The Commercial Banking Firm: A Simple Model
Author(s): James Tobin
Source: The Scandinavian Journal of Economics, Vol. 84, No. 4 (1982), pp. 495-530
Published by: Wiley on behalf of The Scandinavian Journal of Economics
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Scand. J. ofEconomics84 (4): 495-530,1982
The CommercialBankingFirm:
A SimpleModel
James Tobin
USA
Yale University,
New Haven,Connecticut,
Abstract
without
full
A commercial
bankmustdecide its volumeof illiquidloans and investments
exceeditsdefensive
knowledge
ofitsdeposits.In case withdrawals
positionin liquidassets,
or losses
reserverequirements,
thebankincursextracostsin meeting
penaltiesinborrowing
in disposingof illiquidassets. In case of good luck on deposits,an unnecessarily
large
butless liquidassets.The paper
defensive
positionsacrifices
profits
frommoreremunerative
bank.
decisionbya profit-maximizing
presents
a simplemodelofthisprecautionary
portfolio
and competitive
structure
in monetary
The effects
of variations
policy,bankingregulations,
rateceilings,
are traced:reserverequirements,
access to centralbankcredit,depositinterest
of capitaland time
monopolistic
powerin loan and depositmarkets,relativeimportance
anddegreeofdeposituncertainty.
size and natureofilliquidity
depositliabilities,
penalties,
kindof financialintermediary.
Commercial
banksare the mostimportant
Theirliabilitiesare theclosestprivately
issuedsubstitutes
forgovernment
currency.Demanddepositsserveas meansof paymentgenerally
acceptable formosttransactions.
For severalreasons,depositorsbearverylittle
of therisksof theloans thebanksmake.Thereare economiesof scale in
and in specializedadministrapoolingofdefaultrisksspecificto borrowers
tionandappraisaloftheloans.The banks'shareholders
assumetheresidual risk;onlyaftertheirequityis wipedout woulddepositors'claimsbe
jeopardized.Finally,thegovernment
standsbehindbankdepositliabilities,
bothas ))lenderoflastresort>>
to tidebanksovercrisesofilliquidity
andas
insurerofdepositsagainstthecontingency
ofinsolvency.
Government
monetaryand creditpolicyoperatesmainlythroughthe
commercialbankingsystem,and this is the mainreasontodayto give
special attentionto commercialbanks. Historicallythe firstreasonfor
intervention
in thebankingbusinesswas to protectdepositors
government
(orinoldertimesbanknoteholders)againsttherisksofbankilliquidity
and
insolvency.But thepublicregulations
and institutions
establishedforthis
purposecan also be used to regulatecreditmarketsin the interestsof
economicstabilization,
maintenance
of thevalueofthecurrency,
or other
government
objectives.As theprotective
purposeofgovernment
intervenScand. J. of Economics 1982
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496 J. Tobin
tion has been achieved,these otherobjectiveshave come to dominate
bankingsystem.
to thecommercial
relationsofgovernment
banking
theoryof a singlecommercial
This paperpresentsa primitive
functionof
distinctive
to the traditional
enterprise.It refersprincipally
thantheir
andless liquidity
banks,to buyandholdassetsoflongermaturity
liabilities.In recentyearsbanks,largebanksespecially,have increasingly
assets and liabilitiesmore
becomebrokersbuyingand sellingmarketable
The simplemodelof thispaper
and liquidity.
closelymatchedin maturity
to
does notapplyto thatbusiness.Moreover,themodelrefersprincipally
thereare
on branchbanking,
theUnitedStates.Because oflegallimitations
andcompeand thesystemis moredecentralized
manymorebankingfirms
sense-thanin other
in Chamberlin's
competitive
titive-monopolistically
assumed
and centralbankinginstitutions
economies.The legalframework
I believeand hope that
to Americanpractice.Nevertheless,
also conform
themodelhas widerapplicability.
I. The PortfolioChoices of a Bank
thenatureof thebusinessdecito understanding
In a firstapproximation
bank
a commercialbanker,theassets of a commercial
sionsconfronting
and defensive
maybe dividedintotwo categories:loans and investments
are in theshortruneitherilliquidor unpreassets. Loans and investments
theirfullvalue,thebankmusthold
dictableinvalue.To be sureofrealizing
theseassetscan be availableformeeting
themto maturity.
Consequently,
onlyat someriskofloss. It mayindeedbe impossible
depositwithdrawals
to sellor to borrowagainstcertainloans.Defensiveassetsare,incontrast,
assets of veryhighliquidity.The bankknowsthattheycan be sold, or
loss ordelay.Defensiveassetsincludecurrency,
against,without
borrowed
loans to
depositsin the centralbank,depositsin otherbanks,overnight
call loans,
intheU.S.), well-secured
otherbanks(knownas federalfunds>>
as
Treasurybills, and otherpaper of equivalentqualityand eligibility
collateral.In this usage the termcovers both primaryand secondary
reserves,i.e., boththoseassetsthatqualifyas legalreserves,andthosethat
intolegalreservesthatthebankscan regardthem
are so readilyconvertible
as thevirtualequivalent.
Law or convention
requiresthebankto holda certainquantity
generally
of defensiveassets, the bank's requiredreserves.When the Monetary
institutions
ControlActof 1980becomesfullyoperativein 1987,depository
on
in
or
hold
to
be
will
depositin a
currency
in theUnitedStates
required
Federal Reserve Bank about 12 percentof demanddepositsand other
savingsdepositsand
checkableaccountsand 3 percentof non-personal
than31/2years.
shorter
timedepositsof originalmaturity
Scand. J. of Economics 1982
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Thecommercial
banking
firm:A simplemodel 497
Liabilities
Over
Night
Shareholders'
Equity
Deposits
Assets
Defensive Assets
Bor| ed
-Bor| Bolwed
Net Holdings of
Defensive Assets
|Requred|
Reured
Deensve
Defensive
Reserves
Position
Primary Reserves
Excess
row-
Loans and Investments
-------------------------
Secondary
Reserveson
Required
Reserves
Ex- | Secondary
Reserves
cess
Required
Net
Free
iReserves
1----
Secondary
Reserves
ofbankbalancesheet.
Fig. I Schematicrepresentation
A bank's holdingsof defensiveassets will almostalways exceed its
requiredreserves.But someoftheseholdingsmaybe offset
by shortterm
or overnight
fromotherbanksor fromthe centralbank. The
borrowing
amountby whichthe bank's net holdingsof defensiveassets exceed its
requiredreserveswillbe calleditsdefensive
position.For itis thismargin
whichmeasuresthebank'sabilityto meetreserverequirements
ifitshould
be confronted
withunusualdepositwithdrawals
or extraordinary
demands
forloans.In thiscontingency
thebankcan drawdownitsdepositsinother
banks;orfailto renewovernight
loansof federalfunds>>
tootherbanks;or
presentmaturing
Treasurybillsforcash insteadof new bills; or borrow
moneyon thecollateralofTreasurebills;or sellbillseitheroutright
orwith
agreement
to repurchasethem.
The basic accounting
forthecommercial
bank(abstracting
identity
from
itsphysicalassetsand otherinessential
accounts)is:
Deposits+ Shareholders'equity
=Required reserves
+Defensiveposition
+
Loans and investments
Scand. J. of Economics 1982
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498 J. Tobin
and also sets forth
graphically,
Figure1 displaysthisaccountingidentity
thatwillbe useful.
and classifications
someotherdefinitions
threekindsofbankdecito distinguish
It is a convenientsimplification
sions:(1) thesizes of thetwobroadasset categories,theloan and investof the
and the net defensive
mentportfolio
position,(2) themanagement
or secondaryreserve
position:thekindsand amountsofprimary
defensive
and (3) thecomposiborrowing,
assetsheld,and theamountof short-term
Thispaperconcernsmainlythe
portfolio.
tionof theloan and investment
firstdecision,discussessomeaspectsofthesecond,anddoes nottreatthe
third.
II. The Bank's Deposits
The volumeof depositsin a bankis partlywithinand partlyoutsideits
control.The locationof a bank,in botha geographicand an economic
sense, gives it a naturalclienteleof depositors.The bank can seek to
to depositorsby the interestratesit offerson
increaseits attractiveness
deposits,bythequalityand costsofitsservices,andbytheusualmediaof
theorganmusicandthe
splendorofthebuilding,
indirect
competition-the
with
identification
the
cultivated
advertising,
the
lollipopsfor children,
competiof
community
service,and so on. However,someoftheseforms
tionfordepositsare limitedby law or by explicitor tacitconvention.
on
frompayinginterest
In theUnitedStatestoday,banksare prohibited
for
legal
ceilings
and
from
exceeding
demanddepositsof corporations
on othercheckabledeposits,passbooksavingsaccounts,and cerinterest
taintimedeposits.
Thereis anotherand moredirectway by whicha bankmaybe able to
thelevelofitsowndeposits.Whena bankmakesa loanto one of
influence
account.In the
itsimplycreditstheamountto theborrower's
itscustomers
thebank'sdepositsare increaseddollarfordollar
firstinstance,therefore,
spendstheproceedsbycheck,someofthe
withitsloans. As theborrower
recipientswill leave the moneyon depositwiththe lendingbank,while
otherswill deposittheirreceiptsin otherbanks or converttheminto
spendtheirbalancesand in succeedinggenAs theserecipients
currency.
thelendingbankwilllose moreand moreof the
erationsof transactions,
maybuild
depositcreatedbyitsinitialloan. On theotherhand,a borrower
up hisdepositinthelendingbanktowardstheendofthetermofhisloan,in
forrepayingit.
preparation
ofthe
to holda certainfraction
Sometimesa bankrequirestheborrower
less
has
lent
the
bank
in
of
the
effect,
the
term
loan;
during
on
loan deposit
distinguished
be
should
This
the
loan.
of
amount
practice
thanthenominal
fromthe"you lendto me now,I'll lendto youlater"bargainthatcharacScand. J. of Economics 1982
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Thecommercial
banking
firm:A simplemodel 499
terizes,withvaryingdegreesof formaland explicitunderstanding,
the
relationsof a bank and a continuing
customer.By the termsof such an
understanding,
a borrower
forhisloanwithdepositsinpastor
compensates
future,notby holdingsome of the loan itselfon deposit.Throughthese
continuing
customerrelations,a bank's loans affectits deposits,but at
somefuture
time.It is equallytruethatdepositstodayentailsomecommitmentto providefutureloan accommodation,
Indeed,one way in which
banks competefordepositswitheach other,and withothershort-term
borrowers,
is by offering
to regulardepositorsthereciprocalassuranceof
crediton favorabletermswhentheywantto borrowratherthanlend.
The degreeto whicha bankcan expectto retaindepositsresulting
from
itsown loans dependson its size relativeto otherbanks.It a bankis the
itwillretainmorethanifthelocalpayeesofthe
onlyone initscommunity,
borrower'schecksare scatteredamongseveralbanks. But even a local
monopolybankwilllose depositsas transactions
spreadto otherareas of
the economy.And a nationalmonopolybank,ifone existed,wouldstill
faceleakagesto currency
holding,foreign
balances,and otherassets.The
restrictions
on branchbankingintheUnitedStatesmaketheleakagesfrom
an individualbanktypicallymuchlargerthanin the bankingsystemsof
mostothercountries,
wherea handful
ofbanks,each withmanybranches,
thescene.WhereinEnglandorCanadaor Swedena loanbyone
dominates
branchincreasesdepositsinanotherbranchofthesamebank,intheU.S. a
loan by one bankincreasesdepositsin anotherbank.
The leakage is naturallygreaterand quickerforbank investments
in
nationallymarketable
securitiesthanforloans to local businessmen.
An
individualbank, even a large one, can be assumedto retainvirtually
at all of amountsplaced in defensiveassets.A bank'sdecisionto
nothing
holdcurrency,
or depositsin thecentralbank,or thedebtsofotherbanks,
or Treasurysecuritiesdoes not lead to any significant
increasein the
amountsthatotherschoose to leave on depositwiththebank.
III. Bank Portfoliosand Profits
In considering
thebank'sbroadportfolio
choice,betweenloansand investmentsanddefensive
position,itwillbe simplest
tobeginwitha bankwhose
depositsare costlessand independent
of the bank's own portfolio
decisions. Moreover,the bank is assumedto knowwithcertainty
whatits
depositswillbe overtheperiodoftimethatitsfundsare committed.
This
assumption
abstracts,
ofcourse,fromtheessentialproblemofbanking,
the
unpredictability
ofdeposits,anditwillbe removedshortly.
Beginning
with
theunrealistic
case ofcertainty
is an analyticand expository
convenience.
In Figure2, assetsaremeasuredhorizontally
anddepositsvertically.
The
Scand. J. of Economics 1982
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J. Tobin
500
kD
D+E
Disposable
/
Assets
-
7/
/
Dot
to
/
00
Deposits/
EE/45
?5_
Required
Defensive
Position
Reserves,
Loans and Investments
Fig. 2
basic accountingidentitybehind the diagramis the same one given above.
translatedinto symbolsas follows:
D+E=kD+R +L
Thus the horizontaldistance to the dashed lineD+E representstotalassets,
and the horizontal distance to the solid line kD representsrequired reserves. The horizontaldistance between these lines represents"disposable
assets," the amountthatcan be divided between defensivepositionR and
loans and investmentsL. If loans and investmentsexceeded disposable
assets, the defensiveposition would be negative.
Now suppose that the bank earns a rate of interestr on its defensive
positionR (or pays interestat thisrateon a negativeposition),and earns on
a loan and investmentportfolioof size L a total revenue P(L) (net of
administrativecosts and actuarial allowance fordefault).Its deposits are a
given amountbeyond its control,Do, and its equityis fixedat E. The bank
will seek to maximize the total net revenue on its portfolio,P(L)+Rr,
L+R=E+(1-k)Do. If themarginal
subjectto thebalancesheetconstraint
revenue fromloans P'(L) is constantand greaterthanr, thismaximization
puts no limiton the loan and investmentportfolioother than the bank's
abilityto borrow to finance a negativedefensiveposition. If the marginal
revenuefromloans Pt(L) is always smallerthanr, thebank will simplyhold
all its deposits and equity in defensiveassets.
A situationin whichdecliningmarginalrevenue sets a positive limiton
Sc and. J. of Econo1mic s 1982
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Thecommercial
banking
firm:A simplemodel 501
P (L)
I ~~~Maximum
/
ReveeRevenue
Loans and
Investments
|||
pensive
Position
Po itive
+E]-
r[(l-k)Do
+E]
ition
0
Revenue
from
Defensive
Position
r[(l-k)Do
Negative-sPfensi
0/
(r
-
.0
Loans
and Investments-.r
Fig. 3
is shownin Figure3. Here loans and
the loan and investment
portfolio
L are measuredhorizontally.
investments
L, is thepointat whichloansand
investments
exhaustdisposableassets:thatis, LC=E+(l-k)Do. To theleft
of Lc, defensivepositionR is positive;to the right,negative.Total net
is measuredvertically
revenuefromloansandinvestments
upwardfromthe
origin,and totalrevenuefromdefensivepositionis measuredin theopposite direction,downwardfromthe origin.A negativedefensiveposition
means,of course,negativerevenuefromthissource;in Figure3 thisis
shownwhenthelowerrevenuecurverisesabove thehorizontal
axis. Thus
totalrevenuefromthe two sourcescombinedappearsin Figure3 as the
difference
betweenthe two curves.Indeedthelowerline-revenuefrom
defensiveposition- maybe regardedas thecostcurve,albeitopportunity
revenuecurve.In Figure3 a revenue
cost, to the loans and investments
maximum
occursat Lo, wheremarginal
revenueis thesamefromthetwo
sources.The slopeofa lineorcurveis indicatedinthediagram
inparenthefor
the
line
is
r.
of
C
ses;
example, slope
A higherlevel of depositswill enable the bank to enjoy a stronger
defensive
positionat anygivenvolumeofloansandinvestments.
Suppose,
Scand. J. of Economics 1982
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502
J. Tobin
forexample,thatdepositsrisefromDo in Figure2 to Do, so thatthebank
nowhas,aftermeeting
reserverequirements
ofkD, disposabledepositsof
D'(1 -k). Thenthepointofzero defensivepositionL, in Figure3 willshift
to the right,e.g. to L., by the amount(1-k)(D-Do). Therewill be a
parallelshiftto theright,
fromC to C' in thelinerepresenting
therevenue
fromthedefensivepositionin Figure3. The bankwillhave no reasonto
changeits volumeof loans and investments.
The volumethatequatedthe
marginalrevenueand marginalopportunity
cost of lendingstilldoes so.
Giventherequiredreserveratiok, a dollarincreasein depositsmeansan
increaseof requiredreservesof k and an increasein defensivepositionof
1-k.
111.1 Penalties for Negative Defensive Position
So farithas been assumedthatthebankcan alwaysborrowshorttermat
thesame rater thatpositiveholdingsof defensiveassets yield.If so, the
marginalopportunity
cost of lendingis the same fornegativedefensive
positionsas forpositive.Butmorelikelythebankmustpaya higherrateto
borrow,or-what amountsto the same thing-mustliquidateassets that
wouldhave yieldeda higherratethanr. Let theeffective
ratefornegative
positionsbe r+b. In addition,theremaybe a fixedcost ofnetborrowing,
independent
of theamountborrowed,attributable
to thecostsand inconveniencesofarranging
a loan or to theloss ofprestigeinvolvedindisplayinga shortageof ownedreserves.In Figures4a, 4b and 4c, thecurveC
representing
revenuefromdefensive
positionincreasesinslopeatL, fromr
to r+b. In Figure4a thereis no fixedcost involvedin havinga negative
defensiveposition.Butin Figures4 b and4 c theexistenceandamountofa
fixedcost a are shownby theupwardjumpin thecurveC.
These changesin theopportunity
cost curveat Lo haveno effecton the
bank'sdecisionifthedecisionis inanycase to lendan amountless thanor
equal to L, i.e., to holda positiveor zero defensiveposition.Thiswillbe
truewheneverP'(L,.), marginalrevenuefromloans at the pointL, is
smallerthanor equal to r. But the extracosts of a negativepositionare
relevantifP'(L(.) exceeds r. Here thereare severalpossibilities:
The bestportfolio
forthebankis
(a) r<P'(L.)Gr+b. (Not diagrammed)
theone corresponding
to L4.,a zero defensiveposition.
(b) r-+b<P'(L.), no fixedcost (Figure4a). Here thebankwillchoosea
volumeof loans and investments
higherthanL(. and a negativedefensive
position.
(c) r+b<P'(L.), positivefixedcost (Figures4b and 4c). Here thebank
may,as incase (b) proceedto a pointbeyondL, whereP'(L)=r+b. Thisis
in Figure4 b. However,it-is also possiblethatthe revenueat
illustrated
sucha pointis smallerthantherevenueat Lc becauseof thefixedcost of
in Figure4c.
Thisoutcomeis illustrated
borrowing.
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banking
firm:A simplemodel 503
Thecommercial
P (L)
P (L)
I Maximum
Net
Maximum
Net
Revenue
Revenue
from
Loans and
Investments
Revenue
from
Defensive
Revenue
/r'/
(r
Loans
PositionLonanInetns
Ln
and Investments-
ad
Fig.4 b
Fig. 4a
P(L)
0
(r Lan
and Investments
1
Fig. 4c
earnsinterest
The precedinganalysishas assumedthatthebankneither
on itsdeposits.These assumptions
on requiredreservesnorpays interest
theanalysis.
can be relaxedsomewhatwithoutaltering
to
in respectto thepaymentof interest
Centralbankingsystemsdiffer
bankson requiredreserves.In theU.S. FederalReserveSystem,member
yield
bankscan enjoytheprevailing
bankreservebalancesearnno interest;
r on defensiveassets only to the extentthattheirdefensivepositions
agitation
however,thereis considerable
Currently,
exceedsrequirements.
comforpaymentof intereston reserves,because banksare increasingly
not
petingfordepositswithmoneymarketfundsand otherinstitutions
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504
J. Tobin
requiredto holdreserves.In otherbankingsystems,e.g. theUnitedKingdomandCanada,tworeserverequirements
orconventions
havesometimes
beenin effectsimultaneously,
a liquidassetsconvention
and withinthata
In theU.K., forexample,bankscouldearntheprevailcash requirement.
ingrater on all theirdefensiveassets including
thosequalifiedforthe30
percentliquid-assetsconvention,except for the 8 percentof deposits
requiredto be heldin cash or withtheBankofEngland.
If interest
is earnedon requiredreserves,theseearningsadd a constant
amountto the bank's revenuefromits givenvolumeof deposits.This
theoptimalallocaadditiondoes notaffectthecalculationsthatdetermine
anddefensive
assets.
tionofthosedepositsbetweenloansand investments
Likewise,the analysisapplieswhenthe bankmustpay interestto its
depositorsso longas thedepositinterest
rateis regulated
and fixedbelow
whatthe bank can earn on a dollarof deposits.It is stillin the bank's
interest
to acceptas manydepositsas areavailabletoit.The interest
outlay
is thena fixedcost,thesame whateverthebank'sportfolio.
Although
theseinterest
receiptsand outlaysdo notaffectthemaximumprofitportfolio,
theydo affectthe bank's profits.Theymaybe relevant,
therefore,
to theshareowners'decisionwhether
to stayin thebusinessor
not.
111.2. The Value and Cost of Equity
How muchis a dollarof new equityworthto thebank?More precisely,
how muchadditionalprofitwill the bank earn if an additionaldollarof
in Figure
capitalis subscribed?Withtheexceptionof thecase illustrated
4c, additionalcapitalwill simplybe investedin an increaseddefensive
positionand will thusyieldeitherr or r+b. Since r or r+b is also the
itis equallytrueto saythatthevalueofnew
revenuefromlending,
marginal
capitalis P'(L). In theexceptionalcase, thenewequitywillgo intoloans
it willyieldthe marginalrevenueP'(L), whichexceeds
and investments;
r+b. If the existingshareholdersof the bank can attracta new stock
of a dollarby theprospectof a dividendyieldno largerthan
subscription
itwillbe worththeirwhileto do so.
to profit,
thesemarginal
contributions
themarginal
costofequity
In thelong-run
ofthebankingfirm,
equilibrium
capitalwilljust equal the marginalprofit.The bankmaynonethelessbe
makingmonopolyprofitswhichwouldattractnewfirmsiffreeentryinto
theindustry
werepossible.
111.3. The Value and Cost of Deposits
How muchis a dollarincreasein depositsworthto the bank?As noted
meanan increaseof$1-k
above,an increaseindepositsof$1 willnormally
in thedefensivepositionand an increaseof$kinrequiredreserves.Therefore,assumingno interestis paid on requiredreserves,a dollarmorein
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Thecommercial
banking
firm:A simplemodel 505
depositswillpermitthebankto add (1-k) r to itsearningsifitsdefensive
positionis positive,or (1-k)(r+b) ifitsdefensive
positionis negative.Since
in thesesituations
P'(L) is equal to r or r+b, thevalueofdepositscan also
be expressedas (1-k)P'(L).
The situationpicturedin Figure4 c is againsomething
of an exception.
Therean influxofdepositswillpushthejumpintheopportunity
costcurve
to theright.It willstillbe profitable
forthebankto lendup tothejump,the
pointofzerodefensive
position.Hencean increaseindepositswillincrease
loansandinvestments
ratherthandefensive
position.It will,as intheother
add
revenueoflending,
cases,
(1-k) timesthemarginal
(1-k)P'(L,), to the
bank'srevenue.But thisis greaterthan(1-k)(r+b).
The marginal
value of depositsto thebankis smallerthanthemarginal
valueofequityintheproportion
(1-k), becauseofthereserverequirement
againstdeposits.Butofcourseitis generally
easierto attract
depositsthan
to attractcapital.
Ifinterest
is paidat rater' on requiredreserves,thenthemarginal
costof
depositswillbe equatedto a weighted
of
r'
and
the
average
return
marginal
on disposableassets,withweightsk and 1-k respectively.
So longas r' is
thesmaller,one effectof an increasein therequiredreserveratiok is to
lowerthevalueofdepositsto thebank.
Banksgenerallyacceptmorethanone kindof deposit.SupposeI is the
reserverequirement
on a particular
oftimedepositsT. Thenifr' is
category
zero, theadditionalearningsmadepossiblyby a dollaradditionalof time
depositsis (1-I)P'(L). Assuming1 is smallerthan the reserveratiok
requiredfordemanddeposits,thegrossmarginal
valueoftimedepositslies
betweenthecorresponding
values of equityand demanddeposits.But of
coursethenetvalueoftimedepositsdependsalso on theinterest
thatmust
be paidto attractthem.
111.4.Unrestricted
Competition
forDeposits
So longas competition
fordepositsis effectively
limitedbylawsor gentlemen'sagreements
prohibiting
or limiting
interest
payments
to depositors,
thenthefirmand theindustry
are in disequilibrium.
These devicespermit
the bankingindustry
as a whole to receiverentswhichundereffective
competition
wouldbe paidto depositors.In thelongrunthecompetition
of
newbanking
firms
couldreducethemarginal
revenuefromlending
untilthe
valueofdepositsis brought
downto theircost. Butinpracticeentryis not
free.In theUnitedStatesentryis regulated
byfederaland statechartering
authorities.
For manyyearsthelegalceilingson interest
ratespayableon timeand
savingsaccountsin the UnitedStateswereso low as to preventeffective
ratecompetition
fordeposits.Raisingor lifting
ofceilingssince1962paved
thewayforvigorousandopenratecompetition
fortimedeposits.As would
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506 J. Tobin
be expected,theratesbanksofferfortimedeposits,whichhave smallor
are close to theyieldstheycan earnon defenzero reserverequirements,
sive assets. Althougheffectiveceilings,zero to 51/?percent,apply to
demanddeposits,othercheckabledeposits,and passbook savingsacBy theend
legislation.
raisedunderexisting
counts,thesewillbe gradually
into
of thedecade it is quitepossiblethatthesystemwillbe transformed
rates.
ofdepositinterest
regulation
effective
one without
withoutdepositrateceilings,thebankingfirmwill
In a fullequilibrium,
costofeach class ofdepositsto itsmarginal
value,i.e.,
equatethemarginal
to theadditionto earningson assetswhichan additionaldollarofdeposits
be suchthatthis
willgenerally
willpermit.Moreover,theoptimalportfolio
additionto earningswill be the same no matterin whichasset the new
depositexceedsitscost,the
depositis invested.Ifthevalueofan additional
bank will seek to attractdepositsby raisingthe rate of interestpaid
devices.
depositorsor by othercompetitive
depositsmayexceedtheaverageinterest
costofattracting
The marginal
rateon depositsand mayrisewiththebank'svolumeofdeposits.For one
to deposidepositsare notpayments
thing,someof thecosts ofattracting
and
advertising,
promotion,
torsat all butdiffusecosts of administration,
thebankmaybe a monopsoIn payingdepositors,moreover,
atmosphere.
No
nist,just as ithas somemonopolypowerin sellingloansto borrowers.
can
it
as
depositors,
among
discriminate
doubtthebankcan to someextent
the
Thusifittakes12percenttoattracta newdepositor,
amongborrowers.
bankis notnecessarilyforcedto pay 12 percentto all its depositors.As
waysofgivinga depositorspecialremuneranotedabove thereare indirect
when
treatment
tion,e.g. in ancillaryservicesor in promisesof preferred
thedepositorwishesto borrow.
about Deposits
IV. Uncertainty
IV. 1. TheFunctionofReservesand DefensiveAssets
So farit has been assumedthatthebankknowsforsurewhatitsdeposits
willbe. The bankholdsdefensiveassets beyondits reserverequirements
revenue
withthemarginal
onlyto theextentthattheiryieldis competitive
of defensiveassets-highly
fromlending.In thisanalysisthe properties
otherassets
liquidand predictablein value-were quite inessential.Any
withloans
findtheirwayintobankportfolios
withcomparableyieldsmight
and investments.
assets-generallyat lower
reasonthatbanksholddefensive
The principal
yieldsthanthe marginalreturnsfromlending-is to defendthemselves
foresee.Many
whichtheycannotperfectly
againstdepositwithdrawals
factorsoutsidethebank'spowereitherto controlor to predictcan change
the bank's depositsfromweek to week, and even fromday to day. For
Scand. J. of Economics 1982
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Thecommercial
banking
firm:A simplemodel 507
example,expansionor contraction
of thelendingof otherbankswillspill
depositsand reservesintoa bank,or suck themfromit. In decidingthe
volumeofitsloansand investments,
thebankmustcommititselfto illiquid
assets beforeit knowswhatits depositswill be, and mustconsiderthe
consequencesoflargeifimprobable
withdrawals.
At one timetheconsequencemighthavebeena literalinability
to honor
demandobligations.The bank would "fail" not because its loans and
investments
werebad but simplybecause theywereilliquid.The original
and historic
purposeofreserveswas to protectthebankand itsdepositors
againstthiskindoffailure.
Modernfinancialinstitutions
have virtually
eliminated
thisdanger.As a
"lenderoflastresort"thecentralbank-theFederalReserveSysteminthe
U.S.-can preventbanksfromfailingsimplyfromlack of liquidfundsto
meetdepositwithdrawals.
The possibility
of a contagiousloss of confidencein a particular
bankor inbanksingeneralis greatly
reducedbothby
theavailability
of a lenderof last resortand by government
of
guarantees
deposits.A bankmayfailbecause of poormanagement
or extraordinarily
bad luck in makingloans and investments,
not because theseassets are
illiquidbutbecause theyare of insufficient
value even ifheldto maturity
and beyond.The stockholders'
capitalin theenterprise
is thedepositors',
or depositinsurer's,cushionagainstthiskindofmisfortune.
The historicfunctionof reservesas a cushionagainstinsolvencydue
simplyto illiquidity
has been renderedobsolete.The modernfunction
of
reservesis to providea mechanism
ofmonetary
controlovertheeconomy
by the centralbank. Whydo banks hold reserves?They hold reserves
becausetheyare requiredto do so bylaw orbyconvention
withthevirtual
forceof law. These requiredreservesare-paradoxicallyin view of the
original
function
ofreservesandindeedofreserverequirements-unavailable to meetdepositwithdrawals.
Whydo banksmaintain
a netdefensive
positionin excess of reserverequirements?
Theyholdsecondaryreserves
forfearthattheymightnotpass therequiredreservetestwithout
incurring
thespecialcosts of borrowing
or of liquidating
high-yielding
investments.
The consequenceof depositwithdrawals,
againstwhichthebankprotects
itselfbyexcess reserves,is notthedisasterofinsolvency
buttheadditional
cost,including
perhapsinconvenience
and damageto prestige,
involvedin
meetingthe reservetest.Giventhesecosts,uncertainty
aboutthefuture
level of depositsmaylead to a lowervolumeof lendingand to a higher
defensivepositionthantheprofit
maximization
discussedabove.
The analysiswillassumethatonlydemanddepositsare subjectto uncertainty.If thebankalso has timedeposits,theseare assumedto be no less
illiquidthanloans.Thusthemakingofloansneednotbe deterred
byfearof
losingtimedeposits,fortheloanswillbe repaidsoonenoughto meetsuch
withdrawals.
This meansthattimedepositsplay muchthe same role as
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508 J. Tobin
.4
Required
Reserves
i
Dtfensive
DfniePosition-.4--
/
Loans
Laoans
and Investment
(L)
D+E
and Investments
0~~~~~~~~~~~~~
=
Do
D
0-xi??
DDO- XO
0
Deposits/
~~E5s
O <
Required Reserves, Defensive Position,
Loans and Investments -
-
Fig. 5
and possibly
payments
equity,withtheexceptionthattheyentailinterest
extreme,butit is
is unrealistically
This assumption
reserverequirements.
in the rightdirection.Time depositsare less volatileand
qualitatively
of
thandemanddeposits,and a bankwitha highproportion
unpredictable
withdrawals.
timedepositshas less to fearfromunanticipated
IV.2. ThePortfoliothatMaximizesExpectedProfit
a dollar
Once the bank has chosen a volumeof loans and investments,
changein depositswill meana $(l-k) changein defensiveposition.An
if
positionat leasttemporarily;
ofdepositswillincreasethedefensive
influx
it will
gain presumably
the bank becomes convincedit is a permanent
A loss of
choose a new and highervolumeof loans and investments.
this
the
defensive
Again,
position.
out
depositswilllowerandperhapswipe
If the loss of depositsprovesto be
outcomemay be only temporary.
thebankwillintimeloweritsvolumeofloansandinvestments
permanent,
itsdefensiveposition.
in orderto reconstitute
as in Figure2, the volumeof loans and
In Figure5 let Lo represent,
chosen by the bank. Let Do be the expectedvolumeof
investments
sense,thenextweek.In Figure2 thebankwas
deposits,in theprobability
1982
Scanid. J. of Econiomnics
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Thecommercial
banking
firm:A simplemodel 509
Probability
p
1
p(D)
~
p(DO
~
~
~
/
' (D)
p(Do -XO)
pi
(Do
-
MO)
0
-
-
Do- X0 Do -X1
D0
D
Deposits
D->
Fig. 6
sureofDo; now variousdepositlevelsbothhigherand lowerare possible,
andDo is just themeanofa probability
distribution.
Correspondingly
Ro is
in Figure5 theexpecteddefensiveposition.ThroughpointQ in Figure5 a
450 linehas beendrawn,paralleltothelineD+E. Giventhevolumeofloans
andinvestments
position
Lo, theactualoutcomefordepositsanddefensive
will be somewhereon thisline. If moredepositscome to the bank,the
outcomewillbe on thislineabove andto theright
ofQ. Iftherearedeposit
withdrawals
beyondexpectation,the outcomewill be to the leftof and
below Q. A reductionof depositsby X0, fromDo to Do-Xo, wouldwipe
out thedefensiveposition.
The relevantprobabilities
of futuredepositsare not objectiveones, if
indeedtheseexist,butprobabilities
estimated
bythebankinthelightofits
past experience.In Figure6, curvee(D) represents
thecumulative
probabilitydistribution
of deposits.Any pointon the curveis to be read as
follows:theordinategivestheprobability
thatthefuture
depositswillnot
exceedtheabscissa;forexample,theprobability
is Q(Do-Xo)thatdeposits
will not be greaterthanDo-Xo. As in Figure5, Do-Xo represents
the
depositlevelat whichdefensivepositionis zero; therefore
Q(Do-Xois the
thatthe bank will have to borrowor liquidateinvestments.
probability
Suppose thereis an increasein expecteddeposits,say fromDo to Do in
Figures5 and 6 because the bank has becomethrough
externalcircumstancesor through
itsowncompetitive
measuresa moreattractive
depository.Thisis assumedto shifttheentirecumulative
distribution
totheright,
frome(D) toe'(D) inFigure6. The expecteddefensive
positionis increased
fromRo toR' inFigure5. The depositlevelat whichthedefensive
position
33-824817
Scand. J. of Economics 1982
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510
J. Tobin
a
of encountering
is zero remainsthe same, Do-Xo, but the probability
negativedefensivepositionis reducedto Q'(DO-Xo.
shifts
ina specialwaywhen
distribution
It is assumedthattheprobability
thatactualdeposits
expecteddepositschange,namelythattheprobability
ofexpecteddepositsremains
or multiple
willnotexceedanygivenfraction
x is 1/10
Thus,in theexampleabove,Q(xDo)=Q'(xD') whether
unchanged.
or 1/2or I or 3 or anyotherpositivenumber.
distribution
ofdeposits,Q(D), an increase
Giventheoriginalprobability
position,reduce
willlowertheexpecteddefensive
inloansand investments
the safetymarginit providesagainstlosses of deposits,and increasethe
of a negativedefensiveposition.Consider,forexample,an
probability
fromLo toLI inFigure5. Thislowersthe
increaseinloansand investments
from
expecteddefensivepositionfromRo to RI, reducesthesafetymargin
to
position
defensive
of a negative
XOto X1, and increasestheprobability
Q(DO-X1).
thatthebankwill
Everyincreasein loans thusincreasestheprobability
position.The
a negativedefensive
be subjectto thespecialcostsofmeeting
mustbe reckonedas the
costsofmakingloansandinvestments
opportunity
in
thefunds,takingintoaccountthereduction
expectedcostsofproviding
expecteddefensivepositionand safetymargin.Thereforethe expected
as wellas to
to marginal
specialcosts mustbe included.Theycontribute
totalexpectedcosts.
IV.3. Effectsof Uncertainty
costs introduced
by uncertainty
The natureof the changein opportunity
about depositsis indicatedin Figures7, whichshouldbe comparedto
Figures4. In Figure4a L4.(equal toDO(l-k)+E) is shownas thevolumeof
to a zero defensiveposition.At this
corresponding
loansand investments
costcurveincreasesin slopefromr to r+b.
criticalvolumetheopportunity
AndinFigures4 b and 4 c thecostcurvealsojumpsat thispoint,reflecting
a. In Figures7 thesame volumeof loans
cost of borrowing
a once-for-all
to an expecteddefensivepositionof
and investments
Lo now corresponds
ofa negativedefensivepositionis no longerzero
zero. Buttheprobability
grows
to theleftof thispointand one to its right.Ratherthisprobability
cost
theexpectedopportunity
as L increases.Consequently,
continuously
the
that
the
for
an
allowance
include
of
L
must
probability
foreverylevel
the
costs
on
defensivepositionwill turnnegativeand impose special
thechosenvolumeofloans
bank-an allowancewhichis greaterthehigher
thecostcurve
ofsmoothing
Thisallowancehas theeffect
and investments.
in Figures7
C', as illustrated
as shownin theAppendix,marginal
expectedcost is
Formally,
a - fAX)
r+bF(X)+
( 1- k)Do
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Thecommercial
banking
firm:A simplemodel 511
P(L)
Expected
|~~~~Ne
Revenue
/ |
Expected
Revenue
+b)
from
Loans and
Investments
C O
/
Expected
Revenue
f rom
Defensive
Position
Expected
Position
Defensive
Positive
L
~~~~~~c
Defensive
Negative
Expected
Position
s
L
e
L'
e
Marginal Expected Revenue
from Loans and Investments
~~+
Marginal Expected "Costs"
from Defensive
Position
~~~~~~~~I?
I
MC = R
I
|
L
Loans
c
=r+b
~~~MC
|
==
L
Ll
e
e
and Investments
-M
-
Fig. 7a
Here X is a function
of L such thatDo(1-X) is the level of deposits,
(L-E)I(1-k), at whichthe defensivepositionwouldbe zero, F(X) is the
probabilitythat deposits will not exceed this crucial level, and
cost
flX)=dF(X)IdXis thecorresponding
probability
density.Thismarginal
thebankequatesto themarginal
revenueoflendingP'(L).
How does uncertainty
affectthevolumeofloansand investments?
In Figure7 a, thereis no fixedcost of borrowing,
no jump in thecost
curveundercertainty
(curveC in Figure7 a). Uncertainty
resultsin an
expectedtotal cost curve like C' in Figure7 a above C everywhere,
asymptotic
to C at bothends.The corresponding
changeinmarginal
costis
shownin the lowerpanel. The jump in marginal
cost MC, at Lc, is now
distributed
thenewmarginal
throughout
costcurveMC'. Withuncertainty
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512
J. Tobin
/
t
~~~
~~~~Expected
Net Revenue
I
/
Expected
Position
Defensive
Positive
-~
0O o
Expected
Revenue
from
Defensive
Position
I
/
Expected
Revenue
from
Loans and
Investments
I
g
Expected
-Position
.<
Fi
-~
~~~~~a
~
L L
~~~~~
c e
-C
C,
C
-z
<
r+b)
Defensive
Negative
b
I
Li'
e
_
Marginal Expected Revenue
from Loans and Investments
MC = r+b
Marginal Expected "Costs"
from Defensive
Position
C
NC'
MC=r
IM
~~~~~~~~~~~~~~~~~~~~I
.
o
Loans
and Investments
L L
c e
L'e
Le
Fig. 7b
aboutdeposits,marginal
costrisescontinuously,
andalwaysliesbetweenr
and r+b. Therefore,if the maximum-profit
volumeof loans and investmentswere previouslybelow L, implying
a positivedefensiveposition,
deposituncertainty
leads to a lowervolumeoflendingand a largerexpected defensiveposition.But if the maximum-proflit
volumeof loans and
investments
undercertaintywere greaterthanLo, implying
a zero or
negativedefensive
position,uncertainty
leadsto a highervolumeoflending
and a smallerdefensiveposition,i.e., more expectedborrowing.
This
secondresultis theone illustrated
in Figure7 a by theshiftin equilibrium
lendingfromLe to Le. The thirdpossibility
is thattheequilibrium
under
is exactlyat L4-the marginalrevenuecurvegoes through
certainty
the
jumpin MC at thatpoint.In thiseventuncertainty
mayeitherincreaseor
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Thecommercial
banking
firm:A simplemodel 513
P (L)
/
~~~~~~~~~~~
I~~~~~~~~~~~~~~~~~~~
Expected
Revenue
from
Loans and
Investments
O
Expected
~~~~~~~L
c
L '
L
e
e
c
Revenue
from
Defensive
Position
I
M
MC~r+b
MC'|I
MC =r
_ _ _ _ _ __ _ _ _
C
Loans
_ _ _
_ _
I
L
L'
e
c
and Investments
|
_
_
L
_
_
_
_
_
e
Fig. 7c
revenueis greater
or less
marginal
on whether
decreaselending,depending
thanthenewmarginal
costMC' at thatpoint.
cost of
In Figure7 b thereis a jumpin totalcost C at LC,a once-for-all
borrowing.But this cost is small relativeto the additionalinterestb
the
incurred
on a negativedefensiveposition.The resultsare qualitatively
wereat 4.
undercertainty
sameas in thepreviouscase. If theequilibrium
inFigure4 c, then
butmarginal
revenueP'(L) exceededr+b, as illustrated
willlead to increasedlendingand
ofcoursetheintroduction
ofuncertainty
to a negativedefensiveposition.
In Figure7 c, however,thefixedcostofborrowing,
a, is largerelativeto
the special interestchargeb. As a result,the total cost curve under
C, for
curveundercertainty,
C', lies belowthecorresponding
uncertainty,
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514
J. Tobin
Table 1. Effect of uncertaintyabout deposits on volume of loans and
investmentsand expected defensiveposition
Equilibrium
undercertainty
a) Fixedcostof
borrowing
zero
or small
(Figures7 a
and 7 b)
b) Fixedcostof
borrowing
large
(Figure7 c)
Positivedefensive
position
Zerodefensive
position
Negativedefensive
position
Lending
diminished
Result
uncertain
Lending
increased
Result
uncertain
Defensive
position
reduced
Result
uncertain
Defensive
position
increased
Lending
diminished;
defensive
positionincreased
valuesofL above L, i.e., fornegativeexpecteddefensivepositions.Also,
the marginalcost curveMC' eventuallyrisesabove (r+b), and thenapThe new
proachesr+b, the marginalcost underconditionsof certainty.
rising.If the
marginalcost MC' is smooth,but it is not continuously
thenit will be indefensivepositionwould be positiveundercertainty,
This is thesame concreased-and lendingcurtailed-underuncertainty.
clusionas in thetwopreviouscases. For negativedefensivepositions,the
situationis morecomplicatedthanin theprevioustwocases. Even there
costas to reduce
theintroduction
ofuncertainty
mayso increasemarginal
lendingand increase(algebraically)the defensiveposition.This is the
in Figure7 c.
illustrated
possibility
as shownin Table 1.
These resultsmaybe summarized
increasesthe bank's loans and investthatuncertainty
The possibility
and the cases in whichthis resultoccurs are
mentsseems surprising,
importance.These are cases wherethe
probablynot of greatempirical
banksintheUnitedStates
defensive
positionis negativeor zero.Ordinarily
are not in thissituation.These analyticalpossibilitiesmayhave greater
lend
wherebankscustomarily
relevanceincertainforeign
bankingsystems,
and investfundsborrowedfromthecentralbank.
IV.4. Value and Cost of Deposits
and thevalueofan increaseinexpecteddeposits,
Whatare thedisposition
as fromDo to Do in Figures5 and 6? A shiftof thiskindmovesL. to the
sayL4 in
rightin Figure7. At anygivenvolumeofloansand investments,
thediagrams,thepenaltiesof a negativedefensivepositionare less probthe
cost.Therefore
able and haveless weightinthecalculationofmarginal
Scand. J. of Economics 1982
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Thecommercial
banking
firm:A simplemodel 515
marginal
costcurveis movedto therightandgenerally
down.The resultis
volumeof loans and investments.
an increasein theequilibrium
It is not
as
it
about
that
was
with
an
influx
of
certainty
true,
deposits,
depositsgoes
entirely
intodefensiveassets.
thevaluein addedexpectedearnings
Correspondingly,
ofan additionto
revenueoflending,
expecteddepositsis (1-k) timesthemarginal
or,what
is inequilibrium
thesamething,(1-k) timesthemarginal
expectedrevenue
fromdefensiveposition.Even whenthe expecteddefensivepositionis
positive,depositsare worthmorethanr(I1-k).An increment
in expected
theprobability
depositsacquiresadditionalvaluebylowering
thatrandom
depositloss willinflict
on thebankthepenaltycostsofa negativedefensive
position.
An exceptionmay occur whenthe marginalcost curvehas the hump
shape of Figure7 c. Slidingthiscurveto the rightwillraise it at certain
points.If thepreviousequilibrium
had occurredon a decliningstretchof
the curve(not as depictedin Figure7 c), thenthe new equilibrium
will
involvea lowervolumeofloansand investments.
The increaseinexpected
depositsderivesconsiderablevaluefromlowering
theprobability
ofincurringthefixedpenaltycost,butto exploitthisimprovement
thebankmust
plana muchhigherdefensivepositionthanbefore.
As in the case of certainty,
two situationsmay be distinguished
with
respectto thesupplyofdepositsto theindividual
bank:
(a) The expectedvolumeof depositsis externally
determined,
beyond
thebank'scontrol.A marketinterest
ratemayhaveto be paidon deposits;
butgiventhelimitedsupplyof depositsavailableto thebank,theirvalue
exceeds this interestrate. In these circumstances
the cost of deposits
affectstheexpectedprofits
ofthebankbutnotitsoptimalportfolio.
(b) The bankcan influence
itsexpectedvolumeofdepositsbyitsinterest
payments
and otheroutlays.Totalcostsofdepositsare uncertain
because
depositsthemselves
are uncertain.
A simpleand naturalassumption
is that
theaveragecostofdepositsis a function
ofexpecteddeposits,generally
an
increasing
function.
Giventhisaveragecost, actuallyrealizedtotalcosts
dependon therandomelementin deposits.For example,thebanksetsan
interest
ratebutis uncertain
whatvolumeofdepositsitwillattract.In these
circumstances,
thebankwillequatethemarginal
costofexpecteddeposits
to theirvalue. Decisionsaboutdepositlevelsand aboutportfolio
compositionareintertwined.
For example,ifthemarginal
costsofattracting
a given
volumeof expecteddepositsdecline,thebankwillacceptmoredeposits
and makemoreloans. But as the marginal
revenueof loans declines,the
bankwillshiftrelatively
moreofanynewfundsintodefensiveassets.
In Figure8 thiscompleteequilibrium
is illustrated.
The horizontal
axis
measureseitherthebank'sexpecteddisposablefundsE+(1-k)D0 or their
balancesheetequivalentL+R. CurveMCD represents
themarginal
costof
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516 J. Tobin
C' (D0)
MCD=
P' (L)
1_MC
1-k
r+b
----4
MC for R =0O\A
~~~~~~~~~~~~~~~~~~r
|
z
I
Marginal
Revenue,
Cost
-
l
L21
1.0
E.
o<1-4
l
l
l
(1-k)D0
R2
l
F2
F1
Disposable
Assets
Br
ofbank.
Fig. 8. Fullequilibrium
cost ofa dollarofexpected
thedisposablefunds;ifC'(DO) is themarginal
deposits,thenC'(DO)I1-k is the marginalcost of a dollaravailablefor
lendingor placingin defensiveassets. MC represents,as in previous
cost of lending,measuredby theexdiagrams,the marginalopportunity
pectedreturnfromdefensiveposition.But in thisdiagramthe level of
with
expecteddepositsis notfixed.InsteadtheMC curveslideslaterally,
to theabscisthepointofzero defensivepositionL, alwayscorresponding
sa E+Do(l-k). Thus if disposablefundsare F, the properMC curveis
MC1, whereasif disposablefundsare F2, the applicablecurveis MC2.
SupNow F1, withzero defensiveposition,is clearlynotan equilibrium.
pose lendingis set equal to F1 so thatdefensivepositionis zero. Although
the value of defensiveassets equals the cost of deposits,the marginal
revenueof lendingfallsshortofbothof them.Thereis too muchlending;
and in orderto cut it back depositsshouldbe loweredand defensive
with
positionincreased.This leads towardsF2, whichis an equilibrium,
of
loansofL2 and defensiveposition R2.
V. The Bank's Response to ExternalChanges
V. 1. ExogenousChangesin ExpectedDeposits
Whatis thebank'sresponseto an exogenousdecreaseinexpecteddeposits
The
firstapproximation:
point,considerthefollowing
Do? For a reference
meetsits
bank places all of its capitalfundsin loans and investments,
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Thecommercial
banking
firm:A simplemodel 517
Table 2 a
Assets
Liabilities
Expected requiredreserves
Loans, etc.
Expected defensiveposition
10
64
36
Expected deposits
Equity
100
10
Table 2 b
Assets
Required reserves
Loans, etc.
Defensive position
Liabilities
6
64
0
Deposits
Equity
60
10
expectedreserverequirements,
and dividestheremainder
of itsexpected
depositsin constantproportions
betweenloans and expecteddefensive
position.A bank followingthis policywould respondto a 10% cut in
expecteddepositsby a 10% cut inL-E, theamountofdepositsplacedin
loans. The logicof thepolicyis thatthebankis stillprotected
againstthe
same percentagedownwarddeviationof depositsfromexpectation-the
probability
ofa negativedefensivepositionremainsthesame.Thismaybe
illustrated
by a numerical
example,showingexpectedand contingent
balance sheetsbeforeand afterthe changein expecteddeposits.Here the
requiredreserveratiois assumedto be 10% (Table 2a). Committed
to
loansof64,thebankcouldlose 40% ofitsexpecteddepositsbeforerunning
outofdefensiveposition(Table2 b). Now thebank'sexpecteddepositsare
cutby 10%, andthefirst
approximation
policyindicatesa proportionate
cut
in L-E, originally
54, and in expecteddefensiveposition,originally
36
(Table 3 a). The bankis stillprotectedagainsta 40% "loss" of deposits,
i.e., itsdefensivepositionwillbe positiveunlessdepositsfallas low as 54
(Table 3 b).
Whatmight
cause thebanktodeviatefromthispolicy?Forone thing,
the
marginal
revenueoflendingP'(L) mightriseas L is reduced.By itself,this
would lead the bank to a smallercurtailment
of lendingthanthe first
approximation
strategy
suggests.
On theotherhand,themarginal
valueofdefensive
assetsmayalso rise.
This marginalvalue dependson r, b, and a, all of whichare constant.It
dependsalso on X, thecrucialpercentage
downwarddeviationofdeposits
at whichthe defensivepositionwouldbe negative.This is also constant
undertheassumedpolicy.But it dependsfurther
on thelevelofexpected
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518
J. Tobin
Table 3 a
Liabilities
Assets
Expected requiredreserves
Loans, etc.
Expected defensiveposition
9
58.6
32.4
Expected deposits
Equity
90
10
Table 3 b
Liabilities
Assets
5.4
58.6
0
Required reserves
Loans, etc.
Defensive position
Deposits
Equity
54
10
depositsDo itself,providedthe penaltya is positiveand constant.The
positionlowersthe
reasonis thateach dollarincreaseinexpecteddefensive
expecteddepositsare
critialdepositlevelthesame dollaramountwhether
highor low. (In the tabularexampleabove, reducingloans by 9 and
positionlowersthecrititheexpecteddefensive
increasing
correspondingly
whetherexpected
cal depositlevel by 10, to 50 and 44 respectively,
depositsare 100or 90.) But thesame dollaramountis a largerpercentage
intheprobability
a greaterreduction
whendepositsare low,and therefore
with
that
deposits
expected
it
was
likely
the
equally
trouble.
of
example,
(In
of 100actualdepositswouldbe as lowas 60 andthatwithexpecteddeposits
of90 actualdepositswouldbe as low as 54. Butitis less likelythat90 will
turnoutto be 44 or lowerthanthat100willturnoutto be 50 orlower.)The
presenceof a fixedpenaltyleads to a kindof an economyof scale of
cost oflendingis higherwhenexpected
opportunity
deposits:themarginal
depositsare lower.
revenue
policywouldbe optimalifthemarginal
The firstapproximation
no
fixed
were
penalty(a=0).
of lendingP'(L) were constantand ifthere
thispolicyinthefaceofa declineinexpecteddeposits
following
Otherwise,
cost oflending.The
revenueand themarginal
willraiseboththemarginal
optimalcutin lendingmaybe eithersmalleror greaterthanindicatedby a
in L-E.
pro rata reduction
responsiveto thebank'sownpayments
Whendepositsare endogenously
valueofassetswilllead thebankto
to attractthem,anyriseinthemarginal
"buy" moredeposits.The bankwillnotfullyacquiescein an unfavorable
greater
shiftin expecteddepositsbut will partiallyoffsetit by incurring
coststo attractdeposits.
averageand marginal
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Thecommercial
banking
firm:A simplemodel 519
V.2. OtherChangesinAvailableFunds
Ifadditional
fundsbecomeavailableas equityortimedeposits,theycan be
placed in loans and investments
withoutalteringthe bank's marginof
safety.Thefirstapproximation,
is thatall ofa dollarofadditional
therefore,
equity,and $(1-1) of an additionaldollarof timedeposits,will go into
loans. This ruleleaves the size and marginalvalue of defensiveposition
in a purelycompetitive
unchanged.But unlessthebankis operating
loan
market,
themarginal
revenueoflendingwilldecline.Thusthefirst
approximationoverstatestheloan response.To keep its alternative
investments
equallyvaluable on the margin,the bank will have to use some of its
additionalfundsto improveitsexpecteddefensiveposition.
V.3. The YieldofDefensiveAssets
Withgivenexpecteddeposits,(Figure7) an increasein r will raise the
marginalopportunity
cost of landing.The bankwill substitute
defensive
assetsforloansand investments.
Whenexpecteddepositsand thecostsof
attracting
themare endogenous(Figure8) alonga givensupplycurveof
depositsto thebank,an increasein r willalso inducethebankto seekand
accept moredeposits,at highermarginalcost. Defensivepositiongains
fromnew depositsas well as fromcurtailment
of loans. Reductionsin r
havetheoppositeeffects.
One sourceof variationin r is monetary
policy,and indeedit is largely
through
changesin interestrateson defensiveassets in the 'money market" thattheindividual
bankfeelstheimpactofmonetary
policy.
V.4. PenaltiesforNegativeDefensivePosition
An increasein thevariablepenaltyb willraiseboththelevelandtheslope
of the marginalcost curve, withconsequencessimilarto those of an
increasein r as just discussed.The sameis trueofan increasein thefixed
penaltya-except thatforhighvalues of L relativeto Do thislowersthe
slopeoftheMC curve.
For theindividual
bankincreasesinpenaltiesmayarisefroma number
of
sources: greaterexpectationof risk of a futurerise in interestrates,
meaning
greaterlossesifinvestments
mustbe liquidated
tocovera negative
defensiveposition;greaterinterestchargesand transactions
costs in arrangingloans fromotherbanksor the centralbank; greaterestimateof
damageto futurecredit-worthiness
involvedin near-term
borrowing.
V.5. RequiredReserveRatio
One verydirectwayinwhichthemonetary
authorities
affect
thebankis by
settingthelegal reserveratiok. Withgivenexpecteddepositsa risein k
meansthatthebankmustprovideforhigherexpectedrequiredreserves,by
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520 J. Tobin
Table 4 a
Liabilities
Assets
Expected requiredreserves
Loans, etc.
Expected defensiveposition
20
58
32
Expected deposits
Equity
100
10
Table 4 b
Liabilities
Assets
12
58
0
Required reserves
Loans, etc.
Defensive position
Deposits
Equity
60
10
eitherloans or expecteddefensivepositionor,mostlikely,both.
curtailing
How muchwillloans be curtailed?
approximation
usedinSectionV.1
Consider,to beginwith,thesamefirst
reactiontoa
withdepositinflows.Thissuggeststhefollowing
inconnection
thebankadjustsitsloansso as to maintain
changein reserverequirement:
position.Thismeansthat
ofa negativedefensive
theprobability
unchanged
ofL-E withrespectto I -k is
(L-E)I(l -k) remainsconstant;theelasticity
one. For example,suppose thatthe bank had chosenthe balance sheet
shownin Table 2a above whentherequiredreserveratiowas 10%. The
bankis protected
fordepositsas low as 60 as shownin Table 2 b. Now let
planwillmaintherequiredreserveratiobe raisedto 20%. The following
4
the bank is
again
tain the same protectionas before(Table a). Once
protectedfordepositsas low as 60 (Table 4 b). The bank lends out its
of its
thesame relativedispositions
capital(10) and beyondthatmaintains
disposabledeposits.In Table 2 a theexpecteddisposabledepositsof90 are
divided3/5loans(54), 2/5defensiveposition(36). In Table4 a theexpected
3/5loans
disposabledeposits,now80, are dividedinthesameproportions:
(48), 2/5defensiveposition(32).
wouldbe exactifit maintained
equalityof marginal
This approximation
revenuefromdefensiveposirevenuefromlendingand expectedmarginal
tion.But bothoftheelementsofthisequalitywill,ingeneral,be changed.
Reductionin the volumeof loans L willraise its marginalrevenueP'(L)
loan market.The
unless the bank is operatingin a purelycompetitive
revenuefromdefensiveposition,is equal to
secondelement,themarginal
r+bF(X)+
a
(I -k)DD
fiX), whereX
L-E
(I -k)DD
-I
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Thecommercial
banking
firm:A simplemodel 521
is the percentagedeviationfromexpecteddepositsat whichdefensive
positionwouldbe zero (-40 in thetabularillustration
above),F(X) is the
probability
ofdeviationsas bad as thatorworse,andj(X) is dF(X)I(dX),the
at X.
corresponding
probability
densityfunction
The approximation
keepsX constant,
andr,b, a, andDo areall constant.
But unlessa is zero,therisein k increasesthethirdtermofthismarginal
cost of lending.Whenjumpingover the line to a negative
opportunity
defensive
positionentailsa fixedcost,thefactthata dollarincreaseinloans
and investments
increasesthe probability
of crossingthe line mustbe
chargedagainstit. Furthermore,
theamountby whichan additionaldollar
oflendingraisesthisprobability
is greaterwhenthereserverequirement
is
higher.Each dollarincreasein L raises the criticallevel of depositsby
11(1-k),e.g. by 10/8whentherequiredreserveratiois 20% comparedto
10/9whenit is 10%. Withboththemarginal
revenueand marginal
costof
lendinghigher,
itis notpossibleto sayingeneralinwhichdirection
thefirst
approximation
errs.The firstapproximation-the
ofL-E
unitary
elasticity
withrespectto 1-k-is exact if marginalrevenuefromlendingP'(L) is
constantand ifthefixedpenaltya is zero.
The foregoing
analysisof changesin reserverequirement
refersto the
case of exogenousdeposits.Whatof theothercase, wherethebankcan
determine
its own volumeof expecteddeposits?Clearlyan increasein k
raisesthecost ofobtaining,
via additional
deposits,fundsto place in loans
or in defensiveassets. The bankwillseek a smallervolumeof expected
deposits,and therefore
curtailitsloans evenmorethanin thefirstcase.
VI. Retentionof Deposits
The precedinganalysishas assumedthatthebank'svolumeofdepositsis
independent
ofthesize ofitsloansand investments.
Depositsmaydepend
on the interestratesand servicesthe bank offersdepositorsand on its
andpublicrelations
advertising
expenditures.
Butso fartheydo notdepend
on thebank'sportfolio
choice.ForreasonsoutlinedinSectionII abovethis
is an extremeassumption,
and it is timeto examinetheimplications
ofits
removal.
Suppose insteadthat,giventhe attractions
the bank offerspotential
depositors,
thevolumeofitsdepositsis thesumofan autonomous
element
Do and an inducedelementp(L) whichdependson its loans and investments.Withzero loans and investments
depositsare just DO; thatis,
p(O)=0. The functionp(L) may be called the bank's depositretention
function,
and itsslopeq'(L), whichlies betweenzeroandone,itsmarginal
retention
ratio.
Note thatthe bank is assumedto be unableto affectits depositsby
acquiringdefensiveassets; itsmarginal
retention
ratiofromthispartofits
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522
J. Tobin
portfolio
is zero. Indeed,itis onlybecauseitsdepositsdependon howthe
assets thatdepositretention
amongalternative
bankdividesits portfolio
choices.The factthatloans
maybecomea relevantfactorin its portfolio
are morelikelyto be "redeposited"thanotherbankasset purchasesmay
disposethebankto favorloans.
givesto loansmaybe illustrated
The advantagewhichdepositretention
model,wheredepositsareexogenous,
mostclearlyinthefirstdeterministic
An expansionof $1 of loans and
costless,and foreseenwithcertainty.
deposits
by
$p'(L) and lowersthe defensive
investments
now increases
positionnotby$1 butonlyby$(l-q'). Requiredreservesincreaseby$kU'.
opportuDefensivepositionis reducedby$(l- p'+kp'). Thusthemarginal
nitycost of lending$1 is r[1-p' (1-k)] insteadofr fordefensiveposition
positive,or (r+b)[1 -' (1-k)] insteadofr+b fordefensivepositionnegaratio.
retention
tive.This marginalcost is lower,thehigherthemarginal
and thedefenthevolumeofloans and depositswillbe higher,
Therefore,
ratio.
sivepositionlower,thehighertheretention
Take
analysiscan be easilyreinterpreted.
The previousdiagrammatic
Do-the whollyautonomousvolumedepositsassumedin Figures2, 3, and
4-to be the volumeof depositsthe bankcan have withoutmakingany
is to rotatedownwardthe
loansat all. Thentheeffectofdepositretention
withthevertical
cost curveof Figures3 and 4, pivotedon itsintersection
constant.
axis,andto giveita lowerslope,stillpositivebutnotnecessarily
Withthis amendment,
the previousanalysisand conclusionsstillhold.
the
Note,however,thatthesize oftherequiredreserveratiok nowaffects
whichis higherthelargerthefraction
costoflending,
marginal
opportunity
requiredreof induceddepositsthatmustbe placed in interest-barren
serves.
are partially
or evenwholHowever,theadvantagesofdepositretention
orothercostson thebank.
lylostifthe"retained"depositsimposeinterest
themarginal
cost
rateC'(D) is paidto depositors
Forexample,ifan interest
of lendingis raisedby p'(L)C'(D). If thebankis in completeequilibrium,
and if "retained" depositscost as much as any otherdeposits,their
use. Hence C'(D) is
cost is equal to theirvalueinanyinvestment
marginal
equal to r(1-k)-or (r+b)(l -k)-and theretaineddeposithas no netvalue.
The marginal
cost oflendingreducesonce moreto r,or r+b, and portfolio
ofdepositretention.
choiceis independent
ofthe
and reinterpretation
entailssimilarmodification
Depositretention
aboutfuture
ofuncertainty
deposits.Buttheconcluanalysisoftheeffects
thesame.
thevarietyofpossiblecases, remainqualitatively
sions,including
.
VII. Risk Neutralityor Risk Aversion?
The firmmaxiThe analysisabove assumesthatthebankis risk-neutral.
mizes expectedprofits,and this also maximizesits expectedutilityof
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Thecommercialbanking
firm:A simplemodel 523
affectthe bank's
profits.Of course the hazards of depositfluctuation
on
decision.But theydo so through
theircalculableeffect expectedcosts
To some
ratherthanthrough
ofvarianceofprofits.
anyassumeddisutility
The fixedcost a of a
degreethe two approachesare interchangeable.
negativedefensivepositionhas been treatedabove as a definite
pecuniary
be regardedas a psychological
cost. It mightalternatively
cost, i.e., an
approximate
ofadversefluctuations
of
pecuniary
equivalentofthedisutility
riskaversionon themodelas presentdeposits.In anycase, superimposing
ed wouldtiltthe bank's decisionfurther
towarda conservative
portfolio
relativeto defensive
assets.
policy,reducingilliquidloansand investments
forthefirm.A bankis
Risk-neutrality
seemstheappropriate
assumption
managedbyspecialistsengagedintakinga longsequenceofrisksofdeposit
fluctuation
and can expectbad luckand goodluckto "averageout." That
is, thelong-run
varianceof theprofits
associatedwithanygivenpolicyis
muchsmallerthantheshort-run
variance.It is truethatthefirmmightnot
survivea sequenceof heavylosses. In themodelabove thisdangerenters
thesize oftheparameters
through
a and b; iftheyarelargeenoughrelative
to loan and depositopportunities
the firmwillfollowa cautiouspolicy,
perhapsacceptingno illiquidity
risksat all.
The varianceofprofits
of a risk-neutral
bankwillbe takenintoaccount
by the shareowners,who may be risk-averseindividuals.In a system
withoutdepositinsurance,the risksof illiquidity
takenby a bankwould
also be consideredby depositorsgaugingthechancesthatthebankwillbe
unableto meetitsliabilities.The portfolio
choicesofthebankwouldthen
affectitssupplyofdepositsin a mannerthatworksin a direction
opposite
to the loan retention
mechanismdiscussedabove: a moreconservative
balancesheetwouldattractmoredeposits.Depositinsurancehas largely
eliminated
thisconsideration.
It has notdoneso entirely,
becauseitcovers
onlythe first$100,000of a depositand because theremaybe delaysin
receivingpayment.Moreover,the surveillance
formerly
exercisedby depositorsis in somedegreereplacedby thesurveillance
oftheinsurer.
VIII. ConcludingRemarks
The simplemodelof a commercialbankingfirmpresentedin thispaper
examinesthechoice betweenreservesand otherdefensiveassets,on the
one hand,and less liquidloans and investments,
on theother.The bank
mustmakethischoicebeforeit knowswhatitsvolumeofdepositswillbe.
A drainof depositsto otherbanks,e.g. an adversebalancein interbank
checkclearing,has to be metbydisposalofassetsofone kindor theother
or byborrowing.
The sameproblemarisesforbanksin theaggregate
ifthe
publicwithdraws
depositsin orderto holdcurrency.
Meetinga drainby sellingloans and investments
or by borrowing
is
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524 J. Tobin
typically
moreexpensivethandrawingon reservesor sellingliquiddefensive assets. The additionalexpense of a negativedefensivepositionis
to thesize of
modeledto consistofa fixedcostplusa penaltyproportional
of incurring
such expense
The bank weighsthe probability
the shortfall.
profitby holdingless remunerative
of foregoing
againstthe probability
amountsifdepositexperienceturnsouttobe
defensive
assetsinredundant
favorable.
The behaviorof the bank in thismodel is a primeexampleof what
motiveforholdingmoneyandotherliquid
Keynescalledtheprecautionary
assets. Othereconomicagentsface similarproblemsand exhibitsimilar
behavior.Thusthemodel,inspiritthoughnotininstitutional
detail,applies
andotheragents
institutions,
generally
to households,businesses,financial
about their
in the face of uncertainties
who make illiquidcommitments
futurestreamsof receipts.
aspects of the
In its simplicity
the model ignoresseveral important
banks:(1) It is a staticmodeland does not
decisionproblemsofreal-world
deal withthe intertemporal
structure
of bankliabilitiesand assets. Their
for example,will affectthe bank's defensivepositionsfor
maturities,
and borrowforall-futuredates.(2) Assets,liabilities,
many-inprinciple
ingcapabilitiesare notso neatlydivisibleintotwocategoriesbutconstitute
Consequentlythe model's
a spectralmenu of liquiditycharacteristics.
separationof the bank's decisionon defensivepositionversusloans and
investments
fromportfoliochoices withinthose categoriesis somewhat
otherthan
handleuncertainties
artificial.
(3) The modeldoes notexplicitly
calls
on
lines
of credit
include:
with
These
deposits.
those connected
customerswho use thebankbothas a depository
committed
to long-term
ininterest
ratesandprices
sourceofcredit,fluctuations
andas a contingent
loans.
ofmarketable
securities,and possibledefaultson commercial
the
distinctive
Nevertheless,I believe,the modelcapturesin essence
choicemostrelevantto banks'crucialroleinthe
features
ofbankportfolio
for
a micro-economic
foundation
monetary
system.It providestherefore
effectsof monetarypolicyinstruments
analysisof the macro-economic
and regulatory
regimes.Amongthe policyinundervariousinstitutional
variationsof interestare the proceduresand
and structural
struments
targetsof centralbank open marketoperations,the natureand level of
of interest
on reserve
thepaymentor non-payment
reserverequirements,
existence
the
to
central
bank
terms
of
and
the
costs
lending
banks,
assets,
and
the
number
and
and levels of legal ceilingson depositinterestrates,
A tractable
modelof
offirmsinthebankingindustry.
structure
competitive
and
to economy-wide
firmis a prerequisite
analysisoffinancial
thebanking
and"inside"money.
monetary
systemsthatrelyheavilyon intermediation
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firm:A simplemodel 525
banking
Thecommercial
Appendix
aboutDeposits
A. 1. Certainty
A. 1.1. DepositsExogenousand Costless.In thefirstmodeldepositsare
butpossiblyto dependon theloans
assumedto be knownwithcertainty,
ofthebank.Theyare assumed"exogenous"inthesense
and investments
thatthe bank cannotinfluencethe quantityof depositsby varyingits
depositorservices,or advertising.
outlaysfordepositinterest,
forthebankis:
The balancesheetidentity
(Do+p(L)) (1-k)+E-R-L
=0
(1)
The meaningsofthesymbolsare as follows:
Do+qp(L)deposits
k
therequiredreserveratio
equity
E
shareholders'
R
defensiveposition
L
loansand investments
function:p(O)=0,0. p'(L)< I
thedepositretention
q(L)
The profits
peryearofthebankare:
LI= P(L)+ Y(R)
(2)
and Y(R)thenet
whereP(L) is thenetrevenuefromloansandinvestments
revenuefromthedefensiveposition.MarginalrevenuefromlendingP'(L)
withL.
is assumedpositivebutdeclining
Yf)=rR
=
Y(R)
(r+b)R-a
R ~>0
R<O
3
(3)
position;r+b, with
Herer is theinterest
rateearnedon a positivedefensive
ratepaid on a negativedefensiveposition;and a?O the
bO, theinterest
fixedpenaltycost perannumofa negativedefensiveposition.
thatno interestis
withU.S. institutions,
It is assumed,in conformity
earnedon requiredreserves.
Using(1) and (3), (2) can be rewritten:
= P(L)+r{(DO+q(L))
{fl1(L)
(1-k)+E-L}
r12(L)= P(L)+(r+b){(DO+qp(L))(1-k)+E-L}
(R ? 0)
(R < 0)
The problemis to maximize(4) withrespectto L.
Let L, be thevalue ofL suchthatR=(Do+ q(L)) (l -k)+E- L=O.
34-824817
Scand. J. of Economics 1982
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526
J. Tobin
Let L* be thevalueofL, ifone exists,suchthat
P'(L*)=
-Y'(R)
-R
= Y'(R)(1l-'(L*)(l-k))
(r( - c(L *) (1- k))
- T(L*)
l(r +b)(I
(L*
(I -k))
L(.)
(L* > Ld,
Thereare severalcases:
(a) L*%L,.; L* is thesolution.
(b) L*>L(. and HI2(L*)>l1,(L,); L* is thesolution.
L*>Lc but f12(L*)<II(L(.);Lc. is the solution.
(bc) L*>Lc but HO(L*)= Hl(Lc); bothLc and L* are solutions.
(d) r(l-cp'(L( (l-k))<P'(L,,)
(r+b)(l-q'(L,) (l-k));L(. is thesolutionand
(c)
are HII(Lc).
profits
Whenthe maximizing
value of L is L*, it is notalteredby exogenous
changesin E or Do. The marginalvalues of such changesreflectsimply
theirinvestment
in defensiveassets. aJiI/8DO=(l-k)8FI/SE because k of
everyincremental
depositmustbe placedin interest-barren
reserves.
= {
___
(L*V LC)
b
r~~~~b
(6)
aE
~(L*>Lc)
Whenthemaximizing
valueofL is L, then
an
SE
P (Ld(7
1-11SL,
3LC SE
7
1- cp(LC.)(I -k)
is (1-k) timesthisquantity.
Again31-1/3DO
A.1.2. Deposits Exogenous at a Given Cost. A varianton thismodel
rateon deposits,thoughat
wouldrequirethebankto pay a fixedinterest
thisrateitcouldnotobtainmoredepositsthanDo+q(L). Assumethisrate
thesmaller
d to be smallerthanr(l -k). Then(5) mustbe revisedto reflect
valueofretaineddeposits:
P'(L*) = Y'(R)-q'(L*)
(Y'(R) (1 -k+)-d)
(5')
Ifd exceedsr(l -k) butdoes notexceed(r+b) (l -k) itis clearlynottothe
bank's interestto accept depositsand investthemin defensiveassets.
Insteadthebankwillbe fullyloanedup,withR=O, andwithDUDES+rP(L).
willbe suchthat
The volumeofloans and investments
P'(L*) (I-k)
=
(D = L
d
Sc and. J. of Economics
E<Do+q(L*))
1982
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(5")
Thecommercial
banking
firm:A simplemodel 527
If P'(Lc)(1-k)>d, the bank may findit profitable
to accept all the
lendablefundsbyborrowing
depositsavailableat d andobtainadditional
at
rate(r+b). Thenequation(5') appliesonce more.
If d exceeds(r+b) (1-k), the"bank" willacceptno depositsand simply
borrowat rate(r+b) to financeanylendingbeyonditscapital:
(R = L*-E)
P'(L*) = r+b
(5"')
A modelinwhichtheseveralmargins
are metsimultaneously
is givenin
thenextsection.
A. 1.3. DepositsEndogenous.The secondmodelassumesthatthebank
can chooseitsvolumeofdepositsbyincurring
a costC(D) ofa totalvolume
of depositsD=Do+qp(L). The bankmaynow be regardedas havingtwo
decisionvariablesDo and L. Then(4) becomes
{
1I(L, Do) = P(L)+rR-C(D)
2(L D0)
HA
=
(R 3 0)
P(L)+(r+ b)R- C(D) -a
(8)
(R <0)
AtR=O, HlIcan be written
solelyas a function
ofL, namely:
I'l = P(L)-C
= ?)
LE(R
l1-k
(9)
GivenR =0, themaximizing
L, call itL, is suchthat:
(I-k)P'(L
= C(LC-E)
(10)
As before,L*, ifitexists,willbe a solutionto one ofthefollowing
two
pairsofequations:
=L
aHn
aD0
aL
all12
aD0
= P'(L)+ {r(1-k)-C'(D)>
'(L)-r = 0
(L LC.)
+q(L)) = 0
=
r(1-k)-C'(D
=
P'(L)+ {(r+b) (1-k)-C'(D)}
p'(L)-(r+b)
= (r+b) (1-k)-C'(Do+ p(L)) = 0
=
(11)
0
(L > L)
(12)
These maybe rewritten
as follows:
r
JofEn)
e(1- k) C'(D *+ f(L*)
S
(L*< LC
o
(11')
Sand . J. of Etconomic
s 1982
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528
J. Tobin
P'(L*) = r+b
(L* > Lc)
(r+b) (1-k) = C'(D *+(L*))
(12')
Thereare severalcases, as above:
(a) L*6L,; L*,Do is thesolution.
D*>H1-(L.);L*, Do is thesolution.
(b) L*>L. and F12(L*,
(c) L*>L. but112(L*,
D*)<J11(Lc); L, is thesolution.
(bc) L*>LC butF12(L*,D)=I11(L,.); bothL, and (L*,D) are solutions.
are 11(L).
(d) r<P'(L,.)<r+b; L, is thesolutionand profits
is
Note thatwhendepositsare endogenousthedegreeof loan retention
irrelevant.
On the margina depositcosts as muchas it earns.Hence the
bankdoes notgainby havingpartofa loanendup as a depositforwhichit
costofa depositobtained
mustpay. (Thiswouldnotbe trueifthemarginal
cost ofotherdeposits.)
werelowerthanthemarginal
fromloan retention
In general,
P'(L
)
Y (R)(1p(*) L
(13)
In thefirstmodelaIn/aD0is Y'(R)(1 -k) or Y'(R) (1-k)-d and thisleadsto
(5) or(5'). In thesecondmodelanl/aDOis zero,andthisleadsto(1 '), (12').
A.2. Uncertaintyabout Deposits
A.2. 1. Deposits Exogenous but Random. Now assume that deposits are
DO(1+x)+p(L) wherex:- 1 is a randomvariablewithmeanzero. Let F(X)
thatx<X, and letftx)be thecorresponding
probability
be theprobability
the
depositoutcome.
densityfunction.The value of L is set priorto
Therefore,defensivepositionR is also a randomvariable,equal to
(DO(I+x)+,cp(L))(1-k)+E-L. For givenL thecriticalvalueX at whichR is
zero is given by:
XDo=0
L-E
_(D)_(R)
(I1-k)
(14)
(I1-k)
whereX(R) is expectedreservesand X(D)=Do+rp(L) expecteddeposits.
Everydollarincreasein lendingraisesXD0 by 11(1-k)-p'(L).
Profits
are MI(L,x) = P(L)+ Y(R):
fF11(L,x) = P(L)+rR
I12(L,x) = P(L)+(r+b)R-a
(x X)
(X<X)
Scand. J. of Economics 1982
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(15)
Thecommercial
banking
firm:A simplemodel
529
Therefore:
E (n) = P (L) -r(I - k)XDO-b (I -k)XDOF(X)
(16)
+(1-k)bDo fxf(x)dx-aF(X)
&ala; = P'(L)-r[l -(1-k) p'(L)] -b[ I-(1-k) cp'(L)J
F(X)
-
&L~
a
(17)
+ -x-[-b(1 -kXDOf(X)+b(I -k) XDof(X) -af(X)]
aL
Maximumexpectedprofits
are obtainedwhen
P'(L) = [1-(1-k) Tp'(L)][r+bF(X)+
a{)D f(X)]
(18)
The termon therightis themarginal
expectedrevenuefromdefensive
position,themarginal
cost of makingloans. Since bothF(X)
opportunity
and f(X) will generallybe risingin the relevantrange,marginalcost is
increasing.
ButforhighL andX,f(X) declineswithX, andfromthisarises
the possibility
thatmarginalcost declinesin some range.Assumingthat
f(X) goes to zero at bothextremesofX, marginal
cost approachesr as L
goes to zero,and approachesr+b, fromeitherabove or below,as L andX
becomeindefinitely
large.
The valueof additionalequityE is
SEY(R)
MM= P'(L) aL + a' Y(R) AL?a
SE
aE
aL aE
aE
(19)
From(18) we knowthatthefirsttwotermsadd to zero.To evaluatethe
thirdterm,differentiate
X(JJ)-P(L)in (16) withrespectto E, notingthat
ax
aE
-1n
SE
I_
_
_
(1-k)DO
-
r+bF(X)+
a
fiX)
(20)
(1-k) Do
Similarly,
MaD =r(1- k)+ b(1- k)F(X)+-DF(X)(I+X)
(21)
A.2.2. Deposits Endogenous.As in the case of depositcertainty,
an
alternative
modelwouldpermitthebankto influence
itsown depositsby
meansotherthanmakingloans. In thiscase interestpaymentsor other
Scand. J. of Economics 1982
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All use subject to JSTOR Terms and Conditions
530 J. Tobin
outlaysbythebankshiftthecentralvalueDo aroundwhichtheprobability
distribution
f(X) pivots.But thetotalcost of depositsis, likethevolume
itself,a randomvariable.Supposethatcost ofdepositssatisfies:
DO+q(L)
C = C(D+(L))+
0
(22)
~~D+cp(L)
L andDo, establishesan averagecostofdeposits,e.g.
The bank,by setting
rate.The totalvolumeofdepositsbearingthisestablishedrate
an interest
x. SinceDO+p(L) andC(Do+q (L)) are
thendependson therandomelement
the expectedcost of depositsis just C(Do+p(L)) and the
non-stochastic,
expectedcost is as beforeC'(Do+q(L)). Whenthisis takeninto
marginal
accountthe terminvolvingp'(L) dropsout of (18) and theconditionsof
become:
maximum
profit
P'(L) = r+bF(X)+
k)Do
a
(1 -
f(X) = C'(Do+rp(L))
1-k
Scond. J. of Economics 1982
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(23)