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1 Cty Research Onlne Cty, Unversty of London Insttutonal Repostory Ctaton: Bennour, M. & Falconer, S. (2008). The Optmalty of Unform Prcng n IPOs: An Optmal Aucton Approach. Revew of Fnance, 12(4), pp do: /rof/rfn006 Ths s the accepted verson of the paper. Ths verson of the publcaton may dffer from the fnal publshed verson. Permanent repostory lnk: Lnk to publshed verson: Copyrght and reuse: Cty Research Onlne ams to make research outputs of Cty, Unversty of London avalable to a wder audence. Copyrght and Moral Rghts reman wth the author(s) and/or copyrght holders. URLs from Cty Research Onlne may be freely dstrbuted and lnked to. Cty Research Onlne: publcatons@cty.ac.uk

2 The Optmalty of Unform Prcng n IPOs: An Optmal Aucton Approach Moez BENNOURI HEC Montréal Sona FALCONIERI Brunel Unversty December 12, 2007 Abstract Ths paper uses an optmal aucton approach to nvestgate the condtons under whch unform prcng n IPOs s optmal. We show that the optmalty of a unform prce n IPOs depends crucally on whether the (optmal) allocaton rule s restrcted. These restrctons may stem from the retal nvestors budget constrant and/or from the nsttutonal nvestors preferences. We show that the man determnant of the optmalty of a unform prcng rule s the exstence and the shape of the retal nvestors budget constrant. In contrast, nsttutonal nvestors preferences are shown to manly a ect the optmal allocaton rule. Keywords: Intal Publc O erng, Prce Dscrmnaton, Ratonng, Optmal Aucton. JEL Class caton Numbers: D8, G2. We are very grateful to the edtor Marco Pagano and two anonymous referees for ther nsghtful comments. We also thank Robert Clark, Franços Derren, Paolo Fulgher, Davd Martmort and Jean Charles Rochet for helpful comments. We gratefully acknowledge nancal support from the Drecton de Recherche of HEC Montréal and the Intatve of the New Economy (INE) program of SSHRC (Canada).

3 1 Introducton The lterature on Intal Publc O erngs (IPOs) has grown remarkably over the past few decades. Most of ths lterature, both theoretcal (e.g. Rock, 1986, Allen and Faulhaber, 1989, and Benvenste and Spndt, 1989) and emprcal (e.g. Beatty and Rtter, 1986, Rtter 1987, and Cornell and Goldrech, 2001), has focused on explanng a number of apparent market anomales surroundng IPOs such as underprcng, long-term underperformance, and hot ssues markets. 1 Much less attenton has been devoted to understand whether the current IPO format, whch mposes unform prcng, s actually e cent. Frms that go publc are currently forbdden from usng prce dscrmnaton; they can however quantty dscrmnate by choosng an allocaton rule and potentally ratonng some nvestors. Vrtually all papers n the IPO lterature, to be consstent wth current practce, smply assume unform prcng. The only paper addressng ths ssue (Benvenste and Wlhelm, 1990) challenges the e cency of the current regulatory constrants, and suggests that rms could mprove on ther IPO performance (rase the IPO proceeds) f they were allowed to prce dscrmnate across nvestors. The objectve of ths paper s to nvestgate the condtons under whch unform prcng s optmal, where optmalty s de ned from the pont of vew of an ssuer who wants to maxmze the sale s expected proceeds. Our results show that the optmalty of unform prcng depends on whether the (optmal) allocaton rule s subject to certan restrctons. If ths s the case, then dscrmnatory prcng may be requred to elct nformaton from the nformed nvestor. Otherwse (.e. f the ssuer has enough control over the allocaton rule), quantty dscrmnaton alone su ces to acheve optmalty. We consder two types of restrctons on the allocaton rule. The rst are drect and exogenous allocaton constrants. The second type of restrctons endogenously result from assumng non-lnear preferences of nsttutonal nvestors. Our results ndcate that only allocaton constrants crtcally matter for the optmalty of unform prcng. We also show that the optmalty of unform prcng depends on the exstence of an allocaton constrant as well as on the shape of ths constrant, snce ths wll de ne the extent to whch 1 Rtter and Welch (2002) provde a concse and useful revew of the IPO lterature. For a fully comprehensve revew of the theory and evdence on IPO actvty see Jenknson and Ljungqvst (2001). 2

4 the allocaton rule s restrcted. On the other hand, nsttutonal nvestors preferences only a ect the optmal allocaton rule. Spec cally, we show that wth rsk-neutral preferences, the optmal allocaton rule gves prorty to retal nvestors, wth nsttutonal nvestors beng resdual clamants. The opposte holds when preferences are non-lnear. In our IPO a rm wants to place a xed number of unseasoned shares. There are two potental groups of nvestors: n nsttutonal nvestors and a contnuum of retal nvestors. Each nsttutonal nvestor receves a prvate sgnal about the value of the asset for sale, whereas both the ssuer and the retal nvestors are unnformed. The seller desgns the IPO mechansm n order to elct prvate nformaton from nsttutonal nvestors and maxmze the expected IPO proceeds. We apply an optmal aucton approach to derve the optmal IPO, whch conssts n dentfyng the optmal allocaton and prcng rule. We solve two d erent models. In the rst, nsttutonal nvestors are assumed to be rsk neutral. Ths s n lne wth most of the IPO lterature. In the second, nsttutonal nvestors are assumed to have preferences that are concave n quantty but may place a hgher valuaton on the shares than do retal nvestors. The non-lnear preferences case represents the man novelty of the paper. Each of these models s then solved under three d erent assumptons on the allocaton constrant: () no allocaton constrant; () a quantty constrant,.e. a maxmum (mnmum) amount of share s to be allocated to retal (nsttutonal) nvestors; and () a cash constrant,.e. retal nvestors only have a maxmum amount of cash that they can spend n the IPO. 2 We show that the optmal IPO cannot be mplemented wth a unform prcng rule under the second assumpton (.e. when retal nvestors are quantty constraned). In the other cases, we show that unform prcng s optmal n equlbrum. Addtonally, our results mply that underprcng always occurs when dscrmnatory prcng s needed to acheve optmalty. Conversely, ths need not be the case when unform prcng s optmal. The explanaton behnd these results s that quantty constrants mpose tghter restrctons on the seller s ablty to use quantty dscrmnaton as a tool to elct nformaton from nsttutonal nvestors, whch 2 We do not report the results for the case of non-lnear preferences and cash constrants snce due to complexty of the techncal problem, we cannot produce a closed form soluton. 3

5 renders the complementary use of prce dscrmnaton necessary to acheve optmalty. Another contrbuton of the paper s the full characterzaton of the optmal allocaton rule n all of the d erent envronments consdered, wth nterestng mplcatons for the use of ratonng n equlbrum. The only other paper n the IPO lterature that has dealt wth the ssue of optmal prcng s Benvenste and Wlhelm (1990). However, our paper d ers from thers n many respects, the most mportant of whch s the methodology used. We apply optmal aucton theory, whch enables us to characterze the optmal IPO mechansm n a very general envronment wth respect to both the nformatonal structure and nvestors characterstcs. The paper s also closely related to Maksmovc and Pchler (2006), hereafter MP. Although the focus of ther paper s not on optmal prcng rule, they show that the exstence and the sze of underprcng depends on the exstence of allocaton constrants. In ths paper, we show that allocaton constrants also a ect the optmal prcng rule and we are able to say somethng about the lnk between prcng rules and underprcng. Fnally, we extend ther model on three dmensons ) we consder a contnuous state space whle MP s a dscrete model; ) we assume non-lnear preferences of nsttutonal nvestors, whch mples that, n our model, allocaton constrants may e ectvely arse endogenously whereas n MP they are exogenously mposed and nformed nvestors are rsk neutral; and ) we also look at cash constrants, whle they only consder quantty constrants. Our work yelds a number of emprcal predctons. The most drect mplcaton s that unform prcng and dscrmnatory prcng wll lead to the same IPO proceeds f the constrants on the allocaton rule are not too tght. As dscrmnatory prcng s forbdden n most countres, a drect test of ths predcton s only possble n the small number of countres where dscrmnatory prcng has been used n IPOs. For example, Jagannathan and Sherman (2006) document the use of dscrmnatory IPO auctons n Tawan and Japan. We should observe greater underprcng n IPOs when there are tghter restrctons on 4

6 the allocaton rule, e.g. the requrement that a mnmum quantty of shares be allocated to nsttutonal nvestors. 3 Ths s consstent wth the results of MP. Quantty constrants would result n larger underprcng than cash constrants. Ths rases the ssue of understandng whch of these two types of constrants s more relevant n practce, whch s another nterestng emprcal research queston. The paper s organzed as follows. In the next secton, we set up the model and derve the sellers maxmzaton problem for generc nsttutonal nvestor preferences wth no constrants on the retal nvestors. The followng sectons then solve the model under d erent assumptons regardng nsttutonal nvestors preferences and allocaton constrants. Secton 3 analyzes the case of lnear preferences wth and wthout an allocaton constrant. In Secton 4 the same analyss s conducted n the case of non lnear preferences. Fnally, Secton 5 nvestgates the mpact of cash constrants on retal nvestors wth lnear preferences of nsttutonal nvestors. The last secton concludes wth a dscusson of the results. All proofs are n the Appendx. 2 The Model A rm would lke to sell Q shares n an IPO, wth Q xed and, wthout loss of generalty, normalzed to 1. An ntermedary s n charge of marketng the new shares. He s assumed to act n the rm s best nterest. Hereafter, we wll smply refer to the seller to denote the rm-ntermedary coalton. Ths s a standard way of modellng the role of an ntermedary n the IPO lterature. The seller wshes to maxmze the proceeds from the sale. He faces both n(> 2) large nsttutonal nvestors who hold prvate nformaton about the rm s market value, and a frnge of retal nvestors, who are unnformed. Insttutonal nvestors have prvate nformaton n that each agent receves a sgnal s about the market value of the new shares. Sgnals are..d. accordng to a unform dstrbuton 3 Ths practce s supported by theoretcal arguments (Stoughton and echner, 1998) n that the ssuer wants to have a mnmum nsttutonal ownershp because nsttutonal nvestors tend to montor the rm more closely. The emprcal relevance s more d cult to prove, although t has been, for nstance, observed n some OpenIPOs run by WR Hambrecht. (We thank the referee for provdng us wth ths useful pece of nformaton). 5

7 de ned on = [s; s] wth s > 0; so the cumulatve dstrbuton functon s F (s ) = s s s ; wth s = s s, and the densty functon f (s ) = 1 : Let us also denote by f(s) the jont densty s functon so that f(s) = f(s 1 ; :::; s n ) = Q f (s ); wth s = (s 1 ; :::; s n ) 2 = N = [s; s] n. 4 Each sgnal receved by an nsttutonal nvestor represents a pece of nformaton about the market value of the new shares. We therefore assume that the value of the new shares, v; s a functon of the vector of sgnals receved by nsttutonal nvestors. More precsely, we assume that v() s a functon de ned over and s gven by the average of all the sgnals, that s The above functon has two man each sgnal, @s j v(s) = 1 X s : (1) n > 0;.e. the asset value s ncreasng n for any 6= j;.e. sgnals are equally weghted n the valuaton functon. Ths knd of nformatonal structure s very common n aucton theory (e.g. Bulow and Klemperer, 1996 and 2002) and s a straghtforward generalzaton of the smple bnomal nformatonal structure adopted n other IPO papers (e.g. Benvenste and Wlhelm, 1990; Bas and Faugeron-Crouzet, 2002) to a contnuous sgnal space. 5 It can furthermore be proved that our results hold for any generc valuaton functon v(s) = (s) whch sats es propertes a) and b) above a smple spec caton. > 0 j for any 6= j). For the sake of tractablty, we prefer Gven a vector of sgnals s; each nsttutonal nvestor s preferences are gven by the followng utlty functon: u (p ; q ; v(s)) = z(q ; v(s)) q p for all 2 f1; 2; ::::ng; (2) where q s the quantty assgned to nvestor and p s the prce per share he has to pay. We denote by T = p q the total payment from nvestor to the seller. The utlty functon 4 Note that the unform dstrbuton of prvate sgnals sats es the ncreasng hazard rate assumpton. Indeed, we have that for the unform f (s ) 1 0 for all and all s : We 1 F (s (s s ) later n the paper that several of our results (n partcular those n Secton 3) do not change qualtatvely f we consder a more general dstrbuton of sgnals satsfyng the ncreasng hazard rate assumpton. 5 These papers assume that the sgnals nvestors receve can be ether good or bad and the stock market value s monotonc n the number of good sgnals. We make the same assumpton, but we use a contnuum of sgnals. 6

8 n Equaton (2) s lnear n the transfer T : We make the followng assumptons about the nsttutonal nvestors valuaton functon z (the subscrpts denote dervatves wth respect to varables): A1 z 1 > 0 and z 2 > 0; A2 z 11 0; A3 z(0; v) = 0 for all v; A4 z 12 > 0 (sngle-crossng condton); A5 z and z 112 0; A6 z 12 (0; v) 1: 6 We assume a contnuum of compettve, unnformed, and rsk-neutral retal bdders. The total mass of these retal bdders s normalzed to one. In the followng sectons, we wll consder the possblty that these nvestors face an allocaton constrant, wth the constrant takng a number of d erent forms. Nether retal nvestors nor the seller hold any prvate nformaton about the market value of the asset and only observe the densty f() of sgnals. In order to extract the nformaton from the nsttutonal nvestors, the seller desgns a mechansm specfyng the allocaton and prcng rules for both nsttutonal and retal nvestors. By usng the revelaton prncple, we can focus on drect mechansms n whch the seller asks each nsttutonal nvestor to announce hs sgnal and then xes quanttes and prces as functon of ther announcements n such a way as to nduce them to reveal ther nformaton truthfully (see Fudenberg and Trole, 1991). A mechansm s descrbed by a par of outcome functons (p; q) of the form p :! R n+1 and q :! [0; 1] n+1 where p = (p 1 ; :::; p n ; p R ) s a vector of prces and q = (q 1 ; :::; q n ; q R ) s a vector of allocatons. Thus f s s the vector of sgnals announced by nsttutonal nvestors, each nvestor 2 f1; :::; ng receves q (s) shares and pays a prce per share of p (s), whle retal 6 Most of these assumptons are standard n the mechansm desgn lterature. See Fudenberg and Trole (1991) for a dscusson. 7

9 nvestors receve q R (s) and pay a prce per share of p R (s). We assume that all of the shares ssued must be allocated to ether nsttutonal or retal nvestors. The seller s choce of the vector fq g =1;::n mplctly determnes the number of shares allocated to retal nvestors, q R, whch s gven by: X q R = 1 q : (3) The set of possble strateges for the nsttutonal nvestor wth sgnal s s : Faced wth a mechansm (p; q); hs expected utlty f he msrepresents hs sgnal by announcng bs to the seller rather than hs true sgnal s s U (bs ; s ) = E s fu (p (bs ; s ); q (bs ; s ); v(s ; s ))g = [z(q (bs ; s ); v(s ; s )) q (bs ; s )p (bs ; s )] f (s )ds ; (4) where s s the vector of all of the other nsttutonal nvestors sgnals,.e. s = (s 1 ; ::; s f (s ) = Y j6=f j (s j ): 7 1 ; s +1 ; :::; s n ) and The optmal IPO mechansm for the seller s the soluton to the followng optmzaton program: max (p;q) U F = E s " X # T (s) + T R (s) =! X T (s) + T R (s) f(s)ds ; (5) subject to the followng standard constrants: Retal Investors Partcpaton Constrant (RPC). Ths requres the expected payo for retal nvestors to be greater than ther reservaton utlty, whch equals zero. E s [q R (s)(v(s) p R (s))] = q R (s) [v(s) p R (s)] f(s)ds 0: (RPC) Thus the constrant s condtonal on the ntal dstrbuton of the sgnals whch mples that retal nvestors commt to buyng the share wthout ever learnng the reported sgnals. Ths n turn mples that they do not play strategcally n the IPO game. 7 Note that the expected utlty of agent depends on the mechansm o ered by the seller (p; q); whch s omtted n our notaton for the sake of smplcty. 8

10 Insttutonal Investors Partcpaton Constrant (IPC). The IPC ensures that each nsttutonal nvestor s wllng to partcpate n the o erng condtonal on hs own sgnal. The expected utlty of each nsttutonal nvestor, condtonal on hs sgnal, should be greater than hs expected utlty when he does not partcpate n the IPO. The IPC s then wrtten as follows: U (s ; s ) = E s fz(q (s); v(s)) q (s)p (s)g = [z(q (s); v(s)) p (s)q (s)] f (s )ds 0: (IPC) Ths must be sats ed for all and all s : Insttutonal Investors Incentve Compatblty Constrant (IIC). Ths constrant ensures that each nsttutonal nvestor has no ncentve to msrepresent hs type - the sgnal he receves - to the rm. The IIC then requres that each agent be better o by truthfully announcng hs sgnal. Usng Equaton (4) ths may be wrtten as follows: U (bs ; s ) U (s ; s ) for all s ; bs and ; (IIC) or, equvalently, Full Allocaton Constrant (FAC). s 2 arg max bs U (bs ; s ) for all s ; bs and : (IICa) X q (s) + q R (s) = 1 for all s ; (FAC) and the quantty non-negatvty constrants: q (s) 0 for all and s: (6) Fnally, we wll also ntroduce the approprate allocaton constrant. The above optmzaton program can be smpl ed by re-arrangng the constrants. In the Appendx, we show that the seller s optmzaton program can be wrtten as follows 9

11 8 >< >: R max v(s) + P fq g n z(q (s); v(s)) 1 s:t : () U (s; s) = 0 for all 1 () E n s z 12 (q (s); 1 n (s s )z 2 (q (s); v(s)) v(s)q (s) f(s)ds 0 () q (s) 0 for all and all s (v) P q (s) 1 for all s: for all and all s (P) Note that ths program depends only on quanttes. Once optmal quanttes have been determned, optmal prces can be obtaned from the partcpaton constrants of both the nsttutonal and the retal nvestors. For all s and all ; prces for nsttutonal nvestors satsfy p (s)q (s)f (s )ds = z(q (s); v(s))f (s )ds 1 s z 2 (q (es ; s ); v(s))des f (s )ds : (7) n Lkewse, for retal nvestors, the optmal prcng rule must satsfy the (bndng) partcpaton constrant p R (s)q R (s)f(s)ds = Equatons (7) and (8) above wll typcally admt multple solutons. s v(s)q R (s)f(s)ds: (8) We are nterested n verfyng whether there exsts at least one equlbrum requrng a unform prce for all nvestors,.e. a prce functon p(s) such that p(s) = p (s) = p R (s) for all and all s: It s worth notcng that the seller s program n ths setup s qute d erent from that n a standard aucton desgn problem where an unnformed seller faces usually only nformed bdders. The partcpaton of a class of unnformed bdders n the aucton makes the problem rather d erent and nterestng n the sense that t mtgates the adverse-selecton problem vs-à-vs the nformed nvestors and, thus, lowers the seller s cost of extractng ther prvate nformaton. 8 What really matters, however, s not that retal nvestors are unnformed but 8 In a very smlar framework, Malakhov (2006) nvestgates the mpact of retal nvestors partcpaton on the ssuer s revenues. It s shown that the seller s revenues are ncreasng n the number of unnformed nvestors partcpatng n the o erng, snce more unnformed nvestors lowers the outsde opton of nformed nvestors and, consequently, as n our model, reduces the cost of gatherng nformaton. 10

12 rather that ther nformaton, f any, d ers from that of the nsttutonal nvestors and, more mportantly, that they do not play strategcally,.e. t s mpossble to elct ther nformaton. Ths pont has prevously been made by Maksmovc and Pchler (2006). 3 Rsk-neutral nsttutonal nvestors We start by analyzng the standard case n the IPO lterature of rsk-neutral nsttutonal nvestors. We thus assume that the nformed nvestors valuaton functon s z(q; v(s)) = v(s)q ; (9) and the utlty functon of nvestor s u (p ; q ; v(s)) = [v(s) p ] q : (10) We also assume that the allocaton rule s restrcted. We model ths restrcton as a quantty constrant on retal nvestors,.e. there exsts a maxmum quantty of shares K < 1 they can buy. In other words, q R (s) K: (11) Notce that, n our model where the number of shares ssued s xed, ths constrant s equvalent to requrng that a mnmum quantty of shares be allocated to nsttutonal nvestors. In other words we could just as well wrte t as X q (s) 1 K; whch represents a qute common, though mplct, practce n IPOs. There may be several reasons why ths s the case: due to the montorng role played by nsttutonal shareholders whch potentally enhances the rm value (Mello and Parsons, 1998, Stoughton and echner, 1998), or because of the tght lnks wth the underwrter who tends n turn to favor hs nsttutonal clentele over retal demand n IPOs (Aggarwal, 2003; Aggarwal et al., 2002). In ths case the seller s problem s the followng: 11

13 max v(s) fq g n 1 P 1 n (s s )q (s) f(s)ds s:t : () U (s; s) = 0 for (s) () E s 0 for all and all () q (s) 0 for all and all s (v) P q (s) 1 for all s (v) q R (s) K for all s: (P1 ) In the next proposton we characterze the optmal IPO mechansm n terms of allocaton and prcng rules: PROPOSITION 1 If nsttutonal nvestors are rsk neutral and retal nvestors can buy at most K shares, the optmal IPO s characterzed as follows: 1. (Allocaton rule) For all s 2 ; let s m = maxfs 1 ; :::; s n g: In equlbrum, the ssuer allocates as many shares as possble to retal nvestors and the remanng shares to the nsttutonal nvestor reportng s m, that s q R (s) = K and q m (s) = 1 K (Prcng rule) The optmal IPO requres dscrmnatory prcng such that p m (s) < p R (s) = v(s): In the Appendx, the above Proposton s proved for a generc dstrbuton of prvate sgnals satsfyng the ncreasng hazard rate assumpton. To understand the logc behnd the above result notce that the seller s objectve functon P 1 s decreasng n the term n (s s )q (s) whch represents the nsttutonal nvestors nformaton rents and can be easly shown to be ncreasng n the quantty q (s) and n the sgnal s. These propertes together ensure that the seller optmally allocates as much as possble to retal nvestors,.e. up to ther quantty constrant, wth any resdual quantty gong to the 9 Note that as a result of the contnuous dstrbutons we consder n our model, the probablty of havng more than one agent announcng s m s zero. 12

14 nsttutonal nvestor wth the hghest reported sgnal who then receves a strctly postve nformaton rent. 10 To see why he also pays a lower prce than do retal nvestors, smply note that wth lnear preferences, after replacng the optmal quantty, the ICC becomes [v(s) p m (s)](prob s > s )q m (s) [v(s) p (bs ; s )](prob bs > s )q m (s) gven that q m (s) > 0 and v(s) = p R (s); t must be that p R (s) > p m (s): 1: In the absence of an allocaton constrant, the result below follows drectly from Proposton COROLLARY 1 When nsttutonal nvestors are rsk neutral and retal nvestors are not constraned, the optmal o erng s such that all the shares are sold to retal nvestors rrespectve of the sgnals reported by the nformed nvestors, at a margnal prce of p R = v(s). Ths result s a generalzaton of MP to a contnuous state space and mples that rskneutral nsttutonal nvestors receve no nformaton rents because, n equlbrum, they are actually excluded from the o erng. These results can be shown to hold for all dstrbutons satsfyng the ncreasng hazard rate condton. Wth respect to ths secton s results, t s mportant to note that retal nvestors have prorty n the allocaton. Ths hghlghts the mportant role that unnformed nvestors play n the IPO process. Ther presence mtgates the adverse-selecton problem vs-a-vs the nsttutonal nvestors, and consequently allows the seller to lower the cost of elctng ther prvate nformaton by explotng the competton between the two groups of nvestors. If retal nvestors have unlmted buyng capacty, then the cost of nformaton gatherng s zero (Corollary 1). 11 Ths n turn mples that the exstence of an allocaton constrant s n fact detrmental to the e cency of the sellng mechansm. In realty, however, we never observe an IPO n whch all of the shares are placed wth retal nvestors. In the lght of ths emprcal regularty, our results then suggest that: a) the 10 It should be clear that the assumpton of an ncreasng hazard rate s crucal for ths result because t ensures that nformaton rents ncrease n sgnals. D erent assumptons regardng the behavour of the hazard rate would of course a ect the optmal allocaton rule. 11 The role of retal nvestors has been already stressed by Benvenste et al. (1996), Bas and Faugeron-Crouzet (2002), Maksmovc and Pchler (2006) and Bennour and Falconer (2006). 13

15 most relevant emprcal stuaton s that of maxmum (mnmum) quantty constrants on retal (nsttutonal) nvestors; or b) nsttutonal nvestors preferences may not be lnear. In the next secton, we look at the mpact of assumng that nsttutonal nvestors have non-lnear preferences on our results. 4 Non-Lnear Preferences The prevous secton suggests that the optmalty of unform prcng essentally depends on whether the seller can freely use the allocaton rule to dscrmnate among nvestors. It s then natural to ask whether ntroducng rsk averson or some knd of concavty n nsttutonal nvestors preferences may also a ect optmal prcng, to the extent that ths reduces the seller s dscreton n allocatng shares. We therefore relax the rsk-neutralty assumpton n ths secton, and ntroduce concavty n the nsttutonal nvestors valuaton functon, whch becomes: 12 z(q ; v(s)) = q (v(s) 2 q ): (12) When 1; ths new valuaton functon z(; ) s concave n quantty, sats es Assumptons A1-A6, and produces a margnal valuaton whch s decreasng at an exogenously gven rate, > 0: 13 We consder ths spec caton of the utlty functon because concavty ndrectly restrcts the dscreton of the seller to allocate the shares and, as such, t may a ect the optmalty of unform prcng. Further, concavty n quantty may be nterpreted as averson to nventory rsk,.e. the rsk assocated wth portfolo composton. 14 We also assume > 1; whch means that nsttutonal nvestors value the shares more than do retal nvestors for small quanttes, but wth a decreasng margnal valuaton. That s, at q = 0; ther margnal valuaton s v whereas the margnal valuaton of retal nvestors s v: In other words, nsttutonal nvestors are very keen to partcpate n the IPO and obtan a postve quantty of shares, but ther margnal utlty decreases as the allocaton ncreases, possbly due to nventory rsk. If there 12 Rsk neutralty s the most common assumpton n the IPO lterature. Other papers that have, lke us, assumed non-lnear preferences are Stoughton and echner (1998) and Benvenste and Wlhelm (1997). 13 Notce that the new valuaton functon does not descrbe standard rsk-averse preferences, whch are typcally concave n wealth. 14 Ths s well documented for nstance n the market-mcrostructure or foregn-exchange market lteratures (see for nstance O Hara, 1995, and Lyons, 2003). 14

16 s no cash constrant on retal nvestors, the optmal allocaton rule wth = 1 s the same as that under rsk neutralty,.e. the seller allocates the entre quantty to retal nvestors. In ths case, nsttutonal nvestors preferences do not matter. We rst analyze, n the next secton, the optmal mechansm n the absence of constrants on retal nvestors n order to solate the mpact of nsttutonal nvestors non-lnear preferences. We subsequently add the allocaton constrant. 4.1 NO CONSTRAINTS ON RETAIL INVESTORS The seller s optmzaton program n ths case can be wrtten as: max v(s) + P fq g n q (s) ( 1) v(s) 1 s:t : () U (s; s) = (s) () E s 2 q (s) for all and all s () q (s) 0 for all and all s (v) P q (s) 1 for all s: n (s s ) f(s)ds (P2 ) Thus, ( P2) s dentcal to ( P1) except for the d erent utlty functon and the absence of condton (v). The next proposton descrbes the optmal sellng mechansm. PROPOSITION 2 Assume that nsttutonal nvestors have non-lnear preferences descrbed by Equaton (12) and that there s no allocaton constrant. The optmal IPO s characterzed by the followng allocaton and prcng rules: 1. (Allocaton Rule) For all and all s; let eq () be such that ( ( 1) nv(s) (s s ) eq (s) = for all s s n 0 otherwse where s = s ( 1)v 2 1 ; wth v = P j6= s j then 15

17 If X eq 1; only the nsttutonal nvestors reportng a sgnal above s receve a postve quantty equal to eq (s). Retal nvestors receve the remanng shares; If nstead X eq > 1 (oversubscrpton); the nsttutonal nvestors reportng a sgnal above s receve a postve quantty bq (s) = eq (s) they are ratoned. 15 Retal nvestors obtan no shares. Q, where Q s the amount of shares by whch 2. (Prcng Rule) The optmal IPO mechansm can be mplemented by a unform prcng rule wth p v(s). If X eq 1; then p = v(s): If X eq > 1; then p > v(s): Contrary to the case of rsk neutralty, the nsttutonal nvestors now have prorty n the allocaton rule. The result stems not from the non-lnear preferences of the nsttutonal nvestors, but rather from assumng that ther margnal valuaton of the asset at q = 0 s larger than the margnal valuaton of the retal nvestors. Notce that the threshold s s decreasng n the parameter whch mples that as the nsttutonal nvestors margnal valuaton of the asset at q = 0 ncreases, a larger fracton of them receve a postve number of shares. Furthermore, the optmal quantty eq (s) s decreasng n and ncreasng n the sgnal s reported by the nsttutonal nvestor. In other words, the seller rewards better nformaton about the stock value (.e. hgher sgnals) wth a larger quantty of shares. In concluson, the results of Proposton 2 crucally depend on the assumpton of > 1: Indeed, as prevously noted, wth = 1; non-lnear preferences would lead to the same results as n the case of rsk neutralty. The prcng rule s due to the fact that n ths model where there s no allocaton constrant, all the shares can always be sold to retal nvestors at a prce p = v(s); so that the optmal prce need not be set below ths mnmum value. The next secton analyzes the case of non-lnear preferences wth the addton of an allocaton constrant smlar to the one consdered n Secton Spec cally, as we show n the Proof n the Appendx, Q = (s) assocated to the constrant (v) of (P2 ). where (s) s the Kuhn-Tucker multpler 16

18 4.2 QUANTITY CONSTRAINTS We now ntroduce a quantty constrant as de ned n Equaton (11). The seller s optmzaton problem then becomes max v(s) + P fq g n q (s) ( 1) v(s) 1 s:t : () U (s; s) = (s) () E s 2 q (s) for all and all s () q (s) 0 for all and all s (v) P q (s) 1 for all s (v) P q (s) 1 K for all s: n (s s ) f(s)ds (P3) Problem ( P3) s the same as ( P2) wth the addton of constrant (v). mechansm s then descrbed by the followng proposton: The optmal PROPOSITION 3 When nsttutonal nvestors have non-lnear preferences and retal nvestors are quantty constraned, the optmal IPO s characterzed by the followng allocaton and prcng rules: 1. (Allocaton Rule) there exsts a threshold value of the sgnals s = s ( 1)v 2 1 ; wth v = P j6= s j ; such that, If 1 K P eq (s) 1; all the nsttutonal nvestors reportng a sgnal above the threshold s obtan a postve quantty eq (s) = ( 1) nv(s) (s s ). The n remanng shares are allocated to retal nvestors. If P eq (s) < 1 K; then retal nvestors receve K shares; the remanng 1 K shares are allocated to all the nsttutonal nvestors reportng a sgnal s > s K wth s K < s : 17

19 If P eq (s) > 1 (oversubscrpton), the nsttutonal nvestors wth sgnals above the threshold s obtan a quantty bq < eq (ratonng). The retal nvestors receve no shares.. 2. (Prcng Rule) The optmal prcng rule s such that If the allocaton constrant s not bndng, the optmal IPO can be mplemented by unform prcng; Conversely, f the constrant s bndng, then a dscrmnatory prcng rule s optmal, such that: p (v) = p I < v(s) for all and p R = v(s): The allocaton rule result s qute ntutve. When there s a restrcton on the allocaton rule, the seller may be forced to allocate to the nsttutonal nvestors more shares than would otherwse be optmal. Spec cally, n order to place all the shares, the ssuer wll have to allocate a postve amount of shares to some nsttutonal nvestors reportng a sgnal below the threshold s. As for the prcng rule, n the presence of an allocaton constrant, whether the optmal prce s dscrmnatory or unform depends on whether the constrant s bndng or not n equlbrum, whch n turn depends on the value of the parameters and : For su cently hgh values of and/or low values of the allocaton constrant wll not be bndng, and thus, we are back to the case of no constrants where a unform prce s optmal. An alternatve readng of ths result s that f parameter K were to be determned endogenously, t would be optmally set so that the constrant never bnds. 5 Cash-constraned retal nvestors The results of the prevous sectons suggest that the form of the optmal prcng scheme depends on whether there s a bndng allocaton constrant, whch s n turn a ected by the spec c characterstc of the nsttutonal nvestors preferences. We can also thnk about d erent types of constrants. Hence, n ths secton, we analyze the case of cash-constraned retal nvestors 18

20 that can a ord/are wllng to nvest at most K n the sale, n the context of lnear preferences of nsttutonal nvestors. The new optmzaton program for the seller s smlar to program (P1), except that the last constrant, (v), s replaced by the cash constrant for retal nvestors. max v(s) fq g n 1 s:t : P 1 n (s s )q (s) f(s)ds () U (s; s) = 0 for (s) () E s 0 for all and all () q (s) 0 for all and all s (v) P q (s) 1 for all s (v) 1 P q (s) p R (s ; s ) K for all s: (P4 ) We are aware that wth a contnuum of retal nvestors, t s less plausble to justfy the exstence of such a constrant at the aggregate level, even though the constrant holds ndvdually. However, the purpose of our analyss s to nvestgate whether d erent knds of allocaton constrants a ect d erently the optmal prcng rule. In other words, we want to understand whether t s just the exstence of such a constrant that matters or also ts shape (the functonal form). We expect ths to be the case, as d erent forms of allocaton constrants wll determne the tghtness of the allocaton rule. For nstance, contrary to the maxmum quantty constrant prevously consdered, the above cash constrant depends on both the number of shares purchased and ther prce, thereby provdng, by constructon, more exblty to the ssuer when t comes to IPO desgn, as he can rely on both prce and quantty to acheve optmalty. The optmal mechansm s descrbed n the followng proposton: PROPOSITION 4 Wth rsk neutral nsttutonal nvestors and cash constraned retal nvestors, the optmal IPO s characterzed as follows: 1. (Allocaton rule) The ssuer sats es retal nvestors up to ther cash constrant. The remanng shares are allocated to the nsttutonal nvestor reportng the hghest sgnal 19

21 s m = maxfs 1 ; s 2 ; ::::; s n g. 2. (Prcng rule) There exsts at least one unform prce that mplements the optmal IPO. The above results mply that, as wth lnear preferences, retal nvestors have prorty n the allocaton of shares. In ths case, though, the allocaton rule depends on the prce. We thus need to solve for the optmal prcng rule and then step back to obtan explctly the equlbrum quanttes. Ths suggests that the ssuer has enough leeway to choose an allocaton rule that can be supported by an optmal unform prce. More generally, the cash constrant restrcts less the allocaton rule than the quantty constrant. The exblty ganed by the ssuer s enough to allow unform prcng at the optmum. Agan, we do not obtan a unque optmal prce. We nd that there may be more than one optmal unform prce, and that equlbra may exst wth dscrmnatory prcng. 6 Dscusson Ths paper has nvestgated the condtons under whch the unform prcng rule n IPOs s optmal. The ssuer can potentally use both quantty and prce dscrmnaton to elct nformaton from nformed nvestors and acheve optmalty (.e. maxmze the sale proceeds). Our ndngs show that as long as quantty dscrmnaton s su cently unrestrcted, the ssuer does not also need prce dscrmnaton, so that the optmal IPO can be mplemented wth unform prcng. The allocaton rule may be restrcted because of drect allocaton constrants and/or because nsttutonal nvestors have non-lnear preferences. Allocaton constrants n our model can be of two types, cash or quantty constrants. We show that the most mportant determnant of the optmal prcng rule s the exstence of an allocaton constrant and ts shape. Conversely, nsttutonal nvestors preferences are shown to have a drect mpact on the optmal allocaton rule. Spec cally, when non-lnear preferences are consdered, n equlbrum, contrary to the case of rsk neutralty, nsttutonal nvestors have prorty n share allocaton over retal nvestors who become resdual clamants. Furthermore, the nsttutonal nvestors preferences also ndrectly a ect the optmal prcng rule to the extent that the spec c values of parameters 20

22 and determne whether or not the allocaton constrant s bndng n equlbrum. The model delvers a number of emprcal mplcatons: The most drect mplcaton s that unform prcng and dscrmnatory prcng wll lead to the same IPO proceeds f the allocaton rule s unrestrcted. Seeng as dscrmnatory prcng s forbdden n most countres, a drect test of our results wll only be possble n the few countres that have allowed dscrmnatory prcng n IPOs. For example, both Tawan and Japan have used dscrmnatory IPO auctons (.e. pay-your-bd auctons, see Jagannathan and Sherman, 2006), that could allow a test of our results. However, there are stll no examples of dscrmnatory prcng n bookbuldng. A second emprcal mplcaton predcts that we should observe greater underprcng n IPOs characterzed by quantty constrants rather than cash constrants. Ths rases the ssue of understandng whch of these two types of constrants s more relevant n practce, whch s another nterestng emprcal research queston. 16 Related to the prevous pont, our results predct that underprcng should be more mportant n IPOs where the exstence of a quantty constrant s assocated wth a low demand for the shares by the nsttutonal nvestors. Appendx The seller s optmzaton program (P). From the RPC and the maxmand we can see that the seller s pro t s ncreasng n the retal nvestors payments, therefore at the optmum the RPC bnds. We can then rewrte the RPC as follows: p R (s)q R (s)f(s)ds = v(s)q R (s)f(s)ds = v(s) 1! X q (s) f(s)ds ; (13) where we have replaced q R (s) from the FAC. We now turn to the IIC. By applyng the envelope theorem to the maxmzaton problem n Equaton (IICa) and then takng expectatons over 16 Derren (2005) s, to our knowledge, the only paper to document that n some French IPOs a fracton of the shares on sale s explctly reserved to retal nvestors. 21

23 s, we have: 17 U (s ; s ) = U (s; s) + s s 1 z 2 (q (es ; s ); v(s))f (s )ds des : (14) n Snce the seller s payo s decreasng n the nformaton rents pad to the nformed nvestors, at the optmum he wll set U (s; s) = 0; so that the nformed nvestors wth the lowest evaluatons receve zero rents at the optmum. Invertng ntegrals, takng the expectaton over s and applyng Fubn s theorem 18 to Equaton (14) yelds U (s ; s )f (s )ds = 1 s z 2 (q (es ; s ); v(s))des f (s )ds f (s )ds : (15) n s Integraton by parts of the ntegral of the term n brackets on the l.h.s. yelds the followng U (s ; s )f (s )ds = 1 n (s s )z 2 (q (s); v(s))f(s)ds: (16) Fnally, from the de nton of the expected utlty of nsttutonal nvestors, we have for all and all s p (s)q (s)f (s )ds = z(q (s); v(s))f (s )ds U (s ; s ) 0: (17) Takng expectatons over s and usng Equatons (16) and (13) we obtan the seller s objectve functon as stated n the optmzaton program (P). Constrant () n (P) s a monotoncty condton whch s a su cent condton for truth-tellng to be optmal. 19 For prces pad by nsttutonal nvestors, pluggng Equaton (14) nto Equaton (17) yelds Equaton (7). Prces for retal nvestors satsfy ther (bndng) partcpaton constrant. Proof of Proposton 1. For generalty, we prove the result usng a generc dstrbuton f () satsfyng the ncreasng hazard rate assumpton. Wrtng Equaton (16) wth the general dstrbuton functon gves the followng objectve functon for the seller ( X 1 1 F (s ) max v(s) q (s) ) f(s)ds: (18) fq g n 1 n f (s ) 17 Notce that the equaton below holds at the optmum,.e. for bs = s : 18 Fubn s theorem states that we can nvert ntegrals whenever the ntegrand s nte. 19 The monotoncty condton s derved by usng Assumptons 4 and 5. In the mechansm desgn lterature, t s often referred to as an mplementablty condton. 22

24 Ths objectve functon s decreasng n the quantty allocated to nsttutonal nvestors (q (s)). The cost of allocatng a postve quantty to them, measured by ( X 1 1 F (s ) q (s) ) f(s)ds; (19) n f (s ) s decreasng n the sgnal s because of the ncreasng hazard rate assumpton. Consequently, the allocaton rule maxmzng the seller s revenues conssts n allocatng as much as possble to retal nvestors (.e. up to ther budget constrant) and any resdual quantty to the nsttutonal nvestor(s) havng (and reportng) the hghest announced sgnal (.e. the agent wth the sgnal s m = maxfs 1 ; s 2 ; ::::; s n g). Snce sgnals are contnuously dstrbuted only one agent announces s m. Denote by q m (s) the quantty allocated to ths agent. and Wth rsk neutral nvestors, the prcng condtons (7) and (8) become 1 s p (s)q (s)f (s )ds = v(s)q (s) q (es ; s )des f (s )ds n s for all s and all (20) p R (s)f(s)ds = v(s)f(s)ds: (21) We prove the result by contradcton. Assume the exstence of a unform prce functon, such that p(s) = p m (s) = p R (s) for all s; that mplements the optmal mechansm. Applyng ths to Equaton (20), takng expectatons over s and summng over n gves " # X ( " # ( X X 1 s )) p(s) q (s) f(s)ds = v(s) q (s) q (es ; s )des f(s)ds: n s (22) Snce P q (s) = 1 K and R p(s)f(s)ds = R v(s)f(s)ds, a necessary condton for the exstence of a unform prce s ( X 1 s ) q (es ; s )des f(s)ds = 0: n s However, q (s) s always non-negatve and strctly postve for one nvestor. So ths last equaton never holds whch contradcts the ntal assumpton about the exstence of an optmal unform prce functon. 23

25 Proof of Proposton 2. For the rst part of ths proposton, we consder the relaxed problem (.e. we drop the monotoncty constrant n program (P2 )) and check t ex post. As such, the objectve functon becomes an ordnary maxmand wth the constrants de ned at each pont and can be maxmzed pontwse on : The number of shares, q (s); the seller must assgn to nvestor n order to elct hs nformaton s gven by the followng maxmzaton problem, for each s 2, max fq g =1;::n P s:t : U (s; s) = 0 q (s) 0 q (s) ( 1) v(s) 2 q (s) n (s s ) for all and P q (s) 1 for all and s: The Kuhn-Tucker condtons for ths problem are 8 ( 1)v(s) q n >< (s s ) + (s) (s) = 0; for all (s)q (s) = 0 P (s)[1 q (s)] = 0: >: (23) Wth and beng the Kuhn-Tucker multplers assocated to the feasblty constrant and the FAC, respectvely. Now denote the seller s objectve functon by H(q; v(s)), that s H(q; v(s)) = X q (s) ( 1) v(s) 2 q (s) n (s s ) ; (24) wth q 2 [0; 1] and s 2 = [s; s] n : Ths functon s concave n q for all ; H let s (v ) = s ( 1)v (2 1) for all and v and de ne the followng sets for all s 2 ; N (s) = f 2 N j s s (v )g ; N + (s) = f 2 N j s > s (v 2 0. Now We can easly show that for each 2 N (s) t must hold that q (s) = 0: Then, for each 2 N + (s); de ne the quantty eq by ( 1)v(s) eq (s) n (s s ) = 0: (25) 24

26 P For each 2 N + (s), eq (s) s postve: If, eq (s) 1, then the quantty eq (s) s the soluton 2N + (s) P of our mechansm. 20 If however, eq (s) > 1, whch corresponds to oversubscrpton of 2N + (s) the new shares, then the quantty eq (s) cannot be optmal as t volates the FAC. The optmal quanttes, whch we denote by bq (s); are gven by the soluton to the followng system of equatons, 8>< >: bq (s)[( 1)v(s) bq (s) n (s s ) (s)] = 0 X bq (s) = 1; (s) > 0; 2N + (s) (26) whch mples that bq (s) s ether zero or s postve and solves the followng equaton ( 1)v(s) bq (s) n (s s ) (s) = 0: (27) Clearly, n ths case, all the shares are allocated to nsttutonal nvestors. receve nothng n equlbrum. Retal nvestors To prove the exstence of an optmal unform prce functon, we proceed n three steps. In the rst step (Step 1), we derve a condton for the exstence of a unform prcng rule for nsttutonal nvestors,.e. p (s) = p I (s) for all and s. Subsequently (Step 2), we show the exstence of a unque unform prcng rule; last, n Step 3, we show that the same prcng rule can be appled to retal nvestors,.e. p I (s) = p R (s). Step 1: The lnear transfer for nsttutonal nvestors must satsfy Equaton (7). Consder the followng prce functon p 0 (s) = z(q (s); v(s)) 1 n h R s z 2 (q (es ; s ); v(es ; s ))des s q (s) ; (28) for each s and each q (s) such that q (s) 6= 0: 21 Any prce satsfyng Equaton (7) can also be wrtten as p (s) = p 0 (s) + (s), where sats es the followng equaton (s)q (s ; s )f (s )ds = 0; for all and s : (29) 20 In ths case, the equaton de nng eq (s) s the FOC of our objectve functon H, snce the Kuhn-Tucker multplers, (s) and (s) are both zero. 21 Otherwse p 0 (s) = 0: 25

27 The exstence of a unform prce functon for nsttutonal nvestors mples that the margnal e ects of changes n the prvate sgnal of d erent nvestors on prces are equal, @s j for all and j: 22 Applyng ths unformty condton to an admssble prcng functon 0 j j : (30) Multplyng both sdes by q (s), the fact j = n s n s j ; and consderng from Proposton 2 for each s and each ; allows us to wrte Equaton (30) as follows 3 q (es ; s )des 7 q (s) j (s) n s s 3 q (es ; s )des 7 q (s) 5 + 2n ; (31) s for each and each j: Ths s a partal d erental equaton n (s) generc soluton s gven by n s q (es ; s )des q (s) whose (s) = '(s 1 + ::: + s n ) + n R s s q (es ; s )des q (s) + 2n s ; for all (32) where '() s a twce-d erentable functon de ned on the set [ns; ns] = n: So, we have shown that the optmal mechansm may be mplemented by a unform prce schedule f and only f s p(s) = p (s) = p 0 (s) + '(s 1 + ::: + s n ) + n s q (es ; s )des q (s) + 2n s ; (33) for all ; where ' s a twce d erentable functon satsfyng the followng ntegral equaton '(s 1 + ::: + s n )q (s ; s )f (s )ds = 2 s 3 q (es ; s )des 6 s 4 + (34) n q (s) 2n s 7 5 q (s ; s )f (s )ds = g(s ): 22 Ths s true n our model where all sgnals are equally nformatve. In ths case, the unform prce, f t exsts, must be equally senstve to a change of the prvate sgnal of any nvestor. In other words, what matters s by how much the sgnal has changed and not whose sgnal t was. 26

28 Provng the exstence of a unform prce schedule s equvalent to provng the exstence of ths functon '. STEP 2: We rst show that Equaton (34) can be wrtten as a Volterra ntegral equaton of the rst knd. 23 Then, by smply applyng the general propertes of ths knd of ntegral equaton we can prove the exstence and the unqueness of the functon '. To do so, we rst need to transform Equaton (34) nto a smple ntegral equaton,.e. wth the support de ned on R. Notce that, snce q (s ; s ) = 0 when s s 0 (v ); the support of the ntegral Equaton (34) s equal to 0 = f(s 1 ; s 2 ; ::; s 1 ; s +1 ; ::; s n ) j P j6= s j = v > v 0 (s )g where v 0 (s ) s de ned as the nverse of s 0 (v );.e. v 0 (s ) = s (2 1)s : (35) 1 Also, from the de nton of the optmal quantty, the l.h.s. of Equaton (34) equals R 0 '(s + v )q (s ; v )f (s )ds : By applyng the Generalzed Change Varable Theorem (GVCT) we can set v = (s ) for each ; whch nally mples that ( 0 ) = [v 0 (s ); (n 1)s] 2 R: 24 One key mplcaton of the GVCT s that there exsts a measure de ned over [v 0 (s ); (n such that 0 '(s + v )q (s ; v )f (s )ds = (n 1)s v 0 (s ) 1)s] '(s + v )q (s ; v )(dv ): (36) Last, by applyng the Radon-Nkodým theorem 25, we are also able to prove the exstence of a densty functon assocated wth the measure such that (n 1)s '(s + v )q (s ; v )f (s )ds = '(s + v )q (s ; v )(v )dv : (37) 0 v 0 (s ) From the above result, the ntegral Equaton (34) reduces to (n 1)s v 0 (s ) '(s + v )q (s ; v )(v )dv = g(s ): (38) 23 A Volterra ntegral equaton of the the rst knd s de ned n the followng way: (x) y 0 f(x; y)h(x; y)dy = g(x); n other words, one of the ntegral lmts must depend on the varable x: 24 See Dunford and Schwartz (1988, 3rd Ed.), chapter 3, lemma 8, page See, for example, Dunford and Schwartz (1988, 3rd Ed.), chapter 3, theorem 2, page

29 Ths s a Volterra ntegral equaton of the rst knd whch ensures that, as long as the functon g s well behaved, a soluton n ' always exsts. Ths shows the exstence of a unque unform prcng rule for nsttutonal nvestors. We denote such a prcng rule by p I (s) n the followng. Step 3: To complete the proof t remans to show that the unform prce for nsttutonal nvestors also apples to retal nvestors. Ths s equvalent to showng that p I (s) sats es the retal nvestors partcpaton constrant. Ths problem s relevant only n the cases for whch both retal and nsttutonal nvestors receve postve amounts of shares at the optmum. We start by de nng the followng sets: = fs j q = 0 for all g;.e. all the quantty s dstrbuted to retal nvestors; n = fs j q (s) 6= 0 for at least one g: Recall that the retal nvestors partcpaton constrant can be wrtten as follows! X p R (s)q R (s)f(s)ds = v(s) 1 q (s) f(s)ds: (39) Provng that unform prcng apples to all nvestors s then equvalent to demonstratng the exstence of a prce functon p R (s) solvng the above ntegral equaton and such that: ( p (s) for all s 2 p R (s) = (40) p I (s) for all s 2 n : whch requres that retal nvestors are charged d erent prces dependng on whether they receve the whole quantty or not. The problem then bols down to provng the exstence of a prce p (s) such that! X! X p (s)f(s)ds = v(s) 1 q (s) f(s)ds p I (s) 1 q (s) f(s)ds: (41) n We then prove the followng: LEMMA 1 A prce functon p (s) as de ned n Equaton (41) exsts f and only f t sats es the followng equaton p (s)f(s)ds = fv(s) + H(q; v(s))gf(s)ds p I (s)f(s)ds ; (42) n where the functon H(q; v(s)) de nes the seller s payo at the optmum. 28