A joint initiative of Ludwig-Maximilians University s Center for Economic Studies and the Ifo Institute for Economic Research

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1 A joint initiative of Ludwig-Maximilians University s Center for Economic Studies and the Ifo Institute for Economic Research Area Conference on Alied Microeconomics - 2 March 20 CESifo Conference Centre, Munich Multiroduct Search Jidong hou CESifo GmbH Phone: +49 (0) Poschingerstr. 5 Fax: +49 (0) Munich office@cesifo.de Germany Web:

2 Multiroduct Search Jidong hou y November 200 Job Market Paer Abstract This aer resents a sequential search model where consumers look for several roducts and multiroduct rms comete in rices. In a multiroduct search market, both consumer behavior and rm behavior exhibit di erent features from the single-roduct case: a consumer often returns to reviously visited rms before running out of otions; and rices can decrease with search costs and increase with the number of rms. The framework is then extended in two directions. First, by introducing both single-roduct and multiroduct searchers, the model can exlain the henomenon of countercyclical ricing, i.e., rices of many retail roducts decline during eak-demand eriods. Second, by allowing rms to use bundling strategies, the model sheds new light on how bundling a ects market erformance. In a search environment, bundling tends to reduce consumer search intensity, which can soften cometition and reverse the usual welfare assessment of cometitive bundling in a erfect information setting. Keywords: consumer search, oligooly, multiroduct ricing, countercyclical ricing, bundling JEL classi cation: D, D43, D83, L3 I thank V. Bhaskar, Syngjoo Choi, Jan Eeckhout, Antonio Guarino, Philie Jehiel, Toru Kitagawa, Guy Laroque, Andrew Rhodes, Jean-Marc Robin, Georg Weizsacker, Chris Wilson, and esecially Mark Armstrong and Ste en Huck for helful comments. The nancial suort from the British Academy and the Economic and Social Research Council (UK) is gratefully acknowledged. y Corresondence address: Deartment of Economics, University College London, Gower Street, London WCE 6BT, UK. jidong.zhou@ucl.ac.uk.

3 Introduction Consumers often look for several roducts during a given shoing rocess. For examle, they buy food, drinks and household roducts together during ordinary grocery shoing, or urchase clothes, shoes and other goods in high street shoing, or buy several resents when they go Christmas shoing. Sometimes a consumer seeks electronic combinations such as comuter, rinter and scanner, or several travel roducts such as ights and accommodation. 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 all 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, 2 literature has been largely concerned with single-roduct search markets. the search There are robably two reasons why multiroduct search is under-researched. First, as we will discuss in more detail later, a multiroduct search model is less tractable than a singleroduct one. Second, eole may also concern how useful a multiroduct search model will be. This aer develos a tractable model to study multiroduct search markets. We nd that multiroduct consumer search actually has rich market imlications, and the develoed framework is useful in addressing several interesting economic issues. First, a multiroduct search market exhibits some qualitatively di erent roerties comared to the single-roduct case. For examle, in a multiroduct search market, rices can decline with search costs and rise with the number of rms. Second, the multiroduct search model can exlain the henomenon of countercyclical ricing, i.e., rices of many retail roducts fall during high-demand eriods such as weekends and holidays. Third, the multiroduct search model rovides an aroriate setting for studying bundling in search markets, and sheds new light on how bundling a ects market erformance. Our framework 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, and the consumer does not need to buy all roducts from the same rm, i.e., they can mix and match after Marketing research also suggests that retailers regard consumer search behavior (such as how many stores they usually visit and how often they read ads and yers) as an imortant determinant in their ricing and marketing decisions (e.g., Urbany, Dickson and Sawyer, 2000). 2 Multiroduct search is also relevant in the labor market when a coule, as a collective decision maker, is looking for jobs in the same industry. 2

4 samling at least two rms (if rms allow them to do so). In the basic model, we assume linear rices are used, i.e., rms set searate rices for each roduct. A distinctive feature of consumer behavior in multiroduct search is that a consumer may return to reviously visited rms to buy some roducts before running out of otions. While in a standard single-roduct sequential search model, a consumer never returns to earlier rms before having samled 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 for the rm s other roducts. We term this the joint search e ect. As a result, even 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. A related observation is that rices can rise with the number of rms. This is because when there are more rms, it becomes more likely that a consumer will return to revious rms to buy some roducts when she stos searching. This weakens the joint search e ect and so the comlementary ricing roblem. Another rediction of our model is that rms set lower rices in the multiroduct search environment than in the single-roduct case. 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 roblem and so increases roducts cross-rice elasticities. There is a substantial body of evidence that rices of many retail roducts dro during high-demand eriods such as weekends and holidays. 3 Our model can o er a simle exlanation for this henomenon of countercyclical ricing. Suose there are both single-roduct searchers and multiroduct searchers in the market, and suose a higher roortion of consumers become multiroduct searchers during high-demand eriods (e.g., many households conduct their weekly grocery shoing during weekends). Then the above result imlies that rices will decline when demand surges. The second art of this aer allows rms to engage in bundling (i.e., selling a ackage of goods in a articular rice). Bundling is a widely observed multiroduct ricing strategy in the market. For examle, many retailers o er a customer a discount or reward (e.g., free delivery) if she buys several roducts together from the same store. 3 See section 3.5 for related literature and other ossible exlanations for countercyclical ricing. 3

5 Bundling is usually exlained as a rice discrimination or entry deterrence device, 4;5 but in a search environment it has a new function: it can discourage consumers from exloring rivals deals. This is because bundling reduces the anticiated bene t from mixing-andmatching after visiting another rm. As such, rms may have a greater incentive to adot the bundling strategy in a search market. Moreover, this search-discouraging e ect works against the tyical ro-cometitive e ect of cometitive bundling in a erfect information scenario (see the related literature below). When search costs are relatively high the new e ect can be such that bundling bene ts rms and harms consumers. 6 Since the seminal work by Stigler (96), there has been a vast literature on search, but most aers focus on single object search. There is a small branch of literature that investigates the otimal stoing rule in multiroduct search. In Burdett and Malueg (98) and Carlson and McAfee (984), consumers search for the lowest rice of a basket of goods among a large number of stores. The former mainly deals with the case of free recall and the latter deals with the case of no recall. In both cases the otimal stoing rule ossesses the reservation roerty. 7 Gatti (999) considers a more general setting with free recall in which consumers search for rices to maximize an indirect utility function. He shows that the reservation roerty holds in multiroduct search if the indirect utility function is submodular in rices, i.e., if a better o er in one dimension (weakly) reduces the search incentive in the other dimension. (The often adoted additive setting is a secial case of that.) This branch of literature has 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, we argue that desite this similarity, consumer search behavior still exhibits substantial di erences between the two cases. More imortantly, in the above works there is no an active suly side, and the rice (or surlus) distribution among rms is exogenously given. According to our knowledge, the only genuine equilibrium multiroduct search model is McAfee (995). 8 It studies 4 See, for instance, Adams and Yellen (976), and McAfee, McMillan, and Whinston (989) for the view of rice discrimination, and Whinston (990) and Nalebu (2004) for the view of entry deterrence. 5 In di erent settings, Carbajo, de Meza and Seidmann (990) and Chen (997) also argue that (asymmetric) bundling can create vertical roduct di erentiation between rms, thereby softening rice cometition. 6 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, However, with free recall consumers urchase nothing until search is terminated, while with no recall consumers may buy some chea goods rst and then continue to search for the other goods. 8 Lal and Matutes (994) also resent a multiroduct search model where each roduct is homogenous across rms and each consumer needs to ay a location-seci c cost to reach rms and discover the rice information. Their setting is subject to the Diamond aradox. That is, no consumers will articiate in the market given that they exect each rm is charging the monooly rices. Lal and Matutes 4

6 multiroduct rice disersion by extending 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, some consumers only learn information from one store while others learn more. As a result, similar to the single-roduct case, 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. 9 Although the model o ers interesting insights, both the multilicity of equilibria and the comlication of equilibrium characterization restrict its alicability. Our aer develos a more alicable multiroduct search framework with di erentiated roducts and the standard sequential search technology, where the symmetric equilibrium is unique. We do not aim to address rice disersion. Instead, we use the develoed framework to address other economic issues such as countercyclical ricing and bundling in search markets. In terms of the modelling aroach, our aer is built on the single-roduct search model with di erentiated roducts. That framework was initiated by Weitzman (979), and later develoed and alied to a market context by Wolinsky (986) and Anderson and Renault (999). 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 all consumers will sto searching at the rst samled rm. So rivalry between rms has no imact on rice. In search models with roduct di erentiation, there are some consumers who are ill-matched with their initial choice of sulier and then search argue that rms can avoid the market collase by emloying loss-leading strategy, i.e., by advertising (and committing to) low rices of some roducts to ersuade consumers to visit the store. However, in equilibrium each consumer still only samles one rm. Shelegia (2009) studies a multiroduct version of Varian (980) in which for some exogenous reasons one grou of consumers visits only one store while the other visits two. The resence of heterogeneously informed consumers can be a consequence of rational search, but without an exlicit search model the main insights from our aer are absent there. Rhodes (2009) 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 unit consumer demand. 9 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. 5

7 further, so that the ro-cometitive bene t of actual search is resent. 0 Recently this framework has been adoted to study various economic issues such as rominence and non-random consumer search (Armstrong, Vickers, and hou, 2009), rms incentive to use selling tactics such as exloding o ers and buy-now discounts (Armstrong and hou, 200), how the decline of search costs a ects roduct design (Bar-Isaac, Caruana, and Cuñat, 2009), and attention-grabbing advertising (Haan and Moraga-González, 2009). Our aer extends this literature to the multiroduct case. Last but not least, our aer contributes to the literature on cometitive bundling. Matutes and Regibeau (988), Economides (989), and Nalebu (2000) have studied cometitive ure bundling, and Matutes and Regibeau (992), Anderson and Leruth (993), Thanassoulis (2007), and Armstrong and Vickers (200) have studied duooly mixed bundling. One imortant insight emerging from all these works is that 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. 2 This aer is the rst to study bundling in a search environment. Our ndings indicate that assuming away information frictions (which usually do exist in consumer markets) may signi cantly distort the welfare assessment of bundling. In articular, when search costs are relatively high, bundling may actually bene t rms and harm consumers. As such, our work comlements the existing literature. 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 in a duooly and conducts comarative statics analysis, and an alication to countercyclical ricing is then discussed. Section 4 studies bundling in a search market and examines its welfare imacts relative to linear ricing. Section 5 discusses the case with more rms and other extensions, and section 6 concludes. Omitted roofs and calculations are resented in the Aendix. 2 A Model of Multiroduct Search There are n 2 multiroduct rms in the market, each selling two roducts and 2. Sulying each roduct involves a constant marginal cost which is normalized to zero. There are a large number of consumers with measure of one. Each of them has unit 0 In the homogeneous roduct scenario, the main aroach to avoid the Diamond aradox is to introduce heterogeneously informed consumers. See, for examle, Burdett and Judd (983) and Stahl (989), where rice disersion arises endogenously and so consumers have incentive to search. Most of these studies adot the two-dimensional Hotelling setting and assume that consumers are distributed uniformly on the square and have unit demand for each roduct. Armstrong and Vickers (200) consider a fairly general setting with arbitrary distributions and elastic demand. 2 The welfare imact of bundling in the monooly case is ambiguous (e.g., Schmalensee, 984). 6

8 demand for both roducts, and each roduct is horizontally di erentiated across rms. We model this scenario by extending the random utility model in Perlo and Salo (985) to the two-roduct case. 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 ). valuations are realized indeendently across rms and consumers (but a consumer may have correlated valuations for the two roducts in a rm). For simlicity, we 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. Following Perlo and Salo (985), we assume that in equilibrium all consumers buy both roducts, i.e., the market is fully covered. 3 (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.) In the basic model, rms must charge a searate rice for each roduct. We refer to this case as linear ricing henceforth. (We will consider bundling in section 4.) We introduce imerfect information and consumer search as Wolinsky (986) and Anderson and Renault (999) did in a single-roduct framework. Initially consumers are assumed to have imerfect information about the rice and match utilities of all roducts. The 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 rices ( ; 2 ) and both match utilities (u ; u 2 ). The cost of search is assumed to be the same no matter how many roducts a consumer is looking for, which re ects economies of scale in search. 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. In articular, consumers do not need to urchase both roducts from the same rm. (Otherwise, our multiroduct search model would degenerate to a single-roduct model with a match utility u + u 2 and a rice + 2.) For simlicity, we also assume away other ossible costs involved in sourcing sulies from more than one rm. Finally, following most of the literature on consumer search, we assume that consumers have free recall, i.e., there is no cost in going back to a store already visited. Both consumers and rms are assumed to be risk neutral. We focus on symmetric equilibria in which rms set the same (linear) rices and consumers samle rms in a random order (and without relacement). 4 We use the erfect Bayesian equilibrium 3 Although the assumtion of full market coverage is often adoted in oligooly models, it is not innocuous for welfare analysis. In our model it highlights the match e ciency but ignores the outut e ciency. 4 As usual in search models, there exists an uninteresting equilibrium where consumers exect all 7

9 concet. Firms set rices simultaneously, given their exectation of consumers search behavior. 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. We have made several simlifying assumtions to make the model tractable. Economies of scale in search. Our assumtion that the cost of search 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 totally divisible among roducts (so no economies of scale in search at all), a multiroduct search roblem degenerates to two searate single-roduct search roblems. In reality, most situations are in between: the shoing rocess involves a xed cost for reaching the store and also variable in-store search costs for nding and insecting each roduct. Our simli cation is both for analytical convenience and for highlighting the di erences between multiroduct and single-roduct search. Free recall. Free recall could be aroriate when a consumer can hone the visited rms (e.g., furniture retailers) to order the roducts she decides to buy, or when shoing online a consumer can leave the browsed websites oen. In the search literature, sometimes we also assume no recall at all (esecially in the job search case). In most consumer markets, however, there are usually ositive returning costs but they are not so high that returning is totally banned. We choose to assume free recall both for tractability, 5 and for facilitating the comarison between our model and the corresonding single-roduct search model in Wolinsky (986) and Anderson and Renault (999) (both of which assume free recall). We will discuss how costly recall or no recall could a ect our results in section 5.2. Two-sto shoing costs. Even if there are no search costs and returning costs, transacting with two rms may involve some other costs (e.g., the cost of aying two bills). But at least in some retail markets, these two-sto shoing costs seem less imortant than search costs. We will discuss the di erence between this market friction and search frictions in section 5.3. joint-urchase discount we will examine in the bundling art.) (Two-sto shoing costs are also similar to the rms to set very high rices which leave them with no surlus, consumers do not articiate in the market at all, and so rms have no incentive to reduce their rices. We do not consider this equilibrium further. 5 According to our knowledge, Janssen and Parakhonyak (200) is the only aer in the economics literature which studies the otimal stoing rule in single-roduct search with costly recall. They nd that when there are more than two (but a nite number of) rms, the stoing rule is non-stationary and deends on the historical o ers in an intricate way. The otimal stoing rule in multiroduct search with costly recall and an arbitrary number of rms is still an oen question. 8

10 2. The otimal stoing rule We rst derive the otimal stoing rule (which has been formally 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 (u i x)df i (u i ) = ui x 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. (The second equality is from integration by arts.) Note that i (x) is decreasing and convex. Then the otimal stoing rule in a symmetric equilibrium 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 we did not assume u and u 2 are indeendent of each other. The searability of the incremental bene t in (2) is because of the additivity of match utilities and the linearity of the exectation oerator.) Now ste back and consider the situation when the consumer is at the rm before that. If (2) is violated, then samling one more rm is always desirable. By contrast, if (2) holds, then even if the consumer continues searching, she will sto at the next rm no matter what she will nd there. So the bene t from keeing searching is the same as samling one more rm. Execting that, the consumer should actually cease searching now. (This stoing rule also carries over to the case with an in nite number of rms.) Figure below illustrates the otimal stoing rule. 9

11 z 2 a z 2 = (z ) A z r rz _ u B ru a 2 z Figure : The otimal stoing rule in multiroduct search with erfect recall A is the set of (z ; z 2 ) which satis es (2) and let us refer to it as the accetance set. Then a consumer will sto searching if and only if the maximum utility air so far lies within A. We 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. One can show that A is a convex set, and the reservation frontier is decreasing and convex. 6 Let B be the comlement of A. Note that a i on the grah is just the reservation utility level when the consumer is only searching for roduct 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. It is worth mentioning that from () and (2), one can see that only the marginal distributions matter for the exected bene t of samling one more rm. This imlies that if the marginal distributions are xed, the correlation of the two roducts match utilities does not a ect the reservation frontier. Search behavior comarison. It is useful to comare consumer search behavior between single-roduct search 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 6 From the equality of (2), we have 0 (z ) = F (z ) F 2 ((z )) < 0 ; and this derivative increases with z. 0

12 search behavior exhibits some di erences between the two cases, which have not been noticed 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 actually never exercises the recall otion. However, in multiroduct search, a consumer s search decision may deend 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). As a result, in multiroduct search (even with an in nite number of rms), although a consumer will buy at least one roduct at the rm where she stos searching, she may return to a revious rm and buy the other roduct (even if there are unsamled rms left). In the above examle, the consumer will go back to some revious rm to buy roduct 2. We summarize the discussion in the following table. Will revious o ers a ect the current search decision? Will a consumer retrieve revious o ers before searching through all rms? Single-roduct search no no Multiroduct search yes yes Table : Search behavior comarison These di erences will comlicate the demand analysis in multiroduct search. In articular, unlike the single-roduct search case, considering an in nite number of rms does not simlify the analysis (mainly because various tyes of returning consumers still exist). However, the comlication can be avoided if there are only two rms. Moreover, as we will discuss in section 5., such a simli cation does not lose the most imortant insights concerning rm ricing in a multiroduct search setting. Hence, in the following analysis, we restrict our attention to the duooly case. (A detailed analysis of the general case is rovided in the online sulementary document at htts://sites.google.com/site/jidongzhou77/research.)

13 3 Equilibrium Prices 3. The single-roduct benchmark To facilitate comarison, we 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. That is, in a symmetric equilibrium, a consumer will kee searching if and only if the maximum match utility so far is lower than a i, and a higher search cost will make the consumer less willing to search on. In the following analysis, we will mainly focus on the case with a relatively small search cost: s < i (u i ), a i > u i for both i = ; 2 : (4) This condition ensures an active search market even in the single-roduct case. The symmetric equilibrium rice 0 i is then determined by the rst-order condition 7 0 i ai = f i (a i )[ F i (a i )] + 2 {z } 0 u i f(u) 2 du : (5) (Its intuition will be clear soon.) It follows that 0 i increases with the search cost s (or decreases with a i ) if f i (a i ) 2 + fi(a 0 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 the search cost if the match utility has an increasing hazard rate f i =( F i ). 3.2 Equilibrium rices in multiroduct search We 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, the rm in question, and (v ; v 2 ) be the match utilities of rm 2, the rival rm. In equilibrium, for 7 The second-order condition is a delicate issue in the search model with di erentiated roducts. One su cient condition is that the densify f i is weakly increasing (see Aendix B in Anderson and Renault, 999, for further discussion). But that condition is too stringent. For examle, one can check that even in the exonential case the rst-order condition is su cient for the equilibrium rice. 2

14 a consumer who samles rm rst, her reservation frontier u 2 = (u ) is determined by (u ) + 2 ((u )) = s ; (6) which simly says that the exected bene t of samling rm 2 is equal to the search cost. Note that (u ) is only de ned for u 2 [a ; u ] (see Figure 2 below). For convenience, we extend its domain to all ossible values of u, but stiulate (u ) > u 2 for u < a. We derive equilibrium rices by alying the following logic. Starting from an equilibrium, suose rm decreases 2 by a small ". How will this adjustment a ect its ro t? Since the equilibrium demand for its roduct 2 is =2 (remember the assumtion of full market coverage), this deviation rst leads to a ( rst-order) loss "=2. On the other hand, rm gains from boosted demand. This includes two arts: (i) more consumers who visit rm rst will sto searching and buy both roducts immediately; (ii) the consumers who have already samled both rms will buy roduct 2 from rm more likely. In equilibrium, the loss and gain should be such that rm has no incentive to deviate, which generates the rst-order condition for 2. Let us analyze in detail the ( rst-order) gain from the roosed small rice reduction. We rst need to understand how such a rice adjustment a ects the stoing decisions of those consumers who samle rm rst. (Note that the consumers who samle rm 2 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 ) ". That is, the reservation frontier moves downward everywhere by ", and the stoing region A exands (i.e., more consumers buy immediately at rm ) as illustrated in the gure below. 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 3

15 For a small ", the number of consumers who switch from keeing searching to buying immediately at rm (i.e., the robability measure of the shaded area between (u ) and (u j")) is " u f(u; (u))du : (7) 2 a (Remember that half of the consumers samle rm rst. The integral term is the line integral along the reservation frontier in the u dimension.) These marginal consumers now buy both roducts from rm for sure, while before the rice deviation they only bought one or two roducts from rm with some robability (i.e., when they search on and nd worse roducts at rm 2). This is the rst source of gain from the rice reduction. To be seci c, for a marginal consumer with (u ; (u )), if she searched on, she would come back and buy roduct if v < u (of which the robability is F (u )) and buy roduct 2 if v 2 < (u ) (of which the robability is F 2 ((u ))). Hence, for a small " the net bene t of the rice reduction from the boosted demand for roduct 2 is 2 " 2 u a [ F 2 ((u))]f(u; (u))du ; (8) and the net bene t from the boosted demand for roduct is " 2 u a [ F (u)]f(u; (u))du : (9) The rst own-rice e ect is a standard search e ect, similar to the rst term in (5) in the single-roduct case (where the mass of marginal consumers around the reservation oint a i is f(a i )). The second cross-rice e ect is a new feature of our multiroduct search model, and it makes the two indeendent roducts like comlements. This e ect occurs because each consumer is searching for two roducts and the cost of search is incurred jointly for them. So we refer to it as the joint search e ect henceforth. Also note that the mass of marginal consumers in (7) (and so both search e ects) deends not only on the density function f but also on the length of the reservation frontier as indicated in Figure 2, which is another new feature of the multiroduct search model. As we shall see below, both new features lay an imortant role in rms ricing decisions. Now consider the second source of gain from the rice reduction: the consumers who have samled both rms. When rm lowers 2, more of them will buy its roduct 2. This e ect includes two arts: (a) The half of consumers who samle rm 2 rst will come to rm if (v ; v 2 ) 2 B, and then buy rm s roduct 2 if u 2 + " > v 2. For a given (v ; v 2 ) and a small ", the consumers who will be a ected by the rice adjustment are those who value rm s roduct 2 at u 2 v 2, and the density of them is f 2 (v 2 ). Hence, R the demand increment from these consumers is " f 2 B 2(v 2 )df (v). (b) The other half of consumers who samle rm rst will continue to visit rm 2 if (u ; u 2 ) 2 B(") (which converges to B as "! 0), and then come back to buy rm s roduct 2 if v 2 < u 2 + ". 4

16 R The demand increment from these consumers is " f R 2 B(") 2(u 2 )df (u) " f 2 B 2(u 2 )df (u), which is the same as in (a). Therefore, the small rice reduction generates a bene t 2 " f 2 (u 2 )df (u) (0) B from those consumers who samle both rms. (The counterart in the single-roduct case is catured in the second term in (5).) In equilibrium, the ( rst-order) loss "=2 from the small rice adjustment should be equal to the sum of the ( rst-order) gains (8), (9) and (0). This yields the rst-order condition for 2 : 8 [ F 2 ((u))]f(u; (u))du = 2 2 u f 2 (u 2 )df (u) + 2 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 e ect of a roduct s rice adjustment on its own demand (which is similar to the single-roduct search case), and the last term catures the new joint search e ect. Similarly, one can derive the rst-order condition for as: = 2 B f (u )df (u) + u2 where is the inverse function of. + 2 u2 : () a 2 [ F ( (u))]f( (u); u)du (2) a 2 [ F 2 (u)]f( (u); u)du ; Both () and (2) are linear equations in rices, and the system of them has a unique solution. 9 Thus, the symmetric equilibrium, if it exists, will be unique. 20 Notice that if 8 One can also derive the rst-order conditions by calculating the demand functions directly. For examle, when rm 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 ) and (u j") is the reservation frontier associated with the deviation. (Consumers who samle rm rst will buy its roduct if they sto searching immediately or if they search on but nd rm 2 s roduct is worse. Consumers who samle rm 2 rst will urchase rm s roduct if they come to rm and nd rm s roduct is better.) The deviation demand for roduct 2 is similar. However, this direct method will become less alicable in the case with more roducts, more rms, or mixed bundling. 9 One can show that the standard-e ect coe cient in each rst-order condition is greater than the joint-search-e ect coe cient. Then the claim follows. 20 The issue of equilibrium existence is not our central concern. In our multiroduct search model, 5

17 rms ignored the joint search e ect, then the ricing roblem would be actually searable between the two roducts. Let ~ i denote the rice of roduct i in this hyothetical case. A secial case is when s = 0 (so a i = u i and B equals the whole utility domain). Then both search e ects (8) and (9) disaear, and we obtain i = 2 ui u i f i (u) 2 du : In this case, the multiroduct model yields the same equilibrium rices as the singleroduct model. In the following analysis, we will often rely on the case of 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 () and (2) 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: Before roceeding to the comarative statics analysis, let us rst study 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 the search cost condition (4) requires s =2. The reservation frontier satis es ( u) 2 + ( (u)) 2 = 2s ; so the stoing region A is a quarter of a disk with a radius 2s. Then (3) imlies 2 = 2 (=2 )s ; where 3:4 is the mathematical constant. it is rather comlicated to investigate the second-order condition in general. However, in the uniform and exonential examles (which we will often use below), we can verify that a rm s ro t function is locally concave around the equilibrium rices. In addition, one can also show that the rst-order conditions are su cient for the equilibrium rices at least when the search cost is close to zero (i.e., when the ricing roblem is almost searable between the two roducts) and F i is logconcave. 2 The standard e ect is = 2 s=2: the rst term in (3) is 2 R du, so it equals two times the B area of region B, i.e., 2( s=2) = 2 s; and the second term is R a [ (u)]du, which is the area of region A and so equals s=2. The joint search e ect is = R a ( u)du = s according to the de nition of a. 6 (3)

18 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 22 = + s 3 =6 : The rices in these two examles are deicted as the thick solid curves in Figure 3 below. The rice increases with search costs in the uniform examle, but it decreases with search costs in the exonential examle. As we will see below, the result that rices can decline with search costs is not excetional in our multiroduct search model (a) uniform examle s (b) exonential examle s Figure 3: Prices and search costs (symmetric roducts) 3.3 Search cost and rice This section investigates how search costs a ect rices in our multiroduct search model. We rst introduce a regularity condition: h i (u i ju j ) F i (u i ) increases with u i for any given u j. (4) If the two roducts have indeendent match utilities, this is just the standard increasing hazard rate condition. When search costs rise, fewer consumers will samle both rms and become fully informed, which always induces rms to raise their rices. At the same time, when search costs rise, the mass of marginal consumers who distribute on the reservation 22 One can check that the standard e ect is = and the joint search e ect is = s 3 =6. 7

19 frontier also changes. This is another determinant for rices. In articular, the latter can work against the former if the mass of marginal consumers increases with search costs. In multiroduct search, this occurs more likely than in single-roduct search because the reservation frontier often becomes longer as search costs rise (in the ermitted range). Moreover, the marginal-consumer e ect is also stronger in multiroduct search due to the joint search e ect (i.e., stoing a marginal consumer from searching can boost demand for both roducts). Our rst observation is that if the joint search e ect were absent, the marginalconsumer e ect would be usually insu cient to outweigh the rst. (Recall ~ i is the rice of roduct i if both rms ignore the joint search e ect. All omitted roofs can be found in Aendix A.) Lemma 2 Suose the search cost condition (4) and the regularity condition (4) hold. Then ~ i ; i = ; 2; weakly increase with the search cost. That is, without the joint search e ect, the relationshi between rices and search costs would be actually similar to that in the single-roduct scenario. For instance, in the uniform examle, we have ~ i = =(2 s=2) which increases with s; and in the exonential examle which has a constant hazard rate, we have ~ i = which is indeendent of s (so i decreasing with s in this case is urely due to the joint search e ect). However, taking into account the joint search e ect can qualitatively change the icture. We ursue this issue by considering two cases. Symmetric roducts. Suose rst the two roducts are symmetric, and so the equilibrium rice is given in (3). Lemma 2 imlies that the standard e ect indicated in (3) usually decreases with s. However, the joint search e ect can vary with s in either direction even under the regularity condition. If also decreases with s, then the joint search e ect will make the rice increase with search costs even faster. Conversely, if increases with s, then the joint search e ect will mitigate or even overturn the usual relationshi between rice and search costs. As shown in the roof, d ds = f(a; u) u a h 0 (uj(u))f((u))du : (5) This derivative is ositive, for examle, when the (conditional) density function is weakly decreasing. This is true in both the uniform and exonential examle. As a result, the standard hazard rate condition is no longer enough to ensure that rices increase with search costs in our model. The following result gives a new condition. Proosition 2 Suose the search cost condition (4) holds, and the two roducts are symmetric. 8

20 (i) The equilibrium rice de ned in (3) increases with the search cost if and only if u a for all a. f((u)) F ((u)) ff(u)h(uj(u)) + [2 F (u) F ((u))]h0 (uj(u))g du f(a; u) > 0 (ii) If the two roducts further have indeendent valuations, a su cient condition for (6) is that the marginal density f(u) is (weakly) increasing. (6) Condition (6) can be easily violated by some distributions having a decreasing or non-monotonic density (but still having an increasing hazard rate). 23 By continuity, this is true for su ciently small deartures from the exonential case. Other relatively simle examles include: the distribution with a decreasing density f(u) = 2( s 2 [0; =3]; and the logistic distribution f(u) = e u =( + e u ) 2 for s less than about. u) for On the other hand, 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. We can extend our two-roduct model to the case with m roducts (see the details in Aendix A). In articular, in the uniform case, the equilibrium rice is given by = 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 ( 2s) is the volume of an m-dimensional shere with a radius 2s. 24 One can see that increases with s if and only if m < + =2 2:6. Then we have the following result. Proosition 3 Suose the search cost condition (4) holds, and each rm sulies m symmetric roducts with indeendent valuations u i s U[0; ]. Then the equilibrium rice is de ned in (7), and it increases with s if m 2 and decreases with s if m 3. In this examle, if the joint search e ect were absent, the rice would increase with the search cost for any m. But its resence makes the rice decline with search costs whenever consumers are searching for more than two roducts. Asymmetric roducts. Another force which could in uence the relationshi between rices and search costs is roduct asymmetry. Intuitively, when one roduct has a lower 23 One may wonder, if f(a; u) is bounded away from zero, whether the condition always fails to hold f((u)) as a! u (i.e., as s! 0). This is not true because F ((u)) may converge to in nity at the same time. For examle, in the uniform case, the left-hand side is equal to 2 > 0, indeendent of a. 24 The volume formula for an m-dimensional shere with a 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. This is simly because for a xed search cost, if each consumer is searching for a large number of roducts, almost all of them will samle both rms. 9

21 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 more ro table roduct). Then this roduct s rice may go down with the search cost. We con rm this ossibility in a uniform examle in which roduct is a small item and has match utility uniformly distributed on [0; ], and roduct 2 is a big item and has match utility uniformly distributed on [0; 4]. Figure 4 below deicts how (in the left anel) and 2 (in the right anel) vary with search costs. This examle suggests that when the two roducts are asymmetric, search costs can a ect their rices in di erent directions. (This discussion also oens u the ossibility that loss leading might occur in a multiroduct search model. We further discuss this issue in section 6.) (a) small item s (b) big item s Figure 4: Prices and search costs (asymmetric roducts) Discussion: larger 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. In some circumstances, however, consumers conduct multiroduct search just because it is not worthwhile to search for each good searately. We now discuss this case. (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 s is beyond the condition (4), so s > i (u) and a < u. (But s cannot exceed 2 i (u) in order to ensure an active search market.) Then the reservation frontier is shown in Figure 5 below, where c = (u). The key di erence between this case and the case of small search costs is that now the 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.) 20