To Fed Watch or Not to Fed Watch: Equilibrium Analysis of Bank System Dynamics

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1 To Fd Wach or No o Fd Wach: Equilibrium Analysis of Bank Sysm Dynamics by William A. Brock and Josph H. Haslag * Absrac: W build a modl conomy in which Fd waching occurs. Thr is a hug numbr of blogs, financial lrs, and nws rporing ha alks abou wha h Fdral Rsrv is likly o do. W modl his bhavior by allowing for banks o Fd wach, maning ha h bank will apply a cosly forcasing chnology o prdic nx priod s pric lvl. Hr, h banks accp dposis o insur agains idiosyncraic liquidiy shocks. Wihin his modl conomy, w characriz h pric-lvl dynamics. Our findings hav dirc implicaions for h noion of banking criss hough rlad prcisly o h rol of insuranc, no oupu flucuaions. W driv condiions in which hr ar ndognous oscillaions bwn pric lvl spiks and pric-lvl falls; in ohr words, h modl conomy gnras boom-andbuss cycls as ral balancs flucua from high o low valus. W xnd h modl conomy o considr how hrogniy xiss wih h s of cnral bankrs as hy could hav hrognous forcass of h nx-priod pric lvl. JEL cods: C6, E3, E44, E5 Kywords: random rlocaion, hrognous forcass, banks, fd waching * Brock: Dparmn of Economics, Univrsiy of Wisconsin-Madison and Dparmn of Economics, univrsiy of Missouri-Columbia; Haslag: Dparmn of Economics, Univrsiy of Missouri-Columbia. P a g

2 . Inroducion Fd waching xiss. Thr is a hug numbr of blogs, financial lrs, and nws rporing ha alks abou wha h Fdral Rsrv is likly o do. Howvr, hr has bn vry lil rsarch dircd a undrsanding how Fd waching affcs h macroconomy. Thus, par of our moivaion for wriing his papr is o build, from microfoundaions, a modl conomy in which Fd waching is xplicily xamind. In an arlir papr, Balk and Haslag (99) sudid Fd waching in a monary policy gam. As you rcall, priva informaion was valuabl o h policymakr in hs gams. Th objciv was o giv h policymakr a vil bhind which monary policy could rduc h oupu gap a h lows possibl inflaion cos. Canzonri (985), for xampl, prmid h cnral bank o possss a priva forcas of h dmand for mony. Wih is priva mony dmand forcas, h cnral bank could rduc h inflaion bias inhrn in h monary policy gam by a vil of h signal-o-nois; in ohr words, h public could no ll if h oupu gain from unxpcd inflaion was du o cnral bank acion or unxpcdly low dmand for mony. Balk and Haslag xndd Canzonri by allowing agns o xpnd rsourcs o improv hir forcas of mony dmand, hrby rducing h priva informaion. Hr, Balk and Haslag basd Fd waching as a counrmasur o h asymmric informaion problm ha xisd. In ohr words, Fd waching was a sragic rspons in noncoopraiv gam. Afr h financial crisis in 8, rsarchrs wr askd wih dpning our undrsanding of h causs. Why do booms and buss occur? Wha vns would hlp prdic ipping poins? As Goron and Ordoñz (4) pu i, financial criss ar hard o xplain wihou rsoring o larg shocks. (p.343) Ulimaly, h financial criss ows o basic informaion problm in which small shocks srv as ipping poins. Rcnly, Baison, al. (6) usd incrasing corrlaion across financial insiuions whr h individual insiuions wr rprsnd as nods in a nwork o accoun for ipping poins ha lad o financial criss. Thr is hrogniy in h sns ha ach insiuion rprsns a nod in a nwork ha can b diffrn. Ovr im, h ipping poin rflcs shocks, possibl small, ha build up and h individual nods bgin o look alik and in a bad way. In his papr, w wri down a sandard random-rlocaion modl o provid quilibrium bnchmarks. W hn considr a modificaion o h bnchmark modl conomy. Th cnral bank offrs a mssag o vry bank a zro cos. Howvr, i is cosly o apply h mssag in a forcas of h nx priod s pric lvl. I is h prsnc of comping forcasing a coslss on and a cosly on ha Inrsingly, a googl sarch for [ Fd waching monary policy] yilds 8, rsuls. W nd monary policy in h sarch or w will includ cis o Chainz Fds Waching song. Howvr, a sarch for waching Fdral Rsrv yilds.85 million his. A prfc xampl is h Fd waching blog by Tim Duy a hp://conomissviw.yppad.com/imduy/ Th srucur usd a modifid vrsion of Cukirman and Mlzr (986). P a g

3 accouns for h possibiliy of non-uniform banking dposi conracs. W call h mploymn of cosly forcasing chnology by a bank, Fd waching by ha bank. Banks ha do no us such forcasing chnology forcas nx priod by simply xrapolaing wha happnd las priod. Mor prcis dails will follow. W will driv condiions in which all banks will b Fd wachrs, no banks will b Fd wachrs, and som masur of banks will b Fd wachrs and som masur will no. Th conomics is inrsing. A rprsnaiv dposior will b indiffrn bwn a bank ha Fd wachs and on ha dos no bcaus h non-fd waching bank can afford o offr a highr rurn han a bank who Fd wachs. Rurns, howvr, ar no h only considraion. Th Fd-waching bank compnsas dposiors by providing a mor accura forcas of nx priod s pric lvl. Thrfor, h quilibrium consising of boh Fd wachrs and non Fd wachrs is a balancing ac; offrd a highr sa-coningn rurn ha is uncrain agains a lowr sa-coningn rurn ha is accura. Pu his way, hr is a riskrward radoff ha h dposior is willing o accp. Thr ar ohr ways o analyz Fd waching ha do no involv sragic acions. In our sup, banks ar h Fd wachrs. Banks provid insuranc for dposiors facing idiosyncraic liquidiy shocks. 3 In ordr o maximiz h xpcd lifim uiliy of hir dposiors, banks mus forcas h nx priod s pric lvl. In our firs modl, h cnral bank obligs by frly offring informaion. I is cosly o ransform h cnral bank informaion ino a prfc-forsigh poin forcas of nx priod s pric lvl. I is h cosly procssing ha w inrpr as Fd waching. Bcaus w sudy Fd waching as an applicaion of xpcaions formaion, our papr is rlad o work ha is covrd by Brock and Homms (997), Grandmon (998), Homms (3), Sims (3, 6) and h raional inanion modls. Following in his radiion, w considr how an quilibrium wih hrognous forcass ha is, allowing for banks o Fd wach or no is affcd. In hr an quilibrium in which Fd waching occurs. Th answr is ys. In addiion, w characriz h quilibrium dynamics in modl wih Fd waching. Bcaus banks play a faur rol in h modl conomy, our findings hav dirc implicaions for h noion of banking criss hough rlad prcisly o h rol of insuranc, no oupu flucuaions. W driv condiions in which risk avrsion and h siz of h liquidiy shock play imporan rols. Our rsuls indica ha givn vry risk-avrs dposiors, banks facing small nough liquidiy shocks ar associad wih unsabl pric-lvl dynamics. This is somwha counrinuiiv. Firs, h dynamics characriz ndognous oscillaions bwn pric lvl spiks and pric-lvl falls; in ohr words, h modl conomy gnras boom-and-buss cycls as ral balancs flucua from high o low valus. In our sup, h swings ovr im do no dpnd on unanicipad incrass in h dmand for liquidiy. 3 Th classic rfrnc for hs modls is Diamond and Dybvig (983), S also, Champ, Smih, and Williamson (996). 3 P a g

4 Mor imporanly, our rsuls do no spak o boom-and-buss cycls in oupu. Rahr, our rsuls bar on h boom-and-buss consumpion parns by movrs rlaiv non-movrs. Mor gnrally, whn h pric lvl riss sharply, hos who nd liquidiy will consum lss compard wih priods in which h pric lvl is in h low par of h cycl. Sinc oupu is fixd a h ndowmn plus h rurns o capial, h boom-and-bus cycl rfrs o h parn of consumpion by hos nding liquidiy. No ha h cycl is rvrsd for hos no nding liquidiy; ha is whn movrs consum mor goods, non-movrs will consum lss and vic vrsa. Hnc, our boom-and-bus cycls rfr o h disribuion of consumpion no incom. W xnd h basic modl conomy o considr h impac ha hrogniy wihin h FOMC would hav on h banking sysm and h pric lvl dynamics. Mor spcifically, w sudy an conomy in which h mony supply procss rsponds o movmns in h pric lvl. 4 To g a hrogniy wihin h FOMC, w considr a cas in which hr ar diffrn forcass of nx priod s pric lvl usd by h cnral bank whn choosing nx priod s mony supply ha ar usd o choos h mony supply procss. In ohr words, h ndognous mony supply rul is lik a McCallum Rul (s McCallum (988)). Th local dynamics ar inrsing. In an conomy in which h cnral bankrs rly xclusivly on h pas in paricular, on las priod s valu of mony o prdic nx priod s valu w find ha h dynamics xhibi xplosiv oscillaions. Th inrmdia valu horm allows us o driv condiions in which local dynamics xhibi dampd oscillaions. In addiion, w driv condiions in which hr is a coninuum of quilibrium oucoms. Our rsuls bar on h indrminacy of h cnral bank s acions on h quilibrium pah. Wih a coninuum of quilibria, h pah of h quilibrium dynamics will no ncssarily b a symmric parn of boom and buss wih rspc o h pric lvl and h disribuion of consumpion. Wih rgular parns, h qualiy of h insuranc offrd is now a condiional samn; in low-pric priods, movrs rciv good insuranc as rsourcs ar shifd from non-movrs o movrs. Convrsly, in high-pric priods, h consumpion loss suffrd by movrs bcaus of ra-of-rurn dominanc is vn grar. This parn happns from priod o priod. Wih a coninuum of quilibria, howvr h insuranc qualiy is ponially mor volail. Th papr is organizd as follows. Th bnchmark modl conomy is dscribd in Scion. W modl Fd waching in Scion 3. In Scion 4, w characriz h pric-lvl dynamics of h modl conomy whn on banks Fd wach, driving h condiions in which h pah is unsabl. W considr a cnral bank wih hrognous forcass in Scion 5. A brif summary and conclusion is prsnd in Scion 6. 4 In his scion, h mony supply procss is spcifid as a funcion of h valu of mony, which is h invrs of h pric lvl. 4 P a g

5 . Th modl conomy In h conomy, hr ar an infini numbr of discr im priods. L =,,, dno h indxd valu for ach da. Th physical nvironmn consiss of wo spaially-sparad islands. A ach da, hr is a coninuum of masur on of wo-priod livd agns born on ach island. In addiion, hr is a on gnraion, lablld h iniial old who liv on ach island for on priod bu only xis a da. Th iniial old also consiss of a coninuum of masur on of agns. Thr is a singl, prishabl consumpion, rprsnd by c. Th consumpion good can b ransformd ino a sorag good a a on-for-on ra. For ach consumpion good sord a da, hr ar x unis of h da-+ consumpion good. Young agns valu consumpion whn young and whn old. W assum ha prfrncs ar im-addiivly sparabl. Agns rciv an ndowmn of y unis of h consumpion good a h bginning of h priod whn young and nohing whn old. A h nd ach da, young agns rciv noic of whhr hy ar going o b rlocad or no. Th probabiliy of bing rlocad is rprsnd by, whr. Suppos h valu of h rlocaion probabiliy is consan ovr im. 5 By h law of larg numbrs, also rprsns h masur of young agns who ar rlocad. Th ky fricion in his conomy is ha nihr goods nor claims agains goods can b vrifid across islands. 6 Only fia mony is rcognizabl across h wo locaions. Th cnral bank ndows h iniial old wih M is h aggrga supply of fia mony a h bginning of da. Th law of moion for h sock of fia mony is characrizd by M M for. h budg consrain for h cnral bank is givn by M M p, whr mony craion (dsrucion) is usd o mak a lump-sum ransfr (ax) ha applis o ach young prson. This conomy has bn xnsivly sudid in paprs by Smih and Bncivnga (99) and ohrs. 7 For h sak of saving spac, a wlfar-improving bank will aris in a compiiv mark for dposis. As long as movrs and non-movrs ar idnifiabl, a sa-coningn dposi conrac is offrd m n ha pays movrs a ra, dnod r, and non-movrs a ra, dnod r. 5 W us a modifid vrsion of h Champ, Smih and Williamson modl. S Haslag and Marin (7) for dscripion of h random-rlocaion modl. 6 No ha his unrcognizabiliy assumpion is a form of imprfc rcord-kping. 7 S Smih and Schrf (997) or Marin and Haslag (7). 5 P a g

6 Th bank will sk o maximiz h xpcd uiliy of h dposior. Th young agn maks h consumpion-saving dcision, dposiing all goods wih h bank. Taking h quaniy of goods dposid as givn, h bank maximizs xpcd uiliy subjc o a balanc-sh consrain, a payoff consrain ha applis o movrs and a payoff consrain ha applis o nonmovrs. Formally, hs consrains ar wrin as d m s () * p Hr, * d sands for h quaniy of goods dposid by h young agn and m is h nominal quaniy of dollars ha h bank chooss o hold. 8 In addiion, h bank facs wo payoff consrains. Th bank mus hav sufficin rsrvs o payoff movrs wih h promisd rurn, which is wrin m * r d m p and nough rurns from sorag o saisfy h promisd rurn o non-movrs, which is wrin as * r n d xs. A compiiv bank will sk o maximiz h xpcd wlfar of h rprsnaiv (young) dposior, which is wrin as m * * n U y * d V r d V r d subjc o h balanc sh idniy and h wo payoff consrains. Dfin h inflaion ra as p p. m p L. Thn h wo payoff consrains ar wrin as * d r p () m p r n x. (3) 8 In his sup, h quaniy of dposis saisfis h firs-ordr condiion for h young agn, wrin as * m m * n n * U ' y d r V ' r d r V ' r d. 6 P a g

7 r r is h gross ral rurn offrd o movrs (nonmovrs). No ha quaion () m n Hr, rprsns h bank s rurn o movrs as a funcion of h fuur inflaion ra. Afr subsiuing, w can rwri h bank s problm as * * * maxu y d V d V xd. Th firs-ordr condiions for h bank s problm is * ' d ' * V. V. xd. ' m * ' n * Or, furhr simplificaion yilds V r d V r d R whr R. x, Th maximizaion problm is frqunly spcifid in rms of h mony-o-dposi raio. Th oucom offrs a sraighforward conomic inrpraion. In our cas, i is convnin o xprss h bank s problm as max{ V ( M / ( p )) ( ) V ( xs / ( ))} max{ V ( m p / ( p ) ( ) V ( x( d m ) / ( ))} m (4) whr m M / p and d m s. (Rmmbr ha h quaniy of goods dposid is akn as givn.) Th firs insigh w obain from (4) is ha h bank s problm for h spcial cas wih log uiliy, h maximizaion problm rducs o max{ ln[( m p / ( p )] ( )ln[ V ( x( d m ) / ( ))]} max{ ln( m ) ( )ln( d m ) rms} m (5) whr rms dos no dpnd upon h choic variabl m. In paricular h bank s dmand for mony dos no dpnd upon is forcas of nx priod s pric lvl, p. Hnc a bank wih log uiliy funcion, h bank has no incniv o buy Fd waching srvics o g a mor accura forcas of h fuur pric lvl or h fuur pric of mony / p 7 P a g

8 w.r.. For h gnral class of powr uiliy funcions ha is, for h CRRA prfrncs h FONC m rduc o (afr a bi of algbra), ( d m )( p / p ) ( ) m x (6) ( )/ ( )/ Equaion (6) lls us ha h xpcd rurn from holding mony is qual o h xpcd rurn from holding sorag. 9 A saionary compiiv quilibrium consiss of (i) agns choosing dposis and consumpion whn young and whn old o maximiz xpcd lifim uiliy, aking prics and h cnral bank s policis as givn, (ii) banks choos sa-coningn rurns o movrs and nonmovrs o maximiz xpcd lifim uiliy of dposiors, aking h cnral bank s acions as givn, and (iii) h mark for h consumpion good, dposis, h sorag good, and mony clar. Givn a consan mony growh ra, hr xiss a saionary quilibrium in his modl conomy. Th quilibrium laws of moion ar drivd as follows. Th mony mark claring condiion is wrin as M p d. (7) M Solv quaion (7) for h pric lvl, yilding p. d Thus, h quilibrium inflaion ra is wrin as p M d p M d. Wih h mony supply rul, his rducs o, x, d, x,, y,,,,, p d. (8) p d x d x y 9 As a chck on his formula no ha whn, i.. h log cas, (6) implis ha h opimal choic of m dosn dpnd upon h forcas of nx priod s pric lvl and also dosn dpnd upon h rurn x from giving up h convninc of liquidiy. Equaion (5) may b usful o kp in mind for radrs who prfr a concr xampl whil rading h absrac discussion ha follows. Afr giving a gnral absrac discussion w rurn o Equaion (5) for som of our rsuls. 8 P a g

9 Equaion (8) is a firs-ordr nonlinar diffrnc quaion in h inflaion ra. quaions for h rsrv-o-dposi raio and h dposis ha is, Combind wih h,, and x d,,, d x y --w hav a characrizaion of h quilibrium in his conomy. Smih (99) confirms ha h saionary quilibrium in his random-rlocaion modl is unsabl. In gnral, h numbr of bank in his modl conomy is indrmina. For h purposs, w assum ha banks comp for dposis from among h coninuum of young consumrs. Morovr, hr is mor han on bank opraing. Indd, as w procd h radr will noic ha w nd som hrogniy among banks so ha hr ar diffrn yps of banks opraing. Th numbr is larg in h sns ha ach bank will ak wha h ohr bank is doing as givn and no on bank is capabl of xring any mark powr. 3. An conomy wih cnral bank informaion Th purpos of his scion is o inroduc informaion producd by h cnral bank ha affcs h forcass ino h conomy. Suppos h cnral bank producs informaion. For h sak of concrnss, l h cnral bank produc a mssag. Th mssag, rprsnd by, is coslss o produc. Thr is valu o h mssag. Th mssag is frly offrd by h cnral bank and hrfor w avoid daling wih asymmric informaion issus. W assum ha h bank can acquir h mssag and procss i of h mssag is ha h bank can us i o improv is forcas of fuur pric lvl. W assum ha c is h quaniy of h consumpion good usd o produc h mssag, wih c c c,.,.. W sar wih a simpl vrsion in which hr ar wo chnologis; on ha uss h mssag and h ohr ha dos no. Thrfor, for simpliciy, w assum ha producd). if h mssag is producd (no Th valu of h mssag coms from applying i o forcasing nx priod s pric lvl. To mak his mor concr. Considr an conomy in which hr ar wo forcasing chnologis. In on, Hr, w subsiu h soluions o h firs-ordr condiions for h consumr and for h bank. No ha d d r, r,, y m n 9 P a g and xd,,. Using quaions () and (3), subsiu for h rurn banks pay o movrs and nonmovrs o rprsn h soluions o h FOCs as in Equaion (5). Th dynamics in his modl conomy hav bn sudid xnsivly. S (Brock (99) for a horough discussion of h dynamics in an ovrlapping gnraions conomy in which fia mony is valud.

10 h bank can procss a mssag a cos c c and h chnology producs h prfcforsigh soluion so ha hi, E p I, p, whr, h I sands for h xpcaion opraor for nx priod s pric lvl condiional on h informaion s posssss by all banks and on h valu of h cnral bank mssag usd by h bank. In addiion, hr is fr chnology availabl in which h da-+ pric lvl forcas is simply h prvious priod s pric lvl. Formally, h I, E p I, p. Hr, E rprsns h condiional xpcaion opraor for nx priod s pric lvl whn hr is no Fd waching chnology usd by h bank. Th cos of producing his forcas is assumd o b zro. Th upsho of his srucur is characrizd by h forcas rror varianc ha ach bank is willing o olra. Bcaus of h hrognous forcas chnologis, vn hough ach bank is x an idnical, h x pos disribuion of banks is no ncssarily dgnra. As long as som posiiv masur of banks us h mssag o improv h forcas, hn h disribuion of yps is nondgnra. W will g o a formal vrsion of ha dcision in h nx scion. Th fundamnal ida is ha diffrn forcasing chnologis ar rlad o diffrn lvls of Fd waching. In h cas of h prfc-forsigh chnology, h bank innsivly wachs h cnral bank, xpnding rsourcs ha improv h forcas accuracy of nx priod s pric lvl. Bcaus no cnral bank mssag is procssd in h ohr chnology, w assum i is fr o forcas nx priod s pric lvl by using las priod s pric lvl. 3. A rvisd bank problm Th physical nvironmn is modifid such ha h bank chooss which of wo forcasing chnologis o us. Th bank s dcision is basd on maximizing h xpcd wlfar of h rprsnaiv dposior. Bcaus Fd waching is cosly, h bank passs h cos of h mor accura forcas on in h form of lowr rurns o h dposior. Thus, hr is a radoff for h dposior bwn diffrn rurns and mor accura prdicion of h rurn on mony. W us h dualiy of consumr s problm so ha h fd-waching rsourc xpndiur is quivaln o an xpcd uiliy cos. By doing so, w can borrow from Brock and Homms (997) by spcifying a finss masur. In ohr words, h valu of ach forcas is drmind by a finss funcion. Considr a cas in which a bank can ihr choos a chnology ha yilds a prfc-forsigh prdicion of h nx priod s pric lvl or us a coslss chnology in which nx priod s pric lvl is forcasd from a hisory of pric lvl. Th finss masur is dsignd o wigh h uiliy loss of diffrn forcas rrors. For banks a da, h mos rcn forcas rror obsrvd by ach bank is P a g

11 p For ach bank ha mad a dcision a da, rmmbr ha using p 3 is h forcas of. p3 da pric lvl. Consqunly, h mos rcn forcas rror is p p3 ha h da- bank can obsrv. Now w can consruc h finss masur. L h finss funcion b dnod by L. For a bank h I, p using and comparing ha wih h bank using h prfc-forsigh chnology, whr h prfc-forsigh forcas chnology is rprsnd by p h I, whr I, mak up h informaion s wih Fd waching. L h fracion of prfc-forsigh banks b dnod by n and n dno h fracion of banks using las priod s pric lvl as h forcas chnology. Th following quaions dscrib h procss ha drmins h fracion of banks using h hi, chnology: n L L F (9) whr F is h cos of using h prfc-forsigh chnology for h pric lvl forcas. Th implicaion is ha p whr q. Afr collcing rms and simplifying, w obain p p q q () 6 p p 3 q q 3. () Wih h squard forcas rror in plac, w spcify h objciv funcion. Following Brock and Homms (997), our goal is o drmin h facion of banks using h chnology ha rlis on h prvious priod s pric obsrvaion and hos using h prfc-forsigh chnology. Th bank s finss funcion dpnds on h on-sp ahad forcas rror. Throughou our analysis, w assum ha. Th las sp is o consruc h bank s binary choic problm. Each bank facs a discr, binary choic ovr h wo forcas chnologis. No ha F is masurd in h sam unis as h capializd squard forcas rrors in h finss masur for h backwards-looking chnology. P a g

12 Following Andrson, Di Palma and Thiss (99), using hir spcificaion of h logi disribuion o characriz h fracion of young agns using h prvious priod s and h prfc-forsigh chnologis. Wih h Equaion (9), w hav h choic probabiliy for h banks using h hisory of h pric lvl. Hr, h paramr capurs h innsiy of choic. Considr, a bank would choos h prdicor wih h lows ralizaion of L in quaion (9). By h law of larg numbrs, h fracion of banks choosing h prvious priod s (prfc-forsigh) chnology is qual o h probabiliy ha any on bank chooss h prvious priod s (prfc-forsigh) chnology. Th logi modl is usful hr bcaus h innsiy-of-choic paramr,, provids a convnin way of rprsning confidnc in h choic of h chnology. For xampl, if, h bank chooss prdicors ha ar random whil if hn ach bank is choosing h chnology wih h highs finss scor. Th soluions for h fracions of banks choosing h prfc-forsigh and coslss chnologis ar givn by h following quaions n F L F () n L L F (3) whr n n. Th soluion for h fracion of banks of ach yp is usd by h bank agn o choos h quaniy of dposis. Sp in h bank s opimizaion program is prsnd blow, aking as givn h yp of forcas chnology slcd in sp. Th soluion o h bank s problm rsuls in quaniy of dposis ha is rlad o h xpcd rurns on mony. Bcaus a fd-waching bank will us up rsourcs o acquir h prfc-forsigh m chnology, h rvisd vrsion of quaion () is wrin as d c s. In shor, h firs-ordr p condiions will yild rducd-from funcions for h quaniy of dposis rprsnd by p d d r,r ;, c n m p (4) P a g

13 d d r n m,r p ; p (5) whr sands for all h ohr variabls ha h bank aks as givn whn solving hir problm. No ha h -chnology forcass h nx-priod s pric lvl using only h prvious priod s obsrvd pric lvl. Nx, h bank will choos o h rsrv-o-dposi raio o maximiz xpcd uiliy for h agns, aking h dposis as givn. Formally, ach bank maximizs xpcd dposior uiliy m n U y d c V r d c V r d c, max m n U y d EV r d V r d (6) subjc o h balanc-sh consrain, h liquidiy consrain for movrs and h liquidiy consrain for non-movrs. In Equaion (6), h xpcaion opraor accouns for h xpcd uiliy for movrs aking ino accoun ha h gross ral rurn on mony is a random variabl. For h cas w hav bn using, h rsrv-o-dposi raio can b wrin as p, p, p ;. (7) In ohr words, opimal rsrv-o-dposi raio is a scond-ordr nonlinar diffrnc quaion in h pric lvl whr agns ar forming xpcaions of nx priod s rurn. 3. Equilibrium For h fd-waching conomy, w dfin a saionary compiiv quilibrium as follows: (i) wo-priod livd consumrs choos plans for dposis and consumpion whn young and whn old o maximiz xpcd lifim uiliy, aking h pric lvl and sa-coningn rurns offrd by banks as givn; (ii) banks choos h forcas chnology hy us o minimiz loss of forcas accuracy, aking h (uiliy) cos of Fd waching as givn; (iii) banks, condiiond on h forcas chnology hy us, choos sa-coningn rurns o movrs and nonmovrs o maximiz xpcd lifim uiliy of dposiors, aking h cnral bank s acions as givn; and (iv) h mark for h consumpion good, dposis, h sorag good, and mony clar. Toghr, quaion (6) hlps us undrsand ky faurs of h quilibrium in a modl wih Fd waching. Banks ar dividd ino wo groups: hos ha xpnd rsourcs hrough fd wach in ordr o 3 P a g

14 improv forcas accuracy (hrafr FW) and hos ha do no (hrafr NFW banks). In gnral, hr ar hr oucoms. If c, hn hr is a dominan sragy; ha is, h cos of forcas rrors associad wih NFW banks rducs xpcd lifim uiliy and vry bank chooss h FW forcas chnology. Alrnaivly, for a givn xpcd-uiliy loss du o forcas accuracy, if Fd waching is xpnsiv for xampl, c Lasly, hr xiss valus of d -- hn xpcd wlfar is grar undr h NFW chnology. c and wlfar loss du o forcas accuracy in which h bank is indiffrn bwn h FW forcas chnology and h NFW forcas chnology. In his cas, hr is an indrminacy; hr ar infini combinaions ha saisfy n n n / so hr s NOT an infini numbr of combinaions ha saisfy n. In h discr choic modl n n. Th boom lin is ha hr ar wo facors affcing xpcd lifim uiliy. Th opimal bank dcision is o choos h FW or h NFW chnology ha maximizs xpcd lifim uiliy, balancing h lowr rurns o dposiors associad wih xpnding rsourcs on Fd waching o improv forcas accuracy agains h highr rurns and forcas rror associad wih h NFW chnology. Pu mor formally, h radoff is sraighforward. Wih h FW chnology, boh movrs and non-movrs ar subjc o lowr rurns. Each young prson dposis promisd a sa-coningn rurn qual o m, r and d goods wih h bank and is n, r for movrs and non-movrs, rspcivly., Ths rurns ar paid afr h fd-waching cos is xpndd by h bank, yilding r m d c n, for movrs and r d c for non-movrs. If you valuad xpcd uiliy for quaion (7) a c, hn i follows ha xpcd lifim uiliy for dposiors is grar wih Fd waching han wihou. Convrsly, considr c d, whr all h rsourcs availabl for paying rurns o dposiors vanish. In his sing, h NFW chnology will yild xpcd lifim uiliy ha is grar han xpcd lifim uiliy wih h FW chnology. By h man valu horm, h coninuiy of h uiliy funcions implis ha hr xiss a valu of c such ha xpcd lifim uiliy is qual wih h FW and h NFW chnologis. Th cos of fd-waching lowrs h rurns ha banks can afford o pay hir dposiors. Wih h NFW chnology, wlfar is lowr bcaus hr is a disribuion of forcas rrors associad wih ach chnology; indd, h disribuion of forcas rrors wih rspc o rurns wih h FW chnology is dgnra and dominas h disribuion of forcas rrors wih rspc o rurns wih h NFW chnology in h sns of firs-dgr sochasic dominanc 4 P a g

15 To mak hings mor concr, considr h cas wih log uiliy. Formally, h bank choosing h FW chnology is maximizing max ln ln ln y d c d c x d c and h objciv for h bank choosing h NFW chnology is m, maxln y d E ln r d ln xd. I is sraighforward o show ha d d y. In ohr words, dposiors will pu half of hir afr-ax incom ino dposis. Considr only quilibrium in which h xpcd lifim uiliy is qual across h wo yps of banks. Basd on h diffrn sa-coningn rurns offrd, dposiors will know whhr h bank is a fd-waching bank or a non-fd-waching bank. Dposiors ar compnsad for aking on grar risk wih banks ha us h NFW forcas chnology. W assum h bank has alrady mad is commimn o bing a fd wachr or no and hn offrs a mnu of sa-coningn rurns o dposiors. Th fracion of banks offring highr sa-coningn rurns ha is, h banks using h NFW forcas chnology ar fixd bfor dposiors mak hir choic. Bcaus h dposior is indiffrn bwn h wo chnologis in his cas, h mark offrs wo producs wih diffrn aribus ha yild h sam xpcd uiliy o dposiors. 3.3 A Concr Analyical Exampl Th discussion abov was rahr absrac and on can also quarrl wih h snsibiliy of h loss funcion () ha is clairvoyan and hrfor hr is no way ha banks can ac on i. W assum an xognously givn sram of dposis { d } and ra h banking scor as dividing hs dposis bwn liquid forms for h movrs and lss liquid bu highr rurns for h sayrs. Sinc h acual amoun of dposis rcivd by h banking scor is drmind by macro-growh forcs a a lowr frquncy han highr frquncy problms ypically dal wih by a compiiv banking scor, his simplificaion is dfnsibl. 5 P a g

16 p wri W rurn, as promisd bfor o Equaion (5) h log uiliy cas--abov. For a forcas J( d, p, p ) J( d, p, p ),( for p p ) (8) for h loss whn h forcas of h fuur pric lvl is no h sam as h acual fuur pric lvl. Whil (8) can b xplicily compud for h log uiliy cas, on can s by a Taylor sris xpansion of (8) around p and around c and wriing p p ( p p ) crud approximaion is proporional o h squard forcas rror minus a consan ims h cos c k p( p p ) kcc, which w wri as, im and ar xognously givn and mony supply is consan in im. Taking a closr look, for h powr funcion xampl, w hav ha a. Considr h cas whr dposis ar consan in J( d, p, p ) J( d c, p, p ) { ( m ( d, p ; p ) p / p ) ( ) ( x( d m ( d, p ; p ))) }/ ( ) * * { ( m ( d c, p ; p ) p / p ) ( ) ( x( d c m ( d, p ; p ))) }/ ( ) * * (9) If h cos of obaining prfc forsigh, p p. Hr c is zro, h diffrnc (9) would b ngaiv for m ( d, p ; p ) ( p / p ) d /[ ( p / p ) ( ) x ] * ( )/ ( )/ ( )/ () kp, kc Of cours h consans, dpnds on h da via h dpndnc on ( d, p, p ) and ohr paramrs dscribing h uiliy funcion. Throughou our analysis, w considr valus of h risk avrsion paramr ha ar consisn wih an upward-sloping rlaionship bwn h quaniy of mony hld by banks and h rurn on mony. In our viw, h confins of h Hicks-Slusky-Marshallian dmand sup ar oo rsriciv whn a dposior is also facing rlocaion risk. Going back o Hall (978), S Smih (99) for an xampl of h assumpion on rlaiv risk avrsion paramr ha would guaran h downward sloping rlaionship bwn mony dmand and h rurn on mony. 6 P a g

17 h ulr quaion is capuring boh h incom-subsiuion ffcs and h risk prfrncs of h consumr in raional xpcaions sings. W will dvia from h simpl raional xpcaions sup and also considr diffrn valus of h risk-avrsion paramr as w procd. Mor spcifically, w considr, corrsponding o cas in which h dsir for liquidiy is so srong ha h dposior is willing o hold mor ral balancs vn hough h rurn has incrasd. W procd using h finss funcions and choic ruls n k c k c k p p xp( c ) /[xp( c ) xp( p( 3) )] n n xp( k ( p p ) ) /[xp( k c ) xp( k ( p p ) )] p 3 c p 3 () W firs invsiga local sabiliy a sady sa whn n. L h uiliy funcion b h powr funcion. W bgin by analyzing h quilibrium dynamics whn n a all das; ha is, all banks us h zro-cos, backward-looking chnology o forcas nx priod s pric lvl. Th quilibrium dmand for mony is rprsnd by m M / p ( p / p ) d /[ ( p / p ) ( ) x ] M / p ( )/ ( )/ ( )/ d d s p p d M [ ( p p ( ) x ] / ( )/ ( )/ ( )/ ( )/ s () dp / dp W ak diffrnials of h las lin of (4) and compu o loca sufficin dp condiions for sabiliy (insabiliy) a sady sa / dp, ( dp / dp ). To mak h noaion mor compac, w dfin and wri d m ( p / p ) d */[ p / p ( ) x ] d * f ( p / p ) (3) Thr ar hr cass: (i) iff, (ii) iff, (iii) iff. For a s of I banks wih hrognous forcass p,,,..., i, i I wih consan fracion n i of h banks making h forcas pi,, and a cos c for a forcas of yp i, hn h mark claring condiion, whn valuad a sady sa, is 7 P a g i

18 I * d ( ) i () i s i M p p n f d M. (4) W ak h drivaiv of Equaion (4), obaining I I * * i () i i '()( i, ) i i i dp n f d n f dp dp d I I i i i i, i i i dp n f ()( d * c ) n f '()( dp dp )( d * c ) I I I i i i i, i i i, i i i d *( f () f '()) dp ( f () f '())( n c ) dp f '() d * n dp f '() n c dp I will b usful o kp h following s of rsuls: f() / ( ( ) x ) f '() ( ) x / ( ( ) x ) f f x x () '() ( ( )( ) ) / ( ( ) ) / Rmmbr ha h funcion, f p p, is a shorhand way o wri h fracion of dposis hld in h form of ral balancs. No ha h rsrv-o-dposi raio is ngaiv whn valuad a sady sa and ; in ohr words, wih sufficinly risk-avrs consumrs, h bank will rduc is liquidiy provision for a givn lvl of dposis in rspons o incras, for xampl, in h rurn on liquidiy. Cas : All banks us p p Wih n, n, w compu h pric lvl dynamics in h nighborhood of h sady sa. Afr som algbra, w obain dp / dp f '() /[( f () f '())] (5) Thr ar hr subcass corrsponding o,,. Th cass, ar boh locally sabl and h convrgnc is monoonic. Considr a spcial cas in which dposiors ar no risk avrs ha is, --and Fd waching is prohibiivly xpnsiv, h saionary quilibrium is sabl. Our rsuls conras h convnional raional xpcaions finding ha ral balancs convrg o zro or 8 P a g

19 divrg o infiniy. For h cas in which fia mony asympoically loss is valu, h bank no longr offrs any insuranc. Rahr, h consumrs acquir capial wihou inrmdiaion. By having Fd waching b xpnsiv, dposiors ha ar no vry risk avrs convrg o an quilibrium in which banks offr liquidiy insuranc bcaus of h naïv pric-lvl forcas. Th rmaining cas is. For ha cas w hav, no ha h dynamics xhibi oscillaions. Th condiion for xplosiv oscillaions is rndrd dp / dp f '() /[( f () f '())] iff f x x f f '() ( ) / ( ( ) ) [( () '())] To driv h sufficin condiions, w xpand h xprssion o obain ( ) x / ( ( ) x ) ( ( )( ) x ) / ( ( ) x ) iff ( ) x ( ( )( ) x ) In h limiing cas wih, h xplosiv oscillaion condiion simplifis o ( ) / x. Thus, w hav a rlaionship bwn h fracion of movrs and h rurn on capial. In his xrm cas, w find ha h smallr h fracion of movrs, h mor likly ha xplosiv oscillaions will rsul. W do no wan o mak oo much of hs rsuls bcaus h limiing cas prsums an unralisic lvl of risk avrsion. Howvr, h conomics ar prhaps counrinuiiv a a surfac lvl; namly, whn liquidiy shocks ar smallr in his modl conomy, xrm risk avrsion will rsul in insabiliy. Insofar as banks ar prsn as a mans of insuring again h idiosyncracic liquidiy shocks, i sms odd ha whn liquidiy shocks ar small, as masurd by h fracion of movrs, h conomy is unsabl. Nx, w considr h condiions ha saisfy insabiliy for a mor gnral class of risk avrsion paramrs. W can summariz h sufficin condiion in h following proposiion. Proposiion : x, whr dp dp if and only if. Proof: This is a simplificaion of Equaion (5). 9 P a g

20 Hr is whr h algbra from h limiing cas is usful. For a givn rurn on capial, Proposiion indicas ha h lf-hand-sid of h inqualiy dpnds on h fracion of movrs and on how risk avrs consumrs ar. Grar risk avrsion mans ha as h risk avrsion paramr approachs infiniy. No ha as h fracion of movrs approachs zro. Thus, h righ-hand sid is posiiv and ging largr as h fracion of movrs gs smallr and smallr. As w obsrvd in h limiing risk-avrsion cas, h smallr h liquidiy nds in h modl conomy, h mor likly i is ha all banks using simplisic backwards looking xpcaions insad of paying h coss of forward looking xpcaions, i.. Fd waching, coincids wih unsabl, oscillaing pric lvl dynamics. 4. Equilibrium wih Hrognous forcass by h cnral bank Suppos h cnral bank uss diffrn forcas chnologis. Th mpirical basis for his xprimn is way in which h Fdral Opn Mark Commi opras. Wih h Board of Govrnors and h fiv rgional bank prsidns, i is dfnsibl o suppos ha ach paricipan is bringing a diffrn forcas o of h Unid Sas conomy o h abl. In his scion, w considr how ha approach would affc h quilibrium, spcially h dynamics. In his vrsion, h cnral bank adops a McCallum rul, ling h mony supply rspond o movmns in h valu of mony. W us h following noaion, is rprsnd as follows:. p Th mony supply funcion, s M c c, c (6) wih. Hr, h McCallum rul spcifis ha mony supply rsponds posiivly whn h da- valu of mony is grar han h criical valu of mony. Alrnaivly, if h valu of mony is oo low, h mony supply is s qual o zro. Th dmand for mony is h fracion of afr-ax incom. W can rprsn h dmand for mony as a funcion of h valu of mony as follows: P a g

21 M d y for h log uiliy cas. Nx, considr h cnral bank having accss o wo forcasing chnologis. In h cas in which no rsourcs ar usd, h cnral bank forcass h currn priod valu of mony is qual las priod s valu of mony; ha is,. W firs vrify ha a saionary monary quilibrium xiss. Lmma : In sady sa, h quilibrium quaniy of mony is posiiv. Proof: Sady sa is characrizd by y c. 3 Rwri his xprssion as y. c (7) Thr ar wo soluions for h sady sa rprsnd by c D 4, (8) D 4 8. whr c y Thrfor, c iff collc rms, and subsiu for D o obain c. c D 4 Squar boh sids of h quaion, 4 c 8 y 4 y c c. c Bcaus h mark claring condiion is a nonlinar firs-ordr diffrnc quaion, w considr h local dynamics in h nighborhood of h sady sa. Wih h mony mark claring condiion, h drivaiv valuad a sady sa is 3 I is asy o vrify ha M which holds for all das. c P a g in sady sa. Wih M w subsiu using, M

22 y. (9) Afr rarranging, w obain h following quaion y. (3) Nx subsiu for from quaion (8), yilding 4 c cd 4 c 8y y 6. (3) Lmma : If, hn cnral bank using las priod s valu of mony as h forcas for nx priod s valu of mony will rsul in xplosiv oscillaions in h valu of mony. Proof: Equaion (3) rprsns h local dynamics for h valu of mony in h nighborhood of h sady sa. Bcaus hr is a minus sign in from of h xprssion in quaion (3), h local dynamics can only b oscillaions ha ar xplosiv or dampd. Proposiion : Th local dynamics ar markd by xplosiv oscillaions. Proof: Upon xpanding and simplifying quaion (3), w obain h following xprssion c D c. y y No ha h analysis is prformd for a cnral bank ha uss h prvious priod s valu of mony as h forcas for nx priod s valu of mony. Now ha w hav a workd ou an xampl w urn o an analysis of dynamics whr raional poin xpcaions, in his cas i is prfc forsigh, ar availabl a a cos and backwards looking xpcaions ar coslss. Th moivaion is ha raional poin xpcaions rquir a complx srucural modl ha rquirs cnral bank (CB) rsourcs o formula, upda, and mainain whras simpl backwards looking xpcaions ar much chapr. Wih log uiliy, h dynamics ar bing drivn by h mony supply procss. Givn ha h lvl of risk avrsion is fixd, w find is ha h cas for insabiliy dpnds criically on h rsponsivnss of h mony supply procss. In shor, pric lvl dynamics xhibi xplosiv oscillaions whn i is oo xpnsiv o forcas h fuur wih prfc forsigh. P a g

23 Considr h quilibrium dynamics Md ( ) n Ms( ) ( n ) M s( ) (3) F F ( ) n n (, ) / ( ) (33) whr h noaion is, w hop, slf-xplanaory and w us Assumpion : n n. W assum, F, M ' ' s( ) / Md( ) Linariz (34) and (35) a h sady sa o obain, ' ' ' / ( n ) Ms( ) /[ Md ( ) n Ms( )] (34) No ha h rms involving n (, )' cancl ou whn valuad a sady sa. Undr Assumpion, h quilibrium valu of mony,, is locally unsabl. Hnc, if F is infini, n and h forcas a da, is, insad of h prfc forsigh valu. Th dynamics ar locally unsabl bcaus (34) implis. ' ' / Ms( ) / Md ( ). n Wih, howvr, quaion (34) yilds Undr h choic rul (33), w know ha h maximum fracion choosing prfc forsigh a sady sa is n /. Th implicaion is ha h Cnral Bank chooss a mix of forcass ha sabiliz his conomy w nd h following condiion: ' ' ' / (/ ) Ms( ) /[ Md ( ) (/ ) Ms( )]. (C) I is asy o s ha (C) always holds sinc M ( ). ' d To chck whhr condiion is saisfid, w considr h poin a which / whn valuad a sady sa. Pu anohr way, w know hr is a such ha h bordrlin condiion is saisfid. Th qusion is whhr C is saisfid. 3 P a g n

24 Afr a bi of algbra, w can rwri quaion (33) as F F n / ( ) / ( ) (35) whr ' ' ( ) Ms( ) / Md( ). No ha condiion C can b rwrin, using M ' s, as ' M d. (36) Proposiion 3: If ' M d, hn Condiion C is saisfid. Proof: Wih ' M d, i follows ha h dnominaor in Equaion (36) is largr in absolu valu han h numraor. Hnc, Condiion (C) is saisfid. Th conomics mbddd in Proposiion 3 ar sraighforward. As long as h McCallum rul spcifis ha mony supply is no oo rsponsiv rlaiv o h rsponsivnss of mony dmand o changs in h valu of mony, whn valuad a sady sa, hn h valu of mony xhibis dampd oscillaions. 5. Pric lvl dynamics wih cosly forcass Th purpos of his scion is o bring oghr h mony supply rspons whn i is cosly for h cnral bank o forcas nx priod s pric lvl and for h dposi bank o forcas nx priod s pric lvl. Hr, w modify hings slighly, rlaxing h assumpion ha Fd waching, which is cosly, rsuls in prfc forsigh. Rahr, h cos involvs building a srucural modl of h conomy and compuing h sady sa pric lvl. In ordr o mainain a small dposi bank logic whil incorporaing a fricion ha allows banks o b diffrn nough from ohr banks in h mark, w assum ha hr ar a larg numbr of physical marks in which a larg numbr of wo-priod livd consumrs liv. Th ky aspcs of idiosyncraic risk for ach dposior in a mark is raind; ha is, l sand for h fracion of consumrs ha will b movrs o anohr island whn old. Each consumr offrs hir dposis o h bank and h bank maximizs xpcd wlfar for h consumrs in hir physically sgrgad mark, aking h consumrs wlfar as givn. Th bank aks h prfrncs of h consumrs as givn, also 4 P a g

25 aking h prfrncs of all ohr consumrs as givn. In our cas, his mans ha h bank can compu h pric lvl ha would solv hir srucural dmand for mony basd on h consumr s prfrncs bu would no know if hir pric lvl would solv h mark claring condiion. To b mor concr, ach bank would b abl o solv for h sady sa pric lvl basd on h prfrncs in hir physical dposi mark, bu would hav o ak h ru sady sa valu as givn. If w aggrga ovr h paricipaing dposi banks o obain a dmand for mony, w can sar wih h following quaion as h basis for our dynamic analysis. M p p n p p d p p x d ( ) { ( / ) */[ ( / ) ( ) ]} p n { ( p / p ) ( d * c ) /[ ( p / p ) ( ) x ]} p n { ( p / p) ( d * c ) /[ ( p / p) ( ) x ]} M ( p ) s s (37) whr s n sands for h fracion of banks ha us h srucural modl o compu h sady valu of h pric lvl. In ohr words, h bank compus wha sady sa pric would b by aking h prfrncs of h dposiors in hir physical mark as givn. No ha h sady sa pric lvl for all ohr physical marks is akn as givn. In quilibrium, h sady sa pric lvl across marks will b qual. s Hr w assum h coss of gnraing h forcass ar c c. No ha w ar using h suprscrip s o dnos h coss associad wih forming h srucural modl. Th ida is ha in quilibrium, h adding up is don prcisly bcaus consumrs prfrncs ar h sam across all h physically sparad marks. If w ak oal drivaivs of quaion (37) valuad a sady sa p for h cas whr h fracion of banks n s ar fixd, w will hav o us Bindr Psaran or quivaln o pick ou h righ ignvalu and ignvcor in ordr o sudy local insabiliy a p and o loca sufficin condiions on s h n s for sabilizaion of h local insabiliy. Th cas n n is much simplr o sudy. W can impos h sufficin condiions for insabiliy whn sabiliz h sysm holding n hroughou. Rwri (37) in h form, n s s, n and ask how larg n mus b o M p p n d f p p n f p p d c n f p p d c M p (37 ) s s d ( ) { * ( / ) { ( / )( * ) ( / )( * )} s( ) 5 P a g

26 Tak diffrnials of (37 ) and valua a sady sa o obain, (38) dp n d f n f d c n f d c s s { * () { ()( * ) ()( * )} p n d f dp dp p n f dp dp p d c n f d c dp p s s { * '()( ) / { '()( ) / )( * ) '()( * )( / } M ( p) dp ' s s Now, considr a spcial cas in which all prdicors ar fr, c c c, vn if h fracion of popl using ach yp of prdicion ha is, h n s--dpnd upon prvious payoffs as in Brock and Homms (997), h diffrnials of h n s drop ou a sady sa bcaus d n n n d n n n. s s ( ) ( ) W may rwri (38) in h usful form a dp a dp a dp a n d f * '() s s a ( f () f '()){ d * n c n c } a n ( d * c ) f '() (38 ) For h cas whr h fracion of ach yp of bank using h prdicor is xognously fixd, w hav hr a mix of backwards dynamics and forwards dynamics as in Grandmon s much mor gnral classical paprs on his yp of dynamics in conomics (Grandmon (985), (998)). Homms (3) offrs a sa-of-h-ar modlling approach which includs h gnral hory whr h ndognous. W borrow frly from h Homms hory in wha follows. Th quaion for h characrisic roos is nsar ' a a a (38 ) I is usful o brak h analysis of (4 ) ino a s of spcial cass. 5. Prfc-Forsigh is oo xpnsiv s If n,.g. prfc forsigh is oo cosly, and if c, hn w hav dp / dp n f '() / ( f () f '()) (39) 6 P a g

27 Rcall ha w locad sufficin condiions for local insabiliy a sady sa p whn all prdicors ar backwards looking i.., dp / dp, quivalnly, f '() / ( f () f '()). Hnc i is asy o s from (39) how o find a criical valu, * n such ha h dynamics ar locally sabl (locally unsabl) a h sady sa p. * * n n, ( n n ) implis Thr s inrsing conomics in h cas whr all prdicors ar prfc forsigh, n. In ordr o firs build som inuiion for hinking abou forward looking dynamics rcall h quaion for h pricing of a financial ass ha pays a consan arnings y sram forvr, i.. h quaion p (/ ( r))( p y) (/ R)( p y) (4) Th fundamnal soluion of his quaion is p* y / r. If R, for ach p hr is a uniqu soluion p ( p ) ha convrgs o y/ r which is ngaiv sinc r. Of cours h sady sa y/ r is non-conomic in a world of limid liabiliy. If y h quilibrium is zro for boh cass R and h usual cas R. Now urn o diffrnials and h quaion a dp adp (4) which occurs whn n. From (4) i is clar ha if dp / dp a / a, hn for vry iniial dp, h forward dynamics (4) convrgs o zro as. W sum up his discussion ino h following proposiion, Proposiion 4: Forward dynamics produc a coninuum of quilibria whn backwards dynamics ar unsabl. Proof: If dp / dp hn hr ar a coninuum of soluions o (4) and all of hm convrg o zro. If dp / dp, hr is a uniqu no bubbl soluion which is zro. If c, and all prdicors ar prfc forsigh, h cas dp / dp occurs whn, in h cas all prdicors ar backwards looking, h backwards dynamics is unsabl, i.. dp / dp. Hr, hr is conomic conn associad wih h condiions ha yild a coninuum of quilibrium. Thr is a clar diffrnc associad wih Fd waching. Why do w car abou h issu of whhr hr ar a coninuum of quilibria undr prfc forsigh dynamics as would b for h cas if 7 P a g

28 all h banks indulgd in Fd waching giving hm prfc forsigh? Suppos hr is a small prurbaion d from h sady sa valu of mony, a da zro. Considr h normal cas in which h forward dynamics ar unsabl, i.. dp / dp. If h sysm did no jump back o sady sa prurbaions wih dp, h iniial dparur dp from h sady sa would caus h valu of mony o go o infiniy so fas ha far-sighd consumrs would raliz such pahs mak no conomic sns. In h opposi cas dp h valu of mony would bcom ngaiv which is impossibl in a limid liabiliy world. Hnc in h normal cas, dp / dp, limid liabiliy nsurs ha h usual argumns agains dparurs from h sady sa in ass pricing modls apply o mony as wll. Howvr, in h cas, dp / dp, h disciplin associad wih far-sighd consumrs via h sandard argumns do no apply in cass wih mulipl quilibrium ass pric pahs. Basically his siuaion ariss whn, raing mony as an ass, h usual disciplin of forward-looking bhavior fails o anchor h soluion indpndnly of iniial condiions. Hnc ach iniial condiion dp givs ris o is own uniqu quilibrium pah of h mony ass. Buir and Sibr (7) characriz h rlad liraur on mulipl quilibria in monary modls as an unruly liraur and who amp o assss h conomic imporanc of such yps of quilibria. Wih a coninuum of quilibria, hr ar wo snss in which h banks abiliy o provid liquidiy insuranc is marially affcd. Firs, pah dpndnc is associad wih volailiy in h bank s insuranc provision. Bcaus of h indrminacy, vry possibl monoonic convrgnc pah has implicaions for h valu of mony balancs and, hus affcs h consumpion by dposiors nding liquidiy. Mor gnrally, h coninuum of quilibria mans ha h banks provision of consumpion in h fac of liquidiy insuranc is indrmina. Scond, bcaus all pahs convrg o a saionary quilibrium in which fia mony has zro valu, h banks will no b providing liquidiy insuranc asympoically. So, w g noisy and vnually no liquidiy provision by banks in a modl conomy in which hr is posiiv amouns of Fd waching dircd a forcasing h sady sa pric lvl. Th coss o h cnral bank and h dposi banks ar in h form of ffor o compu h sady sa from a srucural modl of h conomy. 6. Summary and conclusions In his papr, w considr Fd waching in a modl conomy. Hr, Fd waching is dfind as h rsourcs dvod o forcasing how cnral bank affcs fuur conomic oucoms. In conras o prvious sudis, Fd waching dos no dpnd on asymmric informaion. Rahr, h conomics 8 P a g

29 focuss on condiions in which improvd forcasing accuracy is worh h rsourc coss. Spcifically, Dposi banks choos bwn wo forcas chnologis: on ha is fr wih imprfc forcas accuracy and on ha is cosly and producs prfc-forsigh poin forcass. W driv condiions in which hr is hrogniy among h dposi banks. Wih hrognous forcass, nonlinar dynamics ar ponially applicabl. Th dynamical analysis has significan implicaions for h liquidiy insuranc ha dposi banks offr. Consumrs fac idiosyncracic risk rgarding fuur liquidiy nds. Banks provid insuranc agains consumpion risk whn consumrs fac hs idiosyncraic shocks. Morovr, liquidiy shocks ar masurd by h probabiliy ha consumr will nd fia mony in h fuur and hus wihdraw arly from h dposi bank. Dposi banks alloca hir porfolio rsourcs fficinly. Wih xpnsiv Fd waching and banks do no Fd wach, w driv condiions in which risk-avrs consumrs rsul in oscillaing dynamics. W driv condiions in which small liquidiy shocks rsul in xplosiv oscillaions. Th implicaion is liquidiy insuranc changs qualiy ovr im. Or pu anohr way, in a priod wih a high valu of mony, consumrs rquiring liquidiy consum mor. In conras, in a priod wih a low valu of mony, consumrs rquiring liquidiy will consum lss. As such, h oscillaions imply ha h qualiy of liquidiy insuranc is volail. Th unsabl pah implis ha in h long run, banks provid zro liquidiy insuranc as fia mony is no valud. Evn in h sabl pah, liquidiy insuranc is volail ovr im. In addiion, w considr a modifid vrsion of h conomy in which h cnral bank applis diffrn forcas chnologis o prdic h valu of mony. Th forcas mars bcaus h mony supply dpnds on h valu of mony. W ar rying o g a h ida ha hr is hrogniy wihin h cnral bank rgarding h fuur conomic oucoms. In his modl conomy, w show ha hr xiss a coninuum of quilibrium pahs. No ha Fd waching is also xndd o considr a forcas chnology ha uss h sady sa valu of mony as h prdicor of nx priod s valu of mony. Th sufficin condiions for a coninuum of quilibria includ a posiiv masur of Fd waching in h form of consrucing a srucural modl conomy o compu h sady sa valu of mony. Howvr, h indrminacy mans hr anohr sourc of volailiy in h sns ha h on canno assss how wll banks ar providing consumpion insuranc agains h liquidiy shocks. Thr ar svral lins for fuur rsarch. In our dynamical analysis, w focus on conomis in which hr ar xognous fracions of dposi banks using a paricular forcasing chnology. How would h dynamical pahs b affcd if on considrd changs ovr im in h masur of banks ha 9 P a g

30 ar Fd waching? Hr, Fd waching is modld in rms of h ffcs on liquidiy insuranc. Anohr qusion would b o sudy h paymn sysm ffcs associad wih Fd waching. 3 P a g

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