NET Institute*

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1 NET Institute* Working Paper #07-5 Septeber 2007 undling and Copetition for Slots Doh-Shin Jeon Universitat Popeu Fabra Doenico Menicucci Università degli Studi di Firenze * The Networks, Electronic Coerce, and Telecounications ( NET ) Institute, is a non-profit institution devoted to research on network industries, electronic coerce, telecounications, the Internet, virtual networks coprised of coputers that share the sae technical standard or operating syste, and on network issues in general.

2 undling and Copetition for Slots Doh-Shin Jeon y and Doenico Menicucci z Septeber 29, 2007 bstract We study copetition aong upstrea rs when each of the sells a portfolio of distinct products and the downstrea has a liited nuber of slots (or shelf space). In this situation, we study how bundling a ects copetition for slots. When the downstrea has k nuber of slots, social e ciency requires that it purchases the best k products aong all upstrea rs products. We nd that under bundling, the outcoe is always socially e cient but under individual sale, the outcoe is not necessarily e cient. Under individual sale, each upstrea r faces a trade-o between quantity and rent extraction duetothe coexistence of theinternal copetition (i.e. copetition aong its own products) and the external copetition (i.e. copetition fro other rs products), which can create ine ciency. On the contrary, bundling reoves the internal copetition and the external copetition aong bundles akes it optial for each upstrea r to sell only the products belonging to the best k. This unabiguous welfare-enhancing e ect of bundling is novel. Key words: undling, Copetition aong Portfolios, Liited Slots (or Shelf Space) JEL Code: D4, K2, L3, L4, L82 This research was partially funded by the NET Institute ( whose nancial support is gratefully acknowledged. y Universitat Popeu Fabra and CRE. doh-shin.jeon@upf.edu z Università degli Studi di Firenze, Italy. doenico.enicucci@dd.uni.it

3 Introduction In vertical relations, very often each upstrea fir sells a portfolio of distinct products which copete for liited slots (or shelf space) of downstrea firs. In this situation, upstrea firs ay eploy bundling as a strategy to win over the copetition for the liited slots. The practice of bundling has been a ajor antitrust issue and a subject of intensive research in the past. However, to the best of our knowledge, the theoretical Industrial Organization literature sees to have neglected copetition aong portfolios of distinct products and, in particular, no paper has studied how bundling affects copetition for liited slots in this fraework. In this paper, we attept to provide a new perspective on bundling by analyzing how it affects copetition aong portfolios of distinct products for liited slots, and social welfare. Exaples of the situation we described above are abundant. For instance, in the ovie industry, each ovie distributor has a portfolio of distinct ovies and buyers (either ovie theaters or TV stations) have liited slots. More precisely, the nuber of ovies that can be projected in a season (or in a year) by a theater is constrained by tie and the nuber of roos. Likewise, the nuber of ovies that a TV station can show at prie tie during a year is also liited. ctually, allocation of slots in ovie theaters has been one of the ain issues of the last presidential election in France regarding the ovie industry. Furtherore, bundling in this industry (known as block booking 2 ) was declared illegal in two supree court decisions in U.S.: Paraount Pictures, where blocks of fils were rented for theatrical exhibition, and Loew s, where blocks of fils were rented for television exhibition. In addition, recently in MC Television Ltd. v. Public Interest Corp. (th Circuit, pril 999), the court of appeals reaffired the per se illegal status of block booking. nother exaple we have in ind is that of anufacturers copetition for a retailer s shelf space. Typically, each anufacturer produces a range of different products (for instance, think about all the products sold under the brand nae of Nestle) and anufacturers copete for a retailer s liited shelf space. In this context, anufacturers having a large portfolio of products ay practice bundling (often called full-line forcing) for their Cahiers du Cinea (pril, 2007) proposes to liits the nuber copies per fil since certain ovies by saturating screens liits other fils access to screens and asks the presidential candidates opinions about the issue. 2 lock booking refers to the practice of licensing, or offering for license, one feature or group of features on the condition that the exhibitor will also license another feature or group of features released by distributors during a given period (Unites States v. Paraount Pictures, Inc., 334 U.S. 3, 56 (948)).

4 advantage and there has been antitrust cases related to this practice: Procter & Gable / Gillette 3 and Société des Caves de Roquefort. 4 Moreover, since we consider a general odel in which a fir can bundle any nuber of products, our setup can be applied to bundling a large nuber of inforation goods, a coon practice in the Digital era (for instance, bundling of electronic acadeic journals). In our odel, we will assue away any asyetric inforation or any uncertainty about values of products. This will allow us to depart fro the existing literature on block booking or bundling (see the review of the related literature later on in this section) and to identify what sees to us a first-order effect of bundling by focusing on the downstrea fir s slot constraint. More precisely, we consider copetition between two upstrea firs selling to a downstrea fir who has k(> 0) nuber of slots. In our setting, a product needs to occupy a slot to generate soe value (i.e. a profit) to the downstrea fir. The upstrea firs products are heterogenous in ters of the value that each of the generates to the downstrea fir. Therefore, social efficiency requires the downstrea fir to purchase and use the best k products aong all products owned by the two upstrea firs. We focus on studying how bundling affects the set of the products that are purchased and consued by the downstrea fir. s the ain result, we find that under bundling the outcoe of copetition is always socially efficient, while this is not necessarily the case under individual sale. Under individual sale, each upstrea fir faces a trade-off between quantity and rent extraction due to the coexistence of the internal copetition (i.e. copetition aong its own products) and the external copetition (i.e. copetition fro other firs products): as a fir increases the nuber of products it induces the downstrea fir to buy, it should abandon ore rent for each product it sells. This trade-off can ake the outcoe inefficient. On the contrary, bundling reoves the internal copetition and the external copetition aong bundles akes it optial for each upstrea fir to sell only his own products which belong to the best k. We think that this unabiguous welfare-enhancing effect of bundling is pretty novel. Furtherore, we show that the efficiency property of bundling is very general in that it holds regardless of whether we consider a sequential or siultaneous gae or whether or not we allow firs to contract directly on exclusive use of slots. The existing literature analyzing bundling in a second-degree price discriination fraework often justifies the rule of reason standard regarding bundling. Our result has strong policy iplications that go beyond the rule of reason. 3 DG Copetition case COMP/M Conseil de la Concurrence, Decision 04-D-3, 8th pril

5 There are only a few papers on block booking. The leverage theory, on which the Supree Court s decisions were based, that block booking allows a distributor to extend its onopoly power in a desirable ovie to an undesirable one was criticized by Stigler since the distributor is better off by selling only the desirable ovie at a higher price. Instead of the leverage theory, Stigler (968) proposed a theory based on second-degree price discriination. However, Kenney and Klein (983) point out that siple price discriination explanation since block booking is inconsistent with the facts of Paraount and Loew s for the prices of the blocks varied a great deal across arkets and argue that block booking ainly prevents exhibitors fro oversearching, i.e. fro rejecting fils revealed ex post to be of below-average value fro an ex ante average-valued package. Their hypothesis is epirically tested in a recent paper by Hanssen (2000) but the author finds little support for the hypothesis 5 and proposes that block booking was priarily intended to cheaply provide fils in quantity. Most papers on bundling study bundling of two (physical) goods in the context of second-degree price discriination and focus on either surplus extraction (Schalensee, 984, Mcfee et al. 989, Salinger 995 and rstrong 996, 999) in a onopoly setting or entry deterrence (Whinston 990 and Nalebuff 2004) in a duopoly setting. akos and rynjolfsson (999, 2000) s papers are an exception, in that they study bundling of a large nuber of inforation goods, but they aintain the second-degree price discriination fraework. Their first paper shows that bundling allows a onopolist to extract ore surplus (since it reduces the variance of average valuations by the law of large nubers) and thereby unabiguously increases social welfare; 6 the second paper applies this insight to entry deterrence (we do not address the entry deterrence issue). Since our novelty is that we assue coplete inforation and hence full surplus extraction is possible under the onopoly setting, the rent extraction issue does not arise in our environent and there is no use in applying the law of large nuber. In Jeon-Menicucci (2006), we took a fraework siilar to the one in this paper to study bundling electronic acadeic journals. More precisely, publishers owning portfolios of distinct journals copete to sell the to a library who has a fixed budget to allocate between journals and books. Publishers are assued to have coplete inforation about the value that the library obtains fro a journal and about the budget. We found that bundling is a profitable strategy both in ters of surplus extraction and entry deterrence. Conventional wisdo says that bundling has no effect in such a setting and this is true without the budget constraint. However, when the budget constraint binds, we found that 5 ut Kenny and Klein (2000) do not agree with Hanseen s analysis. 6 See also rstrong (999). 3

6 each fir has a strict incentive to adopt bundling but bundling reduces social welfare by reducing the library s consuption of journals and onographs. In this paper, instead of focusing on the budget constraint of the buyer, we focus on his slot constraint. nother difference is that Jeon-Menicucci (2006) focus on products (journals) of hoogeneous value while in this paper we consider products of heterogenous value. In spite of siilarities of the fraeworks, the result we obtain here is contrary to the one in the previous paper since we find that the allocation under bundling is socially efficient. Finally, to our knowledge, Shaffer (99) is the only paper that explicitly odels the downstrea fir s liited shelf space. 7 He considers an upstrea onopolist selling two substitutable products with variable quantity. He finds that brand specific two-part tariffs alone do not allow the onopolist to capture the axiu rent fro the downstrea fir but full-line forcing (equivalent to bundling) does. We consider products of independent values and hence the rent extraction issue Shaffer considers does not arise. In a general setup of copetition in which seller i has n i nuber of products and the buyer has k(< i n i ) nuber slots, we study how bundling affects the set of the products that occupy the slots. lthough products have independent values, copetition arises because of the slot constraint as long as there are at least two sellers. In what follows, section 2 presents the odel. Section 3 (4) characterizes the equilibriu under individual sale (bundling). Section 5 copares individual sale with bundling in ters of social welfare. Section 6 shows that when bundling is allowed, firs have an incentive to bundle their products. Section 7 shows that the efficiency property of bundling is very general. Section 8 concludes. Most proofs are gathered in the ppendix. 2 The setting 2. Model There are two upstrea firs, denoted by i =,, and a downstrea fir, denoted by D. Each fir i has a portfolio of n i ( ) products for i =,, and all products are distinct; let n n + n. Fir D has a liited nuber of slots (or shelf space) to distribute the upstrea firs products: the nuber of slots is given by k( ). Given (n, n, k), we assue for siplicity that the cost of producing each product is zero for i =, and the cost of distributing each product is zero for D. D s distribution of a product requires one unit of slot. Therefore, D can distribute at ost k nuber of products and we assue n > k. In this setup, we consider products of 7 See also Vergé (200) who perfors the social welfare analysis in the setup of Shaffer (99). 4

7 heterogenous value and study how bundling affects the set of the products occupying the liited slots. More precisely, we are interested in knowing when D distributes the best k nuber of products. Let u j i denote the gross profit that D obtains fro distributing the j-th best product of i; thus u i u 2 i... u n i i > 0 for i =,. Let u j denote the gross profit (or surplus) that D obtains fro the j-th best product aong all products owned by both upstrea firs, thus u u 2... u n. We assue u k > u k+. Let i denote the nuber of i s products in the set of the k best products aong all products owned by both upstrea firs: by definition, + = k. We assue i for i =,. 8 this paper we study the case of n k n, but we can extend easily the analysis to the case of n < k or k < n. Then, without loss of generality, we suppose n = k. Under individual sale, fir i chooses p j i > 0 for its product with value u j i, and we define w j i u j i p j i as the net profit that D obtains fro buying this product. Let p i (p i, p 2 i,..., p n i i ) and w i (wi, wi 2,..., w n i i ) denote the vectors of prices and of net profits for the products of fir i, respectively. It is clear that there is a one-to-one correspondence between p i and w i, and therefore we can equivalently express fir i s decision proble in ters of either p i or w i. However, when we use w i we need to recall that w j i < u j i for i =, and j =,..., n i. In particular, we will soeties refer to the condition for fir. w < u, w 2 < u 2,..., w k < u k () Under bundling, fir i sells a bundle i at a price P i (> 0). In 2.2 Nonexistence of equilibriu in a siultaneous gae efore presenting the tiing of the sequential gae that we study, we below illustrate that equilibriu often does not exist in a siultaneous gae under individual sale. Suppose that and choose siultaneously p and p. s usual, we adopt as a tie-breaking rule that if indifferent aong different products, D buys the product which generates the highest gross profit. Suppose that () has two (one) products and k = 2. ssue (u, u 2, u ) = (3,, 2). Without loss of generality, we can assue that chooses p such that 3 p ax {0, p 2 }: the net profit that D akes fro buying s best product is positive and larger than the one it akes fro buying s second best product. Given p satisfying 3 p ax {0, p 2 }, s best response is to choose p such that 2 p = 8 The analysis of individual sale applies to i = 0. i bundling. siplifies the exposition of the analysis of 5

8 ax {0, p 2 }: can find such a price since p 2 > 0 and hence ax {0, p 2 } cannot be larger than. Consider first the case in which p 2 > 0. In this case, s best response is p = + p 2. However, then can deviate by charging p 2 = p 2 ε for ε(> 0) sall enough and sell both products. Consider now the case in which p 2 >. In this case, s best response is p = 2. However, then can deviate by charging p 2 = ε for ε(> 0) sall enough and sell both products. Therefore, there is no equilibriu (in pure strategy). 9 The above exaple illustrates well the coitent issue that faces. On the one hand, if can coit not to sell its second best product, then and can sell their best products at the prices that extract D s whole surplus. However, this outcoe cannot be an equilibriu since has an incentive to deviate by undercutting s price (i.e. charging p 2 = ε) to sell both products. On the other hand, there cannot be an equilibriu in which D buys both products of and does not buy s product. Therefore, in what follows, we will consider a sequential gae in which fir can coit to its prices before chooses its prices. 2.3 Tiing and tie-breaking rules We consider the following sequential tiing. When bundling is prohibited (i.e. under individual sales), Stage. chooses p ; Stage 2. after observing p, chooses p ; Stage 3. D akes its purchase decision. When bundling is allowed, at stage (stage 2), () decides whether or not to bundle his products and p or P (p or P ) accordingly: in addition, if fir i decides to bundle his products, it also decides which products to include into the bundle. In what follows we use the concept of subgae perfect Nash equilibriu (SPNE) to deterine the outcoe of this gae. Thus we start with D s purchases at stage three. In the case of individual sales, D chooses the k products yielding the highest non-negative net profits. However, it is necessary to specify how D deals with ties, i.e. with products which have the sae net profit. Therefore we introduce the following tie-breaking rules. T: If D is indifferent between buying a product fro and a product fro (both with non-negative net profits), and cannot buy both of the, then D buys s product. 9 Even if we allow firs to charge zero price, the equilibriu does not exist. Suppose now p 2 = 0: this together with 3 p ax { 0, p} 2 iplies p 2. In this case, s best response is p =. However, then can deviate by charging (p, p2 ) = (3, 2) for instance. Then, sells only its best product but at a higher price. 6

9 T is otivated by the fact that in our sequential gae, given the price of s product in question,, as the follower, can always lower by ε > 0 its price to break D s indifference. Forally, in soe cases has no best reply without this assuption. T2: If D is indifferent aong different products offered by the sae fir, and cannot buy both of the, D buys the product that generates the highest gross value. T2 is a standard tie-breaking rule. Finally, we introduce the following tie-breaking rule for the upstrea firs. T3: Under bundling, if including a product into the bundle that i sells does not strictly increase i s profit, i prefers not to include the product into the bundle. T3 akes a perfect sense when there is an infinitesial cost of production. lthough we do not odel any production cost for siplicity, T3 captures the essential effect that would result fro a positive production cost. 3 Individual sale 3. preliinary result Recall that we have set w j i = u j i p j i for i =, and j =,..., n i, and w i = (w i, w 2 i,..., w n i for i =,. In ŵ i (w () i, w (2) i,..., w (n i) i ) we order instead the net profits in a decreasing way, which eans that w () i w (2) i... w (n i) i. We now prove a siple and intuitive result: there is no loss of generality in assuing that w (j) i = w j i for all j. Lea Without loss of generality, we can restrict our attention to the case in which w i w 2 i... w n i i (i.e., w i = ŵ i ) for i=,. In particular, lea iplies the following onotonicity condition for fir, which we will use repeatedly in the reaining of the paper: w w 2... w k (2) The lea also iplies that when fir i is selling i nuber of products, i is actually selling its products with the i highest gross values. i ) 7

10 3.2 Stage two Now we apply backwards induction to fir, by exaining his decision at stage two. Precisely, we take w as given and consider the following questions: given {,..., n }, is it feasible for to sell units? If so, what is the highest profit can ake by selling units? Lea 2 Given w and {,..., n }, it is feasible for to sell units if and only if u > w k +. In this case, the highest profit can earn by selling products is u u ax { w k +, 0 }. The basic idea of the lea is that D buys units of if and only if products of are aong the k products with the highest net profits. For instance, consider the case of =. If w k u, then cannot sell any product because the inequality w k > w necessarily holds and therefore D will buy k products fro and none fro. If instead w k < u, succeeds in selling his best product by charging a sufficiently low price p such that w k < u p and p j large enough for j 2. Precisely, fro T, the highest price which induces D to buy s best product is p = u ax { w k, 0 }. In words, can sell his best product only if the k-th best product of gives D a net profit that is saller than the gross profit of the best product of. In short, it ust be possible for to push out the k-th best product of by pricing aggressively enough his own best product. For an arbitrary value of in {,..., n }, the sae arguent shows that the inequality w k + < u is necessary, i.e. it ust be possible for to block out the (k + )-th best product of by pricing suitably his own best products. Otherwise, w k + > w and therefore D will buy at least k + units fro, and at ost k (k + ) = fro. When w k + < u, succeeds in selling products by charging prices p,..., p such that w =... = w = ax { w k +, 0 } (again, recall T), or equivalently p j = u j ax { w k +, 0 } for j =,..., and p j large for j = +,..., n ; the resulting profit for is u u ax { w k +, 0 }. In view of lea 2 we define as follows the profit can ake by selling units, for {,..., n }: 0 π () u u ax { w k +, 0 } if u > w k + ; 0, otherwise. In order to exaine how π depends on, we begin by noticing that the higher is, the ore restrictive is the inequality u > w k +. Thus, if is unable to sell units because u w k +, he is a fortiori unable to sell > units. 0 This profit depends also on w, even though we do not ephasize this fact in the notation. 8

11 Now we consider a case in which u + > w k > 0, so that is able to sell + products (and also fewer than + ) and we below exaine how increasing his sale by one ore product affects s profit. When sells units, we have seen that he earns a profit of u u w k + by charging prices p j = u j w k + for j =,..., ; these prices are deterined by the fact that needs to block out the (k + )-th best product of. If instead he sells + units, needs to push out the (k )-th best product of, which is ore valuable than the (k + )-th. Prices are then ˆp j = u j w k for j =,..., +, and ˆp j < p j for j =,...,. This generates a loss for, on his best units, equal to (w k w k + ). However, now gains ˆp + = u + w k > 0 fro the sale of the ( + )-th unit. Whether prefers selling + units to units depends on the coparison between the loss (w k w k + ) and the gain u + w k. In other words, (2) akes face a trade-off between quantity and (per unit) rent extraction: as increases the nuber of products he sells, he ust leave ore surplus per unit to D. 3.3 Stage one We first study the optial pricing conditional on that sells k units. nd then, we study the optial that axiizes s profit s profit when he sells k units Now we consider the first stage of the gae in order to deterine the profit can ake as a function of the nuber of products he sells. Hence, suppose that wants to sell k units for {0,,..., n }. Then, we inquire whether (i) there exists w such that, taking into account the best response by, induces D to buy k units fro ; (ii) within the set of w which allow to sell k units, we identify the vector that axiizes s profit. Forally, the conditions that allow to sell k products can be stated by using the following incentive constraints: (IC, ) π () π ( ) for any and {,..., n } (3) Condition (3) eans that prefers to sell units rather than. In particular, (3) iplies that is not going to push out the (k )-th best unit of (nor any better product of ), and therefore D will buy k nuber of products fro. Then s ctually, it suffices to satisfy the constraints (IC, ) for all >. However, it turns out that it is costless for to satisfy also the constraints (IC, ) for < (see the proof of Proposition ). 9

12 profit is given by: π (k ) k j= (u j w j ) [ w j 0 ] which, we note, is not affected by (w k +,..., w). k We investigate below whether there is a set of w which satisfy (3) and, if so, we axiize π (k ) in this set. We start by observing that it is certainly possible for to sell k n units, and that he can do so without leaving any surplus to D on these products. In order to show the details, suppose that chooses p j = u j for j =,..., k n and p j high enough for j = k n +,..., k. In this way, s n worst products are not copeting with s products while s best k n products give D zero surplus. Then, will reply by charging p j = u j for j =,..., n, and D will buy k n products fro and n products fro, earning no profit. When s objective is to induce to sell only (< n ) products, as it will becoe clear later on, has two strategies: accoodation or fight. ccoodation eans that contents hiself with occupying slots. Fight eans that tries to occupy ore than slots by blocking out soe extra units of. Obviously, to achieve his goal, ust choose prices such that prefers accoodation to fight, which is equivalent to the property that (IC, ) is satisfied for all >. What akes the case of = n straightforward is that sells all his n units by accoodating, and thus he will not fight. The next proposition characterizes the condition under which is able to sell k units and the profit axiizing vector w (hence, the optial prices) conditional on selling k units. For expositional facility, we introduce the following notation. Given {0,,..., n }, let µ k+ (u u ) for = +,..., n (4) Proposition For a given {0,,..., n }, (i) a. can find w that induces D to buy k units fro if and only if u k+ > µ k+ for = +,..., n (5) b. Let ˆ {0,,..., n } denote the sallest for which (5) is satisfied [we set ˆ = n if (5) fails to hold for any {0,,..., n }]. Then, (5) is satisfied also for = ˆ +,..., n. (ii) If ˆ, the profit axiizing w for is as follows: a. when = 0, w =... = w k = u ; 0

13 b. when {,..., n }, w k + = w k +2 =... = w k = 0 (6) w k + = ax{w k +2, µ k + } for = +,..., n (7) w = w 2 =... = w k n = w k n + (8) We below give the intuition of the results in Proposition ; we focus on explaining the profit axiizing w conditional on selling k units for, described in Proposition (ii)b. 2 Given s objective to sell k units, should structure his prices for the best k products (the ones to sell) very differently fro the prices for the worst units (the ones not to sell). On the one hand, regarding the worst products, it is optial to charge very high prices (higher than their values) so that does not face any copetition fro the; precisely, (6) reveals that choosing w k + = w k +2 =... = w k = 0 is optial. The reason is that this pricing axiizes s profit fro accoodation and hence reduces s teptation to fight. In fact, the pricing allows to extract full surplus u u fro his best products if he wants to sell only products. Then, obviously, strictly prefers selling units to selling less than, and hence downward incentive constraints (i.e. (IC, ) for < ) are trivially satisfied. On the other hand, regarding the best k units to sell, the prices should be copetitive enough to ake it unprofitable for to sell ore than units. In particular, cannot extract full surplus fro these products since if he attepts to do that, can sell all of his n products by leaving no surplus per product to D, given T. To explain the optial pricing of the best k units, suppose that wants to sell + units instead of units. Lea 2 shows that can sell + products only if w k < u +. In this case, akes a profit equal to π ( + ) = u u + ( + )w k and we have π ( + ) π () = u + s we discussed after Lea 2, u + ( + )w k ( + )w k is coposed of the loss w k s best units (with respect to selling the at full prices) plus the gain u + selling the ( + )-th unit. Therefore, w k π ( + ): note that it is less restrictive than w k w k u+ = + µk u + w k on fro allows to satisfy π (). Hence, the sallest value of, as described in (7). In order to deter fro u +2, but satisfying (IC,+ ) is w k = µ k selling + 2 units, we can argue as before. sufficient condition is w k 2 Proposition (ii)a is straightforward, as the best way for to sell k products is to set w =... = wk equal to the value of s best product, u, provided that uk > u.

14 when w k < u +2 we ust have: which is equivalent to w k satisfied if w k in{µ k π( + 2) π() = u + + u +2 ( + 2)w k 0, µ k = +2 (u+ + u +2 ). Therefore, (IC,+2 ) is, u +2 }. However, w k should also satisfy the ono- w k ). Fro w k in{µ k, u +2 }, we find that the sallest value of w k satisfying (IC,+2 ) is tonicity condition (2) (in particular, w k and w k w k w k = ax{w k, µ k }, as described in (7). 3 y iterating the arguent we obtain the sallest values of w k, w k,..., w k n + which satisfy (3), as described in (7). This explains the pricing of the worst n units of aong the k units to sell. Finally, regarding the pricing of the best k n units to sell, we observe that the variables in (w,..., w k n ) do not affect (3) and thus each of the can be set equal to w k n + to satisfy the onotonicity condition (2), as described in (8). In this way we have found the sallest values of w,..., w k which satisfy (2) and (3). s we entioned in section 2, the values in w are feasible only if they satisfy () since otherwise there exist no prices p > 0,..., p k > 0 such that w j = u j p j for j =,..., k. Hence, u j ust be larger than the profit-axiizing w j characterized in Proposition (ii). This is why (5) is necessary and sufficient for to be able to sell k units. Notice that Proposition (i)b iplies that there is an ˆ between 0 and n such that is able to sell any nuber of units between 0 and k ˆ, but out arguents above iply that will always sell at least k n units, if k > n Maxiizing s profit with respect to Since Proposition allows to copute π (k ) for any ˆ, the profit-axiizing can be found by coparing π (k n ), π (k n + ),..., π (k ˆ). efore seeing a few exaples and a useful property of π, we can iprove our understanding of the proble of by coparing π (k ) with π (k + ), in order to exaine the incentives of to increase his supply. Let us use here w(),..., w k () to denote D s net profits fro buying s products, as deterined by (7)-(8), when sells k products. Then we find w k () = µ k, w k () = ax{µ k, µ k },..., () = ax{µ k, µ k,..., µ k n + } = w() =... = w k n (). w k n + 3 ctually, w k (i) when in{µ k in{µ k than u +, u +2, u +2 } = u+2 ust be equal to the highest between w k } = µk, we can write w k, wk = ax{µ k and it turns out that this iplies that w k 2 and in{µ k, u +2 }, but, µ k }; (ii) when is uch saller = ax{µ k, µ k } still holds because u +2 = µ k is larger than both u +2 and µ k.

15 When instead sells k + products, we have: w k + ( ) = µ k +, w k ( ) = ax{µ k +, µ k w k n + },..., ( ) = ax{µ k +, µ k,..., µ k n + } = w( ) =... = w k n ( ). It is straightforward to see fro (4) that µ k+ > µ k+ for any { +,..., n }, () for any { +,..., k}. thus we have w k+ ( ) > w k+ The latter inequality is very intuitive: in order to sell one extra unit, (i.e. k + rather than k units). ust increase the rent it abandons to D for all the k initial units. Thus, when we copare π (k + ) = k + j= (u j w( j )) with π (k ) = k j= (u j w()), j we see that π (k + ) contains the additional ter u k + w k + ( ) > 0, which is s profit on the (k + )-th unit sold, but s profit on each of his first k units is reduced fro u j w() j to u j w( j ), as we just proved that w j ( ) > w j () for j {,..., k }. In words, as it is the case with, also faces a trade off between quantity and rent extraction: as sells ore units, it should leave ore surplus per unit to D. Precisely, as increases its sales fro k to k +, inducing to accoodate becoes ore difficult for two reasons. First, s ability to fight is now stronger since he can use his -th best unit, with value u, which was previously sold. Second, has now less to lose by trying to push out a product of, since selling products akes the profit fro accoodation (described just after Lea 2) saller than when selling. Therefore, when sells one extra unit, in order to induce not to fight, should ake his units ore copetitive by leaving D a higher surplus for each unit. We now present a result which siplifies the task of finding the optial. Precisely, we prove a concavity-like property of π which states that the arginal profit for fro selling one extra unit is decreasing: the profit increase fro selling k + 2 products instead of k + is saller than the profit increase fro selling k + products instead of k. Proposition 2 (i) Suppose that it is feasible for to sell k +2 units (i.e. 2 ˆ). Then π (k + 2) π (k + ) π (k + ) π (k ). (ii) The optial for, denoted by, is characterized as follows: π ( ) ax{π ( ), π ( + )} if k n + k ˆ, π ( ) π ( ) if = k ˆ, π ( ) π ( + ) if = k n. Notice that the concavity-like property of π described in Proposition 2(i) iplies iediately Proposition 2(ii): in order to test the optiality of, it suffices to copare the profit as the nuber of products to sell for is decreased by one unit or increased by one unit. In what follows, to give further insight, we study soe specific settings. 3

16 3.3.3 When only the local incentive constraint (IC,+ ) atters Let us present first the siple case in which only the local incentive constraint (IC,+ ) atters. We saw that when wants to sell k units, downward incentive constraint are trivially satisfied but satisfying upward constraints requires to abandon soe surplus to D. We below present a special case in which satisfying only (IC,+ ) is sufficient to satisfy (3), and this akes it straightforward to derive π (k ). Corollary Given such that ˆ n 2, if u +2 + u+ then (5) is equivalent to u k > + u+. When this condition is satisfied, (6)-(8) iply w =... = w k = + u+... µ k n + > 0 = w k + =... = w k ; thus π (k ) = u u k Precisely, if u +2 is sufficiently saller than u + and then (5) is satisfied if and only if u k+, it turns out that µ k k + u+. µ k holds for = +, > µ k+. If this condition is satisfied, then the optial prices for or equivalently u k > + u+ are such that the products he wants to sell give a constant net profit to D equal to + u+, the profit satisfying (IC,+ ) with equality. If the condition u +2 + u+ holds for every { ˆ,..., n 2}, then we have π (k + ) π (k ) = u k + u (k )( u + u+ ). Note however that the conditions u ˆ+ ˆ+ u ˆ+2, u ˆ+2 ˆ+2 u ˆ+3,..., n un u n are soewhat restrictive, since they iply that the values of s products decrease quite quickly. This also suggests that in general ore than one upward incentive constraints atter, as in the exaples below Exaple : When n = 3 Suppose that n = 3. In order to sell k 3 units, sets p = u, p 2 = u 2,..., p k 3 = u k 3, p k 2 u k 2, p k u k, p k u k. and then chooses p = u, p 2 = u 2, p 3 = u 3. Hence, π (k 3) = u + u u k 3. In order to find π (k 2) we have to consider (IC 2,3 ), which is given by (IC 2,3 ) w k 2 3 u3. Therefore, chooses p = u 3 u3, p 2 = u 2 3 u3,..., p k 2 4 = u k 2 3 u3, p k u k, p k u k.

17 which is feasible only if u k 2 > 3 u3. Then, plays p = u, p 2 = u 2. Hence, π (k 2) = u + u u k 2 k 2 3 u3. In order to find π (k ) we need to consider both (IC,2 ) and (IC,3 ), which are given by: (IC,2 ) w k 2 u2. (IC,3 ) w k 2 ax{ 2 u2, 3 (u2 + u 3 )}. Hence, satisfying the incentive constraints is feasible if u k u 3 )}. Then, chooses p j = u j ax{ 2 u2, 3 (u2 + u 3 )} for j =,..., k 2; p k = u k 2 u2, p k u k. > 2 u2 and u k 2 > ax{ 2 u2, 3 (u2 + Then π (k ) = u + u u k 2 + u k (k 2) ax{ 2 u2, 3 (u2 + u 3 )} 2 u2. Finally, is able to sell k units if and only if u k > ku, and then π (k) = u + u u k 2 + u k + u k ku. In order to fix the ideas, suppose that u 2 > 2u 3, so that ax{ 2 u2, 3 (u2 +u 3 )} = 2 u2. Then, fro Proposition 2(ii), we see for instance that it is optial for to sell k 2 products if π (k 2) ax{π (k ), π (k 3)}, which is equivalent to u k 2 k 2 3 u3 and u k k 2 2 (u2 u 3 )+ 2 u2. The first inequality iplies that the gain on the (k 2)-th sold by, u k 2 3 u3, is larger than his loss on the k 3 units, k 3 3 u3, with respect to selling the at full prices. The second inequality eans that selling the (k )-th unit yields a profit of u k u k 2 u2. 2 u2 but results in a loss of k 2 2 (u2 u 3 ), which is larger than Exaple 2: When all s products have the sae value Suppose that u = u 2 =... = u n (= +,..., n ), we find that µ k+ Given, the profit-axiizing w,..., w k u > 0. In this case, for (=,..., n ) and = u. Thus µ k+, deterined by (7)-(8), are is increasing in. w k = + u, w k = u,..., w k n +2 = n u, n = n u = w =... = w n. w k n + n If ˆ, we have that π (k ) = u +...+u k [ n + )]u n +2 n + n n (k

18 In order to find the optial, we exploit lea 2. Thus, = n is optial if π (k n ) π (k n + ), i.e. if uk n + u k n +. Finally, for between and n, is optial if π(k ) π(k ) 0 and π(k + ) π(k ), i.e. n n + k n + n uk + u and 4 undling u k u n + k n + n In this section we initially assue that each fir practices bundling. Precisely, at stage one fir chooses q of his products to include into his bundle, and a price P for. t stage two, after observing the ove of, chooses q of his products to include into his bundle and a price P for. In order to specify the value of a bundle for D, it is not enough to specify the nuber of units it contains, but it is necessary to know the precise products in the bundle. However, it is intuitive that if i inserts q i of his products in M i, it is optial for hi to choose the best q i products aong the ones he can sell. For i =,, let U i (q i ) denote the gross value of M i for D if it includes the q i best products of i: U i (q i ) = q i j= u j i. Let U (q, q ) denote D s gross profit fro buying both bundles, taking into account the capacity constraint of D; thus U (q, q ) U (q ) + U (q ), with equality if and only if q + q k. Therefore, the net profit of D fro buying only M i is U i (q i ) P i, while D s profit fro buying and is U (q, q ) P P. s we did for the gae with individual sales, we apply backwards induction starting with stage three. Clearly, D deterines his purchase by axiizing his own payoff. bout tie-breaking, we assue that D buys both bundles if U (q, q ) P P ax{u (q ) P, U (q ) P }, while he buys if U (q ) P = U (q ) P > U (q, q ) P P (this is consistent with T). t stage two, given (q, P ), wants to choose q and (the axial) P such that D decides to buy. In order to achieve this objective, can choose between two strategies as under individual sale: accoodation and fight. can try to induce D to buy both bundles or try to induce D to buy only (and block out ). Recall fro section 2 that i is the nuber of fir i s products of aong the k best products overall, thus + = k. efore starting the analysis, it is useful to introduce the function in{n, k q } q(q ) = 6 if q < if q

19 The interpretation of q(q ) is as follows: if D has purchased which includes s best q units, then q(q ) is the axial nuber of products of which D would effectively distribute under the slot constraint in case D buys as well. The next lea characterizes s optial strategy at stage 2. Lea 3 t stage two, given a pair (q, P ), (i) fights by choosing q = n and P = U (n ) U (q ) + P if P > U (q, n ) U (n ); (ii) accoodates by choosing q = q(q ) and P = U [q, q(q )] U (q ) if P U (q, n ) U (n ). Not surprisingly, ends up fighting (accoodating) when P is large (sall). Precisely, in order to fight, includes all its products into since this decreases the relative value to D of buying both bundles against buying only, and at the sae ties axiizes the value of. Then it is feasible for to block out when D s profit fro buying only is saller than D s profit fro buying only, which is equivalent to P > U (q, n ) U (n ). Suppose now that accoodates. Notice that for any q, finds it optial to induce D to buy and distribute at least his best units since q(q ). Furtherore, if q <, it is optial for to sell ore than units (if n > ) since his profit is equal to U (q, q ) U (q ). However, if q, it is optial for to sell only units since including ore than units does not affect his profit. Notice also that as long as q and accoodates, D buys and distributes at least best products of. Corollary 2 Suppose that accoodates. Then, (i) For any q, can induce D to buy and distribute at least his best units; hence can never induce D to buy and distribute ore than his best units. (ii) Suppose q. D always buys and distributes at least the best units of ; hence can never induce D to buy and distribute ore than his best units. The next proposition describes the equilibriu under bundling and shows that each fir i chooses q i = i along the equilibriu path. Proposition 3 When both firs practice bundling, there exists a unique SPNE and equilibriu strategies are as follows: (i) chooses q = and P = U (, n ) U (n ); 7

20 (ii) plays as described by Lea 3, and along the equilibriu path chooses q = and P = U (, ) U ( ) = U ( ). (iii) D buys both bundles and hence consues the k best aong both firs products. Proof. t stage one, fir will choose (q, P ) in a way which induces to accoodate (otherwise akes no profit), hence P = U (q, n ) U (n ) and q is selected in order to axiize U (q, n ). Then it follows that q = because q < iplies U (q, n ) < U (, n ), while q > iplies that soe units are included in but do not increase the profit of. Given q = and P = U (, n ) U (n ), will choose as described by Lea 3(ii); thus q = q( ) = and P = U (, ) U ( ) = U ( ). Given s best response described in Lea 3, chooses the largest P which induces to accoodate (i.e. P = U (q, n ) U (n )) and sets q =, since U (q, n ) increases with q up to q = and then becoes constant. Then, the highest P at which induces D to buy both bundles is P = U (, q ) U ( ), fro Lea 3(ii), and this is axiized at q = since U (, q ) increases with q up to q = and then becoes constant. Therefore, D ends up consuing the k best aong both firs products. 4 5 Social welfare coparison In this section, we copare the outcoe under individual sale and the one under bundling in ters of the social welfare. In our odel, social welfare is equivalent to the gross profit of D and thus it is axiized if D consues the k best products aong both firs products. In the previous section we found that in the unique SPNE under bundling, D consues precisely these products. Therefore, under bundling, we always have the socially efficient outcoe. On the contrary, under individual sale, there is no particular reason for copetition to lead to the efficient outcoe. The analysis in section 3 shows that each fir faces a trade-off between quantity and rent extraction. In our sequential gae, the profile of products effectively consued by D is deterined by the first over, fir. However, there is no reason that the trade-off for fir induces hi to sell nuber of products. lthough we copletely characterized each fir s strategy in equilibriu, having a general characterization of the condition under which copetition under individual sale leads to the efficient outcoe is essy since it depends on the vectors (u,..., u n ) and (u,..., u n ). 4 higher P would ake D buy only M. 8

21 In order to provide the intuition for why the copetition under individual sale does not necessarily leads to the efficient allocation, we below provide two siple exaples. Let i denote the nuber of products sold by fir i under individual sale. Obviously, we have + = k. In what follows, we give an exaple of < and another exaple of >. Exaple of < Suppose n = and u k > u, so that = 0. We have π (k ) = U (k ); π (k) = U (k ) + u k ku. Therefore, π (k ) π (k) > 0 if and only if u k < ku. Hence, if u k < ku, we have = k < = k. The intuition for this result is siple. If sells only k = k n products, he can extract full surplus fro his k best products since will not fight. However, if chooses to sell all k products, it has to leave D a net surplus equal to u for each of his product in order to block out s product. This trade-off between quantity and rent extraction akes sell only his k best products when u is not too saller than u k. Exaple of > Consider the setting of exaple 2 in section 3: we have u >... > u > u > u +... > u k, u =... = u = u. Suppose = n. We have: π (k ) = U (k ); > Therefore, π (k + ) = U (k ) + u + (k + ) +. u π (k + ) π (k ) > 0 iff u + > (k + ) u. + For instance, if u + is close to u, we have π (k +) π (k ) > 0 if > k/2. To sharpen the intuition, suppose = 0, = k 2, u u Then, π (k ) = 0. Hence, has to sell at least one product to generate a profit. Suppose that charges p = ε(> 0) very sall and very high prices on the other products. If accoodates s product, s profit is (k )u. Instead, if blocks s product out, s profit is 9

22 ku k(u ε) kε. Therefore, prefers accoodation and hence can sell his inferior product. This exaple is syetric to the previous exaple: takes advantage of s trade-off between quantity and rent extraction in order to sell his inferior product. The next proposition suarizes the ain finding in ters of social welfare coparison: Proposition 4 (social welfare) (i) Under bundling, the outcoe is socially efficient: D always consues the k best products aong both firs products. (ii) Under individual sale, it is not necessarily the case: D can consue soe products which are not the k best either fro or fro. (iii) Therefore, social welfare is higher under bundling than under individual sale. The intuition for the result in Proposition 4 is that under individual sale, fir i faces not only copetition fro fir j( i) s products but also copetition aong its own products while under bundling there is no copetition aong its own products. This iplies that the trade-off between quantity and rent extraction which creates the inefficiency under individual sale does not exist under bundling. ctually, the price that fir i can coand for its bundle weakly increases as it includes ore products. Furtherore, this price strictly increases as long as the additional product that is included into the bundle belongs to the best k products aong all products included in both firs bundles. This is why fir i includes exactly his best products into the bundle and D consues the k best products by purchasing both bundles. 6 Incentives to bundle We have exained above the two different regies of no bundling and bundling. Now we inquire which regie will endogenously eerge when each seller can choose between bundling and no bundling. In short, we find that bundling is weakly doinant for fir and, given that bundles, also for it is weakly doinant to practice bundling. Proposition 5 (i) If fir can ake a profit by pricing his products independently, then can ake at least the sae profit by bundling. (ii) Given that chooses to bundle, if fir can ake a profit by pricing his products independently, then can ake at least the sae profit by bundling. While Proposition 5(i) suggests that never loses fro bundling Proposition 5(ii) establishes the sae result given that bundles, as established by Proposition 5(i). Thus, bundling eerges endogenously when it is not forbidden. In order to iprove our understanding of this fact, it ight be useful to exaine the benefits of fro bundling 20

23 when uses individual sales. In the case in which also uses individual sales and wants to sell products (this objective is attainable if and only if u > w k + ) we know fro Lea 2 that his profit is u u w k +. On the other hand, we show in the proof of Proposition 5 that he can ake U () (w k + Therefore, with individual sales he leaves D a net profit equal to w k + the units he sells, 5 for a cobined value of w k + to leave to D the net value of the worst products of, w k + +w k w) k by bundling. on each unit of. With bundling, instead, he needs +w k w, k which. The reason for the result is that with individual sales, is (weakly) saller than w k + D can replace each single product of with the k + -th product of if the product of yields D a profit saller than w k +. On the other hand, D has less flexibility when bundles as he and can substitute only with the worst units of. This gives an edge to and allows hi to extract a (weakly) higher price fro D. 7 Robustness In this section, we show that the efficiency property of bundling is robust in various settings. 7. Exclusive contracts In all previous sections, after buying a nuber of products, D has the freedo to choose the products to occupy the slots. In this subsection, we allow firs to sign exclusive contracts on the use of slots such that if D buys q i nuber of products fro fir i, i =,, D should allocate exclusively q i nuber of slots on i s products. Introducing exclusive contracts does not affect the analysis of individual sale since under individual sale, D buys only the products that he will effectively distribute. However, introducing exclusive contracts ight affect the analysis of bundling. For instance, we know fro corollary 2 that without exclusive contracts, fir i can never induce D to distribute ore than i units. However, if firs can sign exclusive contracts, for instance, fir can force D to distribute ore than units. Then, the question is to know whether it is profitable for to do that. The next proposition shows that the equilibriu outcoe under bundling is the sae regardless of whether firs use exclusive contracts or not. 6 Proposition 6 Under exclusive contracts, there exists a unique SPNE and equilibriu strategies are as follows, in which ˆq (q ) in{k q, n }: 5 w k + is the value for D of the best product of he does not purchase. 6 Notice however that s best response is different fro the one described in lea 3. 2