Multiproduct Search. Jidong Zhou y. August Abstract

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1 Multiroduct Search Jidong hou y August 202 Abstract This aer resents a sequential search model where consumers look for several roducts from multiroduct rms. In a multiroduct search market, both consumer behavior and rm behavior are di erent from the single-roduct case. For examle, a consumer may return to reviously visited rms before running out of otions, and rices can decrease with search costs. The framework is extended by allowing rms to use bundling strategies. Bundling tends to reduce the intensity of consumer search, which can soften cometition and reverse the usual welfare assessment of cometitive bundling in a erfect information setting. Alications to countercyclical ricing, loss leader ricing, and endogenous retail market structure are also discussed. Keywords: consumer search, oligooly, multiroduct ricing, countercyclical ricing, bundling JEL classi cation: D, D43, D83, L3 I am grateful for their helful comments to Mark Armstrong, Heski Bar-Isaac, Simon Board, Jan Eeckhout, Antonio Guarino, Johannes Horner, Ste en Huck, Philie Jehiel, Preston McAfee, Eric Rasmusen, Regis Renault, Andrew Rhodes, John Riley, Jean Tirole, Michael Whinston, Larry White, Chris Wilson, and seminar articiants in Berkeley, Cambridge, Chicago Booth, ECARES in Brussels, Essex, Indiana Kelley, Michigan, NYU Stern, UPenn, Rochester Simon, TSE, UCL, UCLA, the CESifo Conference in Alied Microeconomics, the Second Worksho on Search and Switching Costs in Groningen, and the Tenth Annual Columbia-Duke-Northwestern IO Theory Conference. y Deartment of Economics, NYU Stern School of Business, 44 West Fourth Street, New York, NY jidong.zhou@stern.nyu.edu.

2 Introduction Consumers often look for several roducts on a given shoing tri. For examle, during ordinary grocery shoing they buy food, drinks and household roducts; in high street shoing they urchase clothes, shoes and other goods; in the Christmas season they look for several resents. Sometimes a consumer seeks electronic combinations such as comuter, rinter and scanner; when furnishing a house they need several furniture items; when going on holiday or attending a conference they book both ights and hotels; and new arents look for many baby roducts. On the other side of the market, there are many multiroduct rms such as suermarkets, deartment stores, electronic retailers, and travel agencies which often suly most of the roducts a consumer is searching for in a articular shoing tri. Usually the shoing rocess also involves non-negligible search costs. Consumers need to reach the store, nd out each roduct s rice and how suitable they are, and then may decide to visit another store in ursuit of better deals. In e ect, in many cases a consumer chooses to sho for several goods together to save on search costs. Desite the ubiquity of multiroduct search and multiroduct rms, the search literature has been largely concerned with single-roduct search markets. In art, this is because a multiroduct search model is less tractable than a single-roduct one. In this aer, I develo a tractable model for multiroduct search markets, and show that a multiroduct search market exhibits some qualitatively di erent roerties comared to the single-roduct case. I then argue that multiroduct search has rich market imlications, and the develoed framework can be used to address interesting economic issues such as countercyclical ricing, the welfare imact of cometitive bundling, loss leader ricing, and endogenous retail market structure. The basic framework of this aer is a sequential search model in which consumers look for several roducts and care about both rice and roduct suitability. Each rm sulies all relevant roducts, but each roduct is horizontally di erentiated across rms. By incurring a search cost, a consumer can visit a rm and learn all roduct and rice information. In articular, the cost of search is incurred jointly for all roducts (i.e., there are economies of scale in search), and the consumer does not need to buy all roducts from the same rm (i.e., they can mix and match after samling at least two rms if rms allow them to do so). Both features are realistic and imortant in multiroduct search markets. In the basic model, I assume that rms use linear ricing strategies (i.e., set searate rices for each roduct). A distinctive feature of consumer behavior in multiroduct search is that a consumer may return to buy from reviously visited rms before running out of otions. By contrast, in a standard single-roduct sequential search model, a consumer never returns to earlier rms before samling all rms. As far as ricing is concerned, with multiroduct consumer search, if a rm lowers one roduct s rice, this will induce more consumers who are visiting it to terminate search and buy some other roducts as well. That is, a reduction of one roduct s rice also boosts the demand 2

3 for the rm s other roducts. I term this the joint search e ect. As a result, even hysically indeendent roducts are riced like comlements. Due to the joint search e ect, rices can decline with search costs in a multiroduct search market. When search costs increase, the standard e ect is that consumers will become more reluctant to sho around, which will induce rms to raise their rices. However, in a multiroduct search market, higher search costs can also strengthen the joint search e ect and make the roducts in each rm more like comlements, which will induce rms to lower their rices. When the latter e ect dominates rices will fall with search costs. Another rediction is that rms in a multiroduct search environment tend to set lower rices than in a single-roduct search environment. This is for two reasons: rst, due to economies of scale in search, consumers on average samle more rms in the multiroduct search case than in the single-roduct search case, which tends to increase each roduct s own-rice elasticity; second, multiroduct search causes the joint search e ect, which gives rise to the comlementary ricing henomenon and so increases roducts cross-rice elasticities. This observation can rovide a ossible exlanation for the henomenon of countercyclical ricing rices of many retail roducts dro during high-demand eriods such as weekends and holidays. During high-demand eriods, it is often the case that a higher roortion of consumers become multiroduct searchers (e.g., many households conduct their weekly grocery shoing during weekends), and so retailers have incentives to reduce their rices. In multiroduct markets, bundling is a widely observed ricing strategy. Bundling is often used as a rice discrimination or entry deterrence device. 2 This aer argues that in a search environment, bundling has a new function: it can discourage consumers from exloring rivals deals. This is because bundling reduces the anticiated bene t from mixing-and-matching after visiting another rm. This search-discouraging e ect works against the tyical ro-cometitive e ect of cometitive bundling in a erfect information scenario. 3 When search costs are relatively high the new e ect can be such that bundling bene ts rms and harms consumers. 4 Therefore, our ndings indicate that assuming away information frictions may signi cantly distort the welfare See relevant emirical evidence documented in, for examle, Warner and Barsky (995), MacDonald (2000), and Chevalier, Kashya, and Rossi (2003). 2 See, for instance, Adams and Yellen (976), and McAfee, McMillan, and Whinston (989) for the view of rice discrimination, and Whinston (990) for the view of entry deterrence. 3 Matutes and Regibeau (988), Economides (989), and Nalebu (2000) studied cometitive ure bundling, and Matutes and Regibeau (992), Anderson and Leruth (993), Thanassoulis (2007), and Armstrong and Vickers (200) studied cometitive mixed bundling. The main insight emerging from all these works is that comared to linear ricing, bundling (whether ure or mixed) has a tendency to intensify rice cometition, and under the assumtions of unit demand and full market coverage (which are also retained in this aer) it tyically reduces rm ro ts and boosts consumer welfare. 4 In di erent settings, Carbajo, de Meza and Seidmann (990) and Chen (997) argue that (asymmetric) bundling can create vertical roduct di erentiation between rms, thereby softening rice cometition. 3

4 assessment of bundling. 5 Related literature. Since the seminal work by Stigler (96), there has been a vast literature on search, but most aers focus on single object search (see, for examle, Diamond, 97, Burdett and Judd, 983, and Stahl, 989 for consumer search models). There is a small branch of literature that investigates the otimal stoing rule in multiroduct search (Burdett and Malueg, 98, Carlson and McAfee, 984, and Gatti, 999). 6 They have emhasized the similarity between single-roduct and multiroduct search in the sense that in both cases the stoing rule often features the static reservation roerty. However, I argue that desite this similarity, consumer search behavior still exhibits substantial di erences between the two cases. More imortantly, the above works do not consider an active suly side. McAfee (995) studies a multiroduct search model with endogenous rices. It extends Burdett and Judd (983) to the multiroduct case. Each roduct is homogenous across stores, and by incurring a search cost a consumer can learn rice information from a random number of stores. In articular, there are consumers who learn information from only one store. As a result, similar to Burdett and Judd (983), rms adot mixed ricing strategies, re ecting the trade-o between exloiting less informed consumers and cometing for more informed consumers. However, multiroduct search generates multile tyes of (symmetric) equilibria. In articular, there is a continuum of equilibria in which rms randomize rices on the reservation frontier such that one roduct s rice decline must be associated with the rise of some other rices. 7 The model o ers interesting insights, but both the multilicity of equilibria and the comlication of equilibrium characterization restrict its alicability. This aer develos an alternative multiroduct search framework with di erentiated roducts, where the symmetric equilibrium is unique and rices are deterministic. Lal and Matutes (994) also resent a multiroduct search model where two rms locate at the two ends of a Hotelling city and each roduct is otherwise homogenous across rms. Each consumer needs to ay a location-seci c cost to reach rms and discover the rice information. Their setting is, however, subject to the Diamond aradox: each rm charges the monooly rices and no consumers articiate in the 5 The Euroean Commission has recently branded all bundled nancial roducts as anti-cometitive and unfair. One of the main reasons is that the ractice reduces consumer mobility. See the consultation document On the Study of Tying and Other Potentially Unfair Commercial Practices in the Retail Financial Service Section, In Burdett and Malueg (98) and Carlson and McAfee (984), consumers search for serval roducts among a large number of multiroduct stores that suly homogenous roducts and set random rices according to an exogenous rice distribution. The former mainly deals with the case with free recall, and the latter deals with the case with no recall. Gatti (999) considers a more general setting in which consumers search for rices to maximize a general indirect utility function. 7 In the other tye of equilibria, rms randomize rices over the accetance set (not just on its border). They are, however, qualitatively similar to the single-roduct equilibrium in the sense that the marginal rice distribution for each roduct is the same as in the single-roduct search case, and so is the ro t from each roduct. 4

5 market. Lal and Matutes show that rms can avoid the market collase by advertising rices of a subset of roducts. In one tye of equilibrium, each rm advertises a low rice (even below marginal cost) of one roduct to ersuade consumers to visit the store (i.e., loss leading occurs), but charges the monooly rice for the other unadvertised roduct. In equilibrium, consumers do not search beyond the rst visited rm and so two-sto shoing never haens. In my model with roduct-level di erentiation, consumers search for both low rices and high roduct suitability, and an equilibrium with an active market exists even without advertising and two-sto shoing takes lace as we often observe in the real market. 8 In terms of the modelling aroach, this aer is built on the single-roduct search model with di erentiated roducts (Weitzman, 979, Wolinsky, 986, and Anderson and Renault, 999). Recently, this framework has been adoted to study various economic issues. 9 Comared to the homogeneous roduct search model, models with roduct di erentiation often better re ect consumer behavior in markets that are tyically characterized by nonstandardized roducts. Moreover, they avoid the well-known modelling di culty suggested by Diamond (97), who shows that with homogeneous roducts and ositive search costs (no matter how small) all rms will charge a monooly rice and no consumers search beyond the rst samled rm. In search models with roduct di erentiation, there are some consumers who are ill-matched with their initial choice of sulier and then search further, so that the ro-cometitive bene t of actual search is resent. The rest of the aer is organized as follows: Section 2 resents the basic model with linear ricing and analyzes consumer search behavior. Section 3 characterizes equilibrium linear rices and conducts comarative statics analysis, and an alication to countercyclical ricing is also discussed. Section 4 studies bundling in a search market and examines its welfare imact relative to linear ricing. Section 5 concludes, and discusses other alications to loss leader ricing and endogenous retail market structure. Omitted details are resented in the Aendix. 8 Ellison (2005) uses Lal and Matutes s framework to study add-on ricing by assuming that the base roduct s rice information is erfect while the add-on s rice information is not. (In the end of this aer, I discuss a related variant in which consumers only need to search for one roduct s information.) Shelegia (202) studies a multiroduct version of Varian (980) in which for exogenous reasons one grou of consumers visits only one store while the other visits two. Rhodes (20) rooses a multiroduct monooly model in which each consumer knows her rivate valuations for all roducts but needs to incur a cost to reach the rm and learn rices. He shows that selling multile roducts can solve the Diamond hold-u roblem, which would unravel the market in a single-roduct case with inelastic consumer demand. 9 See, for instance, Armstrong, Vickers, and hou (2009) and hou (20) for the market imlications of rominence and non-random consumer search, Bar-Isaac, Caruana, and Cunat (202) for how the decline of search costs a ects roduct design, and Haan and Moraga-González (20) for attentiongrabbing advertising. 5

6 2 A Model of Multiroduct Search There are a large number of consumers in the market with measure normalized to one. Each consumer is looking for two roducts (e.g., two furniture items, or clothes and shoes), and they have unit demand for each roduct. There are n 2 multiroduct rms in the market, each sulying both roducts at a constant marginal cost, which is normalized to zero. Each roduct is horizontally di erentiated across rms. For examle, di erent rms may suly di erent brands of furniture or clothes and shoes with di erent styles, and consumers have idiosyncratic tastes. Seci cally, a consumer s valuations for the two roducts in each rm are randomly drawn from a common joint cumulative distribution function F (u ; u 2 ) de ned on [u ; u ] [u 2 ; u 2 ] which has a continuous density f(u ; u 2 ). The match utilities are realized indeendently across rms and consumers (but a consumer may have correlated match utilities for the two roducts in the same rm). For simlicity, I assume that the two roducts are neither comlements nor substitutes, in the sense that a consumer obtains an additive utility u +u 2 if roduct i has a match utility u i, i = ; 2. Let F i (u i ) and H i (u i ju j ) denote the marginal and conditional distribution functions; f i (u i ) and h i (u i ju j ) denote the marginal and conditional densities. I assume that all consumers buy both roducts in equilibrium, i.e., the market is fully covered. 0 (This is the case, for examle, when consumers have no outside otions or when they have large basic valuations for each roduct on to of the above match utilities.) Consumers do not need to urchase both roducts from the same rm. This ossibility of multi-sto shoing is realistic and also imortant for our model. Otherwise, the model would degenerate to a single-roduct one with a comosite roduct with match utility u +u 2. In the basic model, rms use linear ricing strategies and charge a searate rice for each roduct. Initially consumers are assumed to have imerfect information about the (actual) rices rms are charging and match utilities of all roducts. But they can gather information through a sequential search rocess: by incurring a search cost s 0, a consumer can visit a rm and nd out both of its rices ( ; 2 ) and both of its match utilities (u ; u 2 ). At each rm (excet the last one), the consumer faces the following otions: sto searching and buy both roducts (maybe from rms visited earlier), or buy one roduct and kee searching for the other, or kee searching for both roducts. The cost of search is assumed to be indeendent of the number of roducts a consumer is looking for, which re ects economies of scale in search. Finally, I suose that consumers have free recall/return, i.e., there are no extra costs in buying roducts from 0 The assumtion of full market coverage is often adoted in oligooly models. Our main insights carry over to the case without this assumtion (though the analysis will become more involved). If consumers urchase roducts frequently, they may know both rice and roduct information before search. However, in reality both rices and roduct variants in many retailers change over time, such that imerfect information might be still a lausible resumtion. 6

7 a reviously visited store. The timing is as follows: Firms set rices simultaneously rst, and then consumers start to search without observing rms actual rices (though they hold the equilibrium belief about rms ricing strategy). After visiting each rm, consumers decide whether to sto searching or not. Both consumers and rms are assumed to be risk neutral. I focus on symmetric equilibria in which rms set the same rices and consumers samle rms in a random order (and without relacement). 2 The equilibrium concet I use is the erfect Bayesian equilibrium. Each rm sets its rices to maximize ro ts, given its exectation of consumers search behavior and other rms ricing strategy. Consumers search otimally, to maximize their exected surlus, given the match utility distribution and their rational beliefs about rms ricing strategy. At each rm, even after observing o -equilibrium o ers, consumers hold the equilibrium belief about the unsamled rms rices. 3 I have made several simlifying assumtions to make the model tractable, and it is useful to discuss them at this oint. Economies of scale in search. The assumtion that the search cost is indeendent of the number of roducts a consumer is seeking is an aroximation when the search cost is mainly for learning the existence of a seller or for reaching the store. In the other olar case where the cost of search is solely from roduct insection and so totally divisible among roducts, the multiroduct search roblem degenerates to two searate single-roduct search roblems. 4 In reality, most situations are in between (a tyical shoing rocess involves a xed cost for reaching the store and also variable in-store search costs for insecting each roduct). Our simli cation is made both for analytical convenience and for highlighting the di erence between multiroduct and single-roduct search. Free recall. Free recall is often assumed in the consumer search literature. It could be aroriate, for instance, when a consumer can hone reviously visited rms (e.g., furniture stores) to order the roducts she decides to buy, or when shoing online a consumer can leave the browsed websites oen. In most consumer markets, however, there are ositive returning costs. I choose to assume free recall both for tractability, and for facilitating comarison with the single-roduct search model in Wolinsky (986) 2 As usual in search models, there exists an uninteresting equilibrium where consumers exect all rms to set very high rices and do not articiate in the market at all, and so rms have no incentive to reduce their rices. I do not consider this equilibrium further. The issue of ossible asymmetric equlibria will be discussed later. 3 Notice that in our setting there are no correlated economic shocks (e.g., aggregate cost shocks) across rms and so their ricing decisions are indeendent of each other. 4 The case with divisible insection costs among roducts will be non-trivial if consumers have to one-sto sho (e.g., due to bundling). After insecting a roduct at one rm, a consumer needs to decide whether to continue to insect the other roduct at the same rm or to insect roducts in other rms. (This case is also equivalent to a search roblem with multi-attribute roducts and searable insection costs among attributes.) 7

8 and Anderson and Renault (999) (both of which assume free recall). Indeendent roducts. In reality the roducts that a consumer is seeking in a articular shoing tri are rarely indeendent. In many circumstances (e.g., when shoing for clothes and shoes), they are more or less intrinsic comlements in the sense that a higher valuation for one roduct increases the consumer s willingness to ay for the other (e.g., the utility function takes the form of u + u 2 + u u 2 with > 0). As I discuss later, considering comlementary valuations of this kind will comlicate analysis but not generate imortant new insights. In addition, given the assumtion of full market coverage, the two roducts in my model can also be regarded as erfect comlements in the sense that consumers can obtain ositive utility only by consuming both of them together. 2. The otimal stoing rule I rst derive the otimal stoing rule (which has been roved in Burdett and Malueg, 98, or Gatti, 999 in a rice search scenario). The rst observation is that given the indivisible search cost and free recall, a consumer will never buy one roduct rst and kee searching for the other. Hence, at any store (excet the last one) the consumer faces only two otions: sto searching and buy both roducts (one of which may be from a rm visited earlier), or kee searching for both. Denote by i (x) ui x (u i x)df i (u i ) = ui x [ F i (u i )]du i () the exected incremental bene t from samling one more roduct i when the maximum utility of roduct i so far is x and all roduct i have the same rice. (The second equality is from integration by arts.) Then the otimal stoing rule when all rms charge the same rices is as follows: Lemma Suose rices are linear and symmetric across rms. Suose the maximum match utility of roduct i observed so far is z i and there are rms left unsamled. Then a consumer will sto searching if and only if (z ) + 2 (z 2 ) s : (2) The left-hand side of (2) is the exected bene t from samling one more rm given the air of maximum utilities so far is (z ; z 2 ), and the right-hand side is the search cost. This stoing rule seems myoic at the rst glance, but it is indeed sequentially rational. It can be understood by backward induction. When in the enultimate rm, it is clear that (2) gives the otimal stoing rule because given (z ; z 2 ) the exected bene t from samling the last rm is E[max (0; u z )+max (0; u 2 z 2 )], which equals the left-hand side of (2). (Note that I did not assume u and u 2 are indeendently distributed. The searability of the incremental bene t in (2) is because of the additive utility function and the linearity of the exectation oerator.) Now ste back and 8

9 consider the situation when the consumer is at the rm before that. If (2) is violated, samling one more rm is always desirable. By contrast, if (2) holds and if the consumer continues to search, the udated maximum match utility air will be no worse than (z ; z 2 ) no matter what she will nd at the next rm, and so she will sto searching there. This imlies that if (2) holds, the bene t from keeing searching is the same as samling just one more rm. Execting that, the consumer should actually cease her search now. Figure below illustrates the otimal stoing rule. A is the set of (z ; z 2 ) which satis es (2) and let us refer to it as the accetance set. Then when there are no rice di erences across rms, a consumer will sto searching if and only if the maximum utility air so far lies within A. Let B be the comlement of A. z 2 a z 2 = (z ) A z r rz _ u B ru a 2 z Figure : The otimal stoing rule in multiroduct search De ne the border of A as z 2 = (z ) (i.e., (z ; (z )) satis es (2) with equality) and call it the reservation frontier. The reservation frontier is decreasing and convex. From the de nition of (), one can check that 0 (z ) = F (z ) F 2 ((z )) < 0 ; and so () is a decreasing function. Then it is also easy to see that 0 (z ) increases with z, i.e., () is convex. 5 5 If we consider two intrinsic comlements, the reservation frontier may no longer be decreasing. For examle, when the utility function takes the form of u + u 2 + u u 2 with > 0, one can check that in the duooly case, for instance, the reservation frontier satis es 2 ( u ) ( u 2) 2 + (u u 2 ) 2 + ( u u 2 ) 2 = s ; 4 and it is not monotonically decreasing. This is because now nding a better matched roduct may strictly increase a consumer s incentive to nd a better matched roduct 2. When there are more than 9

10 Notice that a i on Figure is just the reservation utility level when the consumer is only searching for roduct i. (If all roduct i have the same rice, a consumer will sto searching if and only if the maximum utility so far is no less than a i.) It solves i (a i ) = s ; (3) and satis es (a ) = u 2 and (u ) = a 2. This is because when the maximum ossible utility of one roduct has been achieved, the consumer will behave as if she is only searching for the other roduct. Search behavior comarison. It is useful to comare consumer search behavior between single-roduct and multiroduct search. The early literature has emhasized that in both cases (given additive utilities in the multiroduct case) the otimal stoing rule ossesses the static reservation roerty. Desite this similarity, consumers search behavior exhibits some imortant di erences between the two cases, which have not been discussed before. In single-roduct search with erfect recall, the stoing rule is characterized by a reservation utility a. When a consumer is already at some rm (excet the last one), she will sto searching if and only if the current roduct has a utility greater than a. Previous o ers are irrelevant because they must be worse than a (otherwise the consumer would not have come to this rm). As a result, a consumer never returns to reviously visited rms until she nishes samling all rms. In articular, if there are an in nite number of rms, the consumer never exercises the recall otion. However, in multiroduct search, a consumer s stoing rule deends on both the current rm s o er u and the best o er so far z. This can be seen from the examle indicated in Figure, where the current o er u lies outside the accetance set A but the consumer will sto searching because z _ u 2 A (where _ denotes the join of two vectors). When she stos searching, she will go back to some revious rm to buy roduct 2. This has two imlications for the demand analysis. First, with multiroduct search, a rm s rice adjustment will not only a ect a consumer s search decision at this rm, but will also a ect her search decisions at subsequent rms if she leaves this rm. Second, a consumer often returns to a reviously visited rm to buy one roduct even if there are rms left unsamled. This is true even if there are an in nite number of rms. These di erences will comlicate the demand analysis in multiroduct search. Moreover, unlike the single-roduct search case, considering an in nite number of rms does not achieve any simlicity. In e ect, with multiroduct search, the simlest case is when there are only two rms. Hence, in the following analysis, I mainly deal with the duooly case. As I will discuss in section 5., such a simli cation does not lose the most imortant insights concerning rm ricing in a multiroduct search market. (A detailed analysis of the general case with more than two rms is rovided in the online two rms, considering intrinsic comlements will even render the otimal stoing rule non-stationary (see Gatti, 999, for a related discussion). 0

11 sulementary document at htts://sites.google.com/site/jidongzhou77/research.) 3 Equilibrium Prices 3. The single-roduct benchmark To facilitate comarison, I rst reort some results from the single-roduct search model (see Wolinsky, 986 and Anderson and Renault, 999 for an analysis with n rms). Suose the roduct in question is roduct i, and the unit search cost is still s. Then the reservation utility level is a i de ned in (3), and it decreases with s (i.e., a higher search cost will make consumers less willing to search on). In the following analysis, I mainly focus on the case with a relatively small search cost: s < i (u i ), a i > u i for both i = ; 2 : (4) (Remember that i (u i ) is the exected bene t from samling another roduct i when the current one has the lowest ossible match utility.) This condition ensures an active search market even in the single-roduct case. The symmetric equilibrium rice 0 i in the duooly case is then determined by 0 i = f i (a i )[ F i (a i )] + 2 ai {z } 0 u i f i (u) 2 du : (5) It follows that 0 i increases with the search cost s (or decreases with a i ) if f i (a i ) 2 + f 0 i(a i )[ F i (a i )] 0 : This condition is equivalent to an increasing hazard rate f i =( F i ). Then we have the following result (Anderson and Renault, 999 have shown this result for an arbitrary number of rms): Proosition Suose the consumer is only searching for roduct i and the search cost condition (4) holds. Then the equilibrium rice de ned in (5) increases with search costs if the match utility has an increasing hazard rate f i =( F i ). 3.2 Equilibrium rices in multiroduct search I now turn to the multiroduct search case. Let ( ; 2 ) be the symmetric equilibrium rices. For notational convenience, let (u ; u 2 ) be the match utilities of rm I, the rm in question, and (v ; v 2 ) be the match utilities of rm II, the rival rm. In the symmetric equilibrium, for a consumer who samles rm I rst, her reservation frontier u 2 = (u ) is determined by (u ) + 2 ((u )) = s ; (6)

12 which simly says that the exected bene t of samling rm II is equal to the search cost. Note that (u ) is only de ned for u 2 [a ; u ] (see Figure 2 below). But for convenience, let us extend its domain to all ossible values of u, and stiulate (u ) > u 2 for u < a. Instead of writing down the demand functions and deriving the rst-order conditions for the equilibrium rices directly, I use the following economically more illuminating method. Starting from an equilibrium, suose rm I unilaterally decreases 2 by a small ". How does this adjustment a ect rm I s ro ts? Let us focus on the rst-order e ects. First, rm I su ers a loss from those consumers who only buy roduct 2 from it because they are now aying less. Since in equilibrium half of the consumers buy roduct 2 from rm I (remember the assumtion of full market coverage), this loss is "=2. Second, rm I gains from boosted demand: (i) For those consumers who visit rm I rst, they will be more likely to sto searching since they hold equilibrium beliefs that the second rm is charging the equilibrium rices. Once they sto searching, they will buy both roducts from rm I immediately. (ii) For those consumers who eventually samle both rms, they will be more likely to buy roduct 2 from rm I due to the rice reduction. In equilibrium, the loss and gain should be such that rm I has no incentive to deviate, which generates the rst-order condition for 2. Now let us analyze in detail the two ( rst-order) gains from the roosed small rice reduction. The rst gain is from the e ect of the rice reduction on consumers search decisions. How many consumers who samle rm I rst will sto searching because of the rice reduction? (Note that the consumers who samle rm II rst hold equilibrium beliefs and so their stoing decisions remain unchanged.) Denote by (u j") the new reservation frontier. Since reducing 2 by " is equivalent to increasing u 2 by ", (u j") solves (u ) + 2 ((u j") + ") = s ; so (u j") = (u ) " according to the de nition of (). That is, the reservation frontier moves downward everywhere by ", and the stoing region A exands (i.e., more consumers buy immediately at rm I) as illustrated in Figure 2 below. 6 For a small ", the number of consumers who originally continued to search but now cease searching and buy immediately at rm I (i.e., the robability measure of the shaded area between (u ) and (u j") in Figure 2) is " u f(u; (u))du : (7) 2 a (Remember that half of the consumers samle rm I rst. The integral term is the line integral along the reservation frontier in the u dimension.) 6 More recisely, (a j") = u 2 " and so the reservation frontier has a small vertical segment at u = a. But this does not a ect our analysis as " is small. 2

13 u 2 a B(") A(") u 2 = (u ) u 2 = (u j") a 2 a 2 " u Figure 2: Price deviation and the stoing rule What is rm I s net bene t from these marginal consumers? Realize that these marginal consumers now buy both roducts from rm I for sure, while before the rice deviation they only bought each roduct from rm I with some robability less than one (i.e., when they search on but nd worse roducts at rm II). To be seci c, consider a marginal consumer on the reservation frontier with match utilities (u ; (u )). If she chooses to samle rm II, she will nd a worse roduct at rm II (i.e., v < u ) with robability F (u ), in which case she will return to rm I and buy its roduct. Similarly, if she continues to samle rm II, she will nd a worse roduct 2 at rm II (i.e., v 2 < (u )) with robability F 2 ((u )), in which case she will return to rm I and buy its roduct 2. Hence, the net bene t from inducing this marginal consumer to cease searching is [ F (u )] + 2 [ F 2 ((u ))]. We then sum this bene t over all marginal consumers on the reservation frontier. By using (7), this total bene t is " u f [ F (u)] + 2 [ F 2 ((u))]g f(u; (u))du : (8) 2 a The second gain is from those consumers who samle both rms. They will now more likely buy roduct 2 from rm I due to the rice reduction. Consider rst a consumer who visits rm I rst and nds match utilities (u ; u 2 ) 2 B("). She will then continue to visit rm II, but will return to rm I and buy its roduct 2 if v 2 < u 2 + ". The robability of that event is F 2 (u 2 + ") F 2 (u 2 ) + "f 2 (u 2 ). So the small rice adjustment increases the robability that this consumer buys roduct 2 from rm I by "f 2 (u 2 ). Then the total increased robability from all such consumers is R " f R 2 B(") 2(u 2 )df (u) " f 2 B 2(u 2 )df (u). (Since B(") converges to B as "! 0, we can discard all higher order e ects.) Similarly, one can show that the increased robability that those consumers who samle rm II rst and then come to rm I buy roduct 2 3

14 at rm I is " 2 R f B 2(v 2 )df (v). Adding them together gives us the second gain, which is 2 " f 2 (u 2 )df (u) : (9) B In equilibrium, the ( rst-order) loss "=2 from the small rice reduction should be equal to the sum of the two ( rst-order) gains in (8) and (9). This yields the rst-order condition for 2 : u = 2 2 f 2 (u 2 )df (u) + 2 [ F 2 ((u))]f(u; (u))du (0) B a {z } standard e ect + u a {z } [ F (u)]f(u; (u))du joint search e ect The rst two terms on the right-hand side cature the standard e ect of a roduct s rice adjustment on its own demand: reducing 2 increases demand for roduct 2. (This is similar to the right-hand side of (5) in the single-roduct search case.) The last term, however, catures a new feature of the multiroduct search model: when rm I reduces its 2, more consumers who samle it rst will sto searching and buy both roducts, which increases the demand for its roduct as well. This makes the two roducts sulied by the same rm like comlements even if they are hysically indeendent. 7 This e ect occurs because each consumer is searching for two roducts and the cost of search is incurred jointly for them, and so I refer to it as the joint search e ect henceforth. Also notice that the size of the joint search e ect (which determines the degree of comlementarity between the two roducts in each rm) relies on the mass of marginal consumers on the reservation frontier, i.e., (7). It deends not only on the density function f but also on the length of the reservation frontier as indicated in Figure 2. For examle, in the uniform distribution case, when the search cost increases, the reservation frontier becomes longer such that the mass of marginal consumers rises and thus the two roducts in each rm become more like comlements. As we shall see below, this observation lays an imortant role in rms ricing decisions. Similarly, one can derive the rst-order condition for : u2 = 2 f (u )df (u) + [ F ( (u))]f( (u); u)du () B a u2 a 2 [ F 2 (u)]f( (u); u)du ; 7 Notice that the comlementarity caused by the joint search cost is di erent from intrinsic comlementarity. If information is erfect and the two roducts are intrinsic comlements, then reducing the rice of a rm s one roduct will not in uence consumers decisions of where to buy the other roduct. Hence, considering a erfect information setting with intrinsic comlements cannot reroduce the main results in this aer. 4 :

15 where is the inverse function of. The rst two terms on the right-hand side re ect the standard e ect of adjusting rice, and the last term catures the joint search e ect. I summarize the results in the following lemma: 8 Lemma 2 Under the search cost condition (4), the rst-order conditions for and 2 to be the equilibrium rices are given in (0) and (). Both (0) and () are linear equations in rices, and the system of the two rices has a unique solution. Thus, the symmetric equilibrium, if it is characterized by the rst-order conditions, will be unique. Notice that if rms ignored the joint search e ect, then the ricing roblem would be searable between the two roducts. A secial case is when s = 0 (so a i = u i and B equals the whole match utility domain). Then the e ect of a rice adjustment on consumer search behavior (i.e., (8)) disaears, and the rst-order conditions simlify to ui = 2 f i (u) 2 du : (2) i u i In this case, the multiroduct model yields the same equilibrium rices as the singleroduct model. In the following analysis, I will often rely on the case with two symmetric roducts. Slightly abusing the notation, let the one-variable functions F () and f() denote the common marginal distribution function and density function, resectively. Let a be the common reservation utility in each dimension. In articular, with symmetric roducts, we have f(u ; u 2 ) = f(u 2 ; u ) and the reservation frontier satis es () = (), i.e., it is symmetric around the 45-degree line in the match utility sace. If is the equilibrium rice of each roduct, then both (0) and () simlify to u = 2 f(u i )f(u i ; u j )du + [ F ((u))]f(u; (u))du B a {z } standard e ect: u + [ F (u)]f(u; (u))du : a {z } joint search e ect: 8 One can also derive the rst-order conditions by calculating the demand functions directly. For examle, when rm I unilaterally deviates to ( " ; 2 " 2 ), the demand for its roduct is u [ H 2 ((u j")ju )( F (u + " ))] df (u ) + u H 2 ((v )jv )( F (v " ))df (v ) ; 2 u 2 u where " = (" ; " 2 ), (u j") = (u + " ) " 2 is the reservation frontier associated with the deviation, and H i (j) is the conditional distribution function. Consumers who samle rm I rst will buy its roduct if they sto searching immediately or if they continue to search but nd rm II s roduct is a worse deal. Consumers who samle rm II rst will urchase rm I s roduct if they come to rm I and nd rm I s roduct is a better deal. The deviation demand for roduct 2 is similar. (3) 5

16 Discussion: the second-order conditions and asymmetric equilibria. In our multiroduct search model, it is di cult to investigate the second-order conditions in general. In the online sulementary document, I show that in the case of symmetric roducts and indeendent match utilities, each rm s ro t function is locally concave around the rice de ned in (3) under fairly general conditions. In the examles with uniform and exonential distribution (which are used for illustration below), it can be numerically veri ed that a rm s ro t function is globally quasi-concave, and thus the rst-order conditions are su cient for the equilibrium rices. A related issue is the ossible existence of a tye of asymmetric equilibrium where rms ut di erent roducts on sale. For examle, in the case with symmetric roducts, one rm may charge rice L for its roduct and rice H > L for its roduct 2, and the other rm sets rices in the oosite way. However, as shown in the online sulementary document, this tye of equilibrium cannot be sustained at least when the two symmetric roducts have indeendent match utilities and f is logconcave. For illustration of the equilibrium rices, I resent two examles: The uniform examle: Suose u and u 2 are indeendent, and u i s U[0; ]. Then i (x) = ( x) 2 =2. So a = 2s, and condition (4) requires s =2. The reservation frontier satis es ( u) 2 + ( (u)) 2 = 2s ; so the stoing region A is a quarter-disk with radius 2s. Then (3) imlies = 2 ( ; (4) )s 2 where 3:4 is the mathematical constant. (The standard e ect is = 2 and the joint search e ect is = s.) 9 s=2, The exonential examle: Suose u and u 2 are indeendent, and f i (u i ) = e u i for u i 2 [0; ). Then i (x) = e x. So e a = s, and the search cost condition (4) requires s. The reservation frontier satis es e u + e (u) = s ; so (u) is one branch of a hyerbola. Then (3) imlies = + 6 s3 : (The standard e ect is =, and the joint search e ect is = s 3 =6.) The rice increases with search costs in the uniform examle, but it decreases with search costs in the exonential examle. As I will exlain below, the result that rices can decline with search costs is not excetional in the multiroduct search model. 9 The rst term in (3) is 2 R du, so it equals two times the area of region B, i.e., 2( s=2) = 2 s. B The second term in (3) is R a [ (u)]du, which is the area of region A and so equals s=2. The joint search e ect is = R ( u)du = s according to the de nition of a. a 6

17 3.3 Price and search cost This section investigates how rices vary with search costs. When search costs rise, there are two e ects: First, consumers will become more reluctant to sho around, and so fewer of them will samle both rms (i.e., the region of B shrinks). This always induces rms to raise their rices. Second, when search costs rise, the mass of marginal consumers on the reservation frontier also changes. If the number of marginal consumers increases with search costs (which is true, for instance, in the uniform distribution case where the reservation frontier becomes longer as search costs rise in the ermitted range of (4)), rms have an incentive to reduce rices. This incentive is further strengthened in the multiroduct case due to the joint search e ect: stoing a marginal consumer from continuing to search can boost demand for both roducts. The nal rediction deends on which e ect dominates. I introduce the following regularity condition: h i (u i ju j ) F i (u i ) increases with u i for any given u j. (5) In articular, if the two roducts have indeendent match utilities, this condition is just the standard regularity condition of increasing hazard rate in the single-roduct case. In the following, I focus on the case with two symmetric roducts, and so the equilibrium rice is given in (3). 20 One can see that increases with search costs if and only < 0, where is the standard e ect and is the joint search e ect as < 0 if the regularity condition de ned in (3). As the following (5) holds. This means that if the joint search e ect were absent, rices would increase with search costs under the condition (5), similar as in the single-roduct scenario. However, taking into account the joint search e ect can qualitatively change the icture. As indicated in the following roosition, the joint search e ect can vary with s in either direction. < 0, the joint search e ect will reinforce the e ect such that the rice increases with search costs even faster. Conversely, > (e.g., when the conditional density is weakly decreasing), the joint search e ect will mitigate or even overturn the usual relationshi between rice and search costs. As a result, the regularity condition (5) is not enough to ensure that rices increase with search costs in our model. The following result gives a new condition (all omitted roofs can be found in Aendix A): 20 Product asymmetry is another force that could in uence the relationshi between rices and search costs. Intuitively, when one roduct has a lower ro t margin than the other, the joint search e ect from adjusting its rice is stronger (i.e., reducing its rice can induce consumers to buy the other more ro table roduct). Then this roduct s rice may go down with the search cost. For examle, when roduct has a match utility uniformly distributed on [0; ] and roduct 2 has a match utility uniformly distributed on [0; 4], one can show that decreases while 2 increases with s when s =2. 7

18 Proosition 2 Suose the search cost condition (4) holds, and the two roducts are symmetric. Then de ned in (3) increases with search costs if and only if u a f((u)) F ((u)) ff(u)h(uj(u)) + [ F (u)]h0 (uj(u))g du u > f(a; u) h 0 (uj(u))f((u))du : a {z } (6) If the two roducts further have indeendent match utilities, a su cient condition for (6) is that the marginal density f(u) is (weakly) increasing. The condition (6), however, can be easily violated (such that rices decrease with search costs) even under the regularity condition of (5). 2 For examle, in the exonential examle with indeendent @ = @s > 0, and so the oosite of (6) holds. Other simle examles include the distribution with decreasing density f(u) = 2( u) and the logistic distribution f(u) = e u =( + e u ) 2 when search costs are relatively small. As we have argued, this surrising result is due to the joint search e ect, the new economic force in our multiroduct search model. If rms suly (and consumers need) more roducts, the joint search e ect could have an even more ronounced imact such that rices fall with search costs more likely. In Aendix A, I extend the two-roduct model to the case with m roducts. In articular, in the uniform case with m symmetric roducts, the equilibrium rice has a simle exression: = 2 V m ( 2s) {z 2 m } standard e ect + (m )V m( 2s) ; (7) 2 m {z } joint search e ect where s 2 [0; =2] and V m (r) is the volume of an m-dimensional shere with radius r. 22 One can check that increases with s if and only if m < +=2 2:6. Therefore, when consumers search for more than two roducts, even in the uniform examle, rices start to decrease with search costs. Discussion: large search costs. Our analysis so far has been restricted to relatively small search costs such that it is even worthwhile to search for one good alone. I now discuss the case with larger search costs beyond condition (4). (In some circumstances, 2 If the oosite of (5) holds, the left-hand side of (6) is negative, and meanwhile h 0 must be negative such that the right-hand side is ositive. Thus, (6) will fail to hold and so the rice will decrease with search costs. This is not surrising, because even in the single-roduct search case (5) imlies that the equilibrium rice decreases with search costs if the match utility has a decreasing hazard rate. What is more surrising in the multiroduct search model is that rices can decrease with search costs even under the regularity condition. 22 The volume of an m-dimensional shere with radius r is V m (r) = (r ) m (+m=2), where () is the Gamma function. One can show that for any xed r, lim m! V m (r) = 0. Then as m goes to in nity, will aroach the erfect information rice =2 in exression (2). This is because for a xed search cost, if each consumer is searching for a large number of roducts, they will almost surely samle both rms. 8

19 a consumer conducts multiroduct search because it is not worthwhile to search for each good searately.) As we shall see later, this discussion will also be useful for understanding the results in the bundling case. For simlicity, let us focus on the case of symmetric roducts. Suose the condition (4) is violated such that s > i (u) and a < u. (But s cannot exceed 2 i (u) to ensure an active search market.) Then the reservation frontier is shown in Figure 3 below, where c = (u). u 2 c A u 2 = (u ) B c u Figure 3: The otimal stoing rule with large search costs The key di erence between this case and the case of small search costs is that now the reservation frontier becomes shorter as search costs go u. This feature has a signi cant imact on how rices vary with search costs. For examle, in the uniform case, a higher search cost now leads to fewer marginal consumers on the reservation frontier, which rovides rms with a greater incentive to raise rices. (In this case, the joint search e ect strengthens the usual relationshi between rices and search costs.) I show in the online sulementary document that if the two roducts are symmetric and they have indeendent match utilities, and if search costs are relatively high such that i (u) < s < 2 i (u), then the equilibrium rice increases with search costs if each roduct s match utility has a monotonic density and a (weakly) increasing hazard rate. 3.4 Price comarison with single-roduct search As the end of this section, I comare the multiroduct search rices in section 3.2 with the single-roduct search rices in section 3., and discuss one emirical imlication of the comarison result. Proosition 3 Suose the search cost condition (4) and the regularity condition (5) hold. Then i 0 i ; i = ; 2; i.e., each roduct s rice is lower in multiroduct search than in single-roduct search. (The equality holds when s = 0.) 9