University of Toronto Department of Economics. Search Intermediaries

Transcription

1 Uiversity of Toroto Departmet of Ecoomics Workig Paper 46 Search Itermediaries By Xiawe Shi ad Aloysius Siow April 05, 011

2 Search Itermediaries Xiawe Shi ad Aloysius Siow March 14, 011 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. This is a substatial revisio of our earlier paper circulated uder the title Iformatio Exteralities ad Itermediaries i Frictioal Search Markets. 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 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. Departmet of Ecoomics, Uiversity of Toroto, xiawe.shi@utoroto.ca Departmet of Ecoomics, Uiversity of Toroto, siow@chass.utoroto.ca 1

3 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 trasactig 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. Two potetial differeces betwee labor markets which use itermediaries i hirig versus those which do ot have to do with the degree of heterogeeity across workers ad across employers, ad the screeig costs ivolved to ascertai the match betwee a worker ad a employer. 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. 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. To see the advatage a itermediary may have, first cosider a market without brokers. Both houses ad buyer prefereces are fixed ad horizotally differetiated. Suppose a buyer values a house oly if the characteristics of the house seller type) fit his prefereces 1 The assumptio of imperfect advertisig of homes for sale is a commo oe i search models of housig market e.g., Wheato 1990)).

4 buyer type). Both house characteristics ad buyer 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, some socially useful iformatio geerated i private search is lost ad ot efficietly utilized. Now cosider such a market with sellers 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. Thus employig a broker to complete a trasactio icurs additioal resource 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. 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 search costs for both parties. Note that this advatage ca be materialized oly whe the broker has additioal goods i his ivetory which are of potetial iterest to the buyer. Otherwise, the seller may as well show the good herself. That is, each seller hires a costly broker because the broker has other types of goods i his ivetory to attract potetial buyers. actively maage both the level ad mix of goods that they represet. Thus brokers have to We first study the perfect commuicatio ad learig case where the broker lears the 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. Differet from his model, the goods i our model caot be easily relocated to a cetralized locatio. 3

5 buyer s type perfectly after a costly first showig ad ispectio for both parties. We the exted the aalysis to the case where the commuicatio ad learig of the buyer s type is imperfect. That is, after the buyer turs dow the first good, the broker is imperfectly iformed about the buyer s type. We restrict our brokers to represet at most two sellers 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, 1) brokers will search ad choose to have two differet types of goods i their ivetory before they search for buyers; ) there has to be sufficiet heterogeeity i buyers ad goods types for brokers to exist; 3) imperfect commuicatio ad learig raise the amout of heterogeeity eeded for brokers to exist; 4) brokers reduce the expected total ispectio ad showig costs by all parties eeded to complete a trasactio. 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. Therefore, our model predicts that the real-estate brokers reduce the expected time for a seller to sell a house ad buyers 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), Elder, et. al. 000), Hedel, et. al. 009), Berheim ad Meer 008), amog others). The literature o itermediaries i frictioal search markets starts with Rubistei ad Wolisky 1987) where itermediaries meet potetial tradig parters at a faster rate tha potetial tradig parters ca meet each other directly. 3 The role of ivetory for itermediaries has bee ivestigated by other researchers. Johri ad Leach 001) show that itermediaries ca improve match quality ad reduce delay costs if they ca carry two uits i a settig with heterogeeous goods ad tastes. The matchig quality i their model is assumed to be idiosycratic betwee ay pair of buyer ad seller, while i our model the prefereces of buyers ad types of goods are persistet. 4 Therefore, the iformatio exteralities arisig i private search idetified i this paper are abset i their settig. I a model with heterogeous agets, Shevcheko 004) assumes that a itermediary ca icrease the level of ivetory with a covex cost fuctio. The probability of a match with a potetial buyer icreases with the level of ivetory. He studies both the optimal level of ivetory ad the equilibrium price distributio of goods. Our paper is complimetary to his. He assumes a reduced form specificatio of the cost of holdig ivetory. 5 While we fix 3 A similar assumptio is made i Yavas 199) who builds a oe-period search model of itermediaries with edogeous search itesity. 4 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. 5 Aother strad of literature studies the role of cetralized matchig agecies i a decetralized matchig 4

6 maximum ivetory capacity, we explicitly derive the cost ad beefit of holdig ivetory from the structure i which the market operates. Fially, iformatioal exteralities i frictioal matchig markets without itermediaries uderlies the social learig literature see a survey by Bikhchadai, et. al., 1998). But to our best kowledge, the role of itermediaries i iteralizig iformatio exteralities through learig ad ivetory maagemet has ot bee explored. The remaiig of the paper is orgaized as follows. Sectio first studies a two-sided search model without brokers, which will serve as a bechmark for our aalysis of brokers. Sectio 3 itroduces brokers ito the search model. We show that, uder the assumptio of perfect learig, brokers improves social welfare by reducig expected total umbers of showigs ad ispectios ecessary to complete a trasactio. Sectio 4 relaxes the assumptio of perfect learig, ad demostrates that the emergece of brokers still improves social welfare as log as there is sufficiet heterogeeity i goods ad buyer prefereces. Sectio 5 discusses a few extesios of the model ad cocludes. 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. All participats discout the future with a commo discout rate r. Followig the stadard two-sided 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, at a poit of time, if there are B buyers ad S sellers i the market, the i each istat, M B, S) buyers will radomly match with the same umber of sellers. The flow of cotacts or the matchig fuctio, M B, S), is assumed to be icreasig i both argumets ad has costat retur to scale. If we use θ = B/S to deote the market tightess, the we ca write the arrival rate of a match for a seller as m θ) M B, S) /S = M θ, 1) ad the arrival rate of a match for a buyer as M B, S) /B = m θ) /θ. Therefore, whe is small, each period a buyer radomly meets a seller with probability m θ) /θ, ad a seller meets a buyer with probability m 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 prefereces) 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 markets see for example, Bloch ad Ryder 000) ad refereces therei). 5

7 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. 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 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 ) t ) ] 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 t; 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 t) ) ] 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 t. 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 mθ) ). θ 6 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). 6

8 The price t is egotiated by the two tradig parties. Differet bargaiig protocols may result i differet trasactio prices, but the total social welfare is idepedet of prices. Sice we primarily cocer 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 all market participats must be willig to participate i the market. I the ed of this sectio, we use Nash bargaiig as a example to illustrate how a specific bargaiig protocol ca determie the expected equilibrium payoff for each market participat. Defiitio 1 A trasactio price t is feasible if all tradig parters are willig to participate. We complete the model by imposig a free etry coditio for sellers. Let K be the cost to a home builder to build a house. The free etry of sellers implies V = K. I order for the market to exist, we assume throughout of the paper that 1 c b c s > K. 3) Defiitio A steady-state search equilibrium without itermediaries is defied by the stocks of market participats B, S), cotiuatio payoffs U, V ), ad market price t such that a) steady-state coditios 1) ad ) hold; b) price t is feasible; c) free etry coditio V = K holds. We are primarily iterested i the limit equilibrium with r 0 where delay cost is egligible. Sice the expected umber of ispectios for producig a successful match is, the total welfare U +V ) i the limit equilibrium for a pair of buyer ad seller is 1 c b c s. Propositio 1 I the limit equilibrium without itermediaries, the total social welfare for a pair of seller ad buyer is U + V = 1 c b c s. 4). Nash Bargaiig If we assume that the trasactio price t is egotiated through Nash bargaiig as i the literature, the we ca completely solve the model ad study how the surplus is divided betwee buyers ad sellers. To avoid the hold-up problem ad a o-existece of equilibrium with trade whe r approaches zero, we assume that the price t is egotiated before 7

9 ispectio, 7 [ t = arg max c b + 1 p ] [ 1 1 p) + U U c s + 1 p + 1 V V ]. 5) To uderstad the formula, ote that if a buyer agrees to price p ad proceeds to ispect the house by icurrig cost c b, with probability 1/ he likes the house ad obtais et payoff 1 p), ad with probability 1) / the house does ot match his prefereces, so he returs to the market. If a buyer does ot agree, he has to retur to the market with cotiuatio value U. The ituitio for the seller s part is similar. It follows that the trasactio price t is set at t = V U + c s c b ). 6) The two fuctioal equatios 1) ad ) ca be rearraged ito rv = m θ) [ c s + 1 ] t V ), ru = m θ) [ c b + 1 ] θ 1 t U). Substitutig t ito the fuctioal equatios, we ca solve U ad V as Lettig r 0, we have U = V = m θ) m θ) + θm θ) + θr 1 c b c s ), θm θ) m θ) + θm θ) + θr 1 c b c s ). U = θ 1 c b c s ), 7) V = θ 1 + θ 1 c b c s ). 8) Ituitively, the social surplus is shared accordig to market tightess θ: the seller gets a larger share if the market coditio is more favorable to the seller i.e., a higher θ). The market tightess θ is recovered from equatio 8) ad the free etry coditio V = K. 7 Both the hold-up problem i this class of models see Spulber 009) ad the o-existece problem of the equilibrium with trade as r approaches zero are well kow e.g. Camera ad Delacriox 003). Dealig with these problems detract from our cocer here. 8

10 3 Search with Brokers: Perfect Learig Agai suppose there are types of houses ad buyers with equal proportio. We add sellerbrokers heceforth brokers) who first cotact sellers to seek exclusive represetatio ad the sell houses o the sellers behalf to buyers. 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. A broker ca represet at most two sellers. 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 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. The ext sectio will relax the assumptio of perfect learig. 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. 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. The i the sellers market, the arrival rate of a match for a seller is give by m θ s ) M A s, S) /S = M θ s, 1), ad the arrival rate of a match for a broker is M A s, S) /A s = 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. The 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. To simplify otatio, i what follows we write m s = m θ s ) 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 9

11 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. 3.1 Equilibrium Welfare We derive the cotiuatio values for a seller, a broker with oe house, a broker with two 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 = r [ m s c s + 1 ) V a φ) + 1 m s 1 ) ] V. 9) 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 t c s) + 1 ) V b + 1 m b ) ] V a. 10) 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 prefereces. 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 matches the buyer prefereces with probability 1/), she icurs cost c s to show her house to the buyer ad obtais price t; if the other house of the broker matches the buyer prefereces 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. 10

12 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 [ 1 ms V b = r θ s V a + 1 m s 1 ) ] V b. 11) θ s The term ms θ s is the probability that the broker meets aother seller i the sellers market is the probability that the secod house is differet from the type of the first house ad 1 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 W 1 = r [ ms c b + 1 ) θ s W + φ) + 1 m s 1 ) ] W 1. 1) 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 ]. 13) 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 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. 11

13 Fially, let U be the cotiuatio value of a buyer who remais i the market at the ed of period. The we have U = r [ mb θ b c b + 1 t 3 )) c b + 1 m b ) ] U. 14) 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 t. 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 t. 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 defer specifyig the bargaiig protocol that determies the trasactio price t ad the commissio φ. We oly require that the price ad the commissio are such that both sellers ad buyers are willig to participate i the markets ad the payoffs for brokers are bouded i the limit equilibrium. Defiitio 3 A pair of commissio φ ad price t are feasible if a) both buyers ad sellers are willig to participate i the market, ad b) the expected profit for a broker from each trasactio is zero as r 0. Recall that each trasactio cosists of a purchase ad a sale. If commissio φ ad price t are equilibrium commissio ad price, brokers must ear zero profit from each trasactio. Otherwise, they will make ifiite total profits because they are ifiitely lived. I particular, whe r 0, the commissio φ just covers the expected search costs icurred by a broker to complete a trasactio, that is, φ = + 1 c s + 1 c b. 15) To see this, otice 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/. 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. 16) 1

14 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 = K. 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. Defiitio 4 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 t such that a) steady-state coditios 9)-16) hold; b) commissio φ ad price t are feasible; c) free etry coditios for sellers ad brokers hold; d) icetive coditios i)-iv) are satisfied. The followig propositio characterizes the equilibrium total welfare. Propositio I the limit equilibrium with itermediaries ad perfect learig, the total expected payoff for a pair of buyer ad seller is Proof. U + V = 1 1 c b + c s ) + 3 c b + c s ) 17) Sice the price ad the commissio are feasible, the brokers ear zero profit from each successful trasactio. Therefore, the total expected payoff of buyers ad sellers coicide with the total social welfare. 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 brokers o average i order to fid a good match because the probability that a buyer to 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 1 extra ispectio whe a broker meets a buyer because coditioal o a match, the 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 claim the follows from the fact that the value of a good match is 1. By comparig the social welfare with brokers 17) ad without brokers 4), we obtai the welfare improvemet due to brokers: U + V ) = ) c b + c s ). 18) 13

15 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. 3. 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). Seller s Icetives 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 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 Buyer s Icetives φ = + 1 c s + 1 c b c s c b ) c s, ) c s A buyer ca preted to be a broker with oe house ad buy directly from a seller at price t φ) i the followig way. If the house fits the buyer s preferece, the buyer pays t φ) 14

16 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 )) c b 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 )) c b That is c b c s ) Broker s Icetives To characterize the broker s icetive coditios, we focus o the potetial gai of the broker from each trasactio by deviatig from the assumed optimal strategy. Recall that a trasactio for a broker cosists of a successful represetatio for a seller ad a successful sale to a buyer. If a broker caot gai from deviatio for oe trasactio, the it caot gai by deviatig for more tha oe trasactios. Therefore, although i priciple the broker ca deviate for ay umber of trasactios, 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 1 c b c 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 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 15

17 which reduces to c b ) c s Now 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 1), 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 1) 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. 3.3 Summary It is easy to see that the broker s icetive coditio 1) are implied by the seller s icetive coditio 19) for all. Therefore, all parties icetive coditios are satisfied if 6 ad c b ) c s Notice that the set of cost ratio c b /c s that satisfies above costraits is o-empty as log as 9. Moreover, as is large, the seller s IC costrait 19) is always satisfied, while the buyer s IC costrait is also satisfied 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 a discussio whe the reverse is true). The followig propositio summarizes the mai result of this sectio. Propositio 3 Suppose the broker s learig of buyers prefereces is perfect. A search equilibrium with brokers exists ad improves social welfare if 9 ad coditio ) holds. 3.4 Nash Bargaiig This sectio shows that a Nash bargaiig protocol for determiig the commissio ad the price of the house ca support the above equilibrium with sellers brokers. Moreover, the bargaiig protocol provides uique idividual equilibrium payoffs. 16

18 Suppose the commissio φ betwee a seller ad a broker is established by Nash bargaiig before the broker ispects the house: φ = arg max c b + 1 p p + W ) + 1 ) W 1 W 1 c s + 1 p + V a) + 1 ) V V φ = 1 V a V W + W 1 + ) 1 c b c s ) The iterpretatio of the objective is aalogous to 5) i the previous sectio. We assume that after the broker meets with the buyer, the broker first egotiates a price t with the buyer o the sellers behalf before performig costly ispectios. Specifically, the trasactio price t betwee a seller ad a buyer is determied by Nash bargaiig as follows: [ 1 t = arg max p p c s) + 1 V b + ] ) [ V a V a c b + 1 p 3 ) c b + ] ) U U. From a seller s perspective, if she agrees to price p, with probability 1/ her house may match the buyer s prefereces i which case she gets p c s ), ad with probability 1/ the other house the broker is represetig matches the buyer s prefereces i which case she gets V b. If oe of the two houses match the buyer s prefereces which happes with probability ) /), her cotiuatio value will be V a. If she does ot agree to price p, she gets her outside optio V a. From a buyer s perspective, if he agrees to price p ad icurs cost c b, with probability 1/ he will fid out that the first of the two houses matches his prefereces i which case he gets 1 p c b ), ad with probability 1/ the secod house the oe the broker did t show) of the two houses matches his prefereces i which case he gets 1 p c b ). With probability ) / either house matches his prefereces ad he obtais cotiuatio value U. If he rejects price p, he gets outside optio U. Therefore, the price is give by t = V a V b U + c s + 3 ) c b. Substitute φ ad t ito the fuctioal equatios. We ca show with some algebra that V = θ s W 1 ad V a = 1θ bu. Takig r 0, we solve U ad V i the limit equilibrium: 8 U = ) c b c s θ b + θ b V = ) c b c s + 1 c s θ b + 1 c b + c s ) It follows that U + V = 1 1 c b + c s ) + 3 c b + c s ) 8 I limit equilibrium, we also obtai W 1 = V/θ s ad W = W 1 +1 c s. Together with the free etry coditios ad equatio 16), oe ca pi dow the market tightess θ, θ b ad θ s. 17

19 which is cosistet with Propositio. Moreover, oe ca easily verify that the equilibrium commissio is ideed give by 15). 4 Search with Brokers: Imperfect Learig I the previous sectio, we assume that after the costly first showig, the broker lears perfectly the buyer s type. This sectio aalyzes the case where the broker s learig is imperfect. If the buyer likes the first house that he is show by the broker, he will meet with the seller to buy it. If he does ot like the first house, the broker is imperfectly iformed as to what is his preferred type of house. Assume that the broker lears a set of houses, Θ, of cardiality k that the buyer s preferred house lies i. The set Θ cotais the buyer s preferred house ad each broker also radomly draws k 1 other types of houses from the house types excludig the preferred type ad the first house). The learig of the broker is more precise for a smaller k, ad we have the special case of perfect learig if k = 1. We assume that the set Θ is radom ad idepedet across buyer-broker pairs. As i the previous sectio, we assume for ow that i) buyers ad sellers will ot trade directly, ii) sellers will ot preted to be brokers with two houses, iii) buyers will ot preted to be brokers with oe house, ad iv) brokers will ot go to the buyers market uless they have two differet types of houses i their ivetory. Later we will specify coditios uder which the icetive coditios i)-iv) hold. 4.1 Equilibrium Welfare We use the same otatios V, V a, V b, W 1, W, U) to deote cotiuatio values of sellers, brokers, ad buyers at the ed of period. Sice imperfect learig is relevat oly i the buyers market, the fuctioal equatios relatig to the sellers markets, V, V b, ad W 1 are the same as i the previous sectio. The fuctioal equatios pertaiig to the buyers market, V a, W ad U, may be differet uder imperfect learig. Cosider a broker with two houses who search for buyers i the buyers market. Oce the broker meets a buyer, they icur ispectio ad showig costs to see the first house. If the buyer likes the first house, he will proceed to meet with the seller to buy the house. If he does ot like the first house, the broker lears that the buyer type may be oe of the k types i set Θ. If the broker s secod house is i set Θ, she will show it to the buyer. If the buyer likes the secod house, he will buy it. Otherwise they will separate. They will also separate if the secod house is ot i Θ. First, we argue that the seller s fuctioal equatio whe she has a broker, V a, is uchaged as i the case of perfect learig. The reaso is that she does ot icur ay extra 18

20 showig cost associated with imperfect learig. Her oly showig cost i the buyers market occurs whe the broker has foud a match of her house with a buyer which is the same as i the perfect learig case. So as before, V a is: [ V a = m b r 1 t c s) + 1 V b ) + 1 m b ) ] V a Next cosider a buyer who radomly meets a broker with two houses. First, he icurs a cost c b to talk to the broker ad ispect the first house. If he likes the house, he meets with the seller ad buys it. If he does ot like the first house ad the secod house is i Θ, the buyer icurs aother cost c b to see the house. If he likes it, which happes with probability 1/k, he will meet the seller ad buy the house at a egotiated price t. separate. Thus the fuctioal equatio for a buyer i the buyer s market, U, is: [ 1 mb U = c b r θ b 1 t c b) + 1 k 1 c b + 1 ) k 1 t c b)) + Otherwise, they 1 m b ) ] U θ b Compared with the perfect learig case, the buyer icurs a additioal expected ispectio cost of k 1c b where k 1 is the probability that he will see a secod house which does ot fit. Fially, cosider the problem of the broker with two houses who meets a buyer. He icurs c s to show the first house. The house fits the buyer with probability 1. With probability 1, the first house does ot fit. I this case, he shows the secod house with probability k. The secod showig will succeed with probability 1. If the buyer does ot buy a house 1 k from the broker s cliets which occurs with probability ), the broker has to look for aother buyer. Thus the fuctioal equatio for the broker with two houses is: [ 1 W = m b 1 + k 1 + r )c s + ) W m b ) ] W Compared with the perfect learig case, the broker icurs a additioal expected showig cost of k 1c s where k 1 fit the buyer. is the probability that he will show a secod house which does ot As i the previous sectio, whe r 0, the brokers must be makig zero expected profit per completed trasactio. Thus the commissio a broker receives to sell a house is equal to the expected search costs icurred by the broker. Assumig the broker already has the first house, the commissio will be: φ = 1 c b k 1 ) c s = 1 c b + + k c s 3) The ituitio behid φ is as follows. The broker still eeds to meet / 1) sellers i order to pick up the secod house ad each time he icurs a ispectio cost c b, which explais the first term. Whe the broker goes to the buyers market, he eeds to meet / 19

21 buyers o average i order to make a sale. Whe a broker meets a buyer, he icurs a showig cost c s to show the first house, ad if the first house does ot fit but the secod house lies i the set Θ which icurs with probability 1 k ), the broker icurs aother showig cost 1 to show the secod house, which explais the secod term. I the steady state, the umber of brokers pickig up houses successfully must be equal to the umber of brokers sellig houses successfully. We also impose free etry coditios for sellers ad brokers. The followig propositio characterizes the equilibrium total welfare whe the broker s learig is imperfect. Propositio 4 I the limit equilibrium with itermediaries ad imperfect learig, the total expected payoff for a pair of buyer ad seller is U + V = 1 1 c s + c b ) + k + c s + c b ) 4) Proof. As i the previous sectio, a successful trasactio takes 1 ispectios o average i the sellers market because a broker picks up a ew seller with probability 1. As we argue above whe we calculate the commissio 3), the expected umber of searches with brokers for a buyer to fid a house he likes is +k. For each oe of these searches the buyer ad broker icurs a cost. Fially there is the fial showig of house by the seller to the buyer, durig which oe more cost of each type is icurred. Thus, we arrive at the above equatio. By comparig the two expressios 4) ad 4), we coclude that brokers with imperfect learig are welfare improvig if A sufficiet coditio is k k + k + 5) + k Icetive Coditios As i the case of perfect learig, we eed to check icetive compatibility coditios for sellers, buyers ad brokers. Notice that whe k +5 the joit payoffs of buyers ad sellers are higher i dealig with brokers compared to tradig directly. Therefore, either the seller or the buyer will be worse off by tradig directly. Therefore, we oly eed to worry about icetive coditios ii)-iv). Sice the aalysis here is aalogous to the oe i the previous sectio, we report the results directly ad omit the details. A seller will ot preted to be a broker with two cliets 0

22 ad approach buyers directly if c b k + 5) + k +. 5) c s A buyer will ot preted to be a broker with oe cliet ad search sellers directly if c b c s + k) 1) k + 5) + k +. 6) Fially, a broker with oe house will ot immediately go to the buyers market to look for a buyer if 4.3 Summary c b c s k) 1). 7) Agai, the broker s icetive coditio 7) is implied by the seller s icetive coditio 5) for all. Therefore, all parties icetive coditios are satisfied if k + 5 ad + k) 1) k + 5) + k + c b k + 5) + k +. 8) c s The set of cost ratio c b /c s that satisfies above costraits is o-empty as log as is large relative to k. Whe is large relative to k, the secod iequality i 8) is always satisfied, while the buyer s IC costrait is also satisfied if c b is relatively higher tha c s. To summarize our aalysis with imperfect learig, we have Propositio 5 Suppose the broker s learig is imperfect. A search equilibrium with brokers exists ad improves social welfare if k + 5 ad coditio 8) holds. It is ituitive that if the broker lears less from costly ispectios i.e., a higher k) we eed a higher heterogeeity i.e., a higher ) i order for brokers to exist. As is clear from our aalysis, our results with imperfect learig are qualitatively similar to the case with perfect learig. 4.4 Nash Bargaiig As i the case with perfect learig, with Nash bargaiig we ca completely solve idividual payoffs to market participats. The commissio φ betwee the seller ad the broker is established by Nash bargaiig before ispectio: φ = arg max c b + 1 p p + W ) + 1 ) W 1 W 1 c s + 1 p + V a) + 1 ) V V φ = 1 V a V W + W 1 + ) 1 c b c s ) 1

23 Similarly, we assume that after the broker meets with the buyer, the broker first egotiates a price t with the buyer o the sellers behalf before performig costly ispectios. Specifically, the trasactio price t betwee a seller ad a buyer is determied by Nash bargaiig as follows: 1 t = arg max p p c s) + 1 V b + ) V a V a c b k c b + 1 p c b) + ) U U t = 1 1 U V b + V a + k ) c b + c s c b ) The seller s part is same as i the case with perfect learig. For the buyer s part, ote that the expected search costs icurred by the buyer, as derived i the proof of Propositio 4, are subtracted from the expected surplus for the buyer. By substitutig φ ad t ito the fuctioal equatios, agai we obtai that V = θ s W 1 ad V a = 1θ bu. Takig r 0, we obtai the value of U ad V i the limit equilibrium: U = 1 + k ) c b c s + c b ) + θ b θ b V = 1 + k ) c b c s + c b ) + k c s + θ b 1 c b + c s ) Therefore, the total social welfare derived from each completed trasactio is give by U + V = 1 1 c s + c b ) + k + c s + c b ) which coicides with expressio 4). Moreover, oe ca verify that the equilibrium commissio charged by the brokers are ideed give by 3). 5 Cocludig Remarks We used a stylized model to illustrate how search itermediaries ca iteralize iformatio exteralities arisig i the two-sided frictioal matchig market. There are several possible extesios. I the paper, we model the imperfect learig as a specific learig process. Our results should geeralize to other learig processes as log as they are memoryless across brokerbuyer pairs i the sese that a buyer caot commuicate to a ew broker the iformatio he gathers i meetigs with previous brokers. With memoryless learig processes, the additioal expected umber of ispectios a buyer has to perform i order to complete a trasactio relative to perfect learig is also equal to the additioal umber of showigs a broker has to do. The zero profit costrait for brokers the implies that the broker s commissio will pick up these additioal expected showig costs. Therefore, we ca proceed