Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків
The comparative analysis of the neoclassical Solow’s model and the modified Solow’s model in the implementation of technological progress has shown undeniable advantages of the modified Solow’s model. A modified version of the Solow’s economic growth model, based on an n-step production function in...
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| description | The comparative analysis of the neoclassical Solow’s model and the modified Solow’s model in the implementation of technological progress has shown undeniable advantages of the modified Solow’s model. A modified version of the Solow’s economic growth model, based on an n-step production function in the form of n S-shaped functions for the implementation of technological progress, ensures the growth of the economy on a sufficiently large time interval comparable to the duration of the life cycle of the economy under study. In this interval, referred to as the “technology gap”, intensive output y (t) can be carried out according to the following options: monotonic decrease (stable 1-cycle) of the considered model; oscillations (stable n-cycles, n=2,4,16,…), “the economy marks time”; chaotic fluctuations. This result for the models of economic growth has not been described in the literature. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2021.3.08 |
| first_indexed | 2025-07-17T10:27:35Z |
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A.K. Lopatin, 2021
Системні дослідження та інформаційні технології, 2021, № 3 99
UDC 519.86
DOI: 10.20535/SRIT.2308-8893.2021.3.08
TECHNOLOGY PROGRESS IMPLEMENTATION BASED
ON A MODIFIED VERSION OF R.M. SOLOW ECONOMIC
GROWTH MODEL: WITH PRODUCTION S-CURVE
CONSISTING OF n-STEPS
A.K. LOPATIN
Abstract. The comparative analysis of the neoclassical Solow’s model and the
modified Solow’s model in the implementation of technological progress has shown
undeniable advantages of the modified Solow’s model. A modified version of the
Solow’s economic growth model, based on an n-step production function in the
form of n S-shaped functions for the implementation of technological progress,
ensures the growth of the economy on a sufficiently large time interval comparable
to the duration of the life cycle of the economy under study. In this interval, referred
to as the “technology gap”, intensive output y (t) can be carried out according to the
following options: monotonic decrease (stable 1-cycle) of the considered model;
oscillations (stable n -cycles, ..=,2,4,16,.n ), “the economy marks time”; chaotic
fluctuations. This result for the models of economic growth has not been described
in the literature.
Keywords: modified Solow’s model, technology gap, periodic cycles and chaos.
INTRODUCTION
In this work, we have obtained a further development of a modified version of the
Solow economic growth model, based on an n-step production function in the
form of n S -shaped functions for the implementation of technical progress,
proposed in [1]. The substantiation of the dependence of macroeconomic
dynamics on technological investment priorities in the form of S -shaped
production functions has been considered in numerous works (see the review
article [2] and the monograph [3]. In this work, we follow [4] and [5] The
development process of each technological paradigm, in general, is described by a
logistic curve, which expresses the most general laws of the dynamics of
progressive cyclic processes. At the beginning of the life cycle of each
technological paradigm, significant costs for its development give insignificant
results — the first flat section of the logistic curve corresponds to this period.
Then, with the development and practical mastering of the corresponding
technical and technological principles, small costs begin to bring a significant ef-
fect and the curve rises steeply. Further, as the technologies of this paradigm are
approaching their technological limits, this technological paradigm again enters a
gentle section of the curve, and no, even large-scale investments in its
development are no longer able to bring a significant effect. At the disposal of the
investor at any given time, there is a limited number of technologies that could
potentially be supported by his investment resources. The meaning of overcoming
the period of technological gap, which is repeated from time to time in each
A.K. Lopatin
ISSN 1681–6048 System Research & Information Technologies, 2021, № 3 100
branch of the economy, is to quickly and as little loss as possible change from one
logistic curve to another, corresponding to a more progressive technological
order. In this case, the correct choice of a substitute technology entirely depends
on the correctness of estimates of the upper technological limits of several
competing technologies designed to solve the same technical problem. There is a
need for a transition to a new technological paradigm. This is one of the most
difficult problems of technical and economic forecasting. A modified version of
the Solow’s economic growth model, based on an n-step production function in
the form of n S -shaped functions for the implementation of technological
progress, proposed in [1], gives an adequate mathematical apparatus for solving
this problem. This work develops the results of work [1] in the following
directions:
comparative analysis of neoclassical Solow’s model and the modified
Solow’s model in the implementation of techno- logical progress;
development of a constructive algorithm based on the apparatus of
nonlinear discrete dynamical systems;
development a modified model, generating periodic and chaotic cycles.
Solow’s model
The founder of modern growth theory was R.M. Solow ( [6], Nobel Prize Winner
in Economics 1987). Until now, due to its combination of simplicity and richness,
the Solow’s model is considered as the basic model of economic growth in most
economics textbooks. For modern post-Solow growth theories see the overview
article [7].
Solow’s model description
Here we follow [8]
Assumptions:
closed economy, producing one good using both labor and capital;
technological progress is given and the saving rate is exogenously
determined;
no government and fixed number of firms in the economy, each with the
same production technology;
output price is constant and factor prices (including wages) adjust to
ensure full utilization of all available inputs.
Neoclassical production function:
Four variables considered:
flow of output, Y ;
stock of capital, K ;
number of workers, L ;
effective labor, LA .
Aggregate production function given by,
),(= ALKFY . (1)
A and L enter multiplicatively, where AL is effective labor, and
technological progress enters as labor augmenting or Harrod neutral.
Technology progress implementation based on a modified version of R.M. Solow economic …
Системні дослідження та інформаційні технології, 2021, № 3 101
Assumed characteristics of the model:
0>
)(
0,>0,>0,>
2
2
2
2
AL
F
K
F
AL
F
K
F
.
Constant returns to scale (CRS) in capital and effective labor:
),(=),( ALKFALKF . (2)
Intensive-form production function: output per unit of effective labor, y , and
capital per unit of effective labor, k , are related by setting
AL
1
= in (2),
),(
1
=1, ALKF
ALAL
K
F
.
Let
AL
Y
y
AL
K
k =,= and ,1)(=)( kFkf . Equation (1) is then written as
intensive-form production function
0=(0)),(= fkfy .
A MODIFIED VERSION OF SOLOW’S ECONOMIC GROWTH MODEL WITH
n-STAGE S-LIKE PRODUCTION FUNCTION
The idea of using S -curves in the Solow’s equation was first (as it seems to us)
expressed in [9] using the example of Richards S -curves. The main result of this
work as applied to the continuous Solow equation is reduced to the classification
of trajectories, similarly to what has been done in the Solow’s equation in [1].
A modified Solow’s model description
Let there be some S -curve depending on the 2p parameters, and the m
parameter is responsible for the shift along the abscissa axis, the u parameter —
along the ordinate axis:
),,,...,( 1 umvvS ppf .
Ecxample. Verholst’s S -curve
,
))((exp1
=)( u
mkaB
A
kS
t
tpf
(3)
here
t
t
t L
K
k = , 0=4,=1,=5,,0=5,= umaBA .
The new production function, which will be called the n-stage production
function ( 1n ), is given by formula:
L
K
LSY pf= . (4)
Let
L
Y
y
L
K
k =,= and )(=)( kSkf pf . Equation (4) is then written as
intensive-form production function
)(= kfy .
A.K. Lopatin
ISSN 1681–6048 System Research & Information Technologies, 2021, № 3 102
It is assumed that the growth of working in economy is described by
formula:
tt LntL )1(=)(1 .
Economic growth determined by the behavior of the capital stock K :
t
t
t
pftt Kd
L
K
SLsK )(1*=1
. (5)
Dividing both sides of equation (5) by tL , after simple transformations we
obtain the second s to eliminate
ttt k
n
d
ksf
n
s
k
1
1
)(
1
=1 . (6)
It is non-linear, first-order difference equation. Here 2,0=d -depreciation
rate, 3,0=s -a fixed fraction of output, 1<<0 s , Nt 0,1,...,= .
The equation (6) will be called the modified Solow’s model with 1-stage
production function. The model with N stages production function was considered
in [1] and with 2 stages production function is considered in section 4 of this
article.
A modified Solow’s model characteristics
All clauses of Assumption in subsection 2.1 are accepted.
The production function can be both increasing and decreasing Growing
production function (see Fig. 1 and 2).
From three cases of
equilibrium for phase trajectory we
choose Case 1: Developed world
(good equilibrium). The phase
curve is located above the bisector
and has one intersection point with
it (a stable equilibrium position)
[1].
Production function (3) has
more parameters than neoclassical
production function.
When changing the
parameters m and u, the
production function can be shifted
to any point in the first quarter.
The condition 0)0( f is not
met.
The need to use the theory
of nonlinear dynamical systems
to study the quantitative and
qualitative characteristics of
solutions to the modified Solow model. For the neoclassical model, this device
has not been used in full before.
y f(k)
)(kf
)(kf
k
Fig. 1. Increasing production function f (k)
y
f(k)
)(kf
)(kf
k
Fig. 2. Decreasing production function f (k)
Technology progress implementation based on a modified version of R.M. Solow economic …
Системні дослідження та інформаційні технології, 2021, № 3 103
The S-shaped production function describes the life cycle of each
technology: origin, leapfrogging, and gradual achievement of the stage of full
maturity of a technological process or product. For the S -curve to be of practical
value, it must predict impending technological change:
.<,0<)(0,>)(
,,0>)(0,>)(
pointinflectionxifkfkf
pointinflectionxifkfkf
.<,0<)(0,>)(
,,0>)(0,>)(
pointinflectionxifkfkf
pointinflectionxifkfkf
A MODIFIED MODEL, GENERATING PERIODIC AND CHAOTIC CYCLES
A modified model construction algorithm
Consider two production functions shown in Fig. 3.
Here
37<0,
))((exp1
=)(1 t
t
tpf ku
mkaB
A
kS
; (7)
47}<37,
))((exp1
=)( 5
555
5
2 t
t
tpf ku
mkaB
A
kS
(8)
in equations (7), (8) the parameters have the following values.
Summary of parameters used
A B a m u A5 B5 a5 m5 u5
117 6,9 0,1 0 -14,1 72 10 -0,5 52 10
The technique for finding the variables Nkt 2,...1,, will be discussed
below.
37<,0
11
=)( 11 ttpftph kk
n
dn
S
n
s
kS
. (9)
Here 2,0=d -depreciation rate, 3,0=s -investment share, 004,0=n is
a coefficient of the growth of person employed in economy:
33=,)1(=)(,)1(=)( 001 LnLtLLntL N
Nttt .
1
S1pf S2pf
yt
kt
Fig. 3. Intensive production functions pfS1 and pfS2
A.K. Lopatin
ISSN 1681–6048 System Research & Information Technologies, 2021, № 3 104
It is easy to see that the phase curve )(1 tph kS (4) is a right-hand side of the
modified Solow’s equation
)(= 11 tpht kSk .
In a similar way, we write down the equation for phase curve )(2 tph kS
47<37
11
=)( 22 ttpftph kk
n
dn
S
n
s
kS
(10)
and corresponding modified Solow’s equation
)(= 21 tpht kSk
and )(2 tss kS which are used to find steady states:
)(=)(,)(=)( 2211 tfptsstfptss ksSkSksSkS .
To find the steady state points of the variable tk of the modified Solow
equation (9) , we construct a diagram.
The Fig. 4 shows that steady
state point of tky = lies outside
where equation (9) is considered
and should not be taken into
account. To control the correctness
of the construction, you need to
make sure that the steady state
points generated by the phase curve
phS1 and ssS1 coincide.
Fig. 5 shows that steady state
point of tky = lies within where
equation (10) is considered and should be taken into account. To control the
correctness of the construction, you need to make sure that the steady state points
generated by the phase curves phS2 and ssS2 coincide.
Dynamics of the modified Solow equation
Let us introduce the 2-stage production function )(2 tkS given by equation
.4737if,)(
,37<0if,)(
=)(2
2
1
ttpf
ttpf
t kkS
kkS
kS
yt, kt+1
S1pf
S1ph
S1ss
kt
Fig. 4. Finding steady state points. Dotted line —
bisector tky = ; dotted line — linear function
tkdny )(=
Fig. 5. Shows phase curves phS2 and ssS2 coincide
yt, kt+1
S2pf
S2ph
S2ss
kt
y=(n+d)kt
k
y=kt
Technology progress implementation based on a modified version of R.M. Solow economic …
Системні дослідження та інформаційні технології, 2021, № 3 105
Output per worker )( tky can be found from the relationship
)(2=)( tt kSky .
Let us introduce the 2-stage phase curve )2( tkS given by equation
.4737if,)(
,37<0if,)(
=)(2
2
1
ttph
ttph
t kkS
kkS
kG
For the function )(2 tkG , we can write down the modified Solow equation
)(2=)( 11 tt kGky (11)
This is a nonlinear discrete first order equation with piecewise continuous
coefficients. The theory of first-order nonlinear discrete dynamical systems is
well developed [10]. Numerical simulation is crucial in the investigation of
nonlinear systems.Let us solve equation (11) and carry out a qualitative study of
the equation. For this we use the E&F Chaos software package [11] (see Figs 6–12).
In fig. 8. As parameter 5a increases from -1 to 0, one observes: each stable
cycle goes through an infinite sequence of period-doublings.
Fig. 6. The stable fixed point (stable 1-cycle) of equation of (11)
100
75
50
25
0
25 50 75 100
Fig. 7. The time series plots of stable 1-cycle of equation (11)
44
33
22
11
0
6 12 24 36
k1
t
Fig. 8. Period doubling of flip bifurcation, with creation of a 2-cycle of (11)
59
53
47
41
35
-2 -0,8 -0,6 -0,4 -0,2 0
k1 C
B
A
A.K. Lopatin
ISSN 1681–6048 System Research & Information Technologies, 2021, № 3 106
For example, the stable fixed point gives rise to a stable 2-cycle just as the
fixed point becomes unstable.
This stable 2-cycle gives rise to a stable 4-cycle just as the 2-cycle becomes
unstable.
This stable 4-cycle gives rise to a stable 8-cycle just as the 4-cycle becomes
unstable.
In general, any stable n -cycle gives rise to a family
cycle16cycle8cycle4cycle2 nnnn .
The amplitude of kn -cycle goes to zero when k goes infinity. Systems with
two properties: bifurcation and period doubling, sensitive dependence to initial
conditions or the largest Lyapunov exponent is positive are considered to be
chaotic in a certain sense. These are termed as the routes to chaos.
In Fig. 9. The largest Lyapunov exponent is the average growth rate of an
infinitesimal state perturbation along a typical trajectory (orbit). For periodic
regimes the largest Lyapunov exponent is negative. This means that they are
asymptotically stable by Lyapunov with respect to small changes in the initial
conditions. For chaotic regimes the largest Lyapunov exponent is positive. This
means that they are asymptotically unstable by Lyapunov with respect to small
changes in the initial conditions (sensitive dependence to initial conditions).
Fig. 9. The largest Lyapunov exponent 5a increases from -1 to 0 for equation (11)
0,3
-0,4
-1,1
-1,8
-2,5
-1 -0,8 -0,6 -0,6 -0,2 0
Fig. 10. Time series of stable 2-cycle at 5,0=5 a of (11)
48
36
24
12
0
6 12 18 24
k1
t
Fig. 11. Time series of stable 4-cycle at 6,0=5 a of (11)
52
39
26
13
0
k1
t
10 20 30 40
Technology progress implementation based on a modified version of R.M. Solow economic …
Системні дослідження та інформаційні технології, 2021, № 3 107
The diagram shows two trajectories under different initial conditions: blue
1,1=0k and red 11,1=0k despite the small difference in the values of the initial
conditions, the trajectories first coincide and then diverge — the effect of the
sensitivity of the solutions to the initial conditions.
An example of a model based on real statistical data
The intervals at which the drop in production occurs, for example, BC, DE, FG
(Fig. 13), will be called the “technological gap” intervals. The very effect of this
behavior is a “technological gap”. This process is the result of a combination of
economic, technological, socio-cultural, political and other events. The task of a
healthy economy is to overcome this gap as quickly as possible and move to
production growth.
Fig. 12. Chaos:sensitive dependence on initial conditions at 85,0=5 a of (11)
k1
t
52
39
26
13
0
10 20 30 40
Fig. 13. GDP per person employed (current US $) of Germany 1972–2018 by year numbers
y1
t
Fig. 14: Approximation of real data using the model in fig. 13
A.K. Lopatin
ISSN 1681–6048 System Research & Information Technologies, 2021, № 3 108
The figures above explain the mechanism of behavior during the transition
from a growing S-shaped production function to a decreasing S -shaped
production function at a certain interval of functioning of the economic system
under study.
CONCLUSIONS
Comparative analysis of the neoclassical Solow model and the modified Solow
model in the implementation of technological progress has shown undeniable
advantages of the modified Solow model. The exogenously introduced S-shaped
production function describes the life cycle of technology at every: origin,
leapfrogging, and gradual achievement of the stage of full maturity of a
technological process or product at every stage. A modified version of the Solow
economic growth model, based on an n-step production function in the form of n
S-shaped functions for the implementation of technological progress, ensures the
growth of the economy by a sufficiently large time interval comparable to the
duration of the life cycle of the economy under study.
The use of the theory of nonlinear dynamical systems to study solutions of
the modified Solow model significantly improves their quantitative and
qualitative characteristics. For the neoclassical model, this technique has not been
used in full before.
A modified model, generating periodic and chaotic cycles for 2 S-shaped
production functions, is created. It explains the mechanism of behavior during the
transition from a growing S-shaped production function to a decreasing S-shaped
production function at a certain interval of functioning of the economic system
under study. In this interval, referred to as the “technology gap”, intensive output
)(ty can be carried out according to the following options (see Fig. 13, 14):
Monotonic decrease (steady-state (stable 1-cycle) of the considered model.
Oscillations (stable n -cycles, ..1,2,4,16,.=n ), “the economy marks time”.
Chaotic fluctuations. This result for the models of economic growth is
fundamental and has not been described in the literature.
REFERENCES
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Future Conference 2007, Procedia Engineering, 9 , 2011, 559572.
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econometric tools for modeling and forecasting evolutionary processes: a
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tehnologicheskie-razryvy
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Russia”, Voprosy Ekonomiki, no. 1, pp. 11–36, 2019.
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6, no. 1, pp. 65–70, 2009.
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Received 30.06.2021
INFORMATION ON THE ARTICLE
Alexey K. Lopatin, ORCID: 0000-0002-0832-3600, Educational and Scientific Complex
“Institute for Applied System Analysis” of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: lopatinalexey142@gmail.com
РЕАЛІЗАЦІЯ ТЕХНОЛОГІЧНОГО ПРОГРЕСУ НА ОСНОВІ МОДИФІКО-
ВАНОЇ ВЕРСІЇ МОДЕЛІ ЕКОНОМІЧНОГО ЗРОСТАННЯ Р.М. СОЛОУ:
S-ПОДІБНА КРИВА ВИРОБНИЦТВА, ЩO СКЛАДАЄТЬСЯ З n-КРОКІВ /
О.К. Лопатін
Анотація. Порівняльний аналіз неокласичної моделі Р. Солоу і її модифікова-
ної версії в реалізації технологічного прогресу показав незаперечні переваги
модифікованої моделі Солоу. Модифікована версія моделі економічного зрос-
тання Р. Солоу, що заснована на n-ступінчастій виробничій функції у вигляді n
S-подібних функцій для реалізації технічного прогресу, забезпечує зростання
економіки на достатньо великому часовому інтервалі, порівнянному з тривалі-
стью життєвого циклу досліджуваної економіки. В інтервалі, званому «техно-
логічним розривом», інтенсивний випуск )(ty може здійснюватися у відповід-
ності з наступними варіантами: монотонне зниження продуктивності
(стабільний 1-цикл) розглянутої моделі; коливання (стійкі n -цикли , n = 2,4,16,
...), «економіка тупцює на місці»; хаотичні коливання. Перераховані результа-
ти для моделей економічного зростання в літературі не описані.
Ключові слова: модифікована модель Р. Солоу, технологічний розрив,
періодичні цикли і хаос.
РЕАЛИЗАЦИЯ ТЕХНОЛОГИЧЕСКОГО ПРОГРЕССА НА ОСНОВЕ
МОДИФИЦИРОВАННОЙ ВЕРСИИ МОДЕЛИ ЭКОНОМИЧЕСКОГО РОСТА
Р.М. СОЛОУ: S-ОБРАЗНАЯ КРИВАЯ ПРОИЗВОДСТВА, СОСТОЯЩАЯ ИЗ
n-ШАГОВ / А.К. Лопатин
Аннотация. Сравнительный анализ неоклассической модели Р. Солоу и ее
модифицированной версии в реализации технологического прогресса показал
неоспоримые преимущества модифицированной модели Солоу. Модифи-
цированная версия модели экономического роста Р. Солоу, основанная на
n-ступенчатой производственной функции в виде n S-образных функций для
реализации технического прогресса, обеспечивает рост экономики на достаточ-
но большом временном интервале, сопоставимом с продолжительность жиз-
ненного цикла исследуемой экономики. В интервале, называемом «технологи-
ческим разрывом», интенсивный выпуск )(ty может осуществляться в
соответствии со следующими вариантами: монотонное снижение производи-
тельности (стабильный 1-цикл) рассматриваемой модели; колебания (устойчи-
вые n-циклы, n = 2,4,16,…), «экономика топчется на месте»; хаотические коле-
бания. Перечисленные результаты для моделей экономического роста в
литературе не описаны.
Ключевые слова: модифицированная модель Р. Солоу, технологический раз-
рыв, периодические циклы и хаос.
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| id | journaliasakpiua-article-244603 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:35Z |
| publishDate | 2021 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/f7/719ed965f63154f5fece553977ab87f7.pdf |
| spelling | journaliasakpiua-article-2446032022-02-09T17:33:09Z Technology progress implementation based on a modified version of R.M. Solow economic growth model: with production s-curve consisting of n-steps Реализация технологического прогресса на основе модифицированной версии модели экономического роста Р.М. Солоу: s-образная кривая производства, состоящая из n-шагов Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків Lopatin, Alexey модифікована модель Р. Солоу технологічний розрив періодичні цикли і хаос модифицированная модель Р. Солоу технологический разрыв периодические циклы и хаос modified Solow’s model technology gap periodic cycles and chaos The comparative analysis of the neoclassical Solow’s model and the modified Solow’s model in the implementation of technological progress has shown undeniable advantages of the modified Solow’s model. A modified version of the Solow’s economic growth model, based on an n-step production function in the form of n S-shaped functions for the implementation of technological progress, ensures the growth of the economy on a sufficiently large time interval comparable to the duration of the life cycle of the economy under study. In this interval, referred to as the “technology gap”, intensive output y (t) can be carried out according to the following options: monotonic decrease (stable 1-cycle) of the considered model; oscillations (stable n-cycles, n=2,4,16,…), “the economy marks time”; chaotic fluctuations. This result for the models of economic growth has not been described in the literature. Сравнительный анализ неоклассической модели Р. Солоу и ее модифицированной версии в реализации технологического прогресса показал неоспоримые преимущества модифицированной модели Солоу. Модифицированная версия модели экономического роста Р. Солоу, основанная на n-ступенчатой производственной функции в виде n S-образных функций для реализации технического прогресса, обеспечивает рост экономики на достаточно большом временном интервале, сопоставимом с продолжительность жизненного цикла исследуемой экономики. В интервале, называемом “технологическим разрывом”, интенсивный выпуск y(t) может осуществляться в соответствии со следующими вариантами: монотонное снижение производительности (стабильный 1-цикл) рассматриваемой модели; колебания (устойчивые n-циклы, n = 2,4,16,…), “экономика топчется на месте”; хаотические колебания. Перечисленные результаты для моделей экономического роста в литературе не описаны. Порівняльний аналіз неокласичної моделі Р. Солоу і її модифікованої версії в реалізації технологічного прогресу показав незаперечні переваги модифікованої моделі Солоу. Модифікована версія моделі економічного зростання Р. Солоу, що заснована на n-ступінчастій виробничій функції у вигляді n S-подібних функцій для реалізації технічного прогресу, забезпечує зростання економіки на достатньо великому часовому інтервалі, порівнянному з тривалістью життєвого циклу досліджуваної економіки. В інтервалі, званому “технологічним розривом”, інтенсивний випуск y(t) може здійснюватися у відповідності з наступними варіантами: монотонне зниження продуктивності (стабільний 1-цикл) розглянутої моделі; коливання (стійкі n -цикли , n = 2,4,16, ...), “економіка тупцює на місці”; хаотичні коливання. Перераховані результати для моделей економічного зростання в літературі не описані. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2021-09-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/244603 10.20535/SRIT.2308-8893.2021.3.08 System research and information technologies; No. 3 (2021); 99-109 Системные исследования и информационные технологии; № 3 (2021); 99-109 Системні дослідження та інформаційні технології; № 3 (2021); 99-109 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/244603/242418 |
| spellingShingle | модифікована модель Р. Солоу технологічний розрив періодичні цикли і хаос Lopatin, Alexey Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title | Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title_alt | Technology progress implementation based on a modified version of R.M. Solow economic growth model: with production s-curve consisting of n-steps Реализация технологического прогресса на основе модифицированной версии модели экономического роста Р.М. Солоу: s-образная кривая производства, состоящая из n-шагов |
| title_full | Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title_fullStr | Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title_full_unstemmed | Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title_short | Реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання Р.М. Солоу: s-подібна крива виробництва, щo складається з n-кроків |
| title_sort | реалізація технологічного прогресу на основі модифікованої версії моделі економічного зростання р.м. солоу: s-подібна крива виробництва, щo складається з n-кроків |
| topic | модифікована модель Р. Солоу технологічний розрив періодичні цикли і хаос |
| topic_facet | модифікована модель Р. Солоу технологічний розрив періодичні цикли і хаос модифицированная модель Р. Солоу технологический разрыв периодические циклы и хаос modified Solow’s model technology gap periodic cycles and chaos |
| url | https://journal.iasa.kpi.ua/article/view/244603 |
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