Reserve Price Signaling

Transcription

1 Reerve Prie Signaling Hongbin Cai, John Riley and Lixin Ye Abtrat Thi paper tudie an aution odel in whih the eller ha private inforation about the objet harateriti that are valued by both the eller and potential buyer. We explore the role of reerve prie in ignaling thi private inforation. We firt haraterize the unique Pareto doinant eparating equilibriu ( the Riley outoe ). Then we derive oparative tati reult and diu an appliation to ignaling in the Leon Market. The eond part of the paper onider equilibriu refineent. The Cho-Krep Intuitive Criterion annot be diretly applied to our etup. Intead we apply a Loal Credibility Tet (LCT) whih i baed on, but i lightly weaker than, Groan and Perry Strengthened Intuitive Criterion. For a ore general ignaling odel in whih reerve prie ignaling i one peial ae, we identify neeary and uffiient ondition for the LCT to be atified in equilibriu. In the reerve prie ignaling odel, uh ondition require that the ignaling effetivene be uffiiently large. UCLA, UCLA, and Ohio State Univerity. We would like to thank In-Koo Cho, David Cooper, Maio Morelli, Jae Pek, and einar partiipant at Illinoi Workhop on Eonoi Theory, Ohio State Univerity, Rutger Univerity, UC Riveride, and Cae Wetern Reerve Univerity, for helpful oent and uggetion. All reaining error are our own.

2 1. Introdution In thi paper we onider an aution environent in whih a eller of one objet ha private inforation about the objet harateriti. Thee harateriti deterine the eller expeted valuation of the objet and the expeted oon valuation for a group of potential buyer, eah of who alo ha an independent private value for the objet. Sine the harateriti an be ulti-dienional and the eller and the bidder ay plae different weight on the relative iportane of different dienion, the eller expeted value and the buyer oon value oponent are likely to be poitively but not perfetly orrelated. For exaple, a eller of an artwork (e.g., an aution houe) ay know it ondition (quality, rarity, hitory, et.) a well a it eondary arket value better than potential buyer. While he i otly onerned with the artwork eondary arket value, potential buyer (who buy for elf onuption) ay alo are about it ondition. By the linkage priniple (Milgro and Webber, 198), it i well known that the eller an inreae her expeted revenue by truthfully revealing her private inforation about the objet harateriti. If diret verifiation of the eller inforation i otle, it i indeed inentive opatible for the eller to truthfully reveal her inforation for the following reaon. Seller with private inforation indiating high oon value for the buyer have an inentive to reveal their inforation ine buyer will then be willing to bid ore for the objet. Sine the ae arguent hold for any et of type, the eller with high type within the et alway have an inentive to reveal and the only Nah equilibriu i full revelation of the eller private inforation. However, in any aution etting, a otle revelation tehnology (e.g., a perfetly neutral and objetive evaluation ethod of a third party) ay not be available to the eller. In uh ae, the eller announing her inforation to the potential buyer ay not be redible a he fae the tandard advere eletion proble, that i, he alway want to lai the highet oon value for the buyer. A natural way to redibly reveal the private inforation i through ignaling, and a natural ignaling intruent in thi environent i the reerve prie: a eller with a high type ha an inentive to try to ignal thi to the buyer by etting a high reerve prie. 1

3 In etion we deribe the reerve prie ignaling odel that ha the natural interpretation of the aution etting deribed above. However, the odel doe not fit into the tandard ignaling fraework. We reforulate the odel in two way. Firt, we redefine the eller type a her own valuation for the objet, not her private inforation about the objet harateriti. Through variable tranforation and by the poitive orrelation between the eller value and the buyer oon value, the eller expeted payoff funtion an be expreed in the tandard for. We then reforulate the odel by fouing on the eller reerve arkup a the key ignaling variable, rather than the reerve prie itelf. With thi variable tranforation, the reforulated odel belong to the faily of reening odel exained by Riley (1979). We tart with the analyi of direte type in Setion 3. With the tandard tehnique, it i eay to haraterize the eparating equilibriu in whih the lowet type eller hooe her optial reerve prie under oplete inforation (the Pareto doinant equilibriu). It i well known that with only two type, thi eparating equilibriu i the unique equilibriu atifying the Cho-Krep Intuitive Criterion. However, with ore than two type, and epeially in our etting where the ultidienional harateriti of the objet an often lead to poitive but not perfet orrelation between the eller value and the buyer oon value, the Intuitive Criterion ha no bite. The Intuitive Criterion annot be diretly applied to the ae of ontinuou type either. A tronger equilibriu refineent onept baed on Groan and Perry (1986a,b) an be ore effetive in eleting the Pareto doinant equilibriu under ertain ondition. Here we weaken lightly the Groan-Perry Strong Intuitive Criterion, and propoe a Loal Credibility Tet in whih a poible deviation i interpreted a oing fro one or ore type whoe equilibriu ation are nearby. Thi tet work equally well whether there are finite or a ontinuu of type. In Setion 4, we onider the ae of a ontinuu of type and haraterize the unique Pareto doinant eparating Nah equilibriu. In thi equilibriu, the reerve prie hedule fully reveal the eller type (her own valuation for the objet). Thu a reerve prie an play a ore entral role than pereived by the traditional literature. In the tandard private value aution odel, the eller optial reerve prie i et to apture additional revenue when there i only one buyer who ha a valuation uh higher

4 than her own. Thi optial reerve prie i independent of the nuber of bidder. Therefore, unle the nuber of bidder i very all, the probability that the reerve prie i binding i all and hene the extra profit aptured by etting a reerve prie i alo low. In the Pareto doinant eparating equilibriu in our odel, we how that the reerve prie inreae with the nuber of bidder and in the iportane of buyer private inforation. At the end of Setion 4, we alo briefly diu an appliation of our reult to ignaling in the Leon Market. Even though not in an aution etting, our reult an be readily applied to the well-known Akerlof Leon Market (Akerlof, 1970) to give rie to a eparating equilibriu in whih eller with different qualitie et different prie and their private inforation i fully revealed to the arket. Setion 5 tudie equilibriu refineent in the ontinuou type odel. We onider a faily of ontinuou type odel of whih the Spene eduation ignaling and the reerve prie ignaling are both eber. We begin by forulating the onept of Loal Credibility Tet (LCT) for the ontinuou type odel. An equilibriu urvive LCT if no deviation-pereption pair i redible in the following ene: for any poible deviation ignal (on- or off-equilibriu), if it i interpreted a fro type of a all neighborhood of the iediate equilibriu type, it i profitable for the type in thi neighborhood to deviate to thi ignal, but unprofitable for type out of thi neighborhood to do o. Clearly, if an equilibriu atifie LCT, it ut be the Pareto doinant eparating equilibriu. We identify neeary and uffiient ondition for the eparating equilibriu in our general ignaling odel to atify LCT. Thee ondition are applied to the Spene eduation ignaling odel and the reerve prie ignaling odel. It i hown that the required ondition are intuitive and an be atified with reaonable paraeter value. In partiular, a long a a eaure of ignaling effetivene i uffiiently high for every type, then the eparating equilibriu an urvive our LCT tet. We alo note that one ae in whih no equilibriu an urvive LCT, both in finite and ontinuou type odel, i when the arginal oial ot of hooing her optial ation i zero for the lowet type. If that i the ae, the ender with uffiiently low type will profit by pooling at the lowet ignal when thi i orretly pereived by the reeiver. 3

5 A paper by Jullien and Mariotti (003) i loely related to our. They olve for the eparating equilibriu of a ignaling gae iilar to our, and opare the equilibriu outoe with the optial ehani for a onopoly broker who buy fro the eller and ell to the buyer. Without uing the reforulation that ake the arkup the key variable and hene enable tranforing the odel into the tandard fraework, their olution ethod i quite different fro our. Moreover, they only onider the ae with bidder and aue effetively γ < 1 in our perfet orrelation forulation. Finally, an iportant part of our paper i foued on equilibriu refineent, while Jullien and Mariotti iply fou on the Pareto doinant eparating equilibriu.. The Model We onider a eond-prie ealed bid aution in whih n buyer bid for a ingle, indiviible objet. 1 The objet harateriti are repreented byθ Θ, where Θ an be any finite dienional pae. Only the eller oberve θ prior to the aution, thu θ i the eller private inforation, or her type. The eller valuation for the objet i given by ( θ ). Buyer j valuation for the objet i given by V = t( θ ) + X, where t( θ ) i the oon value oponent, and oberved by buyer j. Ex ante, X j i the private value oponent whih i only X j i ditributed with.d.f. F( ) and upport X = [ xx, ], where F'( x ) > 0 for all x [ xx, ]. We aue that θ and X j are all independent, whih iplie that ( θ ) and X j are all independent. The above peifiation of inforation truture i rather general, inluding ot oon exiting odel a peial ae. For the iplet ae, θ R i onedienional, and = t = θ ; that i, the eller oberve her valuation for the objet, whih oinide with the buyer oon value oponent. An extenion of thi ae i that θ R i one-dienional and = θ, but t = γθ whereγ > 0 ; that i, the buyer oon value oponent i proportional to the eller valuation. In a ore general ae, the eller private inforationθ Θ i ultidienional, and both her valuation and the j j 1 By the Revenue Equivalene Theore (Myeron 1981, Riley and Sauelon 1981), all our reult ontinue to hold if the objet i old in any other tandard aution. 4

6 buyer oon value oponent are tohati funtion of θ Θ. Forally, the eller valuation i W ( θ, η) and the buyer oon value i V0 ( θ, ω ), where η and ω are rando variable. Sine the eller and the buyer are aued to be rik neutral, only their expeted valuation atter in the analyi. The eller expeted valuation i ( θ ) = E [ W( θη, )], and the buyer expeted oon value oponent i η t( θ ) = E V ( θω, ). However, ine the buyer do not oberveθ, they have to infer fro ω 0 the eller ignal the expetation about the oon value oponent. Upon oberving the eller ignal and foring a belief that ( θ ) =, the expeted oon value i τ ( ) = E [ V ( θω, ) ( θ) = ] = E [ t( θ) ( θ) = ]. { θω, } 0 Throughout we will aue the following auption: θ A1: Diinihing arginal revenue (regularity ondition): 1 F( x) J( x) = x i tritly inreaing in x. F ( x) A: Seller valuation and the buyer oon value oponent t are poitively orrelated. Thu τ ( ) = E[ t = ] i an inreaing funtion of. Note that Auption A1 hold a long a the hazard rate funtion of X i inreaing (whih i atified for alot all the oon ditribution, e.g., unifor, noral, exponential, et.). Auption A i quite natural. A peial ae i when and t are perfetly orrelated, t = γ, γ > 0, o τ ( ) = γ. The eller ove firt by announing a reerve prie. The buyer then ubit ealed bid. Suppoe the buyer, upon oberving a reerve prie r, believe that the value of the oon value oponent i ˆt. Then a buyer of type x j ha an expeted value for the ite of tˆ + x j. Suppoe that all the buyer follow their doinant trategie to bid their valuation tˆ + x j, j = 1,..., n. Then the poible outoe of the aution are a follow: If ˆt + x(1) < r, the objet i not old, whih our with probability of F ˆ (1)( r t) ; if ˆt + x(1) > r and ˆt + x() < r, the objet i old at the reerve prie, whih our with 5

7 probability of F ˆ ˆ ()( r t) F(1) ( r t) ; if ˆt + x() > r, the ite i old at the prie ˆt + x(). Here the ditribution of the firt and eond order tatiti are given by n n 1 F ( x) = Pr{ X x} = F ( x) and F ( x) = Pr{ X x} = F ( x) + n(1 F( x)) F ( x). (1) (1) () () (1) The eller expeted payoff i therefore given by u = F ( r tˆ) + r[ F ( r tˆ) F ( r tˆ)] + ( tˆ + x) df ( x) (.1) (1) () (1) () r tˆ Fro the above expreion, it i lear that even though the eller priitive type i θ, all that atter are her valuation (her true type), = ( θ ), and the buyer belief about the oon value oponent ˆt. In any equilibriu, eller of different priitive type but a ae atual type ut hooe the ae ignal and hene lead to the ae ˆt ; otherwie thoe eller obtaining aller expeted payoff would hange to the ignal that lead to higher expeted payoff. In equilibriu, for a ignaling trategy r, () the buyer pereived oon value oponent i tˆ = E[ t θ :(()) r θ = r]. With thi reforulation, the eller payoff funtion i in the tandard for of ignaling odel, and i expreed without diret referene to her private inforation about the objet harateriti θ. A a funtion of the reerve prie r, Equation (.1) i diffiult to analyze. In our next reforulation, we define the eller reerve arkup r tˆ. Then we an rewrite the eller expeted payoff a follow. x where ut (,, ˆ ) = F ( ) + tˆ(1 F ( )) + B ( ) (.) (1) (1) x B( ) = ( F ( ) F ( )) + xdf ( x) (.3) () (1) () Sine it will be ueful below we note that B ( ) = F ( ) F ( ) f ( ) = f ( ) J( ) (.4) () (1) (1) (1) If buyer believe that the eller type i ŝ, the expeted oon valuetˆ = τ () ˆ. 6

8 DefineU(,, ˆ ) = u(, τ (), ˆ ), then the eller expeted payoff an be expreed in ter of the buyer pereption about the type, ŝ, intead of their pereption about the oon value oponent, τ () ˆ. Differentiating the eller expeted payoff, U (,, ˆ ) = τ ()(1 ˆ F ( )) (.5) (1) U (, ˆ, ) = ( τ ( ˆ)) F ( ) + B ( ) 3 (1) = ( τ ( ˆ ) J( )) F ( ) (.6) Sine U i independent of and U 3 i inreaing in, we have (1) dˆ U 0 (.7) d U 3 = < U Therefore the ingle roing property hold. Benhark: Full Inforation If were diretly obervable to buyer, their pereption ŝ =, o the eller will hooe her arkup to axiize U(,, ). Let reerve arkup, then by Auption A1 and Equation (.6), we have () be the optial full inforation x, τ ( ) < J( x), () = 1 J ( τ( )), τ( ) J( x) (.8) In the perfet orrelation ae t = γ, when γ = 1, the optial arkup i independent of ; when γ > 1 the optial arkup i tritly dereaing in. The intuition i lear. With γ > 1, the bigger i the aller i the proportion of the private value oponent out of the buyer total valuation. Hene the eller optial trategy i to ark up the reerve prie le. In the unifor ae with upport [ x, x ], J( ) = ( x x), and we have 7

9 () = Max{, x ( x x ( γ 1))} 1 3. Finite Type Cae Two type ae Suppoe the objet for ale ha only two poible type of harateriti: either low ( θ 1 ), or high ( θ ). Correpondingly, the eller valuation are and, and = ( θ ) < ( θ ) = ; the buyer oon value oponent are t and t, and 1 t = t( θ ) < t( θ ) = t. With ayetri inforation, a high type θ eller ut hooe 1 a arkup that i inentive opatible. Thu it annot be in the preferred et for a type θ 1 (the haded region in Figure 3.1)., () I 1 I I () R x () () K S () Fig: 3-1: Separating Nah Equilibria 8

10 Obviouly there i a ontinuu of eparating Nah equilibria, e.g., S {( ( ), ), ( ( ), )} a depited. Aong the, there i a Pareto doinant eparating equilibriu (i.e., the Riley outoe), in whih the low type hooe her full inforation optial arkup while the high type hooe a lowet arkup that ake the low type jut indifferent. Thi outoe i indiated in Figure 3.1 by the pair R {( ( ), ), ( ( ), )}. Cho and Krep howed that for the tandard two- type ignaling odel, the Pareto doinant eparating equilibriu i the only equilibriu for whih there i no redible out-of-equilibriu ignal, where redibility i defined a follow: Intuitive Criterion (Cho and Krep): Suppoe that when oe type θ ake an out of K equilibriu hoie, her type i orretly pereived and, a a reult, type θ i better off. If no other type θ Θ i better off iiking typeθ, the pereption of the buyer i redible. In Figure 3.1, it i eay to ee why any Pareto inferior eparating equilibriu uh S a {( ( ), ), ( ( ), )} fail the Intuitive Criterion. If the high type θ (or equivalently ) deviate to K S fro ( ), and it i orretly pereived by the buyer, he i learly better off. Thi deviation would not be profitable for the low type, thu it i indeed redible for the high type to deviate, aking {( ( ), ), ( ( ), )} fail the Intuitive Criterion. S Three type ae While the Cho-Krep Intuitive Criterion work well in the two-type ae, it i well known that it an be diffiult to apply to ae when there are any type, and loe ot of it power in the ontinuou type ae. We now argue that it run into further proble in etting like our where the eller valuation i not perfetly orrelated with the buyer oon value oponent. 9

11 Conider the ae in whih the objet ha three type of harateriti: θ1, θ, θ 3. Let (, t ) = ( ( θ ), t( θ )). Suppoe the eller valuation and the buyer oon value i i i i oponent are given below: Type Probability Seller Coon value oponent valuation of buyer valuation p t θ 1 1 θ θ 3 3 p t p t Given any buyer pereption about the oon value oponent ˆt, the payoff funtion for eah type of eller i given by: u (, t ˆ, ) = F ( ) + tˆ(1 F ( )) + B ( ) (3.1) i i (1) (1) Note that type θ 3 ha the ae oon value oponent a type θ 1 but alo the ae ignaling ot a type θ. Thi iplie that type θ and θ 3 have the ae indifferene urve in tˆ pae, that i, they are of the ae atual type. However, if one applie the Intuitive Criterion to thi three type exaple, it ha no power: for any ignalpereption pair that i tritly preferred by type θ, it will be preferred by type θ 3 a well. Thu, no deviation i redible and o there i a ontinuu of eparating equilibria that urvive the Intuitive Criterion. Note that thi three-type exaple i obervationally equivalent to the two-type ae above beaue type θ and θ 3 are behaviorally idential and an be treated a equivalent. Forally, thi an be een by defining a grouped typeθ 3, uh that θ 3 = and t θ3 E t θ θ θ3 pt p3 t p p3 ( ) ( ) = [ {, }] = ( + ) ( + ). Thu it i highly unatifatory that the Intuitive Criterion elet one equilibriu in the two-type ae while leave in a ontinuu of eparating equilibria in the obervationally equivalent three-type ae. It i natural to eek a refineent that elet the ae ubet of equilibria 10

12 in either ae. It an be verified that the Cho and Sobel (1990) refineent onept of divinity, whih i built on the idea of tability of Kohlberg and Merten (1986) and an be onidered a a logi offpring of the Intuitive Criterion, doe not have power either in the above three-type exaple. Like the Intuitive Criterion, the divinity fae the ae proble of ditinguihing type θ and θ 3 to interpret a poible deviation, while thee type have the ae inentive to deviate. Suh ituation are oon when θ i ultidienional and and tare poitively orrelated. The three-type exaple point out the need to onider deviation not only by a ingle type but alo by a pool of type. Thi idea i inorporated in the trengthened Intuitive Criterion baed on Groan and Perry (1986a,b): Strengthened Intuitive Criterion (Groan and Perry): Suppoe that when eah type θ in a et Θ0 Θ ake an out of equilibriu hoie, the buyer pereption i that the expeted oon value oponent i tˆ = E[( t θ) θ Θ ] and, a a reult, eah type θ Θ o i better off. If no other type θ Θ i better off iiking and hooing, the ignal-pereption pair ( t, ˆ) i redible. o Riley (001) diue in greater detail thee and other refineent onept. We will tudy equilibriu refineent otly in the ontinuou type ae. Raey (1996) extend the Cho and Sobel divinity onept to the ae of a ontinuu of type. 11

13 t I 1 I t t 3 Expeted Coon value Coponent for Type and 3 t C K Fig. 3.: Three type exaple In our iple exaple it i lear how thi an be done. Suppoe buyer oberve the out-of-equilibriu ignal θ C. Knowing type and 3 θ have idential preferene, the pereption i that the expeted oon value oponent i t 3 = E[ t θ { θ, θ 3 }]. Given thi pereption, K u( θ, t, ) > u( θ, t, ), i =,3. i C 3 i 3 C However u( θ1, t3, ) < u( θ1, t, ) o type θ 1 ha no inentive to ii. Thu the Pareto doinated eparating equilibriu in whih type θ and θ 3 hooe K fail the trengthened Intuitive Criterion (SIC). However, with the SIC, we ay run into the proble of non-exitene of equilibriu a illutrated below: 1

14 t I 1 I t 3 Expeted Coon value Coponent for all 3 type t 13 t ˆ S Fig. 3.3: Separating equilibriu fail the Strengthened Intuitive Criterion t = E[( t θ ) θ { θ, θ, θ }] i uffiiently high, then a depited above all type would If have an inentive to deviate to an out-of-equilibriu ignal ˆ and the pair ( t ˆ, 13) i redible aording to Groan and Perry riterion. Thu no eparating equilibriu urvive the SIC. Sine no pooling equilibriu urvive the weaker Cho-Krep Intuitive Criterion, no pooling equilibriu urvive the SIC either. We ugget a loal redibility tet (LCT) whih weaken the pooling requireent of the trengthened Intuitive Criterion. For any deviation, intead of interpreting it a fro any ubet of type a in the SIC, the LCT ugget that the ignal reeiver interpret it a only oing fro one of the nearet type or both, and then hek whether it i redible for the to deviate. For exaple, in a three-type ae withθ 1, θ, θ 3, and 13

15 1 < < 3, onider a eparating equilibriu with ( 1) < ( ) < ( 3). To hek whether it atifie the LCT, uppoe there i an out-of-equilibriu ignal ( ( 1), ( )). The LCT i atified if neither 1 (, t ), (, t ) nor (, t3) i redible. In ontrat, the SIC require to hek all other poible pool of type (and a ingle deviation byθ 3 ). Naturally, the next tep i to explore under what ondition an equilibriu atifie the LCT. Sine our priary interet i on the ontinuou type odel, we will forulate the definition of LCT in the ontinuou type ae and then haraterize the neeary and uffiient ondition for the exitene of equilibriu atifying the LCT. Before doing that in Setion 5, we haraterize the Pareto doinant eparating equilibriu for the ontinuou type odel and analyze it propertie in Setion Continuou Type Cae In the odel with a ontinuu of type, we aue that, indued by the ditribution of θ, ex ante i ditributed a.d.f. G( ) with upport [., ] If there exit a eparating equilibriu, denoted by the invere arkup hedule, ( ) then it ut atify the following ondition: U (,, ) ( J( ) + τ ( ) ) f(1) ( ) U (,, ) τ ()(1 F ( )) 3 ( ) = = (1) (4.1) That i, given any eparating equilibriu hedule, type eller will optially hooe reerve arkup aording to the olution of (4.1). Thi ondition erely ay that the lope of the equilibriu hedule hould equal the arginal rate of ubtitution between the reerve arkup and the arket pereption about the type. 14

16 Indifferene urve for type ˆ= ( ˆ) θ ' z () 3 (, ) = = ( ) U(,, ) MRS U (,, ) ˆ Fig. 4-1: Separating Equilibriu θ ( ) Following arguent paralleling thoe in Riley (1979), there i a unique olution through the full inforation optiu for the lowet eller type ( ( ), ). Call thi R ( ). We next how that thi olution i inentive oparable and i hene a eparating equilibriu. Suppoe that the buyer pereption i given by ˆ = ( ), whih i the olution to (4.1) above. A eller of type thu hooe to axiize U(, ( ), ). Differentiating by, d U (, ( ), ) = U (, ( ), ) ( ) + U 3(, ( ), ) d U3(,( ), ) = U(, ( ), )[ ( ) + ] U (,( ), ) U (( ),( ), ) U (,( ), ) = + ] (,( ), ) U 3 3 (, ( ), )[ U (( ),( ), ) U By the ingle roing property, the ter in the braket above only hange ign one and U(, ( ), ) take on it axiu at where ( ) =. Therefore we have inentive opatibility. Letting = (), we have the following reult: Propoition 1: The olution to the differential equation (4.1): 15

17 ( J( ) + τ ( ) ) f(1) ( ) ( ) = τ ()(1 F ( )) (1) through the full inforation optiu for the lowet eller type (, ) haraterize the unique Pareto doinant eparating equilibriu (the Riley outoe). Auing an interior olution for the full inforation optiu, (4.1) an be rewritten a ( ) = ( J( ) J( ( )) f(1) ( ) τ ()(1 F ( )) (1) Thi iplie that () > () for all >. Due to the reforulation in Setion, the derivation of our haraterization reult Propoition 1 i traightforward. Note that Propoition 1 hold for any inreaing funtion τ ( ) (i.e., poitive orrelation between and t ). In the peial ae of perfet orrelation, tˆ = E[ t ] = τ () = γ, where γ > 0, (4.1) beoe: ( ) [ J( ) (1 γ ) ] f ( ) (1) = (4.) γ (1 F ( )) (1) whih an be rewritten a: 1 1 (1 F(1) ( )) d (1 ) f(1) ( ) = f(1) ( ) J( ) d γ γ Multiplying both ide by (1) 1 (1 F ( )) γ, we have γ γ (1) (1) (1) d [(1 F ( )) ( )] = f ( )(1 F ( )) J( ). d γ Integrating we obtain: γ γ γ (1) (1) (1) (1) γ (1 F ( )) ( ) (1 F ( )) = f ( y)(1 F ( y)) J( y) dy 16

18 Therefore, the invere arkup hedule in equilibriu an be written a 1 (1 ) 1 1 (1 1 ) γ γ γ ( ) = (1 F(1) ( )) f(1) ( y)(1 F(1) ( y)) J( y) dy+ (1 F(1) ( )) γ (4.3) Whenγ 1, (4.3) opletely haraterize the olution for the eparating equilibriu. A an exaple, when (i) X i unifor with upport [0,1]; (ii) n = ; (iii) γ = 1; and (iv) = 0 ; we an integrate (4.3) analytially to obtain where 1 = ( ) = 4( ) + 3ln( ) ln( ) 1+ 1 When 0< γ < 1, the equilibriu reerve prie hedule ay be trunated at oe ritial type, beaue the eller an be better off holding the ite unold a her own valuation get uffiiently large to exeed the equilibriu reerve prie. For the eller to be willing to ell the ite through ignaling, the reerve prie ut be greater than : r ( ) = ( ) + γ ; or, ( ) (1 γ ) (4.4) Taking thi ontraint into aount expliitly, the equilibriu reerve prie hedule an be haraterized ore peifially in the ae of perfet orrelation: Propoition : In the ae of perfet orrelation uh that tˆ = E[ t ] = τ () = γ, Equation (4.3) haraterize the olution for the Pareto doinant eparating equilibriu when γ 1 or when γ < 1 but x > (1 γ ). When γ < 1 and x (1 γ ), the equilibriu hedule deterined by (4.3) i trunated at ( = x, = x (1 γ )); thoe type of x [, ] will withdraw fro the arket. 1 γ 17

19 Proof: We have proved the propoition forγ > 1. When γ < 1 but x > (1 γ ), learly the ontraint of (4.4) doe not bind ine (1 γ ) (1 γ ) < x. Now we onider the ae in whih γ < 1 and x (1 γ ). Define = inf{ : ( ) (1 γ ) }, and = ( ). Then 1 F ( ) 1 F ( ) J( ) (1 γ) = (1 γ) = 0 F ( ) F ( ) By inpeting (4.), for ( ) to be an inreaing equilibriu hedule, we ut have F ( ) = 1, i.e., = x. A a reult, the hedule i trunated at = x (1 γ ). It reain to verify that thi new endpoint ondition (iplied fro the ontraint (1 1 ) (1 1 ) (4.4)) i atified in (4.3). Firt, a x, (1 F(1) ( )) γ γ (1 F(1) ( )) 0. Seond, applying L Hopital rule, we have 1 (1 ) 1 1 γ γ (1) (1) (1) γ li(1 F ( )) f ( y)(1 F ( y)) J( y) dy x = li x 1 1 γ f(1) ( y)(1 F(1) ( y)) J( y) dy γ (1 F ( )) (1) (1 1 ) γ 1 γ (1) (1) 1 γ (1) (1) f ( )(1 F ( )) J( ) = li x ( γ 1)(1 F ( )) ( f ( )) J ( ) = li x 1 γ x = 1 γ Therefore, taking liit on both ide of equation (4.3), we have x ( ) = li ( ) = = x/(1 γ ), whih onfir that for 0 < γ < 1, the eparating x equilibriu i given by (4.3) with a trunation at the eond endpoint ( = x, = x (1 γ )). Q.E.D. 18

20 Propoition give a oplete haraterization of the Pareto doinant eparating equilibriu for the ae of perfet orrelation. Jullien and Mariotti (003) tudy a reerve prie ignaling odel in whih the eller valuation i θ and the buyer valuation are λθ + (1 λ) ε, where [0,1] i λ. Their etup orrepond to the perfet orrelation ae in our odel withγ < 1. Coparative Stati We now derive the oparative tati reult for the unique Pareto doinant eparating equilibriu. Firt we have: Propoition 3: In the eparating equilibriu, for every reerve prie r () = τ () + () i higher for larger n. >, the arkup and hene the Proof: See the Appendix. Thi reult i intuitive. When there are a larger nuber of bidder, the ignaling ot fro higher reerve prie are aller beaue the probability of no ale i lower. A ignaling ot go down, reerve prie will be higher in equilibriu. Thi reult i in ontrat with the well known reult that optial reerve prie i independent of n ; when the ignaling role i taken into aount, a reerve prie will in general depend on the nuber of bidder. The odel an be readily extended to ituation where the relative iportane of oon value and private value oponent in bidder valuation an take on any arbitrary degree. Let Vi = t+ β Xi where β (0, ) eaure the relative iportane of the private value oponent. Clearly β = 1 orrepond to our bai odel. A before let = r t but now all = / β the relative arkup. The intereting quetion in thi ae i how the relative arkup hange a β hange. 19

21 Propoition 4: Suppoe τ ( ) i inreaing in. For any, the relative arkup i inreaing in β. Conequently, the reerve arkup and hene the reerve prie r () = τ () + () are inreaing in β at an aelerating rate. Proof: See the Appendix. The intuition for thi reult i the following. When β inreae, the private value oponent beoe ore iportant while the oon value oponent beoe le o. When τ ( ) i inreaing in, and the oon value oponent beoe aller, the relative arkup for the lowet type atually inreae, beaue the ignaling ot fro no ale i relatively all. It an be hown that the relative reerve hedule follow the ae differential equation a before. A a reult, a larger β iplie a higher initial ondition, thu iplie a higher relative arkup hedule everywhere. Note that the peial ae with t = γ, γ >1 atifie the ondition that τ ( ) i inreaing in. 3 Outide Certifiation We now onider ituation where in addition to ignaling through reerve prie, the eller an redibly reveal θ to the bidder through an outide ertifiation ageny at a fixed ot of > 0. The quetion i when the eller i willing to pay for uh a ervie. For eae of analyi we onider the peial ae in whih t = γ, γ >1. Let u U () = (,, ()) be type eller expeted revenue under full inforation, and let u () = U (,, ()) be type eller expeted revenue in the eparating equilibriu. Alo let W() = u () u(). Iediately, W() = 0 and W() 0for all. To further iplify notation, let Theore, we have = ( ) and = ( ). By the Envelope 3 When τ ( ) i dereaing in, no definite onluion an be ade about whether the relative arkup hedule will ove down everywhere a β inreae. 0

22 dw du du = d d d = F ( ) + γ (1 F ( )) F ( ) (1) (1) (1) = γ + (1 F(1) ( )) ( γ 1)( F(1) ( ) F(1) ( )) Sine, we have dw / d > 0 ertifiation ervie if W( ). Clearly the eller i willing to pay for the. The following reult i iediate. Propoition 5: For any > 0, there exit a utoff type uh that for all [, ], the eller hire the outide ertifiation ageny; for all ignal through reerve prie r () = γ + (). [, ), the eller An Appliation to the Leon Market Even though our analyi o far ha foued on aution, our reult an be readily applied to tudying ignaling in the Leon Market. Conider the following arket ituation for a good (e.g., ued ar). To keep thing iple, uppoe there i a unit a of buyer eah with unit deand, and there i alo a unit a of eller eah with one ite to ell. Eah eller knowθ, the quality of the ite for ale, whih deterine the eller own valuation. Suppoe the oon value oponent of the ite i given by γ, γ >0. However, the buyer do not oberve the quality of the good, but know that the population ditribution of (indued by the ditribution of θ ) i given by.d.f. G( ) with upport S = [, ]. Buyer j valuation for a good with quality θ i Vj = γ + X j where X j i a private value oponent only obervable to buyer j. The population ditribution of X j i given by.d.f F( ) with upport [ x, x ]. What i jut deribed i a ontinuou type verion of the Akerlof Leon Market odel (Akerlof 1970). The fundaental idea of Akerlof analyi i that the prie-taking Walraian equilibriu annot ahieve effiient reoure alloation in the preene of advere eletion proble. In the above odel, abent the advere eletion proble (i.e., if quality i known to the buyer), the firt bet alloation i eaily ahieved by etting a prie of ( θ ) for the good with qualityθ. When θ i not known to the 1

23 buyer, for any fixed prie p hoen by the Walraian autioneer, only thoe eller with valuation p are willing to ell their good, reulting in a total upply of G( p ). Aordingly, the expeted oon value of the good in the arket i τ = E[ γ p] = γe[ p]. Sine only thoe buyer with valuation V = τ + x p are willing to buy, the total deand i1 F( p τ ). For a arket-learing prie, we et 1 F( p τ ) = G( p). In general, the equilibriu prie that lear the arket lead to le than effiient level of trade. For exaple, when both F(.) and G (.) are unifor on [0,1], the arket-learing prie i p = /(4 γ ) if 0< γ, and 1 if γ >, whih iplie a trade volue of in{ /(4 γ ),1}. Trade i effiient only whenγ. When γ <, equilibriu trade i le than the effiient level and i inreaing in γ. The onept of Walraian equilibriu aue prie-taking behavior on both ide of the arket and that prie i publi inforation. In any real life ituation uh a the ued ar arket, neither of thee auption fit: eller et prie for their good and buyer earh for what they want. To odel thee feature in the iplet way, we onider the following ituation: the eller et prie for their good, and without knowing the prie in the arket, eah buyer randoly goe to one eller. In other word, we onider a ituation with pair-wie rando athing in whih the eller et prie. What i the equilibriu outoe in thi arket? Oberve that in our previou analyi of reerve prie ignaling in the aution ontext, we an reinterpret the ingle eller with a type drawn fro the ditribution G(.) a a unit a of eller with unit upply whoe type have a population ditribution of G (.). Then it hould be lear that our previou haraterization reult Propoition 1 applie to the urrent pair-wie athing arket with n = 1. When n = 1, Equation (4.) beoe Aordingly, Equation (4.3) beoe d F ( )[ J ( ) + ( γ 1) ] = d γ (1 F( )) 1 (1 ) 1 1 (1 1 ) γ γ γ ( ) = (1 F ( )) F ( y)(1 Fy ( )) J( ydy ) + (1 F ( )) γ (4.5)

24 where = (). Therefore, thi haraterize a eparating priing equilibriu in whih a eller with valuation hooe a poted prie p = γ + ( ) and the buyer orretly infer the true type fro thi prie hedule and deide whether to buy at thi prie. When F (.) i unifor on [0,1] and = 0 (hene = 0.5 ), the equilibriu arkup i given by 1 1 γ (1 1 ) ( 1) (1 ) ( 1 γ γ + ) (1 ), 0, 1, γ > γ γ γ ( ) = 1 log(1 ) log, γ = 1 ( 1) 4(1 )[log(1 ) + log ], γ = 1 Again, for the ae 0 < γ < 1, the equilibriu hedule i given by the above olution with the undertanding that it i trunated at = 1/(1 γ ) if 1 (1 γ ) <. In thi peifi exaple, it an be verified that both the oial welfare and volue of trade are greater in the Walraian equilibriu than in the ignaling equilibriu. 4 However, thi differene ainly reult fro the auption that the earhing tehnology i extreely priitive and otly in the ignaling equilibriu --- only one round pair-wie athing i allowed --- while on the other hand, the earhing ot i zero in the Walraian equilibriu. Note alo that unlike in the Walraian equilibriu, the prie hedule in the ignaling equilibriu doe not depend on the ditribution funtion G (.), aking the two equilibria ore inoparable. In the appliation to the Leon Market, everal extenion are deirable and worth further reearh. One traightforward extenion i to onider arket in whih eah eller fae ultiple buyer, e.g., houing arket. In thi ae our previou reult diretly apply. Another extenion i to onider heterogeneou buyer preferene over quality. For exaple, it ay be reaonable to uppoe that buyer j valuation for a good with quality θ i Vj = t( θ ) Z j + X jwhere Z j i buyer j preferene for quality and i 4 Our oputation reult how that the eller expeted revenue an be higher in the ignaling equilibriu. 3

25 only known to hielf. In the preeding exaple we overiplified ituation by auing that eah buyer an only aple one eller. It i deirable for future reearh to tudy a ore realiti odel in whih buyer an earh ore than one period and perhap have heterogeneou preferene for quality. 5. Equilibriu Refineent: Loal Credibility Tet In thi etion we tudy equilibriu refineent for the ontinuou type ae uing the onept of Loal Credibility Tet. We haraterize neeary and uffiient ondition under whih the Pareto doinant eparating equilibriu atifie LCT in a general ignaling odel that inlude the reerve prie ignaling odel and the wellknown Spene eduation ignaling odel a peial ae. Then we apply the general reult to thee two odel. A General Signaling Model To begin, let u fix the notation. A before, θ Θ denote the private inforation of the ender. Let (): Θ R, S = [, ], be the true type of the ignal ender. ŝ i the type of the ender pereived by the ignal reeiver(). A ignal hoen by the ender i denoted by y Y, where Y i the et of feaible ignal. Let z(): i S Y be a tritly onotone ignaling funtion that fully reveal the true type of the ender. Let U(,, ˆ y ) denote the utility of the ender whoe true type i and who end out a ignal of y and i pereived to be type ŝ. Aordingly, U () = U (,,()) z i the utility of the ender of true type in the eparating equilibriu z. ( ) We aintain the following tandard auption: (a) U(,, ˆ y) i third order differentiable in all it eleent; (b) U ˆ (,, y ) > 0; () The ingle roing ondition hold: dˆ U U U U U = = < 0 dy U U U

26 In addition, we ake the following tehnial auption: B1: U 1 = 0 for all (,, y ˆ ); B: U = 0 for all (,, y ˆ ). Exaple 1: The reerve prie ignaling odel In the odel tudied in previou etion, the eller expeted payoff an be expreed a U(,, ˆ y) = y+ τ ()(1 ˆ y) + H( y), where we adopt the tranforation y = F (1) ( ) [0,1] and H ( y ) = B ( ( y )). Uing (.3) we an ee that H ( y) = B ( ) y ( ) = F(1) ( ) J( ) F(1) ( ) = J( ( y)) H ( y) = J ( ) F ( ) With thi tranforation, the derivative of (1) U(,, ˆ y ) are U1(,, ˆ y) = y, U(,, ˆ y) = τ ()(1 ˆ y), U3(,, ˆ y) = τ () ˆ + H ( y) U11( y, ˆ, ) = 0, U1( y, ˆ, ) = 0, U13( y, ˆ, ) = 1 U (,, y ˆ ) = τ ()(1 ˆ y), U (,, y ˆ ) = τ (), ˆ U (,, y ˆ ) = H ( y) 3 33 The tandard auption and B1 are all atified. B i atified when τ () i linear in. By the tandard reult, when U 3 (,, z ( )) = τ ( ) + H ( z ( )) 0, a eparating equilibriu atifie U (,, z()) τ ()(1 z()) U (,, z( )) τ ( ) + H ( z( )) = = z () 3 Exaple : The eduation ignaling odel In a oon forulation of the Spene eduation ignaling odel, a worker expeted payoff i U(,, ˆ y) = ˆ C(, y), where i the worker produtivity unknown to fir, ŝ i the worker produtivity pereived by fir and hene i alo the wage offered to her by opeting fir, and y i the eduation ignal the worker an hooe. It i typially aued that for all ( y, )(i) C 1 (, y ) < 0; (ii) C (, y ) > 0 ; and (iii) C1 (, y ) < 0. The derivative of U(,, ˆ y) are 5

27 U (, ˆ, y) = C (, y), U (, ˆ, y) = 1, U (, ˆ, y) = C (, y) U (,, y ˆ ) = C (, y), U (,, y ˆ ) = 0, U (,, y ˆ ) = C (, y) U (,, y ˆ ) = U (,, y ˆ ) = 0, U (,, y ˆ ) = C (, y) 3 33 The tandard auption and B1 and B are all atified. By the tandard reult, and ine by auption U3(,, ˆ y) = C(, y) 0for all (,, y ˆ ), a eparating equilibriu atifie U (,, z()) 1 U(,, z ()) C(, z ()) () = = 3 z Loal Credibility Tet (LCT): In Setion 3, we propoed a refineent onept LCT for the finite type ae. The idea i to weaken the pooling requireent of the Groan and Perry SIC by interpreting a deviating ignal a fro a ingle type or a et of type that are nearby the ignal. The onept an be eaily extended to the ae of ontinuou type. Loal Credibility Tet: Conider any eparating equilibriu z (): S [, zz] Y. Conider any ignal ŷ Y. 1. When ŷ < z, ( y ˆ, ) i a redible deviation if U(,, z) < U(,, yˆ ).. When ŷ > z, ( y ˆ, ) i a redible deviation if U(,, z) < U(,, yˆ ). 3. When yˆ [ z, z], let z ( ˆ 0) = yand onider a all neighborhood of 0, So S. Let ˆ = E[ S0]. If there exit ε > 0 uh that ( i) U(, ˆ, yˆ) > U(,, z( )) + ε, for all int So ( ii) U (, ˆ, yˆ) < U (,, z( )) + ε, forall So then the out-of-equilibriu ignal-pereption ( y ˆ, ˆ) i redible. The above definition of LCT apture the idea that the reeiver interpret (potentially) deviation a fro the nearby type. In partiular, Part 1 ay that any out- 6

28 of-equilibriu deviation below the lowet equilibriu ignal i pereived a fro the lowet type. Thi iplie Lea : If a eparating equilibriu atifie LCT, it i the Pareto doinant equilibriu (the Riley outoe), that i, z ( ) = y( ), where y ( ) axiize U(,, y ). Proof: If z ( ) > y( ), then the out-of-equilibriu ignal-pereption pair ( y ( ), ) i redible, violating Part 1 of the LCT requireent. If z ( ) < y( ), by the ingle roing ondition, the lowet type would want to deviate to type, violating equilibriu ondition. Q.E.D. y ( ) and be pereived a a higher Part of the LCT definition ay that any out-of-equilibriu deviation above the highet equilibriu ignal i pereived a fro the highet type. Thi redibility requireent i alo atified autoatially by the Pareto doinant eparating equilibriu. Therefore, to hek whether there exit an equilibriu atifying LCT, we only need to hek whether the unique Pareto doinant eparating equilibriu atifie Part 3 of the LCT definition. Conider any on-equilibriu ignal, and the type of ender for thi ignal in equilibriu. Suppoe the nearby type all deviate to thi ignal, and thi i orretly pereived by the reeiver, and all the deviating type an gain at leat ε relative to their equilibriu payoff while all other type annot. Then thoe nearby type an redibly deviate to the partiular on-equilibriu ignal by throwing away ε aount of oney. 5 The following reult give an equivalent, but ore operational, requireent for Part 3 of the LCT definition. Propoition 6: Suppoe that the ingle-roing property hold.. For in Part 3 of the LCT definition, if there i an < ˆ uh that y ˆ, S, ˆ a defined 0 5 With a finite type pae and ontinuou ignal paey, the et of ignal that are not eleted in a eparating equilibriu i dene in Y. Thu there i no need to ignal by throwing oney away. 7

29 () i U (,, yˆ ) > Uz (,, ()),forall intso. ( ii) U(,, yˆ ) < U(,, z( )), for all So. then the out-of-equilibriu ignal-pereption ( yˆ, ˆ) i redible. By Propoition 6, to hek whether the Pareto doinant eparating equilibriu atifie LCT, we only need to hek whether there are any redible interior deviation and any redible boundary deviation, in the ene that will be ade preie below. The equilibriu urvive LCT if and only if there i no redible deviation of either kind. Loal Credibility Tet: Interior Deviation For any two type, uppoe thoe in the interval [, ] pool at a ertain ignal y. Let v(, ) be the expeted type of thi pool, that i, v(, ) = 1[ G( ) G()] zdg() z, where G(.) i the.d.f. of. Let v (, ) [, ] and y (, ) Y be a olution to U(, v, y) = U(,, z( )) U(,, v y) = U(,, z()) (5.1) The point ( y (, ), v (, )) i depited below in Figure 5.1. Given thi ignal-pereption pair, all thoe type in (, ) prefer the pool to their eparating equilibriu payoff. 8

30 I I ( yv, (, )) ( z ( ), ) z () ( z ( ), ) z Fig. 5.1: Pool of type in [, ] Fro Propoition 6, in order for the eparating equilibriu haraterized by z ( ) to atify LCT, it ut be that for loe to, any uh ignal-pereption pair of ( y (, ), v (, )) i not redible. That i, for any [, ) and >, v (, ) < v (, ) a. Note that for any [, ), v(,) = v(,) =. Thu, if v(, ) v(, ) < 0a, then v(, ) < v(, ) a. Lea 3: For any [, ), (i) v (,) = 1/ ; (ii) 1 G''( ) v(,) =. 6 G'( ) Proof: See the Appendix. Lea 4: For any uh that U3 (,, z()) 0, (i) under Auption B1, v (,) = 1; (ii) under Auption B1-B, v 1U 1 4U + U U z 1 U z (,) = + '() [ '()] 6 U U U U where all funtion are evaluated at (,, z ()). Proof: See the Appendix. 9

31 The propoition below follow iediately fro Lea 3 and 4. Propoition 7: For any uh that U3 (,, z()) 0, the eparating equilibriu haraterized by z() doe not have redible loal interior deviation a defined in LCT if and only if U U 13 + U U ( ) () U [ ()] G + z z U13 + U U + > 13 U G ( ) (5.) U(,,()) z where z () =. U (,, z()) 3 Proof: We know that v(,) v(,) = 0. Fro Lea 3 and 4, we have v (,) v (,) = 0. If the ondition of the propoition i atified, then v(,) < v(, ). Thu, for a neighborhood around, it ut be v(, ) < v(, ), and o v(, ) < v(, ). Q.E.D. It i eaier to undertand the intuition for Propoition fro the finite type ae. Conider the exaple depited in Figure 3.3. Sine the arginal rate of ubtitution between and t i iilar for the different type in that exaple, the indifferene ap are iilar and o indifferene urve are loe together. A a reult, all type are better off if buyer believe that all ay be hooing to deviate, thu violating the requireent of no redible deviation. However if the arginal rate of ubtitution deline uffiiently rapidly with type, the differene in the lope of the two indifferene ap i greater a depited below. Now only type 1 i better off if the buyer think that all type ay be deviating. Thi belief i therefore no longer redible. 30

32 t I 1 I t 13 t Expeted 3 Coon value Coponent for all 3 type t ˆ S Fig. 5.: Separating equilibriu atifie the Loal Credibility Tet Thi intuition i refleted in Equation (5.). In the ae of a ontinuu of type, the lope of the indifferene ap i given by MRS(,, z()) U (,, z()) 3 =. U(,, z()) U13 Firt note that MRS(,, z()) U3 = by B1 and MRS(,, z()) = U U = = B. Thu the larger i U 13 and U 3 the ore rapidly the MRS deline. Moreover, by U U + U U MRS(,, z) = z ( U), hene the larger iu 3 and U 33 the ore rapidly the MRS deline a z inreae. Clearly, the inequality of (5.) i eaier to be atified for largeru 13, U 3 andu 33. Intuitively, the rate at whih the arginal rate of ubtitution deline with i a eaure of ignaling effetivene. Thu Propoition 7 ugget that when ignaling effetivene i uffiiently large, the eparating equilibriu will urvive 31

33 the LCT tet. Figuratively, when the indifferene urve I i far fro I 1 in Figure 5., there will be no redible deviation with the pereption at t 13. The right hand ide of Equation (5.) i the onavity of the ditribution funtion of, G, () noralized by it denity funtion. Intuitively, the ore onave Gi () (i.e., the aller G i), the ore probability a on aller in any et of type, thu the aller the expeted value of any et of type. Conequently, the aller G i, the le likely a deviation i redible. Figuratively, when the expeted value of t for all three type, t 13, i lower in Figure 5., there will be no redible deviation. Below we apply Propoition 7 to the two exaple introdued above. Exaple 1 (ontinued): The reerve prie ignaling odel We now analyze when the eparating equilibriu haraterized in Propoition 1 atifie the ondition of Propoition 7. We fou on the peial ae t = γ, γ 1. Firt onider (1 γ ) < J( x) o that the eparating equilibriu goe through Sine U 3 (,, z ()) = (1 γ ) J (()) z i dereaing in, 3 U (,, z ()) < (1 γ ) J ( x ) < 0for all [, ]. In thi odel, ineu 113 (,, v y) = U 133 (,, v y) = 0, (5.) beoe () = x at. 1 G ( ) [(4 γ ) z ( ) + H ( z( ))( z ( )) ] >, γ (1 z ( )) G ( ) where γ (1 z ( )) z () =. When thi ondition hold, the eparating (1 γ ) + H ( z( )) equilibriu haraterized in Propoition 1 doe not allow redible interior deviation. For the ae(1 γ ) J( x), then U3 (,, z()) = 0, o Propoition 7 doe not apply. We will onider thi ae later. Exaple (ontinued): The eduation ignaling odel 3

34 In the oon forulation of the odel, U3(,, v y) = C(, y) < 0for all (,, v y ). Fro the derivative of U(, v, y ) derived before, we have U113(,, v y) = C11(, y) and U133(,, v y) = C1(, y). Condition (5.) beoe C "( ) C C C > G C1 C C C1 C G'( ) where all funtion are evaluated at ( z,()). If the above ondition hold, then the U(,,()) z 1 eparating equilibriu given by z () = = and goe through U(,, z ()) C(, z ()) z () = 0doe not allow redible interior deviation. 3 Loal Credibility Tet: Boundary Deviation Finally, for the eparating equilibriu haraterized by zto () atify the LCT in the general odel, we need to onider the following kind of boundary deviation, in addition to the interior deviation repreented by (5.1). For any type >, uppoe thoe in the interval [, ] all hooe the ignal y = z(), the equilibriu ignal by. Let 1 v(, ) be the expeted type of thi pool, that i, v(, ) = zdg() z G ( ). Let v ( ) [, ] be a olution to U(, v, y) = U(,, z( ')) (5.3) In order for the eparating equilibriu haraterized by z () to atify the LCT, it ut be that for loe to, any uh ignal-pereption pair of ( yv, (, )) i not redible. That i, v(, ) < v( ) a. Note that v(,) = v() =. Fro (5.3), total differentiating give 33

35 U1(,, z( )) U1(, v, y) v ( ) = U (, v, y) A, v and z ( ) y. Therefore, v ( ) 0. However, it i eay to how that a, v (, ) 0.5. So in the neighborhood of, v (, ) > v ( ). Therefore, there i alway a redible boundary deviation at the lowet ignal y = z(). The above lower endpoint proble an be overoe if we odify the odel o that given the ignaling hedule z, ( ) the ender of oe lowet type doe not atively partiipate in the arket beaue of oe partiipation ot. Propoition 8: There i no redible boundary deviation if there i a uffiiently large ubet of type [, ] whih do not ignal in the eparating equilibriu. Let u uppoe that due to oe partiipation ot, the lowet type who partiipate in the arket i >, o the ender of all type < tay out of the arket. Now uppoe for oe > thoe in the interval [, ] all hooe the ignal y = z( ), the equilibriu ignal by expeted type a higher than. If the ignal reeiver pereive their and thu pay the aordingly, then all thoe lowet type < will now find it profitable to partiipate in the arket and join the pool of [, ]. But when the (probability or population) a of type [, ] i large, the orret pereption of the expanded pool for thoe who hooe ignal y = z( ) will be aller than, aking it unattrative for type of [, ] to deviate. Let u onider the two peifi exaple tudied earlier. In the aution odel, uppoe the eller need to invet a fixed ot of to run the aution. With the invetent, the eller expeted payoff fro running the aution i (1) (5.4) u () = u() + F ( t ()) dt ; 34

36 while if the eller doe not invet, her payoff i 0. Sine the payoff in (5.4) i tritly inreaing in, there i a unique utoff type o that the eller of type aller than will not be willing to invet and partiipate in the gae, while the eller of type greater than will pay the invetent ot and hooe reerve prie aording to. () When the aution ot i uffiiently large, the a of the type exluded [, ] will be uffiiently large o that there i no redible boundary deviation. Moreover, when i uffiiently large, then (1 γ ) J( x) < 0, hene U3 (,, z()) < 0for all [, ]. Therefore, in the reerve prie ignaling odel, a long a LCT only require (5.) for [, ]. i uffiiently large, the The analyi of the tandard eduation ignaling odel i very iilar. Introdue a reervation wage funtion (alternative job opportunity) w 0 () uh that for uffiiently all, w 0 () >. Suppoe that the worker of type aller than will not be willing to partiipate in thi arket, ine the highet payoff he an get in a eparating equilibriu i < w 0 (). By onotoniity, the worker of type greater than will partiipate in thi labor arket and hooe eduation aording to z ( ) tarting fro z ( ) = 0. When the outide opportunity i uffiiently large, the a of the type exluded [, ] will be uffiiently large o that the lower endpoint proble doe not arie. In uary, when the ondition of Propoition 8 hold, in order to hek whether a eparating equilibriu atifie the LCT, we only need to hek whether (5.) i atified for [, ]. 6. Conluding Reark In thi paper we onider aution in whih the eller valuation i orrelated with a oon value oponent of eah buyer valuation. Only the eller know her valuation and the oon value oponent i not diretly obervable to anyone. We haraterize the unique Pareto doinant eparating equilibriu in whih a higher reerve prie et by the eller i a ignal of her greater valuation. 35