Bena i, Luca; Nicolini, Juan Pablo
Wo king Pape
The wel a e cos s o in la ion econside ed
Discussion Pape s, No. 25-08
P o ided in Coope a ion wi h:
Depa men o Economics, Uni e si y o Be n
Sugges ed Ci a ion: Bena i, Luca; Nicolini, Juan Pablo (2025) : The wel a e cos s o in la ion
econside ed, Discussion Pape s, No. 25-08, Uni e si y o Be n, Depa men o Economics, Be n
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Facul y o Business, Economics
and Social Sciences
Depa men o
Economics
The Wel a e Cos s o In la ion Reconside ed
Luca Bena i, Juan-Pablo Nicolini
25-08
Oc obe , 2025
Schanzenecks asse 1
CH-3012 Be n, Swi ze land
h p://www. wi.unibe.ch
DISCUSSION PAPERS
The Wel a e Cos s o In la ion Reconside ed∗
Luca Bena i
Uni e si y o Be n†
Juan-Pablo Nicolini
Fede al Rese e Bank o Minneapolis
and Uni e sidad Di Tella‡
Abs ac
Mode n analysis o he wel a e e ec s o mone a y policy is based on moneyless
models and he e o e igno es he e ec o in la ion on he e iciency o ansac ions. A
jus i ica ion o his s a egy is ha hese wel a e e ec s a e quan i a i ely e y small,
as a gued by I eland (2009). We e isi I eland’s esul using ecen da a o he Uni ed
S a es and se e al o he de eloped coun ies. Ou compu a ions a e in luenced by he
expe ience o e y low sho - e m a es obse ed since I eland’s wo k in he coun ies
we s udy. We es ima e he wel a e cos o a s eady s a e nominal in e es a e o 5%
o be a leas one o de o magni ude highe han in I eland (2009), which ques ions
he alidi y o pe o ming mone a y policy e alua ion in cashless models.
∗We wish o hank Fe nando Al a ez and Pe e I eland o commen s on a p e ious d a and o e y
help ul sugges ions. The iews exp essed in his pape do no necessa ily e lec hose o he Fede al Rese e
Bank o Minneapolis, o o he Fede al Rese e Sys em.
†Depa men o Economics, Uni e si y o Be n, Schanzenecks asse 1, CH-3001, Be n, Swi ze land. Email:
luca.b[email p o ec ed]e.ch
‡Fede al Rese e Bank o Minneapolis, 90 Hennepin A enue, Minneapolis, MN 55401, Uni ed S a es.
Email: juanpa@minneapolis ed.o g
1
1 In oduc ion
We p o ide new es ima es o he wel a e cos o in la ion. We ollow he adi ion o Bailey
(1956), F iedman (1969), Lucas (2000), and I eland (2009) in ha we es ima e he wel a e
cos using he a ea unde he eal money demand cu e. Fo a s eady-s a e in e es a e o
5%, Lucas (2000) calcula es he cos o be 1.1% o li e ime consump ion, which is a signi ican
amoun . Howe e , I eland (2009) challenges Lucas’s in e p e a ion o he da a and ob ains
an es ima e o a me e 0.037%.
Ou main con ibu ion is o b ing mo e da a o he deba e. We do so in wo ways. Fi s ,
we use he addi ional da a a ailable since I eland’s wo k. This is a pa icula ly abno mal and,
a he same ime, e y in e es ing pe iod, since i was cha ac e ized by se e al obse a ions
wi h e y low in e es a es. Thus, i helps iden i y he beha io o money demand a e y
low a es, which, as we will discuss, is highly ele an . Second, we also s udy e idence om
de eloped coun ies whose in la ion his o ies a e simila o hose o he Uni ed S a es. This
addi ional e idence is eassu ing, since he Uni ed S a es wen h ough egula o y changes
du ing he 1980s and 1990s ha blu ed he dis inc ion be ween ypes o deposi s. This
s a ed a deba e ega ding he p ope way o measu e mone a y agg ega es, an issue ha
I eland emphasized. These issues a e absen in he o he coun ies we s udy.
The e a e wo key aspec s o he money demand ela ionship ha a ec he compu a ion,
as bo h Lucas and I eland no e. The i s is he unc ional o m o he money demand a
e y low a es. Fo example, a unc ion wi h a sa ia ion alue a ze o, like he semi-log, will
end o deli e lowe es ima es han one in which desi ed money balances inc ease wi hou
bound as he nominal in e es a e app oaches ze o, as in he log-log. The second is he
alues assigned o i s pa ame e s.
Ou es s using he en i e sample end o p e e a unc ional o m wi h a ini e sa ia ion
poin , as a gued by I eland. Howe e , ou -o -sample es s o e y low alues o he in e es
a e p e e he log-log speci ica ion, he one p e e ed by Lucas, in all bu one case. To be
conse a i e, we choose he speci ica ion wi h a ini e sa ia ion poin as ou benchma k, bu
2
we also epo he esul s o he log-log speci ica ion. Rega ding he es ima ed pa ame e s,
ou esul s s ongly suppo he numbe s p e e ed by Lucas, bo h o he Uni ed S a es and
o all he o he coun ies.
Fo ou benchma k case in he Uni ed S a es, we ob ain a cos o 0.35%, almos en imes
ha o I eland. Al e na i e scena ios deli e highe alues, bu o e all, we ind ha 0.8% is
a likely uppe bound.
Mode n mone a y policy analysis is based on moneyless models, and i he e o e igno es
he e ec o in la ion on he e iciency o ansac ions. Fo example, he well-known ”di ine
coincidence” case, in which s abilizing p ices also s abilize he ou pu gap, is only op imal
a he cashless limi .1The accu acy o his s a egy in compu ing wel a e e ec s o policy is
a quan i a i e issue. I he wel a e cos o dis o ions on ansac ions is e y small ela i e
o he one a ising om p ice igidi ies, hen he cashless limi is a sensible app oxima ion.
Ou calcula ions sugges ha his is no he case.
Nakamu a e al. (2018) showed ha he wel a e cos o in la ion is qui e small in mon-
eyless New Keynesian models, a ound 0.02% o consump ion o an in la ion a e o 3%
pe cen . Gi en a eal a e o 2% his is consis en wi h a nominal in e es a e o 5%, he
alue conside ed by Lucas and I eland. Mo e ecen ly, A ouzi e al. (2024) show ha wi h
ne wo k e ec s, he cos can be much highe , close o 0.4% o consump ion. Coibion e al.
(2012) s udied a model wi h ecu en , hough no e y equen , episodes wi h he nomi-
nal in e es a e a he ze o lowe bound. They compu ed he wel a e e ec o an in e es
a e o 5% o be close o 0.6% o li e ime consump ion.2Rela i e o hese las wo igu es,
he 0.037% es ima ed by I eland may appea negligible. Bu 0.35%, he lowes numbe we
es ima e, is ce ainly no .
Ou s a ing poin is he e idence suppo ing he no ion o a downwa d and s able long-
1Khan e al. (2003) s udy he ade-o be ween dis o ions esul ing om p ice ic ions and he ones
esul ing om lack o money sa ia ion.
2Coibion e al. (2012) explici ly acknowledge ha hey do no ake in o accoun he cos s de i ed om
lack o money sa ia ion. Taking in o accoun he e ec ha we s udy would ac ually ein o ce he a gumen
o hei pape .
3
un eal money demand discussed in Lucas and Nicolini (2015) and Bena i e al. (2021).
In his pape , we go u he in se e al ways. Fi s , we p o ide o mal es s o compa e
di e en unc ional o ms. Second, we use ecen da a wi h e y low nominal in e es a es
o discipline bo h he unc ional o m and he pa ame e es ima es. Finally, ela i e o Lucas
(2000) and I eland (2009), we b ing e idence om coun ies o he han he Uni ed S a es
o shed ligh on he ques ion. On he heo y side, we inno a e by cons uc ing uppe and
lowe bounds o he es ima e o he wel a e cos o in la ion. Thus, we do no need o
ely on linea app oxima ions. The a ea unde he money demand cu e is an almos exac
measu e o he wel a e cos o a e y gene al class o mone a y models in he neighbo hood
o ze o, as Al a ez e al. (2019) show. Fo a qui e gene al subclass o he models hey
analyze, we compu e exac lowe and uppe bounds o he es ima es o he cos s, using he
a ea unde he money demand cu e, o any alue o he in e es a e. As we show, he
di e ence be ween he uppe and lowe bounds is ex emely small o he ange o in e es
a es obse ed in he Uni ed S a es.
The pape p oceeds as ollows. In Sec ion 2, we discuss a amily o mone a y models o
which we de i e e y igh lowe and uppe bounds o he wel a e cos o in la ion, using he
a ea unde he eal-money demand cu e. In Sec ion 3, we p o ide a discussion o ou main
esul s, using simple plo s, as Lucas (2000) and I eland (2009) did. Sec ion 4 p esen s he
o mal s a is ical analysis o h ee di e en empi ical speci ica ions used in he li e a u e,
including hose Lucas and I eland explo ed. Besides es ima ing he key pa ame e s o each
speci ica ion, we also de elop and pe o m o mal pai wise es s o e alua e he di e en
speci ica ions. Sec ion 5 p esen s ou compu a ions o he wel a e cos unc ions o he
benchma k case, in which he lowe bound o he in e es a e is ze o. The explo a ion o
coun ies o he han he Uni ed S a es highligh s a ea u e ha we b ing o he analysis: he
assump ion ega ding he ue lowe bound on he sho - e m nominal in e es a e. This
is ele an since i de e mines he lowe limi o he in eg al unde he eal money demand
cu e. Bo h Lucas and I eland assumed he lowe bound o be ze o, as did mos o he
4
mone a y economics li e a u e un il 2010. And so did we in Sec ions 4 and 5. Howe e ,
he nega i e in e es a es obse ed in he eu o a ea, Denma k, Sweden, and Swi ze land
mo i a e us o econside ha assump ion. We do so in Sec ion 6. Sec ion 7 concludes.
2 The Model
We s udy a labo -only economy wi h unce ain y, in which making ansac ions is cos ly.
The economy is inhabi ed by a uni mass o iden ical agen s wi h p e e ences gi en by
E0
∞
X
=0
β U(c ),
whe e Uis di e en iable, inc easing, and conca e.
E e y pe iod, he ep esen a i e agen chooses a numbe o po olio ansac ions n
ha allow he o exchange in e es -bea ing bonds o money, which is needed o buy he
consump ion good. The o al cos o hose ansac ions, measu ed in uni s o ime, is gi en
by a di e en iable unc ion θ(n , ν ),whe e ν is an exogenous s ochas ic p ocess. This
o mula ion gene alizes he linea unc ion assumed by Baumol (1952) and Tobin (1956).
The p oduc ion echnology o he consump ion good depends linea ly on ime de o ed
o p oduc ion. The e is a uni o ime o each pe iod ha can be used o p oduce goods o
make ansac ions. Thus, equilib ium in he labo ma ke and easibili y imply
1 = l +θ(n , ν ),
c =z (1 −θ(n , ν )),
whe e z is an exogenous s ochas ic p ocess. The eal wage is hen equal o z .
Pu chases a e subjec o a cash-in-ad ance cons ain
P c ≤n M ,
5
whe e M is a e age money balances.
We allow money o pay a nominal e u n m
. A he beginning o each pe iod, he agen
s a s wi h nominal weal h W ,which can be alloca ed o money o in e es -bea ing bonds
B . This decision, oge he wi h he ime alloca ion and consump ion decisions, aces he
ollowing cons ain :
M +B ≤W ,
W +1 ≤M (1 + m
) + B (1 + b
) + T + [1 −θ(n , ν )] z P −P c ,
whe e b
is he e u n on bonds and T is a ans e made by he mone a y au ho i y.
The uncons ained e icien ou come is o alloca e all he labo inpu o he p oduc ion
o he consump ion good so as o se c =z . Thus, he wel a e cos o making ansac ions,
as a ac ion o consump ion, is gi en by θ(n , ν ).
I is s aigh o wa d o show (see Online Appendix A o de ails) ha an in e io solu ion
o n mus sa is y
n2
θn(n , ν )
(1 −θ(n , ν )) = b
− m
.(1)
As long as b
− m
>0, he cash-in-ad ance cons ain is binding, so eal money demand,
as a p opo ion o ou pu , is equal o he in e se o n .In wha ollows, we le he in e es a e
di e en ial be ween bonds and money be ≡ b
− m
.No e ha equa ion (1) is independen
o z , so he heo y implies a uni income elas ici y o eal money demand.
Fo he maximum p oblem o he agen o be well de ined, i has o be he case ha
= b
− m
>0. The popula ze o-bound es ic ion on policy a es is ob ained using he
s anda d assump ion in he li e a u e ha m
= 0.Bo h Lucas (2000) and I eland (2009)
made his assump ion. Recen expe iences wi h nega i e policy a es in Eu opean coun ies
aise he issue o inco po a ing a nega i e lowe bound. We will do so below, bu in wha
ollows, we main ain, o wo easons, he s anda d assump ion ha he lowe bound on
in e es a es is ze o. The i s one is ha we wan o b ing in new da a bo h om he
6
US and om o he coun ies, o shed ligh on he disc epancies be ween Lucas (2000) and
I eland (2009). The second is ha nega i e policy a es do no necessa ily imply nega i e
a es on deposi s o households and i ms, which a e he ones ele an o ou compu a ions.
We b ie ly discuss hese issues below.
The unc ional o m o he eal money demand unc ion depends on he unc ional o m o
he ansac ions echnology θ(n , ν ), and a his le el o gene ali y, he model is consis en
wi h many di e en possibili ies. In wha ollows, o cla i y he main di e ence be ween
Lucas (2000) and I eland (2009), we conside h ee well-known unc ional o ms ha ha e
been used in p e ious empi ical wo k. All h ee exhibi a uni income elas ici y, as implied
by he model. The i s speci ica ion is he log-log one,
ln M
P y
=a1−ηln +u1
,(2)
which exhibi s a cons an in e es a e elas ici y equal o η. No ice ha as i →0, eal
money demand goes o in ini y. I is his asymp o e a ze o ha Lucas used o a gue ha
he wel a e cos o in la ion is sizable, e en a low alues o he in e es a e. The o he wo
o mula ions ha we explo e a e he semi-log,
ln M
P y
=a2−γ +u2
,(3)
which exhibi s a cons an semi-elas ici y γ, and he Selden-La an´e,
M
P y
=1
a3+ϕ +u3
.(4)
Bo h o mula ions imply a ini e le el o he demand o eal money balances when he
in e es a e di e en ial becomes ze o. This ea u e is emphasized by I eland, who uses (3)
in his e ision o Lucas’s es ima e.
The wel a e cos implica ions o he las wo unc ional o ms a e simila . We choose o
7
b eak would become appa en a ound 1980, e en i M1 was adjus ed by he sweep p og ams.
The e o e, he a gumen goes, he p e-1982 e idence was no e y use ul o es ima ing he
money demand cu e. Using pos -1980 da a alone, he hen made wo poin s. The i s was
ha semi-log was he p e e ed speci ica ion. The second was ha he semi-elas ici y was
close o 1.8, much lowe han Lucas’s p e e ed alue o 7.
I eland’s a gumen can clea ly be seen in Figu e 6, whe e we plo he same da a as be o e,
excep ha we adop ed he de ini ion o he money supply used by I eland - M1 plus he
sweep p og ams.
Figu e 6: US eal money demand, 1915-2008
The igu e clea ly shows he b eak in he beha io o he mone a y agg ega e chosen by
I eland. The da a ollowing 1982 - deno ed by ed X’s - line up ema kably well along a
semi-log money demand cu e wi h a semi-elas ici y o only 1.8.
The main di e ence be ween ou analysis and ha in I eland is he measu e o money,
highligh ed in Figu e 7. The igu e shows he heo e ical cu e co esponding o he log-log
speci ica ion wi h an elas ici y o 0.3, which ma ches he beha io o M1 om 1915 ill 1981.
We also show I eland’s measu e in ed X’s and NewM1 in g een squa es. While a b eak in
he slope is clea using I eland’s measu e, his is no he case when we use NewM1.
As desc ibed ea lie , NewM1 adds o he s anda d M1 measu e he Money Ma ke De-
mand Accoun s (MMDA), which we e c ea ed in 1982 and, in a couple o yea s, became
14
Figu e 7: US money-income a io and sho - e m in e es a e, 1915-2019
a ound 10% o o al ou pu . The jus i ica ion o doing so, as a gued in Lucas and Nicolini,
is ha he MMDA p o ided ansac ion se ices ha we e e y simila o he ones pe o med
by checking accoun s. In addi ion, hey paid in e es , which explains why hey g ew so much
a he expense o s anda d checking accoun s, which we e banned om paying in e es by
Regula ion Q.6
A new de elopmen occu ed by he ea ly 90s, when one bank adop ed a so wa e ha
au oma ically ans e ed unds om checking accoun s o MMDAs o he same clien in he
same bank a ew minu es be o e closing ime, and would ans e hem back o he checking
accoun a ew minu es a e opening he ollowing day. The p o i abili y o hese ”sweeps”,
as hey we e called, is explained by he ac ha he MMDA ese e equi emen was only
1%, while i was 10% o he checking accoun s. As ese e equi emen s a e compu ed o e
end o day balances, he bank could make subs an ial sa ings on ese es. Thus, he sweeps
we e jus a way o a oid he ese e equi emen .
Conce ned abou he implica ions o his p ac ice, o he inancial ins i u ions aised he
issued o he au ho i ies, who did no eac . Gi en he passi e esponse, he p ac ice ex-
ended o many o he banks in a ma e o a ew yea s. In a nu shell, he sweeps we e a way
o bypass he ese e equi emen wi hou ac ually changing he egula ion. Fo a de ailed
6A easonable in e p e a ion is ha he c ea ion o hese new accoun s was a way o lessen he bi e o
Regula ion Q wi hou epealing i , which would ha e equi ed cong essional suppo .
15
accoun o his p oblem, see Cynamon e al.(2006).
The sweep p og ams comple ely blu ed he dis inc ion be ween demand deposi s and
a ac ion o he s ock o MMDAs as epo ed by he Fede al Rese e, since ha ac ion
o he MMDAs we e, om he poin o iew o he holde s, jus demand deposi s. Thus,
he decision o I eland (2009) is o ally jus i ied: om he poin o iew o he holde , he
amoun s in he sweep p og ams ha e he same ansac ional cha ac e is ics as he checking
accoun s. I eland’s choice is ee om con o e sy: any a emp o measu e pu chasing
powe mus include he sweeps.
The a gumen in Lucas and Nicolini (2015) is ha I eland’s adjus men is no enough.
They make he poin ha all he MMDAs, no only he ones a i icially c ea ed by he sweep
p og ams, a e close enough subs i u es o he demand deposi s ha he igh p ac ice is o
include hem in M1.
I eland in e p e ed he egula o y changes o he 80s as pe manen changes in he elas-
ici y o eal money demand. Thus, in o de o e alua e he wel a e cos o in la ion in
he 21s cen u y, only da a om 1980 onwa d should be used - which explains he i le o
his pape .7Lucas and Nicolini (2015), on he con a y, a gue ha he egula o y changes
changed only he composi ion o ansac ional asse s on he supply side, by c ea ing a new
asse ha closely compe ed wi h demand deposi s. Bu , acco ding o Lucas and Nicolini
(2015)’s heo y, his change had minimal impac on he elas ici y o money demand.8
To es ima e he model, we adop he p oposal in Lucas and Nicolini (2015). Tha is
ce ainly a deba able choice. Hoping ha c oss-coun y compa isons help shed ligh on he
issue, we now b ie ly conside he expe ience o coun ies simila o he Uni ed S a es, o
which we use he measu e o M1 epo ed by hei cen al banks o bo h be o e and a e
1982. Fo space easons, we es ic his in o mal discussion o h ee addi ional coun ies.
7The da a o he o al s ock o MMDAs was no eadily a ailable when I eland w o e his pape - hough
he da a on sweeps had been collec ed by Cynamon e al. (2006). The measu e es ima ed by Lucas and
Nicolini (2015) was ob ained om he call epo s.
8The new deposi s could - and did - pay in e es ha depends on he sho - e m in e es a e on go e n-
men deb . And he elas ici y o eal money demand does change when some deposi s pay in e es . Howe e ,
he e ec is quan i a i ely e y small, so we igno e i in his discussion.
16
Ou main in e es in explo ing hese o he coun ies is o see he ex en o which hey
shed ligh on bo h he choice o unc ional o m and he alue o he pa ame e s. Thus, in
he cases ha ollow, we show he da a oge he wi h he log-log heo e ical cu e wi h an
elas ici y o 0.5 - he one p e e ed by Lucas - and wo heo e ical cu es o he semi-log,
one wi h a semi-elas ici y o 7 and one wi h a semi-elas ici y o 1.8, as p e e ed by Lucas
and I eland, espec i ely.
Figu e 8: Canada eal money demand, 1947-2019
In Figu e 8, we show annual da a o Canada om 1947 un il 2019. Da a un il 1981 a e
shown as blue ci cles, da a om 1982 o 2008 a e shown as ed X’s, and da a om 2009
o 2019 a e ep esen ed wi h g een squa es. The i s ea u e we would like o highligh is
ha he Canadian case di e s om he US one in ha he e is no appa en b eak in he
se ies in he 1980s, e en hough we use o he whole pe iod M1 as de ined by he Bank o
Canada. The second ea u e is ha he semi-log cu e wi h he pa ame e equal o 1.8 does
qui e badly ela i e o he one wi h he pa ame e equal o 7. Finally, he log-log does much
be e han he semi-log a acking he alues wi h e y low a es and high money o ou pu
a ios. Inciden ally, no e ha as in he US, he da a wi h low in e es a es ha ollowed
he inancial c isis o 2009 ( he g een squa es) beha e qui e in he same ashion as he ones
igh a e WWII ( he blue ci cles wi h low in e es a es). As in he US, he pos wa yea s
do no appea o be special in Canada.
17
Figu e 9: UK eal money demand, 1922-2016
In Figu e 9, we show simila e idence o he Uni ed Kingdom. The da a om 1922 ill
1981 a e depic ed wi h blue ci cles, he da a om 1982 ill 2008 wi h ed X’s, and he da a
om 2009 ill 2016 wi h g een squa es. As in he case o Canada, no b eak is appa en in
1981. Howe e , he UK case di e s om he o he ones in ha he semi-log appea s o do a
be e job han he log-log, as long as he semi-elas ici y is chosen o be 7. The one wi h a
semi-elas ici y o 1.8 does e y badly. The UK case is also di e en om ha o bo h he US
and Canada in ha , as conjec u ed by I eland, he wa yea s ( he blue ci cles wi h money
o ou pu a ios abo e 0.45) a e e y di e en om he yea s a e 2008 ( he g een squa es).
I is p ecisely i one igno es hose obse a ions ha he semi-log cu e wi h elas ici y o 7
pe o ms pa icula ly well.
Figu e 10: Aus alia eal money demand, 1960-2019
18
Finally, in Figu e 10, we plo he da a o Aus alia s a ing in 1960. As was he case
be o e, blue ci cles co espond o p e-1981 da a, and g een squa es o he 1982-2019 pe iod.
Aus alia did no ha e in e es a es e y close o ze o a e 2009, so we do no di e en ia e
ha sub-pe iod om he o he s. As in Canada and he UK, no b eak in beha io is appa en
in 1981. The semi-log wi h an elas ici y o 1.8 does no ma ch he da a well. Finally, in
o de o ack he yea s o lowe a es, he log-log speci ica ion pe o ms be e , al hough a
slope highe han 0.5 would p obably ma ch he da a be e .
In summa y, once he mone a y agg ega e o he US is adjus ed o ake in o accoun
new liquid deposi s c ea ed in 1982, he e is no appa en e idence o a change in beha io
o money demand. In he o he h ee coun ies, no e idence o a b eak is isible in he da a
using he s anda d M1 measu e.
We ind ambiguous e idence ega ding he supe io i y o he log-log e sus he semi-log
speci ica ions. The log-log clea ly appea s as he be e speci ica ion o Aus alia and he
semi-log o he UK. Fo Canada and he US, bo h speci ica ions do a easonable job a
ma ching he da a, excep o obse a ions wi h in e es a es e y close o ze o, whe e he
log-log pe o ms be e . The WWII yea s appea o be e y special in he UK, bu no in
he US and Canada. When he log-log speci ica ion is p e e ed, he elas ici y ha bes
ep esen s he da a appea s o be smalle han 0.5, he alue p e e ed by Lucas. Finally, o
he semi-log speci ica ion, he cu e wi h a coe icien equal o 7 acks he da a e y well,
while he cu e wi h a coe icien o 1.8 does no .
So a , we ha e ocused on he wo unc ional o ms discussed by Lucas and I eland. In
wha ollows, we b ie ly discuss he pe o mance o he unc ional o m o iginally p oposed
and s udied by Selden (1956) and La an´e (1960), desc ibed in (4). As we show below, his
unc ional o m pe o ms qui e well in he econome ic es s discussed below.
In Figu e 11, we show h ee cu es co esponding o he SL case: one wi h coe icien b
equal o 25, one equal o 35, and one equal o 45. As was he case be o e, in each case, he
cons an is adjus ed so ha he cu e c osses he g and mean o he da a. I is appa en
19
Figu e 11: US Selden-La an´e eal money demand, 1915-2019
ha he cu e wi h coe icien b equal o 35 bes ma ches he da a.
Figu e 12: US eal money demand, 1915-2019
In Figu e 12, we plo he da a, oge he wi h he h ee p e e ed speci ica ions. Essen-
ially, Figu e 12 is he same as Figu e 3, wi h he p e e ed SL speci ica ion added.
The SL speci ica ion does sligh ly be e han he semi-log a e y low alues o in e es
a es, and i does be e han bo h he semi-log and he log-log o alues highe han, say,
2%. As we men ioned abo e, he log-log appea s o unde es ima e he money- o-ou pu a io
o in e media e alues o he in e es a e, and he semi-log appea s o o e es ima e hem.
An a ac i e ea u e o he SL speci ica ion is ha i is be ween he o he wo speci ica ions
in ha ange, so i p o ides a be e app oxima ion o he da a. Below, we o mally show
ha he SL has nice s a is ical p ope ies.
20
4 Es ima ion and Tes ing
We s udy and compa e he h ee unc ional o ms, (2), (3), and (4) using o mal econome ics.
Fo he analysis o his sec ion, we assume ha he lowe bound on in e es a es is ze o.
The me hodology o es ima ing he h ee al e na i e speci ica ions o he money demand
cu es closely ollows he analysis by Bena i e al. (2021). We i s es o uni oo s in he
se ies. The e idence is o e whelming, as shown in Table A1 in Appendix D, which epo s
esul s om Ellio e al.’s (1996) uni oo es s o ei he he le els o he loga i hms o M1
eloci y and he sho a e.
In sea ching o a coin eg a ion ela ionship be ween eloci y and he sho a e, we
i s ake he uni oo es s li e ally and use Johansen’s es s. A plausible al e na i e
in e p e a ion o he esul s in Table A.1 is ha he se ies a e local- o-uni y. So, we also
sea ch o coin eg a ion based on W igh ’s (2000) es , which is alid o bo h exac uni
oo s and oo s ha a e local- o-uni y.9Resul s o bo h he Johansen and he W igh es
a e p esen ed in Tables A.2 and A.3 in Appendix D.
4.1 Pa ame e es ima es
Nei he Johansen’s no W igh ’s es s di ec ly p o ide poin es ima es o he pa ame e s
o he eal money demand unc ion.10 We he e o e es ima e he money demand equa ions
using S ock and Wa son’s dynamic OLS p ocedu e, which deli e s poin es ima es o he
pa ame e s.
Table 1 shows he poin es ima es, as well as 90% con idence in e als, o he coe icien s
ϕ o he Selden-La an´e speci ica ion, γ o he semi-log, and η o he log-log.
The poin es ima e o he semi-elas ici y pa ame e o he US, 9.1,is much close o 7,
he alue adop ed by Lucas (2000), han o 1.8, he one es ima ed by I eland (2009). The
9All o he echnical de ails abou he implemen a ion o he es s a e iden ical o hose o Bena i (2020)
and Bena i e al. (2021), which he eade is e e ed o.
10Fo he Johansen es , he co esponding money demand equa ion is es ima ed in i s VECM o m, om
which he money demand pa ame e s can be indi ec ly ob ained. W igh ’s es , on he o he hand, does no
p oduce poin es ima es, bu a he con idence in e als a he x% le el o he pa ame e s.
21
Table 1 Poin es ima e and 90%-co e age boo s appedacon idence in e al o
he coe icien on ( he loga i hm o ) he sho a e based on S ock and Wa son’s
(1993) es ima o
Money demand speci ica ion:
Coun y Pe iod Selden La an´e Semi-log Log-log
Uni ed S a es 1950Q1-2024Q2 -37.4 [-45.6 -25.8] -9.1 [-11.6 -5.9] -0.17 [-0.26 -0.11]
Uni ed Kingdom 1955Q1-2024Q2 -39.0 [-49.1 -23.7] -8.5 [-11.6 -6.0] -0.28 [-0.40 -0.18]
Canada 1947Q3-2006Q4 -40.4 [-50.6 -26.2] -8.0 [-10.3 -5.2] -0.38 [-0.47 -0.24]
1967Q1-2024Q2 -39.2 [-51.9 -27.0] -8.0 [-11.4 -5.1] -0.31 [-0.41 -0.21]
Aus alia 1969Q3-2024Q2 -61.0 [-74.1 -37.7] -11.4 [-14.1 -7.4] -0.43 [-0.53 -0.29]
New Zealand 1988Q2-2024Q2 -36.7 [-53.3 -16.1] -6.3 [-9.5 -3.0] -0.27 [-0.36 -0.17]
Sou h Ko ea 1964Q1-2024Q2 -44.0 [-47.7 -36.1] -7.4 [-8.8 -4.9] -0.48 [-0.56 -0.31]
Japan 1960Q1-2024Q2 -34.5 [-43.3 -20.4] -18.0 [-23.3 -12.0] -0.37 [-0.51 -0.19]
Hong Kong 1985Q1-2024Q2 -62.8 [-84.3 -41.7] -15.8 [-22.1 -9.9] -0.17 [-0.25 -0.11]
Swi ze land 1972Q1-2024Q2 -26.7 [-32.5 -20.2] -13.1 [-16.7 -9.4] –b
Sweden 1998Q1-2024Q2 -26.8 [-38.4 -16.6] -11.6 [-17.1 -6.9] –b
Eu o a ea 1999Q1-2024Q2 -32.1 [-41.6 -19.7] -14.6 [-19.8 -9.5] –b
Denma k 1991Q1-2024Q2 -15.3 [-20.4 -7.1] -6.6 [-9.6 -2.8] –b
aBased on 10,000 boo s ap eplica ions.
bThe las obse a ions o he sho a e a e ei he ze o o nega i e.
lowe bound o he 90% con idence in e al o ou es ima e is 5.9, subs an ially highe han
he p e e ed alue o I eland.
An inspec ion o he esul s o he o he coun ies shows ha alues close o he es ima e
o he US a e qui e common. In 6 cases, he poin es ima e is mo e han 10, and only o
New Zealand and Denma k i is less han 7 - hough ba ely. The alue I eland ob ains o
he US, 1.8, is subs an ially below he lowes bound o he 90 pe cen con idence in e al o
all he coun ies.
4.2 Which speci ica ion i s he da a be e ?
I is no possible o nes he h ee speci ica ions in o a single encompassing one. Howe e ,
we ound a way o nes he semi-log wi h he log-log on one hand, and he semi-log wi h he
Selden-La an´e on he o he .
We s a om he compa ison be ween he semi-log and he log-log. Fo each coun y,
22
we eg ess ln (M /Y ) on a cons an , plags o i sel , and plags o ei he he le el o he
sho a e o i s loga i hm. A na u al way o in e p e ing hese eg essions is he ollowing.
Unde he assump ion ha coin eg a ion is indeed he e o all coun ies,11 and based on
ei he speci ica ion, bo h YSL
= [ln (M /Y )R ]′and YLL
= [ln (M /Y ) ln (R )]′ha e a
coin eg a ed VECM(p-1) ep esen a ion, which maps in o a es ic ed VAR(p) ep esen a ion
in le els (whe e he es ic ions o igina e om he coin eg a ion ela ionship). The equa ions
we a e es ima ing can he e o e be hough o as he co esponding un es ic ed o m o he
equa ions o ln (M /Y ) in he VAR(p) ep esen a ion in le els o ei he YSL
o YLL
. I
is impo an o s ess ha he wo speci ica ions we a e es ima ing a e in ac nes ed. The
easies way o seeing his is o hink o hem as wo pola cases—co esponding o ei he
θ= 1 o θ= 0—in he ollowing ep esen a ion based on he Box-Cox ans o ma ion o R :
ln M
Y =α+
p
X
j=1
βjln M −j
Y −j+
p
X
j=1
δj Rθ
−j−1
θ!+ε .(8)
We es ima e (8) ia maximum likelihood, s ochas ically mapping he likelihood su ace ia
Random-Walk Me opolis (RWM). The only di e ence be ween he “s anda d” RWM algo-
i hm, which is ou inely used o Bayesian es ima ion, and wha we a e doing he e is ha
he jump o he new posi ion in he Ma ko chain is accep ed o ejec ed acco ding o a
ule ha does no in ol e any Bayesian p io s, as i uniquely in ol es he likelihood o he
da a.12 So one way o hinking o his is as Bayesian es ima ion ia RWM wi h comple ely
unin o ma i e p io s, so ha he log-pos e io collapses o he log-likelihood o he da a. All
11I his assump ion did no hold, he en i e model compa ison exe cise would ob iously be meaningless.
12So, o be clea , he p oposal d aw o he pa ame e ec o β,˜
β, is accep ed wi h p obabili y min[1,
(βs−1,˜
β|Y,X)] and ejec ed o he wise, whe e βs−1is he cu en posi ion in he Ma ko chain and
(βs−1,˜
β|Y, X) = L(˜
β|Y, X)
L(βs−1|Y, X),
which uniquely in ol es he likelihood. Wi h Bayesian p io s, i would be
(βs−1,˜
β|Y, X) = L(˜
β|Y, X)P(˜
β)
L(βs−1|Y, X)P(βs−1),
whe e P(·) would encode he p io s abou β.
23
Figu e 15: Es ima ed wel a e cos unc ions based on he Selden-La an´e and log-log speci i-
ca ion: poin es ima es o he lowe and uppe bounds, 5 h and 16 h pe cen iles o he lowe
bounds, and 84 h and 95 h pe cen iles o he uppe bounds o he boo s apped dis ibu ions
The eason o he disc epancy is ha ou es ima e o he semi-elas ici y is subs an ially
la ge han he one used by I eland (2009).
In he op panel o Figu e 16, we show equi alen esul s o he UK, Japan, Hong Kong,
and Aus alia. In his case, equi alen es ima es ange om 0.45% (UK and Hong Kong) o
0.80% o consump ion (Japan), subs an ially la ge han he ones we ob ained o he i s
g oup.
The bo om panel o Figu e 15 shows he esul s o he log-log case. The igu e highligh s
he heo e ical poin made by Lucas (2000): as he log-log speci ica ion implies ha he
wel a e cos is a con ex unc ion o he in e es a e, i implies subs an ially highe cos s a
e y low in e es a es. This is clea ly he case o Canada, New Zealand and Sou h Ko ea,
whe e he ange o wel a e cos o a 5% in e es a e goes om 0.3% o 0.4%, 0.2% o 0.3%,
30
and 0.35% o 0.85% pe cen , espec i ely.
Figu e 16: Es ima ed wel a e cos unc ions based on he Selden-La an´e and log-log speci i-
ca ion: Poin es ima es o he lowe and uppe bounds, 5 h and 16 h pe cen iles o he lowe
bounds, and 84 h and 95 h pe cen iles o he uppe bounds o he boo s apped dis ibu ions
Howe e , he poin a which he cu es c oss each o he - and he e o e a which he log-
log deli e s lowe wel a e cos s o highe in e es a es - depends on he es ima ed slope and
le el pa ame e s, and hose di e ac oss coun ies signi ican ly. Fo he US, he es ima ed
wel a e cos o a 5% a e o he log-log is lowe han o he Selden-La an´e. Wha explains
his ac is ha ou poin es ima e o he elas ici y (0.17) in he Uni ed S a es is much
smalle han he one used by Lucas (0.5). In ac , he US is he coun y o which he
es ima ed elas ici y is he lowes . In ac , he es ima ed elas ici y is also subs an ially lowe
han 0.3, he alue we chose when discussing he e idence in Figu e 2.
The bo om panel o Figu e 16 shows co esponding esul s o Aus alia, Japan, Hong
Kong, and he UK. The log-log implies highe cos s o Aus alia, Japan, and he UK, while
he case o Hong Kong is like ha o he US in Figu e 15. As i happens, Hong Kong and
31
he US a e he wo coun ies wi h he lowes poin es ima e o he elas ici y in he log-log
case (which in bo h cases is equal o 0.17).
6 Allowing o Nega i e Sho -Te m In e es Ra es
So a , we ha e ollowed he li e a u e in assuming ha as he nominal e u n on bonds goes
o ze o, so does he nominal e u n on money. Unde his condi ion, hen, he lowe bound
on b
is ze o. The ecen expe ience o p olonged nega i e sho - e m in e es a es in se e al
coun ies challenges his no ion. As he oppo uni y cos o money mus be non-nega i e,
he in e es a e on bonds can be nega i e only i he own e u n on money is nega i e, a
leas when b
becomes small.
The ele an oppo uni y cos o he ep esen a i e agen is he di e ence b
− m
.
Ou model does no explici ly model banks, bu i s equilib ium can be decen alized wi h a
compe i i e banking sec o in which nega i e a es a e passed o deposi o s.19 An al e na i e
model, in which banks ha e monopoly powe , may ha e banks ha do no pass he nega i e
a e o hei households, and collec income h ough highe ees.
Did he nega i e policy a es ansla e in o nega i e a es o deposi o s in hese expe i-
ences? The e is e idence ha small deposi s did no pay nega i e a es, e en in Swi ze land,
whe e in e es a es we e he lowes . Bu he e is also e idence ha o la ge deposi s - a ec -
ing mos ly i ms - he nominal e u n was nega i e.20 The e is also e idence o he e ogenei y
among cus ome s and banks. Fo ins ance, Michaelis (2022) shows ha by ea ly 2018, while
40% pe cen o Ge man banks we e paying nega i e a es on a e age on o e nigh deposi s
o non- inancial co po a ions, only 10% we e doing so o households. Howe e , by 2022,
close o he end o he nega i e policy a e pe iod, app oxima ely hal o he banks we e
paying nega i e a es o bo h co po a ions and households. Michaelis also shows ha ee
income subs an ially inc eased du ing his pe iod.
19See, o example, P esco (1987).
20See h ps://www. eu e s.com/business/ inance/c edi -suisse-g oup-ending-nega i e-in e es - a es-
p i a e-clien s-2022-06-29.
32
A one ex eme, one could assume ha he nega i e policy a es we e jus a ax on banks
and i ele an o deposi o s. I his we e he case, he esul s o he p e ious sec ion would
be he alid ones. The pu pose o his sec ion is o illus a e he obus ness o hose esul s
o al e na i e assump ions.
To accoun o nega i e policy a es, we p oceed as ollows. As we iden i y ou measu e
o money wi h M1 in he da a, i is na u al o hink o he e u n on money as an a e age
o he e u n o he wo componen s o M1, cash and demand deposi s. Fo cash, a nega i e
e u n can be a ionalized by he isk o being los o s olen, as Al a ez and Lippi (2009)
measu e using su ey da a.21 Fo deposi s, we use a linea ela ion be ween hei nominal
e u n and he in e es a e on bonds. Ku la (2019) p o ides e y s ong empi ical suppo
o such a ela ionship. These assump ions, aken oge he , a e consis en wi h he e u n
on money sa is ying
m
=−a+b b
, (10)
o a≥0 and b < 1.22 This linea ela ionship implies ha m
will be nega i e o small
enough alues o b
,and i implies ha b
≥ −a/(1 −b).
Thus, o a > 0, he lowe bound on he sho - e m a e is nega i e. The s anda d
assump ion in he li e a u e is ob ained by imposing ha a=b= 0. Ku la (2019) es ima es
b o be close o 0.15, e y p ecisely using mic o-da a om he US. We adop ha alue. We
hen le a= 1, which co esponds o a lowe bound on he sho - e m in e es a e o oughly
−1.2% pe cen .
This can accoun o he obse a ions on sho - e m a es in Denma k, he eu o a ea,
and Sweden. I canno accoun o Swi ze land, o which he lowes alue o he sho - e m
in e es a e was a ound −1.8%, so we do no discuss he log-log case o ha coun y.23
21Al a ez and Lippi (2009) calib a e his e u n a -0.02, using su ey da a om I aly.
22Fu he de ails a e p o ided in he Online Appendix F.
23Swi ze land is special no only because i s low sho a e, bu also in ha i expo s banking se ices.
I is likely ha i s measu e o M1 and i s abili y o implemen nega i e a es in deposi s may e y well be
coun y-speci ic.
33
In addi ion o being based on empi ical e idence, he linea ela ionship has he ad an age
ha he ele an oppo uni y cos becomes
= b
− m
=a+ (1 −b) b
,
which is a linea ans o ma ion o he obse able sho - e m in e es a e b
. As he las
wo unc ional o ms we adop ed o he money demand, equa ions (3) and (4) ,a e ei he a
linea unc ion o o he in e se o a linea unc ion o ,one needs only o es ima e hose
wo speci ica ions unde he benchma k case o a=b= 0, hen adjus he es ima es by he
co esponding linea ans o ma ion. Then, we use hose es ima es o compu e he wel a e
cos .
Howe e , o he log-log speci ica ion, his is no he case, and bo h he coin eg a ion
es s and he es ima es will depend on he speci ic assump ion ega ding he lowe bound.
As i u ns ou , bo h a e qui e sensi i e o he assumed lowe bound, pa icula ly so o he
case o he Uni ed S a es.
We discuss he e ec s o he assumed lowe bound on he coin eg a ion es s and he
compa ison be ween he log-log and he semi-log o Appendix G. In a nu shell, all coin eg a-
ion es s uni o mly imp o e o he log-log speci ica ion when he lowe bound is educed.
In es ing be ween he log-log and he semi-log, he pe o mance o he log-log also imp o es
uni o mly, bu only o Canada does he esul e e se so ha he log-log ou pe o ms he
semi-log. Finally, o he h ee coun ies wi h nega i e a es, he semi-log ou pe o ms he
log-log.
6.1 Es ima ion esul s and wel a e compu a ions
Table 3 p esen s he es ima ion esul s o he log-log case, unde he wo assump ions
ega ding he ze o bound. We i s show he esul s o he h ee cases in which in e es
a es isi ed nega i e e i o y and hen he es o he coun ies. Fo all hese coun ies,
34
Table 3 Poin es ima e and 90%-co e age boo s appedacon idence in e al o
he coe icien on he loga i hm o he sho a e based on S ock and Wa son’s
(1993) es ima o
Coun y Pe iod a=0, b=0 a=-1, b=0.15
Sweden 1998Q1-2019Q4 –b0.250 [0.212 0.291]
Eu o a ea 1999Q1-2019Q4 –b0.398 [0.341 0.465]
Denma k 1991Q1-2019Q4 –b0.298 [0.183 0.396]
Uni ed S a es 1959Q1-2019Q4 0.165 [0.087 0.235] 0.406 [0.255 0.531]
Uni ed Kingdom 1955Q1-2019Q4 0.284 [0.155 0.404] 0.468 [0.259 0.630]
Canada 1947Q3-2006Q4 0.373 [0.236 0.468] 0.544 [0.357 0.676]
1967Q1-2019Q4 0.305 [0.200 0.382] 0.467 [0.295 0.561]
Aus alia 1969Q3-2019Q4 0.749 [0.518 0.892] 0.916 [0.640 1.083]
Sou h Ko ea 1964Q1-2019Q4 0.477 [0.401 0.539] 0.655 [0.565 0.722]
Japan 1960Q1-2019Q4 0.328 [0.172 0.440] 0.646 [0.281 0.917]
Hong Kong 1985Q1-2019Q4 0.171 [0.096 0.241] 0.587 [0.363 0.824]
aBased on 10,000 boo s ap eplica ions. bThe las obse a ions o he in e es a e a e ei he ze o o
nega i e.
he poin es ima es o he in e es a e elas ici y inc ease subs an ially as he lowe bound
is educed.
Fo easons o space, we only epo he wel a e compu a ions o he i s h ee cases
and o he US. Figu e 17 p esen s he wel a e cos s o Denma k, he eu o a ea, Sweden and
Swi ze land o he Selden-La an´e unc ional o m. Fo he case o a ze o lowe bound, he
wel a e cos s a e somewha highe han o he US: be ween 0.4 and 0.6 pe cen age poin s
o consump ion. The ange inc eases o 0.5% o 0.8% when he lowe bound is assumed o
be -1.2%.
In Figu e 20, we epo es ima es o he log-log speci ica ion when we assume a lowe
bound equal o -1.2%. In his case, we ob ain subs an ially highe numbe s: close o 0.6%
o Denma k and Sweden, a ound 0.8% o he eu o a ea, and close o 1% o he Uni ed
S a es.
35
Figu e 17: Es ima ed wel a e cos unc ions based on he Selden-La an´e speci ica ion: Poin
es ima es o he lowe and uppe bounds, 5 h and 16 h pe cen iles o he lowe bounds, and
84 h and 95 h pe cen iles o he uppe bounds o he boo s apped dis ibu ions
Figu e 18: Es ima ed wel a e cos unc ions based on he log-log speci ica ion, a=−1 and
b= 0.15: Poin es ima es o he lowe and uppe bounds, 5 h and 16 h pe cen iles o he
lowe bounds, and 84 h and 95 h pe cen iles o he uppe bounds
36
7 Conclusion
How la ge is he cos o de ia ion om he F iedman ule i he nominal in e es a e is se a
5% in he s eady s a e? A well es ablished adi ion, s a ed by Bailey (1956) and F iedman
(1969), es ima es hose cos s by compu ing he a ea unde he eal money demand cu e.
Lucas (2000) ollows his adi ion and, a guing ha a log-log speci ica ion is a good i o
he US da a du ing he 20 h cen u y, compu es ha cos o be 1.2% o li e ime consump ion.
Howe e , I eland (2009) a gued ha a speci ica ion wi h a sa ia ion poin a he lowe
bound p o ides a much be e i . A dis inc ea u e o he ini e sa ia ion poin when he
oppo uni y cos o money is ze o implies ha he in eg al unde he eal money demand is
no as la ge as wi h he log-log. He also a gues ha he elas ici y is much lowe han he
one used by Lucas. When bo h hings a e conside ed, I eland es ima es he wel a e cos o
be a me e 0.036% o consump ion.
We use new da a o he US and also s udy he beha io o eal money demand o se e al
o he de eloped coun ies. Ou analysis qui e s ongly suppo s Lucas’s es ima es. When
using he ull suppo , a unc ional o m wi h a ini e sa ia ion poin pe o ms be e wi h
he da a. Howe e , he analysis o e y low alues o he oppo uni y cos wo ks be e wi h
he log-log speci ica ion. Finally, ou sample con ains coun ies ha expe ienced nega i e
policy a es, sugges ing he possibili y o a nega i e lowe bound on he oppo uni y cos o
money.
These conside a ions p o ide he wo mos ex eme scena ios. Ou lowes se o es ima es
is ob ained wi h a ini e sa ia ion poin , co esponding o he Selden-La an´e speci ica ion,
and assuming he lowe bound is ze o. This case deli e s a wel a e cos o a 5% nominal
in e es a e o abou 0.35% pe cen o pe manen consump ion o he US. Fo he log-log
case and a nega i e lowe bound compa ible wi h he expe ience o he coun ies in ou
sample, he wel a e cos is abou 1% o pe manen consump ion.
37
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