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. 24-08
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 (2024) : The wel a e cos s o in la ion
econside ed, Discussion Pape s, No. 24-08, 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
24-08
Sep embe , 2024
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
We e isi he es ima ion o he wel a e cos s o in la ion o igina ing om
lack o liquidi y sa ia ion o 11 low-in la ion and 5 high-in la ion coun ies, and
o Weima Republic’s hype in la ion. Ou e idence sugges s ha , con a y o
he implici assump ion in much o he li e a u e, hese cos s a e a om
negligible. Fo he U.S. ou poin es ima es a e equal o abou one- hi d o
hose compu ed by Lucas (2000), and an o de o magni ude la ge han hose
ob ained by I eland (2009). C ucially, he mos empi ically plausible money-
demand unc ional o m poin s owa ds sizeable ‘upwa d isks’ o hese cos s,
wi h he 90% con idence in e al associa ed wi h a 4% nominal in e es a e
s e ching beyond 0.5 pe cen o GDP. The wel a e cos s o in la ion in he
Eu o a ea a e abou wice as la ge as in he U.S., hus sugges ing ha , ce e is
pa ibus, hein la ion a ge should be ma e ially lowe . A he peak o he
in la ion episodes, wel a e cos s had anged be ween 0.3 and 1.9 pe cen o
GDP o low-in la ion coun ies; be ween 4 and nea ly 7 pe cen o high-
in la ion ones; and be ween 26 and 36 pe cen o Weima ’s hype in la ion.
∗We wish o hank pa icipan s o he EEA 2024 mee ing in Ro e dam o use ul commen s and
discussions. 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
The wel a e cos s o in la ion o igina ing om lack o liquidi y sa ia ion–as discussed
in he classic wo k o Bailey (1956), F iedman (1969), Lucas (2000), and I eland
(2009)– end o be consis en ly dis ega ded in he cu en policy deba e. A case in
poin is he p oposal o inc ease in la ion a ge s in o de o dec ease he p obabili y
ha he Ze o Lowe Bound (ZLB) on mone a y policy a es may become binding.
Following Blancha d, Dell’A iccia and Mau o (2010), a as li e a u e has explo ed he
cos s o a binding ZLB on mone a y policy a es wi hin cashless economies. Al hough
his li e a u e has documen ed he bene i s ha inc easing he in la ion a ge deli e s
o socie y, such inc ease also comes a a cos , o igina ing om mo ing he economy
u he away om liquidi y sa ia ion.
A likely eason o he li e a u e’s dis ega d o hese cos s is he widesp ead belie
in he absence o s able money-demand ela ionships, which a e a necessa y condi-
ion o he compu a ion o such cos s. Following Gold eld (1973, 1976), who i s
documen ed he (alleged) ins abili y o he U.S. demand o M1, a la ge li e a u e
has con i med his indings, and i has p oduced quali a i ely he same e idence o
diffe en coun ies. The ecen wo k o Lucas and Nicolini (2015) o he U.S., and
Bena i, Lucas, Nicolini, and Webe (2021) o 38 coun ies since Wo ld Wa I, has
shown howe e ha he no ion o ins abili y o he long- un demand o M1 is, in ac ,
inco ec . On he con a y, he exis ence o a s able long- un demand o M1 appea s
o be one o he mos obus s ylized ac s in he en i e ield o mac oeconomics. As
discussed by Bena i e al. (2021), and as we discuss below in de ail, c ucial issues
in being able o iden i y a s able long- un demand o M1 a e ()imposinguni a y
income elas ici y, which is implied by heo y and i is in ac suppo ed by he da a;
and () adop ing he co ec unc ional o m o he demand o eal money balances.
A second eason o he li e a u e’s consis en dis ega d o he wel a e cos s o
in la ion o igina ing om lack o liquidi y sa ia ion is he (by now) long-s anding
adi ion o e alua ing mone a y policies wi hin cashless economies, wi h mone a y
agg ega es playing no ole wha soe e . As we show below, howe e , he e is no sense
in which mode n ad anced economies a e becoming cashless. Fo coun ies such as
he U.S., he U.K., Sweden, and Swi ze land, o o he Eu o a ea, he a io o M1 o
GDP has luc ua ed in ecen yea s be ween 40 and 100 pe cen . As we show, his
implies ha o empi ically plausible pa ame e iza ions o he demand o money
balances, he poin es ima es o he wel a e cos s o in la ion a e in gene al non-
negligible. Fu he , we show ha once aking in o accoun o s a is ical unce ain y,
in se e al cases (no ably, he U.S.) he mos plausible unc ional o m o he demand
o eal M1 balances poin s owa ds non-negligible ‘upwa d isks’ o he wel a e cos s
o in la ion, in he sense ha (e.g.) 90 pe cen con idence in e als o hese cos s
include alues ha a e de ini ely non-negligible, and in ac o en sizeable.
A inal likely eason o he li e a u e’s dis ega d o hese cos s is ha hey a e
o en hough o be negligible. Fo example, Felds ein (1997, p. 145), in his e iew o
2
he cos s o in la ion, cha ac e ized he wel a e cos s o igina ing om lack o liquidi y
sa ia ion as ‘small ela i e o he o he effec s ha ha e been discussed in his pape ’.
As we show in his wo k, his p esump ion is in gene al inco ec .
The numbe o s udies p esen ing empi ical es ima es o he wel a e cos s o in la-
ion o igina ing om lack o liquidi y sa ia ion is qui e su p isingly limi ed. Fu he ,
p e ious s udies a e sub-op imal along se e al dimensions. Fi s , se e al o hem ( i s
and o emos Lucas, 2000) a e based on calib a ed, as opposed o es ima ed models.
Second, p e ious es ima es a e uni o mly based on unc ional o ms o he demand
o eal money balances ha Bena i e al. (2021) ha e shown o be empi ically highly
implausible.1As we show, in gene al he money-demand speci ica ion does ma e o
he compu a ions o wel a e cos s along se e al dimensions. Thi d, o he bes o ou
knowledge he only p e ious analysis o he wel a e cos s o hype in la ions is Ba o
(1972). Las , bu no leas , he pe iod ollowing he ou b eak o he inancial c isis
has p o ided p e iously una ailable in o ma ion on he beha io o money demand a
e y low in e es a es, which as discussed by Lucas (2000) should play a c ucial ole
in he de e mina ion o he wel a e cos s o in la ion. In p inciple his should allow
us o ob ain be e and mo e p ecise es ima es o hese cos s.
In his pape we e isi he es ima ion o he wel a e cos s o in la ion o ele en
low-in la ion and i e high-in la ion coun ies, and o Weima Republic’s hype in la-
ion. In ou analysis we ollow he adi ion o conside ing he mos liquid mone a y
asse s, which include cash and ansac ional deposi s. We abs ac om a de ailed
discussion o he demand o each o he componen s, an issue ecen ly add essed by
Ku la (2019).2
Ou e idence sugges s ha , con a y o he explici o implici assump ion in
much o he li e a u e, hese cos s a e o en a om negligible. Fo he U.S. ou
poin es ima es a e equal o abou one- hi d o hose compu ed by Lucas (2000), and
an o de o magni ude la ge han hose ob ained by I eland (2009).3C ucially, he
mos empi ically plausible money-demand unc ional o m poin s owa ds sizeable
‘upwa d isks’ o hese cos s, wi h he 90% con idence in e al associa ed wi h a 4%
nominal in e es a e s e ching beyond hal a pe cen age poin o GDP. A he peak
o he in la ion episodes, wel a e cos s had anged be ween 0.3 and 1.9 pe cen o
GDP o low-in la ion coun ies; be ween 4 and nea ly 7 pe cen o high-in la ion
ones; and be ween 26 and 36 pe cen o Weima ’s hype in la ion.
Ou e idence sugges s ha igno ing money in analyzing op imal mone a y poli-
cies can be se iously misleading. Fo ins ance, Coibion, Go odnichenko and Wieland
1No able cases in poin a e Lucas (2000) and I eland (2009).
2He shows ha add essing hese conside a ions in a model wi h impe ec compe i ion subs an-
ially inc eases he es ima es o he wel a e cos , ela i e o models ha igno e he c ea ion o inside
money.
3Fo a s eady-s a e in e es a e o 5 pe cen , Lucas (2000) compu ed he cos o be a ound
1.1 pe cen o li e ime consump ion. I eland (2009) challenged Lucas’ in e p e a ion o he da a,
and es ima ed a me e 0.04 pe cen o li e ime consump ion.
3
(2012)4make a compelling a gumen agains inc easing he in la ion a ge in coun-
ies like he U.S., based on a model wi h ic ions in p ice-se ing and wi h ecu en ,
hough no e y equen , episodes wi h he nominal in e es a e a he ZLB. Based
on hei p e e ed speci ica ion hey compu e he wel a e effec o an in e es a e o
5 pe cen o be close o 0.6 pe cen o li e ime consump ion. Such an es ima e,
combining he cos c ea ed by p ice ic ions and he p obabili y o be a he ZLB,
is o he same o de o magni ude o he es ima es we ob ain o he U.S., and i is
in ac sligh ly smalle han he uppe bound o ou 90 pe cen con idence in e al.
In any o he low-in la ion coun ies he demand o M1 as a ac ion o GDP
a e y low, o e en nega i e in e es a es has (so a ) exhibi ed no ob ious diffe -
ence compa ed o i s beha io a highe in e es a es. These esul s con as wi h
hose o Mulligan and Sala-i-Ma in (2000), who, based on U.S. households mic o
da a, p o ided e idence ha money demand becomes compa a i ely la e a low
in e es a es. We p o ide a s aigh o wa d explana ion o Mulligan and Sala-i-
Ma in’s (2000) inding, by showing ma hema ically ha i he ue money demand
speci ica ion is he one ha , as shown by Bena i e al. (2021), is he mos empi i-
cally plausible a low in la ion a es, Mulligan and Sala-i-Ma in’s (2000) app oach
au oma ically p oduces spu ious e idence o a la e demand cu e a low in e es
a es.
F om a heo e ical s andpoin , ollowing Al a ez, Lippi and Roba o (2019) we
cons uc uppe and lowe bounds o he wel a e cos s o in la ion. As hey show, he
a eaunde hemoneydemandcu eisanalmos exac measu eo hewel a ecos
o a e y gene al class o mone a y models in he neighbo hood o ze o. We ex end
hei esul s o a qui e gene al sub-class o he models hey analyze and compu e
exac lowe and uppe bounds 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 diffe ence be ween he uppe
and he lowe bound is ex emely small o he ange o in e es a es e e obse ed
in low-in la ion coun ies such as he U.S.
Fo policy pu poses, ou main inding is ha he wel a e cos s in la ion in he
Eu o a ea a e abou wice as la ge as in he U.S.. This sugges s ha , ce e is pa ibus,
he in la ion a ge should be ma e ially lowe .
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 discuss
he da a and se e al igu es ha , in ou iew, p esen e y solid e idence in a o
o s able money demand ela ionships o he coun ies we analyze. Sec ion 4 makes
o mally his s a emen by analyzing uni oo and coin eg a ion p ope ies o he
se ies. Sec ion 5 p esen s ou compu a ions o he wel a e cos unc ions. Fo each
in e es a e le el, we compu e he boo s apped dis ibu ion o he wel a e cos s (and
he e o e median es ima es, and con idence in e als) exp essed in pe cen age poin s
4Coibion 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.
4
o GDP. Sec ion 6 concludes.
2TheModel
We s udy a labo -only economy wi h unce ain y in which making ansac ions is
cos ly.5The economy is inhabi ed by a uni mass o iden ical agen s wi h p e e ences
gi en by
0
∞
X
=0
()(1)
whe e is diffe 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
ha allow he o exchange in e es -bea ing illiquid asse s o money, ha is needed
o buy he consump ion good. The o al cos o hose ansac ions, measu ed in uni s
o imes, is gi en by a unc ion ()whe e 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 is gi en by
==
whe e is ime de o ed o he p oduc ion o he inal consump ion good and is an
exogenous s ochas ic p ocess.
The ep esen a i e agen is endowed, in each pe iod, wi h a uni o ime ha is
used o p oduce goods and o make ansac ions. Thus, equilib ium in he labo
ma ke implies ha
1=+()
and easibili y is gi en by
=(1 −())
I ollows ha he eal wage is equal o .
Pu chases a e subjec o a cash in ad ance cons ain
≤(2)
whe e a e a e age money balances and is he numbe o po olio adjus men s
wi hin each pe iod. The a iable is he only economically ele an decision o be
made by he ep esen a i e agen . We allow o money o pay a nominal e u n ha
we denomina e
which in wha ollows, in line wi h he li e a u e, we will se o
ze o.
5The baseline model is discussed a leng h in Bena i e . al. (2020).
5
A he beginning o each pe iod, he agen s a s wi h nominal weal h ha
can be alloca ed o money o in e es bea ing bonds, soa es ic ion o heop imal
p oblem o he agen is
+≤(3)
Nominal weal h a he beginning o nex pe iod, in s a e +1will hen be gi en by
+1 ≤(1 +
)+(1 +
)+(4)
+[1−()] −
whe e
is he e u n on go e nmen bonds and is a ans e made by he mone a y
au ho i y.
No ice ha he uncons ained efficien ou come is o alloca e all he labo inpu
o he p oduc ion o he consump ion good so as o se = Thus, a measu e o
he wel a e cos o making ansac ions, as a ac ion o consump ion, is gi en by he
alue o ()in equilib ium.
In he Online Appendix 1, we show ha as long as he cos unc ion ()is
diffe en iable, an in e io solu ion o mus sa is y
2
()
(1 −()) =
−
(5)
We also show ha as long as
−
0 he cash in ad ance is binding, which implies
ha
=1
(6)
so eal money demand, as a p opo ion o ou pu , is equal o he in e se o No e
ha equa ion (5) is independen o . Thus, secula inc eases in p oduc i i y do no
affec he op imal solu ion o so he heo y implies a uni income elas ici y o
eal money demand.
No e ha he solu ion o and he e o e he solu ion o eal money demand,
depends on he in e es a e diffe en ial be ween bonds and money. As men ioned
abo e, since we assume ha
=0, eal money demand only depends on he in e es
a e on bonds. Fo u he e e ences, we le he in e es a e diffe en ial be ween
bonds and money o be ≡
−
Fo he maximum p oblem o he agen o be well de ined, i has o be he case
ha
=
−
≥0(7a)
which is he well-known lowe bound on he in e es a es in bonds.6The popu-
la ze o-bound es ic ion on policy a es is ob ained om (7)plus he s anda d
assump ion in he li e a u e ha
=0The analyses o bo h Lucas (2000) and
I eland (2009) a e done unde his s anda d assump ion.
6In ui i ely, whe e ()−() o be nega i e, he ep esen a i e agen would ha e incen i es
o bo ow om he go e nmen unbounded quan i ies and hold money.
6
2.1 The unc ional o m o he demand o eal money bal-
ances
The unc ional o m o he eal money demand unc ion depends on he unc ional
o m o he ansac ions echnology (), and a his le el o gene ali y he model
is consis en wi h many diffe en possibili ies. In wha ollows, and o cla i y he
main diffe ence be ween Lucas (2000) and I eland (2009), we conside h ee well-
known unc ional o ms ha ha ebeenusedinp e iousempi icalwo k. Allo he
h ee unc ional o ms exhibi a uni income elas ici y, as implied by he model. The
i s speci ica ion is he log-log one,
ln
=1−ln +1
(8)
ha exhibi s a cons an in e es a e elas ici y equal o .No ice ha as→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 sizeable, e en a low alues o he in e es a e.
The o he wo o mula ions ha we explo e, he semi-log
ln
=2−+2
(9)
ha exhibi s a cons an semi-elas ici y, , and he Selden-La ané
=1
3++3
(10)
bo h imply a ini e le el o he demand o eal money balances when he in e es
a e becomes ze o. This ea u e is emphasized by I eland, who uses (9) in his e ision
o Lucas’s es ima e. By exploi ing ecen da a ha include, o a ew coun ies,
se e al yea s o e y low (o e en nega i e) in e es a es, we can p o ide a sha pe
compa ison o he empi ical pe o mance o he h ee al e na i e unc ional o ms.
As we show below, he wel a e cos s implica ions o he las wo unc ional o ms
a e simila . We do howe e choose o include he Selden-La ané speci ica ion, oge he
wi h he o he s since i does ha e an o e all be e pe o mance han he o he wo,
as ou econome ic analysis shows.7
In he nex Sec ion we show how o build igh uppe and lowe bounds o he
wel a e cos o in la ion, using he a ea unde he es ima ed eal money demand
unc ion.
2.2 The wel a e cos s o in la ion and he a ea unde he
money demand cu e
In his sec ion we apply he echniques de eloped in Al a ez, Lippi and Roba o
(2019) o a class o models ha is mo e es ic i e han he ones hey used. Speci i-
7This is in line wi h he e idence in Bena i, Lucas, Nicolini, and Webe (2021).
7
Table 1 Resul s om W igh ’s es s: 90% boo s apped con idence in e -
al o he second elemen o he no malized coin eg a ion ec o , based on
sys ems o (log) M1 eloci y and ( he log o ) a sho - e m a e
Money demand speci ica ion:
Selden-
Coun y Pe iod La ané Semi-log Log-log
Low-in la ion coun ies:
Uni ed S a es 1959Q1-2019Q4 [-0.587 -0.343] [-0.148 -0.068] [-0.347 -0.067]
1959Q1-2023Q2 [-0.694 -0.438] [-0.175 -0.103] [-0.529 -0.181]
Uni ed Kingdom 1955Q1-2023Q2 [-0.564 -0.371] [-0.115 -0.079] [-0.393 -0.196]
Canada 1947Q3-2006Q4 [-0.713 -0.543] NCD [-0.512 -0.440]
1967Q1-2023Q1 [-0.634 -0.397] [-0.119 -0.035] [-0.409 -0.245]
Aus alia 1969Q3-2023Q1 [-0.858 -0.764] [-0.181 -0.045] [-1.275 -0.974]
Swi ze land 1972Q1-2023Q1 [-0.457 -0.336] [-0.233 -0.128] —
Sweden 1998Q1-2023Q1 [-0.614 -0.299] [-0.153 -0.121] —
Eu o a ea 1999Q1-2023Q1 [-0.586 -0.409] [-0.217 -0.165] —
Denma k 1991Q1-2023Q1 [-0.373 -0.187] [-0.139 -0.043] —
Sou h Ko ea 1964Q1-2023Q1 [-0.579 -0.513] [-0.148 0.027] NCD
Japan 1960Q1-2023Q2 [-0.452 -0.366] [-0.317 -0.033] [-0.610 -0.122]
Hong Kong 1985Q1-2023Q2 [-1.128 -0.769] [-0.257 -0.100] [-0.495 -0.091]
High-in la ion coun ies:
Boli ia 1980-2019 — — [-0.696 -0.260]
Chile 1946-2019 — — [-0.443 -0.095]
Ecuado 1980-2019 — — [-1.217 -1.021]
Is ael 1982Q1-2019Q4 — — [-0.428 -0.392]
Mexico 1982Q1-2019Q4 — — [-0.621 -0.325]
Hype in la ions:
Weima Republic Sep. 1920-Oc . 1923 — — [-0.526 -0.349]
Based on 10,000 boo s ap eplica ions. NCD = No coin eg a ion de ec ed.
The las obse a ions o he in e es a e a e ei he ze o o nega i e.
Bena i (2024) shows ha o 20 hype in la ions e idence in a o o he log-log is o e -
whelming. As o low-in la ion coun ies, he e idence in Bena i e al. (2021) sugges s
ha he Selden-La ané speci ica ion is he mos plausible one, bu he e idence is less
clea -cu . Fo his g oup o coun ies we he e o e now u n o a sys ema ic model
compa ison exe cise.
4.2 Which speci ica ion do he da a p e e ?
Since i is no possible o nes he h ee money demand speci ica ions in o a single
encompassing one, we p oceed as ollows. We s a om he compa ison be ween
he semi-log and he log-log. In ui i ely, he compa ison be ween (9)and(8)boils
down o whe he he dynamics o log M1 balances as a ac ion o GDP (i.e., minus
log eloci y) is be e explained by he le el o he sho a e, o by i s loga i hm.
Fo low-in la ion each coun y we he e o e eg ess ln ()on a cons an , lags
o i sel , and lags 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,15 and based on ei he speci ica ion, bo h
=[ln()]0and
=[ln()ln()]0ha e a coin eg a ed VECM(-
1) ep esen a ion, which maps in o a es ic ed VAR() 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 ()in he VAR() ep esen a ion in le els o ei he
o
. 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: he easies way o seeing his is o hink o hem as wo pola cases–
co esponding o ei he =1o =0–in he ollowing ep esen a ion based on he
Box-Cox ans o ma ion o :
ln µ
¶=+
X
=1
ln µ−
−¶+
X
=1
Ã
−−1
!+(14)
We es ima e (14) ia maximum likelihood, s ochas ically mapping he likelihood su -
ace ia Random-Walk Me opolis (RWM). The only diffe ence be ween he ‘s an-
da 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 based on a ule which does no in ol e any Bayesian p io s, as i uniquely
in ol es he likelihood o he da a.16 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 o he o he es ima ion de ails a e
iden ical o Bena i (2008), o which he eade is e e ed o.
15I his assump ion did no hold, he en i e model compa ison exe cise would ob iously be mean-
ingless.
16So, o be clea , he p oposal d aw o he pa ame e ec o ,˜
, is accep ed wi h p obabili y
min[1, (−1,˜
|,)], and ejec ed o he wise, whe e −1is he cu en posi ion in he Ma ko
12
Table 2aModel compa ison exe cise, semi-log e sus log-log: mode o he log-likelihood
in eg essions o log eloci y on lags o i sel and ei he he sho a e o i s loga i hm
p=2 p=4 p=8
Semi- Log Semi- Log- Semi- Log-
Coun y Pe iod log log log log log log
Uni ed S a es 1959Q1-2023Q2 766.1394 756.6280 763.2818 751.3266 765.1439 740.3543
Uni ed Kingdom 1955Q1-2023Q2 879.6821 877.9350 898.6224 893.7504 892.1970 887.1920
Canada 1947Q3-2006Q4 820.2401 807.8379 813.8001 804.9218 802.7001 794.7403
1967Q1-2023Q1 775.0890 767.0845 775.9595 766.4531 771.9264 766.1943
Aus alia 1969Q3-2023Q1 650.7331 656.0624 649.9510 655.1057 642.6046 650.3903
Sou h Ko ea 1964Q1-2023Q1 630.9515 633.8825 628.2222 634.6372 623.8333 628.0991
Japan 1960Q1-2023Q2 845.3632 850.7677 841.5156 848.6520 832.2577 840.2434
Hong Kong 1985Q1-2023Q2 328.0148 325.5701 326.1339 324.9236 319.8478 325.2641
Fo Swi ze land, Sweden, Eu o a ea, and Denma k he e is no compa ison because he las obse a ions o he
sho a e a e nega i e.
Table 2bModel compa ison exe cise, Selden-La ané e sus semi-log: mode o he log-like-
lihood in eg essions o he sho a e on lags o i sel and ei he eloci y o i s loga i hm
p=2 p=4 p=8
Selden- Semi- Selden- Semi- Selden- Semi-
Coun y Pe iod La ané log La ané log La ané log
Uni ed S a es 1959Q1-2023Q2 -22.9102 -24.0809 -5.8335 -7.3440 12.9347 10.3522
Uni ed Kingdom 1955Q1-2023Q2 -85.7350 -84.1422 -85.4391 -83.9044 -83.6446 -82.1970
Canada 1947Q3-2006Q4 -72.0532 -71.7812 -64.4770 -66.2576 -62.2194 -64.2760
1967Q1-2023Q1 -65.0057 -65.9778 -56.1253 -59.0260 -50.7916 -53.9112
Aus alia 1969Q3-2019Q4 -136.4591 -137.1389 -132.5116 -133.5144 -116.7487 -118.2407
Swi ze land 1972Q1-2023Q1 -45.6989 -45.8396 -39.9744 -40.8984 -20.5636 -22.4888
Sweden 1998Q1-2023Q1 65.5876 65.4372 66.9821 66.9083 70.5126 68.9721
Eu o a ea 1999Q1-2023Q1 63.8008 64.3157 64.5967 65.3372 74.7778 75.5777
Denma k 1991Q1-2023Q1 50.9544 50.7969 60.7088 60.1085 65.6600 64.4409
Sou h Ko ea 1964Q1-2023Q1 -131.5950 -135.8924 -118.2770 -131.1253 -86.1317 -93.5032
Japan 1960Q1-2023Q2 -141.5147 -141.6026 -140.6631 -140.7865 -129.5219 -130.1270
Hong Kong 1985Q1-2023Q2 -65.6601 -65.7389 -60.9537 -61.4880 -50.8999 -51.6665
Fo Swi ze land, Sweden, Eu o a ea, and Denma k he e is no compa ison because he las obse a ions
o he sho a e a e nega i e.
Table 2 epo s, o ei he speci ica ion, and o ∈{248} hemodeo he
log-likelihood. The main esul in he able is ha whe eas he semi-log appea s as he
p e e ed unc ional o m o he U.S. he U.K., Canada, and Hong Kong, he log-log
p oduces a la ge alue o he likelihood o Aus alia, Sou h Ko ea, and Japan, so
ha nei he o he wo speci ica ions clea ly domina es he o he one.17
Tu ning o hecompa isonbe ween hesemi-log and he Selden-La ané, we adop
he same logic as be o e, bu his ime we ‘ lip’ he speci ica ions o eloci y on
hei head, by eg essing he in e es a e on lags o i sel and o ei he he le el o
he loga i hm o eloci y. Once again, hese wo eg essions can be hough o as
pa icula cases o he nes ed eg ession
=+
X
=1
−+
X
=1
⎡
⎢
⎣³−
−´
−1
⎤
⎥
⎦+(15)
wi h ei he =1(co esponding o Selden-La ané) o =0(co esponding o he
semi-log).
A i s sigh his app oach migh appea as ques ionable: since we a e he e dealing
wi h he demand o eal M1 balances o a gi en le el o he sho - e m nominal
in e es a e, why would i make sense o eg ess he sho a e on M1 eloci y? In
ac , his app oach is pe ec ly legi ima e, o he ollowing eason. As shown by
Bena i (2020), M1 eloci y is, o a i s app oxima ion (and up o a scale ac o ), he
pe manen componen o he sho - e m a e,18 so ha ocusing (e.g.) on he Selden-
La ané speci ica ion, =+
,whe eis eloci y, 0a e coefficien s,
and
is he uni - oo componen o he sho a e (), wi h =
+
,and
being he ansi o y componen .19 This can be seen qui e clea ly in Figu e 2 o
Aus alia, Canada, he Eu o a ea, Hong Kong, Sweden, Swi ze land, and he U.K..
chain, and
(−1˜
|)= (˜
|)
(−1|)
which uniquely in ol es he likelihood. Wi h Bayesian p io s i would be
(−1˜
|)= (˜
|)(˜
)
(−1|)(−1)
whe e (·)would encodes he p io s abou .
17This c ucially hinges on he ac ha we a e he e exclusi ely ocusing on low-in la ion coun ies.
As we discuss in Sec ion ??, o high-in la ion coun ies, and especially hype in la iona y episodes,
he da a’s p e e ence o he log-log is o e whelming.
18This exp esses in he language o ime-se ies analysis Lucas’ (1988) poin ha eal M1 balances
a e e y smoo h compa ed o he sho a e.
19A simple a ionaliza ion o his ac is p o ided by a ‘p e e ed habi a ’ model (see Modigliani
and Su ch, 1966, and Vayanos and Vila, 2021) in which ‘long’ in es o s such as pension unds play
an impo an ole in money demand. The in ui ion is ha whe eas pe manen shocks o he sho
a e shi he en i e e m s uc u e o in e es a es, and he e o e affec he demand o M1 coming
13
Reg essing on he e o e amoun s o eg essing he sho a e on i s ( escaled)
s ochas ic end, i.e. he dominan d i e o i s long-ho izon a ia ion, and i is
he e o e concep ually akin o (e.g.) eg essing GDP on consump ion.20
The esul sa e epo edinTable2. The e idence is much sha pe han o he
compa ison be ween he semi-log and he log-log: in pa icula , o equal o ei he
4 o 8 he Selden-La ané speci ica ion is p e e ed o he semi log o all coun ies
excep he Uni ed Kingdom and he Eu o a ea.
Summing up, whe eas he Selden-La ané unc ional o m appea s o be qui e
clea ly p e e ed o he semi-log, he semi-log and he log-log seem o be, om an
empi ical s andpoin , on a oughly equal oo ing.
We d aw wo main conclusions om he e idence so a . Fi s , in line wi h he
e idence in Figu es 1 and 2, he da a p o ide subs an ial suppo o he exis ence
o a s able long- un demand o M1, as p edic ed by he heo y. Second, o low-
in la ion coun ies he Selden-La ané speci ica ion appea s o exhibi he bes o e all
pe o mance among he h ee.
Based on his, o low-in la ion coun ies we choose o use he Selden-La ane as
ou benchma k unc ional o m. Bu we will also p o ide es ima es o he o he wo
speci ica ions.
4.3 Explo ing s abili y and non-linea i ies
A main conce n in wo king wi h es ima ed money demand cu es pe ains o he s a-
bili y o he long- un ela ionship o e ime. As p e iously men ioned, e en wi hou
he econome ic e idence p oduced (e.g.) by F iedman and Ku ne (1992), he sim-
ple isual e idence had been sufficien o disc edi , long ago, any no ion o s abili y
o he U.S. demand o eal M1 balances. As ou esul s make clea , he solu ion p o-
posed by Lucas and Nicolini (2015) has e-es ablished s abili y o he U.S. demand
o M1. Howe e , since o all o he o he coun ies in ou da ase we wo k wi h
he ‘s anda d’ M1 agg ega e, i is a legi ima e ques ion whe he o (some o ) hese
coun ies, oo, some adjus men o he s anda d agg ega e migh be equi ed in o de
o ob ain s abili y o he long- un demand o M1.
4.3.1 Tes ing o s abili y in he coin eg a ion ec o
Table 3 epo s e idence om Hansen and Johansen’s (1999) es s o s abili y in
he coin eg a ion ec o 21 o ou da ase , based on any o he h ee money demand
om all in es o s, ansi o y shocks only impac he sho end o he yield cu e, and he e o e ha e
a much smalle (and in he limi negligible) effec .
20See Coch ane (1994) on consump ion being he pe manen componen o GDP.
21On he o he hand, we do no es o s abili y o he loading coefficien s, since hey pe ain o
he sho - e m adjus men dynamics o he sys em owa ds i s long- un equilib ium, and hey a e
he e o e i ele an o he pu pose o compu ing he wel a e cos s o in la ionin hes eady-s a e.
Finally, we eschew Hansen and Johansen’s (1999) luc ua ion es s because, as shown by Bena i e
14
speci ica ions. Only in wo ins ances, Denma k, and Japan based on he Selden-
La ané speci ica ion, he es s de ec e idence o ins abili y.22
Table 3 Boo s apped p- alues o Hansen and Johansen’s
(1999) es s o s abili y in he coin eg a ion ec o o (log)
M1 eloci y and ( he log o ) a sho - e m a e
Money demand
speci ica ion:
Selden- Semi- Log-
Coun y Pe iod La ané log log
Low-in la ion coun ies:
Uni ed S a es 1959Q1-2023Q2 0.5875 0.8030 0.9940
Uni ed Kingdom 1955Q1-2023Q2 0.5905 0.5480 0.9365
Canada 1947Q3-2006Q4 0.3535 0.6710 0.6910
1967Q1-2023Q1 0.6900 0.7945 0.6070
Aus alia 1969Q3-2023Q1 0.7835 0.7880 0.6950
Swi ze land 1980Q1-2023Q1 0.6378 0.8102 —
Sweden 1998Q1-2023Q1 0.2335 0.1690 —
Eu o a ea 1999Q1-2023Q1 0.4880 0.2915 —
Denma k 1991Q1-2023Q1 0.0085 0.2605 —
Sou h Ko ea 1964Q1-2023Q1 0.1460 0.5835 0.4485
Japan 1960Q1-2023Q2 0.0030 0.2600 0.4030
Hong Kong 1985Q1-2023Q2 0.5280 0.4510 0.8465
High-in la ion coun ies:
Boli ia 1980-2019 — — 0.1020
Chile 1946-2019 — — 0.3740
Ecuado 1980-2019 — — 0.1335
Is ael 1982Q1-2019Q4 — — 0.5330
Mexico 1982Q1-2019Q4 — — 0.3940
Hype in la ions:
Weima Republic Sep. 1920-Oc . 1923 — — 0.2105
Based on 10,000 boo s ap eplica ions. Null o 0 e sus 1
coin eg a ion ec o s. The las obse a ions o he in e es
a e a e ei he ze o o nega i e.
O e all, he e is e y li le e idence o a b eak in he eal money demand ela-
al. (2021) ia Mon e Ca lo, hey exhibi , o e all, a signi ican ly in e io pe o mance compa ed o
he es s o s abili y in he coin eg a ion ec o and loading coefficien s.
22This is in line wi h he e idence in Bena i e al.’s (2021) Sec ion 6.2. The main inding he e was
ha e idence o b eaks in ei he he coin eg a ion ec o o he loading coefficien s ec o is weak
o non-exis en . The es ima ed b eak da es o he coin eg a ion ec o a e 2008Q1 o Denma k
and 1979Q4 o Japan. The second elemen o he no malized coin eg a ion ec o o he i s and
second sub-pe iods is equal o -0.37 and -0.66 o Denma k, and o -0.41 and -0.74 o Japan.
15
ionship de i ed om he heo y. This is eassu ing in i sel , bu also in e e ence o
he issue aised by I eland and ha has p e ailed he discussion in he Uni ed S a es,
ela ed o a s uc u al b eak in his ela ionship somewhe e be ween he la e 70s and
he ea ly 80s. I is he assump ion o such a b eak ha jus i ies ocusing he analysis
using only he ecen da a. These es s show, on he one hand, ha once we ake in o
accoun Uni ed S a es speci ic egula o y changes, he e is no b eak in he money
demand ela ionship o e he pos -WWII pe iod. On he o he hand, hey show ha
in o he simila de eloped coun ies ha did no expe ience egula o y changes, he
high in la ion episode o he la e 70s and ea ly 80s is consis en wi h a s able demand
o eal M1 balances, jus based on he s anda d M1 mone a y agg ega e.
4.3.2 A e he e non-linea i ies in money demand a low in e es a es?
A concep ually ela ed issue pe ains o he possibili y ha , a low in e es a es,
money demand migh exhibi sizeable non-linea i ies, due o he p esence o ixed
cos s associa ed wi h he decision o pa icipa e, o no o pa icipa e, in inancial
ma ke s (see e.g. Mulligan and Sala-i-Ma in, 2000).23 Based on his a gumen , a
sufficien ly low in e es a es money demand (and he e o e money eloci y) should
be la gely un esponsi e o changes in in e es a es, since mos (o all) households
simply do no pa icipa e in inancial ma ke s. The implica ion is ha i should no be
possible o eliably es ima e money demand unc ions (and he e o e he wel a e cos s
o in la ion) based on agg ega e ime se ies da a, as only he use o mic o da a allows
o meaning ully cap u e he non-linea i ies associa ed wi h he cos o pa icipa ing
in inancial ma ke s.
Al hough Hansen and Johansen’s (1999) es s de ec li le e idence o ins abili y
in he coin eg a ion ec o , o he speci ic pu pose o es ing whe he money demand
cu es migh be la e a low in e es a es hese esul s should be discoun ed o (a
leas ) wo easons.
Fi s , as discussed by Bai and Pe on (1998, 2003), when a coefficien expe iences
wob eaksinopposi edi ec ions(e.g., i s an inc ease, and hen a dec ease), b eak
es s which ha e no been explici ly designed o sea ch o mul iple b eaks may ha e
aha d imeinde ec ing he i s b eak o begin wi h. Wi hin he p esen con ex
his could be ele an o h ee coun ies, he U.S., he U.K., and Canada. In any o
hese cases he sho a e had been below 5% (which, ollowing Mulligan and Sala-i-
Ma in, 2000, we ake as he ele an h eshold) a he beginning o he sample; i hen
23The in ui ion is s aigh o wa d. Suppose ha he in e es a e, , is ini ially equal o ze o, and
conside a household wi h nominal asse s , which a e en i ely held in ei he cash o non-in e es -
bea ing deposi s. C ucially, suppose ha i he household wan s o swi ch a ac ion o i s asse s
in o bonds ,i has opaya ixed cos .Asinc eases om ze o o 0, unless he
household will keep all o i s weal h in ei he cash o deposi s o m, and only when he inequali y is
sa is ied i will ha e an incen i e o buy bonds. This implies ha , unde he plausible assump ion
ha is he e ogenous ac oss he popula ion, money demand should exhibi sizeable non-linea i ies
( a he han a s ic discoun inui y) a low in e es a es.
16
24
Figu e 3 In o mal e idence on he possible p esence o non-linea i ies a low in e es a es
25
Figu e 4 M1 eloci y and sho - e m nominal in e es a es: obse a ions wi h
he sho a e abo e and below 5 pe cen (qua e ly da a)
signi ican ly inc eased abo e 5% du ing he G ea In la ion; and i has p og essi ely
dec eased since he ea ly 1980s. Unde he assump ion ha money demand cu es
a e compa a i ely la e a low a es, his implies ha he slope o he cu e should
ha e i s inc eased, and hen dec eased, which is p ecisely he kind o ci cums ance
in which hese es s may ha e p oblems in de ec ing a b eak.
Second, Hansen and Johansen’s (1999) a e es s o b eaks a unknown poin s
in he sample. In p inciple, i should be possible o pe o m mo e powe ul es s i
we had s ong easons o choosing a speci ic h eshold o he sho a e, which, as
men ioned, we ake i o be 5%.
Be o e del ing in o he econome ic e idence, howe e , i is o in e es o see wha
a simple isual inspec ion o he da a sugges s. Figu e 3 shows in o mal e idence on
he possible p esence o nonlinea i ies o i e coun ies o which bo h sub-samples
wi h he sho a e abo e, and espec i ely below 5% a e sufficien ly long. In o de
o p o ide sha pe e idence, o ou coun ies ( he U.S., he U.K., Canada, and
Aus alia) we conside long samples o annual da a ha we do no u he analyze.24
The igu e shows he aw da a o M1 eloci y and a sho a e. The e idence speaks
o i sel , and i p o ides no suppo o he no ion ha eloci y–and he e o e money
demand–may be less esponsi e o in e es a e changes a low in e es a es. The
only possible excep ion is he U.S. un il WWII.
O e all, he ‘big pic u e’ eme ging om Figu e 3 sugges s ha he ela ionship
be ween M1 eloci y and he sho a e is i ually he same a all in e es a e
le els. Al hough we will sho ly discuss he econome ic esul s, in ac we ega d
his e idence, because o i s simplici y, as he s onges a gumen agains he no ion
ha money demand cu es may be la e a low in e es a es.25
Figu e 4 shows e idence based on qua e ly da a o he ou coun ies wi h su -
icien ly long con inuous samples wi h he sho a e bo h abo e and below he 5%
h eshold. The op ow shows sca e plo s o M1 eloci y and he sho a e, wi h he
obse a ions wi h he sho a e abo e and below he h eshold being shown in black
and ed, espec i ely.26 (The sub-samples wi h he sho a e below and abo e 5%
a e epo ed in Table ??.) The panels also show an ho izon al ed line co esponding
o an ex eme e sion o he non-linea i y hypo hesis, in which when he sho a e
alls below 5% by an a bi a ily small quan i y 0, eloci y becomes comple ely
insensi i e o in e es a e luc ua ions (and he e o e pe ec ly la ). The eason o
24This is because, hese being annual se ies, o all o hem a leas one o he sub-samples wi h
he sho a e ei he abo e o below 5% ea u es oo ew obse a ions o p oduce eliable esul s.
25This is in line wi h Summe s’ (1991) poin ha he mos con incing ype o e idence, and he
one ha , his o ically, had he mos impac in e ms o changing he p o ession’s iews, is simple
e idence based on ei he aw da a, o da a ha ha e been subjec ed o e y simple manipula ions.
26Fo Canada (1947Q3-2006Q4) i would seem ha he e is a discon inui y in he ela ionship
be ween eloci y and he sho a e. In ac , his is no he case: a he , in o de o ob ain ‘clean’
samples wi h he sho a e almos en i ely below o abo e 5% we had o elimina e he pe iod
1967Q4- 1973Q1, du ing which he sho a e luc ua ed a ound 5%. By he same oken, o he
U.S. we exclude he pe iod 1991Q4-2000Q4.
17
29
Figu e 7b Compa ing he es ima ed wel a e cos unc ions p oduced by al e na i e speci ica ions: 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
In wha ollows, we discuss in de ail ou esul s using he S ock and Wa son DOLS
es ima es, and lea e o he appendix he analysis wi h W igh ’s (2000) es s, whe e,
as ‘poin es ima es’, we pick he alue ha is mos difficul o ejec a he 10%
le el. Resul s based on W igh ’s (2000) app oach a e e y simila o hose p oduced
by S ock and Wa son’s es ima o , excep o a ew cases in which he es ima ed
wel a e cos a e sligh ly highe .32 The me hodology we use in o de o es ima e he
demand o eal M1 balances ollows Lue kepohl (1991, pp. 370-371). Speci ically,
we s a by es ima ing ia OLS he coin eg a ing eg ession co esponding o any o
he h ee speci ica ions, i.e. o ei he (8), (9), o he in e se o (10).33 This gi es
us he poin es ima es o he pa ame e s we need in o de o compu e he poin
es ima es o he wel a e cos unc ions. We hen es ima e he ele an VECM ia
OLS by imposing in es ima ion he p e iously es ima ed coin eg a ion ec o , and
we cha ac e ize unce ain y abou he poin es ima es o he wel a e cos unc ion by
boo s apping he VECM as in Ca alie e e al. (2012).
In line wi h he p e ious discussion, his p ocedu e is alid i he se ies con ain
exac uni oo s. Unde he al e na i e possible in e p e a ion o he esul s om
uni oo es s, i.e. ha he se ies a e local- o-uni y, we p oceed as in Bena i e al.
(2021, Sec ion 4.2.1). Speci ically, we compu e, based on he jus -men ioned VECM,
he co esponding VAR in le els, which by cons uc ion ea u es one, and only one
exac uni oo , and we u n i in o i s co esponding nea uni oo VAR by sh inking
heuni oo o=1-0.5·(1/), whe e is hesampleleng h.
34 The boo s apping
p ocedu e we implemen o he second possible case, in which he p ocesses ea u e
nea uni oo s, is based on boo s apping such a nea uni oo VAR. In sho , he
wo boo s apping p ocedu es p oduce nume ically nea -iden ical esul s, and in wha
ollows we will he e o e exclusi ely epo and discuss hose based on boo s apping
he VECM ( he al e na i e se o esul s is howe e a ailable upon eques ).
Figu e 5 shows, o any o he ele en coun ies, he es ima ed wel a e cos unc ions
basedon heSelden-La anéspeci ica ion, which based on he p e ious discussion we
ake as ou benchma k unc ional o m o his g oup o coun ies. We plo he poin
es ima es o he lowe and uppe bounds–which, in line wi h he discussion in Sec ion
2, a e nea ly indis inguishable– he 5 h and 16 h pe cen iles o he lowe bounds, and
he 84 h and 95 h pe cen iles o he uppe bounds o he boo s apped dis ibu ions.
Assuming ha he in la ion a ge and he na u al a e o in e es a e bo h equal
o 2 pe cen , he mone a y policy a e will be equal, on a e age, o 4 pe cen . In
u n, by a bi age he same will app oxima ely hold o sho - e m nominal a e. On
he o he hand, he hough expe imen conside ed by bo h Lucas (2000) and I eland
(2009) in ol ed a s eady-s a e nominal a e o 5 pe cen . In o de o be able o d aw
32These cases a e Japan and he U.S. o he semi-log, he Eu o a ea o he Selden-La ané and
Swi ze land o he log-log.
33So, o be clea , when we wo k wi h he Selden-La ane speci ica ion we un he coin eg a ing
eg ession o M1 eloci y, a he han i s in e se, M1 as ac ion o GDP.
34Fo de ails see Bena i e al.’s (2021) oo no e 24.
21
a compa ison wi h hei esul s we he e o e plo he wel a e cos unc ions o e he
domain om 0 o 5 pe cen o he sho a e.
S a ing om he U.S., which has been he ocus o mos p e ious esea ch, based
on a calib a ed log-log Lucas (2000) compu ed a wel a e loss o a 5 pe cen nominal
a e equal o 1.1% o pe manen consump ion. Based on a semi-log es ima ed o e
he pe iod 1980Q1-2006Q4, on he o eh hand, I eland (2009) es ima ed a wel a e loss
o jus 0.04 pe cen o li e ime consump ion. Ou poin es ima e in Figu e 5 is equal
o 0.3 pe cen o GDP, abou one- hi d o he loss compu ed by Lucas (2000), and
an o de o magni ude la ge han ha es ima ed by I eland (2009). So, e en jus
ocusing on poin es ima es, ou e idence o he U.S. sugges s ha hese losses a e
no negligible, since–i is impo an o ecall– hese losses a e suffe ed e e y single
yea .Fu he ,andc ucially, he90pe cen con idence in e al in Figu e 5 s e ches
be ween 0.19 and 0.71 pe cen . This poin s owa ds he exis ence o sizeable ‘upwa d
isks’ po en ially associa ed wi h he wel a e cos s o in la ion, which policymake s
ob iously should ake in o accoun when choosing in la ion objec i es. Su e enough,
his is no he case o all coun ies: e.g., Denma k, he Eu o a ea, and especially
Sweden ha e qui e igh con idence bands. Fo all o he emaining coun ies, howe e ,
hese bands a e uni o mly wide, o e y wide: his is he case, in pa icula , o he
U.K., Aus alia, and Hong Kong. A policymake aiming a implemen ing obus
policies would ake such ‘upwa d isks’ in o accoun , and ce e is pa ibus would se
he in la ion a ge lowe han he alue jus implied by he poin es ima e.
A second obus inding eme ging om Figu e 5 is a non-negligible ex en o
he e ogenei y ac oss coun ies. Fo example, ocusing on poin es ima es, and ou
benchma k hough expe imen o a na u al a e o in e es and an in la ion a ge
bo h equal o 2 pe cen , wel a e cos s in he Eu o a ea, equal o 0.4 pe cen , a e
abou wice as la ge as hose in he U.S., hus sugges ing ha , ce e is pa ibus, he
in la ion a ge should be ma e ially lowe . Mo e gene ally, poin es ima es ange
be weeen 0.07 pe cen o Japan and 0.45 pe cen o Aus alia, whe eas he 95 h
pe cen iles o he boo s apped dis ibu ion anges be weeen 0.13 pe cen and 1.3
pe cen o he same wo coun ies. Again, hese a e sizeable igu es.
Figu e 6 b ings his home in he s a kes possible way. The igu e epo s he
poin es ima es o he lowe and uppe bounds o hese cos s a each poin in ime
along he sample, oge he wi h he 5 h pe cen iles o he lowe bounds and 95 h
pe cen iles o he uppe bounds o he boo s apped dis ibu ions. Once again, he
poin es ima es o he lowe and uppe bounds a e nea ly indis inguishable. A i s
inding eme ging om he igu e is a d ama ic ex en o a ia ion in hese cos s,
due o a co esponding la ge a ia ion in he le el o sho - e m nominal in e es
a es o e he sample pe iods. The main inding in Figu e 7, howe e , is ha i we
uniquely ocus on coun ies whose samples include he G ea In la ion, a he peak o
he in la ion episodes poin es ima es o he wel a e cos s had anged be ween 1.2 pe
cen o GDP o he U.S. and 1.9 pe cen o Aus alia, whe eas he 95 h pe cen iles
o he uppe bounds o he boo s apped dis ibu ions had anged be ween 2.1 pe
22
cen o Canada and 3.5 pe cen o Aus alia. These numbe s a e e y a om
negligble, and in ac hey a e uni o mly sizeable, ei e a ing onec again one o ou
main poin s: he assump ion, explici o implici in he p e ious li e a u e, ha hese
cos s can be igno ed is unwa a ed.
A key implica ion o ou esul s is ha he common p ac ice in he li e a u e
o igno ing money in he analyses o op imal mone a y policies is equally unwa -
an ed, and i can be se iously misleading. Fo Example, Coibion, Go odnichenko
and Wieland (2012)35 ad ance a compelling a gumen agains inc easing he in la-
ion a ge in coun ies like he U.S., based on a model wi h ic ions in p ice-se ing
and wi h ecu en , al hough no e y equen episodes wi h he nominal in e es
a e cons ained by he ZLB. Based on hei p e e ed speci ica ion hey compu e he
wel a e effec o an in e es a e o 5 pe cen o be close o 0.6 pe cen o li e ime
consump ion. Such an es ima e, combining he cos c ea ed by p ice ic ions and
he p obabili y o be a he ZLB, is o he same o de o magni ude o he es ima es
we ob ain o he U.S., and i is in ac sligh ly smalle han he uppe bound o ou
90 pe cen con idence in e al in Figu e 5.
Finally, Figu es 7-7explo e whe he he h ee unc ional o ms o he demand
o eal money balances do, o do no p oduce ma e ially diffe en wel a e cos unc-
ions. O e all, e idence is mixed. S a ing om he U.S., a a 4 pe cen sho a e
poin es ima es a e equal o abou 0.2 pe cen o all unc ional o ms. This shows
ha he con en ional-wisdom no ion ha , ce e is pa ibus (e.g., o gi en obse a ions
o eloci y and he in e es a e), a log-log speci ica ion should be expec ed o p o-
duce compa a i ely highe wel a e cos s is, in gene al, inco ec . The eason o his
is s aigh o wa d: Al hough he log-log speci ica ion does no ha e a ini e sa ia ion
le el o money balances a =0 , empi ical es ima es o he in e cep s and he coeffi-
cien on he (loga i hm o he) in e es a e do in ac ma e . The e o e, ocusing e.g.
on he semi-log and he log-log, i is pe ec ly possible ha he pa ame e s es ima es
a e such ha
−log()=2
µ1−1+
¶≥1
1−1−=log −log()
In se e al ins ances, howe e , he s anda d assump ion appea s o be alida ed: his
is he case o Canada, Japan, and Sou h Ko ea, whe eas e idence o Hong Kong
uns, once again, agains con en ional wisdom.We now u n o he wel a e cos s o
high in la ions and Weima Republic’s hype in la ion
35Coibion 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.
23
30
Figu e 8 Es ima ed wel a e cos unc ions and wel a e losses a each poin in ime o high-in la ion coun ies and Weima :
Republic’s hype in la 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
5.2 High in la ion coun ies and Weima Republic’s hype -
in la ion
As i has been ex ensi ely documen ed, e y high in la ions ha e uni o mly been
associa ed wi h mac oeconomic mayhem and he des uc ion o weal h held in nom-
inal asse s. E idence is especially s a k o hype in la ions. Fo Weima Republic’s
episode, o example, he da a epo ed in Table XL o G aham (1930, p. 317) show
ha he unemploymen a e among ade union membe s, which in 1922 had oscil-
la ed be ween 0.6 and 3.3 pe cen , inc eased apidly ollowing he in asion o he Ruh
on he pa o F ance in Janua y 1923, which as poin ed ou by B esciani-Tu oni
(1937) ‘ga e hecoupdeg âce o hena ional inances and he Ge man ma k’, hus
inaugu a ing he mos ex eme phase o he hype in la ion. Unemploymen eached
6.2 pe cen in May, 9.9 in Sep embe , and i u he inc eased o a ema kable 28.2
pe cen in Decembe , he las mon h o he hype in la ion.
In his sec ion we explo e he wel a e cos s o hese episodes o igina ing om lack
o liquidi y sa ia ion, which, qui e su p isingly, ha e been nea -uni o mly o e looked
by he p e ious li e a u e. The only excep ion we a e awa e o is Ba o (1972), which
epo s e idence o he Weima Republic’s episode and ou o he hype in la ions
ha had been s udied by Cagan (1956). Fo all coun ies we es ima e Mel ze ’s
(1963) log-log speci ica ion. Ou main inding is ha o e y high in la ions and
hype in la ions hese cos s a e e y a om negligible, as a he in la ion peaks o
he espec i e episodes hey ange om abou 4 pe cen o ou pu o Ecuado o
be ween 26 and 36 pe cen o Weima ’s hype in la ion.
Figu e 8 epo s he e idence. The op ow shows he es ima ed wel a e cos
unc ions (in pe cen age poin s o GDP) o alues o he oppo uni y cos o money
omze o o hemaximum alue ha i had akeno e hesamplepe iod. The
bo om ow shows he es ima ed wel a e losses a each poin in ime. The hick black
lines a e he poin es ima es o he lowe and uppe bounds, whe eas he ed lines a e
he 84 h and 95 h pe cen iles o he boo s apped dis ibu ion o he uppe bound,
and he 5 h and 16 h pe cen iles o he co esponding dis ibu ion o he lowe bound.
Se e al ac s clea ly eme ge om he igu e. In pa icula ,
i s , in line wi h he p e ious e idence o low-in la ion coun ies, he poin es-
ima es o he uppe and lowe bounds o bo h he wel a e cos unc ions, and he
wel a elossesa eachpoin in imea e e y igh , o hepoin ha ina ewins ances
(in pa icula , he wel a e losses in he second ow) hey a e nea ly indis inguish-
able. The only excep ion is Weima ’s hype in la ion, o which o high alues o he
oppo uni y cos hey can be clea ly dis inguished.
Second, ocusing again on he poin es ima es, he e idence in Figu e 8 uni o mly
sugges s ha o all o hese episodes he wel a e cos s had been sizeable- o-la ge. In
pa icula , a he peaks o he in la ion episodes hese cos s had been equal o abou
4 pe cen o GDP o Ecuado ; be ween 4 and 5 pe cen o Boli ia, Chile, and
Is ael; nea ly 7 pe cen o Mexico; and be ween 26 and 36 pe cen o income o
24
he Weima Republic. This p o ides a s a k illus a ion o how, beyond he al eady
well known and widely documen ed cos s o e y high in la ions and hype in la ions in
e ms o economic mayhem and he des uc ion o weal h held in nominal asse s, hese
episodes ha e consis en ly imposed non-negligible, and some imes la ge cos s uniquely
in e ms o lack o liquidi y sa ia ion, by compelling agen s o hold compa a i ely low
le els o eal money balances.
Thi d, unce ain y is nea -uni o mly subs an ial, some imes ema kably so. The
only excep ions a e he his o ical wel a e losses o Chile, Is ael, and Mexico: o any
o hese coun ies, he in la ion peaks pe ained o compa a i ely small ac ions o
he espec i e samples, wi h he esul ha o mos o he sample he oppo uni y
cos had been compa a i ely low, which, as he op ow o Figu e 8 shows, is as-
socia ed wi h compa a i ely igh e con idence in e als. S a is ical unce ain y is
excep ionally la ge o Weima ’s hype in la ion, o which s a ing om he second
hal o 1922 nea ly any alue o he wel a e losses is in p inciple plausible.
6Conclusions
In his pape we e isi he es ima ion o he wel a e cos s o in la ion o ele en low-
in la ion and i e high-in la ion coun ies, and o Weima Republic’s hype in la ion.
In ou analysis we ollow he adi ion o conside ing he mos liquid mone a y asse s,
which include cash and ansac ional deposi s. We abs ac om a de ailed discussion
o he demand o each o he componen s, an issue ecen ly add essed by Ku la
(2019).36
Ou e idence sugges s ha , con a y o he explici o implici assump ion in
much o he li e a u e, hese cos s a e o en a om negligible. Fo he U.S. ou
poin es ima es a e equal o abou one- hi d o hose compu ed by Lucas (2000), and
an o de o magni ude la ge han hose ob ained by I eland (2009).37 C ucially, he
mos empi ically plausible money-demand unc ional o m poin s owa ds sizeable
‘upwa d isks’ o hese cos s, wi h he 90% con idence in e al associa ed wi h a 4%
nominal in e es a e s e ching beyond hal a pe cen age poin o GDP. A he peak
o he in la ion episodes, wel a e cos s had anged be ween 0.3 and 1.9 pe cen o
GDP o low-in la ion coun ies; be ween 4 and nea ly 7 pe cen o high-in la ion
ones; and be ween 26 and 36 pe cen o Weima ’s hype in la ion.
Ou e idence sugges s ha igno ing money in analyzing op imal mone a y poli-
cies can be se iously misleading. Fo ins ance, Coibion, Go odnichenko and Wieland
36He shows ha add essing hese conside a ions in a model wi h impe ec compe i ion subs an-
ially inc eases he es ima es o he wel a e cos , ela i e o models ha igno e he c ea ion o inside
money.
37Fo a s eady-s a e in e es a e o 5 pe cen , Lucas (2000) compu ed he cos o be a ound
1.1 pe cen o li e ime consump ion. I eland (2009) challenged Lucas’ in e p e a ion o he da a,
and es ima ed a me e 0.04 pe cen o li e ime consump ion.
25
(2012)38 make a compelling a gumen agains inc easing he in la ion a ge in coun-
ies like he U.S., based on a model wi h ic ions in p ice-se ing and wi h ecu en ,
hough no e y equen , episodes wi h he nominal in e es a e a he ZLB. Based
on hei p e e ed speci ica ion hey compu e he wel a e effec o an in e es a e o
5 pe cen o be close o 0.6 pe cen o li e ime consump ion. Such an es ima e,
combining he cos c ea ed by p ice ic ions and he p obabili y o be a he ZLB,
is o he same o de o magni ude o he es ima es we ob ain o he U.S., and i is
in ac sligh ly smalle han he uppe bound o ou 90 pe cen con idence in e al.
In any o he low-in la ion coun ies he demand o M1 as a ac ion o GDP
a e y low, o e en nega i e in e es a es has (so a ) exhibi ed no ob ious diffe -
ence compa ed o i s beha io a highe in e es a es. These esul s con as wi h
hose o Mulligan and Sala-i-Ma in (2000), who, based on U.S. households mic o
da a, p o ided e idence ha money demand becomes compa a i ely la e a low
in e es a es. We p o ide a s aigh o wa d explana ion o Mulligan and Sala-i-
Ma in’s (2000) inding, by showing ma hema ically ha i he ue money demand
speci ica ion is he one ha , as shown by Bena i e al. (2021), is he mos empi i-
cally plausible a low in la ion a es, Mulligan and Sala-i-Ma in’s (2000) app oach
au oma ically p oduces spu ious e idence o a la e demand cu e a low in e es
a es.
F om a heo e ical s andpoin , ollowing Al a ez, Lippi and Roba o (2019) we
cons uc uppe and lowe bounds o he wel a e cos s o in la ion. As hey show, he
a eaunde hemoneydemandcu eisanalmos exac measu eo hewel a ecos
o a e y gene al class o mone a y models in he neighbo hood o ze o. We ex end
hei esul s o a qui e gene al sub-class o he models hey analyze and compu e
exac lowe and uppe bounds 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 diffe ence be ween he uppe
and he lowe bound is ex emely small o he ange o in e es a es e e obse ed
in low-in la ion coun ies such as he U.S.
Fo policy pu poses, ou main inding is ha he wel a e cos s in la ion in he
Eu o a ea a e abou wice as la ge as in he U.S.. This sugges s ha , ce e is pa ibus,
he in la ion a ge should be ma e ially lowe .
38Coibion 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.
26
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28
o nominal GDP in U.S. dolla s (“P oduc o in e no b u o (PIB), Miles de dóla es”),
a ailable o he pe iod 1965-2011, is om Chap e 4 o “85 Años”. An impo an
poin o s ess is ha since we a e wo king wi h M1 eloci y–de ined as he a io
be ween nominal GDP and nominal M1– he speci icuni inwhich he wose iesa e
exp essed (US dolla s, o Ecuado ian suc es) is i ele an .
B.2.4 Is ael
Qua e ly seasonally adjus ed da a on nominal GDP and he CPI a e om he Cen al
Bu eau o S a is ics, whe eas a se ies o M1 is om Is ael’s cen al bank. A se ies
o he T easu y bill a e is om he In e na ional Mone a y Fund’s In e na ional
Financial S a is ics.
B.2.5 Mexico
Qua e ly seasonally adjus ed da a on nominal GDP a e om Mexico’s s a is ical
agency, INEGI. Qua e ly seasonally adjus ed da a o he CPI, M1, and a 3-mon h
go e nmen bond yield a e all om he Banco de México.
Fo all coun ies, in wha ollows we wo k wi h money eloci y (i.e., he in e se
o money balances as a ac ion o GDP), and a se ies o he oppo uni y cos o
money, which we compu e as he maximum, a each poin in ime, be ween in la ion
and he se ies o he nominal sho - e m in e es a e ( o A gen ina we we e no
able o ind an in e es a e se ies, and we he e o e wo k wi h in la ion).
B.3 Weima ’s Republic
Mon hly da a on he eloci y o ci cula ion o money based on wholesale p ices a e
om Table XXII o B esciani-Tu oni (1937). The se ies had been no malized by 1913
(i.e. o he yea 1913 i ook a alue o one). Based on Bena i, Lucas, Nicolini and
Webe ’s (2021) da a, howe e , in 1913 Ge man money eloci y had been equal o
7.49. Consequen ly, we ha e escaled B esciani-Tu oni’s money eloci y se ies by
mul iplying i by 7.49.39 A se ies o he in la ion a e is om Cagan (1956). A se ies
o he money ma ke a e (‘Tägliches Geld’) is om Table 23 o Hol e ich (1980).
The sample pe iod is Sep embe 1920-Oc obe 1923.
C Why We Do No Use Di isia Agg ega es
Th oughou he en i e pape we wo k wi h ‘simple-sum’ M1 agg ega es. In his
appendix we b ie ly discuss why we ha e chosen o igno e Di isia indices. A i s
p oblem is ha , o he e y bes o ou knowledge, such indices a e only a ailable
39In ac , wo king wi h B esciani-Tu oni’s o iginal se ies p oduces mani es ly absu d esul s, wi h
he wel a e cos s o in la ion e en aking alues in excess o 100 pe cen o GDP.
35
o he Uni ed S a es ( om he Cen e o Financial S abili y, hence o h CFS)and
o he Uni ed Kingdom ( om he Bank o England). A second p oblem is ha , o
he Uni ed S a es, he Di isia M1 se ies cons uc ed by he CFS does no ea u e
MMDAs (which a e ins ead included in Di isia M2). This means ha al hough he
esul ing index o mone a y se ices has been cons uc ed by op imally weigh ing
he unde lying indi idual asse s, i suffe s om he c ucial sho coming ha i is no
including a key componen o he ansac ion echnology. As a esul , al hough Di isia
M1 is in p inciple supe io o he s anda d simple-sum M1 agg ega e, i ul ima ely
suffe s om he same sho coming o no including MMDAs.
So he key ques ion is: Wha is mo e impo an ? Including MMDAs, o op imally
weigh ing he unde lying asse s? Figu e C.1 p o ides e idence on his, by showing
he same e idence shown in Figu e 2 in he main ex o he pape , bu his ime
wi h eloci y being compu ed based on Di isia agg ega es. The igu e speaks o
i sel , and p o ides no e idence o a s able ela ionship be ween he eloci y o any
Di isia agg ega e and i s oppo uni y cos (compu ed based on he use cos se ies
om he CFS). In pa icula , a compa ison be ween he i s panel o Figu e C.1,
and he second panel in Figu e 2, clea ly shows ha , o he pu pose o de ec ing a
s able long- un demand o M1 in he Uni ed S a es, he c ucial issue is including
MMDAs in he de ini ion o M1, a he han compu ing he agg ega e by op imally
weigh ing he unde lying asse s. So al hough, in heo y, Di isia M1 possesses op imal
p ope ies, because o he speci ic way in which is has been cons uc ed, wi hin he
p esen con ex such op imal p ope ies a e umped by he ac ha , exac ly as i s
simple-sum coun e pa , i does no include MMDAs.
36
Table A.1aBoo s apped p- alues o Ellio , Ro henbe g, and S ock uni oo es s
M1 eloci y sho a e
p=2 p=4 p=6 p=8 p=2 p=4 p=6 p=8
Low-in la ion coun ies:
Uni ed S a es 1959Q1-2023Q2 0.9809 0.9323 0.9236 0.9207 0.3122 0.1794 0.1053 0.2918
1959Q1-2019Q4 0.8633 0.8362 0.9048 0.8764 0.4382 0.2861 0.1903 0.4334
1959Q1-2001Q4 0.3529 0.2989 0.4112 0.3768 0.3238 0.2756 0.1428 0.2979
Uni ed Kingdom 1955Q1-2023Q2 0.9257 0.8719 0.8042 0.8490 0.2896 0.3225 0.4118 0.4673
1955Q1-2008Q3 0.8187 0.8012 0.7262 0.7740 0.1416 0.1640 0.2386 0.2600
Canada 1947Q3-2006Q4 0.4641 0.6307 0.3987 0.5405 0.2298 0.2466 0.2224 0.3600
1967Q1-2023Q1 0.9772 0.9682 0.9557 0.9554 0.3996 0.3831 0.3605 0.6097
Aus alia 1969Q3-2023Q1 0.9665 0.9600 0.9580 0.9213 0.4143 0.3218 0.5442 0.7099
1969Q3-2008Q4 0.9883 0.9823 0.9834 0.9679 0.3078 0.2719 0.4116 0.5810
Swi ze land 1980Q1-2023Q1 0.8459 0.7684 0.7566 0.6917 0.3639 0.4453 0.1953 0.2024
Sweden 1998Q1-2023Q1 0.5217 0.4343 0.5006 0.6716 0.3254 0.4637 0.5665 0.5411
Eu o a ea 1999Q1-2023Q1 0.1096 0.0847 0.0515 0.0075 0.4403 0.2511 0.2468 0.3625
Denma k 1991Q1-2023Q1 0.1027 0.2238 0.2289 0.1194 0.0999 0.0433 0.0212 0.0090
Sou h Ko ea 1964Q1-2023Q1 0.0195 0.0000 0.0150 0.0535 0.5079 0.4863 0.3828 0.0867
Japan 1960Q1-2023Q2 0.7305 0.7323 0.8163 0.5827 0.2394 0.3962 0.4375 0.4192
Hong Kong 1985Q1-2023Q2 0.6293 0.6881 0.5964 0.6065 0.3103 0.1820 0.1338 0.3085
Based on 10,000 boo s ap eplica ions o es ima ed ARIMA p ocesses. Tes s a e wi h an in e cep and
no ime end.
Table A.1bBoo s apped p- alues o Ellio , Ro henbe g, and S ock uni oo es s
Loga i hm o :
M1 eloci y sho a e
p=2 p=4 p=6 p=8 p=2 p=4 p=6 p=8
Low-in la ion coun ies:
Uni ed S a es 1959Q1-2023Q2 0.9850 0.9688 0.9712 0.9759 0.1579 0.0883 0.2274 0.2411
1959Q1-2019Q4 0.9749 0.9550 0.9784 0.9611 0.4835 0.3839 0.4577 0.2198
1959Q1-2001Q4 0.3054 0.2694 0.3626 0.3169 0.4250 0.3985 0.3023 0.3665
Uni ed Kingdom 1955Q1-2023Q2 0.9793 0.9496 0.9284 0.9224 0.2758 0.2210 0.2902 0.7804
1955Q1-2008Q3 0.8090 0.8838 0.8299 0.8706 0.1484 0.2162 0.3100 0.4094
Canada 1947Q3-2006Q4 0.1103 0.2339 0.1159 0.2931 0.0590 0.0474 0.0229 0.0275
1967Q1-2023Q1 0.9951 0.9898 0.9803 0.9902 0.1001 0.1006 0.3021 0.7613
Aus alia 1969Q3-2023Q1 0.9898 0.9863 0.9770 0.9609 0.1160 0.0509 0.4413 0.8191
1969Q3-2008Q4 0.9978 0.9946 0.9953 0.9923 0.3915 0.3312 0.5893 0.6785
Swi ze land 1980Q1-2023Q1 0.8647 0.8136 0.7686 0.7438 ————
Sweden 1998Q1-2023Q1 0.7177 0.5494 0.5106 0.6041 0.2756 0.2584 0.3197 0.3893
Eu o a ea 1999Q1-2023Q1 0.4266 0.3080 0.2638 0.0954 0.4084 0.6636 0.6686 0.6779
Denma k 1991Q1-2023Q1 0.2648 0.3825 0.4415 0.2066 ————
Sou h Ko ea 1964Q1-2023Q1 0.3815 0.1906 0.4337 0.4606 0.6510 0.6106 0.5806 0.3996
Japan 1960Q1-2023Q2 0.9694 0.9607 0.9528 0.8539 0.5903 0.6472 0.6553 0.6345
Hong Kong 1985Q1-2023Q2 0.7613 0.7888 0.7563 0.7412 0.1532 0.1217 0.1317 0.2013
High-in la ion coun ies:
Is ael 1982Q1-2019Q4 0.8997 0.8346 0.7721 0.5814 0.5826 0.5698 0.5049 0.4309
Mexico 1982Q1-2019Q4 0.4687 0.4159 0.2106 0.1188 0.2463 0.1085 0.0400 0.0147
p=1 p=2 p=1 p=2
Boli ia 1980-2019 0.7971 0.8018 0.7588 0.7822
Chile 1946-2019 0.6862 0.5399 0.7214 0.6946
Ecuado 1980-2019 0.9309 0.8715 0.8018 0.8928
Hype in la ions:
p=1 p=2 p=3 p=4 p=1 p=2 p=3 p=4
Weima Republic Sep. 1920-Oc . 1923 0.9309 0.9601 0.8592 0.8569 0.5006 0.5459 0.7271 0.5317
Based on 10,000 boo s ap eplica ions o es ima ed ARIMA p ocesses. Tes s a e wi h an in e cep and
no ime end. The sho a e has a ew nega i e obse a ions a he end o he sample.
32
Figu e C.1 Uni ed S a es: money eloci y based on Di isia agg ega es,
and he co esponding oppo uni y cos s
