Key words – discrete dynamic system (semi-cascade), CGE model, parametrical regulation, parametric identification.
Abdykappar A. ASHIMOV*, Bahyt T. SULTANOV*, Zheksenbek M. ADILOV*,
Yuriy V. BOROVSKIY*, Dauren K. SUISSENBAYEV*, Askar A. ASHIMOV*
PARAMETRICAL REGULATION OF ECONOMIC GROWTH BASED ON THE
CGE MODEL WITH KNOWLEDGE SECTOR
The work presents some results of development of the parametrical regulation theory which take into
consideration peculiarities of computable general equilibrium (CGE) models. The method of parametrical
identification of macroeconomic models with large quantity of evaluation parameters is proposed and tested.
There has been proved a statement on the existence of solution of variations calculus task on finding optimum (in
sense of some criterion) values of controlled parameters in some given set for discrete dynamic system. The
application efficiency of obtained results for a choice and realization of the state economic policy in the sphere of
science, education and innovative activity is shown. The task of finding the optimum, in sense of some criterion,
evolution of state economy development is defined. The optimum values of economic tools on the level of 15
parameters of the examined model are defined.
1. INTRODUCTION
As it is known [24], the evaluation of values of economic tools providing the uniform growth
(dynamic equilibrium), which in its turn enables to reach such economic development, where
demand and supply values increasing over time in macroeconomic markets are always equal to each
other when labour and capital are fully used, is required within the framework of running a
macroeconomic policy. The mentioned above condition for economic tools in some sense is the
requirement for mathematical models, applied for evaluation of rational values of economic tools of
a government policy in the field of economic growth.
Today, the economic growth issue is covered by a large quantity of phenomenological and
econometric models [19].
Based on the basic regression equation for estimation of determinants of economic growth
g  a 0   a l xl   b p z p   c r f r   ,
l
p
r
(where g - economic growth rate of basic indicators of national product (GDP, GNP); a0 – constant;
al – coefficients of economic variables; хl – economic variables; bp – coefficients of additional
*
The Kazakh National Technical University named after K. Satpayev. Address: 22 Satpaev str., 050013, Almaty city,
Kazakhstan. Email: [email protected].
variables; zp – additional variables (political, social, geographical and other); cr – coefficients of
dummy variables; fr – dummy variables reflecting the batch effect; ε – random component), various
econometric models of economic growth dependence on different types of determinants were
obtained for the estimation of wide spectrum of hypothesis, proposals of its influence on economic
growth and econometric dynamic intersectoral models [20; 1], econometric macroeconomic models
[7; 8; 12] which are mainly used for the purpose of forecasting and do not comply with the
requirements stated before. Wide variety of phenomenological models [19], starting from Solow’s
mathematical model of the neoclassical theory [21], Swan’s model [23] (supplemented with
dynamic optimization models based on the inclusion of Ramsey’s problem) ending with such
mathematical models of endogenous economic growth, which for example, present: innovation
production as a good, produced by a special sector of economy (for example the model of Grossman
and Helpman, [10]); activity directed at development of a person himself (for example the model of
Robert Lucas [15]); international trade and technology spreading (for example the model of Lucas
[16]) and other provide answers to the questions on sources of economic growth, but also do not
comply with the stated above requirements for mathematical models for the assessment of rational
values of economic tools of a government policy in the field of economic growth.
Within the framework of balanced models [14; 11] where intersectoral connections are
presented by the system of material balances for some set of products covering all national economy
in the aggregate, it can be noticed that the system of material balances expressing intersectoral
connections is formed without market relationships of agents, and they do not contain descriptions
of such important agents as a government, a banking sector, and an aggregate consumer. Therefore
balanced models comply with the requirements to a lesser extent.
The book [17] presents the set of computable general equilibrium models (CGE models), that
comply better with the stated above requirement for mathematical models, applied for the
assessment of rational values of economic tools of a government policy in the field of economic
growth.
Solving the problems of parametrical identification within the framework of CGE models is
connected with the estimation of a large quantity of values of unknown parameters and variables
(more than one thousand); the efficient algorithm for solving such problems is unknown [22].
Evaluation of efficient values of economic tools in the framework of a government policy in
the field of economic growth requires justification of the parametrical regulation approach (which
was proposed and showed its efficiency for continuous dynamic systems [6]) for the class of
discrete dynamic systems.
The present work presents:
- new algorithm of parametrical identification of macroeconomic models with a large quantity
of estimated parameters;
- results of the parametrical regulation theory development for the class of discrete dynamic
systems;
- results of regulation of economic growth of national economy based on the CGE model
considering constraints imposed on the price level, which enables to take into account requirements
of antiinflationary policy to some degree.
2.
DESCRIPTION OF CGE MODELS
The CGE model [17] can be generally expressed by the system of relations, which can be
divided into subsystems of the following types.
1) The subsystem of difference equations, connecting the values of endogenous variables for
two consecutive years:
xt 1  F ( xt , y t , z t , u,  ) .
(1)
Here t – number of a year, discrete time, t  0,1,2,... ; ~
xt  ( xt , y t , z t )  R n - vector of the
systems’ endogenous variables;
xt  ( xt1 , xt2 ,..., xtn1 )  X 1 , y t  ( y t1 , y t2 ,..., y tn2 )  X 2 , z t  ( z t1 , z t2 ,..., z tn3 )  X 3 .
(2)
Here n1  n2  n3  n ; xt variables involve the values of capital stocks, remained cash in
agents’ bank accounts and other; yt 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  - vectors of exogenous parameters;
u  (u 1 , u 2 ,..., u l ) W  R l - vector of controlled (regulated) parameters; X1, X2, X3, W – compact
Int (W )
sets
with
nonempty
interiors
and
respectively;
Int ( X i), i  1,2,3
  1 , 2 ,..., m    R m - vector of uncontrolled parameters;  - open connected set;
F : X 1  X 2  X 3  W    R n1 - continuous mapping.
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 yt in
terms of exogenous parameters and remaining endogenous variables:
yt  G( xt , zt , u,  ) .
(3)
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 ( zt [Q], yt [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 reaches the equilibrium level faster,
however
the
danger
that
price
go
to
negative
domain
increases.
n3
n3
Z : X 2  X 3  (0,)  W    R - continuous mapping which is contracting for given fixed
xt  X 1 , u W ,    and some fixed L. In this case Z mapping has a single fixed point, where
the iteration process (4, 3) converges.
Given fixed values of exogenous parameters for every moment of time t, the CGE model (1,
xt , corresponding to the equilibrium of demand
3, 4) determines values of endogenous variables ~
and supply prices in the markets of agents’ goods and services in the framework of the following
algorithm.
1) On the first step it is assumed that t=0 and the initial values of x0 variables are set.
2) On the second step the initial values of z t [0] variables are set for the current t in different
markets and for different agents; with the help of (3) the values of y t [0]  G( xt , z t [0], u,  ) are
calculated (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 current values of
demand and supply for every Q are found from (3): yt [Q]  G( xt , z t [Q], u,  ) through the
refinement of market prices and budget shares of economic agents.
The condition for completion of iteration process is the equality of demand and supply values
in different markets. As a result equilibrium values of market prices are determined in every market
and budget shares in markets with government prices for different economic agents. Q index is
omitted for such equilibrium values of endogenous variables.
4) On the next step values of xt 1 variables for the next moment of time are found in
accordance with the obtained equilibrium solution for t moment with the help of difference
equations (1). The value of t increases by unity. Transition to the step 2.
The number of reiteration of steps 2, 3, 4 is defined according to the problems of calibration,
forecasting and regulation for time intervals selected in advance.
3.
AGORITHM OF PARAMETRICAL IDENTIFICATION OF THE ECONOMIC
SYSTEMS MODELS
In this case the problem of identification (calibration) of exogenous parameters of the model
adds up to finding the global minimum of some objective function set by the CGE model itself.
Meanwhile constraints on set of optimizations are also set with the help of the model. The problem
of search of the global extremum in general case of high dimensionality is quite complex, random
search methods, parallel algorithms of calculations and other are applied for its solving [9; 13]. The
overview of numerous publications on searching the global extremum is shown in [22]. The present
work presents unmentioned before in the literature the algorithm of parametrical 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 a large quantity of variables
(more than one thousand). Two objective functions (two identification criterions – main and
complementary) are used in the algorithm; this enables to achieve the exit of values from the
neighbourhood of points of local (and non-global) extremums, continue searching the global
extremum while holding the conditions of corresponding movement to the global extremum.
For the assessment of possible values of exogenous parameters, as a range of   W    X 1
the
range
of
type

l  m  n1
 [ai , bi ]
i 1
was
considered,
where
[ai , bi ]
-
intervals
of
i , i  1  (l  m  n1 ) parameter’s possible values. Meanwhile, the evaluation 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 were
chosen with the help of indirect evaluations of their possible values. The Nelder-Mead [18]
algorithm of the directed search was applied in computing experiments for finding the minimum
values of a continuous function with several F :   R variables with additional constraints on
endogenous variables of type (2). Application of this algorithm for the starting point  1   can be


interpreted as converging to the local minimum  F0  arg min F of F function  1 ,  2 ,  3 ,...
, ( 2)
sequence. Here F ( j 1 )  F ( j ) ,  j  , j  1, 2, ... While describing the following algorithm it
was assumed that  F0 point can be found sufficiently accurately.
For the assessment of the quality of retrospective forecasting based on the data of the
economy of Republic of Kazakhstan for the period of 2000-2008 for some starting point  1   the
problem (problem A) of the model’s parameters assessment and assessment of initial conditions for
difference equations was solved with the help of finding K IA criterion’s minimum, which
characterizes the mean square deviation of calculated values from observed basic macroeconomic
indicators of the model, for example GDP and Consumer Price Index.
Along with the problem A the analogous problem (problem B) is solved by applying extended
criterion K IB instead of K IA criterion for the point  1 . K IB criterion characterizes the mean square
deviation of calculated values from observed ones for a larger quantity of endogenous variables than
in K IA criterion. Along with the variables of K IA criterion, variables of producing sectors of the
CGE model (for example: capital funds, GVA, and number of employees of three producing
sectors) are added to this K IB criterion.
The task of parametrical identification for the model (1), (3), (4) is assumed to be solved, if
point  0   is found where K IB ( K0 IA )   for sufficiently small  .
While solving the task of parametrical identification for each of these criterions separately,
due to the existence of several local minimums of functions K IA and K IB , it was quite complex to
achieve values of these criterions that are sufficiently close to zero.
Therefore the final algorithm of solving the task of parametrical identification of the model
was chosen with the help of the following steps.
1. A and B problems are solved simultaneously for some vector of initial values of  1  
parameters. As a result points  K0 IA and  K0 IB are found.
2. If K IA ( K0 IB )   , then the problem of parametrical identification of the model (1, 3, 4) is
solved.
3. Otherwise, the problem A is solved taking  K0 IB point as initial  1 point and the problem 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 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.
4.
DEVELOPMENT OF THE PARAMETRICAL REGULATION THEORY FOR THE
CLASS OF CGE MODELS OF THE TYPE (1, 3, 4)
Elements of the theory of effective parametrical regulation of market economy development
described by the system of ordinary differential and algebraic equations are presented in [3, 6].
Application efficiency of the parametrical regulation approach based on the set of models is shown
[2, 4, 5].
Within the framework of the proposed approach, optimal (in sense of some criterion) values
of parameters were found with the help of the family of functions dependent on endogenous
indicators of the mathematical model and adjustable coefficients.
The summary of elements of the parametrical regulation theory with consideration of
peculiarities of determining CGE models is presented below.
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).
t
x 0 )  Int ( X 1 ) is true
It is assumed that for selected point x 0  Int ( X 1 ) inclusion x t  f ( ~
X1
under fixed u  Int (W ) and    for t  0  N (N – fixed natural number). This f mapping
determines a discrete dynamic system (semi-cascade) in the set X.
f
t

, t  0, 1,...
(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.
xt  f t ( ~
x 0 ) of the semiFor the selected u*  Int (W ) points of the corresponding trajectory ~
cascade is expressed in terms of ~x . Let us denote the closed set in the R ( n l )( N 1) space ((N+1) set
*t
of variables ( ~
xt , ut) for t  0  N ), determined by constraints
~
xt  X , u t W , ~
xt j  ~
x*tj   j ~
x*tj ,
(6)
by  . The last inequalities in (6) are used for some values of j  1  n , and when x*jt are positive,
 j 0.
For the assessment of efficiency of economic system evolution during the period t  0  N ,
(N is fixed) the criterion of type K  K ( ~
x0 , ~
x1 ,..., ~
x N ) is used, where K – continuous function in
N+1
X .
The statement of the problem of finding the optimal value of the controlled vector of
parameters for the semi-cascade (5) is of the following type. Given fixed    , find the set from N
values of controlled parameters ut , t  1  N which provides the lower boundary of the criterion’s
values (7) –
K  inf
ut , t 1 N
(7)
under constraints (6). The analogous problem is set for the case maximization of K criterion.
The following theorem is true.
Theorem. For the indicated semi-cascade (5) under constraints (6) the solution of the
problem (5-7) of finding the lower boundary of K criterion exists.
The proof of this theorem with some alterations, that take into account special features of
CGE models, repeats the proof of the theorem for the case of the flow [6]. The proof is based on the
existence of the upper margin of values of continuous function determined in some compact.
5.
EXAMPLES
5.1
PARAMETRICAL IDENTIFICATION OF THE CGE MODEL WITH KNOWLEDGE
SECTOR
The considered model is presented by the following nineteen economic agents (sectors)
Sector № 1 — sector of science and education (knowledge), which provides educational and
knowledge production services.
Sector № 2 — innovative sector, representing the set of innovative-active enterprises and
organizations.
Sector № 3 — other sectors of economy.
Sector № 4 — aggregate consumer, uniting households.
Sector № 5 — government.
Sector № 6 — banking sector.
This model is presented within: relations (1) by twelve expressions ( n1  12 ); relations (3) by
88 expressions ( n2  88 ); relations (4) by ten expressions ( n3  10 ). The examined model contains
86 exogenous parameters and 110 endogenous variables.
The estimations of unknown values of the model’s 86 parameters, providing the value of the
criterion KIB  0.0073 , were obtained as a result of the application of the presented above
algorithm for solving the problems of parametrical identification of CGE model with knowledge
sector based on the statistical data of the Republic of Kazakhstan for the period of 2000-2008. This
implies that the mean square deviation of calculated values of the basic macroeconomic indicators
from corresponding observed values for the indicated model did not exceed 1%.
5.2
FINDING OPTIMAL VALUES OF CONTROLLED PARAMETERS BASED ON THE CGE
MODEL WITH KNOWLEDGE SECTOR
There has been made the forecast of the main macroeconomic factors for the period of 2010 2014, which shows that from 2008 to 2014 GDP of the country increases by 10.4% in real terms
from 5.26 to 5.81 trillions tenge (tenge – national currency of Kazakhstan, hereinafter in prices of
the year of 2000).
The distribution of additional investments in 2010-2014 over three producing sectors with
total volume of 6.7 trillions tenge was considered during the following computer experiments.
In the case of the scenario realization (the uniform distribution of the stated volume of
investments over 5 years (2010-2014) and 3 producing sectors) the GDP of the country will increase
by 22% in real terms to 6.43 trillions tenge.
In computing experiments K criterion (value of GDP for 2014 in prices of 2000) was used as
the optimization criterion: K  max . For the basic calculated variant (with the application of
exogenous parameters’ values, obtained as a result of the parametrical identification of the model)
the value of this criterion is equal to 5.81 trillions tenge.
During all experiments with the optimization criterion K constraints of the following type
imposed on the growth of consumer price level were added to the constraints of type (2):
Pt  1.09 Pt , t  2010  2014.
(8)
Here Pt and Pt are estimated levels of consumer prices without and with parametrical regulation
respectively.
The following problem of finding controlled vectors’ optimal values was considered.
Based on the CGE model with knowledge sector find values of additional investments
( G j .t ; j  1, 2, 3; t  2010  2014 ) assigned to the sectors with numbers j  1, 2, 3 during the five
year period for t  2010  2014 , which would provide the upper boundary of K criterion under
additional constraints of the following type
2014
3
  G j.t  6.7  1012 .
t  2010 j 1
The solution of this problem was obtained with the help of the Nelder-Mead algorithm [18].
After the application of parametrical regulation of values of stated additional investments, the
value of K criterion increases by 7.6% as compared to the scenario variant. In other words, at the
optimal distribution of investments with total volume of 6.7 trillions tenge the value of K criterion
rises from 6.43 trillions to 6.92 trillions tenge as compared with the scenario providing the uniform
distribution of the same volume of investments (see figure 1).
Fig. 1. Graphs of the calculated values of GDP (
- uniform distribution of additional investments,
investments).
- basic variant (without additional investments),
- optimal distribution of additional
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