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MANAGEMENT SCIENCEVol. 51, No. 10, October 2005, pp. 14941504issn 0025-1909 eissn 1526-5501 05 5110 1494

informs

doi 10.1287/mnsc.1050.0400 2005 INFORMS

Two-Sided Network Effects: A Theory of

Information Product DesignGeoffrey G. Parker

Tulane University, New Orleans, Louisiana 70118, [email protected]

Marshall W. Van AlstyneBoston University, and Massachusetts Institute of Technology, Boston, Massachusetts 02215, [email protected]

How can firms profitably give away free products? This paper provides a novel answer and articulatestrade-offs in a space of information product design. We introduce a formal model of two-sided networkexternalities based in textbook economicsa mix of Katz and Shapiro network effects, price discrimination,and product differentiation. Externality-based complements, however, exploit a different mechanism than eithertying or lock-in even as they help to explain many recent strategies such as those of firms selling operatingsystems, Internet browsers, games, music, and video.

The model presented here argues for three simple but useful results. First, even in the absence of competition,a firm can rationally invest in a product it intends to give away into perpetuity. Second, we identify distinctmarkets for content providers and end consumers and show that either can be a candidate for a free good.Third, product coupling across markets can increase consumer welfare even as it increases firm profits.

The model also generates testable hypotheses on the size and direction of network effects while offeringinsights to regulators seeking to apply antitrust law to network markets.

Key words : network effects; network externalities; two-sided markets; free information; business models;strategic complements; product design

History : Accepted by Rajiv D. Banker, information systems; received April 29, 2003. This paper was with theauthors 5 months for 2 revisions.

1. IntroductionThis paper seeks to explain strategic pricing behav-ior and product design decisions in network markets.Given the seemingly anomalous practice of free-pricing, how is it possible that firms are willing tosubsidize information and related products, appar-ently expecting neither future consumer exploitationnor tying? Why do firms give away applicationsdevelopment toolkits, portable document readers, andInternet browsers without metering tie-ins to thosesame consumers? What theory explains this economiclogic?

In answer, we find that designing matched prod-uct pairs and discounting one relative to the inde-pendent goods case changes the shape of demandin markets joined by network effects. The insight isthat characterizing network markets may require notonly product standardization, essential to demandeconomies of scale (Katz and Shapiro 1985, Farrell andSaloner 1986), it may also require recognizing sharpdistinctions between consumer types. This is essentialfor managing demand interdependence, implement-ing price discrimination, and raising barriers to entry.

Contributing to the recent two-sided network exter-nality literature, we introduce a novel mechanism

(i) that explains firms unbundled component sales

and pricing strategies, (ii) that shows which half of atwo-sided network market to discount, (iii) that is dis-tinct from tying and traditional multimarket price dis-crimination, and (iv) whose formulation is robust inresults to multiple specifications of demand. Of man-agerial interest is the proposition that discountingan unbundled component can increase profits to thepoint where negative prices become optimal. Thishelps not only the monopolist but also the oligopolyfirm seeking barriers to entry. Profits increase con-ditionally, however, on promoting network effectsthrough clever product design. They depend also onthe correct choice of market to discount. Of legal and

economic interest is the implied industry concentra-tion and difficulty applying antitrust tests of preda-tion. Pricing below marginal cost maximizes profitseven in the absence of competition. Platform interme-diaries that internalize these externalities can improveconsumer welfare as well as profit.

The paper proceeds as follows. Section 2 providesa review of related literature in network externalities,multiproduct pricing, and bundling. In 3, we firstdevelop a model of externality-based complements.We find the conditions for a subsidy market to exist

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Parker and Van Alstyne: Two-Sided Network EffectsManagement Science 51(10), pp. 14941504, 2005 INFORMS 1495

and identify which market receives the subsidy. Wealso demonstrate how the key insight is robust to sev-eral modeling generalizations. Section 4 offers furtherapplications and extensions and 5 concludes.

2. Historical Context andRelated Literature

Several branches of literature yield insight forthis research, including network externalities, multi-product pricing, and to a lesser extent bundling. In theclassic network externality story (Katz and Shapiro1985, Farrell and Saloner 1986, Arthur 1989), demandeconomies of scale cause growth in an existing stockof consumer value as new consumers join the net-work. Various authors have used network effects toexplain the popularity of QWERTY and VHS (Arthur1994), ad subsidies of content in circulation indus-tries (Chaudhri 1998), and the importance of stan-

dards and switching costs in network economies(Katz and Shapiro 1985; Shapiro and Varian 1998,1999). In contrast, we consider a matched-market,chicken-and-egg problem, supporting an alternateinterpretation of conventional wisdom. We distin-guish between intramarket and intermarket networkexternalities. Producers want consumers and con-sumers want producers before either prefers a newformat.

An incumbent firm producing content for one for-mat probably does not welcome entry by a com-peting firm producing similar content if there is noprofitable exchange between them. This mitigates a

network effect within one side of the market. Buyersin the end-consumer market, however, welcome entry

Table 1 Two-Sided Market Examples

Product category Market 1 Intermediary Market 2

Portable documents Document reader Adobe Document writerCredit cards Consumer credit Issuing bank Merchant processingOperating systems Complementary applications Microsoft, Apple, Sun Systems developer toolkits

Plug-ins Applications software Microsoft, Adobe Systems developer toolkits

Ladies nights Mens admission Bars, restaurants Womens admission

TV format Color UHF, VHF, HDTV Sony, Phillips, RCA Broadcast equipmentBroadcast & publishing Content Magazine publishers, TV, Advertisements

radio broadcasters

Computer games Game engine/player Ubisoft, ID, valve, electronic arts Level editors

Auctions Buyers E-Bay, Christies, Sothebys SellersAcademic journals Articles Management Science Author submissions

Recruiting Applicants Monster.com EmployersReservation systems Travelers Expedia, Travelocity, Orbitz Hotels, airlines, rental carsShopping malls Shoppers Mall of America StoresStreaming audio/video Content Real audio, Microsoft, Apple ServersPaid search Searchers Google.com MarketersStock exchange Equity purchasers NYSE, NASDAQ Listed companiesHome real estate Home buyers Real estate agents Home sellers

Notes. This table shows how one side of a two-sided network market receives a discounted, free, or even subsidized good (indicated with ). In generalthough not always, Market 1 can be interpreted as the user/consumer market and Market 2 can be interpreted as the producer/developer market. We providea test for which side receives the free good below.

because it increases the prospect of a viable formatshould the incumbent fail. It also increases varietywhile possibly lowering prices. This increases boththe value to individuals and the number of individ-uals willing to switch formats. Thus, in the presentexample, the externality runs from content creators to

end consumers.Conversely, consider the end consumers originalchoice of VHS versus Beta. Initially, at least, con-sumers probably care less about adding another con-sumer to a new formatit could even bid up pricesif supply is limitedthan they care about the numberand diversity of firms who provide content for thatformat. Producers, however, care immensely aboutthe size of the consumer market. The existence of alarger consumer base makes production under anygiven format more attractive. Again, in the presentarticulation, the externality runs across markets fromconsumers to providers of content.

Importantly, the externality benefit can run acrossmarkets and back again. Both content creators andconsumers do value growth in their own markets,

but this may be mediated by the indirect effect ofthe internetwork externality. At issue is whether own-market entry expands participation on the other sideof each transaction. Content creators may not objectto other content-providing firms if effective consumerdemand rises instead of falls.

Consider, finally, a third participant, the focus ofour attention here, who produces tools to support

both content creators and end consumers. Theseare platform intermediaries. Examples in Table 1

include Sun, Apple, and Microsoft, who support soft-ware developers as well as private and business

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Parker and Van Alstyne: Two-Sided Network Effects1496 Management Science 51(10), pp. 14941504, 2005 INFORMS

buyers; Sony and Phillips, who support the enter-tainment industry as well as households; and firmslike Adobe, who produce portable document writ-ers as well as portable document readers. Applied toservices, eBay coordinates buyers and sellers, whileVisa coordinates merchants and card holders. For

these firms, the chicken-and-egg profit-maximizationproblem is how to grow matched markets. A straight-forward and widely observed solution is to discountone market in order to grow both, and to profit morefrom the other.1 This model favors an intermediarywho, straddling both markets, can set prices more effi-ciently by internalizing these two-sided externalities.Independent firms serving either market separatelylose this advantage.

A key contribution of a two-sided network modelis determining which side receives a discount. Differ-ent firms choose different beneficiaries. In streamingvideo, portable documents, and advertising, for exam-

ple, the industry norm is to subsidize content con-sumers and charge content developers. The opposite,however, holds true for operating systems and mul-tiplayer games in which content developers receivesubsidies and consumers pay to join the network. Weshow how this depends on cross-price elasticities aswell as the relative sizes of the two-sided networkeffects.

This analysis adds to recent literature on two-sidednetwork effects (Armstrong 2002, Caillaud and Jullien2003, Rochet and Tirole 2003) that make precise aform of indirect network effect (Katz and Shapiro1985, Liebowitz and Margolis 1994). Indirect effects

are consumption externalities from purchasing com-patible products such as hardware and software. Thekey distinction for two-sidedness is that networkeffects must cross market populations. When they donot, the same consumer populations choose systemsof compatible products only for themselves. Thesesystems lead to pecuniary externalities, efficientlyhandled through the pricing system (Liebowitz andMargolis 1994). In contrast, two-sided networks yieldtrue externalities in which one population chooses agood affecting another populations choice of a dif-ferent good. In a recent working paper, Rochet andTirole (2004, p. 3) state, To use a celebrated example,

the buyer of a razor internalizes in his purchase deci-sion the net surplus that he will derive from buyingrazor blades. The starting point to the theory of two-sided markets by contrast is that an end-user does notinternalize the welfare impact of his use of the plat-form on other end-users. As in Farrell and Saloner

1 The consequences of mispricing internetwork markets can also besevere. In 2001, the number two auction site, Yahoo, raised sellerfees and listings unravelled, dropping 90% (Hansell 2001). Sellersconcluded that for such fees, they preferred the larger number of

buyers on eBay.

(1985), coordinating purchases becomes essential toavoid inertia, but in two-sided networks, coordinationacross markets matters. Coordination within marketsmay have little effect.

Rochet and Tirole (2003), which focuses on com-peting credit card markets, parallels our work.2 They

model two-sided network effects using multiplicativedemand and symmetric spillovers, and nicely cap-ture multihomingthe decision to carry multiplecredit cards from competing networks. Our frame-work, independently developed, differs in severalimportant respects. (1) We show that key results gen-eralize to alternative demand specifications. (2) Weinclude models that correct for boundary conditionproblems. (3) We account for asymmetric and evennegative network effects so that we can model mar-kets with very different properties.

Caillaud and Jullien (2003) consider a matchmak-ing intermediary, as for dating services. Using linear

demand and a Bertrand pricing model, they explainwhy agents register with more than one service, as inthe case of multihoming credit card services. Undercompetition, two-sided network effects lead one firmto corner the market, or multiple firms to share themarket with zero profits. They also show how trans-action costs reduce total surplus. Our work focuses oninformation goods and product unbundling and gen-eralizes to nonlinear demand. Our problem is also dis-tinct from matching distinct agent pairs as in Mongelland Roth (1991), instead representing an effort toincrease total matched transactions occurring on acommon platform. A useful survey of our results and

others is provided in Armstrong (2002), examiningwhich side of a market is subsidized and whether theoutcome is socially efficient, and Rochet and Tirole(2004), which establishes general definitions for two-sided markets.

2.1. Multiproduct PricingThe fact that lowering price on one half of a matchedpair sells more of the other has precedent in multi-product pricing. This parallels Cournots (1838) obser-vation that discounting zinc sells more copper in the

brass market but without the novelty that sales accrueto different buyers. DeGraba (1996) also presents an

example of firms willing to enter a zero-profit market,relying on declining average cost curves to increasedsales. These mechanisms, however, differ in that anal-ysis proceeds from a model of tying, like that in Whin-ston (1990).

2 The two papers were independently and concurrently developedand were conceived to answer related but different questions.Rochet and Tirole cite an earlier version of this paper, titled Infor-mation Complements, Substitutes, and Strategic Product Design,which was presented at the 1999 Workshop on Information Systems& Economics and was posted to ssrn.com in 2000.

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Two-sided network externality coupling is not tra-ditional second-degree price discrimination. It differsfrom tying razors and blades in that one market neednever consume the complementary good. It also dif-fers from penetration pricing in which a good is sub-sidized initially on expectation of future exploitation.

In both cases, a single consumer internalizes his ownvalue calculation such that price changes affecting onematched item are reflected in willingness to pay forthe other. This property fails for two-sided networkmarkets. Consumers of a portable document readermay, at their discretion, forgo the cost of the portabledocument creator now and forever.

Similarly, two-sided network externality couplingdiffers from traditional third-degree price discrimina-tion. Firms offering nonlinear prices to mixed marketsforce heterogeneous consumers to self-select. Busi-ness travellers choose different options than tourists;heavy users of phones and electricity choose differ-

ent tariffs than light users. Such mechanisms differen-tially extract consumer surplus and transfer it to theseller (Varian 1989). In contrast, platform intermedi-aries operating in two-sided markets seek to profit bytransfering surplus from seller to consumer. Growthon one side of a matched market then induces growthon the other, creating exploitable surplus. This effectcan prove unusually vexing to antitrust regulators

because producer and consumer surplus can movetogether, as we show in Proposition 3.

Further, Rochet and Tirole (2004) note that demandcreationas distinct from surplus divisionviolatesthe Coase theorem (1960). This theorem states that,

regardless of externalities, transactions volume will beefficient as long as property rights are clearly definedand there are no information asymmetries or transac-tion costs. Buyers and sellers will bargain their wayto efficiency; pollution trading rights come to mind asan example.

The Coase theorem fails in the case of two-sidednetwork effects. Property rights, symmetric informa-tion, and zero-cost transactions do not suffice for effi-cient trading volume when it is the presence of oneconsumer type that itself creates value for the othertype. Ladies nights, developer toolkits, ad-supportedcontent, as well as cash-back bonuses and frequentflyer points on credit cards constitute continuing sub-sidies to one side of the market.

A product design strategy that discounts price tozero is aided by the unique properties of information.The key, however, relies less on nonrivalry than onlow marginal costs. When these are negligible, a firmcan subsidize an arbitrarily large market based solelyon fixed initial costs. Providing each new sale or ser-vice then costs the clever product designer nearlynothing in incremental costs. Increased consumptionof information may therefore have the potential to

increase the attractiveness of the proposed designstrategy.

3. MarketsThis section adapts a standard externality model toshow why a firm spends resources creating a prod-

uct it distributes for free when two-sided networkexternalities cross market pairs. It also shows whichmarket receives the discount.

Let the first market represent the general consumeror end-user market, C, and let the second market rep-resent the content creator, developer, or joint-producermarket J. Index price p and quantity q terms by mar-ket such that pc and pj describe prices for consumersand content creators, respectively. Let marginal costsfor information goods be negligible with profit given

by = c + j = pcqc + pjqj. To provide a standardconcave profit function, let this be twice differen-tiable in both choice parameters, have 2/p2i < 0,

i cj, and have a positive Hessian determinant.Accordingly, first-order conditions yield prices in bothmarkets.

Demand is also standard but generalizes to allowa cross-market relationship. Each market has a con-tinuum of consumers willing to buy one discreteunit of a good. If v is arbitrary willingness to pay,

then Dp, with upper bound V, is simplyV

vpdDdp.

We require the distribution of values to be boundedabove and well defined over negative values to avoidcorner solutions. In the linear demand case of Katzand Shapiro (1985), this is simply v V . Inour model, this also has a natural interpretation of

allowing price subsidies, and we generalize to non-linear demand. For i jc we assume Di is twicecontinuously differentiable and decreasing in pi. Wedo not require concavity or convexity of demand butonly restrict it sufficiently to ensure concavity of theprofit function via the Hessian. Paralleling Katz andShapiro, we also model total consumption as the sumof a goods intrinsic demand incremented by a func-tion of network size in the other market. That is, weallow consumption q to rise with network effects aswell as value v as specified next.

Our main point of departure is having externalitiescross markets. The internetwork externality term ejc

measures how much effect purchases in the devel-oper market have on the consumer market. Moreprecisely, ejc is the partial derivative of demand inthe C market with respect to demand in the J mar-ket, or qc/qj. Conversely, ecj determines how mucheffect purchases in the consumer market have on the

joint-producer market. Together, these terms describea pair of demand equations:

qc = Dcpc + ejcDjpj (1)

qj = Djpj + ecjDcpc (2)

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Figure 1 Markets Linked by Two-Sided Network Externalities

qcqj qj qc

pc

pc

pj

pj

*

*

*

*

*

Vc

Qc Qc+ ejcQj

(qc, pc)

Price

Price

Price

QuantityQuantity

(consider Jexternality) (forgo Cprofit)(increase Jprofit)

cj

j

Market C Market J Market C

Vc 1+ejcQj

Qc1

pj

Vj

1pj

Vj

Notes. In the linear case, the left panel shows a network externality that shifts the demand curve out. The optimal independent pair qc pc increases from theno-externality case Qc/2 Vc/2. In the middle panel, a monopolist sets p

j > pj and reaps profits

j . In the right panel, price in the Cmarket falls p

c < pc

in order to take profits in the Jmarket. A higher price in market J then somewhat reduces the total externality impact in market C. Discounting the Cmarketis rational when net profits rise.

Several notational conventions help develop usefulintuition. We refer to the marginal cross-price con-tribution to sales qj/pc = ecjD

cpc as the two-sided

network externality or simply spillover effect. Theimportance of network spillovers, as given by the

ratio r qj/pc/qc/pj, plays an important role insubsequent analysis and will later draw attention towhich market generates relatively more surplus. Forconvenience, we define p, p, and p as the optimal(coordinated across both markets) monopoly, uncoor-dinated (two independent firms making independentdecisions), and no-externality prices, respectively. Ingeneral, p = p = p unless ejc = ecj = 0, and qc = Dcpcunless ejc = 0. Where the context is clear, we simplifynotation by suppressing parameters and write Dcfor Dcpc.

For illustration, the linear demand curves Dcpc =Qc1 pc/Vc and Djpj = Qj1 pj/Vj yield simple

and elegant results in terms of consumer surplus. Letexogenous parameters Qi and Vi, i cj representthe maximum market size and maximum productvalue in the absence of externalities, respectively.A consumption externality from J creates an outwardshift in C demand, raising value by an amountproportional to D1c ejcDj; thus,

Vc rises from Vc toVc1 + ejcQj/Qc1 pj/Vj. In the C market, inter-pretation in terms of surplus space can then be found

by integrating demand3 and defining substitutions.

Two-sided network

Native surplus externality surplusSc QcVc Scj ecjQcVjSj QjVj Sjc ejcQjVc

Native terms are proportional to consumer surplusin each market individually, whereas network terms

3Vc

0Qc1 pc/Vc + ejc Qj1 pj/Vjdpc =

12

QcVc + ejcQjVc1

pj/Vj + 1/2QcVce2jcQ

2jV

2c 1 pj/Vj

2, where Vc = Vc1 +ejcQj/Qc1 pj/Vj. The area of the triangles in Figure 1 gives asimilar expression.

are proportional to the surplus created by consump-tion externalities in the complementary market.

3.1. Monopoly ChoiceA simple linear example then illustrates how the

model functions, although linearity is not required.Consider first a monopolists profit-maximizing deci-sion in the absence of network externalities, ecj =ejc = 0. A straightforward maximization on c =pcQc1 pc/Vc yields p

c = Vc/2, q

c = Qc/2, with prof-

its ofc = VcQc/4 = Sc/4 and similarly for j as in theleft-most panel of Figure 1. Optimal prices and quan-tities are simply half the consumer value and half theconsumer market, respectively.

Allowing for positive spillover, the J market exter-nality shifts the demand curve in the C marketup and to the right relative to the no-externalitycase. Increasing pj, however, reduces the total out-

ward shift in the C market as shown in the right-most panel. Taking externality effects into account,the decision of how much to charge in each mar-ket is analyzed below. This graphic captures the ideathat content providers stimulate demand among con-sumers, just as consumers stimulate interest from con-tent providers.

The number of content providers who actuallyenter the market, however, governs the degree ofshift. The externality or coupling between these twomarkets implies that the more a monopolist can coaxconsumers into adopting one of his products, themore it can charge forand sell ofthe other. Ifthe increment to profit on one complementary goodexceeds the lost profit on the other good, then adiscount or even subsidy becomes profit maximiz-ing. Free-goods markets can therefore exist wheneverthe profit-maximizing price of zero or less generatescross-market network externality benefits greater thanintramarket losses. Firms do, in fact, offer subsi-dized goods when they offer free service and tech-nical expertise. Historically, for example, Microsoftoffered technical help on applications program inter-faces (API) in addition to free systems development

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Figure 2 Intersecting Price Reaction Curves Show Consumer Price pc Rising and Developer Price pj Falling as the Developer-to-Consumer

Externality ejc Rises from 0 to34

to 1110

in the Linear Case with ParametersQc = Qj = Vc = Vj = 1 ] and ejc =

13

1 pc pc

1

pj pj pj3

1 1

1

1 32

1

1

2

32

pc32

1

1

III

III IV

III

III IV III IV

III

2 2

1

21

2

1

2

1

2

1

2

2

2

1

2

2

32

32

121

toolkits, and this may have increased the number ofuser applications developed for its operating system.Apple did not and has since changed its strategy. Thisform of subsidy also addresses an interesting problemof adverse selection arising from a subsidy to a gen-eral audience. Only developers request API support,so the subsidy is not wasted on nondevelopers.

In general, the optimal price pair for markets linkedby network externalities is not obvious. Figure 2 illus-

trates that a pricing decision falls into one of fourquadrants. Regions II and IV represent a free-goodsmarket because a firm charges nothing or spendsmoney to place product in the hands of consumersor developers, respectively. In region I, both marketsare charged positive prices while region III repre-sents a subsidy to both markets and is never profitmaximizing.

In principle, the markets are symmetric; in practice,computer games manufacturers charge consumers butgive developers free toolkits, while streaming mediacompanies give consumers free software players butcharge developers to create content. Interestingly,

banks selling credit card services charge both mer-chant and consumer fees, illustrating region I. Thedecision about which market to subsidize, if any, andwhich to charge, then rests on Corollary 2 below.Before analyzing this case, we first develop condi-tions for optimality in terms of elasticities. Definitionsfor own- and cross-price elasticities yield c = pcD

c/

Dc + ejcDj and cj = ejcpjDj/Dc + ejcDj. Recall also

that = c + j = pcqc + pjqj.

Proposition 1. Network effects on one side of the mar-ket render the matched side more inelastic. Optimal prices,however, also depend on cross-price elasticities and the ratio

of surplus created on both sides. That is, p

c depends oncross-market elasticity of C sales with respect to J pricetimes the spillover ratio. Equivalently, pc depends on cross-market elasticity of J sales with respect to C price timesthe profit ratio:

1 = c + cjr (3)

= c + jcj

c (4)

The condition for a free-goods market to exist is optimalsubsidy price pc 0, which occurs at the point 1 jcr.

Proof. To see that ejc makes c = pcDc/Dc + ejcDj

more inelastic, note that increases push c towardzero, decreasing sensitivity to changes in pc. To estab-lish the unit elasticity condition, use the first-ordercondition on total profit with respect to price in theC market to find

pc= Dc + ejcDj + pcD

c + ecjpjD

c = 0 (5)

which rearranges to 1 = c + ecjpjDc/Dc + ejcDj.

Recall that r= qj/pc/qc/pj = ecjDc/ejcD

j. Mul-

tiplying the final term by 1 = ejcDj/ejcD

j and substi-

tuting for cj provides Equation (3). Then, when pricedecreases through zero pc 0, we have c 0, prov-ing the condition on existence of a free-goods market.Note that although r is undefined when ejcD

j 0,

cjr is still defined because the multiplicative term,ejcD

j cancels out. To produce the final expression,

insert definitions for elasticities and simplify to con-firm that jc/cj/j/c = r. Substituting this equal-ity in Equation (3) produces Equation (4). Note that

symmetry implies a parallel set of expressions in the Jmarket where the optimality condition is 1 = j +jc1/r.

Intuitively, rising network effects from J to C allowprice to rise in C. If this were the only effect, we mightexpect network effects to increase prices on both sidesof the market in small neighborhoods around theno-externality prices pc and pj. A firm, however, mustmanage cross-price effects and the ratio of spillovers.These effects are so strong that at the point a sub-sidy becomes optimal, a firm is choosing prices oppo-site those implied by own-price elasticities consideredseparately.

To see this, let pc < 0, and thus necessarily pj > 0with c < 0 and

j > 0. Then signing terms in

the definition of c implies c > 0. At the same

time, Proposition 1 establishes the relation j = 1 jcr = 1 ecjpcD

c/Dj + ecjDcecjD

c/ejcD

j < 1.

In short, pc < 0 implies that one-sided elasticities bythemselves are j < 1 < 0 <

c . When subsidies are

optimal in the C market, price hikes rather than subsi-dies would (locally) increase sales in the Cmarket andprice drops in the J market would (locally) increasesales in the J market.

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Relative cross-price effects are dramatic. Furtherrearranging Equation (3) shows that it is not one-sided prices alone that must be unit elastic, c = 1,

but rather the one-sided prices adjusted for an exter-nality adjusted price ratio c 1 + ecjp

j/p

c = 1.

Graphically, the intuition is again that of Figure 1.

These results are surprisingly robust and are truemore generally for each of the following.

Corollary 1. The demand elasticity properties ofProposition 1 hold for any of the following alternativedemand specifications (equations are symmetric for qj):

Additive qc = Dc + ejcDj

Additive recursive qc = Dc + ejcqj

Multiplicative qc = DcDejcj

Shifted multiplicative qc = Dc1 + Djejc

Multiplicative recursive qc = Dcq

ejc

j

(6)

Proof. Define M 1/1 ecjejc as a market mul-tiplier, noting that ecjejc < 1 must hold to render thedefinition meaningful. Resolving the simultaneouspairs qc qj for the two recursive cases results in qc =

MDc + ejcDj and qc = DcDejcj

M. These differ only bythe constant term M from nonrecursive cases and thedifferences disappear in the optimality calculations.Applying the sequence of steps from Proposition 1yields equations identical to Equations (3) and (4).Moreover, elasticities have the structure ecjejccj =cjjc . A minor exception occurs for the shifted

case which also shifts, becoming ecjejccjDcDj/1 + Dc1 + Dj = cjjc .

The recursive formulations allow reciprocal exter-nalities on both sides of the market. An additiveformulation has a natural interpretation in allowingdemand shifts as in Figure 1 whereas a multiplica-tive formulation pivots around the point of max-imum value Vc. The latter is used in Rochet andTirole (2003), which makes the specialized assump-tion that ecj = ejc = 1. The multiplicative formula-tion has an advantage in forcing high price, zerodemand in the native market to cancel externality-driven demand from the cross-market. Introducing

the shifted multiplicative case corrects for the exoticcondition where 0 < Dj < 1 causes qc to fall in a risingexternality. Equivalence of these five functional formsshows that the main insight is robust to nonlineardemand as well as several alternative combinationsof demand. For completeness, we will occasionallycompare results using the notations pc+ and pc forthe additive and shifted multiplicative formulations,respectively.

Proposition 1 implies that cross-price elasticitiesmatter in both directions. The proposition constrains

price through cj times the spillover ratio. The bidi-rectional nature of spillovers demonstrated in Propo-sition 1 implies that pricing decisions depend heavilyon their relative sizes and, in fact, no difference insize can imply no difference in price. In the case ofadditive demand, this leads to the unusual result that

prices will be optimal, even if set under the myopicassumption of no network effects, when spilloversbetween markets are equal.

Proposition 2. For additive demands, a monopo-list selling to complementary markets with symmetricspillovers sets optimal price independent of the level oftwo-sided network effects. Mathematically, r = 1 if andonly if pc+ = pc+ and p

j+ = pj+. In the (shifted) mul-

tiplicative case, a monopolist always sets p less than orequal to the independent and no-externality prices. That is,pc pc = pc.

Proof. Recall that M = 1/1 ecjejc. First-order

conditions (Equation (5)) indicate p

c = Dc + ejcDj +ecjpjD

c/D

c. For reference, we also calculate inde-

pendent firm and no-externality prices pc and pc. Inthe case of independent firms the final term in thenumerator of pc is zero, so that pc = Dc + ejcDj/D

c,

and in the case of no externalities of either sort, pc =Dc/D

c. For optimal profits, simultaneous solution

of pc and pj yields implicit price

pc = M

Dc + ejcDj

Dc ecj

Dj + ecjDc

Dj

(7)

with a symmetric expression for pj. More intuitively,this can be interpreted as the market multiplier timesthe difference in independent prices M pc ecjpj,showing that own-price generally declines in ownexternality. To complete the proof, note that r = 1requires ejcD

j = ecjD

c. Combining fractions and twice

substituting ejcDj for ecjD

c collapses the equation to

pc = Dc/Dc = pc. Symmetry forces p

j = pj, which

is the required forward result. If both no-externalityprices are optimal, then reversing each step of theequality requires ejcD

j = ecjD

c.

For the (shifted) multiplicative case, FOCs give:

pc= Dc1 + Dj

ejc + pcDc1 + Dj

ejc

+ ecjpjDj1 + Dcecj1D

c = 0 (8)

It is then straightforward to establish that pc = pc =Dc/D

c and that p

c = Dc/D

c ecjpjDj1 + Dc

ecj/1 + Dj

ejc 1 + Dc = pc ecjj/1 + Djejc 1 + Dc.

Because all terms in the rightmost expression arepositve, pc must fall relative to pc.

The proposition states that if the demand external-ity is additive and network effects across markets areapproximately equal in sizethe presence of devel-opers is as attractive to consumers as the presence

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of consumers is to developersthen a monopolistcan safely ignore these effects in all pricing deci-sions. A firm can set prices as ifthe network terms ecjand ejc were both zero. This does not imply that net-work externalities have no effect on quantity sales.In general, output exhibits substantial increases due

to the market multiplier M= 1/1 ecjejc. This resultis unusual in that one might have expected prices torise with increased demand, but the monopolist takesthe gain exclusively in terms of increased sales ratherthan higher prices. The point is that firms need toconsider a joint pricing decision that accounts for net-work externalities only when the difference betweenthe spillover effects on sales, ecjD

c ejcD

j, is substan-

tial. Proposition 2 shows that for additive demandexternalities, optimal price equals no-externality pricepc = pc whenever r = 1, while for the multiplicativecase, independent price equals no-externality pricepc = pc always.

Building on this analysis, we can now answerthe more general question of which market a profit-maximizing firm discounts when spillover levels areunequal.

Corollary 2. A monopolist selling to complementaryadditive demand externality markets discounts the prod-uct with the greater spillover effect. That is, r < 1 pc < pc. The optimal C market price falls relative to theno-externality price if and only if market C contributesmore network externality surplus to market J, or pc < pc Scj > Sjc.

Proof. We have that r < 1 implies ecjDc = ejcD

j

for > 0. An identical sequence of substitutions asthose above yields

pc =Dc

Dc M

Dc

Dj + ecjDc

Dj

(9)

Only Dc Dj < 0 so that p

c falls. Alternatively, this

can be interpreted as pc = pc /DcMpj, and we

note that conclusions are similar for ecjDc = ejcD

j + .

To establish the condition on linear surplus, observethat when Dcpc = Qc1 pc/Vc then qj/pc is sim-ply ecjQc/Vc, with a similar expression for qc/pj.Arrange both terms as an inequality. The result,ecjQc/Vc < ejcQj/Vj, then simplifies to ecjQcVj >ejcQjVc, which completes the proof. Discounting theC market must create more J market surplus than viceversa for Cs price to fall.

A few additional points follow from this analysis.First, for additive demand externalities, a firm canrationally discount only one market at a time. Thismeans that a discount in one market implies a pricepremium in the other, or pc < pc if and only ifp

j > pj.

The reason is the fact that pc < pc implies ecjDc < ejcD

j

and the markets are symmetric. An immediate corol-lary shows that in the linear case, optimal prices fall

in the market that contributes more externality sur-plus to the other market.

While the preceding analysis compares no-exter-nality and optimal prices, it is straightforward to com-pare independent and optimal prices. In an exampleof a classic result on the uncoordinated pricing of

complements, we show below that factoring networkexternalities into the pricing decision, but failing tocoordinate the prices across markets, leads to ineffi-cient price setting. Where p is again optimal, let prepresent independent firm price as distinct from theno-externality price p.

Corollary 3. In the case of symmetric spillovers, ifindependent firms set prices in additive demand externalitymarkets without coordination, then prices vary with thelevel of spillover and are inefficiently high, p p.

Proof. Proposition 2 established that for r = 1,prices are independent of network effects, but for

independent prices, we havepc

=Dc

+ejcDj/

D

c.The difference pc pc is then ejcDj/Dc. For indepen-

dent firms, prices are always positive and increasingin the size of cross-market network effect.

Without externalities, prices pc and pc are identi-

cal and the inefficiency disappears. With externalities,however, independent firms err in direct proportionto their own level of network effect on their marketcomplement. One consequence of Corollary 3 is thatindependent firm prices are strategic substitutes anda price increase by one firm causes a price decrease

by the other.

3.2. Consumer SurplusA firms ability to manipulate two-sided networkexternalities to increase profits raises a concern aboutwhether its gains represent consumer losses. Does theability to link markets, particularly when two prod-ucts are combined under monopoly ownership, hurtconsumers?

In the case of symmetric spillovers, Proposition 2and Corollary 3 imply that the answer is clearly no.Consumer surplus improves under monopoly owner-ship. Because two independent firms both price toohigh, integrated ownership reduces prices even as itincreases profits.

Asymmetric spillovers may leave consumer welfareambiguous because pc = 1/1 ecjejc pc ecjpj anda careful choice of ecj, ejc can force p

c > pc. For lin-

ear demand, however, welfare analysis of asymmetricspillovers provides a particularly strong result. Wel-fare exhibits a strict Pareto improvement in the sensethat even the market paying the increased premium-goods price prefers monopoly control. This result isstronger still in the case of multiplicative demandexternalities. Let superscripts on consumer surplusCS and CS represent the independent and joint opti-mization values from the firms perspective.

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Proposition 3. Consumers and producers benefitwhen firms merge or coordinate prices in marketswith linear demand and two-sided network externalities.That is, CSc CSc and CS

j CSj and thus necessarily

CSc + CSj CSc + CSj.

, so total welfare improves.This is always true for multiplicative demands.

Proof. The upper bound onVc

pcqcrdr is cho-

sen to avoid negative quantities and is given byVc = D

1c qc ejcDj. For qc = 0, this becomes Vc1 +

ejcQj/Qc1 pj/Vj. Integrating Equation (1) and

substituting for Vc gives consumer surplus in theC market as

CSc =ejcQjVcpj Vj + Qcpc VcVj

2

2QcVcV2

j

(10)

In computing optimal pc pj and independent pc pj

prices, we condense several steps. We also simplifythese expressions by converting to surplus notation.This collapses six apparent degrees of freedom to fourtrue degrees. Substituting for pc p

j in (10) yields

consumer surplus at optimal prices.

CSc =2Sc + Scj+ Sjc

2ScSj ScjSjc2

2ScScj+ Sjc2 4ScSj

2 (11)

A similar substitution for pc pj yields consumer sur-plus at independent prices.

CSc = ScScjSjc Sj2Sc + Sjc

2

2ScjSjc 4ScSj2

(12)

Note that squares imply these are always positive.Imposing the Hessian condition 4ScSjScj+Sjc

2 > 0and simplifying the difference in total surplus CSc CSc always proves nonnegative. Equality is achievedonly when the markets are independent (ecj = ejc = 0implying Scj = Sjc = 0), in which case, CS

c = CSc =

18

Sc.The case for CSj is symmetric. In the case of multi-plicative demand, Proposition 2 proves p p = palways, thus CS improves. Finally, from the pro-

ducers perspective, a monopolist can always chooseto set prices independently, thus and total wel-fare Pareto improves.

The underlying intuition is that linking the marketsallows a firm to manipulate the markets total sizevia the cross-market network externality. Consumersthen benefit to the extent that a self-interested firmsets prices more efficiently but cannot capture all con-sumer surplus from the efficiency gains. A firm inter-nalizes the market externality but wins only a fractionof this effect.

4. Applications and ExtensionsTwo-sided markets coupled by network externalitiesare distinguished from other market types by het-erogeneous matched consumers with interdependentdemands. This has a number of implications beyondthose of the present model.

One implication is that firms can deploy networkeffects to stimulate new surplus in contrast to pricediscrimination strategies that extract static surplus(Parker and Van Alstyne 2001). This provides counter-point to key theories of bundling (Adams and Yellen1976, Salinger 1995, Bakos and Brynjolfsson 1999), theadvantage of which is reduced demand heterogeneity.As long as buyer values are not perfectly correlated,

bundled demands converge to a point mass basedon the central limit theorem. With zero marginalcost goods, firms can costlessly smooth idiosyncraticdemands simply by combining items in a bundle. Thisallows a firm to predict and therefore extract con-

sumer surplus. Rather than seeking to reduce mar-ket heterogeneity, two-sided product design seeks toleverage it. Unbundling together with a one-sidedsubsidy sacrifices the ability to predict transactionsvalue in favor of growing transactions volume.

Introducing competition, analysis of two-sided net-work effects can also answer the question of why aduopolist might voluntarily enter into Bertrand pricecompetition, forcing profits to zero in one market seg-ment (Parker and Van Alstyne 1999). Even with anundifferentiated product, an entering firm can use atwo-sided product complement to generate profits onthe far side of a market. A firm can soften an incum-

bents first-mover advantage and survive a marketthat becomes completely competitive post-entry. Theproduct complement makes an entry threat credible.As in the case of the Internet browser wars, consumersurplus rises from the arrival of a zero price substi-tute and entrants can be unusually fierce competitors(Jackson 1999).

Empirically, a two-sided model has the advantageof suggesting new approaches for estimating net-work effects. Models of intramarket externalities suf-fer from endogeneity of demand estimation in thatinstruments can barely distinguish demand shocksfrom the network externality ripple effects they create.If, on the other hand, network effects cross markets,then instrumenting demand shocks in one market canproceed while holding demand constant in the cou-pled market, thus tracing a demand relationship. Theproposed framework might therefore simplify esti-mation of network effects relative to earlier models.Testable propositions include which internetworkedmarket to discount in markets with pricing power,which of several products firms will favor discount-ing in markets without pricing power, and whetherdiscounting lasts longer than predicted by models of

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penetration pricing with lock-in. An empirical test ofParker and Van Alstyne (1999) and Rochet and Tirole(2000, 2003) appears in Gallaugher and Wang (2002).They find significant positive externalities betweenthe Web browser and Web server markets.

Failing to account for externality-based comple-

ments might therefore lead to pricing and prod-uct design errors. Using a product tying model,for example, firms might meter sales to both con-sumer types, missing the opportunity to manage net-work effects between them. Using a lock-in model,they might injudiciously discontinue discounts forthe market stimulating cross-market sales, chokingnetwork effects. Using uncoupled multimarket pricediscrimination, they might fail to offer discountsat all. Each model suggests subtle differences in thelocus and timing of pricing decisions. Suitable mar-ket divisions fall along several different lines basedon demand attributes, product features, or time. Some

practical applications of these divisions are givenbelow.(1) Tangible goods and services with externalities

between tangible and intangible goods, eit > 0 and eti > 0.Example: Historically, the first color TV broadcastoccurred in 1951 but no firm realized a profit on colorTV sets for almost a decade. Consumers delayed buy-ing color TVs until the arrival of substantial content.RCA solved the chicken-and-egg problem in 1961

by subsidizing Disneys Wonderful World of Color(Shapiro and Varian 1998). In 2002, the Federal Com-munications Commission resolved an identical prob-lem for HDTV by mandating the provision of HDTV

sets by electronics firms.(2) Temporal complements with externalities betweentime 1 and time 2. Example: Vendors of precedentand case law databases commonly subsidize studentlawyers in time 1 while charging premium pricesto these same people once they become professionallawyers in time 2. The time 1 version is fully func-tional, so there is no need to upgrade, nor is accessephemeral because it runs the full three years of lawschool. The spillover benefit is to save law firms the(re)training costs of having new lawyers learn thecommand structure and capabilities of a principalresearch tool. The benefit in the other direction is that,ceteris paribus, law students prefer to adopt the pack-age adopted by more law firms.

(3) Upgrades with externalities between novice andpro users, enp > 0. Example: Many firms give awaynovice versions of simulation or CAD software butcharge for a professional version. The less functionalor time-limited version encourages experimentationand purchase of the full-featured version. In con-trast, conventional price discrimination uses a menuof different prices to force consumers with intrinsi-cally different demand profiles to self-select on the

basis of willingness to pay. The standard price dis-

crimination model, however, leaves open the questionof why a firm should incur development costs on agood for which it never anticipates revenues.

(4) Advertising market with externalities between con-sumers and advertisers, eca > 0 and eac < 0 or eca > 0 andeac > 0. Example: Magazine publishers, TV and radio

stations, software vendors, and Web portals offer freeor discounted consumer content in sponsored modein which consumer market C enjoys low prices whileadvertiser market A buys ad placement from the ven-dor in order to reach the C market. This has alsolong been an economic model supporting the freeinformation content found in circulation industries(Chaudhri 1998). A larger consumer market increasesthe attractiveness of buying ads such that eca > 0,

but the presence of ads may or may not increase theattractiveness of consuming content. For fashion mag-azines, the network externality term may be positive,eac > 0. But for many forms of advertising, consumers

may experience clutter costs such that eac < 0.In effect, these points illustrate different dimensionsalong which to cleave the halves of a two-sided mar-ket. These dimensions include tangibility, time, skilllevel, and content. As noted in Table 1, the force oftwo-sided network effects can also extend to near zeromarginal cost services such as credit cards and auc-tions. Future research will likely enumerate variousplatforms across which providers meet consumers,and transactions volume is governed by tipping theplatform to favor one or the other.

5. ConclusionsThe model presented here argues for several simpleand intuitive results. First, a firm can rationally investin a product it intends to give away into perpetu-ity even in the absence of competition. The reason isthat increased demand in a complementary premium-goods market more than covers the cost of investmentin the free-goods market. In this case, market com-plementarity arises from an internetwork externality.This strategy also takes advantage of informationsnear zero marginal cost property as it allows a firmto subsidize an arbitrarily large market at a modestfixed cost.

Second, we identified distinct markets for con-tent providers and end consumers and showed thateither market can be a candidate for discounting orfree distribution. Deciding which market to subsidizedepends on the relative network externality benefits.At a high level of externality benefit, the market thatcontributes more to demand for its complement is themarket to provide with a free good. At lower levels,firms may charge positive prices in both markets butkeep one price artificially low.

Third, we show that, in general, consumer welfareis not harmed when firms set prices across markets

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with positive complementarities. Firms can manipu-late total market size through choice of price in eachmarket. Consumers then benefit to the extent thata self-interested firm sets prices more efficiently butcannot capture all consumer surplus from the effi-ciency gains. A firm internalizes the market external-

ity but wins only a fraction of this effect.The modeling contribution is distinct from tyingor second-degree price discrimination in the sensethat consumers need never purchase both goods.Unlike razors and blades, the products are stand-alone goods. It also differs from multimarket or third-degree price discrimination in the sense that the firmmay extract no consumer surplus from one of the twomarket segments, implying that this market wouldhave traditionally gone unserved.

AcknowledgmentsThe authors are grateful to participants of the 1999 Work-

shop on Information Systems and Economics, the 2000Association for Computing Machinery SIG E-Commerce,the 2000 International Conference on Information Sys-tems, the 2002 Stanford Institute for Theoretical Economics(SITE) workshop on Internet Economics, the 2003 InsitutDEconomie Industrielle second conference on The Eco-nomics of the Software and Internet Industries, as well asnumerous participants at university seminars. They thankTom Noe for helpful observations on oligopoly markets;Lones Smith, Kai-Uwe Kuhn, and Jovan Grahovac for cor-rections and model generalizations; Jeff MacKie-Mason forvaluable feedback on model design and bundling; and HalVarian for helpful comments on firm strategy and modelimplications. Frank Fisher provided helpful advice on, and

knowledge of, the Microsoft trial. Jean Tirole provided use-ful suggestions and examples, particularly in regard tocredit card markets. Paul Resnick proposed the descriptiveterm Internetwork externality to describe two-sided net-work externalities. Tom Eisenmann provided useful feed-

back and examples. They also thank Robert Gazzale, MotiLevi, and Craig Newmark for their many helpful obser-vations. This research has been supported by NSF CareerAward IIS 9876233.

References

Adams, W. J., J. L. Yellen. 1976. Commodity bundling and the bur-den of monopoly. Quart. J. Econom. 90(August) 475498.

Armstrong, M. 2002. Competition in two-sided markets. Mimeo,Nuffield College, Oxford University, Oxford, UK.

Arthur, W. B. 1989. Competing technologies, increasing returns, andlock-in by historical events. Econom. J. 99 116131.

Arthur, W. B. 1994. Increasing Returns and Path-Dependence in theEconomy. University of Michigan Press, Ann Arbor, MI.

Bakos, Y., E. Brynjolfsson. 1999. Bundling information goods: Pric-ing, profits, and efficiency. Management Sci. 45(12) 16131630.

Caillaud, B., B. Jullien. 2003. Chicken & egg: Competing match-makers. RAND J. Econom. 34(2) 309328.

Chaudhri, V. 1998. Pricing and efficiency of a circulation industry:The case of newspapers. Inform. Econom. Policy 10 5976.

Coase, R. H. 1960. The problem of social justice. J. Law Econom. 3

144.Cournot, A. A. 1838. Recherches sur les principes mathematiques

de la theorie des richesses. Libraririe des sciences politiques etsociales M. Rivere & cie., Paris, France.

DeGraba, P. 1996. Why lever into a zero profit industry: Tying,foreclosure, and exclusion. J. Econom. Management Strategy 5(3)433447.

Farrell, J., G. Saloner. 1985. Standardization, compatibility and inno-vation. RAND J. Econom. 16(1) 7083.

Farrell, J., G. Saloner. 1986. Installed base and compatibility:Predation, product preannouncements and innovation. Amer.Econom. Rev. 76(5) 940955.

Gallaugher, J. M., Y. M. Wang. 2002. Understanding network effectsin software markets: Evidence from Web server pricing. MISQuart. 26(4) 303327.

Hansell, S. 2001. Red face for the Internets blue chip. New YorkTimes (March 11).

Jackson, T. P. 1999. U.S. v. Microsoft: Findings of Fact. United StatesDistrict Court for the District of Columbia, Washington, D.C.

Katz, M. L., C. Shapiro. 1985. Network externalities, competition,and compatibility. Amer. Econom. Rev. 75(3) 424440.

Liebowitz, S. J., S. E. Margolis. 1994. Network externality: Anuncommon tragedy. J. Econom. Perspect. 8(2) 133150.

Mongell, S., A. E. Roth. 1991. Sorority rush as a two-sided matchingmechanism. Amer. Econom. Rev. 81(3) 441464.

Nalebuff, B. 1999. Bundling. Yale ICF working paper series 99-14,http://ssrn.com/abstract=185193.

Parker, G., M. Van Alstyne. 1999. Information complements, sub-stitutes, and strategic product design. Workshop on Informa-tion Systems and Economics, Charlotte, NC, http://ssrn.com/

abstract=249585.Parker, G., M. Van Alstyne. 2001. Unbundling in the presence of

internetwork externalities. Mimeo, University of Michigan,Ann Arbor, MI.

Rochet, J. C., J. Tirole. 2000. Platform competition in two-sided mar-kets. Working paper, Institut dEconomie Industrielle, France.

Rochet, J. C., J. Tirole. 2003. Platform competition in two-sided mar-kets. J. Eur. Econom. Assoc. 1(4) 9901029.

Rochet, J. C., J. Tirole. 2004. Two-sided markets: An overview.Working paper, Institut dEconomie Industrielle, France.

Salinger, M. 1995. A graphical analysis of bundling. J. Bus. 68(1)8598.

Shapiro, C., H. Varian. 1998. Information Rules. Harvard BusinessSchool Press, Boston, MA.

Shapiro, C., H. Varian. 1999. The art of standards wars. California

Management Rev. 41(2) 832.

Varian, H. 1989. Sequential provision of public goods. J. PublicEconom. 53 165186.

Whinston, M. 1990. Tying, foreclosure, and exclusion. Amer. Econom.Rev. 80(4) 837859.

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