```Keywords: dynamical systems, systems modelling, parametrical regulation,
task of variational calculation, structural stability
Abdykappar A. ASHIMOV*, Kenzhegaly A. SAGADIYEV*, Yuriy V. BOROVSKIY**,
Nurlan A. ISKAKOV*, Askar A. ASHIMOV*
DEVELOPMENT AND USAGE OF THE MARKET ECONOMY PARAMETRICAL
REGULATION THEORY ON THE BASIS OF ONE-CLASS MATHEMATICAL
MODELS
The paper offers further development and applications of the theory of a parametrical regulation of market
economy evolution. This theory consists of the following sections: formation of a library of economic systems’
mathematical models; of rigidness (structural stability) of mathematical models; development of parametrical
regulation laws etc. The work contains new results of the considered one class models’ rigidness research with
and without parametrical regulation.
1. INTRODUCTION
Many dynamic systems [1] including economic systems of nations [2, 3], 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 )  x0 .
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 ( X )
Int (W )
and
respectively;
a
vector
of
uncontrolled
parameters
  (1 , 2 ,  , m )    R m ,  is an opened connected set; maps f ( x, u,  ) : X  W    R n and
f f f
,
,
are continuous in X W   , [t 0 , t 0  T ] - a fixed interval (time).
x u 
____________________________
* Laboratory of Systems Analysis and Control, Institute of Problems of Informatics and Control, 221 Bogenbay batyr
str., Almaty, 050026, Republic of Kazakhstan, e-mail: [email protected]
** Kazakh-British Technical University, 59 Tole be str., Almaty, 050000, Republic of Kazakhstan, e-mail:
[email protected]
As is known [4], 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
controlled (u) and uncontrolled (λ) parameters. This is why the result of evolution (development) of
a nonlinear dynamic system under the given vector of initial values x0 is defined by the values of
both controlled and uncontrolled parameters.
Main components of market economy parametrical regulation theory have been developed and
offered in [4, 5, 6] based on the fact of a possible description of a country’s economic system with
the help of a mathematic model of type (1). Thus, in [3, 2] an approach is offered to finding the
vector u values in the form of extremals of variation calculation task on the choice of parametrical
regulation laws from the given set of parameters’ dependences on these or those significant
endogenic indicators of economic system’s evolution. These laws are deduced from the conditions
of economic processes optimal evolution, when certain parameters (e. g., price level) are preserved
within the given scales. In [7] a dependence of the deduced optimal regulation laws on a nonregulated vector λ change has been investigated. An assumption has been proved on the existence of
solutions to the above mentioned variation calculation tasks. A definition is provided and an
assumption is proved on the sufficient terms of existence of extremals bifurcation points of the
above-mentioned variation calculation tasks under the parametrical perturbations [4].
Development and usage of this theory in solving of certain tasks of parametrical regulation of
market economy development depend on the choice of mathematic models meeting the
requirements of economic system evolution. This requires additional study of roughness (structural
stability) of the models chosen [1, 12]. The results of this research allow considering an adequate
nature of mathematic models as well as a structural stability of the economic system described by
this model. Development and application of a theory for the chosen models require that definite
parametrical regulation laws should be chosen, and the dependence of the chosen laws on the values
of the non-controlled parameter λ should be studied as well. The paper presents latest results of
development and application of the parametrical regulation theory to a mathematical model of an
economic system for research of influence of the consumer charges of the state on development of
economy [10].
2. RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF A MATHEMATIC
MODEL WITHOUT PARAMETRICAL REGULATION
2.1. MODEL DESCRIPTION
The considered task is solved on the basis of the parametrical regulation theory and on the
sample of the following mathematical model [10], which presents the phase restrictions and
restrictions in the permitted form of the researched variational calculus task at the choice of
parametric regulations laws by the following relationships:
dM  I

 M ,
dt
pb
(2)
dQ

 Mf  ,
dt
p
(3)
dLG
 rG LG   G  n p   n L sR L  nO (d P  d B ) , (4)
dt
dp
Q
 
p,
dt
M
 Rd  RS  L
ds s
d
S
 max 0,
, R  min{ R , R } ,
S
dt 
R


Lp 
dp 
1

1

(5)
(6)
LG ,
(7)
r2 LG ,
(8)
d B  r2 LG ,
(9)
1


 s  

x
1   
1    p 



,


(10)
R d  Mx ,
(11)
O  0 pMf ,
(12)
 G  pMf ,
(13)
1
I 
 1    1
f  1  1 
x ,



(14)
 L  (1  n L ) sR d ,
(15)
1

  (1   )n p
(1  n
p )
G
 

(16)
 n0 (d  d )  n p   nL  (1  n L )n p sR  (   rG ) L ,
B
P
O
L
*
  O  G   L   I ,
R S  P0A exp(  p t )
1
L
, 
.
1  
pP0 exp(  p t )
p
(17)
(18)
G
L
Here: M stands for total production capacity; Q – the general stock of the goods in the market;
– 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; ξ - norm of reservation; β - 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; O - coefficient of owners
disposition to consumption; π - state's consumer expenses share in gross domestic product;
n p , n0 , n L - accordingly rates of taxes on cash flow, dividends and income of employees; b - rate of
fund-capacitance of power unit; μ - factor of power unit leaving which is caused by degradation;  *
- norm of amortization; α - time constant; Δ - time constant, which is set of characteristic time scale
of process relaxation of wage; P0 , P0A - initial number of workers and an aggregate number ablebodied accordingly;  p >0 - appointed tempo of demographic growth; ω – per capita consumption
in group of employees.
The model parameters and initial conditions for the differential equations (2)-(6) were
obtained on the basis of economic data of the Republic of Kazakhstan [17] at 1996-2000 ( r2 = 0.12;
rG = 0.12; β = 2; n p = 0.08; nL = 0.12; s = 0.1; n0 = 0.5; μ = μ* = 0.012; Δ=1) or are assessed
through the solution of the parametric identification task (ξ=0.1136; π=0.1348; δ=0.3; ν=34; O =
0.05; b = 3.08; α = 0.008; Q(0) = - 125000).
2.2. RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF THE MATHEMATIC
MODEL
Let us conduct statement of rigidness (structural stability) of a model under consideration in a
closed field Ω, basing on the definition of rigidness and the theorem on sufficient conditions of
rigidness [12]. The conditions are as follows.
Let N  be some set and N be such a compact subset N  that the closure of interior of N would
be N. Let some vector field be given in the area of set N in N  , then this field would determine the
C1 flow f in this area. The chain recurrent set of the flow f on the N is marked as R( f , N ) .
Let the R( f , N ) be contained inside the N and have a hyperbolical structure; besides this, the
f on the R( f , N ) would satisfy the transversality conditions on stable and unstable manifolds. Then
the flow f on the N is weakly structurally stable. Particularly, if the R( f , N ) is an empty subset,
then the flow f weakly structurally stable on the N.
We will assume further that R d > R S . In this case differential equations (6) are replaced with
the conditions of stability of variable s.
Statement 1. Let N be a compact set in area ( M  0, Q  0, p  0) or ( M  0, Q  0, p  0) ,
of the phase area of differential equations system obtained from the (2-18), i.е. four-dimension area
of variables ( M , Q, p, LG ) ; closure of interior of N coincides with the N. Then the flow f
determined by the (2-18) is weakly structurally stable on the N.
F.i.,
a
parallelepiped
could
be
chosen as
the
N, with
boundaries
p  pmin , p  pmax , LG  LG min , LG  LG max . Here
M  M min , M  M max , Q  Qmin , Q  Qmax ,
0  M min  M max , Qmin  Qmax  0 or 0  Qmin  Qmax , 0  pmin  pmax , LG min  LG max .
The proof. To start with, let us make sure that the half-trajectory of the flow f beginning in
any point of the set N under a certain value of t (t>0) comes out from the N. Let us consider any
half-trajectory starting in the N. There are two cases possible for it if t  0 : all the half-trajectory
points are left in the N, or for a certain t a point of the trajectory does not belong to the N. It follows
dp
Q
 
p, that the variable p(t) for all the t  0
in the first case from equation (5) of the system
dt
M
has got a derivative that is either more than some positive constant under the
N  ( M  0, Q  0, p  0) or less some negative constant under the N  (M1  0, Q1  0, p1  0) ,
that is, the p(t) unlimitedly rises or tends to zero under the unlimited increase of the t. This is why,
the first case is impossible, the orbit of any point from the N comes out from the N.
As far as any chain recurrent set R( f , N ) is an invariant set of this flow, in case of its nonemptiness it should consist of whole orbits. Consequently, in our case the R( f , N ) is empty. The
assumption has been proved.
3. RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF THE MATHEMATIC
MODEL WITH PARAMETRICAL REGULATION
3.1. THE TASK OF CHOICE OF EFFECTIVE PARAMETRICAL REGULATION LAW
The possibility of a choice of the optimum law of parametrical regulation at a level of one of
two parameters ξ (j = 1) and π (j = 2), and in the interval of time [t0 , t0  T ] were investigated in the
environment of the following algorithms.
1)U 1 j (t )  k1 j
M  M0
 const j ,
M0
3)U 3 j (t )  k 3 j
p  p0
 const j ,
p0
2)U 2 j (t )   k 2 j
M  M0
 const j ,
M0
4)U 4 j (t )   k 4 j
p  p0
 const j .
p0
(19)
Here: U ij - is a i-law of regulation of j-parameter, i  1  4, j 1  2 ; t 0 – is the time of the
start of regulation, t  t 0 , t 0  T  ; k ij is a tuned factor of the relevant law ( kij  0 i ); M 0 , p0 are
initial values of the corresponding variables; const j is a constant that is equal to the assessment of
the values of j-parameter by the results of parametric identification. Handling the law U ij under the
fixed k ij in the system (2-18) means the substitution function U ij from (19) in equations of the
system (2-18) instead of a parameter ξ or π.
The task of selection of an optimal parametrical regulation law for the economic system at the
level of one of the economic parameters (ξ, π) was set in the following form: to find on the basis of
mathematical model (2–18) an optimal parametrical regulation law in an environment of the set of
algorithms (19), i.е. to find an optimal law (and its factor k ij ) out of the set { U ij }, which would have
provided a maximum for the criteria
K
1 t0 T
 Y (t )dt ,
T t0
(20)
where Y  Mf is a gross domestic product.
The closed set in the space of continuous vector-functions of target variables of the system (2)
- (18) and regulating parametrical influences is defined by the following conditions
pij (t )  p ** (t )  0.09 p ** (t ),
( M (t ), Q(t ), LG (t ), p(t ), s (t ))  X ,
0  u j  a j , i  1,4, j  1,2, t  [t 0 , t 0  T ].
(21)
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.
The formulated task is solved in two stages:
- at the first stage optimal values of factors k ij for each law U ij are defined by way of their
values sorting in relevant intervals (quantized with small step), which provide maximum K under
the restrictions (21);
- at the second stage an optimal parametrical regulation law of a parameter based on the first
stage results for the maximum value of criterion K will be chosen.
The results of the numerical solution of the problem of choosing an optimum parametrical
regulation law at a level of one of economic parameters for economic system of the state show that
the best result K  177662 can be received via the usage of the following regulation law
  0.095
M  M0
 0.1136 .
M0
(22)
Let us notice that the value of the criterion without usage of the parametrical regulation equals
to K  170784 .
3.2. RESEARCH OF DEPENDENCE OF THE OPTIMUM LAW OF PARAMETRICAL
REGULATION ON THE VALUES OF UNCONTROLLED PARAMETERS
The calculating experiment [6] brought about the graphs of dependence of the optimal value
of criterion K on parameter values (r2 , nO ) for each of the 8 probable rules U ij , i  1,4, j  1,2 .
Figure 1 presents the given graphs for the rules U 21 and U 41 , which give the biggest value of a
criterion in area Λ, the intersection of corresponding surfaces and the projection of this intersection
upon the plane of values λ, consisting of the bifurcation points of this parameter. This projection
divides the rectangle Λ into two parts, where the controlling rule
U 21(t )  k 21
M  M0
 const1
M0
(23)
is optimal for one part, and
U 41(t )  k 41
p  p0
 const1
p0
(24)
is optimal for the other; at the projection itself both rules are optimal.
Fig. 1. Graphs of dependences of the criterion’s K optimal values on the parameters of interest rates on deposits r2
and rates of taxes on dividends nO.
3.3. RESULTS OF RESEARCH OF RIGIDNESS (STRUCTURAL STABILITY) OF THE
MATHEMATICAL MODEL WITH PARAMETRICAL REGULATION
Application of the found above optimal laws of parametrical regulation U 21 and U 41 , means
replacement of parameter ξ by the relevant functions in the equations (7, 8, 16), the rest equations of
the model remain unchanged. The proof of the weak structural stability of the mathematical model
given in i. 2.2. and basing on equation (5), allows to find the following.
Statement 2. Let N be a compact set in area ( M  0, Q  0, p  0) or ( M  0, Q  0, p  0) ,
of the phase area of the differential equations system obtained from the (2-18), i.e. four-dimension
area of variables ( M , Q, p, LG ) ; closure of interior of N coincides with the N. Then the flow f
determined by the (2-18) and (23, 24) is weakly structurally stable on the N.
4. CONCLUSIONS
This paper gives the following latest results of the development and usage of the parametrical
regulation theory for a mathematical model of an economic system for research of influence of the
consumer charges of the state on development of economy [10]:
- the model’s weak structural stability for compact areas of its phase space has been proved;
- the task of choice of optimal parametrical regulation laws has been solved for maximizing
an average VAT of a country under some limitations upon price level increase;
- a dependence of the found optimal laws on the values of non-controlled parameters has been
defined and a set of optimal laws bifurcation points has been found;
- a weak structural stability of a model preserved under the usage of the chosen parametrical
regulation laws influenced by the non-controlled parameters of the model has been proved.
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