Kapu , Basan
A icle
A e-conside a ion o Money Demand Theo y
Ge man Economic Re iew (GER)
P o ided in Coope a ion wi h:
Ve ein ü Socialpoli ik / Ge man Economic Associa ion
Sugges ed Ci a ion: Kapu , Basan (2025) : A e-conside a ion o Money Demand Theo y, Ge man
Economic Re iew (GER), ISSN 1468-0475, De G uy e , Be lin, Vol. 26, Iss. 2, pp. 71-92,
h ps://doi.o g/10.1515/ge -2024-0055
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ge 2025; 26(2): 71–92
Basan K. Kapu *
A Re-Conside a ion o Money Demand
Theo y
h ps://doi.o g/10.1515/ge -2024-0055
Recei ed May 23, 2024; accep ed Decembe 19, 2024; published online Janua y 29, 2025
Abs ac :Po olio models ypically igno e p ecau iona y ansac ions demands
o liquid asse s, and models o p ecau iona y demands ypically igno e asse a e-
o - e u n isk. I asse -holde s a e isk-a e se, howe e , bo h ansac ions isk and
a e-o - e u n isk a ec demands o bo h liquid and illiquid asse s, e en when he
wo isks a e independen o each o he . We demons a e his in a ou -asse ame-
wo k, and show ha ou in eg a ed ea men p oduces unexpec ed and ins uc i e
esul s and insigh s. Fo example, (a) an inc ease in he expec ed e u n o isky
secu i ies inc eases he demand o M1, e en when M1 is used en i ely o ans-
ac ions pu poses, (b) an inc ease in he a iance o secu i ies e u ns educes he
demand o M1, and (c) an inc ease in he asse -holde s’ weal h educes he demand
o M1. A b oade amewo k o he s udy o money demand is hus called o .
Keywo ds: money demand; asse demands; non-sepa abili y; expec ed u ili y
JEL Classi ica ion: E4; G1; G2
1 In oduc ion
Conside a isk-a e se weal h-holde who can alloca e he weal h ac oss ou asse s
– a isky long- e m bond o equi y, a iskless long- e m asse , a lowe -yielding isk-
less sho - e m asse , and cash. He in es men ho izon is sho e han he ma u i y
pe iod o he isky long- e m asse , and she wishes o maximize he expec ed u il-
i y o he e minal weal h. Howe e , ‘hal -way’ h ough he pe iod she aces an
*Co esponding au ho : Basan K. Kapu , Eme i us P o esso , Depa men o Economics, Na ional
Uni e si y o Singapo e, Ken Ridge, Singapo e 117570, Singapo e, E-mail: [email p o ec ed].
h ps://o cid.o g/0000-0002-0685-8425
Open Access. ©2025 he au ho (s), published by De G uy e . This wo k is licensed unde he C ea i e
Commons A ibu ion 4.0 In e na ional License.
72 —B. K. Kapu
unce ain ansac ions equi emen , which could be xwi h a specified p obabil-
i y, and 0 o he wise. He long- e m asse s a e oo cos ly o liquida e o mee his
equi emen , which can be me ei he om cash o om liquida ing pa o all o
he sho - e m asse , o which she has o incu a ansac ions cos o , say, pe
dolla liquida ed.
As we discuss below, he abo e se ing is, in a s ylized way, cha ac e is ic o
many eal-wo ld asse alloca ion p oblems. We a gue in his a icle ha his ai ly
elabo a e amewo k, wi h ou asse s and wo quali a i ely di e en sou ces o
isk, leads o s iking new insigh s, and p o ides he basis o a e-conside a ion o
money demand heo y.
The iskless long- e m asse could be fixed deposi s o CD’s (ce ifica es o
deposi ), he in e es -bea ing sho - e m asse could be sa ings deposi s o o he
ins umen s, and ‘cash’ could ep esen demand deposi s (possibly paying mini-
mal in e es ) and cash-on-hand. In ui ion migh sugges ha one could pa i ion
he asse -holde ’s decision p oblem in o wo componen s: he choice among he
long- e m asse s on he one hand, and ha among he sho - e m asse s on he
o he , wi h he ob ious es ic ion being ha he sum o he sho - e m asse -
holdings no exceed x plus he amoun o ansac ions cos s incu ed in he e en
o a ansac ions shock. Wi h his es ic ion holding, in ui ion migh u he sug-
ges ha , o example, a mean-p ese ing change in he a iance o he e u n on
he isky long- e m asse (we e e o his as a bond hence o h) would a ec he
choice be ween bonds and CD’s, bu would ha e no bea ing on he choice be ween
cash and sho - e m sa ings ins umen s (we e e o hese ins umen s as sa ings
deposi s hence o h). Vice e sa o a change in, say, he in e es a e on sa ings
deposi s.
In his a icle, we show ha his la e in ui ion is inco ec . A mean-p ese ing
change in he a iance o bond e u ns will a ec he choice be ween cash and sa -
ings deposi s, no wi hs anding ha he sum o holdings o cash and sa ings deposi s
p io o he ansac ions shock emains a xplus equi ed ansac ions cos s, and
ha he dis ibu ion o he ansac ions shock is independen o ha o he bond
e u n. Mo eo e , he di ec ion o change is qui e unexpec ed. A mean-p ese ing
inc ease in bond e u n a iance dec eases he demand o cash, and inc eases he
demand o sa ings deposi s, as seen in ou simula ion analysis below. Pe haps
e en mo e su p isingly, an inc ease in he mean e u n o bond-holding induces
a subs i u ion owa ds cash and away om sa ings deposi s.
Risk a e sion is a necessa y condi ion o hese esul s. The e is a posi i e p ob-
abili y ha he ansac ions shock xwill no ma e ialize, and he e minal weal h
will include he holdings o cash and sa ings deposi s. Randomness o he e mi-
nal weal h is hus due o bo h he andomness o he bond a e o e u n, and he
andomness o he ansac ions equi emen . Being isk-a e se, she will hus need
A Re-Conside a ion o Money Demand Theo y —73
o conside bo h sou ces o andomness in he expec ed-u ili y-o -weal h calculus,
no wi hs anding ha hey a e independen o each o he .1Fo conciseness, we e e
o his as he ‘non-sepa abili y p ope y’ o isk-a e se expec ed u ili y.
Ou ou -asse amewo k hus gene a es in e es ing pa e ns o subs i u abil-
i y and complemen a i y ac oss hese asse s, which a e no induced by pa e ns
o co ela ion ac oss asse e u ns as in o he models. Ou analysis should also
p o ide cau iona y ad ice o asse -holde s, including chie financial o ice s o co -
po a ions, agains engaging in he pa i ioning o he decision p oblem desc ibed
abo e, howe e con enien o logical i migh appea o be.
2 Li e a u e Re iew
We seek he e o syn hesize wo hi he o sepa a e s ands o li e a u e, albei in a
simplified se ing. The fi s is a discussion o igina ing om James Tobin’s classic
1958 a icle, ‘Liquidi y P e e ence as Beha io owa ds Risk,’ which was ollowed
by a c i ical a icle by Chang, Hambe g, and Hi a a (1983) wi h he sel -explana o y
i le ‘Liquidi y P e e ence as Beha io owa d Risk is a Demand o Sho -Te m
Secu i ies – no Money.’ Chang e al. co ec ly poin ou ha o he ‘sa e’ asse s,
wi h fixed capi al and in e es a e alues, domina e nonin e es -bea ing cash o
po olio di e sifica ion pu poses alone. In e ms o ou ou -asse amewo k, long-
e m asse s such as fixed deposi s would domina e sho - e m asse s such as sa ings
deposi s o po olio di e sifica ion pu poses.
This lea es unanswe ed wo ques ions, o which he second ollows om he
fi s . Fi s , how hen do we accoun o people’s significan holdings o M1 (cash ou -
side banks plus demand deposi s) obse ed all o e he wo ld? The ob ious answe
is ha M1 enables ansac ions demands o be me mo e cheaply han h ough
he d awing down o o he sho - e m, fixed capi al- alue asse s such as sa ings
deposi s. The e has de eloped an ex ensi e li e a u e on his, s a ing om he clas-
sic a icles o Baumol (1952) and Tobin (1956), and p oceeding h ough a sequence
o complex models allowing o de e minis ic and s ochas ic ansac ions demands
o a ying o ms, as well as a ious o ms o ansac ions cos s. Al a ez, Lippi and
Roba o (2019, Sec ion 5) p o ide a comp ehensi e o e iew o his li e a u e (see
also Al a ez and Lippi 2017). This li e a u e has indeed assumed ha he (single)
sho - e m asse ha is al e na i e o M1 is o fixed capi al alue and o e s a fixed
in e es a e, while i s liquida ion o mee ansac ions demands incu s ansac ions
cos s. The asse -holde ’s p oblem is cas as one o expec ed-cos -minimiza ion o e
he asse -holde ’s infini e li e ime, subjec o mee ing he ansac ions demands,
1I owe his in e p e a ion o John Quah.
74 —B. K. Kapu
wi h exogenous income inflows being ep esen ed as nega i e ansac ions ou -
lays.2
This o mula ion e ec i ely implies isk-neu ali y on he pa o asse -holde s.
Howe e , he assump ion o isk-neu ali y aises u he complica ions, which do
no appea o ha e been adequa ely app ecia ed. In eali y, o cou se, an asse -
holde has access o mul iple asse s, a ious o which o e s ochas ic e u ns. Unde
isk-neu ali y she would, ce . pa ., channel all he asse holdings in excess o he
sho - e m asse s equi ed o mee ansac ions needs o he asse wi h he highes
expec ed e u n: he long- e m po olio holdings would become degene a e, which
is clea ly coun e ac ual.
Second, suppose ha asse -holde s a e no isk-neu al, bu ins ead isk-a e se.
As men ioned in he In oduc ion, and shown below, i is hen no longe he case
ha he p ecau iona y ansac ions demands (adap ing he e minology o F enkel
and Jo ano ic (1980), and o he s) o indi idual sho - e m asse s is independen
o , say, he isk o and e u n on long- e m bonds, which implies he necessi y o a
mo e inclusi e app oach o he de e mina ion o op imal asse alloca ions. This is
he syn hesis ha we seek o e ec in his s udy, in ha we allow o bo h a e-o -
e u n isk and ansac ions isk, whe eas ea lie s udies ha e abs ac ed om one
o he o he o hese (and, when conside ing ansac ions isks, ha e assumed he
special case o isk-neu ali y).
Ou analysis also di e s om h ee o he s ands o he li e a u e. The fi s
is ha o ‘backg ound isk’ (see in pa icula Fage eng, Guiso, and Pis a e i 2018)
– i , o example, an asse holde is also con on ed wi h labou income isk.
By defini ion, his e e s o uninsu able isk – isk ha ‘canno be di e sified
o a oided’ (ibid., p. 437). In ou model, howe e , he asse -holde can adjus he
amoun o liquidi y isk she e ec i ely bea s by changing he a io o money o
sa ings deposi s in he po olio. We show his mo e p ecisely below.
The second is he issue o op imal po olio alloca ion ac oss wo o mo e isky
asse s (see, o example, Hada and Seo 1990), whe ein a majo conce n is es ablish-
ing he condi ions unde which a s ochas ically domina ing shi in he e u ns o
one o he isky asse s unambiguously inc eases he amoun in es ed in ha asse .
In ou model, we also ha e mul iple ( wo) isks, bu one o hem is a liquidi y isk,
which is quali a i ely di e en om a e-o - e u n isk. This is in ui i ely e iden ,
and we explici ly discuss he di e ences below.
Thi dly, and o e lapping somewha wi h he Baumol-Tobin- ype ansac ions
models discussed ea lie , he e ha e been s udies seeking o dis inguish be ween
2Fo example, Al a ez, Lippi, and Roba o (2019): ‘The p oblem o he agen is o minimize he
(expec ed li e ime) cos incu ed o finance an exogenous consump ion s eam’ (p. 211), and ‘The
agen wi hd aws om and deposi s o an asse accoun wi h eal a e o e u n ’(p.212).
A Re-Conside a ion o Money Demand Theo y —75
he ansac ions oles o cu ency ou side banks, bank deposi s, and possibly
o he sho - e m mone a y ins umen s such as MMDA’s (money ma ke deposi
accoun s). F eeman and Kydland (2000) assume a single kind o in e es -bea ing
bank deposi ( his is cha ac e ized as a demand deposi , and included in hei
defini ion o M1), bu he e is a fixed cos o using hese o paymen s pu poses
(which ‘may be hough o as a check-clea ing cos o a cos o e i ying he iden i y
o he pe son w i ing a check o making a wi hd awal’ (ibid., p. 1126)). By con as ,
cu ency ou side banks pays no in e es , bu incu s no fixed cos when used o pay-
men s. They hen show ha i is op imal o use cu ency o small pu chases, and
checks d awn on demand deposi s o la ge ones,3wi h he pu chase h eshold
be ween he wo de e mined endogenously. A puzzling, and possibly inconsis en ,
ea u e o hei s udy is ha hey assume infini ely-li ed indi iduals wi h s ic ly
conca e pe -pe iod u ili y unc ions, as well as echnology and money supply shocks
(which may be au o-co ela ed), bu do no inco po a e any isk p emia in o hei
analyses.4
Belongia and I eland (2019) pos ula e a linea homogeneous mone a y se ices
agg ega o o cu ency and a single ype o in e es -ea ning bank deposi , and adop
a ‘simplified, pe ec o esigh pa ial equilib ium amewo k’ (p.3).Lucas (2000)
also adop s a de e minis ic amewo k in analyzing money demand, wi h money
ei he en e ing he u ili y unc ion o being explici ly used o ansac ions pu -
poses. Lucas (1980) allows o andom ansac ions demands, and, wo king wi h
gene al u ili y and ansac ions isk unc ions, ocuses on cha ac e izing he gen-
e al equilib ium o he economy wi hou de o ing much a en ion o s udying he
p ope ies o indi idual money demand, o he han showing ha indi idual Engel
cu es o eal money balances a e upwa d-sloping. He abs ac s om asse a e-
o - e u n isk. Finally, Lucas and Nicolini (2015) cons uc a de e minis ic model
wi h h ee mone a y asse s – cu ency ou side banks, demand deposi s, and money-
ma ke deposi accoun s – wi h a ying ansac ions cos s, ese e equi emen s,
and in e es a es, and show ha hese indi idual asse s a e used o ansac ions
o di e ing sizes. Plo ing hei NewM1 measu e, including money-ma ke deposi
accoun holdings, and a ios o i s indi idual componen s agains sho - e m T ea-
su y Bill a es, hey find ha ‘while he abili y o he model o ma ch he a ios
3I is assumed ha he e a e n, appa en ly exogenously de e mined, sub-pe iods wi hin each
pe iod, and household s ocks o cu ency and deposi s a e eplenished a e each sub-pe iod, so
ha cu ency- and deposi - holdings a e cons an o e he pe iod as a whole.
4They e e o ea lie wo ks by Lacke (1988) and F eeman and Hu man (1991), bu hese mod-
els ha e o e lapping gene a ions o wo-pe iod-li ed indi iduals, wi h isk-neu al second-pe iod
u ili y unc ions. The la e explici ly no e (p. 650) ha isk p emia become ele an i second-
pe iod u ili y is no linea in consump ion. The e is a simila puzzle in he infini e-ho izon model
o Bena i e al. (2021), which has p oduc i i y, in e es a e, and in e media ion echnology shocks.
76 —B. K. Kapu
be ween he componen s o NewM1 is mixed, he beha io o he agg ega e is
ema kably close o he da a’ (p.60).
5
The e is abundan empi ical e idence ha he ou -asse classifica ion we p o-
pose co esponds well wi h eali y. In ac , he plausibili y o such a co espondence
can be es ablished deduc i ely, as well as shown empi ically. Since sa ings deposi s
o e a highe in e es a e han demand deposi s, asse -holde s would op en i ely
o he o me unless he e is some o se ing disad an age. Indeed, in he US he e
is ypically a limi o six cheque o debi -ca d wi hd awals pe mon h om sa -
ings deposi s,6any u he wi hd awals equi ing a isi o he bank o o an ATM
machine, and minimum-balance equi emen s a e o en imposed as well. Banks a e
obliged o impose such es ic ions gi en ha hey ace lowe ese e equi emen s
agains sa ings deposi s han agains demand deposi s: he lowe ese e holdings
a e p ecisely wha enables hem o o e highe e u ns on sa ings deposi s. Simi-
la ly, a penal y is o en imposed on p ema u e cash wi hd awals om fixed deposi s
and CD’s,7and again he lowe ese e equi emen s a e a majo ac o , along wi h
in es men s in longe - e m asse s, enabling banks o o e highe e u ns on hese
han on sa ings deposi s. Las ly, isk-a e se indi iduals will equi e a isk p emium
o compensa e hem o holding bonds o equi ies, a he han fixed-capi al- alue
fixed deposi s and CD’s.
The e a e also es ic ions o ees o a ying kinds on ‘linked sa ings accoun s’
(Kagan 2020a), ‘high-yield sa ings accoun s’ o e ed by bo h online and b ick-and-
mo a ins i u ions (Ka l 2020c), and money ma ke deposi accoun s and money
ma ke mu ual unds (Ba ba 2023). Bank a e (June 21 2020) lis s a o al o 121 o he
‘bes a ailable a es ac oss di e en accoun ypes,’ and he APY (Annual Pe cen -
age Yield), ac oss sa ings accoun s and money ma ke accoun s, o he op 15 o
hese ange om 1.15 % o 1.36 %. In con as , he bes CD a es a ha ime anged
om 1.50 % APY upwa ds o deposi s o 6 mon hs o longe (Ka l 2020d).
3 The Model
Ou analysis is explici ly pa ial-equilib ium in na u e, as in ou iew a i-
ous ( hough no all) gene al-equilib ium analyses adop , o ac abili y easons,
5New-Mone a is , sea ch- heo e ic app oaches o he s udy o money demand include Kim and
Ma chesiani (2024) and Be en sen, Hube , and Ma chesiani (2015,2018), and abs ac en i ely om
a e-o - e u n isk.
6Fon inelle (2020). We abs ac om NOW accoun s, which ypically ha e minimum balance
equi emen s and o he es ic ions. Edmondson (2021) obse es, ‘Today, he e a e e y ew NOW
accoun s used anymo e as many ypes o checking accoun s can bea (low) in e es .’
7Ka l (2020b).
A Re-Conside a ion o Money Demand Theo y —77
simpli ying assump ions ha do no do ull jus ice o he ichness o he de e -
minan s o indi idual asse demands. A gene al-equilib ium ex ension o ou
elabo a ely-specified pa ial-equilib ium analysis is pos poned o u u e esea ch.
We conside a isk-a e se asse -holde who has an ini ial weal h endowmen o W0
a ime 0, and seeks o maximize he expec ed u ili y o he e minal weal h, W2,a
he end o pe iod 2.8She ecei es no income in pe iod 1. A he beginning o pe iod
1 she aces an unce ain liquidi y demand, which is x>0 wi h p obabili y p and 0
wi h p obabili y 1 −p𝓁. This is an essen ial ‘main enance’ expense (e.g. a medical
expendi u e), bu does no o he wise enhance he u ili y: i could simply be iewed
as o es alling a sha p decline in he u ili y ha would o he wise occu .
She can in es he ini ial weal h ac oss ou asse s: (1) cash o demand deposi s
M0(assumed nonin e es bea ing o con enience9), which can be liquida ed (u i-
lized o pay o he expense) wi hou any ansac ions cos a ime 1, (2) a sho - e m
sa ings deposi S0, on which, ollowing he discussion ea lie , an incon enience o
ansac ions cos o pe dolla wi hd awn a ime 1 is incu ed, (3) a long- e m fixed
deposi o CD F0, which ma u es a ime 2, and (4) a isky secu i y B0, which yields
an in e es paymen a ime 2, as well as any capi al gain o loss hen.10 Fo simplic-
i y, we assume ha he penal y o a p ema u e wi hd awal om he fixed deposi
a ime 1, inclusi e o he o egone in e es , is su icien ly high ha he asse holde
does no en e ain his possibili y. Simila ly, we assume ha o egoing any in e es
paymen i he isky secu i y is liquida ed a ime 1, oge he wi h he b oke age
o ansac ions cos incu ed, is su icien ly cos ly ha his oo is no a wo hwhile
op ion o he .11 We belie e hese las wo specifica ions acco d well wi h eali y.
In p inciple, should he liquidi y demand no anspi e, he asse -holde may
wish o con e he Mand Sholdings in o holdings o Fand Ba ime 1. Howe e ,
he e u ns om con e ing in o Fwould be sha ply educed on accoun o he
educed holding pe iod, as well as he b oke age cos incu ed. This is pa icula ly
8This is specified as he e minal pe iod o no a ional con enience. Below, we poin ou ha ou
analysis is equally applicable in a sui ably-specified infini e-ho izon op imiza ion amewo k.
9As will be e iden om he subsequen analysis, in oducing a u he dis inc ion be ween cash
holdings ou side banks – which pay no in e es and incu no ansac ions cos s – and demand
deposi s – which pay lowe in e es han sa ings deposi s, and incu posi i e bu smalle ansac-
ions cos s han he la e – will no a ec ou quali a i e insigh s.
10 As men ioned we assume ha he bond’s ma u i y da e is a some da e beyond ime 2, so ha
i is no edeemable a pa a ime 2.
11 We also assume a p ohibi i ely high cos o sho sales o any asse (bo owing), so as no o
unduly complica e he discussion. Ou assump ion ha he liquidi y demand can only occu once,
a ime 1, is simply designed as an app oxima ion o he ac ha in eali y he e may be my iad
small liquidi y demands be ween imes 0 and 2, and he asse holde may well ha e o incu fixed
‘a en ion cos s’, as well as b oke age cos s, each ime she has o liquida e small amoun s o bonds.
As such, she does no en e ain his op ion.
78 —B. K. Kapu
so i , ollowing he a gumen in n. 11 abo e, she is unce ain as o p ecisely when
all liquidi y demands migh be incu ed. Rega ding B, heclose sheis o ime2a
he ime she con e s om Mand Sin o his asse , he lowe will be he gain as
he p ice o he secu i y would, wi h adjus men o isk, end o con e ge owa ds
he expec ed g oss e u n, inclusi e o in e es paymen , a ime 2 so as o ule ou
‘abno mal’ expec ed capi al gains o losses.12 In addi ion she would s ill ha e o
incu he b oke age cos o such con e sion. Thus, we ule ou he possibili y o
e-op imiza ion o asse holdings a ime 1. Allowing o e-op imiza ion would sig-
nifican ly complica e he analysis wi hou gene a ing new insigh s, since Mwould
s ill command a liquidi y ad an age o e Sgi en ha she would fi s ha e o incu a
ansac ions cos o con e om S o Mbe o e in es ing in ei he Bo F.Ou spec-
ifica ion is equi alen o assuming, as in infini e-ho izon models such as he one
we ou line b iefly below, ha asse -holdings a e only op imized a he beginning
o each pe iod (whe e he ‘pe iod’ co esponds o he dual pe iods in ou cu en
model).
He weal h cons ain a ime 0 is
W0=M0+S0+F0+B0(1)
Since Mpays no in e es , i s g oss e u n (1 plus in e es ) is simply 1, and he
g oss e u ns on S,F,andB, ne o any b oke age cos s incu ed in in es ing in hem
a ime 0 (assumed p opo ional o he amoun s in es ed), a e, espec i ely, s, ,
and, o B, 1>1 wi h p obabili y pband 2<1 wi h p obabili y 1 −pb. Fo ob ious
easons we assume:
1< s< <pb 1+(1−pb) 2.(2)
Gi en hese inequali ies, i would clea ly no be op imal o he o hold any
Mo Sabo e he sums equi ed o mee he liquidi y demand a ime 1. This does
no imply he es ic ion M0+S0=x, howe e , as we also ha e o accoun o he
ansac ions cos s incu ed i any Sis u ilized o mee ing he liquidi y demand a
ha ime. In e es ingly, i can be shown ha , p o ided ha <1, i is a ma e o
indi e ence whe he hese ansac ions cos s a e incu ed ou o Mo ou o u he
d awdown o S,13 and o conc e eness we assume he o me . Le M1deno e he
12 This ‘no-a bi age’ a gumen would also ensu e ha bond p ices could no be o e ly high
a ime 1 (as he e would hen be high expec ed capi al losses going o wa d), again se ing o
dissuade he use o bonds o mee liquidi y demands a ha ime.
13 I can be shown ha o any gi en M0 he amoun o S0held is he same in ei he case. Fo -
mally, i Sis used o de ay he ansac ions cos s, hen we ha e (1− )S0=x−M0,andi Mis
used o de ay he ansac ions cos s hen we ha e S0=x−M1=x−(M0− S0),whichgene -
a es he same equa ion o S0. The es ic ion <1 ensu es ha holdings o Sdo no ‘explode’ i S
A Re-Conside a ion o Money Demand Theo y —85
Di e en ia ing (18) wi h espec o 1and se ing d 2∕d 1=−1i is eadily
shown ha dB∗
0∕d 1<0, as expec ed, and hence dF∗
0∕d 1>0.
Al hough he e is sepa abili y in ha 1and 2do no a ec M∗
0and S∗
0
unde CARA, he same is no ue o .F om(18) we ha e ha dB∗
0∕d <0,
and om (17) i is eadily shown ha dM∗
0∕d >0. The a ionale o his
la e esul is no el. Manipula ing (3) and (4) we easily ob ain ha S∗
0+M∗
0=
x− M∗
0
1− . Thus, when M∗
0goes up S∗
0+M∗
0goes down, and so some u he unds
a e eleased o in es men in F0. This is he main sou ce o non-sepa abili y
in he CARA case.21
I should be no ed ha in he p esen expe imen o a mean-p ese ing
inc ease in bond e u ns, he e a e no weal h e ec s. Thus, he key di e ence
be ween CARA and CRRA is no he DARA (Dec easing Absolu e Risk A e sion)
en ailed by CRRA u ili y (which plays some pa below), bu he much mo e
limi ed deg ee o non-sepa abili y unde CARA u ili y.
(B) Re u ning o he CRRA case, he nex compa a i e-s a ic expe imen is pe -
haps e en mo e s a k. Le he e ins ead be a ma ginal inc ease in he expec ed
bond e u n. We aise 1 o 1.1102, and 2 o 0.9502. As expec ed, B∗
0 ises, o
0.7829, and F∗
0 alls, o 0.0167. Wha is en i ely unexpec ed is ha M∗
0inc eases,
o 0.0795, and S∗
0dec eases, o 0.1208. D∗is again ex emely low, a 1.3204e-14.
In his case, he ealized a iance o o al bond e u ns has gone up, o 0.0039,
and heu is ically he asse -holde mi iga es he inc ease in he o e all isk
exposu e by subs i u ing away om S0and owa ds M0in mee ing p ecau-
iona y demands.22 Rema kably, a posi i e ela ionship be ween US money
(M1) demand and he 10-yea Go e nmen Bond Ra e was indeed obse ed
in he 1990–2019 pe iod o in e es a es abo e 4 % p.a.,asKim and Ma ch-
esiani (2024) documen . No only does ou esul u he exempli y he non-
sepa abili y p ope y o isk-a e se expec ed u ili y, i also shows ha he
con en ional wisdom ha expec ed bond e u ns should en e nega i ely in
he demand unc ion o M1 need no always hold, and he e ec is ins ead
model-dependen . Mo eo e , we ob ain he same quali a i e ou comes when
𝜎=2. Hence o h, we main ain 𝜎a uni y.23
21 Ano he sou ce, which esul s a he mechanically om he weal h cons ain , has o do wi h
he e ec o son F∗
0, he analysis o which is e y simila o he e ec in he CRRA case ( n.25
below).
22 The e is an o se ing weal h e ec : e alua ed a he ini ial le el o bond-holding, an inc ease
in he mean bond e u n inc eases he asse -holde ’s weal h, and wi h DARA she would wish o
inc ease he holding o Sand educe he holding o M. I u ns ou ha his only pa ially o se s
he e ec desc ibed abo e.
23 A seemingly puzzling ea u e o ou esul is he e y high in e es -elas ici y o money demand
ha is gene a ed, o he o de , in ac , o 155. Be en sen, Hube , and Ma chesiani (2015) ob ain an
86 —B. K. Kapu
A hi d expe imen yields somewha less clea -cu bu none heless highly use ul
addi ional insigh s. We e u n o ou o iginal calib a ion, excep ha we inc ease
ma ginally o 0.01152: a he same ime, in o de o keep M∗
0+S∗
0unchanged, a
0.2004, we inc ease x, and o 4 decimal places he equi ed alue o xis also 0.2004.
Now M∗
0 ises d as ically, o 0.198, and S∗
0 alls d as ically, o p ac ically 0 (ac ual
alue is 0.0024). Following ou ea lie a gumen , one migh expec B∗
0 o inc ease
and F∗
0 o dec ease, since he asse -holde can now ‘a o d’ o ake on addi ional isk.
Ins ead, howe e , B∗
0dec eases ma ginally, o 0.7540, and F∗
0inc eases ma ginally,
o 0.0456. These e y small e ec s a e explained by he weal h e ec : e alua ed
a he ini ial le els o M0and S0, inc eases in and x educe he asse -holde ’s
expec ed e minal weal h ne o expec ed p ecau iona y cos s, and wi h DARA she
chooses o ma ginally dec ease he holdings o Band inc ease he holdings o F.
Any u he inc ease in would esul in a co ne solu ion o S0, which would make
compa a i e-s a ic expe imen s di icul o in e p e . E iden ly, he op imal le el o
S0is highly sensi i e o .24
One may examine he e ec s o a ying o he pa ame e s, such as he a es o
e u n on Sand F, bu he esul s a e simila .25 Ou final hough -p o oking expe -
imen is o examine he e ec s o changes in W0 ela i e o changes in x.The ea e
wo no ewo hy cases he e:
(a) Suppose we inc ease only W0in ou baseline calib a ion, o 1.005. The esul s
a e, qui e ema kably, ha M∗
0dec eases qui e significan ly o 0.0345, and S∗
0
inc eases o 0.1664. B∗
0and F∗
0inc ease sligh ly, o 0.7587 and 0.0454 espec-
i ely. We no e ha M∗
0has dec eased, by o e 50 %, no wi hs anding ha
es ima e o his elas ici y in he upwa d-sloping egion o money demand o 0.16. Howe e , in ou
model he inc ease in money demand is mo e han o se by a concomi an dec ease in demand o
S( om (3) and (4) abo e we ha e ha (M0+S0)=(x− M0)∕(1− )), while he esul an dec ease
in in e es ea nings om Sis ‘compensa ed o ’ by he inc eased bond expec ed e u n. As u he
discussed below, his esul unde sco es he necessi y o a ‘sys ems’ o s uc u al app oach o he
s udy o money demand.
24 In o de o gene a e mo e a ia ion in we changed he ini ial calib a ion o x=0.38874 and
=0.00878, esul ing in an ini ial alue o S∗
0o 0.3526 (we also inc eased o 1.0265 o gene a e an
in e io solu ion o F0). Howe e , e en he e we could only inc ease and x e y sligh ly be o e S∗
0
app oached he co ne a 0, and he esul s a e quali a i ely unchanged.
25 We inc eased sma ginally o 1.0151, keeping all o he pa ame e s a hei baseline alues,
and he e occu s a la ge inc ease in S∗
0, a i ually ze o (0.0023) solu ion o M∗
0, and e y small
dec eases in B∗
0and F∗
0. These small declines a e mainly due o he la ge alue o M∗
0+S∗
0owing
o he highe ansac ions cos s incu ed when S∗
0goes up. Simila ly, a small inc ease in , o
1.025, gene a es as expec ed an inc ease in F∗
0and dec ease in B∗
0, bo h o which a e ai ly la ge
( o 0.0948 and 0.7050 espec i ely), as well as a mode a e dec ease in S∗
0( o 0.1085) and inc ease in
M∗
0( o 0.0917). (Wi h F∗
0+B∗
0inc easing, M∗
0+S∗
0has o dec ease, which is e ec ed by educing S∗
0
and inc easing M∗
0((3) and (4) abo e).)
A Re-Conside a ion o Money Demand Theo y —87
he ini ial o e all po olio size W0has inc eased. The e a e a numbe o
no ewo hy implica ions o hese esul s:
(i) Wi h DARA he demand o he isky asse s has gone up. Howe e , s ikingly,
he demand o S ises much mo e han he demand o B, in bo h absolu e
and ela i e e ms. This is due o a ea u e o he model ha has been alluded
o ea lie : in es ing in S educes he demand o M o p ecau iona y pu -
poses almos pa i passu, and he unds hus eleased can be u ilized o o he
in es men s. No such o se ing elease occu s upon inc easing holdings o B.
(ii) The ac ha he demand o Mhas gone down ma ks a no able depa u e
om he well-known esul exp essed in Theo em 1 o Cass and S igli z (1972)
and in ea lie li e a u e, in which, in a pu e ‘po olio’ analysis o he demand
o one isky asse and money, he weal h elas ici y o he demand o money
is uni y unde cons an ela i e isk a e sion. Wi h ou mo e elabo a e asse
s uc u e, he use o M o po olio di e sifica ion pu poses is domina ed by
F, and he demand o M, is-à- is S, is go e ned by p ecau iona y equi e-
men s and expec ed-u ili y maximiza ion. We hus suppo he opinion o Cass
and S igli z (ibid. p. 331): ‘wi hou s ingen condi ions i does no appea o be
possible o de i e a simple heo y o he demand o money om po olio
analysis’ (p. 331).
A Small Dig ession, Con inued: W0does no appea in (17) and (18), and hence
M∗
0(and S∗
0)aswellasB∗
0a e in a ian o any changes in W0,whichsimplychange
F∗
0pa i passu. This esul clea ly exhibi s he ole played by DARA in he analysis
o weal h e ec s abo e. In iew o he well-known analy ical limi a ions o CARA
u ili y (exemplified, in ac , by he esul he e), we will con inue o wo k wi h CRRA
u ili y.26
(b) Suppose ha he inc ease in W0is accompanied by an inc ease in xsuch ha
he ini ial po olio size, ne o he expec ed size o he p ecau iona y shock,
is unchanged. In ou example, wi h p𝓁=0.5, his equi es ha xinc eases
by wice he inc ease in W0, o 0.209. The asse -holde ’s expec ed ini ial ne
weal h has no changed, bu she con on s highe isk. In his case, we find
ha , no wi hs anding he small absolu e size o hese changes, S∗
0dec eases,
o such an ex en ha he ze o lowe bound on i becomes binding. The final
solu ion hen is ha M∗
0inc eases subs an ially, also o 0.209 (since wi h S∗
0=0
26 Kim and Ma chesiani (p. 1223) obse e ha assuming a CRRA specifica ion o he u ili y unc-
ion ‘is a s anda d p ac ice in he li e a u e.’ Obi e dic um ou analysis should p o ide a cau iona y
no e agains unc i ical adop ion o specifica ions (such as CARA u ili y) which gi e ise o ‘nea ’
closed- o m solu ions, in p e e ence o CRRA u ili y, he analysis o which is mo e ‘messy’ bu also
mo e ealis ic.
88 —B. K. Kapu
he e a e no ansac ions cos s o be incu ed), and wi h inc eased p o ision
o he possible p ecau iona y shock owing o he inc ease in xbo h B∗
0and
F∗
0dec ease ma ginally. We discuss he implica ions o hese esul s below.
The analysis hus a has been conduc ed wi hin a single- (o dual-) pe iod op i-
miza ion amewo k. Unde simpli ying assump ions, he esul s also hold in an
infini e-ho izon amewo k, no ing ha as men ioned we confine ou sel es in his
s udy o a pa ial-equilib ium analysis o asse demands. We assume ha he asse -
holde ecei es an exogenous endowmen each pe iod, which, consis en wi h he
da a, may be g owing o e ime, and is su icien ly impa ien ha she wishes o
consume he en i e end-o -pe iod weal h each pe iod. (F om he defini ion o he
s ochas ic discoun ac o , such impa ience is e en mo e p onounced i consump-
ion is gene ally ising o e ime.) We u he assume ha she aces a bo owing
cons ain ha does no pe mi any bo owing in any pe iod. Finally, we assume
ha he a e-o - e u n isk and he p ecau iona y isk a e independen o e ime,
and independen o each o he . Unde hese specifica ions, i is easy o see ha
ou one-pe iod analysis is eplica ed e e y pe iod, wi h possibly di e en alues
o beginning-o -pe iod weal h, and o and xeach pe iod. I desi ed, he a ious
asse a es o e u n can also be ea ed as exogenously a ying o e ime.
Replica ion, mu a is mu andis, o ou one-pe iod analysis may help o explain
ce ain no ewo hy b oad ends o e ime.27 Fede al Rese e da a (Fede al Rese e
2020) show a p onounced decline in he M1/GDP a io in he US om abou 0.275 in
1959 o abou 0.09 in 2007, p io o he onse o he Global Financial C isis. (The e
we e upwa d mo emen s in he la e-1980s and mid-1990s.) Absolu e le els o M1
ose h oughou his pe iod (ibid.). Basing on ou a ious esul s abo e, ou model
is in p inciple capable o accoun ing o hese, and concu en de elopmen s in
o he asse -holdings, by sui able combina ions o ises in GDP (a p oxy o yea ly
endowmen s), smalle ises in x, some declines in in acco dance wi h he da a,
and, al hough his is no essen ial, some adjus men s in asse e u ns. The fi s wo
o hese changes could p oduce a alling M1/GDP a io and a ising M1,28 and he
27 We a e abs ac ing he e om sa ings o e ime, implici ly assuming, as do Lucas (2000)
and Lucas and Nicolini (2015), ha all income is consumed. The US na ional sa ings a e has
been below 10 % om he mid-1960’s ill 2007, excep o one yea in he ea ly 1970’s (Pe e -
son 2020). The pe sonal sa ings a e has been below 11.4 % (8.8 %) since 1980 (1990) up o 2007
(S a is ica Resea ch Depa men 2020), and pa o hese sa ings, e.g. hose placed in Indi idual
Re i emen Accoun s, a e o longe - e m e i emen pu poses.
28 The e is a possible addi ional eason o hese fi s wo de elopmen s. In 2012 he Fede al
Rese e in oduced he annual Dia y o Consume Paymen Choice, one o he findings o which
has been, ‘His o ically, cash has been he mos used paymen ins umen o small- alue paymen s
(paymen s unde $25)’ (Cubides and O’B ien (2023),p. 11). One migh in ac ex end he a gumen
o M1 as a whole, since small- alue paymen s can also be de ayed h ough debi ca ds and ATM
A Re-Conside a ion o Money Demand Theo y —89
hi d o hese could help o be e ma ch he gene ally ising beha iou o So e
ime. By con as , he amewo k o Belongia and I eland (op. ci .), building on he
ea lie wo k by Lucas (2000,1980), abs ac s comple ely om a e o e u n and
p ecau iona y isks, and imposes an unchanging equilib ium alue o M1/GDP o e
ime i use cos s o holding money a e cons an . This does no appea o do jus ice
o he complexi y o he de e minan s and beha iou o money demand.29
Beyond he o egoing obse a ions, o mal econome ic modelling o money
demand o e ime is ou side he scope o his essen ially concep ual s udy. Such
modelling would ha e o ake in o conside a ion changes in financial egula ions,
‘ he in oduc ion o mo e inno a i e financial p oduc s’ (Kim and Ma chesiani, op
ci .), measu emen issues, mechanism design issues (ibid.), and o he ac o s. I is
unclea as o wha ex en a ious o hese conside a ions can simply be subsumed
unde he ub ic o changes in ou pa ame e . Changes in bo h and xo e ime
would ha e o be iden ified and es ima ed, as also would possibly ime- a ying
pa e ns o co ela ion, i any, be ween di e ing sou ces o isk. The e is also a u -
he conside a ion, which none o he s udies ci ed abo e ha e aken in o accoun :
money demand is a ec ed, no only by he a e age le el o bond e u ns, bu also,
we ha e shown, by a mean-p ese ing inc ease in he a iance o bond e u ns.
Rela edly, one mus be cognizan o he ac ha in ou se up i is no possible o
a y M0independen ly o S0, and, again, a s uc u al app oach o financial asse
demands ( iewing an asse -holde ’s po olio in i s en i e y) is called o . Las ly, o
he ex en ha pa simony is an ad an age, he ac ha ou model can pa simo-
niously ma ch he obse ed posi i e ela ionship, o e a ce ain ange, be ween
money demand and he mean bond e u n gi es i an edge o e he ex ao dina -
ily complica ed (and highly s ylized) New-Mone a is -Mechanism-Design a icle by
Kim and Ma chesiani (op ci .), he only o he s udy among hose ci ed ha can
gene a e such a ela ionship.30 Ou model hus, we belie e, p o ide a aluable
‘sp ingboa d’ o u he s udies, bo h heo e ical and empi ical.
ca ds. One could o con enience hen model he use o M1 o his pu pose as a conca e de e min-
is ic unc ion o GDP, which could easily be in oduced in o ou model as an addi ional sou ce o
demand o M1, hus helping o accoun o he fi s wo de elopmen s men ioned abo e. Un o u-
na ely, da a limi a ions p eclude an empi ical examina ion o his hypo hesis o e he 1959–2007
pe iod.
29 Lucas (2000) himsel no es (looking a he 1900–94 pe iod), ‘The money-income a io is essen-
ially endless o e he en i e cen u y, al hough he e has been a s ong downwa d end since
Wo ld Wa II’ (p.249).
30 The ollowing commen by Kim and Ma chesiani (p. 1232) should, howe e , be acknowledged:
‘O cou se, we do no claim ha he mechanism is he only de e minan o he obse ed money
demand beha io a e he 1990s. We only a gue ha he mechanism, in he o m o MI (Ma ke
In elligence), could ha e, oge he wi h o he ac o s s udied in he li e a u e ... explained pa o
he beha io o he money demand. Ou pape complemen s his li e a u e.’
90 —B. K. Kapu
5 Conclusions
Why is a ou -asse amewo k mo e sui able o he s udy o money demand, and
demand o o he asse s, han a wo- o h ee- asse amewo k? The e a e wo inde-
penden sou ces o isk in he model – a longe - e m po olio o a e-o - e u n
isk, and a sho e - e m ansac ions isk. Wi h ou asse s, wo o hem can be
‘assigned’ o deal wi h each sou ce o isk, in he specific senses ha Band Fa e
no deployed o handle ansac ions isk, while he e would be no demand o M
and Si xwe e 0. By con as , i he e we e o example only h ee asse s, one
o hem, namely S, would ha e o help sa is y bo h po olio di e sifica ion and
ansac ions mo i es. As such, one would expec he demand o S o go down
by less (and hence he demand o M o go up by less) when goes up, since
Sis also held o po olio di e sifica ion pu poses. Asse -holde s a e in eali y
awa e o he benefi s o op imizing ac oss ou asse s, as e idenced by posi i e
eal-wo ld holdings o all hese asse s, and i one ins ead wo ks wi h a h ee- o
wo- asse amewo k, one is likely o ob ain an inaccu a e cha ac e iza ion o asse
demands.
Qui e ema kably, e en hough he demands o Mand Sa ise only when x
is posi i e, he op imal alloca ion be ween M and Sdoes depend on cha ac e is-
ics o long- e m asse s, such as he a e o e u n on bonds and he a iance o
bond e u ns, as we ha e demons a ed. Simila ly cha ac e is ics o sho - e m
asse s such as he uni ansac ions cos do a ec he demands o long- e m
asse s. Changes in he asse -holde ’s weal h also a ec s he demands o a ious
asse s in unexpec ed ways. These esul s a e due o he non-sepa abili y p ope y
o isk-a e se expec ed u ili y o e minal weal h unde DARA, explained ea lie .
The jux aposi ion o his p ope y wi h a ou -asse amewo k hus yields no el
and impo an esul s and insigh s.
In an in e empo al con ex , ou model is po en ially capable o ma ching he
p onounced decline in he M1/GDP a io o e he 1959–2007 pe iod, which a ious
o he wo ks ci ed abo e do no . The e is, finally, a need o ex end he analysis o a
dynamic, s ochas ic gene al-equilib ium amewo k, wi hou imposing simpli ying
assump ions ha yield, o en coun e ac ually, an unchanging equilib ium M1/GDP
a io o e ime.
Acknowledgmen s: I would like o hank wo anonymous e e ees o hei in alu-
able commen s, John K H Quah and Denis Tkachenko o enligh ening discussions,
Ren Jie, Lee Ming Xuan, and Wang Yu Han o supe b esea ch assis ance, and he
Depa men o Economics, Na ional Uni e si y o Singapo e, o financial suppo .
The usual disclaime applies.
Compe ing in e es s: None decla ed.
A Re-Conside a ion o Money Demand Theo y —91
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