bundling electronic journals and competition among...

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BUNDLING ELECTRONIC JOURNALS ANDCOMPETITION AMONG PUBLISHERS

Doh-Shin JeonUniversitat Pompeu Fabra and CREA

Domenico MenicucciUniversità degli Studi di Firenze

AbstractSite licensing of electronic journals has revolutionized the way academic information isdistributed. However, many librarians are concerned about the possibility that commercialpublishers might abuse site licensing by the practice of bundling. In this paper, we analyze howbundling affects journal pricing in the market of scientific, technical, and medical electronicjournals and offer a novel insight on the bundling of a large number of information goods. Wefind that (i) when bundling is prohibited, surprisingly, industry concentration does not affectprices; (ii) when bundling is allowed, each publisher finds bundling profitable and bundlingincreases industry profits while reducing social welfare; and (iii) any merger among publishersalready active in the market is profitable but reduces social welfare. (JEL: D4, K21, L41, L82)

1. Introduction

Site licensing of electronic journals (e-journals, henceforth) has revolutionizedthe way academic information is distributed. With site licensing, there is no needto spend time looking for a paper in a library, and many people can download, read,and print a paper simultaneously from their offices at any given time. Furthermore,e-journals’ web sites provide additional services such as search tools, hypertextlinking, remote access, and so forth. Therefore it seems that, sooner or later,e-journals will supplant print journals as the norm.

However, many librarians are concerned about the possibility that commercialpublishers might abuse site licensing for private gain. First, commercial pub-lishers have aggressively raised prices at a rate that is disproportionate to anyincrease in costs or quality. According to the Association of Research Libraries(ARL), in the United States, during the period 1986–2002, the unit cost for journal

Acknowledgments: We thank seminar participants at University of Bristol, University of Malaga,University of Montpellier, Universitat Pompeu Fabra, EARIE 2004, ESEM-EEA 2003, IIOC 2004,and the Toulouse conference on the economics of the software and Internet industries 2005. We aregrateful to Roberto Burguet, Antonio Cabrales, Patrick Rey, Jean-Charles Rochet, Joel Shapiro, JeanTirole, and an anonymous referee for useful comments. The first author gratefully acknowledgesfinancial support from the Spanish Ministry of Science and Technology under a Ramon y Cajal grantand from DGES and FEDER under BEC2003-00412.E-mail addresses: Jeon: [email protected]; Menicucci: [email protected]

Journal of the European Economic Association September 2006 4(5):1038–1083© 2006 by the European Economic Association

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Jeon and Menicucci Electronic Journal Bundling and Competition 1039

Figure 1. Monograph and serial costs in ARL libraries, 1986–2002. Source: ARL.

subscriptions has grown at the rate of 7.7% per year, which is more than twice the3.6% growth rate in the unit cost of monographs (books).1 As Figure 1 shows,through 2000 the increase in library budgets could not match the increase injournal prices, which resulted in a continuous decrease in the number of jour-nals purchased during most of the period. High subscription prices charged bycommercial publishers even induced some academic societies whose journalshad been published by such publishers to start new competing journals, as withthe launch of the Journal of the European Economic Association.2 Second, sitelicensing of e-journals allows commercial publishers to employ powerful pric-ing strategies such as price discrimination based on usage3 and bundling; withprint journals, publishers practiced neither bundling nor discrimination betweenlibraries in terms of subscription prices.4 In particular, librarians are concernedabout bundling. For instance, according to Kenneth Frazier (2001), director oflibraries at University of Wisconsin—Madison, “The content is ‘bundled’ so that

1. See “Monograph and Serial Costs in ARL Libraries 1986–2002” at http://www.arl.org/stats/arlstat/.2. In other disciplines, there are several cases in which the editorial board of a journal ownedby a commercial publisher resigned and founded an alternative journal. See Theodore Bergman’sweb site: http://www.econ.ucsb.edu/˜tedb/Journals/alternatives.html.3. For instance, Derk Haank (2001), the CEO of Elsevier Science, says, “What we are basicallydoing is to say that you pay depending on how useful the publication is for you—estimated byhow often you use it.” See also Bolman (2002) and Key Perspectives (2002) with regard to pricediscrimination.4. In the case of print journals, arbitrage through resale has to some extent limited publishers’ability to practice price discrimination. In contrast, for e-journals the access to a journal is simplyleased and hence resale is impossible.

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1040 Journal of the European Economic Association

individual journal subscriptions can no longer be cancelled in their electronic for-mat. (The Academic Press IDEAL program and the full ScienceDirect packageoffered by Elsevier are examples of such licensing agreements).”5

Moreover, U.S. and U.K. competition authorities approved four years agoone of the biggest-ever science publishing mergers, between Reed-Elsevier (REhenceforth) and Harcourt, in spite of many librarians’ protests. Indeed, the reportof the U.K. Competition Commission (2001) shows concern about potential wel-fare losses due to the merging publishers’ bundling of their e-journals. Before themerger, RE’s ScienceDirect was the most developed web site and offered accessto about 1,150 journals and Harcourt’s IDEAL offered access to 320 journals.

In this paper, we analyze publishers’ incentives to practice bundling and theensuing effects on social welfare, and we derive implications for merger analysis.To isolate the effect of bundling under price discrimination, we consider a maturestage of e-journals in which publishers practice price discrimination based onusage.6 Therefore, we assume away heterogeneity among libraries and build amodel in which each competing publisher offers a set of journals to a library thatwants to build a portfolio of journals and monographs under a budget constraint.7

We analyze how bundling affects journal pricing through its impact on the library’sallocation of budget between journals and books. Although we assume that thereis no direct substitution among journals (i.e., the value the library derives from ajournal is determined independently of whether or not it buys any other journal),there can be an indirect substitution among journals and among journals andmonographs8 through the budget constraint. The utility that the library derivesfrom spending money on books is assumed to be strictly increasing and strictlyconcave.

We first consider independent pricing (no bundling) and show an irrelevanceresult that industry concentration does not affect prices. For instance, in the sim-ple case of homogeneous journals (in which every journal has the same value)we show that, regardless of the level of industry concentration, there exists aunique equilibrium in which all the journals are sold at the same price. In the gen-eral case of heterogeneous journals, we show that there is a unique equilibrium

5. He further argues that “the push to build an all-electronic collection can’t be undertaken at therisk of (1) weakening that collection with journals we neither need nor want, and (2) increasing ourdependence on publishers who have already shown their determination to monopolize the informationmarket place.”6. This implies that the pricing schemes studied in this paper might not correspond to what weobserve now. In fact, the transition implies a change from subscription-based pricing models tousage-based models. Because a sudden switch in the pricing models generates a large change inthe total price required for a library to maintain its subscription to a given collection of journals,publishers are introducing a progressive change (Bolman 2002).7. Typically, an academic library’s material budget is spent on journals and monographs (Gooden,Owen, and Simon 2002).8. As a result of journal price increases, many university libraries have been forced to reallocatedollars from monographs to journals (Kyrillidou 1999).

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candidate regardless of the level of industry concentration and that the equilibriumalways exists both under maximum concentration (the monopoly case) and underminimum concentration, where each publisher sells only one journal.9 Therefore,the outcome under minimum concentration is equivalent to the outcome undermaximum concentration. The irrelevance result is related to the fact that, underindependent pricing, each journal is priced according to what we call “marginalopportunity cost pricing” in the following sense: When a publisher sells a journal,it expects that this journal is the marginal journal (i.e., the last journal purchasedby the library) and so chooses a price p, to match the library’s opportunity cost ofusing p instead on books, such that the library is indifferent between buying thejournal at p and spending p on buying extra books. A monopolist cannot realizea higher profit than the one under marginal opportunity cost pricing because, inorder to realize a higher profit, he would have to increase the price of the marginaljournal and this induces the library not to buy the journal.

When bundling is allowed, we show that each publisher has an incentive tobundle all its journals. We identify two effects of bundling. First, bundling hasthe direct effect of softening competition from books. To provide intuition, weconsider a monopolist who publishes two journals of the same value u. Underindependent pricing, this publisher expects that each of these journals is the lastjournal to be purchased and chooses the same price p for them. Suppose now thatthe publisher bundles the journals and chooses the price 2p for the bundle. Thenthe library is strictly better off buying the bundle than spending 2p on books;because the marginal utility from spending money on books strictly decreases,the utility from spending 2p on books is strictly smaller than twice the utilityfrom spending only p on books. Therefore, the monopolist can charge 2p + ε

(i.e., more than 2p) for the bundle and still induce the library to buy it. This directeffect of bundling increases with the size of bundle, which implies that a largepublisher gains more than a small publisher in terms of the direct effect.

Second, a publisher’s bundling has an indirect effect of inflicting negativepecuniary externalities on all other publishers. The very fact that bundling allowsa publisher to increase its own profit implies that, after a publisher’s bundling,there is less budget left for books and for all the other publishers’ journals. This inturn implies that, for all the other publishers, competition from books is tougherand hence they must lower the prices of their journals in order to sell them. Inparticular, a small publisher with only a few journals does not gain much fromthe direct effect of bundling, but it may lose a lot from the indirect effect if bigpublishers bundle their journals. Therefore, bundling is a profitable and crediblestrategy: It not only increases the bundling publisher’s profit but also decreasesthe profits of rivals and can even make them unable to sell their journals.

9. For the intermediate case of oligopoly, we give a sufficient condition for equilibrium existencein Theorem 2 (see Section 4). The irrelevance result holds as long as the equilibrium exists.

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The direct and indirect effects of bundling suggest that any merger increasesthe merging publishers’ profits because of the direct effect while reducing rivals’profits because of the indirect effect. We also show that bundling (or any merger)increases industry profits. This result implies that the library purchases fewerbooks after bundling. Because bundling can make small publishers unable to selltheir journals, we conclude that bundling decreases social welfare by reducingboth book and journal consumption. For the same reasons, any merger amongactive publishers reduces social welfare. Our finding is consistent with the pre-diction of Kyrillidou (1999) that, if the current trend continues, the budget formonographs will be the resource depleted most rapidly and that, by 2019, onlyabout 10% of the materials budget will be spent on purchasing monographs.Finally, when we examine a publisher’s incentive to acquire a journal from athird party, we find that in the absence of bundling all publishers have the samewillingness to pay for the journal, whereas under bundling the largest publisheralways has the highest willingness to pay. This suggests that bundling might seri-ously affect industry dynamics, with the largest publisher becoming even largerthrough the purchase of titles sold by small publishers that are forced to exit themarket.

Most of the papers on bundling study bundling of two (physical) goods inthe context of second-degree price discrimination and focus on either surplusextraction (Schmalensee 1984; McAfee et al. 1989; Salinger 1995; Armstrong1996, 1999) or entry deterrence (Whinston 1990; Nalebuff 2004). The papersof Bakos and Brynjolfsson (1999, 2000) are an exception in that they studybundling of a large number of information goods while maintaining the frame-work of second-degree price discrimination. Their first paper shows that bundlingallows a monopolist to extract more surplus because it reduces the varianceof average valuations by the law of large numbers,10 and their second paperapplies this insight to entry deterrence. Our paper also studies the bundling ofany number of information goods. The novelty is that we show that bundlingis a profitable and credible strategy—in terms both of surplus extractionand entry deterrence—even when sellers have complete information on thebuyer’s valuation for each product (so that the law of large numbers playsno role) and there is no interdependency among valuations of different prod-ucts. Conventional wisdom says that bundling has no effect in such a settingand this is true if the budget constraint is not binding. However, if the con-straint is binding then we show that each firm has a strict incentive to adoptbundling.

Our paper is related to McCabe (2002b), who studies the transition from printjournals (no price discrimination and no bundling) to e-journals (prefect price

10. See also Armstrong (1999).

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discrimination and bundling).11 Although his setting is similar to ours, there areimportant differences. First, he considers the transition whereas we consider thesituation when this transition is completed. Second, he does not provide the com-parative statics of the transition; we provide the comparative statics of bundlingversus no bundling in the digital world. Furthermore, McCabe assumes bundlingin the case of e-journals whereas we show that, in equilibrium, all publishersadopt bundling. Finally, he does not consider the substitution between books andjournals, and his results often rely on numerical examples. Edlin and Rubinfeld(2004) also argue that bundling of academic journals builds strategic barriers toentry, but they do not present any formal model. McCabe and Snyder (2004) ana-lyze the market for academic journals from a two-sided market perspective butdo not study bundling.

The comparison between bundling and independent pricing in our paper isalso related to the comparison between patent pooling and independent licensingof patents in Lerner and Tirole (2004). In particular, publishers’ pricing decisionsin our paper are driven by what they call the “competitive margin” in the sensethat each publisher cannot raise its prices without triggering an exclusion of itsjournal(s) from the portfolio of journals bought by the library. However, there aretwo important differences. First, Lerner and Tirole consider a simple case in whicheach owner owns only one patent and hence there is no issue of patent poolingat the individual owner level, whereas we consider a general case in which eachfirm owns multiple journals and therefore bundling is decided at the firm level.Second, patent pooling consequently implies a change from the minimum industryconcentration to a monopoly in their paper, whereas we compare bundling withindependent pricing for any given level of industry concentration.

The rest of this paper is organized as follows. Section 2 describes the model.In Section 3, we consider the simple case of homogeneous journals and explainall our main results with minimum technical details. In Section 4, we con-sider the general case of heterogeneous journals. Section 5 provides concludingremarks. All the proofs that do not appear in the main text are relegated to theAppendix.

2. Model

As we said in the introduction, we consider the mature stage of site licensing inwhich journal prices depend on usage and assume that publishers have completeinformation about the value that a library attaches to a journal. This assumptionallows us to focus on the effects of bundling that arise when buyers have no private

11. McCabe (2002a) provides an empirical analysis showing that mergers significantly contributedto journal price increases.

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information. Therefore, we consider only one library with a budget M > 0 thatis assumed to be known to all publishers.12

2.1. Journals and Publishers

There are N publishers; publisher j is often denoted simply by j . We con-sider only profit-maximizing publishers. Let nj ≥ 1 be the number of journalsthat publisher j publishes (j = 1, . . . , N) and let n ≡ ∑N

j=1 nj (≥ N) be thetotal number of journals. Let uij > 0 represent the utility (or the surplus) thelibrary obtains from journal i = 1, . . . , nj of publisher j . Let Uj ≡ ∑nj

i=1 uij andU ≡ ∑N

j=1 Uj . Journals are said to be homogeneous when uij = u > 0 for all i

and j .Because we focus on how bundling affects which journals are sold and at

what prices, we assume that n number of journals are already produced (i.e., thefixed cost of having the first copy of each journal has already been incurred). Weassume that the marginal cost of distributing a journal is zero.

When each journal is sold independently (i.e., in the absence of bundling),publisher j chooses price pij > 0 for each journal i that it owns. Let p ≡(p11, . . . , pn11, . . . , p1N, . . . , pnNN) ∈ R

n++ represent the price vector underindependent pricing. Under bundling, publisher j chooses price Pj > 0 for thebundle of all its journals, which we denote by Bj . Let P ≡ (P1, . . . , PN) ∈ R

N++denote the price vector under bundling.

2.2. The Library

The library’s budget M > 0 is given, and we study how bundling affects thelibrary’s allocation of that budget between journals and books. The library’s payoffis given by the sum of three components: the utility it obtains from the journals itpurchased, the utility it obtains from the books it bought, and the money left afterall purchases. We define a reduced-form utility for books by using an indirectutility function v : [0, +∞) → R+ such that v(m) is the library’s utility frombooks when it spends m ≥ 0 on buying books. Here v(m) satisfies v(0) = 0 andv′(m) > 0 > v′′(m) for any m ≥ 0. We further assume that v′(m) > 1 for allm ≤ M; thus the library prefers buying books to keeping money.

When each journal is sold independently, we let xij ∈ {0, 1} represent thelibrary’s choice about journal ij : xij = 1 (resp. xij = 0) means that the librarybuys (resp. does not buy) this journal. When all publishers use bundling, welet Bj denote bundle j and Xj ∈ {0, 1} represent the library’s choice about

12. Considering only one library is without loss of generality in our framework because publisherscan price discriminate with respect to uij and M .

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Bj : Xj = 1 (Xj = 0) means that the library buys (does not buy) this bundle. Letx ≡ (x11, . . . , xn11, . . . , x1N, . . . , xnNN) ∈ {0, 1}n and X ≡ (X1, . . . , XN) ∈{0, 1}N . Under independent pricing and given (p, M), the library chooses x andm ≥ 0 to maximize its payoff,13

N∑j=1

nj∑i=1

uij xij + v(m) +⎡⎣M −

N∑j=1

nj∑i=1

pijxij − m

⎤⎦ , (1)

subject to the budget constraint∑N

j=1∑nj

i=1 pijxij + m ≤ M .14 The library’smaximization problem under bundling is similarly defined using (U1, . . . , UN),P, and X.

2.3. Social Welfare

Social welfare is defined as the sum of the payoff of the library, the profits ofthe journal publishers, and the profit of the book industry. The cost of producinga book is composed of a fixed cost and a marginal cost, about which we makea simplifying assumption: The fixed cost incurred by the book industry is notaffected by the library’s choice of m, and the marginal cost of producing a bookis zero.15 Then, social welfare is equal to the total utility the library obtains fromjournals and books up to a constant.

2.4. Timing and Equilibrium Selection

We consider the following game, denoted by �, in which each publisher simul-taneously decides (i) whether to be active or not and, if active, (ii) whether or notto bundle its journals and (iii) the price(s) of its bundle or journals. If a publisheris not active, then it does not offer any journal or bundle and hence the librarycannot buy that publisher’s journal(s) or bundle.

For equilibrium selection, we assume in Section 3 (where we consider homo-geneous journals) that a publisher decides not to be active if it expects to make

13. As a tie-breaking rule, we assume that (i) if the library is indifferent between buying a journal(or a bundle) and not buying it, then it buys the journal/bundle; and (ii) if the library is indifferentbetween two or more combinations of journals and/or bundles, then it chooses the combination withthe highest aggregate value of journals.14. In Section 2.4 we describe how each publisher should first decide whether or not to be activebefore choosing prices. Hence, (1) is correct if all publishers become active. If some are not active,then j runs over the set of active publishers.15. This is only a simplifying assumption. Our social welfare analysis is not qualitatively affectedwhen the cost incurred by the book industry, denoted by c(m), depends on m—as long as consumingmore books is desirable from the social point of view (i.e., v′(m) − c′(m) > 0 for m ≤ M).

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zero profit (i.e., if the publisher expects the library will not buy its bundle or anyof its journals). The assumption can be justified if a publisher would incur a verysmall but positive cost of contracting with the library. Without this assumption,the prices of some items (journals/bundles) the library buys may depend on theprices of the items the library does not buy, as shown by an example in Section 3.2.

We first study in Section 3.1 the game in which no publisher bundles itsjournals; we use �I to denote this game. Then, in Section 3.2, we analyze thegame �B in which all active publishers bundle their journals; in Section 3.3, weexamine each publisher’s incentive to choose between bundling and not bundlingin game �.

3. The Simple Case of Homogeneous Journals

In this section we derive all our main results in the simple case of homogeneousjournals, which means that uij = u > 0 for all ij .

3.1. Independent Pricing

We begin our analysis by examining �I , the game in which each active publisherprices its journals independently. It turns out that, in equilibrium, the competitivemargin (Lerner and Tirole 2004) binds in the sense that each publisher cannotraise the price of any of its journals without inducing the library to exclude thejournal from the portfolio of journals it buys. For expositional facility, we firstintroduce the concept of a marginal bundle of books as follows. Consider thedecision problem that the library faces with respect to the marginal journal (i.e.,the last journal it purchases) with price p. If the library does not buy this journal,then it can use p to buy extra books. Let π (≤M) denote the journal industryprofit when the marginal journal is bought. We define the marginal bundle ofbooks corresponding to price p as all the books that the library wishes to buywith p after spending M −π on books. Then, the utility from the marginal bundleof books is given by

UMB(p, π) ≡ v(M − π + p) − v(M − π) > 0.

Hence, whenu = UMB(p, π)holds, the publisher selling the marginal journalcannot raise its price without triggering an exclusion of the journal from the list ofthe journals bought by the library. We describe in the next lemma some propertiesof UMB that will be used frequently in the rest of the paper.

Lemma 1. (i) UMB strictly increases both with p and with π . (ii) UMB is strictlyconcave in p.

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The proof of this lemma is omitted because it follows directly from the fact thatv(·) is strictly increasing and strictly concave.

As a benchmark, we consider the case of minimum industry concentration inwhich each publisher owns only one journal (N = n). Suppose that all publisherscharge the same price p ≤ M/n. Then, the library prefers buying n′ number ofjournals (with 1 ≤ n′ ≤ n) to buying n′ − 1 journals if and only if the followinginequality holds:

n′u + v(M − n′p) ≥ (n′ − 1)u + v(M − n′p + p),

which is equivalent tou ≥ UMB(p, n′p).

Lemma 1(i) implies that UMB(p, n′p) strictly increases with n′; therefore, it isoptimal for the library to buy all the journals if and only if it prefers buyingn number of journals to buying n − 1, a condition that is equivalent to u ≥UMB(p, np).

We now prove that, if u < UMB(M/n, M) = v(M/n), then the unique p∗satisfying u = UMB(p∗, np∗) is an equilibrium. Suppose that all publishers exceptj charge p∗. If j charges p∗, then all journals are sold and j ’s profit is p∗; hencepublisher j has no incentive to choose a price lower than p∗. If instead j choosespj > p∗, then j ’s journal is the most expensive one and the library will not buyit, since u < UMB(pj , (n − 1)p∗ + pj ) holds by Lemma 1(i).

In case u ≥ UMB(M/n, M) holds, p∗ = M/n is an equilibrium. Still, pub-lisher j has no incentive to choose a price lower than p∗ because it can realizeprofit p∗ by charging p∗. If it chooses pj > p∗, then the library cannot afford tobuy all the journals and, once again, will drop j ’s journal because it is the mostexpensive.

The next proposition states that, regardless of the level of industry concentra-tion (with the exception of monopoly for some parameters), there exists a uniqueequilibrium and it is such that all publishers are active and all journals are sold atthe same price p∗ as just determined.

Proposition 1. (Irrelevance result). Suppose that journals are homogeneous(i.e., uij = u > 0 for all ij ) and are priced independently. Then the followingstatements hold.

(i) For any level of industry concentration, there exists an equilibrium in whichall publishers are active and all the journals are sold at the same price p∗, whichis determined as follows: if M ≤ nv−1(u) then p∗ = M/n, and if M > nv−1(u)

then p∗ is such that np∗ < M and

u = UMB(p∗, np∗). (2)

(ii) The equilibrium is unique unless the industry is a monopoly and M <

nv−1(u).

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Proof. Here we prove (i); the proof of (ii) is given in the Appendix. Suppose thateach publisher except j charges price p∗ (as determined by the statement of theproposition) for any of its journals. We show that choosing p∗ for each journal isa best response for publisher j having nj number of journals; the monopoly caseis a special case with nj = n. We first note that, for any pj ≡ (p1j , . . . , pnj j ),the library will purchase any journal with price lower than or equal to p∗ becauseit is willing to buy n number of journals at price p∗; this follows from (2) ifM > v−1(u) and from u ≥ UMB(p∗, M) = v(M/n) if M ≤ nv−1(u). Hence,for any pj , all the journals of any publisher j ′ (= j) will be purchased and j ’sprofit is equal to njp

∗ if it chooses pj ≡ (p∗, . . . , p∗). Therefore, our proofis done if we show that j cannot achieve a profit higher than njp

∗. This factis obvious if M ≤ nv−1(u), because for any pj the library buys all the journalsof the other publishers and therefore spends at most M − (n − nj )p

∗ = njp∗

for the journals of j . For the case of M > nv−1(u), suppose that j can realizea profit πj > njp

∗. Then it is necessary that the library buys a journal of j

with price p′ > p∗ and this requires u ≥ UMB(p′, (n − nj )p∗ + πj ), but (2)

and Lemma 1(i) imply u = UMB(p∗, np∗) < UMB(p′, (n − nj )p∗ + πj ), a

contradiction.

Proposition 1 establishes several results. First, each publisher is active indepen-dently of the number of journals it owns; because each journal has the samevalue, any publisher can make a positive profit by pricing its journals low enoughto undercut the prices of rivals. Furthermore, all the journals have the same pricep∗ such that the library buys all of them. In the case of a monopolist, chargingthe same price for all journals minimizes the competition from books.16 In thecase of oligopoly (i.e., when N ≥ 2), if p(1) is the price of the most expensivejournal that the library buys and if pij < p(1), then publisher j can increase itsprofit by suitably increasing the price of journal ij and reducing the prices of allits other journals in a way that induces the library to buy the same journals of j

it purchased before but at a higher total price. A similar argument can be usedto show that publisher j can increase its profit when some of its journals are notsold; therefore, all journals are sold in equilibrium.

Second, in equilibrium the competitive margin binds so that each publishercannot raise the price of any of its journals without inducing the library to excludethat journal from the list of journals that it buys. We can further distinguish twocases depending on the way the margin binds. When the journal industry profit issmaller than M , the equilibrium price p∗ is determined by what we call marginalopportunity cost pricing in the following sense: The price p∗ is such that, afterpurchasing n − 1 journals at price p∗, the library is indifferent between buyingan extra journal at price p∗ and spending p∗ on buying books instead. This is

16. More precisely: Given a profit π , for any price vector p with∑n

i=1 pi1 = π and p =(π/n, . . . , π/n), we have that maxi UMB(pi1, π) is strictly larger than UMB(π/n, π).

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Figure 2. Equilibrium under independent pricing when M > nv−1(u) holds.

shown in Figure 2, where the area delineated by ABCD is equal to u.17 When thejournal industry profit is equal to M , the equilibrium price p∗ (= M/n) satisfieswhat we call generalized marginal opportunity cost pricing in that each journalleaves to the library the same extra positive surplus with respect to its opportunitycost, which is equal to u − v(M/n) ≥ 0. Generalized marginal opportunity costpricing includes marginal opportunity cost pricing as a special case in which theextra surplus is equal to zero. In both cases, if a publisher increases the price ofa journal from p∗, the library does not purchase it any more.

Finally, the irrelevance result states that the equilibrium price p∗ is the samefor any level of industry concentration, except for the case of monopoly whenM < nv−1(u). This result comes mainly from the two following facts. First (aswas previously explained) regardless of the level of concentration, some publishercan increase its profit unless all journals have the same price. Second, the sym-metric equilibrium price is uniquely determined by the condition that makes thecompetitive margin bind, and this condition does not depend on industry concen-tration. Observe that the uniqueness result does not hold in the case of monopoly

17. The fact that each publisher regards its journal as the marginal one when choosing the price issimilar to what happens in the literature on multilateral bargaining (Stole and Zweibel 1996a, 1996b;Chemla 2003). For instance, Chemla studies competition among downstream firms buying from anupstream one and finds that each downstream firm pays the price that the marginal firm would payto the upstream one. However, none of these papers studies the issue of bundling.

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with M < nv−1(u). Then, the monopolist can achieve profit M not only bycharging the uniform price p∗ = M/n but also with any price vector such that (i)the sum of the prices is M and (ii) the highest price p′ satisfies u ≥ UMB(p′, M).Finally, we note that the equilibrium price p∗ depends on the number of journalsin the industry and their value.

Example 1. Suppose v(m) = 31m − m2, M = 10, u = 42, and n = 3; thenUMB(p, π) = p(31−p−2(M −π)). By Proposition 1, the equilibrium price p∗under independent pricing is such that UMB(p∗, 3p∗) = 42, since v(M/n) > u;thus, p∗ = 2.

Remark 1 (Robustness of the irrelevance result). Suppose that the library’sutility from buying k number of journals is u(k), with u(·) increasing and concave.Then we can show that the irrelevance result holds as long as u(·) is not veryconcave.

From the irrelevance result, we have the following corollary.

Corollary 1. When journals are homogeneous and under independentpricing:

(i) no merger has any impact on (merging or non merging) firms’ profits andthus firms have no strict incentive to merge; and

(ii) no merger affects social welfare unless it creates a monopoly and(n − 1)v−1(u) ≥ M .

Corollary 1(ii) deserves some explanation. If kv−1(u) ≥ M for some k ≤ n − 1,then the monopolist can achieve profit M by selling just k journals instead of n.Therefore, we cannot rule out the possibility that a merger which creates a monop-olist reduces social welfare, because fewer than n journals may be sold if M isrelatively small.

3.2. Bundling

In this Section we analyze �B , the game in which each active publisher bundlesits journals and chooses a price for the bundle. Without loss of generality, we sup-pose that U1 ≥ U2 ≥ . . . ≥ UN . Let A∗ represent the equilibrium set of activepublishers, P∗ ≡ {P ∗

j : j ∈ A∗} the equilibrium prices charged by the activepublishers, and πB∗ ≡ ∑

j∈A∗ P ∗j the equilibrium industry profit in �B .

Notice that our analysis does not depend on whether journals are homoge-neous or heterogeneous, because only the values U1, . . . , UN of the differentbundles matter; therefore, the results in this section apply to the case of

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heterogeneous journals as well. The next theorem characterizes the uniqueequilibrium of �B .

Theorem 1. Under bundling, there exists a unique equilibrium and it ischaracterized as follows.

(i) If M ≤ v−1(U1 − U2), then only the largest publisher is active andrealizes profit P ∗

1 = M .(ii) If M is such that there exists a k ∈ {2, . . . , N} satisfying

∑k−1j=1 v−1(Uj −

Uk) < M ≤ ∑kj=1 v−1(Uj − Uk+1) (with UN+1 ≡ 0), then only the k largest

publishers are active; they charge prices that satisfy πB∗ = ∑kj=1 P ∗

j = M and

Uj − UMB(P ∗

j , M) = Uj ′ − UMB

(P ∗

j ′, M)

≥ Uk+1 for any {j, j ′} ⊆ A∗. (3)

(iii) If M >∑N

j=1 v−1(Uj ) then all the publishers are active; they charge

prices that satisfy πB∗ = ∑Nj=1 P ∗

j < M and

Uj − UMB

(P ∗

j , πB∗) = 0, j = 1, . . . , N. (4)

We first note that the case of Proposition 1 in which nj = 1 for any j (i.e.,each publisher owns only one journal) is a special case of parts (ii)–(iii) of thistheorem with Uj = u for all j and N = n. Observe also that all the bundlesare sold if and only if the library’s budget is large enough (i.e., if and only ifM >

∑N−1j=1 v−1(Uj − UN)). Otherwise, bundling entails that small publishers

are unable to sell their journals whereas, under no bundling, all journals are soldfor any value of M . In what follows, we provide the main intuition about theequilibrium in �B by examining a special case with two publishers such thatU1 > U2.

Consider first the case in which the journal industry profit πB∗ is smallerthan M; this implies that both publishers are active, since otherwise an inactivepublisher could make a profit by choosing a small price. Then, the equilibriumprices P∗ = (P ∗

1 , P ∗2 ) are determined by marginal opportunity cost pricing as in

�I when M is large:

Uj = UMB(P ∗

j , P ∗1 + P ∗

2

)for j = 1, 2. (5)

Publisher j considers its bundle to be the marginal one and chooses P ∗j such that

the library is indifferent between buying Bj and spending P ∗j on buying extra

books. In the special case where P ∗1 +P ∗

2 is equal to M , we have UMB(P ∗j , P ∗

1 +P ∗

2 ) = v(P ∗j ); hence, by (5), P ∗

j = v−1(Uj ). This suggests that a solution to (5)

exists if and only if M > v−1(U1) + v−1(U2), as Theorem 1 (iii) states.

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Second, consider the case in which the journal industry profit πB∗ is equalto M and both bundles are sold. Then P∗ is determined by generalized marginalopportunity cost pricing as in �I :

U1 − UMB(P ∗

1 , M) = U2 − UMB

(P ∗

2 , M) ≥ 0. (6)

In other words, there is a kind of Bertrand competition such that the extra surplusis the same for all the bundles. The vector P∗ constitutes an equilibrium, becauselowering Pj is obviously suboptimal for publisher j and because, if Pj > P ∗

j ischosen, then the library cannot afford to buy both bundles; it prefers dropping Bj

because at P∗ it is indifferent between dropping B1 and dropping B2.Finally, for a budget small enough we see that publisher 1’s profit is equal to

M , implying that publisher 2 cannot sell its bundle. This happens if U1 −v(M) ≥U2 holds because then, for any P2 > 0, the library prefers buying B1 ratherthan B2 even if P1 = M , because the payoff U1 from buying B1 is greater thanU2 + v(M − P2), the payoff from buying B2 at P2 and spending M − P2 > 0 onbooks. The inequality U1 ≥ U2 + v(M) is equivalent to M ≤ v−1(U1 − U2), thecondition in Theorem 1(i).

Example 2. Consider the parameters of Example 1: v(m) = 31m − m2, M =10, u = 42, and n = 3. Then, under bundling, the following statements hold.

(i) If N = 2, and if n1 = 2 and n2 = 1, then P ∗1 and P ∗

2 satisfy

UMB(P ∗

1 , P ∗1 + P ∗

2

) = 84,

UMB(P ∗

2 , P ∗1 + P ∗

2

) = 42,

since M > v−1(2u) + v−1(u); hence P ∗1 = 4.3655 and P ∗

2 = 1.9382. Noticethat P ∗

1 > 2p∗, p∗ > P ∗2 , and P ∗

1 + P ∗2 > 3p∗.

(ii) If N = 1 and n1 = 3, then the monopolist chooses P m satisfy-ing UMB(P m, P m) = 126, since M > v−1(3u); hence P m = 7. Notice thatP m > P ∗

1 + P ∗2 > 3p∗.

Theorem 1 and the discussion following the theorem show that marginal or gener-alized marginal opportunity cost pricing determines the prices under bundling inthe same way as under independent pricing, and therefore the competitive marginbinds.

The competition between each bundle of journals and the marginal bun-dle of books implies that a large publisher (i.e., a publisher with high Uj ) hasa competitive advantage over a small publisher. Given πB∗, because v′(·) isstrictly decreasing, it follows that, as the number of books in the marginal bundleincreases, the average surplus of the books in this bundle decreases. Therefore,the marginal bundle of books competing with the bundle of a large publisherhas a lower average surplus than the marginal bundle of books competing with

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the bundle of a small publisher. This implies a sort of economies of scale underbundling. The next corollary formalizes this intuition in two different (thoughrelated) ways. The first result shows that publisher j ’s profit per value of bun-dle P ∗

j /Uj strictly increases with the value Uj of its bundle; the second resultestablishes that a large publisher obtains a relatively large share of the industryprofit.18

Corollary 2. Under bundling, we have:(i) {j, h} ⊆ A∗ and Uj > Uh imply P ∗

j /Uj > P ∗h /Uh; and

(ii) if A ⊂ A∗ is such that U1 ≥ ∑h∈A Uh, 1 /∈ A, and U1 > U2, then

P ∗1 >

∑h∈A P ∗

h .

We now use an example to discuss the role of our assumption (introduced inSection 2.4) about active and inactive publishers.

Example 3. Consider the setting with N = 3, U1 = 10, U2 = 2, U3 = 1.5,M = 5, and v(m) = m + √

m. From the proof of Theorem 1 we know that thereexists an equilibrium in �B in which publisher 3 makes a profit if and only ifM > v−1(U1 − U3) + v−1(U2 − U3).19 Because this inequality is violated withour parameters, there exists no equilibrium in which publisher 3 makes a positiveprofit. Without our assumption in Section 2.4 that makes publisher 3 inactive(i.e., publisher 3 stays out of the market), we find infinitely many equilibria in�B . For instance, P = (5 − x, x, x) is an equilibrium for any x ∈ (0, 0.1], andin all these equilibria the library buys B1 and B2.20 There also exists an equi-librium P = (5, 5, 5) in which the library buys only B1. Our assumption aboutactive publishers eliminates all the infinite equilibria in which B1 and B2 arepurchased, and it does not allow the price chosen by a publisher making zeroprofit to affect the equilibrium outcome. The consequence, for this particularexample, is that only publisher 1 is active—as predicted by Theorem 1, becauseM < v−1(U1 − U2).

18. The proof of the corollary is straightforward and so is omitted.19. See Lemma 3(i) in the Appendix.20. That P is an equilibrium can be verified as follows. (i) If publisher 1 increases P1 above 5 − x,then the library obtains a higher payoff by purchasing B2 and B3 (3.5 + 5 − 2x + √

5 − 2x) ratherthan B1 (10 + 5 − P1 + √

5 − P1); (ii) if publisher 2 increases P2 above x, then the library buysB1 and B3 (payoff 11.5) rather than B2 and B3 (3.5 + 5 − x − P2 + √

5 − x − P2) or B2 only(2 + 5 − P2 + √

5 − P2); (iii) if publisher 3 reduces P3 to almost 0, then the library buys B1 and B2(payoff 12) rather than B1 and B3 (11.5 + x − P3 + √

x − P3) or B2 and B3 (3.5 + 5 − x − P3 +√5 − x − P3).

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3.3. Incentive to Bundle

In the previous sections, we examined the two different regimes of no bundlingand bundling (games �I and �B , respectively). In this section, we inquire which ofthese regimes will emerge endogenously in game � by examining each publisher’sincentive to bundle. We obtain the following result.

Proposition 2. (i) If publisher j realizes profit πj > 0 by pricing journalsindependently, then j can earn the same profit by bundling its journals at pricePj = πj .

(ii) In any equilibrium of �, any active publisher bundles its journals.

This result states that any profit publisher j can make without bundling its journalscan also be obtained by bundling the journals; therefore, bundling is a weaklydominant strategy in � for each publisher; however, this fact might be consistentwith the existence of an equilibrium in � in which one (or more) publisher(s)does not bundle. The second part of the proposition establishes that this is not thecase.

Publishers have an incentive to bundle their journals because bundling has thedirect effect of softening competition from books. In order to provide intuition,we consider a simple case of a monopolist owning two journals. Suppose that hisprofit under independent pricing is smaller than M . In this case, by Proposition 1,in �I the monopolist chooses the same price p∗ for both journals, which isdetermined by marginal opportunity cost pricing:

u − UMB(p∗, 2p∗) = 0. (7)

Suppose now that the monopolist bundles his journals. Consider first the casein which he charges price 2p∗ for the bundle. Then, (7) and Lemma 1(ii) imply

2u − UMB(2p∗, 2p∗) = 2UMB(p∗, 2p∗) − UMB(2p∗, 2p∗) > 0. (8)

Under independent pricing, both journals compete with the same marginal bundleof books that give utility UMB(p∗, 2p∗) to the library. In contrast, under bundlingit is as if the first journal competes with the marginal bundle of books givingutility UMB(p∗, 2p∗) while the second journal competes with the marginal bundleof books giving utility UMB(2p∗, 2p∗) − UMB(p∗, 2p∗) = UMB(p∗, p∗) <

UMB(p∗, 2p∗). This explains why the inequality in (8) holds and hence whythere exists an ε > 0 that satisfies

2u − UMB(2p∗ + ε, 2p∗ + ε) > 0.

This inequality shows that the library will buy the bundle if the monopolist chargesP = 2p∗ + ε as the price for the bundle. Thus, bundling allows the monopolist to

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increase his profit with respect to independent pricing; the same intuition appliesto the case where there is an oligopoly.

Remark 2.21 We derived the direct effect of bundling in a setting in which the

utility from money is concave while the marginal utility from the consumptiongood (i.e., journals) is constant. The mechanism behind this direct effect is isomor-phic to the mechanism behind two-part tariffs in a standard setting in which themarginal utility from money is constant and the utility from the good is concave.For instance, consider a consumer with payoff u(q) − t , where u(·) is strictlyincreasing and strictly concave, q is the quantity of the good she consumes, andt is her monetary payment. Suppose that a monopolist produces the good at aconstant marginal cost c (>0). In this case, independent pricing is equivalent tolinear pricing and thus leaves some surplus to the consumer, but the monopolistcan extract the full surplus with a suitable two-part tariff in which the marginalprice is equal to c.

3.4. Comparative Statics

Industry profit and social welfare. Now we study the effect of bundling on indus-try profit and social welfare. Let πI∗ denote industry profits under independentpricing.

Proposition 3. (i) If M > nv−1(u) then bundling strictly increases industryprofits: πB∗ > πI∗. If M ≤ nv−1(u) then bundling does not affect industryprofits: πB∗ = πI∗ = M .

(ii) If M > nv−1(u), then bundling strictly reduces social welfare by strictlyreducing book consumption and weakly reducing journal consumption. If M ≤nv−1(u), then bundling weakly reduces social welfare by weakly reducing journalconsumption.

Proof. (i) From Proposition 1 we know that πI∗ = M if M ≤ nv−1(u). In con-trast, Theorem 1 shows that πB∗ = M when M ≤ ∑N

j=1 v−1(Uj ). Becausev−1(0) = 0 and v−1 is strictly convex, nju = Uj implies njv

−1(u) < v−1(Uj ) andin turn nv−1(u) <

∑Nj=1 v−1(Uj ); this proves the second part of Proposition 3(i).

Assume now M > nv−1(u), so that πI∗ < M . If M ≤ ∑Nj=1 v−1(Uj ) holds,

then πB∗ = M by theorem 1 and so the first part of Proposition 3(i) holds triv-ially. Suppose on the other hand that M >

∑Nj=1 v−1(Uj ), so that πB∗ < M by

Theorem 1. In order to prove that πB∗ > πI∗ we notice that, for each publisher j ,

njUMB(p∗, πI∗) = nju = Uj = UMB(P ∗

j , πB∗). (9)

21. We thank the referee for providing us with this idea.

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Define Pj (π) by Uj ≡ UMB(Pj (π), π) and observe that Pj (·) is strictly decreas-ing by Lemma 1(i). Furthermore, Lemma 1(ii) and the first two equalities in(9) imply that Pj (π

I∗) > njp∗ for any nj ≥ 2. We now prove πB∗ > πI∗

by contradiction. Suppose πB∗ ≤ πI∗. Because Pj (·) is strictly decreasing, wemust have P ∗

j = Pj (πB∗) ≥ Pj (π

I∗); this implies πB∗ ≡ ∑Nj=1 Pj (π

B∗) ≥∑Nj=1 Pj (π

I∗) >∑N

j=1 njp∗ = πI∗, which is a contradiction.

(ii) If M > nv−1(u) then bundling strictly increases industry profits, imply-ing that the library purchases fewer books in �B than in �I . Furthermore, underno bundling the library buys all the journals, whereas bundling can make smallpublishers unable to sell their journals (as stated in Theorem 1).

The result of Proposition 3(i) is due to the direct effect of bundling in termsof softening competition from books. To provide intuition, we consider thecase in which πI∗ < M and suppose that bundling does not increase industryprofits: πB∗ ≤ πI∗. Then, the marginal bundle of books corresponding to anygiven price p has a lower value under bundling than under independent pricingbecause UMB(p, πB∗) ≤ UMB(p, πI∗) holds by Lemma 1(i). Therefore, fromthe direct effect, each publisher can make a higher profit under bundling thanunder independent pricing and hence we obtain a contradiction.

If publishers are symmetric in the sense that U1 = · · · = UN , then bundlingincreases the profit of each publisher. If instead publishers are asymmetric, thenthe fact that bundling increases industry profits is bad news for small publishers,who cannot benefit much from the direct effect of bundling because a publisher’sbundling has the indirect effect of inflicting negative pecuniary externalities onrival publishers. To provide intuition, we consider competition between a bigpublisher with U1 = (n − N + 1)u (and n > N) and N − 1 small publish-ers with U2 = · · · = UN = u. We focus on the case in which p∗ < M/n

in �I and compare �I with �B . Obviously, no small publisher as defined herecan benefit from bundling since it has only one journal. However, by Proposi-tion 3, the big publisher’s bundling increases industry profits: πB∗ > np∗. Thisinflicts negative externalities on all the small publishers, because the marginalbundle of books corresponding to a given price of journal has a higher sur-plus after 1’s bundling than before. For instance, if πB∗ < M then each smallpublisher’s profit in �B is P ∗

2 with UMB(P ∗2 , πB∗) = u = UMB(p∗, np∗);

hence, by Lemma 1(i), P ∗2 is smaller than p∗ because πB∗ > np∗. Further-

more, as we have seen in Theorem 1, these pecuniary externalities make thesmall publishers unable to sell their journals if U1 is large enough to sat-isfy M ≤ v−1(U1 − u). Because the fact that bundling increases industryprofits implies that the library buys fewer books in �B than in �I , it fol-lows that bundling reduces social welfare by reducing purchases of books andjournals.

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Remark 3 (Independent budget for journals). When the budget M can be eitherused for the purchase of journals or kept in cash, we have a setting that is describedformally by v(m) = m. Because most of the effects of bundling are based onthe strict concavity of v, one might expect that bundling has no effect in thisenvironment. In fact, this is true as long as U ≤ M because in this case theequilibrium price of a journal or a bundle is simply equal to its value. However,if U > M then bundling does not affect industry profits, which are equal to M ,but it can reduce social welfare by making small publishers unable to sell theirjournals. As Theorem 1 shows, when industry profits are equal to M , there isa kind of Bertrand competition among bundles that makes the extra surplus thelibrary gets from a bundle with respect to its opportunity cost the same for all thebundles sold, creating an advantage for large publishers.

Mergers. We have seen that there is no incentive to merge under independentpricing. Here we study how bundling affects this incentive by considering a mergerof two publishers. Let πAM∗ (πBM∗) denote industry profits after the merger(before the merger) under bundling, and let πBM∗

j be publisher j ’s profit before themerger. Let ABM∗ denote the set of active publishers before the merger and let j ≡max{j : j ∈ ABM∗} denote the active publisher with the lowest-valued bundle.

Proposition 4. Under bundling, consider a merger of any two publishers j andk such that {j, k} ⊂ ABM∗ and πBM∗

j + πBM∗k < M .

(i) The merger strictly increases the joint profit of the merging publishersand strictly decreases the profit of any other publisher in ABM∗.

(ii) The merger strictly increases industry profits if πBM∗ < M; otherwise,πAM∗ = πBM∗ = M .

(iii) The merger weakly reduces both book consumption (strictly ifπBM∗ < M) and journal consumption. Hence, it always weakly reduces socialwelfare.

The proposition claims that a merger between any two active firms is strictlyprofitable unless the two firms already monopolize the market. As mentionedpreviously, in �B each bundle of journals competes with the marginal bundle ofbooks, and the average surplus of this bundle decreases as the number of booksincreases. Therefore, a large bundle of journals faces relatively soft competitionfrom books. In this way, a merger increases profits of the merged publishers as wellas industry profits. That the library spends more money on journals of the mergingpublishers, however, imposes negative pecuniary externalities on all the other pub-lishers, who thus suffer a reduction in profits because of the merger. With respectto social welfare, because industry profits weakly increase as a consequence of themerger it is obvious that book consumption decreases. Furthermore, the mergermay drive out of the market some publishers that were active before the merger

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(suggesting that the merger between RE and Harcourt is likely to be anticompeti-tive), yet it cannot induce previously inactive publishers to become active. Hence,any merger among active publishers weakly reduces social welfare.

Remark 4. The only role played by a merger in our setting is that it allows themerging publishers to create a larger bundle. Therefore, if two or more publishersagree to create a bundle of all their journals then this will have the same impactas a merger. As long as bundling is allowed, small publishers have an incentive toform a big bundle by making such an agreement. This can improve social welfareif big publishers’ bundling would make the small publishers unable to sell theirjournals without the agreement.22

3.5. Bundling and Incentive to Acquire a Journal

In this section, we study how bundling affects publishers’ incentives to acquirea journal sold by a third party. Our previous results have shown that bundlingcan make small publishers unable to sell their journals. This may induce themto exit the market and to sell their journals to other publishers. Alternatively, thejournal on sale can be interpreted as a new journal. Under this interpretation, westudy how bundling affects the incentive to introduce a new journal by examiningwhich publisher has the highest willingness to pay for it.

There are n number of journals before a third party sells a journal with valueu through a second-price auction; bj represents the bid of publisher j . Wheneach bidder knows before the auction the value that he attaches to the good onsale (the so-called setting of private values), it is well known that there existsa unique weakly dominant strategy for him: bidding his own valuation for thegood. In our setting, however, a bidder’s value for the auctioned journal is givenby the difference between his profit if he wins the auction and his profit if someother publisher acquires the journal. Because the latter profit may depend onthe identity of the winning publisher, a bidder may have no dominant strategyand this makes the analysis more complicated with respect to a standard second-price auction. However, we know that under independent pricing the equilibriumprices do not depend on the industry structure; therefore, a dominant strategyequilibrium exists. Likewise, under bundling, when there are two publishers itis common knowledge that, if publisher 1 (resp. 2) does not win the journal,then publisher 2 (resp. 1) wins23 and so a dominant strategy exists here as well.For simplicity, we assume that before the auction all publishers are active under

22. The agreement does not reduce book consumption, because the fact that some publishers wereinactive before the agreement implies that industry profits before the agreement be equal to M .23. We do not consider reserve prices or other instruments that may leave the journal in the handsof the auctioneer.

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bundling, πB < M and U1 > U2. Obviously, industry profits (weakly) increaseafter the auction. Under bundling, let πBj denote industry profits when publisherj wins the auction.

We obtain the following result.

Proposition 5. Suppose that a third party sells a journal of value u through asecond-price auction. Then the following statements hold.

(i) In the unique undominated equilibrium under independent pricing, allpublishers make the same bid.

(ii) If N = 2 then, in the unique undominated equilibrium under bundling,we have b1 > b2 when πB2 < M and b1 = b2 when πB2 = M . If N ≥ 3 then,under bundling, bidder 1 wins the auction in any undominated equilibrium.

Proof. We prove (i) here; see Appendix for the proof of (ii). Let p∗(n) denote theequilibrium price in �I described in Proposition 1 as a function of the numberof journals. If publisher j wins the auction, we know from Proposition 1 that itwill sell to the library all of its nj + 1 journals at the uniform price p∗(n + 1),thus realizing a profit of (nj + 1)p∗(n + 1). If instead publisher j loses theauction, another publisher will win the journal but the equilibrium price willstill be p∗(n + 1); in this case j ’s profit will be njp

∗(n + 1). Therefore, theincrease in publisher j ’s profit from winning the auction as compared to losingit is p∗(n + 1) for j = 1, . . . , N , regardless of the identity of the winner. Thisimplies that publishers j ’s valuation is p∗(n + 1) for j = 1, . . . , N . Hence eachpublisher has a (unique) weakly dominant strategy, which is bj = p∗(n + 1) forj = 1, . . . , N .

Proposition 5 implies that bundling could have a serious impact on the evolutionof industry concentration. In the absence of bundling, publishers have the samewillingness to pay for an auctioned journal. In contrast, under bundling the largestpublisher always has the highest willingness to pay for such a journal. Although amore careful analysis needs to be undertaken before predicting industry dynamics,our result suggests that bundling might create a vicious circle whereby big pub-lishers induce the exit of small publishers and become even bigger by purchasingtheir titles.

4. The General Case of Heterogeneous Journals

In this section we consider the general case in which journals can have differentvalues. Because Theorem 1 applies to �B , we need only study �I . Concerningthe analysis of �I , we find that the irrelevance result holds when we consider thetwo extreme cases of maximum industry concentration (when there is a monop-olist) and minimum industry concentration (when each publisher owns only one

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journal). For the intermediate setting of oligopoly, the problem mentioned at theend of Section 3.2 arises: The prices of journals that are bought by the librarymight be affected by the prices of journals that are not bought. In particular, apublisher may choose prices for its unsold journals in a way that maximizes itsprofit from the journals it is able to sell; this makes the analysis very compli-cated. For the sake of tractability, we eliminate this problem by assuming (i) thateach publisher chooses, from the set of its journals, a subset of active journals andmakes a journal active only if it expects that journal to be sold at a strictly positiveprice and (ii) that the library can purchase only active journals. This assumptionis stronger than the one introduced in Section 2.4, because in that case we allowa publisher to post prices for all its journals as long as the library buys at leastone of them.

Under this assumption on active journals, there exists a unique equilibriumcandidate (in pure strategies) regardless of the level of industry concentration;therefore, if the equilibrium exists, the irrelevance result holds. The equilibriumexists under the minimum and the maximum industry concentration but, for inter-mediate levels of concentration, it may not exist; we provide a sufficient conditionfor existence as well as an example of non existence.24

The equilibrium of �I when each publisher owns only one journal (i.e.,N = n) can be obtained from Theorem 1 by replacing Uj with u1j , where u1j

represents the value of the unique journal owned by publisher j .

Corollary 3. Under independent pricing, in the n-publisher, n-journal settingthere exists a unique equilibrium and it is characterized as follows:

(i) If M ≤ v−1(u11 − u12), then only the largest publisher is active andrealizes profit p∗

11 = M .(ii) If M is such that there exists a k ∈ {2, . . . , n} satisfying

∑k−1j=1 v−1

(u1j − u1k) < M ≤ ∑kj=1 v−1(u1j − u1k+1) (with u1n+1 ≡ 0), then only the k

largest publishers are active; they charge prices such that∑k

j=1 p∗1j = M and

u1j − UMB(p∗

1j , M) = u1j ′ − UMB

(p∗

1j ′, M)

≥ u1k+1 for any {j, j ′} ⊂ {1, 2, . . . , k}.

(iii) If M >∑n

j=1 v−1(u1j ), then all publishers are active; they chargeprices that satisfy π∗ = ∑n

j=1 p∗1j < M and

u1j − UMB(p∗

1j , π∗) = 0, j = 1, . . . , n.

24. Observe that nonexistence in the example does not depend on the assumption about activejournals.

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Let u(k) be the value of the journal with the k-highest value; hence, u(1) ≥ . . . ≥u(n). The next theorem covers the general case in which at least one publisherowns two or more journals.

Theorem 2. Under independent pricing, the following statements hold.(i) When N = 1, the monopolist’s profit is smaller than M if M >∑n

i=1 v−1(ui1) and is equal to M otherwise. The equilibrium prices in the n-publisher, n-journal setting (Corollary 3[ii]–[iii]) maximize the monopolist’sprofit for any value of M; they are the unique profit-maximizing prices if andonly if M ≥ ∑n

i=1 v−1(ui1).(ii) In the case of oligopoly and under the assumption about active journals:(a) there exists a unique equilibrium candidate regardless of the level of

industry concentration, which is the equilibrium in the n-publisher, n-journalsetting; and

(b) the equilibrium exists if journals are nearly homogeneous.

Theorem 2(i) states that the level of industry concentration does not affect the out-come in the two extreme cases of minimum and maximum concentration as long asthe industry profit is lower than M .25 Furthermore, Theorem 2(ii)(a) establishesthat the outcome does not depend on the level of industry concentration when1 < N < n as long as �I has an equilibrium. In the proof of Theorem 2(ii)(b),we show that the equilibrium exists if journals are nearly homogeneous (henceproposition 1, which deals with the case of homogeneous journals, is a specialcase of Theorem 2). The equilibrium may not exist because a multi journal pub-lisher may change several prices at the same time and this deviation from theunique candidate price vector is sometimes profitable, as in our next example.

Example 4. Suppose that v(m) = m+4√

m, M = 12, N = 2, n1 = 2, n2 = 1,u11 = u21 = 1, and u12 = 24. Because v(11) > 24 and v(1/2) > 1, we maydeduce that (respectively) 11 > v−1(24) and 1/2 > v−1(1). Therefore, M =12 >

∑3h=1 v−1(u(h)) and, in the candidate equilibrium, all journals are active

with prices such that p11 + p21 + p12 < 12, 1 = UMB(p11, π) = UMB(p21, π),and 24 = UMB(p12, π); this yields p∗

11 = p∗21 ≈ 0.17 and p∗

12 ≈ 11.566;however, a profitable deviation for publisher 1 is to set p11 = p∗

11 + 0.05 andp21 = p∗

21 + 0.05 because then the library’s payoff is maximized by purchasingonly publisher 1’s journals.26

25. As in the case of homogeneous journals, if πI∗ = M then for some parameters it is possiblefor the monopolist to realize a profit equal to M by selling a strict subset of the journals that arepurchased by the library in the n-publisher, n-journal setting.26. After the deviation of publisher 1, it is infeasible to buy all the journals becausethe sum of prices is greater than M . The payoff from buying journals 11 and 21 is2 + 12 − 2 × 0.22 + 4

√12 − 2 × 0.22 = 27.16; the payoff from buying 11 and 12 (or 21 and 12)

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This example is somewhat counterintuitive, because the library buys the threejournals under prices p∗ yet it buys journals 11 and 21 and not journal 12 afterp11 and p21 increase while p12 does not change. Note first that, conditional onthe fact that journals 11 and 21 are purchased at prices p11 = p∗

11 + ε1 > p∗11

and p21 = p∗21 + ε2 > p∗

21, pecuniary externalities imply that the library doesnot buy journal 12 because u12 = UMB(p∗

12, p∗11 + p∗

21 + p∗12) while u12 <

UMB(p∗12, p

∗11 +ε1 +p∗

21 +ε2 +p∗12). However, given that it is suboptimal to buy

all three journals, it is puzzling that the dropped journal is the one whose price isunchanged. Comparing the payoffs from the different alternatives sheds light onthis issue; consider ε1 = ε2 = ε. If the library buys journals 11 and 21, its payoffis reduced by v(M−π+p∗

12)−v(M−π+p∗12−2ε) > 0 with respect to the payoff

before the changes in prices; if journals 11 and 12 (or 21 and 12) are purchased,the library’s payoff decreases by v(M − π + p∗

11) − v(M − π + p∗11 − ε) > 0.

Therefore, journal 12 is eliminated if v(M −π +p∗12)−v(M −π +p∗

12 −2ε) <

v(M −π +p∗11)−v(M −π +p∗

11 −ε). Even though 2ε > ε > 0, if p∗12 > p∗

11 itis possible that the inequality holds for some ε owing to the strict concavity of v;in particular, it holds for ε = 0.05. In words, there is much more money left forbooks when an expensive journal like 12 is dropped than when a cheap one like11 or 21 is dropped. Therefore, the utility loss from spending 2ε less money onbooks in the former case can be smaller than the utility loss from spending ε lessmoney in the latter case.

As we mentioned before, Theorem 1 on �B is valid regardless of whetherjournals are homogeneous or heterogeneous. Furthermore, in � (the game inwhich each publisher chooses between bundling and no bundling) there alwaysexists an equilibrium in which every active publisher bundles its journals and A∗and P∗ are determined by Theorem 1.27 Therefore, conditional on the existence ofan equilibrium in �I , most of the results we obtained in the case of homogeneousjournals hold also in the general case of heterogeneous journals: The resultsconcerning the incentive to bundle, the effect of bundling on profits and mergers,and the incentive to acquire a journal. With regard to how bundling affects socialwelfare, we must distinguish its impact on book consumption from its impact onjournal consumption. Bundling decreases book consumption because the directeffect of softening competition from books increases industry profits. Whetherbundling decreases or increases journal consumption depends on the degree ofsymmetry among publishers. For instance, in the extreme case of symmetricpublishers with U1 = · · · = UN , bundling may increase journal consumptionbecause all bundles will be sold, whereas journals of small value might not be sold

is 25 + 12 − 0.22 − 11.566 + 4√

12 − 0.22 − 11.566 = 27.064; and the payoff from buying only12 is 24 + 12 − 11.566 + 4

√12 − 11.566 = 27.069.

27. Observe also that any equilibrium of � in which a publisher does not bundle its journals requiresthe use of a weakly dominated strategy, since a result similar to Proposition 2(i) holds.

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under independent pricing when journals are very heterogeneous. This scenariois not realistic however, because in fact a small number of commercial publishersown a large number of journals;28 for instance, RE alone has about 1,800 journals.Because asymmetry among bundles is likely to be larger than asymmetry amongindividual journals it follows that bundling should reduce journal consumption,making it hard for small publishers to sell their journals even if their quality ishigh.29

5. Concluding Remarks

Our analysis reveals that there is a strong conflict between private and social incen-tives in the bundling of e-journals: Each publisher wants to bundle its journals,and bundling increases industry profits but reduces social welfare. In particular,big publishers’ bundling not only reduces consumption of monographs but alsocan make small publishers unable to sell even the high-quality journals they own.In this respect, it is noteworthy that Wolters Kluwer, which is the sixth-largestplayer in the industry by revenues, recently opted to exit scientific publishing andto focus solely on medical publishing, citing lack of scale as the reason for theexit (Gooden, Owen, and Simon 2002).

We found that bundling has two other important effects. First, bundling createsincentives for mergers. However, mergers among active publishers reduce socialwelfare by reducing book and journal consumption. In contrast, mergers among(small) publishers who would not otherwise be able to sell their journals mightincrease social welfare. Alternatively, it would be desirable for small publisherswho have high-quality journals to sell their journals through a common agency,as in the case of JSTOR. Second, bundling can have a serious impact on theevolution of industry concentration by affecting the incentives to acquire otherjournals. We have shown that in the absence of bundling each publisher has thesame willingness to pay for a journal, whereas under bundling the largest publisheralways has the highest willingness to pay. Hence, bundling might create a viciouscycle that enables big publishers to force the exit of small ones and hence becomeeven bigger by purchasing their titles.

We studied how bundling affects a library’s purchase of journals and bookswhen its budget is given. It would be interesting to study how bundling affectsthe choice of the budget. For instance, one can consider the case in which theuniversity of a library is a first mover and can set the budget before publishers

28. Measured by revenue, in 2001 Elsevier Science had a 16.0% industry share, with Kluwer at8.2% and Thomson-Scientific & Healthcare at 7.5% (Edlin and Rubinfeld 2004).29. Some publishers believe that, if they are below no. 5 on the shopping list of libraries, then thereis no guarantee that there will be any money left in the library budgets for purchasing their journals(Key Perspectives 2002).

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make any decision. Although a complete analysis of this question is beyond thescope of this paper, we have found that in the case of heterogeneous journals(conditional on equilibrium existence under independent pricing) bundling hasan ambiguous effect on the university’s incentive to increase its library’s budgetwith respect to independent pricing (and so it is possible that bundling inducesthe university to increase the budget).30

Finally, it would be interesting to extend our framework to other economicsituations such as bundling (or block booking) in the distribution of movies and ofTV or radio programs.31 A rationale for the per se illegal status of block booking32

comes from the concern that block booking of high-quality movies with low-quality ones would make it difficult for small producers to get their high-qualitymovies into theaters. Our analysis shows that such concern is justified, at least inthe market for e-journals.

Appendix

A.1. Proof of Proposition 1(ii)

We note first that, in the case of the monopolist with M ≥ nv−1(u), charging theuniform price p∗ is the only way to achieve the maximum profit np∗, otherwisewe have a contradiction: The most expensive journal among the ones the librarybuys has price p(1) strictly higher than p∗, but it would not be purchased becauseu = UMB(p∗, np∗) < UMB(p(1), np∗) holds. Therefore, in what follows we focuson the case of N ≥ 2. The proof is composed of three claims. It is useful to recallthat, given the prices chosen by active publishers, it is optimal for the library tobuy the n′ (≤n) cheapest journals, where n′ is endogenous.

Claim 1. All publishers are active in any equilibrium of �I .

Proof. Suppose that publisher h is not active. Then we can easily show that hecan make a positive profit if he becomes active. Given the prices of the journalsof active publishers, let publisher h choose the same price ε > 0 for each ofhis journals, small enough to make them cheaper than any journal of other active

30. Under independent pricing and under bundling, as the library’s budget increases its payoff ispiecewise constant; it jumps upward at a number of points and then increases in a continuous way.In the case of bundling, the jumps occur less frequently but are larger.31. Because we assume price discrimination that is based on usage, our explanation of bundlingis quite different from the one given by Stigler (1968), which is based on second-degree pricediscrimination.32. For the antitrust cases, see United States v. Paramount Pictures, Inc. et al. (1948) and UnitedStates v. Loew’s, Inc. et al. (1962). In MCA Television Ltd. v. Public Interest Corp. (11th Circuit,April 1999), the court of appeals reaffirmed the per se illegal status of block booking.

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publishers. Because u > UMB(ε, nhε) for a small ε, the library buys all the journalsof publisher h and he makes a profit of nhε > 0.

Claim 2. If N ≥ 2 then, in any equilibrium of �I , every journal has the sameprice (denoted by p) and all the journals are purchased by the library.

Proof. For any j , let Rj (Zj ) be the set of j ’s journals that are sold (notsold); R ≡ ⋃

Nj=1Rj and Z ≡ ⋃N

j=1 Zj . We first prove that all journals in R havethe same price. Let p(1) ≡ maxij∈R{pij } denote the price of the most expensivejournal the library buys; then u ≥ UMB(p(1), π). We prove that if pih < p(1) forsome ih ∈ R, then publisher h can increase his profit by increasing the price ofjournal ih by ε > 0 small enough and reducing by ε/|Rh| the price of each otherjournal in Rh, where |Rh| is the number of journals in Rh. Let pih = pih + ε

and pi′h = pi′h − ε/|Rh| for any i′h ∈ Rh\{ih}.33 At the new prices, it is obvi-ous that the library buys all the journals in Rh if at least one journal with pricep(1) is still purchased, because each journal in Rh has a price smaller than p(1).If instead no journal with price p(1) is purchased, then also no journal in Z ispurchased because pij ≥ p(1) for any ij ∈ Z. Therefore, an upper bound for thenew industry profit is π −p(1) +ε, where π is the industry profit before publisherh’s prices change. This implies that all the journals in Rh are purchased becausep

(1)h ≡ maxi′h∈Rh

{pi′h} < p(1), π − p(1) + ε < π , and u ≥ UMB(p(1), π) imply

that u > UMB(p(1)h , π − p(1) + ε).

Now we prove that all the journals are sold. Suppose that ih ∈ Z for some i

and h. Then publisher h can increase his profit by setting pih = ε > 0 and reducingthe price of each journal in Rh by ε/(1 + |Rh|). In this way, all the journals inRh ∪ {ih} are purchased because they are cheaper than any journal in

⋃j =h Rj ,

and the logic of the proof in the previous paragraph applies.

Claim 3. If N ≥ 2 then, in any equilibrium of �I , p is equal to p∗ describedby the proposition.

Proof. The library buys all the journals at price p if and only ifu ≥ UMB(p, np) and np ≤ M . Consider first the case of M > nv−1(u). Thenu < UMB(M/n, M) [=v(M/n)] holds and so we must have p < M/n. Wenow show that p < M/n implies u = UMB(p, np). To prove this, suppose thatp < M/n and u > UMB(p, np) hold. Then, let a publisher h increase the price ofone of his journals to p + ε with ε > 0 small enough. In this case, all journals of allpublishers are still sold becauseu > UMB(p, np) impliesu > UMB(p+ε, np+ε).

33. Note that we can assume without loss of generality that some publisher j = h owns a journalwith price p(1) that is purchased by the library.

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Finally, u = UMB(p, np) has a unique solution smaller than M/n, given thatM > nv−1(u).

In the case ofM ≤ nv−1(u), p must be equal toM/nbecauseu > UMB(p, np)

for any p < M/n.

We have therefore proved Claims 1–3 and thus Proposition 1(ii).

A.2. A Useful Lemma

Lemma 2. Let S denote a set of items34 with PS = ∑s∈S ps ≤ M . If only the

items in S are considered for purchase, then all of them are purchased if andonly if

us ≥ UMB(ps, PS) for all s ∈ S. (A.1)

Proof. Let US ≡ ∑s∈S us .

(⇒) If buying all the items in S is optimal, then buying all gives a higherutility than buying all except a particular item s for any s ∈ S. Therefore, thefollowing condition must be satisfied:

US + v(M − PS) ≥ US − us + v(M − PS + ps) for all s ∈ S,

which is equivalent to (A.1).(⇐) Let S′ ⊆ S denote a subset of S with PS′ = ∑

s∈S′ ps and US′ ≡∑s∈S′ us . Suppose that (A.1) holds, which implies that

US′ =∑s∈S′

us ≥∑s∈S′

UMB(ps, PS) ≥ UMB(PS′, PS).

Here the second inequality holds because UMB(ps, PS) is concave in the firstargument and UMB(0, PS) = 0. The inequality US′ ≥ UMB(PS′, PS) implies thatbuying all items in S gives a higher utility than buying all except the subset S′,because

US + v(M − PS) ≥ US − US′ + v(M − PS + PS′) ⇐⇒ US′ ≥ UMB(PS′, PS).

Because S′ can be any subset of S, we conclude that buying all the items in S isoptimal when (A.1) is satisfied.

34. An item can be either a journal or a bundle, because Lemma 2 applies to the general case inwhich some publishers bundle their journals whereas others do not.

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A.3. Proof of Theorem 1

We first prove two lemmas that together allow us to prove Theorem 1. In the firstlemma, we take the set A of active publishers as given and show that there existsa unique candidate equilibrium price vector. The second lemma describes how A

is uniquely determined depending on the level of M . Therefore, the two lemmasidentify a unique equilibrium candidate as a function of M . Finally, we prove thatthe candidate is indeed an equilibrium. Let j = max{j : j ∈ A} denote the activepublisher with the lowest-valued bundle; obviously, j depends on A even thoughwe do not indicate this in the notation. Furthermore, let A = A\{j} denote theset of active publishers excluding j .

Lemma 3. For a given A∗, the following statements hold.(i) A candidate equilibrium price vector P∗

A∗35 satisfying πB∗ = M exists ifand only if

∑j∈A∗ v−1(Uj − Uj) < M ≤ ∑

j∈A∗ v−1(Uj ); furthermore, P∗A∗ is

unique and satisfies

Uj − v(P ∗

j

) = Uj ′ − v(P ∗

j ′) ≥ 0 for any {j, j ′} ⊆ A∗. (A.2)

(ii) A candidate equilibrium price vector P∗A∗ satisfying πB∗ < M exists if

and only if∑

j∈A∗ v−1(Uj ) < M; furthermore, P∗A∗ is unique and satisfies

Uj − UMB(P ∗

j , πB∗) = 0 for any j ∈ A∗. (A.3)

(iii) If (A∗, P∗A∗) is an equilibrium of �B , then it is necessary that∑

j∈A∗ v−1(Uj − Uj) < M holds.

Proof of (i). In any equilibrium of �B, P∗A∗ is such that Bj is sold for any j ∈ A∗;

otherwise publisher j will not be active. Because πB∗ = M , Lemma 2 implies

Uj ≥ UMB(P ∗

j , M) = v

(P ∗

j

)for any j ∈ A∗. (A.4)

In order for no publisher to have an incentive to increase the price of his bundleabove the price in P∗

A∗ , the following condition must be satisfied:

Uj − v(P ∗

j

) = Uj ′ − v(P ∗

j ′)

for any j, j ′ ∈ A∗. (A.5)

If instead Uj − v(P ∗j ) > Uj ′ − v(P ∗

j ′) for some j and j ′ in A∗, then we can showthat publisher j can increase Pj slightly from P ∗

j to P ∗j + ε and sell his bundle.

After j ’s price change, the library cannot afford to buy all the bundles in A∗

35. For clarity we use the notation P∗A∗ in this Appendix instead of the P∗ used in the main text.

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because the sum of the prices is greater than M . However, Lemma 2 and (A.4)imply that the library will drop exactly one bundle.36 Given UA∗ = ∑

h∈A∗ Uh,the alternative of dropping Bj is suboptimal because it gives the library a smallerpayoff as compared to dropping Bj ′: UA∗ −Uj +v(P ∗

j ) < UA∗ −Uj ′ +v(P ∗j ′ −ε)

for a small ε. This establishes that Bj will be purchased and that publisher j ’s profitincreases. Hence, (A.5) must hold. Finally, (A.4) and (A.5) imply that (A.2) holds.

Now we prove that a (unique) P∗A∗ satisfyingπB∗ = M, (A.2), andP ∗

j > 0 forall j ∈ A∗ exists if and only if

∑j∈A∗ v−1(Uj − Uj) < M ≤ ∑

j∈A∗ v−1(Uj ).Use (A.2) to write Pj as a function of Pj as follows: Pj = v−1[Uj −Uj +v(Pj )]for any j ∈ A∗. After combining this with πB∗ = M , we obtain

F(Pj ) ≡∑j∈A∗

v−1[Uj − Uj + v(Pj )] + Pj − M = 0. (A.6)

Note that F is strictly increasing in Pj

and that v−1(Uj) is the highest

value of Pj

consistent with (A.4). Because F(0) = ∑j∈A∗ v−1(Uj − Uj) − M

and F [v−1(Uj )] = ∑j∈A∗ v−1(Uj ) − M , it follows that a (unique) solution

P ∗j

∈ (0, v−1(Uj )] to (A.6) exists if and only if∑

j∈A∗ v−1(Uj − Uj) < M ≤∑j∈A∗v−1(Uj ) is satisfied. Notice that P ∗

j = v−1[Uj −Uj + v(P ∗j )] ≥ P ∗

j> 0

for any j ∈ A∗.

Proof of (ii). Because πB∗ < M , Lemma 2 implies Uj + v(M − πB∗) ≥ v(M −πB∗ + P ∗

j ) for any j ∈ A∗. We prove that if this inequality holds strictly forone j ∈ A∗ then publisher j can increase his profit by choosing Pj = P ∗

j + ε

with ε > 0 small. By Lemma 2, if the library does not buy Bj then it buys allother bundles because Uh > UMB(P ∗

h , πB∗ − P ∗j ) for any h ∈ A∗\{j}. Its pay-

off is then UA∗ − Uj + v(M − πB∗ + P ∗j ). If the library also buys Bj then its

payoff is UA∗ + v(M − πB∗ − ε), and this is larger than UA∗ − Uj + v(M −πB∗ + P ∗

j ) because Uj + v(M − πB∗) > v(M − πB∗ + P ∗j ) and ε is close to

0.37 Therefore, (A.3) must be satisfied by P∗A∗ .

Now we prove that a (unique) P∗A∗ satisfying πB∗ < M, (A.3), and P ∗

j > 0for all j ∈ A∗ exists if and only if

∑j∈A∗ v−1(Uj ) < M . From (A.3) we obtain

Pj = π − M + v−1[Uj + v(M − π)] for all j ∈ A∗, and adding this equalityover j yields π = ∑

j∈A∗ v−1[Uj + v(M − π)] + |A∗|(π − M), where |A∗|

36. Suppose that Bk is not purchased for some k ∈ A∗\{j}. Then (A.1) is satisfied for S = A∗\{k}because (A.4) implies that Uh > UMB(P ∗

h , M − P ∗k + ε) for any h ∈ A∗\{k, j} and that Uj >

UMB(P ∗j + ε, M − P ∗

k + ε) for ε > 0 small enough. If Bj is not purchased, then (A.1) is satisfiedfor S = A∗\{j} because (A.4) implies Uh > UMB(P ∗

h , M − P ∗j ) for any h ∈ A∗\{j}.

37. We are not proving that the library will buy all the bundles after the increase in the price of Bj .Rather, we prove that the (only) alternative in which Bj is not purchased is suboptimal. Hence thelibrary will buy Bj , but it may not buy Bj ′ if Uj ′ + v(M − πB∗) = v(M − πB∗ + P ∗

j ′ ).

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is the number of publishers in A∗. Hence, we need to find π ∈ (0, M) such thatG(π) = 0, where

G(π) =∑j∈A∗

v−1[Uj + v(M − π)] + (|A∗| − 1)π − |A∗|M. (A.7)

Observe that (i) G strictly decreases in π ; (ii) G(0) > 0 because Uj > 0 for any j ;and (iii) G(M) = ∑

j∈A∗ v−1(Uj )−M . Thus, a (unique) solution πB∗ ∈ (0, M)

to (A.7) exists if and only if M >∑

j∈A∗ v−1(Uj ). Notice that P ∗j = πB∗ −M +

v−1[Uj + v(M − πB∗)] > M − M + v−1[Uj + v(M − M)] = v−1(Uj ) > 0because ∂P ∗

j /∂πB∗ < 0.

Proof of (iii). Parts (i) and (ii) of the lemma show that M >∑

j∈A∗ v−1(Uj −Uj)

is a necessary condition for (A∗, P∗A∗) to be an equilibrium of �B .

Our next lemma is about determining the set A∗.

Lemma 4. Suppose that (A∗, P∗A∗) is an equilibrium of �B . Then:

(i) publisher 1 is active for any M > 0, and if∑k−1

j=1 v−1(Uj −Uk) < M forsome k ∈ {2, . . . , N} then {1, . . . , k} ⊆ A∗;

(ii) if M ≤ v−1(U1 −U2) then A∗ = {1} and if∑k−1

j=1 v−1(Uj −Uk) < M ≤∑kj=1 v−1(Uj − Uk+1) for some k ∈ {2, . . . , N − 1} then j /∈ A∗ for any j > k.

Proof of (i). Suppose that∑k−1

j=1 v−1(Uj − Uk) < M for some k ∈ {2, . . . , N}and that h /∈ A∗ for some h ≤ k. We now show that publisher h can make a profitby choosing a suitable Ph. First, if πB∗ < M then it is trivial to see that Bh ispurchased at Ph > 0 close to 0, because Uh + v(M −πB∗ −Ph) > v(M −πB∗)holds.

Consider now πB∗ = M . Then it is enough to prove that the inequalityUh > Uj − v(P ∗

j ) holds, because we can apply the argument in the proof of

Lemma 3(i) to show that publisher h can sell his bundle by setting Ph > 0 closeto 0. To show that Uh > Uj − v(P ∗

j ), we distinguish the case of j > h from

the case of j < h. For j > h, the inequality Uh > Uj − v(P ∗j ) follows simply

from Uh ≥ Uj . When j < h, we know from Lemma 3 that P∗A∗ satisfies (A.2) and

therefore P ∗j solves (A.6). The inequality P ∗

j > v−1(Uj − Uh), which is equiv-

alent to Uh > Uj − v(P ∗j ), holds because F is strictly increasing and because

F(v−1(Uj − Uh)) = ∑j∈A∗ v−1(Uj − Uh) − M is strictly negative, because

A∗ ⊂ {1, . . . , k}, ∑k−1j=1 v−1(Uj − Uk) < M (by assumption), and Uh ≥ Uk .

Notice that if h = 1 then only the case j > h may arise, and therefore 1 ∈ A∗ for anyM > 0.

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Proof of (ii). In this proof, let k = 1 if M ≤ v−1(U1−U2); otherwise, k is definedas in the statement of Lemma 4(ii). From Lemma 4(i) we know that {1, . . . , k} ⊆A∗. We will show that a contradiction arises if j > k. Because, from Lemma 3(iii),

the inequality∑

j∈A∗ v−1(Uj − Uj) < M should be satisfied and because we

assume M ≤ ∑kj=1 v−1(Uj − Uk+1), it follows that

∑j∈A∗ v−1(Uj − Uj) <∑k

j=1 v−1(Uj − Uk+1) must hold. However, the last inequality fails to hold ifj > k, because this implies {1, . . . , k} ⊆ A∗ and Uk+1 ≥ Uj .

Proof of Theorem 1(i). Lemma 4(ii) and Lemma 3 imply (respectively) that A∗ ={1} and that P ∗

1 = M .

Proof of Theorem 1(ii). Lemma 4 and Lemma 3 imply (respectively) that A∗ ={1, . . . , k} in any equilibrium and that P∗

A∗ is the unique solution to (A.2). Wenow demonstrate that (A∗, P∗

A∗) is indeed an equilibrium of �B by proving thetwo following claims.

Claim 1. Publisher h ∈ A∗ cannot make a profit larger than P ∗h given A = A∗

and Pj = P ∗j for any j ∈ A∗\{h}.

Proof. Let h ∈ A∗. We know from Lemma 2 that all the bundles in A∗ are soldif Ph ≤ P ∗

h ; we shall prove here that Bh is not purchased if Ph > P ∗h . Clearly,

when Ph > P ∗h the library cannot afford to buy all the available bundles. If it buys

Bh, let Z = ∅ denote the set of the bundles in A∗ that it no longer buys, withUZ ≡ ∑

z∈Z Uz and P ∗Z ≡ ∑

z∈Z P ∗z . Then the library’s payoff is UA∗ − UZ +

v(P ∗Z − (Ph − P ∗

h )), where UA∗ = ∑z∈A∗ Uz. If instead the library buys all the

bundles except Bh, then its payoff is UA∗ − Uh + v(P ∗h ). We prove that the latter

payoff is strictly larger than the former for any Ph > P ∗h by showing that the weak

inequality holds at Ph = P ∗h : v(P ∗

Z) − v(P ∗h ) ≤ UZ − Uh. Given any z ∈ Z,

(A.2) implies that Uh − v(P ∗h ) = Uz − v(P ∗

z ) and hence the latter inequality isequivalent to

v(P ∗

Z

) ≤ v(P ∗

z

) + UZ − Uz. (A.8)

If Z\{z} = ∅, then (A.8) holds trivially. If Z\{z} = ∅, then (A.8) holds becausev(P ∗

z ) + UZ − Uz = v(P ∗z ) + ∑

j∈Z\{z} Uj ≥ ∑j∈Z v(P ∗

j ) > v(P ∗Z), where

the first inequality comes from (A.2) and the second from the strict concavityof v(·).

Claim 2. Publisher h /∈ A∗ cannot make a positive profit given A = A∗ andPj = P ∗

j for any j ∈ A∗.

Proof. Let h /∈ A∗. We now prove that at no price Ph > 0 will the bundle Bh besold. For this purpose, we first show that Uj −v(P ∗

j ) ≥ Uh for any j ∈ A∗. Given

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A∗ = {1, . . . , k} we have j = k, and the solution P ∗k of (A.6) is weakly smaller

than v−1(Uk−Uk+1) because F(v−1(Uk−Uk+1)) = ∑kj=1 v−1(Uj −Uk+1)−M

and∑k

j=1 v−1(Uj −Uk+1) ≥ M by assumption. The inequality P ∗k ≤ v−1(Uk −

Uk+1) is equivalent to Uk − v(P ∗k ) ≥ Uk+1 and this implies, from (A.2) and

Uk+1 ≥ Uh, that Uj − v(P ∗j ) ≥ Uh for any j ∈ A∗. Using this inequality, we can

argue as in the proof of Claim 1 to show that at no price Ph > 0 will the bundleBh be sold; hence, it is a best reply for publisher h to be non active.

Proof of Theorem 1(iii). Lemma 4(i) and Lemma 3(ii) imply (respectively), thatA∗ = {1, . . . , N} and that P∗

A∗ is the unique solution to (A.3). The proof that nopublisher h has an incentive to choose Ph = P ∗

h is quite similar to the proof ofClaim 1 and so is omitted.

A.4. Proof of Proposition 2

We use I ⊆ A∗ to represent the set of active publishers who sell their journalsindependently; B ≡ A∗\I is the set of active publishers who bundle their journals.Let πj denote the profit of publisher j (j ∈ A∗); π ≡ ∑

j∈A∗ πj is the industryprofit. In any equilibrium, it is obvious that Bj is sold for any j ∈ B and that atleast one journal of publisher j is sold for any j ∈ I . Let Rj and Zj (j ∈ I ) havethe same meanings as in the proof of Claim 2 of Proposition 1; R ≡ ⋃

j∈I Rj

and Z ≡ ⋃j∈I Zj .

Proof of Proposition 2(i). Suppose that j ∈ I and that the library optimallyspends πj > 0 on buying some journals of publisher j . Then it is still optimal forthe library to buy Bj at price πj , for otherwise the library would have improvedits payoff by not buying any journal from publisher j when his journals were soldindependently.

Proof of Proposition 2(ii). We prove the result after proving two claims, whichtogether establish that all journals of all publishers in I are sold.

Claim 1. In any equilibrium of � with |I | ≥ 2, each journal of each publisherin I has the same price, denoted by p, and the library buys all journals; henceZ = ∅.

Proof. The proof of this claim is the same as that of Claim 2 in the proof ofProposition 1.

Claim 2. There exists no equilibrium of � with |I | = 1 and Z = ∅.

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Proof. Without loss of generality, let I = {1} and suppose that Z1 = ∅. Let p(1)1 ≡

maxi1∈R1{pi1} be the price of the most expensive journal among the journals ofpublisher 1 that the library buys. First, notice that if π < M then publisher 1 canincrease his profit by charging a uniform price p1 = (π1 + ε)/n1 for all hisjournals, where π1 is his profit before this deviation. Then all journals of 1 aresold because u ≥ UMB(p

(1)1 , π) and p

(1)1 > p1 imply that u > UMB(p1, π + ε)

for ε > 0 small enough.Second, when π = M we distinguish the case of Uj − v(Pj ) ≤ u − v(p

(1)1 )

for some j ∈ B from the case of Uj − v(Pj ) > u − v(p(1)1 ) for all j ∈ B. If

Uj − v(Pj ) ≤ u − v(p(1)1 ) for some j ∈ B then publisher 1 can increase his

profit by choosing a small price ε > 0 for one of his journals in Z1—denoted byi1—and reducing by δ the price of each journal in R1 whose price was equal top

(1)1 , so that the sum of prices of journals in R1 ∪ {i1} is larger by α > 0 than

before. The library cannot afford to buy i1 and all the items it purchased beforethe deviation, but by Lemma 2 it will buy all of these items except one. Droppinga journal of 1 with price p

(1)1 − δ yields a payoff of U ′ + v(p

(1)1 − δ − α), where

U ′ is the total surplus from journals and bundles that the library obtains before1’s price changes. Dropping Bj yields U ′ + u − Uj + v(Pj − α), which is largerthan U ′ + v(p

(1)1 − δ − α) at α = 0 and thus (by continuity) also at some α > 0.

Now suppose that Uj −v(Pj ) > u−v(p(1)1 ) for all j ∈ B. Then a necessary

condition for a particular j ∈ B not to profitably deviate by slightly increasingthe price of Bj is that there exist a t ≥ 1 such that

U ′ − Uj + tu + v(Pj − t p) ≥ U ′ − u + v(p

(1)1

), (A.9)

where p is the average price of the t cheapest journals in Z1. We now provethat a profitable deviation for publisher 1 is to bundle all his journals at a priceP1 = π1 + ε > π1. Once again, only one bundle will be dropped. The library’spayoff from dropping Bj is U ′ + |Z1|u − Uj + v(Pj − ε) whereas the payofffrom dropping B1 is U ′ − |R1|u + v(π1). The latter payoff is smaller than theformer because now we prove that U1 + v(Pj − ε) > Uj + v(π1) for a smallε > 0. Inequality (A.9) is equivalent to tu + v(Pj − t p) ≥ Uj − u + v(p

(1)1 ).

Hence |Z1| ≥ t implies that U1 + v(Pj − ε) ≥ |R1|u + v(Pj − ε) + Uj − u +v(p

(1)1 ) − v(Pj − t p)—and this right-hand side is larger than Uj + v(π1), or

equivalently (|R1| − 1)u + v(Pj − ε) + v(p(1)1 ) − v(Pj − t p) ≥ v(π1), by the

following argument. First notice that v(Pj − ε) > v(Pj − t p) for a small ε. Sec-ond, if |R1| = 1 then p

(1)1 = π1 and the result is straightforward. If |R1| ≥ 2, then

(|R1| − 1)u + v(p(1)1 ) ≥ |R1|v(p

(1)1 ) > v(|R1|p(1)

1 ) ≥ v(π1).

By Claims 1 and 2, in the rest of the proof we may assume that all journals of allpublishers in I are sold and that |I | ≥ 2.

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Consider first the case of π = M . Then the equality us − v(ps) = us′ −v(ps′) = b ≥ 0 must hold for any pair of items s and s′ (journal or bundles).In particular, b ≥ 0 by Lemma 2 and, if us − v(ps) > us′ − v(ps′), then thepublisher of item s can increase his profit by slightly increasing the price of s.This statement is proved by arguing exactly as in the proof of Lemma 3(i) if s isa bundle. If instead s is a journal, then s′ is a bundle because pij = p for anyjournal ij (by Claim 1 in this proof) and therefore uij −v(pij ) > us′ −v(ps′) forany ij . The argument of the proof of Lemma 3(i) shows that a publisher in I stillsells all his journals if he increases the price of one of them to p + ε with ε > 0and small. Now let publisher j ∈ I bundle his nj ≥ 2 journals at price πj +ε withε > 0 sufficiently small; the library does not have enough money to buy all theavailable items, but after arguing as in the proof of Lemma 3(i) we see that it willpurchase all items except one. The payoff from dropping Bj is U −unj +v(πj ),and the payoff from not buying a different item s is U −us + v(ps − ε). Becauseπj = nj p and nju − v(nj p) > njb ≥ b, the inequality nju − v(πj ) > us −v(ps − ε) holds for a small ε. Therefore, the library prefers dropping item s todropping Bj .

Consider now the case in which π < M . Then we can prove that us =UMB(ps, π) for any item s. Precisely, us ≥ UMB(ps, π) for any s by Lemma 2,and if us > UMB(ps, π) for some s then the publisher of item s can increase hisprofit by slightly increasing the price of s. The proof of this result mimics thearguments just given for the case of π = M and in the proof of Lemma 3(ii).Now let publisher j bundle his nj ≥ 2 journals at price Pj = πj + ε withε > 0 small enough. We prove that Bj is sold and hence that publisher j

increases his profit. Lemma 2 implies once again that at most one item is notpurchased. Suppose by way of contradiction that it is Bj ; then the library’s pay-off is U − nju + v(M − π + πj ). If Bj is added to the other items, the payoffincreases by

nju − UMB(πj + ε, π + ε). (A.10)

From u = UMB(p, π), pnj = πj , and Lemma 1(ii), we have nju > UMB(πj , π).Hence (A.10) is positive for ε = 0 and also for a small ε > 0.

A.5. Proof of Proposition 4

Suppose that all publishers are active before the merger. Without loss of gener-ality, we assume that publisher 1 merges with publisher 2. Let P BM∗

j and P AM∗j

denote the prices before and after the merger, respectively, of Bj (j = 1, . . . , N);P ∗

1&2 is the price charged by publisher 1&2 after the merger. Consider the case

in which∑N

j=1 v−1(Uj ) < M , so that πBM∗ < M , and assume that v−1(U1 +U2)+∑N

j=3 v−1(Uj ) < M; this implies πAM∗ < M . To prove that πAM∗ > πBM∗,

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we suppose by contradiction that πAM∗ ≤ πBM∗. By the virtue of (4) weobtain:

U1 + U2 = UMB(P BM∗

1 , πBM∗) + UMB(P BM∗

2 , πBM∗); (A.11)

U1 + U2 = UMB(P ∗

1&2, πAM∗). (A.12)

BecauseUMB(P BM∗1 , πBM∗)+UMB(P BM∗

2 , πBM∗)>UMB(P BM∗1 +P BM∗

2 ,πBM∗),(A.11)–(A.12) imply that UMB(P ∗

1&2, πAM∗) > UMB(P BM∗

1 + P BM∗2 , πBM∗).

This inequality and πAM∗ ≤ πBM∗ imply that P ∗1&2 > P BM∗

1 + P BM∗2 . Further-

more, (4) for j = 3, . . . , N implies

dPj

dπ= 1 − v′(M − π)

v′(M − π + Pj )< 0, (A.13)

which means that the merger (weakly) increases the profit of any non mergedpublisher. Because the merger increases each publisher’s profit, this contradictsthe assumption πAM∗ ≤ πBM∗. Thus we must have πAM∗ > πBM∗, which impliesthat the profit of publisher j (j = 3, . . . , N) is reduced because of (A.13) andhence that the profit of publisher 1&2 is larger than P BM∗

1 + P BM∗2 .

If πAM∗ = M then it is obvious that πAM∗ > πBM∗. Then (A.13)implies again that the profit of publisher j (j = 3, . . . , N) is reduced and, asa consequence, the profit of publisher 1&2 is larger than P BM∗

1 + P BM∗2 .

The result can be similarly proved when∑k−1

j=1 v−1(uj − uk) < M ≤∑kj=1 v−1(uj − uk+1) for some k ≥ 3, so that A∗ = {1, . . . , k} and πBM∗ = M .

A.6. Proof of Proposition 5(ii)

Let P kj denote the equilibrium price for Bj when publisher k wins the auction and

bundles the new journal with all his existing journals. Furthermore, let πBk ≡∑Nj=1 P k

j .

Claim 1. πB1 ≥ πB2 ≥ · · · ≥ πBN : The industry profit (weakly) increasesmore as the new journal is integrated into a larger bundle.

Proof. In order to prove that πBj ≥ πBk whenever j ≤ k, observe firstthat the result is straightforward if πBk = M because then Uj ≥ Uk andM ≤ v−1(U1) + · · · + v−1(Uk + u) + · · · + v−1(UN) imply that M ≤v−1(U1) + · · · + v−1(Uj + u) + · · · + v−1(UN); therefore, πBj = M .If instead πBk < M then the result is still obvious if πBj = M .Hence, we need to deal with the case of πBk < M and πBj < M . With-out loss of generality, we show that πB1 > πB2 when M > πB1 and

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M > πB2. Suppose that πB1 ≤ πB2. Then Lemma 1(i) and (4) implyP 1

h ≥ P 2h for all h ≥ 3; we now prove that P 1

1 + P 12 > P 2

1 + P 22 to obtain

a contradiction. Notice that P 11 and P 1

2 are such that

UMB(P 1

1 , πB1) = U1 + u; UMB(P 1

2 , πB1) = U2.

Define f1(π) and f2(π) as follows:38

UMB(f1(π), π) = U1; UMB(f2(π), π) = U2 + u.

We now prove that f1(πB1) and f2(π

B1) satisfy P 11 −f1(π

B1) > f2(πB1)−P 1

2 .Because v′ is strictly decreasing, it follows that

u = UMB(P 1

1 , πB1) − UMB(f1(πB1), πB1) =

∫ P 11

f1(πB1)

v′(M − πB1 + z) dz,

u = UMB(f2(πB1), πB1) − UMB

(P 1

2 , πB1) =∫ f2(π

B1)

P 12

v′(M − πB1 + z) dz.

These equations and the inequality f1(πB1) > P 1

2 together imply that P 11 −

f1(πB1) > f2(π

B1) − P 12 . Hence, P 1

1 + P 12 > f1(π

B1) + f2(πB1) and finally

f1(πB1) + f2(π

B1) ≥ f1(πB2) + f2(π

B2) = P 21 + P 2

2 because f1 and f2 aredecreasing. This gives P 1

1 + P 12 > P 2

1 + P 22 and thus the contradiction.

Claim 2. If N = 2 then each publisher has a weakly dominant bid; the domi-nant bids are such that b1 > b2 if πB2 < M and b1 = b2 if πB2 = M .

Proof. Suppose that N = 2. Then publisher j ’s (unique) weakly dominantstrategy is bj = P

jj − P k

j for j = 1, 2 and k = j . Because

b1 − b2 = πB1 − πB2,

we deduce that b1 > b2 if πB2 < M because this implies πB1 > πB2 and thatb1 = b2 if πB2 = M because this implies πB1 = πB2.

Claim 3. If N ≥ 3 and j = 1, then P 1j < P h

j for any h /∈ {1, j}.

Proof. Notice that, if πBh < M , then πB1 > πBh and pecuniary externalities (i.e.,(3) or (4)) imply P 1

j < P hj . If instead πBh = M , then πB1 = πBh and we prove

P 1j < P h

j as follows. Suppose (without loss of generality, but only to simplify

38. The logic of the proof here mimics the ideas in the proof of Proposition 3.

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notation) that j = N and h = N . If publisher 1 wins the journal, then P 1N solves

(A.6) in the proof of Theorem 1:

v−1[U1 + u − UN + v(PN)] + · · · + v−1[Uh − UN + v(PN)]+ · · · + PN = M. (A.14)

If h wins the journal, then P hN solves

v−1[U1 − UN + v(PN)] + · · · + v−1[Uh + u − UN + v(PN)]+ · · · + PN = M. (A.15)

For a given PN , the left-hand side in (A.14) is larger than the left-hand side in(A.15) because

v−1[U1 + u − UN + v(PN)] − v−1[U1 − UN + v(PN)]> v−1[Uh + u − UN + v(PN)] − v−1[Uh − UN + v(PN)], (A.16)

given that U1 > Uh and because v−1 is convex. Let P 1N satisfy (A.14) and let P h

N

satisfy (A.15). Given (A.16), the left-hand side of (A.14) at PN = P hN is larger

than M; hence P 1N < P h

N . From Pj = v−1[Uj −UN +v(PN)] we obtain P 1j < P h

j

for all h /∈ {1, j}.

Claim 4. If N ≥ 3, then b1 > bj for any j = 1 in any undominated equilibrium.

Proof. We first prove that, for publisher j ≥ 2, any bid larger than Pjj − P 1

j is

weakly dominated by bj = Pjj − P 1

j (hence, in any undominated equilibrium,

publisher j bids bj or less). The difference between the profit upon winning the

auction and the one upon losing it for publisher j is Pjj − P h

j when publisher

h = j is the winner. Because Pjj − P h

j < Pjj − P 1

j = bj for any h /∈ {1, j} by

Claim 3, it follows that the difference for publisher j between a bid bj (>bj ) andthe bid bj is that the first bid makes j win also in some cases when he must pay aprice larger than bj ; but in such cases j prefers losing the auction to winning it.

Now suppose that publisher j makes the highest bid among publishers otherthan 1 and that he bids bj . If P 1

1 − bj > Pj

1 , then 1 is happier when he wins theauction at price bj than when he loses it, and he can win at price bj by bidding anynumber larger than bj . Therefore, if 1 does not win the journal then it is necessary

that bj ≥ P 11 − P

j

1 . However, we have already proved that any bid larger than

bj = Pjj − P 1

j is weakly dominated for bidder h. We show that P 11 − P

j

1 > bj ,which implies that 1 does not win the auction only if some other publisher is

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playing a weakly dominated strategy. Claim 3 and the inequality πB1 ≥ πBj

imply P 11 − P

j

1 > bj .

A.7. Proof of Theorem 2

Proof of Theorem 2(i). Consider first the case of M >∑n

i=1 v−1(ui1). Then weknow from Corollary 3 that the industry profit πI∗ under the n-publisher, n-journalsetting is smaller than M . We prove by contradiction that the monopolist’s profitis smaller than πI∗ if he chooses prices different from p∗. Suppose that themonopolist can realize a profit π ≥ πI∗ with p = p∗. This implies that, amongthe journals sold, there must be at least one journal i1 whose price pi1 is strictlyhigher than p∗

1i , the price of journal 1i in the n-publisher, n-journal setting. Thisyields the contradiction

ui1 = u1i = UMB(p∗

1i , πI∗) < UMB(pi1, π),

given Lemma 1 and that π ≥ πI∗ and pi1 > p∗1i .

If M ≤ ∑ni=1 v−1(ui1), the monopolist can obtain profit M by choosing the

prices p∗ as under the n-publisher, n-journal setting because, by Lemma 2, thoseprices induce the library to buy all the journals 11, . . . , n1.

Proof of Theorem 2(ii)(a). In this proof, let A∗J denote the set of active journals;

ij is the journal with the lowest value in A∗J and A∗

J ≡ A∗J \{ij}. The existence

of a unique equilibrium candidate is established by proving the following twolemmas, which parallel Lemmas 3 and 4 in the proof of Theorem 1. Lemma 5shows that, given A∗

J , there exists a unique candidate equilibrium price vector;Lemma 6 proves that, given M , there is a unique A∗

J .

Lemma 5. For a given A∗J , the following statements hold.

(i) A candidate equilibrium price vector p∗A∗

Jsatisfying πI∗ = M exists if

and only if∑

ij∈A∗Jv−1(uij − uij ) < M ≤ ∑

ij∈A∗Jv−1(uij ); furthermore, p∗

A∗J

is unique and satisfies

uij − v(p∗

ij

) = ui′j ′ − v(p∗

i′j ′) ≥ 0 for any {ij, i′j ′} ⊆ A∗

J . (A.17)

(ii) A candidate equilibrium price vector p∗A∗

Jsatisfying πI∗ < M exists

if and only if∑

ij∈A∗Jv−1(uij ) < M; furthermore, p∗

A∗J

is unique and satisfies

uij − UMB(p∗

ij , π) = 0 for any ij ∈ A∗

J . (A.18)

(iii) If (A∗J , p∗

A∗J) is an equilibrium of �I , then it is necessary that∑

ij∈A∗Jv−1(uij − uij ) < M holds.

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Proof of (i). In any equilibrium of �I , p∗A∗

Jis such that each active journal is sold.

Because πI∗ = M , Lemma 2 implies uij ≥ v(p∗ij ) for any ij ∈ A∗

J . Let bij =uij − v(p∗

ij ) for any ij ∈ A∗J and b = minij∈A∗

J{bij } ≥ 0; we prove that if

bij > b for some ij ∈ A∗J then there exists a profitable deviation for publisher j .

Suppose without loss of generality that j = 1 and that b11 > b and bi1 = b forany other i1 ∈ A∗

J . Let publisher 1 increase slightly the price of journal 11 to p′11

and reduce pi1 slightly to p′i1 for any other i1 ∈ A∗

J in such a way that the sumof prices of his journals in A∗

J does not change. Then, b′i1 = ui1 − v(p′

i1) > b

for all i1 ∈ A∗J and, by Lemma 2, all active journals are still purchased. Now let

publisher 1 increase the price of journal 11 to p′11 + ε with ε > 0 and small. We

prove that all active journals of 1 are still purchased and so his profit increases.Now the library cannot afford to buy all the active journals, but it can afford (andby Lemma 2 is willing) to purchase all active journals except one. Given UA∗

J=∑

ij∈A∗Juij , the library’s payoff if it drops journal 11 is UA∗

J− [u11 − v(p′

11)]whereas if it drops i1 ∈ A∗

J \{11} the payoff is UA∗J

− [ui1 − v(p′i1 − ε)]; if

the library eliminates a journal ij (j = 1) such that bij = b, the payoff isUA∗

J−uij +v(p∗

ij −ε) = UA∗J−b−[v(p∗

ij )−v(p∗ij −ε)]. Because ui1−v(p′

i1) > b

for any i1 ∈ A∗J , it follows that for a small ε it is better to drop journal ij rather

than some journal i1 ∈ A∗J . In this way we deduce that uij − v(p∗

ij ) = b forany active journal and that (A.17) holds. The proof that there exists a (unique)p∗

A∗J

satisfying (A.17), πI∗ = M , and p∗ij > 0 for any ij ∈ A∗

J if and only if∑ij∈A∗

Jv−1(uij − uij ) < M ≤ ∑

ij∈A∗Jv−1(uij ) mimics closely the proof of

Lemma 3(i) and so is omitted.

Proofs of (ii)–(iii). These proofs are omitted because they are very similar to theproofs of Lemma 3(ii)–(iii) and of part (i) above.

In the rest of this proof of Theorem 2, u(k) is the value of the journal with thek-highest value; hence u(1) ≥ · · · ≥ u(n).

Lemma 6. Suppose that (A∗J , p∗

A∗J) is an equilibrium of �I . Then:

(i) the journal with the highest value is active for any M > 0, and if∑k−1j=1 v−1(u(j) − u(k)) < M for some k ∈ {2, . . . , n} then A∗

J includes thek journals with the highest values;

(ii) if M ≤ v−1(u(1) −u(2)) then A∗J includes only the highest-value journal,

and if∑k−1

j=1 v−1(u(j) − u(k)) < M ≤ ∑kj=1 v−1(u(j) − u(k+1)) for some k ∈

{2, . . . , n−1} then A∗J does not include any journal with value smaller than u(k).

Proof of (i). Suppose that∑k−1

j=1 v−1(u(j) − u(k)) < M for some k ∈ {2, . . . , n}and that a journal with value u(h) ≥ u(k) is not active; then the publisher of this

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journal can increase his profit. If π < M , he can make the journal active at a smallprice and decrease the price of all his other active journals such that the sum ofprices of his journals in A∗

J is slightly larger than his profit before the change inprices; then, arguments similar to those in the proof of Lemma 3(ii) show thatall his journals will be sold. Thus, his profit will increase. For the setting withπ = M , we can prove that u(h) > uij − v(p∗

ij ) for any ij ∈ A∗J by arguing as

in the proof of Lemma 4(i). Furthermore, the publisher of the journal with valueu(h) can increase his profit by pricing it close to zero and decreasing the prices ofall his other journals in A∗

J such that the sum of prices is slightly larger than hisprofit before the deviation. The details are quite similar to those in the proof ofLemma 5(i).

Proof of (ii). Let k = 1 if M < v−1(u(1) − u(2)); otherwise, let k be defined asin the statement of Lemma 6(ii). From Lemma 6(i) we know that the k journalswith the highest values are active; by arguing as in the proof of Lemma 4(ii), wefind a contradiction if a journal with value smaller than u(k) is active.

Proof of Theorem 2(ii)(a) (continued). Combining Lemmas 5 and 6 allows us todeduce that, if (A∗

J , p∗A∗

J) is an equilibrium of �I , then A∗

J is equal to the set of

active publishers and p∗A∗

Jis the price vector found in the unique equilibrium for

the n-publisher, n-journal environment described by Corollary 3. Hence, thereexists a unique candidate equilibrium for the oligopoly setting and it is equal tothe equilibrium under the minimal industry concentration.

Proof of Theorem 2(ii)(b). Let u ≡ (u11, . . . , un11, . . . , u1N, . . . , unNN) ∈ Rn++

be the vector of the values of the single journals. Given u > 0, letuu ≡ (u, . . . , u) ∈ R

n++ represent the vector of values when journals are homo-geneous such that uij = u for all ij . We prove that if u is close to uu forsome u > 0 then the unique candidate equilibrium (A∗

J , p∗A∗

J) for �I deter-

mined by Corollary 3 is indeed an equilibrium of �I . First notice that, givenany M > 0, the inequality M >

∑nh=1 v−1(u(h) − u(n)) is satisfied if journals

are sufficiently homogeneous; this implies that all the journals are sold in theunique candidate equilibrium or, equivalently, that A∗

J = A ≡ {1, . . . , n} andp∗

A∗J

= (p∗11, . . . , p

∗n11, . . . , p

∗1N, . . . , p∗

nNN) ∈ Rn++. For expositional simplic-

ity, we use p∗ instead of p∗A∗

Jbecause A∗

J = A throughout this proof; it shouldbe clear that p∗ depends on u even though the notation does not indicate this.We use p∗

j (p∗−j ) to denote the prices in p∗ of the journals owned (not owned) bypublisher j .

Given u > 0, we now prove that there exists a δ > 0 such that, if the distanced(u, uu) between u and uu is smaller than δ (i.e., if journals are approximatelyhomogeneous), then (A, p∗) is an equilibrium of �I . Suppose by contradictionthat this statement is false. Then, for t = 1, 2, . . . , there exist: (i) ut ∈ R

n++such that d(ut , uu) < 1/t ; (ii) the unique candidate equilibrium (A, p∗) given

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ut , determined by Corollary 3; and (iii) a publisher j (t), a set of active journals

Atj (t) ⊆ {1j (t), 2j (t), . . . , nj (t)j (t)} for him, and prices pt

j (t) ∈ R|At

j (t)|

++ for hisactive journals such that, given that the other publishers make all their journalsactive and choose prices p∗t

−j (t), the library buys all the journals in Atj (t) and

j (t) makes a profit∑

ij (t)∈Atj (t)

ptij (t) that is larger than his profit

∑nj(t)

i=1 p∗tij (t) if

he makes all his journals active with prices p∗tj (t). This last claim requires that,

given Atj (t) and prices pt ≡ (pt

j (t), p∗t−j (t)), the library does not buy all the journals

of the other publishers.39 Let Qt be the set of all the possible combinations ofjournals that the library can afford to buy given (At

j (t), pt ), and let St−j (t) ⊂

S−j (t) ≡ {ih : i = 1, . . . , nh and h = j (t)} be the set of the journals ofpublishers different from j (t) that the library buys in order to maximize its ownpayoff. Observe that, by definition, (At

j (t) ∪ St−j (t)) ∈ Qt for any t .

Because there are finitely many publishers and journals, there exists a subse-quence of the original sequence {ut }+∞

t=1 along which j (t),St−j (t),A

tj (t), andQt are

all constant. Without loss of generality, we assume that (i) the subsequence is theoriginal sequence: j (t) = j , St

−j (t) = S−j , Atj (t) = Aj , and Qt = Q for all t ; and

that (ii) j = 1. Let UtA1

= ∑i1∈A1

uti1, πt

A1= ∑

i1∈A1pt

i1, UtS−1

= ∑ij∈S−1

utij ,

and π∗tS−1

= ∑ij∈S−1

p∗tij . Then, for any t , the library’s payoff from buying the

journals in A1 ∪ S−1 is UtA1

+ UtS−1

+ v(M − πtA1

− π∗tS−1

), which by definitionis larger than the payoff from any other feasible combination of journals in Q.We prove that for a large t this leads to a contradiction. We start by observingthat, as t tends to infinity, p∗t

−1 tends to pn−n1u ≡ (p∗, . . . , p∗) ∈ R

n−n1 , wherep∗ is determined by Proposition 1 (this fact is both intuitive and simple to prove).Then, we can show that publisher 1 makes all his journals active in any profitabledeviation and that limt→+∞ pt

1 = pn1u ≡ (p∗, . . . , p∗) ∈ R

n1 .

Claim 1. A1 = {11, 21, . . . , n11} and limt→+∞ pt1 = pn1

u .

Proof. First notice that a subsequence of pt1 converges; without loss of gener-

ality, we suppose that the subsequence is the original sequence. Let p1 denotelimt→+∞ pt

1. Because (A1, pt1) is a profitable deviation for 1, the inequality

πtA1

>∑n1

i=1 p∗ti1 holds for any t . Furthermore, limt→+∞

∑n1i=1 p∗t

i1 = n1p∗ and

thereforelim

t→+∞ πtA1

=∑

i1∈A1

pi1 ≥ n1p∗. (A.19)

39. This fact is obvious if the industry profit π∗t before j (t)’s deviation is equal to M . If insteadπ∗t < M , then ut

ij = UMB(p∗tij , π∗t ) for any ij with j = j (t). If j (t)’s profit increases by ε > 0 and

if all journals of the other publishers are purchased, then the industry profit increases to π∗t +ε > π∗t .But ut

ij ≥ UMB(p∗tij , π∗t + ε) cannot hold for any ij with j = j (t), and Lemma 2 implies that the

library does not buy all journals of publishers different from j (t).

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Now assume by way of contradiction that p1 = p|A1|u . Notice that p1 = p|A1|

u

implies pi1 > p∗ for at least one i1 ∈ A1, since otherwise pi1 ≤ p∗ for any i1 ∈ A1

and this (together with p1 = p|A1|u ) violates (A.19). Without loss of generality,

we suppose that 11 ∈ A1 and p11 > p∗. Because S−1 is a strict subset of S−1, leti′j ′ ∈ (S−1\S1) = ∅ be a journal of publisher j ′ with j ′ = 1 that the library doesnot buy given (At

j (t), pt ). Because limt→+∞ p∗ti′j ′ = p∗ and limt→+∞ pt

11 =p11 > p∗, it is obvious that (A1\{11} ∪ S−1 ∪ {i′j ′}) ∈ Q for any large t . Thelibrary’s payoff from buying the journals in A1\{11}∪S−1 ∪{i′j ′} is Ut

A1−ut

11 +Ut

S−1+ut

i′j ′ + v(M −πtA1

+pt11 −π∗t

S−1−p∗t

i′j ′), and for any large t this is largerthan the payoff from buying the journals in A1 ∪ S−1: A contradiction. Becausepi1 ≤ p∗ for any i1 ∈ A1, (A.19) implies that A1 = {11, 21, . . . , n11} and thatpi1 = p∗ for i = 1, . . . , n1.

Claim 1 says that (for a large t) any profitable deviation of publisher 1 is such thatall of his journals are active, the library buys all of them, and the price vector pt

1is close to pn1

u ; hence pt1 is also close to p∗t

1 . The next claim establishes a resultabout the journals of the other publishers the library buys given 1’s deviation.

Claim 2. The set S−1\S−1 includes only one journal.

Proof. Because limt→+∞ pt1 = pn1

u , it follows that∑n1

i=1 pti1 + ∑

ij∈S−1p∗t

ij

is only slightly larger than∑n1

i=1 p∗ti1 + ∑

ij∈S−1p∗t

ij for a large t ; there-

fore,∑n1

i=1 pti1 + ∑

ij∈(S−1\i′j ′) p∗tij <

∑n1i=1 p∗t

i1 + ∑ij∈S−1

p∗tij if i′j ′ is an

arbitrary journal of a publisher j ′ with j ′ = 1. Furthermore, recall thatp∗t is such that ut

ij ≥ UMB(p∗tij ,

∑n1i=1 p∗t

i1 + ∑ij∈S−1

p∗tij ) for any ij . Hence,

utij > UMB(p∗t

ij ,∑n1

i=1 pti1 + ∑

ij∈(S−1\i′j ′) p∗tij ) for any ij = i′j ′ with j = 1.

Because the library buys all journals of 1, this means (by Lemma 2) that it willdrop exactly one journal of the other publishers.

Claims 1 and 2 imply that, for a large t , at prices pt the library buys all the journalsof publisher 1 and all but one journal of the other publishers. However, our nextclaim shows that this is impossible given approximate homogeneity.

Claim 3. If journals are almost homogeneous, then no price vector for thejournals of 1 induces the library to drop exactly one journal of the other publisherswhile buying all journals of 1.

Proof. Let publisher 1 modify the prices of journals 11 to n11 by ε1, . . . , εn1 ,respectively, with ε = ε1 + · · · + εn1 > 0;40 the new prices for journals of 1 arepi1 = p∗

i1 + εi (i = 1, . . . , n1).

40. Observe that we allow the prices of some journals of 1 to be unchanged.

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The library’s payoff if it drops a journal ij (j = 1) is U − uij + v(M − π +p∗

ij −ε), reduced by v(M−π+p∗ij )−v(M−π+p∗

ij −ε)+b (where b = 0 if π <

M and b ≥ 0 if π = M) with respect to the payoff U+v(M−π) before the changein prices by 1. Because v is strictly concave, this reduction in payoff is minimizedfor the journal ij (j = 1) with the highest price, which we denote by p∗−1. Ifinstead the library does not buy a journal i1, its payoff is U−ui1+v(M−π+p∗

i1−ε+εi), reduced by v(M−π+p∗

i1)−v(M−π+p∗i1−ε+εi)+b with respect to U+

v(M −π). Therefore, the library prefers to drop the highest-priced journal of theother publishers rather than one journal of 1 if v(M−π+p∗−1)−v(M−π+p∗−1−ε) < v(M−π+p∗

i1)−v(M−π+p∗i1−ε+εi) for i = 1, . . . , n1; this set of condi-

tions is equivalent to v(M −π + p∗−1)−v(M −π + p∗−1 −ε) < φ(ε1, . . . , εn1) ≡min{v(M − π + p∗

i1) − v(M − π + p∗i1 − ε + εi), i = 1, . . . , n1}.41 This shows

that, given ε, publisher 1 is interested in choosing ε1, . . . , εn1 that maximize φ.The optimal values of ε1, . . . , εn1 are denoted by ε1, . . . , εn1 and satisfy:

v(M − π + p∗

i1

) − v(M − π + p∗

i1 − ε + εi

)= v

(M − π + p∗

n11

) − v(M − π + p∗

n11 − ε + εn1

)for i = 1, . . . , n1 − 1;

(A.20)

ε1 + ε2 + · · · + εn1 = ε,

provided that M − π + p∗i1 − ε + εi > 0 for i = 1, . . . , n1. If this is the case,

then φ(ε1, . . . , εn1) = v(M − π + p∗11) − v(M − π + p∗

11 − ε + ε1); otherwise,φ(ε1, . . . , εn1) ≤ v(M −π +p∗

11)−v(M −π +p∗11 −ε+ ε1). We shall prove that

v(M − π + p∗−1

) − v(M − π + p∗−1 − ε

)> φ(ε1, . . . , εn1) (A.21)

when M −π +p∗i1 −ε+ εi > 0 for i = 1, . . . , n1; hence, we conclude that (A.21)

holds a fortiori when M − π + p∗i1 − ε + εi = 0 for some i. Given that journals

are approximately homogeneous, p∗i1 is close to p∗

n11 for i = 1, . . . , n1 − 1 andtherefore (A.20) implies that εi is close to εn1 , which means that εi is close toε/n1 > 0 for i = 1, . . . , n1. Then (A.21) holds because p∗−1 is close to p∗

11 andbecause the left-hand side of (A.21) is close to v(M−π+p∗

11)−v(M−π+p∗11−ε)

whereas the right-hand side is close to v(M − π + p∗11) − v(M − π + p∗

11 − ε +ε/n1).

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