Zakha o a, K is ina Alekseye na; Mu a ye , Danil Ana olye ich; Ka agulian, Egine
A a a o na
A icle
Modeling o complex s a e inancial suppo o small and
medium-sized en e p ises
Economies
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Sugges ed Ci a ion: Zakha o a, K is ina Alekseye na; Mu a ye , Danil Ana olye ich; Ka agulian,
Egine A a a o na (2024) : Modeling o complex s a e inancial suppo o small and medium-sized
en e p ises, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 12, Iss. 7, pp. 1-18,
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Ci a ion: Zakha o a, K is ina
Alekseye na, Danil Ana olye ich
Mu a ye , and Egine A a a o na
Ka agulian. 2024. Modeling o
Complex S a e Financial Suppo o
Small and Medium-Sized En e p ises.
Economies 12: 188. h ps://doi.o g/
10.3390/economies12070188
Academic Edi o : Ka a ina
Valasko a
Recei ed: 5 June 2024
Re ised: 12 July 2024
Accep ed: 14 July 2024
Published: 18 July 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
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4.0/).
economies
A icle
Modeling o Complex S a e Financial Suppo o Small and
Medium-Sized En e p ises
K is ina Alekseye na Zakha o a * , Danil Ana olye ich Mu a ye and Egine A a a o na Ka agulian
Depa men o Economics and Finance, Finance and Economics Ins i u e, Uni e si y o Tyumen, 625003 Tyumen,
Russia; [email p o ec ed] (D.A.M.); [email p o ec ed] (E.A.K.)
*Co espondence: k [email p o ec ed]
Abs ac : This a icle desc ibes a new app oach o de e mining he op imal amoun o s a e inancial
suppo p o ided o business en i ies. I is shown ha he e a e h ee a ailable me hods o suppo
economic agen s. The mos cos -e ec i e op ion is subsidizing business en i ies o expand hei
cu en asse s. I has been e ealed ha he e a e no jus op imal amoun s o go e nmen inancial
suppo bu also op imal no - o-exceed amoun s ha make i possible o iden i y he bounda ies
o he so-called highly p oduc i e s a e o he economy. In his case, when he economy is highly
p oduc i e, he p ices o goods (se ices) all, wo ke s spend hei sa ings, and he olume o
p oduc ion inc eases. This ul ima ely leads o an inc ease in he well-being o he popula ion. The
di e en ial equa ions a e he basis o he model, which is simila o he model o a simple wo-sec o
single-p oduc economy. The Mon e Ca lo me hod is used o de e mine he op imal no - o-exceed
amoun o go e nmen inancial suppo . The iden i ica ion o such in e als allows us o de e mine
he amoun o s a e inancial suppo ha will lead o a highly p oduc i e s a e and will no con ibu e
o an un easonable expansion o he budge expendi u e. This s udy’s esul s can be u ilized by
go e nmen au ho i ies o he de elopmen o a comp ehensi e sys em o s a e inancial suppo o
en ep eneu ship. Business en i ies can use he esul s o his esea ch conce ning he calcula ion o
he op imal no - o-exceed amoun o inancial suppo .
Keywo ds: s a e inancial suppo ; no - o-exceed amoun ; subsidies; ma hema ical modeling; wo-
sec o economy
1. In oduc ion
One o he common p oblems in he unc ioning o companies is he lack o inancial
esou ces o hei u he de elopmen . Business en i ies s i e o minimize ope a ing
expenses and u n o ax planning and op imiza ion o expenses o o e come his obs acle
(Rowse 1985;Assidi e al. 2016). Recei ing inancial suppo om he go e nmen is consid-
e ed o be an al e na i e op ion. Types o s a e inancial suppo o business en i ies a y
depending on he legal amewo k and cha ac e is ics o na ional economies. Subsidies,
gua an ees, concessional inancing on a e undable basis, g an s, and ax suppo a e he
mos p e alen and common ypes o business suppo in mos coun ies.
The scien i ic communi y and public adminis a ion ecognize he impo ance o public
inancial suppo due o di e en easons. On he one hand, he key asks o he s a e’s
inancial suppo o economic en i ies a he mac o le el include s imula ing business
de elopmen , inc easing g oss ou pu , c ea ing addi ional jobs, imp o ing he wel a e o he
popula ion, and inc easing ax con ibu ions o he budge . On he o he hand, one o he
pu poses o public inancial suppo is o s imula e he c ea ion o o ganiza ions, especially
in c isis si ua ions ( o example, business suppo du ing he COVID-19 pandemic (Honda
e al. 2023;Nguyen e al. 2023)). The inancial condi ion and sus ainabili y o he en i y can
be posi i ely impac ed by go e nmen al inancial suppo a he mic o le el.
Economies 2024,12, 188. h ps://doi.o g/10.3390/economies12070188 h ps://www.mdpi.com/jou nal/economies
Economies 2024,12, 188 2 o 18
De e mining he op imal no - o-exceed amoun o s a e inancial suppo and i s
impac on he company’s key indica o s is a challenge in p o iding s a e inancial suppo
o economic en i ies. Fi s ly, he expendi u e po ion o he consolida ed s a e budge
should no be un easonably inc eased by p o iding public inancial suppo . Secondly,
he economic impac o public inancial suppo should be assessed. The dis ibu ion o
inancial esou ces should be balanced among economic agen s.
In his s udy, we de eloped a ma hema ical model o a simple wo-sec o (p i a e
closed) economy ha includes he amoun o s a e inancial suppo gi en o en ep eneu s.
The aim o his s udy is o de elop a me hod o de e mining he op imal no - o-exceed
amoun o s a e inancial suppo gi en o economic en i ies, which will enable he e i i-
ca ion o he e iciency and e ec i eness o suppo measu es o di e en sec o s o he
economy. The ollowing asks will be ca ied ou o achie e his objec i e:
1.
Desc ibe he equi ed ools used o ma hema ical modeling o complex s a e inancial
suppo o business en i ies.
2.
Cons uc a comp ehensi e model o s a e inancial suppo o economic en i ies
unde condi ions o unce ain y and isk, allowing he de e mina ion o he socio-
economic de elopmen o egions and he conside a ion o he in e es s o he s a e
and business en i ies.
3.
Assess he adequacy o he buil model o he complex inancial suppo o busi-
ness en i ies.
The esul s o his s udy will con ibu e o he me hodology o p o iding s a e inan-
cial suppo o business en i ies and be used bo h by go e nmen bodies and economic
agen s o calcula e he op imal no - o-exceed amoun o inancial suppo planned o be
ecei ed. The di e ence be ween he de eloped app oach and he exis ing s udies lies
in he comp ehensi eness o he p esen ed ma hema ical model, i.e., i s applicabili y o
di e en sec o s o he economy, aking in o accoun one c i e ion. The no el y o he
app oach is de e mined h ough he calcula ion o no jus some op imal amoun o s a e
inancial suppo bu h ough he iden i ica ion o in e als o amoun s wi hin which he e
is a simul aneous sa is ac ion o he in e es s o he s a e and business en i ies.
2. Li e a u e Re iew
A p esen , he e is no single uni ied heo e ical app oach o calcula ing he op imal
no - o-exceed amoun le el o go e nmen inancial suppo o en e p ises. On he one
hand, his is due o di e en legal amewo ks o p o iding go e nmen suppo in
di e en coun ies. On he o he hand, his is due o he he e ogenei y o he suppo
measu es p o ided. The analysis o he scien i ic publica ions ela ed o he esea ch opic
made i possible o iden i y wo app oaches o embedding go e nmen al inancial suppo
o i ms in economic–ma hema ical models. Fi s , he amoun o inancial suppo can be
conside ed o be explained a iable in he model. Acco ding o esea che s, his app oach
dis inguishes a ew ac o s ha de e mine suppo le els wi h some ce ain y (Yu e al.
2023;Fan e al. 2024).
The second app oach is he mos common one. I analyzes he impac o p o iding
s a e inancial suppo o business en i ies on ce ain economic a iables.
Mos o he li e a u e ha deals wi h he impac o public subsidies ocuses on g an s
o esea ch and de elopmen . The e a e some no able na ional impac analyses o g an s
o young companies (Cza ni zki and Delano e 2015;P ei e and Reize 2000;Dec ame
and Vano melingen 2016;Colombo e al. 2014). Liu e al. (2016) and Calo i e al. (2018)
examined how he p o ision o R&D subsidies in luences business ac o s who in es in
R&D. Some pape s (Bu le e al. 2010;C epon and Dugue 2003;P ei e and Reize 2000)
assessed he impac on i m ou comes, such as su i al and i m pe o mance, while o he s
assessed he p obabili y o ecei ing a subsidy (González and Pazó2008). Mos o he
e idence shows posi i e e ec s on su i al and/o pe o mance. E en hough pe iods o
economic down u n a e ecognized as being pa icula ly dange ous o young i ms, he e
is li le empi ical e idence on he impac o g an s du ing ecessions. Bu ge and Rojec
Economies 2024,12, 188 3 o 18
(2018), A is ei e al. (2016) and Hud and Hussinge (2015) assessed he impac o R&D
g an s on R&D spending du ing he las ecession and ound a posi i e e ec .
The s udy by Sha e al. (2024) e alua ed he impac o ag icul u al subsidies on he
o e all wel a e o a me s. Kállay and Takács (2023) in es iga ed he impac o he p o ision
o go e nmen subsidies on he economic g ow h o i ms.
Se e al s udies ha e been de o ed o he analysis o he impac o p o iding s a e
inancial suppo o business en i ies on employmen and unemploymen le els in he
economy and on he labo ma ke in gene al (Cingano e al. 2023;B ans e e e al. 2023;
Tübbicke 2024). Job g ow h a es o small i ms a e assis ance a e highe han in la ge
i ms; howe e , la ge i ms p o ide many mo e jobs a a lowe cos . Hube e al. (2017)
claimed ha young i ms con ibu e mos o ne job c ea ion despi e high ela i e exi a es.
This is due o he high g ow h a es o young su i ing i ms. The job c ea ion a es o
small i ms a e simila o hose o la ge i ms, condi ional on hei su i al. Howe e ,
smalle companies con ibu e less o job c ea ion due o highe exi a es (Hube e al. 2017).
McKenzie e al. (2016) a gued ha go e nmen in e en ion is po en ially jus i ied
because small i ms unde in es in lea ning o easons such as unde es ima ing bene i s,
spillo e s o compe i o s, o c edi ma ke impe ec ions. To acili a e lea ning, business
de elopmen g an s o en include unding o aining and pu chasing consul ancy se ices.
The esul ing inc ease in business skills and manage ial capi al is a possible mechanism
h ough which hese business de elopmen g an s can gene a e addi ionali y. Highe
manage ial capi al can lead o be e s a egic and ope a ional decisions and o mo e
e icien use o o he ac o s such as physical capi al and labo (B uhn e al. 2012). The e
is also e idence ha highe le els o human capi al ha e an impac on he su i al o
young i ms h ough he e ec on hei ini ial size (Colombo and G illi 2005). Fo example,
speci ic business skills acqui ed h ough de elopmen g an s ha e a pa icula ly high alue,
a ec ing bo h su i al and g ow h and business pe o mance (Backman e al. 2016).
O he s udies ha e a emp ed o assess he impac o p o iding di ec inancial suppo
o i ms on he ac o p oduc i i y (Lin and Zhang 2024), while o he s epo ed a limi ed ole
and hus an impac ha is di icul o iden i y (B ans e e e al. 2023). S udies analyzing he
impac o s a e inancial suppo o economic en i ies on popula ion wel a e a ac a en ion.
I is a well-known ac ha he economic ca ego y o “ he wel a e o he popula ion” is a
complex one, and i can be conside ed as a ious economic indica o s (Walke e al. 2024).
Howe e , i is impo an o some esea ch o assess he in luence o go e nmen inancial
suppo on people’s wel a e and, a pe capi a income le els, social inequali y (Vigani e al.
2024;Sha e al. 2024;Oko ie and Wesseh 2024).
Some scien is s a e in e es ed in examining he p o ision o go e nmen inancial
suppo o business en i ies a he meso le el (Sha e al. 2024;Be nini and Galli 2024;
Soepa na and Tao iqu ohman 2024;Zhao e al. 2024;Lin and Zhang 2024;Guo e al. 2024;
Rajpu and Venka a aman 2024). Recen ly, he e has been an inc eased publica ion ac i i y
ega ding he impac o go e nmen inancial suppo o en e p ises on he sus ainable
de elopmen o indi idual egions o sec o s ela ed o he de elopmen o he “g een”
economy (Guo e al. 2024;Zhao e al. 2024;Be nini and Galli 2024).
The impac o p o iding ce ain ypes o inancial suppo has been assessed by se e al
esea che s. Fo example, s udies by Be ge and Ho en o (2021), Cha eddine and Zaouali
(2022), Muelle (2023), and Tübbicke (2024) examined he impac o go e nmen subsidies
on i ms in he ea ly phases o hei li e cycles.
The s udy by S hoj e al. (2020) p esen ed new mic oeconome ic e idence on he
impac o business de elopmen g an s on ou pu and inpu addi ionali y. P e ious esea ch
by he au ho s has mainly ocused on he impac o R&D g an s, while he e is li le
empi ical e idence on he impac o business de elopmen g an s despi e he policy in e es
o his ype o g an (D oule ýe al. 2021;Cadil e al. 2020).
The ole o business de elopmen g an s was examined by S hoj e al. (2020). Business
de elopmen g an s a e smalle g an s. The e o e, hey may con ibu e o he isk o
subs i u ing public money o p i a e money a he han c ea ing an addi ional bene i . As a
Economies 2024,12, 188 4 o 18
esul , a numbe o esea che s ha e ques ioned he e ec i eness o such subsidies, i.e., he
wisdom o using public unds, pa icula ly in de eloping coun ies (OECD e al. 2016).
The esul s o S hoj e al. (2020) showed, on a e age, s ong posi i e e ec s o business
de elopmen g an s on capi al s ock, bank loans, in e media e inpu s, and alue added,
bu no empi ical e idence o a posi i e e ec on p oduc i i y measu es, sales, employmen ,
a e age wage, and in en o ies. Howe e , he signi ican esul s we e based on he posi i e
e ec s on smalle i ms. The posi i e e ec on he g ow h o small i ms is in line wi h
he s udies o Bia and Ma ei (2012), Munch and Schau (2018), Be oni e al. (2019), and
C iscuolo e al. (2019).
Mo e a o able e ec s o business de elopmen g an s on capi al, bank loans, employ-
men , alue added, sales, and labo p oduc i i y o smalle i ms a e heo e ically jus i ied.
This is because he speci ic g an amoun is be e designed o ease he capi al cons ain s
o smalle i ms compa ed o la ge i ms.
Looking a he mechanism o he e ec s sugges s ha he main e ec o he subsidy
has been o enable i ms o aise mo e capi al. The s ong ce i ica ion e ec also means ha ,
on a e age, subsidized i ms ecei e an e en la ge bank loan han he amoun conside ed
when g an ing he subsidy. O e all, public unding o smalle i ms inc eases capi al s ock
by almos wice he g an amoun , unlocks bank loans o 1.5 imes ha amoun , sa es jobs,
and c ea es economic alue abo e he g an amoun (Sk inja ic e al. 2021;Pelleg ini and
Muccig osso 2016;Ogujiuba e al. 2022,2023). To u he explo e he ques ion o wha le el
o subsidy migh be oo low o whe e he u ning poin om subs i u ion o addi ionali y
lies, we calcula e i . As in Biagi e al. (2015), inc easing he amoun o g an s induces la ge
e ec s on small i m pe o mance measu es.
The esea che s ound ha subsidies a his s age in he de elopmen o a business
ha e a posi i e e ec on he subsequen de elopmen o he business uni s. I should be
no ed ha he esea che s mos o en used an econome ic app oach and less game heo y
and ma hema ical models when modeling go e nmen inancial suppo o business uni s.
Thus, in he e iewed s udies, he mechanism o s a e inancial suppo was analyzed
in a agmen ed manne depending on he di ec ion o suppo , he sec o o he economy,
he s age o he li e cycle o he objec o suppo , and o he c i e ia. We ha e been gi en a
new challenge. This is he cons uc ion o a complex model o s a e inancial suppo o
subs an ia e he condi ions and e ec i eness o suppo measu es o di e en sphe es o
economic ac i i y when deciding on hei p o ision o business en i ies.
3. Me hodology
In addi ion o he p e ious s udies, we de eloped a model o h ee op ions o go -
e nmen inancial suppo and assessed hei impac on wo ke s’ sa ings, he p ice o he
p oduc (se ice), and he olume o ou pu .
Fo his pu pose, we use nume ical ma hema ical modeling o economic p ocesses
and he Mon e Ca lo me hod. The de eloped app oach allows he iden i ica ion o such
amoun s o anges o amoun s o s a e inancial suppo o business en i ies ha will
achie e a simul aneous inc ease in he sa ings o employees and he olume o p oduc s
p oduced in he economy and a dec ease in he p ice o goods (se ices).
In his esea ch, we conside ed a model o a simple wo-sec o (p i a e closed) econ-
omy. The model is a single p oduc :
dUm
d =n
mnp( )QhUn
p( )i−Pno
dp
d =γnnQhUn
p( )i+mQhgUm
p( )i−F[V, n′, p, τ]o
Q(ξ)=QI.0 ξ
ξ
0+ξ +εξ
F[V, p, n′,τ]=(V
p( )τ,V
p( )n′τ<Fmax
n′Fmax,V
p( )n′τ≥Fmax
Pn=hm
nF[(1−g)Um, p, n′,τ]
Un=M−m′Um
n′
(1)
Economies 2024,12, 188 5 o 18
The assump ions o he p esen ed ma hema ical model a e as ollows: This model
cha ac e izes a socie y composed o wo g oups such as he owne s o en e p ises and
he employees. En ep eneu s a e economic agen s. They ecei e income om he sale o
goods p oduced in hei en e p ises and pay wages P
n
o hei employees. The numbe o
employees includes no only he di ec p oduce s o goods bu also se ice wo ke s and
wo ke s employed in he s uc u al di isions o he i m.
The sa ings o he employe s and employees a e di e en . Howe e , o he sake o
he law o conse a ion o money, we assume ha hey a e he same in each g oup:
n′+m′=N, (2)
n′Un+m′Um=M, (3)
n=n′
N; m =m′
N; n +m=1, (4)
n·Un+m·Um=U=M
N, (5)
whe e n
′
is he popula ion size o wo ke s, m
′
is he popula ion size o business owne s,
U
is he a e age sa ings, and M is he money supply.
The sa ing dynamics o he en ep eneu s U
m
is de e mined by he balance o income
and expendi u e, which in a single-p oduc model is educed o he wages o he wo ke s
Pn. The wages o wo ke s in he model a e piecewo k wages.
The dynamics o wo ke s’ sa ings U
n
is no an independen a iable, and i is de e -
mined by he law o conse a ion o money:
n′Un+m′Um=M⇒Un=M−m′Um
n′, (6)
The p ice o a p oduc (se ice) p is based on he supply and he demand and he
speed o i s in oduc ion o he ma ke
γ
. The demand (consump ion) o employe s and
employees is desc ibed by he Tö nq is unc ion. The ac i i y o he en e p ise is desc ibed
by a p oduc ion unc ion F, which depends on he cu en asse s V, he popula ion size o
p oduce s (employees) n
′
, he p ice o goods (se ices) p, and he du a ion o he in en o y
u no e ( u no e ime)
τ
, calcula ed acco ding o Wilson’s o mula. The p oduc ion
unc ion has an uppe limi , F
max
, which is de e mined by he le el o echnology, he
o ganiza ion o p oduc ion, and he physical abili ies o he wo ke s.
The me hodology o calcula ing he coe icien h is no explici ly p esen ed in he
scien i ic li e a u e. Based on he seman ic con en o he indica o ela ed o he p esen ed
model, we unde s and i as a coe icien cha ac e izing he addi ional p oduc o an eco-
nomic uni . The sys em o equa ions can be sol ed using he ou h-o de Runge–Ku a
me hod (Wilkie 2004). This me hod is cha ac e ized by a highe accu acy compa ed o
o he nume ical modeling me hods. We use he inancial and managemen epo ing da a
o he business uni o de e mine he inpu pa ame e s o he model. A ma hema ical model
speci ied by a sys em o equa ions is a basic model. I s pu pose is o cla i y he na u e o
he desc ibed phenomenon o p ocess.
4. Da a Analysis and Resul s
In he i s s age o his s udy, we analyze he ime dependence o sa ings o en-
ep eneu s and employees wi hou including hem in he model o s a e inancial suppo .
Figu e 1shows he ime dependence o he sa ings made by en ep eneu s and employees
o an economic uni ha ope a es unde condi ions o unce ain y and isk. The sampling
s ep is 10−1. I he s ep is hal ed, he simula ion esul s change by less han 10%.
Economies 2024,12, 188 6 o 18
Economies 2024, 12, x FOR PEER REVIEW 6 o 19
me hod (Wilkie 2004). This me hod is cha ac e ized by a highe accu acy compa ed o
o he nume ical modeling me hods. We use he inancial and managemen epo ing da a
o he business uni o de e mine he inpu pa ame e s o he model. A ma hema ical
model speci ied by a sys em o equa ions is a basic model. I s pu pose is o cla i y he
na u e o he desc ibed phenomenon o p ocess.
4. Da a Analysis and Resul s
In he i s s age o his s udy, we analyze he ime dependence o sa ings o en e-
p eneu s and employees wi hou including hem in he model o s a e inancial suppo .
Figu e 1 shows he ime dependence o he sa ings made by en ep eneu s and employees
o an economic uni ha ope a es unde condi ions o unce ain y and isk. The sampling
s ep is 10−1. I he s ep is hal ed, he simula ion esul s change by less han 10%.
Figu e 1. Time dependence o sa ings o business owne s Um( ) and wo ke s Un( ). No e: n = 0.9056,
m = 0.0944, γ = 7.94, QI.0 = 13,919, g = 0.8 (ma ginal p opensi y o business owne s o consume), V =
2,816,137,000, Fmax = 500,033, τ = 33, h = 0.8, M = 34,141,693,993.05, n′ = 240, m’ = 25, U =
32,400.19, p = 107.28.
The sa ings o business owne s he eina e a e calib a ed by 10−10 imes, and he p ice
o goods (se ices) is calib a ed by 10−5 imes.
A mo e de ailed analysis shows ha sa ings a e posi i ely ime-dependen . This is
because he owne s o an economic uni end o sa e h oughou he yea . In o he wo ds,
owne s end o sa e h oughou he inancial yea .
Howe e , a mo e de ailed analysis shows ha , in he i s segmen o he modeling,
he sa ings cu e o he owne s is loca ed below he abscissa cu e (Figu e 2, be ween he
wo g een pa allel lines). Only la e does i mo e in o he i s quad an o he coo dina e
plane (a he 42nd ime uni ).
Figu e 1. Time dependence o sa ings o business owne s U
m
( ) and wo ke s U
n
( ). No e: n = 0.9056,
m = 0.0944,
γ
= 7.94, Q
I.0
= 13,919, g = 0.8 (ma ginal p opensi y o business owne s o consume),
V = 2,816,137,000
, F
max
= 500,033,
τ
= 33, h = 0.8, M = 34,141,693,993.05, n
′
= 240, m
′
= 25,
Umini ial
=
32,400.19, pini ial = 107.28.
The sa ings o business owne s he eina e a e calib a ed by 10
−10
imes, and he p ice
o goods (se ices) is calib a ed by 10−5 imes.
A mo e de ailed analysis shows ha sa ings a e posi i ely ime-dependen . This is
because he owne s o an economic uni end o sa e h oughou he yea . In o he wo ds,
owne s end o sa e h oughou he inancial yea .
Howe e , a mo e de ailed analysis shows ha , in he i s segmen o he modeling,
he sa ings cu e o he owne s is loca ed below he abscissa cu e (Figu e 2, be ween he
wo g een pa allel lines). Only la e does i mo e in o he i s quad an o he coo dina e
plane (a he 42nd ime uni ).
Economies 2024, 12, x FOR PEER REVIEW 6 o 19
me hod (Wilkie 2004). This me hod is cha ac e ized by a highe accu acy compa ed o
o he nume ical modeling me hods. We use he inancial and managemen epo ing da a
o he business uni o de e mine he inpu pa ame e s o he model. A ma hema ical
model speci ied by a sys em o equa ions is a basic model. I s pu pose is o cla i y he
na u e o he desc ibed phenomenon o p ocess.
4. Da a Analysis and Resul s
In he i s s age o his s udy, we analyze he ime dependence o sa ings o en e-
p eneu s and employees wi hou including hem in he model o s a e inancial suppo .
Figu e 1 shows he ime dependence o he sa ings made by en ep eneu s and employees
o an economic uni ha ope a es unde condi ions o unce ain y and isk. The sampling
s ep is 10−1. I he s ep is hal ed, he simula ion esul s change by less han 10%.
Figu e 1. Time dependence o sa ings o business owne s Um( ) and wo ke s Un( ). No e: n = 0.9056,
m = 0.0944, γ = 7.94, QI.0 = 13,919, g = 0.8 (ma ginal p opensi y o business owne s o consume), V =
2,816,137,000, Fmax = 500,033, τ = 33, h = 0.8, M = 34,141,693,993.05, n′ = 240, m’ = 25, U =
32,400.19, p = 107.28.
The sa ings o business owne s he eina e a e calib a ed by 10−10 imes, and he p ice
o goods (se ices) is calib a ed by 10−5 imes.
A mo e de ailed analysis shows ha sa ings a e posi i ely ime-dependen . This is
because he owne s o an economic uni end o sa e h oughou he yea . In o he wo ds,
owne s end o sa e h oughou he inancial yea .
Howe e , a mo e de ailed analysis shows ha , in he i s segmen o he modeling,
he sa ings cu e o he owne s is loca ed below he abscissa cu e (Figu e 2, be ween he
wo g een pa allel lines). Only la e does i mo e in o he i s quad an o he coo dina e
plane (a he 42nd ime uni ).
Figu e 2. Time dependence o sa ings o owne s U
m
( ) and wo ke s U
n
( ) a he ini ial segmen o
he simula ion. No e: n = 0.9056, m = 0.0944,
γ
= 7.94, Q
I.0
= 13,919, g = 0.8, V = 2,816,137,000,
Fmax = 500,033
,
τ
= 33, h = 0.8, M = 34,141,693,993.05, n
′
= 240, m
′
= 25,
Umini ial
= 32,400.19,
pini ial = 107.28.
F om he poin o iew o mac oeconomic heo y, especially Keynesian, i is obse ed
ha a he beginning o he inancial yea , he owne s o a gi en economic uni spend
hei sa ings o sa is y consume demand. Fu he , wi h he g ow h o disposable income,
he sa ings cu e in he speci ied ime uni c osses he abscissa axis. A he in e sec ion
poin , he e is equali y o disposable income mul iplied by he ma ginal p opensi y o sa e,
au onomous consume spending.
Economies 2024,12, 188 7 o 18
Acco ding o equa ion U
n
( he six h equa ion in he sys em o equa ions in Equa-
ion (1)), he sa ings made by employees a e in e sely ela ed o hose made by en-
ep eneu s. F om he poin o iew o economic physics, his is explained by he law
o conse a ion o money. Acco ding o co po a e inance heo y, i is explained by he
agency con lic . Thus, employees accumula e sa ings in he ini ial phase o he simula ion
and spend hem in he inal phase (a he end o he inancial yea ), as shown in Figu es 1
and 2; while en ep eneu s spend hei sa ings wi hin a sho pe iod, employees spend o
a inancial yea .
To assess he impac o go e nmen al inancial suppo on he model a iables, we
add he indica o Sub o he model, which cha ac e izes he amoun o suppo p o ided in
mone a y e ms. In his case, he e a e h ee op ions o he dis ibu ion o he inancial e-
sou ces: (1) p o iding s a e inancial suppo o bo h g oups o economic agen s (undi ec ed
expendi u e); (2) gi ing s a e inancial suppo o employees (s imula ion o employee
demand); (3) p o iding s a e inancial suppo o en ep eneu s (s imula ion o supply)
(op ions (2) and (3)—di ec ed expendi u e). Then, we explo ed each o hese op ions.
I inancial suppo is p o ided by he go e nmen o bo h g oups o economic agen s
(business owne s and employees), he money supply in he economy will inc ease by he
amoun Sub, so ha M
′
= M + Sub. In his case, he inal sa ings o en ep eneu s and
employees will no change o will change sligh ly (less han 0.001%) only i a signi ican
amoun o unds is p o ided (wi h Sub > 11,000,000,000). This is no compa able o he
concep o “ inancial suppo ”. This allows us o conclude ha he p o ision o go e nmen
inancial suppo in he o m o a non-add essed issue does no ha e he expec ed economic
e ec in a model o a simple wo-sec o closed economy (based on cu en model da a).
Con e sely, i go e nmen inancial suppo is p o ided o wo ke s in o de o s im-
ula e demand and he subsequen expec ed inc ease in supply, he equa ion o wo ke s’
sa ings (6) is modi ied:
Un=M−m′·Um
n′+Sub, (7)
Howe e , his ype o suppo does no ha e he necessa y e ec on he economy.
The inal alues o he sa ings o en ep eneu s and employees change only in he case o
signi ican olumes o inancial esou ces p o ided (wi h Sub > 43,000,000). Ob iously, he
limi ed budge does no allow such inancial suppo o en ep eneu s.
P o iding go e nmen inancial suppo o business owne s can be ca ied ou in wo
di ec ions. Fi s ly, inancial suppo can be p o ided h ough di ec suppo o owne s in
o de o s imula e he demand o business owne s. Howe e , his op ion is conside ed
socially unaccep able. I inc eases social ensions and imbalances and, he e o e, does no
con ibu e o he expec ed economic e ec . Secondly, inancial suppo can be p o ided
h ough in es men s (subsidies, sub en ions, and o he simila o mula ions, depending
on he legal and egula o y amewo k o go e nmen inancial suppo in a gi en coun y)
in he company’s cu en asse s so ha V
′
= V + Sub. In his case, i is o emos necessa y o
de e mine he le el o go e nmen al inancial suppo ha will allow sus ainable social
and economic e ec s o be achie ed, such as educing he sa ings made by employees,
educing he p ice o he p oduc (se ice), and inc easing he olume o p oduc ion in
compa ison wi h he ini ial si ua ion, i.e., in he absence o go e nmen inancial suppo
o economic en i ies.
To de e mine he op imal amoun o go e nmen al inancial suppo , we will use he
Mon e Ca lo me hod. The Mon e Ca lo me hod allows, by changing one o a numbe o
inpu pa ame e s (i e a ions), he alues o he dependen a iable o be ob ained unde
o he equal condi ions. The esul s o he simula ion o he inal sa ings o business owne s
and employees, he p ice o goods (se ices), and he Mon e Ca lo ou pu a e p esen ed in
Figu e 3.
Economies 2024,12, 188 8 o 18
Economies 2024, 12, x FOR PEER REVIEW 8 o 19
employees, educing he p ice o he p oduc (se ice), and inc easing he olume o p o-
duc ion in compa ison wi h he ini ial si ua ion, i.e., in he absence o go e nmen inan-
cial suppo o economic en i ies.
To de e mine he op imal amoun o go e nmen al inancial suppo , we will use he
Mon e Ca lo me hod. The Mon e Ca lo me hod allows, by changing one o a numbe o
inpu pa ame e s (i e a ions), he alues o he dependen a iable o be ob ained unde
o he equal condi ions. The esul s o he simula ion o he inal sa ings o business own-
e s and employees, he p ice o goods (se ices), and he Mon e Ca lo ou pu a e p e-
sen ed in Figu e 3.
(a) (b)
(с) (d)
Figu e 3. Simula ion modeling o sa ings o business owne s (panel (c)) and employees (panel (a)),
p ices o goods (se ices) (panel (b)), and olume o ou pu (panel (d)). No e (1): n = 0.9056, m =
0.0944, γ = 7.94, QI.0 = 13,919, g = 0.8, V = 2,816,137,000, Fmax = 500,033, τ = 33, h = 0.8, M =
34,141,693,993.05, n′ = 240, m′ = 25, U = 32,400.19, p = 107.28; (2): le6 is equi alen o 10 o
he six h powe .
The amoun o go e nmen al inancial suppo a ies om 0 o 5·106 con en ional
mone a y uni s. The disc e iza ion s ep is 103 con en ional mone a y uni s. Black lines
co espond o he bounda ies o he 90% con idence in e al. The alues o he model a -
iables ha a e ou side he con idence in e al a e he esul o he hys e esis p ope y o
he model. In pa icula , he e a e amoun s o go e nmen suppo a which he alues o
modeled pa ame e s change sha ply. Mo eo e , wi h a sligh inc ease in he amoun o
go e nmen suppo , he economy e u ns o he con idence in e al zone.
Looking a he dynamics o sa ings by p op ie o s and employees, we ind ha in a
simple wo-sec o (p i a e closed) economy, due o he law o conse a ion o money, an
inc ease in he le el o sa ings by p op ie o s educes he le el o sa ings by employees
and ice e sa. The effec o his law is shown in Figu e 3 as a mi o image o he sa ings
o owne s and employees.
Figu e 3. Simula ion modeling o sa ings o business owne s (panel (c)) and employees (panel
(a)), p ices o goods (se ices) (panel (b)), and olume o ou pu (panel (d)). No e (1):
n = 0.9056
,
m = 0.0944,
γ
= 7.94, Q
I.0
= 13,919, g = 0.8, V = 2,816,137,000, F
max
= 500,033,
τ
= 33, h = 0.8,
M = 34,141,693,993.05
, n
′
= 240, m
′
= 25,
Umini ial
= 32,400.19,
pini ial
= 107.28; (2): le6 is equi alen o
10 o he six h powe .
The amoun o go e nmen al inancial suppo a ies om 0 o 5
·
10
6
con en ional
mone a y uni s. The disc e iza ion s ep is 10
3
con en ional mone a y uni s. Black lines
co espond o he bounda ies o he 90% con idence in e al. The alues o he model
a iables ha a e ou side he con idence in e al a e he esul o he hys e esis p ope y
o he model. In pa icula , he e a e amoun s o go e nmen suppo a which he alues
o modeled pa ame e s change sha ply. Mo eo e , wi h a sligh inc ease in he amoun o
go e nmen suppo , he economy e u ns o he con idence in e al zone.
Looking a he dynamics o sa ings by p op ie o s and employees, we ind ha in a
simple wo-sec o (p i a e closed) economy, due o he law o conse a ion o money, an
inc ease in he le el o sa ings by p op ie o s educes he le el o sa ings by employees
and ice e sa. The e ec o his law is shown in Figu e 3as a mi o image o he sa ings
o owne s and employees.
To assess he social and economic impac , we compa e wo ke s’ sa ings, he p ice
o goods (se ices), and he olume o p oduc ion wi h he ‘ e e ence’ alue. By de aul ,
we ake he alues o hese a iables in he absence o any inancial suppo om he
s a e o he economic en i ies in he economy. The esul s o his compa ison a e shown in
Figu e 4.
Economies 2024,12, 188 15 o 18
p inciples o he e icien use o budge unds, a ge ing, and he a ge ed na u e o budge
unds as p inciples o he cons uc ion o he s a e budge sys em.
Howe e , we do no claim ha using a ma hema ical model o s a e inancial suppo
should eplace checking i s e ec i eness. In addi ion, he in oduc ion o he de eloped
ma hema ical model should no cancel he p esco ing o economic en i ies o he igh o
ecei e s a e inancial suppo . I he coun y has high ins i u ional ba ie s o ecei ing
s a e inancial suppo , no model o calcula ing he amoun o s a e inancial suppo o
business en i ies will wo k and help ensu e a mul iplie e ec .
Such ins i u ional ba ie s include he la ge numbe o documen s equi ed o ob ain
s a e inancial suppo , he long ime i akes o conside a company’s applica ion o
suppo , which leads o a delay in he g ow h o economic ac i i y, and he lack o elec onic
documen low be ween he s a e and business en i ies. These ins i u ional ba ie s can
be elimina ed by a uni ied in elligen in o ma ion sys em ha , on he one hand, p o ides
in o ma ion on he condi ions and p ocedu es o ob aining all measu es o s a e inancial
suppo in he coun y.
On he o he hand, a pe sonal accoun o he business en i y is es ablished. The abili y
o di ec ly submi an applica ion and ack i s s a us should be among he ea u es o he
pe sonal accoun . I is di icul o business en i ies o use a ma hema ical model wi hou
access o he app op ia e so wa e o an in elligen in o ma ion sys em wi hou in eg a ed
p og am code. We unde s and ha he p obabili y o hi ing a p og amme o en e ing in o
a sepa a e con ac o he nume ical implemen a ion o a ma hema ical model is educed
o ze o. As a esul , we emphasize he need o in eg a ion o he p esen ed ma hema ical
model in o he s a e in elligen in o ma ion sys em. In his case, an en e p ise can easily
calcula e he a ailable amoun o s a e inancial suppo a e en e ing he inpu da a om
inancial and managemen epo ing.
A modi ica ion and complica ion o he in elligen in o ma ion sys em can also be
he in eg a ion o machine lea ning algo i hms in o i . These algo i hms make i possible
o gene a e managemen ecommenda ions o each speci ic business en i y in o de o
inc ease he possible amoun o suppo i can ecei e. In gene al, he anspa ency o
economic policy and con idence in i can be inc eased by using a single me hodology o
calcula ing he amoun o s a e inancial suppo o all o ganiza ions.
As a esul , he model allows he de e mina ion o he op imal no - o-exceed amoun
o suppo o business en i ies, he iden i ica ion o he e ec i eness o he measu es ha
we e used in he pas , and he possibili y o hei use in he u u e (in he case o he o ecas
o inpu pa ame e s).
Thus, he p oposed algo i hm is a new app oach o he calcula ion o he op imal no -
o-exceed amoun o go e nmen al inancial suppo o business en i ies. The p inciples
o his app oach include p ac ical easibili y, conside ing ex e nal and in e nal en i on-
men al ac o s, uni e sali y, and anspa ency. The esul s ob ained du ing his s udy a e
bidi ec ional. Fi s ly, hey can be used by go e nmen au ho i ies in he de elopmen o
me hods o he calcula ion o he op imal no - o-exceed amoun o go e nmen al inancial
suppo and hei subsequen implemen a ion in in elligen digi al in o ma ion pla o ms
o he suppo o business en i ies. Secondly, when de e mining he maximum op imal
no - o-exceed amoun o go e nmen al inancial suppo , he esul s o ma hema ical mod-
eling can also be used by business en i ies. Based on he conclusions d awn, i is possible o
o mula e app op ia e o ganiza ional and manage ial ecommenda ions on he mechanism
o op imal dis ibu ion o go e nmen al inancial suppo in he coun y in condi ions o
an uns able ex e nal en i onmen .
The p ospec s o u he esea ch a e seen in he ex ension o he model o a model o
a h ee-sec o economy h ough he inclusion o he ax a e. Fu he mo e, he model can
conside no only di ec axes (on p o i o income) bu also indi ec axes (VAT, excise, and
o he s, depending on he composi ion o a coun y’s ax sys em). This ans o ma ion o
he model will allow he assessmen o he e ec i eness o he p o ision o no only di ec
inancial suppo measu es bu also a ange o ax suppo measu es o business uni s.
Economies 2024,12, 188 16 o 18
Mo eo e , wi hin he amewo k o he p esen ed model, i is possible o di ide socie y
no only in o en ep eneu s and employees bu also in o pensione s, employees o s a e
and municipal en e p ises, employees o budge a y o ganiza ions (kinde ga ens, schools,
hospi als), go e nmen machine y, e c. On he one hand, a sepa a e a ea o he model is
de o ed o he assessmen o he e ec i eness o he p o ision o di ec inancial suppo
measu es, as well as a ange o ax suppo measu es.
On he o he hand, he iden i ica ion o me hods o calcula ing he mul iplie e ec o
s a e inancial suppo and i s calcula ion h ough he example o speci ic business en i ies
using he de eloped model can be a sepa a e a ea o esea ch.
Au ho Con ibu ions: Concep ualiza ion, K.A.Z. and D.A.M.; me hodology, K.A.Z.; so wa e,
D.A.M.; alida ion, K.A.Z., D.A.M. and E.A.K.; o mal analysis, E.A.K.; in es iga ion, K.A.Z., D.A.M.
and E.A.K.; esou ces, K.A.Z. and D.A.M.; da a cu a ion, D.A.M.; w i ing—o iginal d a p epa a ion,
K.A.Z., D.A.M. and E.A.K.; w i ing— e iew and edi ing, K.A.Z., D.A.M. and E.A.K.; isualiza ion,
K.A.Z. and D.A.M.; supe ision, K.A.Z.; p ojec adminis a ion, K.A.Z.; unding acquisi ion, K.A.Z.
All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch was unded by Russian Science Founda ion, g an numbe 23-28-01690,
h ps:// sc . u/p ojec /23-28-01690/ (accessed on 16 July 2024), Uni e si y o Tyumen.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : The da a used o suppo he indings o his s udy a e a ailable om
he co esponding au ho upon eques .
Con lic s o In e es : The au ho s decla e no con lic s o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip ; o
in he decision o publish he esul s.
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