Application of number theory methods for task solution of intertemporal balance in economy
Economic model is presented in which two agents are taking place, which are producer-company and owner-user. They are aggregated presenting productive and non-productive economic fields. Mission is to learn a principal of exploitations availability of intertemporal balance for explanations of change...
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Інститут економіки промисловості НАН України
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irk-123456789-240132022-09-29T21:21:17Z Application of number theory methods for task solution of intertemporal balance in economy Zhykhareva, Y.I. Economic Theory Economic model is presented in which two agents are taking place, which are producer-company and owner-user. They are aggregated presenting productive and non-productive economic fields. Mission is to learn a principal of exploitations availability of intertemporal balance for explanations of change processes in economics. The problem was fully solved at any initial conditions. Key words: agent, property asset, intertemporal balance, balance income, profitableness. У статті розглядається модель економіки, в якій діють два агенти: фірма-виготовник та власник-споживач, які агрегійовано представляють виробничу та невиробничу сфери економіки. Мета — вивчення питання про принципову можливість використання моделі міжчасової рівноваги для описування перехідних процесів в економіці. Отримано повне розв’язання задачі при будь-яких початкових умовах. Ключові слова: агент, власний капітал, міжчасова рівновага, балансовий прибуток, доходність. Рассматривается модель экономики, в которой действуют два агента: фирма-производитель и собственник-потребитель, которые агрегировано представляют производственную и непроизводственную сферы экономики. Цель — изучение вопроса о принципиальной возможности использования модели межвременного равновесия для описания переходных процессов в экономике. Получено полное решение задачи при любых начальных условиях. Ключевые слова: агент, собственный капитал, межвременное равновесие, балансовая прибыль, доходность. 2010 Article Application of number theory methods for task solution of intertemporal balance in economy / Zhykhareva Y.I. // Економічний вісник Донбасу. — 2010. — № 4(22). — С. 49-54. — Бібліогр.: 9 назв. — англ. 1817-3772 http://dspace.nbuv.gov.ua/handle/123456789/24013 338.1:51-7 en Економічний вісник Донбасу Інститут економіки промисловості НАН України |
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Economic Theory Economic Theory Zhykhareva, Y.I. Application of number theory methods for task solution of intertemporal balance in economy Економічний вісник Донбасу |
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Economic model is presented in which two agents are taking place, which are producer-company and owner-user. They are aggregated presenting productive and non-productive economic fields. Mission is to learn a principal of exploitations availability of intertemporal balance for explanations of change processes in economics. The problem was fully solved at any initial conditions. Key words: agent, property asset, intertemporal balance, balance income, profitableness. |
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Zhykhareva, Y.I. |
| author_facet |
Zhykhareva, Y.I. |
| author_sort |
Zhykhareva, Y.I. |
| title |
Application of number theory methods for task solution of intertemporal balance in economy |
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Application of number theory methods for task solution of intertemporal balance in economy |
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Application of number theory methods for task solution of intertemporal balance in economy |
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Application of number theory methods for task solution of intertemporal balance in economy |
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Application of number theory methods for task solution of intertemporal balance in economy |
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application of number theory methods for task solution of intertemporal balance in economy |
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Інститут економіки промисловості НАН України |
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2010 |
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Economic Theory |
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http://dspace.nbuv.gov.ua/handle/123456789/24013 |
| citation_txt |
Application of number theory methods for task solution of intertemporal balance in economy / Zhykhareva Y.I. // Економічний вісник Донбасу. — 2010. — № 4(22). — С. 49-54. — Бібліогр.: 9 назв. — англ. |
| series |
Економічний вісник Донбасу |
| work_keys_str_mv |
AT zhykharevayi applicationofnumbertheorymethodsfortasksolutionofintertemporalbalanceineconomy |
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2025-07-03T03:28:59Z |
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1836594852133863424 |
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49
Економічний вісник Донбасу № 4 (22), 2010
In this article an intertemporal balance is being
researched in one-commodity model of economy, which
consists of two aggregated agents: consumer-owner and
manufacturer. The peculiarity of this model is that the
balance is researched not only at goods market, but also
at stock market. To research the model till the very end
we don’t consider market resources, other words we
ignore the necessity to use labour and natural resources
in manufacturing.
Also, building models, which allow to reveal causal
relations between variables, is a very important problem
of economics modeling. Such models are assigned for
definition of quantitative effect of independent variables
on dependent one [1]. The solution of this task is rather
difficult. Discrepancy between the results of modeling
to correlations, which are taking place in reality, appears
due to the number of reasons, as a matter of fact
because of breaking the main prerequisites of regression
analysis. Most often they are broken because of
multicollinearity [2]. This phenomenon brings to
estimation of parameters with great dispersion, which,
in most of cases, doesn’t allow to interpret them
intensionally because of wrong signs, for instance. The
usage of prior information is a powerful means of
building regression models mentioned above. As a matter
of fact it decreases the influence of multicollinearity on
the estimation of parameters. But formalization of the
prior information is not very easy because of its
ambiguity. One of the possible approaches to the solution
of this task is being researched in this article.
1. Necessary and sufficient conditions of regular
optimality.
In conditions of regular balance, in Lagrange
functional it is possible to make the integration by parts,
as a result, the functional will take the form:
£ ),([,,, tQΦξϕψ && X )(t , ),(tP Y )](t ++= )0()0()0()0( QX ξψ
( )
( )∫ ∫
∫∫
+−+
+
∂
∂
++
+
+
∂
∂
+++ −
T T
TT
t
dttYtsttdttXtt
t
trt
dttQtt
t
dttPtptetPf
0 0
00
)()()()()()(~)(~)()(
)()(~)()()()())((
ξψϕψξ
ρξξχ
(1)
( ) ( ) )()(~)()( TXTaTTQ ψξ −Φ+−Φ+
It is clear that this expression reaches the
maximum in accord with piecewise differentiable
functions X )(t , Q(t) with specified initial conditions and
in accord with piecewise continuous functions P(t) ≥ 0,
Y(t) only that time when almost everywhere at [0,Т]
derivatives turn to zero by Q(t),X(t),P(t),Y(t) of
integrand in (2.1), and also derivatives (2.1) by
Q(T),X(T)
),()())((' tptetPf t ξχ =− (2)
0)(~)( =+
∂
∂ tt
t
ρξ , (3)
,0)(~)(~)()( =+
∂
∂
+ tt
t
trt ϕψξ (4)
)()()( tstt ξψ = , (5)
)(~ Tξ=Φ (6)
)(~ Ta ψ=Φ (7)
[3] contains the rigorous proof of this fact.
Thereby, to make the regular solution of agent’s
task [4] by the collection of straight lines
),(~[ tQ X )(t , ),(~ tP Y )(t ] and dual ones
[ Φ~),(~),(),(~),( tttt ρξϕψ ], it is necessary that at [0,Т]
equalities (2) — (7) and slackness conditions must be
accomplished [4].
Derived conditions can be considerably simplified.
Notice, firstly, because of the non-negativeness of
dual variables р(t),Ф from (3), (6) appears from the above
that the continuous function )(tξ is non-negative when
all ],0[ Tt ∈ .
Let us consider condition (2). As far as function
P(t) — is a piecewise continuous one, then it is limited
per [0,Т], and then, under the function’s properties of
usefulness (.)f , inequality is accomplished P{t)>0, and
the left part of the equation (2) is positive and is separated
from 0 per [0,Т]. It means, that )(tξ is also separated
from 0, and, under (6),
Ф>0. (8)
From (8), because of the last condition in [4], it
follows that terminal condition in regular balance is being
done like equation:
УДК 338.1:51-7
Y. I. Zhykhareva,
National Pedagogical Dragomanov University,
Kyiv
APPLICATION OF NUMBER THEORY METHODS FOR TASK SOLUTION
OF INTERTEMPORAL BALANCE IN ECONOMY
Y. I. Zhykhareva
50
Економічний вісник Донбасу № 4 (22), 2010
.0)()( =+ TaXTQ (9)
As far as functions 0)( >tξ and )(tψ — are
absolutely continuous, from (5) it follows that
informational variables s(t) at regular balance must
be almost everywhere equal to absolutely continuous
functions. It is obvious, that overdetermination s(t)
at the set of measure 0 doesn’t break the conditions
of balance, it can be considered that the functions
s(t) at regular balance are absolutely continuous. It
means that the path of stock price s(t) and price p(t),
which compose vector-functions s(t), must be also
absolutely continuous. But then from (2) it follows
that the price p(t) is positive and is separated from 0
per [0,Т].
When s(t) are absolutely continuous the equation
(5) is done for all ],0[ Tt ∈ . When t = Т it gives the
relation )()()( TsTT ξψ = , from which because of (6),
(7), (8) it follows that the agent’s task has a regular
solution, if only the coefficients of the terminal condition
are coordinated with the prices:
a=s(T).
This result saves us from the necessity to discuss
the question about the way agents evaluate their funds at
the end of the process. In the bounds of the researched
model they must evaluate them just in accord with existing
in the final courses.
Notice, if we didn’t set the terminal limit, then we
would have conditions of optimality 0)( =Tξ and
0)( =Tψ , to provide which, it is easy to check, we
would have to allow unlimited threads into the momentт
Т: either real P(t), or nominal p(t)P(t).
After we have received these conclusions, it is
possible to exclude the dual variable )(Tψ , and also
the auxiliary straight variable Y(t) from the system of
conditions of optimality. It is also possible to exclude
the dual variables )(tξ and Ф. In order to do this instead
of )(~),(~ tt ρϕ we will introduce normalized quantity
(см. (3)):
)(
)(~
)(
t
tt
ξ
ϕϕ = ,
).(
)(
1
)(
)(~
)( t
ttt
tt ξ
ξξ
ρρ
∂
∂
−==
The equation (4) in these variables after exclusion
of )(Tψ will look like 0)()()()()( =+
∂
∂
+− tts
t
tsttr ϕρ ,
and as long as 0)( >tξ , in inequations in a set of
conditions of complementary slackness [4] one can
substitute )(~),(~ tt ρϕ for )(),( tt ρϕ . Finally, due to the
fact that the first part (2) is absolutely continuous, the
function P(t) can also be considered absolutely
continuous. Taking logarithmic derivative from both parts
of the equation (2), we will have:
),()()(
)(
tttP
ttP
ιρχν
+−=−
∂
∂
− (10)
Where
0)(
))(('
))(("
>== consttP
tPf
tPfν — is the
agent’s odium on risk ( B=ν for an enterprise and
βν = for an owner), and
)(
)(
1)( tP
ttp
t
∂
∂
=ι — the rate
of inflation. Constant of integration in equation (10)
ultimately will be defined from the terminal condition (9),
so there is no need to identify )(tξ and Ф for the agent’ss
task solution.
Thus, we arrive at the following statement.
Statement 2.
1) If a regular balance exists, the path of price p(t)
and the stock price s(t) are absolutely continuous,
and 0)( >tp where ],0[ Tt ∈ .
2) The agents’ conduct in a regular balance is
described with the following conditions:
)()()()()()()( tPtptX
t
tstXtrtQ
t
−
∂
∂
−=
∂
∂ ; (11)
)())()(()( tPtttP
t ν
χιρ −−
=
∂
∂
; (12)
)()()()()( ts
t
trtstt
∂
∂
−−= ρϕ ; (13)
,0)()( =tXtϕ 0)( ≥tϕ , 0)( ≥tX ; (14)
,0)()( =tQtρ 0)( ≥tρ , 0)( ≥tQ ; (15)
0)()()( =+ TXTsTQ ; (16)
where the initial values 0)0(,0)0( =≥ QX are given.
2. The integral of the owned capital.
The equation of financial balance (11) can be
rewritten the following way:
( ) )()()()()()()()( tPtptXts
t
trtXtstQ
t
−
∂
∂
+=+
∂
∂
(17)
On the agent’s optimal path, due to (13), (14), (15):
=+=
∂
∂
+ )()()()()()()()( tXttXtsttXts
t
tr ϕρ
( ).)()()()()()()()()()()( tXtstQttXttQttXtst +=++= ρϕρρ
Thus, for the value
)()()()( tXtstQt +=Ω (18)
From (17), (16) we have
)()()()()( tPtpttt
t
+Ω=Ω
∂
∂ ρ , 0)( =Ω T (19)
In terms of the theory of optimal management,
Y. I. Zhykhareva
51
Економічний вісник Донбасу № 4 (22), 2010
the value )(TΩ is a counterpart of the integral of motion
of Hamiltonian system of optimum condition.
According to Nether theory, the most interesting
integrals of motion are connected with the task’s
symmetry. As it is shown in [3], integral )(TΩ has the
same origin — it emerges as a result of the fact the
agent’s task is homogeneous in relation to the funds
).(),( tXtQ
From the economic point of view, as we will see
below, and as in a more common case it is shown in [3],
the valueу )(TΩ can be interpreted as the agent’s owned
capital. The correlation (18) is in fact a report balance in
remains — it presents the owned capital through
pecuniary valuation of funds (assets and liabilities).
The correlations (19) show that on the optimum
path the owned capital must coincide with the given
sum of the planned “useful expenses” p(t)P{t) (dividends
for the enterprise and consumer expenses for the
owner):
∫
∫
=Ω
−T
t
dvv
dueuPupt
u
t .)()()(
)(ρ
(20)
Other words, as it will be shown below, the term
)()( tt Ωρ corresponds to the agent’s balance profit — the
difference of the income and expenses taking into account
the profit on reevaluation of the funds. Then the dual variable
)(tρ expresses the agent’s capital profitability, and the
right part of the equation in (19) shows that the capital is
derived from the undistributed profit.
Finally, notice, that the formula (20) is positive
when Tt ≠ . It is known, that the negativity of the
owned capital is a characteristic feature of the “financial
pyramid”. Thus, the terminal limits applied above, which
result in the formula (20), in fact, play the role of
condition of pyramid’s absence (no ponzi game
condition), which is usually applied in tasks of financial
planning. Besides, the next statement emerges from
(20), (18).
Statement 3.
Regular balance exists only if the initial conditions
for both agents meet the inequations:
).0()0()0()0( XsQ +=Ω
However, one must bear in mind that in these
inequations 0)0( ≥X are given, and S(0) are defined
from the conditions of balance, where some
components of these vectors are knowingly negative
(which follows from the definition of the agent’s task
regular solution).
3. Regimes realized in a regular balance.
Hereon we conclude the general overview and move
on to the construction of a regular balance [4]. Conditions
of complementary slackness (14), (15) describe
alternative regimes of optimum agents’ functioning.
Provided to be in a regular balance at all moments of
time a definite set of regimes can be realized.
First of all, it is worth mentioning that if the money
funds of agents are zero in the beginning, they will
remain the same in a balance [4]. That is why, in a
regular balance the condition (15) comes to the
condition:
0)( ≥tρ , 0)( =tQ (21)
As it was shown above, in a balance p(t) > О, C(t)>О
так, что потребительские расходы собственника по-
ложительны. such a way, that the consumer expenses
of the owner are positive. When the fund of money
is absent, [4], the positive expenses require positive
share stock (see [4]), and the positive consumption
requires the positive issue stock in such a way, that
in a balance
0)(,0)()( >>= tYtAtS where ),0[ Tt ∈ .
Due to [4] these inequations mean that in the balance
conditions of complementary slackness (14) for both
agents are realized a
0)( =tϕ , 0)( >tX . (22)
4. Solution of the enterprise task.
According to statement 2, the solution of enterprise
task is described with system (11) — (16), for sets:
)(),()(,)(),()(),()( tY
t
tA
t
tYtYtAtXtWtQ
∂
∂
∂
∂
=== ;
)()(,)(),()( ttttt WYA ρρϕϕϕ == where )(,0)( tptr = ,
)(),()( tbptsts −= .
Taking into account (21), (22), this system
transforms into:
;0)()()()()()()( =−
∂
∂
−
∂
∂
+ tZtY
t
tbptA
t
tstYtp (23)
( )
;
)(
)()()(
)(
)(
∆−−
=
∂
∂
tp
tZ
B
tt
tp
tZ
t
W ιρ
(24)
0)()()( =
∂
∂
+− ts
t
tstWρ ; (25)
( ) 0)()()( =
∂
∂
−− tp
t
btptbptWρ ; (26)
0)()()()( =+− TYTbpTATs . (27)
Excluding (25), (26) )(tWρ , we get the connection
between informational variables: stock rate s(t) and the
price p(t):
Y. I. Zhykhareva
52
Економічний вісник Донбасу № 4 (22), 2010
).(
)(
1)(,)0()( 0
)(
tp
ttp
teests
t
duu
b
t
∂
∂
=
∫
=
ι
ι
(28)
Thus, the system (24) — (26) is degenerated: when
the correlation (28) is done, it has multiple solutions, and
when it is broken, it has none. The ambiguity of optimum
enterprise conduct can be interpreted as continuous
flexibility of stock offer function A(t). When the
connection (28) is broken between s(t) and p(t) the
enterprise either doesn’t issue safety stock or tries to
place them as much as it is possible. When the connection
(28) is stable the volume of issue is indifferent for the
enterprise.
Due to the dependence of (24) — (26) system,
the equation (23) is better to change with the following
from (24) — (26) equation (19) for the enterprise’s
capital:
),()()()()( tAtstbYtpt −=Ω (29)
and the terminal condition (27) — with condition
0)( =Ω T . Differential equations (19), (24) can be solved
easily and with (26), give the terms for Z(t) and )(tΩ :
( )
( ) ;
1
1)0()(
1(
)()1(
0
beB
eebBTZ
Bb
TbB
duu
Bb
tb
t
−
∫
∆−−Ω
=
∆−−
∆− ι
(30)
( )
( ) .
1
11)0()( 1(
1(
)(
0
−
−
−
∫
Ω=Ω ∆−−
∆−−
Bb
TbB
Bb
tbB
duu
b
t
e
eeet
t
ι
(31)
From the terms for the capital (31), (29), with (28),
we have the connection between the stock offer A(t)
and the product offer Y{t):
( )
( ) ).()0()()0(
1
11)0( 1(
1(
tYbptAes
e
ee b
t
Bb
TbB
Bb
tbB
b
t
+−=
−
−
−Ω ∆−−
∆−−
(32)
Individually values )(tA and )(tY cannot be
defined from the task solution, but the requirement of
the task solution of the enterprise fixes the connection
(28) between the price of the product and the stock
price.
From the conditions of optimal behavior of the
enterprise one more condition for the price condition
appears. As far as 0)( ≥tWρ , from (26) it is an inequation:
,1)(
)(
1)(
b
tp
ttp
t −>
∂
∂
=ι (33)
which means that there can’t be a very strong
deflation on the equilibrium path.
5. The owner’s task solution.
According to the statement 2, the owner’s task
solution is described by the system (11) — (16):
;)()(),()(,)()(),()( tS
t
tYtCtPtStXtMtQ
∂
∂
====
)()(,)(~)( tttt MS ρρϕϕ ==
where )()( trtr = , )()( tsts = .
With (21), (22), this system starts to look like:
;0)()()()()()( =−
∂
∂
− tCtptS
t
tstStr (34)
( ) ;)()()()( tCtttC
t
M
β
ιρ ∆−−
=
∂
∂
(35)
( ) 0)()()( =
∂
∂
−− ts
t
trtstMρ (36)
.0)()( =TSTs (37)
This system solution asks the question on the
consumer market C(t) and on the equity market S(t). Let
us remind that due to (21) the return on capital of the
owner )(tMρ is nonnegative in the balance. From (36) it
follows that even in case when the entitlement payment
is zero r(t) = 0 the capital grows due to the growth of the
stock course price s{t).
6. The description of the balance.
The price path p(t), the stock course s(t) and
the dividend rate r(t) must be formally defined from
the balances [4]. Substituting the owner’s demand
into the stock S(t), with (28), and the stream of
dividends Z(t) (30) in the partition condition and
supposing
,01
)0()0(
)0()0(
)0()0(
)0(,)( 0
)(
)(
>−=
Ω
=
∫
=
As
bYp
As
etG
t
du
us
ur
ω (38)
we will get for G(t) a nonlinear integral differential
equation:
−
∆−−=
∂
∂
∫∫
−−−
∆−−
∆−−
−−− T u
b
b
T
Bb
bB
t
Bb
bB
T
t
u
b
b
dueuG
beB
ebBdueuG
t
tG
0
11
1
1
11
)(
1
)1()()( β
δβ
β
β
β
δβ
β
β ω (39)
and the original condition 1)0( =G . (40)
Statement 4. On the initial conditions that
)0()0(,0)0(,0)0()0( WMYSA =>>= ,
the regular balance could exist, it is necessary and
enough, that at some 0>ω the equation (39) had a
positive and absolutely continuous solution G(t), which
satisfies (40) conditions.
Y. I. Zhykhareva
53
Економічний вісник Донбасу № 4 (22), 2010
The proof. The necessity has been already proved.
Let’s try to prove the sufficiency, other words let’s build
the regular balance, starting from the positive and
absolutely continuous function G(t), which satisfies (39),
(40) when some 0>ω . First of all, when G(t) > 0 from
(39), (40) it is the positivity of the second factor in the
left part of (39), and from 0>ω the positivity of the
right part of (39) is obvious. Such way,
0)( >
∂
∂ tG
t ,
and from the definition G{t) (38) we can find the real
rate of paying dividends:
0)(
)(
1
)(
)(
>
∂
∂
= tG
ttGts
tr
(41)
Let’s initialize now the absolutely continuous price
path randomly p(t) > 0 in such a way to satisfy the
condition (33). Then, from (28) it is possible to identify
the nominal stock rate s(t) accurate within the initial value.
The value s(0) can be defined due to and the initial
conditions from (38). Meanwhile the nominal value of
the capital of the enterprise will be positive.
.0)0()0(
1
)0()0()0()0()0(
,0
)0()1(
)0()()(
>
+
=−=Ω
>
+
=
bYpAsbYp
A
bYetpts
b
t
ω
ω
ω (42)
Thus, the informational variables are identified in
the balance. They define the positive profitability (see
(25), (36)):
.01)(
)(
)(
)(,01)()( >+∂
∂
=>+=
b
t
tG
tG
tt
b
tt WM ιριρ
Submitting the term for pw(t) in (35), we will have
the term for consumption accurate within the factor:
,))()(0()(
11
t
b
b
etGCtC
δ
β
−
= (43)
and from (34), (37) with (41), (43), (42) — the
term for the volume of outstanding stock:
∫ >== −
−T
t
udueuGtG
s
pCtStA .0))(()(
)0(
)0()0()()(
1
δβ
β
(44)
The initial value р(0) we have assigned above, and
s(0) we have found in a such way that the initial
consumption A(0) > 0 is being defined categorically from
(44) in accord with assigned С(0)> 0.
Now, from (32) the issue Y{t), will be defined, which
will be positive due to the positivity of the capital of the
enterprise along all the path.
7. The existence and the uniqueness of the balance:
logarithmic utility.
The question about the solvalibility of system (39),
(40) and also about the existence of the system is still
open. The equation (39) is reduced to a differential
equation of the form with standard methods:
),()()(2)()()( 22 xgxfycxxbyxayxyxy
x
++++=
∂
∂
which, as far as we know, cannot be solved in
quadratures. To achieve function у(х) from function G(t),
we have to execute two quadratures and to solve two
final transcendental equations.
However, it is possible to pay attention to the
case when the system (39), (40) is solved easily —
this is the case of the logarithmic utility function of
the consumer 1=β (see (1.13)). When 1=β the
solution of the equation (39) reduces to quadratures,
and for the real rate of dividend payment (41) we
have the term:
( )
( ) ( )( )∫ −∆−−
−
+
−
−
= −
∆−−
∆−−
−−
∆−−
−−t
T
T
Bb
bB
t
Bb
bB
Tu
u
Bb
bB
Tt
ebB
eBb
du
eee
eee
tr
ts
0
1
1
1
,
11
1
)(
)(
δ
δδ
δδ
ω (45)
which in fact shows that if 1=β , then the balance
exists with all positive δ,,,, bTB ∆ for any 0>ω .
Statement 4 shows, that regular balances in the
model are not singular, it being known that this non-
uniqueness has the dual character. For one thing, if the
equation exists under some path of price change p(t), so
it will exist and under another function p(t), which
satisfies (33). It is not surprising. As far as agents don’t
keep money, inflation in the model is almost equivalent to
the denomination, which should not change the
substantive behavior of economic agents. When we
change p(t) the equilibrium paths of real variables Y(t)
and C(t), it is easy to check, do not change. Restriction
(33) on deflation appears because in the model agents
can want to stash money and it brings a profit, when the
rate of price declination becomes higher than the
profitability of the enterprise, which in our model is value
1−b . The non-uniqueness of the balance, connected with
the possibility to size the prices, con be considered as
immaterial.
More important ambiguity 0>ω в (39). As follows
from general considerations, and as it is evident from the
logarithmic utility (45), system (39), (40), if it has any
solution, then a whole range of values ω .
At the same time, the trajectory of real values for
different ω are different.
From the statement 4 it can be seen that the choice
of the value ω actually equal to the choice of the initial
Y. I. Zhykhareva
54
Економічний вісник Донбасу № 4 (22), 2010
value of the real stock price
)0(
)0(
p
s .
If we assume that this value is inherited from the
history of the economic system, then the resulting
equilibrium trajectory can be regarded as an idealized
description of the transition process.
References
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во Э. Лекции по микроэкономическому анализу /
Э. Маленво. — М. : Наука, 1973. 3. Никайдо Х. Вы-
пуклые структуры и математическая экономика / Х. Ни-
кайдо. — М. : Мир, 1972. 4. Андрияшин А. В. Ди-
намическая модель общего равновесия при наличии
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рия полезности для принятия решений / П. С. Фиш-
берн. — М. : Наука, 1978. 7. Brock W. A.,
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Economic Review. Vol. 22. — Is. 1 (Feb., 1981). —
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Zhykhareva Y. I. Application of number theory
methods for task solution of intertemporal balance
in economy
Economic model is presented in which two agents
are taking place, which are producer-company and
owner-user. They are aggregated presenting productive
and non-productive economic fields. Mission is to learn
a principal of exploitations availability of intertemporal
balance for explanations of change processes in
economics. The problem was fully solved at any initial
conditions.
Key words: agent, property asset, intertemporal
balance, balance income, profitableness.
Жихарєва Ю. І. Застосування методів теорії
чисел для вирішення завдання міжтимчасової
рівноваги в економіці
У статті розглядається модель економіки, в якій
діють два агенти: фірма-виготовник та власник-спо-
живач, які агрегійовано представляють виробничу та
невиробничу сфери економіки. Мета — вивчення пи-
тання про принципову можливість використання мо-
делі міжчасової рівноваги для описування перехідних
процесів в економіці. Отримано повне розв’язання
задачі при будь-яких початкових умовах.
Ключові слова: агент, власний капітал, міжчасо-
ва рівновага, балансовий прибуток, доходність.
Жихарева Ю. И. Применение методов тео-
рии чисел для решения задачи межвременного
равновесия в экономике
Рассматривается модель экономики, в которой
действуют два агента: фирма-производитель и соб-
ственник-потребитель, которые агрегировано представ-
ляют производственную и непроизводственную сфе-
ры экономики. Цель — изучение вопроса о принци-
пиальной возможности использования модели меж-
временного равновесия для описания переходных про-
цессов в экономике. Получено полное решение зада-
чи при любых начальных условиях.
Ключевые слова: агент, собственный капитал,
межвременное равновесие, балансовая прибыль, до-
ходность.
Received by the editors: 24.11.2010
and final form in 01.12.2010
Y. I. Zhykhareva
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