Systemic banks and the lender of last resort

1688-7565

004 - 2012

Jorge Ponce Marc Rennert

 

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to liqthe osociabailoprovnon-imparesor

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quidity shooptimal alally optimaout rule whides a ratsystemic bact of consrt responsi

En este

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                     hank Bruno Biaispants at the Touloauthors and do noo Central del Urugsche Bundesbank

 

mic ban

opose a moortfalls suchlocation of

al to move hen the shtionale forbanks in asidering sysbilities for

artículo selidad de e importann banco ceueños de lrte irrestinido. La ea que el basistémicos

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                    s,  Fany Declerck, ouse School of Ecot necessarily repguay. k and IAE Toulouse

nks and

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odel wherebh that a lenf lender ofresponsibi

hortfall is lr the centa larger rastemic risksystemic b

utiliza un prestam

ntes coexisentral actúiquidez, y

tricto cuanexistencia anco centras en un m

consideraprestamist

ntes es amb

c banks, sy

     Dominik Grafenonomics for theirresent the views 

e.

d the l 

Jorge PoncMarc Renn

ovember 2

Abstract

by systemicnder of lastf last resolities fromlarge enoutral bank nge of the

k on the opbanks is am

Resumen

modelo paista de sten con búe como pque esa re

ndo los prde bancosal actúen cayor rango

ar el riesgoa de út

bigua.

stemic risk

hofer, David Heir comments. All eof the institution

lender

ce2 nert3

2012

t

c and non-st resort port respons the centra

ugh. The exto act as

eir liquiditptimal allo

mbiguous.

n

ara analizaúltima in

bancos no prestamistaesponsabiliroblemas d sistémica

como presto de sus po sistémiclima inst

k, lender of

lmann,  Sebastienerrors remain ourns to which they a

of last

systemic blicy is requibilities anal bank to axistence of

lender ofty shortfal

ocation of t

ar la distribnstancia sistémicos

a de últimidad sea sude liquidezmente imptamista de problemas o sobre la

tancia pa

f last resort

n Pouget, Markusr own. The views are affiliated.

t resor

anks are exuired. We and find thaan uncondf systemic f last resols. Howevethe lender

bución óptcuando b

s. Es sociala instanci

ustituída pz sobrepasportantes última insde liquide

a distribucra con b

t policy.

s Reisinger and aexpressed herein

1 

rt1

xposed analyze at it is

ditional banks

ort for er, the of last

ima de bancos lmente a para or una

san un provee stancia ez. Sin ión de bancos

all  seminar n are those 

 

manycollaoperinstitprobrespoassisdurininstitfor ppreveaim resolmanimpllendeliquiacad

modeilliqufromin mlong-modeimpllende

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y other cripse so thaations. 4 Stutions we

blems rapidonse to thestance and ng the Sututions witproviding tent furtherto enhan

lution of dner.7 Althoications ofer of last rdity dry upemics. Thi

We presel, externa

uid, althoum a lender omaturity tr-term asseeled as aications ofer of last

                     on and Metrick(20al repo transactioargue  that  also  tprovide  evidencrmeier (2008) andarya et al.  (2010)ze  and  interconnelar, financial instit. example  in  Euroeuropa.eu/legislaBasel Committee oting  specific  rewww.financialstaant institutions, tsion. withdrawal of deppaper. ecent crisis provit al (2009), Flannnk markets only 

 

troduct

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nce the reistressed s

ough it prof systemicaresort policp has not rs paper aim

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ope  liquidity  inttion_summaries/on Banking Superequirements  forbilityboard.org/phe elaboration in

posits is only mod

ide several examery (1996), Freixaachieve a secon

tion

ations from past, motvent banksrge or hicenter of

over non-syolve the prructural refisis governy independity supporn effects.6

esilience osystemicallyoved to beally importcy to providreceived mms to contrmal model of funding

solvent, brt (LLR) cation by inidity short withdraw

ence of systolicy.9 In

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ystemic risk of  indood determinantman Brothers, Mer

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deled in reduced f

ples of interbankas and Jorge (200d‐best allocation

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rovision of forms of thnments andently of trt was to sStructural

of financiay importane a very imtant financde funding

much attentribute towa

which is (e.g. the in

banks so thn ensure th

nvesting detfall occurswal of detemically iour mode

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nterbank mhat only anhe bank’s cemand des at an inemand demportant

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‐2008 crisis the rthe funding capaount of  funding dovernight  interbof the financial c  institution durinutions of  individutearns or AIG imp

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centives of depos

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though shaank marketg to financstemic) finsystem andThird, the

mergency liqwork. Manypported finion. The raal system alatory framto promo

ons in an ong the cristhe design xternal souakers nor an the literato (2000) . not availa

ncy liquiditon. Banks eto risky, ile date, whWe analyzthe designcoexists w

psed. Increasing ng sector.    Copely. Acharya  and Mg  the  subprime events. Subprime crisis. Ttutions  to  systememic risk for the U

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ared by ts may e their nancial d their

e policy quidity y times nancial ational and to

mework ote the orderly sis, the

of the rces of among ture. In the

able for ty loan engage lliquid, hich is ze the of the with a

haircuts of eland et al. Merrouche crisis.  See   

They show mic  risk.  In US financial 

r  instance, m. ess releases 4.htm  and ystemically re efficient 

f the scope 

or example   imply that rove  social 

 

non-non-provlow-aimplassetto anprefiallocsociamakeassisis thappliprovbankcondcanddoescons

showresposecontwo iact arule bankbecathe cof illlargewidemanyof thunco

qualilendeshouexistlendebankoverruncosyste

         welfare

systemic bsystemic bide emergeasset-qualitement thists and thern agency wxed policyation of th

al welfare. er may allostance. We e lender oied. In theiding an e

ker is instdition of thdidates to a not impridering the

In a benw that it onsibilitiesnd-best optintervals. Fas lender oshould be

k is concernuse it incu

central baniquid bank

e enough lely exceedsy banks. H

heir solvenonditional b

When witative resuer of last re

uld be apptence of syer of last r

k. The resuriding the

onditional emic bank.

                     e.

 bank. The bank but noency liquidty banks shs first-bestrefore its s

which obsey rule. The he lender ofWe assume

ocate respoconcentrat

of last resoe first cas

emergency ructed to he illiquid act as lendrove the oese two altechmark cais second

s between timal alloc

For banks sof last reso

applied. Tned about

urs monetanker, in proks increaseliquidity sh the first b

Hence, the cy conditiobailout rule

we considerults are asesort for smplied whenystemic baresort in a

ult can be ee central bailout rulSince the

                    

failure of ot vice versdity assistahould be ct policy beolvency corves the soproblem o

f last resore that the

onsibilities te the anal

ort and anose it is th

loan to thprovide abank. In

der of last ptimal alloernatives.se with on

d-best socithe centraation of th

showing smort. For largThe rationaits expectery losses a

oviding an s in propohortfalls th

best social social planon, which e. r that systes in the bemall liquidn shortfal

anks provida larger raexplained i

banker’s le for the failure of t

                    

the systemsa. We findance to banclosed dowecause it condition. Holvency sigof the policrt responsibliquidity shconditionaysis on two

other in we central

he illiquid n emergean extensresort (e.g

ocation of

ly one typeially optim

al bank ande lender of

mall liquidige liquidityale for thised utility frand politicaemergency

ortion to thhe solvencyoptimal, i.

nner preferis implem

emic banksenchmark dity shortfals exceed des a rationge of liquin the follo

lending non-systemthe system

                    

mic bank md that it is fnks with an. Howeve

cannot verHence, it ei

nals throucy maker ibilities in ohortfall is val on the so cases: onhich an unbanker wbank. In tncy loan ion we sh

g. the depof responsib

e of bank (emal to shd the uncof last resortty shortfally shortfallss allocationrom the lenal costs why loan, req

he size of thy requireme. the cent

rs to suppomented thro

s coexist wcase: the clls and thea certain

onale for tuidity shorowing way.decision

mic bank dic bank dec

                     

may hurt tfirst-best sossets of hi

er, the poliify the quther deleggh superviis to annoorder to maverifiable. Hize of the r

ne in whichnconditionho makes

the second regardless ow that coosit insurabilities that

e.g. non-sysare lende

onditional t responsibls the centrs the uncon is as follnder of las

hen a bank uires that their liquid

ment of thtral bankerort illiquid ough the a

with non-sycentral ba

e unconditin thresholdthe centrartfalls for t. The optimthrough tepends oncreases the

                    

the return ocially optigh qualitycy maker cality of a ates the deision or apunce ex anaximize exHence, therequired liq

h the centraal bailout the deciscase the cof the so

onsideringnce corpot is obtain

stemic banr of last bailout rul

bilities consral banker

onditional bows. The ct resort actfails. As a

the assets qity shortfae central

r closes dowbanks rega

application

stemic bannk should ional bailod. Howevel bank to the non-symal threshthe use o

n the state e expected

                    

3 

of the imal to , while cannot bank’s

ecision pplies a nte the xpected e policy quidity al bank rule is

sion of central olvency g other ration) ned by

nks) we resort

le. The sists of should bailout central tivities result,

quality lls. For banker wn too ardless

n of an

nks the act as

ut rule er, the act as

ystemic old for of the of the return

              

 

of thilliqubencfirst-shouliqui

of sybecasystewith uncobankgranEverymoreoutco

the rpreseexist5 whcandcorpoSecti

and borroal. (2the oincom(2011whetdepoagen         10  The c

central Goodfriunnecethat theinferior hazard. Flannerrationaland presurvivinOverall,

he non-sysuid, non-sychmark cas-best over uld be the ledity shortf

Howeveystemic riskuse there

emic risk ithe system

onditional ker itself wting of lastything elsee responsiome depen

The restrelated liteent the bets, for furthhere we shodidates to oration in ion 7 we of

2 Re Our mod

on the iowing exte2011), Poncoptimal inmplete con1) extend ther or notosit insuranncies. We b                     concept of the lebank  should  aciend  and  King  (1ssary. Goodhart (e  lender of  last re to the market alHowever, Roche

ry (1996), Freixas le for lender of laevent inefficient cng banks with nec, Bagehot’s (1873

 stemic banystemic base. Hence,a larger s

ender of lafalls. r, we are nk on the oare two complies tha

mic bank. Obailout ru

will be less t resort loae being conbilities as

nds on the t of the paperature. In enchmark cher referenow the maact as lenaddition t

ffer some f

elated li

del builds oinstitutionaensively froce (2010) astitutional ntracts framRepullo´s

t a unified nce in a sinuild on the                    

ender of last resoct  as  the  lender 1988)  argue  that(1999) points outesort might not blocation. Castiglioet (2004) provideet al. (2000), Roast resort interveclosure of solventcessary liquidity t3) doctrine is wide

nk the mink is from, the centset of liquast resort fo

not able toptimal alloounteractinat the sociOther thingule for sys

strict becaans to the nstant this

lender ofrelative strper is orgaSection 3 case, in wces. We inin findings

nder of lasto the centfinal remar

terature

on the preval allocati

om its insigand Repullo

allocationmework of(2000) moregulator,

ngle agenceir insights     rt can be traced bof  last  resort  let  the  existence t that it is difficultbe better  informonesi and Wagnees a rationale for chet and Vives (2entions. In these pt banks. Moreoveto acquire the illiely accepted amo

nimum som a first-be

ral bankeridity shortor the non-

o prove a nocation of ng effects. al planner

gs equal, thstemic banause it antnon-system

s implies thf last resorengths of tanized as fo

we introduwhich only

troduce sys of this past resort bytral bank aks.

e

vious literaon of len

ghts.10 Closo (2000). R

n of lenderf Dewatripoodel by ini.e. the len

cy, is supers and analy

back to the workending  to  solvenof  an  interbank t for the central bed than the markr (2012) show thaa  lender of  last r2004) focus on copapers the existeer, as in Acharya aiquid banks’ asseong academics and

olvency reqest point or’s lendingtfall. As a -systemic b

non-ambiguresponsibiOne the o

r will be bhis implies nks. On thticipates himic bank what the cen

ort for thethese two eollows. Secuce the baon type o

ystemic riskaper. In secy consider

and the un

ture on thender of lasely related

Repullo firsr of last reont and Titroducing nder of lasrior to an ayze the opti

k by Bagehot (187nt  banks,  at  a  pmarket  makes 

bank to distinguisket. Therefore, that under some coresort  in a frameoordination failurence of a lender oand Yorulmazer (ets and avoid efficd policymakers.

quirementof view higg decision

result, thebank on a la

uous effectlities for thone hand,

biased towaa more fre

he other higher expe

when the syntral banke systemic effects. ction 2 proasic model.of bank (e.k into the ction 6 we ring the dconditiona

e lender ofast resort d papers ast consideresort respoirole (1994systemic

st resort coarchitecturimal institu

73) and Thorntonpenalty  rate  andthe  liquidity  prosh between solvehe lender of  last onditions penalty ework with sophires in interbank mof last resort can(2008) emergencyciency losses due

t to suppogher than is closer

e central arger rang

t of the exihe systemithe existe

ards forbeequent useand, the ccted lossesystemic oner should rbank. The

vides a rev. In Section.g. non-sysmodel in Sextend theeposit ins

al bailout r

f last resortresponsib

re by Espinrs the quesonsibilities 4). Espinosarisk and a

ombined wre with seputional allo

 (1802). They sta  requiring  good ovision  to  individnt and insolvent resort allocationrates increase basticated  interbanmarkets and prov assure market py liquidity loans pe to misallocation

4 

ort the in the to the banker e of its

istence c bank

ence of arance

e of the central s in its

ne fails. receive e final

view of n 4 we stemic) Section e set of urance rule. In

t policy bilities, nosa et tion of in the

a et al. analyze ith the

parated ocation

te that the collateral.   

dual  banks banks, and n should be anks’ moral nk markets. vide further participants provide the n of assets. 

 

of leunco(2010

was aboufromthe lperfelendeagenin thdo ncomploan lossecorpoliquiemersignasociashocseconin chwhileliqui

meriinsurliquifunctconsliquiits pr

uncobankoptimlargethe manbailodeterfirst-the c

modecentr

ender of onditional 0) in order

The optinitially stu

ut the provm a liquidit

lender of ect but noer of last r

ncies have teir mandatnot coincipensate deit can liq

es from theoration is dation valrgency loaal. Repullo ally optimaks, but toond-best opharge of the the depodity shocks

Kahn ats of centrance functdity shockstions separidering thedity shock rivate infor

Ponce onditional bk in troublemal allocate liquidity scentral baipulate the

out rule shr him from-best allocacentral ban

Espinoel. As in Kralization o

 last resortbailout ruto consideimal institudied by Rvision of emty shortfalllast resort nverifiable

resort: the the objectites so that ide. The epositors inuidate ban

e lender of biased toue of the bn. It grantshows tha

al. The ceno restrictiveptimal alloche lender oosit insurans.

and Santos tralization tion.They fs and leadsrated causee existencethey show

rmation. (2010) ex

bailout rulee regardlestion consisshocks andanker for e size of thould be com manipuation can bker. sa et al. (

Kahn and Sof regulato

t responsile. In so d

er systemictutional allRepullo (20mergency . The bankis given t

e signal abcentral bave to maxitheir indivdeposit in

n case of anks in troulast resort owards prbank. The ts the emet the depos

ntral bank e for largecation invoof last resonce should

(2005, 200of lender

find that ces to inefficies softer lee of inform

w that the c

xtends Repe meaningss of the b

sts of the ad the alloca

small liqhe liquidityomplementulating thebe achieved

(2011) intrSantos (200ory arrange

bilities bedoing we ac risk. location of

000). In his liquidity a

ks’ solvencthe author

bout their snk and themize their

vidual lendnsurance a bank’s fauble, realiz

activities. rompt liqucentral bargency loasit insuranon the co

liquidity solves both ort respondecide ab

06) use Repof last res

entralizatioient investmending decmational fricentral ban

pullo´s (20g that an embank’s solvapplication ation of theuidity shoy shortfall, ted by a pe liquidity d with an a

roduce a s05,2006), thements on

etween theare extend

f lender ofmodel a le

assistance tcy is privatrity to evasolvency. Te deposit inr expected fing decisiocorporatio

ailure. Wheze the liquFor this re

uidation inank’s engagan conditioce corpora

ontrary is tshortfalls. agencies. Tsibilities fo

bout the liq

pullo´s (20sort respoon inducesment into tisions for sictions abok does not

000) frammergency lvency. He s of the unc

e lender of ocks. Since

the applicunishmentshortfall.

appropriate

systemic bheir objectthe incent

e central ding the an

f last resorender of lato banks tte informataluate bankTwo agencnsurance cfinal wealtons as a lenon has then refusingidation va

eason the dn order togement is

onal on theation is alwtoo soft foIn RepulloThe centraor small liqquidity assi

00) framewnsibilities more forbthe risky asmall liquidout the bant have an in

ework by loan will beshows thatconditiona

f last resorte Banks mcation of tht to the baMoreover

e compens

bank into ive is to sttives of reg

banker annalysis by

rt responsiast resort dhat are sution so thaks and rec

cies may accorporationh. But they

nder of lasthe obligatig the emelue and li

deposit inso maximizrestricted

e bank’s soways tougheor small liq’s framewo

al bank shoquidity shoistance for

work to stuand the d

bearance fosset. Keepidity shortfank’s solvenncentive to

introducie providedt the seconal bailout rt responsibmay be ahe uncondanker in orr, he showsation sche

Repullo´s tudy the efgulatory ag

5 

nd the Ponce

bilities decides ffering at only ceive a ct as a n. Both y differ t resort ion to rgency mit its urance ze the to the

olvency er than quidity ork the ould be ortfalls

r larger

udy the deposit or large ing the alls. By cy and

o share

ng an d to the nd-best rule for bility to able to ditional rder to

ws that me for

(2000) ffect of gencies

 

to exmandtowaincencondrisk v

coexfailunon-formsyste

entirdepoinsurfirst

randsuccebank

0L

bankhas impa

RR =~

(2011refersagencan bsystesystein orcollaincluframfinan

         11    Esp

non‐sysbank. Wmakes t

xert forbeadate to ex

ards systemntive to s

ditions, an vis-a-vis a m

In this pists with are of the ssystemic b

mer.11 Our emic risk fo

3 Th We prop

rely by deosits at thered by the or the seco

The banom returneed, RR =

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k can be liq0,1 . As in

k (S) and a a negative

act reducesR . We

1), but ours to the pro

nt through fbe interpreemic bank em. E.g. in rder to mapse the no

uding claimmework Frncial institu

A bank f

                     pinosa et al. (2011stemic bank. We While both assumthe algebra easie

 rance and plicitly ove

mically imhare it wunified re

multiple repaper, we e non-system

systemic babank, and t

objective or the optim

he mode

pose a momand dep

e beginnindeposit in

ond period nks invest n R

~ for ea

R , or fail, Rquidated a

Espinosa enon-system

e effect ons the returndiffer in thr approachopagation ofinancial treted as losand is theinterbank

nage liquidon-systemicms againstreixas00 shutions and failure can

                    1) model this poinmodel  it differen

mptions allow capr.

to share inersee systeportant in

with other egulatory agulatory arextend Ponmic bank. ank leads tthat the fais differen

mal allocat

el

odel inspireposit contrg of their

nsurance aof operatiotheir depo

ach unit in0=

~R . The t the inter

et al. (2011mic one (Nn the non-n of the nohe modelinh follows of an agentransactionsses from in

erefore relak markets bdity preferc bank’s ast the syst

how that thcan triggeoccur beca

     nt by assuming thntly: we assume tpture of the exter

nformationemic risk, rstitutions regulatorsrrangemenrrangemennce (2010)As in Espito a lower

ailure of thnt: we stuion of lend

ed by Repracts. More

operationnd that thon. osits into anvested aft

asset is exrmediate d

1) we consN). A bank -systemic n-systemicng of the cRochet an

t’s economis". From thnterbank oated to thebanks are crences. As asset e.g. a temic bankhe failure oer liquidatioause after t

hat the failure of that the failure ornality that syste

n. They shoregulators and that

s. They cont can rednt.

by considnosa et al. expected

he latter doudy the imder of last r

pullo (2000e preciselys. We assuey can be

an illiquid ter two pex ante profate. The li

sider two tis considerbank. We

c bank’s asscontagion nd Tirole (ic distress tohis point oor paymente counterpconnected a consequeportfolio c

k yields aof a systemons of non-the first pe

f the systemic banof the systemic bmic banks may  im

ow that, unwould be regulators

onclude thuce the de

ering that (2011) we

return on oes not havmplicationsresort respo

0) where by banks raume that dwithdrawn

risky asseriods. The

fitable: ~

(REquidation

types of bred as systeassume th

set in the seffect from

(1996) wheo other age

of view the t system clarty risk wthrough in

ence of theconsisting

a lower remic bank sp

-systemic beriod of ope

nk reduces the prank reduces the mpose on non‐sy

nder an expmore forb may have

hat, underegree of sy

a systemie assume th

the assets ve effects s of consionsibilities

banks are faise one udeposits arn either aft

et which yasset can

1>)~R . Thevalue is eq

banks: a syemic if its hat the syuccessful s

m Espinosaere "systements linked

systemic ilaims again

within a finnterbank lee systemic of several

eturn. In spills over tobanks. eration a fr

robability of succreturn of the noystemic ones, ou

6 

panded bearing e little

r some ystemic

c bank hat the

of the on the idering s.

funded unit of re fully fter the

ields a either

e entire qual to

ystemic failure

ystemic state to a et al.

mic risk to that impact nst the nancial ending bank’s assets

such a o other

raction

cess for the on‐systemic r approach 

 

0vbe in(1983of thThe withdis puliquicumu

illiquemersociaeffecshocliqui

bank

non-vassetassignecesupeit mproceante

betwsociaincluTirolmotidimedecisbe prin caof diWe ithe oweal

wher

bankthe cwe a

0,1 of bannterpreted 3). Howeve

his paper isdepositorsdrawal behublicly obsdity shockulative dist

Since buid, banks rgency liqual costs inccts on the ks of bothdity situati

Additionk’s asset in

verifiable st at maturgned with ssary info

ervision in oay be basess. This aallocation

The polween the ceal welfare. uding the ale (1994) tvate agenc

ensions mision makinrivate info

ase of a faimensions ncorporateobjective fth, net of i

U

re I corre

k fails and central banssume tha

 nks’ deposias deposit

er, we do ns on the o

s’ behaviorhavior of deservable so

k v correstribution Ganks do nface bank

uidity assisclude, for economy b banks areion and donally, there

the model

signal iu rity is realithe LLR rermation aorder to fued on soft

assumptionof respons

licy makerentral bankIn the pubachievementhis does ncies if two ght be diff

ng of the agrmation. Flure underthe allocate Tirole´s function foncurred po

1= {IU fesponds to

zero othernker in caset the politi

its are withtors’ consunot model doptimal allor is beyondepositors lo that theponds to tG with su

not hold akruptcy if stance. A cexample, bbeyond thee independ

o not conside exists unl. Simultan

with Si ,zed. The sesponsibilind the ab

ulfill this taft informatn is decisivsibilities har can allocker and the lic sector mnt of the anot prevenconcerns a

ficult. Whilgency to en

For this rear his mandtion of we(1994) bas

or the centolitical cost

,}cfailure

o the agen

rwise, and e of a bankical cost of

hdrawn. Thmption prdepositors’ocation of d the scopike queuin

e liquidity the realiza

upport in [0any liquid

0>v unleclosure of bankruptcye banking dent. This der contagincertainty neously wit

N, about signal is pities, becaubility to assk. The soltion obtainve for the leas to be concate the lunconditio

many agencagencies’ ant the policare considee the failurnsure the s

ason the ceate. Secondights to thsic ideas antral banket from a ba

cy’s net in

is the k’s failure. f a bank’s f

he sudden references ’ behavior lender of

e of this png in front

shock vtion of a r

0,1]. reserves a

ess the lena bank cauy costs andsector. Weimplies th

ion effects about the

th the liqui

the succesrivately obuse it has sses the qvency sign

ned duringender of landitional oender of onal bailoucies have m

aims at reacy maker dered. First,re of a banstability of ntral bankd, due to te differentnd follow r so that i

ank failure:

ncome, {1 fa

weight givLike Repu

failure for t

withdrawaas in Diamin detail, blast resort

paper. We of banks dis publicly

random va

and assetsder of lastuses sociald costs rele assume that we focof system success p

dity shock

ss probabilbserved onthe autho

quality of nal is non-vg asset quast resort pon the liqui

last resortut rule in omultidimenasonable codesigning the quant

nk is publicf the financker has to bthe existent dimensioPonce (201it cares ab

}ailure is equ

ven to the llo (2000) athe central

al of deposmond and Dbecause thet responsibassume thuring a bay verifiablariable v~

s are comt resort pr costs of ated to nethat the liqcus on indliquidity cr

probability

iv a perfe

lity of the ly by the arity to colbanks’ ass

verifiable buality assespolicy becadity shortft responsirder to mansional maost. Accorda mechan

tification ocly observacial systembear politicce of mult

ons is of co10) by settbout its fin

ual to one

political cand Ponce l banker do

7 

its can Dybvig e focus bilities. hat the nk run e. The with a

pletely rovides c . The egative quidity ividual risis. of the

ect but

bank’s agency lect all sets by ecause ssment ause ex fall iv . bilities ximize ndates

ding to ism to f some ble the might

cal cost iplicity

oncern. ting up nancial

(

e if the

cost for (2010)

oes not

(1)

 

exceefor ahold

respoamotroubloan.

bankcentrneganot iappli

the fstartbankeven

systeand

last

decidan ethe n

         12    Esp

failure. first‐beslevel co

ed the socia fraction o

the centraThe ce

onsibilitiesunt of the ebled bank .

As in Poker the polral banker

ative effect incur any ied.

The tim

following. Fts to operatk N is dets and faci

At date emic bank invests it in

At date

resort obs

des about tmergency

non-systemAt date

                     pinosa et al.  (201By construction, st policy. Under oonditional on the 

 ial cost (of the soci

al banker rentral ban

s is determemergencyfails after

once (2010icy maker is instructon its util

political co

ing of the For simplifte at date 0layed to dalitates the 0 the policS and the nto a risky 1 bank S’s

erves in a

the provisiloan and c

mic bank N 2 bank N’s

                    1) and Kahn and this assumption our assumption, regulators’ incent

1< ). We aral cost cau

esponsible nker’s netmined by y loan whe

being sup

0) apart frcan impleted to provity in caseost from a

Figure 1:

model is sfication but0 while theate 1. Thisanalysis of

cy maker anon-systemasset.

s liquidity s

addition pr

on of the econtinues traises one s public liq

      Santos  (2005) aleads to considehowever, regulattive structure and

rgue that tused by a for the rea

t income its mandan it is enga

pported the

rom allocatment an uvide liquid

e of default failure wh

Timing of

summarizet without loe starting d sequentiaf the lendernnounces tmic bank N

shortfall v

rivately the

emergencyto operateunit of dep

quidity sho

ssume  instead ther regulators thattor’s level of forbd the bank’s solve

the central bank failu

alized sociafrom th

ate. Its expaged in lique central b

ting the reuncondition

ity to the t. In this cahen the un

f the model

d in figureoss of genedate of opel structurer of last resthe lender N. Bank S

Sv is public

e solvency

y liquidity l or bank Sposits and

ock Nv is r

hat the  regulatort are always biasebearance can exceency. 

banker canre because

al cost at me lender posure coruidity provbanker lose

esponsibilitnal bailouttroubled base the cenncondition

l.

e 0 and wilerality the seration for e avoids thesort policieof last resraises one

cly observe

y signal Su

oan. EitheS is closed.invests it irealized. B

r’s political cost eed towards forbeeed or fall short w

n only be be the socie

most.12 of last

rresponds vision. In caes its eme

ty to the ct. In this cabank withontral bankeal bailout

l be explaisystemic bthe non-sye simultan

es for both ort policy f

e unit of de

ed. The len

S of bank

r bank S re. Simultanento a riskyank N’s so

exceeds  the sociaearance with respwith respect to t

8 

blamed ety will

resort to the

ase the rgency

central ase the out any er does rule is

ined in ank S

ystemic neity of banks. for the eposits

nder of

S and

eceives eously,

y asset. olvency

al cost of a pect to the he optimal 

 

signain chremawas retur

realiz

decistherethe bwith facesthe llendeassetsimilcons

liqui

whecontperioliquiliqui

the uR

If themer

bank

al Nu is pharge applains open inot liquid

rn is realizeIf bank

zed.

4 Be In our

sion withine is no contbank collec

a randoms a randomender of laer of last ret quality tlar to the mider the de

4.1 Fi In order

dity shock The expe

[= EWN

ere LLR1 isinuation vods of opedated aftedation is (

Since thbank if

)(1 Lcu

u

he solvencyrgency liqu

4.2 Se We anal

k starting

 privately oblies the lef the lende

dated befoed. N is still o

enchma

benchmarn a framewtagion effects one un

m return afm but publast resort pesort respoo decide w

model studeposit insu

irst-best

r to determv as well

ected socia(({1 uRLLR

s equal to value of theration is r one perio

)( cL . e bank’s liq

the ban,cL

*

R

Luu

y signal fauidity assist

econd-be

lyze the sewith the

bserved by nder of la

er of last rere bank S

operating

rk case

rk case wework consect on otheit of deposfter two peicly observprovides anonsibility uwhether oied in Ponrance corp

lender o

mine the fl as the solval welfare f

(1))(1 cu 1 if the ba

he bank in(1( uuR

od of opera

quidation vnk’s solve

.c

L

lls short oftance.

est lende

econd-best lending de

the lenderst resort psort provid

S’s risky as

at date 3

e analyze isting of or financial sits and ineriods. Aftvable liquidn emergen

uses a perfer not to sce (2010). T

poration in

of last res

first-best levency signafrom the ba

)(11 LLLR ank is suppncluding th

))c . In casation the b

value is conency signa

f the thres

er of last

lender of ecision of

r of last respolicy. Bandes an emesset matur

the return

the first-only one sinstitution

nvests themter one perdity shock

ncy loan. Tect but nonupport theThe main dour analys

sort polic

ending decal u are bank is:

[{1=)] Ec LL

ported andhe social cse the banbank’s valu

nstant it isal is ab

hold *u t

resort po

last resortthe centr

sort. The renk N is eitergency loares simulta

n of bank N

and secoingle bank

ns. As descrm into a ilriod of ope

v and caThe agency n-verifiablee bank. Oudifference sis.

cy

cision we both verifia

)(( cRuLR 0 otherwicost of a fnk is not sue net the s

s socially opove the

he bank sh

olicy

t policy foal banker

egulatory ather closed

an. In case aneously a

N’s risky a

ond-best lek. In this sribed in secliquid riskyeration the

an only sur in charge

e signal abour benchmis that we

assume thable.

() cLL se. The exfailure aftesupported social cost

ptimal to sthreshold

hould not r

r the bencfollowed

9 

agency d or it bank S

and its

asset is

ending section ction 3 y asset e bank rvive if of the out the

mark is do not

hat the

)],c (

xpected er two and is

t of the

upport *u :

(

receive

chmark by the

(2)

(3)

 

prov

emerbankwith supplost. failuemerdoesincurin tro

Otheliqui

expre

Accsuppsigna

ision of liq 4.2.1 C Assume

rgency loak exceeds th

an amouported ban

In additiore. It follorgency liqu not provirs the politouble if the

u

erwise thedated.

4.2.2 U The lend

essed in th u

cording to port banks al u .

 uidity acco

Central ban

that the cen to the bahe utility frunt of v k is succesn the cent

ows that tuidity assides the em

tical cost e solvency

vuu CB

central

Unconditio

ding decisie following

0 UBRuu the unconin trouble

ording to th

nker as th

entral bankank in trourom closingis providessful. Otheral banker he centralstance is e

mergency lc . Conseq

signal is ab

.c

v

banker ref

onal bailou

ion given tg way: . nditional be with an e

he uncond

e LLR

ker is the leuble if the g the banked the emrwise the has to bea banker’s equal to loan the bquently, thebove the th

fuses the

ut rule

the uncond

bailout ruleemergency

itional bail

ender of laexpected

. If the ememergency lo

amount var the politexpected

)((1 vu ank is close central b

hreshold u

emergenc

ditional ba

e the centy loan inde

lout rule.

st resort. Itutility fromergency liqoan will b

v of the emtical cost utility fro

)c . If thesed and thanker will CB : )((1 u

cy loan a

ilout rule i

ral bankerependently

t will provm supportiquidity assibe repaid mergency

c of the m providine central

he central support th

)( ccv

nd the b

is applied

r is instrucy of the so

10 

ide the ng the istance if the

loan is bank’s ng the banker banker e bank ,c

(

ank is

can be

(cted to olvency

(4)

(5)

 

Figubenc(CB)provnon-d

rule

imm

plane*u i

The liquitougthrouprovshortreasothe

of liqemerrefusintuicentrdoescentrcentrthanmorelarge

first-origilenditoug

first-

expre

re 2: Lendichmark ban provides ide socialdesirable e

(UBR). L

ediate that Figure 2

e. The firstndependencentral badity shockher so thatugh the oriision a cotfall. But ton the unco

),( vu planThe cent

quidity, toorgency loases to proition of thiral banker so the exral bankerral banker cu )(1 .e severe soe.

The unc-best lendinn of the gring decisioher and req

4.2.3 O Followin

-best thres

essed as:

 ing decisionks with sosocially n

lly desirabemergency

Let R

v A

t 1<<0 Av

2 plots tht-best emernt of the sianker’s threk. With inct the centrigin. The uonstant levthe minimonditional

ne. tral bankero soft for sns. For lar

ovide the s observathas an incpected cos

r refuses thwill incur

. For a largo that liqui

conditionang decisioraph the unon. For poquires a po

Optimal al

ng Ponce

shold u *

ons in the bolvency sig

non-desirabble emergloans are

cL

cL

be

1.

e differentrgency liquize of the eshold of tcreasing lial banker’sncondition

vel of solvum asset bailout ru

r’s lending mall liquidrger liquidsocially d

ion is that centive to lst from prohe emerger the politiger liquidiidity is on

l bailout rn because

nconditionaositive liqositive solv

location o

e10 the ex

cR

L

for

benchmarkgnals abovble emergegency loanprovided b

e the valu

t lending uidity provliquidity shthe solvencquidity sh

s lending dnal bailout ency indequality reqle lending

decision isdity shortfaity shocks

desirable efor very sm

end to the oviding theency loan ical cost ity shock tly provided

rule is alwthe requir

al bailout ruidity sho

vency signa

of LLR resp

xpected so

the provi

k case. It isve *u . In rency loansns. In reby followin

ue for v

decisions vision requhock. It is cy signal d

hortfalls thdecision is rule requirpendent oquirement decision co

s, compareall and prov

the centraemergencymall liquidbank in tr

e emergenthe bank c with pr

the exposud if the so

ays too sored asset qrule coincidortfalls theal.

ponsibilitie

ocial welfa

sion of an

s socially oregion a ths; in regioegions a g the unco

so that u

derived aires minimtherefore adepends on

he central a concave res as the f

of the size is equal t

oincide wit

ed to the firvides sociaal banker

y liquidity dity shocks rouble. If thncy loan iswill be liqrobability 1re of the clvency sign

oft in comquality is zde with the central b

es

re functio

n emergen

ptimal to lhe central bon c it do

and b, sonditional b

*= uvuCB

bove in amum asset q

a horizontan the size banker befunction p

first-best liqof the liq

to zero. Foth the absc

rst-best proally non-deis too toug

assistanceclose to ze

he central cu )(1 .

quidated a1 which is central bannal is suffi

parison wizero. Only e central babanker is

on (2) give

ncy loan c

11 

lend to banker es not

socially bailout

. It is

a ),( vu quality al line. of the

ecomes passing quidity quidity or this cissa in

ovision sirable gh and e. The ero the banker . If the nd the larger

nker is ciently

ith the in the

anker’s always

en the

can be

 

To welfa

the cis ap

follow

v if

(2) w

Figuoptimfollow

Wmaximize

are: w

From 7 central banpplied:

w

w

Followinwing prope

Lemma

fR

vv A <

UCB ww =0 Proof. Se

re 3: Normal allocatws the upp

 ([1= EW LLR

(6) it is

([1= uEw LLR

we can denker acts as

=)(1

vw Cu

CB (=

1

0uwUBR

ng Ponce erties summ

1 Assum

cLR

cL

, de

UBR ; and, (3)

ee Appendi

rmalized etion of theper envelop

)](( * Ruu sufficient

)].*uu

erive the ns the lende

d)( *

)(uu

vCB

).(d)* uFu

(2010) wemarized in

me |~

uuE

ecreasing if

) >(0)CBw

ix 8.1.

expected soe lender ope of solid

)() cLc to maxim

normalizeder of last re

),(d uF

e can sho Lemma 1.

>(1)uCB

if Avv > , a

0>1CBw .

ocial welfaof last reso functions

).mize the n

expected esort or the

ows that .

*u . Then,

and has a g

are for thort activity: for *< vv

normalized

social welfe unconditi

these fun

(1) vwCB

global max

he benchmy for the * the cent

expected

fare given ional bailo

ctions hav

is increas

ximum at

mark bank.benchmarkral banker’

12 

(social

(

either ut rule

(

(

ve the

sing in

Avv = ;

. The k bank ’s (CB)

(6)

(7)

(8)

(9)

 

decisw .

Theysociaincreof thassisreaso. To diffewhile

lowehas onorm

allocliquiimpl

the lbelowunco

rand

qualiprovshortlendibailothe emerthe orespoenha

sion maxim

Figure 2

y are preseal welfare easing for he first-besstance of ton the normthe left anr from thee on the rig

r than Cwover the wh

malized expSince on

ate the ledity shock ies the follo

Proposi

lender of law the threonditional b

The con

om bank is

ity (i.e. uide sociallytfall. But ting decisio

out rule forcentral bargency loanone closest onsibilitiesancing.

5 Fin

 mizes w ; fo

2 visualizesnted as a ffunction g

Avv < andst and the the centralmalized exnd the righe first-bestght the cen

ACB v . Thehole suppopected socinly the liqunder of lato maxim

owing seco

ition 1 A

ast resort reshold *v bailout rule

ndition E

s more like

][0, *u ). Iy desirable

the averageon. For thisr large liquanker will ns. For smto the first

s to the

nancial

or *vv t

s the propefunction of given the d decreasincentral ba banker copected socht of Av tt requiremntral banke

e solvency ort of liquidal welfare f

uidity shockast resort ize the exp

ond-best op

Assume tha

responsibili,1)( Av . O

e.

(|~

uuu CB

ely to be of

In the intee emergene bank hass reason, it uidity shock

be too rall liquidityt-best solutcentral b

system

the uncond

erties of fuf the liquid

central bang otherwianker coinorresponds

cial welfarethe solvencent. On ther is too to

requiremedity shocksfunction isk v is pubresponsibipected socptimal alloc

t |~

uuE

ities to the Otherwise,

*>(1) u

i

f average q

erval [0, Cucy loan des a sufficieis welfare-

ks because restrictive y shocks thtion so thaanker for

with a

ditional ba

nction (8) ity shortfalanker is thise. At Avcide so ths to the fie function hcy requiremhe left the ough. There

ent of the s constant s a horizontblic informlities condial welfarecation:

>(1) uuCB

central bait is soci

mplies tha

uality (i.e.

(1)]CB the epending oent quality -enhancingfor these sand not

he central bt the alloca

small liq

system

ilout rule (

and (9) stall. The normhe lender the solven

at the emerst-best prhas an maxment of th

central baefore, wCB

unconditioto zero. Fotal line.

mation the ditional one. As in P

*u . It is op

anker for lially optim

at the ass

,[ * CBuuucentral ba

on the sizeaccording

g to apply tshocks it isprovide s

banker’s leation of lenquidity sh

ic bank

(UBR) max

ated in lemmalized exof last re

ncy requireergency liq

rovision. Foximum for he central banker is to v for v onal bailouor this reas

policy makn the size Ponce10 lem

ptimal to a

iquidity shmal to app

set quality

(1)]B ) than

anker mige of the liqg to the firthe unconds more likesocially dending deci

nder of lasthocks is w

13 

ximizes

mma 1. xpected sort is

ements quidity or this

Avv =banker oo soft

Av is

ut rule son the

ker will of the

mma 1

allocate

ortfalls ply the

y of a

of low

ht not quidity rst-best ditional ely that sirable ision is t resort welfare

 

systeto desolvesyste

the b

S suc

bank

respoare bN is:

whesucceemer

failu

suppThe c

liquiemer

R

the erealiz

solve

In this se

em with a setermine te the modeemic bank.

We definbelow-ment

• 1=1SSS

• 1=1SFS

• 1=1S

• 1=1SSN

cceeded. • 1=1SF

N

k S failed at 5.1 Le 5.1.1 F For the

onsibilitiesboth public

[11

{= EWSFN

SFS

N

ere the firessful systergency loa

re occurs w

ported the bclosure cau

The secodated at dargency loan

. The ass

emergency zed. A liqu

For the dency signa

 

ection we ssystemic anthe optimael backwar

ne the follotioned con

if systemi

if system

if LLR loa

if LLR loa

if LLR loat date 2 or

ender of

First-best

e determins we assumc informat

)((

([{11

Ru

u

N

NSSN

SSS

rst term ofemic bank n the bank

with a prob

bank will buses social ond term oate 1 or its n its risky a

et fails wit

loan is refidation caudeterminatl Nu are d

study the ond a non-syal allocatiords startin

owing indicditions hoic bank S s

ic bank S f

n is provid

an is provid

an is providwas closed

f last reso

nation of me that the

ion and ve

))(1

)(1

cu

uR

N

N

f this expr(case SS).

k succeeds

bability (1be liquidatecost of c .of (10) is trisky asset

asset succe

th probabi

fused bankuses social tion of the derived sep

optimal lenystemic ba

on of respog with the

cator variald: ucceeds at

fails at date

es to syste

ded to non-

ded to nond at date 1.

ort policy

the sociae liquidity serifiable. T

)(1(1

1(1))

L

cSFN

SSN

ression is . If the nonwith prob

)Nu and

ed and the the expectet failed at deeds with p

ility (1 Nuk N is liquid

cost of c .first-best l

parately fo

nder of last ank as desconsibilitiese non-syste

ables with a

date 2.

e 2 or was c

mic bank S

-systemic b

-systemic b

y for the

ally optimshock Sv ahe expecte

)]},

)])(

cL

cL

the expecn-systemic bability Nu

causes soc

liquidatio

ed social wdate 2 (case

probability

)N which c

dated and . ending dec

or both sta

resort policribed in se for the syemic bank

a value equ

closed at d

S.

bank N give

bank N giv

non-syst

al allocatiand the so

ed social w

ted social bank is su

N and yield

cial cost c

n value L

welfare in e SF). If ba

Nu but y

causes soc

a liquidatio

cision the tates of the

icy for a finection 3. Inystemic ba

followed

ual to one i

ate 1.

en systemi

ven systemi

temic ban

ion of tholvency signwelfare from

welfare gupported wds a return

. If bank N

will be re

case bank ank N receiield only a

ial costs o

on value of

thresholds systemic b

14 

nancial n order ank we by the

in case

c bank

ic

nk

e LLR nal Su

m bank

(1

given a with an n R . A

N is not

ealized.

S was ives an return

f c . If

f L is

on the bank S

10)

 

(caseto prN’s coptim

whithe semer

assis

In ethe nreturthres

cost immhas tseconnot ppolitcentrthe scentr

whicase.emer

e SS and carovide an econtinuatiomal lending

u

u

ich is equivsolvency srgency liqu

If the systance to ba

u

u

equation (1non-systemrn the firsshold for ca

5.1.2 C The cent

from provediately. Ttwo compond the polprovide thetical cost ral banker’successful ral banker’

B

The cent u

u

ich is equi. If the sorgency loan

 ase SF). Firemergencyon exceedg decision

(1 uRu NN

Ruu SS

NN

valent withsignal Nu uidity assistystemic baank N if:

)(RuN

R

uu SFNN

12) we obsemic bank’s st-best lenase SS in eq

Central ban

tral bankerviding the eThe centralonents: Firitical cost e liquidity b

c for the’s expectednon-systems expected

[{11=B SN

SSSN

(([11 SFN

SFS

tral banker)( NN vcu

(vuu NCBNN

ivalent to lvency sign and the n

rst, the casey loan to bs the socito bank N

,) cLcN

,cR

L

h the first-bis below

tance. ank S fails

)(1 cuN

. cR

L

erve the neasset retu

ding decisquation (11

nker as th

r will only semergencyl banker’s rst, the exp

c due tobank N wil

e bank failud cost, becmic bank w utility from

)(1( uNSSN

)(1 cuN

r will provi,) Nv

)cv

v

N

NN

the centranal Nu is non-system

e of a succeank N if thal welfare in case SS

best lendinSSNu it is

s it is opt

,cL

egative imurn in thrsion in ca1).

e LLR

support they loan is loexpected cpected losso a failure ll be closedure. The stcause Rwill be ablm the lend

)))( vc N

(1))vN

de the eme

,

al banker’sbelow CB

Nu

mic bank wi

essful bankhe expecte

of bank Nis:

ng decisionnot social

imal to pr

pact of thereshold SF

Nu

ase SF is

e non-systeower than cost of an ses of the of bank N

d and the ctate of ban

1> . Evene to repayer of last re

)(1(1 SSN

)].)(1 cSFN

ergency liq

s lending dCBN the centll be closed

k S is analyed social wN’s liquida

n in our bely optima

rovide eme

e systemic F . Due to tougher c

emic bank the cost oemergencyliquidity in. If the cenentral bannk S has nn if the sysy the emeresort activ

)]c

uidity if:

decision intral banked.

yzed. It is owelfare from

ation. The

enchmark cl to provid

ergency liq

bank’s failthe lower

compared

N if the exf closing by loan to bnjection v

ntral bankeker will inc

no impact stemic banrgency loaities is:

n the bencer will refu

15 

optimal m bank

social

(1

case. If de the

quidity

(1

ure on r asset to the

xpected bank N bank N

Nv and er does cur the on the

nk fails n. The

(1

(1

chmark use the

11)

12)

13)

14)

 

expre

It im

inde

Figuto nothe cc shesociabailostate

v AN

the v

N deis inliquisolve

abovindedecis

5.1.3 T The lend

essed in th u

mplies that

pendent of

re 4: Lendion-systemiccentral bane does notally non-deout rule (Ue SF the

cLR

cL

value for v

Figure 4

efined abovndependentdated the

ency signa

ve the thrpendent ofsion for a

 The uncon

ding decisie following

0 UBNN uu

t banks wi

f the solven

ing decisioc banks wit

nker (CB) pt provide sirable emBR). In sta systemic

be the valu

Nv so that

4 shows theve. The first of the sfirst-best lel SF

Nu . For

reshold inf bank S’s liquidity s

ditional b

ion given tg way:

.BR

th a positi

ncy signal

ons for the th solvency

provides sosocially deergency loate SS th bank eit

ue for Nv

NCBN uvu =

e liquidity st-best lendize of the ending dec

r this reaso

n case SSstate. It o

shock of si

ailout rule

the uncond

ve liquidity

Nu .

non-systemy signals abocially non-esirable emans are proe systemicther was

so that u

SFNu I t is im

provision ding decisio

liquidity cision is mon the firs

S. The ceonly coincid

ze ANv ( C

Nv

e

ditional ba

y shock Nv

mic bank. Ibove i

Nu f-desirable e

mergency lovided by fc bank’s as

liquidated

SSNN

CBN uv =

mmediate t

thresholdson dependshortfall v

more restrict-best lend

ntral bandes with thCN ) in case

ilout rule i

N will alwa

t is sociallyfor ,{SSiemergencyloans. In rfollowing tset was su

d or its aS

N and vCN

hat <0 ANv

s for the nos on the st

Nv . If banctive and rding decisi

ker’s lendhe sociallySS (SF). S

is applied

ays be sup

y optimal t}, SF . In re

y loans; in regions a the unconduccessful wasset faile

cLR

cL

1<< CNv .

on-systemiate of bankk S fails orequires a on in case

ding decis optimal leSince the c

16 

can be

(1

pported

to lend egion a region and b,

ditional while in

d. Let

be

c bank k S but or was higher

e SF is

sion is ending central

15)

 

bankthe cliquiloan

benc

bankliquidassistlendidecisbank

betwrespoapprthe b

thres

It is

in o

normof las

papefunct

ker’s expectcentral badity shockso that the

The comchmark cas

Proposi

k is more dated or ftance in thing decisionsions of thek are identic

Proof. Se 5.1.4 O Since th

ween betweonsibilitiesroach to debenchmark

shold to pro

W

s sufficient w

order to obtAs the a

malized expst resort or

w

w

Lemma er and protions (18) a

Lemma

 ted utility nker’s lend

ks. The une lending dmparison oe yields the

ition 2 Trestrictive

failed i.e. she benchmn for non-sy central bacal to with

ee Appendi

Optimal al

he first-beeen cases s for both cefine the opk case the

ovide emer

[1= EW SSN

SSN

t to maximi[1= SS

NSSN Ew

tain the mpproach fo

pected socir the uncon

=)(, vw NCBN

(=1

0

, uwUBRN

2 follows Poves someand (19).

2 Assum

is decreasiding decisconditiona

decision in of the poe following

The first-besthan in t

some bankmark case ystemic bananker and tthe benchm

ix 8.2.

location

est lendingSS and S

cases separptimal secoexpected s

rgency liqu

)](( uu SSNN

S ize the nor

)]( SSNN uu

aximum ofor SF=ial welfare nditional ba

(1

)(uN

NvCBNu

(d) Fuu NN

Ponce (201e propertie

me that E

ng with thsion becomal bailout the (v,u) plicies for

g propositio

st lender ofthe benchmks that wheare not sunk is identicthe uncondmark case.

g decisionSF we wilrately. On tond-best alsocial welfa

uidity uSSN

()( LcR rmalized so

f the socialis analogofunctions ailout rule

),(d) uFuN

).(u

10) results es of the

CNNN uuu |

~

e size of thmes more rule alwaylane coincithe non-syon:

f last resortmark case ere suppor

upported nocal to the b

ditional bai

for the l study thethe basis oflocation ofare in (10)

cR

L

ca

).cocial welfar

welfare inous it followgiven the cis applied

,

adapted tonormalize

SFN

CBN u>(1)

he requiredrestrictive

ys provideside with thystemic b

t policy forif the sys

rted by emow. Otherw

benchmark lout rule to

non-systeme optimal af case SS f LLR respo) given the

n be expre

re:

n equation (ws for {central banare:

o the modeed expecte

F . Then, (1

d emergencwith incr

s the emehe abscissa.ank N wi

r the non-systemic ban

mergency liqwise the ficase. The l

o the non-sy

mic bank allocation we illustra

onsibilitiese socially o

ssed as:

(16). },{ SSSF th

nker is the

el studied d social w

1) if the sy

17 

cy loan reasing rgency . th the

ystemic nk was quidity rst-best ending ystemic

differs of LLR ate our s. As in optimal

(1

(1

hat the lender

(1

(1

in this welfare

ystemic

16)

17)

18)

19)

 

bank

) is i

decre

Nv =

{S

Figuoptimfollowis su

Nv <

unco

systewas

(19) socialiquibankliquidecrethe f

k succeeded

ncreasing i

easing if vANv (respec

},SFSS . Proof. Se

re 5: Nomal allocatws the uppccessful. O

iNv the c

onditional b

emic bank’liquidated

Figure 4

and explaial welfare dity shortf

ker’s lendindity shockeases becafirst-best pr

 d, SS= (

in Nv if v

ANN vv > (re

ctively at v

ee Appendi

rmalized etion of theper envelopOtherwise icentral ban

bailout rul

s asset waor its asset

4 presents tns the progiven the fall smalleng decisioks above tuse the ce

rovision of

(respectively

Rvv A

NN <

espectively CNN vv = ); (2

ix 8.3.

expected se lender ofpe of solid fit follows tnker’s (CB

e (UBR) m

as successft failed.

the normaoperties pro

central baer than A

Nv

on converghese thres

entral bankf liquidity. T

ly failed,

cLR

cL

(re

CNN vv > ),

(2) , 0CBNw

ocial welfaf last resorfunctions ithe upper eB) decision

maximizes

ful while in

lized expecoven in theanker is thAN ( C

Nv ) inges to the sholds the ker becomeThe norma

SF= ), the

espectively

and (iii)

,= UBRNw ; (

are for thert activity n case the envelope on maximi

Nw for in state SF

cted sociale Lemma 2he lender on case SS

first-best normalize

es more realized expec

en (i) SCBNw ,

vv CNN <

has a glob

(3) 0,CBNw

e non-systefor the nosystemic b

of the dashzes Nw ;

},{ SFSS .

F the syste

welfare fu2. The normof last res

(SF) becaprovision

ed expectestrictive ancted social

NSS v ( CB

Nw

cLR

cL

bal maxim

1>0 ,CBNw

emic bankon-systemicbank survivhed functio

for N vv In state S

emic bank

unctions (1malized exort increasause the cof liquidit

ed social wnd diverge welfare fu

18 

NSFB v,

), (ii)

mum at

0>

. The c bank

ves and ons: for

iNv the

SS the

either

8) and xpected ses for central ty. For welfare es from unction

 

givenuncothe lwelfathe nappli

the shortfollow

succeliquid

that non-sto ap

centr

NuE~

bank). Wwhicthe cwith uncothe uncoshoc

the cProp

largefailed

n the unonditional liquidity share functionormalizedied in case

Since onlender of tfall in ordwing secon

Proposi

eeded, =dity shortfa

it is optisystemic ba

pply the unc Proposit

ral banke

CBNNN uu|

~,

k is of averith increas

ch liquiditycentral ban

low quaonditional central ba

onditional k are alloca

We can central ba

position 4 s Proposi

er range ofd than whe

Proof. Se This res

 conditionabailout rulhock. Due

on given thd expected

SS (SF) fo

nly the liqulast resort

der to maxnd-best opt

ition 3 A

SS= (respeall of the n

imal to aank to the cconditional

tion 3 caner’s lend

SFNu>(1)

age qualitysing liquidy support isnker complity assetsbail out ru

anker’s thrbailout ruated to theshow that nker to acummarizes

ition 4 Tf liquidity sen it succeed

ee Appendi

sult can be

al bailout e providesto the co

e central bsocial welfr two liquid

uidity shockt responsiximize the timal alloca

Assume tha

ectively faion-systemic

llocate thecentral banl bailout ru

be explaiing decis

it is more

y (i.e. [uity shocks s social opt

pared to ans. Therefoule for largreshold is le. Therefo

e central bathe existen

ct as a lens this findi

The central shortfalls ofded (i.e. S

Nv

ix 8.4.

e explaine

rule is s an emergncave func

banker is thfare functiodity shocks

k Nv is vebilities acexpected

ation:

t NN uuE |~

iled, SF=c bank SS

Nv

e lender oker for liqu

ule for liquid

ned as folsion is

e likely tha

(1),[ CBN

SFN uu

it is moretimal does

n unconditore, the pge liquidity

closer to ore, the LLanker. nce of the snder of lasng:

banker shof the non-s

SFN

SSN v< ).

d in the f

applied isgency loan ction the nhe lender oon if the us: 0 and S

Nv

erifiable thcording tosocial welf

CBNN u (1)

F ), there ,1)( A

NS v (

of last resuidity shortfdity shortfa

llowed. Fotoo restr

at the asset

] ) than of e likely thas not receivtional bailopolicy maky shortfalls

the sociaLR respons

systemic bast resort w

ould act assystemic ba

following w

s horizontindepende

normalizedof last resorncondition

SSN (0 and

e policy mo the size fare. Lemm

SFNu> . If th

exists a t(respectively

sort responfalls below alls above it

or large liqrictive. G

t of a rand

low qualityat a non-syve an emerout of a noker chooses. For smaally optimasibilities fo

ank providwith an ext

s a lender oank when t

way. With

tal becausent of the d expectedrt, intersectnal bailout

SFNv ).

aker will aof the liq

ma 2 impl

he systemic

threshold fy ( C

NSFN vv

nsibilities fthe threshot.

quidity shoGiven con

om non-sy

y (i.e. [uystemic bargency loanon-systemices to appll liquidityal one thaor small liq

es a rationtended ma

of last resothe systemi

a failure

19 

se the size of social ts with rule is

allocate quidity ies the

c bank

for the ,1)C

N ) so

for the old and

ock the ndition

ystemic

][0, SFNu

ank for n from c bank

ply the y shock an the quidity

nale for andate.

ort in a c bank

of the

 

systeliquiloan liquibankliquibailo

emerfrom

well

wheemersyste

RuS givenbecaon tsociaoper

whithe sthe nis uN

liquisocianon-bank

net o

bankAs bsyste

emic bank dity shockincreases.

dity shockker’s one fdity shock

out rule stil 5.2 Le 5.2.1 F As for t

rgency liqum supportin

as the solv

W

W

ere the firrgency liqemic bank

cuS )(1 ) n the systeuse as we

the emergeal welfare ate is:

W

ich consistsystemic banon-system

uR NN (1dated so t

al cost duesystemic b

k failure is

of the sociaThe seco

k is closed. before the emic bank w

 the expec

s the socia. Since theks the socfor a large

ks the centll maximize

ender of

First-best

the non-syuidity assisng and not

vency shock

[{1= SS uEW

[{1= SS uEW

st term is uidity assi

k’s expecteand the

emic bankshowed in

ency liquidfrom the n

{= EuW SSC

N

{)(1 EuSts of the exank is succ

mic bank’s ec) . If the

that the so to bank faank’s expe

RuN )( al cost of a ond term oThe liquidexpected

was closed

cted returnally optimae central bcially optimer intervaltral bankeres the expe

f last reso

ystemic bastance by supportin

k Su are b

(1 SS uRu

)(S LcRu the social

istance. Ined continuexpected

k continuen section 5.dity provisnon-system

(({1 RuNSSN

((1 RuNSFN

xpected socessful or fexpected coemergency

ocial welfafailure Lcted contin

cuN )(1 .

bank failurof equationation valuesocial we

has to be c

n of the noal thresholdbanker becmal lendinl of liquidr is still toected socia

ort policy

ank we dethe compag the bank

both verifia

]) SCNS Wc

SCN WWL

l welfare gn this caseuation valsocial welfs to opera.1 the statesion for th

mic bank g

())1 cuN

(1) uNocial welfarfails at dateontinuationy loan is rere is equac . In case nuation va The liquid

re is cL .n (20) is the net of socelfare fromconsidered

on-systemid for the pcome less fng decisiondity shocksoo severe, l welfare in

y for the

etermine tarison of tk given tha

able. The ex

)[1(1 S L

]SLN cLW

given the se the sociue at datfare of th

ate. The lae of the syhe non-sysiven the s

)(1(1 LSSN

1(1))c SFN

re from the 2. If the n value netefused the l to the liqthe system

lue net of tdation valu

. he social wcial cost of

m the nond. This is:

c bank falprovision oforbearingn is closes. However

so that thn this inter

systemic

the first-bthe expecteat the liquid

xpected soc

]},SLNWc

},SLNW

systemic bal welfare

te 2 net e non-syst

atter has tystemic banstemic bansystemic ba

)}c

)},)( cLF e non-systsystemic bt of the soc non-systequidation

mic bank fathe expecte

ue of the no

welfare in cf a failure an-systemic

ls so that of the eme with incrr to the cr, for veryhe uncondrval.

c bank

est provised social wdity shock

cial welfare

bank receive consists of social temic banto be consnk has an ink. The exank contin

temic bankbank is succial cost of mic bank wvalue net ails at dateed social coon-systemi

case the syat date 1 is

bank give

20 

for all rgency

reasing central y large ditional

sion of welfare

Sv as

e is:

(2ves the of the cost (k SC

NW sidered impact

xpected nues to

(2

k given cessful failure will be of the

e 2 the ost of a c bank

ystemic cL .

en the

20)

21)

 

whereceisituasupp

RuN (

net o

so t

syste

Is tsocia

If thebankpolitsystethe scentrIf thenot bankfromthe cthe s

bank

W

ere the firives emergation whenported its e

uR (1) of the socia

We defin W

that W

Given (2emic bank i

u

u

he solvencally not opt

5.2.2 C Suppose

e central bak fails thetical costs emic bank systemic bral banker’e central bprovide an

ker’s cost cm the non-scentral bansystemic ba

B

B

The centk continues

 {1= EW SF

NSL

N

st term cogency liqun the emexpected co

cuN ) . If the

al cost of a ne:

{(1= SSNN EW

= SLN

SCN WW

20) and (24if:

( WcRuS

* SS Ruu

cy signal betimal to pro

Central ban

e the centraanker enga central b

c . In addcontinues

bank influe’s responsibanker refuny liquiditonsists on

systemic banker’s expecank S is:

[{1= SS EB

[{1= SS uEB

tral bankers to operate

)(( RuNF

N orrespondsuidity assiergency lo

ontinuatione non-syste

bank failur

()(1 NSFN

SN u

.= NSWu

4) it is socia

,) LWN

. NWcL

elow *Su t

ovide the e

nker as th

al banker iages in the banker losdition the to operateences the bilities as auses to supty. Bank Sly of the poank N givected utility

)((1 S cu )( SS cvu

r’s utility fre is:

))(1) cuNs to the sitstance whoan is refn value netemic bank

re at date 2

))( LcR

al optimal

he systemiemergency

e LLR

s the lendeemergenc

ses the liqutility from

e SCNB has

expected a lender ofpport the sS will be colitical cosen the closy from its le

) SCNS Bvc

SCNS Bv

rom the no

)(1(1) SFN

tuation whhile the sefused. If tt of the socis not supp

2 is cL .

}1 NSFN u

to provide

ic bank sholoan.

er of last rey liquidity

quidity injem the non-to be consprofitabilit

f last resortsystemic baclosed. In st c and sure of theender of las

)[1(1] S

]SLN cB

on-systemic

)},( cL hen the noecond termthe non-sycial cost ofported the

0,

e the emerg

ould be clo

esort for thassistanceection Sv -systemic bsidered becty of the t for the noank the cethis situathe centra

e systemic st resort re

]}SLNBc

}.SLNB

c bank N g

on-systemicm refers ystemic bf a bank faliquidation

gency loan

osed becau

he systemice but the sy

and incubank N givcause the sbank N an

on-systemicentral bankation the cal banker’sbank SL

NB .esponsibilit

,

iven the sy

21 

(2c bank to the ank is ilure is n value

(2

(2

to the

(2

se it is

c bank. ystemic urs the ven the state of nd the c bank. ker will central utility . Thus, ties for

(2

ystemic

22)

23)

24)

25)

26)

 

Theseconthe olendethe cbankthres

centrfailubank

bank

whebankloan the ncost thresbe re

so t

The(CB

S vu

thresinto

)(1

=

S

SSCN

u

uB

e first termnd term reoptimal seer of last rcentral banker will onshold CB

Nu .

ral banker re c . Be

k N and incThe util

k S is:

B

ere SFNv is

ker is the lif the non

non-systemc . The c

shold (CBNu

epaid and tWe defin

B

that B

Given (2 u

u

e central ba)Sv . The s

shold in ththe lending

 

)(

00

(

00

NvCBNuSF

Nv

NvCBNuSS

Nv

m reflects tefers to theecond-best resort for tker is resply support. In case

will lose thelow the socur the poliity from t

=0

SFNvSL

NB

the non-slender of l

n-systemic mic bank w

entral ban)( Nv . If the

the central ne:

=0

SFNv

SSNvNB

= SLN

SCN BB

26) and (29( SS cvu

)( SCBSS vuu

anker will solvency thhe benchmg decision

()(

()(

)

)

N

N

dFc

dFc

the situati situation liquidity s

the non-sysonsible fort the bankthe non-sy

he emergenolvency thitical cost he non-sys

()(

0

NvCBNu

c

systemic bast resort.bank’s solvill be closeker suppor non-systebanker wi

)()(

0

NvCBNu

dc

.NS Bu

) the centr,) SN vB

)

S

S

cvv

refuse thehreshold ofark case infor the sys

)(

)(

1

(

1

)(

NvCBNu

NvCBNu

u

u

ion when twhen the sshocks belstemic banr the provisk in troubleystemic ba

ncy loan v

reshold thc .

stemic ban

)()1

uudFc

bank’s liqu The centrvency sign

ed and the rts bank N mic bank fll incur add

)(1

( CB

NuudF

ral banker l

. NB emergencf the centrn equationstemic ban

)((1

)((1

) N

N

u

u

the systemsystemic balow which nk as definsion of the e if the soank fails w

Nv and inche central

nk N given

(11

)( NvCBNu

idity shockral banker al is belowcentral bagiven the sfails the emditionally t

)((1)(

NNv

u

ends to ba

cy loan if tral banker (4). The ck the effect

()

()

N

N

dFvc

dFvc

mic bank isank fails. v

the centrned in Prop

emergencyolvency sigwhile bein

ur the polibanker wi

n a closure

)( NN vcu

k below wwill refuse

w )( NCBN vu .

nker will insolvency simergency lthe politica

)( N dFvc

nk S if:

he solvencis lower c

central bant of its beh

)()(

)()(

N

N

vdGu

vdGu

s successfuSSNv and v

ral banker position 3.y loan the c

gnal is abog supporte

tical cost oill never s

e of the sy

)()N dGudF

which the ce the emeIn this sit

ncur the pignal is aboloan Nv w

al cost c .

)()(

NvdGuF

cy signal iscompared

nker incorphaviour in r

22 

).

(2

ul. The SFNv are is the

When central

ove the ed the

of bank upport

ystemic

),( NvG (2

central rgency tuation olitical ove the will not

0,)

(2

(3

below to the

porates respect

27)

28)

29)

30)

 

of itsmoreless s

expre

Ban

the s

decis

is sofin thlendicase

bankexpeis nothe lof thnot sthe spolic

This non-the cbankactivdecrebiase

equa

be ex

s responsibe responsibstrict with

5.2.3 U The lend

essed in th u

nks with a

solvency sigProposit

sion compa Proposi

fter comparhe benchmaing decisionwhile the u

Proof. Se The intu

ks can be ected return

ot only driviquidation e non-systesupported social welfcy is more f

The cenis due to

systemic bcentral banker is hencvities becaueases. In oed towards

5.2.4 O As abov

ation (??) g

xpressed as

 bilities towbilities for the latter o

Unconditio

ding decisie following

0 UBSS uu

positive li

gnal Su . tion 5 sumared to the

ition 5 Thred to the bark shouldn for the sy

uncondition

ee Appendi

uition of thexplained bn of the noven by the

value of themic bank the expect

fare is harforbearing tral banke the secon

bank. A closnker as a lece exposeduse his marder to avo

s forbearan

Optimal al

ve the expe

given the fi

s:

wards the nthe non-sy

one in orde

onal bailou

ion given tg way:

.BR

iquidity sh

mmaries th benchmar

The first-besbenchmark

d receive suystemic ba

nal bailout

ix 8.5.

he softer fiby the negaon-systemiccompariso

he systemicin both stated profita

rmed. For for the sys

er is also mnd-best opsure of theender of lad to a largeandate is eoid this negce for the s

location

ected socia

irst-best liq

non-systemystemic baner to avoid

ut rule

the uncond

hock Sv w

he effect ork case.

t lender of k case, i.e. support if thnk is also lrule remain

rst-best lenative impac bank. Th

on betweenc bank but ates of the ability of ththis reaso

stemic banmore lenienptimal alloc

systemic bst resort foer expectextended. Agative impsystemic ba

al welfare

quidity pro

ic bank. Sink if the sythe extend

ditional ba

ill always

of the syste

f last resort some bankshey are sysless strict cns unchang

nder of lasct of a syse first-best

n the expecalso considsystemic b

he non-systn, the firsk comparent comparcation of tbank implior the non-ed loss fromAs a conseqpact on its ank.

function f

vision thre

ince the ceystemic baded mandat

ilout rule i

be support

emic bank

policy for ts that do nostemic. Thecompared tged.

st resort potemic ban

t lender of cted continders the co

bank. If thetemic bant-best lendd to the be

red to the the respones more re-systemic bm the lendquence theutility the

for the sys

eshold Su =*

entral banknk fails it te.

is applied

ted regard

k on the le

the systemiot receive s

e central bato the benc

olicy for syk’s failure last resortuation val

ontinuatione systemic bk is reduceder of last enchmark cbenchmar

nsibilities fesponsibilitbank. The cder of last e expected central ba

stemic ban

NWcR

L=

23 

ker has will be

can be

(3

dless of

ending

ic bank support anker’s chmark

ystemic on the

t policy ue and n value bank is ed and

resort case. k case. for the ties for central resort utility

nker is

nk S in

NW

can

31)

 

in functbailo

in Le

incre

globa

bencsystenon-

the cshortthe follow

shortf

respo

the it

W

It is suff w

order to mtions given

out rule is a

w

w

We can emma 3.

Lemma

easing in v

al maximum

Proof. Se The sha

chmark andemic banksystemic b

The policentral bantfall in ordliquidity swing propo

Proposi

tfall {*S vv

onsibilities

t is optimal

 ([1= SS uEW

ficient to m([1= SS uEw

maximize n the centrapplied are

=)(vwuN

CBS

(=1

0uw S

UBRS

show that

3 Assume

Sv if S vv <

m at S vv =

ee Appendi

ape of the d as for thk the globank. cy maker w

nker and ther to maxihortfall is osition:

ition 6

,1}ASv so t

to the centr

l to apply th

)](*SN Ruu

maximize th)]*

SN uu equation

ral banker e stated bel

(1

)(uuS

SvCBSu

)(d)* uFuS

these func

e that ~uE

A

S LR

cLv

(

ASv ; (2) CB

Sw

ix 8.6.

normalizedhe non-systbal maxim

will allocatehe uncondmize the npublicly a

Assume E

that it is

ral banker f

he uncondi

() cLc

he normaliz

(32). The is the lendow:

),(d)* uFuS

).

tions have

|~

CBSSS uuu

N

N

Wc

B ), (i

UBRS

B w=0 ;

d expectedtemic bank

mum at v

e the lendeitional bai

normalizedavailable a

|~

SS uuuE

optimal t

for all liqu

itional bailo

).SLNWc

zed expecte

normalizeder of last

the follow

*>1 Su

. Th

i) decreasin

; (3) 0CBSw

d social welk. The diff

ASS vv = is

er of last reilout rule c

d expected and verifia

*>1 SCBS uu

to allocate

uidity shock

out rule.

ed social w

ed expecteresort or t

wing proper

Then, (1)(i)

ng if S vv >

>1>0 CBSw

lfare functiference her

also dete

esort respoconditionasocial welfble. Lemm

. There e

e the lend

k smaller th

welfare:

ed social whe uncond

rties summ

SCBS vw is

ASv , and (iii)

0> .

ions are asre is that fermined b

nsibilities al on the liqfare becaus

ma 3 impli

exist an liq

der of last

han *Sv . Abo

24 

(3

(3welfare ditional

(3

(3

marized

s

) has a

s in the for the by the

among quidity se only ies the

quidity

resort

ove *Sv

32)

33)

34)

35)

 

implqualithe csystedecisuncooptimcentrcentrrespo

solvepointliquibailonon-othe ewith decis

bankuncocentrFirstcompconsnon-moreshocSecobanknon-equarespowherthe mparaamballoc

The intu

ies that theity (i.e. ucentral ba

emic bank sion is tooonditional mal to supral banker ral banker’onsibilities

If the co

ency is on t of view. dated. In t

out rule beoptimal em

entire set oa sufficien

sion of the Proposit

k where tonditional ral banker t, the centpared to tequences systemic be responsibks the cent

ond, the firk because systemic b

al the lowonsibilitiesre the centmodel defimeters foriguous so tation for th

6 Ex

 

uition of pr

e asset of a ]1,[ * CB

SS uunker does if the liquid

o restrictivebail out r

pport illiquis still mo

’s thresholds for small

ondition E

average inInstead of

this situatioecause too mergency lof liquidity nt first-bestcentral ba

tion 6 defihe responbailout ruand the un

tral bankerthe benchmof the sy

bank. Keepbilities for ttral bankerrst-best len

the negabank’s prof

wer solvencs for the ctral bankernes which r the effecthat the ovhe systemi

xtension

roposition 6

a random s] ) than of l

not provdity shortfae. For thisule for laruid banks ore restrictd is closer liquidity sh

|~

SS uuuE

nsufficient being sup

on the polimany low

liquidity loshocks bect solvency nker for smnes the th

nsibility is le. For thenconditionr’s lendingmark case stemic ba

ping the firthe centralr’s behavio

nding decisative effecfitability iscy requirecentral banr is too stri

of the twoct of the

verall effectc bank is u

n

6 is as follo

systemic blow qualityide the soall is larger reason th

rge liquiditunconditio

tive than thto the soc

hock are all

*>1 SCBS uu

to receivepported the

cy maker pw quality syoans. The ccause welfaare overco

mall liquidireshold on

transferee determinnal bailout g decision because nk’s collaprst-best lenl banker be

or is closer sion itself ict of thes taken intment of t

nker becaungent incro effects psystemic

t of the sysundetermin

ows. Condi

ank is mory (i.e. [uocially optir because t

he policy mty shocks. onally. Forhe uncondcially optimlocated to t

is not sa

e an emerge systemic prefers notystemic bacentral banare losses fompensatedty shocks.

n the liquided from thnation of tthere existis less str

the centrapse into inding deciecause for ato the firstis more fo systemicto considethe first-be

use the intreases. Therevails. Burisk on b

stemic risk ned.

tion ~SuE

re likely to ]0, *

Su ). It isimal emerthe central maker choo

As above r small liqitional bai

mal one. Ththe central

tisfied the

gency loan bank shou

t to apply tanks wouldnker will bfrom closind by the re

dity shock he centralthe range ot two counrict for thal banker its responssion consta larger intt-best provrbearing w

c bank’s cration. Otest solutioerval of liq paramete

ut the imploth lendinon the opt

1| CBSS uu

be of an as more likergency loa

banker’s leoses to app

it is not quidity sho

lout rule bherefore, thl banker.

e systemic

from a firuld be closethe uncondd receive se responsig systemic

estrictive le

for the syl banker tof action f

nteracting ee systemicincorporatsibilities ftant this leterval of liqision of liq

with the sycollapse oher thingson leads tquidity shor constellalications ofng decisiotimal secon

25 

*>1 Su

verage ely that n to a ending ply the always

ock the but the he LLR

bank’s

rst-best ed and

ditional socially ble for

c banks ending

ystemic to the for the effects. c bank tes the for the eads to quidity

quidity. ystemic on the being to less ortfalls tion of f these ns are nd-best

 

respoIn hiavailthat bankcons

obligfor tcan rinsurprem

solvemanddepocost

in thprese

the pdepoexceelendisuccethe bcompbankto incurdepoinsurif:

Theshortamo

Until n

onsibilitiesis section able policythe optim

k derived aidered abo

The depges it to cohe compenrealize the rance pre

mium is norWhen ap

ency signadate it inc

osit insurerincurred in

In the fohe benchment the effe

6.1 Le Suppose

provision oosit insureeds the uting to theessful or nbank fails tpensate thk’s failure

)(1(1 u r the poli

ositors but rer will be

u

e deposit intfall becauunt of depo

 

ow the ps only betwwe introduy instrume

mal allocatibove was n

ove. posit insurempensate nsation payliquidatio

miums. Formalized toppointed al u . In caurs politicr cares abon case of a ollowing w

mark case ect for the

ender of

e that the dof an emer will sup

tility from bank the ot. If the b

the deposithe remainin

c . The ex)c . If the d

tical cost can realiz

cL 1 .

)(1(1 u

1uu DI

nsurer’s lenuse the liaosits. The e

policy makween the ceuce the deents for theon of respnot determ

er has to depositorsyments. Fin value Lor simplico zero. as the lendase the baal cost cout the exbank failur

we will anaas well asoptimal se

f last reso

deposit insergency loapport the

liquidatinamount

bank is suct insurer long depositxpected utideposit insof the ba

ze the liqu. The depo

1) Lc

.c

L

nding decisability of texposure is

ker could entral bankeposit insue policy maponsibilitie

mined by th

carry out s if a bank first, it has . Second, i

city, we a

er of last rank in trouc . As for thpected valre. lyze the le for the ncond-best a

ort policy

urer has toan to a babank if thg the banv depends

ccessful theoses the emtors )(1 vility from ssurer does ank’s failu

uidation vaosit insurer

,c

sion does nthe deposis only redu

allocate tker and therer into thaker is enl

es for the she truncate

the deposifails. It hasaccess to

it is fundedassume th

resort the duble fails ohe central bue of its f

nding decinon-systemallocation.

y in the b

o decide inank hit byhe expectek. The deps on whete emergen

mergency lo and incu

supporting not suppor

ure c analue L . Sor will lend t

not dependit insurer ced by the

the lende unconditi

he model sarged. Wesystemic a

ed set of po

it insurancs two optiothe failed d by banks

hat the de

deposit insor is liquibanker we final wealth

ision of theic and sys

benchma

n the benchy a liquidited utility fposit insurher the sucy loan v

oan v . In aurs the pol

the bank irt the banknd has to o the utilitthe amoun

d on the sizis boundeliquidatio

r of last onal bailou

so that the do this to

and non-syolicy instru

ce functionons to raisebank’s ass

s through deposit ins

surer obserdated durassume th

h net of p

e deposit istemic ban

rk case

hmark casety shock vfrom supprer’s utilityupported b

is repaid.addition it itical cost s therefore

k in troublecompensa

ty of the dnt v to th

ze of the liqed above n value in c

26 

resort ut rule. set of

o verify ystemic uments

n. This e funds set and deposit urance

rve the ing its hat the olitical

insurer nk and

e about v . The porting y from bank is . When has to of the

e equal e it will ate all deposit e bank

(3

quidity by the case of

36)

 

a cloComwith thaninsur

Figubenc(CB)prov(DI) socia

bailo

vB

More

with rule. plottbailorangdepowhilecentrshort

osure. It is paring the(36) it is

the first-rer is more

re 6: Lendi

chmark ban provides side sociallydoes not

ally non-de

out rule (U

cL

cL

1

eover, c <

Figure 6

that of theOn the ho

ted on theout rule doe of liquid

osit insurere the uncoral banker tfalls comp

 not affecte

e first-best obvious thbest lendi

e restrictive

ing decisionks with sosocially nony desirableprovide s

sirable em

BR). Let v

the value

L

L1 imp

6 presents te first-best orizontal axe ordinate.o not coindity shocks.r is always nditional bis too soft pared to t

ed by the lending d

hat the deng decisioe and does

ons in the bolvency sign-desirablee emergencsocially desergency lo

LR

cLv A

for v so

lies that v

the lendingsolution, txis we find. The agencide with . They alsotoo stringebailout rulefor small lihe first-be

substitutiodecision in posit insur

on in the not provid

benchmarkgnals above emergenccy loans. Insirable emans are pro

c be the

that CB vu

1<Bv .

g decision the central d the liquidncy’s lendieach othe

o do not ment compare is alwaysiquidity shost provisio

on of depothe bench

rer requirebenchmark

de socially o

k case. It isve *u . In rcy loans; inn regions c

mergency loovided by f

value for

DIuv = . It

of the depbanker an

dity shock wing decisior. Both ar

match with red to the fs too lenienocks and to

on. Ponce

sits by an hmark casees a higherk case. Heoptimal em

s socially oregion a thn regions c c and e theoans. In rfollowing t

v so that

is obvious

posit insured the uncowhile the sons and thre constant

the first-bfirst-best lint. As menoo tough fo(2010) poi

emergencye in equatr solvency ence the d

mergency lo

ptimal to lhe central b

and d it doe deposit iregions a the uncond

t = uvuCB

that v<0

er in componditional bsolvency sihe uncondt over the

best solutioquidity protioned abo

or larger liqints out th

27 

y loan. ion (3) signal

deposit oans.

lend to banker oes not insurer and b,

ditional *u and

BA vv < .

parison bailout gnal is

ditional whole

on. The ovision ove the quidity hat the

 

mainthe ddepothe lthe efailu

insurgiven

welfa

Lemsociaequa

if v <

CBw

wDI

of thwelfainsurrule.

allocliquifollow

lendethe thbailo

n reasons fdiffering i

osit insureriquidationemergencyre ( , )

In orderrer is consn the depo

are functio

w

mma 4 proval welfare fation (??) a

Lemma

LR

cv A

<

UBRw >=0

vwCB ; an Proof. Se Lemma

he central are. Otherwrer as lend

Since onate the ledity shockwing secon

Proposi

er of last rehreshold v

out rule.

 for the divempacts of

r has to com value the

y loan. Adddrive the l

r to determsidered weosit insurer

on in (2) an

(=1

uw DIu

DI ves some pfunctions ond (??):

4 Assum

cL

cL

, decre

DIw> ; (3)

nd, (4) Dw

ee Appendi

4 implies tbanker is wise it is ider of last

nly the liqunder of la

k to maximnd-best opt

ition 7 A

esort respon),(* BA vvv

ergence bef an emergmpensate a central baditionally, ending dec

mine the see derive thr acts as th

d the first-

)(d)* uFuu

properties oof the centr

me |~

uuE

easing if v

) If vv <

>1> CBDI w

ix 8.7.

that for liqdominatinin the inveresort is a

uidity shockast resort mize the etimal alloca

ssume that

nsibilities to. Otherwise

etween the gency loanall depositoanker’s expthe differecisions apaecond-best he normalihe lender o

best thresh

).

of (??) andral banker

*> uuDI

.

Avv > , and

L

cLvB

1

0> .

quidity shong that of terse. Furthalways dom

k v is pubresponsibi

expected sation:

t |~

uuE

o the centre, it is socia

agencies’ n on the aors of a collposure is rent weightart.

optimal azed expect

of last reso

hold R

u *

relates it tand the un

Then, (1)

has a glob

c, then

ocks below the deposi

hermore, wminated by

blic informlities condocial welfa

*> uuDI

.

ral banker fally optima

lending deagencies’ ulapsed banrestricted tts of the p

allocation wted social rt from the

cR

L

:

to the normncondition

vwCB is

bal maximu

ww CDI <

Bv the liqit insurer

we observe y the unco

mation the ditional onare. Lemm

It is optim

for liquidityal to apply t

ecisions is utility. Whk, but can

to the amopolitical cos

when the dwelfare fue expected

malized exnal bailout

s increasing

um at v =

vCB , oth

quidity proin term ofthat the d

nditional b

policy makn the size ma 4 impli

al to alloca

ty shortfallsthe uncond

28 

due to ile the realize

ount of st of a

deposit unction d social

(3

xpected rule in

g in v

Av ; (2)

herwise

ovision f social deposit bailout

ker will of the

ies the

ate the

s below ditional

37)

 

affecexistthe rbailofirst-mand

exist

point

rand

qualiprovemersupp

[0ubailoinsurclosebecolast r

non-the enon-returthat state

bank

repay

and

the lthe ncost can bank

Proposit

ct the allotence of theresponsibil

out rule. Ab-best liquiddate should

Furthermtence of the

ts out that

om bank i

ity (i.e. uide sociallyrgency loa

ports grant]0, *u . For

out rule insrer. For smest to the fomes too toresort respo

6.2 Le Suppose

systemic bexpected utsystemic brn is high ethe deposi

e. If the d

k will fail w

y Nv . The

incurs the

iquidity is non-system

c . In adrealize the

k is L 1

 

tion 7 showocation of e deposit inlities are tbove Bv tdity provisd be restricmore, the e allocation

t the condi

s more lik

][0, *u ). Sy desirablean is not ted through

this reasostead of allmall liquidfirst-best soough, so thonsibilities

ender of

e that thebank. The dtility from bank. Dueenough to it insurer’seposit insu

with probab

deposit in

political co

)((1( uNmic bank widdition thee liquidatio

c . The de

)(1(1 uN

ws that thresponsib

nsurer setstransfered the centralsion than cted to Bv

deposit in of respon

tion |~uE

kely to be o

Since in thee emergencprovided ch the uncoon, it is wocating len

dity shocksolution. What for smas to the cen

f last reso

e deposit deposit insproviding

e to the arepay all d

s liquidity purer provi

bility of (1

nsurer has

ost of a ba

))(1 c . Ifill be closed

e deposit inon value Leposit insur

) Lc

e introducbilities in s an upper

from the banker is the depos. insurer sensibilities d

>| uu DI

of average

e interval cy loan, it icompared onditional welfare-enhnder of lasts the centhen liquidill liquidity

ntral banke

ort policy

insurer issurer will o

liquidity iassumptiondeposits in provision ides the em

)1 Nu . It fo

to compen

nk failure

f the deposd and the dnsurer hasL . So the urer will len

,1 c

ction of ththe benchlimit Bv fcentral bamore stric

sit insurer

ets the codefined in p

*u implies

quality (i.e

],[ * DIuu tis more likto a sociabailout ruhancing tot resort res

tral bankerty shocks shocks the

er is welfare

y for the

s the lendonly suppos superior n 1>Rcase systes independmergency

ollows that

nsate the re

c . The e

sit insurer deposit ins to compeutility fromd the amou

e deposit hmark casefor the threanker to thct and lessr. So the

ondition wproposition

s that the a

e. ,[ * uuuthe depositkely that a ally non-dele for liquio apply thsponsibilitir’s lendingincrease the allocatione enhancin

non-syst

der of lastrt the nonto the util the non

mic bank fdent of theloan Nv t

t the bank

emaining d

xpected ut

refuses thesurer will iensate all dm closing tunt Nv if:

insurer doe. Howeveeshold *v he unconds in line wcentral ba

which insun 7. Ponce

asset quali

]DIu ) than

t insurer wsocially deesirable liqidity shortfhe uncondies to the dg decision he central n of the lenng.

temic ban

t resort fo-systemic bity of closi-systemic

fails. This ie systemic the non-sy

will be una

depositors

tility of pro

e emergencncur the pdepositors,the non-sy:

29 

oes not er, the where

ditional ith the anker’s

re the (2010)

ity of a

of low

will not sirable quidity falls in

ditional deposit

is the banker nder of

nk

for the bank if ing the bank’s mplies bank’s

ystemic

able to

Nv1

oviding

cy loan olitical but it

ystemic

 

If thloan.indebenc

provand tsocialende

whersomecentrequa

succe

Nv

Nv >

(resp

DINw ,

bailoliquihigheliquiwelfa

accosocia

u

he solvency. As in thpendent o

chmark casHaving

ision of ththe fist-besal welfare fer of last re

w

re ,{SSe propertieral banker

ation (19).

Lemma

eeded, =

if vv ANN <

ANv (respe

pectively at

NCBN vw ,<

Proof. Se As for th

out rule alwdity shocker normaldity shockare given t

The polrding to th

al welfare. T

 

1uu DI

NN

y signal is he benchmof the lique in equatidefined t

e deposit ist solvency function foesort:

(=1,w DINu

DIN

}SF indic

es of (39), in equatio

5 Assum

SS (respe

cLR

cL

ectively Nv

CNN vv = );

, otherwis

ee Appendi

he benchmways doms below Nv

lized sociak is above he deposit licy makerhe size of thThe follow

.1 c

L

below DINu

mark case uidity shocion (36) shothe threshinsurer wesignal thre

or the non

d) Fuu NN

cates the sthe norm

on (18) an

e that uE

ectively faile

(respecti

CNN v> ), an

(2) , 0CBNw

se DIN ww ,

ix 8.8.

ark case Leinates the BN the cent

al welfare above B

Nv

insurer acr will alloche verifiabing second

I the depothe thres

ck. Compaows that boold on th

e derive froesholds in (-systemic b

),(uF

state of thealized exp

nd the one

DINNN uuu |

~

ed, SF=

vely vvN <

nd (iii) h

, >=0 UBRNw

NCBN vw , ; (4

emma 5 imdeposit in

tral bankethan app

the orderts as the lecate the lle liquidity

d-best optim

osit insurershold for ring (38) oth lending

he solvencom the soc(11) and (1bank given

e systemicected socia of the un

SFN

I u>

. Th

F ), then (i)

LR

cvC

N

has a glob

,> DINw ; (3)

4) >,DIN ww

mplies that nsurer as ar as the le

pointing thr inverses ender of lasender of

y shock Nv

mal allocat

r will refusthe liquidwith the tg decisionsy signal f

cial welfare2) the norm

n the depo

c bank. Leal welfare

nconditiona

hen, (1) if t

NSSCB

N vw ,

c

cL), (

bal maxim

If vv BNN <

0>1,CBNw

applying ta lender of

ender of lahe depositand the n

st resort is last resort to maximion results

se the emedity provisthreshold s are equivafor the liqe function malized ex

osit insurer

emma 5 prfunctions

al bailout

the systemi

is increas

(ii) decreas

mum at v

cL

cL

1

{SS

the uncondf last resost resort yt insurer. normalizedhigher. t responsi

mize the exfrom lemm

30 

(3

rgency sion is in the alent. quidity in (10)

xpected r is the

(3

resents of the

rule in

ic bank

sing in

sing if

ANN vv =

c, then

},SFS .

ditional rt. For

yields a If the social

bilities xpected ma 5:

38)

39)

 

succeliquid

so thnon-sto ap

limitresporule. depo

cond

more[uu

non-an emqualiapplythe L

the uncoliquicentrmaxi

chanthe ecentrcentr

largefailed

centrthe s

Proposi

eeded, =dity shortfa

hat it is osystemic ba

pply the unc The intr

t BNv for t

onsibilitiesThis thre

osit insure

dition whic

e likely th], DI

NSFN uu )

systemic bmergency lity asset isy the uncoLLR respon

It is not central ba

onditional dity shocksral banker’imizes the

As statenge the alloextended sral banker ral banker’

Proposi

er range of d than whe

Proof. Se The intu

ral banker systemic ba

 

ition 8 A

SS= (respeall of the n

optimal to ank to the cconditional

roduction othe threshs is handedeshold is er is consid

ch ensure P

at a randothan of l

bank for whloan from ts bailed ouonditional bnsibility.

always opanker’s thr

bailout rus are alloca’s lending expected s

ed in propocation of tset of polic

given the s mandate

ition 9 Thliquidity sh

en it succeed

ee Appendi

uition for tand the u

ank the exp

Assume th

ectively faion-systemic

allocate tcentral banl bailout ru

of the depohold whered over fromequivalent dered. The

Proposition

om non-syow qualityhich liquidthe depositt unconditbail out ru

ptimal to sureshold is ule. For thated to the decision is

social welfaposition 8 the lender cy instrumstate of th

e in both ca

he central bhortfalls of tded (i.e. S

Nv

ix 8.9.

this result nconditionpected retu

at N uuE |~

iled, SF=c bank SS

Nv

the lender ker for liqu

ule for liquid

osit insurere the man

m the centrato the on

e existence

n 8. Given

ystemic bay (i.e. u

dity support insurer thtionally. Thle instead

upport illiqcloser to

his reasoncentral ba

s too restriare. the existeof last reso

ments does he systemiases.

banker shouthe non-sys

SFN

SSN v< ).

is equivalenal bailout urn of the

DINN uu >

F ), there ),( B

NAN

S vvof last res

uidity shortfdity shortfa

r into the mndate for al banker t

ne in the be of the d

condition

nk’s asset ][0, SF

Nu ). rt is sociallhan that a nherefore, thof assignin

quid banksthe socia

n the LLR nker. If theictive. The

nce of theort responsnot affectc bank. Pr

uld act as astemic bank

ent to the rule are conon-system

SFNu . If th

exists a t(respective

sort responfalls below alls above it

model implthe lende

to the uncobenchmarkdeposit ins

NN uuE |~

is of averIt is mo

ly optimal non-systemhe policy mng the dep

s. For smalally optima

responsibe liquidity s unconditi

e deposit isibilities. Wt the resporoposition

a lender of lk when the

explanatioonsidered. mic bank fa

he systemic

threshold fely (SF

N vv nsibilities fthe threshot.

ements aner of last onditional bk case whesurer affec

SN

DIN uu >

rage qualitore likely

does not rmic bank wmaker chooposit insure

l liquidity al one thabilities forshock is laronal bailo

insurer doWe can shoonsibilities 9 summar

last resort isystemic ba

on when onWith a fai

alls so that

31 

c bank

for the ), B

NCN vv )

for the old and

n upper resort

bailout en the cts the

SFN it is

ty (i.e. that a receive ith low oses to er with

shocks an the r small rge the ut rule

oes not ow that

of the ies the

in a ank

nly the lure of

t for all

 

liquiloan liquibankliquiout r

systeactiv

If tsucce

depo

syste

has

insurinfluthe dto op

syste

the lfrom

provbear depoof dnon-depoemercompbankdepodepofailuinsurcorre

dity shockincreases.

dity shockker’s one fdity shock

rule maxim 6.3 Le Suppose

emic bank. vities for th

D

D

the depositessful with

osit insure

emic bank f

to compen

rer incurs ences the

deposit insperate SC

ND

emic bank

iquidationm the non-sy

Accordinision of emany politic

osit insurerdistress. Wsystemic b

ositors net rgency loapensate theker is not osit insurerosit insurerre. Thereforer from tesponds to

 s the socia. Since theks the socfor a larges the centr

mizes the ex

ender of

e now thatThe depose systemic

[{1= SS ED

[{1= SS uED

t insurer ph a probab

r does no

fails. The d

nsate the

the politicprofitabiliturer from

C enters in

is not supp

value of tystemic bang to propmergency local cost in r still has t

When the bank is no

of the liqun but the e remainininsured byr is (1 Nur has to coore, the expthe non-sy:

ally optimae central bcially optimer intervalral banker xpected soc

f last reso

t the deposit insurer’

c bank is giv)(1(1 Su

)(1S cu provides tbility Su a

ot suffer a

deposit insu

remaining

cal cost ty of the nthe non-sy

nto the exp

ported the

the systemnk in case

position 8 oans to thecase the noto compencentral b

ot supporteuidation vanon-system

ng depositoy the depo

))(1 NN v . Iompensate pected costystemic ba

al thresholdbanker becomal lendinl of liquidis still too

cial welfare

ort policy

osit insurer’s expectedven by:

]) SCNDc

( SCNDL

he emergeand repays

any losses.

urer loses t

g deposito

c . The denon-systemystemic banpected util

expected c

ic bank anthe systemthe depos

e non-systeon-systemisate the noanker is ed the dealue (1 Lmic bank ors becauseosit insuraIf the uncoall deposit

t is (1 Nuank if the

d for the pomes less

ng decisiondity shockso stringente in this int

y for the

r is the led utility fro

)[1(1] S L

)]SLN LD

ency loan s the liquid

. With a

the provide

rs ( Sv1 ).

ecision to smic bank. C

nk given thity of the d

costs are L

d SLND is t

mic bank is sit insurer emic bank. c bank failon-systemithe lendeposit insu)L . If the cfails the d

e the emergnce. Thus,onditional tors in cas)N . This saisystemic

provision oforbearingn is closes. Howevert so that uterval.

systemic

nder of laom the len

1 DcL 1 SL

NDc the systemdity assista

probability

ed emerge

. Addition

support thConsequenthe systemicdeposit ins

DcL 1

the deposiclosed. is not resFor this re

s or is closc bank’s dr of last rer has to

central bandeposit insgency loan, the expecbailout ru

se of a nond the utilitbank S i

of the emeg with incrr to the cr, for veryncondition

c bank

ast resort fder of last

]},SLND

}.L mic bank wance so th

y of (1 Suncy loan v

ally, the d

he systemictly the utic bank consurer. In ca

SLND where

t insurer’s

sponsible feason it do

sed. Howevepositors iresort an

o compensnker providsurer only n from the ccted cost f

ule is applin-systemic ty for the ds not liqu

32 

rgency reasing central y large nal bail

for the t resort

(4will be hat the

)S the

Sv and

deposit

c bank lity for

ntinues ase the

e L is

utility

for the oes not ver, the in case nd the sate all des the has to central for the ed the bank’s

deposit uidated

40)

 

SCND

0

SFNv

wher

is liq

whisuppcompequafails the eand expe

so th

Giv

whersystesignathe s

welfa

=

00

SSNv

Su

(

)(

0

NvCBNuF

re SSNv and

The depuidated is

D

ich followsport bank pensates a

al to (1the depos

expected cobank N facted costs

We defin

D

hat D

ven (40) and u

u

re ND re

emic bank oal is belowsystemic ba

Using th

are functio

 

(1)(

0

NvCBNu

(11

0

1

SSNv

)()(1 udFL

(11

0

1 SF

Nv

d SSNv are

osit insureequal to:

=0

SFNvSL

ND

(11

0

1

SFNv

s the same

N the dll deposito

)L . If the it insurer osts are ails the deequal to

ne:

=0

SFNv

SSNvND

= SLN

SCN DD

d (43) the d(1 DcuS

1 DI

SS uu

presents ton the expe

w DISu the

ank will be he solvency

on from e

)() udFL

)(1 NN vu

)1

)( NvCBNu

)() N dudFu

the seconder’s utility f

(1)(

0

NvCBNu

)(1 NN vu

reasoningdeposit insrs. In this scentral bahas to com

)(1(1 Nu eposit insu

)(1(1 Nu

(1)(

0

NvCBNu

L

.= NS Du

deposit ins,) LDN

, NDcL

the impacected cost deposit inclosed.

y threshold

equation

(11

)( NvCBNu

)() N dGudF

)(1(1 Nu

,)(

NvdG

d-best thresfrom the no

)() udFL

()()N dGudF

g as above.surer liqusituation thnker provi

mpensate o)Nv . If the

rer compe)Nv .

)((1) NuL

surer lends

,

ct of the related to tsurer will

of the dep

(20) and

)(1 NN vu

(1)( NvG

))(1 NN vv

shold as deon-systemi

(1

)( NvCBNu

),Nv

. In case thidates thehe expecteided the emonly the ree unconditiensates all

()(1 N udFv

to bank S

deposit inthe non-sysnot provid

posit insure

S cR

Lu =*

)()N dGudF

)Su

()() dGudF

efined in pric bank if t

)(11 N vu

he central e non-systd cost for dmergency maining dional bailodepositors

),() NvdGu

if:

nsurer’s bstemic ban

de the eme

er defined

NW

L we

)( NvG

)( Nv

roposition he systemi

)()N dGudFv

banker dotemic bandeposit insloan but bepositors sut rule is as which le

behavior tonk. If the soergency loa

in (44), the

can deriv

33 

(4

8. ic bank

)( NvG (4

oes not k and

surer is bank N so that applied eads to

(4

(4

owards olvency an and

e social

ve the

41)

42)

43)

44)

 

normof las

Lemexpethe le

incre

globa

then

lendeunco

bankdepowhen

betwcondwelfa6:

*Sv

the c

to ap

lendeuppecentr

malized expst resort fo

w

mma 6 procted sociaender of la

Lemma

easing in v

al maximu

CBS

DIS ww <

Proof. Se Lemma

er of last onditional

ker. If the osit insurern the centr

The polween the ceditional on are. The fo

Proposi

},{ BS

AS vv so

central ban

pply the unc The exte

er of last rer limit B

Sv

ral banker

 pected socior the syste

(=1

uw DISu

DIS

oves somel welfare fust resort or

6 Assu

Sv if S vv <

m at Sv =

Sv , otherw

ee Appendi

6 shows thresort respbailout ru

liquidity sr is the lenral banker alicy makerentral bank the liquid

ollowing pro

ition 10 A

that it is o

ker for all

conditional

ended set resort respoBS for the to the unc

ial welfare mic bank:

(d)* uFuSS

e propertieunction for the uncon

ume that

A

S LR

cLv

(

ASv ; (2) C

Sw

wise DISw

ix 8.10.

hat the norponsibilitiele and for

hocks are nder of lasacts as the r will allocker, the depdity shortfaoposition i

Assume E

optimal to a

liquidity sh

l bailout ru

of policy onsibilitiesthreshold onditional

function g

).u

es of (45) r the systenditional b

|~

SS uuE

N

N

Wc

B ), (i

UBRS

CB w=0

SCBS vw ; (4

malized soes to the d

liquidity s

above BSv

st resort exlender of lcate the lposit insurall in ordeis derived f

|~

DSSS uuu

allocate the

hock smalle

ule.

instrumens for the sywhere thebailout ru

given the d

and relateemic bank bailout rule

*> SDIS uu

i) decreasin

DISw> ; (3)

4) >DIS ww

ocial welfardeposit insshocks belo

the normxceeds theast resort.ender of

rer and ther to maximfrom the p

*> SDIS u

. Th

e lender of

er than *Sv .

nts does nystemic bae mandatele because

deposit insu

es it withgiven the

e is applied

. Then, (

ng if S vv >

If BSS vv <

0>1CBSw .

re from thesurer is doow B

Sv als

malized soce normalize

last resorte unconditimize the nroperties p

here exist a

f last resort

. Above *Sv

not affect tnk. It prov

e is hande from a soc

urer is the

the normcentral ba:

(1)(i) CBSw

ASv , and (iii)

cL

cL

1

(

e allocationominated so by the c

ial welfareed social w

t responsiional bailonormalizedproven in L

liquidity sh

responsibil

the it is o

the allocatvides howed over frocial welfare

34 

lender

(4

malized nker is

Sv is

i) has a

N

N

Dc

B ),

n of the by the central

e if the welfare

bilities ut rule social

Lemma

hortfall

lities to

optimal

tion of ever an om the e point

45)

 

of vieinsur

deter

implqualimoreemerassetthe uLLR

liquithanliquicentrmaxi

resorfailuvise-votheof las

betwemerresulprovcentrassisbecomighrequshou

non-the non-even

the c

ew the cenrer’s liquid

As for th

rmines the

ies that theity (i.e. ue likely thrgency loats is bailedunconditioresponsibi

As abovdity shock the uncondity shocksral banker’imizes the

7 Co This pa

rt responsire of the versa. Botr source ofst resort is

We showween the cergency liqult is as foides sociaral bank castance is coomes moreht even refired emerg

uld be appliWe find

systemic bnegative isystemic bt that the s

For the central ban

 ntral bankeity provisiohe benchm

e condition

e asset of a],[ * DI

SS uu ) hat the dn to a sys

d out uncoonal bail oulity.

ve it is nothe centra

nditional bs are alloca’s lending expected s

onclusio

per analysbilities in asystemic bh banks af external fnecessary

w that theentral banuidity assisllows. On lly undesian improve

onditional oe restrictivfuse to provgency loan ied for larg that the

bank shouldimpact of

bank the cesystemic basystemic b

nk and the

er’s lendingon.

mark case a

n ensuring

a random sthan of lo

eposit insstemic bannditionallyut rule ins

ot always al banker’sbailout ruleated to the decision is

social welfa

on

ses the opa framewobank hurtsre exposedfunding arto avoid so

e lender ok and an ustance regathe one hrable emee expectedon the ban

ve with incvide socialis too larg

ge liquidityallocation d be condi

f a systementral banank collaps

bank howevunconditio

g decision f

and the no

propositio

systemic bow quality surer doesnk than thy. Thereforstead of as

optimal ts thresholde. Therefocentral ba

s too restriare.

timal instrk with a sy

s the returd to a liqure not avaiocially inef

of last resounconditioardless of

hand, the ergency lod social wek’s solvenccreasing lily optimale. For this shortfalls.of lender tional on t

mic bank’s k should b

ses. ver the detonal bailou

for BSS vv >

on-systemi

on 10. Con

ank is mor(i.e. [0u

s not proat a system

re, the polissigning th

to support d is closer tre, the LLRnker. If theictive. The

itutional aystemic an

rn of the nuidity shoclable, publ

fficient andort responsonal bailou

their solveunconditioans. For s

elfare becaucy. On the oiquidity sh emergencreason the

of last resthe state o

failure obe given m

terminationt rule is am

BS is worse

c bank the

ndition E

re likely to ]0, *

Su ). Forovide the mic bank icy maker

he deposit

illiquid bto the sociR responsie liquidity s unconditi

allocation ond a non-synon-systemk. Taking lic intervend detrimentsibilities st rule whe

ency. The onal bailousmall liquuse its emother handhortfalls. Tcy liquiditye uncondit

sort responof the systen the pro

more respo

n of the rambiguous b

than the d

e deposit i

|~

DSSS uuu

be of an ar this reaso

socially owith low qchooses toinsurer w

banks. Forially optimibilities forshock is laronal bailo

of lender ystemic banmic bank b

for grantention by a tal bank fahould be

ere banks rintuition f

ut rule tooidity shocergency liq

d the centraThe centray assistanceional bailo

nsibilities femic bank.ofitability onsibilities

ange of actbecause the

35 

deposit

insurer

*> SDI u

verage on it is optimal quality

o apply ith the

r small mal one

r small rge the ut rule

of last nk. The but not ed that lender ilures. shared receive for this o often cks the quidity al bank l bank e if the

out rule

for the Given of the in the

ion for ere are

 

counmorehaveitselfbankwher

f is

posit

vuCB

so th

1

(0 CBu

~

uE

~uE

=*u

nteracting e forbearane less lendef will be lesk. This leadre the unco

8 Ap 8.1 P (1) The

s the densi

tive for all

*> uv , and

hat vu ACB

(2) Since

(3)(a)

*

0) dFuu

(0)|~

CBuu

(0)| uuCB

(b) w

1

1<

cR

L

8.2 P (1) u =*

8.3 P (1) The f

 effects. Onnce for syser of last ress restrictivds to moreonditional

ppendix

roof of L

first deriv

ity functio

v and u

d has a glob

*= u (see

e CBu 0=0

Ass

1

(1) uu CBu

(1)

CBuu

(1)uu CB

=11

CBu

CB

1=1 CBu

c

roof of P

SFN

SSN uu <

roof of L

first deriva

n the one stemic banesort respove in ordere responsibbailout rule

x

Lemma 1

ative of w

n of the ra

u , vwCB

bal maxim

Figure 2), UBRu=0 , th

sume

* udFuu

*

CBuFu

*u . A con

*

1dFuu

1< both f

Propositi

because

Lemma 2

ative of CNw

hand, fromnk is desironsibilities.r to limit itsbilities for te should be

vwCB is: w

andom var

is increasi

um for uC

the result f

hen CBw 0

(0CBw

0

(1) CCB uF

ntradiction

~

= uEu

factors are

on 2

0> (2) u

NCBN v, is:

m the sociable so th. On the ots potential the centrae applied.

uvwCB =

riable u . S

ng in v i

*= uvCB . S

follows.

UBRw= .

(1)0) CBw

, u

0(0) CB

.

1>| CBuu

positive, th

CBN

CB uu = (3

vw NCBN

=,

ial optimuat the centher hand, losses for l bank and

vuvu CBCB

Since vuCB

if <vuCB

Since vuCB

0

*(1)

(0) uu

CBu

CBu

1* Fu

hen 1CBw

3) UBR uu =

vu NCBN =

m point otral bank the centrathe non-sy

d a smaller

ufu* ,

v and f*u , decrea

0>v and

.

0udF

,

1CBuF .

0> .

UBRNu

uvu NNCBN

36 

of view should

al bank ystemic r range

where

u are

asing if Av is

Then

,

and

Since

ufN ,

 

wher

uf

CBN vu

CBN vu

(resp

1

, CBNu

uE

A con

Since

,SCBNw

non-

decre

wher

wher

uf

re f is th

are pos

NN uv < ,

NN uv = . S

pectively u

(2) Since

(3)(a)

(0)

duu NN

|~

CBNNN uuu

ntradiction

(b) w

e 1

1<Nu

8.4 P Given

)( CBNN

SS wv

increasing

easing. It f 8.5 P

The min

re =N EW

8.6 P (1) The

re f is th

are posit

 

he density

sitive for

decreasing

Since CBN vu

SFN

CN

CBN uvu =

e CBNu 0=0

Ass

1

, udF CBNu

(1),

N

B u

n.

=11

,CBNu

CBNw

1= CBNu

c

roof of P

equation

(=)(,N

SFB uv

in Nv , (d

follows that

roof of P

nimum so

)(1{(1 SFN

SSN

roof of L

first deri

he density

tive for all

function o

all Nv a

g if CBNu

0>Nv andSFN ) (see Fig

UBRNu=0 , th

ume

(1), uu NN

(1),

CB

NuF

1

NN uu

1< both fa

Propositi

(18) an

)[1SSN

SFN uu

) To the ri

t SFN

SSN vv <

Propositi

olvency req

)(( N cRu

Lemma 3

ivative of

function o

Sv and u

of the rand

and Nu ,

NN uv > ,

d ANv (resp

ure 3), the

hen , 0CBNw

(0,CBNw

0udF

(0) , CBNuF

= EudF

actors are

on 4

d (19) (

(( NCBN vuF

ght of CNv

.

on 5

quirement

1) NSFN uL

SCBS vw i

of the rand

Su , SCBS vw

dom variab

NCBN vw ,

and has

pectively v

result follo

,=0 UBRNw .

(1)0) , CBNw

, u

00) , an

>|~

NN uuu

positive, th

(a) SUBRNw ,

))] , (c)

both CBNw

in the fi

0} . It fo

s: vw SCBS

dom variab

is increa

ble Nu . Sin

is increa

a globalCNv ) is so t

ows.

0

(1),

(0), uN

CBNu

CBNu

d N uuE

|

~

1 N

CBN uu

hen 1CBNw

SFUBRN

SS w =,

(,N

SSCBN vw

)(,N

SSB v and

irst-best is

ollows that

vu SCBS=

ble ~Su . Si

asing in Sv

nce NCBN vu

asing in

l maximu

that AN

CBN vu

.

udFuN

CBNN u (1),

1 ,CBNuF

0> .

SSN

SFN uu =

() ,SFCBN vw

d (,N

SFCBN vw

s

S cRu*

** uuS .

uvu SSCBS

nce SCBS vu

S if SCBS vu

37 

N and

Nv if

m for

SSN

AN u=

Then

0 ,

Nu

) .

1 .

, (b)

)Nv is

)N are

NWc

L

ufS* ,

S and

*< SS u ,

 

decre

CBS vu

1

(0 CBSu

~

uE

~SuE

UBRSw

1CBSu

f is

posit

vuCB

such

.

~

uE

imply

easing if

0>Sv and

(2) Since

(3)(a)

*

0) dFuu SS

(0)|~

CBSS uu

(|~

CBSSS uu

CBSw> .

(b) CSw

1

1=1

c

|~

SS uuE

8.7 P The proo(1) The

s the densi

tive for all

*> uv , and

that vuCB

(2) (a) S

Then

|~

DIuuu

(3) Since

y that wDI

  >S

CBS uvu

d ASv is su

e CBSu 0=0

Ass

1

(1) uF CB

Su

(1) CBS uu

*1) Su

. A

=11

1CBSu

CB

1<NB

an

>(1)CBSS u

roof of L

of of lemmfirst deriv

ity function

v and u

d has a glob

*= uvA , th

ince CBu 0

*1

0 duu

*

DIuFu

e Bv is so

wwUBRI <

*Su , and ha

uch that CSu

UBRSu=0 , th

sume

* dFuu SS

) *

S uFu

A contradi

*

SS dFuu

d assumpt

*> Su both

Lemma 4

a 4 is takenative of w

n of the ra

u , vwCB

bal maxim

e result fol

UBRu=0=0

1 uudF DIu

0 , and E

o that CB vu

vwCB for

as a globa

*= SAS

CBS uv ,

hen CBSw 0

(0CBSw

0u

(1) CBS uFu

ction. Tog

~

= SuEuF

tion |~SuE

factors are

n from Po vwCB is: w

andom var

is increasi

um for uC

lows. R , then CBw

* udFuu

|~

uuuE D

DIB uv = , tBvv < and

al maximu

the result

UBRSw= .

(1)0) CBSw

, u

0(0) CBSu

gether wit

1>| CBSSS uu

(1)CBSS uu

e positive, t

once10 . uvwCB =

iable ~u . S

ng in v i

*= uvCB . S

UBRB w=0

0 ,

*uDI

. A c

hen CB vw

d that wDI

um for CSu

follows.

0

(1)

(0) uuS

CBSu

CBSu

h property

11 *Su

*>) Su

im

then 1CBSw

vuvu CBCB

Since vuCB

if <vuCB

Since vuCB

. (b) Assum

0

uDIu

contradicti

DIB wv = . P

vwCBI f

*= SSCB uv .

.

0* udFS

,

y (2) this

1CBSuF .

plies

0>1 .

ufu* ,

v and f*u , decrea

0>v and

me UBR ww

* udFu

on.

Properties 1

for Bvv .

38 

Since

Then

,

and

imply

Since

where

u are

asing if Av is

0DIw

0 ,

1 and 2

 

CBw

=*u

wher

uf

CBN vu

CBN vu

(resp

UBRNw

and

for v

CBNw

,SCBNw

(4) Si

=1

1

1CBuu

1

1<

cR

L

8.8 P (1) The f

re f is th

are pos

NN uv < ,

NN uv = . S

pectively u

(2) (a)

0,, DINw

0 uN

DINu

N uuE

|

~

(3) Sinc

2a imply tBNN vv .

(4) Sin

=1

1

1,CBNu

u

1

<Nu

8.9 P Given

)( CBNN

SS wv

 nce <Bv

* udFu

1=1 CBu

c

roof of L

first deriva

he density

sitive for

decreasing

Since CBN vu

SFN

CN

CBN uvu =

Since u

0 . Then 1

0

udFuN

DINN uu

ce BNv is s

that DINw ,

nce <BNv

NN dFuu

1=1 CB

Nuc

roof of P

equation

(=)(,N

SFB uv

1< , pr

>|~

= uuE

1< both f

Lemma 5

ative of CNw

function o

all Nv a

g if CBNu

0>Nv andSFN ), the resu

CBNu =0=0

1

0 dFuu NN

0 , ~

uE

Nu . A cont

so that CBNu

UBRN ww ,<

1< , pro

~

= NuEu

1<1 both

Propositi

(18) an

)[1SSN

SFN uu

roperty

1> CB uu

factors are

NCBN v, is:

of the rand

and Nu ,

NN uv > ,

d ANv (resp

ult follows.UBRNu= , th

1 uuF DI

Nu

|~

DINNN uuu

tradiction.

DIN

BN

CBN uv = ,

NCBN vw , fo

operty 3

>| CBNNN uu

factors are

on 9

d (19) (

(( NCBN vuF

3 implie

1* CBuFu

positive, th

vw NCBN

=,

dom variab

NCBN vw ,

and has

pectively v

.

hen ,CBNw

udFuNN

N

I uFu

, then CBNw

for BNN vv <

implies

11 Nu

e positive, t

(a) SUBRNw ,

))] , (c)

es that

1B

hen 1CBw

vu NCBN =

ble ~Nu . Sin

is increa

a globalCNv ) is so t

,=0 UBRNw

0u ,

0DINu , and

, = DN

BN

CBN wv

and that

s that

1,CBNuF

then 1CBNw

SFUBRN

SS w =,

(,N

SSCBN vw

> CDI ww

.

0> .

uvu NNCBN

nce NCBN vu

asing in

l maximu

that AN

CBN vu

. (b) A

d

,DIN . Prope

t CN

DIN ww ,

>, CBN

DIN ww

. Since

0>1 .

SSN

SFN uu =

() ,SFCBN vw

39 

1CB .

Since

ufN ,

N and

Nv if

m for

SSN

AN u=

Assume

rties 1

NCBN v,

1,B .

, (b)

)Nv is

 

non-

decre

wher

uf

decre

CBS vu

. The

2a im.

CBSw

CBSw

(2010and tRichUSA.

LiquPape

increasing

easing. It f 8.10 P (1) The

re f is th

are posit

easing if

0>Sv and

(2) (a) S

en 10

uuS

|~

SuE

(3) Since

mply that

(4) Si

=1

1

1 SCBSu

u

=*S R

u

0>1 .

Refere Acharya

0): “Measuthe New Arardson, a.

Acharyaidity and In

er 16395, NAcharya

  in Nv , (d

follows that

Proof of

first deri

he density

tive for all

SCBS uvu >

d ASv is so

ince CBSu 0

* udFuuS

DI

SS uu

e BSv is so

UBRS

DIS ww <

nce <BSv

*SS udFu

<NWc

L

ences

, V. V., C. Buring Systemrchitecture oand I. Walt

, V. V., annter-Bank

National Bu, V. V., and

) To the ri

t SFN

SSN vv <

Lemma 6

ivative of

function o

Sv and uSu , and ha

o that CBS vu

UBRSu=0=0

1 duu SSDI

Su

*

DI

SS uFu

o that CBSu

SCBS vw

1< , pr

|~

= S uuE

1

1

NBc

Brownlees, mic Risk,” of Global F

ter, pp. 87–

nd O. MerrMarkets: E

ureau of Ecd T. Yorulm

ght of CNv

.

6

SCBS vw i

of the rand

Su , SCBS vw

as a globa

*= SAS uv , th

R , then CBSw

0udF ,

0 , and E

DIS

BS uv = ,

for BSS vv <

roperty

1> CBSS uu

1<1= CBSu

R. Engle, Fin Regulat

Finance, ed–119. John W

rouche (201Evidence froonomic Re

mazer (2008

both CBNw

s: vw SCBS

dom variab

is increa

al maximu

he result fo

UBRS

B w=0

*

0 uu SS

DISu

|~

SS uuuE

then CBSw

BS and that

3 implie

1*S Fu

1 both fa

F. Farazmaing Wall St. by V. V. AWiley & So

10): “Precaom the Subesearch. 8): “Cash-in

)(,N

SSB v and

vu SCBS =

ble ~Su . Si

asing in Sv

um for CSu

llows.

. (b) Assum

0* udFS ,

*S

DIS uu

. A

DIS

BS wv = .

t CBS

DIS ww

es that

1CBSu . Si

actors are

and, and Mtreet: The D

Acharya, T. ons, Inc., H

utionary Hb-Prime Cr

n-the-Marke

d (,N

SFCBN vw

uvu SSCBS

nce SCBS vu

S if SCBS vu

SS

CB uv = .

me UBRS ww

,

A contradic

Properties

SCB v for

> CS

DIS ww

nce

e positive,

. P. RicharDodd-Frank

F. Cooley, oboken, NJ

Hoarding ofrisis,” Work

et Pricing a

40 

)N are

ufS ,

S and

SS u< ,

Since

0DISw

ction.

1 and BSS vv

1CBS .

, then

rdson Act M. P. J,

f king

and

 

Opti

Cent

Lond

2007

LendCred

fromYork

The

and

“SystWP/1

Disco

Polic1151

Relatand B

Polic74(3)

Finan

Repo

LendRevie

ResoBank

Fina

Nece

Cont

mal ResoluAllen, F.

tral Bank InBagehot

don: HenryBrunner

7-08,” WorkCastiglio

ding at Penit and Bank

Copelanm the Tri-Pa

. Dewatri

MIT Press.Diamon

Liquidity,”Espinosa

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Flanneryount Wind

Freixas, cy: A Mode–1176.

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42 

sk,”

er of

edit of

rnal