Key words: economic system, mathematical model, parametrical regulation, task of variational calculation, extremal,
functional, bifurcation.
Abdykappar ASHIMOV, Kenzhegaly SAGADIYEV, Yuriy BOROVSKIY, Nurlan ISKAKOV,
Askar ASHIMOV
ON THE MARKET ECONOMY DEVELOPMENT
PARAMETRICAL REGULATION THEORY
The paper offers the theory of a paramterical regulation of market economy development that is represented via a
system of ordinary differential equations. This theory consists of the following sections: forming of a library of
economic systems mathematical models; researches on rigidness (structural stability) of mathematical models;
development of parametrical regulation laws; finding of bifurcation points of the extremals of one class of
variational calculation tasks, etc. The report delivers some of the results obtained in the course of developing the
theory under consideration. The work contains also example of a finding of bifurcation points for one model of
economic system.
1. INTRODUCTION
Many dynamic systems [1] including economic systems of nations [2], after certain
transformations can be presented by the systems of non-linear ordinary differential equations of the
following type:
dx
 f ( x, u,  ), x(t 0 )  x 0 ,
dt
(1)
Here x  ( x1 , x 2 ,..., x n )  X  R n is a vector of a state of a system; u  (u1 , u 2 ,..., u l )  W  R l - a
vector of (regulating) parametrical influences; W, Х are compact sets with non-empty interiors Int (W ) and Int ( X ) respectively;   (1 , 2 ,, m )    R m ,  is an opened connected set;
f f f
maps f ( x, u,  ) : X  W    R n and
,
,
are continuous in X W   , [t0 , t0  T ] - a
x u 
fixed interval (time).
As is known [3], solution (evolution) of the considered system of the ordinary differential
equations depends both on the initial values’ vector x0  Int ( X ) , and on the values of vectors of

Institute of Problems of Informatics and Control. Pushkin str., 125, Almaty, 050010, Republic of Kazakhstan. E-mail:
[email protected].
1
controlled (u) and uncontrolled ( ) parameters. This is why the result of evolution (development) of
a nonlinear dynamic system under the given vector of initial values x 0 is defined by the values of
both controlled and uncontrolled parameters.
In the recent years the dynamic trends of such parameters are being actively researched as:
various tax rates, state expenditures, accounting rate, reservation norm, currency rate and others.
The influence of these parameters upon the evolution of economic processes is also in the focus of
researches. Thus, in [4] econometric methods are used for modelling the dynamic lines and
statistical forecasting of tax revenues. In [5] econometric methods are used for the analysis of
dependences between the parameters of cash and loan policy (refinance rate, reservation norm) and
economic development indicators (indexes of investment activity in the real sector etc.). Paper [3]
considers the impact of state expenditures’ share in the gross domestic product and of the state
loans’ interest upon the average real income of the workers, upon average state expenses in stable
prices and on the average gross domestic product. This is based on the mathematical model
suggested by the authors [2] and conducted after solving the task of parametrical identification.
Currently, due to the development of the theory of dynamic systems [6], the parametrical
influence has become applied to the regulation of economic systems. The dynamics of these
economic systems, according to the opinion of a number of experts, is described with the help of
nonlinear models [10], whose behaviour can be chaotic. Thus [8], the parametrical influence that is
defined by the Otto-Gregory-Yorke method [19], was used in order to stabilise the unstable
solutions in the models of a neoclassical theory of optimal growth.
In a number of papers [10-15] the parametrical influences are being offered in order to
effectively regulate the market economy development in the given range of the changes of main
endogenic indicators of an economyc system and to hinder its expansion outside the given range.
The suggested parametrical influences are the extremals of the corresponding variational
calculations tasks on the choice of optimal laws of parametrical regulations within the given finite
set of algorithms. The functionals in the given tasks of variational calculation express certain
(global, intermediate or tactical) goals of economic development. Phase restrictions and permitted
resatrictions are represented by mathematical models of economic systems from [2]. Mathematical
models from [2] contain a range of indexes, whose chage within a certain interval leads to the
deformation (perturbation) of the considered variational calculus tasks.
Nowadays, the parametrical deformation of the variational calculus tasks is being widely
researched. Thus, the parametric perturbation in [16] is used in order to obtain the sufficient
extremum conditions through construction of the corresponding S-functions and usage of the
principle of restrictions removal. In [17] the problem of the terms of stability of variations calculus
tasks solutions is set (Ulam’s problem). The study of this problem is brought to finding the terms of
regularity under which the functional of the perturbation task has the point of a minimum that is
close to the point of a minimum of a non-perturbation task’s functional. In [18] the theorem about
the terms of bifurcation point existence for one variations calculus task is proved. The functional of
0
this task is considered in the Sobolev’s space W pm () (2  p  ) and depends on the scalar
parameter   [0,1] . Thus, it could be noted that among the existing sources there has been no
evidence as to the terms of existence of the solutions to variational calculus tasks on the choice of
optimal laws of parametrical regulation in an environment of the given finite set of algorithms.
There are no researches on the influence of parametrical perturbations upon the solution of the
given tasks.
Existence of the solutions to the variational calculus tasks on the choice of optimal laws of
parametrical regulation in an environment of the given finite set of algorithms is under study in [19-
2
21]. These research works also focus on the influence of parametrical perturbation (changes of
uncontrolled parameters) upon the results of solving the tasks under consideration. In other words,
the researches are conducted, in particular, as to the bifurcation of extremals of these tasks under
parametrical perturbations.
The approaches suggested in [10-15] and the results obtained in [19-21] could be treated as
components of the parametrical regulation theory developed by the authors.
2. ELEMENTS OF THE MARKET ECONOMY DEVELOPMENT PARAMETRICAL
REGULATION THEORY
On the whole, the following components could characterise the initial version of the market
economy development parametrical regulation theory:
1. Methods of formation of a macroeconomic mathematical models library. These methods
are oriented towards a description of a variety of certain socio-economic situations with the account
of environmental safety conditions.
2. Rigidness (structural stability) conditions assessment methods of mathematical models of
an economic system out of the library without the parametrical regulation. At this, it is studied if the
mathematical models under consideration belong either to the class of Morse-Smale system or to
 -rigid systems, or to the uniformly rigid systems, or to the class of У-systems.
3. Methods of control or suppression of non-rigidness (structural instability) of mathematical
models of an economic system. The choice (synthesis) of algorithms of control or structural
instability suppression of a relevant mathematical model of a country’s economic system. [10]
4. Methods of choice and synthesis of the laws of parametrical regulation of market economy
mechanisms based on the mathematical models of a country’s economic system [12-15].
5. Rigidness (structural stability) conditions assessment methods of mathematical models of
an economic system out of the set/store with the parametrical regulation. At this, it is studied if the
mathematical models under consideration belong either to the class of Morse-Smale system or to
 -rigid systems, or to the uniformly rigid systems, or to the class of У-systems.
6. Methods of clarification of the limits on the parametrical regulation of market economy
mechanisms in case of a structural instability of mathematical models of a country’s economic
system, with parametric regulation. Clarification of the limits on the parametrical regulation of
market economy mechanisms.
7. Methods of study and research on bifurcation of the extremals of variational calculus tasks
on the choice of optimal laws of parametrical regulation [19-21].
8. Econometrical analysis, political and economic interpretation and coordination of the
analytical researches findings and calculation experiments with the preferences of decision-makers.
9. Development of an informational system for the research and imitational modelling of
market economy mechanisms with parametrical regulation.
10. Development of recommendations on elaboration and implementation of an effective state
economic policy based on the theory of parametrical regulation of market economy mechanisms
with the account of certain socio-economic situations.
Putting this newly-developed theory of parametrical regulation of market economy
mechanisms into practice - elaboration and implementation of an effective state economic policy could be as follows:
1. Choice of a state economic development direction (strategy), on the basis of a relevant
assessment of a country’s economic status in the framework of economic cycle stages.
2. Selection of one or several mathematical models of economic systems that correspond to
the tasks of economic development out of the library.
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3. Assessment of suitability of the mathematical models to the given tasks: calibration the
mathematical models (parametrical identification and retrospective prognosis according to the
current evolution indicators of the economic system) and an additional verification of the selected
mathematical models with the help of econometric analysis and political and economical
interpretation of the sensibility matrixes.
4. Assessment of structural stability (rigidness) of mathematical models without the
parametrical regulation according to the above given methods of assessment of rigidness conditions
(see item 2). Structural stability of a mathematical model reflects the stability of the economic
system itself. In this case a mathematical model can be used after the econometrical analysis,
political and economic interpretation of the rigidness research findings, in order to solve the task of
choice of optimal laws of regulating the economic parameters and forecasting the macroeconomic
indicators.
5. If a mathematical model is structurally unstable, it is necessary to choose the algorithms
and methods of stabilisation of the economic system in accordance with the methods pointed out in
section 3 of the theory under development. The result obtained in the course of the corresponding
econometrical analysis, and political and economic interpretation could be approved and ready for
practical implementation.
6. Choice of optimal laws of regulating the economic parameters.
7. Assessment of structural stability (rigidness) of mathematical models with the chosen
parametrical regulation laws according to the above given methods of assessment of rigidnesss
conditions (it.2). If the mathematical model is structurally stable under the selected parametrical
regulation laws, then the results obtained in the course of the corresponding econometrical analysis,
and political and economic interpretation, and after coordination with the preferences of decisionmakers could be adopted for their practical implementation.
If the mathematical model under the selected parametrical regulation laws is structurally
unstable, then the decision on the choice of parametrical regulation laws is clarified. The clarified
decisions on the choice of parametrical regulation laws are also subject to be considered according
to the above given scheme.
8. The study of dependence of the chosen optimal laws of parametrical regulation on the
changes of uncontrolled parameters of an economic system. Here it is possible to replace some
optimal laws by the others.
The given enlarged scheme on making decisions in the sphere of elaboration and
implementation of an effective state policy through the choice of optimal values of economic
parameters should be supported with the modern informational technologies of research and
imitational modelling.
At the present time, the above given sections 1, 2, 3, 5, 6 of the theory of parametrical
regulation are developed in the framework of the modern approaches of the theory of identification
[22, 23] and the theory of dynamic systems [24, 25].
This report presents some findings of the research on the conditions of existence of one
variational calculus task and in the framework of the above given section 7 of the theory of
parametrical regualtion of market economy in progress.
3. ON MAPPING OF THE CONJUNCTION OF SOME AREAS OF A PHASED SPACE
FOR NON-PERTURBED AND PERTURBED DYNAMIC SYSTEMS
The following theorem has been formulated and proved in the framework of sections 2 and 5
of the parametrical regulation theory.
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Theorem 1. Let x0  X , u W and    be fixed, and the inequality f i ( x, u,  )  C  0 is
x (t ) be a solution to
fulfilled in area X of a phased space of system (1) for some i (1  i  n) . Let ~
Cauchy’s task (1), determined for t  [0, T ] and located in Int(X). Let f1 ( x) be a perturbed vector
field under X that meets the requirements laid out in the Introduction; f ( x, u,  )  f1 ( x)   for
x (t ) be the solution to the corresponding Cauchy’s task for the field
x  X and some   0 . Let ~
1
f1 ( x) .
~
~
~
x (t )  Int ( X ) and
Then (closed) areas X  X and X 1  X will be found, where ~
~
~
~
~
x1 (t )  Int ( X 1 ), 0  t  T , for which there exists a homeomorphism H : X  X 1 , that at the same
~
~
time realizes one-to-one correspondence between the phase curves in areas X and X 1 keeping
their direction (for the fields f and f1 respectively).
4. CONDITIONS OF EXISTENCE OF A SOLUTION TO THE ONE VARIATIONAL
CALCULUS TASK
The variational calculus task on the choice of an optimal set of parametrical regulation laws at
the multitude of combinations from p parameters on r in an environment of the given finite set of
algorithms and the assertion on existence of a solution to the variational calculus task in an
environment of the given finite set of algorithms looks like this:
Let x (t ) be a solution of the given task (1) in the interval [t 0 , t 0  T ] under the constant
values u W and    . Let x (t )  Int ( X ) . Let us mark the solution (1) for the selected
u*  (u*1 , u*2 ,..., u*l )  W through x* (t ) . Further u* is fixed.
Let us mark through  the closed set in the space of continuous vector-functions
C [t 0 , t 0  T ] , which consists of all the continuous vector-functions x(t ), u(t ) that satisfy the
following restrictions.
n l
x  X , u W , x j (t )  x*j (t )  x*j (t ) , t  [t0 , t0  T ] , j  1  n,   0 .
(2)
Let {F i ( x) : i  1  r} and G ( x)  0 be the finite set of continuous for x  X real value
F i
are also continuous in X . An opportunity of choice of an
x j
optimum set of parametrical regulation laws at the set of combinations from the p parameters on r
and in the time interval [t0 , t0  T ] is explored in the following algorithms (laws of control):
functions. All the functions
U
ij

 kij F i ( x)  u*j , i  1  r, j  1  l .
(3)
Here, k ij  0 are adjusted factors.
Handling of the set of r ( 1  r  l ) laws U ij from (3) under the fixed k ij in the system (1)


means the substitution set of functions u js  U is js in the right parts of the equations of the system
j
for r different indexes js (1  s  r , 1  j s  l , 1  is  p ). Herewith, the rest values of u , where j
5
does not enter into the specified set of indexes j s , are considered as constant and equals to the
j
values u * .
For the solutions of system (1) under the usage of the set of r laws of control the following
functional (criterion) is considered:
K
t0 T
 G( x (t ))dt .
(4)
t0
Setting the task of a choice of a dynamic system parametric regulation set of laws in the
environment of the finite set of algorithms looks like the following.
Under the fixed    find the set of r laws U  U is js , s  1  r from the set of algorithms


(3), which provides the supremum of the values of criterion (4)
K  sup
(5)
U
under the fulfilled conditions (1,2) for the given time interval.
Theorem 2. There is a solution to the task of finding the supremum of criterion K under the
usage of any selected set of laws U from the set of algorithms (3) with the restrictions (1) and (2):
t0 T
 G( x (t ))dt 
t0

.
sup
(6)
( ki1 j1 , ki2 j2 , , kir jr )

Herewith, if the set of possible values ki1 j1 , ki2 j2 , ..., kir jr of factors of laws of the considered task is
limited, then the indicated supremum for the selected set of laws is reached. The task (1)-(5) has got
the solution for the finite set of algorithms (3).
The proof of the theorem 2 is present in the [19].
5. CONDITIONS SUFFICIENT FOR THE EXISTENCE OF A BIFURCATION POINT OF
EXTREMALS OF ONE VARIATIONAL CALCULATION TASK
Let us suggest the following definition, which characterizes the values of parameter  , under
which the replacement of one optimal law for another becomes possible.
Definition. Value *   would be called a bifurcation point of the task (1)-(5) extremal, if
under    there were, as minimum, two different optimal sets of the laws from (3), that would
differ at least by one law U ij , and if in each neighborhood of the point  there is such value
   , for which the task (1)-(5) would have a single solution.
The following theorem provides sufficient conditions for the existence of a bifurcation point
of the extremals for the considered variations calculus task at the choice of parametrical regulation
set of laws in the given finite set of algorithms.
Theorem 3 (about the existence of a bifurcation point). Let for the values of the parameter 1
and 2 , ( 1  2 , 1 , 2   ) task (1)-(5) has the relevant unique solutions for the two different
sets of r laws from (3) that would differ at least by one law U ij . Then there is at least one point of
bifurcation    .
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The proof of the theorem 2 is present in the [19].
Let us consider a possibility of finding the bifurcation points for one variations task at the
choice of parametric regulation laws of the market economy mechanism in an environment of finite
fixed set of algorithms on a mathematic model basis [3] of a country’s economic system. The paper
[3] presents the phase restrictions and restrictions in the permitted form of the researched variations
calculus task at the choice of parametric regulations laws by the following relationships:
dQ

dLG
 rG LG   G  n p   n L sR L  nO (d P  d B ) ,
 Mf  ,
dt
dt
p
 Rd  RS  L
dp
Q
1 G
ds s
p
d
S
 
p,
L ,
 max 0,
, R  min{ R , R } , L 
S
dt
M

dt 
R


1 


 s   

1
d
0
dp 
r2 LG , d B  r2 LG , x 
1     , R  Mx ,   0 pMf ,
1    p  



dM  I

 M ,
dt
pb
(7)
1
 G  pMf ,
I 

 1    1
f  1  1 
x ,



 L  (1  n L ) sR d ,
 

1
(1  n p ) G  n0 (d B  d P )  n p  O  nL  (1  nL )n p sR L  (  *  rG ) Lp ,
  (1   )n p
   O   G   L   I , R S  P0A exp(  p t )
1
,
1  

L
.
pP0 exp(  p t )
Here: M – total production capacity, Q – the general stock of the goods in the market; LG –
total amount of a public debt; p – price level; s – average real wages; LP - volume of manufacture
debts; d P and d b - accordingly owner’s and bank’s dividends; R d and R S - accordingly demand
and supply of labor;  , - parameters of function f(x); x - solution of f x   s p equation;  L and
 O - accordingly consumer expenses of employees and owners;  I - investment flow;  G consuming government spending;  - norm of reservation;  - the attitude of average rate of return
from commercial activity to the rate of return of the investor; r2 – interest rate on deposits; rG interest rate on government bonds;  0 - coefficient of owners disposition to consumption;  state's consumer expenses share in gross domestic product; n p , nO , nL - accordingly rates of taxes
on cash flow, dividends and income of employees; b - rate of fund-capacitance of power unit;  factor of power unit leaving which is caused by degradation;  * - norm of amortization;  - time
constant;  - time constant, which is set of characteristic time scale of process relaxation of wage;
P0 , P0A - initial number of workers and an aggregate number able-bodied accordingly;  p >0 appointed tempo of demographic growth;  – per capita consumption in group of employees.
Possibility of a choice of the optimum set of laws of parametrical regulation of type (3) at a
level of one of two parameters  (j = 1) and  (j = 2) and on an interval of time [t 0 , t 0  T ] were
investigated in the environment of the given six algorithms [19].
7
In the considered task the criterion K 
1 t 0 T
 Y (t )dt of the (4) type (was used average GDP
T t0
Y  Mf ). The closed set in the space of continuous vector-functions of target variables of the
system (7) and regulating parametrical influences is defined by the following relationships.
pij (t )  p ** (t )  0.09 p ** (t ),
( M (t ), Q(t ), L (t ), p(t ), s(t ))  X , 0  u j  a j , i  1,4, j  1,2, t  [t 0 , t 0  T ].
G
(8)
Here a j - is the biggest possible value of j-parameter, pij (t ) are the values of price levels under the
U ij regulation law; p ** (t ) are model (counting) values of price levels without parametrical
regulation, X is a compact set of allowable values of the specified parameters.
In the given variation calculus task its dependence on a two-dimensional factor   (r2 , nO )
of a mathematical model was under consideration, whose values could possibly belong to a certain
area (a rectangle)  lying in the plane.
Fig. 1. Graphs of optimal values of a criterion K.
The calculating experiment brought about the graphs of dependence of the optimal value of
criterion K on parameter values (r2 , nO ) for each of the 12 possible laws U ij , i  1,6, j  1,2 . Figure
1 presents the given graphs for laws U 21 and U 41, which give the biggest value of a criterion in
8
area  , the intersection of corresponding surfaces and the projection of this intersection upon the
plane of values  , consisting of the bifurcation points of this parameter. This projection divides the
rectangle  into two parts, where the controlling law U 21 is optimal for one part, and U 41 is
optimal for the other; at the projection itself both laws are optimal.
6. CONCLUSIONS
1. The first version of the components of the market economy development parametrical
regulation theory is represented.
2. Main stages of handling the parametrical regulation theory for the purpose of elaboration
and realization of an effective state economic policy are pointed out.
3. The following results are provided in the framework of the market economy development
parametrical regulation theory under development:
3.1. The approach to parametrical regulation of a nonlinear dynamic system’s development in
the form of setting and solving the task of variational calculation on the choice of the optimum laws
of parametrical regulation within the given finite set of algorithms is offered.
3.2. An assumption about the existence of solution to the task of variational calculation on the
choice of the optimum laws of parametrical regulation within the given finite set of algorithms is
formulated and proved.
3.3. The definition of a bifurcation point of an extremal of a task of variational calculation on
the choice of the optimum laws of parametrical regulation within the given finite set of algorithms
is given.
3.4. An assumption about the conditions sufficient for the existence of an extremal’s
bifurcation point of a task of variational calculation on the choice of the optimum laws of
parametrical regulation within the given finite set of algorithms is formulated and proved.
3.5. The example of finding the bifurcation points of the extremals of one task of variational
calculation on the choice of the optimum laws of parametrical regulation within the given finite set
of algorithms is provided.
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