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Co-Creation with Production ExternalitiesNiladri B. Syam, Amit Pazgal

To cite this article:Niladri B. Syam, Amit Pazgal (2013) Co-Creation with Production Externalities. Marketing Science 32(5):805-820. http://

dx.doi.org/10.1287/mksc.2013.0791

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Vol. 32, No. 5, SeptemberOctober 2013, pp. 805820ISSN 0732-2399 (print) ISSN 1526-548X (online) http://dx.doi.org/10.1287/mksc.2013.0791

2013 INFORMS

Co-Creation with Production Externalities

Niladri B. SyamDepartment of Marketing, C. T. Bauer College of Business, University of Houston, Houston, Texas 77204,

[email protected]

Amit PazgalDepartment of Marketing, Jones Graduate School of Business, Rice University, Houston, Texas 77005,

[email protected]

Co-creation, the participation of customers in the design and production of goods and services, has beengaining popularity in recent years. In this research we incorporate firm pricing into the joint production pro-cess allowing us to study (1) production externalities between firm and customers, (2) production externalitiesamong customers, and (3) optimal pricing by firms.

We show that given a choice, a monopoly firm will opt for co-creation with customers rather than deal withpassive price-taking consumers. Furthermore, the firm will increase the effort it devotes to co-creation as the

number of potential co-creating customers increases.We show that the profit of a firm facing a centralized pattern of externalities among customers (with an expert,or lead user, in the center) can be higher than its profit when facing a decentralized pattern of externalitiesamong customers and clearly dominates its profit when customers do not have any cross externalities. Thus, weprovide a different justification for the use of lead users, one that depends on their network centrality and noton having lower cost, more information, or greater ability than the firm. Because the decentralized pattern hasmore links than the centralized pattern, our results demonstrate the importance of the pattern of links betweencustomers, and not just their number, in determining the profitability of co-creation.

Furthermore, we find that the lead users externality spillover to other connected users, her neighbors, actsas a force multiplier on the efforts exerted by all participants in equilibrium. Specifically, a higher spillover fromthe lead user increases the efforts of the firm, the neighbors, and the lead user herself, and this may lead to

beneficial outcomes for all.Finally, we show that in co-creation environments, a monopolist firm may benefit by committing to a single

price rather than exercising price discrimination. This is because the pricing structure affects customers incentiveto invest effort in the innovation-production stage.

Key words : co-creation; customization; networks; externalities; game theoryHistory : Received: May 3, 2012; accepted: March 16, 2013; Preyas Desai served as the editor-in-chief and Yuxin

Chen served as associate editor for this article. Published online in Articles in AdvanceJuly 26, 2013.

1. IntroductionInstances of co-creation, the participation of potentialcustomers in the production of goods and services,continue to increase in number and importance (Pineand Gilmore 1999). Collaborating with customers toco-create products has always been commonplace in

business-to-business (B2B) settings, well before the

current proliferation of business-to-consumer (B2C)instances. Specifically, user participation in the designof products, especially in business-to-business con-texts, is the earliest and most prominent manifesta-tion of product co-creation (Griffin and Hauser 1993,Thomke and von Hippel 2002, von Hippel and Katz2002). Recent advances in information technologyand flexible manufacturing have greatly enhancedthe ability of firms and customers to co-create prod-ucts such as cameras, copier finishers, laptop bags,laundry products, etc. (Redstrom 2006, Sawhney

et al. 2005). Paris Miki, a Japanese eyewear retailer,has developed a system where designs are createdin close collaboration with its customers; customerscomplete the design task by selecting options for thenose bridge, hinges, and arms (Gilmore and Pine1997). eMachineshop.com provides individuals theability to create their own tools by allowing access to

an infrastructure that was hitherto available only toreal manufacturers.

Academic researchers have also turned their atten-tion to co-creation, with an emphasis on viewingthe customer as a coproducer of goods and ser-vices (Jaworski and Kohli 2006; Kalaignanam andVaradarajan 2006; Prahalad and Ramaswamy 2000,2004; Vargo and Lusch 2004). Our goal in this paperis to rigorously model co-creation and to determinethe optimal cooperative productive efforts of the firmand the consumer as well as the pricing by the firm.

805


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Syam and Pazgal: Co-Creation with Production Externalities806 Marketing Science 32(5), pp. 805820, 2013 INFORMS

There are two types of co-creation: proprietary andnonproprietary (Bessen 2006, Rossi and Bonaccorsi2006). The types differ in the ownership structureand distribution of the final product. Nonproprietaryco-creation, which has the flavor of public goods,describes environments wherein the firm and its cus-

tomers invest in creating a better product that is thendistributed freely to all users (open source software,for example). Bramoulle and Kranton (2007), Ballesteret al. (2006), and Calv-Armengol and Zenou (2004)interpreted the efforts of the agents as product inno-vations and treated such innovations as strict pub-lic goods. Proprietary co-creation, on the other hand,refers to situations where customers participate in thedevelopment of the product but the firm retains own-ership and can sell the final product. Proprietary co-creation has flavors of both public and private goods.

In this paper, we study co-creation environmentswhere the price mechanism is crucial in mediating

the value exchanged between the firm and the cus-tomer, i.e., the proprietary setting. Customers can ben-efit not only from the effort of the firm but alsofrom the efforts and resources of their fellow cus-tomers. Specifically, some users may be experts1 inthat their needs are similar to others and their effortscan substantially benefit other customers. Considerthe phenomenon of lead users in business markets.2

Lead users are at the forefront of emerging trendsand can help other users with their product solutions.In this paper, we broaden the definitions and denoteas experts/lead users those customers who are able tosubstantially benefit other users through their inno-

vation efforts. Consider, for example, the oil and gasindustry. Original equipment manufacturers of gen-erators and turbines (e.g., GE, Simmons, Mitsubishi)develop their products jointly with big advancedmultinational oil companies (e.g., ExxonMobil, RoyalDutch Shell, Chevron). After the development stage,the manufacturer aims to sell the equipment to allpotential end users (including many less sophisticatedusers, such as Petronas or Sonangol) with the appro-priate fine-tuning and customizations. This forms anoil and gas network of co-creators where the firmis connected to its lead users and regular customers.The former help in the development of the new prod-

uct, and their expertise benefits all potential users.The efforts of the latter are concentrated on workingwith the manufacturers to fine-tune and customizethe products for their own use.

This paper captures three different aspects ofco-creation: (1) Customers and firms both contribute

1 We use the terms expert and lead user interchangeably todenote those customers whose innovative efforts are universally

beneficial.2 For extensive discussions, see von Hippel (1986, 1998) and Urbanand von Hippel (1988).

to the design of products. Thus there are productionexternalities between customers and the firm. (2) Somecustomers have capabilities or expertise that are ofvalue to other customers in the market. Thus there areproduction externalities among customers. (3) Firmscharge for products co-created with the help of

customers.We compare and contrast optimal effort levels aswell as firm prices and profits in several represen-tative environments. We compare the basic no co-creation case to the intermediate case, where there areno lead users but every customer can improve his orher product directly with the firm (no links betweenusers), to the advanced cases that include lead userswhose efforts improve the products of all users ormore diffused connections between similar users. Wefurther investigate the optimality of price discrimina-tion by the firm when there are lead users (i.e., whenthey charge different prices to lead and regular users).

The rest of the paper is organized as follows. Sec-tion 2 describes the relevant literature; 3 presents ourgeneral findings and the intuition behind them. Thegeneral model is developed in 4. Section 5 providesrobustness checks for the main results of 4. Section 6is devoted to the analysis of the benefits of a singleprice over price discrimination, and 7 concludes.

2. Literature ReviewMuch of the research that deals with the produc-tion externalities of the type that we study hasappeared in the networks literature. The existing lit-erature that examines networks and externalities can

be broadly divided into three streams. The first andearliest research stream covers only network external-ities. Crucially, this literature (implicitly) assumes thatevery agent (be it a firm or a customer) is connected toevery other agent. In the terminology of graph theory,these papers assume a complete graph and ignorethe specific pattern of connections between agents(e.g., Katz and Shapiro 1985, Farrell and Saloner 1985)The second stream of work examines network topol-ogy, but it tackles only the problem of agent contribu-tions (or efforts) and how agents optimally determinetheir contributions given the network of connections.Thus, it has the flavor of contributions to a pub-lic good, where the pattern of externalities is given

by the topology of local network effects. Prominentpapers in this group include Bramoulle and Kranton(2007), Ballester et al. (2006), and Calv-Armengoland Zenou (2004). We differ in two respects. First,and most importantly, we investigate proprietary sit-uations where firm pricing plays an important role.Second, we focus on specific patterns of externali-ties (i.e., network structures) that incorporate the factthat certain agents, the lead users, are experts who

benefit their fellow users. Both aspects are critical


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Syam and Pazgal: Co-Creation with Production ExternalitiesMarketing Science 32(5), pp. 805820, 2013 INFORMS 807

to co-creation and have not been examined by theabove public goods literature. A recent paper, Chen-namaneni and Desiraju (2011), investigates these pub-lic good inefficiencies in the context of comarketingalliances and offers a contractual solution to theseproblems, though the issues of network topology

and ownership and the selling of a product are notinvestigated.The third and nascent research stream examines

pricing in networks and accounts for local networkeffects, but the firms and customers play completelydistinct and different roles. Firms set prices, and thecustomers determine their contributions. There is noeffort contribution on the part of firms, and the onlydecision for consumers to make is whether to buyor not buy the offered products. Thus, these paperscan be thought of as investigating consumption exter-nality rather than production externality (Banerji andDutta 2009, Bloch and Qurou 2013).

Co-creation involves both the joint contributionof the firm and customers in designing a prod-uct and the firms pricing. This paper takes a firststep towards understanding the importance of bothnetwork connections and pricing in the co-creationprocess.

3. Results and IntuitionIn this paper, we analyze co-creation between a firmand a set of users, where the users can further beconnected among themselves according to specificpatterns of externalities. We analyze three canonical

externality patterns. In the network literature, thesepatterns correspond to different networks: the firsthas no connections among users (null pattern), thesecond has an expert/lead user at the center who isconnected to the other users called neighbors (central-ized pattern), and the third has no recognized expertamong the users, all of whom have symmetric con-nections (decentralized pattern). See Figure 1 for adepiction of these externality patterns.

We have three main results. First, a centralized pat-tern of externalities, with a lead user in the cen-ter, can be more profitable than both the null and

Figure 1 Externality Patterns

Firm

User User User User

User

User

UserUserUser User

User

User

Firm Firm

Lead

user

CentralizedDecentralizedNull

decentralized patterns of externalities, even thoughfor a given number of customers, the decentralizedpattern has more links than the centralized pattern.The networks literature has shown that the patternof links as well as the number of links matters, butmore importantly, we demonstrate in our specificcontext exactly how the pattern of links affects co-creation efforts and profits. Specifically, a firm facinga centralized pattern of links can be more profitablethan one facing a decentralized pattern, even whenall spillovers among customers are the same in bothcases. Thus, we provide a novel justification for theuse of a lead user by a profit-seeking firm. The bene-fit of the lead user does not depend on her having alower cost or greater ability compared with the firm(the usual reasons advanced in the trade press to jus-tify a lead user) but on the fact that she is centralin the production network. When the spillover fromthe lead user to other customers increases, the firmincreases its effort and therefore can increase its price.This force is absent in a decentralized network. There-fore for high spillover, a centralized network is alwaysmore profitable than a decentralized network.

Our second result states that if users are connectedby a centralized externality pattern, then when thelead user increases her spillover to the other users, allthe involved parties benefit: the firms profit, the leadusers expected utility, and the other users expectedutilities all increase. The spillover of the lead user cap-tures the extent to which her effort can benefit theother users connected to her. Thus we show that whenthe lead users effort is more beneficial to the otherusers, both she and the firm benefit. As mentioned

above, a higher spillover from the lead user increasesthe effort the firm exerts, and this benefits the leaduser. Even though the price increases, the positiveeffect dominates, and the lead users expected utilityincreases in her spillover.

An interesting implication of the above is that thelead user and the firm are not mere substitute expertsin the creation of value. In fact, they complement eachother. Having a better (more representative or bet-ter connected) lead user should not cause the firmto reduce its creative effort; instead, the firm shouldincrease it.


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Our third finding shows that when customers areconnected by a centralized pattern of externalities(with a clear distinction between lead users and reg-ular users), it is better for the firm to charge a sin-gle price than to price discriminate, even if it is amonopolist. It is well documented that through price

discrimination, the firm can better extract the surplusof heterogeneous customers. However, in co-creationenvironments, this turns out to be problematic forthe firm. Price discrimination leads to reduced con-sumer surplus, which in turn causes consumers torationally reduce their co-creation efforts. Because thefirms profit increases in the total efforts exerted, pricediscriminating reduces profit compared with charg-ing a single price. In a sense, by agreeing to notprice discriminate, the firm signals to the customersthat they will retain more of their co-creation efforts,leading them to increase their contribution enoughto offset the firms losses from not customizing the

prices. The extant literature has shown the subopti-mality of price discrimination when firms competefor customers (especially in behavior-based schemes).Our result is similar to the potential extra profitabilityof a monopolists ability to commit to a single pricefor a good over several time periods (see Fudenbergand Villas-Boas 2006). The point of difference lies inthe mechanism by which this occurs in the two sit-uations: in co-creation, the commitment to a singleprice changes consumers incentives to expend efforton product development; in the behavior-based pric-ing literature, it changes the consumers timing ofpurchase.

4. A Model of Co-Creation withProduction Externalities

Consider a situation where several agents engagein joint production (of a product or service) with amonopolist firm. In co-creation each agent is a user-creator who will participate in joint production withthe firm and perhaps with other fellow user-creators,and this agent may purchase the final product for itsown use. We conceptualize cooperative production asa network of contributors, with each user-creator rep-resented by a node. The links in the network of exter-

nalities represent the potential effect of one userseffort on the others overall utility. In the remainderof the paper, we will refer to the user-creators simplyas users. We also assume that the firm is (by defini-tion) connected to all users. Thus, the specific patternsof externalities that we will investigate (null, central-ized, and decentralized) refer only to the pattern oflinks between the users.

4.1. No Externalities Among UsersIn this section we analyze the case where there areno links between the users themselves, so that each

users effort does not benefit other users. We can thinkof each user as exerting effort to better fit the productfor his or her own specific circumstances. We refer tothis base case as the null network.

As a base case, we investigate an environment witha single firm and U symmetric users. Let the pro-

duction effort of the firm be ef and that of a userbeeu. It is important that our use of the term effortdoes not confuse the reader; we use the term here tocapture the amount of innovation or product designthat an agent contributes. The literature on contribu-tions to open source software uses the term in thesame spirit (Hindricks and Pancs 2002, Johnson 2002).The newer literature on public goods in networks,e.g., Bramoulle and Kranton (2007) and Ballester et al.(2006), also interprets an agents efforts as innovationattempts. Agents could innovate by experimentingwith new technology or by generating new informa-tion that is shared among their network of partners in

a collaborative venture. In this sense the innovationefforts of various agents are observable.3

Each user benefits from the combined total effortof all agents connected to him. Specifically, we definethe total effort eu at a users node as

eu= eu+ fef (1)

where f1 represents the degree of effort substi-tutability between the firm and all the users. Theusers benefit is thus a functionb of the above totaleffort. We also assume a convex cost of effort c with c0= 0. The total consumer surplus is given

by CSu = beu ce

2

u pu+ , where pu is the pricecharged to a user for the co-created product, and is a zero mean random variable representing uncer-tainty the user has about the fit of the products toher needs. This fit parameter is unknown to the con-sumer during the development stage and is revealedto her once the product is available in the market.4

The firm only knows the distribution of the fit param-eters in the population, never its actual realizations.Throughout the paper we assume that is distributeduniformly on [ ]. To concentrate on the networkeffect, we assume that the cost coefficient c is also thesame across all users.

The competitive environment unfolds in threestages. In the first stage, both the customers and thefirm determine their innovation effort levels and thenexert that effort. In the second stage, the prototype

3 We do not use the term effort in the sense of the prin-cipalagent models where the agents efforts are unobservable(e.g., Holmstrom 1979).4 In many new product development cases, consumers can articu-late their needs but cannot fully evaluate the benefit of the productor their willingness to pay for it until a prototype is developed andis shown to them.


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Figure 2 Timeline of the Competitive Environment

Stage 1 Stage 2 Stage 3

Firm and customers determineand exert efforts Firm sets prices

Consumers evaluate product fitand decide whether to buy or not

is available, and by observing effort levels, the firmdetermines the prices. In the third stage customersevaluate the fit of the product and make their pur-chase decisions. Figure 2 depicts this timeline.

Note the importance of the first stage in the co-creation game. When users exert their innovationeffort, they do not know the fit of the final productto their needs. Thus, depending on the magnitudeof the fit parameter, there is a positive probabilitythat the customers will not purchase the co-createdproduct (despite exerting a costly effort). Rational

consumers must take into account the probabilityof non-purchase when making their effort decisionsin the first stage. Moreover, at the last stage, whenagents make their purchase decisions, their cost ofeffort has already been incurred in the first stage andis considered to be sunk for the buying decision.

To solve for the optimal prices and effort level, weuse the concept of a subgame-perfect equilibrium and

backwards induction. Starting at the last stage, givena pricepu, a user will purchase the product with prob-ability given by Pr > pubeu. The probability thata user purchases is thus

Pr[Useru buys] =+ beu pu2

(2)

Notice that the cost, ceu, does not figure in the prob-ability calculation because when the user makes thepurchase decision, the cost of co-creation is alreadysunk (determined in the first stage).

The expected profit of the firm is

f=U

+ beu pu

2

pu cef (3)

In the second stage, the firm choosespuoptimally; thefirst-order condition (FOC) gives the price in terms of

the efforts:

pu=+ beu

2 (4)

Note that the price charged to a user is increasing inthe benefit a user gets from joint production with thefirm, beu, and so it is increasing in the users owneffort, eu. Intuitively, the firm can extract the benefitthat the user enjoys by consuming the product, andthe products benefit increases in eu.

We now turn to the first stage and determine theoptimal efforts of the firm and the users. The expected

profit of the firm after price has been incorporatedfrom (4) is

=U+ beu

2

8 cef (5)

Next, we need to calculate the expected consumer sur-plus of the users. A given user,u, will realize the exactvalue of the product fit only after the effort choice.Thus if the product is ultimately a poor fit, the usermay not buy in the future despite the efforts exertedin the development stage. This fact needs to be incor-

porated in the calculation of the expected consumersurplus:

ECSu = ECSu ifu buys ubuys Prubuys

+ECSu ifu doesnt buy udoesnt buy

Prudoesnt buy ceu

Notice that the user incurs the cost of effort, regard-less of whether she buys the product in the futureor not. Also, as already noted, the probability of pur-chase in the future is independent of the cost of effort.Thus,

ECSu = Ebeu pu+ pu beu

Pr pu beu ceu

=beu pu+E pu beu

+ beu pu

2

ceu

This gives us ECSu=1/4+ beu pu2 ceu.

After substituting the price from (4), we get

ECSu= 1

16+ beu

2 ceu (6)

The optimal effort profile e = ef eu is solved bymaximizing (5) and (6) simultaneously.

The users FOC ECSu/eu = 0 gives 1/8beu + b

eu = ceu, and the firms FOC

E/ef = 0 gives U + beu/4fbeu

cef= 0. For the sake of tractability, we assume thatthe benefit function is linear and the cost is quadratic.Yet, as 5 shows, the results hold for more gen-eral functions. Specifically, we assume that beu =eu = eu+ fef and ceu= ce

2u. The solutions for the

users and the firms efforts are shown in (7) and (8),


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respectively;5 all optimal quantities are denoted witha superscript that corresponds to the network type:

eNullu =

16c 22fU 1 (7)

e

Null

f =

2fU

16c 22fU 1 (8)

For participants to exert an effort (an interior solu-tion), we need eNullf >0 and e

Nullf >0. For these to be

satisfied, we need

16c 22fU 1 > 0 (9)

In the rest of the paper, we maintain this assump-tion.6 Note that the parameter restriction is satisfiedas long as the number of users is not too large; that is,U < 16c 1/2f.

Clearly, the optimal effort of the firm increases inU,

the number of users. This is due to the pricing effect.Without pricing, the firm would be just another agentcreating a product with other economic agents; thisis akin to a public goods situation. The public goodsliterature has found free riding to be prominent, andas the number of agents increases, the problem of freeriding is exacerbated. We show that allowing the firmto own the product and set its prices reverses thefree-riding effect caused by the public goods natureof co-creation. Finally, the optimal price and profit ofthe firm can be obtained by suitable substitutions. Westate these criteria as Proposition 1.

Proposition 1.

When a monopolist firm co-creates aproduct with U homogeneous users with no connec-tions between them ( forming a null pattern of exter-nalities), then the optimal effort of each user is eNullu =/16c22fU 1, and the optimal firm effort is e

Nullf =

2fU/16c 22fU 1. The firms profit is given by

Null = 4c2U 8c 2fU/16c 22fU 1

2.

It is straightforward to check that if 8c 1 > 0, theoptimal profit of the firm increases in the number ofusers U that it is connected to.7 This is an immedi-ate consequence of the price effect. Even though thefirms own effort increases in the number of users,leading to a higher cost of co-creation, the increase inprice with an increase in U more than makes up forthat increased cost.

5 The second-order condition (SOC) for the users effort requires16c 1 > 0, and the SOC for the firms effort requires8c 2fU > 0. We assume they are satisfied.

6 We focus on interior solutions; otherwise, the parties would put inzero effort, and this would not be a true co-creation environment.7 That is,

Null

U =

8c24c16c 1+ 2fU 8c 1

16c 22fU 13

As a benchmark we analyze the case without co-creation, where the users are merely price takers as intraditional markets.

4.1.1. Benchmark Without Co-Creation. A mono-polist firm creates and sells a product to U homoge-neous users who do not exert effort. Suppose the firm

first chooses its design effort and then its price. Theoptimal effort and profit of the firm are

eNoCo-creationf =fU

8c2fU NoCo-creation=

c2U

8c2fU

Proof. See the appendix.

A direct comparison of the monopolists profitswith and without co-creation yields the followingresult.

Corollary. Suppose that a monopolist has the optionof co-creating products with its customers. The monopo-lists profit from choosing co-creation, even when there are

no links among users (Null

), is higher than that with noco-creation (NoCo-creation).

This result follows from the convexity of the costfunction, but it is still useful to point out. In sub-sequent sections we analyze the more challengingcases where there are different networks of external-ities between the users, and we perform profit com-parisons between networks under different functionalforms of the cost and benefit functions. Because usersdo not innovate or create in a vacuum, an analysis ofthe patterns of externalities between users is a salientaspect of co-creation.

4.2. Centralized Pattern of ExternalitiesAmong Users

Although theoretically there could be many patternsof links between users, the most common in prac-tice is that with an expert user whose productiveinput benefits all other users. A prominent example isthe lead user phenomenon, which has been the mostwell established and widely studied method for userdesign, especially in B2B markets (von Hippel 1986,1998; Urban and von Hippel 1988).

Though very common in business markets, theexpert user phenomenon finds resonance in consumermarkets as well. For example, at Threadless.com, cus-tomers, usually amateur designers at the forefront offashion trends, can send in their own T-shirt designs.Popular designs are then printed and sold as a newlycreated product. In software design, beta testers areoften expert users.

In this subsection we analyze the case where thereare links among the users themselves. These linksform a centralized network with the expert/lead userat the center. In terms of the network literature, thecentralized network is an important canonical one,which captures situations where one expert agent has


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more connections, and more impact, than all otheragents.

The analysis of this case is more complex than thatpresented in 4.1 because the agents are not symmet-ric with respect to their numbers of links, and so wehave to allow for an asymmetric equilibrium in effort

and choice. We denote the users linked to the leaduser as her neighbors, and the number of neighbors isexogenously fixed atN.

The total efforts at a neighbors node en and at thelead users node el are, respectively,

en= en+ fef+ lel andel= el+ fef+ nNen

(10)

where l1 is the benefit to a neighbor from a uniteffort exerted by the lead user, andn 1 denotes the

benefit to the lead user from a unit effort exerted byhis neighbor. We assume that the lead user has higherspillover than her neighbors:

l

n

.For purposes of comparison with 4.1 and 4.3, we

will setN+1= U. We assume that the firm can chargedifferent prices to the neighbors and the lead user.It is straightforward to assume different benefit func-tions b l and bn for the lead user and neighbors,respectively, but we lay these aside to focus on thenetwork structure.

Proceeding similarly to 4.1, the probability that acustomer of typei purchases is Pr[User i buys] = +bei pi/2, where i = 1 n. The expected profit ofthe firm is

=+ bel pl

2

pl

+N

+ ben pn

2

pn cef (11)

The firm will choose p1 and pn optimally, and theFOCs give the two prices in terms of the efforts:

pi=+ bei

2 i= 1 n (12)

As in 4.1 (see Equation (6)), each neighbors and leadusers surpluses, after prices are incorporated intoEquation (11), areCSi= 1/16+bei

2cei i =

1 n. Finally, the expected profit of the firm after priceshave been incorporated is

=+ bel

2

8 +N

+ ben2

8 cef (13)

To characterize the effort profilee = ef el enin equi-librium, we first focus on the problem of customers 1andn. The two FOCs for the lead user and neighbor,CSi/ei= 0 fori = 1 n, give us, respectively,

+bel=8cel and +ben=8c

en (14)

The firms FOC,

E

ef=

+bel

4

bel

ef+N

+ben

4

ben

ef

=c ef

becomes

celbel+Nc

enben =

1

2fcef (15)

For the linear benefit and quadratic cost case, thethree FOCs solved simultaneously give the optimalefforts of the lead user, neighbors, and the firm,respectively, as

eCentrl =16c 1+nN

16c 116c 1 22fN+ 1

22fN l+ nlnN1

(16)

eCentrn =16c 1+l

16c 116c 1 22fN+ 1

22fN l+ nlnN1

(17)

eCentrf =2f16c 1N+ 1+N l+ n

16c 116c 1 22fN+ 1

22fN l+ nlnN1

(18)

Substituting the optimal efforts in (16)(18) in theprices given in (12) gives the optimal prices:

pCentrl = 8ceCentrl p

Centrn = 8ce

Centrn (19)

From (16), (17), and (19), we get the following propo-sition regarding optimal efforts exerted and pricespaid by the lead user and by each neighbor.

Proposition 2. When the spillover from the lead userto the neighbors is not too large (l< N n), the lead userexerts a greater effort and also faces a higher price than the

neighbors.From (12) we see that the price paid by a user

increases in the total benefit she receives, and there-fore in the total effort accruing at the user s node. Thetotal benefit received by the lead user is higher thana neighbor if the spillover from the lead user to theneighbors (1) is smaller than the total spillover fromall the neighbors to the lead user (N n). By contrast,the neighbor pays a higher price and the lead user alower price when the spillover from the lead user ishigher than the spillover from the neighbor.


9/17


In equilibrium, after substituting the optimalefforts, the probabilities of purchase by the lead userand neighbor are given by

Pr[Lead user buys] = 4ceCentrl

Pr[Neighbor buys] = 4ceCentrn

The optimal profit of the firm in the centralizednetwork can be expressed conveniently in terms ofefforts as

Centr =32c2eCentrl 2+32c2NeCentrn

2ceCentrf 2 (20)

The following result states the main finding of thissection.

Theorem 1. (a)When the users have a centralized pat-tern of externalities among them, the lead users and theneighbors expected surplus as well as the firms profit all

increase in the lead users spillover, l.(b) The firms equilibrium profit and effort are higher if

it co-creates with users who are connected to each other bya centralized pattern of externalities than if it co-createswith users who do not have any connections with eachother (Centr > Null andeCentrf > e

Nullf ).

Part (a) of Theorem 1 provides one of the maininsights of the paper: when the lead users spilloverincreases, i.e., when her effort is more beneficial to herneighbors,all partiesthe lead user, the neighbor, and the

firmbenefit. Interestingly, the symbiotic relationshipbetween the lead user and firm occurs independentlyof their costs and abilities and also independently ofthe spilloverf between them. It can be argued that alead user is more effective when her efforts are more

beneficial to the other users. Our analysis shows thateveryone benefits when the lead user becomes moreeffective, even though effective is defined here onlyas being advantageous for other users (the neighbors)and not for the lead user or the firm directly.

A further implication of Theorem 1, part (a) is thatthe efforts of the firm and the lead user are not sub-stitutes: a better (more representative or better con-nected) lead user is not a reason for the firm to lower

its creative effort. Thus, the lead user and the firm arenot mere substitute experts in the creation of value.In fact, they complement each other.

The intuition is as follows. In a centralized network,as the spillover of the lead user (l) increases from agiven initial level, the benefit to the neighbors goesup as a result of two different effects: the direct ben-efit increases and the probability of buying increases.The price charged by the firm will increase as well butonly by half the benefit (see Equation (12)); overall,the neighbors will benefit. In fact, the neighbors will

now choose to exert a greater effort themselves.8 Asa result of increased spillovers by the lead users, theneighbors are not content to just receive the extra ben-efit; they actually increase their own effort as well togain even more utility. In response to this increase, thefirm will not only increase its price to the neighbors

but will also increase its effort to make even higherprofits (using a similar argument to the neighborsincrease). Now, the lead user is faced with greaterfirm and neighbor efforts and thus finds it beneficialto increase her own effort as well. Greater efforts byall parties create a larger benefit for the lead user, andthough the price increases as well, it increases by onlyhalf of the extra benefit. This leads to a higher surplusfor the lead user.

The endogenous efforts also explain why the profitfrom a centralized network of externalities, with alead user in the center, is higher than co-creation witha null network of externalities. Because the efforts of

the firm, lead user, and neighbors in the centralizednetwork are greater than the corresponding efforts inthe null network, the total efforts are greater for thecentralized network. This results in higher prices andthus higher profits for the centralized network com-pared to the null network. Thus the firm benefits fromthe lead user.

The rationale for using the lead user describedin the previous paragraph is novel. We have notassumed any differential cost or information betweenthe firm and the lead user, the usual argumentpresented in the trade press for why lead usersmay be valuable.9 The business press holds that

lead users are privy to information that firms donot have, especially product use information sincethey are users themselves. Lacking the knowledgerequired for product-use-related innovations, firmsfind it beneficial to co-opt the lead users innovationson that dimension. This information asymmetry is notneeded to obtain our result, which is driven purely

by the network centrality of the lead user.Note that there is an increase in the number of links

when we go from a null network to a centralized net-work; part of the reason why the centralized networkis more profitable could be attributed to the num-

ber of links. We demonstrate below the importance of

the pattern of links as opposed to just the number oflinks. We prove the stronger result that the centralizedexternality network can be more profitable than thedecentralized externality network even though thedecentralized network has more links between users.

8 Recall that CSn = 1/16+ ben2 cen. So the marginal

benefit equals marginal cost equation is 1/8+ benben=

cen. Since increasing 1 unequivocally increases the left-handside, the optimal choice ofen increases.9 By assumption, the marginal cost of effort,c, is the same for boththe firm and the lead user.


10/17


4.3. Decentralized Pattern of ExternalitiesAmong Users

We now analyze the case where links between theusers form a decentralized network. We focus onlyon links among users. For U users, a decentralized(circle) network has U links among users, whereas

a centralized (star) network has U1 links amongusers (if we include links to the firm as well, then thenumber of links is 2Uand 2U 1, respectively). Themain reason for analyzing a decentralized network isto establish that a centralized network of externalitieswith a recognized expert (lead user) can yield higherprofits than the decentralized network, despite havingfewer links than its counterpart.

In the networks literature, the decentralized net-work is one of the important canonical networks. Itcaptures situations where the agents are symmetricand connected to each other but there is no recog-nized expert in the network. A decentralized net-

work can emerge when there is social learning amongagents who are close to each other in terms oftheir tastes and/or preferences, and there is only localobservation among agents (e.g., Goyal 2007, p. 97;Banerji and Dutta 2009).

We number the U nodes from 1 through U. Thetotal effort eu at a user us node (1 < u < U ) is

eu= eu+ ueu1= ueu+1+ fef

Each node u benefits from the efforts expended byher two neighbors, (u 1) and (u+ 1). Here, u is theeffort spillover between users. As in the two previous

sections, the firms optimal price is pu=+ beu/2,and a users consumer surplus after incorporating thefirms optimal price is CSu =1/16 + beu

2

ceuwithbeu = beu+ ueu1+ ueu+1+ fef.The FOC for user u is 1/8 + beu= c

eu.Using symmetry, linear benefits and quadratic costsyield the optimal efforts of the firm and the users witha decentralized network:

eDecentru =

16c 22fU 1 2u

eDecentrf =2fU

16c 22f

U 1 2u

(21)

The firms profit is given by

Decentr =4c2U 8c 2fU

16c 22fU 1 2u2

(22)

Comparing the profit with the null network given inCorollary 1, we immediately see that the firms profitwith a decentralized network is higher.

The main result of this section compares the firmsprofit from a centralized network with that from a

decentralized network. To make the comparison, weassume that all users in the decentralized networkhave the same spillover as the neighbors in the cen-tralized network; that is, u=n. In this way we canfocus on the lead users spillover l, since it is thepresence of the expert/lead user that is the major dif-

ference between the two networks.Theorem 2. There exist critical values 1 and n of the

lead users and neighbors spillovers such that, for n< nand 1 > 1, environments with centralized externality

patterns among users are more profitable for the firm thanenvironments with decentralized externality patterns.

Proof. See the appendix.

To understand the above, first note that if n= 0,the decentralized network reduces to the null net-work. We have already shown in part (b) of Theo-rem 1 that the firm is better off under a centralizednetwork than with the null network. Using continu-

ity arguments, the same holds for small n. Now, thedecentralized network does not have a lead user, andthe firms profit clearly does not depend on l. On theother hand, the firms profit with the centralized net-work increases in l, as seen in Theorem 1, part (a).So if we let u= n and allow l to increase, the prof-its of the centralized network will eventually exceedthat of the decentralized one. The only question is,would we get the same result for the allowed val-ues of the spillover parameter, l 1? If we set l= 1,one can see that for values of n that are not toolarge, the firms profit under the decentralized net-work is indeed lower than under a centralized one.

The decentralized network has more links than thecentralized network, which shows that the centralizednetwork can be more profitable because of the pat-tern of links and not the number of links. Our findingis consistent with the networks literature, which hasshown that the pattern of links matters. More impor-tantly, we demonstrate exactly how it matters in ourspecific context. In co-creation it can be more prof-itable for the firm when the customers form a cen-tralized pattern of links among themselves than whenthey form a decentralized pattern. As shown below,this is true even when the spillovers among users arethe same in both cases.

Note that it is possible for the centralized net-work of externalities to dominate the decentralizedone even when the lead user and the neighborshave the same spillovers on each other, l=n. Thisoccurs when the minimum of the profit differenceCentr Decentr is positive, where the minimumdifference corresponds to l = n (recall that, byassumption, l n). Thus the centralized networkcan dominate the decentralized network even whenall the spillovers in both networks are exactly thesame; u =n =l. For example, with c = 1, = 1,


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Figure 3 Difference in Profit Between Centralized and Decentralized

Networks vs. l

0.01

0.04

0.09

0.14

0.19

0.24

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Centr

Decentr

l

n= 0.2

n= 0.05

n= 0.15

u= l= n= 02, and U ranging from 10 to 25, theprofits of the centralized structure exceed those of thedecentralized one, with improvement values rangingfrom 0.4% to 0.8%.

Figure 3 depicts the profit difference as a functionof the lead users spillover l for several values of theneighbor spillover. Since ln, each graph starts atthe pointl= n. The graph shows that the profit dif-ference is indeed increasing inl, but it is also higherfor lower values ofn.

5. Different Cost and BenefitFunctions

To check the robustness of our previous results, wesolved the equilibria of our model allowing for dif-ferent specifications of the cost and benefit functions.We began by assuming a linear benefit function andgeneralizing the cost to the family of convex func-tions cx=cxp with p > 1. The appropriate FOCs do

not allow for a closed-form analytical solution, butthey provide for a unique numerical solution. Thegraph in Figure 4 shows the difference between thefirms profits in centralized and decentralized exter-nality networks as a function ofl for the followingparameter values:N= 10,f= 025, andn= u= 01.This clearly demonstrates that Theorem 2 still holds.

Figure 5 demonstrates that, for the same parame-ter values, Theorem 2 holds even when we assumequadratic cost and change the benefit function.

Figure 4 Difference in Profits for Various Cost Functions vs.l

0.05

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

CentrDecentr

l

c(x) = x3

c(x) = x2.5

c(x) = x2

c(x) = x1.5

Figure 5 Difference in Profits for Various Benefit Functions vs.l

0.02

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

b(x) = x2/3

Centr

Decentr

l

b(x) = x1/2

b(x) = x1/3

6. Centralized Pattern of Externalitieswith a Single Price

In 4.2 we analyzed the centralized externality pat-tern where the firm price discriminates between thelead user and neighbors based on its knowledge ofthe degrees of the nodes. In this section we examinethe centralized network where the monopolist firmsets a single price for both types of users. The mainpurpose here is to compare the potential advantageof price discrimination in a co-creation setting.

Though the main result of this section (see The-orem 3) holds for the general centralized networkwhere both the lead user and neighbors exert effortas in 4.2, for analytical simplicity and to clearly

bring out the forces involved, we will present the casewhere only the firm and lead user exert effort and theneighbors do not.

The total efforts at a neighbors node and at thelead users node en and el are, respectively,

en= fef+ lel andel= el+ fef

(23)

The consumer surpluses are CSi=biei ce2i p+ ,

i = l n. The single price charged to all the agentsis p. Proceeding similarly to 4.2, the probabilitythat a customer of type i purchases the product is+ bei p/2), where i = 1 n.

The expected profit of the firm is10 SP = +bel p/2p+ N+ ben p/2p ce

2f. The

optimal price in terms of the efforts is

p=N+ 1+ bel+Nben

2N+ 1 (24)

10 We denote the various quantities in the single-price case with thesuperscript SP.


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The lead users surplus in terms of price is the sameas in 4.2, and after incorporating the optimal pricefrom (24), the expected surplus becomes

ECS1= 1

4

2N+ 1

2N+ 1bel

N

2N+ 1ben+

2

2ce2l

The expected profit of the firm after incorporatingprice isSP = N+1+bel+Nben

2/8N+1ce2f. The FOC for the lead user is

1

4N+1

2N+1

2N+1el+fef

N

2N+1fef+lel+

2

2N+1lN 2cel =0

The firms FOC is

1

4N+ 1N+ 1+ 1+ lN el+ N+ 1fef

N+ 1f 2cef= 0 (25)

Solving the FOCs simultaneously gives the optimalefforts eSPl and e

SPf . These expressions are complex

and are recorded in the appendix in (35) and (36),respectively. Substituting back gives the firms opti-mal profit in the centralized network when it doesnot price discriminate and charges only a single price,SP. This is given in the appendix in Equation (37).

To compare the monopolists profit with andwithout price discrimination, we need to provide thesolution for an environment where the firm price dis-

criminates between the lead user and the neighborsin a centralized network of externalities, but whereonly the lead user and the firm itself exert effort (notethat the analysis in 4.2 had the neighbors also exert-ing effort). The details are straightforward and arerelegated to the appendix. The optimal efforts of thelead user and firm when the firm price discriminates

between lead users and neighbors are, respectively,11

ePDl = 8

1202f16+ 15+lN and

ePDf =f16+ 15+ lN

120 2

f

16+ 15+ lN

(26)

The optimal profit is

PD =2048+815+l

2N2f16+15+lN 2

1202f16+15+lN2

(27)

The following theorem states the main result of thissection.

11 To simplify expressions, we set c= 1 and= 1, and we denote thevarious quantities in the price discrimination case with the super-script P D.

Theorem 3. When a monopolist co-creates a productwith customers who form a centralized pattern of external-ities, its optimal profit is higher if it charges a single priceto all agents than if it price discriminates between the leaduser and neighbors. Both the lead user and firms optimalefforts are also higher with a single price.

Proof. See the appendix.We find that even in the more general central-

ized network, with both the lead user and neighborsexerting co-creation effort (as in 4.2), the monop-olist is better off without price discrimination com-pared with a situation where it price discriminates.This result stands in contrast to the situation in tradi-tional markets without co-creation, where customersdo not exert product development effortsin thosecases, price discrimination is always better for themonopolist.12 The reason for this result is that pricediscrimination by the monopolist has two opposite

effects: First, it can better extract the surplus of het-erogeneous customers; this is the standard positiveeffect of price discrimination on profits, as is welldocumented in the literature. Second, because of thereduced surplus left with them, consumers rationallyreduce their co-creation efforts; this reduces the ben-efit of the product as captured by the total effortsexerted by all network partners. Because the firmsprice is increasing in the efforts exerted, this secondforce has a negative effect on firm profits, and Theo-rem 3 shows that the negative effect dominates. Thesecond effect is clearly seen from the fact that for thelead user, eSPl > e

PDl . In fact, in so far as the efforts

are concerned, there is a double effect when the firmprice discriminatesnot only does the lead user exertless effort but so does the firm. That is, eSPf > e

PDf (see

the proof of Theorem 3 in the appendix).Our finding above has practical ramifications.

Despite the opportunity to price discriminate, inmany B2B settings, firms often have a strict no-discount book price policy. Our analysis showsthat for co-creation, this self-imposed lack of pric-ing flexibility may actually be profit enhancing. Thisresult is reminiscent of similar game-theoretic find-ings where a commitment by a monopolist to acertain action (thus limiting its flexibility) causes a

change in customer behavior that is profitable to themonopolist. For example, a monopolistic firm thatcan profitably employ behavior-based price discrimi-nation may wish instead to commit to a single priceover time. This commitment will change the pur-chase timing of the customers and lead to higher

12 In the appendix we analyze the case of a centralized networkwhereonlythe firm exerts effort, so there is no co-creation with thecustomers. We analyze both the cases with and without price dis-crimination and show that if there is no co-creation, the monopolistis better off with price discrimination, as is to be expected.


13/17


profits (see Fudenberg and Villas-Boas 2006 for asurvey and Fudenberg and Tirole 2000 for a sim-ilar result in a competitive setting). However, themechanism through which this occurs in co-creation(changing consumers incentives to expend effort onproduct development) is different than the mecha-

nism through which it occurs in the literature onbehavior-based pricing, changing consumers timingof purchase.

7. ConclusionThe idea that producers and consumers collabo-rate to co-create goods and services has a long his-tory in business and marketing thought. In B2Bcontexts, co-creation has always been a commonphenomenon. Recently, however, with increases ininformation technology support, flexible manufactur-ing, and advanced user interfaces, it has becomeincreasingly feasible for customers to participate in

the production process even in the B2C world.Despite the great deal of interest in co-creation, therehas been little academic research in this field, espe-cially with a view to developing economic models ofthis phenomenon. This paper takes a first step towardaddressing this gap.

We model co-creation and the interaction betweenfirm and users/consumers as a network of external-ities. Each user is located on a node of the network.Any two users whose innovation efforts affect eachother are depicted as connected nodes. This particularnetwork topology can be used to capture the specificpattern of production externalities between different

users; of course, the firm is assumed to be connectedto all of them. This setting allows us to model theimportant role played by lead users in the innova-tion process. Lead users can be modeled in the net-work simply as central nodes with many connectionsto other nodes, each with a user on it.

Variants of our model can be used to analyze manyother phenomena that involve team production or

joint production. In team selling, especially for com-plex products, some team members are more likelyto benefit other members. These asymmetric effects insales teams can be captured by the pattern of links ina network and by different spillover coefficients. Asanother example, it is well known that people oftenconstruct their preferences for novel products ratherthan having preformed preferences. Such constructionoccurs over social networks wherein people discussnovel product features with each other. Clearly, thesesocial connections can be modeled using our networkapproach, and the disproportionate influence of somemembers can be captured by the number of connec-tions and the size of the spillover effects.

In our specific model, we find that the firmsprofit from co-creation dominates its profit in a no

co-creation environment. Also, pricing by the firmreverses the usual free-riding effect found in publicgoods, so that in co-creation, the firms effort increasesin the number of users it is connected to. If there areconnections among users such that they form a cen-tralized pattern of externalities with a lead user in the

center, the optimal profit is higher than that withoutuser connections. Furthermore, the centralized exter-nality network can generate higher profit than thedecentralized one even though the latter has morelinks, even in cases where the spillovers among usersare identical in both cases. This highlights the impor-tance of the pattern of links in co-creation produc-tion networks as well as the number of links. Anotherinsight our model provides is that as the spilloversfrom the lead user to the neighbors increase, all par-ties can benefit, including the neighbors, the firm, andthe lead user herself. Finally, we find that it is bet-ter for the monopolist firm to charge a single price

in a centralized network than to price discriminatebetween the lead user and her connected neighbors.

Our analysis has clear managerial implications.First, firms should try to identify and developalliances with users that are central to the network,i.e., users who have production externalities withmany other users,even whensuch central users do nothave lower marginal costs or more product knowl-edge compared to the firm. The user innovation lit-erature exhorts firms to identify and build allianceswith lead users who can foresee emerging trendsin the marketplace (von Hippel 1986). Another eco-nomic reason for a firm to ally with customers is that

customers may have lower costs or higher innova-tion ability than the firm, especially in B2B contexts.Clearly, it is difficult to identify customers who canforesee future trends and who have lower costs orhigher abilities. Thus identifying lead users accordingto the usual prescriptions may be difficult. The leaduser literature views the identification of lead users asa major obstacle. We show that it may be beneficial to

just identify users who are linked to many other cus-tomers informally or through strategic alliances andother types of research and development networks. Insome circumstances a users network of connectionsmay be more publicly observable than its costs and

abilities, which are internal to its functioning.Second, in terms of pricing regimes, firms should

consider their effect on the effort exerted in the inno-vation and production stages. It is sometimes betterfor the firm to commit to charging a single price thanto price discriminate in the presence of lead usersin the network. In B2B settings, despite the oppor-tunity to price discriminate, many firms often havea strict no-discount book price policy. Our analysisshows that for co-creation, this self-imposed lack ofpricing flexibility may actually be profit enhancing.


14/17


Appendix

Proof of a Benchmark Without Co-CreationBecause there is no co-creation by the users, the effort ata given users node only comes from the firm. Thus, eu=fef, and the surplus a user receives is beupu+. Givena price pu, a user will purchase the product with probabil-

ity given by Prpubeu. The probability that a userpurchases is + beu pu/2. The expected profit of thefirm is

f=U

+ beu pu

2

pu ce

2f (28)

The firm will choose pu optimally, and the FOC gives theprice in terms of the efforts:

pu= + beu

2 (29)

As in 4.1, the profit after incorporating price is

=U + beu

2

8 ce2f (30)

Specializing the benefit function to be = e, the firms FOC

givesE/ef=U+fef/4f2cef= 0. This givesthe optimal effort of the firm as

eNoCo-creationf =U

8

f

c2fU /8 (31)

After simplifications, the optimal profit of the firm withoutco-creation is

NoCo-creation =U

8

1+

U

8

2f

c 2fU /8

(32)

Clearly, for the effort and the profit to be positive, we needc > 2fU /8, and we make this assumption. We can seethat the optimal effort increases in U.

Proof of Theorem 1. (a) The lead users expected sur-plus isECS1 = 1/16+bel

2 ce2l . After substitutingthe optimal efforts, the derivative of the expected surpluswith respect to (w.r.t.) 1 is

ECSl

l=

1

8+ eCentrl + fe

Centrf +N ne

Centrn

eCentrl

l+f

eCentrf

l+N n

eCentrnl

2ceCentrleCentrl

l

Collecting terms involving eCentrl /l, we can rewrite thederivative as

ECSl

l=

1

8+ eCentrl +fe

Centrf +N nle

Centrn 2ce

Centrl

eCentrl

l+

1

8+ eCentrl + fe

Centrf +N le

Centrn

f

eCentrf

l+N n

eCentrnl

(33)

From the lead users FOC given in (15), the term in curlybraces on the right-hand side (RHS) of (33) is zero. Also,from (18) and (19), it is clear that eCentrn /l > 0 andeCentrf /l> 0. Since the efforts are positive by assumption,

we can therefore conclude that ECSl/l> 0. Using a sim-ilar approach, it can be shown that ECSn/l> 0; i.e., theneighbors surplus increases in l.

Now consider the derivative of the firms profit given in(14), after substituting the optimal efforts w.r.t. l:

Centr

l

= 1

4

+ eCentrl +feCentrf +N le

Centrn

eCentrl

l+ f

eCentrf

l+N n

eCentrnl

+N

4+ eCentrn + fe

Centrf +le

Centrl

eCentrn

l+ f

eCentrf

l+ eCentrl + l

eCentrll

2ceCentrfeCentrf

l

Rearranging and combining all terms involving eCentrf /l,we get

Centr

l = 1

4 + eCentrl + feCentrf +N leCentrn f

+N

4+ eCentrn + fe

Centrf + 1ne

Centrl f 2ce

Centrf

eCentrf

l+

1

4+ eCentrl +fe

Centrf +N nle

Centrn

eCentrl

1l+N n

eCentrnl

+

N

4+ eCentrn + fe

Centrf

+leCentrl

eCentrn

l+ eCentrl + l

eCentrll

(34)

The entire term in curly braces on the RHS of (34) is zeroby the firms FOC given in (16). Again, from (17) and (18),

it is clear thateCentrl /l> 0 ande

Centrn /l> 0. Since all the

efforts are positive by assumption, we therefore proved thatCentr/l> 0.

(b) In part (a) we have shown thatCentr/l> 0. Usingthe same technique, it can also be shown that Centr/n> 0. Thus the minimum profit with the centralized net-work (minimum w.r.t. l and n corresponds to l 0and n 0. This minimum profit is

Centrl0 n0 =

4c2U 8c 2fU/16c 22fU 1

2. From Lemma 1, this

is the same as the profit with a null network, Null. Thus,for all positive l and n, we have

Centr > Null.Also, eCentrf /l >0 and e

Centrf /n > 0 imply that the

minimum effort of the firm with the centralized networkcorresponds tol 0 andn 0. WithU= N+ 1, the min-

imum effort is eCentrf l0 n0 = 2fU /16c 2

2fU1.

From Lemma 1, we can see that this is the same as the firmseffort with a null network, eNullf . Thus, for all positive 1nand nl, we have e

Centrf > e

Nullf . In fact, it can be similarly

shown that both the lead user and the neighbors in the cen-tralized network exert more effort than a typical user in thenull network; i.e., e Centrl > e

Nullu and e

Centrn > e

Nullu .

Proof of Theorem 2. In the proof of Theorem 1, wealready established that Centr/l > 0; that is, the profitof the firm with the centralized network increases in thespillover from the lead user to her neighbors. Also, it isclear that the firms profit with the decentralized network


15/17


is independent of1n. If Centr > Decentr even at the min-

imum value of l, which is l = n, then we are done.If Centr < Decentr at l = n, then the difference

Centr Decentr increases as l increases from its minimum valuen. We will show that

Centr > Decentr when Centr is eval-uated at l= 1. This will establish that there exists a valuel of the lead users spillover such that the centralized net-

work is more profitable than the decentralized network forevery l> l.

Since the expressions are cumbersome, we will simplifythem by setting c=1 and =1. These parameters are notof interest to us and do not affect any results in the paper.

ConsiderCentr11, the firms profit evaluated at l= 1.We need to show that Centrl1 >

Decentr. Now, usingthe same techniques as in Theorem 1, we can show thatCentr/nl > 0. Since this holds for any l, includingl = 1, therefore

Centrl1n0 is clearly a lower boundof Centrl1. In other words,

Centrl1n0< Centrl1.

Hence, we are done if we can show that Centrl1n0>Decentr. Substituting l= 1 and n= 0, the profit from thecentralized network reduces to

Centrl1n0=42048U 248 2f16U 12

225 22f16U 12

The firms profit with the decentralized network isDecentr =4U 82fU/152

2fU2n

2. Observe that

Centrl1n0

does not depend on n, which appears only in

the denominator ofDecentr. Therefore, Centrl1n0

> Decentr

if n is not too large. Specifically, a sufficient, but by nomeans necessary, condition is

n< n 1

2

15 22fU 225 2

2fU 16U 1

U 82fU

2048U 248 2fU 16U 12

Proof of Theorem 3. We will prove this result in threesteps.

Step1. We will show thateSPl > ePDl ande

SPf > e

PDf . That is,

the efforts of the lead user and firm are higher when charg-ing a single price than they are when using price discrimi-nation. We will also show that eSPl /f> e

PDl /f.

Step2. Using the inequalities established in part (a)above, we will show that SP PD /f> 0. That is, thedifference (SPPD ) increases in f, and so the minimumdifference corresponds tof= 0.

Step3. Finally, we will show that the difference of profitsevaluated atf= 0 is positive. That is, the minimum differ-enceSPPD f=0> 0. Then, by (b) above, the differenceis always positive.

Proof of Step 1. We first record the optimal efforts andfirm profit with the centralized network when the firmcharges a single price:

eSPl =4N+ 11+ 2 lN

416N+ 12 + 1+ 2 lN 2fN+ 1

8N+ 12 1lN 1+ 2lN1

(35)

eSPf =fN+ 18N+ 1

2 1 lN 1+ 2 lN

4

16N+ 12 + 1+ 2 lN 2fN+ 1

8N+ 12 1 lN 1+ 2lN1

(36)

Substituting back gives the firms optimal profit as

SPf =

N+ 182fN+ 1

8N+ 12 1 l

N 1+ 2lN 2

416N+ 12 + 1+ 2lN 2fN+ 1

8N+ 12 1lN 1+ 2lN21

(37)

The optimal efforts and firm profit when the firmprice discriminates, but only the lead user and firm exerteffort, are

ePDl = 8

1202f16+ 15+lN

ePDf =f16+ 15+ lN

120 2f16+ 15+ lN

(38)

The optimal profit is

PD =2048+ 815+l

2N 2f16+ 15+lN 2

120 2f16+ 15+lN2

(39)

For notational simplicity, we write the efforts as eSPl =4N+ 11+ 2 lN/D1 and e

PD1 =8/D2, where D1 =

416N+ 12 + 1 + 2 lN 2fN+ 18N+ 1

2

1 lN 1+ 2lNand D2= 1202f16+ 15+lN .

Since the numerators ofeSPl andePDl are positive, the denom-

inators areD1> 0 and D2> 0. Simple algebra shows that

eSPl ePDl =

41lN 17+18l82fN+1

D1D2> 0 (40)

Similarly,

eSPf ePDf =

4f1 lN 17 + 18l1+ lN

D1D2> 0 (41)

The inequality in (40) follows because, by assumption from4.1, 8c2fU >0. With c=1, =1, and U=N+ 1 the

condition becomes 82fN+ 1 > 0.

The derivative ofe SPl w.r.t. f is

eSPl

f

=8fN+1

21+2lN8+15+lN+6+3llN2

D21. (42)

The derivative ofe PDl w.r.t. f is

ePDlf

=16f16+ 15+ lN

D22 (43)

We can show that /1neSP1 /f


16/17


difference between this minimum value ofeSPl /fand theRHS of (43) gives us

eSPlf

l=1

ePDl

f

=4f1 lN 900

4fN+ 116+ 15+ lN

1202f

16+ 15+ lN 215 22

fN+ 12

(44)

Clearly, the RHS of (44) will be positive, and we will haveshown that eSPl /f > e

PDl /f if the expression in curly

braces in the numerator is positive. We know from 4.1that 16c 1 22fU > 0. Sincec= 1, = 1, and U= N+ 1,

this condition therefore reduces to 2fN+1 < 7 5. The

expression in curly braces will be positive if4fN+ 116+15+lN is less than 900. Setting 1n = 1 increases thevalue of the expression, which now becomes 164fN+ 1

2.If this is less than 900, we are done. Since, by assumption,2fN+ 1 ePDl , e

SPf > e

PDf ,

and eSPl /f > ePDl /f, by inspecting the first set

of curly braces, we can see that it is positive if 1+lN

2/N+ 1eSPl > 1+2l N e

PDl . We can show that

this is indeed the case. The term in the second set of curlybraces is clearly positive. Thus, we have established thatSP PD /f> 0.

Proof of Step 3. We will now evaluate the profits atf= 0. The firms profit with a single price is

SPf=0=N+ 18+N 16+ l+ 6N+ 3llN

2

25+ 6lN 23+ 2+ lN

2 (51)

With price discrimination, it is

PD f=0=256 + 15+l

2N

1800 (52)

Simple algebra and some rearrangement shows that

SPf=0PD f=0=

A+BN+CN2+DN3+EN4

36005+6lN 23+2+lN 2 (53)

where A = 7935 + 222l B = 45759 + l2386 +

15l C = 210734 + l13658 + 101 13ll D =41584 + l6150 + l629 l209 + l andE= 18 ll360 + l162 l41 + l. All these areclearly positive, and we have thus established the minimumvalue ofSP PD > 0. Theorem 3 is proved.

Price Discrimination in a Centralized Network WhereOnly the Lead User and Firm Exert EffortThe consumer surpluses are CSi = bie

2i ce

2i pi + ,

i= 1 n. The prices charged to the neighbor and lead userare pn and pl, respectively. Furthermore, let ben = en =lel+ fef andb el=el= el+ fef.


17/17


The expected profit of the firm is

=

+ bel pl

2

pl+N

+ ben pn

2

pn ce

2f (54)

The two prices in terms of the efforts are

pi= + bei

2 i= 1 n (55)

The lead users surplus after price is CS1 =1/16bel+

2ce2l . The neighbor does not exert effort;the expected profit of the firm after prices have been incor-porated is =+ bel

2/8 + N+ ben2/8 ce2f.

The firms FOC is

1

4el +fef+f+

N

4fef+lel +f2cef= 0 (56)

The lead users FOC is similarly obtained. Solving thesesimultaneously, we get the optimal efforts as in (26), andsubstituting back gives us the firms profit as in (27).

Centralized Network with Only Firms Effort: With andWithout Price Discrimination

Price Discrimination: The consumer surpluses are CSi =bieice

2i pi+ i= 1 n. The prices charged to the neigh-

bor and lead user arepn and pl, respectively. Furthermore,let ben= en = fef and bel=l + el = l +fef l > 0. Weintroduce l >0 merely to create heterogeneity between thelead user and neighbor so that there is a role for price dis-crimination. Because this is just an additive constant, noneof the results in the paper would change if we had assumedbel = l+el throughout. The expected profit of the firm andthe prices in terms of effort remain the same as (54) and(55). Following the usual method, the optimal effort of thefirm and the optimal profit are

eTwoPricesf =fl + U

8c 2fU

TwoPricesf =8cl2 + 2l+ 2U 2fl

2U 1

88c 2fU

(57)

Single Price: The consumer surpluses and benefit func-tions remain as above. We record the optimal effort andprofit of the firm:

eOnePricef =fl + U

8c 2fU OnePricef =

cl + U 2

U 8c 2fU (58)

Simple algebra shows that TwoPricesf OnePricef =

l2U 1/8 U > 0. Thus, when there is no co-creation, themonopolist firm is better off exercising price discriminationthan committing to a single price.

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