Elements of the Parametrical Regulation Theory of
the National Economy Evolution Based on
Computable General Equilibrium Models
Abdykappar A. Ashimov, Bahyt T. Sultanov, Zheksenbek M. Adilov, Yuriy V. Borovskiy, Nikolay Yu. Borovskiy,
Askar A. Ashimov
Parametrical regulation
Kazakh National Technical University
Almaty, Kazakhstan
[email protected], [email protected]
Abstract — The paper presents some results on the
development of the parametrical regulation theory based on
computable general equilibrium (CGE) models, that are discrete
dynamic systems (semi-cascades), and shows the efficiency of
application of this theory. We propose methods of parametrical
identification, assessment of weak structural stability of the
discrete dynamic system and the method of parametrical
regulation of the dynamical system evolution in the form of
formulating and solving the problem of variational calculus. We
present some theorems about the existence and continuous
dependence of the criterion’s optimal value on the parameter of
the stated variational calculus problem. The effectiveness of the
theory of parametrical regulation is illustrated with the help of a
CGE model.
property of invariability of the qualitative image of the
trajectories under small in a certain sense disturbances of the
right-hand part of system. In other words, system must possess
the robustness or structure stability property.
Keywords — discrete dynamic
identification, parametrical regulation.
- The methods for assessing the state of the national
economy and the choice of areas of economic policy;
I.
system,
For the reason of aforesaid, the theory of parametrical
control of the market economics development is proposed in
[3], [4], [5], and [6]. This theory consists of the following
components:
- The methods for forming the set (library) of the
macroeconomic mathematical models. These methods are
oriented to description of various specific socio-economic
situations;
parametrical
- The methods for estimating sustainability indicators and
weak structural stability (robustness) of the mathematical
model;
INTRODUCTION
As is well known, the state implements one of its prime
economic functions, namely, budget and fiscal, as well as
monetary and credit policy, by way of normative establishing
such economic parameters as various tax rates, public
expenses, discount rate, norm of reservation, credit rate,
exchange rate, and others.
- The methods for estimating the values of (the laws of
change of) economic instruments of economic policy in this
direction based on a set of considered models;
- The methods for estimating the weak structural stability
(robustness) of the model with the selected optimal values of
economic instruments;
Nevertheless, the modern economic theory does not have a
unified and clear approach to determining optimal values of the
aforementioned parameters – instruments of state economic
policy with regard to the requirements of the evolution of a
state’s economic system.
- The methods for refinement of limitations of the problems
of choosing the optimal economic policy based on the unstable
(non-robust) mathematical models;
Many of the dynamic systems including the national
economic systems [1] after some transformations can be
described by the systems of nonlinear ordinary differential
equations (continuous dynamic system). As is well known, the
solution (evolution) of the considered system of ordinary
differential equations depends on both the vector of initial
values (x0) and the vector values of controlled (u) and
uncontrolled (λ) parameters. Therefore the result of evolution
(development) of the nonlinear dynamic system with a given
vector of the initial values x0 is defined by the values of the
vectors of both controllable and uncontrollable parameters.
- The methods for study of the effects of external socioeconomic indicators on the solution results of problems of
choosing optimal economic policy;
- The recommendations on the choice of values (the laws of
change) of economic instruments of economic policy in the
selected direction.
This paper contains the results of development of the
parametrical regulation theory and its illustration for the case of
one CGE model.
It is also known [2] that one can judge the object described
by system by its solutions, this system must possess the
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II.
Given fixed values of exogenous parameters for every
moment of time t, the CGE model (1, 3, 4) determines values
of endogenous variables ~x t , corresponding to the equilibrium
of demand and supply prices in the markets of agents’ goods
and services in the framework of the following algorithm.
DESCRIPTION OF CGE MODELS
The CGE model [7, Ch. 3] can be generally expressed by
the following system of relations:
1) The subsystem of difference equations, connecting the
values of endogenous variables for two consecutive years:

xt 1  F ( xt , yt , zt , u, λ) .
The CGE model under consideration of type (1, 3, 4) can be
presented as continuous mapping f : X  W    R n , which
sets conversion of values of endogenous variables of the
system for the null year into corresponding values of the
proceeding year according to the referred above algorithm.
Here the compact X in the phase space of endogenous variables
is determined by the set of possible values of x variables (the
compact X1 with nonempty interior) and corresponding
equilibrium values of y and z variables calculated with the help
of relations (3) and (4).

Here t – number of a year, discrete time, t = 0, 1, 2, …;
~
xt  ( xt , yt , zt )  R n – vector of the systems’ endogenous
variables;

xt  ( xt1 , xt2 ,..., xtn1 )  X 1  yt  ( yt1 , yt2 ,..., ytn2 )  X 2 
zt  ( zt1 , zt2 ,..., ztn3 )  X 3 

It is assumed that for selected point x0  Int( X 1 ),
inclusion xt  f t ( ~
x0 ) X  Int( X 1 ), is true under fixed
Here
; x t variables involve the values of
capital stocks, remained cash in agents’ bank accounts and
other; y t involves 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 λ – vectors of exogenous
parameters; u  (u 1 , u 2 ,..., u l ) W  R l – vector of controlled
(regulated) parameters; X 1 , X 2 , X 3 ,W – compact sets with
nonempty interiors – Int( X i ), i = 1, 2, 3 and Int(W)
1
u  Int(W ), and λ   for t = 0, 1, 2, …,N (N – fixed natural
number). This f mapping determines a discrete dynamic system
(semi-cascade) in the set X:

For the selected u*  Int(W ), points of the corresponding
x0 ) of the semi-cascade is expressed in
trajectory xt  f t ( ~
terms of ~
x .
2) The subsystem of algebraic equations, describing the
behavior and interaction of agents in different markets during
the selected year, these equations allow the expression of
variables in terms of exogenous parameters and remaining
endogenous variables:
*t
III.

ELEMENTS OF THE PARAMETRICAL REGULATION
THEORY FOR THE CLASS OF CGE MODELS
A. Algorithm of CGE Model Identification
In this case the problem of identification (calibration) of
exogenous parameters (u, λ) is reduced to finding the global
minimum in some closed region Ω of some objective function
defined by the CGE model itself. The constraints on the set of
optimization are also given by the model.
Here G : X 1  X 3  W    R n2 – continuous mapping.
3) The subsystem of recurrent relations for iterative
computations of equilibrium values of market prices in
different markets and budget shares in markets with
government prices for different economic agents:
zt [Q  1]  Z ( xt [Q], yt [Q], L, u, λ) 
(5)
Such description of economic system (1), (3), (4), (5) of the
country differs from the description of economic system with
the help of continuous dynamic system [6] and justifies the
necessity of parametrical regulation theory development for the
discrete case of semi-cascade.
respectively; λ  (λ1 , λ 2 ,..., λ m )    R m – vector of
uncontrolled parameters; Λ – open connected set;
F : X 1  X 2  X 3  W    R n1 – continuous mapping.
yt  G( xt , yt , u, λ) 
{ f t , t  0,1,...} .
For the parametrical identification of CGE models based on
the sufficiently admissible evidences of local extrema
mismatch from general arguments ( ω   ) two criterions of
the following type were offered:

Here Q=0, 1, 2, … – number of iteration; L – set of positive
numbers (adjustable constants of iterations). When their values
decrease, economic system reaches the equilibrium level faster,
however the danger that price go to negative domain increases.
Z : X 2  X 3  (0, ) n3  W    R n2 – continuous mapping
(which is contracting for given fixed xt  X 1 , u  W , λ  
and some fixed L). In this case Z mapping has a single fixed
point, where the iteration process (4), (3) converges.
2
K A (ω) 
1 T n A  yti  yti* 
 
 α i 
nαT t 1 i 1  yti* 
K B (ω) 
1 T nB  yti  yti* 
 
 β i 
nβT t 1 i 1  yti* 
2

2

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where y ti , y ti* – calculated and observed values of model’s exit
endogenous
variables respectively;
–
the
K A (ω)
complementary criterion; K B (ω) – the main criterion;
nB  n A ; α i  0 and β i  0 are some weight coefficients,
values of which are determined during the process of parameter
identification for some dynamic system;
nA
nB
i 1
i 1
continuous line L consequently connecting the points
{ f t ( x0 ), t  0,..., T } . One can choose a piecewise linear line
with nodes in the points of the mentioned discrete-time
trajectory of the semi-cascade as such line.
1. The invertibility test for the limitation of the mapping
f: N→N' onto the line L, namely, f: L→f(L). This test reduces to
the ascertainment of the fact that the line f(L) does not have the
points of self-crossing, that is ( x1  x2 )  ( f ( x1 )  f ( x2 )) ;
x1 , x2  L . For instance, one can determine the absence of the
self-crossing points by means of testing monotonicity of the
limitation of the mapping f onto L along any coordinate of the
phase space of the semi-cascade f.
 α i  nα ; βi  nβ .
Algorithm of the parametrical identification of CGE model
has the following type:
Step 1. Based on the considered CGE model A and B
problems of criterion K A (ω) and K B (ω) minimization are
solved simultaneously for some vector ω1   of initial values
of parameters. As a result points ω0K A and ω0KB are found. A
sufficiently small parameter ε is set.
2. The invertibility test for the mapping f in the
neighborhoods of the points of the line L (local invertibility).
On the base of the inverse function theorem, such test can be
carried out as follows. For sufficiently great number of the
chosen points x ∈ L one can estimate the Jacobians of the
mapping
f
using
the
difference
derivations:
 f i

J ( x)  det j ( x)  , i, j=1,…,n. Here i, j are the coordinates
 x

of the vectors, n is the dimension of the phase space of the
dynamic system. If all the obtained estimates of Jacobians are
nonzero and have the same signs, one can make a conclusion
that J(x) ≠ 0 for all x ∈ L and, hence, that the mapping f is
invertible in some neighborhood of each point x ∈ L.
Step 2. If K B (ω0K B )  ε , then the problem of parametrical
identification of CGE model presented by (1), (2), (3), (4) is
solved.
Step 3. Otherwise, based on CGE model, A problem of
criterion minimization is solved taking ω0KB point as initial ω1
point and B problem of criterion minimization is solved taking
ω0K A point as initial ω1 . Transition to the step 2.
Quite large number of iterations of 1, 2, 3 steps provides an
opportunity for desired values to come out from the
neighborhoods of points of non-global minimums of one
criterion with the help of another criterion, thereby the task of
parametrical identification can be solved.
The enlarged algorithm of estimation of weak structural
stability of the discrete-time dynamic system (semi-cascade
defined by the mapping f) with the phase space N ' R n defined
by the continuously differentiable mapping f can be formulated
as follows.
B. Methods of Structural Stability Analysis of Mathematical
Models of National Economic Systems
The analysis of the weak structural stability of CGE models
can be made on the basis of the Robinson theorem [8], which
guarantee the weak structural stability of the cascade, set by
homeomorphism f: N→N' in the compact N in the case of
emptiness of the chain-recurrent set R(f, N).
1. Find the discrete-time trajectory { f t ( x0 ), t  0,..., T } and
line L, in the closed neighborhood N of which it is required to
estimate weak structural stability of the dynamic system.
Let us further propose an algorithm of localization of the
chain-recurrent set for a compact subset of the phase space of
the dynamic system described by the system of ordinary
differential (or difference) and algebraic system. The proposed
algorithm is based on the algorithm of construction of the
symbolic image [9].
3. Estimate (localize) the chain-recurrent set R(f, N). At
that, by virtue of the evident inclusion R( f,N 1 )  R( f,N 2 )
upon N1  N 2  N ' , one can use any parallelepiped belonging
to N' and containing L as the compact set N.
2. Test the invertibility of the mapping f in the
neighborhood of the line L using the algorithm described
above.
4. In the case when R(f,N)=∅, make a conclusion on weak
structural stability of the considered dynamic system in N.
In the case when the considered discrete-time dynamic
system is a priori the semi-cascade f, one should verify
invertibility of the mapping f defined upon N (since in this case
the semi-cascade defined by f is the cascade) before applying
Robinson theorem A [8] for estimating its weak structural
stability.
This enlarged algorithm can be applied also to estimate
weak structural stability of the continuous-time dynamic
system (the flow f), if the trajectory L  { f t ( x0 ), 0  t  T } of
the dynamic system is considered as the line L. In this case,
item 2 of the enlarged algorithm is omitted.
Let us give the numerical algorithm for estimating
invertibility of the differentiable mapping f: N→N' where some
closed neighborhood of the discrete-time trajectory
{ f t ( x0 ), t  0,..., T } in the phase space of the dynamic system
is used as N. Suppose that N contains inside itself the
C. The Method of Parametrical Regulation of the National
Economy Evolution Based on CGE Models
For discrete dynamic systems an estimate of optimal (in
sense of some criterion) values of regulated parameters in a
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given set of values is of practical interest. Let us give the
corresponding formulation of the problem of finding the
optimal values of the criterion and the theorem about the
existence of the solution for this problem.
The model under consideration is represented in the
frameworks of the general expressions of relations (1), (3), (4)
respectively n1 = 67, n2 = 597, n3 = 34 expressions, with the
help of them the values of its 698 endogenous variables are
calculated. This model also contains 2045 estimated exogenous
parameters.
For the trajectory of semi-cascade (5) on the time interval
t=0,…,N, (N – fixed) let us consider the following objective
functional (criterion):
K  K (~
x0 , ~
x1 ,..., ~
xN ) 
As a result of the application of the stated above algorithm
of solving the problems of parametrical identification of CGE
model of sectors of economy, estimation of values of unknown
parameters of models providing the value of the criterion
KB < 0.01 were obtained based on the statistical data of the
Republic of Kazakhstan for the period of 2000 – 2008. This
implies the mean square deviation of calculated values of basic
macroeconomic indicators from corresponding observed values
for indicated models did not exceed 1%.

where K – continuous function in X N 1 .
The statement of problem of finding the optimal value of
the controlled vector of the parameters (the problem of
synthesis of the parametrical control laws) for semi-cascade (5)
is as follows. For the fixed λ ∈ Λ, find the set of N values of the
controlled parameters ut, t=1,…,N that provides the supremum
of the values of criterion (7):
K  sup 
B. Analysis of Structural Stability of the Mathematical CGE
Model of Economic Sectors without Parametrical
Regulation
According to the parametrical identification and calculation
of the model for the period of 2000 – 2015 the trajectory L was
obtained in the phase space X and the compact N⊂X containing
this trajectory was selected.

ut ;t 1,...,N
under the following constraints (for t=0,…,N and some j;
α j  0 ):
~
xt j  ~
x*tj  α j ~
x*tj 
xt  X  ut  W  ~
Based on the above algorithm the reversibility of the
mapping f into N was tested. By applying the numerical
algorithm of estimation of weak structural stability of the
discrete-time dynamic system for the chosen compact set N, we
obtain that the chain-recurrent set R(f,N) is empty. This means
that CGE model of sectors of economy is estimated as weakly
structurally stable in the said compact set N.

The similar problem can be stated also for the case of
minimization of the criterion K.
The following theorem holds true.
C. Parametrical Regulation of Economic Growth Based on
the CGE Model of Economic Sectors
We consider the problem of regulation of the evolution of
the national economy through parametrical regulation of the
annual additional investments from the state budget into the
following six priority sectors (1 – Agriculture, hunting and
forestry; 3 – Mining; 4 – Manufacturing; 5 – Production and
distribution of electricity, gas and water; 6 – Construction;
9 – Transport and communications) totaling 6700 billion of
tenge at current prices during the period 2010 – 2015 (tenge –
the monetary unit in Kazakhstan).
Theorem 1. For mentioned semi-cascade (5) under
constraints (9), there exists the solution to the problem (5), (8),
(9) of finding the supremum of the criterion K.
The proof is based on the existence of the supremum of the
values of the continuous function defined on some compact set.
The following theorem sets the conditions of continuous
dependence of K criterion optimal values of the stated problem
of finding optimal values of regulated parameters on
uncontrolled parameter λ.
Theorem 2. Let functions f and K satisfy the Lipschitz
condition in the domain X×W×Λ. Then the function K=K*(λ),
(K*(λ) is optimal value of the criterion of the problem (5), (8),
(9) for the fixed λ) is continuous in Λ.
Based on the CGE model of economic sectors find values
of additional investments G t j ( j {1,3,4,5,6,9} ; t = 2010,…,
2015) from the state budget into the budgets of mentioned
sectors, which would provide the upper bound of criterion
IV. APPLICATION OF THE PARAMETRICAL REGULATION
THEORY BY THE EXAMPLE OF CGE MODEL OF ECONOMIC
SECTORS
K
A. Results of Parametrical Identification of CGE Model of
Economic Sectors
The model under consideration according to the statistics of
the Republic of Kazakhstan economy is represented by
nineteen economic agents (sectors), among which there are 16
agents-producers, and also the sector – aggregate consumer, the
sector – the government and the banking sector.
1 2015 g
Yt 
6 t 2010

(average GDP for the period of 2010 – 2015 in fixed prices of
2000) under the additional constraint. The total volume of these
investments should not exceed 6700 billion of tenge:
2015

 G  6700 
j
t
t  2010 j{1, 3, 4 , 5 , 6 , 9}
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Estimated consumer price level with parametrical control
(Pt) in this period does not exceed 1.09 of its calculated level of
consumer prices without parametrical regulation ( Pt ):
REFERENCES
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Differential Equations. New York: Springer-Verlag, 1988.
[3] A. A. Ashimov, K. A. Sagadiyev, Yu. V. Borovskiy, N. A. Iskakov, and
As. A. Ashimov, “On the market economy development parametrical
regulation theory,” Kybernetes, The Intl. J. of Cybernetics, Systems and
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models,” Proc. of 19th Intl. Conf. on Systems Engineering ICSEng 2008,
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As. A. Ashimov, “Development of the market economy evolution
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IASTED Intl. Conf. on Modelling, Identification and Control, Innsbruck,
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[6] A. A. Ashimov, B. T. Sultanov, N. A. Iskakov, Yu. V. Borovskiy, and
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Economical System Evolution of a Country. Moscow: Physmathlit, 2009
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Pt  1.09 Pt t = 2010,…, 2015.
As the result of the numerical solution of this problem
applying the Nelder–Mead algorithm [10] the value of the
criterion (10) appeared to be equal to K = 6133. As the result of
the application of the parametrical regulation approach this
value increased by 5.06% as compared to the basic variant
(uniform distribution of the specified amount of additional
investment to 6 priority sectors).
D. Dependence Analysis of a Criterion’s Optimal Values on
Uncontrolled Parameters
In this section we analyzed the dependence of optimal
values of K criterion on values of uncontrolled parameters of
the model by example of bidimensional parameter
λ  (T vad , T p r ) where T va d – VAT rate and T p r – income tax
rate for individuals. The range of variation of these parameters
was determined based on the observed values of T va d and T p r :
Λ = [0.113; 0.15]×[0.225; 0.2875].
The Fig. 1 presents some results of the analysis: the graphs
of dependences of K criterion on bivariate parameter
λ  (T vad , T p r ) (where λ ∈ Λ) for the basic variant and for the
considered above problem of parametrical regulation.
Figure 1. Graphs of dependences of K criterion on bivariate parameter
( – basic variant, – regulation of additional investments, directed to
subsidize 6 priority sectors of economy).
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