯θ

R(θ) f (θ) dθ

s.t. q(θ) ≤ qH , q′(θ) ≥ 0.(A.29)

Since the sorting condition, uθq(θ, q) > 0 holds, andu(θ, 0)� 0, for any q > 0, the individual rationality constraintis satisfied everywhere if it holds at the lowest type. Sincev(θ)�

∫ θ

θuθ(t , q(t)) dt. Standard transformation yields

max{q(θ), p(θ)}

∫ θ̄

θ

(u(θ, q(θ))− 1−F(θ)

f (θ) uθ(θ, q(θ))− c(q(θ)))

· f (θ) dθs.t. q′(θ) ≥ 0

q(θ) ≤ qH(A.30)

The firm essentially needs to find q that maximizes thesurplus extracted from each consumer of type θ; i.e., we caneasily employ pointwise maximization to the term inside theintegrand in Equation (A.30).

Our aim is to show that versioning can be the optimalstrategy only if

Rqq(θ, q) < 0 for some θ ∈ [¯θ, θ̄]. (A.31)

In other words, the marginal rent to the firm needs to bestrictly concave for some θ ∈ [

¯θ, θ̄] to make versioning an

optimal monopolist strategy.Suppose that the payoff function satisfies Rqq(θ, q) � 0 for

all θ ∈ [θ, θ̄]. It implies that the first-order derivative of thefirm’s payoff function uq(θ, q) − ((1 − F(θ))/ f (θ))uθq(θ, q) −cq(q) is invariant with respect to q. Then the firm willsimply assign qH to consumers of type θ with uq(θ, q) −((1 − F(θ))/ f (θ))uθq(θ, q) − cq(q) ≥ 0, whereas assigningzero quality to consumers of type θ with uq(θ, q) −((1− F(θ))/ f (θ))uθq(θ, q) − cq(q) < 0.

Given uθq(θ, q)> 0 and the monotone hazard rate, we have

Rθq(θ, q) �∂∂θ

[uq(θ, q) −

1− F(θ)f (θ) uθq(θ, q) − cq(q)

]�

(1− d

dθ

(1− F(θ)

f (θ)

))uθq(θ, q) > 0. (A.32)

Because the marginal rent is increasing in θ for all val-ues of q, it must be unique if there exists some θ such that

uq(θ, q) − ((1 − F(θ))/ f (θ))uθq(θ, q) � 0. Denote this uniquepoint by θ̂. Then q(θ) � qH for all θ ∈ [θ̂, θ̄] and q(θ) � 0otherwise. It implies that, if there is a marginal type θ̂ of con-sumers the firm finds profitable to serve, then all consumerswith θ > θ̂ will be served with quality qH . This is the optimalschedule with Rqq(θ, q)� 0 for all θ ∈ [

¯θ, θ̄].

Since versioning by definition requires that there exist atleast two different positive qualities, the proof above showsthat versioning is not the optimal strategy if Rqq(θ, q) � 0 forall θ ∈ [θ, θ̄].

Since R( · ) is defined by Equation (A.28), Equation (A.31)can be true if

(a) consumer utility function itself is strictly concave, i.e.,uqq(θ, q) < 0; or/and

(b) if u(θ, q) is monotonically increasing concave or linear,the marginal cost c(q) suffered by the firm is strictly convexi.e., cqq(q) > 0.

From a monopolist’ point of view (a) and (b) are duals ofeach other, since the net surplus that can be extracted is thesame.

Endnotes1Note that extant models of vertical segmentation use the form θqfor the utility function. In our paper, we introduce the nonmonotonicform given in Equation (1). The traditional monotonic form assumesfree disposal, while the form of utility used in this paper abstractsno-free-disposal. By setting λ � 0, we are able to easily accommodateand compare with the traditional form.2A stylized form (quadratic) for the utility function is used here tocompare our results directly with those from extant literature on ver-sioning, which commonly uses the linear form. Note, however, thatin Section 3.3 we consider a general form of the utility functionthat accommodates the quadratic form as well as any other functionthat satisfies nonmonotonicity properties.3http://answers.microsoft.com/en-us/office/forum/office_2010-outlook/search-results-will-not-print-as-a-table/5f62908c-8b02-4880-9c4c-19d411761ffb (accessed August 1, 2016).4A discrete distribution with point masses lends itself to an easierand lucid managerial interpretation. For example, consider a dis-crete distribution with sufficient granularity of represented typesthat approximately conforms to the density curve in a continuousversion. In this setting the shifting threshold of consumer types canbe interpreted to mean that the vendor offers more or fewer versions.5Along the lines of standard principal-agent models, second-best hererefers to the situationwhere themonopolist (principal) has to accom-modate the consumers’ (agents’) incentive constraints as opposed tothe full information case (referred to as first-best).

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