The Optimality of Uniform Pricing in IPOs: An Optimal Auction Approach

Transcription

1 The Optmalty of Unform Prcng n IPO: An Optmal Aucton Approach October 4, 2005 Abtract Th paper provde theoretcal upport to the ue of the unform prcng rule n the Intal Publc Offerng of hare. By ung an optmal aucton approach, we are able to how that the optmalty of a unform prce n IPO crucally depend whether the eller ha full decreton on the ue of quantty dcrmnaton. Th n turn depend on two man dmenon of the IPO game: the nttutonal nvetor preference and extence or not of budget contrant on retal nvetor. Specfcally, we how that n the tandard cae, wth rk neutral nttutonal nvetor and cah contraned retal nvetor, the optmal IPO mplemented by a unform prce. Keyword: Intal Publc Offerng, Prce Dcrmnaton, Ratonng, Optmal Aucton. JEL Clafcaton: D8, G2. 1

2 1 Introducton The lterature on Intal Publc Offerng (IPO) ha been growng remarkably n the lat decade. Mot of th lterature, both theoretcal (e.g. Rock, (1986), Allen and Faulhaber, (1989), Benvente and Spndt, (1989)) and emprcal (e.g Beatty and Rtter, (1986), Rtter (1987) Cornell and Goldrech, (2001), Derren and Womack (2003)) ha partcularly focued on the explanaton of ome apparent market anomale urroundng IPO, uch a underprcng, long-term underperformance and hot ue market. 1 Much le attenton ha been devoted to undertand whether the current IPO format whch mpoe unform prcng effcent or not. Vrtually all paper mply aume unform prcng to be content wth the current practce of IPO. A a matter of fact, the only paper addreng th ue (Benvente and Wlhelm, (1990)) challenge the effcency of the current regulatory contrant and ugget that frm could mprove on the IPO performance (rae the IPO proceed) f allowed to prce dcrmnate acro nvetor. 2 Th paper nvetgate more cloely th ue,.e. whether unform prce retrcton lead to neffcent outcome. The frt reult we obtan to how that n the tandard IPO et up wth rk neutral nttutonal nvetor and cah contraned retal nvetor unform prcng the optmal prcng rule. However. the man contrbuton of the paper to how that the optmalty of the unform prcng rule crucally depend on two man dmenon of the problem: the nttutonal nvetor preference and the extence or not of budget contrant on retal nvetor. Specfcally, we fnd that a long a retal nvetor are not cah contrant -.e. they can buy all the hare - the optmal IPO can alway be mplemented by a unform prce. Converely, when retal nvetor are cah contrant, the nttutonal nvetor preference become determnant: prce dcrmnaton needed n addton to quantty dcrmnaton f nttutonal nvetor are rk avere. The reaon for th that, f the eller enjoy full dcreton on how to allocate hare, then there no need to alo employ 1 Rtter and Welch (2002) provde a handy revew of the IPO lterature, wherea for a fully comprehenve revew of the theory and evdence on IPO actvty we refer to Jenknon and Ljungqut (2001). 2 Brley (2003) n a note how that Benvente and Wlhelm reult hold only when regular nvetor are, on average, le nformed than retal nvetor. Otherwe, the unform prce retrcton doe not affect IPO proceed. 2

3 prce dcrmnaton to acheve optmalty. The abence of cah contrant on retal nvetor enure the eller enough leeway on the allocaton of hare ndependently of the nttutonal nvetor preference. When ntead nttutonal nvetor are cah contraned, the eller may tll have enough leeway on the allocaton rule f nttutonal nvetor are rk neutral, but not f they are rk avere. In th latter cae quantty dcrmnaton alone not enough to acheve optmalty and the eller need to employ alo prce dcrmnaton. The methodology we employ, whch explot the tool of optmal aucton and mechanm degn theory for nformaton gatherng (Myeron, 1981; Makn and Rley, 1989), repreent by telf one of the contrbuton of th paper. There are few other paper n the IPO lterature ung the ame approach (Ba, Boaert and Rochet (2002), Ba and Faugeron-Crouzet, (2002), Makmovc and Pchler, (2002)). Our paper dffer from each of them n everal repect, the mot mportant of whch that, a antcpated, all of them aume unform prcng wherea we do not put any retrcton ex ante on the prcng rule. We endogenouly derve the optmal prcng rule and how that t unform. In addton to th, Ba, Boaert and Rochet (2002) focu on IPO n whch the ntermedary act n the bet nteret of the nttutonal nvetor ntead of the frm. They derve the optmal ellng mechanm that enable the eller to extract all the nformaton from the coalton ntermedary-nttutonal nvetor and fnd that th mechanm pecfe a prce chedule decreang n the quantty allocated to retal nvetor. Th prce chedule on one hand elmnate the wnner cure for the unnformed nvetor and on the other hand exhbt underprcng (n the form of nformatonal rent) to nttutonal nvetor. Ba and Faugeron-Crouzet ntead compare the performance of alternatve IPO procedure and fnd that n general aucton-lke mechanm uch a Me en Vente and Bookbuldng are more nformatonally effcent and can lead to prce dcovery. They alo how that pure aucton mechanm mght lead to tact colluon among the bdder and, conequently, to neffcent outcome. Makmovc and Pchler (2002) look at the optmal amount of nformaton gatherng by comparng prvate and publc offerng wth a partcular attenton to the determnant of 3

4 underprcng. They fnd that n abence of any allocaton contrant and under rk neutralty, the eller able to degn a ellng mechanm wth zero underprcng (no rent to the nformed nvetor). We prove a mlar reult n a partcular ubcae of our model. Takng th reult a a benchmark, the author then lnk the extence and magntude of underprcng to varou contrant eventually mpoed on the bookbuldng procedure. In our IPO a frm want to place a fxed number of hare. Two group of nvetor take part n the IPO: n nttutonal nvetor and, a group of atomtc nvetor (retal nvetor). The latter typcally partcpate only occaonally n the offerng. Inttutonal nvetor have uperor nformaton about the market value of the aet becaue each of them receve a prvate gnal, wherea retal nvetor have no nformaton about the value of the aet on ale. The eller degn the IPO mechanm n order to elct the nformaton from nttutonal nvetor and maxmze the IPO proceed. Dervng the optmal IPO mean dervng the optmal allocaton rule and the optmal prcng rule. We tart by lookng for the optmal IPO n the tandard IPO et up where nttutonal nvetor are aumed to be rk neutral and retal nvetor cah contraned. In th context we prove that unform prcng the optmal prcng rule and furthermore there ext a multplcty of unform prce mplementng the optmal IPO. We next move away from the tandard IPO et up by challengng the aumpton of rk neutral nttutonal nvetor. We ntead analyze how the optmal IPO change f nttutonal nvetor are rk avere,wheretherkaveronwehavenmndanaverontothenventory rk. The reult we obtan ugget the optmalty of the unform prcng rule crucally depend on two dmenon of the problem, the extence of budget contrant to retal nvetor and the nttutonal nvetor preference. In partcular, a long a retal nvetor are not cah contraned the unform prcng rule alway upport the optmal IPO allocaton no matter what the preference of nttutonal nvetor are. When ntead retal nvetor are cah contraned then nttutonal nvetor preference play a role and, pecfcally, we can retore the optmalty of the unform prcng rule only f they are rk neutral (but not 4

5 necearly when they exhbt rk averon). In other word, for the unform prce to be optmal, t mut be that the eller can freely chooe the allocaton rule,.e. he not ubject to any retrcton on how to allocate the hare, a condton whch clearly not atfed when retal nvetor are cah contraned. Therefore, n th cae, he need to gan ome leeway elewhere and th acheved when nformed buyer preference offer uffcent flexblty (.e. the cae of rk neutralty). The paper organzed a follow. In the next ecton we et up the model and the IPO degn problem for the eller n the tandard cae of rk neutral nttutonal nvetor and cah contraned retal nvetor. The reult are preented n Secton 3. The cae of rk avere nttutonal nvetor analyzed n Secton 4. Some concludng remark are dcued n the lat ecton. All the proof are preented n the Appendx. 2 The Model A frm want to ell Q hare n an IPO, wth Q fxed and, wthout lo of generalty, normalzedto1. Anntermedarynchargeofmarketngtheue. Heaumedtoactn the frm bet nteret, o, hereafter, we wll mply refer to the eller to denote the coalton frm-ntermedary. Th a tandard way of modellng the role of ntermedary n the IPO lterature. The eller want to maxmze the proceed from the ale. He face n(> 2) large, nttutonal nvetor wth prvate nformaton about the frm market value, and a frnge of retal and unnformed nvetor. Inttutonal nvetor preference are decrbed by the followng utlty functon: u (p,q,v)=z(q,v) q p for all {1, 2,...n} where q the quantty agned to nvetor and p the prce per hare nvetor ha to pay. We denote by T = p q the total payment from nvetor to the eller. v the value of new hare. Notce that, the above utlty functon lnear n the tranfer T. 3 We aume that the valuaton functon z for nvetor atfy ome tandard aumpton (ubcrpt are 3 Th knd of utlty functon exhbtng lnearty n the total payment, are very common n the aucton lterature. (See for ntance Crèmer and McLean, 1988). 5

6 for dervatve wth repect to varable): 1) z 1 > 0 and z 2 > 0; 2) z 11 0; 3) z(0,v)=0for all v; 4) z 12 > 0 (ngle-crong condton); 5) z and z 112 0; and 6) z 12 (0,v) 1 (ee Fudenberg and Trole, 1991). Inttutonal nvetor have prvate nformaton n that each of them receve a gnal about the valuaton of the frmbythemarketvalue. Thevalueoftheharev gven by the average of all the gnal, that v() = 1 n Our choce of the nformatonal tructure very common n aucton theory (e.g. Bulow and Klemperer, 1996, 2002). Addtonally and more mportantly, our nformatonal tructure a traghtforward generalzaton of the mple bnomal nformatonal tructure adopted n other paper on IPO (e.g. Benvente and Wlhelm, 1996; Ba and Faugeron-Crouzet, 2002). 4 Sgnal are..d. accordng to a unform dtrbuton defned on Ω =[; ], o the cumulatve dtrbuton functon F ( )=, and the denty functon f ( )= 1. Let u alo denote by f() the jont denty functon o that f() =f( 1,..., n )= Q f ( ), X wth =( 1,..., n ) Ω = N Ω =[, ] n. We aume a contnuum of compettve, rk neutral retal bdder. The total ma of thee retal bdder normalzed to one. We aume that they face a cah contrant o they cannot aborb the whole ue. In the followng ecton we wll analyze the effect of the extence of th cah contrant on the mplementaton of optmal IPO procedure wth a unform prce. We denote by q R the quantty they receve n equlbrum; by p R the prce per unt of hare they are aked and by T R = q R p R the total tranfer to the eller. Budget contrant for unnformed agent tpulate that q R p R K. 4 Thee paper aume that the gnal nvetor receve can be ether good or bad and the tock market value monotonc n the number of good gnal. Wth contnuou gnal, a we aume, our pecfcaton of v preerve th monotoncty property,.e. hgher gnal lead to hgher market value of the tock. Furthermore, our reult are robut to a generalzaton of th functon to a generc v = Ψ() wth Ψ ncreang n. We choe however not to ue th pecfcaton becaue t make notaton and computaton much heaver wthout addng anythng n term of nght. 6

7 Retal nvetor a the eller do not have any prvate nformaton about the market value of the aet and only oberve the denty f() of gnal. In order to extract the nformaton from the nttutonal nvetor, the eller degn a contract pecfyng the quantty and the prce per hare to pay to each nttutonal nvetor and to retal nvetor. By ung the revelaton prncple, we can focu on drect mechanm n whch the frm ak each nvetor to announce h gnal and then fxe the quantty and the prce a a functon of ther announcement n uch a way to nduce them to truthfully reveal ther nformaton (ee Fudenberg and Trole, 1991). We aume that all the hare ued mut be allocated between nttutonal and retal nvetor. Conequently, by choong the vector {q } the frm mplctly determne the number of hare to allocate to retal nvetor, whch gven by: q R =1 X q The contract the eller propoe can thu be wrtten n the followng way: Γ = {p (b, b ),q (b, b ); p R (b, b ),q R (b, b )} for all and all (b, b ) Ω wth b beng nttutonal nvetor 0 announcement and b the vector of the announcement of all the other nvetor. The optmal IPO mechanm for the frm the oluton to the followng optmzaton program: " # X max U F = E T (b, b )+T R (b, b ) q ubject to the tandard contrant (ee Fudenberg and Trole, 1991): Retal Invetor Partcpaton Contrant (RPC): E S [q R (, )(v p R (, )] 0 whch requre ther expected payoff to be larger than ther reervaton utlty whch, wthout lo of generalty, may be et equal to zero. 7

8 Inttutonal Invetor Partcpaton Contrant (IPC): U (b, b )=E {z(q (b, ),v(b, )) q (b, )p (b, )} 0 for all and whch ha the ame meanng a the RPC. Note however that the expected proft of partcpaton for each nformed nvetor condtonal on h gnal. Inttutonal Invetor Incentve Compatblty Contrant (IIC): U (b, ) U (, ) for all, b and, wth U (b, )=E {z(q (b, ),v(, )) q (b, ))p (b, )} whch enure that nttutonal nvetor do not have ncentve to mrepreent ther type - the gnal they receve - to the frm. Lat, the Full Allocaton Contrant (FAC): X q (, )+q R (, )=1, the feablty contrant: q (, ) 0 for all, and and the budget contrant for unnformed nvetor: q R (, )p R (, ) K for all Alo notce that the monotonc hazard rate condton trvally atfed by the unform f ( ) 1 dtrbuton functon,.e. = 1 F ( ) ( ) 0. The next Lemma how how to mplfy the above maxmzaton program. 8

9 Lemma 1 The eller optmal IPO degn problem can be rewrtten n the followng way: ( max v + X ) z(q (),v) 1 {q } n 1 Ω n ( )z 2 (q (),v) vq () f()d.t : U (,)=0 for all 0 1 n E z 12 (q (),v) q () q () 0 for all and all P q () 1 for all. µ 1 P q () p R (, ) K for all and all for all where the econd contrant the monotoncty contrant whch enure that the SOC are met and the mechanm mplementable. Proof: See the Appendx. The eller program n th et-up qute dfferent from a tandard aucton degn problem where an unnformed eller face uually only nformed bdder. The partcpaton to the aucton of a cla of unnformed bdder, make the problem qute dfferent and nteretng becaue t mtgate the advere electon problem and, thu, lower the cot for the eller to extract the nformed nvetor prvate nformaton. 5 Furthermore, notce that prce are preent n the eller program only n the budget contrant of unnformed nvetor. 3 Rk neutral nttutonal nvetor and cah contraned retal nvetor. We tart by analyzng a cae mlar to Benvente and Wlhelm (1990) where nttutonal nvetor are uppoed to be rk neutral and retal nvetor cannot afford to buy the whole quantty of hare,.e. are budget contraned. Notce that th alo the mot common et up n the IPO lterature. Aume that the valuaton functon of nformed nvetor z(q, v) =vq, o that the 5 See Bennour and Falconer (2005) for a careful analy of th ue. 9

10 utlty functon of nvetor u (v,q )=v()q () p ()q () and addtonally, that retal nvetor are ubject to the followng cah contrant p R ()q R () K for all. We aume alo that K o that the budget contrant matter. The eller program become then ( max v X {q } n 1 Ω ) 1 n ( )q () f()d.t : U (,)=0 q () E 0 for all q () 0 for all and all P q () 1 for all. µ 1 P q () p R (, ) K for all and all for all Notce that the eller objectve functon decreang n q (.) for each. Th mple that at the optmum, the eller wll allocate a much a poble to retal nvetor,.e. up to ther budget contrant. The propoton below formalze the reult about the optmal allocaton rule. Propoton 1 Wth rk neutral nttutonal nvetor and cah contraned retal nvetor, the optmal IPO uch that retal nvetor are atfed frt up to ther budget contrant wherea nttutonal nvetor get the remanng hare. So n the optmal mechanm, retal nvetor get prorty n the allocaton (ee alo Ba and Faugeron-Crouzet, 2002). Th reult hghlght the mportant role played by the unnformed nvetor n the IPO proce. Ther preence mtgate the advere electon problem v-a-v the nttutonal nvetor and, conequently, allow the eller to lower the 10

11 cot to extract ther prvate nformaton. The ntuton for th that the eller can alway threaten the nttutonal nvetor to allocate the hare to retal nvetor although n th pecfc tuaton, the effectvene of th threat mght be lmted by the fact that retal nvetor are ubject to cah contrant. For the prcng rule, we know that retal nvetor budget contrant hould be bndng n equlbrum,.e. p R ()q R () =K for all So we now focu on the optmal prcng rule whch, for each nttutonal nvetor mut atfy the followng equaton: p ()q ()f ( )d = Ω Ω ½ v()q () 1 q (e, )de ¾ f ( )d n for all and all (1) and for unnformed nvetor the followng p R ()q R ()f()d = Ω We can how that the next reult hold: Ω v à 1 X! q () f()d = k Propoton 2 Wth rk neutral nttutonal nvetor and cah contraned retal nvetor, the optmal IPO can be mplemented wth a unform prcng rule. Proof: ee the Appendx. The ntuton behnd th reult very mple. Snce aymmetrc nformaton n the model n one-dmenonal, the eller need only one tool to dcrmnate the nformed nvetor. In other word, quantty dcrmnaton (and ratonng) alone uffcent to allow the eller to gather nformaton and thu acheve an optmal performance. For th to be true, however, t mut be that the eller enjoy enough dcreton on how to allocate hare among nvetor. In th pecfc cae, the eller partly lmted on the ue of the allocaton rule becaue retal nvetor are cah contraned, but th offet by the flexblty ganed from the nttutonal nvetor rk neutralty. Th trengthened by the followng Corollary: 11

12 Corollary 1 Wth rk neutral nttutonal nvetor and cah contraned retal nvetor, the unform prce mplementng the optmal IPO not unque. The fact that there are multple unform prce that can mplement the optmal IPO confrm the large leeway the eller ha on how to degn the ale. In concluon, the above reult together ugget that n the preence of cah contrant on retal nvetor, nttutonal nvetor preference become crucal n order to enure the eller enough leeway wth repect to the ue of the allocaton rule and, thu, make unneceary the ue of prce dcrmnaton. We hall ee more n the Secton 4 how cah contrant on retal nvetor and nttutonal nvetor preference nteract to determne the optmalty of a unform prcng rule n IPO. A we know, mot of the IPO lterature devoted to undertand whether and why underprcng occur. Although th not the focu of th paper, our model ha mplcaton for the extence of underprcng a hown n the next propoton. Propoton 3 The optmal IPO mechanm characterzed by underprcng,.e. for all, hare are prced below ther expected value v. Proof: ee the Appendx. Our defnton of underprcng the ame a n Benvente and Spndt (1989) and Rock (1986). The ratonale for the extence of underprcng n our model the ame a n Benvente and Spndt (1989),.e. the eller underprce the hare to lower the cot of elctng nformaton from the nttutonal nvetor. No cah contrant on retal nvetor The cae of rk neutral nttutonal nvetor and no cah contrant partcularly nteretng. Becaue of what we ad before, n th cae the eller ha full dcreton wth repect to the choce of the allocaton rule, the conequence of th tated n the next corollary. Corollary 2 When nformed nvetor are rk neutral and retal nvetor are not cah contraned,.e. they could buy all the hare on ale, then the optmal offerng uch that 12

13 all the hare are old to retal nvetor rrepectve of the gnal reported by the nformed nvetor, at a margnal prce equal to p R = v. The above reult mple that rk neutral nttutonal nvetor would get zero rent n equlbrum becaue the eller would optmally chooe to exclude them from the offerng. In other word, the nttutonal nvetor are alway excluded from the offerng and th drven by the fact that nformed nvetor compete agant unnformed nvetor (Bennour and Falconer, 2005). 4 Rk averon and no cah contrant Although the benchmark IPO n the lterature ha alway condered nttutonal nvetor to be rk neutral, there no reaon to exclude the poblty that even nttutonal nvetor may exhbt ome knd of rk averon. Specfcally, we thnk t not unreaonable that nttutonal nvetor exhbt and averon to the, o called, nventory rk, that the rk aocated to the compoton of ther portfolo. A a matter of fact, th well recognzed by the market mcrotructure lterature or the foregn exchange market lterature (ee for ntance O Hara (1996), Lyon (2003)). Therefore n th ecton we analyze how the optmal IPO and the optmal prcng rule affected by th aumpton. In Secton 3 we have already antcpated that the optmalty of the unform prcng motly depend on the degree of freedom the eller ha wth repect to the allocaton of the hare, thu, ntutvely, we mghtexpectthatthenttutonal nvetor rk averon wll reduce the eller freedom. We hall how that th not necearly the cae. For the tme beng we wll aume ntead that retal nvetor do not bear any cah contrant. The nttutonal nvetor rk averon nterpreted a we ad a an averon to the o called nventory rk, and modeled, mlarly to Benvente and Wlhem (1990), by aumng that ther preference are concave n the quantty,.e. z(q,v)=q (αv δ 2 q ). Our functon z a varant of a tandard Mean-Varance utlty functon n whch the expected value v weghed more than the tranfer T. The parameter α > 1 meaure the regular nvetor aggrevene: the hgher α, the more aggreve the nvetor n the IPO 13

14 procedure. The ntuton that, depte ther rk averon, nttutonal nvetor are very much ntereted n the ale and, therefore, they wll bd aggrevely n the IPO. If α =1 then the problem would be equvalent to havng rk neutral nttutonal nvetor and, n th cae, we know that the optmal IPO uch that all the hare wll be allocated to retal nvetor no matter the nformaton reported by the nttutonal nvetor. 6 The next propoton derve the optmal allocaton rule. For the detal of the proof the reader can refer to Bennour and Falconer (2004) Propoton 4 The optmal IPO characterzed by the followng allocaton rule: there ext a threhold value of the gnal (v ), wth v = P j6= j, uch that, If all the nformed nvetor report gnal below th threhold, all the hare are awarded to the unnformed nvetor; otherwe all the nttutonal nvetor reportng a gnal below (v ) get zero, wherea the other get a potve quantty whch equal to eq = (α 1) nv α ( ), f the ue not overubcrbed, wth retal nvetor nδ recevng the remanng hare; bq < eq f the ue overubcrbed, wth bq () =eq () β()/δ where β()/δ the amount of hare by whch they are ratoned. Retal nvetor get nothng. The nttutonal allocaton rule uch that the optmal quantty eq () ncreang n the gnal reported by the nttutonal nvetor. In other word, the eller reward better nformaton wth a larger quantty. Th n lne wth the extng lterature on bookbuldng (Benvente and Spndt (1989) and Cornell and Goldrech (2001)), whch how that nttuton receve more hare n hot ue,.e. IPO wth trong pre-market demand and alo wth the tradtonal Rock argument accordng to whch nttuton alway try to avod lemon. In a recent paper, Aggarwal et al. (2002) alo provde ome emprcal 6 In addton, the aumpton reflect the practce of IPO underwrter to urround themelve of a core of regular nvetor who repeatedly take part nto IPO (Benvente and Spndt, 1989). 14

15 upport to thee predcton, confrmng that nttutonal allocaton are ndeed larger n better performng IPO. We can alo how that the optmal mechanm obtaned above atfe the SOC a repreented by the monotoncty condton. We omt the proof for the ake of mplcty. 7 Prevou reult n the aucton lterature ((Makn and Rley (1984), Eő (2005)) have hghlghted that ntroducng rk averon for nformed bdder generally create a trade-off becaue, on the one hand, the nformed buyer mut be compenated for the rk they bear n the aucton 8 and the cot of creenng whch mght ntead be reduced by expong them to ome rk. For th reaon, t nteretng to ee what happen to the optmal prce chedule when nformed nvetor are rk avere. We can however how, that depte the rk avere nttutonal nvetor, the optmal IPO can be mplemented by an unform prcng chedule a n the tandard cae. The reult tated n the next propoton. Propoton 5 The optmal IPO mechanm can be mplemented by a unform prce chedule to all nvetor. Proof: (See the Appendx). To gve an dea of the reaonng behnd the proof of th Propoton, frt recall that the optmal prce for nttutonal and retal nvetor mut atfy, repectvely, the followng ntegral equaton p ()q ()f ( )d = Ω and Ω Ω ½ z(q (),v) 1 z 2 (q (e, ),v(e, ))de ¾ f ( )d ; n (2) p R ()q R ()f()d = Ω Ã v 1 X! q () f()d 7 See Bennour and Falconer (2005) for a detaled proof n a more general aucton framework. 8 The rk come from the fact that the payment of each bdder depend on the announcement of the other bdder (Eő, 2005). 15

16 The proof then artculated n two tep. The frt tep cont n provng a prelmnary reult,.e. we how the extence and unquene of a unform prce chedule atfyng Equaton (2) for all nttutonal nvetor. Next, we how that the ame prce chedule alo olve the partcpaton contrant of retal nvetor. The ratonale for the above reult that, although the nttutonal nvetor rk averon may mpoe retrcton on the eller n how to allocate hare, the fact that retal nvetor are ntead not cah contraned warrant hm wth enough leeway. More generally, the reult mple that a long a retal nvetor are not cah contraned, the eller enjoy enough dcreton on how to allocate the hare no matter what nttutonal nvetor preference are. A a conequence, there wll not be any need to ue prce dcrmnaton n combnaton wth quantty dcrmnaton to upport the optmal mechanm. To fully undertand th, n the next Secton we conder the cae n whch nttutonal nvetor are rk avere and retal nvetor are cah contraned. Th the only cae n whch the optmalty of the unform prcng rule break down. Rk averon and cah contrant Let u now aume that retal nvetor are now budget contraned. Notce that for the ake of tractablty, n th cae, we wrte th contrant a a quantty contrant,.e. we aume that retal nvetor can buy a maxmum quantty of hare K<1,.e. 9 : q R () K<1, for all Ω whch mple that retal nvetor cannot buy all the hare on ale. The quantty contrant on retal nvetor modfe the eller optmzaton problem a follow: 9 For the ake of tractablty, we aume quantty contrant ntead of a more general budget contrant. Notce that f anythng th contrant more retrctve than the general cah contrant gven that we loe one degree of freedom (the prce dmenon). 16

17 ( max v + X {q } n 1 Ω µ ) q () (α 1) v δ 2 q () α n ( ) f()d.t : U (,)=0 q () 0 for all and all q () 0 for all and all P q () 1 K for all P q () 1 for all. The oluton to the econd problem gven n the followng theorem. Propoton 6 The optmal IPO characterzed by the followng allocaton rule: there ext a threhold value of the gnal b (v ), wth v = P j6= j, uch that, 1. Informed nvetor reportng the lowet gnal get no hare; 2. All nttutonal nvetor wth uffcenly hgh gnal get a potve quantty eq () = (α 1) nv α ( ) provded that 1 K P eq () 1 wth the remanng hare nδ allotted to retal nvetor. If ntead P eq () < 1 K, then K hare are allocated to retal nvetor and all ntutonal nvetor wth a gnal above the lowet one,, get a potve quantty eq K() 6= eq (); 3. In the cae of overubcrpton, all nttutonal nvetor wth uffcently hgh gnal are ratoned and retal nvetor get nothng. Proof: (See the Appendx). 10 The above reult very ntutve. Whenever retal nvetor quantty contrant ht, the eller forced to allocate nformed nvetor more hare than what would be otherwe optmal, n partcular he forced to allocate ome hare alo to nttutonal nvetor wth relatvely bad gnal (below the threhold). 10 We can alo prove that th mechanm mplementable that t atfed the SOC (monotoncty contrant). 17

18 Gven the optmal allocaton chedule we can derve the optmal prcng rule and by ung the ame argument a n Propoton 5 we can prove the followng reult: Propoton 7 If retal nvetor are ubject to quantty contrant and nttutonal nvetor are rk avere, the optmal ellng mechanm a derved n Propoton 6 can be mplemented only by a dcrmnatory prcng chedule for all nvetor, uch that each cla of nvetor pay the ame prce. The proof very much lke the proof of Propoton 5. We tart by howng that an optmal unform prce ext for nformed nvetor and that th cannot be appled to retal nvetor a well becaue t would volate ther partcpaton contrant. The reult not urprng: whenever the uer face a tuaton n whch the quantty contrant on retal nvetor ht he wll be forced to allocate to nformed nvetor more hare than what would be otherwe optmal for hm to do and th, n turn, requre to adjut downward the hare prce. However, t mportant to notce that nttutonal nvetor all pay the ame prce,o,nomeene,thereextaunform prce for each cla of nvetor. Depte the fact that the ntroducton of budget contrant on retal nvetor weaken our reult on the optmal of unform prcng, t mut be notced that aumng that retal nvetor are not budget contraned not that retrctve for two man reaon. Frt, they mght be cah contraned ndvdually but not n the aggregate (Ellul and Pagano, 2003). Secondly and mot mportantly, t mportant to undertand that t a mplfyng aumpton becaue n practe the eller can regan flexblty n other way, by for ntance wthholdng hare whenever t not optmal to allocate them to nformed nvetor and retal nvetor do not have the capacty to aborb them; or alternatvely by ung a frmcommtment contract of underwrtng, whch a common practe n the US and through whch the underwrter commt to buy all the hare not old durng the offerng. It eay to how that ncorporatng th tuaton n our IPO framework would lead to the ame reult a wth no cah contrant. 18

19 5 Concluon and Remark In th paper, we adopt an optmal aucton approach to analyze the functonng of ntal offerng wth the focu on the choce by the eller of the optmal ntrument to allocate effcently the ue, th choce beng prce veru quantty dcrmnaton (ratonng). Our reult hghlght that the eller doe not need prce dcrmnaton to acheve an optmal performance. Intead, the optmal IPO can alway be mplemented through a unform prce chedule. Th reult hold very generally. In partcular, we how that t hold n the tandard IPO framework wth rk neutral nttutonal nvetor and cah contraned retal nvetor. A deeper nvetgaton how that a a general rule, a long a the eller can fully rely on the allocaton rule,.e. no contrant mpoed to hm on how and to whom to allocate the hare, then unform prcng enough to acheve an optmal performance. The dcreton n the allocaton rule n turn depend on two man dmenon: the extence or not of cah contrant on to retal nvetor and the nttutonal nvetor preference. In partcular, retal nvetor appear to play a very mportant role n the offerng nce, a long a they are able to buy the whole quantty then ratonng alone enough to upport the optmal IPO mechanm. When th not the cae, then nttutonal nvetor preference tart to matter. 11 Th ugget that an mportant emprcal queton to nvetgate whether and to what extent retal nvetor are actually cah contraned epecally n the lght of the recent debate and emprcal evdence whch ndcate dtorton n the allocaton of new ue due to the ntermedare wll to favor nttutonal nvetor at the expene of retal nvetor. Smlarly, the ue of whether nttutonal nvetor may exhbt ome knd of rk averon and the extent of t defntely deerve more emprcal nvetgaton. From a theoretcal pont of vew, the central role of unnformed nvetor n IPO ha been recently tuded, n a very mlar framework, by Malakhov (2004). The author how that the eller revenue 11 Th alo explan the dfference between our reult and the Benvente and Wlhelm one. In ther paper, the underwrter not able to freely raton regular nvetor. He can only apply a dcrete ratonng cheme,.e. ether he fll the full order or he gve nothng. Therefore, prce dcrmnaton needed n addton to ratonng to allow the underwrter to elct the nformaton from the regular nvetor. 19

20 are ncreang n the the number of unnformed nvetor partcpatng nto the offerng, the reaon beng that a larger number of unnformed nvetor lower the outde opton of the nformed one. 6 Appendx ProofofLemma1: By the RPC and the maxmand we know that the eller proft ncreang n the retal nvetor payment, therefore at the optmum he wll make the RPC bndng. We can then rewrte the RPC n the followng way: p R ()q R ()f()d = Ω Ω vq R ()f()d = Ω v à 1 X! q () f()d (3) wherewehaveput =(, ) and replaced to q R () from the FAC. Now, let u conder the IPC whch can be rewrtten a follow: p ()q ()f ( )d Ω = z(q (),v)f ( )d U (, ) 0 Ω for all and. (4) by ntegratng over Ω ( that takng the expectaton over ) we fnd: p ()q ()f()d = z(q (),v)f()d U (, )f ( )d. (5) Ω Ω Ω Let u now conder the IIC. By applyng the envelope theorem to the IIC, we have: U 0 (, )= 1 n Ω z 2 (q (),v)f ( )d (6) thu, ( ) 1 U (, )=U (,)+ z 2 (q (e, ),v())f ( )d de. (7) n Ω For the IPC to be atfed we can mply et U (,)=0whch mean that the eller leave zero rent to the nformed trader wth the lowet evaluaton. 20

21 Applyng Fubn theorem to equaton (7) yeld: U (, )= 1 n Ω ½ z 2 (q (e, ),v())de ¾ f ( )d. Integrate over Ω, and apply agan Fubn theorem to get the followng Ω U (, )f ( )d = 1 n Ω ½ Ω z 2 (q (e, ),v())de f ( )d ¾ f ( )d. Lat, ntegraton by part of the ntegral the followng equaton: RΩ h R z 2(q (e, ),v())de f ( )d, yeld Ω ½ z 2 (q (e, ),v())de f ( )d = ¾ z 2 (q (e, ),v())de (F ( ) 1) whch can be rewrtten a follow: Ω U (, )f ( )d = 1 n where ( ) the nvere of the hazard rate. Ω z 2 (q (),v)(f ( ) 1)d. Ω ( )z 2 (q (),v)f()d. (8) By replacng equaton (8) nto equaton (5) and ung equaton (3) we get the eller objectve functon. Lat, n order to how the prevalence of the monotoncty condton, notce that the (IIC ) may be wrtten a follow arg mn [U (b, b ) U (, b )] for all and. b Wrtng the SOC for th problem and ung equaton (6) gve the reult. ProofofPropoton2 In order to make the reaonng n the proof clear, we plt out the proof n three tep. In the frt tep, we derve condton for the extence of a unform prce for nformed nvetor. In tep 2 we derve condton for the extence of a unform prce for all nvetor. Fnally, 21

22 we how the extence of a mechanm (allocaton and prcng functon) atfyng thee condton. Step 1: The lnear tranfer for the nttutonal nvetor mut atfy equaton (1). Now, conder the followng prce functon p 0 () =v h 1 R n q (e, )de q () (9) for each and each q () uch that q () 6= Any prce atfyng equaton (1) can alo be wrtten a p () =p 0 ()+Φ () wth Φ uch that Φ ()q (, )f ( )d =0, for all and. Ω A unform prce functon ext f and only f t atfe the followng partal dfferental equaton p () = p () j for all and j whch equvalent to p 0 () + Φ () = p0 () j + Φ () j. (10) wth, p 0 () = q () p 0 ()+v q () (11) and p 0 () q () = q () p 0 j ()+v + 1 q () j n q (e, ) de. (12) j Multplyng both de of equaton (10) by q () and from equaton (11) and (12) yeld Φ () j q () 1 R q (e, )de n [q ()] 2 1 n + Φ () = + 1 n q () j R q (e, )de [q ()] 2 1 n q (e, ) de j q () (13) yet 12 Otherwe p 0 () =0. q () 1 R q (e, )de n [q ()] 2 1 n = 1 n "R q # (e, )de q () 22

23 and q () 1 j R q (e, )de n [q ()] 2 1 n q (e, ) de j q () = 1 n j "R q # (e, )de q () Subttuton n equaton (13) gve " R Φ () 1 q # (e, )de n q () " R = Φ () 1 j n q # (e, )de q () Th a partal dfferental equaton n Φ () α n th PDE gven by q (e, )de. A generc oluton for q () Φ () =ϕ( n )+ 1 n R q (e, )de, forall q () where ϕ( ) a twce dfferentable functon defned on the et [n,n] =nω. Summng up, o far we have hown that the optmal mechanm may be mplemented by a unform prce chedule for nformed nvetor f and only f p () = p 0 ()+ϕ( n )+ 1 n = v()+ϕ( n )=v()+ϕ(v) q (e, )de q () (14) where ϕ a twce dfferentable functon atfyng the followng ntegral equaton Ω ϕ( n )+ 1 n q (e, )de q () q (, )f ( )d =0. or alternatvely ϕ( n )q (, )f ( )d = Ω 1 q (e, )de f ( )d = g( ). n Ω The extence of a unform prce functon acro all nformed nvetor then equvalent to the extence of the functon ϕ a a oluton to the ntegral equaton defnednequaton(15). (15) 23

24 Step 2: Let now turn to derve the condton for a unform prce functon for all nvetor. For unnformed agent, the unform prce functon hould atfy ther partcpaton contrant a well a ther budget contrant. Th gve the followng condton à v 1 X! q () f()d = K (16) Ω and à (v()+ϕ(v)) 1 X! q () = K for all (17) Th latter equaton allow to wrte ϕ a follow: ϕ(v) = K (1 P q v for all (18) ()) Therefore, provng the extence of a unform prce functon equvalent to prove the extence of a et of functon (q ()) =1,...,n that the oluton to equaton (16) and ½ K Ω (1 P q ()) v q (, )+ 1 q (e, )de ¾ f ( )d =0 n for all and all (19) Notce that we can re-wrte equaton (19) a a partal dfferental equaton. Equaton (16) mply defne a fnal condton for th partal dfferental equaton. Addtonally, from equaton (17) t mut be that (1 P q ()) alo a functon of v a ϕ, o n what follow we denote by ξ(v) =1 X q () =q R () (20) Step 3: The remanng of the proof cont n howng the extence of functon q and ξ atfyng the uffcent condton for the extence of a oluton to equaton (16), (19) and (20). A trval oluton to equaton (19), after replacng 1 P q () by ξ(v) obtaned by ettng the value of the ntegrand equal to zero n each pont,.e. 13 K ξ(v) v q (, )+ 1 q (e, )de =0 for all and all n 13 Th condton reduce ubtantally the et of poble oluton to our equaton. In a more general et 24

25 rearrangng term yeld the followng dfferental equaton n q (e, )de q (, ) = q (e, )de 1 h n v K ξ(v) and the oluton where, H(v) uch that mple computaton lead to µ ln q (e, )de = H(v)+C(v ) (21) H(v) = 1 h n v K ξ(v) q (, )= L (v ) h exp(h(v)) (22) n v K ξ(v) where L(v )=exp(c(v ) an arbtrary contnuouly dfferentable functon. Now ummng up over and ung the fact that 1 P q () =ξ(v) yeld 1 ξ(v) = 1 h R(v)exp(H(v)) (23) n v K ξ(v) where 14 R(v) = P L (v ). Let u take the dervatve wth repect to v and ue equaton (23) to ubttute [1/n(v K ξ(v) )]R(v)exp(H(v)) by 1 ξ(v) and we fnally get the followng dfferental equaton ξ0 (v) 1 ξ(v) + Kξ 0 (v) ξ(v)[vξ(v) K] = R0 (v) (24) R(v) We now need to determne the oluton of the lat dfferental equaton n ξ n order to contruct the optmal allocaton to nformed nvetor. Note that equaton (24) depend on the arbtrarly choen functon R(.) and t poble to chooe uch functon o that the oluton of equaton (24) atfe equaton (16). Then from the choen functon R we can up we can conder the followng dfferental equaton K ξ(v) v q (, )+ 1 q (e, )de = z (, ) for all and all n where z (.) uch that z (, )f ( )d =0. Snce we need jut to prove the extence of a oluton, Ω and n order to mplfy the preentaton, we retran our analy to the homogenou dfferental equaton. 14 From equaton (23) and the fact that ξ afunctonofv, then R hould alo be a functon of v. 25

26 contruct ndvdual L (v ) for each and fnally the allocaton for each nformed nvetor (q ()) by ung equaton (22). Thee quantte atfy by contructon equaton (19) and equaton (16) [q.d.e.] Proof Propoton 3: From equaton (14), (18) and (20), the unform prcng functon equal to k ξ(v). In order to check for underprcng we jut need to compare v and k ξ(v). From equaton (22), we know that (v k ξ(v) ) and L (v ) have the ame gn. However, form the defnton of L (v ) derved mplctly n equaton (22) we have that L (v ) > 0 and o (v Proof Propoton 4: k ξ(v) ). We conder the relaxed problem,.e. we drop the monotoncty contrant n Lemma?? and check t ex pot. In th way, the objectve functon become an ordnary maxmand wth the contrant defned n each pont and can be maxmzed pontwe on Ω. The amount of hare, q (), the frm mut agn to nvetor n order to elct h nformaton mut olve the followng maxmzaton problem, for each Ω: µ P max q () (α 1) v δ {q } =1,..n 2 q () α n ( ).t : U (,)=0 q () 0 and for all P q () 1 for all and. The Kuhn-Tucker condton of th maxmzaton program are the followng: (α 1)v δq α n ( )+λ () β() =0, for all λ ()q () =0 β()[1 P q ()] = 0. (25) wth λ and β are the Kuhn-Tucker multpler aocated repectvely to the feablty contrant and to the FAC. Now let u denote by H(q, v) our objectve functon, that : 26

27 H(q, v) = X µ q () (α 1) v δ 2 q () α n ( ) wth q [0, 1] and Ω =[, ] n. Th functon concave n q for all, nce 2 H aumpton (2) and (5). Now let conder (v )= α (α 1)v (2α 1) the followng et for all Ω N () = { N (v )} N + () = { N > (v )} q 2 0 by for all and v and defne Wecanealyhowthatforeach N () t mut be that q () = 0. Then, for each N + (), defne the quantty eq a the one olvng the equaton below, (α 1)v δeq () α n ( )=0 By the concavty of functon H (wth repect to q ), eq () trctly potve (for each N + ()). Now, f P N + () eq () 1 the quantty eq () the oluton of our mechanm. 15 P N + () eq () > 1, th cae correpond to a tuaton of overubcrpton of the new hare. The quantty eq () cannot thu be the optmal oluton becaue t volate the FAC, o the optmal mechanm gven by the oluton of the followng ytem of equaton that we denote by bq (): bq ()[(α 1)v δbq () α n ( ) β()] = 0 X bq () =1; β() > 0, N + () whch mple that bq () ether zero or potve and olve the followng equaton (α 1)v δbq () α n ( ) β() =0 15 In th cae, the equaton defnng eq () the FOC of our objectve functon H, nce the Kuhn-Tucker multpler, λ () and β() are both zero. 27

28 Clearly, n th cae, all the hare are allotted to nttutonal nvetor. Retal nvetor get nothng n equlbrum. ProofofPropoton5: Th proof contructed n the ame way a Propoton 2. We dvde the proof n three tep. In the frt tep, we derve condton for the extence of a unform prcng rule among nttutonal nvetor. Then n tep 2, we how extence of a unque prcng rule atfyng thee condton. Fnally, n tep three, we how that the optmal unform prcng rule may be appled to retal nvetor. Step 1: The lnear tranfer for the nttutonal nvetor mut atfy the followng equaton p ()q ()f ( )d = Ω Ω ½ z(q (),v) 1 z 2 (q (e, ),v(e, ))de ¾ f ( )d ; n (26) Then, conder the followng prce functon h R z(q (),v) 1 p 0 n z 2(q (e, ),v(e, ))de () = q () (27) for each and each q () uch that q () 6= Any prce atfyng equaton (26) can be alo wrtten a p () =p 0 ()+Φ () wth Φ atfe the followng equaton Φ ()q (, )f ( )d =0, for all and. Ω After characterzng unform prcng a n Propoton 2 (ee equaton (10)), we can how that the optmal mechanm may be mplemented wth a unform prce f p 0 ()+αv δq () µ q () 16 Otherwe p 0 () =0. q () α q () j n q (e, ) de + j Φ () q () = Φ () q (). (28) j 28

29 We can ealy how from Propoton 4 that the optmal quantte allocated to nttutonal nvetoruchthat q () q () j = α nδ for each and each. Th allow u to re-wrte equaton (28) a follow α p 0 ()+αv δq () q (e, ) de j 1+ + Φ () n δq () q () Replacng p 0 () by t value a defnedbyequaton(26)yeld q (e, ) q (e, )de de α j n [q ()] 2 + q () α nδ + Φ () = Φ () j. = Φ () j. By replacng α nδ by q () q () and after ome mple computaton, we get j Φ () α q (e, )de n q () = j Φ () α q (e, )de n q () + α 2n, (29) for each and each j. Th a partal dfferental equaton n Φ () α n whoe generc oluton gven by Φ () =ϕ( n )+ α n R q (e, )de + α q () 2n, for all q (e, )de q () where ϕ( ) a twce dfferentable functon defned on the et [n,n] =nω. Summng up, o far we have hown that the optmal mechanm may be mplemented by a unform prce chedule f and only f p () =p 0 ()+ϕ( n )+ α n q (e, )de q () + α 2n where ϕ a twce dfferentable functon atfyng the followng ntegral equaton ϕ( n )+ α q (e, )de + α n q () 2n q (, )f ( )d =0. Ω 29

30 or alternatvely Ω α n ϕ( n )q (, )f ( )d = Ω q (e, )de + α q () 2n q (, )f ( )d = g( ). (30) Henceforth, provng the extence of a unform prce chedule equvalent to prove the extence of uch a functon ϕ. STEP 2: We frt how that equaton (30) can be wrtten a a Volterra ntegral equaton of the frt knd. 17 Then, by mply applyng the general properte of th knd of ntegral equaton we can prove the extence and the unquene of a functon ϕ. To do th, we frt need to tranform equaton (30) nto a mple ntegral equaton,.e. wth the upport defned on R. Notce that, nce q (, )=0when 0 (v ), the upport of the ntegral equaton (30) equal to Ω 0 = {( 1, 2,.., 1, +1,.., n ) P j6= j = v > v 0 ( )} where v 0 ( ) defned a the nvere of 0 (v ),.e. v 0 ( )= α (2α 1) α 1 Alo, by ung the defnton of the optmal quantty, the LHS term of equaton (30) equvalent to R Ω 0 ϕ( + v )q (,v )f ( )d. Now, by applyng the Generalzed Change Varable Theorem (GVCT) we can et v = γ ( ) for each, whch fnally mple that γ (Ω 0 )=[v0 ( ); (n 1)] R. 18 A man mplcaton of the GVCT that, there ext a meaure λ defned over [v 0 ( ); (n 1)] uch that ϕ( + v )q (,v )f ( )d = Ω 0 (n 1) v 0 ( ) ϕ( + v )q (,v )λ(dv ). 17 A Volterra ntegral equaton of the the frt knd defnednthefollowngway: τ(x) y 0 f(x, y)h(x, y)dy = g(x) n other word, one of the ntegral extreme mut depend on the varable x. 18 See Dunford and Schwartz (1988, 3rd Ed.), chapter 3, lemma 8, page

31 Lat, by applyng the Radon-Nkodým theorem 19, we are alo able to prove the extence of a denty functon ρ aocated to the meaure λ uch that Ω 0 ϕ( + v )q (,v )f ( )d = (n 1) v 0 ( ) By ung the reult above, the ntegral equaton (30) reduce to (n 1) v 0 ( ) ϕ( + v )q (,v )ρ(v )dv = g( ) ϕ( + v )q (,v )ρ(v )dv. Th a Volterra ntegral equaton of the frt knd whch enure that, a far a the functon g well behaved, a oluton n ϕ alway ext. Th how the extence of a unque unform prcng rule for nttutonal nvetor. We denote n the followng by p I () uch prcng rule. Step 3: To complete the proof we are left to how that the unform prce for nttutonal nvetor alo applcable to retal nvetor. Th equvalent to how that p I () atfe the retal nvetor partcpaton contrant. Notce that the problem relevant only n thoe cae n whch both retal and nttutonal nvetor receve a potve amount of hare at the optmum. We tart by defnng the followng et: Ω = { q =0for all },.e. all the quantty dtrbuted to retal nvetor; Ω\Ω = { q () 6= 0 for at leat one } Recall that, retal nvetor partcpaton contrant wrte a follow à p R ()q R ()f()d = v 1 X! q () f()d Ω Ω Then, provng that the prcng rule unform to all nvetor equvalent to how the extence of a prce functon p R () olvng the above ntegral equaton and uch that: ( p () for all Ω p R () = p I () for all Ω\Ω 19 See, for example, Dunford and Schwartz (1988), chapter 3, theorem 2, page 176 n the thrd edton. 31

32 whch requre that retal nvetor are charged dfferent prce dependng on whether they get the whole quantty or not. The problem then bol down to prove the extence of a prce p () uch that à p ()f()d + p I () 1 X Ω Ω\Ω! q () f()d = Ω v à 1 X! q () f()d or equvalently p ()f()d = Ω Ω Ã v 1 X Then we prove the followng: q ()! à f()d p I () 1 X Ω\Ω! q () f()d (31) Lemma 2 A prce functon p () a defnedaboveextfandonlyftatfe the followng equaton p ()f()d = {v + H(q, v)}f()d Ω Ω p I ()f()d. Ω\Ω (32) where the functon H(q, v) defne the eller payoff at the optmum (ee the proof of Propoton 4). ProofofLemma2: = Suppoe there ext a prce p () whch verfe the partcpaton contrant of the retal nvetor. Th mple that the eller expected payoff at the optmum are gven by p ()f()d + p I ()f()d = {v + H(q, v)}f()d Ω Ω\Ω Ω becaue of the unform prcng rule and the fact that all the hare are alway old. = Conder the condton atfed by the prce to the nttutonal nvetor. Summng over and over we get X p ()q ()f()d = Ω Ω X ½ z(q (),v) 1 z 2 (q (e, ),v(e, ))de ¾ f()d n ung the fact that nttutonal nvetor all pay the ame prce and addng R Ω p I()f()d and R Ω v (1 P q ()) f()d to both de yeld the followng R Ω p()(1 P q ()) f()d R Ω v (1 P q ()) f()d = R Ω p()f()d R Ω {v + H(q, v)}f()d. 32

33 If the unform prce ext then the RHS of the above equaton equal to zero whch mmedately mple that the retal nvetor partcpaton contrant, on the LHS of the equaton, met. Th end the proof of Lemma 2 Notce that the rght hand de of equaton (32)doe not depend the agent gnal and o the ntegral equaton defned over p () ha many oluton. Th prove that the eller could fnd a prcng functon where a unform prcng rule for all nvetor may be appled at the optmal mechanm. Note fnally that the eller may chooe among dfferent prcng rule that may be appled for retal nvetor when belong to Ω. ProofofPropoton6 Let u defne the followng three et: N () = { N (v )} N + () = { N > (v )} Ω + () = { N 6= } and $ + = Card(Ω + ()). The oluton of the relaxed problem very mlar to that of Propoton 4. Pontwe maxmzaton lead to the followng Kuhn-Tucker condton: (α 1)v δq α n ( )+λ () β()+γ() =0, for all λ ()q () =0 β()[1 P q ()] = 0 (33) γ()[1 P q () K] =0 where λ, β and γ are the Kuhn-Tucker multpler aocated to the three remanng contrant. Note that β and γ cannot be dfferent from zero at the ame tme. If we conder that γ =0, then we get the ame problem a n the cae wthout budget contrant and prove pont (1), (2) and (4) of the Propoton. 33

34 P For the thrd pont, uppoe that eq () 1 K. Sonthcaeβ() =0and λ () N + () for all hould be equal to zero otherwe ome hare wll reman unold. In th cae, retal nvetor receve K hare and the optmal allocaton rule for nformed nvetor the one atfyng the followng FOC: (α 1)v δq α n ( )+γ() =0 for all Ω + () wth γ() olvng the equaton below: γ() = δ(1 K) $ + + α n α n$ + X j Ω + () j (α 1)v By replacng γ() nto the prevou FOC, we fnally obtan that the optmal quantty to nformed nvetor equal to eq K () = 1 δ$ + δ(1 K)+α n ($+ 1) α n X j Ω + (), j6= j 34

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