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Parametrical Regulation of Economic Growth
Based on the Stochastic Computable Model of
General Equilibrium with Knowledge Sector
Abdykappar A. Ashimov, Bahyt T. Sultanov, Zheksenbek M. Adilov, Yuriy V. Borovskiy, Dauren K. Suissenbayev,
Askar A. Ashimov
Parametrical regulation
Kazakh National Technical University
Almaty, Kazakhstan
[email protected], [email protected]
Abstract — The paper aims to present one new method of
parametrical identification, the results of development of the
parametrical regulation theory and to show the application
efficiency of the proposed method of identification and
theoretical results for solving the respective problems of
application. The paper describes the proposed method of
parametrical identification, we formulated and proved the
theorems about the conditions for the existence of one of the
problems of variational calculus on the synthesis of the optimal
law of parametrical regulation and about the conditions for
continuous dependence of optimal values of the criterion of the
given problem on unregulated parameters applying the
mathematical objects of a discrete stochastic control system. The
problems of parametrical identification of the calculated general
equilibrium model with knowledge sector were solved effectively.
Also we show the efficiency of determining the optimal values (in
sense of the criterion, that characterizes economic growth) of the
parameters of deterministic and stochastic versions of the
computable general equilibrium model with knowledge sector.
dynamic equations of a corresponding mathematical model of
economic system.
Keywords — stochastic control, modelling and simulation,
identification and estimation, theory of parametrical regulation
- the theorem about sufficient conditions of continuous
dependence of a criterion’s optimal values of a problem of
variational calculus on synthesis of regulation optimal laws on
values of uncontrolled parameters.
I.
INTRODUCTION
As it is known [7], presented in the literature models of
national economy reflect in the mathematical form important
properties of economic system and do not account for a set of
appearing shocking effects, for example as irregularities from
the aspect of supply (productivity and supply of labor),
irregularities on the demand side (preferences, investment
specificity, government spending), effect of increasing costs or
margins (markups, salary supplements, risk premium) and
others.
According to the theory of business cycles, factors that
define economic growth and business cycles are factors of the
same nature [8]. Therefore the study of economic growth
against the background of business cycles should be conducted
on the basis of one mathematical model. In the present work
the study of economic growth against the background of
business cycles the following assumption was made:
mentioned above and other possible irregularities in a
deterministic mathematical model of national economy can be
approximated by adding additive noise to the right sides of
In this paper the theory of parametrical regulation of
national economy evolution (efficiency of which was shown on
the class of models as a continuous or discrete dynamic
systems [1], [2], [3], [4]) develops to the class of discrete
dynamic systems with additive noise, the important subclass of
which are computable models of general equilibrium (so called
CGE-models [5]) with additive noise. In particular, in the
framework of development of the parametrical regulation
theory in this paper for discrete dynamical systems with
additive noise theorems about sufficient conditions were
formulated and proved:
- the theorem about sufficient conditions of a solution
existence to the problem of variational calculus on a choice of
the system parameters’ optimal values in the given set of their
values (on synthesis of optimal laws of regulation).
Fulfillment of the conditions of solution existence of the
stated problem of variational calculus guarantees, in particular,
finiteness of the expectation of the phase trajectories process on
a finite time interval.
In this paper the discrete stochastic model, obtained from
deterministic computable model of general equilibrium of
economic sectors of Makarov [5] by adding additive noise to
the right sides of the model’s dynamical equations, was
considered as an example of theoretical results applications.
The problem of parametrical identification of the researched
model was solved based on statistical data on economy
evolution of the Republic of Kazakhstan. The problem of
optimal (in sense of some criterion) economic growth of
national economy of the Republic of Kazakhstan was
formulated and solved applying methods of the parametrical
regulation theory based on deterministic and stochastic variants
of a computable model of general equilibrium with knowledge
sector.
II. ELEMENTS OF THE PARAMETRICAL REGULATION
THEORY BASED ON THE DISCRETE DYNAMIC SYSTEM WITH
ADDITIVE NOISE
We consider discrete stochastic regulated system
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑎) + ξ(𝑡), 𝑡 = 0, … , 𝑛 − 1, 

𝑥(0) = 𝑥0 

Here 𝑡 – time, taking nonnegative integer values; 𝑥 = 𝑥(𝑡) ⊂
𝑅𝑚 – function of the system (1), random vector-function of
discrete argument (a vector random process); 𝑢 = 𝑢(𝑡) ⊂ 𝑅𝑞
– regulation, vector-function of discrete argument; 𝑎 –
deterministic vector of uncontrollable parameters, 𝑎 ∈ 𝐴; 𝐴 –
given set, 𝐴 ⊂ 𝑅 𝑠 ; ξ = ξ(𝑡) = (ξ1 (𝑡), … , ξ𝑚 (𝑡)) – known
vector random process, which expresses the noise (as such may
be, for example, an additive Gaussian white noise); 𝑓 – a
known vector-function of its arguments; 𝑥0 – initial state of the
system, a deterministic vector.
We define the optimality criterion to be the maximization
for 𝑎 fixed:
𝐾𝑎 = 𝐄{∑𝑛𝑡=1 𝐹𝑡 [𝑥(𝑡)]}

Here 𝐹𝑡 – are known functions; 𝐄 – mathematical expectation,
𝑥(𝑡) – a solution to the system (1), (2) for a given 𝑎.
We introduce the phase constraints for the system (1), (2):
𝐄[𝑥(𝑡)] ∈ 𝑋(𝑡), 𝑡 = 1, … , 𝑛

where 𝑋(𝑡) – given set. Hereinafter the mathematical
expectation of a vector chance quantity means the vector of
mathematical expectations of the origin of this value.
where 𝑥(𝑡) is the solution to the system (1), (2) corresponding
to the regulation 𝑢(𝑡) and set 𝑎 parameter value. Then the
Problem 1 is reduced to maximizing the functional 𝐾 = 𝐾𝑎 (𝑢)
which is defined by the formula (3), on the set 𝑈𝑎 of allowable
regulations of the system. We call the Problem 1 non-trivial, if
the set 𝑈𝑎 is non-empty and contains some open set.
The theorem on sufficient conditions for existence of
solution to the Problem 1 (set in accordance with the method of
parametrical regulation) and the theorem of continuous
dependence of the optimal value of 𝐾𝑎 criterion of the Problem
1 on the parameter 𝑎 are presented below.
Theorem 1. Suppose given fixed 𝑎 ∈ 𝐴 in non-trivial
Problem 1 for any 𝑡 = 1, … , 𝑛 chance quantity ξ(𝑡) are
absolutely continuous and have zero mathematical
expectations, functions 𝑓 and 𝐹𝑡 satisfy the Lipschitz condition.
Functions 𝑓 (for 𝑢 ∈ 𝑈𝑎 ) and 𝐹𝑡 by module are constrained by
some linear functions of |𝑥 𝑖 | variables (𝑥 𝑖 – coordinate of x
vector). Then the Problem 1 has a solution.
Theorem 2. Suppose when 𝑎 ∈ 𝐴 and 𝑡 = 1, … , 𝑛 for nontrivial Problem 1 chance quantity ξ(𝑡) are absolutely
continuous and have zero mathematical expectations, functions
𝑓 and 𝐹𝑡 satisfy the Lipschitz condition. Functions 𝑓 and 𝐹𝑡 do
not exceed a linear function with respect to |𝑥 𝑖 | function in
some neighborhood of a point 𝑎 . Then the mapping 𝑎 →
max 𝐾𝑎 (𝑢) is continuous in 𝐴.
𝑢∈𝑈𝑎
Proofs of Theorems 1 and 2 are presented in Appendix.
III. REPRESENTATION OF COMPUTABLE GENERAL
EQUILIBRIUM MODELS IN THE FORM OF DISCRETE STOCHASTIC
DYNAMIC SYSTEMS AND THE PROBLEM OF PARAMETRICAL
REGULATION OF THE NATIONAL ECONOMY EVOLUTION ON THE
BASIS OF COMPUTABLE MODELS
In the below considered tasks, explicit constraints on
regulation are assumed:
Deterministic computable general equilibrium model in
general form is represented by the following system of
relations [5, Ch. 3].

1) The subsystem of difference equations, binding the
values of endogenous variables for two consecutive years:
where 𝑈(𝑡) – given set, 𝑈(𝑡) ⊂ 𝑅𝑞 . Sets 𝑋(𝑡) and 𝑈(𝑡) for
all determined above 𝑡 values are the closures of limited open
sets.
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝑎), 𝑥(0) = 𝑥0  
𝑢(𝑡) ∈ 𝑈(𝑡), 𝑡 = 0, … , 𝑛 − 1
The method of parametrical regulation is used in the
formulation and solution of the following variational problem,
so called the problem of variational calculus of the synthesis of
the optimum law of parametrical regulation.
Problem 1. Given the vector of unregulated parameters
𝑎 ∈ 𝐴 find such regulation 𝑢, that satisfies the condition (5), so
that corresponding to it the solution of dynamical system (1),
(2) satisfies the condition (4) and provides the maximum of the
functional (3).
For the fixed 𝑎 ∈ 𝐴 let us define the set of allowable
regulations for the system (1), (2) in the following manner:
𝑈𝑎 = {𝑢|𝑢(𝑡) ∈ 𝑈(𝑡), 𝑡 = 0, … , (𝑛 − 1); 𝐄[𝑥(𝑡)] ∈ 𝑋(𝑡), 𝑡
= 1, … , 𝑛}
Here 𝑡 – year number, discrete time, 𝑡 = 0, … , 𝑛 − 1; 𝑥̃(𝑡) =
(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) ∈ 𝑅𝑚 – vector of the system’s endogenous
variables;
𝑥(𝑡) ∈ 𝑋1 (𝑡) ⊂ 𝑅𝑚1 ; 𝑦(𝑡) ∈ 𝑋2 (𝑡) ⊂ 𝑅𝑚2 ;
𝑧(𝑡) ∈ 𝑋3 (𝑡) ⊂ 𝑅𝑚3 

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 values of market prices and budget shares in
the markets with state prices for different economic agents;
𝑢(𝑡) and 𝑎 are vectors of exogenous parameters, 𝑢(t) ∈
𝑈(𝑡) ⊂ 𝑅𝑞 – vector of controlled (regulated) parameters;
𝑋1 (𝑡), 𝑋2 (𝑡), 𝑋3 (𝑡), 𝑈(𝑡) – compact sets with nonempty
interior; 𝑚1 + 𝑚2 + 𝑚3 = 𝑚 ; 𝑎 ∈ 𝐴 ⊂ 𝑅 𝑠 – vector of
uncontrolled parameters; 𝐴 – open connected set; 𝑓: 𝑋1 (𝑡) ×
𝑋2 (𝑡) × 𝑋3 (𝑡) × 𝑈(𝑡) × 𝐴 → 𝑅𝑚1 – continuous mapping for
𝑡 = 0, … , 𝑛.
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).
Here 𝑔: 𝑋1 (𝑡) × 𝑋3 (𝑡) × 𝑈(𝑡) × 𝐴 → 𝑅𝑚2
mapping; 𝑡 = 0, … , 𝑛.
–
continuous
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
the different economic agents:
𝑧(𝑡)[𝑄 + 1] = ℎ(𝑦(𝑡)[𝑄], 𝑧(𝑡)[𝑄], 𝐿, 𝑢(𝑡), 𝑎).

Here 𝑄 = 0, 1, 2, … – 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 →
𝑚3
𝑅
– 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 (8), (9) converges); 𝑡 = 0, … , 𝑛.
Computable model (6), (8), (9) 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 algorithm below.
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 (8) the values of 𝑦(𝑡)[0] =
𝑔(𝑥(𝑡), 𝑧(𝑡)[0], 𝑢(𝑡), 𝑎) are calculated (initial values of
demand and supply of agents in markets of goods and
services).
3) On the third step the iteration process (9) is run for
current 𝑡. Meanwhile current values of demand and supply for
every
𝑄
are
found
from
(8):
𝑦(𝑡)[𝑄] =
𝑔(𝑥(𝑡), 𝑧(𝑡)[𝑄], 𝑢(𝑡), 𝑎) 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 𝑥(𝑡 + 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 (6). 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.
Stochastic computable general equilibrium model
(stochastic computable model) obtained from the deterministic
model (6), (8), (9), is a model to the dynamical equation’s (6)
right side of which we added an additive noise ξ(𝑡):
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝑎) + ξ(𝑡),
𝑡 = 0, … , 𝑛 − 1; 𝑥(0) = 𝑥0 

i.e., the model of type (10), (8), (9).
We formulate the problem of variational calculus on the
synthesis of the parametrical regulation optimal law for a
stochastic computable model.
Problem 2. Given the vector of unregulated parameters
𝑎 ∈ 𝐴 find such regulation law 𝑢(𝑡) , which satisfies the
condition (5), so that the corresponding to it the condition of
the dynamical system (10), (8), (9) satisfies the following
condition
𝐄[𝑥(t)] ∈ 𝑋1 (𝑡) × 𝑋2 (𝑡) × 𝑋3 (𝑡), 𝑡 = 1, … , 𝑛

and provides maximum to the functional
𝐾𝑎 = 𝐄{∑𝑛𝑡=1 𝐹𝑡 [𝑥(t)]}.

It is not difficult to check, that formulated above Theorems
1 and 2 hold true for the computable model with the continuous
mappings 𝑓, 𝑔, ℎ and with convergent iterative process (8), (9).
IV. COMPUTING EXPERIMENTS OF FINDING OPTIMAL
VALUES OF REGULATED PARAMETERS BASED ON THE
PARAMETRICAL REGULATION THEORY
A. The Results of the Parametrical Identification of a
Deterministic Computable Model with Knowledge Sector
The considered model is presented by the following six
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.
Here the economic sectors № 1 – 3 are producing agents.
This model is presented within: relations (6) by twelve
expressions (𝑚1 = 12); relations (8) by 88 expressions (𝑚2 =
88); relations (9) by ten expressions (𝑚3 = 10). The examined
model contains 86 exogenous parameters and 110 endogenous
variables.
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 (7), 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
𝑞+𝑠+𝑚
domain of Ω = ∏𝑖=1 1[𝑎𝑖 , 𝑏 𝑖 ] type, where [𝑎𝑖 , 𝑏 𝑖 ] – 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 , 𝑏 𝑖 ] intervals with centers
in corresponding observed values (in case if there is one such
value). Other [𝑎𝑖 , 𝑏 𝑖 ] intervals for parameters searching were
chosen with the help of indirect assessments of their possible
values. In computing experiments the Nelder–Mead algorithm
[6] 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 (7).
Application of this algorithm for the starting point ω1 ∈ Ω can
TABLE I.
Year
𝑌∗
be interpreted as a sequence {ω1 , ω2 , … } which converges to
the point of local minimum ω0 = argmin 𝐹 of function 𝐹
Ω,(7)
where 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)
𝐾𝐵 (ω) = √
1
𝑛β (𝑡2 −𝑡1 +1)
2
𝐴
∑𝑡𝑡=𝑡
∑𝑛𝑖=1
α𝑖 (
1
𝑦 𝑖 (𝑡)−𝑦 𝑖∗ (𝑡)
2
𝐵
∑𝑡𝑡=𝑡
∑𝑛𝑖=1
β𝑖 (
1
𝑦 𝑖∗ (𝑡)
𝑦 𝑖 (𝑡)−𝑦 𝑖∗ (𝑡)
𝑦 𝑖∗ (𝑡)
2001
2002
2003
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.
2004
2.60
2.95
3.24
3.54
3.88
𝑌
2.60
2.95
3.24
3.54
Δ𝑌
-0.04
0.00
0.04
0.03
𝑌1∗
0.107
0.102
0.116
𝑌1
0.109
0.102
Δ𝑌1
1.80
-0.76
𝑌2∗
0.033
𝑌2
2005
2006
2007
2008
4.26
4.72
5.14
3.88
4.26
4.72
5.15
5.26
0.04
-0.01
0.07
0.26
-0.73
0.125
0.134
0.157
0.164
0.165
0.200
0.116
0.126
0.134
0.158
0.164
0.164
0.199
-0.01
1.32
0.46
0.49
-0.02
-0.59
-0.34
0.0364
0.041
0.045
0.050
0.049
0.068
0.072
0.061
0.041
0.0343
0.039
0.044
0.050
0.049
0.067
0.071
0.061
Δ𝑌2
25.00
-5.86
-4.73
-2.65
-0.57
-0.19
-0.13
-0.88
0.16
𝑌3∗
2.23
2.46
2.80
3.07
3.36
3.68
4.03
4.48
4.88
𝑌3
1.9
2.463
2.8
3.07
3.36
3.68
4.03
4.48
4.89
Δ𝑌3
𝑃∗
-14.63
𝑃
Δ𝑃
0.00
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 identification of the dynamic system;
𝑛𝐵
𝐴
∑𝑛𝑖=1
α𝑖 = 𝑛α ; ∑𝑖=1
β𝑖 = 𝑛β .
OBSERVED, CALCULATED VALUES OF OUTPUT VARIABLES OF THE MODEL AND CORRESPONDING DEVIATIONS
2000
2
) 
5.30
0.08
0.07
0.03
0.02
0.02
0.00
0.11
0.28
106.40
106.60
106.80
106.70
107.50
108.40
118.80
109.50
106.57
106.81
106.95
106.83
107.64
108.50
118.90
109.50
0.16
0.20
0.14
0.12
0.13
0.09
0.09
0.00
2. If 𝐾𝐵 (ω𝐵0 ) < ε , then the problem of parametrical
identification of the model (6), (8), (9) 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 value 𝐾𝐵 = 0.0073 was 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 0.73%.
The results of calculation and retrospective forecasting of
the model for 2008 that are partially presented in table 1
demonstrate calculated (𝑌, 𝑌1 , 𝑌2 , 𝑌3 , 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 – gross domestic product
( × 1012 tenge, in prices of 2000); 𝑌𝑖 – gross value added of
i-th sector ( × 1012 tenge, in prices of 2000; 𝑖 = 1,2,3); 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.
B. Finding Optimal Values of Regulated Parameters Based
on the Stochastic Computable Model with Knowledge
Sector
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 a discrete
Gaussian noise with independent components to the right side
of all dynamical equations (6) of the model. Such equations
involve the equations for calculating the following endogenous
variables:
- gross added values (𝑌1 , 𝑌2 , 𝑌3 ) of three producing sectors
with the help of the corresponding productivity functions;
- capital funds (𝐾1 , 𝐾2 , 𝐾3 ) of three producing sectors;
- annual budgets (𝐵1 , 𝐵2 , … , 𝐵5 ) of sectors 1 – 5.
Additive noises added to 𝑌𝑖 expressions can initiate
corresponding cyclical fluctuations, caused by sudden shifts
(shocks) in the development of technological progress and
random changes in population growth. Additive noises added
to 𝐾𝑖 expressions characterize random changes in budget shares
of agents-producers assigned for a purchase of investment
goods, and a random character of retirement funds coefficients.
Additive noises added to 𝐵𝑖 expressions describe a random
character of income, obtained by a sector in a current period.
In this study of estimates of the average squared deviations
of the generated Gaussian random variables defining the
specified noise were obtained on the basis of the analysis of
relevant statistical data of the Republic of Kazakhstan economy
development for 2000 – 2008 in the following way. For each
time series of observed values of the stated above variables we
calculated sample mean square deviation of the differences
between observed values and the trend of these values. Found
in this way values were taken as estimates of standard
deviations of components of generated discrete Gaussian noise
added to the right side of the above 11 dynamic equations.
In solving the stated above Problem 2 of parametrical
regulation based on the stochastic computable model with
knowledge sector as the optimization criterion we used the
following criterion of the form (12)
1
𝐾𝑠 = 𝐄 [ ∑2015
𝑡=2010 𝑌(𝑡)] → max
6

Here 𝐾𝑠 is mathematical expectation of the average GDP of the
country in prices of 2000 for 2010 – 2015. In computing
experiments the calculation of the criterion 𝐾𝑠 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 (6), average values of
1
expressions ∑2015
𝑌(𝑡) on these 𝑁 simulations were taken
6 𝑡=2010
as the value of 𝐾𝑠 criterion. Similarly, the conditions of the
type (11) of endogenous variables belonging to the given
domains of the phase space of the model were checked.
During the experiments with the optimization criterion (14)
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 computational experiments we made regulation of 18
controlled parameters 𝑂𝑖 (𝑡) (𝑡 = 2010, … ,2015; 𝑖 = 1, 2, 3) –
consolidated budget shares assigned to subsidize the three
agents-producers in 2010 – 2015. In this case the natural
limitation must hold: the sum of the consolidated budget’s all
eight shares used in the model (including stated above shares
𝑂𝑖 (𝑡); 𝑖 = 1, 2, 3) must not exceed unity:
∑8𝑖=1 𝑂𝑖 (𝑡) ≤ 1𝑡 = 2010, … ,2015

The following problem of finding controlled parameters’
optimal values was considered. On the basis of a stochastic
computable model with knowledge sector find 𝑂𝑖 (𝑡) values of
consolidated budget’s shares assigned to three agentsproducers, which would provide an upper bound of the
criterion with additional constraints (16) on these shares.
The solution to this optimization problem was obtained
with the help of Nelder–Mead algorithm [6]. After application
of the stochastic model’s parametrical regulation of budget
shares, the value of the criterion turned to be 𝐾𝑠 =
7.2341012 its value increased by 24.93% compared to the
basic variant 𝐾𝑠 = 5.791012 .
An analogous problem of parametrical regulation with
corresponding constraints was solved on the basis of a
deterministic CGE model with knowledge sector applying 𝐾𝑑
criterion (deterministic analogue of the criterion (15)):
1
𝐾𝑑 = ∑2015
𝑡=2010 𝑌(𝑡)
6
After application of parametrical regulation of consolidated
budget’s shares assigned to subsidize the three agentproducers, the value of the criterion of the deterministic model
turned out to be 𝐾𝑑 = 9.23 ⋅ 1012 , 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.
V.
CONCLUSION
1. The paper presents the results on development of the
parametrical regulation theory for the class of discrete
stochastic dynamical systems with additive noise, and the
application efficiency of the obtained results was shown by the
example of one stochastic computable model with knowledge
sector.
2. The effectiveness of the proposed method of
parametrical identification of the deterministic model was
demonstrated.
3. The method of estimating the optimal values of
economic policy controlled parameters was proposed based on
stochastic and deterministic variants of the computable model
with knowledge sector and the corresponding estimates of the
optimal control parameters were found.
The obtained results can be used in the development and
implementation of effective state economic policy.
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[1]
[2]
[3]
[4]
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Novikov, R. V. Nizhegorodcev, and As. A. Ashimov, Macroeconomic
Analysis and Economic Policy Based on Parametrical Regulation.
Moscow: Physmatlit, 2010 (in Russian).
А. А. Ashimov, N. A. Iskakov, Yu. V. Borovskiy, B. T. Sultanov, and
As. А. Ashimov, “Parametrical regulation of economic growth on the
basis of one-class mathematical models,” Systems Science, vol. 35, no.
1, 2009, pp. 57–63.
A. A. Ashimov, K. A. Sagadiyev, Yu. V. Borovskiy, N. A. Iskakov, and
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parametrical regulation theory,” Proc. of the 9th IASTED International
Conf. on Control and Applications, 2007 – Montreal, Quebec, Canada,
pp. 296–301.
A. A. Ashimov, K. A. Sagadiyev, Yu. V. Borovskiy, N. A. Iskakov, and
As. A. Ashimov, “On the market economy development parametrical
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systems and management sciences, vol. 37, no. 5, 2008, pp. 623–636.
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A. A. Samarskyi and A. P. Mikhailov, Mathematical Modeling: Ideas.
Methods. Examples. Moscow: Physmatlit, 2002.
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APPENDIX
Proof of Theorem 1. According to the Weierstrass
theorem, a continuous function on a nonempty closed limited
set reaches its maximum. Thus, we need to show that a defined
with the help of (3) the function of many variables 𝐾 = 𝐾𝑎 (𝑢)
is continuous and the set 𝑈𝑎 is closed and limited. Its nonemptiness is a part of the theorem.
We show that there are mathematical value expectations in
the phase constraint (4). Indeed, according to equation (1), we
have
𝐄[𝑥(𝑡 + 1)] = 𝐄[𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑎)] + 𝐄[ξ(𝑡)]
The second term on the right side of this equation makes
sense under the conditions of the theorem, and the first is
calculated by the formula
𝐄[𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑎)] = ∫𝑅𝑚 𝑓(ω, 𝑢(𝑡), 𝑎)𝑝𝑥(𝑡) (ω)𝑑ω,
if the last integral is absolutely convergent (here the probability
density function of a random variable 𝑥(𝑡) is expressed in
terms of 𝑝𝑥(𝑡) ). The latter fact is indeed the case due to
constraints on the growth of 𝑓 function and due to the existence
of the mathematical expectation of 𝑥(𝑡) value for any 𝑡 =
1, … , 𝑛 (this fact is verified by induction).
The existence of mathematical expectation on the right side
of this equation (3) follows from the constraints on growth of
𝐹𝑡 function and the existence of the mathematical expectation
of 𝑥(𝑡) value. We prove its continuous dependence on 𝑢 .
Suppose we have the convergence of vectors 𝑢𝑘 → 𝑢, 𝑢𝑘 ∈ 𝑈𝑎 .
From equation (1) it follows that
|𝑥𝑘 (𝑡 + 1) − 𝑥(𝑡 + 1)|
= |𝑓(𝑥𝑘 (𝑡), 𝑢𝑘 (𝑡), 𝑎) − 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑎)|
where 𝑥𝑘 and 𝑥 are problem (1), (2) solutions under regulations
𝑢𝑘 and 𝑢 respectively. Then the following relation is true
|𝑥𝑘 (𝑡 + 1) − 𝑥(𝑡 + 1)|
≤ 𝐿𝑓 [|𝑥𝑘 (𝑡) − 𝑥(𝑡)| + |𝑢𝑘 (𝑡) − 𝑢(𝑡)|]
where 𝐿𝑓 is a Lipschitz constant of the function 𝑓. Repeating
similar arguments and noting that, in virtue of condition (2),
𝑥𝑘 (0) = 𝑥0 , we have
|𝑥𝑘 (𝑡 + 1) − 𝑥(𝑡 + 1)| ≤ (𝐿𝑓 )2 |𝑥𝑘 (𝑡 − 1) − 𝑥(𝑡 − 1)|
+(𝐿𝑓 )2 |𝑢𝑘 (𝑡 − 1) − 𝑢(𝑡 − 1)| + 𝐿𝑓 |𝑢𝑘 (𝑡) − 𝑢(𝑡)|
≤ ∑𝑡𝑠=0(𝐿𝑓 ) 𝑠+1 |𝑢𝑘 (𝑡 − 𝑠) − 𝑢(𝑡 − 𝑠)| ≤ ε𝑘 
where ε𝑘 → 0 when 𝑘 → ∞.
Denoting the maximum of the Lipschitz constant of 𝐹𝑡
functions, for 𝑡 = 1, … , 𝑛 through 𝐿𝐹 we get the estimate
|𝐹𝑡 [𝑥𝑘 (𝑡)] − 𝐹𝑡 [𝑥(𝑡)]| ≤𝐿𝐹 ε𝑘 
Taking into account f-continuity of the family of sets {𝑋𝑎 }
and continuity on 𝐴 × 𝑋 of the function 𝑓𝑎 (𝑥), we conclude
that for any value of 𝜀 > 0 there is such a number of 𝑘0 , so that
when 𝑘 > 𝑘0 there are points 𝑥′𝑘 ∈ 𝑋𝑎0 for which the
inequalities |𝑓𝑎𝑘 (𝑥′𝑘 ) − 𝑓𝑎𝑘 (𝑥𝑘 )| ≤ ε hold
and
also
max|𝑓𝑎𝑘 (𝑦) − 𝑓𝑎о (𝑦)| ≤ ε.
𝑦∈𝑋
As a result, when 𝑘 > 𝑘0 , we get the following inequalities
𝑓𝑎0 (𝑥0 ) ≥ 𝑓𝑎0 (𝑥 ′ 𝑘 ) ≥ 𝑓𝑎𝑘 (𝑥 ′ 𝑘 ) − 𝜀 ≥ 𝑓𝑎𝑘 (𝑥𝑘 ) − 2ε 
Having calculated the mathematical expectation of both
sides of this inequality, we obtain the inequality
The following relation is checked in the same way:
𝑓𝑎𝑘 (𝑥𝑘 ) ≥ 𝑓𝑎0 (𝑥0 ) − 2ε
Е{|𝐹𝑡 [𝑥𝑘 (𝑡)] − 𝐹𝑡 [𝑥(𝑡)]|}≤𝐿𝐹 ε𝑘 
It follows that Е{|𝐹𝑡 [𝑥𝑘 (𝑡)] − 𝐹𝑡 [𝑥(𝑡)]|}→0 and
convergence of the considered sequence 𝐄{𝐹𝑡 [𝑥𝑘 (𝑡)]} →
𝐄{𝐹𝑡 [𝑥(𝑡)]}. Hence, (3) implies the continuity of the functions
𝐾𝑎 from 𝑢. Limitation of the set 𝑈𝑎 follows from the limited
set 𝑈(𝑡). Closure of the set 𝑈𝑎 follows from the continuous
mapping 𝑈𝑎 → 𝑋, set with the help of definition of the set 𝑈𝑎
and compactness of the set 𝑋 (the theorem on the closure of a
complete preimage of a compact under the continuous
mapping). Now the existence of problem solution follows from
the Weierstrass theorem. The theorem is proved.
Tentatively we formulate the following definition and
auxiliary statement used in the proof of Theorem 2.
Definition. Let the family of function 𝑓𝑎 (𝑥), 𝑎 ∈ 𝐴, 𝑥 ∈ 𝑋
be defined for the family of subsets {𝑋𝑎 } of some set 𝑋 of
Euclidian space with the parameter 𝑎 from some subset 𝐴 of
Euclidian space. The family {𝑋𝑎 } is f-continuous on the set 𝐴,
if for any ε > 0 there is a number δ > 0 , so that when
performing the inequality |𝑎 − 𝑏| ≤ δ, 𝑎, 𝑏 ∈ 𝐴 for any 𝑥𝑎 ∈
𝑋𝑎 there is such point 𝑥𝑏 ∈ 𝑋𝑏 , so that |𝑓𝑎 (𝑥𝑎 ) − 𝑓𝑎 (𝑥𝑏 )| < ε
inequality holds.

From (14) and (15) it follows that under sufficiently large
values of 𝑘 the relation |𝑓𝑎𝑘 (𝑥𝑘 ) − 𝑓𝑎0 (𝑥0 )| ≤ 2ε , which
provides the convergence of the sequence 𝑓𝑎𝑘 (𝑥𝑘 ) → 𝑓𝑎0 (𝑥0 ) ,
is true. The lemma is proved.
Proof of Theorem 2. In the proof of Theorem 1 we
established that sets 𝑈𝑎 are closed and bounded and 𝐾𝑎
function is continuous. There we also established continuity of
the mapping 𝑢 → 𝐄[𝑥𝑎𝑢 (𝑡)] from which the 𝐾𝑎 -continuity of
the family of sets follows. Here 𝑥𝑎𝑢 (𝑡) denotes the solution of
the system (1), (2) for selected 𝑎 and 𝑢.
We show the uniform on 𝑢 continuity of the mapping
𝑎 → 𝐾𝑎 (𝑢). Suppose that there is convergence 𝑎𝑘 → 𝑎 From
the condition (1) we get the following estimate
|𝑥𝑎𝑢𝑘 (𝑡 + 1) − 𝑥𝑎𝑢 (𝑡 + 1)| = |𝑓(𝑥𝑎𝑢𝑘 (𝑡), 𝑢(𝑡), 𝑎𝑘 )
−𝑓(𝑥𝑎𝑢 (𝑡), 𝑢(𝑡), 𝑎)| ≤ 𝐿𝑓 [|𝑥𝑎𝑢𝑘 (𝑡) − 𝑥𝑎𝑢 (𝑡)| + |𝑎𝑘 − 𝑎|]
where 𝐿𝑓 is the Lipschitz constant of 𝑓 function. Given the
initial state of the system, we obtain the inequality
|𝑥𝑎𝑢𝑘 (𝑡) − 𝑥𝑎𝑢 (𝑡)| ≤ ∑𝑛𝑠=1(𝐿𝑓 )𝑠 |𝑎𝑘 − 𝑎|.
According to this definition the family of sets {𝑋𝑎 } is 𝑓continuous, if in the case of enough proximity of the parameter
𝑏 to 𝑎, for any element from the set 𝑋𝑏 there is arbitrarily close
(for a function value) to it element of the set 𝑋𝑎 .
Let 𝐿𝐹 denote the maximum of Lipschitz constants of 𝐹𝑡
function. Then the following estimate is true
The following lemma is auxiliary to prove the continuity of
optimal values of the problems of variational calculus on the
synthesis of optimal laws of parametrical regulation.
|𝐹𝑡 [𝑥𝑎𝑢𝑘 (𝑡)] − 𝐹𝑡 [𝑥𝑎𝑢 (𝑡)]| ≤ 𝐿𝐹 |𝑥𝑎𝑢𝑘 (𝑡) − 𝑥𝑎𝑢 (𝑡)| ≤
𝐿𝐹 ∑𝑛𝑠=1(𝐿𝑓 )𝑠 |𝑎𝑘 − 𝑎|.
Lemma 1. Let 𝐴 and 𝑋 be some subsets of Euclidian
spaces, 𝑋 compact; the family of closed subsets {𝑋𝑎 } belongs
to 𝑋 . Let the mapping (𝑎, 𝑥) → 𝑓𝑎 (𝑥) be continuous in the
product of 𝐴 × 𝑋 . Let the family of subsets {𝑋𝑎 } be 𝑓 continuous in 𝐴 . Then the mapping 𝑎 → max 𝑓𝑎 (𝑥) is
Having proceeded to the mathematical expectations of the
right and left sides in the last inequality, we obtain that
Е{|𝐹𝑡 [𝑥𝑎𝑢𝑘 (𝑡)] − 𝐹𝑡 [𝑥𝑎𝑢 (𝑡)]|}→0. It follows that, when 𝑘 → ∞
for
any
𝑡 = 1, … , 𝑛
there
are
convergence
Е{𝐹𝑡 [𝑥𝑎𝑢𝑘 (𝑡)]}→Е{𝐹𝑡 [𝑥𝑎𝑢 (𝑡)]} uniformly on 𝑢 from the sum of
all sets 𝑈𝑎 and hence, uniform on 𝑢 continuity of the mapping
𝑎 → 𝐾𝑎 (𝑢) and continuity on (𝑎, 𝑢) of the function 𝐾𝑎 (𝑢) .
Having applied Lemma 1 we obtain the desired result. The
theorem is proved.
continuous in 𝐴.
𝑥∈𝑋𝑎
Proof of Lemma 1. Suppose we have a convergence of a
sequence 𝑎𝑘 → 𝑎0 , where 𝑎0 , 𝑎𝑘 ∈ 𝐴 Let 𝑥𝑘 denote the
maximum point of the function 𝑓𝑎𝑘 on the set 𝑋𝑎𝑘 , and 𝑥0 – the
maximum point of the function 𝑓𝑎0 on the set 𝑋𝑎0 .