Who Searches? Department of Economics, University of Helsinki Discussion Paper No 601:2004 ISSN ISBN May 5, 2004.

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1 Who Searche? Klau Kultti Univerity of Helinki Tuoma Takalo Bank of Finland Antti Miettunen Helinki School of Economic Juha Virrankoki Univerity of Helinki Department of Economic, Univerity of Helinki Dicuion Paper No 601:2004 ISSN ISBN May 5, 2004 Atract Thi paper anwer the quetion whether uyer eek for eller or vice vera, in a directed earch model with no explicit earch cot. We focu on a ymmetric mixed-trategy equilirium where oth uyer and eller can earch and wait. The mixed trategy tell the likelihood of earching veru waiting, and it provide inight into the relevance of tandard practice of potulating the identity of earcher and waiter. We conider two trading mechanim, auction and argaining. The Department of Economic, P.O. Box 17, FIN-00014, Univerity of Helinki, Finland. Tel: , klau.kultti@helinki.fi Bank of Finland, Reearch Department, P.O. Box 160, FIN-00101, Helinki, Finland. Tel: , tuoma.takalo@of.fi Correponding author. Reearch Unit on Economic Structure and Growth, Department of Economic, P.O. Box 17, FIN-00014, Univerity of Helinki, Finland. Tel: , juha.virrankoki@helinki.fi 1

2 main reult i that the memer of the larger population earch and thoe of the maller population wait. JEL Claification: J41, J64, C78, D44, D83 Keyword: earch, auction, argaining 1 Introduction Search theoretic model are widely ued with variou level of detail a to the actual earch and meeting proce. In the implet form agent meet each other with exogenouly given rate, while in more detailed model the agent may chooe their earch effort and whether to earch or wait. The very quetion of who wait and who earch i addreed in a earch model of endogenou money y Burdett, Cole, Kiyotaki and Wright (1995). In the model ome agent are money holder, or uyer, and other are commodity holder, or eller. Searching involve an explicit cot which may e different for different kind of agent, while waiting involve no explicit cot. The author determine the equiliria of the model when the agent aic trategic choice i whether to earch or wait. Not too urpriingly there are many equiliria depending on the cot. But matter of coordination alo generate multiple equiliria. Roughly put, if all eller wait, then all the uyer mut earch, and vice vera. To gain ome inight into the likelihood that it i the uyer (or eller) who earch, the author compare the ize of the parameter pace upporting variou equiliria. Their reult i that the larger the numer of different commoditie, or the more difficult the doule coincidence prolem of arter, the more likely it i that the money holder, or uyer, earch. The link etween the ize of the parameter pace upporting an equilirium and the likelihood of that equilirium i tenuou at et. In thi article we try to offer a more traightforward link etween the likelihood of the earch deciion and the fundamental oftheeconomyuingamodelwherethereareonlytwokindofagentcalleduyerand eller. We do not introduce any cot of earching, ut the earch deciion turn out to depend on the trading mechanim and the ratio of uyer to eller. We ue auction and argaining a alternative trading rule. Herreiner (1999) tudie alo the quetion aout 2

3 who earche, ut he ignore price formation. Given a trading mechanim we are intereted in how the ratio of uyer to eller affect the likelihood of earching for oth type. We focu on ymmetric (mixed trategy) equiliria and ignore the two co-ordination game like equiliria where all uyer earch and all eller wait or vice vera. The mixed trategy then ha a traightforward interpretation a the likelihood that a particular type of agent earche rather than wait. It turn out that mixed trategy equiliria do not exit when the numer of uyer and eller differ a lot. In thi cae we potulate that the reaonale equilirium i the one where the mixed trategy equilirium converge to when the numer of uyer and eller change appropriately. Kultti and Takalo (1999) alo invetigate thi kind of model ut their focu i on the evolutionary taility of market with different trading rule and earch pattern. The main reult of the preent article i that the memer of the larger population earch and thoe of the maller population wait. We check that thi equilirium i immune againt coalitional deviation. If the population are equal enough, there are two market: uyer earch in one market and eller in the other. In thi cae we how that auction i more efficient trading rule than argaining, ecaue the agent are plit into the market in more advantageou proportion. The ret of the article i organied a follow: Section 2 preent a matching model with deciion to earch or wait, and in Section 3 and 4 we incorporate the trading rule into the model. In Section 5 we derive the main reult of the paper: who earch and who wait. The relative efficiency of auction and argaining i tudied in Section 6. In Section 7 we extend the model to a one where agent do not decide whether to earch or wait ut they chooe which of the many marketplace to go, and we compare the efficiency of that market tructure to that of the aic model. Section 8 conclude. The derivation of the reult are in the Appendix. 3

4 2 The Model There are B uyer each with a unit demand and S eller each with one indiviile oject for ale. The uyer get utility normalied to one from conuming the oject, and the eller get utility normalied to zero from conuming it. The economy extend to infinity, and time proceed in dicrete period. The agent dicount future with factor δ (0, 1). When the agent trade they exit the economy and are replaced y identical utunmatchedagent. Thimeanthattheratioofuyertoellerremaintheame in every period. To model the meeting proce we ue an urn-all model (ee for example Lu and McAfee, 1996). The agent who decide to wait are in fixed poition, one agent in each location, and the agent who decide to earch are randomly and independently allocated on the waiting agent. Thi meeting technology i well defined and tractale. Further, ince multiple meeting are poile one can meaningfully tudy a variety of trading mechanim. If w agent wait and m agent move, the numer of agent a waiting agent meet i a inomial random variale with parameter m and 1/w. Tractaility i achieved y auming that w and m are infinite, ince in thi cae one can approximate the inomial with a Poion ditriution with parameter m/w. Then the proaility that a waiting agent meet exactly k moving agent i e m/w (m/w) k /k!. We are going to invetigate two different trading mechanim: auction and argaining. Auction i modelled a a econd price ealed id procedure in which everyone in a meeting participate. For concretene, conider a eller who wait and meet exactly one uyer. Then the eller id hi reervation value, which i the ame a ignoring the uyer and waiting for new trading opportunitie in the next period; and the uyer id one minu the eller reervation value. The uyer win the oject and pay the eller reervation value. If the eller meet two or more uyer, oth uyer id the ame and one of them get the oject. It doe not matter which uyer get the oject ince the uyer are indifferent etween getting the oject and earching for new trading opportunitie. Another way to look at the auction i that in the firt cae the uyer make a take-itor-leave-it offer, and in the econd cae the uyer engage in a Bertrand-type idding. 4

5 Bargaining i alway pairwie, and if a eller meet everal uyer he jut pick one of them at random for hi trading partner. To make thing imple we aume that the uyer and eller jut plit the availale urplu in half. One would proaly alo want to conider the cae of poted price. Kultti (1999) how that thi i equivalent to auction. Since oth uyer and eller can decide to wait or earch, we conider two market. Inonemarketproportionx of uyer earch and proportion y of eller wait. In the other market proportion 1 x of uyer wait and proportion 1 y of eller earch. In equilirium the uyer have to do equally well regardle of whether they wait or earch. The ame mut hold for the eller. Of coure, it i not necearily the cae that thi condition i conitent with there eing two market, and in thee cae there will e only one market in equilirium. Let u denote the ratio of total numer of uyer to that of eller y B/S = θ. In the market where the eller wait the ratio of earcher to waiter i xb/ys = xθ/y γ, andinthemarketwheretheuyerwaittherelevantratioi(1 y) S/(1 x)b = (1 y) / (1 x) θ ϕ. In other word, the numer of agent that arrive in a eller or uyer location i governed y a Poion proce with parameter γ or ϕ, repectively. Let u next determine the agent expected utilitie in each market with the two trading mechanim. 3 Auction Conider the market where eller wait. Let u determine a eller expected life time utility V and a uyer expected life time utility V, evaluated in the very end of a period, a V = δ e γ V + γe γ V + 1 e γ γe γ (1 V ), (1) V = δ e γ (1 V )+ 1 e γ V. (2) In (1) the firt term in the quare racket i the proaility of the eller meeting no-one in which cae he get hi expected life time utility from waiting, V. The econd term i the proaility of meeting exactly one uyer. In thi cae the eller alo get V ince the uyer make a take-it-or-leave-it offer to the eller. The take-it-or-leave-it offer 5

6 aumption give the mover a poitive proaility to get the whole urplu of the trade (like the tayer have, too), treating the tayer and mover a equally a poile. The third term i the proaility of meeting two or more uyer in which cae the uyer get their reervation utility and the eller get the ret of the urplu, 1 V. It hould e noted that a uyer alway meet a eller. Thu, the proaility that no other uyer meet theameellerie γ, and in thi cae the uyer get all the urplu from the meeting, 1 V. Thi i the firt term in the quare racket in (2). If other uyer appear, each of them get hi expected utility from earching, V. Solving (1) and (2) yield V = δ (1 e γ γe γ ) 1 δγe γ, (3) V = δe γ. (4) 1 δγe γ In an analogou fahion one can calculate the uyer and eller expected life time utilitie in the market where the uyer wait. In thi cae, if only one eller arrive, he make a take-it-or-leave-it offer to the uyer. The life-time value are W = δe ϕ, (5) 1 δϕe ϕ W = δ (1 e ϕ ϕe ϕ ) 1 δϕe ϕ. (6) 4 Bargaining There i aically only one point that may not e immediate in deriving the expected utilitie. When there are two or more uyer in a meeting with a eller, one of the uyer i elected at random, and the proaility that a uyer get to trade with a eller i e γ X k=0 γ k k! 1 k +1 = 1 e γ. (7) γ Conider firt the market where eller wait and ue the ame notation for the expected utilitie a in the auction etting. The life time utilitie are determined y V = δ e γ V + 1 e γ µ V (1 V V ), (8) 6

7 µ 1 e γ V = δ V + 1 γ 2 (1 V V ) and olving thee give V = + γ 1+e γ γ V, (9) δγ (1 e γ ) (2 δ δe γ )γ + δ(1 e γ ), (10) δ (1 e γ ) V = (2 δ δe γ )γ + δ(1 e γ ). (11) The correponding utilitie in the market where eller earch are W = W = δ (1 e ϕ ) (2 δ δe ϕ )ϕ + δ(1 e ϕ ), (12) δϕ (1 e ϕ ) (2 δ δe ϕ )ϕ + δ(1 e ϕ ). (13) 5 The Equilirium Market Structure So far we have determined the expected utilitie of the agent under auction and argaining, ut we have not aid anything aout the equilirium of the economy. Since there i an infinite numer of agent, Nah-equilirium doe not provide ufficient retriction a any deviating agent i of meaure zero, and hi ehaviour doe not affect anything ut hi own utility. We require that the equilirium hould e immune againt a deviation y a coalition of agent. In more detail, it hould e impoile for a coalition of uyer and eller to put up another market where oth of them do etter than in equilirium. It i immediate that thi i impoile if there are two market in equilirium. The criterion, however, elect one of the two ymmetric pure-trategy equiliria: under auction it elect the very equilirium to which the mixed trategy equilirium converge when the ratio of uyer to eller increae or decreae without ound. Under argaining the criterion give a harp and not oviou reult a almot urely there are no mixed-trategy equiliria, and it elect from the two pure trategy equiliria. A an example to motivate coalitional deviation, conider the following ituation: eller advertie that they have an oject for ale, uyer oerve the advertiement and chooe which eller to contact. Then it hould not e poile that omeone, ay, put 7

8 up a magazine and manage to induce ome uyer to put ad there and ome eller to uy the magazine (and then contact the uyer). Definition 1 An equilirium i a pair (x, y) [0, 1] 2,wherex i the proportion of uyer who earch and y i the proportion of eller who wait, uch that no coalition of uyer and eller can put up a new market uch that all deviator fare etter than in the equilirium. To determine who earch and who wait i quite traightforward even though the detailed analyi involve a little computation in ome cae. The aim i to determine an equilirium in which oth market are active, i.e. an equilirium where ome uyer a well a eller oth wait and earch. Thi mean that waiting uyer have to e equally well-off a earching uyer. The ame condition mut hold for eller, too. Thi requirement produce two condition V = W, (14) V = W, (15) for the two unknown x and y. Of coure, we have to olve thee for oth trading mechanim. Itinotalwaythecaethattwomarketexitimultaneouly. Thenthere are two equiliria: one in which all uyer earch and all eller wait, and another one where all uyer wait and all eller earch. We return to thi point later, and then we evaluate which equilirium i more likely. One hould notice that thee two equiliria alway exit ut ince they look like equiliria in a pure co-ordination game, we focu on the other equiliria when they exit. Lemma 1 Under auction two market exit only if 1/θ 3 <θ<θ 3,whereθ In equilirium x =(θ 3 (θθ 3 1)) / θ θ and y =(θθ 3 1) / θ Proof. In the Appendix. Thi i already derived in Kultti and Takalo (1999), and it i noticeale that when two market exit, the ratio of earcher to waiter in oth market i contant, namely θ 3. 8

9 Lemma 2 Under argaining two market exit only if θ =1. All configuration x = y (0, 1) are equiliria. Proof. In the Appendix. Thi i alo derived in Kultti and Takalo (1999). Outide the aove region for θ, only uyer or eller earch. Propoition 1 and 2 tate the main reult of thi article: Propoition 1 If trade are conummated y auction, i) all the uyer earch and all the eller wait if θ>θ 3, ii) all the uyer wait and all the eller earch if θ<1/θ 3. Proof. In the Appendix. Propoition 2 When trade are conummated y argaining, i) all the uyer earch and all the eller wait if θ>1, ii) all the uyer wait and all the eller earch if θ<1. Proof. In the Appendix. The utilitie from waiting and moving depend on the Poion parameter that govern the mover arrival to waiter location. With oth trading rule, if all uyer earch and all eller wait, a profitale deviation y uyer require that in the new market the Poion parameter i large enough, wherea a deviation y eller require that it i mall enough. Thee range for the value of the parameter do not overlap if θ i large enough. On the other hand, if all uyer wait and all eller earch and θ i mall enough, we cannot find a value for the parameter that induce oth type of agent to deviate. There are, however, value of θ that enale profitale deviation, indicating that the original market market tructure i not an equilirium. 6 The Relative Efficiency of Trading Rule Here we conduct a rather traightforward comparion of efficiency under argaining and auction. The efficiency meaure i the numer of trade per period (which in teady tate i the ame each period). Thi i intereting only when two market exit ince if there i only one market under either trading rule, the ame type of agent wait, and the numer 9

10 of trade i the ame. Let u denote the numer of trade per period under auction or argaining y M au and M a,andlete = M au /M a. There are everal cae to e tudied, and we lit them elow. Note that under argaining the only cae when there are two market in equilirium i when the numer of uyer equal the numer of eller, i.e. θ =1. But then any configuration x = y [0, 1] i an equilirium, and in all uch equiliria the utility of a given type of agent i the ame. the agent utilitie are the ame. Thu, the cae are ditinguihed y the parameter interval relevant in auction. Cae a) θ<1/θ 3. Alltheuyerwait(whetherauction or argaining i ued), and the numer of matche i M au = M a = 1 e 1/θ B. Cae ) θ>θ 3. All the eller wait (whether auction or argaining i ued), and the numer of matche mirror that of cae a: M au = M a = 1 e θ S. Cae c) 1/θ 3 <θ<1. There are two market if auction i ued. In one market there are ys eller who wait and xb uyer who move, in the other market (1 x) B uyer wait for the (1 y) S eller to viit. The Poion parameter in oth market i equal to θ 3. The numer of matche i the um of the matche in two market: M au = ys 1 e θ 3 +(1 x) B 1 e θ 3, and uing Lemma 1 we have M au = 1 e θ 3 θ (θθ 3 1) S + θ 3 θ θ B. (16) If argaining i ued, all uyer wait and all eller earch, thu M a = 1 e 1/θ B. The relative efficiency of auction to argaining i then 1 e θ 3 (1 + θ) E = > 1, (17) (1 + θ 3 )(1 e 1/θ ) θ and it can e hown that E/ θ > 0: The efficiency of auction relative to argaining increae a the ratio B/S increae. Cae d) 1 <θ<θ 3. In auction, two market exit, and the um of matche i ys 1 e θ 3 +(1 x) B 1 e θ 3, the ame a in cae c. In cae of argaining, all eller wait, and M a = 1 e θ S, and we have E = 1 e θ 3 (1 + θ) (1 + θ 3 )(1 e θ ) 10 > 1, (18)

11 and E/ θ > 0. The maller the value of θ, the more efficient auction i relative to argaining. We can ummarie the reult in Propoition 3 Auction i more efficient than argaining if 1/θ 3 <θ<θ 3.Outidethi region, auction and argaining are equally efficient. Thi reult i aed on the comparion of equiliria that atify Definition 1. The matching function aove have contant return to cale, yet two mall market can perform etter than one large market. The explanation for thi i that in one mall market uyer earch and in the other they wait, and the ratio of mover to tayer in oth market i more advantageou than in the one large market. 7 Extenion: Common Location In the model preented aove, either the uyer go to the eller location or the eller viit the uyer. Thi i, however, only one of many conceivale meeting technologie. For example, one could think that in the eginning of each period, each uyer i in hi location and each eller i in hi location. Then ome uyer leave their location and go to eller location to viit them. The ret of the uyer tay put in their location and wait for viiting eller. Seller, too, have a choice to tay or to go and viit the uyer. Thi model i analyed in Kultti, Miettunen and Virrankoki (2003), where the equilirium choice of taying and viiting are derived. In the equel we conider an environment where neither uyer nor eller have their own location. Intead, there are everal marketplace, or common location. Each agent goe randomly to one location in every period until he trade and leave the economy. Trading within a location i frictionle: the numer of matche in location i i equal to min {B i,s i } where B i and S i are the numer of uyer and eller who happen to chooe location i. Friction arie ecaue the agent are randomly ditriuted on the location. We compare the numer of matche and agent life-time value in thi model to thoe in the aic model analyed aove. 11

12 The tradale oject i indiviile, and there are B uyer and S eller with the ame preference a in the aic model. In order to trade, they have to go to marketplace. The numer of uch location i L, and we aume that L = B + S. In the eginning of each period, each uyer and eller chooe randomly which location to go. Every location then ha more uyer than eller, or more eller than uyer, or equally many of them. Some location may remain empty. The trading mechanim i a mixture of auction and argaining in the following way: If the numer of uyer and eller in a location i unequal, there i an auction. The more numerou type get hi reervation value, and the le numerou type get one minu the other reervation value. If there i an equal numer of uyer and eller, they argain uch that everyone get hi reervation value plu one half of the urplu. In the equal-numered cae we aume argaining ecaue we want to treat uyer and eller ymmetrically. Agent who have traded exit the market, and they are replaced y identical ut unmatched agent. Let a () e the proaility that there are more (le) uyer than eller, repectively, in a location choen y a uyer. Let c (d) e the proaility that there are more (le) eller than uyer, repectively, in a location choen y a eller. The value function are, for a uyer and for a eller, µ = δ av + (1 V )+(1 a ) V (1 V V ), (19) V co V co Solving for V and V give µ = δ cv + d (1 V )+(1 c d) V (1 V V ). (20) V co = V co = δ (1 a + ) 2 δ (a + c d), (21) δ (1 c + d) 2 δ (a + c d). (22) The proailitie a,, c and d cannot e olved analytically. Intead, they are olved numerically y calculating the cumulative Poion ditriution function for the numer of uyer and eller in a location. We have i) a = G 0 F 0 +(G 1 G 0 ) F 1 +(G 2 G 1 ) F , ii) = G 0 (1 F 1 )+(G 1 G 0 )(1 F 2 )+(G 2 G 1 )(1 F 3 )+..., iii) c = F 0 G 0 +(F 1 F 0 ) G 1 +(F 2 F 1 ) G , and 12

13 iv) d = F 0 (1 G 1 )+(F 1 F 0 )(1 G 2 )+(F 2 F 1 )(1 G 3 )+..., where F n i the proaility that at mot n other eller have come to the location, and G n i the proaility that at mot n uyer have arrived. The proailitie a,, c and d are olved for cae θ =1/3, 1/2, 1, 2, and3. Note that the aolute value of B and S do not matter, ut θ (= B/S) doe. We let δ =0.9. We firt compare the efficiency of the common-location model to that of the aic model (with auction and argaining). In the common-location model, the numer of matche i M co = B + ds +(1 a )B, (23) where B i the numer of matche that form in location where there are le uyer than eller, ds i the numer of matche in location where there are le eller than uyer, and (1 a )B matche form in location where there are equally many uyer and eller. We can write M co = ds +(1 a)b, andaameaureofefficiency we ue the numer of matche per eller, M co /S = d +(1 a)θ. Theefficiency meaure in the aic model are M au /S and M a /S, where we take into account that there are two market for certain value of θ. Tale 1 ummarie the efficiency in each model: Tale 1: Efficiency in the common-location model and in the aic model M co /S M au /S M a /S θ =1/ θ =1/ θ = θ = θ = The meeting technology of the aic model i roughly two time a efficient a that of the common location model. Next we compare the life-time utilitie of uyer and eller in the common-location model to thoe in the aic model. In the aic model we conider oth auction and argaining. The identity of mover and tayer i determined a tated in propoition 1 and 2. For example, if trade are conummated y argaining, all the uyer earch and all the eller wait if θ =2or if θ =3, and o on. Let u ue the following notation: 13

14 V au and V au are the life-time utilitie for uyer and eller in the aic model when the trading mechanim i auction; V a ued, and V co the utilitie: and V co and V a are the repective value when argaining i are the utilitie in the common-location model. Tale 2 preent Tale 2: Welfare of uyer and eller in the common-location model and in the aic model V co V au V a V co V au V a θ =1/ θ =1/ θ = θ = θ = Tale 3 how the ordering of utilitie: θ =1/3 and θ =1/2 θ =1 θ =2and θ =3 Tale 3: Ordering of utilitie Buyer Seller V au V au V a >V a >V a >V co >V co >V co >V au V a V au V au >V co >V a >V a >V au >V co >V co We ee that the configuration with common location i dominated y the aic configuration with either auction or argaining. Thi i due to large numer of location, a it i clear that letting the numer of location go to one yield the maximum numer of meeting. We alo ee that when the numer of uyer and eller in the economy i equal, the uyer and eller in the aic configuration prefer auction to argaining. Thi reult hold with all δ 1. 8 Concluion We take a tandard urn-all model with uyer and eller and tudy a ymmetric mixed trategy equilirium when the agent choice et conit of the deciion to wait 14

15 or earch. Focuing on mixed ymmetric trategie look reaonale a in large market pure-trategy equiliria eem to require plenty of co-ordination. On top of that, ymmetric trategie indicate how likely it i for a certain type of agent to earch. Thi way we get a quite clear-cut picture of the earch and wait deciion. The main conlcuion i that the more numerou party i more likely to earch, regardle of how term of trade are determined. If the difference etween the numer of uyer and eller i large, there exit only a pure-trategy equilirium where the more numerou agent earch and the le numerou agent wait. Our reult hould e of interet for modelling purpoe a it i cutomary to aume that one type, eg. employer in laour market, or eller in decentralied good market, or women in marriage market, wait and the other type earche. Thee aumption eem well motivated if the waiting type i not ignificantly more numerou, ut in the oppoite cae the practice i in dout. The model alo ha an oviou empirically tetale implication. The reult are very clear due to the mall numer of equiliria, a we do not aume that there are any cot of earching or waiting, or any kind of heterogeneity. One may want to relax thee aumption, ut it i likely to lead to a huge numer of equiliria while in our imple etting there are practically at mot three configuration that may e regarded a equiliria under any reaonale criterion. 9 Appendix ProofofLemma1 Equilirium condition W = V and W = V yield y (3)-(6) that e γ 1 γ = 1 e ϕ 1 ϕ which i equivalent to e γ = eϕ ϕ e ϕ 1 ϕ + γ. The uyer equilirium condition W = V yield (A1) (A2) e γ 1 γ e γ δγ = 1 e ϕ δϕ (A3) 15

16 which i equivalent to e γ = eϕ δϕ e ϕ 1 ϕ + δγ. (A4) Next we how that there i a unique olution ϕ = γ = θ 3 for (A1) and (A3) to hold where e θ 3 2 θ 3 =0. Equation (A2) and (A4) yield γ = ϕ e ϕ 1 ϕ. (A5) Equation (A2) and (A5) yield e γ /γ = e ϕ /ϕ which i equivalent to e f(ϕ) /f (ϕ) =e ϕ /ϕ where f (ϕ) =ϕ/ (e ϕ 1 ϕ). Still, a more convenient form i f (ϕ) ϕ =ln f (ϕ) ϕ. (A6) The left-hand ide and the right-hand ide of (A6) are plu infinity at ϕ = 0. The derivative of the left-hand ide i f 0 (ϕ) 1= eϕ 1 ϕe ϕ (e ϕ 1 ϕ) 2 1 (A7) which i negative for ϕ>0. The derivative of the right-hand ide i (1 e ϕ ) / (e ϕ 1 ϕ) which i alo negative for ϕ>0. Let u compare the derivative; we how that the derivative of the left-hand ide i maller than that of the right-hand ide for mall value of the argument. Formally, e ϕ 1 ϕe ϕ (e ϕ 1 ϕ) 2 1 < 1 eϕ e ϕ 1 ϕ (A8) which i equivalent to e ϕ 1 ϕ ϕ 2 < 0,whichholdalongaϕ i le than 1.8 (approximately). Thu, there i at mot one zero. One immediately ee that θ 3 > 1 i arootofequation(a6)whereθ 3 atifie 2+θ 3 e θ 3 =0, yielding θ 3 = ϕ = γ Two auction market exit only if γ = xθ y = ϕ = 1 y (1 x) θ = θ 3. (A9) Solving thi yield y =(x/θ 3 ) θ, which inerted into the latter equation give x = θ 3 (θθ 3 1) θ θ 2 3 1, (A10) 16

17 from which we ee that equilirium with two auction market exit if and only if θ 3 > θ>1/θ 3. ProofofLemma2 The equilirium condition W = V and W = V yield y (10)-(13) that γ =1/ϕ. Inerting thi ack to one of the equilirium condition give 1 e γ γ 1 e 1/γ =0. (A11) Next we how that (A11) ha exactly one poitive olution, γ =1. Let u tudy function g(h) =e h 1 he h + he h 1/h that ha the ame zeroe a the left-hand ide in (A11). We immediately ee that g(0) = 0. Thederivativeofg i µ g 0 (h) = he h + e h 1/h h +1+ 1, (A12) h and ince e 1/h /h =0at h =0we ee that g 0 (0) = 0. The econd derivative of g i µ g 00 (h) = e h he h + he h 1/h 2+h + 2 h + 1, (A13) h 3 and g 00 (0) = 1. Thu, at firt g i decreaing. It i alo immediate that g(1) = 0 and g 0 (1) > 0. Nextwehowthatg i not zero etween h =0and h =1. If there were zeroe etween h =0and h =1and if g attained trictly poitive value, there hould e at leat two zeroe. Before the lat zero g would reach a maximum and it derivative would e zero. Let u denote the lat maximum of g (where it i poitive) y k.thu we know that µ g 0 (k) = ke k + e k 1/k k =0, (A14) k and g(k) =e k 1 ke k + ke k 1/k > 0. From thee condition we get g(k) =e k 1 ke k + k 3 e k 1+k + k 2 > 0 (A15) which hold if and only if e k 1 k k 2 > 0. We have e h 1 h h 2 =0at h =0 and e h 1 h h 2 < 0 at h =1. It i eay to ee that e h 1 h h 2 < 0 in interval h (0, 1]. Thu, the aumption that g (h) > 0 at h (0, 1] lead to a contradiction, thu g (h) < 0 at h (0, 1], and (A11) ha no olution at γ (0, 1). 17

18 To how that equation 1 e γ γ + γe 1/γ =0ha no olution at γ > 1 it i enough to how that g 0 (h) = he h + e h 1/h (h +1+1/h) > 0 when h>1, ecaue g(1) = 0 and g 0 (h) > 0 at h =1. The ign of g 0 (h) i poitive if and only if v(h) 1 e 1/h e 1/h /h e 1/h /h 2 < 0. We ee that v(1) < 0 and lim h v(h) =0. Further, v 0 (h) =e 1/h (1/h 3 )(1 1/h) which i poitive if h>1. Thati,v(h) < 0 if h>1, thu g 0 (h) = he h + e h 1/h (h +1+1/h) > 0 when h>1, conequently (A11) ha no olution at γ>1. We have hown that equation 1 e γ γ + γe 1/γ =0ha exactly one trictly poitive olution, γ =1.Ifγ (0, 1), then1 e γ γ + γe 1/γ < 0, andifγ>1, then 1 e γ γ + γe 1/γ > 0. Analogouly, it i eay to ee that there doe exit a zero z uch that g(z) =0i a local maximum. In equilirium γ =1/ϕ which hold if and only if x = y. Comined with γ =1,weconcludethattwoargainingmarketexitifand only if θ =1,foranyx and y uch that x = y. ProofofPropoition1 The proof of part (i) of the propoition i hown here. analogou and i left to the reader. The proof of part (ii) i (i) In cae θ>θ 3 we how that if all the uyer wait and all the eller earch, there i a coalition of uyer and eller who are etter off in a market where uyer earch and eller wait, implying that the former market i not an equilirium. On the other hand, if all the uyer earch and all the eller wait, a profitale deviating coalition cannot exit. All uyer wait and all eller earch Let ψ =1/θ e the parameter of the Poion proce in the original market where uyer wait and eller earch, and let α = ηb/µs e the repective parameter in the new market where eller wait and uyer earch. Becaue θ>θ 3,thenψ<1/θ 3. A eller utility in the original market i V o, and hi utility in the new, deviating market i V n. The correponding utilitie for a uyer are V o a eller require V n >V o : 1 e α αe α 1 δαe α > and V n.aprofitale deviation for e ψ, (A16) 1 δψe ψ 18

19 and condition V n >V o for deviating uyer i e α 1 δαe α > 1 e ψ ψe ψ 1 δψe ψ. (A17) Let α 1 atify V n = V o : 1 e α 1 α 1 e α 1 1 δα 1 e α 1 = e ψ, (A18) 1 δψe ψ and let α 2 atify V n = V o : e α2 1 δα 2 e α 2 = 1 e ψ ψe ψ 1 δψe ψ. (A19) Next we how that a deviating coalition exit if α 1 <α 2,incaewecanchooe α (α 1,α 2 ) that atifie (A16) and (A17). After that we how that α 1 <α 2 for all ψ (0, 1/θ 3 ]. Expreion 1 e h he h / 1 δhe h i increaing in h: µ 1 e h he h 1 δhe h = e h (1 δ) h + δ 1 e h h (1 δhe h ) 2, (A20) and e h / 1 δhe h i decreaing in h: µ e h 1 δhe h h = e h 1 δe h (1 δhe h ) 2. (A21) In Figure 1, h i on the horizontal axi, and ψ, θ 3, α 1 and α 2 are the value of our interet. We ee that if curve 1 e h he h / 1 δhe h i teeper than curve e h / 1 δhe h,thenα 1 <α 2, and we can chooe α (α 1,α 2 ) that atifie (A16) and (A17). If α 1 >α 2, a deviating coalition doe not exit. The um of the right-hand ide of (A20) and (A21) i equal to e h (1 δ)(h 1) (1 δhe h ) 2, (A22) and we note that curve 1 e h he h / 1 δhe h i teeper than curve e h / 1 δhe h if h>1. That i, if ψ i larger than one, then α 1 <α 2. However, the value of ψ i aumed to e le than 1/θ 3 which i maller than one. Do we have α 1 <α 2 for all ψ<1? Inordertohaveα 1 >α 2,wehouldhaveα 1 = α 2 for ome value of ψ. Next we how that thi i not poile. 19

20 1 e h 1 δhe he h h e h 1 δhe h ψ θ 3 α 1 α 2 h Figure 1: A deviating coalition exit. Let α atify 1 e α αe α 1 δαe α = e ψ 1 δψe ψ (A23) and e α 1 δαe = 1 e ψ ψe ψ. (A24) α 1 δψe ψ By umming (A23) and (A24) we get 1 αe α 1 δαe α = (A25) 1 ψe ψ 1 δψe ψ which hold for an aritrary value of δ only if αe α = ψe ψ. Equation (A23) and (A24) then yield e α = 1 e ψ ψe ψ, (A26) e ψ = 1 e α αe α. (A27) Oviouly, α = ψ = θ 3 olve (A26) and (A27). Are there other olution? Solving α 20

21 from (A26) yield α = ln 1 e ψ ψe ψ, and uing thi in (A27) we get ψe ψ 1 e ψ ψe ψ =ln 1 e ψ ψe ψ. (A28) Sutracting the derivative of the left-hand ide from the derivative of the right-hand ide equal ψe ψ 1 e ψ ψe ψ e ψ e ψ + ψ 1 (1 e ψ ψe ψ ) 2 (A29) whichhatheameigna1 e ψ ψe ψ ψ 2 e ψ.thiequalzeroatψ =0,and 1 e ψ ψe ψ ψ 2 e ψ ψ = ψe ψ (ψ 1) (A30) which i negative if ψ<1. Atψ =1.2 (>θ 3 ) we have 1 e ψ ψe ψ ψ 2 e ψ , andweconcludethat1 e ψ ψe ψ ψ 2 e ψ < 0 for ψ (0,θ 3 ]. Thu the left-hand ide of (A28) can cut the right-hand ide of (A28) only from elow in interval ψ (0,θ 3 ], therefore (A28) ha a unique olution in that interval. We conclude that (A26) and (A27) have a unique olution, namely α = ψ = θ 3. That i, there doe not exit ψmaller than θ 3 and a correponding α larger than θ 3 that atify (A23) and (A24). Becaue of that and ecaue α 1 <α 2 for ψ larger than one, we conclude that α 1 <α 2 for all ψ (0, 1/θ 3 ], and therefore a deviating coalition exit for all ψ (0, 1/θ 3 ]. All uyer earch and all eller wait Thi i a revere cae to the aove. The Poion parameter in the original market i θ which i larger than θ 3, and the Poion parameter in the new market i ω which i maller than θ 3.Aprofitale deviation require that ω i maller than ω 1 which atifie V n = V o,giveny and larger than ω 2 which atifie V n e ω 1 1 δω 1 e ω 1 = 1 e θ θe θ 1 δθe θ, (A31) = V o,giveny 1 e ω 2 ω 2 e ω 2 1 δω 2 e ω 2 = e θ 1 δθe θ (A32) Proceeding like in the aove, it can e hown that ω 1 <ω 2 for all θ larger than θ 3,and a deviating coalition doe not exit. ii) In cae θ<1/θ 3 the proof goe imilarly a in cae θ>θ 3, one jut ha to replace uyer for eller and eller for uyer. 21

22 ProofofPropoition2 The logic of the proof i imilar to that of Propoition 1. The proof i hown only for part (i). (i) In cae θ>1 we how that if all the uyer wait and all the eller earch, there i a coalition of uyer and eller who are etter off in a market where uyer earch and eller wait, implying that the former market i not an equilirium. On the other hand, if all the uyer earch and all the eller wait, a profitale deviating coalition cannot exit. All uyer wait and all eller earch The Poion parameter in the original market i ψ =1/θ, and the utilitie are V o for a uyer and V o for a eller. The correponding parameter in the deviating market where uyer earch and eller wait i α = ηb/µs, and the utilitie are V n and V n.conider a deviating coalition where eller wait, uyer earch, and the ratio of uyer to eller i α. Profitale deviation require that V n α (1 e α ) (2 δ δe α )α + δ(1 e α ) 1 e α (2 δ δe α )α + δ(1 e α ) >V o,andv n >V o,thati, > > 1 e ψ (2 δ δe ψ )ψ + δ(1 e ψ ), (A33) ψ 1 e ψ (2 δ δe ψ )ψ + δ(1 e ψ ). (A34) Let α 1 atify V n = V o,andletα 2 atify V n = V o. Expreion 1 e h (A35) (2 δ δe h )h + δ(1 e h ) i decreaing in h, and expreion h 1 e h (A36) (2 δ δe h )h + δ(1 e h ) i increaing in h. A in the proof of part (i) of Propoition 1, a deviating coalition exit if α 1 <α 2. The curve given y (A36) i teeper than the curve given y (A35) if 2(1 δ) 1 e h he h h 2 e h < 0, whichholdifh i ufficiently mall. Thu α 1 <α 2 if ψ i ufficiently cloe to one. A in the proof aove, α 1 >α 2 require that α 1 = α 2 for ome value of ψ. We how that the latter cannot happen. Let α atify α (1 e α ) (2 δ δe α )α + δ(1 e α ) 1 e α (2 δ δe α )α + δ(1 e α ) = = 22 1 e ψ (2 δ δe ψ )ψ + δ(1 e ψ ), (A37) ψ 1 e ψ (2 δ δe ψ )ψ + δ(1 e ψ ). (A38)

23 Dividing the left-hand ide of (A37) y the left-hand ide of (A38), and doing the ame to right-hand ide give α =1/ψ. (A39) Adding the left-hand ide of (A37) to that of (A38), and doing the ame to the right-hand ide lead to ψ 1 e α αe α = α 1 e ψ ψe ψ. (A40) Oviouly, α = ψ =1olve (A40). If thi i the only olution, then α 1 <α 2 for all ψ maller than one. Becaue α =1/ψ, we look for olution to equation of form h 1 e 1/h (1/h)e 1/h =(1/h) 1 e h he h. (A41) At h =1the left-hand ide i decreaing and the right-hand ide i increaing. If another trictly poitive olution exit, the derivative of the right-hand ide mut e maller than the derivative of the left-hand ide. It i however eay to how that in any trictly poitive olution to (A41), the derivative of the right-hand ide i larger than the derivative of the left-hand ide, thu h =1i the only olution. Then α 1 <α 2 for all ψ maller than one, and we can chooe α (α 1,α 2 ) uch that V n >V o and V n >V o. If all eller wait and all uyer earch in the original market, the ame logic i applied to demontrate that a deviating coalition doe not exit. (ii) For θ<1, the proof i analogou to cae θ>1 and i left to the reader. Reference Burdett, K., Cole, M., Kiyotaki, N. and R. Wright, 1995, Buyer and eller: hould I tay or hould I go? AEA Paper and Proceeding, May, Herreiner, D., 1999, The deciion to eek or to e ought? Rheiniche Friedrich- Wilhelm-Univerität Bonn, Dicuion Paper B-462. Kultti, K., 1999, Equivalence of auction and poted price, Game and Economic Behavior 27, Kultti, K., Miettunen, A. and J. Virrankoki, 2003, Phyical earch, Univerity of Helinki, Department of Economic Dicuion Paper No. 573:

24 Kultti, K. and T. Takalo, 1999, Equilirium in auction and argaining market when agent can wait and earch, Bulletin of Economic Reearch 51, Lu, X. and R.P. McAfee, 1996, The evolutionary taility of auction over argaining, Game and Economic Behavior 15,