Crises in Competitive versus Monopolistic Banking Systems

Transcription

1 Cises in Compeiive vesus Monopolisic Banking Sysems John H. Boyd Gianni De Nicoló Buce D. Smih Peliminay Daf, Mach 0, 003 Absac We sudy a moneay, geneal equilibium economy in which banks exis because hey povide ineempoal insuance o isk-avese deposios. A banking cisis is defined as a case in which banks exhaus hei eseve asses. This which may bu need no be associaed wih liquidaion of a soage asse. When such liquidaion does occu, he esul is a eal esouce loss o he economy and we label his a cosly banking cisis. Thee is a moneay auhoiy whose only policy choice is he long-un, consan ae of gowh of he money supply, and hus he ae of inflaion. Unde diffeen model specificaions, he banking indusy is eihe a monopoly bank o a compeiive banking indusy. I is shown ha he he pobabiliy of a banking cisis may be eihe highe unde compeiion o unde monopoly. This is shown o depend on he ae of inflaion. In paicula, if he nominal ae of inees ae of inflaion is below above some heshold, a monopolisic banking sysem will always esul in a highe lowe cisis pobabiliy. Thus, he elaive cisis pobabiliies unde he wo banking sysems canno be deemined independenly of he conduc of moneay policy. We fuhe show ha he pobabiliy of a cosly banking cisis is always highe unde compeiion han unde monopoly. Howeve, his appaen advanage of he monopoly bank is sicly due o he fac ha i povides elaively less valuable ineempoal insuance. Keywods: Banking Cisis Panic; Moneay Geneal Equilibium. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

2 I. Inoducion Banking panics and cises have been elaively fequen and cosly evens. In he hisoical Unied Saes, fo insance, banking panics occued in 89, 837, 857, 873, 893, 907 and These cises wee no only fequen bu nealy always wee associaed wih majo ecessions. In ode o peven alogehe o, a leas, o educe he fequency of banking cises --- many counies ceaed lendes of las eso and deposi insuance sysems. Ealy in hei incepion such banking sysem safey nes wee ofen accompanied by consideable egulaion. Howeve, as bank egulaion saed o be elaxed in he 970 s, banking cises eemeged as fequen and cosly phenomena. Capio and Klingebiel 997, fo insance, documen he occuence of ove 80 banking cises duing he las hee decades. These cises occued houghou he globe and in all ypes of economies; developed, developing, and ansiional. Moden banking cises ofen involve a diffeen ype of cos han hisoical banking panics in paicula, he govenmen ofen bails ou he banking sysem ahe han allowing deposios o bea loses diecly hey have noneheless ofen been cosly. Ineesingly, hee is a vaiey of expeience wih espec o he oupu losses associaed wih moden banking cises. In a sample of 3 economies ha expeienced a single pos-wa C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

3 banking cisis, Boyd, Kwak, and Smih 00 ague ha 4 of hese economies expeienced no oupu losses whasoeve. Howeve, of he emaining counies in he sample, he median value of los oupu is 7% of he discouned pesen value of cuen and fuue GDP. Fo he seven sample economies wih he lages oupu losses, he median value of los oupu is moe han 0% of he discouned pesen value of cuen and fuue GDP. Thus some banking cises ae associaed wih no eal esouce losses whasoeve, while ohe seem o be accompanied by quie lage esouce losses. Among macoeconomic facos ha help o pedic he occuence of banking cises, inflaion is by fa he mos pominen. Demiguc-Kun and Deagiache 99X show ha highe aes of inflaion ae one of he few macoeconomic facos ha obusly incease he pobabiliy of a banking cisis. And, Boyd, Gomis, Kwak, and Smih 00 ague ha economies ha fail o educe hei inflaion aes duing and afe a banking cisis have a much highe pobabiliy of expeiencing subsequen cises. Thee is a consideable heoeical lieaue ha has examined he elaionship beween banking indusial oganizaion and he isk pobabiliy of failue of individual banks. This lieaue has poduced somewha mixed findings and suffes fom seveal majo limiaions. The fis is ha his wok is mosly paial equilibium and heefoe ignoes possibly-impoan feedbacks fom he banking indusy o banking egulaion o he maco economy, Boyd, Fo example, Helmann, Mudoch and Sigliz 000, Allen and Gale 00, o Boyd and De Nicoló 00. The only empiical eseach we have seen on his opic is by De Nicoló e al. 003 and Beck, Demiguc- Kun and Levine 003. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 3

4 Chang and Smih00. Anohe limiaion is ha his eseach has no consideed inflaion --- o he possibiliy ha inflaion is elaed o banking cises --- even hough he exisence of such a elaionship has song empiical suppo. To ou knowledge, none of his lieaue addesses he following basic issues.. Wha ae he elaive pobabiliies of banking cises in compeiive vesus monopolisic banking sysems ceeis paibus?. Wha is he elaive pobabiliy ha banking cises will involve some eal oupu losses in compeiive vesus monopolisic banking sysems? 3. Wha ae he expeced oupu losses fom a cises in compeiive vesus monopolisic banking sysems? This pape addesses each of hese issues. In addiion, we inoduce a govenmen ha chooses a seady sae inflaion ae nominal ae of inees. We can hen ask: 4. How does he inflaion ae affec he pobabiliy of a banking cisis unde he wo sysems? How does i affec he pobabiliy ha some esouce losses will be obseved? To ha end, we need a moneay model wih banks ha ae poenially subjec o cises, and hee we daw on wok by Champ, Smih and Williamson 996 and Smih 00. Those analyses, howeve, conside only economies wih compeiive banking sysems. In his sudy, we compae he siuaion wih monopolisic vesus compeiive banking sysems. As will be shown, he naue of banking sysem compeiion has significan implicaions fo he pobabiliy ha a banking cisis will occu, and fo wha happens if i does. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 4

5 To explain ou main esuls, i is useful o sa wih a descipion of wha we mean by a banking cisis. The specific noion of a banking cisis ha we employ is moivaed pimaily by he hisoical expeiences of banking panics. Fo insance, Noyes 909 lised he following disinguishing feaues of a banking panic: a he suspension of cash paymens o deposios, b he depleion of cash eseves, c he emegence of a pemium on cuency, and d he use of emegency expediens o povide subsiues fo media of exchange. This is how we will hink abou a banking cisis. We employ a model ha is explici abou he ole of cuency in ansacions, and in which banks aise o insue agens agains wha amoun o liquidiy pefeence shocks. In addiion, he wihdawal demand confoned by banks ha is, he demand o conve deposis ino cuency is subjec o aggegae andomness. When wihdawal demand is high enough, banks will exhaus hei opimally chosen level of cash eseves. As noed by Champ, Smih, and Williamson 996, when banks exhaus hei eseves his can be associaed wih a suspension of conveibiliy of deposis. And, when banks exhaus hei eseves, cuency pemia will be obseved, and subsiues fo explici media of exchange may be used. Thus he exhausion of cash eseves poenially involves all of he feaues of hisoical banking cises noed by Noyes. We hen analyze he poenial fo, and he naue of such banking cises unde compeiive vesus monopolisic banking sysems. We now biefly descibe ou majo esuls. Admiedly, in he envionmen we sudy, hee is no disincion beween an hisoical banking panic and a moden banking cisis. Tha is because we do no allow fo he exisence of he egulaoy inevenions lende-of-las-eso, and deposi insuance --- ha have aguable uned he fome ino he lae. Ou hinking is o sudy he effecs of bank compeiion in he simples geneal equilibium envionmen possible. Adding a discoun window and deposi insuance o he model ae woks in pogess. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 5

6 Fis, he elaive pobabiliy of a banking cisis unde compeiion vesus monopoly canno be infeed independenly of moneay policy. If he nominal ae of inees he ae of inflaion is below some heshold, a monopolisic banking sysem will involve a highe cisis pobabiliy of a panic han a coesponding compeiive banking sysem. Howeve, if he nominal inees ae he ae of inflaion is above ha heshold, he cisis pobabiliy will be highe unde compeiion han unde monopoly. Inuiively, a monopolisic bank can geneae highe expeced pofis by limiing is holdings of cash eseves. Ohe hings equal, his will aise he pobabiliy of a banking cisis eseve exhausion elaive o a compeiive banking sysem. Howeve, a monopolisic bank also offes deposios elaively lowe euns, ceeis paibus. This faco ends o educe he pobabiliy of eseve exhausion. The elaive pobabiliy of panic unde monopoly vesus compeiion depends on he senghs of hese wo foces. A low high levels of nominal inees aes, he fome lae foce will dominae, and a monopolisic banking sysem will confon a highe lowe pobabiliy of a cisis han a compeiive banking sysem. Second, we show ha he pobabiliy of some oupu loss due o a banking cisis is always highe unde compeiion han unde a monopolisic banking sysem. This occus because he monopoly bank has a song pofi moive o economize on he liquidaion of any asse excep cash. As will be seen, howeve, his is a so of mixed blessing wih a monopoly banking sysem. The monopoly bank is able o avoid some eal oupu losses only because i povides elaively poo ine-empoal insuance o deposios. This is an impoan poin because i eflecs anohe way ha a banking indusial oganizaion may be evaluaed ---- in C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 6

7 his case, hei efficiency in poviding ine-empoal insuance. This link has eceived no aenion we ae awae of in eihe he heoeical o empiical lieaue. Finally, inceases in he nominal ae of inees he ae of inflaion incease he pobabiliy of a banking cisis unde boh compeiive and monopolisic banking sysems. This is consisen wih he empiical evidence on inflaion and he pobabiliy of a banking cisis cied above. 3 The emainde of he pape poceeds as follows. Secion II descibes he geneal envionmen analyzed. Secion III of he pape inoduces a compeiive banking sysem, and descibes he equilibium behavio of compeiive banks. This secion daws heavily on Smih 00 and is pesened in cusoy manne, fo compleeness. Secion IV conducs a compaable analysis fo a monopolisic banking sysem. Specifically, his secion descibes he behavio of agens who do no use banks. Such an analysis is necessay since ou monopoly banks can exac all he suplus deposios migh ean fom having access o a bank sysem. Secion V. hen consides how he pobabiliies of a banking cisis, he pobabiliies of oupu losses due o a cisis, and expeced oupu losses due o a cisis diffe unde compeiion vesus monopoly in banking. Poofs of all poposiions ae deailed in he Appendix. 3 In addiion, we show ha in economies wih compeiive banking sysems, inceases in he nominal ae of inees he ae of inflaion incease he pobabiliy ha some oupu losses will occu associaed wih a banking cisis. Howeve, when he banking sysem is monopolisic, he pobabiliy of some such oupu loss is independen of he nominal inees inflaion ae, if he nominal ae of inees is below some heshold. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 7

8 II. The Envionmen We conside a discee ime economy populaed by an infinie sequence of wo-peiodlived, ovelapping geneaions. Le =,,... index ime. The economy consiss of wo islands. A each dae a new young geneaion is bon on each island compised of wo ypes of agens. One ype is poenial bank deposios. In each geneaion hee is a coninuum of young deposios on each island wih uni mass. The ohe ype is poenial bank opeaos. Thee ae N of hese agens bon on each island a each dae. Seing N = > allows us o conside a monopolisic compeiive banking sysem. Thee is a single good a each dae. All deposios ae endowed wih w > 0 unis of his good when young, and agens have no endowmen when old. In addiion, fo simpliciy deposios cae only abou second peiod consumpion, denoed by c. Then deposios have he lifeime uiliy level u c ln C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 8 = c. Bankes, on he ohe hand, have no endowmen of goods in eihe peiod of life. They also cae only abou second peiod consumpion, and ae isk neual. All agens bankes and deposios have access o a echnology fo soing he consumpion good. One uni of he good soed a yields > unis of consumpion a +. In addiion, a soage invesmen iniiaed in any peiod can be scapped lae in he same peiod. Scapped soage invesmens yield < unis of consumpion. Following Townsend 980, 987, we geneae a ansacions ole fo money by emphasizing limied communicaion acoss spaially sepaaed makes. In paicula, a each dae agens can ade and communicae only wih ohe agens who inhabi he same locaion. The naue of ade is as follows. Young agens can sell consumpion goods in exchange fo cuency held iniially by old agens. In addiion young agens can soe he good. These ansacions may o may no be inemediaed, as descibed in fuhe deail below.

9 Afe young agens allocae hei savings beween cuency and soage invesmens a, a facion of young agens is seleced a andom o be moved o he ohe locaion. The value is he same in each locaion. Howeve, a each dae is iself a andom vaiable dawn fom he disibuion F. Le f denoe he pdf associaed wih his disibuion, le [ 0, ] be he suppo of f, and le f > 0 hold 0,. The disibuion F is known by all agens, bu he value is no known a he ime agens allocae hei pofolios. The significance of andom elocaion is as follows. If agens inves diecly ha is, if savings ae no inemediaed, hen hey can leave soage invesmens in place unil mauiy. Howeve, agens who elocae will have no access o any invesmens lef in soage. Thus elocaed agens scap any soage invesmens and cay he goods obained along wih any cuency in hei possession o hei new locaion. Cuency held can hen be used o puchase addiional consumpion goods. Thus elocaion acs like a liquidiy pefeence shock ha foces agens o liquidae highe yielding in favo of lowe yielding asses. Agens would heefoe like o be insued agains he even of being elocaed. As in Diamond and Dybvig 983, such insuance can be povided hough banks. If banks opeae, a he beginning of a peiod deposios deposi all hei funds wih a bank. The bank uses deposis o acquie pimay asses: cuency and soage invesmens. Afe banks allocae hei pofolios beween hese asses, he value is ealized, and he specific ideniies of he agens o be elocaed ae evealed. Fo agens who ae elocaed, hey conac hei bank in a decenalized manne, 3 and make wihdawals. When a wihdawal is made, a deposio migh eceive cash enabling him o puchase goods in his new locaion o liquidaed soage invesmens which can be caied C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 9

10 o he agen s new locaion and consumed. Howeve, spaial sepaaion and limied communicaion imply ha agens do no emain in conac wih hei bank afe hey physically elocae. Hence hey canno use checks, cedi cads, o ohe pivae cedi insumens in hei new locaion. On he ohe hand, agens who ae no elocaed do emain in conac wih hei bank, and hence do no need cash in ode o ansac. Bankes diffe fom deposios no only in hei pefeences and endowmens. They also diffe in ha bankes ae neve elocaed, so ha hey can always be conaced by hei deposios. In addiion o bankes and deposios hee is a govenmen, which injecs o wihdaws fia money. Le M be he ime money sock pe deposio. Then he nominal money sock evolves accoding o M + = σ M. The goss ae of money ceaion, σ, is seleced once and fo all in he iniial peiod. Moneay injecions o wihdawals ae accomplished via lump-sum ansfes o young agens. III. A Compeiive Banking Sysem In his secion we assume ha he numbe of bankes, N, exceeds one. In his conex his is sufficien o guaanee ha he banking sysem is compeiive. Wih compeiive banks, a each dae each young deposio deposis his enie afe-ax endowmen, w + τ, wih a bank. Banks use hei deposis o acquie he economy s pimay asses: cuency which banks hold as eseves o pay elocaed agens and soage invesmens. Le m denoe he eal value of cash eseves held by a epesenaive bank pe deposio, and le s denoe he eal value of soage invesmens. Since bankes have no esouces of hei own, he bank faces he balance shee consain. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 0

11 3. m + s w+ τ In addiion o choosing eseve and invesmen levels, m and s, he bank makes seveal ohe choices. Among hese, a bank chooses a schedule of goss, eal euns paid o deposios m who do [do no] elocae a. We denoe hese euns by d d, and noe ha hese euns will be a funcion of he aggegae sae,. 4 Since banks mus give elocaed agens eihe cash o he poceeds of liquidaed soage invesmens o finance consumpion in hei new locaions, hee ae seveal consains ha a bank faces on is choice of deposi eun schedules. In ode o descibe hese consains, le [ 0,] α denoe he facion of is cash eseves ha he bank pays ou a as funcion of he ime sae, le [ 0,] is soage invesmens ha he bank scaps a. Finally, define γ m / w τ δ denoe he facion of + o be a epesenaion bank s eseve-deposi aio. I is hen possible o wie he bank s emaining esouce consains as follows: m 3. d α γ + δ γ p d α γ + δ γ p p + p Espession 3. asses ha he bank mus pay he facion of agens who wihdaw ealy by liquidaing is own cash eseves and soage invesmens, so long as he bank does no exhaus is own eseves. Espession 3.3 says ha he facion of agens who ae no elocaed a ae paid ou of any unliquidaed cash eseves and he poceeds of unliquidaed soage invesmens. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

12 Wih o moe bankes, banks compee agains each ohe fo deposis. This compeiion akes he following fom: each bank announces a se of schedules,, α, δ, m d d b, and a pofolio allocaion summaized by is esevedeposi aio γ, aking he choices announced by ohe banks as given. In a Nash equilibium, he esul is ha he objecs jus descibed mus be chosen o maximize he expeced uiliy of a epesenaive deposio, ln m + ln 0 d d f d, subjec o he consains 3. and 3.3. We now descibe he soluions o his maximizaion poblem. A fomal p + deivaion of he soluion appeas in Appendix A. Define I o be he goss nominal p ae of inees. Define by he vaiable by 3.4 γ γ, I H γ + γ I. Then, fo, 0 δ =, and 3.5 / α =. In paicula, if he facion of agens wihdawing ealy is less han, hen banks do no exhaus hei eseves. O, in ohe wods, is he ciical level of wihdawal demand above which banks do exhaus hei eseves. Moeove, since scapping soage invesmens involves an oppouniy cos, soage invesmens ae neve scapped if a bank has unliquidaed cash eseves. Finally, fo m, d d = holds. Tha is, if banks do no exhaus hei eseves, hey povide complee insuance agains he even of elocaion. Once > holds, C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

13 banks exhaus hei eseves. As discussed in he inoducion, we associae his even wih a banking cisis. Nex, define by 3.6 γ Q γ, I γ + γ I. Then, fo,, α =, 0 m 3.7 d and 3.8 d γ p p + =, γ = > d δ =, m all hold. Fo [,] we have m 3.9 d / d and 3.0 δ =, α =, γ =. I γ In sho, fo,, banks exhaus hei eseves. Howeve, he oppouniy cos of scapping capial invesmens makes i opimal no o do so. In addiion, hee is now a posiive oppouniy cos of poviding complee insuance agains he even of elocaion, and banks cease o do so a sufficienly high levels of wihdawal demand. When > is saisfied, he facion of elocaed agens is so lage ha banks ae willing o scap soage invesmen in ode o augmen he consumpion of agens wihdawing C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 3

14 ealy. Howeve, he oppouniy cos of doing so is posiive, and his again pevens banks fom offeing complee insuance agains he isk of elocaion. Fo ou puposes, he values and ae of paicula impoance. When banks exhaus hei eseves, and a banking cisis esuls. Noe ha, wih a compeiive banking sysem, he pobabiliy of a banking cisis is F. > Howeve, fo,, even hough a banking cisis is undeway, banks do no liquidae invesmens. An implicaion is ha, while some agens may suffe fom he effecs of a m cisis elocaed agens have d d cisis. < if >, hee ae no eal esouce losses fom a Once > holds, howeve, he liquidaion of socially poducive invesmen occus. Since his liquidaion is cosly, hee is no jus incomplee insuance hee is an acual physical esouce loss. Such a loss occus wih pobabiliy F. As we have noed, in pacice some banking cises occu wih no associaed esouce losses, while ohe cises ae associaed wih quie lage losses. Thus ou model can confon hese obsevaions. We have ye o deemine he bank s opimal choice of a eseve-deposi aio. We nex un ou aenion o his ask. The Opimal eseve-deposi aio The analysis above implies ha < and < hold if I > 0. Assuming I > 0, we now analyze he bank s equilibium choice of is eseve-deposi aio. The obsevaions in he pevious secion imply ha he expeced uiliy of a epesenaive deposio as a funcion of γ is C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 4

15 H γ, I 3. ln 0 p γ + γ f d p + p γ Q γ, I p + γ + ln + ln f d H γ, I p + ln γ Q, I γ f d γ + p + + ln f d M, I γ. Q γ I, Appendix A esablishes ha, a an ineio opimum saisfying I > 0, a compeiive bank s equilibium eseve-deposi aio saisfies. 3. M I γ =, I FH γ, I γ γ I + I + { FQ γ, I } + γ + γ I Q γ, I I f d 0 H γ, I γ γ I. To chaaceize soluions o 3., we fis assume ha is unifomly disibued, so ha a. f [ 0,] =. Second, we obseve ha he definiion of implies ha C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 5

16 4. γ = I I + Appendix B poves he following esul. Poposiion. a if I > 0 and a. holds, hen he equilibium value of saisfies 3.4 I + + I + I * I + + I holds as an equaliy if > 0. b M 0, 0 I > holds if I < is saisfied; c If I <, equaion 3.4 a equaliy has a unique soluion fo. Once he equilibium value of is obained, he bank s equilibium eseve-deposi aio can be deduced fom equaion 3.3. Suppose ha < holds hee is a posiive pobabiliy ha capial invesmens will be scapped. Then, when a. holds is unifomly disibued, we have he following esul. Poposiion. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 6

17 a The equilibium value of _ fo a compeiive bank saisfies he condiion 3.5 I = I + G b G 0 = 0, G = I I > hold., and G 0 The poof of poposiion 4 appeas in appendix C. The poposiion implies ha hee is always a. unique soluion fo in he uni ineval so long as I. The equilibium values of γ and can hen be ecoveed fom he definiions 3.4 and 3.5: γ I I +., IV. A Monopolisic Bank We now conside an economy idenical in all especs o he one discussed hus fa, excep ha now we se N =. Thus hee is a monopoly in banking. Wih he same noaion as peviously, a bank eceiving all deposis eans an ex-pos pofi p of α m + δ s d w τ p +, measued in unis of dae + consumpion. Tha is, he second peiod eal pofis of a monopolisic bank consis of he eal C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 7

18 value of unliquidaed cash eseves, plus he value of unliquidaed soage invesmens, less paymens o non-elocaed deposios in he second peiod. As befoe, le γ m w τ + denoe he bank s eseve-deposi. Then consains m coninue o apply o he bank s choices of α, δ, d, d, and γ. In addiion, we allow he bank o impose a minimum deposi equiemen which in his case will clealy be w τ +. This allows he bank o exac he maximum possible suplus fom deposios in effec he bank can impose a wo-pa aiff. Since deposios always have he opion of invesing auakically, i follows ha he bank faces an addiional consain, he paicipaion conain of deposios. We deive auakic agens expeced uiliy nex. Auakic Agens We begin by descibing he behavio of agens if hee ae no banks in opeaion o, equivalenly, if agens choose no o save hough inemediaies. We efe o agens whose savings ae no inemediaed as auakic. Le ˆ f d be he expeced value of, le 0 a γ be he facion of savings an auakic agen holds in he fom of cash eseves so ha is he facion invesed in a γ soage, le p be he ime pice level in each locaion, and le τ be he eal value of he lumpsum ax/ansfe paid/eceived by an agen when young. 5 Then an auakic agen who is elocaed will liquidae his soage invesmens, cay he goods obained o his new locaion, and use he cuency he holds o puchase addiional consumpion goods. Since he goss eal eun C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 8

19 on cuency beween and + is p / p +, he old age consumpion of an agen who is elocaed p + + a a a is γ γ w τ p +. Fo an agen who is no elocaed, soage invesmens can be lef in place unil mauiy. Hence agens who emain in hei oiginal locaion have second peiod consumpion equal o a p γ + γ τ p + a w uiliy of a young deposio a is given by. I follows ha, fo a given pofolio allocaion, he expeced ˆ ln γ a p a a p a ˆ ln p γ γ p γ Young deposios hen choose + + a [ 0,] γ o maximize his expession. I is eviden ha soage will occu a all only if ˆ ˆ p + >, and ha cuency p will be held a all only if p / p+ >. Boh condiions ae assumed o hold fo he emainde of his secion. When hey do, he soluion o he deposio s poblem is o se + 4. γ a p p ˆ p+ p+ = max 0, p p p+ p +. p + a Defining I o be he goss nominal ae of inees, clealy γ > 0 holds iff p 4. ˆ ˆ + > I. In sho, if he nominal ae of inees he oppouniy cos of holding money is oo high, agens will no hold i. Moeove, if 4. holds, 4. implies ha. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 9

20 4.3 γ a I ˆ = I I. Appendix D esablishes he following esul. Poposiion 3. Suppose ha min, + > I 4.4 ˆ ˆ a a γ holds. Then γ > 0 and < 0. Fo values of I a I saisfying 4.4, le agen s aio of cash-eseves o oal savings. Then define γ denoe he opimal choice of an auakic I 4.5 a a v I ˆ ln γ I + γ I I + a a a p a ln γ I ln / ˆ γ I I + + I ln γ I + γ I p + ˆ a p a + ˆ ln γ I + γ I p + o be he indiec uiliy funcion of an auakic agen. Appendix E demonsaes Poposiion 4. Fo values of Clealy v I ln w τ v I < holds. I saisfying 4.4, gives he maximum expeced uiliy aainable by an auakic agen, given he ansfe w + τ. Fo a given value of τ, his uiliy level is deceasing in I. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 0

21 The Monopolis bank poblem The deposios paicipaion conain is 4.6 ln m + ln 0 d d f d v I The bank hen maximizes is expeced pofis, 0 p α γ + δ γ d f d p, subjec o he + consains 3., 3.3, 4.6, and non-negaiviy. α In ode o simplify his poblem, we make he following obsevaions. Fis, if < holds he bank does no exhaus is cash eseves, i does no ake he cosly acion of liquidaing is soage invesmens δ = 0 m 4.7 d Second, if α invesmens, i ses δ = 0. Hence, siuaion, m 4.8 d p = α γ p +. Thus, if α <, = he bank exhauss is eseves and i does no liquidae is soage p = γ. p + I is possible ha he bank will liquidae some soage invesmens. Hence, in his p γ δ γ. p + m 4.9 d = + C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

22 We now anicipae a esul: ha α a monopoly bank we denoe his heshold by < will hold if lies below some heshold. Fo ζ. Similaly, 0 δ > holds if exceeds an addiional heshold, which we denoe by ζ. Hence hee ae wo possibiliies: a ζ < holds, o b ζ =. We now conside each case in un. The Case ζ <. We begin by using o ewie he deposio s paicipaion consain as 4.0 p α γ + ζ p ln + f d 0 p p γ γ + δ γ ζ p p ln + + f d ln + f d ζ ζ ' + ln d f d v I 0 Similaly, he bank s objecive funcion can be wien as ζ 0 p α γ f d p + + γ δ f d ζ ζ d f d γ f d. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc

23 A monopoly bank maximizes his expession, subjec o he consain 4.0. Le λ be he Lagange muliplie associaed wih 4.0. Appendix F poves ha he soluion o he bank s maximizaion poblem has he following popeies. Define ζ and ζ by 4. ζ γ p p +, λ and 4. ζ = ζ. Then 4.3 α δ ζ ; ζ = ; ζ C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 3 δ = ; ζ, ζ γ ζ = ; ζ, γ I ζ. Noe ha ae he exac counepas of equaions Howeve, he deeminaion of ζ and ζ is consideably diffeen han in he compeiive case. We also noe ha a monopoly bank chooses is deposi eun schedules as follows: 4.6 d m 4.7 m 4.8 d = λ; d = λ; 0, γ p ζ p + = ; ζ, ζ

24 m 4.9 d = λ ; ζ,. I emains o deemine ζ, γ and λ. Appendix F demonsaes ha a monopoly bank s opimal choice of ζ saisfies 4.0 { ζ } I = F ζ + F +. ζ f d ζ ζ The values γ and λ ae hen deemined by using all of hese condiions in 4.0, and solving he esuling condiion a equaliy fo λ. γ can hen be deduced fom 4.. If assumpion a. holds ha is, if is unifomly disibued, hen a paiculaly simple expession fo ζ.obains. In his case 4.0 educes o I 4. ζ = A banking cisis occus he monopoly bank exhauss is eseves if > ζ holds. This occus wih pobabiliy F ζ involve a social esouce loss if Finally, we can sae pecisely when. Banks liquidae soage invesmens so ha banking cises > ζ. This occus wih pobabiliy F ζ ζ < holds. Since ζ. = ζ, equaion 4. implies ha ζ < will hold iff 4. I > + > C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 4

25 obains. Thus, liquidaion of soage invesmens will occu a all unde a monopolisic banking sysem iff he nominal ae of inees he ae of inflaion is sufficienly high. We now conside wha happens when 5.9 fails o hold. ζ 0 The Case ζ =. When ζ = he bank s objecive funcion educes o p α γ f d p + + γ d f d 0 and he deposios paicipaion consain becomes 4.3 p α γ + ζ p ln + f d 0 p γ ln f d + ln d f d v I 0 p + ζ. Appendix G shows ha he bank s opimal choice of ζ in his case is given by I = Fζ + f d ζ ζ 4.4. Again λ can be deduced fom he deposios paicipaion consain, and γ can be ecoveed fom 4.. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 5

26 V. Monopoly vesus Compeiion, he Pobabiliy of a Banking Cisis, and he esouce Losses Associaed wih a Banking Cisis In his secion we conside hee issues. The fis is he following: wha is he pobabiliy ha cosly liquidaion of invesmens occus in monopolisic vesus compeiive banking sysems? As we show, he pobabiliy ha some esouce losses aise due o invesmen liquidaion is unambiguously highe in compeiive han in monopolisic banking sysems, ohe condiions equal. Thus compeiion leads o a highe pobabiliy of some esouce loss in each peiod. Second, we conside he pobabiliy of banking cises which may o may no involve invesmen liquidaion unde monopoly vesus compeiion in banking. Hee he esuls ae moe ambiguous. If he nominal ae of inees he ae of inflaion is sufficienly low, he pobabiliy of a banking cisis is highe in monopolisic han in compeiive banking sysems ohe facos being equal. Howeve, if he nominal inees ae he ae of inflaion is sufficienly high his anking is evesed. In paicula, when he nominal ae of inees exceeds some ciical level, banking cises occu wih highe pobabiliy unde compeiion han unde monopoly in banking. Thus he conduc of moneay policy ineacs wih he indusial oganizaion of he banking sysem in influencing he elaive likelihood of banking cises. Finally, we compae he expeced oupu losses condiional on a cisis occuing unde monopolisic and compeiive banking sysems. Invesmen Liquidaion Ou fis esul is ha he pobabiliy of some invesmen liquidaion is highe unde compeiion F han unde monopoly F ζ in banking. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 6

27 Poposiion 5. ζ holds. The inequaliy is sic if I >. The poof of poposiion 5 is given in appendix H. Inuiively, monopolisic banks ean highe expeced pofis o he exen ha hey can avoid liquidaing soage invesmens. Thus a monopoly bank has less incenive o liquidae invesmens ealy han does a compeiive bank. In paicula, compeiive banks ean zeo pofis in any even, and hey face song incenives o liquidae some soage invesmen as a way of enhancing insuance when wihdawal demand is sufficienly high. Thus he pobabiliy of a banking cisis enailing a social oupu loss is highe when he banking sysem is compeiive han when i is monopolisic. The Pobabiliy of a Banking Cisis Ou nex esul concens he elaive pobabiliy of a banking cisis unde monopoly vesus compeiion. Poposiion 6. a Suppose ha I =. Then = ζ =. b Suppose ha + I,. Then ζ = >, and > ζ holds. c Suppose ha I + > holds. Then hee exiss a value + I%, such ha > < ζ holds if I < > I %. Poposiion 6 is poved in Appendix I. The inuiion undelying he poposiion is as follows. Fis, if I =, hen cuency is as good an asse as soage. If follows ha, even if N =, a bank has no monopoly powe. Hence monopolisic and compeiive banks behave C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 7

28 idenically. And, wih I =, compeiive banks se γ =. 6 In paicula, doing so allows banks o povide complee insuance agains elocaion isk, and hee is no oppouniy cos o foegoing soage invesmens. If + I,, equaion 4. implies ha ζ = >. Hee a monopoly bank neve liquidaes invesmens. Moeove, hei incenive o inves in soage is song; hey ean geae expeced pofis by soing han by no soing goods. Hence monopoly banks have elaively low cash eseve holdings, and consequenly hey have a elaively high pobabiliy of exhausing cash eseves. When I > + holds, howeve, he above agumen is oo simple. Monopoly banks no only hold low cash eseves, bu hey pomise low aes of inees o deposios who wihdaw ealy. The pobabiliy of eseve exhausion depends on he elaive sengh of hese wo foces. Poposiion 6 shows ha, when I is sufficienly high, monopoly banks offe sufficienly low deposi euns ha he lae effec dominaes. In his case a monopolisic banking sysem has a lowe pobabiliy of a banking cisis han does a compeiive banking sysem given he pevailing equilibium value of I. Noe ha moneay policy he choice of I o σ ineacs wih he sucue of he banking sysem o deemine which ype of banking sysem faces a highe pobabiliy of a cisis. I is eviden ha ζ I < 0 holds. And Smih 00 shows ha I < 0 holds. Thus inceases in he nominal ae of inees he ae of inflaion aise he pobabiliy of a banking cisis. This is ue whehe o no he banking sysem is compeiive o monopolisic. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 8

29 Demiguc-Kun and Deagiache 997 and Boyd, Gomis, Kwak, and Smih 00 show ha, empiically, highe aes of inflaion do incease he pobabiliy ha banking cises will occu. Expeced Oupu Losses We now compue he expeced oupu loss fo economies wih monopolisic and compeiive banking sysems, condiional on some oupu loss occuing. We conside fis a monopolisic banking sysem. If yields γ δ > ζ a dae, he quaniy of invesmen liquidaed equals γ δ. This unis of consumpion a. A he same ime, γ δ unis of ime + oupu is foegone in he pocess of his liquidaion. Discouning his loss o dae implies a discouned pesen value of los oupu in he amoun γ δ. Using equaion 4.5, he expeced oupu loss condiional on some loss occuing is given by ' ' γ δ d I γ ζ ζ d ζ ζ = ζ ζ. I γ ζ = ζ An analogous expession fo an economy wih a compeiive banking sysem yields an expeced oupu loss condiional on some oupu loss occuing of I γ. In compaing hese expessions, i mus be kep in mind ha he eseve-deposi aios unde monopoly and unde compeiion will geneally be diffeen. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 9

30 Poposiion 5 implies ha ζ ζ <. Clealy, if ζ = condiional expeced losses unde compeiion ae highe han hose unde monopoly. Howeve, if ζ < holds, condiional expeced oupu losses unde he wo banking sysems canno be compaed wihou knowledge of he elevan eseve-deposi aios. We compued such eseve-deposi aions unde compeiion and monopoly fo seveal economies, and found ha in each case he eseve-deposi aio unde monopoly is always lowe hen he aio unde compeiion. Thus, fo such economies he expeced oupu losses unde compeiion ae always lage han hose unde monopoly. VI. Appendix A. The Maximizaion Poblem of a Compeiive Bank m Conside fis he poblem of choosing schedules d, d, α, ' b o maximize ln m + ln 0 δ and d d f d, subjec o consains 3. and Le λ i, i=,3 be he Lagange muliplie associaed wih consains 3. and 3.3. Then, if α < holds, he fis ode condiion fo he choice of α is λ = λ3. choice of 4 Noice ha, a his poin, we ake 0, γ. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 30 γ o be an abiay numbe. Below we analyze he opimal

31 m m The fis ode condiion fo he choice of d is λ d f ode condiion fo he choice of d is λ δ = 0. 5 =. And, he fis d f 3 =. In addiion, i is easy o veify ha m Fom hese condiions, i follows ha d = d if α m δ = 0 in 4. and 4.4 and seing d d A. α In addiion γ = + I = γ p d d. p + m A. = = γ + γ Clealy α holds if. = yields m If α = holds, hen he fis ode condiions fo d and m given by λ d = f and λ The fis ode condiion fo δ is A.3 λ λ3, wih equaliy if δ > 0 holds. d = f. 3 <. Then, seing d coninue o be 5 So long as banks do no exhaus hei cash eseves, hee is no oppouniy cos o fuhe eseve liquidaion. Thee is an oppouniy cos o scapping soage invesmens. Hence α < implies δ = 0. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 3

32 m δ =, hen he peceding condiions imply ha d d If 0 mus be saisfied. Thus banks now povide incomplee insuance agains he even of elocaion. Moeove, seing α = and 0 m δ = in 3., and 3.3, d d holds if, as defined in he ex. 3.7 and 3.8 hen give he deposi eun schedules. If δ > 0 holds >, hen he bank s fis ode condiions imply ha 3.9 holds. Seing α = in 3. and 3.3, and using 3., 3.3, and 3.9 yields 3.0. The Equilibium Choice of eseve-deposi aio Subsiuing he pevious esuls ino he bank s objecive funcion give he expession in 3. fo he maximized value of deposio expeced uiliy as a funcion of γ and I. In addiion, if I > 0 holds, hen < < is saisfied. I is saighfowad o show ha p + A.4 M γ, I ln γ γ F H γ, I p + ln γ p { F Q, I } ln F Q, I p γ + + γ + γ + ' ln { Q γ, I, µ FQ, I } ln F d γ + + Q γ, I Q H γ, I γ, I p γ p γ ln + + ln f d I hen follows ha { } C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 3

33 I M, I = FH γ, I + γ γ I + A.5 γ I F Q γ, I + γ + γ I Q H γ, I γ, I I f d + γ I γ p + H { ln γ γ f H p + p γ p H γ, ln I + f + µ H H γ, I γ γ I H γ, I ln f H H, } γ µ { Q γ, I ln ln γ + γ + Q f Q, I, p p p γ p ln + γ + Q γ, I ln + Q γ, I ln Q γ, I } Q γ, I Howeve A.6 γ γ, = γ + γ and H I I + C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 33

34 A.7 γ γ I γ γ = = H, γ γ + γ I p γ γ γ γ I γ + = + p+ I boh hold, as do A.8 γ p p + p = γ + γ, + Q I p γ and A.9 γ γ = Q γ, I I γ γ I γ + = p I γ γ + γ p + I γ p = γ + p + γ using A.3 A.6 in A. gives he expession fo M I γ in equaion 3.., B. Poof of Poposiion. In ode o pove poposiion, we begin by noing ha, by definiion, A.0 + φ Noe ha φ 0 = 0, φ =, and φ > 0 hold. Noe fuhe ha he definiion of implies ha 4.8 holds. Finally, all of hese obsevaions imply ha C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 34

35 A. γ = I +, A. + µ + I + = p p + γ + γ, and A.3 γ γ p + = p + I + Using hese obsevaions and assumpion a. in 3., i follows ha he condiion I M γ, 0 is equivalen o. A.4 p + µ I p + + I + + d 0 I Now obseve ha A.5 d = I I I Moeove, A.7 implies ha C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 35

36 A.6 = + and A.7 = Finally, using A.7, A.8 = + Using A.5 A.8 in A.0, i follows ha M I γ, 0 is equivalen o A.9 I + I I + + I + + I eaanging ems in A.9 yields 3.4 > 0. I b Seing = 0 and hence 0 eaanging ems gives equaion 4.0 in he ex. c Define γ = in 3.4, we have M 0, 0 I > if C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 36

37 Q Then + I + I + I + Q = + I I + + I I is hen appaen ha he lef-hand side of 3.4 is deceasing in if A.0 I + Q = I I I + I + I + + I 0. I + is saisfied. eaanging ems in A.0 yields ha he lef-hand side of 3.4 is deceasing in if A. I I +. We now obseve ha if > 0 I holds, I I is hen immediae ha A. is saisfied, he lef-hand side of 3.4 is deceasing in and ha M 0, 0 I >. I > 0 implies ha < holds, so 3.4 has a unique soluion. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 37

38 C. Poof of Poposiion. a Smih 00, poposiion 3, shows ha M I A. = I F d γ. γ, = 0 iff Using he assumpion ha is unifomly disibued, and ha-by definiion A.3 holds, A. educes o A.4 I γ In addiion, by definiion, A.5 = γ = I I. + Subsiuing A.5 ino A.4 and eaanging ems gives equaion 3.5 in he ex. b Tha G 0 = 0 is obvious. To evaluae G Thus G I =. = lim In ode o show ha G > 0 holds, noe ha we can wie, noe ha, by L Hopial s ule, C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 38

39 A.6 G Nex, define A.7 T I + + { I } Clealy, if T 0, hen G 0 >. Diffeeniaing A.34 gives I T = > 0. T I This esablishes he poposiion. D. Poof of Poposiion 3. a Tha γ > 0 holds follows fom he discussion in he ex. Diffeeniaing 4.3 yields A.8 a γ = I I ˆ I I a γ I ˆ a Thus γ I < 0 holds if A.9 I ˆ I I < I ˆ a eaanging ems in A., one obains ha γ I < 0 if C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 39

40 A.30 I > ˆ I is saisfied. I is eadily veified ha he igh-hand side of A.30 is inceasing in I, I saisfying 4.4. Thus, since I <, A.30 holds I saisfying 4.4 if > = ˆ. A.3 ˆ Bu A.3 is obviously saisfied, esablishing he esul. E. Poof of Poposiion 4 Diffeeniaing 4.5 and using he envelope heoem yields A.3 Iv I a γ a γ a γ γ I I = ˆ + ˆ = a γ + I γ + I a a a a γ γ ˆ ˆ < 0. a a a a γ + γ I γ γ + I F. The Opimal Behavio of a Monopolisic Bank; ζ <. The bank s fis ode condiion fo α is λ α γ C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 40 p = p + eaanging ems and using he definiion of ζ in he ex gives equaion 4.3. Noe ha α < holds iff < ζ.

41 The bank s fis ode condiion fo δ is p λ γ + δ γ, p + wih equaliy if δ > 0. This equaion implies ha 0 δ = if 4.4 is saisfied. Fo > ζ, solving he above expession fo δ and using he definiion of ζ gives equaion 4.5. The fis ode condiion fo d is d Equaions ae deived fom and The bank s fis ode condiion fo γ is p p 0 + ζ A.33 α f d F ζ ' ζ δ f f d + ζ ζ λ f d ζ γ + λ d + 0 γ p δ p λ + f d ζ p γ δ γ + p + f ζ p ζ α ζ γ + γ p+ { } ζ f ζ γ δ ζ = 0 γ = λ. This is equaion 4.6 in he ex. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 4

42 Now noe ha α ζ = and 0 equaion A.35 educes o ζ ζ = 0 ζ A.34 0 f d IF ζ δ ζ =. I follows ha he fis ode condiion fo γ γ ζ I f d ζ + γ I ζ p + p 0 γ ζ λ { f d + p δ f d f d λ p ζ p p p ζ γ p ζ + } whee we have used o subsiue α, b, and δ ou of A.34. Now noe ha λ f d = f d ζ ζ ζ A.35, 0 0 p γ p + λ f d = f d ζ ζ ζ A.36 and, ζ ζ p γ p + C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 4

43 A.37 γ ζ f d + ζ γ I ζ, δ f d = F ζ ζ Subsiuing A.35 A.37 ino A.34 and eaanging ems we obain he equivalen condiion ζ A.38 I = f d + F ζ + F ζ ζ ζ eaanging ems in A.39 yields equaion 4.0 in he ex. G. Opimal Behavio of a Monopoly Bank; ζ =. Hee he fis-ode condiions fo α, d, and b ae exacly as in Appendix F. The fis ode condiion fo γ is ζ A.39 I = α f d + 0 ζ p p f d f d p + + γ p γ + + λ { 0 ζ Using he fis ode condiions fo α and b see Appendix F in A.40 yields he equivalen condiion I F ζ f d ζ ζ A.40 = + eaanging ems in A.4 gives equaion 4.4 in he ex. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 43

44 H. Poof of Poposiion 5. If I =, hen = ζ =. We now conside wha happens if I > holds. Thee ae wo cases o conside. a ζ =. Hee < ζ necessaily holds if I >, since < is implied by 3.5, > G I see pa b of poposiion 4, and G > 0. b ζ <. If ζ < holds, hen he definiion of he funcion G implies ha I ζ = ζ I =. Moeove, I + G = I + + > I + Thus 3.5 and pa b of poposiion imply ha < ζ holds. I. Poof of Poposiion 6. I is easy o veify ha a compeiive banking sysem has A.4 = J +. Clealy J 0 = 0, J =, and J > 0 hold. Now, define he funcion H by C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 44

45 A.4 H GJ H > 0 holds. veify ha. Poposiion implies ha 0 0 H = and ha H = I. Moeove, clealy a If I =, hen 4. is violaed and ζ =. Moeove, if 4. is violaed i is easy o A.43 ζ = I I. Thus, when I =, ζ =. The esul ha = = = is implied by J G I I when I = holds. =, and b We fis conside he case in which I > + holds. This implies ha ζ <. We begin by obseving ha he equilibium value saisfies A.44 = = H G I. In addiion, given he esicion on I, A.45 ζ holds. I = ζ = The obsevaion ha 0 H > holds implies ha > < ζ holds if C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 45

46 A.46 GJ < > since ζ I saisfies GJ = A.47 J and ha I. We now noe ha I + + { } ζ = I A.48 H ζ ζ ζ ζ I J I = + + I J + J Moeove, A.49 J and + ζ = + + I A.50 J I ζ I = + + I { } Subsiuing A.50 and A.5 ino A.47 and eaanging ems yields C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 46

47 A.5 A.5 H 3 I { I } I + + ζ = I + + We now noe ha H < > ζ I holds if I I + + > < I + + Moeove, algebaic manipulaion esablishes ha A.53 is equivalen o A.53 I I < > I Now define he funcion Z I by A.54 Z I I I is easy o veify ha Z I 0 I I I > >. Moeove, clealy Z 0 > holds. Finally, one can check ha + Z < 0 holds if < +, C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 47

48 which obviously holds. I follows ha hee is a unique value I% +, saisfying Z I = 0 H %. In addiion, Z I < > 0 holds fo I ζ < > I fo I I < > I %. I is hen immediae ha < > %. Hence > < ζ holds if I I emains o conside ha case in which, + * holds, and ζ is given by A.55 I I 0.5 ζ = <. I is easy o check ha, fo I + A.56 I holds. I hen follows ha I I is saisfied fo all I + I. < > I %. I. Hee ζ =, I is now immediae ha G J I ζ G J < holds fo all. Moeove, since I < + + < I%, I GJ ζ G J I < < fo all I, +. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 48

49 Thus H ζ < I = H fo all elevan values of I, and > ζ holds. Endnoes Goods in soage ae no anspoable. Also, goods canno be kep ou of soage and anspoed acoss locaions goods mus go hough he soage pocess in ode o be caied ino fuue peiods. As in Diamond and Dybvig 983, if banks opeae, all savings will be inemediaed. 3 In paicula, no makes opeae afe i is evealed who is o be elocaed. 4 To be moe specific, we allow banks o opimally insue individuals agains he even of elocaion and also agains he ealizaion of. In paicula, banks ae no subjec o a sequenial sevice consain hee. 5 Since agens pofolios ae allocaed pio o he ealizaion of, he equilibium pice level and he equilibium ansfe display no aggegae andomness. 6 See Smih 00 fo a poof. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 49

50 efeences Allen, F. and D. Gale 00, Compaing Financial Sysems, Cambidge, MA: MIT Pess Beck, T., Demiguc-Kun, A. and Levine,., 003, Banking Cises and Bank Concenaion, mimeo, Apil. Boyd, J., Chang, and Smih, B., 00, Deposi Insuance: A econsideaion, Jounal of Moneay Economics, 49, 6, pp Boyd, J., and De Nicoló, G., 00, Bank isk-taking and Compeiion evisied, mimeo, Decembe. Boyd, J., Gomis, P., Kwak, S. and Smih, B., 00, A Use s Guide o Banking Cises, Woking Pape, Univeisy of Texas. Boyd, J., Kwak, S. and Smih, B., 00, The eal Oupu Losses Associaed Wih Moden Banking Cises, Woking Papee, Univesiy of Texas. Capio, J. and Klingebiel, D., 997, Bank Insolvency: Bad Luck, Bad Policy, o Bad Banking?, in M.Buno and B. Pleskovic, eds., Annual Wold Bank Confeence on Developmen Economics. Washingon D.C.: Wold Bank, 997, pp Champ, B., Smih, B. and Williamson, S., 996, Cuency Elasiciy and Banking Panics: Theoy and Evidence, Canadian Jounal of Economics, 9, pp Demiguc-Kun, A. and Deagiache, E.,997, The Deeminan of Banking Cises: Evidence fom Indusial and Developing Counies, Woking Pape, Wold Bank, Washingon D.C. C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 50

51 De Nicoló, G., Baholomew, P., Zaman, J. and Zephiin, M., 003, Bank Consolidaion, Inenaionalizaion and Conglomeaion: Tends and Implicaions fo Financial isk, mimeo, Mach. Diamond, D. and Dybvig, P., 983, Bank uns, Liquidiy and Deposi Insuance, Jounal of Poliical Economy, June, 93, pp Hellmann, Thomas, Mudock, Kevin and Sigliz, Joseph 000 Libealizaion, Moal Hazad in Banking, and Pudenial egulaion: Ae Capial equiemens Enough?, Ameican Economic eview, Mach, 90, Noyes, A. 909, A Yea Afe he Panic of 907, Quaely Jounal of Economics, Febuay, 3, pp Smih, B., 00 Appendix o Moneay Policy, Banking Cises, and he Feedman ule, mimeo, Univesiy of Texas, Febuay, Smih, B., 00 Moneay Policy, Banking Cises, and he Feedman ule, Ameican Economic Associaion Papes and Poceedings, 9,, pp Townsend,., 980, Models of Money wih Spaially Sepaaed Agens. In Models of Moneay Economies, edied by J. Kaeken and N. Wallace, Fedeal eseve Bank of Minneapolis. Townsend,., 987, Economic Oganizaion wih Limied Communicaion, Ameican Economic eview, 77, pp C:\Documens and Seings\DMC.000\My Documens\pojecs\Ongoing\confeences\003\jmcb\DAFT SENT TO WOLD BANK.doc 5