Tussupo a, Kamsha ; Mu zabeko , Zainelkh ie
A icle
Op imal alloca ion o esou ces in an open economic
sys em wi h Cobb-Douglas p oduc ion and ade balances
Economies
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Tussupo a, Kamsha ; Mu zabeko , Zainelkh ie (2025) : Op imal alloca ion
o esou ces in an open economic sys em wi h Cobb-Douglas p oduc ion and ade balances,
Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 7, pp. 1-22,
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Recei ed: 2 May 2025
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Published: 26 June 2025
Ci a ion: Tussupo a, K., &
Mu zabeko , Z. (2025). Op imal
Alloca ion o Resou ces in an Open
Economic Sys em wi h Cobb–Douglas
P oduc ion and T ade Balances.
Economies,13(7), 184. h ps://doi.o g/
10.3390/economies13070184
Copy igh : © 2025 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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licenses/by/4.0/).
A icle
Op imal Alloca ion o Resou ces in an Open Economic Sys em
wi h Cobb–Douglas P oduc ion and T ade Balances
Kamsha Tussupo a * and Zainelkh ie Mu zabeko
Depa men o In o ma ion Sys ems, Al-Fa abi Kazakh Na ional Uni e si y, Al-Fa abi A e. 71/23,
Alma y 050040, Kazakhs an; [email p o ec ed]
*Co espondence: [email p o ec ed]u
Abs ac
This pape de elops a nonlinea op imiza ion model o he op imal alloca ion o labo
and in es men esou ces in a h ee-sec o open economy. The model is based on he
Cobb–Douglas p oduc ion unc ion and inco po a es sec o al in e dependencies, capi al
dep ecia ion, ade balances, and impo quo as. The esou ce alloca ion p oblem is o mal-
ized as a cons ained op imiza ion ask, sol ed analy ically using he Lag ange mul iplie s
me hod and nume ically ia he golden sec ion sea ch. The model is calib a ed using eal
s a is ical da a om Kazakhs an (2010–2022), an open esou ce-expo ing economy. The
esul s iden i y s uc u al h esholds ha de ine balanced g ow h condi ions and esou ce-
e icien con igu a ions. Compa ed o exis ing s udies, he p oposed model uniquely
in eg a es ex e nal ade cons ain s wi h analy ical sol abili y, illing a me hodological gap
in he li e a u e. The de eloped amewo k is sui able o medium- e m planning unde
s able ex e nal condi ions and enables sensi i i y analysis unde al e na i e scena ios such
as sanc ions o p ice shocks. Limi a ions include he assump ion o s a iona i y and he
absence o dynamic o s ochas ic ea u es. Fu u e esea ch will ocus on dynamic ex ensions
and applica ions in o he open economies.
Keywo ds: open economy; esou ce alloca ion; nonlinea p og amming; s a iona y s a e;
sus ainable g ow h
1. In oduc ion
In he con ex o he g owing complexi y o he global economy, accele a ed global-
iza ion and inc easing o eign ade es ic ions, he ask o e icien alloca ion o limi ed
p oduc ion esou ces is becoming inc easingly ele an o open economic sys ems. This
p oblem is especially acu e o de eloping coun ies wi h a aw ma e ial ocus and s uc-
u al dependence on he impo o capi al-in ensi e componen s and echnologies.
Despi e he widesp ead use o he Cobb-Douglas p oduc ion unc ion in economic
modeling, mos o he wo k is ocused ei he on simpli ied one- o wo- ac o models
(Ta asye e al.,2023) o on sol ing p oblems wi hou aking in o accoun he cons ain s
ypical o open economies (Vasyl’ye a,2021). A signi ican pa o applied models use
linea o simula ion app oaches (Sa ae & Sa ae a,2020), which limi s he possibili ies o
decision-making based on op imali y p inciples.
The issues o simul aneous conside a ion o in e sec o al in e ac ions, es ic ions on
in es men and labo , dep ecia ion o ixed capi al, as well as ade es ic ions and quo as
emain poo ly s udied. Classical models, as a ule, lack a lexible s uc u e ha would allow
Economies 2025,13, 184 h ps://doi.o g/10.3390/economies13070184
Economies 2025,13, 184 2 o 22
o ex e nal shocks o sys emic dependence on impo s, which limi s hei applicabili y o
economic policy.
An example o a h ee-sec o model in an open economy is he wo k o Kolemaye
(2008), who cons uc ed a heo e ical s uc u e o balanced g ow h aking in o accoun
in e sec o al lows. Howe e , his model has a numbe o limi a ions: i is s a ic, ocused on
equilib ium dis ibu ion, and does no con ain a o mal op imiza ion s a emen . In addi ion,
he e is no nume ical solu ion scheme, which complica es i s p ac ical applica ion.
In wo ks (Z. Mu zabeko e al.,2020;Z. Mu zabeko & Tussupo a,2024;Z. N. Mu z-
abeko e al.,2022) he au ho s p oposed me hods and algo i hms o sol ing op imal
con ol p oblems wi hin he amewo k o a h ee-sec o model o an economic clus e .
Using a special o m o he Lag ange mul iplie me hod, he au ho s simpli ied he com-
pu a ional p ocedu es, allowing o nume ical calcula ions o a s eady s a e wi hou he
use o hea y i e a i e p ocedu es. These app oaches ha e p o en e ec i e in modeling
closed economic sys ems ha a e no a ec ed by o eign ade. Howe e , mode n eali ies
equi e a ansi ion om closed models o mo e uni e sal sys ems ha ake in o accoun
bo h in e nal cons ain s and ex e nal economic condi ions.
This s udy aims o de elop exis ing models by o mula ing a new op imal con ol
p oblem in a h ee-sec o open economy ha simul aneously akes in o accoun :
• limi ed labo and in es men esou ces;
• dep ecia ion o ixed p oduc ion capi al;
• balance o ma e ials be ween sec o s;
• es ic ions on impo s o capi al-in ensi e componen s;
• and he need o maximize he ou pu o he consume indus y as he inal esul .
The p oposed model is o malized as a nonlinea cons ained op imiza ion p oblem
sol ed using he Lag ange mul iplie me hod and a nume ical one-dimensional sea ch
using he golden sec ion me hod. This ensu es bo h analy ical in e p e abili y and nume i-
cal easibili y, allowing pa ame ic analysis o be pe o med and economically signi ican
scena io esul s o be ob ained.
The Republic o Kazakhs an, an expo -o ien ed coun y wi h a high sha e o he
aw ma e ials sec o and signi ican dependence on impo s o in es men and consume
goods (Hasanli e al.,2024), was chosen as he empi ical basis o he model. Kazakhs an
is cha ac e ized by a high deg ee o openness: acco ding o he Wo ld Bank and he IMF,
he sha e o o eign ade in GDP consis en ly exceeds 50%, and he le el o a i ba ie s
emains low. In addi ion, he s uc u e o he coun y’s economy is ully consis en wi h he
model aking in o accoun he expo o aw ma e ials, he impo o in es men goods and
he cen alized dis ibu ion o esou ces.
Fo nume ical calib a ion, s a is ical da a o 2010–2022 we e used (Tussupo a & Mi za-
khmedo a,2024): indus ial ou pu , numbe o employees and in es men in ixed asse s
by sec o .
The pu pose o his wo k is o de elop and sol e a limi ed nonlinea model o he
op imal alloca ion o esou ces ha e lec s he in e ac ion be ween in e nal p oduc ion
mechanisms and ex e nal ade balances.
In his ega d, he ollowing ques ions a e conside ed wi hin he amewo k o he s udy:
• How do indus y ade es ic ions a ec he op imal alloca ion o capi al and labo ?
•
Wha a e he equilib ium ela ionships be ween he dis ibu ion o labo and
in es men ?
•
How sensi i e is he equilib ium o changes in he elas ici y o p oduc ion and ade
pa ame e s?
Economies 2025,13, 184 3 o 22
Thus, he p oposed model is bo h a de elopmen o economic and ma hema ical
heo y and a ool ha can be used in s a egic analysis and de elopmen o economic policy
in he con ex o esou ce and ade es ic ions.
2. Li e a u e Re iew
The Cobb–Douglas p oduc ion unc ion, p oposed by Cobb and Douglas (1928), e-
mains one o he mos widely used ools in applied economics. Due o i s analy ical
simplici y and in e p e abili y, i is widely used in modeling he ela ionship be ween he
main ac o s o p oduc ion (capi al and labo ) and ou pu . In i s classical o m, his unc ion
is applied o assessing he echnological pa ame e s o indus ies, analyzing he e iciency
o esou ce use, o ecas ing ou pu , and cons uc ing g ow h models.
O e ime, he applica ion o his unc ion was expanded o mul isec o , dynamic, and
op imiza ion models, which made i possible o desc ibe mo e complex economic sys ems.
Mode n esea ch co e s bo h heo e ical aspec s o he p oduc ion s uc u e and applied
p oblems o esou ce alloca ion, sus ainable g ow h, and ex e nal economic impac s. The
e iew o exis ing wo k can be condi ionally di ided in o i e a eas: (1) compu a ional
me hods o sol ing p oblems wi h he Cobb–Douglas unc ion; (2) mul i-sec o and dy-
namic models; (3) wo k on open economies and ex e nal economic es ic ions; (4) applied
and empi ical esea ch; (5) app oaches o designing so wa e sys ems o managemen and
decision suppo .
2.1. Compu a ional App oaches o Sol ing Cobb–Douglas Models
Dinc e al. (2025) p oposed using a gene ic algo i hm o op imize he pa ame e s o
he Cobb–Douglas unc ion. The me hod showed high lexibili y in sea ching o a global
op imum, bu he app oach i sel is echnical — i does no ake in o accoun esou ce
cons ain s, in e -indus y in e ac ions, o he ex e nal economic s uc u e. Simila ly, in
he wo k o (Be ancu -Hines oza e al.,2025), he Cobb–Douglas unc ion was sol ed in
a quan um se ing ia he Clai au di e en ial equa ion. Al hough his me hod expands
compu a ional ho izons, i is no based on a ealis ic economic s uc u e and does no
include p oduc ion o ade balances.
2.2. Mul isec o and Dynamic Models
In wo k (Ma sumo o & Szida o szky,2021), he au ho s conside ed a wo-sec o model
o economic g ow h wi h he Cobb-Douglas unc ion, which akes in o accoun ime lags
in he dynamics o capi al and labo . The au ho s showed ha unde ce ain pa ame e s,
he sys em loses s abili y h ough a Hop bi u ca ion. Howe e , he model is limi ed by
in e nal in e ac ions and does no con ain cons ain s ela ed o o eign ade o in e -sec o
ealloca ion. Mu o (2013) de eloped a h ee-sec o model o GDP using capi al, labo , and
land as ac o s o p oduc ion. He demons a ed he e ec o ela i e p ices on he in e es a e
and land en , which is impo an o s uc u al analysis, bu he wo k does no o mula e
he p oblem o op imal esou ce alloca ion and does no ake in o accoun cons ain s.
2.3. Op imiza ion and Open Economy Models
The wo k (En ique & Ga cia-Salaza ,2023) is de o ed o a wo-sec o model aking
in o accoun capi al lows and subsidies. I emphasizes he complexi ies o economic policy
in an open sys em, bu does no conside p oduc ion echnologies, ma e ial balances, and
he s uc u e o he p oduc ion unc ion. The s udy (Meidu e-Ka aliauskiene e al.,2021)
demons a es he applica ion o objec i e p og amming o he alloca ion o na u al gas
be ween sec o s. Howe e , he app oach is linea and specialized—i is no uni e sal
o he analysis o o al economic ou pu . The wo k (P. Li & Zhong,2020) p oposes an
in e egional esou ce alloca ion model ha akes in o accoun compe i ion. The model is
Economies 2025,13, 184 4 o 22
in e es ing om he poin o iew o geoeconomics, bu is no based on he ac o s uc u e
o p oduc ion and does no use Cobb-Douglas- ype unc ions.
2.4. Applied Empi ical Resea ch and Es ima ion Models
Chi e al. (2021) applied an imp o ed Cobb–Douglas unc ion o analyze he ela-
ionship be ween ene gy consump ion and economic g ow h. The use o coin eg a ion
analysis allowed us o iden i y long- un ela ionships, bu he wo k is ocused on s a is ical
es ima ion a he han op imiza ion. K. Li e al. (2019) used he Cobb–Douglas unc ion o
analyze u ban wa e consump ion, bu he model does no ha e a mul i-sec o s uc u e and
does no conside he op imal esou ce alloca ion. In (Jin & Zhang,2011), hey p oposed an
analy ical solu ion o he g ow h p oblem in a mul i-sec o model, bu wi hou including
ade es ic ions and esou ce balances. And in pape (Kim & Jeon,2025) a model o
consump ion and in es men wi h pa ial bo owing es ic ions in a wo-sec o economy
was conside ed, igno ing he e ms o o eign ade and balance.
In hei s udies (Zmeškal e al.,2023), he au ho s cons uc ed a model o p edic ing
he dis ibu ion o he EVA alue based on Le y p ocesses o a small open ma ke . The
model has a na ow applied ocus and is ocused on pos -p ocessing o indica o s, wi h-
ou o ming a con ollable sys em. The wo k (Zhang,2020) emphasizes he in luence o
anspo a ion cos s on egional imbalances in China, bu i conside s exogenous s uc u al
cons ain s wi hou o malizing he p oblems o op imal esou ce alloca ion.
2.5. A chi ec u al and So wa e App oaches o Managemen
Recen de elopmen s in economic planning and indus ial policy emphasize he g ow-
ing need o in eg a ed decision-suppo sys ems and sus ainabili y-o ien ed amewo ks.
The wo k o Go no e al. (2021) and Muhammad e al. (2023) explo es a chi ec u es o
op imal con ol and en e p ise-le el esou ce managemen , ocusing on da a in eg a ion,
so wa e implemen a ion, and p ocess au oma ion. While hese app oaches a e impo -
an om an enginee ing s andpoin , hey do no in ol e o mal economic models wi h
mul i-sec o s uc u es and ade cons ain s.
Simila ly, ecen e iews on sus ainable manu ac u ing and ci cula economy s a egies
(e.g., Ka uppiah e al. 2024a,2024b) s ess he ole o digi al echnologies, esou ce e iciency,
and supply chain esilience. Howe e , hese wo ks ypically lack quan i a i e op imiza ion
models ha connec policy scena ios wi h sec o al p oduc ion dynamics. In con as , he
model p oposed in his s udy con ibu es a ma hema ically igo ous ool o explo ing
esou ce alloca ion p oblems unde ex e nal cons ain s, b idging he gap be ween high-
le el sus ainabili y goals and ope a ional economic modeling.
3. Ma hema ical Model o an Open Economic Sys em
A mac oeconomic model o an open h ee-sec o economy wi h cen alized esou ce
alloca ion is conside ed and deno e he sec o s o he model as ollows:
•i=
0— he ma e ial sec o , p oduces esou ces used in he o m o aw ma e ials and
semi- inished p oduc s by o he sec o s;
•i=
1— he capi al- o ming sec o , p oduces means o p oduc ion (machine y, equip-
men , e c.);
•i=
2— he consume sec o , p oduces inal p oduc s o domes ic consump ion
and expo .
Each sec o has i s own p oduc ion unc ion and in e ac s wi h o he s h ough in e -
sec o al esou ce lows and o eign ade.
Economies 2025,13, 184 5 o 22
The ou pu olume in each i- h sec o is de e mined by he Cobb-Douglas p oduc-
ion unc ion:
Xi=Fi(Ki,Li) = AiKαi
iL1−αi
i,(i=0, 1, 2). (1)
whe e:
•Xi—ou pu olume;
•Ki— olume o ixed p oduc i e asse s;
•Li— olume o labo esou ces;
•Ai—coe icien o neu al echnological p og ess;
•αi—capi al elas ici y coe icien ;
• (1 −αi)—labo elas ici y coe icien .
This o m allows o e lec bo h di e ences in capi al in ensi y o indus ies and
scalabili y o echnology. Pa ame e s
Ai
,
αi
a e subjec o calib a ion acco ding o indus-
y s a is ics.
Each sec o has i s own ixed p oduc i e asse s (FPA), while labo esou ces and
in es men s can be eely edis ibu ed be ween sec o s. The change in he FPA o he i- h
sec o o e a pe iod consis s o dep ecia ion—µiKiand an inc ease due o in es men s Ii:
dKi
d =Ii−µiKi,Ki(0) = K0
i,(i=0, 1, 2). (2)
Le us in oduce he ollowing no a ions:
•θi=Li
L—sec o al sha es in labo esou ce dis ibu ion;
•si=Ii
X1—sec o al sha es in in es men esou ce dis ibu ion;
• i(ki) = Xi
Li—labo p oduc i i y in he i- h sec o ;
•ki=Ki
Li—capi al-labo a io o he sec o s;
•y1=Y1
L—sha e o impo ed goods o in es men ;
•y2=Y2
L—sha e o impo ed goods o consump ion;
•xi=θi i(ki)—speci ic ou pu o he sec o s.
Then, he Equa ion
(2)
can be ew i en in he ollowing o m o he capi al-labo a io
o he sec o s:
˙
ki=−λiki+si
θi
(x1+y1),ki(0) = k0
i,λi>0, (i=0, 1, 2), (3)
xi=θiAikαi
i,Ai>0, 0 <αi<1(i=0, 1, 2), (4)
he equilib ium o esou ces is ensu ed by he ollowing cons ain s:
• In es men balance:
s0+s1+s2=1, si>0, (i=0, 1, 2), (5)
• Labo balance:
θ0+θ1+θ2=1, θi>0, (i=0, 1, 2), (6)
• Ma e ial balance:
(1−β0)x0=β1x1+β2x2+y0, 0 <βi<1, (i=0,1,2). (7)
• Fo eign ade balance:
q0y0=q1y1+q2y2. (8)
• Indus ial secu i y:
y1≤γ1x1,y2≤γ2x2. (9)
Economies 2025,13, 184 6 o 22
These condi ions model he limi s o pe missible ex e nal p essu e.
He e:
•γ1,γ2—maximum pe missible sha es o impo s o in es men and consume goods;
•y0
—expo o aw ma e ials;
y1
,
y2
— olumes o impo s o in es men and consume
goods, espec i ely;
•q0—wo ld p ice o expo ed ma e ials;
•q1,q2—wo ld p ices o impo ed in es men and consume goods;
•λi
—coe icien o capi al-labo a io educ ion due o capi al dep ecia ion and employ-
men g ow h;
•βi—di ec ma e ial cos s pe uni o ou pu in he i- h sec o .
He e i is assumed ha in es men esou ces come cen ally om he capi al- o ming
sec o and a e supplemen ed by impo s. Such an assump ion is ypical o economies wi h
high impo dependence o capi al goods.
4. Fo mula ion o he P oblem o Op imal Resou ce Alloca ion
The p oblem consis s o inding he op imal alloca ion o esou ces be ween h ee
sec o s in an open economy, conside ing quo as on he impo o in es men goods. The
objec i e is o maximize he speci ic ou pu o he consume sec o
(x2=θ2A2kα2
2)
, which
se es as he p ima y sou ce o p o i .
To achie e his, i is necessa y o ake in o accoun he sha e o ma e ial expo s, global
p ices o expo ed ma e ials, as well as he p ices o impo ed in es men and consume
goods. This leads o a nonlinea p og amming p oblem aimed a de e mining he s able
s a e o he sys em.
In his s udy, indus ial secu i y is in e p e ed as ensu ing condi ions ha main ain
equilib ium and s abili y wi hin he sys em:
y1=γ1x1,y2=γ2x2. (10)
Then, he equa ion o he o eign ade balance (8) can be ew i en as ollows:
y0=q1
q0γ1x1+q2
q0γ2x2. (11)
Thus, he o mula ion o he p oblem o op imal esou ce alloca ion in he economic
sys em educes o a nonlinea p og amming p oblem:
x2−→ max (12)
subjec o he ollowing cons ain s:
s0+s1+s2=1, si>0, (i=0, 1, 2), (13)
θ0+θ1+θ2=1, θi>0, (i=0, 1, 2), (14)
(1−β0)x0=β1+q1
q0γ1x1+β2+q2
q0γ2x2,
0<βi<1, (i=0, 1, 2). (15)
xi=θiAikαi
i,Ai>0, 0 <αi<1(i=0, 1, 2), (16)
−λiki+si
θi
(1+γ1)x1,λi>0, (i=0, 1, 2). (17)
Economies 2025,13, 184 7 o 22
The de eloped simpli ied model enables a mo e e icien s udy o he economic sys-
em’s s abili y, emphasizing he condi ions and cons ain s ha di ec ly a ec he main e-
nance o equilib ium and s abili y in he p oduc ion en i onmen .
5. Solu ion o he P oblem o Op imal Resou ce Alloca ion
Since he sea ch is o a s a iona y solu ion, he capi al-labo a io o he sec o s emains
cons an o e ime. In his case, he ollowing sys em o nonlinea equa ions can be de i ed
om Equa ion (17):
−λ0k0+s0
θ0(1+γ1)θ1A1kα1
1=0, (18)
−λ1k1+s1(1+γ1)A1kα1
1=0, (19)
−λ2k2+s2
θ2(1+γ1)θ1A1kα1
1=0. (20)
Then, ind he o mulas o ki:
k0=A1
λ0(1+γ1)A1
λ1(1+γ1)
α1
1−α1s0θ1
θ0s1s
1
1−α1
1, (21)
k1=A1
λ1(1+γ1)
1
1−α1s
1
1−α1
1, (22)
k2=A1
λ2(1+γ1)A1
λ1(1+γ1)
α1
1−α1s2θ1
θ2s1s
1
1−α1
1. (23)
By subs i u ing he alues o
k0
,
k1
and
k2
ob ained om Fo mulas
(21)
–
(23)
in o he
ini ial o mula o he speci ic ou pu o each sec o
(16)
, ob ain a ans o med sys em o
equa ions in he o m:
x0=ω0s0θ1
s1θ0α0
θ0s
α0
1−α1
1, (24)
x1=ω1θ1s
α1
1−α1
1, (25)
x2=ω2s2θ1
s1θ2α2
θ2s
α2
1−α1
1. (26)
whe e ωj,j=0, 1, 2 deno e he cons an coe icien s o he sys em:
ω0=A0A1
λ0(1+γ1)α0A1
λ1(1+γ1)
α0α1
1−α1,
ω1=A1A1
λ1(1+γ1)
α1
1−α1,
ω2=A2A1
λ2(1+γ1)α2A1
λ1(1+γ1)
α2α1
1−α1.
As can be seen, he sys em o Equa ions
(24)
–
(26)
consis s o six exogenous pa ame e s,
which a e desc ibed by h ee balance Equa ions (13)–(15).
To maximize he speci ic ou pu o he consume sec o , he golden a io p inciple is
applied Jin and Zhang (2011). This p inciple helps de e mine he p opo ions among he
Economies 2025,13, 184 8 o 22
exogenous pa ame e s
θi
,
si(i=
0, 1, 2
)
ha ensu e he maximum speci ic ou pu while
conside ing all cons ain s and in e dependencies be ween hem.
Since
s1
and
θ1
a e ee a iables, he emaining sha e o esou ces o he ma e ial and
consume sec o s will be 1 −s1and 1 −θ1, espec i ely.
Deno ing
m
as he sha e o he consume sec o in he emaining in es men esou ces
and
h
as he sha e o he consume sec o in he emaining labo esou ces, ob ain he
ollowing exp essions o dis ibu ion:
• o labo esou ces:
θ0= (1−h)(1−θ1),θ2=h(1−θ1), (27)
• o in es men esou ces:
s0=m(1−s1),s2= (1−m)(1−s1). (28)
And hus, by ans o ming he o iginal p oblem
(12)
–
(17)
using
(27)
and
(28)
, ob ain
he ollowing nonlinea p og amming p oblem:
x2−→ max (29)
subjec o he condi ions:
(1−β0)x0=β1+q1
q0γ1x1+β2+q2
q0γ2x2,
0<βi<1, (i=0, 1, 2), (30)
x0=ω0m(1−s1)θ1
s1(1−h)(1−θ1)α0
(1−h)(1−θ1)s
α0
1−α1
1, (31)
x1=ω1θ1s
α1
1−α1
1, (32)
x2=ω2(1−m)(1−s1)θ1
s1h(1−θ1)α2
h(1−θ1)s
α2
1−α1
1. (33)
To sol e his nonlinea p og amming p oblem
(29)
–
(33)
, he Lag ange mul iplie s
me hod is applied. The Lag ange unc ion is w i en as ollows:
L(θ1,s1,x0,x1,x2,c1,c2,c3,c4) = −x2+
+c1(1−β0)x0−β1+q1
q0γ1x1−β2+q2
q0γ2x2+
+c2
x0−ω0m(1−s1)θ1
s1(1−h)(1−θ1)α0
(1−h)(1−θ1)s
α0
1−α1
1
+
+c3
x1−ω1θ1s
α1
1−α1
1
+
+c4
x2−ω2(1−m)(1−s1)θ1
s1h(1−θ1)α2
h(1−θ1)s
α2
1−α1
1
(34)
Economies 2025,13, 184 15 o 22
2.
W i e ou he necessa y i s -o de condi ions and sol e he co esponding sys em o
nonlinea algeb aic Equa ions (35)–(42) and (31)–(33);
3.
Calcula e he ela ionships be ween he pa ame e s
θi
and
si
ha sa is y he condi ions
o he equilib ium dis ibu ion o esou ces speci ied by Equa ions (43)–(45);
4.
Plo he unc ion
(s1)
(acco ding o Equa ion
(76)
) and de e mine he op imal alue
o he pa ame e
s1
— he sha e o in es men esou ces alloca ed o he capi al-
o ming sec o ;
5.
Find he alues o he Lag ange mul iplie s:
c1
—using Fo mula
(63)
and
c2
,
c3
,
c4
—using
Fo mulas (60)–(62);
6.
De e mine he labo o ce
h
dis ibu ed be ween he ma e ial and consume sec o s
using Equa ion (71);
7. De e mine he sha e o in es men s mo he same nodes using Fo mula (64);
8. Find he alue o he pa ame e θ1using Equa ion (68);
9.
Calcula e he emaining sha es o esou ces
θ0
,
θ2
and
s0
,
s2
—using no a ions
(27)
–
(28)
;
10.
Check whe he es ic ions (13)–(15) a e me ;
11.
De e mine he s a iona y s a e o sec o al capi al in ensi y based on Equa ions
(21)
–
(23)
;
12.
De e mine he s a iona y s a e o p oduc ion unc ions in ela i e e ms using
Fo mulas (31)–(33);
13.
Calcula e he speci ic ma e ial expo acco ding o Equa ion (11).
Fo he nume ical solu ion o he p oblem, a p ocedu e was implemen ed in he
Maple so wa e (6.04 Build 756) en i onmen , using buil -in op imiza ion and symbolic
di e en ia ion ools. The golden sec ion me hod was used o one-dimensional sea ch
by pa ame e
s1
, wi h he i s -o de condi ions ( om he Lag ange equa ions) sol ed
analy ically and nume ically a each s ep. The calcula ions we e pe o med on a s anda d
compu e (In el Co e i7, 16 GB RAM), he a e age calcula ion ime o one op imal dis ibu-
ion was less han 1 s. The esul ing scheme is accu a e and s able wi h espec o changes
in ini ial app oxima ions.
7. Resul s
7.1. Nume ical Calcula ions
To e i y he p oposed model o op imal esou ce alloca ion in an open economic
sys em, a se o nume ical expe imen s was conduc ed. I should be no ed ha in nume ical
calcula ions he model conside s a s able esou ce alloca ion wi h s a iona y pa ame e s
o he ex e nal economic en i onmen . This is in line wi h he p ac ice o s a egic and
budge planning, whe e a e age o ecas indica o s a e used. In pa icula , in Kazakhs an
s a e p og ams and budge s a e de eloped o a h ee-yea ho izon. The main goal is o
de e mine a s able s uc u e o esou ce alloca ion ha achie es maximum ou pu in he
capi al- o ming sec o while obse ing all speci ied cons ain s.
Fo nume ical calcula ions, he alues o he coe icien s
αi
,
βi
,
λi
and
Ai
we e used
(see Table 1), calcula ed on he basis o s a is ical da a o he Republic o Kazakhs an o
2010–2022, published on he o icial websi e o he Bu eau o Na ional S a is ics h ps://
s a .go .kz (accessed be ween Augus and No embe 2023), and e lec in o ma ion on he
olume o indus ial p oduc ion, he numbe o employees and in es men s in ixed capi al
by indus y (Tussupo a & Mi zakhmedo a,2024).
Economies 2025,13, 184 16 o 22
Table 1. Model coe icien s.
i 0 1 2
αi0.65 0.7 0.68
βi0.4 0.11 0.22
λi0.07 0.08 0.05
Ai1.92 1.3 2.17
A he i s s age o he nume ical analysis, a g aph o he unc ion
(s1)
s cons uc ed,
co esponding o Equa ion
(76)
, which de e mines he op imal sha e o in es men e-
sou ces di ec ed o he capi al- o ming sec o . Calcula ions a e ca ied ou in he ange om
0 o 1, and he esul ing op imal alue is used o calcula e all o he exogenous pa ame e s
o he model (see Figu e 1).
Table 2shows he esul s o calcula ing he exogenous pa ame e s o a closed eco-
nomic sys em.
Figu e 1. G aph o he unc ion (s1).
Table 2. Calcula ion esul s o he s a iona y s a e o a closed economic sys em.
i 0 1 2
θi0.336 0.468 0.194
si0.294 0.51 0.165
ki1058.337 1151.892 1695.374
xi59.812 84.626 66.350
m= 0.602, h= 0.366.
Table 3shows he esul s o calcula ing he exogenous pa ame e s o an open economic
sys em.
Table 3. Calcula ion esul s o he s a iona y s a e o a open economic sys em.
i 0 1 2
θi0.521 0.325 0.153
si0.471 0.37 0.158
ki1390.574 1526.937 2227.593
xi110.485 71.716 62.781
yi33.624 35.858 31.390
m= 0.751, h= 0.224.
Economies 2025,13, 184 17 o 22
Figu e 2illus a es he solu ions o he di e en ial equa ions o capi al in ensi y
ki,(i=0, 1, 2)unde gi en ini ial condi ions.
˙
k0=−λ0k0+s0θ1
θ0(1+γ1)A1kα1
1,k0(0) = 1100, (77)
˙
k1=−λ1k1+s1(1+γ1)A1kα1
1,k1(0) = 1250, (78)
˙
k2=−λ2k2+s2θ1
θ2(1+γ1)A1kα1
1,k2(0) = 1600. (79)
Figu e 2. Capi al in ensi y dynamics in he sec o s o an open economy.
Figu e 3illus a es he changes in sec o al ou pu pe uni ,
xi
,
(i=
0, 1, 2
)
, o e ime
in an open economy. The g aph is cons uc ed based on Equa ion (4).
Figu e 3. Dynamics o sec o al ou pu in an open economy.
Economies 2025,13, 184 18 o 22
7.2. Pa ame ic Analysis
To assess he sensi i i y o he model o changes in pa ame e s, a pa ame ic analysis
was conduc ed using wo key coe icien s-
α0
(see Table 4and Figu e 4) and
β2
(see Table 5
and Figu e 5). These coe icien s e lec he c i ical s ages o he unc ioning o he economic
sys em: he ini ial s age o a ac ing esou ces o he ma e ial sec o and he inal phase
o p oduc elease in he consume sec o . Analysis o he in luence o hese pa ame e s
allows o a comp ehensi e assessmen o he s abili y and e iciency o he en i e model
when a ying he s uc u al cha ac e is ics o he ini ial and inal s ages o he p oduc ion
p ocess (see Figu e 6).
Table 4. In luence o α0on esou ce alloca ion and ou pu in he capi al- o ming sec o .
α0x2s0s1s2θ0θ1θ2
0.1 2.72 0.6 0.17 0.22 0.933 0.025 0.041
0.2 4 0.64 0.18 0.17 0.902 0.048 0.049
0.3 6.88 0.64 0.21 0.14 0.858 0.082 0.059
0.4 14.32 0.61 0.26 0.13 0.778 0.144 0.076
0.5 37.43 0.53 0.34 0.13 0.626 0.264 0.109
0.6 62.781 0.471 0.37 0.158 0.521 0.325 0.153
0.7 111.64 0.38 0.46 0.15 0.379 0.455 0.164
0.8 266.03 0.22 0.58 0.2 0.139 0.627 0.233
The model e eals a s uc u al no m. As can be seen om Table 4, wi h a mode a e
inc ease in he capi al in ensi y o he aw ma e ials sec o (
α0≈
0.4–0.6), he economy
eaches a balanced s a e—ou pu g ows (
x2
14.32 o 62.781), bu no ab up ly, and labo
and in es men emain e enly dis ibu ed.
(a) (b)
Figu e 4. Analysis o he impac o changes in he elas ici y coe icien on he dis ibu ion o in es -
men (a) and labo (b) esou ces ac oss indus ies o he ma e ial sec o .
Economies 2025,13, 184 19 o 22
(a) (b)
Figu e 5. Analysis o he dis ibu ion o in es men s (a) and labo (b) esou ces by consume sec o
indus ies depending on he olume o ma e ial cos s.
(a) (b)
Figu e 6. The in luence o α0(a) and β2(b) on he speci ic ou pu o he consume sec o x2.
Table 5. In luence o β2on esou ce alloca ion and ou pu in he consume sec o .
β2x2s0s1s2θ0θ1θ2
0.1 97.1 0.445 0.37 0.184 0.496 0.32 0.18
0.2 62.78 0.471 0.37 0.158 0.521 0.32 0.15
0.3 70.89 0.494 0.37 0.135 0.547 0.32 0.131
0.4 62.46 0.5 0.37 0.12 0.563 0.32 0.115
0.5 55.82 0.522 0.37 0.107 0.576 0.32 0.103
0.6 50.46 0.532 0.37 0.097 0.587 0.32 0.093
0.7 46.03 0.541 0.37 0.088 0.595 0.32 0.085
0.8 42.32 0.548 0.37 0.081 0.602 0.32 0.078
Simila ly, om Table 5. we can see ha wi h a mode a e ma e ial in ensi y o he
consume sec o (
β2≈
0.2–0.3), high p oduc ion is main ained wi hou an excessi e bu -
den on he aw ma e ials sec o . This indica es an op imum zone in which sus ainable
de elopmen is possible wi h minimal cos s and a balanced economic s uc u e.
8. Discussion
This s udy p oposes a o malized nonlinea model o a h ee-sec o open economy,
in eg a ing p oduc ion, labo , capi al, and ade balances in o a uni ied op imal esou ce
alloca ion p oblem. The ob ained esul s allow o in e p e ing no only he in e nal sec o al
dynamics bu also he sys em’s esponse o changes in echnological and ma e ial cons ain s.
Economies 2025,13, 184 20 o 22
In con as o exis ing wo ks such as he balanced g ow h model o Kolemaye (2008),
which is s a ic and lacks a compu a ional solu ion, o he s udy o Ta asye e al. (2023),
limi ed o CES unc ions and wo- ac o in e ac ions, his pape p esen s a s uc u ally
comple e and compu a ionally implemen able model. In pa icula , i ex ends he s uc u al
amewo k o include sec o al ade es ic ions and dep ecia ion o ixed capi al, while
main aining analy ic ac abili y. Unlike he linea in es men alloca ion in Sa ae ’s model
(Sa ae & Sa ae a,2020), he cu en wo k sol es a ully nonlinea cons ained op imiza ion
p oblem wi h in e p e able equilib ium condi ions. The use o a Lag angian amewo k,
coupled wi h he golden sec ion me hod, makes he solu ion bo h igo ous and e icien .
Fu he , nume ical me hods such as hose p oposed in (En ique & Ga cia-Salaza ,
2023), using gene ic algo i hms o sol e he Cobb–Douglas unc ion, lack s uc u al ealism
and policy ele ance. By con as , ou app oach p ese es he economic meaning o he
cons ain s and balances, making he model sui able o eal-wo ld planning scena ios.
The pa ame e analysis demons a es a s uc u ally meaning ul pa e n: an inc ease in
capi al p oduc i i y in he ma e ial sec o leads o a ealloca ion o labo and in es men
in o he capi al- o ming sec o , he eby enabling a signi ican expansion o consume
ou pu . Con e sely, an inc ease in ma e ial consump ion pe uni o inal p oduc leads o
a measu able con ac ion in ou pu and a eo ien a ion o esou ces owa d he ma e ial
sec o . These pa e ns a e consis en wi h economic heo y and con i m he model’s abili y
o e lec eal s uc u al ade-o s.
Al hough he model assumes s a iona y ex e nal economic condi ions, i s s uc u e
enables sensi i i y and scena io analysis. By a ying ade pa ame e s such as
qi
,
γi
and
yi
, o imposing exogenous cons ain s, one can simula e un a o able ade condi ions,
including sanc ions, a i shocks, o luc ua ions in wo ld commodi y p ices. Thus, while
no explici ly s ochas ic o dynamic, he model se es as a lexible amewo k o applied
economic analysis unde mul iple planning scena ios.
Compa ed o ecen li e a u e on sus ainable p oduc ion and op imiza ion in open
sys ems (e.g., K. Li e al.,2019;Kim & Jeon,2025), his app oach is dis inc in i s combina ion
o economic in e p e abili y, algo i hmic sol abili y, and policy- ele an s uc u al de ail.
In his sense, he model ills a me hodological gap and o e s a scalable ool o examining
s uc u al policy al e na i es in esou ce-cons ained, ade-dependen economies.
While ou model ocuses on p oduc ion e iciency and op imal alloca ion unde ade
cons ain s, i may con ibu e o he b oade sus ainabili y agenda ou lined in ecen e iews
on ci cula economy and ad anced supply chains. I s capabili y o simula e s uc u al
adjus men s and ade- ela ed shocks makes i sui able o e alua ing indus ial esilience
and long- e m esou ce sus ainabili y.
9. Conclusions
This pape p oposes a no el nonlinea op imiza ion model o esou ce alloca ion in a
h ee-sec o open economy, including in es men , labo , ma e ial, and ade cons ain s.
Unlike exis ing models ha a e ei he s a ic, pu ely empi ical, o s uc u ally simpli ied,
his app oach in eg a es in e sec o al dynamics wi h p oduc ion echnologies based on
he Cobb–Douglas unc ion, dep ecia ion o capi al, and impo quo as. The model is
analy ically ac able and compu a ionally e icien , elying on he Lag ange mul iplie
me hod and a golden sec ion sea ch.
The esul s demons a e how changes in capi al p oduc i i y o ma e ial in ensi y can
eshape sec o al s uc u e and o e all ou pu , p o iding insigh in o how p oduc ion sys ems
espond o echnological o ade-induced shi s. This enables p ac ical applica ions in policy
planning, pa icula ly in economies wi h ade dependencies and esou ce cons ain s.
Economies 2025,13, 184 21 o 22
P ospec s o u he esea ch include he de elopmen o a dynamic economic man-
agemen model, wi h a ansi ion om he ini ial s a e o he icini y o he s a iona y
egime. This equi es he implemen a ion o a di e en ial sys em wi h s a e-dependen
con ol pa ame e s and can be sol ed using Lyapuno s abili y heo y and op imal con ol
me hods. Addi ionally, i is p oposed o expand he model o assess in es men e iciency
unde condi ions o s ochas ic dis u bances and uns able global ma ke condi ions.
Au ho Con ibu ions: Concep ualiza ion, K.T. and Z.M.; me hodology, K.T.; so wa e, K.T.; alida-
ion, K.T. and Z.M.; o mal analysis, Z.M.; in es iga ion, K.T.; da a cu a ion, Z.M.; w i ing—o iginal
d a p epa a ion, K.T.; w i ing— e iew and edi ing, K.T. and Z.M.; isualiza ion, K.T.; supe ision,
K.T.; p ojec adminis a ion, K.T.; unding acquisi ion, K.T. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding: This esea ch is unded by he Science Commi ee o he Minis y o Science and Highe
Educa ion o he Republic o Kazakhs an (g an no. AP22684879).
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 aw da a suppo ing he conclusions o his a icle will be made
a ailable by he au ho s on eques .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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