Download 376.00Kb - G

Application of the parametrical regulation theory for reduction the effect of a shadow sector of economy
Ashimov A.A., Sultanov B.T., Adilov Zh.M., Borovskiy Yu.V., Merekeshev T.B., Ashimov As.A
The work presents some results of the parametrical regulation theory which consider peculiarities of CGE
models. The method of parametrical identification of macroeconomic models with large quantity of evaluation
parameters is proposed and tested. The application efficiency of the parametrical control theory for reduction of the
effect of a shadow sector of economy is shown by example of one CGE model. Optimum values have been obtained
for regulation of economic system development based on examined mathematical model on the level of 19
parameters.
Key words: shadow economy, computable general equilibrium model, discrete dynamic system (semicascade), parametrical regulation, parametric identification.
1. Introduction
As it is known, national economies of many countries function with a shadow sector, which activity affects
national economy development negatively [10-12]. A rational direction of evaluation and searching of the effective
economic measures to reduce the effect of a shadow sector on development of a country’s economic system is
application of the mathematical model of a national economy. [8] presents a computable general equilibrium model
(CGE model) with a shadow sector with large quantity of adjustable parameters, which evaluation on the basis of
computational algorithms requires application of effective methods of identification.
The task of identification (calibration) of exogenous parameters of the model comes to the searching of the
global minimum of some objective function, which is set with the help of the CGE model itself. Meanwhile
limitations on set of optimization are also set with the help of the model. The task of searching the global extremum
in general case of high dimensionality is quite complex; random search methods, parallel algorithms of calculation
and other methods are applied for its solving [14, 15]. The overview of numerous publications on searching the
global extremum is shown in [13]. The paper contains unmentioned before in the literature the algorithm of
parametric identification of a model, which considers peculiarities of macroeconomic models of high dimensionality
and in some cases enables to find the global minimum of an objective function with large quantity of variables (more
than one hundred). Two objective functions (two identification criterions – main and complementary) are used in the
algorithm; this enables to obtain the exit of values from points neighborhood of local (and non-global) extremums, to
continue searching the global minimum while keeping the conditions of corresponding movement to the global
extremum.
[1, 3, 5, 4] present the elements of the theory of the effective parametrical regulation of market economy
development described by the system of ordinary differential and algebraic equations. Application efficiency of
parametrical regulation approach based on a number of models is showed in [2, 6, 7]. In the network of the proposed
approach optimum (in sense of some criterion) values of parameters have been obtained with the help of a family of
functions, defined with the help of endogenous factor of the mathematical model and adjustable coefficients.
Development of the theory of parametrical regulation for the case when optimum (in sense of some criterion) values
of controlled parameters are evaluated in some given set is of practical interest.
The present work contains results of development and application of the parametrical regulation theory for
a given case based on the CGE model, in which shadow economy is considered to be of two type: “white-collar” and
“grey” [10]. Such processes of white-collar shadow economy as transfer of asset share from budgets of production
sectors and a consolidated budget to a budget of household were used in the work while modeling some scenarios of
economic development of a state with negative influence of shadow economy and modeling neutralization of
negative outcomes of such processes applying the parametrical regulation theory approach.
The CGE model with a shadow sector of economy, proposed in [8], is presented in the general view with
the help of the following correlation system.
1) Subsystem of difference equations, binding the values of endogenous variables for the two consistent
years.
(1)
xt 1  F ( xt , y t , z t , u,  )
1
Here t - number of a year, discrete time, t  0,1,2,... ; ~
x t  ( x t , y t , z t )  R n - vector of endogenous variables
of the system;
(2)
xt  ( xt1 , xt2 ,...,xtn1 )  X 1 , yt  ( yt1 , yt2 ,...,ytn2 )  X 2 , z t  ( z t1 , z t2 ,..., z tn )  X 3 .
3
Here n1  n2  n3  n ,
other;
xt variables include values of basic funds, agents’ remained cash in bank accounts and
yt include agents’ demand and supply values in different markets and other, z t - different types of market
prices and budget shares in markets with exogenous prices for different economic agents; u and  - vectors of
exogenous parameters, u  (u 1 , u 2 ,..., u l )  W  R l - vector of controlled (regulated) parameters; X1, X2, X3, W-


compact sets with non-empty interiors- Int ( X i), i  1,2,3 and Int (W ) respectively;   1 , 2 ,...,m    Rm vector of uncontrolled parameters,  - open connected set; F : X1  X 2  X 3  W    R
continuous
mapping.
2) Subsystem of algebraic equations, describing agents’ behavior and interaction in different markets during
the selected year, these equations allow expression of yt variables through exogenous parameters and rest
n1
endogenous variables
y t  G( x t , z t , u,  ) ,
(3)
Here G : X1  X 3  W    R n2 continuous mapping.
3) Subsystem of recurrent correlations for iterative calculations of equilibrium values of market prices in
different markets and values of a budget share in the market with state prices for different economic agents.
z t [Q  1]  Z ( z t [Q], y t [Q], L, u,  )
(4)
Here Q  0,1,2,... - number of iteration. L – set of positive numbers (adjustable constants of iterations).
When their values decrease, economic system comes to equilibrium faster, however the risk that prices go to the
negative range increases at that time. Z : X 2  X 3  (0, ) n3  W    R n3 - continuous mapping (contractive when
x t  X 1 , u W ,    and some L are fixed). In this case Z mapping has a single fixed point, where an iteration
process converges (4, 3).
CGE - model (1, 3, 4) when exogenous parameters are fixed in every point of t time defines values of
endogenous ~
x t variables, corresponding to demand and supply prices equilibrium in markets of agents’ goods and
services in the bounds of the following algorithm.
1) On the first step it is assumed that t=0 and initial values of x0 variables are specified.
2) On the second step initial values of z t [0] variables are specified for current t in different markets and for
different agents; values of y t [0]  G( x t , z t [0], u,  ) are calculated with the help of (3) (initial values of demand and
supply of agents in markets of goods and services).
3) On the third step the iteration process (4) is run for current t. Meanwhile for each Q current values of
demand and supply are calculated from (3): yt [Q]  G( xt , zt [Q], u,  ) through specifying market prices and budget
shares of economic agents.
The condition for the iteration process termination is equality of demand and supply values in different
markets. As a result, equilibrium values of market prices in every market and budget shares in markets with state
prices for different economic agents are defined. Q index is omitted for such equilibrium values of endogenous
variables.
4) On the next step values of xt 1 variables for next moment of time are found regarding equilibrium
solution for t moment with the help of difference equations (1). The value of t increases by unit. Transition to the
step 2.
Number of reiterations of steps 1, 3, 4 is defined according to the problems of calibration, forecast and
regulation for time intervals selected in advance.
2
2. Development of parametrical regulation theory for the class of CGE models of a general view (1, 3, 4)
Examined CGE-model of the (1, 3, 4) type can be expressed as continuous mapping f : X  W    R n ,
specifying conversion of values of the system’s endogenous variables for the null year to the corresponding value of
the following year according to the set above algorithm. Here compact X in the phase space of endogenous variables
is determined by a set of possible x variables’ values (X1 compact with non-empty interior) and corresponding
equilibrium values of y and z variables calculable with the help of (3) and (4) correlations.
t
x 0 )  Int ( X 1 ) is right, when u  Int (W )
Let us assume that for x0  Int( X1) point inclusion x t  f ( ~
X1
and    are fixed for t  0  N (N – fixed natural number). This f mapping determines the discrete dynamic
system (semi-cascade) in the X set.
f
t

, t  0,1,...
(5)
Such description of a state’s economic system (1, 3, 4, 5) differs from description of economic system with
the help of continuous dynamic system in [1] and validate necessity of the parametrical regulation theory
development for a discrete case of a semi-cascade.
x for selected
Let us denote points of corresponding trajectory ~
x  f t (~
x ) of semi-cascade by ~
t
*t
0
u*  Int (W ) .
Let us denote the closed set in space R n( N 1) ((N+1) sets of the variables ~
x t for t  0, N ), determined by
limitations
~
xt  X , ~
xt j  ~
x*jt   j ~
x*jt ,
(6)
through  . Last inequality in (6) is used for some values j  1  n , when x*t are positive;  j  0 .
j
For the assessment of efficiency of economic system evolution during the period of t  0  N time, (N fixed), let us use the criterion of type K  K ( ~
x0 , ~
x1 ,..., ~
x N ) , where K - the continuous function in XN+1.
The definition of the problem of finding optimum values of a controlled vector of parameters for the semicascade (5) is of the following type. When    is fixed, to find set out of N values of controlled parameters
u t , t  1  N , which provide the lower boundary of (6) criterion’s values –
K  inf
(7)
ut , t 1 N
under constraints (6). The analogous task is set for the case of the criterion K maximization.
The following theorem is true.
Theorem. For the defined semi-cascade (5) under constraints (6) the problem (5-7) solution of finding the
lower boundary of K criterion exists.
The proof. Matching of set of values ut , t  0  N  (where ut W ) of regulated parameters to the
corresponding output values xt , t  0  N  of the discrete dynamic system (5) (under the regulation by means of
l ( N 1)
this set of parameters) sets the continuous mapping H of some subset R
in the space R n( N 1) .
The complete preimage H
1
() of  set with H mapping is compacted according to the theorem about
compactness of the complete preimage of the compact set under continuous mapping. The set H
as it contains (u* ) N 1  Int(W N 1 ) at which constraints (6) are satisfied.
3
1
() is nonempty,
The function, which sets the value of K criterion for each point of the set H
trajectory of the system (5) is continuous in the compact H
1
1
() by the corresponding
() , and therefore at some point of this set it takes its
minimum value. The theorem is proved.
3. Example
Efficiency of the obtained results is illustrated below by the example of the CGE model with a shadow
sector [8]. This model describes behavior and interaction of the following agents in 13 markets (final goods,
investment and capital goods, and the market of labor force):
Economic agent № 1 - a state sector of an economy. This sector includes entities government share in which
is more than 50%.
Economic agent № 2 - a market sector, which consists of legally existing entities and organizations with
private and mixed ownership.
Economic agent № 3 – a shadow sector. This sector describes the type of economic activities that are not
taken into account in official statistics, i.e. which are hidden from statistical reports. The model considers two types
of shadow economy: “white-collar” and “grey” economy.
Economic agent № 4 - aggregate consumer, which joins households.
Economic agent № 5 - government. Additionally this sector includes non-commercial organizations, serving
household (political parties, labor unions, public associations etc.).
Economic agent № 6 - banking sector.
Examined model contains 144 exogenous parameters (which values require to be evaluated by solving a
task of parametric identification) and 123 endogenous variables. The considered CGE model with shadow sector is
presented in the frameworks: relations (1) – by means of 11 expressions ( n1  11 ); relations (3)- by means of 98
expressions ( n2  98 ); relations (4) - by means of 14 expressions ( n3  14 ).
3.1. Parametric identification and retrospective forecast based on the CGE model with a shadow sector
The task of parametric identification of the examined macroeconomic mathematical model consists of
finding estimates of unknown values of its parameters, that enables to reach minimum value of an objective function,
which defines deviations of output variables’ values from a corresponding observed value (known statistical data).
This task comes to finding a minimum value of a function of several variables (parameters) in some closed domain
 of Euclidean space under constraints of type (2), imposed on values of endogenous variables. In case of high
dimensionality of possible values’ range of desired parameters, standard methods of finding functions’ extremums
are often inefficient due to existence of several local minimums of an objective function. The algorithm which
considers special features of macroeconomic models’ parametric identification task and which enables to bypass the
mentioned problem of “local extrema”.
For the assessment of possible values of exogenous parameters, the range of  
l m n1
[ai , bi ]  W    X
i 1
type has been considered. Here [ai , bi ] - interval of  i , i  1  (l  m  n1 ) parameter’s possible values. Meanwhile,
assessment of parameters, for which observed values existed, were searched in [ai , bi ] intervals with centers in
corresponding observed values (in case if there is one such value) or in some intervals covering observed values (in
case if there are several such values). Other [ai , bi ] intervals for parameters searching have been chosen with the
help of indirect assessments of their possible values. In computing experiments the Nelder-Mead [16] algorithm of
the directed search has been applied for finding the minimum values of a continuous function F :   R with
several variables with additional constraints on endogenous variables of type (2). Application of this algorithm for
the starting point  1   can be interpreted as a sequence  1 ,  2 ,  3 ,... which converges to the point of local


minimum F0  arg min F of function F where F ( j 1 )  F ( j ) ,  j  , j  1, 2, ... While describing the
, ( 2)
following algorithm let us assume that F0 point can be found sufficiently accurately.
4
For the assessment of the quality of retrospective forecasting based on the data of the Republic of
Kazakhstan for the period of 2000-2004 for some starting point the task (task A) of the model’s parameters
assessment and assessment of initial conditions for difference equations has been solved with the help of finding
K IA criterion’s minimum:
2
K IA
 *
1 2004  Yt  Yt


10 t  2000  Yt*

2

 *
   pt  pt

 p*
t






2
.


(8)
Here t – number of a year; basic macroeconomic measures:
Yt - estimated GDP in milliards of tenge, prices of 2000;
pt - estimated level of consumer prices.
Here and below the sign «*» corresponds to observed values of corresponding variables. The task of
parametric identification for the model (1), (3), (4) is assumed to be solved, if point  K0 IA   is found where
K IA ( K0 IA )   for sufficiently small  .
Along with the task A for the point  1 the analogous task (task B) has been solved with application of
extended criterion K IB instead of K IA criterion.
2004  *
Y  Y   p*  p
1
{   t * t    t * t
12 .15 t  2000  Yt
  pt

2
2
2008 
 K* K 
 K* K 
  0.1 1t * 1t   0.1 2t * 2t 
 K

 K

t  2000 
1t
2t



 
2
2
K IB

2
 L*  L1t

  0.1 1t

 L*
1t


 Y * Y
 0.01 1t * 1t
 Y
1t

2

 L*  L1t
  0.1 1t

 L*
1t


2

 Y *  Y2 t
  0.01 2t

 Y*
2t






2
2





 Y *  Y3t
  0.01 3t

 Y*
3t






2

}.


(9)
Here:
L1t – number of employees in a state sector;
L2 - number of employees in a market sector;
K1t - basic funds of a state sector;
K2t – basic funds of a market sector;
Y1t- gross value added of a market sector;
Y2t – gross value added of a state sector;
Y3t – gross value added of a shadow sector.
Values reducing weights in criterion (9) are defined during the process of identification of parameters for
the concrete dynamic system.
While solving the task of parametric identification for each of these criterions separately, due to the
existence of several local minimuma of functions K IA and K IB , it is quite difficult to achieve values of these
criterions that are sufficiently close to zero. Therefore the ultimate algorithm of solving the task of parametric
identification of the model has been chosen with the help of the following steps.
1. The A and B tasks for some vector of initial values of  1   parameters are solved simultaneously, as a
result points  K0 IA and  K0 IB are found.
2. If K IA ( K0 IA )   or K IA ( K0 IB )   , then the task of parametric identification of the model (1, 3, 4) is
solved.
3. Otherwise, the task A is solved taking  K0 IB point as initial  1 point and the task B is solved taking  K0 IA
point as initial  1 . Transition to the step 2.
Quite large number of iterations of 1, 2, 3 steps in some cases provides an opportunity for desired values to
come out from the point neighborhood of non-global minimum of one criterion with the help of another criterion,
thereby the task of parametric identification can be solved.
5
As a result of simultaneous solving of the A and B tasks in accordance with the stated algorithm the values
K IA  0.0025 and K IB  0.12 have been obtained. This implies that deviation’s relative size of variables’ calculated
values used in criterion (8) from corresponding observed values is less than 0.25%.
Results of the retrospective forecast of the model for the period of 2005-2008 presented in the table 1 show
calculated, observed values deviations of calculated values of main output variables of the model from
corresponding actual values.
Table 1. Results of the retrospective forecast of the model.
Year
2005
2006
2007
2008
Yt*
4258.03
4715.65
5136.54
5303.27
Yt
Error (%)
4221.69
-0.861
4586.33
-2.820
5004.12
-2.646
5478.31
3.195
107.6
108.4
118.8
109.5
108.4
0.706
109.5
1.017
112.6
-5.528
112.0
2.240
p t*
pt
Error (%)
For conducting the following experiments the task of parametric identification of a model for 2000 - 2008
time interval was solved repeatedly using solving the tasks of type A and B. As a result of the solution of the task of
parametric identification of a model for the stated time interval, values of criterions of type K IA and K IB turned out
to be 0.015 и 0.15 respectively.
3.2. Scenario approach and finding of optimum values of parameters based on the CGE model with shadow
sector
Result of the shadow economy (in opinion of financial flows) is redirection of funds share, assigned to the
budgets of production sectors of legal economy and consolidated state budget to the budget of households (players of
shadow economy) [8]. This transfer of funds can be made through economic activity in a network of a shadow sector
of economy, as well as directly with the help of some illegal acts (thefts, bribes, kickbacks, etc.)
In the network of examination of analysis of the connection between some effects of shadow economy and
basic macroeconomic measures of a state (GDP and index of consumer prices), a number of computable experiments
has been conducted (scenarios estimations, which consider some possible negative phenomenon in economy of a
state), analogous to experiments in [8].
The work examines the following 6 scenarios.
1) Imitation of a process of cash withdrawal (10%, 20%, 30%) from a consolidated budget of a state and
transfer of it to households from 2003 (scenario 1, 2, 3). The process of theft has been imitated as direct or as quite
legal process of budgetary funds spending (process of kickback).
2) Imitation of cash withdrawal (10%, 20%, 30%) from producer and its redirection to households starting
from 2003 (scenario 4, 5, 6). In this case the process of giving (by producer) and taking a bribe (finally by
households) has been imitated.
Results of the listed 6 scenarios of economic development of a state with negative effect of shadow
economy in comparison with initial variant of evolution are presented in tables 2 and 3.
Table 2. GDP values (in milliards of tenge using prices of 2000) in initial variant and applying scenarios 16.
GDP
Year
Initial variant
Scenario 1
Scenario 2
Scenario 3
2005
4300103
4301026
4301887
4302752
2006
4618653
4623221
4627487
4631527
2007
4963707
4975060
4985442
4994972
6
2008
5337048
5357813
5376495
5393343
Scenario 4
Scenario 5
Scenario 6
4298244
4296520
4294935
4612732
4607483
4602927
4953878
4945717
4939176
5324870
5315665
5309146
Table 3. Values of consumer price indexes (in % to preceding year) in initial variant and applying scenarios
1-6.
Year
Initial variant
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
Scenario 6
2005
107.624
115.575
123.530
131.481
138.576
171.450
206.522
Price index
2006
2007
108.602
109.334
109.706
109.986
109.761
110.470
108.962
111.001
118.506
113.760
123.439
115.029
125.441
114.879
2008
108.816
108.989
109.044
109.006
111.462
111.904
111.508
Analysis of table 2 and 3 shows that examined scenarios affect GDP insignificantly, at the same time the
indexes of consumer prices increase significantly during the first year of 1-6 scenarios application, during the
following years their influence on price indices decreases.
It should be mentioned that aspects of shadow economy (thefts from a budget and bribes) lead to explicit
negative consequences for economy of a state. In both cases demand for consumer goods increase, it leads to natural
increase of consumer prices. Beside that, often a producer announces expenses, which are associated with bribes to a
price of his products, which also leads to increase in prices. Finally, in any case, the majority of a population, which
are not borne out relation to sharing out budget funds and bribes and kickbacks receives, suffers.
The next series of computable experiments aimed to decrease negative influence of each considered
scenario on one of the main macroeconomic factors: price level, using the methods of parametrical regulation. In the
network of application of the parametrical regulation approach , the task has been set to find for 2005-2008 optimum
values of such 19 parameters ( u li , i  1  19 - number of a parameter, l  2005  2008 - number of a year),
controlled by government (and for each considered scenario), as
- different tax rates,
- shares of consolidated budget, assigned to finance state and market sectors, purchase of final goods,
- shares of different type of goods produced by state sector of economy for realization in different markets.
When applying j-scenario, a level of consumer prices of a country for 2008 relative to 2004 has been used
as a minimizing criterion K. Among constraints of the solving variation task constraints of the type (6) have been
used for a country’s GDP:
Yt j  Yt j* , j  1  6 .
Here Yt j* - values of GDP while applying j-scenario without parametric control, Yt j - values of GDP while
applying j-scenario and optimum values of controlled parameters in sense of K criterion.
Constraints in networks (6) for controlled parameters u li are presented in the table 4.
№
1
2
3
4
5
6
Table 4. Controlled parameters of the model and constraints imposed on them.
Interval of a controlled parameter’s
Controlled parameter u i
possible values
Value added tax rate
[0.135, 0.165]
Income tax rate for entities
[0.27; 0.33]
Property tax rate
[0.009; 0.011]
Income tax rate for individuals
[0.135; 0.165]
Single social tax rate
[0.099; 0.121]
Share of consolidated budget assigned for a purchase of final goods
[0.117; 0.143]
7
7
8
9
10
11
12
13
14
15
16
17
18
19
Share of consolidated budget, assigned for financing a state sector
Share of consolidated budget, assigned for financing a market sector
Share of consolidated budget, assigned for financing social transfers
Share of consolidated budget assigned for a purchase of capital goods
Share of consolidated budget assigned for a purchase of investment
goods
Share of produced goods of a state sector assigned to be sold on a
market of final goods for a market sector
Share of produced goods of a state sector assigned to be sold on a
market of final goods for an economic agent № 5 for exogenous
prices
Share of produced goods of a state sector assigned to be sold on a
market of final goods for an economic agent № 5 for market prices
Share of produced goods of a state sector assigned to be sold on a
market of investment goods for exogenous prices
Share of produced goods of a state sector assigned to be sold on a
market of investment goods for market prices
Share of basic funds of a state sector assigned to be sold on a market
of capital goods for exogenous prices
Share of basic funds of a state sector assigned to be sold on a market
of capital goods for market prices
Share of produced products assigned to be sold on markets of final
goods in outer countries
[0.325; 0.398]
[0.028; 0.034]
[0.320; 0.391]
[0.129; 0.158]
[0.068; 0.083]
[0.101; 0.123]
[0.039; 0.048]
[0.039; 0.048]
[0.107; 0.131]
[0.107; 0.131]
[0.200; 0.244]
[0.200; 0.244]
[0.230; 0.281]
The set task of finding a minimum value of K criterion of function with 76 variables (and corresponding
values of controlled parameters u il  arg min K ) for each of the 6 examined scenarios has been solved with the
 
(6)
help of Nelder-Mead algorithm. Below the table 5 presents results of the set task.
Table 5. Results of application of the parametrical regulation approach
j*
j
Year
K criterion
K criterion
Y2008
Y2008
without
under found
parametric
optimum
control
values of
parameters
Scenario 1
1.52
1.32
5.35*1012
5.47*1012
12
Scenario 2
1.63
1.41
5.38*10
5.45*1012
12
Scenario 3
1.73
1.50
5.40*10
5.50*1012
12
Scenario 4
2.08
1.87
5.32*10
5.44*1012
12
Scenario 5
2.72
2.47
5.32*10
5.44*1012
12
Scenario 6
3.32
3.04
5.31*10
5.44*1012
The analysis of the table 4 shows that the parametrical regulation approach enables in case of examined
scenarios to reduce the price level for 2008 by 9.4 - 13.2% and to increase GDP of a country for 2008 by 1.30 2.44% compared to the case with no control.
4. Conclusion
1. Some results of the parametrical regulation theory development based on the one class of CGE models
have been presented.
2. Efficiency of application of the parametrical regulation theory has been shown by example of one CGE
model with shadow sector. Optimum values of controlled parameters of economic politics based on the examined
mathematical model have been suggested.
8
3. Efficiency of one of the methods of parametric identification of a macroeconomic model with large
number of evaluated parameters has been checked.
4. The obtained results can be used in realization of efficient state economic policy.
References
[1] Ashimov A.A., Borovsky Yu.V., Sultanov B.T., Iskakov N.A. & Ashimov As.A., The elements of
parametrical regulation theory of economical system evolution of a country, Physmathlit, Moscow, 2009, (in
Russian).
[2] Ashimov A.A., Iskakov N.A., Borovskiy Yu.V., Sultanov B.T. & Ashimov As.A. On the development
of usage of the market economy parametrical regulation theory on the basis of one-class mathematical models, Proc.
of 19th International Conference on Systems Engineering ICSEng 2008, Las Vegas, Nevada, USA, 43-48.
[3] Ashimov A.A., Sagadiyev K.A., Borovskiy Yu.V, Iskakov N.A. & Ashimov Аs.A. Elements of the
market economy development parametrical regulation theory. Proc. of the ninth IASTED International Conference
on Control and Application, Montreal, Quebec, Canada, 2007, 296-301.
[4] Ashimov A.A., Sagadiyev K.A., Borovskiy Yu.V., Iskakov N.A. & Ashimov As.A., On the market
economy development parametrical regulation theory. Kybernetes, The international journal of cybernetics, systems
and management sciences. Vol. 37, №5, 2008, 623-636.
[5] Ashimov A.A., Sagadiyev K.A., Borovskiy Yu.V, Iskakov N.A. & Ashimov Аs.A. On the Market
Economy Development Parametrical Regulation Theory. Proc. of the 16th International Conference on Systems
Science, Wroclaw, Poland, 2007, Vol. 1, 493-502.
[6] Ashimov A.A., Sagadiyev K.A., Borovskiy Yu.V., Iskakov N.A. & Ashimov As.A., Multi-targeted
parametrical regulation of market economy development with the account of non-controlled parameters influence,
Proc. of the 10th IASTED International Conference on Intelligent Systems and Control, Cambridge, MA, USA,
2007, 280-284.
[7] Ashimov A.A., Sultanov B.T., Adilov Zh.M., Borovskiy Yu.V., Borovskiy N.Yu. & Ashimov As.A.
Development of parametrical regulation theory on the basis of one class computable general equilibrium models.
Proc. of 12th International Conference on Intelligent Systems and Control, Cambridge, MA, USA, 2009, 212-217.
[8] Makarov V.L., Bakhtizin A.R. & Sulakshin S.S. The application of computable models in state
management, Scientific expert, Moscow, 2007, (in Russian).
[9] Peter M.W at alias, MONASH-MRF: A Multi-sectoral, Multi-regional Model of the Australian
Economy, Monash University, Victoria 3800, Australia, 2001.
[10] Latov Yu.V. Economy outside the law: sketches about theory and history of shadow economy.
Moscow: Moscow public scientific fund. 2001, (in Russian).
[11] Sachs J.D., Larrain F. Macroeconomics in the Global Economy, Prentice-Hall, Inc. 1993.
[12] Schneider F., Enste D. Shadow Economies: Size, Causes and Consequences // Journal of Economic
Literature. 2000. Vol. 8. №1. P. 100.
[13] Strongin R.G., Sergeyev Y.D. Global Optimization with Non-Convex Constraints. Sequential and
Parallel Algorithms. Dordrecht/Boston/London: Kluwer Academic Publishers, 2000.
[14] Evtushenko Yu. G., Malkova V. U., Stanevichyus A. A.. Parallel global optimization of functions of
several variables. Journal of calculus mathematics and mathematical physics. 2009. Vol. 49, No 2, pp. 246-260.
[15] Koplyk I.V., etc. The search of a global extremum of the function set by imitating model. Bulletin
SumGU. "Engineering science" series, №2, 2009, P. 105-112 (in Russian).
[16] Nelder J.A., Mead R. A simplex method for function minimization, The Computer Journal, №7, 1965.
P. 308-313.
9