ARE BROKERS' COMMISSION RATES ON HOME SALES TOO HIGH? A CONCEPTUAL ANALYSIS. Paul Anglin. and. Richard Arnott. September 1995.

Transcription

1 ARE ROKERS' COMMISSION RATES ON HOME SALES TOO HIGH? A CONCEPTUAL ANALYSIS Paul Anglin and Richad Anott Septembe 1995 Abstact Many people believe that pevailing commission ates fo esidential eal estate bokes ae "too high" but do not offe a fomal model. This pape pesents a geneal equilibium model of the housing maket in which eal estate bokes seve as matching intemediaies. We use this model to constuct an illustative example which is "calibated" using data epesentative of a typical housing maket. Depatment of Economics Depatment of Economics oston College Univesity of Windso Chestnut Hill, MA Windso, Ontaio N9 3P4 USA CANADA

2 (mailing addess) 2

3 Ae okes' Commission Rates On Home Sales Too High? A Conceptual Analysis * The question posed in the title appeas odd -- not to a layman, but to an economist. An economist tying to explain a pice that was "too high" would look fo the execise of maket powe. ut, at fist glance at least, esidential eal estate bokeage appeas highly competitive: enty and exit ae vitually costless, aveage fim size is small, and licensing equiements ae not oneous. Thee is consideable dispute in the liteatue concening the mechanism by which commission ates ae detemined. oadly speaking, thee ae fou classes of "models" -- none of which has been fully fomalized. In the fist, a competitive, symmetic infomation model, bokes compete ove commission ate schedules based on local maket conditions, offeing bette sevice fo a highe commission ate. A home selle then chooses that commission ate/sevice quality pai fom the competitively-detemined commission ate schedule which maximizes his 1 utility. The stylized facts appea inconsistent with the pedictions of this model. If the model is ealistic, one would expect to see a wide ange of commission ate/sevice quality pais; instead, one obseves that esidential commission ates in Noth Ameica lie in a band between 5% and 7% with little vaiation ove time. The second class of models views the bokeage contact fom a static pincipal-agent pespective, focusing on the inability of the selle (and the buye) to monito the agent's effot and to gauge the agent's ability. These models pedict contactual divesity. A selle who wants to sell his home quickly should base the agent's emuneation on time to sale; a selle who equies a minimum amount of equity out of his home should offe a high-poweed incentive contact which offes the agent a lage bonus if the sales pice is above that level which satisfies the equity equiement; and so on. While thee is a tend towads contactual divesity, this class of model fails to explain why the commission contact emains dominant. The thid class of * Anglin would like to thank the Social Sciences and Humanities Reseach Council of Canada fo its suppot. Anott would like to thank semina paticipants at oston College, Havad Univesity, and the Univesity of Melboune, as well as Mak Avin, fo helpful comments. This daft of the pape was completed when Anott was a Lusk Fellow at U.S.C. in May/June Anott would like to thank the goup thee not only fo being so hospitable but also fo poviding such a stimulating envionment fo "economic ubanologists." 1 Thoughout, we teat buyes and selles as male and eal estate agents as female. 3

4 models is simila, but takes the commission fom of the contact as given. Since the commission contact has only a single paamete, 2 it cannot "sepaate" agents. Since both agent quality and agent effot ae unobsevable, the selle will go with that agent offeing the lowest commission ate. Commission ates will be bid down to the lowest possible quality of sevice, which does not at all accod with obsevation. The fouth class of models is based on collusion. Thity yeas ago, local eal estate boads set commission ates. Due to anti-tust action, they switched to ecommending commission ates; and faced with futhe anti-tust action, they do not even do this now. How then can collusion be sustained? The stoy uns as follows. Most buyes use "selling agents" to assist them in seaching listed popeties. To sell a popety, the selle's agent -- the "listing agent" -- equies the coopeation of selling agents. 3 Thus, the blackballing of discount listing agents by selling agents can be effective in sustaining a collusive commission ate. Discount listing agents claim that such blackballing is common (Fedeal Tade Commission, 1983), and in some juisdictions that they ae also baed fom listing unde the Multiple Listing Sevice. Anothe facto in sustaining a collusive commission ate is that the significant minoity of selles do not ealize -- and ae not infomed -- that the commission ate is negotiable. This class of models is consistent with the stylized facts but does not explain why one commission ate is chosen ove anothe. Apat fom these models, thee is a widely-held view that eputation plays an impotant ole in the maket in poviding agents with an incentive to povide quality sevice. In shot, thee ae no complete and pesuasive models of how the commission ate is detemined. Thus, to investigate whethe the commission ate is too high, a conceptualization is needed that is independent of how the commission ate is actually set. Futhemoe, because of the extenalities endemic to seach models, the socially optimal commission ate will in geneal diffe fom the commission ate thown up by the maket, howeve detemined. 2 This is not stictly coect. Thee ae two othe paametes, the peiod of the listing ageement and the listing pice. Accoding to this class of models, the peiod of the listing ageement is ielevant since, if thee ae any costs to switching agents, the selle will elist with the same agent; the listing pice is ielevant too, since the buye can make an offe at any pice he wishes. 3 Using the data descibed in Anglin (1994) fo Windso, Canada, it is possible to gauge the impotance of coopeation. 36 pecent of popeties wee listed with the top fou fims, and on at least 50 pecent of all tansactions eithe the listing agent o the selling agent o both wee employed by one of these fims. 4

5 Fo this eason, it is necessay to employ a geneal equilibium model. 4 This pape pesents such a model in which eal estate agents ae matching intemediaies, matching buyes with houses. Obseving how the maket pefoms (e.g., popotion of homes fo sale, aveage time to sell, aveage time to buy, numbe of full-time-equivalent agents) one can make infeences concening the matching technology. And with knowledge of the matching technology, the optimal commission ate can be detemined and compaed to the pevailing ate. Section 1 pesents a model of the housing maket with eal estate agents, and investigates the existence and uniqueness of equilibium, as well as its compaative static popeties, taking the commission ate as a datum. Section 2 descibes the nomative famewok fo detemining the socially optimal commission ate and illustates how the model can be calibated o estimated using aggegate maket data. We then illustate how the calibated model can be used to compute the socially optimal commission ate. Section 3 concludes with a discussion of diections fo futue eseach. Although the pape focuses on the housing maket, the same issues aise in othe makets in which seach intemediaies play a ole; fo example, the commissions paid to tavel agents, to job seach sevices, whethe pivate "head huntes" o govenment-sponsoed agencies (Osbeg, 1993, Section VI), and to etail sales pesons ae amenable to the same type of analysis. 1. The asic Model The model adapts the ental housing maket model 5 pesented in Igaashi (1991), which in tun daws on labo-maket matching models, especially 4 In ecent yeas, a lage numbe of papes have been published on the economics of esidential eal estate bokeage, paticulaly on the fom of the bokeage contact (ecent contibutions include Anglin and Anott (1991), Miceli (1992) and Yavas (1994)). Unfotunately, with the exception of Yinge (1981) and Fu (1993), the models in all these papes ae patial equilibium in natue and hence inappopiate fo addessing the question at issue in this pape. A few papes examine empiically the deteminants of the commission ate, such as Simans, Tunbull and enjamin (1991). 5 Fu (1993) independently developed a model that is simila to ous. Using a matching model whee agents can ente feely, Chapte 2 in Fu (1993) chaacteizes the optimal commission ate. His analysis focuses on the effect of the commission ate on the numbe of agents and on the effot pe agent: viz., maximizing the aggegate matching ate at a given cost equies that the elative maginal poducts of an additional agent and of additional effot should equal thei 5

6 Diamond (1981) and Hosios (1990), and on Anott's model of thin ental housing makets with idiosyncasy (1989). The two essential featues of the model ae idiosyncasy and matching. Combining these two featues geneates an analytically neat teatment of housing seach. 1.1 oad-bush desciption Households have idiosyncatic tastes and houses diffe in idiosyncatic ways. Futhemoe, these idiosyncasies ae symmetic in the following sense. Houses all cost the same to constuct. Each household has an ideal house of this constuction cost. If he esides in some othe type of house, he expeiences a mismatch cost. Idiosyncasies ae symmetic in the sense that the pobability distibution of the mismatch cost fo an abitaily-selected house and an abitaily-selected household is independent of the house and the household. Seach is epesented by an aggegate matching function. Let be the stock of home buyes, S the stock of houses fo sale, A the numbe of eal estate agents, and g the effot expended by a epesentative agent. Then seach geneates m = m(,s,a,g) aggegate oppotunities to match pe peiod, o moe pecisely an aggegate Poisson oppotunity-to-match ate of m. y symmety, each home buye eceives oppotunities to match at the Poisson ate m and each house selle at the ate m S. The details of the seach pocess, including what it is that agents do -- apat fom exeting effot -- ae suppessed. Associated with a paticula oppotunity to match is a daw fom the pobability distibution of mismatch costs. The full pice associated with an oppotunity to match equals the maket pice of the unit plus the discounted mismatch costs associated with the oppotunity to match. The housing maket will be assumed to be in a steady state. As a esult, households will adopt a time-invaiant esevation full pice ule unde which they will puchase a house that they have been matched with if its full pice is below the esevation full pice and not othewise. In the basic model, households ente the housing maket but do not exit. The ate of enty is popotional to the housing maket population. This, combined with assumptions that housing constuction and the matching elative maginal social costs. In contast, ou model focuses on the effect of the commission ate on the numbe of buyes, selles, and agents and on how each contibutes to the aggegate matching ate. 6

7 technology exhibit constant etuns to scale, leads to a steady-state equilibium in which the housing maket gows exponentially ove time, without changes in elative magnitudes. Households choose the esevation full pice to minimize the expected pesent value of accommodation costs, the sum of expected discounted seach and homeowneship costs. uildes constuct houses and chage a pice 6 to maximize expected discounted pofits, taking into account that a highe-piced house is likely to take longe to sell. Fee enty and exit in the constuction business leads to zeo expected pofits. Real estate agents choose sales effot, and fee enty and exit in esidential eal estate bokeage foces bokes' utility down to its esevation level. When a house is sold, the builde pays the boke a commission equal to the commission ate times the sale pice. The commission ate is fixed exogenously. This basic model is desciptively unealistic in a numbe of espects: only newly-constucted houses ae sold; thee is no exit of households; households and houses ae identical, apat fom idiosyncatic diffeences; thee is no housing depeciation o maintenance; and thee is no land. The pupose of stating with a desciptively unealistic model is to pesent the stuctue as stakly as possible. Some extensions in the diection of ealism will be discussed in section Full desciption The population of households in the housing maket is nomalized at one: + O = 1, whee is the stock of home buyes and O the stock of homeownes. The housing maket is in steady state with the population gowing at ate n. Thee is enty of households into the housing maket but no exit. 6 We chose this exchange mechanism because it is simple and because Igaashi (1991) and Anott (1989) employed it successfully in analyzing the ental housing maket. Obsevation suggests, howeve, that while take-it-o-leave-it picing is standad in the ental housing maket, bagaining is the nom in the owne-occupied housing maket. We did investigate eplacing take-it-o-leave-it picing with bagaining in ou model and found that doing so geneated moe heat than light. In any event, simply assuming the fom of the exchange mechanism is unsatisfactoy -- it should be made endogenous. See Petes (1994) and the efeences theein fo wok along these lines. 7

8 In equilibium, all buildes choose the same housing pice. The household's esevation full pice ule fo puchasing a home theefoe educes to a esevation mismatch cost ule. Hence, whee X is the pe-peiod mismatch cost, FX the cumulative distibution function of mismatch cost (with f (), which is assumed to be continuous, its deivative, and F() its integal) and X * the esevation mismatch cost, a popotion FX ( * ) of oppotunities to match esult in a successful match -- a home puchase. Thus, whee m = m,s,a,g is the aggegate ate of oppotunities to match, ( X * ) is the aggegate ate of tansactions. The stock of home buyes gows at the ate Ḃ = n ( X* ), since households ente the pool of homebuyes at ate n and exit it at ate ( X * ). A steady-state equiement is that Ḃ = n. Combining these two conditions yields the steady-state equilibium condition n( 1 ) ( X * )= 0. (1) We now tun to the household's esevation full pice decision. Let C be the expected pesent value of accommodation costs fo a homebuye, which includes discounted seach costs pio to home puchase, the discounted home puchase pice, and discounted mismatch costs subsequent to home puchase. In making this decision, the household takes the maket pice of a house, P, as given. Thus, its esevation full pice decision educes to a esevation mismatch cost decision, with the esevation mismatch cost depending on the maket pice. Let be the discount ate, b be pe-peiod seach costs (which include the costs of living in tempoay accommodation), X( X * ) be the expected mismatch cost as a function of the esevation mismatch cost ( X( X * X * )= Xf ( X)dX F( X * )), and T be the time spent seaching, a andom vaiable. Then C T = E T be t dt + Pe T + X( X * )e t dt 0 T, whee E T {} epesents the expectation with espect to time-to-buy. Since home puchase occus accoding to a Poisson pocess with aival ate ( X * ), the pobability that T is in the inteval ( t,t + dt) is ( X * ) X exp ( * )t dt. Then staightfowad calculation yields 0 8

9 C = b + ( X * ) P+ X X*. ( X * ) + The household chooses X * to minimize C, esulting in the intuitive esevation mismatch cost ule that the esevation full pice equal the expected costs of continuing seach: C = P + X*. (The second ode condition fo a minimum is satisfied.) Combining the above two equations yields b + ( X * ) P+ X X* ( X * ) + P X* = 0, (2) which is an implicit function chaacteizing a household's esevation mismatch cost as a function of the maket pice fo a house. Next we examine behavio in the constuction secto. The epesentative builde chooses P to maximize the expected discounted value of pofits fom building a house taking the household esevation full pice, P + X *, as given. Let P o be the pice chaged by the builde. The builde knows that a pospective buye will puchase the house if and only P o + X = P + X *, which occus with pobability FP+X ( * P o ). Theefoe, whee K is the constuction cost of a house, φ the commission ate, and T s the time the house is on the maket, a andom vaiable, his maximization poblem is maxπ=e P T s ( 1-φ)P o e T s o { } K. Since house sale occus accoding to a Poisson pocess with aival ate ( P+ X * P o ) S, this poblem can be ewitten as maxπ= ( 1-φ)P o P o ( P+ X * P o ) S K. S + P+ X * P o 9

10 Expected discounted pofit maximization educes to expected discounted evenue maximization and yields an obvious fist-ode condition. 7 Since all buildes have the same cost and take the households' esevation full pice as given, they all choose the same P o. Consistency then equies that the pice buildes choose is the pice households take as given, i.e. P = P o. Incopoating this condition into the fist-ode condition yields FX ( * )( ( X * )+S) P 2 Sf ( X * )= 0. Fee enty and exit implies zeo pofits: (( 1 φ)p K)( X * ) SK = 0. (3) The above two equations can be combined to yield P = K 1 φ + FX* f X *. (4) Finally, we tun to esidential eal estate agents. It is assumed that the matching ate achieved by a paticula agent is m g o A g, whee g o is the paticula agent's effot and g is the maket level of effot which each agent takes as exogenous. This implies that an incease in the effot of an individual agent inceases he sales but not the aggegate matching ate. The agent's cost of effot is Cg,C( 0)= 0, C > 0, C > 0, which can be intepeted eithe as the disutility of effot o advetising expenditues. An agent peceives he income to be A φmpf X * g o g Cg ( o ). The fist-ode condition of he effot choice poblem is staightfowad. Since each agent faces the same maximization poblem, consistency equies that g o = g. Combining these two conditions yields φmpf( X * ) A C ( g)g = 0. 7 The second-ode condition is 2 f 2 + F f < 0. We assume it to be satisfied. 10

11 An agent's esevation income in an occupation in which she expends minimal ( g= 0) effot is ŵ, which is assumed to be exogenous. Fee enty and exit will dive agents' income to this esevation income. Hence, φmpf( X * ) Cg = ŵ. A The above two equations imply that the equilibium level of g, g *, is the unique solution to ŵ = C ( g)g Cg, which is independent of maket conditions. 8 Letting w ŵ + Cg ( * ), the above equation can be witten as φmpf( X * ) w = 0, (5) A whee w is the income the agent must eceive to be indiffeent between being a eal estate agent and taking esevation employment. Equations (1) - (5) give 5 equations in 5 unknowns,, X *, P,S, and A. entes only in (1), S only in (3), and A only in (5). Thus, the equations can be simply educed to 2 equations in X * and P, (4) and whee X * b 1 ( X * ) ( X * X( X * )) P X * = 0, (2 ) n m = m 1 n X* (( 1 φ)p K) X *, K Substitution of (2 ) into the above equation gives m = m X * X( X * ) b P X ( * ) X* w φmpf X*, (( 1 φ)p K) X *, K,g *. (6) w φmpf X*,,g *. 8 This aspect of the model is unfotunate because it implies that an incease in the commission ate, φ, inceases the numbe of agents but has no effect on the level of effot. Thus, ou model cannot addess the question of whethe an excessive commission ate causes undeemployment of agents. 11

12 Since m() is, by assumption, homogenous of degee one in its fist thee aguments, the above equation educes to, 1 φ 1 FX ( * )m X* X X * b P X * P K, φp K w,g* = 0, (7) except when ( X * )= 0 in (7), in which case (2 ) and (6) must be employed. We shall efe to m() in (7) as the nomalized matching function, and when its aguments ae suppessed shall denote it by m. To simplify subsequent notation, we let define GX ( *,P;,b,φ,K,w,g *,F ()) 1 FX * IX ( *,P;,φ,K,F ()) P K 1 φ FX* f X *, 1 φ m X* X X * b P X * P K, φp K w,g* (8). (9) I()=0 incopoates the builde's fist-ode condition and zeo-pofit condition. G()=0 incopoates the steady-state condition, the household's choice of X *, the builde's zeo-pofit condition, and the eal estate agent's effot-choice decision, and enty equilibium condition. The est of this section is devoted to exploing the popeties of G()=0 and I()=0. We shall employ the following assumptions: Assumption 1: b K 1 φ Assumption 2: i) F f is inceasing 9 in X, implying 10 that F F is inceasing in X ; ii) lim X 0 F f = 0 9 Note that this implies that the second-ode condition of the builde's maximization poblem is satisfied. See fn Stict log-concavity is peseved unde integation (see Caplin and Nalebuff (1991)). Thus, F is stictly log-concave, which geneates the following sting of equivalencies: F stictly log-concave d ( F F ) > 0 F 2 f F>0 d ( X * X) > 0. dx * dx * 12

13 Assumption 3: i) m( 0,.,.,. )= 0, m (.,0,.,.)= 0; ii) m i > 0, i = 1,...,4; iii) m ii < 0, m ij > 0, i, j = 1,K,3, whee subscipts denote patial deivatives. Assumption 1 states that seach costs exceed amotized goss-of-commission constuction costs. Assumption 2 is that F() is stictly log-concave. Assumption 3 states that the matching function has the popeties of a egula poduction function ove "inputs", S, and A, and that and S ae essential in poduction. Assumption 2 implies that I()=0 has the following popeties K I 0, = 0 1 φ dp dx * I =0 = d dx F( X * ) f( X * ) >0. (10a, b) We also have the following popeties fo G()=0: G 0, b = 0 K G b 1 φ, K = 0, (11a, b) 1 φ and fo those X * and P fo which the matching ate is positive ( X * > 0 and P > K 1 φ ) dp dx * G=0 X f m F m * X 1 + d ( X * X) dx * ( b P X = * ) 2 ( b P X * ) ( X F m * X) ( 1 φ) φ 1 + m ( b P X * ) 2 2 K + m 3 w <0 (11c) and ( X * + P b) G()=0 <0. (11d) Eqs. (11a) and (11b) ae obtained fom (2 ) and (6) (since at both points the aggegate matching ate is zeo). The inequality (11c) follows fom Assumption 2i) -- see fn and (11d) follows fom (2 ) and states that expected discounted accommodation costs in the seach pool ae less than the costs of seaching foeve. 13

14 Fom these popeties of G()=0 and I()=0, it is staightfowad to show that equilibium exists and is unique. Figue 1 potays G()=0 and I()=0 in X * P space, using the paametes of the numeical example which ae listed in Table 2. INSERT FIGURE 1 HERE Thee ae numeous distotions in the model. Thee ae the familia seach extenalities. Neithe households when they choose X *, no buildes when they choose to build and to set the pice, no eal estate agents when they choose effot and enty, take into account the effect of thei actions on the aggegate matching ate. uildes exploit thei maket powe in setting pices. Agents ae subject to a at ace extenality in thei choice of effot. And the commission ate itself is distotionay. Igaashi and Anott (1995) povide a easonably full analysis of the analogous extenalities fo the ental housing maket. Fo this pape, we note simply that the optimal commission ate is the optimal ate fo a highly distoted economy, and can be viewed as balancing the "ight" numbe of eal estate agents against mitigating distotions. We now tun to the compaative static popeties of equilibium. The esults ae deived in Appendix 1 and pesented in Table 1. ξ measues the efficiency of the matching pocess, enteing G()=0 as 1 Fξ m = 0. The table pesents no supises and the esults ae eadily explainable. Fo easons of space, we shall discuss only the effects of a change in the commission ate. A ise in φ Table 1 Compaative Statics b φ K w g * n ξ X? +?? P? +? causes the I()=0 cuve to otate counteclockwise; holding X * fixed, to estoe zeo pofits fo buildes equies a ise in pice. Its effect on G()=0 is, howeve, ambiguous. The intuition is that, holding P and X * fixed, the ise in the commission ate equies that the numbe of houses fo sale fall (to continue 14

15 satisfying the zeo pofit condition fo buildes) and causes the numbe of agents to ise. The fome effect deceases the matching ate, the latte inceases it. If the fome effect dominates, then, holding P fixed, X * must ise in ode that (7) continue to hold, so that G()=0 bulges out. In this case, the shifts in I()=0 and G()=0 combine to incease the equilibium pice while the equilibium value of X * can ise o fall. If the latte effect dominates, pice may eithe ise o fall, but the esevation mismatch cost unambiguously falls. 2. Nomative Popeties of the Model 2.1 The optimal commission ate Vaying the commission ate geneates a locus of equilibium points in X ( *, P) space, Γ( X *, P)= 0. We wish to detemine which of these points is optimal and, fom this, what the optimal commission ate is. What can be said about this locus? It is continuous and lies inside the tiangle with vetices ( 0, K 1 φ ),( 0, b ) and b ( K 1 φ, K 1 φ ). Also, since a ise in the commission ate otates I()=0 counteclockwise but has an ambiguous effect on G()=0, the locus cannot intesect itself. These estictions ae not vey estictive. What is the appopiate citeion to judge between altenative commission ates? Since buildes make zeo pofits and eal estate agents ean thei oppotunity wage, social costs equal the expected pesent value of accommodation costs, C, and the optimal commission ate is that which minimizes these costs. We aleady know fom the fist-ode condition with espect to X * of the household's cost-minimization poblem, that C = P + X *. Thus, detemination of the optimal commission ate has a neat geometic intepetation (see Figue 2 which employs the paametes of the example) because social cost contous have a X slope of - 1 in *, P space. INSERT FIGURE 2 HERE The optimum, indicated by Z in the Figue, occus at the point of lowest social cost on Γ( X *, P). Algebaically, the optimal commission ate solves 15

16 min X,P,φ C = X* + P s.t. i) G()=0 (12) ii) I()=0. Substituting out P fom I()=0 and using (8), this poblem becomes min X *,φ X * + K 1 φ + FX* f X * (12 ) s.t. X 1 FX ( * X( X * ) * ( 1 φ)fx * )m, b K 1 φ F X* f( X * ) X* 2 Kf X * φ, K 1 φ + F X* f X * w,g * = Illustative calibation In this subsection, we show how the model can be calibated on the (unealistic) assumption that the model povides a tue desciption of eality. Fo a paticula housing maket, infomation is eadily available on the inteest ate, the population gowth ate, the volume of sales, aveage sales pice, mean time to sale fo houses, mean time on the maket fo buyes, the aveage commission ate, and the numbe of full-time-equivalent eal estate agents. We wish to employ such data to estimate the unknowns -- constuction costs, and chaacteistics of the matching function and of the c.d.f. of mismatch costs. The pocedue is simila to that employed by lanchad and Diamond (1990, 1991) fo the labo maket. Taking dollas and yeas as ou units of measuement, suppose 11 that we obseve a housing maket in which =.05, n =.04, φ =.06, mean time to buy = =.3. (4 months), mean time to sell = S =.25 (3 months), P = $100000, and w = $ Then, fom (1), = , implying that = and S = ; fom (5), A = ; fom (3), K = ; and fom (4), F f = That leaves us with (2). Assume that FX has the fom 12 FX =c 0 X δ. 11 The numbes wee chosen to be boadly consistent with aveage maket data epoted in National Association of Realtos (1991). 12 Anott (1989) povides a theoetical justification fo this assumption, with δ being intepeted as the effective dimensionality of the chaacteistics space. 16

17 Then X = δ δ +1 X*, X * X = 1 δ +1 X*, and F f = X * * X δ. If b is measuable, then (2) and = δ povide two equations in X * and δ. If, fo example, b = , then X * = 1000 and δ = 16.20, implying that X = If b is not measuable, then obsevation of two equilibia would pemit the simultaneous detemination of the X * 's fo both equilibia, as well as b and δ. The concept of an "oppotunity to match" is useful in the development of the theoy but is not empiically opeational. One could make the concept concete by defining an oppotunity to match as the viewing of an advetisement, an inquiy put to a eal estate agent concening a popety, o an inspection. ut data on these magnitudes ae geneally not available. Note, howeve, that the matching function m() always entes as FX ( * )m(). Thus, the theoy can be ecast in tems of a "successful match" o sales function σ( X *,,S, A,g). Sales ae obsevable, as ae, S, and A; g can be suppessed since the theoy pedicts that it is constant; and X * can be infeed pe the pocedue employed in the pevious paagaph. Thus, σ() is estimable empiically. To solve fo the sales function would equie data fo diffeent equilibia. If the sales function wee the same acoss cities, coss-section data could be employed. If the sales function wee constant ove time, times seies data could be used. Suppose, fo the sake of illustation, that the sales function is σ( X *,,S, A) = c( X * ) S.4 A.2, with c = Ã 10-49, which is consistent with the numeical example. Then, with the assumed paametes, fom (12 ) the optimal commission ate is.28%. We attach little pactical significance to this figue since the elasticities of the matching function wee assumed athe than estimated and since the undelying model is unealistically simple. Futhe, this commission ate is fa below the ate offeed in any housing maket in the wold. The example is nevetheless of inteest because it illustates a geneal pocedue that can be employed to estimate the optimal commission ate. It is also instuctive in illustating how a change in the commission ate impacts the housing maket. The esults of the example ae shown in Table 2. The most stiking featue of the example is how little effect the eduction in the commission ate fom 6% to.28% has on the numbe of buyes, the numbe of selles, expected time to buy, and expected time to sell. uyes simply become somewhat less fussy in thei choice housing. 17

18 Table 2 Numeical Examples Exogenous paametes (units: ys., $) =.05 b = 9, γ = 0.2 n =.04 δ = 16.2 c = Ã w = α = 0.4 K = 92,839.5 β = 0.4 Results ase Case Optimum ( φ =.06) ( φ =.0028) C (social cost) 120, ,136 P 100,000 94,381. X * 1, S A (expected time to buy) S (expected time to sell) Concluding Comments We fist discuss ways in which the model can be extended to impove its ealism fo empiical application. 18

19 3.1 Diections fo futue eseach The model can be extended to teat exit fom the maket, as well as mobility within the maket. A simple way to teat these phenomena is as follows. Assume that a household eceives a signal to exit (accoding to a Poisson pocess with ate µ ) and anothe signal which tigges a change in taste (accoding to a Poisson pocess with ate θ ), pehaps due to a change in the household's composition, that equies an owne-occupying household to move within the maket. A household knows only the stochastic pocess geneating these signals. Assume that when a household eceives a signal, it moves immediately and puts its unit on the maket; assume also that housing makets ae symmetic, so that the numbe of households enteing a housing maket equals the numbe exiting the maket plus that maket's shae of newly-fomed households. by 13 Unde these assumptions, eqs. (3) - (5) ae unchanged. Eq. (1) is eplaced Eq. (2) is eplaced by ( n + θ + µ )1 ( ) = 0. +µ +θ b( )+ PΞ+ X + + µ + θ PΞ X* =0 whee Ξ= +µ+θ+ S S ( 1 + φ ( µ+θ) ) > 1. + Reducing this system of equations yields two equations, (4) and ( + µ + θ) b PA X* n + θ + µ n+θ +µ X * X = 0, in two unknowns, whee 13 The deivations ae pesented in Appendix 2. 19

20 A = + µ + θ ( µ + θ)k P and ( X* ) (( m = m 1 n+θ +µ, 1 φ)p K) X * K n φmpf X*,,g *. This teatment of exit and intenal mobility is tactable but not completely satisfactoy. Fist, the decisions to exit the maket o to move within the maket ae in fact endogenous. One would expect µ and θ to depend on the commission ate, since a highe commission ate has a lock-in effect. As well, even with ex ante homogeneous households, one would expect that households would acquie some infomation duing the seach pocess concening thei futue exit o intenal mobility, which would affect thei choice of X *. Second, a homeowne's decision concening whethe to move immediately upon eceiving a signal o to stay until the home is sold (o to adopt some intemediate stategy) should be endogenized. 14 This model is unealistic in anothe way because it ignoes the ental secto. Thee ae seveal linkages between the ental and owne-occupied housing maket. Not all households that ente the maket aim to own thei esidence, at least not immediately; thee is mobility within the maket between the ental and owne-occupied submakets; and eal estate agents opeate in both submakets. The simplest way to teat the two submakets simultaneously is to assume that a household is assigned to a submaket by an exogenous stochastic pocesses. Assume that a popotion q of entants to the housing maket aim to 14 The model with intenal mobility and no exit is simila to that of Wheaton (1990). The model of Wheaton's pape diffes fom ous in the following espects: i) it focuses on the shot un in which the stock of houses is fixed (which is why, in contast to ou model, housing pice is so sensitive to the vacancy ate); ii) it teats only two housing types so that households ae eithe matched o mismatched; iii) housing pice is detemined by bagaining; and iv) when a homeowne becomes mismatched with his home, he does not put his "old" home on the maket until he has found a "new" home, which assumes that he stays in his old home until he has found a new home. With egad to the diffeence between Wheaton's assumptions iii) and iv) and the coesponding assumptions in ou model, each pai of assumptions is equally abitay (though they may diffe in thei ealism). A moe satisfying teatment would make both the pice detemination mechanism and the household's decision concening when to stat seaching fo a new home, to move, and to list the old home endogenous. In contast to ou model (with the assumption of the log-concavity of F ()), in Wheaton's model thee may be multiple equilibia. 20

21 ent, that a household in the ental submaket (whethe enting o in the ental seach pool) switches to the owne-occupied seach pool at the Poisson ate ρ, and that a household in the owne-occupied submaket (whethe owning o in the owne-occupied seach pool) switches to the ental seach pool at the Poisson ate ω. As in the basic model, assume that a household vacates its unit as soon as it eceives a signal to change its state. Let D be the popotion of households that ae entes and E the popotion that ae in the ental seach pool. Account also needs to be taken of seach in the ental housing maket. Modeling this analogously to seach in the owne-occupied housing maket, let ME,V, A, g be the aggegate ate of oppotunities to match in the ental submaket with V the numbe of vacant ental units, A the numbe of eal estate agents opeating in the ental submaket, and g the effot each expends. Also, let Y be ental mismatch costs with cdf HY. Finally, as with the owne-occupied submaket, exit fom the maket (at popotional Poisson ate µ ) and mobility within the ental secto (at popotional Poisson ate λ ) can be teated. The flows between the vaious household states, unde these assumptions, ae given in Figue 3 below. µ µo (1-q)(n+µ) O E θ O O D MH q(n+µ) E D D µe µd Figue 3: Flows between household states with ental and owne-occupie submakets 21

22 We shall omit the analysis which is tedious. Some points bea mention. Conside the modeling of the ental secto. Pospective tenants ae modeled in essentially the same way as in Igaashi (1991) except that, hee, ental eal estate agents chage a finde's fee, which is a popotion of annual ent. Taking the maket ent as given, pospective tenants choose a esevation mismatch cost Y *. Landlods too ae modeled in essentially the same way as in Igaashi (1991). They puchase houses fom buildes at pice P and then choose ent R 0 to maximize the pesent value of pofits, taking the maket ent as given. Imposition of symmety yields an equation fo the maket ent, and the zeo pofit condition gives an equation fo ental housing vacancies. Finally, the ental eal estate agents' effot and enty decisions ae vey simila to the owneoccupie eal estate agents' decisions which wee analyzed ealie. Conceptually the model is no moe complicated than the basic model of the pevious section. Analytically, howeve, it is consideably moe cumbesome because of the complexity of the flows between household states. This extended model could be eniched by having the household choose whethe to switch between states and whethe to emain in its unit while it is up fo sale o ent depending on the poductivity of seach in each situation. Anothe way in which the model is desciptively unealistic is that it teats households as ex ante identical and houses as completely identical, except fo symmetic idiosyncatic diffeences. In fact, households diffe in tastes, income, and demogaphic chaacteistics, and houses diffe in quality and location. The stengths and weaknesses of epesentative agent models ae well-known and the points apply with double foce hee, because thee ae epesentative houses as well. Teating epesentative agents and units esults in consideable algebaic simplification, but, consideing the highly non-linea natue of the model, this teatment may geneate substantial aggegation bias. The seveity of this bias is an empiical matte. The model could be augmented to allow fo housing submakets diffeentiated by quality, location, stuctue type, and numbe of bedooms, fo example. And household tastes ove submakets could be epesented by pobabilistic choice models. Then the popotion of households in a paticula income-demogaphic goup choosing to seach in diffeent ental and owne-occupied submakets would be calculated fom the systematic expected discounted utility fo the vaious submakets as well as the degee and fom of idiosyncasies in tastes ove these submakets. And intenal 22

23 mobility could be modeled as tiggeed by income changes and demogaphic tansitions. These and elated issues 15 ae impotant fo constucting an accuate model but they ae also impotant to undestanding whethe esidential eal estate commission ates should diffe acoss submakets. We wee able to sidestep modeling what it is that eal estate agents do and how thei pesence facilitates home sales, by simply inseting the numbe of agents in the oppotunities-to-match function. This is appopiate if, as in ou model, all sales ae intemediated by bokes, all bokeage contacts ae the same, and all households, houses, and agents ae ex ante identical. As noted in the intoduction, howeve, thee has been a tend towad geate vaiability in the commission ate on popotional commission contacts as well as geate divesity in contact fom. Thee has also been a movement towads buyes' bokes. As well, household, houses, and eal estate agents ae not the same. And finally, a significant popotion -- between 15 and 20 pecent -- of esidential sales ae made without boke intemediaies. When these complications ae accounted fo, the aggegate oppotunities to match function no longe depends simply on the numbe of buyes, houses fo sale, and agents. A iche model will have to come to gips with what it is that agents do, how they allocate thei time ove popeties and ove the duation of a listing, how thei behavio is modified with altenative contactual foms, and how they signal thei ability. These issues have been addessed in the many patial equilibium papes on esidential eal estate bokeage Summay This pape developed a simple, seach-based, geneal equilibium model of the owne-occupied housing maket with the aim of detemining the socially optimal commission ate payable to eal estate agents on thei sales. A boad- 15 A satisfactoy teatment of housing demand should take into account the intetempoal natue of the household budget constaint, as well as moving costs, demogaphic tansitions, and cedit constaints, especially downpayments on owne-occupied housing. And the supply side should incopoate quality depeciation, maintenance, and ehabilitation, as well as land and hence the possibility of edevelopment. 16 These issues have been discussed in vaious papes by Anglin (1993), Chen and Rosenthal (1994), Colwell and Yavas (1994), Geltne, Kluge and Mille (1991), Miceli (1989), Zhang (1993), and Zon and Lasen (1986). 23

24 bush desciption of the model is as follows. Households and houses diffe only in idiosyncatic ways and eal estate agents ae identical. Upon enteing the gowing maket, a household joins the seach pool of pospective home buyes and eceives oppotunities to match accoding to an oppotunities-to-match function à la Diamond/Pissaides/Hosios, which depends on the stock of buyes, homes fo sale, and eal estate agents, as well as agents' effot level. The household chooses the esevation quality of a match, and when it eceives a satisfactoy match moves immediately into its home whee it emains foeve. Competitive buildes choose the pice of housing and competitive eal estate agents thei effot level, with enty diving down buildes' pofits to zeo and agents' income (with an exogenously-specified commission ate) to its oppotunity level. The unique (long-un) equilibium is chaacteized by the popotion of households in the seach pool, the atio of vacant houses to seaches, the numbe of eal estate agents, the esevation match quality, agents' effot, and the pice of housing. ecause buildes make zeo pofits and eal estate agents thei oppotunity income, all suplus accues to households. The socially optimal commission ate theefoe minimizes the expected discounted pesent value of an enteing household's accommodation costs. The tadeoff detemining the optimal commission ate is staightfowad. The lage the numbe of agents, ceteis paibus, the geate the numbe of oppotunities to match, which pemits faste and/o bette matching, but also the highe the aggegate cost of agents' time, which is eflected in a highe housing pice. While thee is consideable scope fo impoving the basic model by adding desciptive ealism, to eject it because of its simplicity would be a mistake. Fom a theoetical pespective, its simplicity is a vitue. The basic model highlights the essential tadeoffs undelying detemination of the optimal commission ate on owne-occupied housing with a minimum of clutte. Fom an empiical pespective, its simplicity has the advantage that the model, augmented to include exit and intenal mobility, can be estimated/calibated on the basis of eadily available data. Estimating eniched vesions of the model should esult in moe confident estimates of the optimal commission ate but, because of data deficiencies which plague econometic wok in housing, the complexity of the estimation execise will be consideably geate. 24

25 While we focused exclusively on the model's application to calculation of the optimal commission ate, the model, appopiately adapted, can be applied to othe public policy issues vis-à-vis esidential eal estate bokeage. An inteesting example is the ecent tend, paticulaly in the westen United States, to buyes' bokes, wheeby the pospective buye hies his own agent, in contast to the conventional system unde which the pospective buye woks with a selling agent who is an agent to the listing agent and hence a sub-agent to the selle. Using coss-sectional state data, one could estimate the aggegate oppotunities-to-match function, including the numbe of buyes' agents and conventional agents as sepaate aguments. A finding that buyes' agents ae moe poductive at the magin than conventional agents would povide evidence favoing laws and egulations which facilitate buye bokeage. 25

26 ILIOGRAPHY Anglin, P., 1993, "A Note concening a Competitive Equilibium in the Maket fo Agents," Economics Lettes 41, Anglin, P. and R. Anott, 1991, "Residential Real Estate okeage as a Pincipal- Agent Poblem," Jounal of Real Estate Finance and Economics 4, Anott, R., 1989, "Housing Vacancies, Thin Makets and Idiosyncatic Tastes," Jounal of Real Estate Finance and Economics 2, lanchad, O. and P. Diamond, 1990, "The Aggegate Matching Function," in P. Diamond, ed., Gowth/Poductivity/Unemployment (Cambidge, MA: The M.I.T. Pess), lanchad, O. and P. Diamond, 1989, "The eveidge Cuve," ookings Papes on Economic Activity 1, Caplin, A. and. Nalebuff, 1991, "Aggegation and Social Choice: A Mean Vote Theoem," Econometica 39, Chen, Y., and R. Rosenthal, 1994, "Asking Pices as Commitment Devices," woking pape, oston Univesity. Colwell, P. and A. Yavas, 1994, "uye okeage: Incentive and Efficiency Implications," woking pape, Univesity of Illinois, Office of Real Estate Reseach, Ubana, IL, USA. Diamond, P., 1981, "Mobility Costs, Fictional Unemployment, and Efficiency," Jounal of Political Economy 89, Fedeal Tade Commission, 1983, The Real Estate okeage Industy (Washington, D.C.: U.S. Govenment Pinting Office). Fu, Y., 1993, "Thee Essays in Real Estate Economics," unpublished Ph.D. dissetation, Faculty of Commece and usiness Administation, Univesity of itish Columbia, Vancouve, Canada. 26

27 Geltne, D.,. Kluge, and N. Mille, 1991, "Optimal Pice and Selling Effot fom the Pespectives of the oke and Selle," AREUEA Jounal 19, Hosios, A., 1990, "On the Efficiency of Matching and Related Models of Seach and Unemployment," Review of Economic Studies 57, Igaashi, M., 1991, "A Random-Matching and Seach Model of Rental Housing Makets," unpublished Ph.D. dissetation, Depatment of Economics, Queen's Univesity, Kingston, Ontaio, Canada. Igaashi, M. and R. Anott, 1995, "Rent Contol, Mismatch Costs and Seach Efficiency," Regional Science and Uban Economics, fothcoming. Miceli, T., 1989, "The Optimal Duation of Real Estate Listing Contacts," AREUEA Jounal 17, Miceli, T., 1992, "The Welfae Effects of Non-Pice Competition among Real Estate okes," AREUEA Jounal 20, National Association of Realtos, 1991, Home uying and Selling Pocess. Osbeg, L., 1993, "Fishing in Diffeent Pools: Job Seach Stategies and Job Finding Success in Canada in the Ealy 1980's," Jounal of Labo Economics 11, Petes, M., 1994, "Equilibium Mechanisms in a Decentalized Maket," Jounal of Economic Theoy 64, Pissaides, C., 1990, Equilibium Unemployment Theoy (Cambidge, MA: asil lackwell). Simans, C.F., G. Tunbull, and J. enjamin, 1991, "The Makets fo Housing and Real Estate oke Sevices," Jounal of Housing Economics 1, Wheaton, W., 1990, "Vacancy, Seach and Pice in a Housing Maket Matching Model," Jounal of Political Economy 98, Yavas, A., 1994, "Economics of okeage: An Oveview," Jounal of Real Estate Liteatue 2,

28 Yavas, A., 1995, "Can okeage Have an Equilibium Selection Role?" Jounal of Uban Economics 37, Yinge, J., 1981, "A Seach Model of Real Estate okeage," Ameican Economic Review 71, Zhang, A., 1993, "An Analysis of Common Sales Agents," Canadian Jounal of Economics 26, Zon, T.S. and J.E. Lasen, 1986, "The Incentive Effects of Flat Fee and Pecentage Commissions fo Real Estate okes," AREUEA Jounal 14,

29 Appendix 1 Compaative Static Popeties of Equilibium Total diffeentiation of (8) and (9) yields X f m F m * X 1 + d ( X * X) dx * ( b P X * ) 2 ( b P X * ) 1 = F m 1 F 2 f X * X 2 b P X * d( F f) 1 dx * P X * X ( b P X * ) 2 m 2 X F m * X 1 φ φ 1 + m ( b P X * ) 2 2 K + m 3 w dx * dp ( 1 φ P K 2 K ) d + F m X * X 1 ( b P X * ) 2 0 db + F m P P 2 K + m 3 w K dφ 1 φ 2 + F m 1 2 ( K ( 1 φ )P K K ) φ dk + F m 3 0 φp w 2 dw + F m 4 0 dg* dn dξ. (A1.1) Some of the compaative static deivatives can be signed on the basis of the sign patten dx * db > 0 dx * dw > 0 dx * dg < 0 dx * * dn = 0 dx * dξ < 0 (A1.2a) dp db > 0 dp dk > 0 dp dw > 0 dp dg * < 0 dp dn = 0 dp dξ < 0. (A1.2b) The signs of the othe compaative static deivatives can be signed on the basis of neithe the sign patten no a fulle analysis. Intepetation will be facilitated by ewiting (A1.1) in tems of elasticities. Note that ε m, = m 1 F ε m, = m S 2 SF ε = m 3ÃF ε = m m, Ã m, g 4gF (A1.3) 29

30 whee, S, and à now denote the nomalized, S, and A -- e.g., = Also, ε m, = ε m, etc. Thus, X * X b P X *. fx * ε m, P X F ε * m, + ε b P X * X * X, X * X* d( F f) dx * = ε 1 P m,( b P X ) ε * m,s ˆ + ε m, F f 0 + ε φp m,s( ( 1 φ )P K)+ε m,a ˆφ + ε m,s Kφ ( 1 φ) 2 ( 1 φ)p ( b P X ) ε * m,s (( 1 φ )P K) ε m,a ˆX * P ˆP b b P X * ( 1 φ)p (( 1 φ )P K) K 1 φ ˆK ˆb (A1.4) + ε m,a ε 0 ŵ+ m,g 0 0 ĝ+ 0 ˆn+ 1 0 ˆξ. The -notation denotes a popotional change in the elevant vaiable. The compaative static deivatives of geatest inteest ae with espect to the commission ate, φ. Denote the deteminant of the matix pemultiplying [ ˆX * ˆP ] T in (A1.4) by <0. Then ε = ˆX * X *,φ ˆφ = 1 φp P ε m,s (( 1 φ )P K)+ε m,a ε m, b P X * Kφ P ( 1 φ) 2 ( 1 φ)p ε m,s (( 1 φ )P K) ε m,a (A1.5a) and ε P,φ = ˆP ˆφ = 1 fx * which educes to φp ε m,s 1 φ X F ε * m, + ε b P X * X * X, X * X* d( F f) dx * (P K)+ε m,a Kφ, (A1.5b) ( 1 φ ) 2 ε = 1 X *,φ ε m, 1 φ PKφ 2 b P X * ε m,s P φ Kφ +ε m,a P+ 1 φ 1 φ 2 (A1.5a*) and 30

31 ε P,φ = 1 fx * Kφ X * ε F( 1 φ) 2 m, b P X + ε * X * X, X * X * dff +ε m, A dx *. P K Kφ φp ε ( 1 φ) 2 m,s 1 φ X* dff dx * (A1.5b*) A fai bit of the complication in the algeba deives fom the geneality of the fom of F(). When F() is unifom, fx * ( 1 φ)p K = X* (fom (9)). Then and dx * F = 1, ε X * X, X * = 1, d F f b P + X * X = P1+ε m, b P X * + ε m,s ( * 1 φ)+ε m,a P = 1, and ε = P X *,φ ε m, 1 φ Kφ 2 b P X * +ε φ m,s 1 φ ε 1+ Kφ m,a P 1 φ 2 (A1.6a) ε P,φ = P Kφ P1 φ +ε b P 2 m, b P X * Kφ P 1 φ 2 +ε m,s φ ε m,a X *. (A1.6b) P 31

32 Appendix 2 Deivation of Analogs to (1) and (2) in Section 3.1 n µ O µo µ ç O The flows ae given in the above diagam. We have that Ḃ = n + µo + θo + O = 1 Ḃ = n (A2.1.1) (A2.1.2) (A2.1.3) (A2.1.1) descibes the evolution of ; (A2.1.2) indicates that population is nomalized at one; and (A2.1.3) is the steady-state condition. Substituting (A2.1.2) and (A2.1.3) into (A2.1.1) yields ( n + θ + µ )1 ( ) = 0, (A2.1.4) which is the analog to (1) pesented in the body of the pape. Let C O be the expected pesent value of accommodation fo a household in owne-occupied housing, fom the pespective of a household in the seach pool. Then C = bdt + C 1 µdt dt + C ( µdt)+ C O + P dt e dt (A2.1.5) which educes to 0 = b + C + CO + P. (A2.1.6) 32