ELEMENTS of the MARKET ECONOMY DEVELOPMENT
PARAMETRICAL REGULATION THEORY
Abdykappar A. Ashimov, Kenzhegaly A. Sagadiyev, Yuriy V. Borovskiy, Nurlan A. Iskakov, Askar A. Ashimov
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.
Keywords: economic system, mathematical model, parametrical regulation, task of variational calculation,
extremal, functional, bifurcation.
I. 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
(1)
 f ( x, u,  ), x(t 0 )  x 0 ,
dt
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; maps f ( x, u,  ) : X  W    R n and
f f f
,
,
are continuous in X W   , [t0 , t0  T ] - a fixed interval (time).
x u 
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 ( ) 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.
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 [5] econometric methods are used for modelling
the dynamic lines and statistical forecasting of tax revenues. In [6] 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 [3] and conducted after solving the task of parametrical identification.
Currently, due to the development of the theory of dynamic systems [7, 8, 9], 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 [11], the
parametrical influence that is defined by the Otto-Gregory-Yorke method [12], was used in order to stabilise the unstable
solutions in the models of a neoclassical theory of optimal growth.
In a number of papers [13-18] 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 [3]. Mathematical models from [3] 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 [19] 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 [20] 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
1
minimum of a non-perturbation task’s functional. In [21] the theorem about the terms of bifurcation point existence for one
0
variations calculus task is proved. The functional of 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 [22-24]. 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 [13-18] and the results obtained in [22-24] could be treated as components of the
parametrical regulation theory developed by the authors.
II. 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. [13]
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 [15-18].
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 [22-24].
8. Econometrical analysis, political and economic interpretation and coordination of the analitical 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.
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
2
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 decision-makers 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 [25, 26] and the theory of dynamic systems [27, 30].
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.
III. Conditions of existence of a solution to the variational calculus task on the choice of an optimal set of
parametrical regulation laws
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 [t0 , t0  T ] under the constant values
   . Let x (t )  Int ( X ) .
Further u* is fixed.
Let us mark through
u W
and
Let us mark the solution (1) for the selected u*  (u*1 , u*2 ,..., u*l )  W through x* (t ) .
 the closed set in the space of continuous vector-functions C nl [t 0 , t 0  T ] , which
consists of all the continuous vector-functions x(t ), u(t ) that satisfy the following restrictions.
x  X , u W ,
x j (t )  x*j (t )  x*j (t ) ,
(2)
t  [t0 , t0  T ] , j  1  n,   0 .
Let {F ( x) : i  1  r} and G ( x)  0 be the finite set of continuous for x  X real value functions. All the
i
F i
are also continuous in X . An opportunity of choice of an optimum set of parametrical regulation laws at
x j
the set of combinations from the p parameters on r and in the time interval [t0 , t0  T ] is explored in the following
functions
algorithms (laws of control):
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  Uis js } in the right parts of the equations of the system for r different indexes js
3
(1  s  r , 1  js  l ,
1  is  p ). Herewith, the rest values of u j , where j does not enter into the specified set of indexes j s , are considered as
j
constant and equals to the 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 i
j
s s
, 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.
The fact of existence of the solution to the task (1)-(5) can be proved with the help of the theorem on continuous
dependence of solution to Cauchy’s task on the parameters and using the theorem on continuous dependence of a definite
integral on a parameter.
Theorem 1.
There is a solution to the task of finding the supremum of criterion K under the usage of any selected set of laws
U  {U i
j
s s
, s  1  r } ( r  l ) 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 ( k i1 j1 , k i2 j2 ,  , k ir 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 theorem 1. The comparison of the relevant target functions and regulating parametrical influences
x (t ), u(t ) of system (1) under its regulation with the help of the set of laws
U  {( U i
1 j1
, ki
1 j1
), (U i
2 j2
, ki
2 j2
),  , (U i
r jr
, ki
r jr
)} with the values of factors ( k i j , k i j ,  , k i j ) from the
11
2 2
r r
U
sets
a continuous map H of some subset of R  [0,) into the space C n  l [t 0 , t 0  T ] .
l

l
A complete prototype H 1 () of the set  under the map
H
is close according to the theorem of closure of a
1
complete prototype of the closed set under continuous map. The set H ij () is not empty, as it contains the beginning of
coordinates of
Rl (with zero factors of laws the pair of functions x(t )  x* (t ), u(t )  u*  obviously satisfies to
restrictions (2)).

1
The comparison of criterion (4) K for the solution of system (1) to the set of factors k  H () defines a
continuous function
K : H1() [0,) .
Hence, with the chosen set of laws
U
the task (1) - (5) is equivalent to the task of determination on the closed set
1
H () of supremum of the continuous limited function
y  K ( k ij )
This function is continuous by virtue of the theorem of continuous dependence of the decision of system of the ordinary
differential equations on parameters, and due to limitation of this decision by virtue of inclusion x  X from (2) and
continuous dependences of definite integral on a parameters. Therefore, for the fixed set of laws U the task (1)-(5) always
4
has the solution including finite optimum value of criterion

K * . For the limited set H 1 () this value of criterion is
achieved under some values of factors k (theorem of achievement of the biggest value of a continuous function on a
compact). For the unlimited set H 1 () , the growing and infinitely large sequence of values of the law factor k ij from
*
H 1 () can be found, the corresponding values of criterion K for which elements has a limit K . Thus, the fact of
existence of the solution of a variational calculus task for a case of the chosen set of the laws of parametrical regulation is
proved. The fact of existence of the solution of task (1) - (5) follows from the finiteness of a set of the possible laws of
regulation (3).
IV. 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 2 (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    .
The proof of theorem 2. Let us connect the points
1
and
2
with a smooth curve S lying in the  :
S  { (s), s [0, 1]},  (0)  1 ,  (1)  2 . Let us designate an optimum value of criterion K of the task (1)-(5) for the
chosen set of laws U  {U i1 j1 , U i2 j2 ,  , U ir jr } and values  (s ) through K U (s ) . Function y  KU (s) is continued at
segment [0,1] in accordance with the theorem on a continuous dependence of the solution to the system of ordinary
differential equations, a continuous dependence of a definite integral and, on the whole, by virtue of the above proved
lemma. Therefore, function y  max K (s)  K * (s) , which provides the solution to the task under consideration (1)-(5), is
U
U
also continuous at segment [0,1] . Let us designate through (U )  [0,1] the set of all those values of parameter s, for which
KU (s)  K * (s) . This set is closed, as a full prototype of the closed set {0} for the continued function
y  KU (s)  K * (s) . Set (U ) could be empty as well. As a result, segment [0,1] is presented as the following finite
combination which consists, as minimum from the two closed sets. (see the theorem conditions).
[0,1]   (U ) .
U
Hence , according to the provisions of the theorem, 0  (U ) for some set of laws
U , corresponding to 1 , and

1 (U ) , there is a boundary point s of set (U ) , in the interval (0, 1) (let us cosider that s  is a infinum of these

boundary points for set (U ) ). Point s is also boundary for another multitude
(U 1 ) and belongs to it. For this value s 
point  ( s  ) is a bifurcation point as far as under  ( s  ) there are at least two sets of optimum laws, and under
0  s  s * there is one optimum law - U . The theorem has been proved.
The following theorem is a direct corollary of theorem 2.
Theorem 3
Let’s assume that under the value   1 regulation with the help of a certain sets of r laws from (3) provides the
solution for the task (1)-(5), and under   2 , ( 1  2 ,
1 , 2   ) regulation through this sets of laws does not

provide the solution for the task (1)-(5). Then there is at least one bifurcation point    .
5
Let us consider a sample of application of the obtained results to finding the bifurcation points of one task of
variational calcuation.
The sample. Finding of the bifurcation points of the extremals of one variational calculation task on the choice of an
optimal law of parametrical regulation of market econopmy mechanisms
A possibility of parametrical regulation of market economy mechanisms with the account of the foreign trade’s
influence will be considered on the basis of the mathematical model suggested in [3] in order to study the impact of foreign
trade. This model after some transformations looks like the following system of differential and algebraical equations:
dM i
 iI

 i M i ;
dt
pi bi
dQi

 M i fi  i ;
dt
pi
(7)
(8)
dLGi
 rG i LGi   Gi  n p i  i  n L i si RiL  nO i (d iP  d iB );
dt
(9)
dpi
Q
  i i pi ;
dt
Mi
(10)
 Rd  RS 
dsi
s
 i max 0, i S i , RiL  min{ Rid , RiS };
dt  i
Ri


(11)
LPi 
LGi ;
(12)
 i r2 i LGi ;
(13)
1  i
d iP 
i
1  i
i
diB   i r2 i LG
i ;

xi  i
1 i

  si i
1  
  pi




(14)
1 i

;



i
(15)
Rid  M i xi ;
(16)
1
 1   i  1 i
f i  1  1 
xi  ;
i


O
 i  0 i pi M i f i ;
(17)
(18)
 Gi   i pi M i f i ;
(19)
  (1  nL i )si R ;
L
i
 iI 

d
i
(20
k qi M i f i
1
{
 (1  n pi ) Gi  n0 i (d iB  d iP )  n p i  Oi 
1 npi
i

(21)
 n L i  (1  n L i )n p i si R  n pi ( ji   i  ij )   i L pi  rG i L };
RiS  P0Ai exp(  p i t )
1
1   ii
L
i
P
i
; 
i
(1  C iL i
6
 iL
pj
pi
;
) P0 i ( pi t )
j  3  i;
(22)
С1L
 12 
p2
p1
p2
p1
С1O
p
1 C  2
p1
 1L 
L
1
p
1 C  2
p1
 1O
;
(23)
O
1
p1
p
С 2O 1
p2
p2

 2L 
 O2 ;
p
L 1
O 1 p1
1
1  С2
1  C2
 p2
 p2
С 2L
 21
(24)
(25)
1  1I  1L  1O  1G   21  12 ;
1
(26)
 2   2I   2L   O2   G2  12   21 ,

Here: i = 1, 2 is for a number of the state; t is for time; M i – total production capacity, Qi – the general stock of
i
the goods in the market; LG i – total amount of a public debt;
P
p i – price levels; s i – average real wages; LPi -volume of
B
d
S
manufacture debts; d i and d i - enterprise and bank dividends respectively; R i and R i –a supply and demand of a
i ,  i
labor respectively;
- parameters of function f i ;
x i – a solution to the equation f i( xi ) 
si
;  i and  i L
pi
O
consumer spending of workers and proprietors;  i - investment flows;  i - consumption government spending;  ij –
I
G
consumer spending of the i-th country on an imported product from the j-th country; θ - exchange rate of the first country in
relation to the exchange rate of the second country (  1   ,  2  1 /  ); CiL (CiO ) - volume of imported product units
consumed by the workers (proprietors) of the i-th country on the domestic product unit;
i
– norm of reservation;
ratio of the average profit rate from commercial activities to the rate of return of the investor;
credit rate; rGi – government bonds rate;
bi
i
- deposits rate ; r1i -
- rates of the tax to flow payments, dividends tax and
i
-factor of power unit leaving which is
ai - time constant;  i - time constant setting the characteristic time-
caused by degradation;  i -norm of amortization;
respectively;
n Pi , nOi , n Li
- norm of a capital intensity of a production capacity unit;
scale of wages process relaxation;
-
 Oi - factor of proprietors’ propensity to consume;  i -share of consumption
government spending from the gross domestic product;
surtax accordingly;
r2 i
i
P0i , P0Ai - initial values of amount of workers and total amount able-bodied population
- per capita consumption in group of employees;
 P  0 - appointed tempo of demographic growth; k qi
- share of the gross domestic products reserved in gold.
The possibility of the choice of an optimal set of laws of type (3) of parametrical regulations was researched: at the
level of one of the two parameters
 i (β=1) ,

i
(β=2); at the time interval
[t0 , t0  T ] and in an environment of the
following algorithms.
1) U 1i,   k1i, 
M i (t )
 const i ;
M i (t 0 )
2) U 2i ,   k 2i , 
3) U 3i ,   k3i , 
M i (t )
 const i ;
M i (t 0 )
pi (t )
 const i ;
pi (t0 )
4) U 4i ,   k 4i , 
Here Ui , - is a α-law of regulation of β-parameter of the i-state,
(27)
pi (t )
 const i .
pi (t0 )
  1  4,  1  2 . Case   1
corresponds to the
parameter  i ,   2   i ; M i (t )  M  ,  ,i (t )  M i (t 0 ), pi (t )  p ,  ,i (t )  pi (t 0 ), t0 – is the time of the start of regulation,
t  t 0 , t 0  T . Here M  , ,i (t ) , p ,  ,i (t ) are values of the production capacity and price levels of the i–state
respectively under the
Ui , -regulation
law.
ki ,
i
is a tuned factor of the relevant law ( k , 
 0 i ); const i is a
constant that is equal to the assessment of the values of β-parameter by the results of parametric identification.
7
The task of selection of an optimal parametrical regulation law for the economic system of the i-country at the level
of one of the economic parameters (ξi,πi,θ) was set in the following form: to find on the basis of mathematical model (7–26)
an optimal parametrical regulation law in an environment of the set of algorithms (27), i.е. to find an optimal law (and its
i
i
factor k , ) out of the set { U , }, which would have provided a maximum for the criteria
Ki 
1 t T
 Yi (t )dt ,
T t
0
(28)
0
where
Yi  M i f i
is a gross domestic product of the i-state. The calculation experiments studied the impact of parametrical
regulation of the first country (i=1).
The closed set
  C 7 t0 , t0  T  in the space of continuous vector-functions of discharge variables of the
system (3.4.1)-(3.4.20) and regulating parametrical influences is defined with the following relationships:
p1 (t )  p i** (t )  0.09 p 1** (t ),

( M i (t ), Qi (t ), LG i (t ), pi (t ), si (t ))  X ,
(29)
0  u   a  ,   1,4,   1,2, t  [t 0 , t 0  T ].
Here
a
is the biggest possible value of a α-parameter, pi

i
(t ) are values of price levels under the U  law of
regulation; p (t ) are model (acccounting) values of price levels in i-state without the parametrical regulation, X is a
compact set of possible values of the given parameters.
The given task of variational calculation considered its dependence on a two-dimensional factor   (r2,1 ,  ) of
i**
the mathematical model, whose possible values belong to some area (rectangle)  on the plane.
As a result of calculation experiment dependence graphs of the optimal value of criterion K on the values of
parameters (r2,1 ,  ) were obtained for each of the 8 possible laws U1 , ,   1,4,   1,2 . Figure 1 demonstrates the given
graphs for two laws,
U 21, 2 and U 41, 2 , which give the biggest value of the criterion in area  , an intersection line of
corresponding surfaces and a projection of this intersection line upon the plane of values  , which contains the bifurcation
points of this two-dimensional parameter. This projection divides the rectangle  into two parts; the regulating law
p1 (t )
M 1 (t )
 const 1 is optimal for the other,
U 1  k 1
 const 1 is optimal in one of these parts, and the law U 1  k 1
2, 2
2, 2
M 1 (t 0 )
4, 2
2
both laws are optimal for the projection line itself.
8
4, 2
p1 (t 0 )
2
Fig. 1. Graphs of dependences of a criterion’s optimal values on the parameters of interest rates on deposits
r2
and
currency exchange rate θ.
Conclusion
The paper gives an account of the main components and findings in the framework of the market economy
development parametrical regulation theory under development.
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