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МАТЕМАТИЧЕСКИЕ И ИНСТРУМЕНТАЛЬНЫЕ МЕТОДЫ
ЭКОНОМИКИ
MATHEMATICAL AND INSTRUMENTAL METHODS
OF ECONOMICS
DOI: 10.18287/2542-0461-2021-12-2-154-170 ÉLJLJ
SCIENTIFIC ARTICLE
Submitted: 28.03.2021 Revised: 30.04.2021 Accepted: 27.05.2021
Equations of nonlinear dynamics of development of industrial enterprises, taking into account the amount of its maximum profit
A.L. Saraev
Samara National Research University, Samara, Russian Federation E-mail: alex.saraev@gmail.com. ORCID: http://orcid.org/0000-0002-9223-6330
L.A. Saraev
Samara National Research University, Samara, Russian Federation E-mail: saraev_leo@mail.ru. ORCID: http://orcid.org/0000-0003-3625-5921
Abstract: In the published article, new modifications of economic and mathematical models of the dynamic development of enterprises are proposed, the production of which is being restored due to the introduction of their own investments. The developed models are presented in the form of systems of differential equations for an arbitrary number of production factors. Stationary solutions of these systems of equations correspond to the equilibrium states of the operation of enterprises and represent the limiting values of the factors of production. Two versions of systems of differential balance equations for enterprises, describing the growth of factors of production and output, have been established. In the first case, the growth of resources and output is limited by the limiting values of the factors of production. In the second case, the growth of resources and output is limited by the pre-calculated values of the factors of production that correspond to the value of the maximum profit of the enterprise. It is shown that the growth of production factors of the enterprise should not exceed the values corresponding to the value of the maximum profit. Otherwise, the company starts to operate at a loss. In the presented models, proportional, progressive and digressive depreciation deductions are considered. The constructed models make it possible to describe various modes of operation of enterprises. Such regimes include a stable output of products by enterprises, a temporary suspension of the work of enterprises during its technical re-equipment, and a temporary partial curtailment of production.
Key words: enterprise; production; resources; production factors; investments; depreciation; production function; labor.
Citation. Saraev A.L., Saraev L.A. Equations of nonlinear dynamics of development of industrial enterprises,
taking into account the amount of its maximum profit. Vestnik Samarskogo universiteta. Ekonomika i
upravlenie = Vestnik of Samara University. Economics and Management, 2021, vol. 12, no. 2, pp. 154-170. DOI:
http://doi.org/10.18287/2542-0461-2021-12-2-154-170.
Information on the conflict of interest: authors declare no conflict of interest.
© Saraev A.L., Saraev L.A., 2021
Alexander L. Saraev - Candidate of Economic Sciences, associate professor of the Department of Mathematics and Business Informatics, Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.
Leonid A. Saraev - Doctor of Physical and Mathematical Sciences, professor, head of the Department of Mathematics and Business Informatics, Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.
НАУЧНАЯ СТАТЬЯ
УДК 330.42
Дата поступления: 28.03.2021 рецензирования: 30.04.2021 принятия: 27.05.2021
Уравнения нелинейной динамики развития производственных предприятий, учитывающие размер его максимальной прибыли
А.Л. Сараев
Самарский национальный исследовательский университет имени академика С.П. Королева,
г. Самара, Российская Федерация E-mail: alex.saraev@gmail.com. ORCID: http://orcid.org/0000-0002-9223-6330
Л.А. Сараев
Самарский национальный исследовательский университет имени академика С.П. Королева,
г. Самара, Российская Федерация E-mail: saraev_leo@mail.ru. ORCID: http://orcid.org/0000-0003-3625-5921
Аннотация: В публикуемой статье предложены новые модификации экономико-математических моделей динамического развития предприятий, производства которых восстанавливаются за счет ввода собственных инвестиций. Разработанные модели представлены в виде систем дифференциальных уравнений относительно произвольного числа производственных факторов. Стационарные решения этих систем уравнений соответствуют равновесным состояниям работы предприятий и представляют собой предельные значения факторов производства. Установлено два варианта систем дифференциальных уравнений баланса для предприятий, описывающих рост факторов производства и выпуска продукции. В первом случае рост ресурсов и выпуска продукции ограничивается предельными значениями факторов производства. Во втором случае рост ресурсов и выпуска продукции ограничивается вычисленными заранее значениями факторов производства, отвечающими значению максимальной прибыли предприятия. Показано, что рост производственных факторов предприятия не должен превышать значений, соответствующих значению максимальной прибыли. В противном случае предприятие начинает работать себе в убыток. В представленных моделях рассмотрены пропорциональные, прогрессивные и дигрессивные амортизационные отчисления. Построенные модели позволяют описывать различные режимы работы предприятий. К таким режимам относятся стабильный выпуск продукции предприятиями, временная приостановка работы предприятий на время его технического переоснащения и временное частичное сворачивание производства.
Ключевые слова: предприятие; производство; ресурсы; производственные факторы; инвестиции; амортизация; производственная функция.
Цитирование. Saraev A.L., Saraev L.A. Equations of nonlinear dynamics of development of industrial enterprises, taking into account the amount of its maximum profit // Вестник Самарского университета. Экономика и управление. 2021. Т. 12, № 2. С. 154-170. DOI: http://doi.org/10.18287/2542-0461-2021-12-2-154-170.
Информация о конфликте интересов: авторы заявляют об отсутствии конфликта интересов. © Сараев А.Л., Сараев Л.А., 2021
Александр Леонидович Сараев - кандидат экономических наук, доцент кафедры математики и бизнес-информатики, Самарский национальный исследовательский университет имени академика С.П. Королева, 443086, Российская Федерация, г. Самара, Московское шоссе, 34.
Леонид Александрович Сараев - доктор физико-математических наук, профессор, заведующий кафедрой математики и бизнес-информатики, Самарский национальный исследовательский университет имени академика С.П. Королева, 443086, Российская Федерация, г. Самара, Московское шоссе, 34.
Introduction
The development and improvement of mathematical methods for predicting indicators of the dynamics of the economic development of industrial enterprises is one of the urgent problems of modern economic theo-
ry. A successful solution to this problem makes it possible, in certain cases, to carry out an adequate analysis of the activities of enterprises, to calculate the limiting values for their resources, production volumes and profits, to describe quite accurately the dynamics of production, costs and profits, etc.
An important and long-term trend towards an increase in the indicators of the national economy is provided by economic growth and the development of manufacturing enterprises. A significant contribution to the theoretical foundations of economic growth is presented in the works [1-7].
On the basis of these theories, a whole range of growth models has been created for economic systems, taking into account the role of technical innovations and information technologies [8-18].
The patterns and features of the dynamics of the development of enterprises are formed from the interaction of the volumes of investments invested in production and the volumes of resources withdrawn as a result of depreciation, the cost of modernizing the means of production. Differential equations and systems of differential equations are widely used as one of the main mathematical tools for constructing models of economic development of enterprises [19-33].
The aim of the published work is to develop new economic and mathematical models of the dynamics of enterprise development, which take into account the influence of accompanying production costs and profits. This accounting makes it possible to predict the output of the enterprise's capacities to such a limiting state of production at which the enterprise's profit becomes maximum.
The scientific novelty and features of these models lie in the fact that they describe the interaction of proportional, progressive and digressive depreciation deductions with investments and costs invested in production and allow calculating the limiting values of factors of production and output.
Variants of stable progressive development of the enterprise, suspension of its work during the re-equipment of production and temporary crisis curtailment of production when replacing equipment are considered.
Statement of the problem
Consider a multifactor enterprise, the output of which is provided by a set of resources (Qx,Q2,---,Qn) • The quantities Q can represent fixed assets, working capital, financial capital, labor resources, materials involved in production, technologies and innovations, etc.
Production factors Q change over time t and are continuous and continuously differentiable functions
Q = Q (t). The units of measurement of a variable t , depending on the economic situation under consideration, can be one month, one quarter or one year.
Limited functions Q = Q (t) are enclosed between their upper and lower boundaries
Q0 ^ Q ^ q: ,
where Q0 = Q. ( 0) are the initial values of the factors of production Q, and Q: = lim Q (t) are their lim-
1 t—>: 1 \ /
iting values.
At the initial moment of time of the considered process t = 0 , the values of the components Q0 are considered known. The limiting values of the quantities Q: follow from the developing economic situation
and are subject to calculation.
The volume of production of the enterprise V is provided by the multifactorial production function of Cobb - Douglas
n
v = p hqls, (i)
ss
s=l
where P is the cost of products produced for unit volumes of resources, a are the elasticities of output with respect to the corresponding resources Qs , (0 < a < 1).
The proportional costs of an enterprise with resources are
TC = YjHs ■ Qs + TFC , (2)
s=1
where Hs are the cost of costs for unit volumes of resources Q, TFC are the fixed costs of the enterprise.
The expression for the profit of the enterprise PR = V — TC is expressed by the difference between formulas (2) and (3)
n
PR - P П О" -У H • Qs - TFC. (3)
^s
s=1 i=1
The maximum profit of the considered enterprise is found from the conditions
QPR P ■ a.
or
n
ПQas - H = 0
öß Q nQs s
n
p п Qas• q » (4)
s=1
where a = Hl .
1 a
The system of equations (4) shows that the values of resources Q and Qk for any indices (/ = l,2,...,n) and (k —1,2,.. ,,/i) are related by the relations
Q= a• Qk. (5)
a
Substituting formulas (5) into the equations of system (4), we find the values of the resources Q™*
_ 1
P
^max _
п
«
« s=1 \as у
1-У as
У . (6)
The maximum profit value is calculated by the formula
n n
PRmax = P ^ QmaX f -S H ^ Qr - TFC . (?)
s=l s=l
Let us now compose the balance equations for the dynamics of the development of the considered enterprise. To do this, we will express the increments of the volumes of resources AqQ = Q (t + At) — Q (t) at
some small time interval At as the sum of two components
AQ (t ) = AQf (t ) + AQ/(t), (8)
where AQf (t) is the partial depreciation loss of a resource Q over a period of time At, AQ. (t) is a partial recovery of a resource Q over a period of time At with the help of internal investments. The quantities AQf (t) can be represented as
aqa (t ) = —e(t )• amt (t) • At,
or
aQA (t) = — a -0(t)• QU (t)• At, (9)
n
where am\ (t) = a ■ qui (t) are the depreciation corresponding to the factor of production Q. at the moment of time t, A are the depreciation coefficients expressing the shares of the lost volumes of resources Q. per unit of time, Ui are the indicators of the intensity of the amortization process.
The value u; = 1 corresponds to the proportional depreciation, the values u; > 1 correspond to the progressive depreciation, the values u < 1 correspond to the regressive depreciation.
The function d = d(t ) in relations (9) determines the options for the development of the enterprise under consideration. For a constant and unitary function Q(t ) = 1 , the development of the enterprise will be stable. Different sizes of the deviation of the value of the function 0( t ) from one in the direction of decreasing will
correspond to a slowdown in the development of an enterprise, its temporary halt during a change in production technologies, and a partial curtailment of production [26].
The values of partial recovery of resources due to internal investment AQ/ ( t ) over time At are expressed by the ratios
AQ(t) = 0(t)• I, (t)• At,
or
AQ (t) = 0(t)• B, • V(t) • At, (10)
where / (t) = B • V(t) is the investment corresponding to the factor of production Q at the moment of
time t, B is the rate of accumulation of these investments.
Using formulas (9) and (10), balance equations (8) take the form
AQ =£•(-A • QU + Bt • V)• At,
and the passage to the limit at At ^ 0 leads to the system of nonlinear differential equations
dql = q\-ai • qu + B • v). (ii)
dt \ ' '
Substitution of the production function (1) into the system of equations (11) gives
Q в
v
- A-QU + BrP ■ Поа
(12)
dt
The initial conditions for the system of equations (7) are the relations
Q,|t=o = Q( 0)=Q0. (13)
The structure of the system of equations (12) shows that the growth of resources and output will continue as
long as the derivatives dQL are positive. If the values dQL turn to zero, then the development of the enterprise dt dt will stop. This will happen when the volume of investments becomes equal to the volume of depreciation charges.
Thus, the limiting state of the development of production Q ( t ) = Q1 will correspond to the conditions
n as
W = -AM, + It = -A, • Q )U + Bt • P • n(ô; ) = 0. (14)
s=1
Obviously, the ideal option for any enterprise is the one in which the enterprise reaches the mode of obtaining maximum profit. This occurs with the values of production factors Q ( t) ^ Q™* . If the limiting values of resources Ql exceed the values of resources Q™ax : (g" > Q™ax ) corresponding to the maximum
profit P^ax , then the solutions of the system of equations (12) at Q (t ) ^ Q" will significantly reduce
the values of the enterprise's profit, worsening its economic condition.
In this case, instead of the system of equations (12), it is advisable to use the system of equations
dt
= в-
-A - QU+ B • pПQ
1 -
Q
Q
(max
i J
(15)
The solutions of the system of equations (15), in contrast to the solutions of the system of equations (12), will always have limiting values for the resources of the enterprise Q. (t) ^ Q™35 .
The forms of the integral curves of the systems of equations (12) and (15) significantly depend on the type of function 0(t) that determines the center of the time interval, its length and the amount of deviation from a single value at which the enterprise operates stably.
If in the interval of time (t* — 7, t* + 7) the enterprise makes a complete or partial replacement of technological equipment, then the function 0(t) can be written in the form [28]
в( t ) = 1 -ю- exp
ti)
2
2-a
2
(16)
where O is the maximum size of the deviation of the function 0(t) from unity, t is the center of the time interval, and 7 is the radius of the time interval.
If (O = 0, then the enterprise will work stably, if 0 <O< 1, then in the vicinity of the point t = t the growth of functions Q(t) slows down, if O = 1, then at the moment of time t = t the growth of functions
Qs(t) stops, and in the time interval (t* — o,t* + there is a re-equipment of production, if o> 1, then in the
(* * \
t — 7, t + 7 I there is a re-equipment of production, accompanied by some curtailment.
Mathematical model of the development of a one-factor manufacturing enterprise
Let us now apply the results obtained to describe a manufacturing enterprise, the output of which is provided by only one resource Q = Q(t |.
A continuous and continuously differentiable function Q = Q(t|is bounded on the number semiaxis
(0 < t < , Qo < Q(t) < Qx, by its hmiting vdues Qo = Q(0), Qx = limQ(t).
Cobb - Douglas function (1) takes the form
V = P ■ Qa . (17)
The proportional costs of the enterprise are described by the formula
TC = HQ ■ Q + TFC . (18)
The expression for the profit of the enterprise is written in the form
PR = P ■ Qa - HQ ■ Q - TFC.
The maximum profit of the enterprise is found from the condition
P ■ a ■ Qa-1 - HQ = 0,
dPR „ oa-1
dQ
(19)
(20)
The value of the resource Qmax, corresponding to the maximum profit has the form
i
Qm
/ л P
V Q J
1-a
(21)
Hr
where Œq =
a
The values of resources Q and Q, at which the profit vanishes, are found from the equation
PR =P Qa -Hq-Q-TFC = 0,
in this case, it is obvious that the inequality takes place Q < Qmax < Q.
(22)
The maximum profit value is calculated by the formula
PR = PQa - H Q - TFC
PRmax P Qmax hq Qmax lrC
or
= P
P
VaQ У
1 - a
- Hr
P
VaQ У
1 - a
- TFC.
The system of equations (12) is reduced to one equation
dQ
dt
= в-(-AQ-Q" + BQ ■ P о ),
and the system of equations (15) is transformed to a single equation of the form
dQ dt
в\-AQQ + Bq-P ■ Qa )■
1 -
Q
Qm
V .simax У
(23)
(24)
(25)
Initial conditions (13) for equations (24) and (25) take the form
Q ( 0 ) = Q|t=o = Qo. (26)
The system of equations (14) for determining the limiting state of the enterprise is reduced to one equa-
tion
whose solution has the form
wq =- AMq + Iq =-aq + Bq-PQ" = О
(27)
Q. =
Bq-P
V Ло У
(28)
Figure 1 shows graphs of the functions of investment , and their differences Wn .
i0
-10
WQ
Q.
^-AM q
50 Q
Figure 1 - Graphs of functions of investment / , depreciation AMQ and their differences Wg . The dot
marks, calculated by the formula (28), the value of the limiting value of the volume of the production factor Q = 43,5857
1
a
0
Figure 2 shows the graphs of the profit function PR and the function of the difference between investment and depreciation Wq .
10
0
PRmax / 1 ^ / 1 / 1 / I
1 1 / 1 Л """Чч^ 1 1 -i- J\
70
Ql
Qm
Q
Qoc
Qr
Figure 2 - Graphs of the profit function PR and the function of the difference between investments and depreciation Wq . The points on the abscissa axis indicate, calculated by formulas (21), (22) and (28), the
values of the volumes of production factors Q = 1,1776 ; Qmax = 19,9596 ; Q = 43,5857 ; Q = 67,2341
Figure 3 shows the graphs of the functions of output volumes V ( t ) constructed from the results of numerical solutions to the Cauchy problem (24), (26) and the Cauchy problem (25), (26).
V
40
0
/ — y--^ ____— —
V.
t
0 40
Figure 3 - Graphs of functions of output volumes V ( t ) constructed from the results of numerical solutions of the Cauchy problem (24), (26) and the Cauchy problem (25), (26). The dashed line corresponds to the solution of the Cauchy problem (24), (26). The solid line corresponds to the solution of the Cauchy problem (25), (26). The value of the volume of production VM = 37,4775 corresponds to the limiting value
of the volume of the production factor Q = 43,5857. The value of the volume of production Vmax = 28,5136 corresponds to the value of the volume of the production factor Q^ = 19,9596
Figure 4 shows the graphs of the functions of the enterprise's profit volumes PR (t) constructed using
formula (19) and the results of numerical solutions to the Cauchy problem (24), (26) and the Cauchy problem (25), (26).
PR
10
0
• ^^^^
7
PR.....
PR,
t
0 40
Figure 4 - Graphs of functions of the enterprise's profit volumes constructed by formula (19) and the results of numerical solutions of the Cauchy problem (24), (26) and the Cauchy problem (25), (26). The dashed line corresponds to the solution of the Cauchy problem (24), (26). The solid line corresponds to the solution of the Cauchy problem (25), (26). The value of the volume of the enterprise's profit PRx = 5,6846 corresponds to the limiting value of the volume of the production factor Q = 43,5857 . The value of the volume of the maximum profit of the enterprise PR^ = 8,5339 corresponds to the value of the volume of the production factor Q^ = 19,9596
Figure 5 shows the graphs of the functions of the output volumes V (t) constructed based on the results of
numerical solutions to the Cauchy problem (25), (26) for cases of stable operation of the enterprise, temporary suspension of the operation of the enterprise and partial curtailment of the enterprise.
V
30
0
о = 0— Уо =1/^ ¿Г^О = 1.5/
t
0 40
Figure 5 - Graphs of functions of output volumes V (t ) constructed based on the results of numerical solutions of the Cauchy problem (25), (26), for cases of stable operation, temporary suspension of work and partial termination of work. The corresponding parameter values C are marked on each curve
When constructing graphs of functions in Figures 1-5, the calculated values were used: P = 10; a = 0,35; u = 0,96; HG = 0,50; A = 0,20; BQ = 0,20; TFC = 10.
Analysis of the graphs of the functions shown in Figures 2-4 shows that an increase in the production factors of the enterprise after the value Qmax leads to a decrease in profits and makes the enterprise less efficient. Thus, the system of equations (15) and, in particular, equation (25) more adequately describes the process of enterprise development through investment.
Mathematical model of the development of a two-factor manufacturing enterprise
Let us now consider a mathematical model of a manufacturing enterprise, the output of which is provided
by two resources, fixed capital Q = K = K (t) and labor resources Q = L = L (t).
Continuous and continuously differentiable functions K = K (t) and L = L (t) are bounded on the number semiax is (0 < t <<»), K0 < K(t)< Kx, L < L(t)< L by their limiting values K0 = K(0), KB = limK(t) and L0 = L(0), LM = limL(t).
Let us introduce the notation a = a ; a2 = b; u1 = u ; u2 = v; H = HK; H = H; A = AK;
A = al ; Bi = Bk ; B2 = bl .
Cobb-Douglas function (1) takes the form
V = P ■ Ka ■ Lb .
The proportional costs of the enterprise (2) are described by the formula
TC = AK ■ K + A ■ L + TFC .
The expression for the profit of the enterprise (3) is written in the form
PR = P ■ Ka ■ Lb - HK ■ K - HL ■ L - TFC.
The maximum profit of the enterprise is found from the conditions
dPR
= P ■ a ■ Ka-1 • L
The values of the resources K„
dK dPR ~dL
and L„.
= P■b ■ Ka ■ L
b-1
Hk = 0,
HL = 0.
(29)
(30)
(39)
(40)
, the corresponding maximum profit, have the form
=
P
a
K
a
L.
P
a
\aL J
a
V aK J
1
1-a-b
1
1-a-b
(41)
where aK = —K
, aL =■
Hr
a b
The maximum profit value is calculated by the formula
PRmax = P ■ KL ■ Llx - HK ■ Kmx - HL ■ L^ - TFC.
(42)
The values of production factors Kmax , Lmax ,
at which profit vanishes are found from the condition
b A ^ A T rrrr^ n (43)
PR = P ■ Ka ■ Lb - AK ■ K - A ■ L - TFC = 0.
<
Equation (43) in the general case can be solved only numerically; it describes some closed curve on the coordinate plane PR = 0.
This time, the system of equations (12) is reduced to two equations
dK -6\-AK ■ Ku + BK ■ P■ Ka ■ Lb) ,
(44)
ät
— = &■(—A ■ L + B ■ P■ Ka ■ Lb),
dt
and the system of equations (15) is transformed to a single equation of the form
dK = £■(-A ■ Ku + BK ■ P ■ Ka ■ Lb )■[ 1
äL = &■(—Al ■ L + BL ■ P■ Ka ■ L)^|l
K >
Kmax j
L ]
Lmax j
(45)
Initial conditions (13) for equations (44) and (45) take the form
Î4= K ( 0 ) =
4=0 = L ( 0 ) = L>- .
(46)
The system of equations (14) for determining the limiting values of the resources of the enterprise is reduced to two equations
W =—Шк + h =—A ■ Ku + BK ■ P ■ Ka ■ L = 0,
ab ab
WL = — AML +1L =— A ■ L + B ■ P ■ Ka ■ L = 0,
whose solution has the form
K„ =
P ■
V AL J
/п \
K
B
—b
L.
f -n, \
P ■
B
A
V Ak J
r ,, \
l
u^—av—bu
B
K
A,
l
uv—av—bu
v V ¿y V К J у
Figure 6 shows a graph of the surface of the profit function (39).
(47)
(48)
200
(Kmax , Lmax , PRmax )
' ' ''m у
(Kœ , Lœ , PRX)
Г
0^-i-' L 1800
0
K\
1800
Figure 6 - Graph of the surface of the profit function (39). The closed line of intersection of this surface with the coordinate plane PR = 0 corresponds to the solution of equation (43). The point with coordinates on the surface of the profit function (39) marks the value of the maximum profit of the enterprise.
<
<
The point with coordinates (Km, Lm,PRœ) on the surface of the profit function (39) marks the limiting value of the enterprise's profit
Figure 7 shows the graphs of the functions of output volumes V ( t ), constructed from the results of numerical solutions to the Cauchy problem (44), (46) and the Cauchy problem (45), (46).
V
1000
0
У / /
у
F
t
0 120 Figure 7 - Graphs of functions of output volumes V ( t ), constructed from the results of numerical solutions of
the Cauchy problem (44), (46) and the Cauchy problem (45), (46). The dashed line corresponds to the solution of the Cauchy problem (44), (46). The solid line corresponds to the solution of the Cauchy problem (45), (46). The value of the volume of production corresponds to the limiting values of the volume of production factors. The value of the volume of products VM = 900,8822 corresponds to the limiting values of the volume of production factors Km= H96,132l; Lm= 841,0418. The value of the volume of products Vmax = 520,0980 corresponds to the values of the volumes of production factors Kmax = 455,0858; Z^ = 416,0784
Figure 8 shows the graphs of the functions of the volumes of profit of the enterprise PR (t) constructed by
formula (39) and the results of numerical solutions to the Cauchy problem (44), (46) and the Cauchy problem (45), (46).
PR
200
0
лк— ï N ч
1
I4i
PR,
t
0 120 Figure 8 - Graphs of the functions of the volumes of profit of the enterprise PR (t ) constructed according
to the formula (39) and the results of numerical solutions of the Cauchy problem (24), (46) and the Cauchy problem (45), (46). The dashed line corresponds to the solution of the Cauchy problem (44), (46). The solid line corresponds to the solution of the Cauchy problem (45), (46). The value of the volume of the enterprise's profit PRx = 97,0386 corresponds to the limiting values of the volumes of production
factors KM =1196,1321; L = 841,0418. The value of the volume of the maximum profit of the enterprise PRmax = 172,0343 corresponds to the values of the volumes of production factors Kmax = 455,0858; Lmax = 416,0784
Figure 9 shows the graphs of the functions of output volumes V ( t ) constructed based on the results of
numerical solutions to the Cauchy problem (45), (46) for cases of stable operation of the enterprise, temporary suspension of the operation of the enterprise and partial curtailment of the operation of the enterprise.
V
600
o[__t
0 120
Figure 9 - Graphs of the functions of the output volumes V (t) constructed based on the results of numerical solutions to the Cauchy problem (45), (46), for cases of stable operation, temporary suspension of
work and partial termination of work. The corresponding parameter values O are marked on each curve
When constructing the graphs of the functions in Figures 5 - 7, the calculated values were used: P = 10; a = 0,35; u = 0,96; v = 0,95; AK = 0,20; AL = 0,15; ^ = 0,20; BL = 0,10; H^ = 0,40; HL = 0,375;
TFC = 10.
Analysis of the graphs of the functions shown in Figures 5-7 shows that an increase in the production factors of the enterprise after the values Kmax =455,0858; Lmax =416,0784 l eads to a decrease in profits and makes the enterprise less efficient. Thus, the system of equations (15) and, in particular, the system of equations (45) more adequately describes the process of enterprise development through investment.
Conclusion
1. Two versions of new systems of differential equations for the balance of dynamic development of enterprises, the production of which are recovering their capacities due to the introduction of their own investments, are proposed.
2. For both options, the conditions are formulated for enterprises to reach an equilibrium state.
3. In the first variant, the growth of resources and output is limited by the calculated marginal values of the factors of production at which the sizes of depreciation volumes and the sizes of investments become the same.
4. In the second option, the growth of resources and output is limited by the calculated values of the factors of production, at which the profit of the enterprise will be maximum.
5. It is shown that in order to ensure the positive dynamics of the enterprise's work, the growth of its production factors should be limited to values corresponding to the value of the maximum profit.
6. In the presented models, proportional, progressive and digressive depreciation deductions are considered.
7. The constructed models make it possible to describe various modes of operation of enterprises. Such regimes include a stable output of products by enterprises, a temporary suspension of the work of enterprises during its technical re-equipment, and a temporary partial curtailment of production.
ю = 0 ю = ХуГ ю = 1.5
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Библиографический список
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