MULTI-TARGETED PARAMETRICAL REGULATION OF MARKET
ECONOMY DEVELOPMENT with the ACCOUNT of NON-CONTROLLED
PARAMETERS INFLUENCE
Abdykappar A. Ashimov*, Kenzhegaly A. Sagadiyev*, Yuriy V. Borovskiy**, Nurlan A. Iskakov*, Askar A. Ashimov*
* Laboratory of Systems Analysis and Control, Institute of Problems of Informatics and Control
125 Pushkin str., Almaty, 050010, Republic of Kazakhstan
** Kazakh-British Technical University
59 Tole be str., Almaty, 050000, Republic of Kazakhstan
email: [email protected]*, [email protected]**
ABSTRACT
The paper presents the findings on working out an
approach to a parametrical regulation of market economy
system developing with a variety of its functioning subareas, and the studies of optimal parametrical impacts
dependences on non-controlled parameters alterations.
The bifurcation lines of extremals are formed up for a task
of variational calculation at the choice of the optimum
laws of parametrical regulation in the framework of the
given finite set of algorithms under a bi-parametrical
perturbation.
KEY WORDS
Mathematical model, economic system, parametrical
regulation, extremal, functional, task of variational
calculation, bifurcation.
1. Introduction
It is well-known that the economic system of a country
can be described with the help of the ordinary differential
equations system of the following type.
dx
(1)
 f ( x, u ,  ) , x(t 0 )  x0 .
dt
Here, a vector of a state of a system x  ( x1, x 2 ,..., x n )  X  R n ; a vector of (regulating)
parametrical influences - u  (u1, u 2 ,...,ul ) W  Rl ; W,
Х – compact sets with non-empty interiors - Int (W ) and
Int (X ) respectively;   (1 , 2 ,, m )    R m ,  opened connected set of non-controlled parameters;
f f
f
mapping f ( x, u, ) : X W    R n and
,
,
x u 
are continuous in X W   ; [t0 , t0  T ] - a fixed
interval (time); t  [t0 , t0  T ] ; x0  Int ( X ) .
It is also known that the solutions to system (1) depend on
x0 , u and  .
In [1] the structure and in [1-7] the results are offered on
the parametrical regulation of market economy system
theory elements development (1). The given theory allows
to choose the laws of parametrical regulation within the
components of vector u in the given multitude of
algorithms. Making the choice of the above-mentioned
laws is stipulated by the extremum of a certain functional.
These functionals characterise the efficiency of an
economic system development (1).
The area of economic system functioning (1) in the
framework of some components of vector х – endogenic
factors can be presented by two sub-areas – acceptable
and unacceptable. For instance, it would be natural to set
the acceptable and unacceptable sub-areas for the
endogenic factor of prices level. The unacceptable subarea could characterize the unacceptable rates of inflation
and deflation. It would be also natural to assume the
existence of corresponding regulation targets for each
sub-area of economic system functioning.
As the offered parametrical regulation theory shows [2],
the can lead to the bifurcation of extremals of the
variational calculation task on the choice of the optimal
set of parametrical regulation laws within the given set of
algorithms [2].
There has been no solution in the existing papers so far to
the task of the choice of optimal parametrical regulation
laws of the market economy development for the
corresponding sub-areas of an object under consideration
functioning with the account of non-controlled factors’
influence.
This paper offers the solution to the above-mentioned task
with the account of vector  which is based on the
application of the parametrical regulation theory.
2. The choice of optimal laws of multitargeted parametrical regulation of the
market economy development with the
account of non-controlled parameters
influence
The considered task is solved on the basis of the
parametrical regulation theory and on the sample of the
following mathematical model [8], which 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:
dM  I

 M ,
dt
pb
dQ

 Mf  ,
dt
p
(2)
(3)
dLG
 rG LG  G  n p  nL sR L 
,
dt
nO (d P  d B )
dp
Q
 
p,
dt
M
d
S
ds s
 R  R 
 max 0,
,
dt 

R S  ,
R  min{ R , R },
1 G
Lp 
L ,
L
d
d B  r1 LG


(5)
(6)
S

1
dp 
r2 LG ,

   s
x
1  
1     p
(4)
1



,


(10)
(12)
  pMf ,
(13)
1
 1    1
f  1  1 
x
,



 L  (1  nL )sR d ,
1
I 

  (1   )n p
(14)
(15)
{(1  n p ) G  n 0 (d B  d P )  n p  0 
(16)
  O  G   L   I ,
(17)
n L  (1  n L )n p sR L }  ( *  rG ) L p ,
The initial conditions for this system are not mentioned
here for the sake of brevity.
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;
product;
n p , n0 , n L - accordingly rates of taxes on cash flow,
  0 pMf ,
G
(18)
(8)
(11)
0
L

.
pP0 exp(  p t )
0 - coefficient of owners disposition to consumption;
 - state's consumer expenses share in gross domestic
R  Mx ,
d
1
,
1  
(7)
(9)



R S  P0A exp(  p t )
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.
The model parameters and initial conditions for the
differential equations (7)-(11) were obtained on the basis
of economic data of the Republic of Kazakhstan at 19962000 (r2=0.12; rG=0.12; β=2; np=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; η0=0.05; b=3.08; α=0.008;
Q(0)=125000).
In the framework of the considered model for prices level
we could conventionally single out two sub-areas: the
acceptable and unacceptable sub-areas of prices level
change. The unacceptable sub-area (А) of prices level
change can be defined with the help of equations:
p(t )  pl (t ) or p(t )  pu (t ) , where pl (t ) is the lower
acceptable border of prices level change, and pu (t ) is the
upper acceptable border of prices level change
( pl (t )  pu (t ) , t  [t 0 , t 0  T ] ). Solution of equation
p(t )  pl (t ) shows the existence of some deflation
process, and solution of p(t )  pu (t ) shows the existence
of some excessive inflation. It is possible to set an
acceptable sub-area (В) of prices level change by means
of inequality pl (t )  p(t )  pu (t ) , t  [t0 , t0  T ] .
Setting the task of choice of optimal parametrical
regulation (influence) laws depends on finding the values
of prices level in the sub-areas А or В and results in the
following tasks:
- in the sub-area А it is necessary to find and realize such
parametrical regulation laws within a given set of
algorithms with the account of non-controlled parameters
impact, which secure the minimum of a criterion that
characterizes the quality of transition processes under the
restrictions set upon possible values of the corresponding
factors of the economic status and regulation parameters
(block А).
- in the sub-area В it is necessary to find and realize such
parametrical regulation laws within a given set of
algorithms with the account of non-controlled parameters
impact, which provide the maximum of average GDP at
the given time interval restrictions upon possible values of
the corresponding factors of the economic status and
regulation parameters (block В);
The suggested approach is realized as follows: initially,
the process of an economic system modelling is launched
according to the results of a parametrical identification
task solution. Primarily, the sub-areas А and В for the
values p(t ) are defined by the results of modelling. The
algorithm of the calculation experiment has a logical
condition which determines finding the value of prices
level in this or that sub-area. If in the process of this
evaluation it will be found that the value p(t ) is in the
sub-area А, then the block А is involved for the solution
of the task of an object’s movement from the
unacceptable sub-area А to the acceptable sub-area В. If
the value p(t ) is found in the sub-area В, then the block
В is involved to provide the achievement of the maximum
average GDP level at the given time interval.
2.1 Block А
Let us consider now an opportunity of an effective state
policy implementation in the framework of block А
through the choice of optimal regulation laws with the
account of influence of vector   (r2 , nO ) within the
following economic parameters: share of the state
consumer expenditures from the Gross Domestic Product
(  ), interest rate on government bonds ( rG ) and
reservation norm (  ).
0, if pl (t )  p(t )  pu (t ),

Let p(t )   p(t )  pl (t ), if p(t )  pl (t ),
 p(t )  p (t ), if p(t )  p (t ).
u
u

In the process of finding the level of price in the
unacceptable sub-area А the choice of optimal
parametrical regulation laws is made within the set of the
following dependences (regulation laws):
p(t )
1) V1 j  k1 j
 const j
p(t0 )
2) V2 j  
k2 j t0  t p(t )
 p(t ) dt  const j ,
0
t
(19)
t0
 p(t ) 1 t 0  t p(t ) 
3) V3 j  k3 j 
 
dt   const j
 p(t0 ) t t 0 p(t0 ) 
Here sample j=1 corresponds to the parameter ξ; j=2 – to
the parameter π; j=3 – to the parameter rG ; kij is a tuned
coefficient of the i-regulation law of the j- parameter,
kij  0 ; constj is a constant equal to the assessment of the
value of the j parameter by the results of parametrical
identification.
It is possible to formulate the task of choice of an optimal
law of parametrical regulation at the level of one of
economic parameters ( , , rG ) taking into account the
influence of non-controlled parameters as follows: to find
on the basis of a mathematical model (2–18) for each
possible value of vector    an optimal law of
parametrical regulation at the level of one of the three
economic parameters ( ,  , rG ) within the set of
algorithms (19), in other words, to find an optimal law
from multitude { Vij }, which would provide the minimum
of criterion
K1 
1
T1
t 0 T
2
.
 p(t ) dt  {Vmin
,k }
t0
ij
(20)
ij
under the restrictions
M (t )  M ** (t )  0,09 M ** (t ) ,
( M (t ), Q(t ), LG (t ), p (t ), s (t ))  X ,
0  Vij (t )  a j , i  1,3, j  1,3,
(21)
t  [t 0 , t 0  T ],   .
Here  is some area on the plane (r2 , nO ) , a j - is the
biggest possible value of j-parameter, M ** (t ) are model
(counting) values of total production capacity without
parametrical regulation, X is a compact set of allowable
values of the specified parameters.
This task is solved in two stages:
- at the first stage the optimal values of factors kij are
2.2 Block B
defined for each law Vij by way of sorting the values of
When finding the prices level in the acceptable sub-area В
possibility of a choice of the optimum set of laws of
parametrical regulation at a level of one of two
parameters  (j = 1) and  (j = 2) with the account of
non-controlled parameters influence and on an interval of
time [t 0 , t 0  T ] were investigated in the environment of
the following algorithms.
M  M0
1)U1 j (t )  k1 j
 const j ,
M0
coefficients in the intervals of type [0, kijm ) quanted with
the step 0.01, and providing the minimum of criterion K 1
under restrictions (21). Here k ijm is the first value of the
factor at which (21) is destroyed;
- at the second stage the law of optimal regulation of a
certain parameter (out of the three) is chosen based on the
results of the first stage on the minimal value of criterion
K1 .
The algorithm of regulation has been approbated on the
model of the economy of the Republic of Kazakhstan for
the prices level change thresholds pl (t )  0.9 and
p u (t )  1.1 .
As a result of the calculation experiment the graphs of
dependences of an optimal value of criterion K on the
parameters values were obtained (r2 , nO ) for each of the
9 possible laws Vij , i  1,3, j  1,3 . Figure 1 presents the
above-mentioned
graphs
for
the
four
laws
( V11 , V12 , V21 , V22 ), which provide the lowest values of
criterion K 1 in area  , the crossing lines of the
corresponding surfaces and the projection of these
crossing lines upon the values plane  . The given lines
projections consist of the bifurcation points of this two
dimensional parameter  , these points divide the
rectangular  into parts, and for each of the part only
one regulation law is optimal. Two or three different laws
are optimal at the projection line itself.
2)U 2 j (t )  k 2 j
3)U 3 j (t )  k3 j
M  M0
 const j ,
M0
p  p0
 const j ,
p0
4)U 4 j (t )  k 4 j
p  p0
 const j ,
p0
(22)
 M  M 0 p  p0 
  const j ,
5)U 5 j (t )  k5 j 

p0 
 M0
 M  M 0 p  p0 
  const j .
6)U 6 j (t )  k6 j 

p0 
 M0
Here M 0 , p0 are initial values of the corresponding
variables, and const j is a value of an appropriate
parameter obtained by the results of parametrical
identification of a model.
The task of the choice of an optimal parametrical
regulation law at the level of one of the economic
parameters ( ,  ) with the account of non-controlled
parameters influence could be formulated like this: to find
on the basis of mathematical model (2–18) for each
possible value of vector    an optimal parametrical
regulation law at the level of one of the two economic
parameters ( ,  ) within the set of algorithms (22), in
other words, to find an optimal law out of multitude
{ U ij }, which would provide the maximum of criterion
K
1 t 0 T
,
 Mfdt  {Umax
T t0
ij , k ij }
(23)
under restrictions
p (t )  p ** (t )  0.09 p ** (t ),
( M (t ), Q (t ), LG (t ), p (t ), s (t ))  X ,
0  U ij (t )  a j , i  1,6, j  1,2,
(24)
t  [t 0 , t 0  T ],   .
Fig. 1. Graph of optimal values of a criterion K 1 .
Here p** (t ) are model (counting) values of price levels
without parametrical regulation.
The formulated task is solved with the same method as a
task from block А.
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
rules U ij , i  1,6, j  1,2 . Figure 2 presents the given
graphs for the rules U 21 and U 41 , which give the biggest
value of a criterion in 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 rule U 21 is optimal for one part, and U 41 is
optimal for the other; at the projection itself both rules are
optimal.
Fig. 2. Graphs of optimal values of a criterion K.
3. Conclusion
Based on the theory of parametrical regulation an
approach is suggested to solve the task of multi-targeted
parametrical regulation of market economy development
with the account of non-controlled parameters influence.
The facts of bifurcation extremals of variational
calculation tasks on the choice of parametrical regulation
laws for the sub-areas of economic system functioning
under the double parametric perturbation have been
established.
A more complicated nature of bifurcation lines at the
choice of a parametrical regulation law for an
unacceptable sub-area has been revealed.
The research findings could be applied to the choice and
realization of an effective state policy.
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