PARAMETRICAL REGULATION OF ECONOMIC GROWTH BASED ON ONE
COMPUTABLE GENERAL EQUILIBRIUM MODEL TAKING INTO
ACCOUNT NOISE EFFECTS
Abdykappar A. Ashimov*, Bahyt T. Sultanov*, Yuriy V. Borovskiy*, Simon Ya. Serovajsky**, Nikolay Yu. Borovskiy*,
Askar A. Ashimov*, Bakytzhan A. Aisakova*
*The Kazakh National Technical University
**The al-Farabi Kazakh National University
*050013, Almaty, 22 Satpaev str, Republic of Kazakhstan
**050012, Almaty, 39/47 Masanchi str, Republic of Kazakhstan
Email: ashimov37@mail.ru
ABSTRACT
The work presents the application results of the
parametrical regulation theory for a computable general
equilibrium model with additive noise. The application
efficiency of one method of parametrical identification of
the examined computable general equilibrium model of
economic sectors was shown. Optimal (in sense
characterizing economic growth) values of regulated
parameters for the deterministic and stochastic versions of
the model were found.
KEY WORDS
Stochastic modelling, identification,
parametrical
regulation, dynamic system with additive noise.
1. Introduction
The computable general equilibrium model of economic
sectors, designed to study the effects of certain agentsproducers on economic growth, describes important
features of behavior of economic agents and their
interactions on macroeconomic markets and do not take
into account the number of random occurring events, for
example, such as shocks from the supply side (production
and supply of labor), irregularities on the demand side
(preferences,
investment
specificity,
government
spending), the effect of increasing costs or margins
(markups supplements to wages and salaries, risk
premiums), irregularities of monetary circulation (interest
rates, etc.).
In the present work we adopted the assumption that
these and other possible irregularities in the mathematical
model of national economy can be simulated by adding
additive noise to the right sides of the dynamical
equations of the corresponding mathematical model of
economic system.
Also in order to find an effective strategy of
economic growth, we applied methods of the theory of
parametrical regulation of national economy evolution,
effectiveness of which is shown on a class of models in
the form of continuous or discrete dynamic systems [1]-
[4], in particular on models of general equilibrium (so
called CGE-models [5]).
2. Representation of computable general
equilibrium models taking into account the
effect of random influences on economic
growth
The determinate computable general equilibrium model in
the general form is presented by the following system of
relations [5, Ch. 3].
1) The subsystem of difference equations, relating the
values of endogenous variables for two consecutive years:
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝑎), 𝑥(0) = 𝑥0 , (1)
Here 𝑡 = 0, 1, … , 𝑛 − 1 – number of a year, discrete
time; 𝑥̃(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) ∈ 𝑅𝑚 – vector of
endogenous variables of the system;
𝑥(𝑡) = (𝑥 1 (𝑡), 𝑥 2 (𝑡), … , 𝑥 𝑚1 (𝑡)) ∈ 𝑋1 (𝑡),
𝑦(𝑡) = (𝑦1 (𝑡), 𝑦 2 (𝑡), … , 𝑦 𝑚2 (𝑡)) ∈ 𝑋2 (𝑡), (2)
𝑧(𝑡) = (𝑧1 (𝑡), 𝑧 2 (𝑡), … , 𝑧 𝑚3 (𝑡)) ∈ 𝑋3 (𝑡).
Here 𝑥(𝑡) variables include values of fixed assets of
producing sectors, balances in agents’ bank accounts, etc.;
𝑦(𝑡) include agents’ demand and supply values in
different markets, etc.; 𝑧(𝑡) – different types of market
prices and budget shares in markets with state prices for
different economic agents; 𝑚1 + 𝑚2 + 𝑚3 = 𝑚; 𝑢(𝑡) and
𝛼 vectors of exogenous parameters, 𝑢(𝑡) =
(𝑢1 (𝑡), 𝑢2 (𝑡), … , 𝑢𝑞 (𝑡)) ∈ 𝑈(𝑡) ⊂ 𝑅𝑞 – vector of
controlled (regulated) parameters; 𝑋1 (𝑡), 𝑋2 (𝑡), 𝑋3 (𝑡),
𝑈(𝑡), – compact sets with nonempty interiors; 𝑎 =
(𝑎1 , 𝑎2 , … , 𝑎 𝑠 ) ∈ 𝐴 ⊂ 𝑅 𝑠 - vector of unregulated
parameters, 𝐴 – open connected set; 𝑓: 𝑋1 (𝑡) × 𝑋2 (𝑡) ×
𝑋3 (𝑡) × 𝑈(𝑡) × 𝐴 → 𝑅 𝑚1 – continuous mapping for 𝑡 =
0, 1, … , 𝑛 − 1.
2) The subsystem of algebraic equations, describing
behavior and interaction of agents in different markets
during the selected year. These equations allow the
expression of variables 𝑦(𝑡) in terms of exogenous
parameters and the remaining endogenous variables:
𝑦(𝑡) = 𝑔(𝑥(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝑎).
(3)
Here 𝑔: 𝑋1 (𝑡) × 𝑋3 (𝑡) × 𝑈(𝑡) × 𝐴 → 𝑅 𝑚2 – continuous
mapping; 𝑡 = 0, 1, … , 𝑛.
3) The subsystem of recurrent relations for iterative
calculations of equilibrium values of market prices in
different markets and of budget shares in markets with
state prices for different economic agents:
𝑧𝑄+1 (𝑡) = ℎ(𝑦𝑄 (𝑡), 𝑧𝑄 (𝑡), 𝐿, 𝑢(𝑡), 𝑎).
(4)
Here 𝑄 = 0, 1, … – iteration number; 𝐿 – set of positive
numbers (adjustable iteration constants, when their values
decrease the economic system reaches the equilibrium
state faster, but the risk that the price go to the negative
domain increases); ℎ: 𝑋2 (𝑡) × 𝑋3 (𝑡) × (0, +∞)𝑚3 ×
𝑈(𝑡) × 𝐴 → 𝑅 𝑚3 – continuous mapping (which is
contracting when 𝑥(𝑡) ∈ 𝑋1 (𝑡), 𝑢(𝑡) ∈ 𝑈(𝑡), 𝑎 ∈ 𝐴 are
fixed and some fixed 𝐿. In this case ℎ mapping has a
single fixed point, to which the iterative process (3), (4)
converges); t = 0, 1, … , n.
Computable model (1), (3), (4) for given fixed values
of exogenous variables for each moment of t time defines
values of endogenous variables 𝑥̃(𝑡) that correspond to
the demand and supply equilibrium in the markets of
agents’ goods and services in the framework of the
following algorithm.
1) On the first step we assume that 𝑡 = 0 and initial
values of 𝑥(0) variables are set.
2) On the second step for a current we set initial
values of variables 𝑧0 (𝑡) for the current 𝑡 in different
markets and for different agents; with the help of (2) we
calculate values of 𝑦0 (𝑡) = 𝑔(𝑥(𝑡), 𝑧0 (𝑡), 𝑢(𝑡), 𝑎)
(agents’ initial demand and supply values in markets of
goods and services).
3) On the third step for the current t we run the
iterative process (4). Meanwhile for each value of 𝑄
current demand and supply values are defined from (3):
𝑦𝑄 (𝑡) = 𝑔(𝑥(𝑡), 𝑧𝑄 (𝑡), 𝑢(𝑡), 𝑎) through the refinement of
market prices and budget shares of economic agents.
A condition for stopping the iterative process is the
equality of the demand and supply in various markets. As
a result, we determine the equilibrium values of market
prices in every market and of budget shares in markets
with state prices for different economic agents. 𝑄 index
for such equilibrium values of endogenous variables is
omitted.
4) On the next step using obtained equilibrium
solution for t time with the help of the difference
equations (1) we find the values of 𝑥(𝑡) variables for the
next moment of time. A value of 𝑡 is incremented by one.
Transition to the step 2.
The number of iterations of steps 2, 3, 4 is
determined in accordance with the problems of
parametrical identification, forecasting and control for the
pre-selected time intervals.
A stochastic computable general equilibrium model
(stochastic computable model) obtained from the
deterministic model (1), (3), (4), is the model to the
dynamical equation’s (1) right side of which we added an
additive noise 𝜉(𝑡) for 𝑡 = 0, … , 𝑛 − 1:
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝛼) + 𝜉(𝑡), 𝑥(0) = 𝑥0 ,
(5)
that is a model of type (5), (3), (4).
Let us formulate the problem of variational calculus
of the parametrical regulation optimal law synthesis for
the stochastic computable model.
Problem 1. Given the vector of unregulated
parameters 𝑎 ∈ 𝐴 find such law of parametrical
regulation
𝑢(𝑡) ∈ 𝑈(𝑡), 𝑡 = 1, … , 𝑛 ,
so
that
corresponding to it solution of the dynamical system (5),
(3), (4) satisfies the condition for specified t values
𝐄[𝑥̃(𝑡)] ∈ 𝑋1 (𝑡) × 𝑋2 (𝑡) × 𝑋3 (𝑡), 𝑡 = 1, … , 𝑛
(6)
and provides maximum to the functional
𝐾𝑎 = 𝐄{∑𝑛𝑡=1 𝐹𝑡 [𝑥̃(𝑡)]}.
(7)
Here 𝐹𝑡 – some continuous functions, 𝐄 – mathematical
expectation.
3. Computing experiments of finding
optimal values of regulated parameters based
on the parametrical regulation theory
3.1 The results of the parametrical identification of a
deterministic computable model of economic sectors
The considered model according to the statistical data of
the Republic of Kazakhstan is presented by the following
nineteen economic agents (sectors):
Sector № 1. Agriculture, hunting and forestry;
Sector № 2. Fishing and fish breeding;
Sector № 3. Mining industry;
Sector № 4. Manufacturing industry;
Sector № 5. Production and distribution of the
electric power, gas and water;
Sector № 6. Construction;
Sector № 7. Trade, car repairs and repair of products
of house using;
Sector № 8. Hotels and restaurants;
Sector № 9. Transport and communication;
Sector № 10. Financial activity;
Sector № 11. Operations with real estate, rent and
services to enterprises;
Sector № 12. Public administration;
Sector № 13. Education;
Sector № 14. Public health services and social
services;
Sector № 15. Other municipal, social and personal
services;
Sector № 16. Services in housekeeping;
Sector № 17. Aggregate
consumer
including
housekeeping activities;
Sector № 18. Government, represented by central,
regional and local governments and also by non-budget
funds. This sector also involves noncommercial
organizations serving households (political parties, trade
unions, public associations etc.);
Sector № 19. Banking sector, involving Central bank
and commercial banks.
Here the economic sectors № 1 – 16 are producing
agents.
The considered model is presented in the framework
of general expressions of relations (1), (3), (4)
respectively by 𝑚1 = 67, 𝑚2 = 597, 𝑚3 = 34
expressions, with the help of which values of 698
endogenous variables are calculated. This model also
contains 2045 estimable exogenous parameters.
The problem of parametrical identification of the
studied macroeconomic mathematical model consists of
finding estimations of unknown values of its parameters
that enable to reach the minimum value of the objective
function which defines deviations of output variables’
values of the model from the corresponding observed
values (known statistical data). This problem reduces to
finding minimum value of the objective function of
several variables (parameters) in some closed domain Ω
of Euclidian space under the constraints of type (2),
imposed on values of endogenous variables. In the case of
a large dimension of the domain of unknown parameters’
possible values, standard methods of finding extremums
of functions are often ineffective due to the presence of
several local minimums of the objective function. Below,
we propose the algorithm that takes into account the
peculiarities of the problem of parametrical identification
of macroeconomic models and allows circumventing the
problem of “local extremums”.
For the estimation of possible values of exogenous
parameters, as the domain Ω ⊂ 𝑈 × 𝐴 × 𝑋1 we considered
the domain of
𝑞+𝑠+𝑚1
Ω = ∏𝑖=1
[𝑎𝑖 , 𝑏𝑖 ]
type, where [𝑎𝑖 , 𝑏𝑖 ] – the interval of 𝜔𝑖 parameter’s
possible values, 𝑖 = 1 ÷ (𝑞 + 𝑠 + 𝑚1 ). Meanwhile,
estimates 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). Other [𝑎𝑖 , 𝑏𝑖 ] intervals for parameters
searching were chosen with the help of indirect
assessments of their possible values. In computing
experiments the Nelder-Mead [6] algorithm of the
directed search was applied for finding the minimum
values of a continuous function 𝐹: Ω → 𝑅 of several
variables with additional constraints on endogenous
variables of type (2). Application of this algorithm for the
starting point 𝜔1 ∈ Ω can be interpreted as a sequence
{𝜔1 , 𝜔2 , … } which converges to the point of local
minimum 𝜔0 = argmin 𝐹 of function F where
Ω,(2)
𝐹(𝜔𝑗+1 ) ≤ 𝐹(𝜔𝑗 ); 𝜔𝑗 ∈ Ω; 𝑗 = 1,2, …. While describing
the following algorithm let us assume that 𝜔0 point can be
found sufficiently exactly.
To solve the problem of parametrical identification of
the considered computable model based on the apparent
assumption of the mismatch (in general case) of minimum
points of two different functions, two criteria of the
following types were proposed:
𝐾𝐴 (𝜔) = √
1
𝑛𝛼 (𝑡2 −𝑡1 +1)
2
𝐴
∑𝑡𝑡=𝑡
∑𝑛𝑖=1
𝛼𝑖 (
1
2
𝑦 𝑖 (𝑡)−𝑦 𝑖∗ (𝑡)
𝑦 𝑖∗ (𝑡)
) ,
(8)
𝐾𝐵 (𝜔) = √
1
𝑛𝛽(𝑡2 −𝑡1 +1)
2
𝐵
∑𝑡𝑡=𝑡
∑𝑛𝑖=1
𝛽𝑖 (
1
𝑦 𝑖 (𝑡)−𝑦 𝑖∗ (𝑡)
𝑦 𝑖∗ (𝑡)
2
) .
Here {𝑡1 , … , 𝑡2 } – the time span of identification;
𝑦 𝑖 (𝑡), 𝑦 𝑖∗ (𝑡) – calculated and observed values of the
model’s output variables respectively; 𝐾𝐴 (𝜔) – auxiliary
criterion, 𝐾𝐵 (𝜔) – basic criterion; 𝑛𝐵 > 𝑛𝐴 ; 𝛼𝑖 > 0 and
𝛽𝑖 > 0 – some weight coefficients, which values are
determined by solving the problem of parametrical
𝑛𝐴
identification of the dynamic system; ∑𝑖=1
𝛼𝑖 = 𝑛 𝛼 ;
𝑛𝐵
∑𝑖=1 𝛽𝑖 = 𝑛𝛽 .
The algorithm of solution to the problem of
parametrical identification of the model was selected in
the form of the following steps.
1. The problems 𝐴 and 𝐵 (problems of a finding of
the specified minima of functions 𝐾𝐴 (𝜔) and 𝐾𝐵 (𝜔)
accordingly) are solved simultaneously for some vector of
initial values of parameters 𝜔1 ∈ Ω. As a result points 𝜔𝐴0
and 𝜔𝐵0 are found.
2. If 𝐾𝐵 (𝜔𝐵0 ) < 𝜀, then the problem of parametrical
identification of the model (1), (3), (4) is considered to be
solved.
3. Otherwise, the problem 𝐴 is solved taking the point
𝜔𝐵0 as a starting point 𝜔1 and the problem 𝐵 is solved
taking the point 𝜔𝐴0 as a starting point 𝜔1 . Proceed to
step 2.
Sufficiently large number of iterations of steps 1, 2, 3
gives an opportunity for searched values of parameters to
come out from neighborhood of points of non-global
minimums of one criterion with the help of another
criterion and thus to solve the problem of parametrical
identification.
As a result of simultaneous solving of the problems A
and B applying the stated algorithm with the help of the
Nelder-Mead algorithm [6] the values 𝐾𝐴 = 0.015 and
𝐾𝐵 = 0.0063 were obtained. Meanwhile the relative
magnitude of the deviations of calculated values of the
variables used in the basic criteria from the corresponding
observed values was less than 0.63%.
The results of calculation and retrospective
forecasting of the model for 2008 that are partially
presented in Table 1 demonstrate calculated (𝑌, 𝑌𝑔, 𝑃),
observed values and deviations of calculated values of
basic output variables of the model from corresponding
observed values. Here the time interval 2000-2007
corresponds to the period of the parametrical
identification of the model; year of 2008 is the period of
retrospective forecasting, 𝑌 – total output ( × 1012 tenge,
in prices of 2000); 𝑌𝑔 – GDP ( × 1012 tenge, in prices of
2000); 𝑃 – consumer price index in percentage of the
previous year, the sign “*” corresponds to the observed
values, the sign “Δ” corresponds to the deviations (in
percentage) of calculated values from the corresponding
observed values.
3.2 Finding optimal values of regulated parameters
based on the stochastic computable model of economic
sectors
The stochastic computable model of economic sectors
was obtained from the corresponding deterministic model
(with found by solving the problem of parametrical
identification estimate values of exogenous parameters)
by adding discrete Gaussian noise with independent
components to the right sides of all dynamic equations (5)
of the model. Values to the order of magnitude smaller
compared to the values of deterministic parts of the
corresponding dynamic equations were taken as the
standard deviations of Gaussian random variables
defining the noise.
Year
Y*
Y
In solving the stated above problem 1 of parametrical
regulation based on the stochastic computable model of
economic sectors as the optimization criterion we used the
criterion of the following form (7)
1
𝐾1 = 𝐄 { ∑2015
𝑡=2010 𝑌(𝑡)} → max.
6
Here 𝐾1 is mathematical expectation of the average
gross output of the country in prices of 2000 for 20102015. In computing experiments the calculation of the
criterion 𝐾1 was carried out in the following way. By
applying the Monte-Carlo method N realizations of the
random process 𝜉(𝑡) were simulated, after N calculations
of the model for all these simulations that are alternately
used in the equations (5), average values of expressions
1 2015
∑
𝑌(𝑡) on these 𝑁 simulations were taken as the
6 𝑡=2010
value of 𝐾1 criterion. Similarly, the conditions of the type
(6) of endogenous variables belonging to the given
domains of the phase space of the model were checked.
The value of 𝐾1 criterion for the basic computational
variant (applying the values of exogenous parameters,
obtained as a result of the model’s parametrical
identification) equals to 𝐾1 = 0.9891 ∙ 1013 .
During the experiments with the optimization
criterion (9) the constraints on growth of consumer prices
of the following type was used:
𝐄(𝑃(𝑡)) ≤ 1.09𝐄(𝑃̅ (𝑡)), 𝑡 = 2010 ÷ 2015.
Here 𝑃̅(𝑡) – calculated consumer price level of the model
without parametrical regulation, 𝑃(𝑡) – consumer price
level with parametrical regulation.
In the computational experiments the regulation of 1536
𝑗
exogenous parameters was carried out 𝑂𝑖 (𝑡)
(𝑡 = 2010 ÷ 2015; 𝑖, 𝑗 = 1 ÷ 16) – of 𝑗-th agent-
Table 1
Observed, calculated values of output variables of the model and corresponding deviations
2000
2001
2002
2003
2004
2005
2006
2007
2008
5.44
6.32
6.47
6.86
7.72
8.52
9.25
9.69
9.84
5.38
6.32
6.47
6.86
7.72
8.52
9.27
9.64
9.82
-1.22
-0.02
0.00
0.00
0.05
0.08
0.21
-0.51
-0.26
2.45
2.78
3.05
3.36
3.72
4.09
4.55
5.01
5.18
𝑌𝑔
2.47
2.78
3.05
3.35
3.72
4.09
4.55
5.01
5.20
Δ𝑌𝑔
0.88
0.07
-0.04
-0.02
-0.02
-0.02
-0.04
-0.15
0.38
P*
P
106.4
106.6
106.8
106.7
107.5
108.4
118.8
109.5
107.6
106.8
106.9
106.7
107.3
108.2
118.6
109.4
Δ𝑃
1.13
0.18
0.08
-0.05
-0.23
-0.22
-0.24
-0.05
𝛥𝑌
𝑌𝑔
∗
(9)
producer’s budget shares, assigned to purchase of goods
and services, that are produced by 𝑖-th agent-producer for
the years of 2010-2015. Here
16
𝑗
∑ 𝑂𝑖 (𝑡) ≤ 1
𝑖=1
for the specified values of i, j and t. Basic values of the
specified shares, obtained as a result of solution of the
parametrical identification problem according to the data
𝑗
for 2000-2008 are expressed in terms of 𝑂̅𝑖 , 𝑖, 𝑗 = 1 ÷ 16.
The following problem of finding the values of
regulated vectors parameters was considered. Based on
the stochastic computable model of economic sectors find
𝑗
values of agent-producers budget shares (𝑂𝑖 (𝑡); 𝑖, 𝑗 =
1 ÷ 16 ; 𝑡 = 2010 ÷ 2015) that would provide the upper
bound of 𝐾1 criterion under the following additional
constraints on these shares:
𝑗
𝑗
𝑗
0.5𝑂̅𝑖 ≤ 𝑂𝑖 (𝑡) ≤ 2𝑂̅𝑖 ; 𝑖, 𝑗 = 1 ÷ 16; 𝑡 = 2010 ÷ 2015.
The solution of this optimization problem was
obtained with the help of Nelder-Mead algorithm [6].
After application of the parametrical regulation of the
stochastic model’s budget shares, the value of the
criterion turned out to be 𝐾1 = 1.2453 ∙ 1013 its value
increased by 25.89% as compared to the basic variant.
The analogous problem of the parametrical regulation
with corresponding constraints was solved on the basis of
the deterministic CGE model of economic sectors by
applying the criterion 𝐾2 (a deterministic analogue of the
criterion (9)):
1
K 2 = ∑2015
t=2010 Y(t).
6
After application of the parametrical identification of
agent producer’s budget shares the criterion value of the
deterministic model turned out to be equal to 𝐾2 =
1.6283 ⋅ 1013 , the value of the criterion increased by
33.14% as compared to the basic variant.
If to compare the results of the problem of the
variational calculus on the basis of stochastic and
deterministic computable models of general equilibrium,
one can say that there is a reduction in the estimated value
of the functional of the variational problem, taking into
account the disturbing violations in the deterministic
computable general equilibrium model in the form of
additive noise.
4. Conclusion
The paper shows the application efficiency of the
parametrical regulation theory by the example of one
stochastic computable model of economic sectors.
The application effectiveness of the proposed method
of parametrical identification was demonstrated.
The method of optimal values estimation of regulated
parameters of economic policy based on the considered
mathematical model was proposed and estimations of
regulated parameters’ optimal values were found.
The obtained results can be used in the development
and implementation of effective state economic policy.
References
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Borovskiy, D.A. Novikov, R.M. Nizhegorodcev, & As.A.
Ashimov, Macroeconomic analysis and economic policy
based on parametrical regulation (Moscow: Physmatlit,
2010, in Russian).
[2] А.А. Ashimov, N.A. Iskakov, Yu.V. Borovskiy, B.T.
Sultanov, & As.А. Ashimov, Parametrical regulation of
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