Tanaka, Yasuhi o
A icle
Budge de ici in a g owing economy and impossibili y o
iscal collapse: A con inuous ime analysis
Cen al Eu opean Economic Jou nal (CEEJ)
P o ided in Coope a ion wi h:
Facul y o Economic Sciences, Uni e si y o Wa saw
Sugges ed Ci a ion: Tanaka, Yasuhi o (2024) : Budge de ici in a g owing economy and impossibili y
o iscal collapse: A con inuous ime analysis, Cen al Eu opean Economic Jou nal (CEEJ), ISSN
2543-6821, Sciendo, Wa saw, Vol. 11, Iss. 58, pp. 305-319,
h ps://doi.o g/10.2478/ceej-2024-0020
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To ci e his a icle
Tanaka Y. (2024). Budge De ici in a G owing Economy and Impossibili y
o Fiscal Collapse: A Con inuous Time Analysis. Cen al Eu opean
Economic Jou nal, 11(58), 305-319.
DOI: 10.2478/ceej-2024-0020
To link o his a icle: h ps://doi.o g/10.2478/ceej-2024-0020
Budge De ici in a G owing
Economy and Impossibili y o
Fiscal Collapse: A Con inuous
Time Analysis
Yasuhi o Tanaka
Open Access. © 2024 Yasuhi o Tanaka, published by Sciendo.
This wo k is licensed unde he C ea i e Commons A ibu ion 4.0 In e na ional License.
Yasuhi o Tanaka
Facul y o Economics, Doshisha Uni e si y, Kamigyo-ku, Kyo o, Japan
co esponding au ho : ya [email protected]
Budge De ici in a G owing Economy and Impossibili y
o Fiscal Collapse: A Con inuous Time Analysis
Abs ac
Using a con inuous ime dynamic model o g owing economy we will show he ollowing esul s. 1) When people
de i e u ili y om hei money holding (o go e nmen bond holding) along wi h hei consump ion, a budge de ici is
essen ial o achie e and main ain ull employmen unde s able p ices o in la ion in a g owing economy. 2) I we ake
in o accoun ha go e nmen spending due o budge de ici s inc eases inancial asse s held by he p i a e sec o ,
and hen consump ion will occu om asse s in addi ion o consump ion om income, e en when he in e es a e on
go e nmen bonds is highe han he eal economic g ow h a e, he a io o go e nmen deb o GDP can no di e ge
and he di e gence is na u ally p e en ed by mild in la ion. The equi ed in la ion a e is such ha he in e es a e o
he go e nmen bonds is smalle han he weigh ed a e age o he a e o e u n on capi al and he nominal g ow h
a e. Since he in e es a e o he go e nmen bonds is usually conside ed smalle han he a e o e u n on capi al,
his is no a e y demanding equi emen . Thus, we need no wo y a all abou he accumula ion o go e nmen deb
o abou he di e gence o he deb o GDP a io, which is o en aken as an indica o o iscal collapse.
Keywo ds
budge de ici | g owing economy | in ini ely li ing consume s | con inuous ime model | impossibili y o iscal collapse
JEL Codes
E12, E24
1. In oduc ion
The pu pose o his pape is o p o e he ollowing
esul s using a ela i ely simple ma hema ical model.
1. When people de i e u ili y om hei money
holding (o go e nmen bond holding) along wi h hei
consump ion, a budge de ici is essen ial o achie e
and main ain ull employmen unde s able p ices o
in la ion in a g owing economy.
2. Wi h he ollowing i ems (a) and (b) in mind,
e en when he in e es a e on go e nmen bonds is
highe han he eal economic g ow h a e, he a io
o go e nmen deb o GDP does no di e ge and he
di e gence is na u ally p e en ed by mild in la ion.
The equi ed in la ion a e is such ha he in e es a e
o he go e nmen bonds is smalle han he weigh ed
a e age o he a e o e u n on capi al and he nominal
g ow h a e. Since he in e es a e o go e nmen
bonds is usually conside ed smalle han he a e
o e u n on capi al, his is no a e y demanding
equi emen .
(a) Go e nmen spending due o budge de ici s
inc eases go e nmen bonds o base money held by
banks and inc eases inancial asse s held by he p i a e
sec o .
(b) As inancial asse s inc ease, consump ion will
occu om asse s in addi ion o consump ion om
income.
I he GDP a io o go e nmen deb di e ges and
becomes in ini ely la ge, hen he GDP a io o p i a e
inancial asse s also becomes in ini ely la ge, and he
GDP a io o consump ion om asse s also becomes
in ini ely la ge. Howe e , since consump ion is a pa o
GDP, such a si ua ion canno occu and a con adic ion
a ises. In such a case, an inc ease in consump ion demand
would cause a ise in p ices, which would p e en he
deb - o-GDP a io om di e ging. This is a na u ally
occu ing phenomenon, no caused by some policy.
Thus, we need no wo y a all abou he
accumula ion o go e nmen deb o abou he
di e gence o he deb - o-GDP a io, which is o en
aken as an indica o o iscal collapse.
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 307
I cu en accoun de ici s con inue and ne
ex e nal deb accumula es, i will e en ually ha e o be
epaid, which will be a majo obs acle o he coun y’s
economy. Howe e , as long as he go e nmen deb
emains domes ic, he e is no p oblem a all.
In Sec ion 2, we explain he me hodology o his
pape and p esen a li e a u e e iew.
Sec ion 3 p esen s his pape ’s model and analyses
o he beha iou o consume s, i ms, and ma ke
equilib ium. We will p o e he necessi y o a budge
de ici o ull employmen unde cons an p ices. We
show also ha la ge budge de ici s cause in la ion,
o we need a la ge budge de ici o ealise ull
employmen unde in la ion. The policy o inc easing
money h ough budge de ici s in line wi h he a e o
economic g ow h as indica ed by he conclusions o
his sec ion is consis en wi h he Mone a is k% ule.
Sec ion 4 p esen s he explici alues o he sa ings
and ha o money holding in he s eady s a e.
In Sec ion 5, a case wi h in e es -p oducing
go e nmen bonds ins ead o money will be
conside ed. I will be shown ha he deb - o-GDP
a io should be cons an in a s eady-s a e g ow h pa h,
and he la ge p opensi y o consume leads o a small
deb - o-GDP a io.
In Sec ion 6, we will p o e ha di e gence o he
deb - o-GDP a io o in ini y canno occu e en i he
in e es a e o he go e nmen bond is la ge han he
eal g ow h a e, ha is, iscal collapse is impossible.
In ha case, in la ion aises he nominal g ow h a e
and p e en s he deb - o-GDP a io om di e ging o
in ini y. The equi ed in la ion a e is such ha he
in e es a e o he go e nmen bonds is smalle han
he weigh ed a e age o he a e o e u n on capi al
and he nominal g ow h a e. Hype -in la ion will
no occu . I is he in e es on go e nmen bonds ha
causes di e gence. Wi hou in e es , di e gence does
no occu .
Sec ion 7 is he concluding sec ion.
2. Me hodology and li e a u e
e iew
A mac oeconomic model ha includes mic oeconomic
ounda ions o consume s’ beha iou and i ms’
beha iou will be used. Consume s a e assumed
o li e in ini ely, and u ili y maximisa ion o e an
in ini e ime is conside ed. People’s u ili y depends on
hei holding o money o go e nmen bonds as well
as hei consump ion. A con inuous ime dynamic
model will be used.
Economic g ow h is no based on he assump ion
ha new gene a ions will be bo n one a e ano he ,
bu a he on he assump ion ha he popula ion o he
same gene a ion will inc ease. This can be in e p e ed
as economic g ow h due o echnological p og ess
ha inc eases labou p oduc i i y. As a esul , he
size o he economy con inuously inc eases, and he
accumula ed inancial asse s mus be discoun ed by
he g ow h a e.
In he abo e discussion, he pe manence o he
s a e o mankind is assumed. Unde his assump ion,
bo h go e nmen deb and p i a e inancial asse s will
con inue o accumula e o e e . I he des uc ion o
he na ion o he ex inc ion o he human ace a e
o eseen, people will y o use up all o hei asse s
by hen, so consump ion will inc ease u he , ull
employmen can be main ained e en wi h budge
su pluses, and bo h p i a e inancial asse s and
go e nmen deb will g adually decline and disappea
on he day o des uc ion.
In Sec ions 3 and 4, we conside budge de ici
due o money issuance, no go e nmen bonds. In
Sec ions 5 and 6, we examine he case o budge de ici
by go e nmen bonds. Please see Ogu i (2011) o he
ela ionship be ween go e nmen bonds and money.
The e is li le li e a u e w i en om he
iewpoin o no conside ing he accumula ion o
go e nmen deb o be a p oblem, excep o hose
who belong o MMT (Mode n Money Theo y, o
example, W ay (2015), Mi chell, W ay and Wa s
(2019), Kel on (2020) o i s pe iphe y. Since MMT
membe s do no like o use ma hema ical models,
such pape s a e e en sca ce . In his sec ion, we would
like o b ie ly men ion some o he li e a u e ha we
consul ed in w i ing his pape .
Mos o he discussions abou he deb - o-GDP
a io use a simple calcula ion abou p ima y budge
balances, he in e es a e, and he g ow h a e. In
his 2022 and 2023 a icles, Blancha d in oduces he
ollowing p oblem: “When does he le el o deb
become unsa e? To answe his ques ion, we need a
de ini ion o ‘unsa e’. I p opose he ollowing: Deb
becomes unsa e when he e is a non-negligible isk
ha , unde exis ing and likely u u e policies, he
a io o deb o GDP will s eadily inc ease, leading o
de aul a some poin . The na u al way o p oceed is
hen s aigh o wa d. The dynamics o he deb a io
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 308
depend on he e olu ion o h ee a iables: p ima y
budge balances ( ha is, spending ne o in e es
paymen s minus e enues); he eal in e es a e ( he
nominal a e minus he a e o in la ion); and he eal
a e o economic g ow h.” (Blancha d, 2002/3)
Howe e , in a s eady s a e unde ull employmen
wi h o wi hou in la ion, he necessa y budge
de ici o ull employmen is de e mined by se e al
pa ame e s o he economy. The la ge budge de ici
aises he in la ion a e, and he la ge (o small)
p opensi y o consume leads o he small (o la ge)
budge de ici equi ed o ull employmen unde
cons an p ice o in la ion. The e o e, he la ge
p opensi y o consume leads o a small deb - o-GDP
a io.
While his pape uses a con inuous ime model ha
assumes people li e in ini ely, we ha e also analysed
p oblems ela ed o budge de ici s using o e lapping
gene a ion models. In doing so, we e e ed o
Diamond (1965), J. Tanaka (2010, 2011a, 2011b, 2013)
and O aki (2007, 2009, 2015). As o economic g ow h,
his pape uses an exogenous g ow h model, bu we
ha e also used an endogenous g ow h model due o
in es men by i ms wi h e e ence o G ossman and
Yanagawa (1993) and Maebayashi and J. Tanaka (2021).
The e a e pape s, o example, Weil (1987, 1989),
ha use con inuous ime models which assume people
li e in ini ely. Howe e , in his esea ch, he budge
cons ain in he disc e e ime model is conside ed
implici ly in ou mind wi h e e ence o Tachibana
(2006), and he budge cons ain in he con inuous
ime model is de i ed by making he ime in e al o
he disc e e ime model in ini esimal.
We ollow Le ne ’s unc ional inance heo y
(Le ne , 1944). He did no conside whe he he
go e nmen should un su pluses o de ici s o be
meaning ul in and o i sel . He belie es ha iscal
policy should be used o ealise ull employmen
a oiding in la ion. Fo mo e on Le ne ’s unc ional
inance heo y, see Fo s a e (1999).
Ou model is a kind o neoclassical model, bu
i s spi i may be pos -Keynesian in ha i does no
abandon he goal o ull employmen ou o a dislike
o budge de ici s o he accumula ion o go e nmen
deb . Lopez-Galla do (2000) is a s udy o budge
de ici s and ull employmen om a pos -Keynesian
s andpoin . I is inspi ed by Minsky (1986) and ela ed
o Mosle (1997-1998), W ay (1998) and K egel (1998).
Mosle , W ay and K egel (M-W-K), as a policy o ull
employmen , p oposed he ollowing, as desc ibed in
Lopez-Galla do ((2000), p. 550):
“Le he go e nmen assume he ole o employe o
las eso a a gi en wage a e, so ha anybody willing
o wo k a ha a e will ge a job om he go e nmen .
Go e nmen expendi u e will hus expand, bu will
no en ail any complica ion because “ he pu chasing
abili y o he go e nmen is limi ed only by wha is
a ailable o sale in exchange o dolla s” (Mosle ,
1997-1998, p.169), while his a ailabili y, we a e old,
is elas ic below ull employmen . Now, go e nmen
expendi u e will g ow p obably o e and abo e ax
eceip s, and a budge de ici will ensue. Howe e ,
M-W-K demons a e, wi h explana ions ich in
heo e ical, his o ical, and ins i u ional de ails, ha
he go e nmen can simply c ea e enough new money,
o o he wise sell secu i ies, o inance he de ici wi h
an unchanging a e o in e es .”
Lopez-Galla do (2000) discusses a ious p oblems
wi h his p oposal, which a e no o in e es o his
pape .
Ano he desc ip ion om a pos -Keynesian
s andpoin , acco ding o Sawye (2020), is ha
“Kalecki (1944) a gued ha he e would be he need
o pe manen budge de ici s in he ace o in en ions
o sa e exceeding in en ions o in es .”
This pape is also an example o an analysis,
using a simple ma hema ical model, o he ollowing
s a emen by J. M. Keynes:
“Unemploymen de elops, ha is o say, because
people wan he moon; — men canno be employed
when he objec o desi e (i.e. money) is some hing
which canno be p oduced and he demand o which
canno be eadily choked o . The e is no emedy bu
o pe suade he public ha g een cheese is p ac ically
he same hing and o ha e a g een cheese ac o y (i.e.
a cen al bank) unde public con ol.” (Keynes (1936),
Chap. 17)
The goal o mac oeconomic policy is, mo o-wise,
“ ull employmen wi hou in la ion”. I is no igh o
be conce ned abou iscal su pluses o de ici s, since
hey a e me ely he means o ha goal and he esul
o ha goal. Whe he o no o epay go e nmen
deb wi h axes should be de e mined based on how i
will a ec p ices and employmen . Repaymen is no
a na u al assump ion, and whe he i is epaid o no
has no alue in i sel .
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 309
3. Holding o money and budge
de ici in a g owing economy
3.1. Consume s’ beha iou
U ili y unc ion
We conside an exogenous g ow h model in which
consume s in ini ely li e and hold money o he
eason o liquidi y and so on.
The consume ’s u ili y o e an in ini e ime is
∫∞
𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚
𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑. (1)
(1)
The u ili y unc ion is
𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�=𝛼𝛼𝛼𝛼ln 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+ (1 −𝛼𝛼𝛼𝛼)ln 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡. (2)
(2)
c is he eal alue o he consump ion by a consume ,
and p is he p ice o he good. m is he nominal alue
o he money holding o he consume . The e o e, m /p
is he eal money holding. The consume ’s u ili y
depends on he consump ion and he eal money
holding.
δ
>0 is he discoun a e. α is he p opensi y o
consume o he consume s. 0<α<1.
Budge cons ain
The budge cons ain o he consume is
=(1-
τ
)w l -p c - m +( -n)b .(3)
b is he pe capi a sa ings o he consume , and
is
he ime de i a i e o b . Gene ally, he ime de i a i e
o a a iable x is deno ed by
. w is he wage a e, l
is an indica o o whe he he consume is employed
o no . 1 i employed, 0 i no . The meanings o his
equa ion a e as ollows.
1.
is he change in he pe capi a alue o he
sa ings.
2. (1-
τ
)w l -p c is he di e ence be ween he pe
capi a disposable labou income and consump ion.
3. b -m is he po ion o he pe capi a sa ings
ha is in es ed in p oduc i e capi al, which gene a es
in e es ( e u n) .
4. We assume ha people li e in ini ely and no new
gene a ion will be bo n, howe e , he cu en gene a ion’s
popula ion will g ow. Hence he sa ings mus be
discoun ed by he g ow h a e as exp essed by -nb .
U ili y maximisa ion
The p esen alue Hamil onian is w i en as ollows.
𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�+𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡−𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡−𝑟𝑟𝑟𝑟
𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]. (4)
(4)
𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�+𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡−𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡−𝑟𝑟𝑟𝑟
𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]. (4)
λ
is he Lag ange mul iplie . The i s o de condi ions
a e
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒
−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡
𝛼𝛼𝛼𝛼
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡−𝜆𝜆𝜆𝜆
𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝
𝑡𝑡𝑡𝑡
= 0, (5)
(5)
and
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒
−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡
1−𝛼𝛼𝛼𝛼
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡−𝜆𝜆𝜆𝜆
𝑡𝑡𝑡𝑡
𝑟𝑟𝑟𝑟
𝑡𝑡𝑡𝑡
= 0. (6)
(6)
The cos a e equa ion is
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝑡𝑡𝑡𝑡
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=(𝑟𝑟𝑟𝑟
𝑡𝑡𝑡𝑡
−𝑛𝑛𝑛𝑛)𝜆𝜆𝜆𝜆
𝑡𝑡𝑡𝑡
=−𝜆𝜆𝜆𝜆
𝑡𝑡𝑡𝑡
. (7)
(7)
By (5) and (6), we ge
𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏
𝑡𝑡𝑡𝑡�,
and
𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏
𝑡𝑡𝑡𝑡�. (8)
They mean
𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿
𝜆𝜆𝜆𝜆𝛿𝛿𝛿𝛿= (1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝛿𝛿𝛿𝛿𝑙𝑙𝑙𝑙𝛿𝛿𝛿𝛿+(𝑟𝑟𝑟𝑟𝛿𝛿𝛿𝛿−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝛿𝛿𝛿𝛿−𝑏𝑏𝑏𝑏
𝛿𝛿𝛿𝛿.
Thus,
𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 =1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡.
Di e en ia ing his wi h espec o ,
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 310
𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡=−𝛿𝛿𝛿𝛿1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 =−𝛿𝛿𝛿𝛿𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡.
Then, om (7) we ind
=n+δ. (9)
This is he equilib ium in e es a e ( a e o e u n).
S eady s a e
Le us conside a s eady s a e. Unde cons an p ices,
w c , m and b a e cons an in he s eady s a e. They
a e he wage a e, and he pe capi a alues o eal
consump ion, nominal money holding and nominal
sa ings. Then,
=0.
On he o he hand, unde in la ion a a cons an a e
π
, c is cons an , bu w , m and b inc eases a he a e
o
π
. Then,
=b
π
.
Deno e he labou supply o he employmen unde
ull employmen by L
. Also, we deno e
B =b L
,C =c L
,M =m L
.
The eal alue o he capi al is
𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡.
Deno e he eal capi al pe labou unde ull
employmen by
𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡=𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡
𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓.
In he s eady s a e unde ull employmen ,
k
=0.
3.2. Fi ms’ beha iou
Le y be he ou pu , K be he capi al, and L be he
employmen o a i m. Then, he p oduc ion unc ion
is w i en as ollows.
y =F(K , L )=L (k )=L F(k ,1).
We assume he cons an e u ns o scale p ope y
o he p oduc ion unc ion. We no malise so ha
he numbe o i ms is one. Each i m maximises i s
p o i . The p o i o a i m is
p y -p K -w L =p L (k )-p K -w L .
The i s o de condi ions o p o i maximisa ion a e
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓′(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡),
and
𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡[𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡)−𝑓𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡)𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡].
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡
and
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
a e he ma ginal p oduc i i y o capi al and
ha o labou . F om hem, we ha e
w L =p [ (k )- ‘(k )k ] L ,
and
p K =p
‘ (k ) K = p
‘(k ) k L
Then, we ob ain
py y =w L +p K .
This is he o al nominal supply o he good.
The eal alue o he capi al inc eases a he a e
o n. Then,
𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡.
𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡
is he ime de i a i e o K . This inc ease in capi al
is he in es men . We assume ull employmen .
The e o e, l =1 o all people.
3.3. Ma ke equilib ium
We conside wo cases, wi h and wi hou in la ion.
Wi hou in la ion
Deno e he cons an p ice by
p
. In he s eady s a e
unde ull employmen and cons an p ice, he o al
consump ion demand is
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 311
p
C =L
p
c =α[(1-
τ
)w +( -n)b ] L
=α[(1-
τ
)w L
+( -n)B ].
The o al money holding is
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�. (10)
(10)
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�. (10)
Le G be he nominal alue o he iscal expendi u e.
The o al nominal demand is
G +α[(1-
τ
)w L
+( -n)B ]+
p
nK .
p
nK is he nominal in es men . The ma ke clea ing
condi ion is
G +α[(1-
τ
)w L
+( -n)B ]+
p
n K =w L
+
p
K .(11)
F om his we ob ain (see Appendix 1)
G -
τ
w L
=nM . (12)
So long as 0<α<1 and n>0, his is posi i e. I is
he budge de ici . The e o e, we ha e shown he
ollowing esul .
P oposi ion 1
I consume s’ u ili y depends on holding o money, a
budge de ici is necessa y o economic g ow h unde ull
employmen and cons an p ices.
Suppose ha M is gi en. I , p io o ha ime, ull
employmen was achie ed, he budge de ici shown
in (12) is necessa y and su icien o con inuous ull
employmen wi hou in la ion.
The policy o inc easing money h ough budge
de ici s in line wi h he a e o economic g ow h as
indica ed by (12) is consis en wi h he Mone a is
(F iedman’s) k% ule (Hal on, 2023).
Unde in la ion a a cons an a e o π
In his case,
=
π
b .
The e o e, (10), (11), (A-1) in Appendix 1 and (12)
a e ew i en as
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�,
𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡+𝛼𝛼𝛼𝛼�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+ (𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡,
𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡+𝛼𝛼𝛼𝛼�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+ (𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡,
𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡−𝜏𝜏𝜏𝜏𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝛼𝛼𝛼𝛼�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡+𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓,
𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡−𝜏𝜏𝜏𝜏𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝛼𝛼𝛼𝛼�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡+𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓,
and
G -
τ
w L
=nM +
π
B .
This is g ea e han he alue in (12). I means he
ollowing esul s.
P oposi ion 2
1. A budge de ici la ge han i s alue unde ull
employmen and cons an p ices in (12) leads o
in la ion.
O ,
2. To achie e ull employmen unde in la ion, a
budge de ici la ge han ha unde cons an p ices in
(12) is equi ed.
4. Explici alues o he sa ings
and money holding in he
s eady s a e
Wi hou in la ion
F om (9) in he s eady s a e, he alue o he in e es
a e is
=n+
δ
.
Deno e his alue wi h . Then, he capi al-labou
a io, k , which is cons an in he s eady s a e, sa is ies
‘(k )= . (13)
Deno e his alue o k wi h k. Then, he s eady-s a e
alue o he capi al is
𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡=𝑘𝑘𝑘𝑘
�𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓,
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 312
also, we ha e
𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡=𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑝 . (14)
(14)
The s eady-s a e alue o he nominal wage a e is
𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝�𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘
�)−𝑓𝑓𝑓𝑓′(𝑘𝑘𝑘𝑘
�)𝑘𝑘𝑘𝑘
��.
The s eady-s a e alue o he money holding is
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�
.
I is ew i en as
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟 𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡. (15)
(15)
F om (14),
𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡 (16)
(16)
By (15) and (16), we ob ain
𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟
�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟
�−𝑛𝑛𝑛𝑛
𝑟𝑟𝑟𝑟
�
. (17)
(17)
This is he explici solu ion o he alue o he sa ings.
By simila calcula ions, we ge
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟
�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟
�−𝑛𝑛𝑛𝑛
𝑟𝑟𝑟𝑟
�−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡 (18)
(18)
=
(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟
=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟.
=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
�𝑡𝑡𝑡𝑡
𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟.
This is he explici solu ion o he alue o he money
holding.
Unde in la ion a a cons an a e o
π
Unde in la ion a a cons an a e o
π
,
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�.
The e o e, we ge
𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡=
(1−𝛼𝛼𝛼𝛼)1
𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡
1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟
(19)
and
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡
𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟) (20)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡
𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟).
𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡
𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟) (20)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡
𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾
�𝑡𝑡𝑡𝑡
𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟).
5. Go e nmen bond holding in
he s eady s a e
In his sec ion, we conside he case whe e inancial
asse s a e held no in money bu in in e es -p oducing
go e nmen bonds ha ha e almos he same liquidi y
as money.
Budge cons ain and u ili y maximisa ion
Le i< be he in e es a e o he go e nmen bonds.
I is usually smalle han he a e o e u n on capi al
o he isk p emium. The budge cons ain o he
consume is1
b =(1-
τ
) w l -p c -( -i) m +( -n) b .
im is he in e es income om he go e nmen bond
holding. Excep o ha , i is he same as (3).
The consump ion and he go e nmen bond
holding a e
𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏
𝑡𝑡𝑡𝑡�,
and
𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=�1−𝛼𝛼𝛼𝛼
𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑖𝑖𝑖𝑖��(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏
𝑡𝑡𝑡𝑡�. (21)
(21)
1 We do no conside axa ion o in e es on go e nmen
bonds, bu i could be included. An in e es ax would
educe he likelihood o a di e gence o he deb - o-GDP
a io.
CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 319
Since, om (26)
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=(𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡)𝜋𝜋𝜋𝜋
, we ob ain
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=�𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋.
This means
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋.
The e o e,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋.
Again by (21),
(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡.
F om his,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋
=��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋.
By
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=(𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡)𝜋𝜋𝜋𝜋,
𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋=𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋.
Thus,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡= ��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋,
The e o e,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋= ��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�(𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋)
=�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡.
Then, by (28)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡��=�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼�𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�− 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋 =�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖
1−𝛼𝛼𝛼𝛼−𝜋𝜋𝜋𝜋� 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�.
Since - n=
δ
,
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡��=�𝑖𝑖𝑖𝑖−𝑛𝑛𝑛𝑛−𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼−(1 −𝛼𝛼𝛼𝛼)𝜋𝜋𝜋𝜋
1−𝛼𝛼𝛼𝛼 �𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡
𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�.
