Information Externalities and Intermediaries in Frictional Search Markets

Transcription

1 Iformatio Exteralities ad Itermediaries i Frictioal Search Markets Xiawe Shi ad Aloysius Siow September 3, 013 Abstract I frictioal matchig markets with heterogeeous buyers ad sellers, sellers icur discrete showig costs to show goods to buyers who icur discrete ispectio costs to assess the suitability of the goods o offer. This paper studies how brokers ca help reduce these costs by maagig the level ad mix of goods i their ivetory. We fid that itermediaries emerge ad improve social welfare whe there is sufficiet heterogeeity i the types of goods ad prefereces. Our aalysis highlights how learig ad ivetory maagemet eable search itermediaries to iteralize iformatio exteralities geerated i uitermediated private search. We thak Ettore Damiao, Jea Guillaume Forad, Robert McMilla, Caroly Pitchik, Shouyog Shi, Matt Turer, Asher Wolisky, Adriy Zapechelyuk, ad semiar participats at Pekig Uiversity, Quee s Uiversity, Uiversity of Toroto, ad Ecoometric Society World Cogress at Shaghai for helpful commets ad suggestios. We are grateful to three aoymous referees for may costructive commets. We also thak Lucas Siow for research assistace. The first author is grateful to the Alexader vo Humboldt Foudatio for fiacial support ad to the Chair of Ecoomic Theory II at Uiversity of Bo for its hospitality where part of this work was completed. Both authors also thak SSHRC for fiacial support. This paper was previously circulated uder the title of Search Itermediaries. Departmet of Ecoomics, Uiversity of Toroto, 150 St. George Street, Toroto, ON M5S3G7, Caada. xiawe.shi@utoroto.ca, siow@chass.utoroto.ca. 1

2 1 Itroductio I may frictioal matchig markets, heterogeous buyers ad heterogeous sellers (characterized by differet types of goods) search to match ad trade with each other. Upo meetig a seller, a buyer has to ispect the good o offer to see if it matches his preferece. If there is o trade, both the buyer ad seller will separate ad cotiue to search for other tradig parters. While a buyer s preferece ad the characteristics of a good are persistet over time, buyers ad sellers exit the market after trade ad they seldom retur. Itermediaries are active i some of these markets ad ot others. What is their role? The leadig example of such a market with itermediaries is the residetial housig resale market where the majority of trasactios are brokered by real estate agets. Houses are differet ad a descriptio i a olie listig service, such as the Multiple Listig Service, does ot describe a house completely. Thus a buyer must icur a costly persoal ispectio of the house i order to ivestigate all its attributes. Similarly, a seller also has to icur a discrete showig cost to show the house to iterested buyers. 1 Therefore, both buyers ad sellers i the housig market would like to reduce ispectio or showig costs by avoidig uecessary but costly home ispectios. Aother example is the market for corporate executives where headhuters, as itermediaries, play a importat role i matchig workers with vacacies. Based o iterviews with headhuters, Filay ad Coverdill (00) coclude that the success of a placemet i this market depeds o both tagible iformatio about cadidates ad employer prefereces revealed through job advertisemets ad resumes, ad itagible oes revealed through subsequet costly iterviews. I other labor markets such as retail sales, workers ad employers ofte match directly without usig employmet agecies. There is oe importat differece betwee labor markets which use itermediaries i hirig versus those which do ot: the degree of heterogeeity across workers ad across employers. The objective of this paper is to ivestigate the role of itermediaries i reducig ispectio ad showig costs i the stadard framework of two-sided sequetial search. We study how itermediaries use both the level ad the mix of ivetory to reduce these costs. The degree of heterogeeity i goods/prefereces plays a crucial role i our theory. Although the model is ot specific to a particular market, we will use the housig market as a ogoig example to provide cotext for the model. 1 The assumptio of imperfect advertisig of homes for sale is a commo oe i search models of housig market (e.g., Wheato 1990).

3 To see the advatage a itermediary may have, first cosider a market without brokers. Both houses ad buyers prefereces are fixed ad horizotally differetiated. Suppose a buyer values a house oly if the characteristics of the house (seller type) fit his preferece (buyer type). Both house characteristics ad buyers prefereces are difficult to articulate or describe completely a priori. Each period buyers ad sellers i the market search for tradig opportuities. Whe a type A buyer radomly meets a type b seller, the buyer icurs a ispectio cost ad the seller icurs a showig cost, the buyer fids out that the house is type b ad decides ot to buy it. The iformatio that the house is type b has o value to this buyer for his ow future search, ad similarly, the iformatio that the buyer is type A has o value to the future search of this seller. The iformatio, however, is valuable for other buyers ad sellers to avoid icurrig uecessary search costs. Sice commuicatio with other potetial tradig parters is costly, either the type b seller or the type A buyer has icetive to pass the iformatio o to others. As a result, there are iformatio exteralities: some socially useful iformatio geerated i private search is lost ad ot efficietly utilized. Such iformatio exteralities are crucial for our theory. Three assumptios are key for iformatio exteralities to arise i uitermediated private search. First, goods ad buyers prefereces are heterogeeous, which creates the possibility of mismatch. Secod, prefereces or product characteristics, especially those itagible oes, caot be commuicated easily amog tradig parters. A seller has to icur a showig cost ad a buyer has to icur a ispectio cost to fid out whether the buyer s preferece matches the type of the good. Moreover, oce a seller icurs a cost ad lears the preferece of a buyer, she ca remember the buyer s preferece, but it is agai costly for her to commuicate the preferece to aother seller. Commuicatio barriers could arise because laguage is vague or there is laguage barrier (Lipma 009, Blume ad Board 01). Third, buyers prefereces are persistet. Hece, iformatio about a particular buyer s preferece, though ot useful for the curret seller i case of mismatch, is valuable for the buyer s future tradig parters to avoid uecessary screeig costs. Now cosider such a market with seller brokers. Suppose these brokers do ot have ay ispectio or showig costs advatage over buyers ad sellers. A broker has to look for sellers to represet. Upo meetig a potetial cliet, the broker has to pay a ispectio cost ad the seller has to icur a showig cost for the broker to ispect the good to determie its type. After a agreemet to represet the seller, the broker also has to pay a showig cost to show the good to ay potetial buyer. After sigig up a seller broker, the broker will carry out the subsequet showigs of the house 3

4 Thus employig a broker to complete a trasactio icurs additioal screeig costs which are abset without such a broker. A broker s advatage i the market comes from the possibility that a broker ca represet more tha oe type of sellers. I this case, it is advatageous for a buyer to cotact a broker, because after the broker lears of the buyer s type through a first showig, the broker ca ecoomize o further showig costs ad ispectio costs by ot showig other goods that the buyer will ot be iterested i. For example, suppose a type A buyer radomly meets a broker who represets two differet houses, say type b ad c. After showig house b to the buyer, the broker lears that the buyer is type A. 3 The the broker ca tell the buyer that there is o eed for him to see the secod house because it does ot fit, savig screeig costs for both parties. This advatage ca be materialized oly whe the broker has additioal goods i his ivetory which are of potetial iterest to the buyer. Each seller hires a costly broker because the broker has other types of goods i his ivetory to attract potetial buyers. 4 Thus brokers have to actively maage both the level ad mix of goods that they represet. We first study the case of seller brokers. A seller broker first picks up seller represetatios from the sellers market ad the sells houses to buyers o the sellers behalf i the buyers market. We restrict our brokers to represet at most two cliets at a time. This is the miimal size of ivetory eeded for brokers to exist. I order to tease out the itermediaries role i reducig discrete search costs from their other roles, our aalysis primarily focuses o the limit equilibrium as the discout rate goes to zero. I the limit equilibrium, brokers exist ad improve welfare if there is sufficiet heterogeeity i the types of buyers ad goods. Furthermore, brokers reduce the expected total ispectio ad showig costs by all parties eeded to complete a trasactio. Because the umber of houses see by a buyer is a good proxy for the buyer search duratio while the umber of showigs by a seller is a good proxy of the time o the market, our model predicts that the real-estate brokers reduce the expected time for a seller to sell a house ad a buyer to buy a house, which is a robust fidig of empirical studies o the role of real estate brokers (see for example, Baryla ad Zumpao 1995, Hedel, Nevo, ad Ortalo-Mage 009, Berheim ad ad thus icur all the subsequet showig costs. If the broker fids a buyer that likes the house, the the seller eeds to icur oe last showig cost (bargaiig cost etc.) i order to fialize the trasactio. 3 It is ot critical that the broker lears the buyer prefereces perfectly. Appedix B exteds the aalysis to the settig where the commuicatio betwee buyers ad brokers is oisy. 4 The argumet is remiiscet of Wolisky s (1983) argumet for why competig retailers locate i shoppig malls i spite of more itese price competitio. By ispectig differet goods i the same locatio, shoppig malls help cosumers save o travel costs. 4

5 Meer 008, amog others). We the exted the aalysis to the case of buyer brokers who first seek buyer represetatios before goig to the sellers market to buy houses o the buyers behalf. Agai with sufficiet heterogeeity, buyer brokers emerge ad improve welfare i the limit equilibrium. However, for fixed model parameters, seller brokers ad buyer brokers caot be both welfare improvig i equilibrium. I particular, welfare improvig buyer brokers ca exist oly if the ispectio cost is less tha the showig cost, while welfare improvig seller brokers ca exist oly if the reverse is true. The atural separatio of the parameter space arises because i the case of buyer brokers the cost of itermediatio (equilibrium commissio fee) is primarily due to ispectio costs i cotrast to the case of seller brokers where showig costs make up the majority of the itermediatio cost. The remaiig of the paper is orgaized as follows. The ext sectio reviews related literature. There we discuss the coectios ad differeces betwee this paper ad related studies o search itermediaries. We the start with a two-sided search model without brokers i Sectio 3, which will serve as a bechmark for our aalysis of brokers. The seller brokers are itroduced ito the model i Sectio 4. We show that seller brokers improve social welfare by reducig expected total umbers of showigs ad ispectios ecessary to complete a trasactio. Sectio 5 briefly ivestigates the case of search with buyer brokers. Sectio 6 discusses a few extesios of the model ad cocludes. Several appedices collect techical results, oe robustess check ad some algebras that are omitted from the text. Related Literature Our paper belogs to the literature o search itermediaries startig with Rubistei ad Wolisky (1987). This literature focuses primarily o the role of itermediaries i reducig delay costs, ad typically assumes that itermediaries ejoy superior search techology tha idividual buyers or sellers. For example, i Rubistei ad Wolisky, itermediaries or middleme emerge ad improve welfare because they ca meet customers faster tha buyers ca meet sellers directly. I cotrast, itermediaries i our model share the same search techology as buyers ad sellers. The two mostly closely related papers are Johri ad Leach (00) ad Shevcheko (004). I settigs similar to ours with heterogeeous goods ad tastes, Johri ad Leach (00) ad Shevcheko (004) assume middleme have o iheret advatage i search techology ad explore how middleme, who ca hold more tha oe uit of ivetory, ca geerate welfare gais over uitermediated search. I Johri ad 5

6 Leach (00), there are may ifiitely lived cosumers ad suppliers. Each supplier ca hold up to two uits of heterogeous goods, ad ca produce oe oly if he does ot curretly hold ay heterogeous goods. A supplier with oe uit of good, defied as a producer, ca become a middlema, defied as a supplier with two uits of ivetory, by pickig up a secod good from aother supplier through radom search, ad a middlema ca covert back to be a producer by sellig oe uit to a cosumer, agai through radom search. Similar to our model, a buyer is better off meetig a middlema tha meetig a producer, because a middlema with two heterogeous goods ca geerate a high matchig value with higher probability for the buyer. I Shevcheko (004), each aget produces oe type of goods, but desires to cosume aother so they search. If a aget radomly meets aother aget, they trade oly if there is a double coicidece. But if a aget radomly meets a middlema, double coicidece is ot required, because middleme are assumed to be edowed with a techology that ca trasform ay good ito a good they desire to cosume. As a result, a trade occurs betwee a aget ad a middlema as log as the middlema carries the good that the aget desires. Middleme ca icrease tradig probability by icrease their ivetory variety, but this comes at a cost which is icreasig ad covex. He derives both the optimal level of ivetory ad the equilibrium price distributio of goods. I Shevcheko (004) ad our model, the buyers prefereces are biary: a good is either acceptable or ot, while Johri ad Leach (00) allows for more geeral prefereces: a good is always acceptable to a buyer at certai price. Both Johri ad Leach (00) ad Shevcheko (004) assume that the realizatio of matchig value is idepedet across both uits ad buyer. Moreover, upo each meetig of a buyer ad a middlema, the matchig value is a fresh radom draw ad it is immediately revealed freely whether ad how the buyer s preferece matches ay goods i the middlema s etire ivetory. I cotrast, i our model, both goods type ad buyer type are persistet, ad it is costly for both tradig parties to fid out whether the buyer likes ay goods i the ivetory. 5 Therefore, the iformatio exteralities idetified i this paper are abset i both Johri ad Leach (00) ad Shevcheko (004), ad learig does ot play a role sice iformatio is always revealed freely upo meetig. I our model, ivetory per se does ot automatically result i a cost advatage for itermediaries. It is the combiatio of ivetory ad learig that allows itermediaries to ecoomize o showig ad ispectio costs. 6 5 There are a large literature of frictioal matchig with vertically differetiated persistet types (e.g. Burdett ad Coles 1997, Smith ad Shimer 000). The iformatioal iefficiecy idetified here is also preset there. 6 Iformatioal exteralities i frictioal matchig markets without itermediaries uderlies the 6

7 Aother strad of literature o itermediaries argue that itermediaries emerge because they ca help mitigate bilateral cotractig problems which arise from the search eviromet, such as moral hazard (Diamod 1984), ad adverse selectio (Biglaiser 1993, Li 1998). For example, Diamod (1984) argues that by aggregatig borrowers, fiacial itermediaries ca reduce the variace of its portfolio payoff, reducig the cost of moitorig, ad thus mitigate the icetive problems betwee borrowers ad leders. Biglaiser (1993) ad Li (1998) show that the emergece of itermediaries ca improve social welfare i markets with adverse selectio if itermediaries have a reputatio cocer ad a lower moitorig cost due to ivetories which are larger tha those carried by idividual buyers. Similar to these papers, the advatage of our itermediaries also comes from larger ivetories. Differet from these papers, however, iformatio is symmetric betwee tradig parters i our model: either the seller or the buyer kows the matchig value before ispectio ad they both kow it after ispectio. Fially, there is also a sizeable literature o the price-settig itermediaries, such as dealers ad market makers, i search markets. For example, Spulber (1996) studies a equilibrium search model with three types of agets: cosumers, producers ad middleme (dealers). I his model, cosumers ad producers caot trade directly with each other. Istead all trades have to be itermediated by the middleme who post bid ad ask prices. He shows that, the bid ad ask prices coverge to Walrasia equilibrium prices as the discout rate goes to zero, ad approach moopoly prices as the discout rate icreases. Rust ad Hall (003) itroduce a moopolist market maker ito Spulber (1996), ad ivestigate whe the market maker ca survive ad improve welfare. Yavas (199) compares market makers ad match makers i a oe-period bilateral search model of itermediaries. I his model, it takes time ad effort for idividual participats to fid a match, while market makers ca trade immediately by postig bid ad ask prices, ad match makers ca provide matchig service to improve the matchig probabilities of idividual participats. Noe of these papers cocers the type of iformatio exteralities that we focus o. social learig literature (see a survey by Bikhchadai, Hirshleifer, ad Welch, 1998). But to our best kowledge, the role of itermediaries i iteralizig iformatio exteralities through learig ad ivetory maagemet has ot bee explored. 7

8 3 Search without Brokers To aid expositio, we preset the model i the familiar housig market settig. Time is discrete with a period legth which is assumed to be small. 7 All participats discout the future with a commo discout rate r. Followig the stadard twosided search literature, we assume that two parties, buyers ad sellers, simultaeously search for tradig opportuities i the market. Each seller has oe house for sale, ad each buyer wats to buy oe house. Buyers ad sellers i the market are matched accordig to a radom matchig techology. Specifically, for a give period, if there are B buyers ad S sellers i the market, the M (B, S) buyers will radomly match with the same umber of sellers. The matchig fuctio M (B, S) is assumed to be icreasig i both argumets ad has costat retur to scale. We use θ = B/S to deote the market tightess ad use m (θ) to deote the arrival rate of match for a seller: m (θ) M (B, S) /S. The, i each period, a seller radomly meets a buyer with probability M (B, S) S = m (θ), ad a buyer meets a seller with probability M (B, S) B = m (θ). θ There are types of buyers ad sellers, with equal fractio of each type i the populatio. Whe a radomly chose buyer meets a radomly chose seller, the value of the match is either 1 or 0. If the buyer s type (preferece) matches the type of the house for sale, the value of the house to the buyer is 1. Otherwise, the house has value 0 to the buyer. Before the buyer sees the house, both the buyer ad the seller do ot kow the match value which ca be foud out oly through costly house ispectio. Every time a buyer ispects a house, the seller has to pay a showig cost c s ad the buyer has to pay a ispectio cost c b. If the seller has the type of the house that the buyer wats to buy ad they are able to egotiate a sale, both parties leave the market permaetly; otherwise, both of them will retur to the market. We assume that there is a icomig flow of ew buyers ad ew sellers such that the stocks ad distributios of buyers ad sellers do ot chage over time. 7 As stadard i the literature, the small assumptio allows us to use Poisso arrival process to approximate the discrete arrival probabilities. 8

9 The goal of the paper is to ivestigate the role of search itermediaries i iteralizig iformatio exteralities by reducig discrete search costs. I order to tease out itermediaries role i reducig discrete search costs, we will primarily focus o the limit (steady-state) equilibrium whe r approaches 0. 8 Note that it is sesible to focus o the limit equilibrium with r = 0 oly if we ca prove the existece of a sequece of equilibria with r > 0 that coverge to the limit equilibrium. I Appedix A, we use Nash bargaiig as a example of bargaiig protocols to illustrate how to costruct such a sequece both for search without brokers ad for search with seller brokers. 3.1 Equilibrium Welfare The cotiuatio payoff V for a seller who remais i the market at the ed of period, whe is small, is give by V = r [ m (θ) ( c s + 1 ) p + (1 1 ) ] m (θ) V. (1) To uderstad the formula, ote that if the seller radomly meets a buyer ext period (which happes with probability m (θ) ), she icurs a showig cost c s to show the house to the buyer, ad if after costly ispectio the buyer likes the house (which happes with probability 1/) she sells the house at a egotiated price p; if the seller does ot meet ay buyer or if the seller meets a buyer but the match value turs out to be 0 after costly ispectio, the seller remais i the market ad receives cotiuatio value V, which happes with probability ( 1 1 m (θ) ). Similarly, the cotiuatio payoff U for a buyer who remais i the market at the ed of period, whe is small, is give by [ 1 m (θ) U = ( c b + 1 ) 1 + r θ (1 p) + ( 1 1 ) ] m (θ) U. () θ If the buyer radomly meets a seller ext period (which happes with probability m (θ) /θ), he icurs a ispectio cost c b to ispect the house, ad if he likes the house (which happes with probability 1/) he will buy the house at the egotiated price p. If the buyer does ot meet ay seller, or if the buyer meets a seller but the match value turs out to be 0 after costly ispectio, the buyer remais i the market ad receives U, which happes with probability (1 1 m(θ) ). θ The price p is egotiated by the two tradig parties. It potetially depeds o the discout rate r ad market coditios such as θ ad m (θ). Differet bargaiig 8 The sequetial search literature worries about delay cost ad is ucocered with ispectio ad showig costs (e.g. Rogerso, Shimer ad Wright 005). A otable exceptio is Ataka (006). 9

10 protocols may result i differet trasactio prices, but the total social welfare is idepedet of prices. Sice we are primarily cocered about the total welfare implicatio of search itermediaries, we do ot assume a particular bargaiig protocol. Istead, we oly impose a weak requiremet that uder the trasactio price both tradig parties are willig to participate i the market, that is, a feasible price must i the core for each meetig. We complete the model by imposig a free etry coditio for sellers. Let F be the cost to a home builder to build a house. The free etry of sellers implies V = F. I order for the market to exist, we assume throughout of the paper that 1 c b c s > F. (3) Defiitio 1 A steady-state search equilibrium without itermediaries is defied by the stocks of market participats (B, S), cotiuatio payoffs (U, V ), ad market price p such that (a) seller ad buyers behave optimally so (1) ad () hold; (b) price p is feasible; ad (c) free etry coditio V = F holds. We are primarily iterested i the limit equilibrium with r 0 where delay cost is egligible. We derive all our equilibrium cotiuatio values for geeral r, ad the take the limit. 9 It is straightforward to derive U ad V from (1) ad (), ad take r 0 to obtai the limit cotiuatio value U ad V as: V = lim r 0 p c s 1 + r/ (m (θ)) = p c s, U 1 p c b = lim r rθ/ (m (θ)) = 1 p c b, It follows from the free etry coditio V = F that the limit price p must be p = F + c s. (4) Therefore, the total social welfare for a pair of seller ad buyer is U + V = 1 c b c s. (5) This will serve as the welfare bechmark for our subsequet aalysis of search with brokers. Ituitively, there are type of goods, so whe a pair of buyer ad seller meets, the probability of havig a successful match is 1/. This implies that the expected umber of ispectios for producig a successful match is. Hece, the total welfare (U + V ) for a pair of buyer ad seller is 1 c b c s. 9 The derivatio is stadard i the frictioal search literature. See, for example, Rogerso, Shimer ad Wright (005). 10

11 4 Search with Seller Brokers Agai suppose there are types of houses ad buyers with equal proportio. We add seller-brokers (heceforth brokers) who first cotact sellers to seek exclusive represetatio ad the sell houses o the sellers behalf to buyers. For simplicity ad tractability, we assume that there are two physically distict markets where trade occurs. I the sellers market, brokers search for sellers to represet them. I the buyers market, brokers meet buyers to arrage trasactios o the sellers behalf. Buyers, sellers ad brokers ca visit either market at ay time. Each participat ca visit oly oe market at a time. We say a broker completes a trasactio after he picks up a seller represetatio i the sellers market ad the successfully sells the house o the seller s behalf to a buyer i the buyers market. For techical tractability, we assume that a broker ca represet at most two sellers. Uder this assumptio, there are oly four possible types of brokers: brokers without a house, brokers with oe house, brokers with two idetical houses, ad brokers with two distict houses. Similarly, there are oly four possible sellers: sellers without brokers, or sellers cotracted to oe of three brokers with houses. I other words, this small capacity assumptio reduces the umber of possible broker types ad seller types we eed to track i equilibrium. If we icrease the capacity of brokers, the umber of possible broker types ad seller types grows expoetially, substatially complicatig the aalysis without addig much isights. Sice a larger capacity will oly stregthe the advatage of itermediated trade over direct trade, our small capacity assumptio will lead to most striget coditios for search itermediaries to arise ad improve welfare. We assume for ow that a broker wats to have two differet types of houses i her ivetory before goig to the buyers market. Upo sellig oe house, the broker will retur to the sellers market to fid aother seller to represet whose house is differet from the oe that the broker has already represeted. After the broker has obtaied represetatio of two differet types of houses, the broker returs to the buyers market ad so o. We assume that, after a seller sigs up a seller broker, all future showig costs are icurred by the broker rather tha the seller, except that, whe the broker fids a perfect match for the house, the ower of the house (the seller) eeds to icur oe last showig cost to the buyer (who the icurs the correspodig ispectio cost) to complete the trade. The last showig (ispectio) cost captures potetial closig or bargaiig cost that the seller (buyer) icurs i order to complete the deal. The assumptio of the last showig cost also has a techical role. As we previously metioed, i order to justify our focus of the limit equilibrium, we eed to prove 11

12 the existece of a sequece of equilibria with r > 0 that coverge to the limit oe. We costruct such a sequece by assumig Nash bargaiig protocol i Appedix A. With Nash bargaiig, if the seller does ot eed to icur the last showig cost, i the limit with r 0, the seller will hold up the broker because, while the seller icurs o cost to bypass the curret buyer ad fid the ext suitable buyer, it is costly for the broker to do so. As a result, the limit of the sequece of equilibria uder Nash bargaiig may ot exist. The last showig cost ca help avoid this hold-up problem. Sice the additioal showig cost cuts i the advatage of search with brokers, this agai will lead to more striget coditios for welfare improvig brokers to exist. We assume that brokers have o cost advatage over sellers ad buyers. First, whe a seller ad a broker radomly meet i the sellers market, a costly ispectio is carried out. After the seller icurs a showig cost c s ad the broker icurs a ispectio cost c b, the broker lears the type of the house. Secod, whe a buyer ad a broker radomly meet i the buyers market, the broker icurs a showig cost to show a radomly chose house to the buyer who icurs a ispectio cost to see the house. After costly ispectio, the buyer figures out if he likes the house, ad the broker also lears the buyer s preferred type of house. I Appedix B, we show that our aalysis remais valid qualitatively if we relax the assumptio of perfect commuicatio. We also assume that brokers have o matchig advatage over sellers ad buyers: the matchig techology betwee brokers ad sellers (or buyers) is the same as the oe i the market without itermediaries. 10 Let A s deote the umber of brokers with oe house i the sellers market, ad A b deote the umber of brokers with two houses i the buyers market. Let θ s deote the market tightess of the sellers market where brokers pick up houses: θ s = A s /S, ad let m (θ s ) deote the the arrival rate of a match for a seller i the sellers market: m (θ s ) M (A s, S) /S = M (θ s, 1). The i the sellers market, The probability for a seller to meet a broker is m (θ s ), ad the probability for a broker to meet a seller is m (θ s ) /θ s. Similarly, defie θ b = B/A b as the market tightess of the buyers market where brokers sell houses to buyers, ad let m (θ b ) deote the the arrival rate of a match for a broker i the buyers market: m (θ b ) M (B, A b ) /A b = M (θ b, 1). The i the buyers market, the probability for a broker to radomly meet a buyer is m (θ b ), ad the probability for a buyer to meet a broker is m (θ b ) /θ b. To simplify otatio, i what follows we write m s = m (θ s ) 10 I practice, however, it is ofte easier for home buyers ad sellers to cotact brokers tha to cotact each other. This gives a exogeous advatage to brokers i meetig buyers ad sellers. We wat to dispese with this advatage which has already bee studied i Rubistei ad Wolisky (1987). Thus we do ot give brokers a exogeous faster arrival rate of cliets to focus o the ivetory advatage. 1

13 ad m b = m (θ b ). We also assume for ow that (i) buyers ad sellers will ot trade directly, (ii) sellers will ot preted to be brokers with two houses, ad (iii) buyers will ot preted to be brokers with oe house. We also assume earlier that (iv) a broker wats to have two differet types of houses i his ivetory before goig to the buyers market. Later we will specify coditios uder which, i the search equilibrium with brokers, these icetive coditios (i)-(iv) are ideed satisfied. That is, we first characterize a limit equilibrium where brokers are active ad the we will derive coditios for equilibrium existece. 4.1 Equilibrium Welfare Now we wat to derive the cotiuatio values for a seller, a broker with oe house, a broker with two distict houses, ad a buyer. A seller i the market could be i oe of the followig three possible states: ot cotracted with a broker, cotracted with a broker who has aother cliet, ad cotracted with a broker who has o other cliet. First, the fuctioal equatio for V, the cotiuatio value of a seller without a broker at the ed of period, is [ V = m s r ( c s + 1 (V a φ) ) + ( 1 m s 1 ) ] V. (6) Here m s is the probability for a seller to meet a broker i the sellers market. Oce they meet, the seller icurs cost c s to show the house to the broker who icurs a cost c b. With probability ( 1) /, the house type is differet from the type of the other house that the broker already represets. I this case, the seller pays a commissio φ to the broker who will show the house o the seller s behalf. Otherwise, the seller cotiues to search for a broker i the sellers market. Secod, the fuctioal equatio for V a, the cotiuatio value of a seller cotracted with a broker who has aother cliet, is [ V a = m b r ( 1 (p c s) + 1 V b ) + ( 1 m b ) ] V a. (7) The seller s broker meets a buyer i the buyers market with probability m b. Oce the broker ad the buyer meet, the broker icurs a cost c s ad the buyer icurs a cost c b through a costly ispectio to fid out the buyer s preferece. If the buyer s type is a perfect match with oe of the broker s house (which happes with probability /, with probability 1/ for each seller), the broker asks the ower of the house to show it to the buyer. Cosider seller 1 who is cotracted with the broker. If her house 13

14 matches the buyer preferece (with probability 1/), she icurs the fial showig cost c s to show her house to the buyer ad obtais price p; if the other house of the broker matches the buyer preferece (with probability 1/), the her broker has to go back to the sellers market to pick up aother house, that is, her cotiuatio value becomes V b which is defied below. If oe of the above happes (with probability 1 m b ), the seller retais cotiuatio value V a. Third, the fuctioal equatio for V b, the cotiuatio value of a seller who is cotracted with a broker but her broker has o other cliet ad thus has to retur to the sellers market to pick up aother seller before sellig her house i the buyers market, is V b = [ 1 ms 1 ( 1 + r θ s V a + 1 m s 1 ) ] V b. (8) θ s The term ms θ s is the probability that the broker meets aother seller i the sellers market ad 1 is the probability that the secod house is differet from the type of the first house that the broker already represets. That is, ms θ s 1 is the probability for the broker to successfully pick up aother seller, ad i this case the cotiuatio value for the seller becomes V a. Otherwise, the broker has to cotiue search i the sellers market for aother period. Let W 1 ad W be the cotiuatio value of a broker with oe house ad a broker with two houses at the ed of period, respectively. The the fuctioal equatio for a broker with oe house is [ ( 1 ms W 1 = c b + 1 ) 1 + r θ s (W + φ) + ( 1 m s 1 ) ] W 1. (9) θ s Here ms θ s is the probability for a broker with oe house to meet aother seller i the sellers market. Oce they meet, the broker performs a costly ispectio. With probability 1, the secod house type is differet from the type of the first house represeted by the broker. I this case, the broker will agree to represet the secod house with commissio φ, ad the cotiuatio value for the broker becomes W. Otherwise the broker has to cotiue search i the sellers market. The fuctioal equatio for a broker with two houses is [ ( 1 W = m b c s r (W 1 1 ) c s) + ( 1 m b ) W ]. (10) The term m b is the probability for the broker to meet a buyer i the buyers market. Upo a meetig, the broker shows a first house to the buyer at a cost of c s. After showig the house, the broker will lear whether the buyer likes the first or secod house, or other types of houses. If the buyer likes the first house (with probability 1/), the buyer will ext deal with the seller to buy the house. If the broker lears 14

15 that the buyer likes the secod house (with probability 1/), he will icur aother showig cost c s to show the secod house to the buyer. After seeig the secod house, the buyer will deal with the seller to buy the house. After showig the first house, if the broker lears that the buyer does ot like either house, they separate ad the broker has to cotiue search i the buyers market for aother period. Fially, let U be the cotiuatio value of a buyer who remais i the market at the ed of period. The we have [ ( 1 mb U = c b + ( 1 p 3 )) ( 1 + r θ b c b + 1 m b ) ] U. (11) θ b The term m b θ b is the probability for a buyer to meet a broker with two houses i the buyers market. Oce a buyer meets a broker, the buyer icurs a ispectio cost c b to see a house. If the first house fits (with probability 1/), the buyer will icur aother ispectio cost to deal with the seller ad buy the house for price p. If the first house does ot fit ad the secod house fits (with probability 1/), the buyer will icur aother ispectio cost to see the secod house, ad a further ispectio cost to deal with the seller ad buy the house for price p. If either house fits, the buyer separates from the broker ad cotiue to search i the buyers market for aother period. As i the previous sectio, we do ot specify the bargaiig protocol that determies the trasactio price p ad the commissio φ. We oly require that the price ad the commissio are feasible i the followig sese: the price p or the commissio φ is i the core for every meetig, ad the expected profit for a broker from each trasactio is zero as r 0. I the steady state, the umber of brokers pickig up houses successfully i the sellers market must be equal to the umber of brokers sellig houses successfully i the buyers market. That is, 1 m s A s = A b θ s m b. (1) Fially, we impose free etry coditios to complete the model. First, with free etry of home builders, sellers must get the same reservatio utility as home builders, that is, V = F. Secod, we impose free etry coditio for brokers. Let L deote the broker s outside optio. The broker without a cliet ca go to the sellers market to pick up a house by icurrig cost c b. But the cotiuatio value for a broker with oe cliet is W 1. Therefore, whe r 0, we ca write the free etry coditio for the broker as: W 1 = L + c b φ. 15

16 Defiitio A steady-state search equilibrium with brokers is defied by the stocks of market participats (B, S), cotiuatio payoffs (V, V a, V b, W 1, W, U), commissio φ ad market price p such that (a) all market participats behave optimally so that equatios (6)-(11) hold; (b) the statioary coditio (1) holds; (c) commissio φ ad price p are feasible; (d) free etry coditios for sellers ad brokers hold; ad (f) icetive coditios (i)-(iv) are satisfied. Agai, as i the previous sectio, we derive all our equilibrium cotiuatio values for geeral r, ad the take the limit. It is straightforward to compute the cotiuatio value U ad V for a fixed r, ad the take r 0 to obtai the limit cotiuatio values: U = 1 p + 3 c b, V = p φ 1 1 c s, where p is the limit of trasactio price p, ad φ is the limit of the commissio fee φ. Note that the limit commissio fee φ must be such that the expected profit for a broker from each trasactio is zero as r 0. Otherwise, brokers will make ifiite total profits because they are ifiitely lived. As ca be verified usig fuctioal equatios (9) ad (10), i order to esure the existece of the limit of W 1 ad W, the limit commissio fee φ must exactly cover the expected ispectio costs ad showig costs that a broker eeds to icur i order to complete a trade, that is, φ = + 1 c s + 1 c b. (13) To gather ituitio, we recall that each trasactio cosists of a purchase ad a sale. Notice that a broker eeds to meet / buyers o average i order to fid a buyer who will like oe of the two houses, ad the broker eeds to meet / ( 1) sellers i order to re-stock a house. Ad whe he fids a buyer who matches a house, the broker may still eed to icur aother showig cost with probability 1/. This gives us the right had side of (13). It the follows from the free etry coditio for the seller V = F that the limit price must be p = F + 1 c b + ( ) c s (14) Sice the brokers ear zero profit from each successful trasactio, the total expected payoff of buyers ad sellers coicide with the total social welfare. Therefore, 16

17 i the limit search equilibrium with seller brokers, the total expected social welfare for a pair of seller ad buyer U + V = 1 1 (c b + c s ) + 3 (c b + c s ) (15) The ituitio for (15) is the followig. First, a successful trasactio takes 1 ispectios o average i the sellers market because a broker picks up a ew seller with probability 1. Secod, a buyer eeds to talk to brokers o average i order to fid a good match because the probability that a buyer likes oe of the two houses maaged by a broker is. Moreover, i case there is a match, the buyer eeds to deal with the seller of the house oe more time to verify the match. As a result, a successful trasactio eeds +1 ispectios i the buyers market. Fially there is a expected extra ispectio whe a broker meets a buyer because coditioal o a match, the 1 secod house, rather tha the first, fits the buyer with probability 1/. Therefore, the total expected search cost for a successful trasactio is (c 1 b + c s ) + +3 (c b + c s ). The expressio (15) the follows from the fact that the value of a good match is 1. By comparig the social welfare with brokers (15) ad without brokers (5), we obtai the welfare improvemet (for a pair of buyer ad seller) due to brokers: ( 1) (c b + c s ). (16) Therefore, the itroductio of brokers improves social welfare as log as 6. Ituitively, itroducig brokers ito the market adds two extra rouds of screeig costs from usig brokers. I order to recover these additioal screeig costs, the degree of heterogeeity must be sufficietly large such that buyers ca avoid ispectig houses that they will reject. 4. Icetive Coditios I order to fully characterize the equilibrium, we eed to fid coditios uder which the followig icetive coditios hold: (i) buyers ad sellers will ot trade directly, (ii) sellers will ot directly go to the buyers market, (iii) buyers will ot directly go to the sellers market, ad (iv) brokers will ot go to the buyers market uless they have picked up two seller represetatios with two differet types of houses. Recall that, whe 6, the joit payoffs of buyers ad sellers are higher i dealig with brokers compared to tradig directly. By tradig directly either the seller or the buyer will be worse off ad thus at least oe of the two parties will refuse to trade directly. Therefore, icetive coditio (i) is satisfied if 6. It remais to fid coditios for (ii)-(iv). 17

18 Let us first cosider the icetive of sellers. A seller ca preted to be a broker with two houses ad approach buyers i the buyers market. If a buyer rejects the house o offer, the seller ca say that the other house i her phatom ivetory does ot fit the buyer s preferece either. Whe a seller goes to the buyers market directly by pretedig to be a broker, she saves the commissio but icurs a icrease i expected showig cost equal to c s ( 1 + 1) c s, where the first term is the expected showig cost without a broker ad the secod term is the expected showig cost with a broker. Therefore, a seller will ot preted to be a broker if which is equivalet to φ = + 1 c s + 1 c b c s ( ) c s, c b (17) c s Next we cosider the icetives of buyers. A buyer ca preted to be a broker with oe house ad buy directly from a seller at price (p φ ) i the followig way. If the house fits the buyer s preferece, the buyer pays (p φ ) to the seller who will happily accept. If the house does ot fit, the buyer tells the seller that the house coicides what he already has. Whe a buyer goes to the sellers market directly by pretedig to be a broker, he gais from price reductio of φ but icurs a icrease ( ( i expected ispectio cost equal to c b )) cb. The first term is the expected ispectio cost without a broker ad the secod term is the expected cost with a broker. Buyers will ot eter the sellers market directly if φ = + 1 c s + 1 c b c b ( 1 + ( )) c b That is c b 1 c s (18) Fially, we cosider the icetives of brokers. Although i priciple the broker ca deviate for ay umber of trasactios, by the uimprovability criterio of dyamic programmig, it is sufficiet to show that the broker caot gai from deviatio for oe trasactio. We first fid coditios uder which a broker with oe house will ot immediately go to the buyers market to look for a buyer. Sice a broker s deviatio caot affect market prices, we oly eed to compare expected cost to complete a trasactio. Cosider a broker with oe house. If he picks up aother house of a differet type before he goes to the buyers market, his additioal expected cost to complete a trasactio is c 1 b + +1c s. If a broker with oe house goes to the buyers market directly, his expected cost to completig a trasactio is c s. Therefore, a 18

19 broker with oe house will ot preted to be a broker with two houses ad search buyers directly if 1 c b c s c s which reduces to c b + 1. (19) c s Next, we look for coditios to isure that a broker will ot go to the buyers market with two idetical houses. Suppose a broker with oe house meets a seller to pick up a secod house ad fids out that it is a duplicate of the first. This broker s expected cost of completig a trasactio is o differet from that of a broker with oe house. We have show earlier that uder coditio (19), a broker with oe house wats to fid aother house which is ot the duplicate of the first. Sice his ispectio of the duplicate secod house is already suk, there is o additioal cost to discardig it ad searchig for a differet house compared with a broker with oe house. Therefore, as log as coditio (19) holds, the broker will reject the seller ad cotiue search i the sellers market rather tha go to the buyers market with two idetical houses. 4.3 Summary It is easy to see that the broker s icetive coditio (19) is implied by the seller s icetive coditio (17) for all. Therefore, all parties icetive coditios are satisfied if We ca plot coditio (0) as follows: c b (0) c s 19

20 Notice that the set of cost ratio c b /c s that satisfies above costraits is o-empty as log as 11. Moreover, as the degree of heterogeeity icreases, the set defied by (0) expads. However, the welfare improvig seller brokers exist oly if c b > c s. The ratioale for seller brokers is to hold ivetory i order to ecoomize o the buyers expected ispectio costs. Thus it should ot be surprisig that such equilibria exists oly whe ispectio cost exceeds showig costs (see Sectio 5 for aalysis of buyer brokers whe the reverse is true). The followig propositio summarizes the mai result of this sectio. Propositio 1 Suppose that broker learig is perfect. A search equilibrium with seller-brokers exists ad improves social welfare if coditio (0) holds. To coclude this sectio, we ote that if brokers eed a licese to operate, the either buyers or sellers ca preted to be brokers. If this is the case, the oly relevat icetive costraits are those for brokers, ad the costrait that buyers ad sellers do ot trade directly. Therefore, brokers exist ad improve welfare if 6 ad (19) holds. 5 The Case of Buyer Brokers We ote that i the previous sectio, the search equilibrium with seller brokers may ot exist if c b < c s. Ca other types of itermediaries exist if c b < c s? This sectio briefly illustrates how we ca adapt our previous aalysis to study buyer brokers who represet buyers i their search for goods. Agai suppose there are types of goods ad buyers with equal proportio. New buyers ad sellers, draw uiformly from the types, arrive at the market sequetially. Istead of seller brokers, cosider a buyer broker (heceforth broker) who first cotacts buyers i the buyers market. Whe a broker meets a buyer, the broker icurs a show cost c s, the buyer pays a ispectio cost c b, ad the broker fids out the buyer s preferece. If they agree o the commissio φ ad represetatio, the the broker will represet the buyer i the sellers market ad receive φ. Oce a broker meets with a seller, the seller pays a show cost c s to show his good ad the broker pays c b to ispect the good. After the broker fids a suitable good for her cliet to buy, the buyer will ispect the good ad egotiate to buy the good from the seller. The buyer ad seller leave the market permaetly after trade. Assume that a broker ca represet at most two buyers. Similar to the case of seller brokers, we assume for ow that (i) buyers ad sellers will ot trade directly, (ii) sellers will ot preted to be brokers with oe cliet, (iii) 0

21 buyers will ot preted to be brokers with two cliets, ad (iv) brokers will ot go to the sellers market uless they have already cotracted with two differet buyers. At the ed of this sectio, we will specify coditios uder which the icetive coditios (i)-(iv) hold. With abuse of otatio, we let A s deote the umber of brokers with two buyers, ad A b deote the umber of brokers with oe buyer. Defie θ s the market tightess of the sellers market where brokers try to fid the right sellers to their cliets: θ s = A s /S. Similarly, defie θ b the market tightess of the buyers market where brokers fid buyers: θ b = B/A b. We assume that the matchig techology betwee brokers ad sellers (or buyers) is the same as the oe i markets without itermediary. Therefore, with abuse of otatio, i the sellers market, we deote the arrival rate of a match for a seller by m (θ s ) M (θ s, 1), ad the arrival rate of a match for a broker by m (θ s ) /θ s. Similarly, i the buyers market, the arrival rate of a match for a broker is m (θ b ) ad the arrival rate of a match for a buyer is m (θ b ) /θ b. Similar to the case of seller brokers, i the steady state, the umber of brokers pickig up buyers successfully must be equal to the umber of brokers who fid the right good for oe of their cliets: m (θ s ) 1 A s = A b θ s m (θ b). (1) To complete the model, we impose the free etry coditio for sellers (V = F ) ad for brokers (W 1 = L + c s φ ). The search equilibrium with brokers ca be defied aalogously to the oe with seller brokers. As i the case of seller brokers, we ca write dow the value fuctios for sellers, buyers ad brokers. We ca solve these fuctioal equatios ad obtai the followig limit solutio (see Appedix C for details): with the limit commissio fee φ V = p + c s, () U = 1 p φ 1 1 c b, (3) φ = ad the limit price p p = F + + c s. Therefore, the total social welfare is U + V = c s + c b, (4) (c s + c b ) 1 (c s + c b ) 1

22 Fially, we verify that all parties icetive coditios are satisfied (see Appedix C for details) if 5 ad 5 + c b c s 3 1. (5) The set of cost ratio c b /c s that satisfies above costraits is o-empty as log as 8. We ca plot coditio (5) as follows: It is clear from (5) ad from the above graph that, as icreases, the set defied by (5) expads. Propositio Suppose that broker learig is perfect. A search equilibrium with buyer-brokers exists ad improves social welfare if coditio (5) holds. Agai broker licesig may relax our icetive costraits so that our welfare improvig brokerage ca exist uder weaker coditios. Suppose a buyer broker eeds a licese to operate, so that either buyers or sellers ca preted to be broker. I this case, the oly relevat icetive costraits are those for brokers, ad the costrait that buyers ad sellers do ot trade directly, amely (39) i Appedix C ad the costrait 5. It is worth otig that welfare improvig buyer brokers ca exist oly if c b is less tha c s, while welfare improvig seller brokers ca exist oly if the reverse is true. Therefore, the parameter space for seller brokers to exist ad for buyer brokers to exist is aturally separated. To uderstad the uderlyig ratioale for this separatio, let us compare the limit commissio fees for the two types of brokers: seller brokers: buyer brokers: + 1 c s + 1 c b, 1 c s + c b.