Akhalaya, Kseniya; Shikhman, Vladimi
A icle — Published Ve sion
Solow-Swan Model o Economic G ow h wi h Allee E ec
Jou nal o Quan i a i e Economics
Sugges ed Ci a ion: Akhalaya, Kseniya; Shikhman, Vladimi (2025) : Solow-Swan Model o Economic
G ow h wi h Allee E ec , Jou nal o Quan i a i e Economics, ISSN 2364-1045, Sp inge India, New
Delhi, Vol. 23, Iss. 4, pp. 1259-1278,
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
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ORIGINAL ARTICLE
Solow‑Swan Model o Economic G ow h wi hAllee E ec
KseniyaAkhalaya1· Vladimi Shikhman1
Accep ed: 30 June 2025 / Published online: 15 July 2025
© The Au ho (s) 2025
Abs ac
In his pape we expand he neoclassical Solow-Swan model o economic g ow h by
in oducing popula ion dynamics wi h Allee e ec . Allee e ec implies he exis ence
o a h eshold o he iabili y o popula ions, i.e. a popula ion below his h eshold
dec eases. Abo e he h eshold, he popula ion g adually sa u a es. We show ha he
co esponding capi al s ock pe capi a may s abilize a wo di e en le els. Bo h can
be exp essed in e ms o equilib ium poin s o he s anda d Solow–Swan model wi h
pa icula cons an popula ion g ow h a es. Su p isingly enough, he capi al s ock
pe capi a pe o ms in he long un be e i he popula ion becomes ex inc , a he
hen i ad ances he sa u a ion le el. Fo his conclusion he dec ease o popula ion
should be ela i ely mode a e compa ed o he capi al dep ecia ion.
Keywo ds Economic g ow h· Solow-Swan model· Allee e ec · Capi al s ock pe
capi a
JEL Classi ica ion C62· O40
In oduc ion
In 1956, Solow and Swan p oposed a celeb a ed g ow h model o an economy
(Solow 1956; Swan 1956). The meanwhile classical Solow–Swan model desc ibes
how g ow h o capi al and labo o ce a ec he o al ou pu o an economy. The
ou pu is gene a ed by a neoclassical p oduc ion unc ion, which exhibi s con-
s an e u ns o scale, posi i e and diminishing ma ginal p oduc s wi h espec o
capi al and labo , and sa is ies he so-called Inada condi ions. Pa o he ou pu
will be consumed and he es will be sa ed wi h a cons an a e. The g adual
dec ease in he economic alue o he capi al s ock dec eases wi h he cons an
a e
𝛿
. The popula ion is assumed o g ow exponen ially wi h some a e
𝜆
. By
* Vladimi Shikhman
ladimi .shikhman@ma hema ik. u-chemni z.de
1 Depa men o Ma hema ics, Chemni z Uni e si y o Technology, Reichenhaine S . 41,
09126Chemni z, Ge many
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
using hese assump ions, a di e en ial equa ion o he capi al s ock pe capi a
can be de i ed. I essen ially de e mines he pe o mance o he economy in he
Solow-Swan model. The main esul in Solow (1956) says ha he e is a unique
globally asymp o ically s able equilib ium poin o he unde lying di e en ial
equa ion. In economic e ms, he economy’s capi al s ock pe capi a always s abi-
lizes a a non i ial le el.
Since i s in oduc ion he Solow-Swan model ha e been gene alized in many
di ec ions, see e.g. Ba o and Sala-i-Ma in (1995). One o he main a emp s o
make he Solow-Swan model mo e ealis ic is o modi y he unde lying popula-
ion dynamics. In Sca pello and Ri elli (2003), he au ho s assumed he popula-
ion dynamics o ollow he logis ic law. Acco ding o he la e , he popula ion
g ows, bu i s g ow h a e dec eases o ze o. The co esponding capi al s ock pe
capi a is shown o s abilize a he equilib ium poin o he s anda d Solow–Swan
model wi h ze o popula ion g ow h a e. In Accinelli and B ida (2006), he same
gene al assump ions on he popula ion size and i s g ow h a e a e made. No
su p isingly, he achie ed s abili y esul s pe sis . Ano he gene aliza ion, which
leads mainly o he quali a i ely same indings, has been p esen ed in Accinelli
and B ida (2005). He e, he so-called Richa ds law o popula ion dynamics has
been inco po a ed in o he Solow-Swan model. In Gue ini (2006), he popula ion
dynamics does no ollow a p esc ibed law. Ins ead, he popula ion g ows, bu i s
g ow h a e has a limi a in ini y. The capi al s ock pe capi a is shown o s abi-
lize a he equilib ium poin o he s anda d Solow–Swan model wi h he cons an
popula ion g ow h a e equal o his limi .
In his pape , we s udy he Solow-Swan model wi h popula ion dynamics
unde lying he so-called Allee e ec . This e ec is an impo an biological phe-
nomenon and was i s in oduced by Allee in 1930, see e.g. Cou champ e al.
(2008). Allee obse ed du ing his esea ch on gold ishes ha a popula ion wi h
low-densi y in a ce ain a ea leads o he ex inc ion o he species in he long
e m. Indi iduals wi hin a popula ion o en equi e he assis ance o ano he indi-
idual o mo e han simple ep oduc i e easons in o de o pe sis . The mos
ob ious example o his is obse ed in animals ha hun o p ey o de end
agains p eda o s as a g oup. In he con ex o labo , we can hink on employees
who a e elying on he colleagues in hei common ac i i ies. I some o hem
depa , he wo king en i onmen becomes abandoned. This may cause e en mo e
people o qui o no o en e he job a all. Allee e ec implies he exis ence o
a h eshold o he iabili y o popula ions, i.e. a popula ion below his h eshold
dec eases and ac ually becomes ex inc o e ime. Howe e , abo e he h eshold,
he popula ion g ows in he way simila o he logis ic law. No e ha he no el y
o ou app oach is wo- old:
• In o he same Solow-Swan model we inco po a e quali a i ely di e en popu-
la ion dynamics. Depending on he ini ial alue, he popula ion may become
ex inc o sa u a e, bu i s g ow h a e is always dec easing.
• The case o dec easing popula ion is new o he li e a u e on Solow-Swan mod-
els. This is i compa ed o he logis ic o Richa ds law, as well as o implici ly
de ined popula ion dynamics men ioned abo e.
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
I is wo h o men ion ha he use o Allee e ec in he con ex o Solow-Swan mod-
els has been sugges ed in Khan e al. (2021). The e, jus a nume ical s udy o he
Cobb-Douglas p oduc ion unc ion wi h pa icula a en ion o uzzy numbe s has
been ca ied ou .
Le us empi ically jus i y he in oduc ion o he Allee e ec in he amewo k o
economic g ow h models. Fo ha , i is c ucial o ocus on he no el case o dec eas-
ing popula ion below he Allee h eshold, i.e. whe e he popula ion g ow h a e is
nega i e. Fo de eloped coun ies his is no unusual i one neglec s he mig a ion
e ec s, e.g. see Fig.1 o Ge many in 1999–2023. Mainly due o he ela i ely low
e ili y a es in Ge many a ound 1.5 child en pe woman, he popula ion would
dec ease wi hou an addi ional mig a ion. Ou in e es he e lies in he modeling o
exac ly his popula ion dynamics and i s consequences o he economic g ow h.
No e ha an i-mig a ion policies a e al eady implemen ed in some de eloped
coun ies, such as e.g. he wel a e educ ion o immig an s in Denma k Age snap
e al. (2020), o a e hough o be implemen ed by a - igh poli ical pa ies all o e
Eu ope Abubaka a e al. (2024). In de eloping coun ies – we hink in i s place
on China – he ise o popula ion has been bounded o decades by go e nmen al
bi h es ic ions. Consequen ly, he one-child policy lead o he nega i e popula-
ion g ow h a es, see Fig.2 o China in 1999–2023. Al hough he Chinese bi h
policies ha e been g adually loosened (e.g. in 2013, China allowed couples o ha e a
second child i ei he pa en is an only child, in 2016, i allowed ma ied couples o
ha e wo child en, and in 2021, i announced suppo o couples who wish o ha e a
hi d child), he demog aphic si ua ion would emain ecessiona y o decades (Yin
2023). Fu he , he in oduc ion o Allee e ec o human socie ies can be explained
by he depopula ion phenomenon om economic geog aphy. The e, i is de ined
as a p ocess in which he popula ion densi y o an a ea dec eases s eadily o e
ime. As b ie ly summa ized in Maya (2024), one dis inguishes se e al easons o
Fig. 1 Popula ion g ow h a e o Ge many in 1999–2023 wi h and wi hou mig a ion p ocessed by UN,
Wo ld Popula ion P ospec s (2024), Ou Wo ldinDa a.o g/popula ion-g ow h
1262
Jou nal o Quan i a i e Economics (2025) 23:1259–1278
depopula ion: "The e a e social easons, since depopula ion is a d i e o inequal-
i y due o he dec ease in se ices ha declining u al popula ions ecei e. The e
a e cul u al and emo ional easons ha link pe sonal and collec i e oo s o a land-
scape ha is disappea ing. The e a e economic easons, since he e a e esou ces in
he e i o y ha a e no longe exploi ed. The e a e en i onmen al easons since he
abili y o con ol ce ain impac s on he en i onmen , such as o es i es, is educed.
The e a e geopoli ical easons, since socie y loses so e eign y o e depopula ed e -
i o ies, opening an oppo uni y window o o he powe ne wo ks, such as c ime
o o he poli ical agen s, o come o con ol hem." O e all, he mo e people aban-
don an a ea he mo e o hei ellows decide o do he same. This posi i e eedback,
which accoun s o depopula ion, mo i a es ou use o Allee e ec o modeling he
popula ion dynamics he e. Addi ionally, we e e o Me ino and P a s (2020) o he
impac o economic and go e nmen al ac o s on depopula ion, as well as o Kulcsá
(2016) o his o ical examples a bo h he na ional and local le els.
Ou indings on he Solow-Swan model wi h Allee e ec a e as ollows. Depend-
ing on he popula ion ini ial alue, he capi al s ock pe capi a s abilizes a wo di -
e en le els. Fi s , i he popula ion dynamics s a s below he Allee h eshold, he
capi al s ock pe capi a is shown o s abilize a he equilib ium poin o he s anda d
Solow–Swan model wi h a pa icula cons an popula ion g ow h a e. The la e con-
s an equals
−𝜆
, he nega i e o he in insic popula ion g ow h a e. Mo e in e es ingly,
he global asymp o ic s abili y is gua an eed i
𝛿>𝜆
. In economic e ms, he capi al
s ock pe capi a may s abilize e en i he popula ion becomes ex inc . Howe e , his
happens jus i he dec ease o popula ion is ela i ely mode a e, i.e. he in insic popu-
la ion g ow h a e
𝜆
does no exceed he a e o capi al dep ecia ion
𝛿
. Second, i he
popula ion dynamics s a s abo e he Allee h eshold, he capi al s ock pe capi a is
shown o s abilize a he equilib ium poin o he s anda d Solow–Swan model wi h
ze o popula ion g ow h a e. This mo e o less ypical beha io is in acco dance wi h
Fig. 2 Popula ion g ow h a e o China in 1999–2023 wi h and wi hou mig a ion p ocessed by UN,
Wo ld Popula ion P ospec s (2024), Ou Wo ldinDa a.o g/popula ion-g ow h
1263
Jou nal o Quan i a i e Economics (2025) 23:1259–1278
he esul s in Accinelli and B ida (2006) and Gue ini (2006). The compa ison o bo h
equilib ium poin s o he Solow-Swan model wi h Allee e ec leads o a su p ising
conclusion. Namely, he co esponding capi al s ock pe capi a pe o ms in he long
un be e i he popula ion becomes ex inc , a he hen i ad ances he sa u a ion le el.
Finally, i is wo h o men ion ha ou model emains neoclassical. He e, he assump-
ion on he popula ion dynamics ia Allee e ec is exogenous. This is in s ong con-
as wi h o e lapping-gene a ion models, see e.g. Galo and Weil (2000); Yin (2023),
whe e he g ow h o popula ion is explained endogenously. In Galo and Weil (2000),
he ansi ion om Ma husian s agna ion o he demog aphic ansi ion and beyond is
conside ed. The au ho s a gue ha in wha hey call he Mode n G ow h Regime by
hen a ound 2000 "many ich coun ies ha e popula ion g ow h a es nea ze o", and
u he – "o e he nex se e al decades much o Wes e n Eu ope is o ecas o ha e
nega i e popula ion g ow h." Ou pape assumes he Allee e ec in o de o analyze he
economic g ow h co esponding o he la e demog aphic end o diminishing popula-
ion. By doing so, we enla ge he scope o he analysis in Galo and Weil (2000).
The pape is o ganized as ollows. In Sec . 2, we in oduce he Solow-Swan
model wi h Allee e ec . Sec ion3 is de o ed o he s abili y analysis o i s equilib-
ium poin s. In Sec .4, we discuss he Solow-Swan model wi h Allee e ec o he
Cobb-Douglas p oduc ion unc ion.
Model Desc ip ion
We conside a closed economy consis ing o a single good Y( ), he communi y’s
eal income. The ou pu Y( ) is he measu e e m o economy g ow h and will be
gene a ed by capi al K( ) and labo o ce L( ). We assume he neoclassical p oduc-
ion unc ion
F
∶ℝ
2
+
→ℝ
+
wi h he ollowing p ope ies:
(P1) The unc ion F exhibi s cons an e u ns o scale, i.e.
(P2) The unc ion F exhibi s posi i e and diminishing ma ginal p oduc s wi h
espec o each inpu
K>0
and
L>0
:
(P3) The unc ion F ul ils he Inada condi ions (Inada 1963):
No e ha he neoclassical p ope ies o he p oduc ion unc ion imply ha he wo
inpu s, K and L, a e each essen ial o p oduc ion, see e.g. Fä e and P imon (2002):
Le he le el o echnological possibili ies
T>0
be cons an . We ha e
F(aK,aL)=aF(K,L) o all a>0.
𝜕
F
𝜕K
>0, 𝜕F
𝜕L
>0 and 𝜕
2
F
𝜕K
2<0, 𝜕
2
F
𝜕L
2<
0.
lim
K
→
0
𝜕F
𝜕K
=lim
L
→
0
𝜕F
𝜕L
=∞, lim
K
→
∞
𝜕F
𝜕K
=lim
L
→
∞
𝜕F
𝜕L
=
0.
F(K,0)=F(0, L)=0.
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
Pa o each ins an ’s ou pu will be consumed and he es wi h cons an a e
s∈[0, 1]
sa ed o in es ed. We also assume he g adual dec ease in he economic
alue o he capi al s ock wi h he a e
𝛿>0
. The ollowing inc ease o capi al s ock
is hus gi en by
We in end o in es iga e he pe capi a income which can be simpli ied by using
(P1):
whe e he capi al s ock pe uni o e ec i e labo is se as
Pe capi a conside a ions help o measu e a sec o s’ a e age incomes and imp o e
he compa ison o di e en economies. The main in e es o he model is he dynam-
ics o k( ). Taking de i a i es wi h espec o ime , we ob ain
whe e
is he g ow h a e o popula ion. The di e en ial equa ion in (4) is he economic key
o he Solow-Swan model.
Now, we u n ou a en ion o he assump ion on he popula ion dynamics. In
he canonical Solow-Swan model (Solow 1956; Swan 1956), he g ow h a e n( ) is
assumed o be cons an , i.e.
whe e
𝜆>0
is an in insic g ow h a e. Hence, he popula ion size yields exponen-
ial beha io o e ime and o any ini ial le el
L0>0
, a ime he le el o labo
o ce is
This beha io ep esen s a use ul model o simple popula ions o e ela i ely
sho pe iods o ime wi hou accommoda ion o g ow h educ ions due o limi ed
esou ces. In Sca pello and Ri elli (2003), a modi ica ion o he canonical Solow-
Swan model was p oposed by assuming a logis ic popula ion g ow h. This is an
(1)
Y( )=T
⋅
F(K( ),L( )).
(2)
K�( )=s
⋅
Y( )−𝛿
⋅
K( ).
(3)
y( )=Y( )
L
(
)=T⋅F
(
K( )
L
(
),1
)
=T⋅F(k( ),1)
,
k
( )=
K( )
L( )
.
(4)
k
�( )=
K�( )
⋅
L( )−K( )
⋅
L�( )
L
2
( )
=s⋅y( )−(𝛿+n( ))⋅k( )
,
n
( )=L
�
(
)
L( )
L�( )=𝜆
⋅
L( ),
L
( )=L
0
e
𝜆 .
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
al e na i e model, which s a es ha a s able popula ion would ha e a cha ac e is ic
sa u a ion le el
M>0
. The co esponding equa ion is he well-known logis ic di -
e en ial equa ion
which can be sol ed by sepa a ion o a iables. The analy ical solu ion is gi en by
Unde his assump ion on popula ion g ow h he economy has a unique non- i ial
equilib ium poin in e ms o he capi al s ock pe capi a.
In his pape , we go a s ep u he and ex end he logis ic popula ion g ow h
wi h he Allee e ec , see e.g. Cou champ e al. (2008). A ma hema ical ep esen-
a ion o he Allee e ec o popula ion size is gi en by
The pa ame e N is he c i ical alue o he so-called Allee h eshold. We couple (7)
wi h he ini ial condi ion
L( 0)=L0>0
o o m he ini ial alue p oblem. Un o u-
na ely, he unique solu ion L( ) o (7) canno be gi en explici ly in he gene al case.
The e o e, we look a i s quali a i e beha io depending on he ini ial alue
L0
, see
Fig.3 wi h
N=1
and
M=2
. We obse e asymp o ically s able equilib ium poin s
M and 0 as well as an uns able equilib ium poin N.
Le us now assume he popula ion law wi h Allee e ec om (7) o he Solow-
Swan model (4). We see ha he economy o his modi ied Solow-Swan model is
desc ibed by he di e en ial equa ion
(5)
L
�( )=𝜆⋅L( )⋅
(
1−L( )
M
),
(6)
L
( )=
M
1+e−𝜆
(
M
L0
−1
).
(7)
L
�( )=𝜆⋅L( )⋅
(
1−L( )
M
)
⋅
(
L( )
N−1
).
Fig. 3 Popula ion g ow h (7)
wi h Allee e ec
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
whe e
In wha ollows, we shall s udy he co esponding ini ial alue p oblem o he
Solow-Swan model wi h Allee e ec o gi en capi al s ock pe capi a
k( 0)=k0>0
.
S abili y Analysis
Le us show ha he di e en ial equa ion (8) is uniquely sol able. Fo ha , we i s
s udy he beha io o he popula ion g ow h a e n( ) depending on he ini ial alue
L0>0
. F om
we ob ain:
By addi ionally using he p ope ies o L( ) and
L�( )
, we may dis inguish he ollow-
ing cases o he popula ion g ow h a e:
(1) I
L0∈(0, N)
, hen i holds:
(2a) I
L
0∈
(
N,
N+M
2]
, hen i holds:
whe e
is he unique solu ion o
L
( )=
N+M
2
.
(2b) I
L
0∈
(N+M
2
,M
)
, hen i holds:
(3) I
L0∈(M,∞)
, hen i holds:
(8)
k�( )=s
⋅
T
⋅
F(k( )
,1
)−(𝛿+n( ))
⋅
k( ),
n
( )=L
�
( )
L( )=𝜆⋅
(
1−L( )
M
)
⋅
(
L( )
N−1
).
n
( )=𝜆⋅
(
1−L( )
M
)
⋅
(
L( )
N−1
)
n
�( )=𝜆⋅L�( )⋅
(
1
M+1
N−2L( )
MN
).
n
( )<0, n
�
( )<0, lim
→∞
n( )=−𝜆
.
n
( )>0, n�( )
⎧
⎪
⎨
⎪
⎩
>0 i <
=0 i =
<0 i >
, lim
→∞
n( )=
0,
n
( )>0, n
�
( )<0, lim
→∞
n( )=
0.
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
(ii) We know ha o
L0∈(M,∞)
he popula ion g ow h a e has he ollowing
p ope ies, see case (3) om abo e:
The p oo o he asse ion he e goes along he same lines as in Theo em4. We jus
add
n(
)
o he dep ecia ion a e and sub ac i om he popula ion g ow h a e:
He e,
is aken su icien ly la ge o gua an ee ha
𝛿+n(
)>0
. Clea ly, (8) can be
equi alen ly w i en as (11), whe e (12) holds wi h
Since he shi ed dep ecia ion a e
𝛿
emains posi i e, he s abili y analysis om
Theo em3 applies. I says ha he capi al s ock pe capi a o (11) s abilizes a
k∗
−n(
)
.
Recalling
𝛿=𝛿+n(
)
, his equilib ium poin co esponds o
k∗
0
, and he asse ion
ollows.
(iii) We know ha o
L0∈{N,M}
he popula ion g ow h a e anishes, see case
(4) om abo e:
Then, (8) becomes he s anda d Solow-Swan model wi h ze o popula ion g ow h
a e. Thus, he asse ion i ially ollows.
◻
F om Theo ems4 and5 he ins abili y o he emaining equilib ium poin s o
(9) ollows.
Co olla y 1 (Ins abili y) The equilib ium poin s
(k∗
0,N)
, (0, M), (0,N), and (0,0) o
he Solow-Swan model (9) wi h Allee e ec a e uns able.
Le us compa e he pe o mance o he capi al s ock pe capi a in he long un
depending on he popula ion g ow h.
Rema k 2 (Compa ison o equilib ia) We assume ha
𝛿>𝜆
. Then, acco ding o
Theo ems4 and5, he Solow-Swan model (8) wi h Allee e ec has wo globally
asymp o ically s able equilib ium poin s
k∗
−𝜆
and
k∗
0
. Since
k∗
n
is s ic ly dec easing
wi h espec o
n∈ (−𝛿,∞)
, we ha e:
F om he economic poin o iew, he capi al s ock pe capi a pe o ms in he long
un be e i he popula ion becomes ex inc , a he hen i ad ances he sa u a ion
le el.
n
( )<0, n
�
( )>0, lim
→∞
n( )=
0.
𝛿=𝛿+n(
)
,
n( )=n( )−n(
).
n=−n(
),n∞=−n(
).
n( )
≡
0.
k∗
−𝜆>k∗
0.
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
Le us es ablish bounds on he capi al s ock pe capi a in e ms o he s anda d
Solow-Swan model wi h cons an popula ion g ow h a es.
Rema k 3 (Bounds on capi al s ock pe capi a) We assume ha
𝛿>𝜆
. F om he
p oo o Theo em4 we see ha in case o
L0∈(0, N)
i holds o all
≥ 0
:
Analogously, om he p oo o Theo em 5 we see ha in case o
L0∈(N,M)
i
holds o all su icien ly la ge
≥ 0
:
In case o
L0∈(M,∞)
i holds o all
≥ 0
:
In economic e ms, he capi al s ock pe capi a is bounded by ha o he s and-
a d Solow-Swan model wi h cons an popula ion g ow h a es
−𝜆
, 0, and
n( 0)
,
espec i ely.
Discussion o Cobb‑Douglas P oduc ion Func ion
In his sec ion, we e i y he esul s om Sec .3 on an illus a i e example. Fo ha ,
he wo-dimensional sys em o di e en ial equa ion (9) will be sol ed nume ically
o a gi en p oduc ion unc ion sa is ying (P1)-(P3). We choose he well-known
Cobb-Douglas p oduc ion unc ion:
wi h
𝛼∈(0, 1)
. The Solow-Swan model (9) wi h Allee e ec hen becomes
Addi ionally, we make a compa ison o (18) wi h he Solow-Swan model which
assumes he popula ion dynamics acco ding o he logis ic law, c . Gue ini (2006);
Sca pello and Ri elli (2003):
k
n(
0
)(
,
0
,k
0)≤
k
(
,
0
,k
0)≤
k
−𝜆(
,
0
,k
0)
.
k
n(
0
)
( ,
0
,k
0
)≤k( ,
0
,k
0
)≤k
0
( ,
0
,k
0
).
k
0
( ,
0
,k
0
)
≤
k( ,
0
,k
0
)
≤
kn
(
0
)
( ,
0
,k
0
).
F(K,L)=K𝛼
⋅
L1−𝛼
(18)
⎧
⎪
⎨
⎪
⎩
k�=s⋅T⋅k𝛼−
�
𝛿+𝜆⋅
�
1−L
M
�
⋅
�
L
N−1
��
⋅k
,
L�=𝜆⋅L⋅
�
1−L
M
�
⋅
�
L
N−1
�
.
(19)
⎧
⎪
⎨
⎪
⎩
k�=s⋅T⋅k𝛼−
�
𝛿+𝜆⋅
�
1−
L
M
��
⋅k
,
L�=𝜆⋅L⋅
�
1−
L
M
�
.
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
Assuming
𝛿>𝜆
, we ge he equilib ium poin s o he capi al s ock pe capi a o
(18) in explici o m, c . Theo ems4 and5:
The equilib ium poin o he capi al s ock pe capi a o (19) is jus
k∗
0
, see (Gue -
ini 2006; Sca pello and Ri elli 2003). We aim o in es iga e he dependence o he
solu ions o (18) and (19) on he popula ion ini ial alue. The ini ial alue
L0
can
be below he Allee h eshold N, be ween he Allee h eshold N and he sa u a ion
le el M, and abo e he sa u a ion le el M. We e e o Figs.4-6 o hese h ee cases.
He e, he solid line co esponds o he Solow-Swan model (18) wi h Allee e ec ,
and he do ed line o he Solow-Swan model (19) wi h logis ic law. The pa ame e s
ela ed o capi al a e chosen as ollows:
The pa ame e s de e mining he popula ion dynamics a e se as
The main di e ence be ween assuming Allee e ec o logis ic law in he Solow-
Swan model is i s a ing below he Allee h eshold N, see Fig.4. I u ns ou ha hen
he capi al s ock pe capi a s abilizes a a highe le el
k∗
−𝜆
o he Allee e ec in com-
pa ison o he lowe le el
k∗
0
o he logis ic law. I s a ing abo e he Allee h eshold
N, see Figs.5 and6, he beha io o he capi al s ock pe capi a o he Allee e ec is
as expec ed simila o ha o he logis ic law. In o de o illus a e he blow-up e ec ,
which occu s i he s abili y condi ion
𝛿>𝜆
is iola ed, le us ake
k
∗
−𝜆=
(
s⋅T
𝛿−𝜆)
1
1−𝛼,k∗
0=
(
s⋅T
𝛿)
1
1−𝛼
.
𝛼=0.3, s=0.025, T=1, 𝛿=0.075.
𝜆=0.025, N=1, M=2.
𝜆=0.085.
Fig. 4 Case
L0∈(0, N)
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
I s a ing below he Allee h eshold N, he capi al s ock pe capi a explodes in his
case, see Fig.7. This is in con as wi h he logis ic law, whe e he co esponding
capi al s ock pe capi a s abilizes a
k∗
0
once again.
Fig. 5 Case
L0∈(N,M)
Fig. 6 Case
L0∈(M,∞)
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Jou nal o Quan i a i e Economics (2025) 23:1259–1278
Acknowledgemen s The au ho s would like o hank he anonymous e e ees o sugges ing aluable
imp o emen s o he pape .
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