Price Comparison Websites

Transcription

1 Warwick Ecoomics Research Paper Series Price Compariso Websites David Roaye October, 2015 Series Number: 1056 (versio 4) ISSN (olie) ISSN (prit)

2 Price Compariso Websites David Roaye 1 7 October 2015 Abstract The large ad growig idustry of price compariso websites (PCWs) or web aggregators is poised to beefit cosumers by icreasig competitive pricig pressure o firms by acquaitig shoppers with more prices. However, these sites also charge firms for sales, which feeds back to raise prices. I ivestigate the impact of itroducig PCWs to a market for a homogeeous good. I fid that itroducig a sigle PCW icreases prices for all cosumers, both shoppers ad o-shoppers. More geerally, i the most profitable equilibrium for competig PCWs, prices ted to rise with the umber of PCWs. (JEL: L11, L86, D43) Keywords: olie markets; price compariso sites; competitio; price dispersio; 1 Uiversity of Warwick; [email protected]. I thak Da Berhardt, Rai Spiegler, Motty Perry, Daiel Sgroi, Kobi Glazer, Adrew Oswald, Alessadro Iaria ad Giulio Trigilia for their helpful discussio. I also thak participats of the RES ad EARIE 2015 cofereces alogside those at various other semiars. 1

3 1. Itroductio Over the past two decades a ew idustry of price compariso websites (PCWs) or web aggregators has emerged. The idustry has eabled cosumers to check prices of may firms sellig a particular service or product simultaeously i oe place. The sites are popular i may coutries, ad i may markets icludig utilities, fiacial services, hotels, flights ad durable goods. 2 These sites commad billios of dollars of reveue aually. 3 I the UK, PCWs for utilities ad fiacial services have bee particularly successful. There are roughly 48 such PCWs i the UK, where over 70% of iteret users have used such a site. The largest four aggregators geerated approximately 800m ($1.2b) i reveue durig 2013, with average aual profit of the group icreasig by 14% that year. 4 The iteret has altered search costs, allowig cosumers to compare prices across firms i a matter of clicks, itesifyig competitive pricig pressure betwee firms. While a cosumer may ot kow all the firms i a market, a PCW ca expose the full list of market offerigs, maximizig iter-firm pressure. However, uderlyig this icreased competitio are the fees paid by firms who sell their products through the websites. I the UK these are uderstood to be betwee ($69-95) for a customer switchig gas ad electricity provider. 5 These fees, i tur, act as a margial cost faced by producers, affectig their pricig decisios. 6 The 2 Examples for utilities ad services iclude Moeysupermarket.com, Google Compare ad Gocompare.com; for flights Skyscaer.et ad Flights.com; for hotels Hotels.com ad Bookig.com; ad for durable goods Amazo Marketplace, Priceruer.com ad Pricegrabber.com. 3 Regardig travel services, Pricelie Group (which ows Bookig.com ad Pricelie.com) ad Expedia Ic. (which ows Expedia.com ad Hotels.com) made approximately $6b i total agecy reveues i Regardig durable goods, Amazo Marketplace sold 2 billio items from third-party sellers. See their 2014 Aual Reports for details. 4 Number of PCWs take from Cosumer Focus (2013) report ito PCWs i the UK. The big four refers to Moey Supermarket, Compare the Market, Go Compare ad Cofused.com. Number of PCWs Usage data from the 2013 Mitel Report o Web Aggregators. Fiacial iformatio take from the compaies ow aual reports where available, otherwise iferred from paret group reports or ewspaper articles. See appedix for details. 5 For example, see BBC (2015). Fees are sigificat i other sectors too e.g., i the hotel-reservatio sector they are reported to be 15-25% of the purchase price (see Daily Mail, 2015). 6 I practice, PCWs i some markets charge per-click, but may per-sale. I abstract from this differece ad 2

4 idustry gleas substatial profits from these fees. As such, it is ot clear whether the cetral premise that PCWs lower prices is valid. This fudametal tesio is ecapsulated i a quote from the BBC (2014): There s aother cost i the bill. It s hidde, it s kept cofidetial, ad yet it s for a part of the idustry that appears to be o the cosumers side. This is the cut of the bill take by price compariso websites, i retur for referrig customers. The recommedatio to switch creates chur i the market, ad it is see by supplier compaies as worth payig high fees to the websites. Whether or ot customers choose to use the sites, the cost to the supplier is embedded withi bills for all customers. 7 This article examies this chur ad addresses the fudametal questio of whe cosumers are better off with a PCW i the marketplace. I homogeeous good markets, I characterize whe all cosumers are made worse off followig the itroductio of PCWs. My model builds o the elegat framework of Baye ad Morga (2001), who ivestigate the strategic icetives of a PCW or iformatio gatekeeper. Without the PCW i their model, each cosumer is served by a sigle local firm which sells at the moopoly price (it is too costly for cosumers to travel to aother store). This leads to the result that cosumers beefit from the itroductio of a PCW, because firms must compete for the busiess of cosumers who ejoy free access to the site. I the moder olie marketplace however, firms also have their ow websites. 8 I the absece of a aggregator, cosumers do ot eed to physically travel to purchase the good; they ca visit aother firm s website just as easily as they could a aggregator s. My model features two types of cosumers: shoppers, who use PCWs i equilibrium, ad model the latter, assumig that the two are correlated. 7 The cocer was also expressed by US seator Amy Klobuchar regardig mergers of hotel-reservatio sites: The whole idea of cheaper hotels is very good, but if it all starts to come uder oe compay, you ca easily foresee the situatio where they ca charge higher commissios that are the passed o to cosumers. (New York Times, 2015) 8 Ofte, a PCW simply re-directs users to the selected firm s ow website to complete the purchase. 3

5 iactive cosumers, who buy directly from a particular firm. Although shoppers are always better off tha iactive cosumers i equilibrium, my primary result is that both types are always made worse off by the itroductio of a sigle PCW. I the provide coditios uder which cosumers are harmed by the itroductio of multiple competig PCWs. This is the first article i this settig to show such results, reversig those i the existig literature, which I show ca be see as a special limitig case. 9 My model supposes each shopper is iformed of the prices from q > 1 of the firms websites i the absece of a PCW, rather tha q = 1 as i Baye ad Morga. Without a PCW, shoppers see q prices; after the itroductio of the PCW, they will see all of them i equilibrium. I show that addig competitio (q > 1) to the settig without a aggregator reverses their fidig: expected equilibrium prices are raised by the itroductio of a sigle PCW, makig all cosumers worse off. What happes is that the equilibrium fee a sigle PCW charges for a sale through its site is so high that it more tha egates ay beefits from the icreased firm competitio. Oe may cojecture that allowig multiple competig aggregators will udo this result, aki to textbook Bertrad competitio. I exted the model to allow for multiple PCWs, ad for shoppers to check ay umber of them. My characterizatio shows that both the umber of PCWs ad the umber of PCWs that shoppers check (or are iformed of) matter. Cocretely, whe shoppers oly check oe of may PCWs, they all effectively remai moopolists ad so cosumer welfare does improve with ay umber of PCWs. At the other extreme, where shoppers check all PCWs, Bertrad-style reasoig at the aggregator level results as a special case: PCWs udercut each other s fees to reach a uique zero-profit equilibrium ad shoppers beefit from their existece. Whe shoppers visit some, but ot all PCWs, cosumer welfare teds to rise with the umber of PCWs checked, but it falls with the umber of PCWs. I particular, i this realistic sceario, there is a critical umber of PCWs beyod which all cosumers ca be worse off tha without ay aggregators at all. I the ivestigate the impact of differet policies ad other market features. Geerally, PCW fees are either ot disclosed or are detailed o a subpage of the websites. Whe PCWs 9 This is i the absece of ay persuasio or directio of cosumers to more expesive products by firms (e.g., Armstrog ad Zhou, 2011), or biased itermediaries (e.g., de Corire ad Taylor, 2014). 4

6 publicly aouce fees so that cosumers are made aware of them, I show that this creates the possibility that firms ad shoppers could coordiate o the cheapest PCW, which ca result i equilibria that beefit shoppers relative to a world without PCWs. My primary focus is o settigs where a firm sets the same price for a product o every website that it is sold. This assumptio describes may markets where PCWs operate icludig gas, electricity, mortgages ad durable goods. However, I exted the aalysis to cosider markets such as the hotel reservatio sector, where a firm s price ca differ across sites. I show that both types of cosumers ca be worse off whe shoppers oly check oe PCW i equilibrium. However, ulike markets without price discrimiatio, whe shoppers check at least two PCWs, competitive pressure betwee aggregators ca work à la Bertrad. I also cosider markets where cosumers may face some o-egligible search cost i order to retrieve prices e.g., home isurace quotes. Here, the umber of shoppig cosumers is edogeously determied by the distributio of search costs ad the expected savigs cosumers ca make by usig it. I show that despite this activity at the extesive search margi, a aggregator ca still make all cosumers worse off, icludig those who decide to start shoppig. Sectio 2 reviews the literature; Sectio 3 presets the model; Sectio 4 coducts comparative statics with a moopolist PCW; Sectio 5 models competig aggregators; Sectios 6-8 cosider settigs with publicly observable fees, price discrimiatio ad search costs; Sectio 9 cocludes. 2. Literature This article cotributes to the literature o clearig-house models, see for example (Salop ad Stiglitz, 1977; Varia, 1980; Rosethal, 1980; Baye ad Morga, 2001, 2009; Baye et al., 2004; Chioveau, 2008; Arold et al., 2011; Arold ad Zhag, 2014). These models ratioalize price-dispersio i homogeeous goods markets. 10 Ideed, price dispersio has persisted 10 The search cost literature also provides explaatios of price dispersio (e.g., Burdett ad Judd, 1983; Elliso ad Elliso, 2009; Elliso ad Wolitzky, 2012; Stahl, 1989; 1996; Stigler, 1961). Motivated by the rise of the iteret, clearig-house models abstract from a direct modelig of cosumer search costs. The frameworks are to some extet isomorphic (see Baye et al., 2006). 5

7 despite the advaces of techology such the iteret ad compariso sites. Early studies documeted marked dispersio i the olie markets for goods (e.g., Bryjolfsso ad Smith, 2000; Baye et al., 2004). A recet study by Gorodicheko et al. (2015) fids substatial cross-seller variatio i prices, ad voices support for clearig-house models that categorize cosumers ito loyal ad shoppig cosumers. The equilibria i my model feature price dispersio regardless of whether there is a aggregator (or clearig-house ). Without a PCW, this is because there is some cosumer search. Producig price dispersio with a PCW that employs the pricig mechaism see i practice, is a challege. The shift i the aggregator idustry away from chargig oe-off fixed fees, toward pay-per-sale fees to firms is ratioalized as profit-maximizig PCW behavior by Baye et al. (2011). However, without the itroductio of some other exogeous fixed cost to a firm of listig o the PCW (e.g., trasactio costs), price-dispersio vaishes i equilibrium. I do ot dey the existece of such additioal costs, but emphasize that dispersio arises i my framework without appeal to fixed listig costs. The larger relevat literature is that of two-sided markets, pioeered by Rochet ad Tirole (2003) (see also Caillaud ad Jullie, 2003; Elliso ad Fudeberg, 2003; Armstrog, 2006; Reisiger, 2014). These articles model platforms where buyers ad sellers meet to trade, focusig o platform pricig ad the effect of etwork exteralities with differetiated products ad platforms. These models do ot explicitly model seller-side competitio, which is cetral to my settig. More recet cotributios do (e.g., Belleflamme ad Peitz, 2010; Hagiu, 2009) but they model the platform as the oly available techology, which is ot appropriate for the questios I address. Edelma ad Wright (2015) allow sellers ad buyers to coduct busiess off platform, but they study eviromets with differetiated product markets, where the itermediary directly offers buyer-side beefits such as rebates. I cotrast, I model a homogeeous good, isolatig price as the determiat of cosumer welfare, where the oly beefit a platform brigs is iformatioal: it lists available prices. The potetial beefit of a PCW to cosumers is that it ca lower prices via the iteractio of strategic, competig firms, ad hece the size of the beefit is determied edogeously via the equilibrium actios of firms ad cosumers. 6

8 3. Model A World Without a PCW There are firms ad a uit-mass of cosumers. Firms produce a homogeeous product at zero cost without capacity costraits. Cosumers wish to buy oe uit ad have a commo willigess to pay of v > 0. Each cosumer is edowed with a default, curret or preferred firm from which they are iformed of the price. This assumptio has may atural iterpretatios. I a market for services or utilities (e.g., gas ad electricity tariffs, mortgages, credit cards, broadbad, cellphoe cotracts, car, home ad travel isurace etc.) cosumers ca be thought of as havig a curret provider for the service for which they kow the price they pay ad the reewal price should they remai with the same provider. 11 I the market for flights, hotels or durable goods, cosumers ca be thought to have a carrier, hotel (or hotel chai) or producer which they prefer, perhaps bookmarked i their browser or for which they receive marketig s that directly iform them of the price or prompt the user to fid out via hyperliks, i a matter of secods. A proportio α (0, 1) of cosumers are shoppers. Casual empiricism suggests that may people ejoy browsig or lookig for a bargai. Shoppers check (are aware of) at least oe rival firm s website ad therefore kow q > 1 of the prices. A shopper sticks with his or her default firm if its price is ot beate by aother price. If a rival firm s price is cheaper, the shopper switches. The remaiig proportio of 1 α cosumers are auto-reewig, loyal, offlie or iactive cosumers who do ot shop aroud. I markets where firms are service providers, such cosumers simply allow their cotract with their existig provider to cotiue with their default firm. Much of the furore surroudig PCWs has bee directed at those operatig i the services ad utilities sectors. I the presetatio that follows I adopt termiology suited to that sector, of curret firms or providers with cosumers referred to as shoppers ad auto-reewers. I assume that each firm has a equal share of curret cosumers of each type ( α shoppers ad 1 α auto-reewers). Firms set prices ad shoppers simultaeously decide which rival- 11 Providers usually provide reewal price iformatio directly to cosumers. 7

9 firm websites to visit. I focus o equilibria i which firms adopt idetical pricig strategies ad shoppers employ symmetric shoppig strategies. Goig forward I refer to such symmetric equilibria simply as equilibria. This setup geeralizes Varia (1980), estig his equilibrium. 12 He motivates the two types as beig completely iformed about all prices, or uiformed. The iformed buy the cheapest o the market, while the uiformed buy from their default firm. I my model, shoppers see q prices where 1 < q ad I derive the uique equilibrium. Whe q =, Varia s equilibrium correspods to that derived i Propositio 1 below. Shoppers ca be characterized by ( q) groups. Each group is a list of the firms checked by that cosumer. For example, cosider q = 2, = 4 with firms idexed 1,2,3,4. The there are 6 possible comparisos shoppers could make: {12, 13, 14, 23, 24, 34}, where the two digits refer to which firms prices are checked. Each firm is ivolved i ( 1 q 1) = 3 price comparisos. Shoppers employ symmetric shoppig strategies, ad hece are evely distributed across these groups: 1 compare the prices of firms 1 ad 2, 1 compare firms 1 ad 3, ad so o. Equivaletly, 6 6 oe could iterpret a cosumer as radomizig uiformly over which rival-firm websites she or he checks. 13 Now cosider the best-respose of firms. Without loss of geerality, let F : p [0, 1] deote the cumulative distributio fuctio of prices charged by firms i equilibrium. Propositio 1 describes the equilibrium without a PCW. Propositio 1. I the uique equilibrium firms mix accordig to the distributio [ ] 1 [ ] (v p)(1 α) q 1 v(1 α) F (p) = 1 over the support p qpα 1 + α(q 1), v. Price dispersio is a cetral feature i clearig-house models ad showcases how they ratioalize price dispersio i homogeeous goods markets. The limited-search assumptio does ot alter the ituitio for why. Bertrad-style reasoig of udercuttig dow to the margial 12 The oly differeces beig that I assume zero costs ad do ot employ his zero-profit coditio. 13 Although I focus o a symmetric distributio across these pairs, this shopper behavior is ot a assumptio per se; it is a best respose i equilibrium to symmetric firm behavior because cosumers are the idifferet to which firm they check. 8

10 cost (here ormalized to 0) does ot play out. A firm ca guaratee itself a profit of at least v(1 α) > 0 from its auto-reewers. Therefore, ay poit mass i equilibrium strategies must be for some ṗ > 0. Ay such mass would always be udercut by firms to gai a discrete umber of shoppers for a arbitrary ɛ-loss i price. Models buildig o Varia s assume that i a world without a clearig-house, cosumers caot check other prices, so the pure moopolistic-price equilibrium of p = v results. As i Stahl (1989), were there o shoppers, I would obtai such a Diamod (1971) equilibrium with each firm chargig v; ad were there oly shoppers, the the Bertrad outcome of p = 0 would result. Ulike Stahl, shoppers do ot ecessarily kow the prices of all firms, but they kow at least two. Propositio 1 shows that with some (but o-zero) search, price dispersio still emerges i equilibrium. It is also istructive to ote that equilibrium pricig does ot vary with the umber of firms,, as log as q <. Each shopper compares q prices, regardless of the total umber of firms. Whe makig pricig decisios, a firm is ot cocered about the umber of other firms per se, but rather about the other prices shoppers kow. As all firms price symmetrically ad idepedetly i equilibrium, it is as if each firm oly faces q 1 rivals. I other words, what matters is the umber of comparisos shoppers make, ot the umber of firms i the market. A World With a PCW Suppose a etrepreeur creates a price compariso website. I add a prelimiary stage to the game at which the PCW sets a click-through fee c R + that a firm must pay to the aggregator per sale made via the site. Cosumers do ot lear c, although they will have correct expectatios i equilibrium. 14 Each firm sees the fee c ad must choose a price, ad whether or ot to post it o the PCW. Models with a clearig-house that does ot charge a fixed fee typically have may equilibria, see for istace Baye et al. (1992) for their characterizatio of the full set of equilibria of Varia (1980). Throughout the paper, I focus o symmetric equilibria i which shoppers oly 14 As the BBC quote i the Itroductio otes, the exact fee is kept cofidetial, ot publicly aouced. 9

11 check PCWs. 15 With a moopoly PCW, this results i a uique equilibrium. I this equilibrium, firms list o the PCW with probability oe ad prices are dispersed, providig shoppers a strict icetive to check the PCW. 16 To fid the equilibrium, I use the followig lemma which takes c as fixed ad characterizes the esuig mutual best-resposes of firms. Defie G(p; c) to be the cumulative distributio fuctio of prices charged by firms for a give click through fee c: G(p; c) = 1 [ ] 1 (v p) (1 α) 1 α(p ( 1)c) Lemma 1. The mutual best-resposes of firms as a fuctio of c: 1. If c [0, v(1 α)), firm best-resposes are described by G(p; c), ad have o poit masses. 2. If c = v(1 α), there are two classes of resposes, oe with o poit masses described by G(p, c), ad those i which all firms charge the same price. 3. If c (v(1 α), v], all firms charge the same price. Where pricig is described by G(p; c) each firm always lists its price o the PCW. Whe c is i the lower iterval described by Lemma 1, firms udercut each other util the poit at which they would be better off chargig the maximum price v ad oly sellig to their auto-reewers. However, if c exceeds v(1 α), udercuttig does ot reach this poit, so o firm would jump to v. Rather, it reaches a poit where all firms charge the same price, sellig to all their cosumers directly; o firm udercuts further or jumps to v. The reaso o further udercuttig occurs is that doig so would mea chargig a price p < c that would wi all rival firms shoppers, but would do so at a loss. The reaso o firm jumps to v is that they are 15 Recall this is i additio to kowig their curret or preferred firm s price, which upo sale, cotiues to cout as a direct purchase e.g., the auto-reewal letter, the bookmarked airlie or the hotel s marketig More geerally there exist other equilibria e.g., trivial equilibria i which o firms or shoppers atted the PCW. The set-up is the idetical to that of the previous sectio ad the equilibrium is give by Propositio 1 with the vacuous additio that the PCW ca charge ay c. There also exist asymmetric equilibria where ot all firms list o the PCW (see Footote 17). 10

12 makig more i this pure-pricig equilibrium where they sell to all their cosumers directly. If c = v(1 α) the both mixed ad pure equilibria obtai as threshold cases. To derive the equilibrium fee set by the PCW, cosider its icetives. The PCW will make zero profit if firms all charge the same price, as the o shoppers switch. I cotrast, the PCW ears positive profit i ay mixig equilibrium wherever c > 0. The PCW thus has a strog icetive to iduce price dispersio because shoppers switch whe they ca obtai a strictly lower price with a ew firm. A feature of distributios with o poit masses is that the probability of a tie i price is zero. As a result, shoppers at 1 of the firms will switch. Give that firms mix i this way, the PCW will raise c as high as possible before reversio to a pure equilibrium, which here happes for c > v (1 α). Propositio 2 characterizes the equilibrium. 17 Propositio 2. I the uique equilibrium where shoppers check the PCW, the PCW sets a click-through fee of c = v (1 α), firms list o the PCW ad mix over prices accordig to G(p; v(1 α)) over the support p [v(1 α), v]. That oly pure equilibria exist for c > v(1 α) is what limits the PCW fee ad allows there to be price dispersio at equilibrium without fixed costs for firms. It is the outside optio of curret shoppers for firms that limits the fee that the PCW charges. Shoppers kow their curret provider s price regardless of where they check prices ad stay with their curret firm if there are o lower prices to be foud o the PCW. Hece, a firm ca always guaratee itself its ow shoppers ad avoid the aggregator s fee by chargig a price low eough to udercut the market ad ot list o the PCW. It is this threat that discourages the PCW from chargig fees that are eve higher i equilibrium. If firms did ot have this outside optio, the PCW could raise its fee to c = v, firms would charge p = v, price dispersio would be lost ad the PCW would be able to extract all the surplus. 17 Note that i the symmetric equilibrium, all firms list o the PCW. I practice, some firms do ot always list o PCWs. Note that here there exist asymmetric equilibria where m 2 firms list o the PCW, mixig over prices [v(1 α), v] i a similar way to the CDF of Propositio 2, with the other m do ot list, chargig v ad sellig oly to their auto-reewers with all shoppers uiformly spread over the mixig firms. The price-risig result of this article also holds i ay of these asymmetric equilibria. 11

13 4. Comparative Statics Comparig the equilibria of Propositios 1 ad 2 reveals how the PCW affects cosumer welfare. Propositio 3. Both types of cosumer are worse off with the PCW tha without. The key to the proof is to show that the expected shopper-price uder F is less tha the lower boud of the support of G (see Figure 1). It immediately follows that shoppers expect to pay more uder G, as the expected lowest price is eve higher. That G first-order stochastic domiates F shows that auto-reewers expect to pay more uder G, as the expected price is higher. Figure 1: Equilibrium price distributios calibrated with q = 2, v = 1, α =.7 1 E F [p (1,2) ] 0 0 p 1 F (p) G(p; v(1 α)) for = 2, 3, 5, 10, 50 With the itroductio of a PCW, the two effects stated i Corollaries 1 ad 2 give rise to Propositio 3: Corollary 1. Withi the mixed-price equilibrium firm resposes of Lemma 1, as c [0, v(1 α)] icreases, the expected price paid by both types of cosumer icreases. 12

14 Corollary 2. As the umber of firms icreases, the expected price paid by shoppers falls, but the expected price paid by auto-reewers rises. Corollary 1 shows that whe the PCW sets a higher fee, the expected price paid by both shoppers ad auto-reewers rise. The fee is passed o by firms to cosumers through a firstorder stochastic shift i the prices set i equilibrium. Upo wiig, a firm must pay the PCW for all of the ( 1 ) shoppers who purchase through the site. Because the amout paid rises with the fee, the price charged i equilibrium also rises with the fee. The secod effect is that the PCW icreases competitive pressure amog firms to fight for all 1 rival firms shoppers. I cotrast, without the PCW, firms effectively competed agaist oly q 1 rivals. Differet models i the clearig house literature offer differet predictios about the effect of o equilibrium prices. Some derive distributios for which a icrease i raises prices for both types of cosumer (e.g., Rosethal, 1980); while i other models it lowers prices for shoppers ad raises prices for captive cosumers (e.g., Varia, 1980; Morga et al., 2006). 18 My model belogs to this secod category. A icrease i has two effects o equilibrium prices. First, icreased competitio pushes probability mass to the high-price extreme of the distributio, as Figure 1 shows. This results i a first-order stochastic orderig i : expected price thus icreases i ad auto-reewers pay more. Secod, shoppers ow pay the lowest of + 1 prices rather tha, which reduces the expected lowest price. Corollary 2 reveals that this secod effect more tha offsets the first, implyig that shoppers pay less i expectatio whe there are more firms. 19 I order to relate these two effects of a PCW to Propositio 3, I make the followig remark. Remark 1. If q =, G(p; 0) = F (p) The etrace of a PCW icreases both the fee firms pay (from 0 to v(1 α)) ad the umber of prices that shoppers compare (from q to ). The first effect raises the expected price auto- 18 Also see Baye et al. (2006) for a discussio o this poit ad a wider review of clearig-house models. 19 Although the result of Corollary 2 is commo to clearig-house models, it may seem a rather uaced predictio of equilibrium pricig. Morga et al. (2006) coduct a experimet with participats playig the role of firms agaist computerized buyers ad foud that whe was icreased, prices paid by ielastic cosumers ideed icreased whereas those paid by shoppers decreased. 13

15 reewers pay, which is compouded further by > q, so they are uambiguously worse off with a PCW. The effects pull shopper welfare i opposite directios, but Propositio 3 shows that o matter how large is, it fails to udo the effect of the optimally chose PCW fee c. Hece, shoppers are also worse off i expectatio with a PCW. Due to the costat-sum ature of the game, welfare ecessarily sums to v i equilibrium. As firms make the same expected profit i both worlds, there is a oe-to-oe relatio betwee the decrease i cosumer welfare ad the profits of the PCW. That the PCW does ot reduce firm profits comes from the fact that (with or without the PCW) firms have ( 1 α) auto-reewers. Firms ca therefore guaratee themselves v ( ) 1 α i both worlds. Although this article focuses o cosumer rather tha producer welfare, it is importat to poit out that the icetives of the PCW ad firms are ot aliged. This is because there is exactly oe cheapest price ad hece α ( ) 1 shoppers who switch from o-cheapest prices, which icreases with. A icrease i however, would of course, squeeze per firm profit. ecourage market etry if it could. Therefore, a PCW would always Oe chael through which all cosumers would gai is by more auto-reewers becomig shoppers: 20 Corollary 3. As the proportio of shoppers α icreases, expected prices paid by shoppers ad auto-reewers both decrease. Oe may cojecture that a PCW would wat to maximize α (the umber of shoppers) i order to obtai more referral fees. However, this logic is icomplete. Expadig the PCWs actio set to iclude the determiatio of α (oe ca thik of the PCW determiig α through advertisig) yields the followig result: 21 Corollary 4. If the PCW ca determie α as well as c i the prelimiary stage, the the PCW sets α = This is a predictio commo to clearig-house models, for which Morga et al. (2006) fid experimetal support. 21 α is determied costlessly for the PCW. If this advertisig costs were a covex fuctio of α, it would ot chage the results qualitatively. 14

16 PCW reveue is hump-shaped i profit. As α 0 it receives less ad less traffic, ad hece vaishig reveue. As α 1, firms have fewer auto-reewers to exploit, which pushes v(1 α), the maximum fee for which firms are willig to mix, to zero. Ideed, whe α = 1 all cosumers are shoppers ad the PCW removes ay icetive for firms to icrease prices as there are o auto-reewers to exploit. As a result, all firms charge the same price, leavig the PCW with zero profit. Thus, eve if the PCW could brig all cosumers olie, it has a strict icetive to esure that oly some do so. 5. Competig Aggregators Now suppose there are K > 1 PCWs idexed by k = 1,..., K. Each PCW moves simultaeously i the first period with PCW k settig a fee c k. A crucial measure of competitive pressure is the umber, r, of aggregators shoppers check where 1 r K. I the settig without PCWs, shoppers kow the prices of q firms. I the world with PCWs, oe ca iterpret r K as the umber of aggregators that shoppers are aware of. I the equilibria derived, firms list o all PCWs so shoppers are idifferet to which set of PCWs they check. However, PCW fees ad firm-pricig strategies deped o r. Hece, there are differet sets of equilibria for each r. This leads me to idex equilibria by r. I first cosider the case where shoppers check just oe of the PCWs. Propositio 4. With K > 1 competig PCWs, if shoppers check r = 1 PCW the both types of cosumer are worse off with the PCWs tha without. Propositio 4 obtais because the high-fee, sigle-pcw equilibrium of Propositio 2 with c = v(1 α) remais the uique equilibrium fee. 22 Itroducig competig PCWs exerts o dowward pressure o fees. There are o equilibria at lower fee levels because shoppers oly check oe PCW, which meas there is o icetive for PCWs to udercut each other s fees. If they did, they would ot icrease their volume of sales, but they would receive lower fees. I cotrast, for ay cadidate equilibrium with c < v(1 α), there is a icetive to raise the fee because PCWs ca maitai the same volume of sales ad ear a higher fee. 22 As i the moopolist case, firms outside optio gives rise to pure-pricig firm resposes at higher fee levels, which bouds equilibrium fee levels from above at c = v(1 α); see the Appedix for details. 15

17 I the simplest textbook Bertrad result, there is a immediate fall i the equilibrium price betwee that of a moopoly firm ad the margial-cost pricig of two firms. At the aggregator level however, Propositio 4 shows that this logic is icomplete: with K > 1 such udercuttig does ot eve ecessarily begi. The persistece of the cosumer-welfare-decreasig equilibrium i Propositio 4 is a artefact of each shopper oly checkig oe PCW which effectively makes each PCW a moopoly, facig o competitive pressure. Because all PCWs offer the same iformatio, shoppers have o icetive to check other PCWs. I this respect, the equilibrium is remiiscet of Diamod (1971), but at the aggregatio level, ot the firm level. Oe might thik that the Bertrad remedy for shoppers would be to require them to check at least two PCWs, causig PCWs to udercut each other util a equilibrium with all PCWs chargig c = 0 is reached. I ow explai how this logic is icomplete. Suppose ow that there are K > 1 aggregators, ad shoppers check r > 1 PCWs. First, I examie the special case of r = K > 1: Propositio 5. With K > 1 competig PCWs, shoppers are guarateed to be better off tha before the itroductio of PCWs if r = K > 1. Further, the uique equilibrium PCW fee-level is c = 0 if ad oly if r = K > 1. The spirit of this result resembles that of Bertrad. To see why there caot be some other equilibrium with c > 0 whe r = K > 1, suppose so ad cosider a udercuttig deviatio by PCW 1 to some ĉ 1 = c ɛ. Shoppers do ot detect the deviatio ad so do ot chage their behavior. As for firms, otice that for ay p they strictly prefer to list exclusively o PCW 1 : Whe a firm is the cheapest, it will sell to all shoppers so log as it lists o some PCW. This is precisely because r = K. Hece by listig o PCW 1 oly, there is o reductio i the umber of shoppers switchig to them whe they are cheapest, but there is a reductio i the fee the firm pays as ĉ 1 < c. The PCW fids this deviatio strictly profitable because it receives a discrete gai i the umber of shoppers switchig through it, for a arbitrarily small loss i price. Where shoppers check more tha oe PCW, but ot all PCWs, we have: Lemma 2. Whe K > r > 1, there exists a equilibrium i which PCWs charge c > 0, ad firms list o all PCWs, mixig over prices by G(p; c) where c = v(1 α)kr(k r) K(1 + r(k 2)) + αr(k 1)(r 1)( 1). 16

18 More geerally, there exist equilibria i which all PCWs charge c [0, c]. To uderstad how much cosumer welfare ca be reduced, I aalyze the highest-fee equilibrium from this set. Because PCWs have a strog icetive to coordiate o this equilibrium, it may be especially relevat i practice. Notice that substitutig r = K i Lemma 2 yields c = 0, ad oe obtais Propositio 5. Oe ca see ow that the Bertrad-style reasoig uderlyig Propositio 5 was a special case. To uderstad why the priciple does ot apply more geerally, cosider a fee level c > 0 ad a udercuttig deviatio by PCW 1 to some c 1 = c ɛ > 0. Ulike whe K = r > 1, whe K > r > 1 it is ot ecessarily better for a firm to oly list o the cheaper PCW 1. By listig a price exclusively o PCW 1, there are ow K r K > 0 shoppers who do ot see the firm s price. These shoppers will ot buy from it eve if it is the cheapest. Firms ow face a trade-off: Exclusively listig o PCW 1 meas that ay sales icur oly the lower fee ĉ 1 upo a sale, but there will be a reductio i sales volume because ot all shoppers check PCW 1. Which force is stroger i this trade-off depeds o the size of the udercut ɛ. If PCW 1 udercuts by eough, firms will deviate to list exclusively o PCW 1, breakig the symmetric equilibrium. Ulike the simpler logic of Propositio 5, it is o loger true that ay ɛ > 0 udercut will attract firms to exclusively list o the cheapest PCW. Hece, PCWs do ot always have a icetive to udercut each other ad higher-price equilibria are sustaied. I ow discuss how the set of equilibria varies with how may PCWs shoppers check (r) ad the umber of aggregators (K). Firstly, as shoppers check more PCWs i equilibrium, the icetive for a PCW to udercut the fees of other PCWs icreases so that oly lower fee-levels ca be sustaied i equilibrium. That is, c is limited by a higher r: d c dr < 0 However, as the umber of aggregators icreases, the icetive is reversed. The umber of shoppers checkig a give PCW ( r ) falls. Accordigly, i equilibrium each firm receives K less of its expected reveue from ay sigle PCW. It the requires a more severe udercut from a PCW to get firms to exclusively list o it ad forgo the busiess available from the other aggregators. For udercuts that are too severe, it is uprofitable for a PCW to deviate, eve if it were to wi exclusive arragemets with all firms as a result. Thus, as K icreases, higher equilibrium fees ca be sustaied i equilibrium: d c dk > 0. This allows for the result that a 17

19 higher umber of aggregators ca lead to higher fees, ad hece higher prices. Furthermore: Propositio 6. For ay r, there exists a K such that as log as there are more tha K aggregators both types of cosumer are worse off tha before the itroductio of PCWs. I the limit, the proportio of firm icome that comes from sales o ay oe aggregator becomes vaishigly small. As this happes, c v(1 α) i.e., sustaiable equilibrium fee levels approach the moopoly-pcw level, agai makig both types of cosumer worse off tha before the itroductio of the sites. 6. Publicly-Observable Fees Oe reaso that competig aggregators do ot drive fees to zero is that shoppers do ot detect chages i the fees set by PCWs i equilibrium. This precludes a coordiated respose betwee firms ad shoppers that could puish a PCW that charges higher fees. If fees were publicly aouced so that shoppers were aware of them, the credible subgame equilibria could follow the fee-settig decisio i which the PCW chargig the lowest fee is atteded by all firms ad shoppers. Ay higher-fee equilibrium would the be udercut util c = 0, leadig to lower shopper-prices. It follows immediately that: Propositio 7. Whe K > 1 ad PCW fees are observed by shoppers, there exists a equilibrium with c = 0. There are multiple equilibria because there are may subgame equilibria that ca follow ay vector of PCW fees, icludig less ituitive oes where coordiatio occurs at more expesive PCWs. 23 If oe adopts the plausible refiemet that firms ad cosumers oly patroize the lowest-fee PCWs, the oe obtais the zero-fee equilibrium as the uique equilibrium. However, eve with sufficiet refiemet criteria to implemet the zero-fee equilibrium, i some markets oe may questio a policy of fee-aoucemets o a more fudametal level. 23 Propositio 7 excludes the moopoly-pcw case (K = 1), where the uique equilibrium is still that of Propositio 2 because of course o coordiatio betwee firms ad cosumers over which PCW to atted is possible whe there is oly oe PCW. 18

20 If fees ca be publicly aouced, the surely so ca firm prices, which would extiguish the role of a PCW i the first place. I reality, PCW fees are ot publicized directly. However, some PCWs do advertise summary statistics of the price iformatio of firms that list o them. PCWs frequetly advertise the average savigs a cosumer usig their site is expected to make, which could direct shoppers to the cheapest PCW. By Propositio 7, this could lead to a shopper-welfare-improvig equilibrium. However, may PCWs do ot advertise based o purchase-relevat iformatio. 24 PCWs sped large sums o such advertisig, which has bee show to correlate with the umber of uique visitors they experiece, suggestig that may shoppers are directed to PCWs based o iformatio other tha price. 25 If such persuasive advertisig caused all shoppers to loyally visit oe site each i.e., r = 1, Propositio 2 applies ad all cosumers would have bee better off without the PCW idustry. 7. Price Discrimiatio So far, I have cosidered the impact of web services that list or aggregate the available iformatio (prices) charged by firms offerig a product or service. I practice, this is ofte the case i the markets for gas, electricity, fiacial products such as mortgages, ad durable goods. 26 However, i other markets, a firm may set a price p 0 for a direct purchase, ad p k differet prices for each PCW k that it lists o. Where a PCW operates by referrig shoppers back to a firm s website to complete the purchase, the fact that the click came from a PCW is recogized by the firm s site, which the offers the price see o the PCW that attracted the click. Whe K r = 1, we have: Propositio 8. With price discrimiatio, if r = 1, there exists a equilibrium i which PCWs set c = v(1 α), firms list o all PCWs, p 0 = v ad p 1 = = p K = v(1 α). 24 See the campaig of Comparethemarket.com, based o a story about meerkats. 25 The big four PCWs i the UK spet approximately 110m i 2010 o advertisig. The evidece here is from Nielse Compay (fidigs reported by the This is Moey 2015 ad Marketig Magazie 2011). 26 For UK gas ad electricity markets, regulatio limits each eergy compay to offerig a maximum of four tariffs i total. 19

21 The ability of firms to price discrimiate does ot prevet PCWs from settig fees at the same high level as i Propositio 4 because r = 1. As before, there is effectively o competitive pressure betwee PCWs. However, price discrimiatio does lead to firms listig a commo price o PCWs i equilibrium because PCWs o loger have a icetive to keep prices posted o it dispersed. This is because firms ca set a high direct purchase price p 0 = v ad a lower price through the PCWs. Give this, shoppers always purchase through a PCW. Because all α shoppers ow purchase through PCWs, rather tha i the case without discrimiatio, where oly α ( ) ( 1 did so, total PCW profit is ow αv(1 α) rather tha 1 ) αv(1 α). As firm profit is uaffected, it follows the that relative to the case without price discrimiatio, total cosumer welfare falls. There are, however, opposig effects o shoppers ad auto-reewers. Auto-reewers are worse off, as firms charge them the moopoly price v. Shopper welfare though improves as they ow face p k = v(1 α) for sure, whereas before this was just the miimum of the support of equilibrium prices. More importatly, shopper welfare does ot improve sufficietly to overtur Propositio 3, which cotiues to hold: all cosumers are worse off tha i a world with o PCWs. The ability of firms to discrimiate allows them to fully extract surplus from their captive auto-reewers; but PCWs ca ow extract the reveue from sales to all shoppers through their sites. 27 I equilibria with r > 1, the icetive for aggregators to udercut is preset. Corollary 5 describes a best respose of firms followig a uilateral dowward deviatio by a PCW 1 from a symmetric fee level. Corollary 5. Whe firms price discrimiate, followig c 1 (0, c 2 ) ad c 2 = c 3 =,..., = c K c (0, v(1 α)] there exist mutual best-resposes of firms such that they list o all PCWs, settig p 0 = v ad p k = = c k for all k. Whe r = 1, there is o icetive for a PCW to make such a udercuttig deviatio as suggested by the equilibrium described i Propositio 8. Whe r > 1 however, PCWs have this icetive to udercut because they ca ejoy a discrete gai i fee reveue from the r 1 K 27 As before, i the equilibrium of Propositio 8, PCWs caot raise fees further, to say c > v(1 α) because of firms outside optio. Followig such a uilateral PCW deviatio, firms would set p 0 = v(1 α) so that their shoppers purchase directly from them, reducig PCW profit to zero. 20

22 of shoppers who were checkig their PCW but buyig though aother site i the symmetric equilibrium. Whe firms ca price discrimiate across websites, firms ca compete i prices o PCW 1 without chagig their prices o other PCWs. This shows how price discrimiatio ca uleash udercuttig at the PCW level wheever cosumers check r > 1 PCWs, which ca i tur lead to zero-fee equilibria. 28 This cotrasts with markets where PCWs aggregate price-iformatio, where Propositio 5 showed that this udercuttig was oly fully ulocked whe r = K. Aggregatio ad Discrimiatio i Large Markets I the settig with a PCW ad price discrimiatio, the equilibrium of Propositio 8 shows the price paid by the two types are as maximally separated: Shopper price is competed dow to firms margial cost c, ad auto-reewers pay v. I the settig with a aggregator ad o price discrimiatio, the equilibrium is give by Propositio 2 where Corollary 2 explais that as the umber of firms icreases, the expected prices paid by shoppers ad that paid by auto-reewers, diverge. This occurs because as the umber of firms icreases, so does the competitive pressure o pricig to wi shoppers. As a result, more probability mass is placed o lower prices. Firms compesate for this by also icreasig the mass placed o higher prices, icreasig their expected profit from auto-reewers. I fact, for arbitrarily large markets, I show that these two settigs are equivalet. Propositio 9. As, followig the itroductio of a PCW, the expected prices faced by both types of cosumer i a settig without price discrimiatio (Propositio 2) approach those i a settig with price discrimiatio (Propositio 8). The result highlights the coectio betwee the two market structures. Ideed, I fid that all cosumers ca be worse off with a PCW with or without the possibility of price discrimiatio. I emphasize therefore that the key differece betwee the settigs lies i their predictios uder competitio at the aggregator level. 28 The existece of zero-fee equilibria whe r > 1 are show i the Appedix. 21

23 8. The Extesive Search Margi This paper utilizes a clearig-house framework, where auto-reewers are iactive ad ca also be iterpreted as beig offlie, loyal, uiformed or as havig high search costs. Here, I focus o a search-cost ratioalizatio, better applied to markets where obtaiig a quote requires more iformatio from the cosumer e.g., home isurace. I a eviromet without a PCW where auto-reewers fid it too costly to eter these details ito a firm s website to retrieve oe extra price, the itroductio of a PCW offers to expose all prices to them, for the same sigle search cost. Depedig o their search cost, the itroductio of a PCW may the cause a autoreewer to egage i compariso via the PCW. Some empirical studies have offered a similar argumet to explai observed icreases i market competitiveess (e.g., Brow ad Goolsbee, 2002; Byre et al., 2014). Their argumets are distict from mie because they cotrast a world with web-based aggregators relative to a world without the Iteret, rather tha a world with the Iteret ad firm websites. This egagemet of ew customers is commoly referred to as the extesive search margi (for a recet discussio see Moraga-Gozález et al., 2015). The beefit of a additioal search for a cosumer i the world without a PCW is the differece betwee the expected price ad the expected lowest of two prices draw (from F ). With a PCW, the beefit of a search o the PCW is the differece betwee the expected price ad the expected lowest of draws (from G). 29 I deote these search beefits with ad without a aggregator respectively as, B 1 = E G [p] E G [p (1,) ], B 0 = E F [p] E F [p (1,2) ]. The model is as before save that each auto-reewer faces a search cost s. I assume these costs are heterogeeous, distributed by S over s [ s, ). 30 I assume s > B 0 which meas that without a PCW, o auto-reewers shop, preservig the equilibrium of Propositio 1. After the itroductio of a PCW, the beefit of a search (B 1 ) may outweigh the cost (s) for some auto-reewers, who the choose to use the site. I use the term coverts ad o-coverts to 29 These terms are aalogous to the value of iformatio i Varia (1980). 30 That there is o upper boud esures that there are always some auto-reewers, ad hece price dispersio, i equilibrium. 22

24 distiguish betwee auto-reewers who decide to shop or ot i equilibrium with a PCW. The total umber of cosumers usig the PCW (shoppers ad coverts) is edogeously determied i equilibrium ad is deoted α = α + (1 α)s(b 1 ). Give α, the PCW sets its profit-maximizig fee c = v(1 α). As c ad α are exogeous to firms, equilibrium pricig is as i Propositio 2 with α replacig α. I tur, pricig determies B 1. There is a equilibrium whe this value of α satisfies S(B 1 ) = α α. Whe there exists such a S, α is said to be 1 α ratioalized. Corollary 3 showed that a higher α icreases the welfare of all cosumers. Corollary 1 showed that a lower c has the same effect. As the equilibrium fee level is v(1 α), both forces work to beefit all types of cosumer. However, whether cosumers actually gai depeds o how may auto-reewers are coverted. I fact, relative to the world without a PCW, the presece of coverts is ot sufficiet to guaratee lower prices for ay cosumer, ot eve coverts themselves: Propositio 10. Shoppers, coverts ad o-coverts ca all be worse off with a PCW tha without. The proof gives a example of such a equilibrium, alog with a distributio S that ratioalizes it. More geerally, there ca be may equilibria, each with a differet α. If the beefit (B 1 ) is small or there are o types with low search costs so that S(B 1 ) = 0, there are o coverts ( α = α) ad the equilibrium of Propositio 2 applies. Propositio 10 shows that whe some auto-reewers covert, all cosumers ca be worse off with a PCW. However, there may also exist equilibria where α is high eough such that some cosumers beefit. Whether these equilibria exist depeds o the distributio of search costs S. For a give PCW search beefit B 1, whe more auto-reewers have low search costs (higher S(B 1 )), α is higher, c is lower ad total cosumer welfare is higher. The umber of firms also determies the size of the beefit the PCW offers (B 1 ), ad hece the umber of coverts. I ow ivestigate which cosumers beefit from a aggregator whe the potetial beefit it offers to cosumers (B 1 B 0 ) is as large as possible. For a give α, a higher umber of firms icreases the equilibrium search beefit B 1 (see Corollary 2). Specifically, as, B 1 v c (see Propositio 9), which is as large as possible. To maximize B 1 B 0, 23

25 I let q = 2 i the world without a PCW, which makes B 0 as low as possible, as B 0 is icreasig i q. 31 Accordigly, I defie, Defiitio. A market has maximum potetial whe q = 2 ad. Propositio 11. If the market has maximum potetial: coverts are better off; shoppers ca be worse off; ad o-coverts are worse off with a PCW tha without. Whe the market has maximum potetial, coverts are guarateed to be better off but shoppers still may ot be. 32 Shoppers may ot be better off because eve with maximum potetial, there may still ot be sufficietly may auto-reewers covertig to PCW use. 9. Coclusio The aalysis shows that the itroductio of PCWs may ot i fact beefit cosumers by reducig expected prices. The itroductio of a sigle aggregator facilitates compariso of the whole marketplace for shoppers, exertig competitive pressure o firm pricig. However, the aggregator charges a fee which, i tur, places upward pressure o prices. The et effect is that prices icrease for all cosumers, who would be better off without the site. Competitio at the aggregator level eed ot lead to a reductio i fees. There are may equilibria, which I parameterize by the umber of PCWs that shoppers check. If shoppers oly check oe PCW each, cosumers are o better off tha i the moopoly settig. More aggregators oly guaratee to beefit shoppers if they check all of them. If shoppers check a itermediate umber, icreasig the umber of aggregators ca lead to higher prices; for a sufficietly high umber, all cosumers ca agai be better off without the idustry. Hece, whe there are may aggregators i the market, how may of them shoppers check is a crucial variable. As a result, regulatory bodies may wish to cosider icetivizig cosumers to check more, alogside stroger actios such as limitig the fees charged by aggregators ad limitig the umber of PCWs i the market. 31 See the ed of the proof of Propositio By Propositio 9, oe ca also use Propositio 11 to cosider the effect of itroducig a PCW ito a market with search costs ad price discrimiatio uder the equilibrium of Propositio 8. 24

26 If competig PCWs publicly aouce fees, this ca result i low-fee equilibria, but it relies o coordiatio betwee firms ad shoppers. I markets with price discrimiatio, if shoppers oly check oe PCW, the moopoly fee level ca still be sustaied, makig cosumers worse off. However, with competig aggregators where shoppers check multiple PCWs, there is the the icetive for the sites to udercut each other s fees. As such, regulators may also wish to cosider whether it is possible to itroduce price discrimiatio ito markets i which it is ot curretly preset. I olie markets with o-egligible search costs, eve those cosumers who ratioally start egagig i price compariso may be worse off followig the itroductio of a PCW. Helpful policies would help erode these costs where possible ad ecourage more iactive cosumers to egage i compariso. Appedix A1. Summary Statistics of UK PCW Idustry The figures quoted for turover ad profit i the itroductio for the UK utilities ad services idustry are estimates for 2013 for the largest four such compaies. The estimate for each site was take from the followig sources: Moey Supermarket: Their ow 2013 aual report. Cofused: The Admiral Group s Aual Reports 2012 ad Go Compare: Newspaper The Guardia 2014 article. Compare the Market: Paret BGL Group s Aual Report 2013 ad BBC article For the first three, the figures were take directly from the cited sources. For Compare the Market, estimated to be the largest of the four sites, the estimate is particularly rough as BGL Group offer o breakdow of their accouts. I assumed that the proportio of BGL s total reveue ad profit due to Compare the Market was the same where the estimate for aual profit due to Compare the Market is take from the BBC article. Eve if the estimated 800m of total turover ad 14% for average aual profit growth for these sites is off by a margi, the idustry ca still be cosidered large ad growig. 25

27 A2. A World Without a PCW The domai of prices is R. The equilibrium pricig strategy ca always be described by its CDF (deoted F i this sectio, ad with other letters later o). I what follows, either equilibrium pricig distributios will be pure (so that F is flat, with oe jump discotiuity at this price); or will have o poit masses so that F is cotiuous, which implies the desity f exists ad f = F, wherever F exists. Lemmas A1-A3 are variats of Varia (1980) s Propositios 1,3 ad 7 respectively. Lemma A1. I ay equilibrium, there are o prices p charged s.t. p 0 or p > v. Proof: Ay p 0 geerates firm profits π(p) 0 which is domiated by p = v which gives profit of at least v (1 α) > 0 because firms always sell to their auto-reewers. For p > v, π(p) = 0 because o oe will buy at such a high price, hece agai p = v domiates. Lemma A2. I ay equilibrium, there are o poit masses. Proof: Suppose ot. The the there is a poit mass i equilibrium, ṗ s.t. pr(p = ṗ) > 0. Note that ṗ (0, v] from Lemma A1. Because the umber of poit masses must be coutable, there exists a ε > 0 small such that ṗ ε > 0 ad is charged with probability zero. Cosider a deviatio of a firm from the equilibrium F to a distributio over prices where the oly differece is that the ew distributio charges ṗ ε with probability pr(p = ṗ) ad ṗ with probability zero. Note that a firm appears i ( ) ( 1 q 1 of the groups of shoppers. Idex these groups z = 1,..., 1 ) q 1 (this is without loss as F is symmetric). Let pr(ṗ; t, z) be the probability uder F that a firm is cheapest i group z alog with t = 0,..., q 1 others. Call the differece i profit due to the deviatio d ad ote that: lim d = ε 0 ( 1 q 1) q 1 z=1 t=0 pr(ṗ; t, z) ( α ( ( )) 1 q (t + 1) 1 + ṗ q) q q(t + 1) The term i paretheses is the differece i the amout of shoppers wo uder the deviatio ad uder F, i give group, whe the firm alog with t others charge ṗ, ad ṗ is the lowest price i that group. Due to symmetry of the groups, pr(ṗ; t, z) is the same for all groups, so let this more simply be termed pr(ṗ; t). Also give pr(ṗ; t) > 0, this simplifies to: ( 1 q 1 ) q 1 ( q) t=1 ( ) t pr(ṗ; t)α ṗ > 0 t

28 hece for some ε > 0 there exists a profitable deviatio, so F could ot have bee a equilibrium. Lemma A3. I ay equilibrium, the maximum of the support of f must be v. Proof: Suppose ot. Defie p as the maximum elemet of the support ad ote that by Lemma A2 the probability of a tie at ay price is zero. By Lemma A1, p (0, v). Cosider a firm called upo to play p < v i equilibrium. They oly sell to their auto-reewers, makig p (1 α) would strictly prefer to deviate to v ad make v (1 α), a cotradictio. but Propositio 1. I the uique equilibrium firms mix accordig to the distributio [ ] 1 [ ] (v p)(1 α) q 1 v(1 α) F (p) = 1 over the support p qpα 1 + α(q 1), v. Proof: By Lemma A2, there is more tha oe elemet of the equilibrium support, ad by Lemma A3 v is the maximal elemet. I equilibrium, a firm must be idifferet betwee all elemets of the support p, hece profit must equal v ( ) 1 α for all p, that is: ( ) [ ] 1 α 1 α (1) v = p + ( α q)x(p) X(p) ( )( ) 1 1 q 1 1 F (p) 0 (1 F (p)) ( q 1 q 1 )( 1 q 1 ) F (p) q (1 F (p)) q 1 The first term o the RHS of (1) is the profit from ARs, who always purchase at p v. The secod term is the expected proportio of shoppers that a firm will wi, chargig price p. Shoppers ca be characterized by ( q) groups, where the set of groups is give by {1,..., } q. X(p) describes the expected umber of groups a firm expects to wi give it charges p. By Lemma A2 there are o ties i price, so label prices by p 1 <... < p. p 1 will be the cheapest i every group i which it appears, ad it appears i ( 1 q 1) of the groups. The probability of beig the lowest price is give by ( 1 1) F (p) 0 (1 F (p)) 1, which accouts for the first term i X(p). The observatio that p i is the cheapest i ( i q 1) groups if i (q 1), zero groups otherwise, accouts for the remaiig terms of X(p). The followig maipulatios to simplify the secod term o RHS of (1) this make use of the biomial theorem: = p α ( q) 1 j=q 1 ( j q 1 )( 1 j ) F (p) j 1 (1 F (p)) j 27

29 which after some maipulatios ca be show to be: = pα q (1 F (p))q 1 rearragig for F (p) gives: Notice that this is a well-defied c.d.f. over: [ (v p)(1 α) F (p) = 1 qpα p Notice that v is strictly preferred to ay p [ v(1 α) ] 1 + α(q 1), v [ ) v(1 α) 0,. 1+α(q 1) ] 1 q 1 A3. A World With a PCW I look for symmetric equilibria where PCWs charge some fee level c 0 ad shoppers check PCWs i equilibrium. I do ot look at equilibria where firms ever list o PCWs, where the settig without a PCW applies. To derive equilibria, oe eeds to kow the mutual bestresposes of firms to uilateral deviatios of PCWs. To do so, take c ad equilibrium shopper strategy as give, ad cosider the firm resposes. Let K 1 deote the umber of PCWs ad r : K r 1 the umber of PCWs checked by shoppers. I symmetric equilibrium, a proportio r K of each firm s shoppers check ay give PCW. Defie a vector of PCW fees as c = (c 1,..., c K ) R K + labeled such that c 1... c K. Let β k [0, 1] be the probability with which a firm eters PCW k ad defie the evet E: all PCWs are empty. Deote (a 1, a 2 ) = (p, K) as a firm s actio, where p is the price charged ad 2 {1,...,K} is set of all combiatios of PCWs they could choose to list i where K a typical elemet, ad deotes ot listig o ay PCW. Defie the followig CDF, which is used throughout: (2) G(p; c) = 1 [ (v p)(1 α) α ( p ( 1) 1 K (c c K ) ) ] 1 1 which is well-defied over the support [ p(c), v] where = p(c) v(1 α)+ 1 K (c 1+ +c K )α( 1). Whe 1+α( 1) c 1 = = c K c, let G(p; c) ad p(c) also be writte G(p; c) ad p(c). 28

30 Lemma A4. I ay equilibrium, there are o prices p charged s.t. p 0 or p > v. Proof: No p 0 or p > v because they yield egative ad zero profit respectively, whereas (v, ) yields v(1 α) > 0. Lemma A5. If c 1 [0, v(1 α)), pr(e) = 0. Proof: Suppose pr(e) > 0. Deote the ifimum of prices charged whe o PCW is listed o ad that of prices ever listed o a PCW as p 0 ad p respectively. [Note the ifima exist because prices are a bouded from below by Lemma A4]. Note that p 0 p because ( p, ) is strictly preferred to ay lower price as a firm faces o competitio for prices below p off the PCWs. Cosider whe the firm is called upo to play (p 0, ) (or a price arbitrarily closely above p 0 ): If p 0 > c 1, a deviatio to (p 0 ɛ, 1) is strictly profitable. This is because with probability pr(e) 1 > 0 PCWs are empty with other firms chargig at least p 0. By listig the firm the has a positive probability of wiig α r K 1 will offset the arbitrary loss i reveue from its ow cosumers. ew shoppers. For a sufficietly small ɛ > 0, this If p 0 c 1, firm profit must be at least p which ca be guarateed by ( p, 1), because p 0 p. I tur, this must be at least as much as v(1 α) which the firm ca guaratee by playig (v, ). Puttig these together, p 0 p v(1 α) > c 1 which cotradicts p 0 c 1. Lemma A6. If c 1 [0, v(1 α)), firm strategies have o poit masses. Proof: Defie p 1 (c 1 ) as the price at which a firm is idifferet betwee sellig to all shoppers exclusively through the cheapest PCW(s) ad chargig v, oly sell to auto-reewers: 1 ) = p1(c v(1 α) + αc 1( 1) 1 + α( 1) By Lemma A5, some (p, K) is played. Note that firms would by costructio ot play (p, K) where p < p 1 (c 1 ), strictly preferrig (v, ). To see that there are o poit masses: If there were a poit mass at ( p 1 (c 1 ), K) the there is a positive probability of beig tied for the lowest price at ( p 1 (c 1 ), K). By defiitio of p 1 (c 1 ) firms would strictly prefer to deviate to (v, ). If there were a poit mass at (ṗ, K) s.t. ṗ > p 1 (c 1 ) the there is a positive probability of beig tied for the lowest price at (ṗ, K). A firm would strictly prefer to shift that probability mass to (ṗ ɛ, K ) where K = K \ {k : c k ṗ}. Here, the firm would sell to α r K 1 other 29

31 firms shoppers at a arbitrary loss i reveue from its ow cosumers. There is always a ɛ > 0 small eough to esure this is profitable because p 1 (c 1 ) > c 1 c 1 < v(1 α). Lemma A7. If there are o poit masses, the maximum of the support f must be v. Proof: This is a variat of Varia (1980) Propositio 7. Lemma A8. If c 1 [0, v(1 α)) ad c 1 < c 2, β 1 = 1. Proof: By Lemma A5 it is ever the case that all PCWs are empty. Suppose β 1 < 1. Lemma A6 implies there is more tha oe price, ˆp, that is listed o some other PCWs. By Lemma A7 there is oe such that ˆp < v. Cosider a firm beig called upo to play this (ˆp, K-1) where 1 / K-1 ad m = max{k-1}. As this price has a positive probability of beig the lowest of all firms, it will geerate sales through the PCWs i K-1. But as PCW 1 is the uique cheapest PCW, there is a strictly profitable deviatio to (ˆp, K-1 1 \ m). Lemma A9. If c 1 =..., c K c (v(1 α), v], firm pricig strategies are pure where p [v(1 α), c]. Ay (β 1,..., β K ) (0, 1] K ca be supported. Proof: Either there is a poit mass or there is ot. 1. Suppose there is a poit mass at price ṗ. If ṗ > c, firms have a strict icetive to shift this mass to (ṗ ɛ, {1,..., K}). If ṗ < v(1 α), firms prefer to shift this mass to (v, ). These leaves ṗ [v(1 α), c] as the oly poits that ca be poit masses. There ca be at most oe poit mass: If ot, the there a secod poit mass p < c, which if played with K would geerate egative profit, so ( p, ) is preferred; if K =, the ( p + ɛ, ) for some sufficietly small ɛ > 0 is preferred. To see that this pure pricig at p [v(1 α), c] ca be part of firm strategies, ote that firm profit is π = p v(1 α), so there is o strict icetive to sell oly to auto-reewers istead. Because shoppers buy directly whe prices are all the same, there are o sales through PCWs ad so firms are idifferet betwee ay (β 1,..., β K ) (0, 1] K. 2. Suppose there is o poit mass. By Lemma A7 the maximum of the support is v, where v is ot the oly elemet of the support, else it would be a poit mass. There is therefore a positive probability of a firm beig the cheapest at some p. There ca be o (p, K) s.t. p < c charged: If K played profit from these sales is egative, so (p, ) is preferred; if K =, the (p + ɛ, ) for some sufficietly small ɛ > 0 is preferred. Give p [c, v], pr(e) = 0. 30

32 This follows because firms strictly prefer (p, {1,..., K}) to (p, ) for all p (c, v). For ay 1 r K firms are cotet to list prices o at least as may PCWs as is ecessary to make sure every shopper sees their price e.g., for r = K = 1 all of them; for r = K, just oe of them. There ca therefore, be differet cofiguratios of β K s depedig o r, K so log as pr(e) = 0. To determie firm pricig strategy it must be that firms are idifferet betwee every p they are called upo to play: ( ) ( ) 1 α 1 α v = p [ ] α + (1 G(p; c)) 1 α( 1) p + (p c) which ca be re-arraged to give G(p; c) from (2). However, p(c) < c because c > v(1 α), so firms would make strictly egative profits at prices p ( p(c), c), preferrig ot to list. This provides a cotradictio, so there do ot exist strategic firm resposes with o poit masses. A4. Results for K = 1 or r = 1 Lemma A10. If r = 1, c 1 = = c K 1 [0, v(1 α)) ad c K [c 1, p(c)): β k = 1 for all k where firms mix by CDF G(p; c). Proof: By Lemma A5 there is always some price posted o some PCW(s). By Lemma A6 these prices are ot poit masses ad by Lemma A7 the maximum of the support is v, where oly auto-reewers are sold to. There is therefore a positive probability of a sale through some PCW(s) at some (p, K). If k K where k < K ad sales there are profitable (whe p > c 1 ), the {1,..., K 1} K; if K K ad sales there are profitable p > c K (whether or ot p > c K is the oly cosideratio because r = 1), it follows that {1,..., K} K. To be part of firm strategy, it must also be that firms prefer to play (p, K) tha to charge v ad sell oly to their auto-reewers. Note that from (2) is the price at which firms are idifferet betwee p(c) sellig through {1,..., K} with certaity ad sellig oly to their auto-reewers. Similarly, deote as the idifferece poit betwee sellig for sure o {1,..., K 1} ad chargig p-k(c) v, ad ote < There will the be o (p, {1,..., K 1}) played s.t. p < p-k(c) p(c). p-k(c) ad o (p, {1,..., K}) s.t. p < For prices close to v, K = {1,..., K}. To determie p(c). firm pricig strategy, it must be that firms are idifferet betwee every (p, {1,..., K}) they 31

33 are called upo to play: ( ) ( ) [ ( ( ))] 1 α 1 α α v = p +(1 G(p; c)) 1 α( 1) K 1 p + p c 1 K + c 1 K K which ca be re-arraged for G(p; c) to give the CDF from (2). Because the miimum of the support is p(c) > c K c 1 (the first relatio follows because c 1 < v(1 α)), all prices i the support geerate profitable sales through all the PCWs, there is o price charged s.t. p [ p-k(c), p(c)). It follows that β k = 1 for all k. For K = 1, let c 1 [0, v(1 α)) ad set K = 1 i the expressios of the Lemma. Lemma A11. If r = 1 ad c = v(1 α) there are the followig firm resposes: 1. Pure-price strategies where p = v(1 α) is the oly price ever charged. Here, ay (β 1,..., β K ) (0, 1] K ca be supported. 2. Mixed-price strategies where β k = 1 for all k ad prices are distributed accordig to the CDF G(p; v(1 α)) where p(v(1 α)) = v(1 α). Proof: Either there is a poit mass or there is ot. 1. Suppose there is a poit mass at price ṗ. If ṗ > c, firms have a strict icetive to shift this mass to (ṗ ɛ, {1,..., K}). If ṗ < c, firms have a strict icetive to shift this mass to (v, ). These leaves ṗ = c as the oly poit that ca be a poit mass. To see that this pure pricig ca be part of firm strategies, ote that firm profit is π = c 1 = v(1 α), so there is o strict icetive to sell oly to auto-reewers istead. Because shoppers buy directly whe prices are all the same, there are o sales through PCWs ad so firms are idifferet betwee ay (β 1,..., β K ) (0, 1] K. 2. Suppose there is o poit mass. By Lemma A6, v is the maximum of the support of prices. No prices p < c are charged because (v, ) is strictly preferred. For p > c, sales through all PCWs are profitable so β k = 1 for all k. Whe v is played, firm profit is π(v) = v(1 α). To determie firm pricig strategy, it must be that firms are idifferet betwee every p they are called upo to play: (3) π(v) = p ( ) 1 α + α (1 G(p; v(1 α))) 1 [p ( 1)v(1 α)]. which ca be re-arraged to give the CDF from (2). 32

34 Lemma A12. If r = 1, c 1 = = c K 1 = v(1 α) ad c K (c 1, v] there exist firm resposes i pure-price strategies where v(1 α) is the oly price ever charged. Here, ay (β 1,..., β K ) (0, 1] K ca be supported. Proof: I such a equilibrium, firm profit is π = v(1 α). Cosider a deviatio to p. If p (v(1 α), v], deviatio profit is ˆπ = p(1 α) π. If p < v(1 α), firms have a strict icetive to shift this mass to (v, ). Because shoppers buy directly whe prices are all the same, there are o sales through PCWs ad so firms are idifferet betwee ay (β 1,..., β K ) (0, 1] K. Lemma 1. The mutual best-resposes of firms as a fuctio of c: 1. If c [0, v(1 α)), firm best-resposes are described by G(p; c), ad have o poit masses. 2. If c = v(1 α), there are two classes of resposes, oe with o poit masses described by G(p, c), ad those i which all firms charge the same price. 3. If c (v(1 α), v], all firms charge the same price. Where pricig is described by G(p; c) each firm always lists its price o the PCW. Proof: See Lemmas A10, A11 ad A9 respectively. Propositio 2. I the uique equilibrium where shoppers check the PCW, the PCW sets a click-through fee of c = v (1 α), firms list o the PCW ad mix over prices accordig to G(p; v(1 α)) over the support p [v(1 α), v]. Proof: If c [0, v(1 α)), Lemma A10 shows that there is a profitable upward deviatio to c (c 1, p(c)). It is profitable because there is a icrease i fee-level but o reductio i the quatity of sales. If c (v(1 α), v], Lemma A9 shows that firms will play pure-price strategies ad so u = 0. If c = v(1 α) ad firms play pure-pricig strategies, the u = 0 agai. I these cases, Lemma A10 shows that a deviatio to c (0, v(1 α)) will geerate u 1 > 0, so there are o equilibria where c (v(1 α), v] or for c = v(1 α) whe firms respod with pure-pricig strategies. 33

35 If c = v(1 α) ad firms mix over prices by G(p; v(1 α)) as i Lemma A11, equilibrium PCW profit is u = v(1 α) α K 1 > 0. Lemma A9 shows that ay upward deviatio would yield u = 0. There ca be o profitable deviatio dowwards because, as Lemma A10 shows, the fee would be reduced for o gai i the quatity of sales. Propositio 3. Both types of cosumer are worse off with the PCW tha without. Proof of Propositio 3: First I show that auto-reewers are worse off uder G(p; v(1 α)) tha F (p) (referred to as G ad F here). Auto-reewers pay the price quoted by their curret firm. To show they are worse off with the PCW, I show that E F [p] < E G [p] by showig that G firstorder stochastic domiates (FOSDs) F. The distributios share the same upper boud o their supports, with F havig a lower lower boud. Hece, G FOSDs F if G(p; v(1 α)) F (p) for p [v(1 α), v], which ca be re-arraged as [ (v p)(1 α) αqp ] 1 q 1 [ (v p)(1 α) αp αv(1 α)( 1) ] 1 1. This holds because the terms i paretheses are i [0,1], the power o of the LHS is larger ad the deomiator o the LHS is larger iff p( q) v(1 α)( 1) which is satisfied because p v(1 α) ad q < 1. To show shoppers are worse off uder G tha F, first show that E F [p (1,2) ] is lower tha the lower boud of the support of g i the case of q = 2. The, I use Propositio 3 of Morga et al. (2006) which correspods to my setup (the oly differece is that they have v = 1), which states that E F [p (1,2) ] is decreasig i q. Hece I show the first step here to prove that E F [p (1,2) ] is below v(1 α) for all q. For q = 2, E F [ p(1,2) ] = v v[ 1 α 1+α] f ( p (1,2) ) pdp where f ( p (1,2) ) = 2 (1 F (p)) f(p) is the desity fuctio of the lower of the two draws shoppers receive from F. Computig yields ( ) [ ] 2 [ 1 α v E F p(1,2) = log α 2 ( ) 1 α + 2α ]. 1 + α 1 α 34

36 The E F [ p(1,2) ] < v (1 α) ca be rearraged to obtai which holds for α (0, 1). log ( ) 1 + α > 2α 1 α Corollary 1. Withi the mixed-price equilibrium firm resposes of Lemma 1, as c [0, v(1 α)] icreases, the expected price paid by both types of cosumer icreases. Proof: From Lemma 1 the pricig strategy for c [0, v(1 α)] is give by, G(p; c) over [ p(c), v]. Differetiatig, dg (p; c) dc = ( ) 1 1 (v p) (1 α) 1. c( 1) p αp α ( 1) c The secod term is 0 else G(p; c) > 1. The first term is 0 c 1 because p c > c 1 < p which is esured whe c v(1 α) (this follows because c v(1 α) c). p(c) The for ay c, c [0, v(1 α)], if c > c the the equilibrium pricig distributio uder c first order stochastic domiates that uder c. Hece the expected price (paid by auto-reewers) ad the expected lowest price from draws (paid by shoppers) are higher uder c. Corollary 2. As the umber of firms icreases, the expected price paid by shoppers falls, but the expected price paid by auto-reewers rises. Proof: Usig a observatio from Morga et al. (2006), idustry profit of firms is give by αe G [p (1,) ] + (1 α)e G [p] = v(1 α), where E G [p (1,) ] deotes the lowest price of draws from G(p; v(1 α)) ad E G [p] deotes the expected price from G(p; v(1 α)). The RHS is the idustry profit of firms if it charged v ad oly sold to auto-reewers. Differetiatig ad rearragig, de G [p (1,) ] d so the derivatives have opposite sigs. G(p, v(1 α)) is stochastically ordered i : dg(p; v(1 α)) d ( ) 1 α deg [p] = α d Now show that E G[p] d 0. To do this, show that ( ) (v p)(1 α) ( 1)(p v(1 α)) 0 log + X(p) 0. α(p v(1 α)( 1) p v(1 α)( 1) 35

37 Note that X(v(1 α)) = 0, ad dx dp < 0: dx(p) dp 0 p v(1 α) [ ] (2 2 + α( 1) 2 ). (2 1 + α( 1) 2 ) Notice that the term o RHS i square brackets is below 1. Recall that p v(1 α) as this is the lower boud of the support, hece this is satisfied. Propositio 4. With K > 1 competig PCWs, if shoppers check r = 1 PCW the both types of cosumer are worse off with the PCWs tha without. Proof: There are o equilibria where c [0, v(1 α)): Lemma A10 shows that there is a profitable upward deviatio to c K (c 1, p(c)). It is profitable because there is a icrease i fee-level but o reductio i the quatity of sales. Note that there ca be o profitable deviatio dowwards give r = 1 because the fee would be reduced for o gai i the quatity of sales. There are o equilibria where c (v(1 α), v]: Lemma A9 shows u k = 0 for all k. Lemmas A6 ad A8 show that a deviatio to c 1 (0, v(1 α)) will geerate u 1 > 0. The oly remaiig optio is c 1 = = c K c = v(1 α), where firm resposes are described by Lemma A11. If firms play a pure-pricig strategy, the as i the previous case, PCWs make zero profit ad there is a profitable deviatio to c 1 (0, v(1 α)). If however, firms respod with the mixed-price strategy, this fee-level is a equilibrium: There ca be o profitable deviatio dowwards for PCWs because the fee would be reduced for o gai i the quatity of sales as r = 1. Followig a upward deviatio from PCW K to c K (v(1 α), v], whe firms respod with a pure-pricig strategy as detailed i Lemma A11, PCW K s profit falls to zero. There is the a equilibrium at this fee level, ad it is the uique such level where there exists a equilibrium. At this fee-level, firms play just as they did whe K = 1 with the same CDF over prices ad all PCWs list all firm prices. Both types of cosumer are therefore left with the same level of surplus they had uder K = 1. Corollary 3. As the proportio of shoppers α icreases, expected prices paid by shoppers ad auto-reewers both decrease. 36

38 Proof: Differetiatig G(p; c) by α, dg (p; c) dα = ( ) 1 1 (v p) (1 α) 1 α( 1)(1 α) αp α ( 1) c The first term is > 0. The secod term is always 0 or G(p; c) > 1. The for ay α, α (0, 1), if α > α the the equilibrium pricig distributio uder α first order stochastic domiates that uder α. Hece the expected price (paid by auto-reewers) ad the expected lowest price from draws (paid by shoppers) are lower uder α. Corollary 4. If the PCW ca determie α as well as c i the prelimiary stage, the the PCW sets α = 1 2. Proof: I expad the PCW s actio set to iclude α (0, 1). Notice that for ay choice of α (0, 1), by the reasoig as i the proof of Propositio 2, the PCW will avoid the pure equilibria of Lemma 1 so c [0, v(1 α)], firms mix ad the PCW is give by cα ( ) 1. The PCWs optimizatio problem ca hece be solved by, max c,α cα ( ) 1 s.t. c [0, v(1 α)] ad α (0, 1) where the solutio is c = v(1 α), α = 1 2. A5. Results for K > 1 ad r > 1 Lemma A13. If r > 1, c 1 = = c K 1 [0, v), c K (c 1, v], u K = 0. Proof: This Lemma says that there is o profitable upwards deviatio for a PCW from ay equilibrium. Cosider such a uilateral deviatio by PCW K. Firm respose ca either be pure or mixed pricig. If pure, the u k = 0 for all k. If mixed, the at ay price p that has positive probability of sales through PCWs where K K, firms have a strict preferece to play (p, {1,..., K 1}) istead. This is because r > 1: Every shopper who sees the prices o PCW K also sees the prices o aother PCW. Firms ca therefore avoid PCW K s higher fee by ot listig there while facig o reductio i the quatity of sales. Lemma A14. If r = K, c 2 = = c K (0, v(1 α)] ad c 1 [0, c 2 ): β 1 = 1, β k = 0 for k = 2,... K ad prices are distributed accordig to the CDF G(p, c 1 ). 37

39 Proof: There are o poit masses by Lemma A6. By Lemma A8 β 1 = 1. β k = 0 for k = 2,..., K follows because all shoppers check every PCW (r = K). Therefore, at ay price p that has positive probability of geeratig sales through PCWs, firms have a strict preferece oly to list o PCW 1 without ay reductio i the quatity of sales. By Lemma A7 the maximum of the support is v. Firms must be idifferet to all (p, 1) they are called upo to play, hece: v ( 1 α ) ( = p 1 α ) + α (1 G(p; c 1)) 1 [p ( 1)c 1 ]. which ca be re-arraged to give G(p, c 1 ) from (2). Propositio 5. With K > 1 competig PCWs, shoppers are guarateed to be better off tha before the itroductio of PCWs if r = K > 1. Further, the uique equilibrium PCW fee-level is c = 0 if ad oly if r = K > 1. Proof: Sufficiecy: Suppose ot. The there exists a equilibrium with c > 0. By Lemma A13, there is o profitable upward deviatio. Now cosider a dowward deviatio. If c [v(1 a), v] ad firms respod with a pure-pricig equilibrium, as detailed i Lemma A9, the u k = 0. A deviatio by PCW 1 to c 1 (0, v(1 α)) would lead to the resposes detailed i Lemma A14 ad deviatio profit of u 1 = cα 1 by G(p, c), as i Lemma A14, the u k = c α K c 1 < c exists s.t. u 1 = c 1 α 1 > c α K > 0. If c (0, v(1 a)] ad firms respod with by mixig 1 1 > 0 for all k. But a deviatio by PCW 1 to = u k. This deviatio is strictly profitable, so c could ot have bee a equilibrium. To see that c = 0 is a equilibrium, recall that by Lemma A13 there is o profitable upward deviatio. Necessity: Lemma A11 shows that for K r = 1 the uique equilibrium fee level is c = v(1 α). Lemma 2 shows that for 1 < r < K there are multiple equilibria. Hece K = r > 1 is the oly case where the uique equilibrium of c = 0 obtais. Cosumer welfare: As oted i the text, as firms make the same expected profit i both worlds, there is a oe-to-oe relatio betwee cosumer welfare ad PCW profit. To see the differece i the chages to shopper ad auto-reewer welfare from a move to a world with a PCW but c = 0, Propositio 3 of Morga et al. (2006) (the oly differece is that they have v = 1) shows that the icrease from q to results i a reductio i the expected price paid by shoppers, ad a icrease for auto-reewers. 38

40 Lemma 2. Whe K > r > 1, there exists a equilibrium i which PCWs charge c > 0, ad firms list o all PCWs, mixig over prices by G(p; c) where c = v(1 α)kr(k r) K(1 + r(k 2)) + αr(k 1)(r 1)( 1). Proof: I show that there exist equilibria such that c [0, c]. Note that c [0, v(1 α)) hece ay c [0, v(1 α)). Take such a c as a cadidate equilibrium fee level. There are o poit masses by Lemma A6. By Lemma A7 the maximum of the support is v. By Lemma A5 pr(e) = 0 ad as all PCWs charge the same fee, firms are cotet to list i all of them i.e., β k = 1 for all k. Firms must be idifferet to all (p, {1,..., K}) they are called upo to play, hece v ( 1 α which ca be re-arraged to give G(p; c) i (2). ) ( = p 1 α ) + α (1 G(p; c)) 1 [p ( 1)c] To cofirm c is a equilibrium fee level, esure there is o profitable PCW deviatio. By Lemma A13, there is o profitable upward deviatio. However, ulike Lemma A14, whe 1 < r < K it is o loger true that ay udercut by PCW 1 to c 1 < c will result i all firms listig o it exclusively. This is because cosumers see r > 1 PCWs: whe a firm sells 1 havig played (p, {1,..., K}) it pays c 1 + c K 1, but were it to have played (p, 1) it would K K have paid c 1 r K i.e., the firm faces a trade-off betwee lower fees ad higher sales volume. By Lemma A8 β 1 = 1. PCW 1 s deviatio profit is therefore determied by β k, k > 1. For smalleough udercuts of c, firms will still be cotet to list o all PCWs. Suppose however, that PCW 1 udercuts by just eough such that such that firms are o loger cotet to list all their prices o the other PCWs. I the best case for PCW 1, β k = 0 for all k so that all shoppers checkig PCW 1 buy oly through PCW 1. If i this case, PCW 1 still does ot make more tha its equilibrium profit u = c α K equilibrium fee level. I ow carry out this logic. 1, the the udercut is ever profitable ad c costitutes a Deote c 1 < c as the udercut of PCW 1 ad c 1 as the threshold level required for c 1 to etice firms to de-list some of their prices from PCWk s.t. k > 1. Give c = (c 1, c,..., c), similarly to the derivatio above, oe ca show firms will respod by G(p; c) over [ p(c), v] as 39

41 i (2). Firm deviatio profit from this respose to (p, 1) is give by ( ) 1 α π(p) = p + (v p)(1 α) [Kp r(p c 1)( 1)]. (Kp ( 1)(c 1 + c(k 1))) Note that this is valid for p [ p(c), v], but p < p 1 give strictly less profit tha p = p(c). Oe ca show that π(p) is covex i p. Together with the observatio that (v, 1) gives the equilibrium profit v(1 α), this says that the optimal deviatio is to ( p(c), 1) ad is profitable if ad oly if π( p(c)) > v(1 α), which ca be rearraged to give c 1 < c(k 1)(K + rα( 1)) Kv(K r)(1 α) rα(k 1)( 1) + K(r 1) c 1. Suppose that the firm respose followig the udercut was such that β k = 0 for all k > 1 whe PCW 1 sets the highest such udercut just below c 1. PCW 1 prefers ot to make the deviatio wheever which ca be rearraged to give u αr 1 c 1 K c r c 1 c v(1 α)kr(k r) K(1 + r(k 2)) + αr(k 1)(r 1)( 1) c. Therefore at fee levels c [0, c] there exist equilibria where β k = 1 for all k ad firms mix by the CDF give i the Lemma. Propositio 6. For ay r, there exists a K such that as log as there are more tha K aggregators both types of cosumer are worse off tha before the itroductio of PCWs. Proof: To show for shoppers, let the equilibrium be give as i Lemma 2 with c = c ad see that (4) lim c = v(1 α). K For shopper welfare otice that for K = r, c = 0 ad (5) E F [p (1,q) ] E G(0) [p (1,) ] where G(c) ad F deote G(p; c) ad F (p). This is because the PCWs effectively icrease the umber of firms competig for each shoppers from q to, ad from the last poit of the proof 40

42 of Propositio 5, this lowers the price paid by shoppers. Propositio 3 shows that both types are worse off with PCWs whe c = v(1 α) for K r = 1, but otice that if c = v(1 α) i equilibrium with K > r 1, the this will be true a fortiori, because it ca be show that G(v(1 α)) first-order stochastically domiates F for all q. By (4) the as K, d c E F [p (1,q) ] < E G(v(1 α)) [p (1,) ]. Because c is cotiuous i K, > 0 ad de G( c)[p (1,) ] dk dk > 0 by Corollary 1, there exists K s.t. for K > K there is always a equilibrium fee level that makes shoppers worse off relative to a world without PCWs. The proof for auto-reewers is similar but simpler to the above steps for shoppers ad has p replacig p (1,q) ad p (1,) : From Propositio 3 of Morga et al. (2006), the effect of icreasig the umber of firms competig for each shopper icreases prices for auto-reewers. The by Corollary 1, their prices oly rise further uder G(c). Hece auto-reewers are worse off for ay c ad hece ay K. A6. Results uder Publicly Observable Fees Defiitio (Coordiated Subgame). Give c, a coordiated subgame is whe shoppers atted ad firms list i oly the cheapest PCWs i.e., shoppers i PCW k if c k = c 1, ad for firms β k = 1 if c k = c 1 else β k = 0. Firms mix by G(p, c 1 ). Lemma A15. Whe fees are observed by shoppers, ad all subgames are coordiated subgames there exists a subgame perfect equilibrium where c = 0, β k = 1 for all k ad prices are distributed accordig to the CDF G(p, 0). Proof: There are o profitable deviatios for firms: G(p, 0) esures they are idifferet betwee listig all prices i the support [ p(0), v], listig ay price less tha p(0) is less profitable tha listig v, ad ay (p, ) geerates at most v(1 α) whe p = v. To see there is o profitable upward deviatio for PCWs, ote that due to the coordiated subgame shoppers ad firms would ot atted this PCW followig such a deviatio, so the PCW receives zero profit. To see that these coordiatio subgames are Nash equilibria: for firms oe ca coduct the same roud of checks as at the begiig of this proof; for cosumers otice that as the o-cheapest PCWs are all empty, there is o icetive to check them. 41

43 Propositio 7. Whe K > 1 ad PCW fees are observed by shoppers, there exists a equilibrium with c = 0. Proof: That there exists a equilibrium with c = 0 follows directly from Lemma A15. For shopper welfare at c = 0, see the last poit of the proof of Propositio 5. A7. Results uder Price Discrimiatio Now assume that differet prices ca be charged directly ad through the PCW. Deote such prices p 0 ad p k respectively, where k = 1,..., K idexes the PCW as before. Lemma A16. If r = 1, c 1 = = c K 1 = v(1 α) ad c K [c 1, v] the there exist firm resposes s.t. p 0 = p 1 = = p K = v(1 α) ad β k = 1 for all k. Proof: Uder these strategies, firm profit is π = v(1 α). There is o profitable deviatio i- volvig p k for ay k: lower would prompt sales at p k < c k, higher would ever attract ay shoppers. There is o profitable deviatio ivolvig p 0 : lower would still sell to auto-reewers ad ow-shoppers but at a lower price, higher would oly sell to auto-reewers for which the optimal such deviatio is to v geeratig profit v(1 α) = π. Propositio 8. With price discrimiatio, if r = 1, there exists a equilibrium i which PCWs set c = v(1 α), firms list o all PCWs, p 0 = v ad p 1 = = p K = v(1 α). Proof: Give c 1 = = c K c = v(1 α), let us cofirm there are o profitable deviatios for firms. Equilibrium firm profit is π = v(1 α). There is o profitable deviatio ivolvig p k for ay k: lower would prompt sales at p k < c k, higher would ever attract ay shoppers. There is o profitable deviatio ivolvig p 0 : lower would either sell oly to auto-reewers at a lower price, or to both auto-reewers ad ow-shoppers but oly for prices at least as low as v(1 α), geeratig profit o greater tha v(1 α) ; higher would be above v ad hece make zero profits. Now cosider a PCW deviatio. Equilibrium PCW profit is u k = αv(1 α) K > 0. There is o profitable deviatio to a lower fee: as r = 1 doig so would at best sell to the same proportio of shoppers ( α ) but at a lower fee. As for a deviatio to a higher fee: assume that K firms mutual best resposes are give by those i Lemma A16 where PCW profit is zero. 42

44 Corollary 5. Whe firms price discrimiate, followig c 1 (0, c 2 ) ad c 2 = c 3 =,..., = c K c (0, v(1 α)] there exist mutual best-resposes of firms such that they list o all PCWs, settig p 0 = v ad p k = = c k for all k. Proof: Give c as i the Lemma, let us cofirm there are o profitable deviatios for firms. Firm profit is π = v(1 α). There is o profitable deviatio ivolvig p k for ay k: lower could prompt sales, but oly at p k < c k, higher would ever attract ay shoppers. There is o profitable deviatio ivolvig p 0 : lower would either sell oly to auto-reewers at a lower price, or to both auto-reewers ad ow-shoppers but oly for prices at least as low as c 2, geeratig profit o greater tha c 2 < π; higher would be above v ad hece make zero profits. Lemma A17. If r > 1, c 1 = = c K 1 = 0 ad c K [0, v] the there exist firm resposes s.t. p 0 = v, p 1 = = p K 1 = 0, β k = 1 for k < K ad β K = 0. Proof: Uder these strategies, firm profit is π = v(1 α). There is o profitable deviatio ivolv- ig p K : p K < 0 would give sell at a loss whereas p K > 0 would ever attract ay shoppers as r > 1. There is o profitable deviatio ivolvig p k for ay k < K: Lower would oly create sales at a loss, higher would ever attract ay shoppers. There is o profitable deviatio ivolvig p 0 : lower would still sell to auto-reewers ad (for p 0 0) ow-shoppers but at a lower price, higher would sell to o-oe. Lemma A18. If r > 1, there exists a equilibrium where c = 0, firms list o all PCWs, p 0 = v ad p 1 = = p K = 0.. Proof: Oe ca follow the proof of Propositio 8 to cofirm there are o profitable deviatios for firms. For PCWs, oe oly eed cosider a upward deviatio i fee level i which case, assume firms respod as i Lemma A17, which yields zero profit for the deviatig PCW. Propositio 9. As, followig the itroductio of a PCW, the expected prices faced by both types of cosumer i a settig without price discrimiatio (Propositio 2) approach those i a settig with price discrimiatio (Propositio 8). Proof: Takig limits, lim G(p; c) = 0 43

45 which shows lim E G [p] = v. The CDF for the lowest of draws is give by for p [ p(c), v]. Takig limits, H(p; c) = 1 (1 G(p; c)) = 1 [ ] (v p)(1 α) 1 α (p ( 1)c) lim H(p; c) = 1 ad p(c) lim = c which shows lim E G [p (1,) ] = c. These prices are the same as those i the equilibrium of Propositio 8. A8. Results with Search Costs I the world without a PCW uder q = 2, oe ca compute E F [p] = v [ ] [ ( )] 1 α 1 + α log 2 α 1 α E F [p (1,2) ] = v [ ] 2 [ ( ) 1 α 1 α log + 2α ] 2 α 1 + α 1 α B 0 = v [ ] [ ( ) ] 1 α 1 + α log 2α. 2 α 2 1 α I the world with a PCW uder, Propositio 9 shows that, E G [p] = v, E G [p (1, ) ] = c, B 1 = v c where c = v(1 α) v(1 α) i equilibrium. Propositio 10. Shoppers, coverts ad o-coverts ca all be worse off with a PCW tha without. Proof: This is a proof by example. Let v = 1, α = 0.7, = 3 ad q = 2. The, i the world without a PCW, it ca be computed that, E F [p] = , E F [p (1,2) ] = , B 0 = I the world with a PCW, assume α = 0.71 (i.e., the PCW attracts 0.01 more cosumers i.e., 44

46 1 auto-reewers, to compare prices), so c = v(1 α) = 0.29 ad 30 E G [p] = , E G [p (1,3) ] = , B 1 = Comparig, oe ca see that if there is a S such that this ca be ratioalized as a equilibrium, the all types of cosumer will be worse off with a PCW tha without. To ratioalize, I costruct a S such that: where s is determied by 0 if B 1 < S(B 1 ) = s B 1 s else B 1 S(0.1808) = 1 30 s = Fially, otice that s > B 0. Lemma A19. Uder maximum potetial, ay α ca be ratioalized by a S. Proof: Note that ay α ca be ratioalized with a S (as i the proof of Propositio 10) because B 0 < B 1. To see this holds, oe ca show that B 0 < vα holds for ay α (0, 1) ad hece so does B 0 < v α. Propositio 11. If the market has maximum potetial: coverts are better off; shoppers ca be worse off; ad o-coverts are worse off with a PCW tha without. Proof: As v(1 α) > v(1 α) whe there are coverts, to show coverts are better off with a PCW uder maximum potetial oe ca show, E F [p] > v(1 α) log ( ) 1 + α > 2α 1 α which is satisfied for α (0, 1). Note that α is ratioalizable by some S (Lemma A19). Shoppers are worse off uder maximum potetial wherever v(1 α) > E F [p (1,2) ]. To show this ca occur, ote that v(1 α) > E F [p (1,2) ] holds for all α (0, 1) (see Propositio 3). Hece for ay α there exists some α > α small eough such that shoppers are worse off. Note that such a α is ratioalizable by some S (Lemma A19). No-coverts are worse off uder maximum potetial because E F [p] < E G [p] = v. 45

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