A Theory of Intermediated Investment with Hyperbolic Discounting Investors

Transcription

1 A Theory of Intermedated Investment wth Hyperbol Dsountng Investors January 3, 04 Abstrat We study the role of fnanal ntermedares n provdng lqudty for eonom agents wth hyperbol dsountng. We show that n a ompettve market wth fnanal ntermedares makng zero profts, sophstated agents are offered wth a ontrat of perfet ommtment, whle naïve ones wll be attrated by a ontrat that offers seemngly attratve return n the long run, but ntrodues a dsontnuous penalty for early wthdrawal. In ompettve equlbrum wth partally naïve types, Pareto Optmalty s not aheved. When types are prvate nformaton, naïve depostors early wthdraw and ross-subsdze sophstated ones. If ontrats are lnear or a seondary market for long-term depost ontrat opens for tradng, welfare of partally naïve agents wll be mproved. We show that arbtrage-free ontrats offered by fnanal ntermedares allow for a unque term struture of nterest rates, whh ontans a premum for naïveté.

2 . Introduton Investment s a omplex ntertemporal deson nvolvng tradeoffs among osts and benefts ourrng at dfferent tmes, whh not only affet one's health, wealth, and happness, but may also determne the eonom prosperty of natons. Due to ther omplexty and market frtons for nvestment, fnanal ntermedares are nvolved n most of the nvestment atvtes. One mportant role of fnanal ntermedares, lke banks, s to provde depostors nsurane aganst preferene shoks, whh tradtonal nsurane market annot provde beause the shoks are prvate nformaton (Damond and Dybvg, 983. A growng lterature on savngs suggests that agents have tme-nonsstent taste for mmedate gratfaton, and are often naïve about ths taste (Frederk, Loewensten and O Donoghue, 00. Ths taste for mmedate gratfaton also generates "preferene shoks" that reate lqudty needs. Ths paper ntends to study the role of fnanal ntermedares when preferene shoks are generated by tme-nonsstent preferenes. Our model bulds on the three-date model n Damond and Dybvg (983 and assumes an llqud long-term asset whh yelds low return f operated for a sngle perod but hgh return f operated for two perods. But there are at least two key dfferenes: Frst, whle n Damond-Dybvg model t s unertan whether an agent s type or type ex-ante, preferene shoks generated by hyperbol dsountng s ertan. For example, the date- utlty funton of an agent wth hyperbol dsountng fator β s u( + βu(. So t s a "shok" only for non-sophstated agents who mspredt ther future utlty. Seond, onsstent wth muh of the lterature, we equate welfare wth self-0 s utlty, so all agents have the same welfare-maxmzng alloaton, and the fnanal ntermedary s role s to offer ommtment aganst early lqudaton rather than to provde rsk sharng among people wth dfferent preferenes. Gven these dfferenes, the preferene shoks n ths paper are mpulsve and rratonal from a long-term welfare perspetve. We start by showng that autarky and prvate market usually annot maxmze welfare, even for fully sophstated onsumers, beause ex-ante ommtment s ostly. Then we study a ompettve equlbrum smlar to Hedhues and Kőszeg (00. We defne a ompettve equlbrum as a set of ontrats suh that eah ontrat earns zero profts, no ontrat an generate strtly postve profts, and ontrats as well as optons n ontrats that do not affet expetatons or behavor are elmnated. Our model shows that n ompettve equlbrum, fnanal ntermedares provde perfet ommtment for sophstated agents. However, even f depostors are slghtly naïve, fnanal ntermedares offer seemngly attratve return n the long run, but ntrodues a dsontnuous penalty for wthdrawng n advane. Consstent wth ths predton, tme deposts, or ertfates of depost (CD usually promse a hgh yeld for a fxed perod, but harge a non-trval early wthdrawal penalty. We show that ths dsontnuty has some adverse welfare onsequenes. Non-sophstated depostors, even f they are very lose to sophstated and only slghtly mspredt ther future behavor, would end up falng to resst the temptaton of mmedate gratfaton and beng harged wth early wthdrawal penalty. A ommonly artulated justfaton for early wthdrawal penaltes n CD ontrats s to offer a ommtment deve for tme-nonsstent onsumers by preventng them from makng mpulsve wthdrawals. (Labson, 997; Ashraf et al., 003. However,

3 empral work shows that early wthdrawal from Certfates of Depost (CD aounts s at eonomally sgnfant level. Glkeson, Lst and Ruff (999 found that depostors wthdraw a sgnfant amount of ther tme deposts before maturty.4% and 6.4% of the depost base eah year for shortest and longest maturty type, respetvely despte an average negatve renvestment nentve. In fat, few people ask about how muh early wthdrawal penalty s when buyng CDs. Gven ths, t s not easy to justfy early wthdrawal penaltes as a ostly nentve deve to dsplne savngs behavor, and suggests some depostors may be naïve about ther tme nonssteny. Another applaton of our theory s penson funds, lke IRAs, 40(k plans, et. Early wthdrawals from these retrement funds are also onsderable: wthdrawals for nonretrement purposes by aount holders under 60 amount to $60 bllon a year, or 40 perent of the $76 bllon employees put nto suh aounts eah year and nearly a quarter of the ombned $94 bllon that workers and employers ontrbute. Moreover, most early wthdrawals from penson funds are made by low nome workers, who assumably are also more naïve about ther tme nonssteny. These fats support our predton that fnanal ntermedares fool naïve depostors by ntrodung penaltes they do not expet. Ths paper also onnets to a number of papers on nonlnear depost ontrats. Ln(996 shows that when people have random dsount fators, the optmal nentve-ompatble rsk sharng ontrat has a onvex struture, that s, the nterest rate s hgher when there s less early wthdrawal and larger depost balane. Ambrus and Egorov (0 show that wthdrawal penaltes (money burnng s optmal f onsumers fae a severe and rare negatve lqudty shok. Ths paper offers another justfaton for early wthdrawal penalty n CD ontrats. Besdes, all of these papers assume people are tme-onsstent or sophstated, so our model s more onsstent wth the unexpeted nature of early wthdrawals. The dea that ompettve market offers ommtment to sophstated agents and explotatve ontrat to naïve agents are prevalent n the lterature on ontratng wth tme nonssteny. Our results are most smlar to Hedhues and Kőszeg (00 n that ontrat and welfare s dsontnuous at full sophstaton, and even slghtly naïve onsumer would swth away from her preferred repayment and be penalzed ex post. DellaVgna and Malmender (004 show that a monopolst frm offers a two-part tarff to partally naïve onsumers, n whh the per-usage pre falls below the frm s margnal ost n the ase of nvestment goods, and les above margnal ost n the ase of lesure goods. These ontrat features have adverse effets on onsumer welfare only f onsumers are naïve. Elaz and Spegler (006 studes a two-perod model n whh the frm sreens agents by the probablty they attah to eah state ex ante, namely ther sophstaton. The optmal menu provdes a perfet ommtment deve for relatvely sophstated types, and explotatve ontrats whh nvolve speulaton wth relatvely nave types. Ths paper spealzes these models to nvestment ontrats offered by fnanal ntermedares, and yelds spef predtons, suh as early wthdrawal penalty, that ondes wth features of depost ontrats and penson funds n realty. Ths s aordng to a report by Hello Wallet, "The Retrement Breah n Defned Contrbuton Plans", publshed n January 03, and was ted on the New York Tmes on February, 03: s.html.

4 We also onsder a lnear restrted market n whh ontrats are restrted to have a lnear struture and onsumers an transfer onsumpton wth more flexblty between two dates at a pre-spefed nterest rate. We show that beause ths nterventon prevents fnanal ntermedares from ndung naïve agents to drastally mspredt ther future behavor, t s welfare-mprovng as long as agents are not too naïve. Demand depost s a natural example of ths lnear nterventon. Dvdng money n multple aounts or tradable CD both requre that the ontrat s arbtrage-free, therefore are also equvalent to restrtng ontrats to be lnear. We show that a lnear depost ontrat mples a unque term struture of nterest rates n our three-date model. Interestngly, our numeral example shows that when there are more naïve people n the populaton, short-term nterest rates derease whle long-term nterest rates nrease, suggestng a term premum for naïveté. Thus our paper offers a new explanaton for term premum. Fnally, we study the role of transpareny of fnanal ntermedares. If the fnanal ntermedary has to dslose ts fnanal reords by the end of eah perod, t an no longer offer an unrealstally hgh nterest rate over a long perod to fool the naïve agents. In equlbrum, the fnanal ntermedary an earn postve profts by explotng naïve people s mspredtons. However, f fnanal ntermedares own aptal, they are not affeted by ths restrton, and would offer the same ontrat as when they are opaque. The rest of the paper s organzed as follows. Seton ntrodues our bas three-date model wth tme-nonsstent preferenes. Seton 3 haraterzes ompettve equlbrum wth fnanal ntermedares and derves unrestrted nonlnear ompettve-equlbrum depost ontrat. In Seton 4, we dsuss lnear nterventon s effet on ontrats and onsumers welfare. Seton 5 extends the results to general forms of partal naïveté, and dsusses the mplatons for term strutures and results under tradable CDs and transpareny requrements. Seton 6 onludes wth some fnal thoughts. All proofs are n the Appendx.. Bas Model In an eonomy there are three dates ( t = 0,, and a sngle homogeneous good. Eah agent s endowed wth unt of good at date 0 and the good s to be onsumed at date and. The good an be stored from one date to the next or an be nvested at t = 0 n a long-run tehnology, whh returns R > unts at t =, and lqudaton s ostless: one an get unt for eah unt lqudated at t =. The agent has tme-nonsstent preferenes. Self 0 s utlty s u( + u(, where 0 and 0 are her onsumpton n dates and, respetvely. Self maxmzes u( + βu(, where 0 < β s the hyperbol dsount fator. The per-date utlty funton u (. s strtly onave and twe dfferentable, wth u (0 = 0 and a suffently large u (0. Followng O Donoghue and Rabn (00, n most of our analyss, we assume that The assumpton of ostless lqudaton s not essental to our man results, beause banks are optmzng and there s no lqudaton n equlbrum. It may affet onsumer s hoes n autarky ase * (Seton.. Our man results also hold f eah unt an be lqudated at value R (, R beause t s * essentally the ostless ase wth eah unt equalng to / R.

5 self 0 beleves wth ertanty that self wll maxmze u( ˆ + βu(, where β βˆ. ˆβ reflets self 0 s belefs about her tme-nonssteny, so that ˆβ = β orresponds to perfet sophstaton about future preferenes, and ˆ β = orresponds to omplete naïveté about tme-nonssteny. In Seton 6 We onsder a more general form of partal naïveté, and shows that our qualtatve results stll hold f self 0 attahes sgnfant probablty to her tme-nonssteny above β. We measure welfare usng long-run self-0 preferenes followng muh of the lterature on tme nonssteny (DellaVgna and Malmender, 004; O Donoghue and Rabn, 006. Although for smplfaton we onsder a three-date model, n realty tme-nonssteny plays out over many short perods, so weghtng eah date equally would be more reasonable, but the major results would reman qualtatvely the same... The Frst Best We frst onsder the frst-best alloaton n our model. Note that n optmum there s no lqudaton n date, so the welfare-maxmzng alloaton s determned by the followng program: The frst-order ondton s: max u( + u( (, s. t. + / R u ( = R u ( fb fb Note that sne the frst-best soluton does not depend on degree of tme-nonssteny ( β or sophstaton ( ˆ β of the agent, a soal planner an maxmze welfare by fb fb nvestng on behalf of all agents and offerng eah agent and n dates and, respetvely. It would be more nterestng to onsder agents welfare when they nvest on ther own or through ompettve proft-maxmzng fnanal ntermedares, whh are studed n the followng setons... The Autarky Case For omparson, we also derve the autarky outomes wthout fnanal ntermedary. Suppose at date 0, all agents antpate ther preferene at tme s u( + βu(. An agent s naïve f hs preferene turns out to be u( + βu( wth β < β. An agent s sophstated f hs preferene s tme-onsstent. Frst, suppose there s no trade between agents. Agents annot ommt, and they wll optmally hoose ther lqudaton at date. All agents maxmze ther date- utlty u( + βu( subjet to budget onstrant: + / R =. So ther atual onsumpton satsfes: u ( = βr u ( Next, we onsder a market where people an trade lams on future goods. At date, the pre of date onsumpton must be / R, otherwse ether (, 0 or (0, R would domnate all other ponts on the budget lne and t annot be an equlbrum. In ( (3

6 equlbrum, supply equals demand, and everyone maxmzes ther date- utlty on the budget lne. Therefore agents onsumpton s the same as n the ase wthout tradng, that s, the exstene of market annot mprove the agents welfare. In Seton 3 and 4, we wll ntrodue ompettve fnanal ntermedares nto the model to see whether fnanal ntermedares an help agents mprove ther welfare. The ompettveness among fnanal ntermedares guarantee all the surplus goes to eonom agents nstead of fnanal ntermedares. When the agent types are known to fnanal ntermedares, we fnd that the exstene of fnanal ntermedary weakly mproves agents welfare. 3 Sophstated people are strtly better off by nvestng through a fnanal ntermedary than nvestng on ther own, beause the fnanal ntermedary an help them ommt aganst taste for mmedate gratfaton, by lmtng ther hoes or hargng an early wthdrawal penalty. Compared to autarky, naïve people are equally well off when the ontrat s nonlnear and strtly better off when the ontrat s lnear, so the fnanal ntermedary an also help partally naïve people to overome ther tme nonssteny problem under the lnear ontrat. Moreover, by restranng people mpatene, the fnanal ntermedary holds more money for a longer perod and make more nvestments to the eonomy. When there are both naïve agents and sophstated agents and fnanal ntermedares annot dfferentate them, the early wthdrawal penalty serves as a ommtment deve for sophstated people whle explotng naïve people s wrong expetatons, and t makes sophstated agents better off but ould make naïve agents worse off ompared wth autarky. 3. Compettve Equlbrum 3.. Defnton of Compettve Equlbrum In a ompettve market, agents nterat wth ompettve, rsk-neutral and proft-maxmzng fnanal ntermedares. Fnanal ntermedares fae the long-term tehnology desrbed above. For smplty, we assume that n date 0, depostors put all ther money n the fnanal ntermedary and an sgn unrestrted nonlnear ontrats regardng repayment shedule. These ontrats are exlusve: one an agent sgns wth a fnanal ntermedary, she annot nterat wth other fnanal ntermedares. We assume there are fnte β s among people and β < β <... < βi, and ˆ β { β,..., β I }. Fnanal ntermedares offer a fnte menu of repayment optons C = {( s, s } s S to eah agent at date 0, where and are onsumptons n date and, respetvely. Defne ( (., (. : β,..., β I R+ as an nentve-ompatble map, f agents wth hyperbol dsountng fator β prefer ( ( β, ( β among all repayment optons, that s, u( ( β + βu( ( β u( + βu( for all (, C. An agent of type ( β, ˆ β beleves n date 0 she would hoose ( ˆ ˆ ( β, ( β from C, but n realty she hooses ( ( β, ( β when onfronted wth C n date. 3 If lqudaton s ostly, that s, L <, then both sophstated and naïve people would be worse off n autarky, and fnanal ntermedares stll weakly mprove people s welfare.

7 We defne a ompettve equlbrum as a ontrat C offered by the fnanal ntermedares and nentve ompatble map ( (., (. that satsfes the followng propertes:. [Zero-proft] For eah fnanal ntermedary C yelds zero expeted profts.. [No proftable devaton] There exsts no ontrat C wth an nentve ompatble map ( (., (. suh that for some ˆβ, u( ( ˆ β, ( ˆ β > u( ( ˆ β, ( ˆ β, and C yelds postve profts. 3. [Non-redundany] For eah repayment opton ( j, j C, there s a orrespondng type ( β, ˆ β suh that ether ( j, j = ( ( β, ( β or (, = ( ( ˆ β, ( ˆ β. j j The frst two ondtons are typal for ompettve markets, sayng that fnanal ntermedares earn zero profts by offerng these ontrats, and they an do no better. The last ondton says that all repayment optons are relevant n that they affet the expetatons or behavors of depostors. Due to the redundany ondton, many optons are exluded from the ompettve-equlbrum ontrats, n partular, non-sophstated onsumers an only hange ther repayment opton by payng a large early wthdrawal penalty as dsussed below. 3.. Unrestrted Contrats when β and ˆβ are known We start by onsderng a fully sophstated depostor wth ˆ β = β <. Sne a tme-onsstent depostor would orretly predt her hoe n date, only her hosen repayment opton s relevant n both perods. We an assume that the ontrat offered by fnanal ntermedares only nlude one repayment opton that she atually hooses. The fnanal ntermedary s problem s: max( R (4, s. t. u( + u( u [PC] PC s the partpaton onstrant, where u s a onsumer s pereved utlty from the perspetve of date 0 f she aepts a purported ompettve-equlbrum ontrat. It s lear that PC bnds; otherwse the fnanal ntermedary ould nrease profts by dereasng. Competton drves the fnanal ntermedary s proft to zero through lftng u. Sne n a ompettve market ( R = 0, the fnanal ntermedary s maxmzng self 0 s utlty subjet to the budget onstrant. Proposton If ˆβ = β, the ompettve-equlbrum depost ontrat has a sngle repayment opton satsfyng u ( / u ( = R, and ( R = 0. The agent gets the same alloaton as the frst best. The stuaton s entrely dfferent for partally naïve agents. Partally naïve ( ˆ β > β agents mspredt ther utlty and thus the repayment opton they would hoose n date. The fnanal ntermedary offers them a former deoy repayment opton ( ˆ ˆ, self-0 expets to hoose and a latter hosen repayment opton (, self- atually hooses subjet to the followng onstrants. Frst, for the onsumer to be wllng to aept the fnanal ntermedary s offer, self 0 s utlty from the deoy

8 opton must be at least u. Ths s a verson of the standard partpaton onstrant ( PC, exept that self 0 may make her partpaton deson based on norretly foreasted future behavor. Seond, f self 0 s to thnk that she wll hoose the deoy opton, then gven her belefs ˆβ she must thnk she wll prefer t to the other avalable optons. These are the pereved-hoe onstrants ( PCC. Thrd, f an agent wth short-term mpatene atually hooses the repayment shedule ntended for her, she has to prefer t to the other repayment optons. Ths s analogous to standard nentve-ompatblty onstrants ( IC for self. Therefore the fnanal ntermedary solves: max ( R (5 ˆ, ˆ,, s. t. u( ˆ ˆ + u( u, [PC] u( ˆ + ˆ βu( ˆ u( + ˆ βu(, [PCC] u( + βu( u( ˆ + βu( ˆ [IC] As before, PC must bnd n equlbrum. In addton, IC also bnds beause otherwse the fnanal ntermedary ould nrease profts by lowerng. Gven that IC bnds and ˆβ > β, PCC s equvalent to ˆ. Intutvely, f self s n realty ndfferent between two repayment optons, then self 0 who overestmates her future patene predts she wll prefer the opton wth more repayment later. We frst derve the optmal soluton wthout PCC below, and onfrm that ˆ s satsfed. The relaxed problem s: max ( R (6 ˆ, ˆ,, s. t. u( ˆ + u( ˆ u, [PC] u( + βu( = u( ˆ + βu( ˆ [IC] The optmal soluton must have ˆ 0 =, otherwse sne β <, the fnanal ntermedary ould derease u( ˆ and nrease u( ˆ by the same amount, keepng PC onstrant unhanged and loosng IC onstrant, allowng t to lower. Proposton Suppose ˆβ > β, the ompettve-equlbrum depost ontrat has two repayment optons, wth the onsumer expetng to hoose ˆ = 0, ˆ > 0, and atually hoosng, satsfyng u ( / u ( = β R, and ( R = 0. Ths ontrat offers an opton for the onsumer to onsume lttle n the short run, but also ntrodues an opton to wthdraw n advane for a sgnfant penalty. Sne the fnanal ntermedary desgns the ontrat to ndue repayment behavor that self 0 does not expet, ts goal wth the hosen opton s to maxmze the gans from trade wth self keepng the IC onstrant satsfed, and t aters fully to self s taste for mmedate gratfaton. Note that there s a dsontnuty at full sophstaton: all non-sophstated depostors, even near-sophstated ones, reeve dsretely dfferent ontrats from and dsretely lower welfare than sophstated depostors. Ths s beause agents are

9 maxmzng utlty from the deoy repayment opton, whle the fnanal ntermedary s proft s determned by the hosen repayment opton. These two optons dffer as long as the depostors are not fully sophstated. By desgnng an unrestrted nonlnear ontrat and aterng to preferenes of self, the fnanal ntermedary exaggerates even an arbtrarly small amount of naïveté, and ndues the agent to make a non-trval mstake. These propertes losely resemble some mportant features of CD ontrats n banks. For example, for a one-year CD, the penalty of early wthdrawal s often three months nterests. CDs promse a favorable return as long as money s loked away for a fxed perod, but f the depostor wthdraws even one dollar before maturty, she would be harged wth a substantal penalty, and nvestng n CD beomes extremely unproftable. Penson funds also share some smlar features wth penalty of early wthdrawal. In the autarky ase, non-sophstated depostor s long-run self-0 utlty s u( + u( where and satsfy u ( / u ( = β R and ( R = 0. Obvously, the welfare of non-sophstated depostor does not hange wth the presene of fnanal ntermedary beause the realzed onsumptons are dental Unrestrted Nonlnear Contrats when β and ˆβ are unknown In ths seton we study ompettve equlbra when ether ˆβ, or β and ˆβ, are unknown to the fnanal ntermedary. We show that the man results from last seton reman: sophstated and non-sophstated depostors are dental ex-ante and sgn the same ontrat, but non-sophstated depostors hoose the repayment opton wth large early wthdrawal penaltes and have dsontnuously lower welfare than sophstated depostors Known ˆβ, Unknown β Suppose that all agents has ˆβ at date 0 and s known to the frm, and at date they have β = β ˆ < β wth probablty p, and β = β ˆ = β wth probablty p. Beause sophstated and naïve agents have the same belef at date 0, they aept the same ontrat. The fnanal ntermedary s problem s: max p (( R + p (( R (7,,, s. t. u( + u( u, [PC] u( + β u( u( + β u(, [IC ] u( + β u( u( + β u( [IC ] Clearly PC bnds, and IC also bnds, otherwse the fnanal ntermedary an derease wthout volatng the onstrants, and nrease profts. Then we have the followng proposton. Proposton 3 Suppose ˆβ s known, and β takes two values, β ˆ < β and β ˆ = β wth probablty p and p, respetvely. In ompettve equlbrum, the repayment optons (, and (, satsfes p (( R + p(( R = 0,

10 and u ( = βr u ( (8 u ( p u ( = R + ( β u ( p u ( Condton (8 s a verson of the standard effeny-at-the-top result ommon n many sreenng problems. It says that repayment shedule of non-sophstated agents s effent from self s perspetve the same result as n the ase wth known ˆβ and β. By ondton (9, sophstated agents repayment shedule s too bak-loaded even from the long-term self 0 s perspetve. When onsderng whether to alloate more of the sophstated agent s repayment to date, the fnanal ntermedary faes a trade-off: on one hand, ths adjustment nreases sophstated agent s expeted utlty n date 0, and nreases fnanal ntermedary s profts; on the other hand, ths adjustment also nreases non-sophstated agent s utlty n date, makng early wthdrawal penaltes less preferable, and dereases fnanal ntermedary s profts. Ths s smlar to the tradeoff n standard sreenng problems between nreasng effeny of less proftable type and dereasng nformaton rent of pad to the more proftable type. Sophstated agent s alloaton s dstorted from date-0 perspetve to keep non-sophstated agent, the more proftable type, nentve ompatble from date- perspetve. As before, the two frst-order ondtons mply dsontnuty at full sophstaton. Even for arbtrarly lose β and β, a non-sophstated agent gets dsontnuously dfferent repayments from a sophstated agent, and s dsontnuously worse off as a result. In other words, the welfare of non-sophstated agents s redued wth the presene of fnanal ntermedary beause n ths ase and satsfes u ( / u ( = β R and ( R < 0. Next we onsder how welfare hanges wth the proporton of non-sophstated agents n the populaton. Proposton 4 (The Cross-Subsdy Effet Suppose ˆβ s known, and β takes two values, β ˆ < β and β ˆ = β wth probablty p and p, respetvely. The sophstated type s welfare n the ompettve equlbrum s strtly nreasng n p. The ntuton for ths result s that when types are unknown, fnanal ntermedares make money on non-sophstated depostors and lose money on sophstated depostors, so an nrease n p would result n postve profts. Competton drves fnanal ntermedares to offer more attratve ontrats, and sne sophstated onsumers orretly antpate ther outomes, ths would make them better off Unknown β and ˆβ As n Hedhues and Kőszeg (00, we provde a ondton under whh agents self-selet aordng to ˆβ n date 0, and then aordng to β n date. Let u be the pereved utlty from ompettve-equlbrum ontrat when ˆ β = β (9

11 s observable, wth probablty she s type β. Condton : u s nreasng n β. p she s sophstated, and wth probablty ( Under ths ondton agents would self-selet nto the ontrat ntended for her true ˆβ beause: Frst, she wouldn t hoose a ontrat ntended for any ˆ β < ˆ β sne t would gve her lower utlty (by Condton ; Seond, whle she prefers the ontrat for ˆ β > ˆ β, she expets to swth away from t n date and get lower utlty at last (otherwse the ontrat desgned for ˆβ s suboptmal ompared to more attratve and proftable ontrat for ˆ β > ˆ β. Proposton 5 If Condton holds, then n the unque ompettve equlbrum wth ˆβ unobserved by fnanal ntermedares, eah agent aepts the same ontrat as when ˆβ s observed by fnanal ntermedares. Even when fnanal ntermedares do not know agent s ˆβ or β, a non-sophstated depostor would self-selet nto a ontrat whh she would pay the early wthdrawal penalty, and our result about dsontnuty of welfare and behavor at full sophstaton remans. p 4. Restrted Lnear Contratng In ths seton, we onsder ontrats that are restrted to have a lnear struture. In date 0, the fnanal ntermedary spefes some R % and T, and agents hoose from all (, that satsfes + / R% = T. We show that preventng fnanal ntermedares from foolng slghtly naïve agents nto dsretely mspredtng ther behavor, restrtng ontrats to be lnear rases naïve agents welfare. 4.. Full Informaton Frst we onsder ompettve equlbrum n a restrted lnear market when both β and ˆβ are known to the fnanal ntermedares. A perfetly sophstated agent s fully aware of her tme nonssteny, so t would be proft-maxmzng to offer her a ontrat wth an nterest rate of R% = R / β whh algns self s nterest wth the long-term welfare. Ths ontrat ounterats self s tendeny for mmedate gratfaton and maxmzes sophstated depostors welfare. More nterestngly, restrted lnear ontrats prevents fnanal ntermedares from settng too hgh early wthdrawal penaltes, so a slghtly naïve agent only mspredts her future behavor by a small amount, and aheves nearly optmal welfare. Proposton 6 Restrtng fnanal ntermedares nvestment ontrats to a lnear form keeps fully sophstated agents equally well off and strtly rases not-too-naïve agents welfare ompared to unrestrted nonlnear ontrat. The welfare of naïve agents s nreased by the presene of fnanal ntermedary when preferenes are observable. Note that ths proposton holds only for not-too-naïve agents sne both lnear and unrestrted nonlnear ontrats would lead very naïve agents to severely

12 underestmate ther onsumpton n date. But f all agents are not too naïve, the lnear nterventon offers a Pareto mprovement upon unrestrted nonlnear ontrats. 4.. Unknown Types Then we onsder the welfare effets of lnear nterventon when β s unknown to the fnanal ntermedares. Proposton 7 Suppose ˆβ s known, and β takes the values β ˆ < β and β ˆ = β wth probablty p and p = p, respetvely. Agents strtly prefer the ompettve equlbrum n the unrestrted market over that n the restrted market. However, f non-sophstated agents are suffently sophstated ( β s suffently lose to β, ther welfare and the populaton-weghted sum of two types welfare, n greater n the restrted lnear market than n unrestrted market. As n the ase of full nformaton, restrted lnear ontrats prevent non-sophstated agents from drastally mspredtng ther behavor and rase ther welfare. But all agents strtly prefer unrestrted market to restrted lnear market, and sne fully sophstated agents orretly predt ther future behavor, they are made worse off by the lnear nterventon. Ths s beause lnear ontrats elmnates dsontnuty n welfare and redues the ross-subsdy from non-sophstated agents to sophstated ones. When types are unknown, restrtng ontrats to have a lnear struture generally does not Pareto-domnate unrestrted nonlnear ontrats. Nevertheless, the beneft of ths nterventon to non-sophstated agents outweghs the harm to sophstated depostors. Sne ths nterventon dereases the dstorton n repayment shedules to both types, t nreases the populaton-weghted sum of welfare. Furthermore, ths nterventon has redstrbutve benefts f non-sophstated agents are poorer than sophstated agents n general. An example of lnear nterventon s the demand depost, whh allows depostors wthdraw any amount at a pre-spefed nterest rate. But to redue rsks of bank runs, nterest rates of demand deposts are usually very low, and depostors welfare under demand depost ontrats may not be hgher than under restrted depost ontrats. A smlar example s enhaned or flex CDs ntrodued n reent years.(brooks, 996; Clne and Brooks, 004 These CDs offer an opton to wthdraw early one wthout penalty. Though enhaned CDs were ntrodued n order to redue lqudty or nterest rate rsks, n some extent they restrt CD ontrats to have a lnear struture by allowng depostors to wthdraw early at the spefed nterest rate. As shown above, enhaned CDs would nrease populaton-weghted sum of welfare even though they have a slghtly lower rate than standard CDs. Penson fund depost ontrats also have a lnear struture. For example, earnngs from money wthdrawn from the new Roth 40(k or 403(b aounts before the age of 59 / are subjet to nome tax and a 0 perent early wthdrawal penalty, and ths 0 perent penalty orresponds to the lnear ontrat n our model Term Premum The restrted lnear ontrat mples a term struture of nterest rates n our three-date model. Suppose the one-perod and two-perod nterest rates are and, respetvely, then the repayment optons under ths term struture nlude all (,

13 that satsfes: ( + ( + + = (0 So a lnear ontrat + / R% = T has the term struture: + = T / + = ( RT % ( When all agents are sophstated ( ˆβ = β, as shown n Proposton 6, fnanal ntermedary offers R% fb fb = R / β, and T = q + r / R % fb fb, where (, s the fb fb fb fb frst-best alloaton. Ths mples + = + β / R, and ( + = R / β +, so when people get more mpatent, fnanal ntermedary would offer a lower one-perod nterest rate and a hgher two-perod nterest rate suh that people are ommtted to onsume the frst-best alloaton. Next we onsder the term struture of nvestment ontrat when types are unknown. The assumptons the same as n Seton 4.. Suppose the fnanal ntermedary offers the ontrat (,, and type- and type- depostors onsume (, and (,, respetvely. Sne every depostor maxmzes her self- utlty among all repayment optons, (, s the soluton to / ( + + / ( + = and u ( / u ( = β( + / ( +, (, s the soluton to / ( + + / ( + = and u ( / u ( = β ( + / ( +, and,,, are all ontnuous funtons of and. Proposton 8 Suppose ˆβ s known, and β takes two values, β ˆ < β and β ˆ = β wth probablty p and p, respetvely. The ompettve equlbrum lnear ontrat has a term struture (, where, are one-perod and two-perod nterest rates, respetvely, and s determned by: ( ( ( R + ( β + + = ( R + ( β and p (( R + p (( R = 0 ( n whh,,, are ontnuous funtons of and as defned above. In partular, when β = β = β, s nreasng n β and s dereasng n β. ( Next we study how the term struture hanges wth the proporton of naïve agents. By the ross-subsdy effet, the fnanal ntermedary earns money on naïve agents to subsdze sophstated agents. When there are more naïve agents, thus more agents to subsdze the fnanal ntermedary, the fnanal ntermedary an reman budget balaned by explotng every naïve agent less. So the fnanal ntermedary would nrease the long-term nterest rate relatve to the short-term nterest rate to ndue naïve agents to onsume less at date 0, and mtgate ther tme nonssteny problem. Ths ndates a term premum for naïveté: the dfferene between long-term and short-term yelds nreases wth the share of naïve agents.

14 The followng proposton onfrms the above ntuton. Proposton 9 Suppose ˆβ s known, β takes two values, β ˆ < β and β ˆ = β wth probablty p and p, and the lnear ontrat has a term struture (, where, are one-perod and two-perod nterest rates. Then, n the ompettve equlbrum, term premum s nreasng n p f xu" ( x / u ( x > β. 5. Extensons and Dsussons 5.. General Consumer Belefs In prevous setons we use the defnton of partal naïveté by O Donoghue and Rabn that self 0 beleves wth ertanty that self s dsount fator s ˆβ, n ths seton we extend the onept of partal naïveté to a more general form whh norporates partal naïveté ( ˆ β as speal ases, and study ontrats under ths general spefaton of onsumer belefs. Let the umulatve dstrbuton funton F( ˆ β represent onsumers belefs about β n date 0. Suppose fnanal ntermedares know onsumers true β.ths s plausble gven that fnanal ntermedares have a lot of nformaton about depostors and spend a lot on researhng ther behavor. Suppose onsumers expet to hoose ( ˆ ˆ ˆ ( β, ˆ ( β n date 0 for eah ˆβ. Denote the support of F by F. Frst we suppose fnanal ntermedares know F (.. Then we an show as n Hedhues and Kőszeg (00 s Appendx that the ontrat s ompettve equlbrum even f fnanal ntermedares do not observe onsumers belefs. The fnanal ntermedary s problem s: max ( R (0,, ˆ ˆ ( β, ˆ ˆ ( β s. t. u( ˆ ˆ ˆ ˆ ( β u( ˆ ( β + df( β u, [PC] u( ˆ ( ˆ β + ˆ βu( ˆ ( ˆ β u( ˆ ( ˆ β + ˆ βu( ˆ ( ˆ β for any ˆ β, ˆ β F, [PCC] u( + βu( u( ˆ ( ˆ β + βu( ˆ ( ˆ β for any ˆ β F [IC] Proposton 0 Ether when fnanal ntermedares know onsumers belefs or not, n a ompettve equlbrum the repayment shedule a onsumer wth belef F (. hooses satsfes: u ( β R = u ( F( β β + ( F( β Note that β R u ( / u ( R. Sophstated onsumers have F( β =, thus the repayment shedule satsfes u ( / u ( = R and s welfare-maxmzng; Partally naïve ( ˆ β > β onsumers have F( β = 0, so the repayment shedule satsfes u ( / u ( = β R. Therefore we replate the results n Seton 3. The onsumer s welfare depends solely on F( β, the probablty she attahes to (

15 unrealstally hgh levels of self-ontrol. Another example s the defnton of partal naïveté n Elaz and Spegler (006. Suppose the agent beleves wth probablty p she s tme-onsstent ( β = and wth probablty p her type s β, and p measures her degree of sophstaton. The repayment shedule satsfes u ( / u ( = β R / (( p β + p and s ontnuous n p : t approahes sophstated agents repayment shedule as p 0, and approahes type- β partally naïve agents repayment shedule as p. 5.. Tradng of CDs Some CDs, lke brokerage CDs, an be traded on a seondary market. So for ths type of CDs, f a depostor wants to lqudate before maturty, she an sell the deposts on a seondary market rather than payng the early wthdrawal penalty. The followng proposton shows that, when there are both naïve and sophstated people, tradable CDs wth unrestrted nonlnear ontrats s equvalent to lnear depost ontrat. Proposton Suppose ˆβ s known, and β takes the values β ˆ < β and β ˆ = β wth probablty p and p = p, respetvely. If nvestment ontrats an be traded on a seondary market at date, n ompettve equlbrum all the agents get the same alloatons as when nvestment ontrats are restrted to be lnear. Intutvely, when agents an trade date- onsumpton for date- onsumpton under a ertan pre, people s hoe set s a lnear set. In equlbrum, supply equals demand, and everyone maxmzes ther date- utlty along the lnear set, therefore tradable CDs are equvalent to restrted lnear depost ontrats Transpareny of Fnanal Intermedares So far we have assumed that the fnanal ntermedary s not transparent. In ths seton we onsder a transparent fnanal ntermedary, that s, the fnanal ntermedary has to dslose ts amount of nvestment at date 0 and needs to be able to satsfy all ustomers needs aordng to the ontrat. For example, the unrestrted nonlnear ontrat wth naïve agents s not feasble for a transparent fnanal ntermedary, beause the expeted repayment shedule s out of the fnanal ntermedary s budget lne, and though t would not be realzed n equlbrum, the fnanal ntermedary annot reman solvent ex-ante when there s possblty that all agents hoose that expeted repayment opton. We reonsder the fnanal ntermedary s problem when all agents are naïve and the fnanal ntermedary s transparent. Sne lqudaton of nvestment s ostless, the fnanal ntermedary an satsfy agents needs as long as all repayment optons le wthn the budget lne. Sne the budget onstrant of the hosen repayment opton s mpled by nonnegatve profts, we only need to add the budget onstrant of the expeted deoy repayment opton, and ths s the transpareny onstrant (TC. The fnanal ntermedary s problem s: max ( R ( ˆ, ˆ,, s. t. u( ˆ + u( ˆ u, [PC] u( ˆ + ˆ βu( ˆ u( + βu ˆ (, [PCC] u( + βu( u( ˆ + βu( ˆ [IC]

16 ( ˆ R ˆ 0 [TC] IC and TC bnd, the deoy repayment opton s the same as the frst-best alloaton, whle the hosen repayment opton satsfes: and u ( = β R u ( (3 u( + βu( = u( + βu( (4 fb fb Note that the fnanal ntermedary makes postve profts, and there s no devaton that an earn hgher profts, beause ex-ante naïve agents only are about the deoy repayment opton, but the deoy repayment opton s onstraned by the transpareny ondton. So the fnanal ntermedary an explot naïve agents usng the atually hosen repayment opton wthout nfluenng ther ex-ante expetaton. The regulaton that requres the fnanal ntermedary to dslose ts fnanal reords and reman transparent n fat hurts the naïve people by restrtng the fnanal ntermedary s hoes of repayment optons. One way to nrease naïve agents welfare when the fnanal ntermedary s transparent s to have aptal. If the fnanal ntermedary has some aptal, t would be able to offer repayment optons beyond the budget lne, and aheve the same welfare as when the fnanal ntermedary s opaque. For example, suppose all depostors are naïve, f the fnanal ntermedary has suffent aptal K, the fnanal ntermedary wll offer ( ˆ, ˆ at date 0, where ( ˆ ˆ, are as defned n Proposton. If everyone hooses the opton ( ˆ ˆ,, the fnanal ntermedary s budget onstrant s: ĉ ( + K R (5 The mnmum aptal requred to aheve the same welfare as an opaque fnanal ntermedary would be: 5.4. Savng Amount K mn = ˆ / R (6 In prevous analyss, we assume eah agent has unt savng. In ths seton, we onsder the ase where nvestors an hoose ther optmal savng amount. For smplty, we assume that savng an amount redues utlty by,.e., self 0 has utlty u( + u(. Therefore, the frst best alloaton n ths ase solvng the followng program: max, (, u + u( (7 s. t. + / R It s easy to know that the optmal I ( s the nverse funton of u (. * = I(, * I R = ( /, = + / R, where * * If ndvdual agents have no hoe but to nvest through ntermedary and the ntermedary offer an unrestrted nonlnear ontrat. When there are only sophstated depostors wth ˆ β = β <, the fnanal ntermedary s problem s:

17 max ( R (8, s. t. u( + u( = u [PC] The frst order ondton s u ( = and ( u = / R. When there are only nave depostors wth ˆβ > β, The problem s: max ( R (9 ˆ, ˆ,, s. t. u( ˆ + u( ˆ = u, [PC] u( + βu( = u( ˆ + βu( ˆ [IC] / u = / R. Ths suggests the savng of a nave agent s hgher than that of a sophstated agent. The frst order ondton s u ( = β and ( 6. Conluson Our model offers a new explanaton of early wthdrawal penaltes n fnanal ntermedares nvestment ontats when people are tme-nonsstent, and dentfes smple welfare-mprovng nterventons. For example, a seondary market n whh long term depost ontrats an be traded ould mprove naïve agents welfare. However, whle these nterventons rase soal welfare, they would not be aepted by agents who all beleve they are ratonal. So t remans to be nvestgated whether there are modfatons of ths nterventon that agents would prefer. For smplfaton, we onsder a model wth three dates throughout the analyss. An nterestng dreton for future researh s to extend ths model to nfnte horzon wth overlappng generatons and onsder the term struture of nterest rates n the nfnte horzon model. To solate the ommtment problem of tme-nonsstent onsumers, we assume that that lqudty need s totally generated by rratonal tme-nonsstent preferenes and all agents dffer only n ther belefs (or degree of sophstaton. Future researh would lkely onsder a more general model ombnng tme nonssteny and real lqudty shoks, as n Amador, Wernng and Angeletos (006, n order to provde a more omplete haraterzaton of fnanal ntermedares role of provdng rsk-sharng and ommtment n ompettve equlbrum. It would also be nterestng to nvestgate whether bank run s an equlbrum n the generalzed model.

18 REFERENCES Amador, Manuel, Iván Wernng and George-Maros Angeletos (006: Commtment vs Flexblty. Eonometra, 74(: Ambrus, Attla, and Georgy Egorov (0: Commtment-Flexblty Trade-off and Wthdrawal Penaltes. Unpublshed. Ashraf, Nava, Nathale Gons, Dean S. Karlan, and Wesley Yn (003: A Revew of Commtment Savngs Produts n Developng Countres. Asan Development Bank Eonom and Researh Department Workng Paper No. 45. Beshears, John, James J. Cho, Davd Labson, Brgtte C. Madran, and Jung Sakong (0: Self Control and Lqudty: How to Desgn a Commtment Contrat. Unpublshed. Bolton, Patrk, and Mathas Dewatrpont (005: Contrat Theory. Cambrdge, MA: MIT Press. Brooks, Robert (996: Computng the Yelds on Enhaned CDs. Fnanal Serves Revew, 5(: 3-4. Clne, Brandon N., and Robert Brooks (004: Embedded Optons n Enhaned Certfates of Depost. Fnanal Serves Revew, 3(: 9-3. DellaVgna, Stefano, and Ulrke Malmender (004: Contrat Desgn and Self-Control: Theory and Evdene. Quarterly Journal of Eonoms, 9(: Damond, Douglas W., and Phlp H. Dybvg (983: Bank Runs, Depost Insurane and Lqudty. Journal of Poltal Eonomy, 9(3: Elaz, Kfr, and Ran Spegler (006: Contratng wth Dversely naïve Agents. Revew of Eonom Studes, 73(3: Frederk, Shane, George Loewensten, and Ted O Donoghue (00: Tme Dsountng and Tme Preferene: A Crtal Revew. Journal of Eonom Lterature, 40: Frexas, Xaver, and Jean-Charles Rohet (997: Mroeonoms of Bankng. Cambrdge, MA: MIT Press. Glkeson, James H., John A. Lst, and Crag K. Ruff (999: Evdene of Early Wthdrawal n Tme Depost Portfolos. Journal of Fnanal Serves Researh, 5(: 03-. Hannan, Tmothy H., and Allen N. Berger (99: The Rgdty of Pres: Evdene from the Bankng Industry Ameran Eonom Revew, 8(4: Hedheus, Paul, and Botond Kőszeg (00: Explotng Naïveté about Self-Control n the Credt Market. Ameran Eonom Revew, 00(5: Labson, Davd (997: Golden Eggs and Hyperbol Dsountng. Quarterly Journal of Eonoms, (: Labson, Davd, Andrea Repetto, and Jeremy Tobaman (007: Estmatng Dsount Funtons wth Consumpton Choes over the Lfeyle. Natonal Bureau of Eonom Researh Workng Paper 334. Ln, Png (996: Bankng, Inentve Constrants, and Demand Depost Contrats

19 wth Nonlnear Returns. Eonom Theory, 8: O Donoghue, Ted, and Matthew Rabn (00: Choe and Prorastnaton. Quarterly Journal of Eonoms, 6(: -60. O Donoghue, Ted, and Matthew Rabn (006: Optmal Sn Taxes. Journal of Publ Eonoms, 90:

20 Appendx : Proofs Proof of Proposton. In the text we have establshed that ˆ = 0. Usng u (0 = 0, the two onstrants n the relaxed problem ombnes nto: u( + β u( = βu (A So the frst-order ondton s: u' ( / u'( = βr. And ( R = 0 omes from the zero proft ondton of ompettve equlbrum. Proof of Proposton 3. Sne PC and IC both bnd, we frst gnore IC. Wth Lagrange multplers λ and µ, the frst order ondtons are: ( p u'( + p / u'( 0 pr + µ u'( = 0 p + µβu'( = 0 pr + λu'( µ u'( = 0 p + λu'( µβu'( = 0 (A λ = / R > and µ = p R / u'( > 0. Elmnatng λ and µ, we then get equaton (8 and (9 n Proposton 3. Sne u( + β u( = u( + βu(, and u' ( / u'( u'( / u /(, we have <, thus verfy that IC holds. Proof of Proposton 4. Let u( p be the pereved outsde opton n the ompettve equlbrum when the proporton of non-sophstated depostors s p, ths s also the sophstated depostors atual welfare wth suh a dstrbuton of types. Take any p > p. Sne the fnanal ntermedary makes money on non-sophstated depostors, f the proporton nreases to p and the outsde opton s u( p, the equlbrum ontrat wth proporton p makes postve profts and satsfes the partpaton and nentve onstrants. Therefore we must have u( p > u( p. Proof of Proposton 5. We have argued n the text that when Condton holds, depostors would self-selet nto the same ontrats as when ˆβ s observed, so t s a ompettve equlbrum. Now we prove t s a unque one. By ontradton suppose there s an equlbrum n whh not all ˆβ types are offered the ompettve equlbrum as when ˆβ s observed. Let u be the pereved utlty of ˆ β n ths equlbrum. Frst we show that there exsts some suh that u < u. Suppose u u for all. Then even f ˆβ s observable, the fnanal ntermedary an only break even and aheve ths by provdng eah type the equlbrum ontrat a ontradton. Now onsder the hghest suh that u < u. For suffently small ε > 0, we an

21 fnd a ontrat wth pereved outsde opton u + ε < u that attrats type ˆ β and makes postve profts on ths type. For j >, u + ε < u < u j < u j, so ths ontrat does not attrat type ˆ β j. Besdes, even f t attrats some ˆ β j for j < (though by self-seleton ths may not happen, they would selet the non-sophstated repayment opton and generates postve profts. So the ontrat makes postve expeted profts. Proof of Proposton 6. The ase of the fully sophstated has already been proved n text. The fnanal ntermedary offers lnear ontrat ( R%, T, where T s the dsounted value of total repayment at date. Suppose fully sophstated agents are offered ( / β, T. The ontrat hosen by agents at date 0 depends only on ˆβ, denote the optmal ontrat offered to agents wth ˆβ by ( R % ( ˆ β, T ( ˆ β. The agents pereved utlty s ontnuous n ˆβ. We have shown that for ˆβ = β, the funton has a unque maxmum at ( / β, T, therefore, as ˆβ β, we must have ( R% ( ˆ ( ˆ β, T β ( / β, T. Therefore the welfare of the unsophstated approahes that of the sophstated as ˆβ β, whle t s not true wth unrestrted nonlnear ontrat by Proposton 3 and 4. Proof of Proposton 7. Frst, we show that the pereved utlty u n ompettve equlbrum n unrestrted market s hgher than that n restrted market. Suppose not, the ontrat n restrted market satsfes PC, IC, PCC, and breaks even, and s therefore a ompettve equlbrum ontrat. But the ompettve equlbrum we derved annot hold n restrted market, beause u' ( / u'( = βr requres R % =, whle at ths nterest rate sophstated onsumers would not hoose,.(they have the same belef ex-ante ( To prove that restrted market rases welfare for suffently sophstated onsumers we use a smlar argument as proof of Proposton 8. Holdng the dstrbuton of onsumers fxed, for any R %, there s a unque T suh that the ontrat ( R, T yelds zero profts, whh n turn ndues a pereved utlty u as a funton of R %. For β = β, all onsumers are sophstated, so the funton has a unque maxmum at ( / β, T. Sne the funton s ontnuous n β, the same argument n Proposton 6 follows. Next we onsder soal welfare. As β β, both types repayment shedules approah the welfare-optmal one n long-term restrted market. Whle n unrestrted market, the repayment shedule of non-sophstated approahes to '( / u'( = β R and s thus neffent. u Proof of Proposton 8. The fnanal ntermedary s problem s:, max,, p (( R + p (( (A 3

22 s t. u( + u( u. Frst order ondtons wth respet to and are: pr pr p p pr pr p p + λu'( + λu'( + λu'( + λu'( Elmnatng λ, and usng u ( / u'( = β ( + /( +, we have: ' = 0 = 0 (A 4 R + ( + + = ( β + R + β + (A whh s equaton (. And equaton ( s the zero proft ondton of ompettve equlbrum. Proof of Proposton 9 The budget lne an be rewrtten as + / R = T, then = T and = ( RT % /. In perod, type ( =, maxmzes hs utlty u( + βu( suh that + / R = T. So u( / u( = βr and = R( T. Smple exerses show that and T βr u''( = β R u''( + u''( β( u''( + u'( =. R β R u''( + u''( Now assume the eonomy s n the equlbrum, the ntermedary slghtly nreases R % and dereases T suh that the equlbrum dereases and nreases, and u ( + u( keeps onstant, that s, u( + u( = u( δ + u( + δ. Sne β <, we have u( + β u( > u( δ + βu( + δ, suggestng δ + + / R < and δ + ( + / R < T. If R% < R, ( T δ = p ( R + δ < δ δ 0, ndatng that the equlbrum R % s larger than R. Otherwse, we an nrease the ntermedary s proft wthout lowerng self 0 s utlty by nreasng R %. By zero-proft ondton p ( R + + p( R + = R, we have R = ( p + p ( R R TR, thus R% > R mples T <. + Now we show that the T and R % must satsfy TR % / T < 0 and R % / T < 0. Beause the optmal T and R % maxmze u + u(, we have ( R R u '( + + u'( + = 0. T R T T R T

23 Thus, R( β T + T = R R R R( β T Smlarly, we have Sne and Rβ( β u'( ( T u''( β Ru'( = 3 Ru''( + β R u'( R / T R( β TR = T T R( β + /( TR R T β R = T β = TR T ( β R u''( + u''( u''( u'( ( u''( + u'( ( β R u''( + u''( we know that R β R u''( + u''( + u'( ( β R / T R( β = < 0 T T β R u''( + u''( and R( β + R f xu ( x / u ( x > ( β. /( TR = u''( + R β ( β u'( TR u + u''( ( ''( + β Ru''( > As shown n Proposton 6, the optmal R% = R / β when p = 0. In ths equlbrum, R + < R + = R. Beause < T <, there always exsts a R < R suh * * that R + =. In other words, for the equlbrum R % and T gven R and R 0 p =, there exsts a orrespondng R < R and p = suh that the equlbrum R % and T are the same. Ths ndates the equlbrum R % when p = should be larger than R / β. If the optmal R % and T are ontnuously hange wth p, they must be monotone. Otherwse, we an fnd a p suh that the perturbaton of p would not hange optmal R % and T. Ths s mpossble beause the zero proft ondton would be volated when p hanges but R % and T keep onstant. Therefore, when p nreases from 0 to zero, the optmal R % s nreasng, T s dereasng and T R s nreasng, that s, the term premum nreases. < 0 0 Proof of Proposton 0. PC bnds beause otherwse the fnanal ntermedary