macroeconomic analysis and parametric control of a national economy

Macroeconomic Analysis and Parametric Controlof a National Economy

Abdykappar A. Ashimov l Bahyt T. SultanovZheksenbek M. Adilov l Yuriy V. BorovskiyDmitriy A. Novikov l Rakhman A. AlshanovAskar A. Ashimov

Macroeconomic Analysisand Parametric Controlof a National Economy

Abdykappar A. AshimovKazakh National Technical UniversityNational Academy of Sciences of theRepublic of Kazakhstan

Almaty City, Kazakhstan

Bahyt T. SultanovKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City, Kazakhstan

Zheksenbek M. AdilovKazakh National Technical UniversityAlmaty City, Kazakhstan

Yuriy V. BorovskiyKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City, Kazakhstan

Dmitriy A. NovikovInstitute of Control Sciences RASMoscow, Russia

Rakhman A. AlshanovKazakh National Technical UniversityAlmaty City, Kazakhstan

Askar A. AshimovKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City, Kazakhstan

ISBN 978-1-4614-4459-6 ISBN 978-1-4614-4460-2 (eBook)DOI 10.1007/978-1-4614-4460-2Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012948194

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Contents

1 Elements of Parametric Control Theory of MarketEconomic Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Components of Parametric Control Theory

of Market Economic Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Methods of Analysis of the Stability and Structural

Stability of Mathematical Models of National

Economic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Development of the Methods for Evaluating

Stability Indicators of Mathematical Models . . . . . . . . . . . . . . . . . . 3

1.2.2 Development of Methods for Evaluating Weak

Structural Stability of a Discrete-Time Dynamical

System (Semi-cascade) Based on the

Robinson Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Approach to Synthesis and Choice (in the Environment

of a Given Finite Set of Algorithms) of Optimal Laws

of Parametric Control of a National Economic System’s

Development. Existence Conditions for a Solution

to Respective Variational Calculus Problems. Conditions

of Influence of Uncontrolled Parameters to These Problems. . . . . . . . . . 9

1.3.1 Analysis of the Existence Conditions for a Solution

of the Variational Calculus Problem of Synthesis

and Choice (in the Environment of a Given Finite

Set of Algorithms) of Optimal Laws of Parametric

Control of a Continuous-Time Deterministic

Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9





Control of a Discrete-Time Deterministic

Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

v





Control of a Discrete-Time Stochastic

Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.4 Analysis of the Influence of Uncontrolled

Parameter Variations (Parametric Disturbances)

on the Solution of the Variational Calculus Problem

of Synthesis and Choice of Optimal Parametric

Control Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Algorithm of Application of Parametric Control Theory

and Rules of Interaction Between Persons Making

Decisions on Elaboration and Realization of the Effective

State Economic Policy on the Basis of an Information

System for Decision-Making Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 Algorithm of the Application of Parametric

Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.2 Rules of Interaction for Decision Makers on the

Formulation and Implementation of an Effective

Public Economic Policy Based on the Information

Decision Support System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Examples for Application of Parametric Control Theory . . . . . . . . . . . . 36

1.5.1 Mathematical Model of the Neoclassical Theory

of Optimal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5.2 One-Sector Solow Model of Economic Growth. . . . . . . . . . . . . . 42

1.5.3 Richardson Model for the Estimation of Defense Costs . . . . . 46

1.5.4 Mathematical Model of a National Economic

System Subject to the Influence of the Share

of Public Expense and the Interest Rate

of Government Loans on Economic Growth . . . . . . . . . . . . . . . . . 50

1.5.5 Mathematical Model of the National Economic

System Subject to the Influence of International

Trade and Currency Exchange on Economic Growth. . . . . . . . 70

1.5.6 Forrester’s Mathematical Model of Global Economy. . . . . . . . 83

1.5.7 Turnovsky’s Monetary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

1.5.8 Endogenous Jones’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

vi Contents

2 Methods of Macroeconomic Analysis and Parametric Controlof Equilibrium States in a National Economy . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.1 Macroeconomic Analysis of a National Economic State

Based on IS, LM, and IS-LM Models, Keynesian

All-Economy Equilibrium. Analysis of the Influence

of Instruments on Equilibrium Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.1.1 Construction of the IS Model and Analysis

of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . 118

2.1.2 Macroeconomics of Equilibrium Conditions

in the Money Market. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.1.3 Macroestimation of the Mutual Equilibrium State

in Wealth and Money Markets. Analysis of the

Influence of Economic Instruments. . . . . . . . . . . . . . . . . . . . . . . . . 124

2.1.4 Macroestimation of the Equilibrium State

on the Basis of the Keynesian Model

of Common Economic Equilibrium. Analysis

of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . 125

2.1.5 Parametric Control of an Open Economic

State Based on the Keynesian Model. . . . . . . . . . . . . . . . . . . . . . . 127

2.2 Macroeconomic Analysis and Parametric Control

of the National Economic State Based on the Model

of a Small Open Country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2.2.1 Construction of the Model of an Open Economy

of a Small Country and Estimation of Its Equilibrium

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2.2.2 Influence of Economic Instruments on Equilibrium

Solutions and Payment Balance States . . . . . . . . . . . . . . . . . . . . . 136

2.2.3 Parametric Control of an Open Economy State

Based on a Small Country Model . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3 Parametric Control of Cyclic Dynamics of Economic Systems. . . . . . . 141

3.1 Mathematical Model of the Kondratiev Cycle. . . . . . . . . . . . . . . . . . . . . . 141

3.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.1.2 Estimating the Robustness of the Kondratiev

Cycle Model Without Parametric Control . . . . . . . . . . . . . . . . . . 143

3.1.3 Parametric Control of the Evolution of Economic

Systems Based on the Kondratiev Cycle Model . . . . . . . . . . . 144

3.1.4 Estimating the Structural Stability of the Kondratiev

Cycle Mathematical Model with Parametric Control . . . . . . 147

3.1.5 Analysis of the Dependence of the Optimal Value

of Criterion K on the Parameter for the Variational

Calculus Problem Based on the Kondratiev

Cycle Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents vii

3.2 Goodwin Mathematical Model of Market Fluctuations

of a Growing Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.2.2 Analysis of the Structural Stability of the Goodwin

Mathematical Model Without Parametric Control . . . . . . . . . 149

3.2.3 Problem of Choosing Optimal Parametric Control

Laws on the Basis of the Goodwin

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.2.4 Analysis of the Structural Stability of the Goodwin

Mathematical Model with Parametric Control . . . . . . . . . . . . . 153

3.2.5 Analysis of the Dependence of the Optimal

Parametric Control Law on Values of the

Uncontrolled Parameter of the Goodwin

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4 Macroeconomic Analysis and Parametric Controlof Economic Growth of a National Economy Basedon Computable Models of General Equilibrium. . . . . . . . . . . . . . . . . . . . . . . 157

4.1 National Economic Evolution Control Based on a

Computable Model of General Equilibrium of

Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.1.1 Model Description, Parametric Identification,

and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.1.2 Macroeconomic Analysis on the Basis of the

Computable Model of General Equilibrium

of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.1.3 Finding Optimal Parametric Control Laws

on the Basis of the CGE Model

of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

4.2 National Economic Evolution Control Based on the

Computable Model of General Equilibrium with the

Knowledge Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211



4.2.2 Estimation of the Macroeconomic Theory Provisions

on the Basis of the Computable Model of

General Equilibrium with the Knowledge Sector . . . . . . . . . . 236

4.2.3 Finding Optimal Parametric Control Laws Based

on the CGE Model with the Knowledge Sector . . . . . . . . . . . . 238

viii Contents

4.3 National Economic Evolution Control Based

on the Computable Model of General Equilibrium

with the Shady Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242



4.3.2 Finding the Optimal Values of the Adjusted

Parameters on the Basis of the CGE Model

in the Shady Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

About The Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Contents ix

Chapter 1

Elements of Parametric Control Theoryof Market Economic Development

1.1 Components of Parametric Control Theoryof Market Economic Development

The application of mathematical models of a national economy is an important

subject area for the analysis of an effective public policy in the area of the economic

growth [73].

Many dynamical systems, including the national economic system [33, 30], after

some transformations, can be described by the following systems of nonlinear

ordinary differential equations:

x� ðtÞ ¼ f xðtÞ; uðtÞ; að Þ; (1.1)

with the initial condition

x t0ð Þ ¼ x0: (1.2)

Here t is the time, t 2 t0; t0 þ T½ �; T>0, is a fixed number;

x ¼ xðtÞ 2 Rm is the state of system (1.1), (1.2);

x0 2 Rm is the initial state of the system (deterministic vector);

u ¼ uðtÞ 2 Rq is the vector of controlled (regulated) parameters; the functions uðtÞand their derivatives are to be uniformly bounded;

a 2 A � Rs is the vector of uncontrolled parameters; and A is an open connected set.

For a solution to system (1.1), (1.2) to exist, let’s assume that the vector function

f satisfies the Lipschitz condition and the following linear constraints on its growth

rate:

jf ðx; u; aÞj � cð1þ jxjÞ;

A.A. Ashimov et al., Macroeconomic Analysis and Parametric Controlof a National Economy, DOI 10.1007/978-1-4614-4460-2_1,# Springer Science+Business Media New York 2013

1

where c is a positive constant.As is well known, the solution (evolution) to the considered system of ordinary

differential equations depends on both the vector of initial values x0 and the values

of vectors of controlled (u) and uncontrolled (a) parameters. Therefore, the result of

evolution (development) of the nonlinear dynamical system, with a given vector of

the initial values x0 , is defined by the values of vectors of both controllable and

uncontrollable parameters.

It is also known [3] that the process described by (1.1) may be judged by the

solutions of this system only if the qualitative image of the trajectories of this

system is invariable under small—in some sense—disturbances of the right-hand

side part of (1.1). In other words, system (1.1) must possess the property of

robustness or structural stability.

For the reason just mentioned, the theory of parametric control of the market

economic development is proposed in [7, 8, 53–55].

This theory consists of the following components:

1. The methods for forming the set (library) of macroeconomic mathematical

models. These methods are oriented toward the description of various specific

socioeconomic situations, taking environmental safety conditions into

consideration.

2. The methods for estimating the conditions for robustness (structural stability) of

the models of national economic systems from the library without parametric

control. Here, the conditions of belonging to the considered mathematical

models of the Morse–Smale class of systems, the class of O-robust systems,

the class of uniformly robust systems, the class of У-systems, or the class of

systems with weak structural stability are verified.

3. The methods of control or attenuation of nonrobustness (structural instability) of

mathematical models of economic systems. This category involves choosing

(synthesis) the algorithm for control or attenuation of structural instability for the

respective mathematical models of a national economic system.

4. The methods of choice and synthesis of the laws of parametric control of market

economic development based on the mathematical models of a national eco-

nomic system.

5. The methods of estimating the robustness (structural stability) of mathematical

models of a national economic system from the library with parametric control.

In this group, the conditions of belonging to the considered mathematical models

with parametric control of the Morse–Smale class of systems, the class of

O-robust systems, the class of uniformly robust systems, the class of Y-systems,

or the class of systems with weak structural stability are verified.

6. The methods of adjustment of constraints on parametric control of market

economic development in the case of structural instability of mathematical

models of the national economic system with parametric control and adjustment

of constraints on the parametric control of market economic development.

7. The methods of research and analysis of bifurcations of extremals of variational

calculus problems of choosing optimal laws of parametric control theory.

2 1 Elements of Parametric Control Theory of Market Economic Development

8. Development of recommendations on the elaboration and implementation of

efficient governmental economic policy on the basis of parametric control theory

of market economic development, taking into consideration specific socioeco-

nomic situations.

1.2 Methods of Analysis of the Stability and Structural Stabilityof Mathematical Models of National Economic Systems

1.2.1 Development of the Methods for Evaluating StabilityIndicators of Mathematical Models

By Orlov’s definition [31], the mathematical model of an economic system in

general form is some function y ¼ f ðpÞ

f : A ! B; (1.3)

transferring values of initial (exogenous) data p 2 A to solutions (values of

endogenous variables) y 2 B:After constructing a mathematical model of some real-life phenomenon or

process f and defining some actual values of the point p by known measured data

or solving the parametric identification problem, the question arises about the

adequacy of the analyzed model. The condition of model stability against admissi-

ble perturbations of the initial data [31] is one of the conditions of the model’s

adequacy. In case of such stability, small perturbations of the model’s initial data

result in small changes in its solution. In monograph [31], the definitions of the

basic stability indicators are introduced (these definitions are presented below). But

Orlov [31] does not propose any algorithm for computing the considered indicators

of the mathematical model’s stability.

In this chapter we present the developed algorithms for evaluating the mathe-

matical model’s stability indicators, which characterize the stability of solutions of

the mathematical model against initial data perturbations. All of the model

parameters and variables must be made dimensionless first.

Let X ¼ ðX1;X2; :::;XkÞ be some vector of values of the model exogenous

parameters for a time interval t 2 f0; :::; Tg. Let X0 ¼ ðX10;X

20; :::; X

k0Þ denote the

respective vector of base values for the same time interval. The vector that

incorporates the values of parameters and initial values of the variables of differen-

tial (or difference) equations is the vector X. The vector of measured statistical data

used for finding the model equation coefficients is denoted by vector X for econo-

metric models.

Let p ¼ ðp1; p2; :::; pkÞ be a vector of the normalized input data of the mathemat-

ical model, where pi ¼ Xi

Xi0

; i ¼ 1; :::; k. The vector p0 ¼ ð1; 1; :::; 1Þ.

1.2 Methods of Analysis of the Stability and Structural Stability. . . 3

Let А be a space of the normalized input data vectors that includes all admissi-

ble sets p. A � Rk is a metric space with the Euclidean metric defined by the space

Rk; po 2 A.Let Y ¼ YðpÞ ¼ ðY1; Y2; :::; YnÞ be a selected vector of the values of endogenous

variables for some chosen interval (or moment) of time obtained for the selected

values of p. The vector that incorporates the values of some selected set of the

model endogenous variables for the aforesaid interval (or moment) of time is

denoted by vector Y for the dynamical models. The vector of coefficients of the

model equations or vector of values of some selected set of the model endogenous

variables for the aforesaid interval (or moment) of time is considered to be vector Yfor econometric models.

In particular, with p ¼ p0, we introduce the notation Y0 ¼ Yðp0Þ ¼ ðY10 ; Y

20 ; :::;

Yn0Þ. The normalized vector of values of the endogenous variables for the moment

of time T1 is denoted by y ¼ yðpÞ ¼ Y1

Y10

; Y2

Y20

; :::; Yn

Yn0

� �; y0 ¼ yðp0Þ ¼ ð1; 1; :::; 1Þ:

Let B � Rn be a region that contains all possible output values y for p 2 A with

the Euclidean metric of space Rn; y0 2 B . The considered model defines the

mapping f of set A into set B.For the selected point p 2 A and number a > 0, letUaðpÞ denote the intersection

of a neighborhood of the point p with radius a with set A:

UaðpÞ ¼ fx1 2 A : pðp1; pÞ � ag:

Here and below, r denotes the Euclidean distance between two points of the

Euclidean space.

For some subset B1 � B, let dðB1Þ denote the diameter of set B1; i.e.,

dðB1Þ ¼ supðrðy1; y2Þ; y1; y2 2 B1Þ:

Definition 1.1 The number

bðp; aÞ ¼ dð f ðUaðpÞÞ (1.4)

is defined as the stability indicator of the econometric model at the point X 2 Afor a > 0.

Algorithm 1.1 for evaluating the model stability indicator bðp; aÞ by the Monte

Carlo method is as follows:

1. Choose sets of input parameters (X) and output variables (Y), and compute their

normalized values.

2. Define the vector of normalized input data p ¼ ðp1; p2; :::; pkÞ, number a > 0,

and set UaðpÞ:3. Generate a set of sufficiently large number M of pseudo-random points (p1,

p2, . . ., pM) uniformly distributed in UaðpÞ:


For this purpose, consecutively generate the coordinates pijði ¼ 1; :::; k; j ¼ 1;:::;MÞof the point pj in numerical intervals ½pi � a; pi þ a�coveringUaðpÞusinga sensor of pseudo-random numbers distributed uniformly. If the inequality

Xk

i¼1ðpij � piÞ2 � a2

holds (i.e., xj 2 UaðpÞÞ, this point is added to the created set.

4. For each point pj of the set, define point yj ¼ f pj� �

, j ¼ 1; ::: ;M, by simulation.

5. Evaluate

b ¼ maxðrðyi; yjÞ : i; j ¼ 1; :::;MÞ:

6. Stop.

With a ¼ 0.01, the obtained number b/2 characterizes the (maximum) percent-

age change of values of the model output variables under the perturbed input

data by 1%.


bðxÞ ¼ inf0<a�a0

bðp; aÞ (1.5)

is called the absolute stability indicator of the econometricmodel at point x 2 A. Here,a0 is the maximal admissible relative deviation of values of the model input data.

Algorithm 1.2 for evaluating the absolute stability indicator bðpÞ of the econo-

metric model is as follows:

For the selected value a0 and numbers j ¼ 0; 1; 2; . . . ; consecutively find

numbers bj ¼ b p; a0=2jð Þ;and then evaluate the number

bðpÞ ¼ infj¼0;1;2;:::

bj

by Algorithm 1.2. If bðrÞ turns out to be less than some a priori given small number

e [i.e., bðrÞ is considered to be approximately zero], then the mapping f defined by

the analyzed model is evaluated at point p continuously depending on the input

values.


g ¼ supp2AbðpÞ (1.6)

is called the maximal absolute stability indicator of the model for region A.


Algorithm 1.3 for evaluating the maximal absolute stability indicator g of the

model by the Monte Carlo method is as follows:

1. Generate the set of sufficiently large numberM of pseudo-random points (p1, p2,. . ., pM) uniformly distributed in A.

2. For each point pj in the set and chosena0>0, find the numbers bðpÞ; j ¼ 1; :::;M;by Algorithm 1.2.

3. Determine the number

g ¼ maxj¼1;:::;M

bðpjÞ:4. Stop.

If the number g turns out to be less than some a priori given small number

e (i.e., g is considered to be approximately zero), then the mapping f definedby the analyzed model is evaluated in set A continuously depending on the

input values.

The developed algorithms were applied for evaluating the econometric model of

correlation of macroeconomic indicators and the CGEmodel of economic branches.

For the CGE model of the economic branches, we consider the set of possible

values of initial statistical data used for parametric identification of coefficients and

initial conditions from the difference equations of that model as set A. The resultsfrom the model’s simulation for some definite moments of time following the

period of model parametric identification are denoted as set B.

1.2.2 Development of Methods for Evaluating Weak StructuralStability of a Discrete-Time Dynamical System(Semi-cascade) Based on the Robinson Approach

The methods for analyzing the robustness (structural stability) of mathematical

models of national economic systems are based on

– Fundamental results on dynamical systems in the plane

– Methods of verification of mathematical models belonging to certain classes of

structurally stable systems (classes of Morse–Smale systems, O-robust systems,

Y-systems, systems with weak structural stability)

At present, the theory of parametric control of market economic development

has a number of theorems about the structural stability of specific mathematical

models (the model of the neoclassical theory of optimal growth; models of national

economic systems, taking into consideration the influence of the share of public

expenses and of the interest rate of governmental loans on economic growth;

models of national economic systems, taking into consideration the influence of

international trade and exchange rates on economic growth; and others formulated

and proved on the basis of the aforementioned fundamental results).


Along with analysis of the structural stability of specific mathematical models

(both with and without parametric control), based on results of the theory of

dynamical systems, one can consider approaches to the analysis of the structural

stability of mathematical models of national economic systems using computer

simulations.

We shall consider below the construction of a computational algorithm

for estimating the structural stability of mathematical models of national economic

systems on the basis of Robinson’s theorem (Theorem A) [69] on weak structural

stability.

LetN0 be some manifold and N a compact subset inN0 such that the closure of theinterior of N is N. Let some vector field be given in a neighborhood of the set N in

N0: This field defines the C1 -flux f in this neighborhood. Let Rðf ;NÞ denote the

chain-recurrent set of the flux f on N.Let Rðf ;NÞ be contained in the interior of N. Let it have a hyperbolic structure.

Moreover, let the flux f upon Rðf ;NÞ also satisfy the transversability conditions of

stable and unstable manifolds. Then the flux f on N is weakly structurally stable. In

particular, if Rðf ;NÞ is an empty set, then the flux f is weakly structurally stable on

N. A similar result is also correct for the discrete-time dynamical system (cascade)

specified by the homeomorphism (with image) f : N ! N0.Therefore, one can estimate the weak structural stability of the flux (or cascade) f

via numerical algorithms based on Theorem A via the numerical estimation of the

chain-recurrent set Rðf ;NÞ for some compact region N of the phase state of the

considered dynamical system.

Let’s further propose an algorithm of localization of the chain-recurrent set for a

compact subset of the phase space of the dynamical system described by a system of

ordinary differential (or difference) equations and algebraic system. The proposed

algorithm is based on the algorithm of construction of the symbolic image [33].

A directed graph (symbolic image), being a discretization of the shift mapping

along the trajectories defined by this dynamical system, is used for computer

simulation of the chain-recurrent subset.

Suppose an estimate of the chain-recurrent setRðf ;NÞof some dynamical system

in the compact set N of its phase space has been found. For a specific mathematical

model of the economic system, one can consider, for instance, some parallelepiped

of its phase space including all possible trajectories of the economic system’s

evolution for the considered time interval to be the compact set N.The localization algorithm for the chain-recurrent set consists of the following:

1. Define the mapping f defined on N and given by the shift along the trajectories of

the dynamical system for the fixed time interval.

2. Construct the partition C of the compact set N into cells Ni. Assign the directed

graph G with graph nodes corresponding to the cells and branches between the

cells Ni and Nj corresponding to the conditions of the intersection of the image of

one cell f(Ni) with another cell Nj.

3. Find all recurrent nodes (nodes belonging to cycles) of the graph G. If the set ofsuch nodes is empty, then Rðf ;NÞ is empty, and the process of its localization


ceases. One can draw a conclusion about the weak structural stability of the

dynamical system.

4. The cells corresponding to the recurrent nodes of the graphG are partitioned into

cells of lower dimension, from which a new directed graph G is constructed (see

item 2 of the algorithm).

5. Go to item 3.

Items 3–5 must be repeated until the diameters of the partition cells become less

than some given number e.The last set of cells is the estimate of the chain-recurrent set Rð f ;NÞ.The method of estimating the chain-recurrent set for a compact subset of the

phase space of a dynamical system developed here allows us to draw a conclusion

about the weak structural stability of the dynamical system when the obtained

chain-recurrent set Rð f ;NÞ is empty.

In the case when the considered discrete-time dynamical system is a priori the

semi-cascade f, one should verify the invertibility of the mapping f defined on N(since, in this case, the semi-cascade defined by f is the cascade) before applying

Robinson’s theorem for estimating its weak structural stability.

Let’s give a numerical algorithm for estimating the invertibility of the differ-

entiable mapping f : N ! N0 , where some closed neighborhood of the discrete-

time trajectory f f tðx0Þ; t ¼ 0; . . . ; Tg in the phase space of the dynamical system

is used as N. Suppose that N contains a continuous curve L connecting the points

f f tðx0Þ; t ¼ 0; . . . ; Tg. We can choose as such curve a piecewise linear curve with

nodes at the points of the above-mentioned discrete-time trajectory of the

semi-cascade.

An invertibility test for the mapping f : N ! N0 can be implemented in the

following two stages:

1. An invertibility test for the restriction of the mapping f : N ! N0 to the curve L,namely, f : L ! f ðLÞ. This test reduces to ascertaining the fact that the curve f ðLÞdoes not have points of self-crossing; that is, ðx1 6¼ x2Þ ) ð f ðx1Þ 6¼ f ðx2ÞÞ; x1;x2 2 L. For instance, one can determine the absence of self-crossing points by

testing the monotonicity of the limitation of the mapping f onto L along any

coordinate of the phase space of the semi-cascade f.2. An invertibility test for the mapping f in neighborhoods of the points of curve L

(local invertibility). Based on the inverse function theorem, such a test can be

carried out as follows: For a sufficiently large number of chosen points x 2 L;one can estimate the Jacobians of the mapping f using the difference

derivations: JðxÞ ¼ det@f i

@xjðxÞ

� �; i; j ¼ 1; . . . ; n: Here i, j are the coordinates

of the vectors, and n is the dimension of the phase space of the dynamical

system. If all the obtained estimates of Jacobians are nonzero and have the

same sign, one can conclude that JðxÞ 6¼ 0 for all x 2 L and, hence, that the

mapping f is invertible in some neighborhood of each point x 2 L.


An aggregate algorithm for estimating the weak structural stability of the

discrete-time dynamical system (semi-cascade defined by the mapping f ) with

phase space N0 2 Rn defined by the continuously differentiable mapping f can be

formulated as follows:

1. Find the discrete-time trajectory f f tðx0Þ; t ¼ 0; . . . ; Tg and curve L in a closed

neighborhood N that are required to estimate the weak structural stability of the

dynamical system.

2. Test the invertibility of the mapping f in a neighborhood of the curve L using the

algorithm described above.

3. Estimate (localize) the chain-recurrent set Rð f ;NÞ . By virtue of the evident

inclusion Rð f ;N1Þ � Rðf ;N2Þ for N1 � N2 � N0, one can use any parallelepipedbelonging to N0 and containing L as the compact set N.

4. IfRð f ;NÞ ¼ Ø, draw a conclusion about the weak structural stability of the

considered dynamical system in N.

This aggregate algorithm can also be applied to estimate the weak structural

stability of a continuous-time dynamical system (the flux f ) if the trajectory L ¼f f tðx0Þ; 0 � t � Tgof the dynamical system is considered to be the curve L. In thiscase, item 2 of the aggregate algorithm is omitted. The mapping f t for some fixed

t (t > 0) can be accepted as the mapping f in item 3.

1.3 Approach to Synthesis and Choice (in the Environmentof a Given Finite Set of Algorithms) of Optimal Lawsof Parametric Control of a National Economic System’sDevelopment. Existence Conditions for a Solutionto Respective Variational Calculus Problems. Conditionsof Influence of Uncontrolled Parameters to These Problems

1.3.1 Analysis of the Existence Conditions for a Solution ofthe Variational Calculus Problem of Synthesis and Choice(in the Environment of a Given Finite Set of Algorithms)of Optimal Laws of Parametric Control of aContinuous-Time Deterministic Dynamical System

1.3.1.1 Statement of the Variational Calculus Problems of Synthesisand Choice of Parametric Control Laws for a Continuous-TimeDeterministic Dynamical System

Let’s consider a nonlinear dynamical system (1.1) given by

x: ðtÞ ¼ f ðxðtÞ; uðtÞ; lÞ

1.3 Approach to Synthesis and Choice (in the Environment of a Given Finite Set. . . 9

with the initial condition (1.2)

xðt0Þ ¼ x0:

We’ll introduce the optimality criterion to be maximized as

K ¼Z t0þT

t0

F½xðtÞ�dt; (1.7)

where the function FðxÞ satisfies the Lipschitz condition.The following state-space constraints are on the system:

xðtÞ 2 XðtÞ; t 2 ðt0; t0 þ T�: (1.8)

The following constraints on control can also be imposed in problems consid-

ered ahead:

uðtÞ 2 UðtÞ; t 2 ½t0; t0 þ T�: (1.9)

Here XðtÞ � Rm;UðtÞ � Rq are some compact sets with nonempty interiors, and the

sets X ¼ [t2ðt0;t0þTÞ

XðtÞ and U ¼ [t2½t0;t0þTÞ

UðtÞ are bounded.Let’s formulate the following variational calculus problem called the variational

calculus problem of synthesis of optimal parametric control law for a continuous-time deterministic dynamical system.

Problem 1.1 From a known vector of uncontrolled parameters, find □ parametriccontrol law u satisfying condition (1.9) such that the respective solution of thedynamical system (1.1), (1.2) satisfies condition (1.8) and maximizes function (1.7).

In the second problem, we again consider a continuous-time dynamical control

system described by (1.1), (1.2) in the presence of state-space constraints (1.8).

However, unlike the previous case, the control is to be selected from this set of

given control laws:

ujðtÞ ¼ Gjðv; xðtÞÞ; t 2 ðt0; t0 þ T�; j ¼ 1; :::; r; (1.10)

where n ¼ ðn1; :::; nlÞ is the vector of coefficients to be adjusted (control parameters)

of the control law. It is assumed that vector function Gjðv; xÞ satisfies the Lipschitzcondition and linear constraints on growth rate:

jGjðn; xÞj � cð1þ jxjÞ; (1.11)

where c is some positive constant.

These adjusted coefficients are imposed by the constraints

v 2 V; (1.12)


where V is some compact subset of the space Rl. Moreover, it is assumed that the

control parameters must be such that the respective control law (1.11) satisfies

condition (1.9); i.e., the inclusion

Gjðv; xjðtÞÞ 2 UðtÞ; t 2 ðt0; t0 þ T� (1.13)

holds true, where xj(t) is the solution to problem (1.1), (1.2) for the selected values

of v,l and jth parametric control law.

The following optimality criteria are under consideration:

Kj ¼ Kjðv; lÞ ¼ðt0þT

t0

F½xjðtÞ�dt (1.14)

Let’s state the following extremal problem, called the variational calculus prob-lem of the choice (in the environment of a given finite set of algorithms) of optimalparametric control law for a continuous-time deterministic dynamical system.

Problem 1.2 From a known vector of uncontrolled parametersl 2 L, for each of rcontrol laws (1.10), find a vector of adjusted coefficients v such that the respectivesolution x ¼ xj of problem (1.1), (1.2) with control law u ¼ uj defined by formula(1.10) satisfies conditions (1.8), (1.12), (1.13) and maximizes function (1.14) withthe subsequent choice of the best obtained control law, which results in themaximum value of the optimum criterion.

1.3.1.2 Solvability Conditions for the Variational Calculus Problemof Synthesis of the Optimal Control Law for a Continuous-TimeDynamical System

Let’s now proceed to considering Problem 1.1.

We’ll consider the parametric control laws u ¼ uðtÞ ¼ ðu1ðtÞ; :::; uqðtÞÞ in subsetW of the Sobolev space ½H1ð0; TÞ�q of vector functions satisfying the relations

uðtÞ 2 UðtÞ; j _uðtÞj � c; t 2 ð0; TÞ: (1.15)

Here U(t) is a compact set in Rq defined above, and c is some positive constant.

We will refer to the set of such control laws u 2 W that yield the existence of the

solution to system (1.1), (1.2) and satisfy inclusion (1.8) as a set of admissible

control Wad.

Problem 1.1 consists of finding such an admissible control law u(t) that

maximizes function (1.7) on the set Wad.

The following theorem is valid.


Theorem 1.1 Under the above assumptions, Problem 1.1 is solvable if the set ofadmissible control Wad is a nonempty set.

The proof is presented in the appendix.

1.3.1.3 Solvability Conditions for the Variational Calculus Problemof Choice (in the Environment of a Given Finite Set of Algorithms)of the Optimal Control Law for a Continuous-Time DeterministicDynamical System

Consider Problem 1.2 of the variational calculus problem of choice (in the environ-

ment of a given finite set of algorithms) of the optimal control law on the basis of a

continuous-time dynamical system (1.1) with initial conditions (1.2).

For jth control law (1.10), system (1.1) is described by the equation

x: ðtÞ ¼ f ðxðtÞ;Gjðv; xðtÞÞ; lÞ (1.16)

with initial condition

xðt0Þ ¼ x0: (1.17)

Here, function f satisfies the constraints assigned in the beginning of Section 2.3,

and functionGj grows in absolute value not faster than a linear function and satisfies

the Lipschitz condition. The vector of adjusted coefficients is imposed by constraint

(1.12): u 2 V.Since, after the redesignation, (1.16) can be reduced to (1.1) with the same

functional properties, we can conclude that the solution to the considered system

above exists and is unique. As it is determined by concrete values of the control law

number j and adjusted coefficient v, hereinafter we shall use the notation x ¼ xvj ðtÞ.For any j ¼ 1; :::; r; we’ll define the set Vj

ad of admissible values of the adjusted

coefficient as a set consisting of such values of v satisfying condition (1.12),

yielding the respective solution to problem (1.16), (1.17) to satisfy the inclusions

Gjðv; xvj ðtÞÞ 2 UðtÞ; t 2 ðt0; t0 þ TÞ (1.18)

xvj ðtÞ 2 XðtÞ; t 2 ðt0; t0 þ TÞ: (1.19)

Problem 1.2 consists of choosing the best control law, namely, such a number jthat yields the maximum of function (1.14) KJ on the set Vj

ad to be the largest.

Theorem 1.2 Under the above assumptions and in the case when the sets Vjad are

nonempty, Problem 1.2 is solvable.This proof is presented in the appendix.


1.3.2 Analysis of the Existence Conditions for a Solutionof the Variational Calculus Problem of Synthesisand Choice (in the Environment of a Given Finite Setof Algorithms) of Optimal Laws of Parametric Controlof a Discrete-Time Deterministic Dynamical System

1.3.2.1 Statement of the Variational Calculus Problems of Synthesisand Choice of Parametric Control Laws for a Discrete-TimeDeterministic Dynamical System

Let’s consider the discrete-time stochastic control system

xðtþ 1Þ ¼ f ðxðtÞ; uðtÞ; lÞ; t ¼ 0; 1; :::; n� 1; (1.20)

xð0Þ ¼ x0; (1.21)

where the time t is a nonnegative integer value, and n is a fixed natural number;

x ¼ xðtÞ 2 Rm is the state vector of system (1.20), (1.21), a vector function of the

discrete-time argument;

x0 2 Rm is the initial state of the system and a deterministic vector;

u ¼ uðtÞ 2 Rq is the control parameters’ vector, a vector function of the discrete-

time argument;

l 2 L � Rs is the vector of the uncontrolled parameters, L is the open connected

set;

f is the defined vector function of its arguments.

The optimality criterion to be maximized is given by

K ¼Xnt¼1

FtðxðtÞÞ: (1.22)

Here Ft(x) are the known functions.

The state-space constraints on the system are as follows:

xðtÞ 2 XðtÞ; t ¼ 1; :::; n: (1.23)

The following constraints on the control can also be imposed in the problems

considered below:

uðtÞ 2 UðtÞ; t ¼ 0; :::; n� 1 (1.24)


Here XðtÞ � Rm; UðtÞ � Rq are some compact sets with nonempty interiors.

Let’s formulate the following variational calculus problem, called the varia-tional calculus problem of the synthesis of the optimal parametric control law fora discrete-time deterministic dynamical system.

Problem 1.3 From the known vector of uncontrolled parameters l, find aparametric control law u satisfying condition (1.24) such that the respectivesolution of dynamical system (1.20), (1.21) satisfies condition (1.23) and maximizesfunction (1.22).

In the second problem, we again consider the continuous-time dynamical control

system described by (1.20), (1.21) in the presence of state-space constraints (1.23).

Here, the control is to be selected from the set of given control laws:

ujðtÞ ¼ Gjðv; xðtÞÞ; t ¼ 0; :::; n� 1; j ¼ 1; :::; r: (1.25)

Here Gj is the vector function of its arguments satisfying the Lipschitz condition,

and v ¼ ðv1; :::; vlÞ is the vector of the coefficients of control law Gj to be adjusted.


v 2 V; (1.26)

where V is some compact subset of space Rl. Moreover, it is assumed that the

control parameters must be such that the inclusion

Gjðv; xvj ðtÞÞ 2 UðtÞ; t ¼ 0; :::; n� 1; (1.27)

holds true. Here xvj ðtÞ is the solution to problem (1.20), (1.21) for the selected values

of v, a, and jth parametric control law.

We consider the following optimality criteria:

Kj ¼ Kjðv; lÞ ¼Xn

t¼1Ftðxvj ðtÞÞ: (1.28)

Let’s state the following extremal problem, called the variational calculusproblem of choice (in the environment of a given finite set of algorithms) of theoptimal parametric control law for a discrete-time deterministic dynamical system.

Problem 1.4 From a known vector of uncontrolled parameters l 2 L, for each of rcontrol laws from set (1.25), find a vector of the adjusted coefficients v such that therespective solution x ¼ xj of problem (1.20), (1.21) with this control law satisfiesconditions (1.23), (1.26), (1.27) and maximizes function (1.28).


1.3.2.2 Solvability Conditions for the Variational Calculus Problemof Synthesis of the Optimal Control Law for a Discrete-TimeDeterministic Dynamical System

We will refer to the set of such the control laws u satisfying (1.24) such that the

solution to system (1.20), (1.21) satisfying inclusion (1.23) is the set of admissible

controls Wad.

Problem 1.4 consists of finding such an admissible control law u that maximizes

function (1.22) on the set Wad.

The following statement is a natural development of the classical Weierstrass

theorem on the existence of the extremum of a continuous functionwithin an interval.

Theorem 1.3 Let the vector function f be continuous with respect to the aggregateof arguments, let the sets X(t), U(t) be closed and bounded for all definite t, and letthe function Ft(x) be continuous. Then, if the set Wad of the admissible control lawsis nonempty, Problem 1.4 is solvable.


1.3.2.3 Solvability Conditions for the Variational Calculus Problemof Choice (in the Environment of a Given Finite Set of Algorithms)of the Optimal Control Law for a Discrete-Time DeterministicDynamical System

Let’s now consider Problem 1.4 formulated above.

Denote by xvj the solution of system (1.20), (1.21) for the selected jth parametric

control law (1.25), its adjusted coefficient v, and parameter l:

xvj ðtþ 1Þ ¼ f ðxvj ðtÞ;Gjðv; xvj ðtÞÞ; lÞ; t ¼ 0; :::; n� 1; (1.29)

xvj ðt0Þ ¼ x0: (1.30)

Similarly to Theorem 1.2, we’ll define the set of admissible valuesVjad consisting

of such values of v satisfying condition (1.26), which yields the respective solution

of problem (1.29), (1.30) to satisfy the inclusions

Gjðv; xvj ðtÞÞ 2 UðtÞ; t ¼ 0; :::; n� 1; (1.31)

xvj ðtÞ 2 XðtÞ; t ¼ 1; :::; n: (1.32)

Theorem 1.4 Assume that in Problem 1.3.4 the functions f, Gj, and Ft are continu-ous with respect to the aggregate of the arguments. Then, if sets Vj

ad are nonempty,Problem 1.3.4 is solvable.


1.3.3 Analysis of the Existence Conditions for a Solutionof the Variational Calculus Problem of Synthesisand Choice (in the Environment of a Given Finite Setof Algorithms) of Optimal Laws of Parametric Controlof a Discrete-Time Stochastic Dynamical System

1.3.3.1 Statement of the Variational Calculus Problems of Synthesisand Choice of Parametric Control Laws for a Discrete -TimeStochastic Dynamical System

Let’s consider the discrete-time stochastic control system

xðtþ 1Þ ¼ f ðxðtÞ; uðtÞ; lÞ þ xðtÞ; t ¼ 0; :::; n� 1; (1.33)

xð0Þ ¼ x0 (1.34)

Here

x ¼ xðtÞ 2 Rm is the state vector of system (1.33), (1.34), a random vector function

of the discrete-time argument (a vector random sequence);

x0 is the initial state of the system and a deterministic vector;

u ¼ uðtÞ 2 Rq is the control parameters’ vector, a vector function of the discrete-

time argument;

l 2 Rs is the vector of the uncontrolled parameters; l 2 L; L � Rsis a given set;

x ¼ xðtÞ ¼ ðx1ðtÞ; :::; xmðtÞÞ is the known vector random sequence corresponding to

the disturbances (e.g., Gaussian white noise);

f is the defined vector function of its arguments.

The optimality criterion to be maximized is given by

K ¼ EXn

t¼1FtðxðtÞÞ

n o(1.35)

Here Ft are known functions, and E is the expectation.

The following state-space constraints are on the system:

E½xðtÞ� 2 XðtÞ; t ¼ 1; :::; n; (1.36)

where X(t) is a given set.

The problems considered below retain the constraints on the control defined

above:

uðtÞ 2 UðtÞ t ¼ 0; :::; n� 1; (1.37)

where U(t) � Rq is the given set.


We’ll formulate the following problem, called the variational calculus problemof the synthesis of the optimal parametric control law for a discrete-time stochasticdynamical system:

Problem 1.5 For the known vector of uncontrolled parameters a, find a parametriccontrol law u satisfying condition (1.37) such that the respective solution ofdynamical system (1.33), (1.34) satisfies condition (1.36) and maximizes function(1.35).

In the second problem of parametric control of a discrete-time dynamical

system, we again consider the discrete-time dynamical control system described

by (1.20), (1.21) in the presence of state-space constraints (1.23). In this problem, as

in the previous Problems 1.2 and 1.3.4, the control is to be selected from the set of

given control laws:

ujðtÞ ¼ Gjðv; xðtÞÞ; t ¼ 0; :::; n� 1; j ¼ 1; :::; r; (1.38)

where Gj is the known vector function of its arguments, and v ¼ ðv1; :::vlÞ is the

vector of parameters of the control law Gj to be adjusted.


v 2 V; (1.39)

where V is some compact subset of the space Rl. Moreover, it is assumed that the

control parameters must be such that the respective control law (1.43) satisfies

condition (1.42); i.e., the inclusion

E[Gjðv; xvj ðtÞÞ� 2 UðtÞ; t ¼ 0; :::; n� 1; (1.40)

holds true. Herexvj ðtÞ is a solution to problem (1.38), (1.39) for the selected values of

v, l, and jth parametric control law.

We consider the following optimality criteria:

Kj ¼ Kjðv; lÞ ¼ EXnt¼1

Ftðxvj ðtÞÞ( )

: (1.41)

Let’s state the following extremal problem, called the variational calculusproblem of choice of the optimal parametric control law for a discrete-timestochastic dynamical system.

Problem 1.6 From a known vector of uncontrolled parameters l 2 L, for each of rcontrol laws, find a vector of adjusted coefficients v such that the respective solutionx ¼ xj of problem (1.33), (1.34) with control law u ¼ uj defined by formula (1.38)


satisfies conditions (1.39), (1.40) and maximizes function (1.41) with the subsequentchoice of the best obtained optimal control laws, i.e., corresponding to the greatestvalue of the optimality criterion.

1.3.3.2 Solvability Conditions for the Variational Calculus Problemof Synthesis of the Optimal Control Law for a Discrete-TimeStochastic Dynamical System

Let’s now consider the solvability of Problem 1.5. Define set Uad of the admissible

control laws for the considered system as the aggregate of control laws u(t)satisfying constraint (1.37) such that the expectation E xðtÞ½ � of the respective

solution of the stochastic system satisfies inclusion (1.36).

Theorem 1.5 In Problem 1.5 with l 2 L and for any t ¼ 1; . . . ; n; let the randomvariables xðtÞ be absolutely continuous and have zero expectations. Let the setsXðtÞ;UðtÞðt0Þ � xvj ðt0Þ ¼ 0 be closed and bounded for all t, let function f satisfy theLipschitz condition, and let function Ft be continuous. Let the absolute values offunction f (foru 2 Uandl 2 L) and Ft not exceed some linear functions in |x|. Then,if set Uad of the admissible control laws is nonempty, Problem 1.3.5 is solvable.


1.3.3.3 Solvability Conditions for the Variational Calculus Problemof Choice (in the Environment of a Given Finite Set of Algorithms)of the Optimal Control Law for a Discrete-Time StochasticDynamical System

Let’s now consider Problem 1.6 formulated above.

Denote by xvj , the solution to system (1.33), (1.33) for the selected jth parametric

control law (1.25), its adjusted coefficient v, and parameter a:

xvj ðtþ 1Þ ¼ f ðxvj ðtÞ;Gjðv; xvj ðtÞÞ; lÞ þ xðtÞ; t ¼ 0; :::; n� 1; (1.42)

xvj ð0Þ ¼ x0: (1.43)

For the considered problem, we’ll define the set of admissible values Vjad

consisting of such values of v 2 V satisfying condition (1.39), which yields the

respective solution of problem (1.33), (1.34) to satisfy the inclusions

E[Gjðv; xvj ðtÞÞ� 2 UðtÞ; t ¼ 0; :::; n� 1; (1.44)


and

E[xvj ðtÞ� 2 XðtÞ; t ¼ 1; :::; n: (1.45)

Problem 1.6 is called nontrivial if the respective setVjad is nonempty and contains

some open set for any j ¼ 1; :::; r.

Theorem 1.6 In Problem 1.6, let l 2 L, let the sets U(t), X(t), and V be compact,and let the functions f, Gj, and Ft satisfy the Lipschitz condition. Let these functionsalso satisfy the following constraints on growth: The functions jf ðx;Gjðv; xÞ; lj andjFtðxÞjdo not exceed some functions linear in jxjuniformly with respect to v 2 V. Letthe random variable xðtÞ be absolute and continuous and have zero expectation.Then, if set Vj

ad is nonempty, Problem 1.6 is solvable.The proof is presented in the Appendix.

1.3.4 Analysis of the Influence of Uncontrolled ParameterVariations (Parametric Disturbances) on the Solutionof the Variational Calculus Problem of Synthesisand Choice of Optimal Parametric Control Laws

Below we present the results of the analysis of the influence of uncontrolled

parameter variations and the bifurcation point changes under parametric

disturbances in the variational calculus problem of choosing optimal parametric

control laws in the environment of given finite sets of the algorithms with phase

constraints and constraints in the allowed form.

The functional or phase constraints, as well the constraints in the allowed form

of the considered problems, often depend on the values of the vector parameter. The

analysis of similar problems requires first finding sufficient conditions for stability

of optimal values of the criteria considered as a function of the uncontrolled

parameters.

With the application for solving the problems of choosing parametric control

laws in the environment of a given finite set of algorithms, we are required to define

the bifurcation point and conditions of its existence and to analyze the bifurcation

value of this parameter. With the application of parametric control of the mecha-

nisms of market economies, finding the extremal solution of the respective problem

and its type can depend on the values of some uncontrollable parameters, and the

task of defining the bifurcation value becomes practical.


1.3.4.1 Analysis of Continuous Dependence of Optimal Values of theCriteria of Variational Calculus Problems of Synthesis and Choiceof Optimal Parametric Control Laws on the Values of UncontrolledParameters

The following auxiliary definition and lemma are useful for proving the continuous

dependence of optimal values of the criterion of the above variational calculus

problem of synthesis of optimal parametric control laws on the values of uncon-

trolled parameters.

Definition Let the family of functions KaðuÞ; a 2 A and u 2 Ua be defined under afamily of subsets fUag of some set U in the Banach space with parameter a in somesubset A of a Euclidean space. The familyfUag is called K-continuous in set A if, forany e>0; there exists a number such that if inequalitiesja� bj � d and a; b 2 Ahold true, then for any point ub 2 Ub, there exists some point ub 2 Ub such thatinequality jKaðuaÞ � KaðubÞj<e holds true.

By this definition, the family of sets fUag is K-continuous if, in the case of

sufficient proximity of parameter b to a for any element from set Ua, there exists an

element of set Ub that is arbitrary close to it (by the value of the functions).

Lemma 1.1 Let A and U be some subsets of Euclidean and Banach spaces,respectively. Let U be compact; all elements in the family of closed subsets fUaglie in U. Let the mapping ða; uÞ ! KaðuÞbe continuous on the productA� U. Let thefamily of subsets fUag be continuous in all points of some neighborhood at pointa0 2 A. Then the mapping a ! maxu2Ua

KaðuÞ is continuous at point a0 2 A.The proof is presented in the appendix.

We’ll now formulate the theorem on sufficient conditions of the continuous

dependence of optimal values of criterion of Problems 1.1, 1.3, and 1.3.5 on the

uncontrolled parameter.

For all aforesaid problems, we will assume that sets X(t) andU(t) are compact, and

sets [t2ðt0;t0þTÞ

XðtÞ and [t2ðt0;t0þTÞ

UðtÞ for Problems 1.1, 1.3, and 1.3.5 are bounded; sets

L are connected and open. It is also assumed that fulfillment of the conditions of

Theorems 1.1, 1.3, and 1.5, which guarantee the continuous dependence of the

considered system, states xu;lðtÞ (or E[xu;l(t)] for the stochastic problems) and the

optimality criterion of Problems 1.1, 1.3, and 1.3.5; dependence of K ¼ Kðu; lÞ onthe respective values of control parameters or control laws (u) and uncontrolled

parameters (l) in the sets determined by theorems stated and with the metrics

indicated above.

Theorem 1.7 Let the conditions of Theorems 1.1, 1.3, and 1.5 hold true for anyl 2 L. Then the optimal values for the criteria of respective Problems 1.1, 1.3, and1.3.5 are continuous functions of parameter l 2 L:



Let’s now analyze the conditions of continuous dependence of optimal values of

the criterion of the variational calculus problems of choice of the parametric control

laws on uncontrolled parameters.

Theorem 1.8 Let the conditions of Theorems 1.2, 1.4, and 1.6 hold true for anyl 2 L . Then for a selected law number j, the optimal values of criteria Kj

corresponding to Problems 1.2, 1.3.4, and 1.6 are continuous functions of parame-ter l 2 L.


Corollary 1.1 If the conditions of Theorem 1.8 hold, the optimal values of thecriteria of Problems 1.2, 1.3.4, and 1.6 are continuous functions of parameterl 2 L:

The proof is given in the appendix.

1.3.4.2 Analysis of Bifurcation Points of the Variational CalculusProblems of Choice of Parametric Control Laws in theEnvironment of a Given Finite Set of Algorithms

Let’s introduce a definition characterizing such values of uncontrolled parameter

l 2 L; which admits a change of one optimal control law to another.

Definition The parameter l 2 L is said to be the bifurcation point for Problem1.2, 1.3.4, or 1.6 if there exist two different numbers of control laws i; j 2 f1; :::; rgsuch that the relation

maxv2Vi

ad;l

Kiðv; lÞ ¼ maxv2Vj

ad;l

Kjðv; lÞ ¼ maxj¼1;:::;r

maxv2Vj

ad;l

Kjðv; lÞ

holds true. An additional requirement is the existence of such a point l1 2 L in anyneighborhood around point l that yields the sole law number j corresponding tomaxj¼1;:::;r

maxv2Vj

ad;l1

Kjðv; l1Þа.

The following theorem establishes sufficient conditions of existence for the

bifurcation point of extremals of the considered variational calculus problems for

choosing a parametric control law in a given finite set of algorithms for the cases of

a continuous-time deterministic or discrete-time deterministic or stochastic

dynamical system.

Theorem 1.9 Let the conditions in Theorem 1.2, 1.4, or 1.6 be fulfilled for any l2 L; where L is an open compact set. Suppose there exist two different values of


parameter l0 and l1 l0 6¼ l1; l0; l1 2 Lð Þ that yield the solutions of the respectiveproblems for choice of the optimal parametric control laws to be attained for twodifferent laws Gj0and Gj1ðj0 6¼ j1Þ in the given finite set; i.e.,

maxj¼1;::;r

j6¼j0

maxv2Vj

ad;l0

Kjðv; l0Þ < maxv2Vj0

ad;l0

Kj0ðv; l0Þ;

maxj¼1;::;r

j 6¼j1

maxv2Vj

ad;l1

Kjðv; l1Þ < maxv2Vj1

ad;l1

Kj1ðv; l1Þ:

Then there exists at least one bifurcation point of extremals in Problem 1.2,

1.3.4, or 1.6.The proof can be found in the appendix.

The following statement immediately follows from Theorem 1.9.

Corollary 1.2 Let the conditions of Theorem 1.2, 1.4, or 1.6 be fulfilled for anyl 2 L; where 0:0012<pt � 0:0480; t ¼ 10; :::; 14; is an open compact set. For l¼ l0; let the control law Gj0 from a given finite set of the parametric control lawsyield the solution of Problem 1.2, 1.3.4, or 1.6. For l ¼ l1; ðl0 6¼ l1; l0; l1 2 LÞ;the control law Gj0 from this set does not result in the solution of the consideredProblem 1.2, 1.3.4, or 1.6. Then there exists at least one bifurcation point of theextremals of the considered problems.


In conclusion, we’ll introduce a numerical algorithm for finding the bifurcation

value of parameter l of one of Problems 1.2, 1.3.4, or 1.6 regarding choice of the

parametric control laws (in the environment of a given finite set of algorithms)

under fulfillment of Theorem 1.9’s conditions.

Connect the points l0 and l1 by a smooth line s � L. Partition this line into nequal parts with sufficiently small steps. For the obtained values bi 2 s; i ¼ 0; :::;n; b0 ¼ l0; bn ¼ l1 define the optimal numbers ji of the control laws solving

Problem 1.2, 1.3.4, or 1.6 with l ¼ bi . Then find the first value of i at which the

respective number of the law differs from j0. In this case, the bifurcation point of

parameter l lies on the arc bi�1; bi of line S.For the obtained section of the line, the algorithm of defining the bifurcation

point with given accuracy e implies the application of the bisection method. As a

result, we find the point g 2 ðbi�1; biÞ, from one side of which, within the limits of

deviation e from value g, the set of lawsGj0 is optimal, but from another side within

the limits of deviation e from the value g, this set is not optimal. From Corollary 1.2,

it follows that there exists a bifurcation point of the solved problem on the

mentioned arc.


1.4 Algorithm of Application of Parametric Control Theoryand Rules of Interaction Between Persons MakingDecisions on Elaboration and Realization of the EffectiveState Economic Policy on the Basis of an InformationSystem for Decision-Making Support

1.4.1 Algorithm of the Application of ParametricControl Theory

The application of the theory of parametric control of market economic evolution

for the definition and implementation of efficient public economic policy developed

here seems to be as follows [7, 11, 53, 54]:

1. The choice of direction (strategy) for economic development of a country on the

basis of estimating its economic state in the context of phases of the economic cycle.

2. The choice of one or several mathematical models addressing the problems of

economic development from the library of mathematical models of economic

systems.

3. The estimate of the adequacy of mathematical models of the problems. The

calibration of mathematical models (parametric identification and retrospective

prediction by the current indices of evolution of the economic system) and

additional verification of the chosen mathematical models via econometric

analysis and political-economic interpretation of sensitivity matrices.

4. The evaluation (if necessary) of indicators of stability and/or structural stability

(robustness) of mathematical models without parametric control in accordance

with the aforesaid methods of estimation of stability indicators and robustness

conditions. If the mathematical model is stable and/or structurally stable, it may

be used after the econometric analysis and political-economic interpretation of

the results of robustness analysis, and for solving the problem of choosing the

optimal control law for economic parameters and the prediction of macroeco-

nomic indicators.

5. If the mathematical model is unstable or nonrobust (structurally unstable), then it

is moved to a reserve section of the model library.

6. The choice of optimal laws of control for economic parameters.

7. The estimation (if necessary) of indicators of stability and/or structural stability

(robustness) of mathematical models with the chosen laws of parametric control

according to the aforesaid methods of estimation of robustness conditions. If the

mathematical model with the chosen laws of parametric control is stable and/or

structurally stable, then from the obtained results, after carrying out the respective

econometric analysis, the completing the political-economic interpretation, and

garnering approval of the decision makers, the obtained results can be put into

practice. If the mathematical model with the chosen laws of parametric control is

unstable and/or structurally unstable, then the choice of parametric control laws

1.4 Algorithm of Application of Parametric Control Theory and Rules. . . 23

must be refined. The corrected decisions on choosing the parametric control laws

are also to be considered according to the above-mentioned scheme.

8. Analysis of the dependence of the chosen optimal laws of parametric control on

the variation of uncontrolled parameters in an economic system. In this regard,

replacing one optimal parametric control law by another is possible.

This aggregate scheme for making decisions on the development and implemen-

tation of an efficient public economic policy via the choice of optimal values of

economic parameters must be maintained by methods of analysis and computer

simulations. This aggregate scheme for making decisions is presented in Fig. 1.1.

As is known, decision making is a process [25, pp. 126, 146] that can be divided

into the following three stages (G. Simon, [23]): information search, searching for

and finding of alternatives, and choosing the best alternative.

The information decision support system (IDSS) for applications based on the

parametric control system is intended for the support decision makers in the field of

economic policy under conditions of small open national economy on the basis of

parametric control theory.

The subjects of interaction of the decision makers (DM) with the IDSS are as

follows:

1. Evaluation of national economic conditions and choice of direction of economic

policy;

2. Choice of one or several econometric models and(or) dynamical models (in the

form of dynamical systems) from the library of mathematical models that are

consistent with economic policy problems;

3. Statement and problem solving of estimation values (variation laws) of eco-

nomic instruments for the economic policy implementation in the chosen

direction;

4. Analysis of the influence of values of exogenous social and economic indicators

on the results of the problem’s solution and estimation of values (variation laws)

of economic instruments of the economic policy implementation in the chosen

direction;

5. Formulating recommendations on the choice of values (variation laws) of

economic instruments of implementation of an economic policy in the chosen

direction.

1.4.2 Rules of Interaction for Decision Makerson the Formulation and Implementationof an Effective Public Economic Policy Basedon the Information Decision Support System

Let’s consider the rules of interaction between the DM and IDSS at the levels of

problems listed above.


Yes No

1. Evaluation of the national economic conditions and choice of direction of economic developmentin coordination with the decision maker

2. Selection of a set consisting of one or several econometric models and (or) dynamic models(models in form of dynamic system) from the library which are consistent with problems of developmentdirection in coordination of the results with the preferences of the decision maker. Parametricidentification of the selected models (if necessary)

3. Evaluation (if necessary) of the stability indicators of each econometric model in the set and/orweak structural stability (robustness) of each dynamic model in the set.

4. Are all models in the setstable or weakly structurally

stable (robust)?

6. Moving of unstable or non-robust models to the reservesection of the model library.

Go to block 2.

5. Statement and solution to problems of evaluating values (law of variation) of economic instruments foran economic policy implementation in the chosen direction based on the set of considered modelsin coordination with the decision maker. Econometric analysis, political-economic interpretation ofthe choice of the optimal economic policy

7. Evaluation (if necessary) of stability indicators of each econometric model in the set with selectedoptimal values of economic instruments. Evaluation (if necessary) of robustness of each dynamical modelin the set with selected optimal values of economic instruments. Econometric analysis, political-economicinterpretation of the results of the analysis of stability and robustness of the mathematical models withthe selected optimal values of economic instruments

No Yes8. Are all models in the

set with selected optimal values ofeconomic instruments stable

or robust?

9. Correction of the constraints in the problems of choice of optimal economicpolicy based on unstable (non-robust) mathematical models. Econometricanalysis, political-economic interpretation of the results of correction ofconstraints. Go to block 7.

10. Analysis of influence of exogenous social and economic indicators on results of solving the problem ofchoice of optimal economic policy in coordination with the decision maker. Econometric analysis,political-economic interpretation of the results of the extremals and adjustment in accordance with thedecision maker

11. Formulation of recommendations on choosing values (variation laws) of economic instrumentsof implementation of economic policy in the chosen direction in coordination with the decision maker

Fig. 1.1 Aggregate scheme of the algorithm for decision making and the implementation

of efficient public economic policy via the choice of optimal values of economic instruments:

(a) Sheet 1; (b) Sheet 2; (c) Sheet 3


1.4.2.1 Rules of Interaction Between the DM and IDSS at the Levelof the Problem “Evaluation of the National Economic Conditionsand Choice of an Economic Policy Direction”

According to G. Simon, a part of the considered problem “evaluation of the national

economic conditions” is connected with the stage of information search for struc-

turing the problem of choosing an economic policy’s direction. It results in forming

the necessary initial information based on the IDSS for the mentioned problem in

the operating model of man–machine procedures of interaction between the DM

and IDSS.

The rules of interaction between the DM and IDSS at the considered stage are

formulated as the following stages:

1. Computations. Administrator(s) of the IDSS presents to the DM the initial

information obtained from the IDSS in the form of, e.g., equilibrium values of

credit rate i, gross national income y in welfare and monetary markets by years

characterizing points of effective demand, and indicators of equilibrium based

on the Keynesian model of yearly all-economic equilibrium; indicators of the

national economic equilibrium based on the model of an open economy of a

small country by years and data on actual annual values of national economy

indicators within the limits of their equilibrium values in their respective mac-

roeconomic markets; causes and characteristics of market cycles in economics;

indicators of activity of economic agents; national economic functioning in the

past period and middle-time forecasting period based on various computable

models of general equilibrium and information obtained on the basis of other

mathematical models of the national economy.

2. DM analysis. Evaluating the presented information, the DM determines if the

information is admissible. If it is, then the procedure of interaction between the

DM and IDSS is executed. Otherwise, the DM requests new information.

3. Computations. If the request for new information can be provided, the adminis-

trator(s) of the IDSS presents the requested information based on working with

the IDSS (possibly, this is made in coordination with the DM).

The process of interaction between the DM and IDSS within the framework of

steps 1–3 continues until the DM receives admissible information.

The other part of the considered problem, “choice of economic policy’s direc-

tion,” is solved in two stages. In the first stage, based on information obtained from

the IDSS, information on the results of monitoring the socioeconomic area

conditions, and political stability and national security protection, the problem of

choosing an economic policy direction is structured in the form of a tree open graph

for multiobjective choice in coordination with the DM in the following steps:

Step 1. Based on the information obtained from the IDSS and the results from

monitoring national economic conditions and properties of a tree open graph

(tree of objectives), the administrator(s) of the IDSS presents a version of

decomposition of the main goal to a subgoal within the framework of hierarchy,


and relative significance [23] to the DM. The formulation of such goals that

cannot be divided and give the final results defined by the main goal is the

complete flag for decomposition of the main goal.

Step 2. Evaluating the presented version of the tree of objectives, the DM

determines whether or not it is admissible. If not, the DM formulates the remarks

and requests adjustment of the presented version of the tree of objectives.

Step 3. The administrator(s) of the IDSS adjusts the initial version of the tree of

objectives in accordance with the remarks and requests of the DM and presents

the renewed version of the tree of objectives for the problem of choosing the

economic policy’s direction.

Steps 2 and 3 must be repeated until the version of the tree of objectives for the

problem of choosing the economic policy’s direction is agreed upon and accepted

by the DM.

The second stage of solving the problem “choice of the economic policy’s

direction” reduces to making the decision based on the tree of objectives via

interaction of the DM with the administrator(s) of the IDSS within the framework

of the following steps, which are realized by a hierarchal analysis method or

paired comparison method (such as ELECTRE I, ELECTRE II, ELECTRE III,

and others), with corresponding means for evaluating the results of coordination

with the DM.

Step 1. Rank the goals of each level by the relative significance of attaining a

higher-level goal.

Step 2. Evaluate the importance of each goal with respect to higher-level goals and

the main goal. The sum of the significance coefficients of goals of each level

must be equal to 1 or 100.

Step 3. Evaluate the significance coefficients of this goal with respect to the main

goal.

Step 4. Choose an economic policy direction based on the results of steps 1–3.

1.4.2.2 Rules of Interaction Between the DM and IDSS at the Levelof Problem “Choice of One or Several Econometric Modelsand (or) Dynamical Models (in Form of Dynamical Systems)from the Library of Mathematical Models That Are Consistentwith Economic Policy Problems”

Solving the stated problem reduces to making a decision via interaction of the DM

with administrator(s) of the IDSS within the framework of the following steps:

Step 1. Based on results of choosing the economic policy’s direction, the adminis-

trator(s) of the IDSS proposes a set of mathematical models (or one model) that

are consistent with the economic policy’s direction from the library of mathe-

matical models.


Step 2. Evaluating the proposed set of mathematical models (one model), the DM

determines whether or not it is admissible. If yes, the interaction between the

DM and IDSS is complete. Otherwise, the DM puts new requests within the

framework of choice from the set of mathematical models (one model).

Step 3. If the DM’s request can be implemented, the IDSS administrator(s) presents

the DM with a new version of the set of mathematical models (one model) from

the library.

The process of interaction between the DM and IDSS continues within the

framework of steps 1–3 until the DM receives the admissible set of mathematical

models (one model).

1.4.2.3 Rules of Interaction Between the DM and IDSS at the Levelof Problem “Statement and Solution of the Problem(s)of Estimation(s) of Values (Variation Laws) of EconomicInstruments for Implementation of the Economic Policyin the Selected Direction”

The considered problem “statement and solution to the problem(s) of estimation(s)

of values (variation laws) of economic instruments for implementation of the

economic policy in the selected direction” is concerned with its agreement at the

level of criterion (criteria) and constraints on the range of variation of economic

instruments. It reduces to making the agreed-upon decisions via the interaction

between the IDSS administrator(s) and the DM at the level of the following steps:

Step 1. Based on the selected direction of the economic policy and agreed-upon

version of the set of mathematical models (one model), the IDSS administrator

(s) proposes the statement of problem(s) of estimation(s) of values (variation

laws) of economic instruments for implementation of the economic policy in the

selected direction to the DM.

Step 2. Evaluating the proposed statement for the problem of estimation(s) of values

(variation laws) of economic instruments for implementation of the economic

policy in the selected direction, the DM determines whether or not it is admissi-

ble with respect to criteria and constraints. If yes, the interaction between the

DM and IDSS is complete. Otherwise, the DM formulates the remarks and

requests with respect to the criterion (criteria) and constraints of the stated

problem.

Step 3. The IDSS administrator(s) presents a new statement for the problem of

estimating values (variation laws) of economic instruments for implementation

of the economic policy in the selected direction, taking into account initial

values and requests of the DM by the criterion (criteria) and constraints of the

initial statement of problem of estimation(s) of values (variation laws) of

economic instruments for implementation of the economic policy in the selected

direction.


The interaction between the DM and IDSS continues until the statement for the

problem of estimation(s) of values (variation laws) of the economic instruments for

implementation of the economic policy in the selected direction is not accepted by

the DM.

Another part of the aforesaid problem, “statement and solution of the problem(s) of

estimation(s) of values (variation laws) of the economic instruments for implementa-

tion of the economic policy in the selected direction,” is solved using two stages.

Based on the interaction between the DM and IDSS, in the first stage, the

methods for solving the formulated problem (if this problem is multiobjective) of

estimation(s) of values (variation laws) of economic instruments for implementa-

tion of the economic policy in the selected direction occur within the framework of

the following procedure:

Step 1. Based on the formulated problem “statement and solution of the problem(s)

of estimation(s) of values (variation laws) of the economic instruments for

implementation of the economic policy in the selected direction” and prelimi-

nary evaluations of characteristics of the methods of decision-making support in

the following classes:

• Methods of solution search with no participation of the DM;

• Methods using the DM’s preferences for constructing the rules of choice of a

unique solution or a small number of Pareto-effective ones;

• Interactive procedures to solving the problem with participation of the DM;

• Methods based on the approximation of the Pareto bound and informing the

DM about this in one or another form. Furthermore, the DM indicates the

most preferable criterion point on the Pareto bound. The preferable solution is

derived for this criterion point.

The methods from the first two classes are based on the construction of a solving

rule, i.e., a rule for finding one or several solutions from an admissible set. The main

difference between the first and second classes is that for the first class, the solving

rule is constructed with no participation from the DM. For the second class,

information about the DM’s preferences is used.

Step 2. Evaluating proposed methods for solving the problem “statement and

solution of the problem(s) of estimation(s) of values (variation laws) of eco-

nomic instruments for implementation of economic policy in the selected direc-

tion,” the DM determines whether or not it is admissible. In it is admissible, the

interaction between the DM and IDSS is complete. Otherwise, the DM suggests

choosing a method for solving the formulated problem “statement and solution

of the problem(s) of estimation(s) of values (variation laws) of economic

instruments for implementation of the economic policy in the selected

direction.”

Step 3. The IDSS administrator(s) illustrates the application of methods proposed

by the DM for solving the formulated problem and evaluating its characteristics.

The interaction between the DM and IDSS continues within the framework of

steps 1–3 until the method for solving the formulated problem “statement and


solution of the problem(s) of estimation(s) of values (variation laws) of economic

instruments for implementation of the economic policy in the selected direction” is

not accepted by the DM.

The second stage for solution of the considered problem is concerned with

solving the formulated problem “solution of the problem(s) of estimation(s) of

values (variation laws) of economic instruments for implementation of the eco-

nomic policy in the selected direction” based on the selected method with partici-

pation of the DM. We’ll illustrate it with the following example.

Example: In the first stage of solving the problem(s) of estimation(s) of values

(variation laws) of economic instruments for implementation of the economic

policy in the selected direction, based on the interaction between the DM and

IDSS, let the methods be selected that use the DM’s preferences for construction

of the rule for choosing a unique solution or a small number of the Pareto-effective

solutions of the multiobjective problem defined here. Let the hierarchal analysis

method [25] be selected as one of these methods.

The interaction between the IDSS administrator(s) and the DM while solving the

“problem(s) of estimation(s) of values (variation laws) of economic instruments for

implementation of the economic policy in the selected direction” according to the

hierarchal analysis method is implemented by the following scheme:

Step 1. For certain values of the exogenous parameters, the IDSS administrator(s)

finds the estimation of the Pareto set P of the considered multiobjective problem

using the method stated in the appendix.

Step 2. For the set of criteria for the considered multiobjective problem, the DM

defines the indicators of relative significance aij:ði; j ¼ 1; :::;m; j>iÞ for all pairsof criteria ðKi;KjÞ of the considered multiobjective problem.

Step 3. The IDSS administrator(s) computes the weight Wi of the linear utility

function UðK1;K2; :::;KmÞ ¼Pmi¼1

WiKi by the method of hierarchal analysis.

Step 4. In the constructed set P, the IDSS administrator(s) determines a point (set of

points) P+ maximizing function U.Step 5. Evaluating the presented point (set of points) P+ and the respective

estimations of values (variation laws) of economic instruments for implementa-

tion of the economic policy in the selected direction, the DM determines whether

it is admissible (Pareto-effective solutions). If it is not, then the DM formulates a

new set of indicators of relative significance of the criterion pair aij.Steps 2–5 are repeated until the version agreed upon with the DM’s Pareto-

effective solutions is accepted.

Step 6.When the set P+ contains more than one point, the DM selects a unique point

in P+ and respective estimations of values (variation laws) of economic instru-

ments for implementation of the economic policy in the selected direction.

We shall illustrate these steps on the basis of the following computational

algorithm for estimation of the Pareto-effective solution of a multiobjective


optimization problem based on parametric control (considering an example of a

computable model with the knowledge sector and a model of a small open

economy).

The combined model, including the computable model of general equilibrium

with the knowledge sector and an econometric model of a small open economy, is

constructed in the following way.

The following two combined models are considered:

Model 1 is a variant of the computable model of general equilibrium with the

knowledge sector. The parameters in this model are determined from known statis-

tical data from the Republic of Kazakhstan for the years 2000–2010 and solution of

the respective parametric identification problem. This model incorporates the

constraints on possible admissible values of its endogenous parameters.

Model 2 is a variant of the model of a small open economy with the coefficients

evaluated on the basis of known statistical data from the Republic of Kazakhstan for

the years 2000–2010 and values of the exogenous economic indicatorsPZ, iZ, and ее andeconomic instruments M and G forecasted in 2011. This model incorporates the

constraints on the possible equilibrium values of its endogenous parameters Y, P, i,and N.

As for the adjusted parameters’ vector, we consider vector p ¼ ðp1; p2; p3Þ wherepi; i ¼ 1; 2; 3, are values of additional investments into budget of 2011 in sectors

1, 2, and 3 of model 1, respectively. Also, the respective valueDG ¼ p1 þ p2 þ p3 isadded to the real volume of public expenses G of model 2.

Let’s define the following three criteria for the two- and three-criteria problems

of parametric control.

1. The calculated normalized value for the GDP of the country from 2011 to 2015

for model 1 is

K1 ¼ YðpÞY0

! max:

2. The calculated normalized value of welfare export in 2011 for model 2 is

K2 ¼ QexðpÞ=Qex0 ! max:

3. The calculated normalized value of the consumption of imported welfare in

2011 (with the negative sign) for model 2 is

K3 ¼ �QimðpÞ=Qim0 ! max:

Here Y0;Qex0; andQim0 are some base values of the respective variables.

Let’s formulate two multiobjective problems of parametric control on the basis of a

combined model.


Problem 1 Find the estimate of the Pareto set on the plane ðK1;K2Þ on the basis ofmodels 1 and 2, criteria K1 and K2, under the following constraints on adjusted

parameters:

p1 0; p2 0; p3 0; p1 þ p2 þ p3 � pm;DG � pm: (1.46)

Here, pm is the maximal value of additional investments from the public budget to

the producing sectors of the economy.

Problem 2 Find the estimate of the Pareto set in space ðK1;K2;K3Þ on the basis ofmodels 1 and 2, criteria K1, K2, and K3, under constraints (1.46) on the adjusted

parameters.

The algorithm to find the estimate of the Pareto sets indicated in the above

problems with the use of methods of concessions is as follows [25]:

Algorithm of Solution to Problem 1 Let the calculated ideal point of values of the

criteria of the stated problem have coordinates ðK1 ;K

2Þ; where K*

1, K*2 are the

maximal values of the criteria for models 1 and 2 for solutions of the respective

monoobjective problems under constraints (1.46). With the attainment of value K1

of the first model criterion, let the value of the second model criterion be equal to

K02 ; whereK

02<K

2 (in caseK02 ¼ K

2 the two-criteria problem is solved). Let’s divide

the numerical interval ½K02;K

2 � into m equal parts with step h ¼ ðK

2 � K02Þ=m .

We obtain the set of points Ki2 ¼ K0

2 þ ih; i ¼ 0; :::;m; where Km2 ¼ K

2 .

For each i ¼ 0; :::;m� 1 , we’ll solve the following auxiliary optimization

problem. Find the maximum of the criterion K1 ¼ K1ðpÞ under constraints (1.46)and K2ðpÞ Ki

2. As a result, we obtain optimum values of the first model criterion

Ki1; i ¼ 0; :::;m; and the set of mþ 1 points

Q ¼ ðKi1; K

i2Þ; i ¼ 0; :::;mg�

, which is

an estimate for the unknown Pareto set.

Algorithm of Solution of Problem 2 Let the calculated ideal point of values of the

criteria of the said problem have the coordinates ðK1 ;K

2;K

3Þ; where K

1 ;K2 ;K

3 are

the maximal values of the criteria for models 1 ðK1Þ and 2 ðK

2;K3Þ for solutions of

the respective monoobjective problems under constraints (1.46). With the attain-

ment of the criterion valueK1 ¼ K1, let the value of the criterionK3 ¼ K01

3 . With the

attainment of the criterion value K2 ¼ K2 , let the value of criterion K3 ¼ K02

3 ;

K03 ¼ minðK01

3 ;K023 Þ andK0

3<K3 (whenK

03 ¼ K

3, the three-criteria problem reduces

to a two-criteria problem).

Let’s divide the numerical interval ½K03;K

3 � into m equal parts with step

h ¼ ðK3 � K0

3Þ=m. We obtain the set of points Ki3 ¼ K0

3 þ ih; i ¼ 0; :::;m; where

Km3 ¼ K

3.

For each i ¼ 0; :::;m� 1, let’s solve the following auxiliary two-criteria optimi-

zation problems. Find an estimate of the Pareto set based on the combined model


for the problem with maximization of criteria K1 and K2 under constraints

(1.46) and

K3ðpÞ Ki3:

The solution to this problem is found by the aforesaid algorithm for estimating

the Pareto set for the two-criteria problem. As a result, for each fixed I, we obtain

the set of spatial points Kij1 ;K

ij2 ; ;K

i3g

n, where j ¼ 0; :::;m:

Finally, the estimation of the unknown Pareto set will consist of all pointsQ¼ Kij1 ;K

ij2 ;K

i3; i; j ¼ 0; :::;m

n odetermined before.

Let the DM’s preferences of the m-criteria problem be defined by the linear

utility function as

UðK1;K2; :::;KmÞ ¼Xmi¼1

WiKi; (1.47)

where Ki are the criteria and Wi>0 are the weights whose values are not known

beforehand.

It is assumed thatPmi¼1

Wi ¼ 1.

According to themethod of hierarchal analysis [25, p. 176], within the framework

of determining the values of the weightsWi, the DM asks to present the indicators ofrelative significance aijði; j ¼ 1; :::;m; j>iÞ for all pairs of criteria ðKi;KjÞ instead ofthe immediate values of the weights. The values of aij are selected from some scale,

e.g., {1,2,. . .,10}, and the number 10 means that the ith criterion is more significant

than the jth criterion. The number 1 means that these criteria are approximately

equivalent with respect to significance assessment.

With absolute logical answers, the DM must receive values

aij ¼ Wi

Wjði; j ¼ 1; :::;m; j>iÞ: (1.48)

It should be noted that for the two-criteria problem (m¼ 2), the values of weights

Wi are uniquely defined by values of the relative significance indicators aij. In this

case, the DM determines the valuea12 ¼ W1

W2. By adding conditionW1 þ W2 ¼ 1, we

obtain the system of two equations with two unknowns, which yields

W2 ¼ 11þa12

;W2 ¼ 1�W1:

According to the hierarchal analysis method, for m > 2, we define a matrix A(m � m) composed of indicators aij. Instead of the values of indicators determined

by the DM, in this matrix we suppose aii ¼ 1 and aij ¼ 1aji

for j<i. Let l be the

maximal eigenvalue of matrix A. Let W be the respective positive identity eigen-

vector. Then the coordinates of vector W ¼ ðW1;W2; :::;WmÞ are considered to be

the desired set of weights.


The unknown Pareto-effective set is estimated as a point (set of points) from

the estimation of the Pareto set which corresponds to the maximal value of the

utility function U.

1.4.2.4 Rules of Interaction Between the DM and IDSS at the Levelof Problem “Analysis of the Influence of Values of ExogenousSocial and Economic Indicators on the Results of the Problemof Solution and Estimation of Values (Variation Laws) ofEconomic Instruments of the Economic Policy Implementationin the Chosen Direction”

The considered problem reduces to decision making based on the interaction

between the DM and IDSS within the framework of the following procedures:

Step 1. Based on the analysis of “statement of the problem(s) of estimation(s) of

values (variation laws) of economic instruments for implementation of the

economic policy in the selected direction,” the IDSS administrator(s) informs

the DM about the planned analysis of the selected set of exogenous socioeco-

nomic indicators (one indicator) for the analysis of their influence on results

of the problem of solution and estimation of values (variation laws) of

economic instruments of the economic policy implementation in the chosen

direction.

Step 2. Evaluating the proposed analysis of the selected set of exogenous socio-

economic indicators (one indicator) for the analysis of their influence on results

of the solution of problem “analysis of influences of the values of exogenous

social and economic indicators on the results of the problem solution and

estimation of values (variation laws) of economic instruments of the economic

policy implementation in the chosen direction,” the DM determines whether the

selected set is admissible. If yes, the procedure is complete. Otherwise, the DM

formulates a suggestion on the set of the exogenous socioeconomic indicators

for the analysis of their influence on the results to the solution of problem

“analysis of influence of the values of exogenous social and economic

indicators on the results of the problem solution and estimation of values

(variation laws) of the economic instruments of the economic policy implemen-

tation in the chosen direction.”

Step 3. Based on the DM’s requests and analysis of the mathematical model of

“statement of the problem(s) of estimation(s) of values (variation laws) of


direction,” the IDSS administrator(s) proposes the new set of exogenous socio-

economic indicators for analysis of their influence on the results of solution to

problem “analysis of influences of the values of exogenous social and economic

indicators on the results of the problem of solution and estimation of values

(variation laws) of economic instruments of the economic policy implementa-

tion in the chosen direction.”


The interaction between the DM and IDSS continues in steps 1–3 until the set

of exogenous socioeconomic indicators for the analysis of their influence on the

results to the solution of problem “analysis of influences of the values

of exogenous social and economic indicators on the results of the problem

solution and estimation of values (variation laws) of the economic instruments

of the economic policy implementation in the chosen direction” is not accepted

by the DM.

1.4.2.5 Rules of Interaction Between the DM and IDSS at the Levelof Problem “Formulation of Recommendations on the Choiceof Values (Variation Laws) of Economic Instruments ofImplementation of the Economic Policy in the Chosen Direction”

The considered problem reduces to making a decision based on the interaction

between the DM and IDSS using the following procedure:

Step 1. Based on the estimation of values of the exogenous socioeconomic

indicators and results of the analysis of dependencies of the solutions to the

problems of estimation of values (variation laws) of economic instruments for

implementation of the economic policy in the selected direction, the IDSS

administrator(s) proposes a recommended version of values (variation laws) of


direction.

Step 2. Evaluating the recommended version of values (variation laws) of economic

instruments for implementation of the economic policy in the selected direction,

the DM determines whether or not the proposed version of values of economic

instruments for implementation of the economic policy in the selected direction

is admissible. If yes, the interaction between the DM and IDSS is complete.

Otherwise, the DM requests new information about possible values of exoge-

nous socioeconomic indicators and respective versions of the values (variation

laws) of economic instruments for implementation of the economic policy in the

selected direction.

Step 3. By request, the IDSS administrator(s) proposes new information about

possible values of exogenous socioeconomic indicators and respective versions

of the values (variation laws) of economic instruments for implementation of the

economic policy in the selected direction to the DM.

The interaction between the DM and IDSS continues within the framework of steps

1–3 until the set of values is accepted by the DM.


1.5 Examples for Application of Parametric Control Theory

1.5.1 Mathematical Model of the Neoclassical Theoryof Optimal Growth

1.5.1.1 Model Description

A mathematical model for economic growth [46] is given by the following system

of two ordinary differential equations containing the time (t) derivative:

dk

dt¼ Aka � c� ðnþ dÞk;

dc

dt¼ c

1� bðaAka�1 � ðdþ pÞÞ:

8>><>>: (1.49)

k is the ratio of capital (K) to labor (L). In this model, the country’s population and

labor force (labor) are not distinguished;

c is the mean consumption per capita;

n is the level of growth (or decrease) of population: LðtÞ ¼ L0ent;

d is the level of capital depreciation, d>0;

p is the discounting level;

e�pt is the discounting function (p>n);

A and a are parameters of the production function y ¼ ’ðkÞ ¼ Aka, where y is theratio of the gross domestic product to the labor force; that is, mean labor productiv-

ity (0<a<1; A>0);

b is a parameter of the social utility function characterizing the mean welfare of

the population:

UðcÞ ¼ Bcb ð0<b<1; B>0Þ:

The first equation in system (1.49) is the fundamental Solow equation taken from

the theory of economic growth. The second equation of this system is derived from

the maximum condition of the objective function

Z10

UðcÞLðtÞe�ptdt ¼ BL0

Z10

eb ln c�ðp�nÞtdt;

characterizing the total welfare of the entire population within time interval 0 � t<1. This function is maximized under the constraints

kð0Þ ¼ k0; k0 ¼ Aka � c� ðnþ dÞk; 0 � cðtÞ � ’ðkðtÞÞ

and constant values of parameters d, n, p, A, B, a, and b.


The solution to system (1.49) will be considered in some closed region Owhose

frontier is a simple closed curve belonging to the first quadrant of the phase plane

R2þ ¼ fk>0; c>0g. kð0Þ ¼ k0; cð0Þ ¼ c0; ðk0; c0Þ 2 O.

1.5.1.2 Analysis of the Structural Stability from a Mathematical Modelof the Neoclassic Theory of Optimal Growth Without ParametricControl

Estimation of robustness (structural stability) of the considered model without

parametric control in a closed region O whose boundary is a simple closed

curve belonging to the first quadrant of the phase plane R2þ ¼ fk>0; c>0g, kð0Þ

¼ k0; cð0Þ ¼ c0; ðk0; c0Þ 2 O relies on the theorem of necessary and sufficient

conditions of robustness [12]. First, let’s prove the following assertion:

Lemma 1.2 System (1.49) has the unique singular point

k ¼ aAdþ p

� � 11�a

;

c ¼ kðnþ dÞð1� aÞ þ p� n

a

� �8>>>><>>>>:

(1.50)

in R2þ. This point is a saddle point of system (1.49).

Proof Setting the right-hand side of the equations of system (1.49) to zero, we

obtain expressions (1.50). Obviously, k>0; c>0. Consider the determinant of the

Jacobian matrix for the right-hand side of (1.50) at the point (k; c):

D ¼ a� 1

ð1� bÞa ðpþ dÞððnþ dÞð1� aÞ þ p� nÞ:

Since for all stated values of the parameters A; a; b; p; n; d of the mathemati-

cal model we haveD<0, it follows that the singular pointk; c is the saddle point ofsystem (1.49).

Theorem 1.10 Let the right-hand sides of the system

x0 ¼ f1ðx; yÞ;y0 ¼ f2ðx; yÞ

((1.51)

be smooth functions in some region O1 � R2, and suppose that system (1.51) has aunique saddle singular point (x; y) in this region. Then system (1.51) is robust inthe closed region O (O � O1) containing the point (x; y).

1.5 Examples for Application of Parametric Control Theory 37

ProofLet’s make sure that system (1.51) does not have cyclic trajectories. Assume the

contrary. Let the region O1 have a cyclic trajectory. Then in its interior there exists

at least one singular point, and the sum of the Poincare indices of the singular points

within this cycle must be 1 [12, p. 117]. But in the regionO1 there is a unique saddle

point with index equal to –1. Thus, we have arrived at a contradiction.

Let’s make sure that the stable and unstable separatrices of the saddle point ðx;yÞ do not form the same trajectory in the region O1. Assume the contrary. Let the

stable and unstable separatrices of the saddle point ( x; y ) constitute the same

singular trajectory g lying in the region O1. Then this trajectory (or, if it exists, the

second trajectory composed of other stable and unstable separatrices), together with

the singular point, is the boundary of the closed cellO2 lying in the regionO1. Let’s

consider the semitrajectoryLþ coming from some point (x1; y1), where (x1; y1) is theinterior point of O2 . Then, by virtue of the absence of cyclic trajectories and the

uniqueness of the equilibrium point, the limit points of Lþ must be the boundary of

the cellO2 (the point (x1; y1) cannot be a unique limit point ofLþ since this point is a

saddle [9, p. 49]). Now, let’s consider the semitrajectory L� coming from the point

ðx1; y1) in the direction opposite Lþ. It is obvious that the boundary ofO2 cannot be

the limit points of L� . Since there are no other singular points and singular

trajectories in the region O2, we have a contradiction.

In accordance with [12, p. 146, Theorem 12], the assertion is proved.

Corollary 1.3 System (1.49) is robust in the closed region O (O � R2þ) contained

inside the point (k; c) for all fixed values of the parametersn; L0; d; p; A; a; B; bfrom the respective ranges of their definition.

In particular, it follows that there are no bifurcations of the phase-plane portrait

of system (1.49) in the regionO under variation of the given parameters within their

range of definition.

1.5.1.3 Choosing Optimal Laws of Parametric Control of MarketEconomic Development Based on a Mathematical Modelof the Neoclassical Theory of Optimal Growth

Consider now the feasibility of the realization of an efficient public policy on the

basis of model (1.49) by choosing the optimal control laws using the capital

depreciation level (d) as an example of the economic parameter.

Choosing the optimal parametric control laws is carried out in the environment

of the set of the following equations:

1:U1ðtÞ ¼ l1DkðtÞkð0Þ þ d; 2:U2ðtÞ ¼ �l2

DkðtÞkð0Þ þ d;

3:U3ðtÞ ¼ l3DcðtÞcð0Þ þ d; 4:U4ðtÞ ¼ �l4

DcðtÞcð0Þ þ d:

(1.52)


Here Ui is the ith law of the control of the parameter d (i ¼ 1; :::; 4);li is the adjustedcoefficient of the ith control law;li 0;d is the constant equal to the basic value ofthe parameter d ; DkðtÞ ¼ kiðtÞ � kð0Þ;DcðtÞ ¼ ciðtÞ � cð0Þ; (kiðtÞ , ciðtÞ ) is the

solution of system (1.49) with the initial conditions kið0Þ ¼ k0; cið0Þ ¼ c0 with useof the control lawUi. Use of the control lawUimeans the substitution of the function

from the right-hand side of (1.52) into system (1.49) instead of the parameter d; t ¼0 is the time of control commencement; t 2 ½0; T�.

The problem of choosing an optimal parametric control law at the level of one

economic parameter d can be formulated as follows: On the basis of mathematical

model (1.49), find the optimal parametric control law at the level of the economic

parameter d in the environment of the set of algorithms (1.52); that is, find the

optimal law from the set {Ui} that maximizes the criterion

K ¼ BL0

ZT0

eb ln ciðtÞ�ðp�nÞtdt ! maxfUi; lig

(1.53)

under the constraints

kiðtÞ � kðtÞj j � 0:09kðtÞ; ðkiðtÞ; ciðtÞÞ 2 O; where t 2 ½0; T�: (1.54)

Here ðkðtÞ; cðtÞÞ is the solution of system (1.49) without the parametric control.

The stated problem is solved in two stages:

– In the first stage, the optimal values of the coefficients li are determined for each

law Ui by the enumeration of their values on the respective intervals (quantized

with a small step) maximizing K under constraints (1.54).

– In the second stage, the law of optimal control of the parameter d is chosen on

the basis of the results of the first stage by the maximum value of the criterion K.

The considered problem was solved under the following conditions:

given parameter values a ¼ 0:5, b ¼ 0:5, A ¼ 1, B ¼ 1, k0 ¼ 4, c0 ¼ 0:8, T ¼ 3,

L0 ¼ 1;

for the following fixed values of the uncontrolled parameters: n ¼ 0:05, p ¼ 0:1;for the basic value of the controlled parameter d ¼ 0:2.The results of the numerical solution of the problem of choosing the optimal

parametric control law at the level of one economic parameter of the economic

system show that the best result K ¼ 1.95569 can be obtained with the use of the

following law:

d ¼ 0; 19DkðtÞ4

þ 0:2: (1.55)

Note that the criterion value without use of the parametric control is equal to K¼1.901038.


1.5.1.4 Analysis of the Structural Stability of the Mathematical Modelof the Neoclassical Theory of Optimal Growth with ParametricControl

Let’s analyze the robustness of system (1.49), where the parameter d is given in

accordance with the solution to the parametric control problem, taking into account

the influence of variations of the uncontrolled parameters n and p by the expression

d ¼ l1k � k0k0

þ d0 (1.56)

with any fixed value of the adjusted coefficient l1>0. Here k0>0 and d0>0 are some

fixed numbers. Substitute (1.56) into the right-hand sides of the system (1.49) and

set them equal to zero. We obtain the following system with respect to the unknown

variables ðk; cÞ (other admissible values of variables and constants are fixed):

Aka � c� nþ l1k � k0k0

þ d0

� �k ¼ 0;

c

1� bðaAka�1 � l1

k � k0k0

þ d0Þ � p

� �¼ 0:

8>>><>>>:

(1.57)

Since the function from the right-hand side of the second equation of system

(1.49) is strictly decreasing as a function of variable k and takes on all values with

k > 0, then it follows that the second equation has a unique solution k . For thissolution, there exists a unique solution c of the first equation in (1.57); that is,

system (1.57) has the unique solution ðk; cÞ . If ðk; cÞ=2R2þ , then, obviously,

system (1.49) with the control law U1 is structurally stable in any closed region

O � R2þ.

Now, let ðk; cÞ 2 R2þ . Let’s find the determinant of the Jacobian of the

functions f1, f2; which are the left-hand sides of the respective equations of system

(1.57) at this point. Since

@f1@c

ðk; cÞ ¼ �1;@f2@k

ðk; cÞ ¼ c

1� bðaða� 1ÞAðkÞa�2 � l1Þ<0;

@f2@c

ðk; cÞ ¼ 0;

the determinant of the matrix isD<0. Therefore, in this case the point ðk; cÞ is thesaddle point of system (1.49) with control law U1. From Theorem 1.10, it follows

that the system is structurally stable in the closed region O � R2þ containing the

point ðk; cÞ inside.In particular, with the use of law (1.55), system (1.49) remains structurally

stable.

The methods above allow the analysis of the robustness conditions for system

(1.49) with the use of the optimal control law d ¼ �l1 c�c0c0

þ d0, when the values of

the parameters ðn; pÞ are within the closed region in R2þ.


1.5.1.5 Finding Bifurcation Points for the Extremals of the VariationalCalculus Problem Based on the Mathematical Modelof the Neoclassical Theory of Optimal Growth with ParametricControl

Let’s analyze the dependence of the results of choosing the parametric control law

at the level of parameter d on the uncontrolled parameters (n, p) with values in some

region (rectangle) L in the plane. In other words, let’s find possible bifurcation

points for the variational calculus problem of choosing the optimal parametric

control law of a given model of economic growth.

As a result of computational experiments, plots of dependencies of the optimal

value ofK in criterion (1.53) on the values of the parameters (n, p) were obtained foreach of four possible laws Ui. Figure 1.2 presents the plots for the laws U1 and U4,

which give the maximum values of the criterion in the region L, the intersection

curve for these surfaces, and the projection of the intersection curve onto the region

of the values of the parameters (n, p) consisting of the bifurcation points of these

parameters. This projection divides the rectangle L into two parts: In one, the

control law U1 is optimal, while in the other one, the law U4 is optimal. Along the

projection itself, both of these laws are optimal.

As a result of this analysis of the dependence of the results of the solution of the

considered variational calculus problem on the values of the uncontrolled parameters

(n, p), one can approach choosing optimal parametric control laws in the following

way: If the values of the parameters (n, p) lie to the left of the bifurcation curve in therectangleL (Fig. 1.2), then the lawU1 is recommended as the optimal law. If the values

of the parameters (n, p) lie to the right of the bifurcation curve in the rectangleL, thenthe lawU4 is recommended as the optimal law. If the values of the parameters (n, p) lieon the bifurcation curve in the rectangle L, then any of the laws U1, U4 can be

recommended as the optimal law.

Opt

imal

val

ues

of c

rite

rion

K

Fig. 1.2 Plots of optimal values of criterion K


1.5.2 One-Sector Solow Model of Economic Growth


The one-sector Solow model of economic growth is presented in the book [19].

The model is described by the system of equations (1.58), which includes one

differential equation and two algebraic equations:

LðtÞ ¼ Lð0Þent;dK

dt¼ �mKðtÞ þ rXðtÞ;

XðtÞ ¼ AKðtÞaLðtÞ1�a:

8>>><>>>:

(1.58)

Here t is the time (in months), L(t) is the number of people engaged in the economy,

K(t) is capital assets, X(t) is the gross domestic product, v is the monthly rate of

increase of the population engaged in the economy, m is the share of basic produc-

tion assets retired for a month, r is the ratio of gross investments to the gross

domestic product, A is the coefficient of neutral process improvement, and a is the

elasticity coefficient of the funds.

1.5.2.2 Estimation of the Model Parameters

In the context of the solution of the problem of preliminary estimation of the

parameters, we are required to estimate the values of the exogenous parameters n,m, r, А, and a by searching for the sense of the minimum of the criterion (sum of

squares of the discrepancies of the endogenous variables).

The parametric identification criterion is as follows:

K ¼ 1

9

Xð0Þ � Xð0ÞXð0Þ

� �2

þ Xð12Þ � Xð12ÞXð12Þ

� �2


� �2

þ


� �2


� �2

þ Kð12Þ � Kð12ÞKð12Þ

� �2

þ


� �2


� �2


� �2!

! min

(1.59)

Here XðtÞ represent data about the gross domestic product of the Republic of

Kazakhstan for the period 2001–2005, KðtÞ are the capital assets of the Republicof Kazakhstan for the period 2002–2005, andXðtÞ andKðtÞ are the calculated valuesof the variables of system (1.59).


In computations, we use a value of L(0) equal to 6.698 and a value of K(0) equalto 4004 (which corresponds to 2001), as well as the mean value of the exogenous

parameter v equal to 0.0017.

The relative value of the mean square deviation of the calculated values of the

endogenous variables from the respective observable values (statistical data) is

equal to 100ffiffiffiffiK

p ¼ 3.8%.

1.5.2.3 Analysis of the Structural Stability of the One-Sector Solow Modelof Economic Growth Without Parametric Control

By applying a numerical algorithm of the estimation of weak structural stability of

the discrete-time dynamical system for the chosen compact set N, defined by the

inequalities 3; 000 � K � 10; 000, 5 � L � 10 in the phase space of the variables

(K, L), we discover that the chain-recurrent setRðf ;NÞ is empty. This means that the

one-sector Solow model of economic growth describing the interaction between the

benefit market and the money market is estimated as weakly structurally stable in

the compact set N.

1.5.2.4 Choosing Optimal Laws of Parametric Control of MarketEconomic Development Based on the Solow Mathematical Model

Let’s consider now the feasibility of the realization of an efficient public policy on

the basis of model (1.58) by choosing the optimal control laws using the ratio of

gross investments to gross domestic product (r) as an example of an economic

parameter.

The choice of optimal parametric control laws is made in the environment of the

following relations:

1Þ rðtÞ ¼ r þ k1KðtÞ � Kð0Þ

Kð0Þ ; 2Þ rðtÞ ¼ r � k2KðtÞ � Kð0Þ

Kð0Þ ;

3Þ rðtÞ ¼ r þ k5XðtÞ � Xð0Þ

Xð0Þ ; 4Þ rðtÞ ¼ r � k6XðtÞ � Xð0Þ

Xð0Þ :

(1.60)

Here ki is the adjusted coefficient of the ith control law, and ki 0; r* is the value ofthe exogenous parameter r obtained as a result of the parametric identification of

the model.

The problem of choosing the optimal parametric control law at the level of one

of the economic parameters d can be formulated as follows: On the basis of

mathematical model (1.58), find the optimal parametric control law at the level of

the economic parameter r in the environment of the set of algorithms (1.60)

maximizing the performance criterion (mean value of the gross domestic product

on the considered time interval)


K ¼ 1

49

X48t¼0

XðtÞ

under the constraints K > 0. The base value of the criterion (without application of

scenarios) is equal to 409.97.

The numerical solution of the problem of choosing the optimal parametric control

law at the level of one economic parameter of the economic system shows that the

best result, K ¼ 511.34, can be obtained with use of the following law:

rðtÞ ¼ r þ 0:268XðtÞ � Xð0Þ

Xð0Þ : (1.61)

The values of the endogenous variables of the model without using scenarios, as

well as with use of the optimal law, are presented in Figs. 1.3 and 1.4.

Years

Scenario 3 is used Without scenario

Fig. 1.3 Capital assets without parametric control and with use of law 3 optimal in the sense of

criterion K

Years

Scenario 3 is used Without scenario

Fig. 1.4 Gross domestic product without parametric control and with use of law 3 optimal in the

sense of criterion K


1.5.2.5 Analysis of the Structural Stability of the One-Sector Solow Modelof Economic Growth with Parametric Control

For carrying out this analysis, the expression for optimal parametric control law

(1.61) is substituted into the right-hand side of the second equation of system

(1.58) instead of parameter r. Then, by applying the numerical algorithm of

estimation of weak structural stability of the discrete-time dynamical system for

the chosen compact set N, defined by the inequalities 3,000 � K � 10,000, 5 � L�10 in the phase space of the variables (K, L), we obtain that the chain-recurrent

set R(f, N) is empty. This means that the one-sector Solow model with the

optimal parametric control law is estimated as weakly structurally stable in

the compact set N.

1.5.2.6 Analysis of the Dependence of the Optimal Value of CriterionK on the Parameter for the Variational Calculus Problem Basedon the Solow Mathematical Model

Let’s analyze the dependence of the optimal value of criterion K on the exogenous

parameter m, the share of the basic production assets retired for a month for

parametric control laws (1.60) with the found optimal values of the adjusted

coefficients ki . Plots of the dependencies of the optimal value of criterion K were

obtained from computational experiments (see Fig. 1.5). Analysis of the presented

plots shows that there are no bifurcation points of the extremals for the given

problem for the analyzed interval of values of the exogenous parameter m.

Opt

imal

val

ue o

f cr

iter

ion

Scenario 1 is used Scenario 3 is used Without parametric control

Fig. 1.5 Plots of the dependencies of the optimal value of criterion K on the exogenous

parameter m


1.5.3 Richardson Model for the Estimation of Defense Costs


The model is described by a system of two linear differential equations with

constant coefficients [20]

dx=dt ¼ ay� mxþ r;

dy=dt ¼ bx� nyþ s:

((1.62)

Here t is the time (in months), x(t) is the defense costs of the first country (group of

countries), y(t) is the defense costs of the second country (group of countries), a is thescale of threat for the first country (group of countries), b is the scale of threat for thesecond country (group of countries),m is the armament costs of the first country (group

of countries), n is the armament costs of the second country (group of countries), r isthe scale of the past damage suffered by the first country (group of countries), and s isthe scale of the past damage suffered by the second country(group of countries).

1.5.3.2 Estimation of Model Parameters

In the context of the solution of the problem of the preliminary estimation of the

parameters, we are required to estimate the values of the exogenous parameters a, b,m, n, r, and s by the searching method in a sense of the minimum of the criterion

(sum of the squares of the discrepancies of the endogenous variables).

The parametric identification criterion is as follows:

K ¼ 1

8

xð1Þ � xð1Þxð1Þ

� �2

þ xð2Þ � xð2Þxð2Þ

� �2


� �2

þ


� �2

þ yð1Þ � yð1Þyð1Þ

� �2


� �2

þ


� �2


� �2!

! min :

(1.63)

Here x*(t) represents statistical data on the armament costs of France and Russia for

the years 1910–1913; y*(t) is statistical data about the armament costs of Germany

and the Dual Monarchy (Austria–Hungary) for the same years; and x(t), y(t) are therespective calculated values of the endogenous variables of system (1.62). The

statistical data (in millions of pounds sterling) are presented in Table 1.1.

The problem of preliminary estimation is solved by the Gauss–Seidel method

with the discrete divisor of the estimation range equal to 100,000. The number of

iterations of the algorithm is 50. To improve the result of parameter estimation, a

series of 1,000 experiments on random settings of the initial values of the estimated

exogenous parameters from the ranges of their estimation was conducted.


As a result of solving the problem of the preliminary estimation of the

parameters, the following values were obtained: a ¼ 0.4846, b ¼ 0.3498, m ¼0.2526, n ¼ 0.4390, r ¼ 0.3387, s ¼ –0.3386.

The relative value of the mean square deviation of the calculated values of the

endogenous variables from the corresponding observable values (100ffiffiffiffiK

p) is 3.2819%.

1.5.3.3 Analysis of the Structural Stability of the RichardsonMathematical Model Without Parametric Control

For obtained values of the parameters of system (1.62), its stationary point has the

coordinates x0 ¼ 0:2625; y0 ¼ � 0:5273ð Þ and does not lie in the first quadrant of

the phase plane R2þ ¼ fx>0; y>0g . Therefore, system (1.62) is robust for any

closed region O � R2þ.

1.5.3.4 Choosing Optimal Laws of Parametric Control of MarketEconomies on the Basis of the Richardson Mathematical Model

Let’s consider now the feasibility of the realization of an efficient public policy on

the basis of model (1.62) by choosing the optimal control laws using the threat level

for the second group of countries, b, as an example of the parameter.


of the following relations:

0Þ bðtÞ ¼ b þ k1XðtÞ � Xð0Þ

Xð0Þ ;

1Þ bðtÞ ¼ b � k2XðtÞ � Xð0Þ

Xð0Þ ;

2Þ bðtÞ ¼ b þ k3YðtÞ � Yð0Þ

Yð0Þ ;

3Þ bðtÞ ¼ b � k4YðtÞ � Yð0Þ

Yð0Þ :

(1.64)

Here ki is the coefficient of the scenario, and b* is the value of the exogenous

parameter b obtained as a result of the preliminary estimation of the parameters.


of the economic parameters can be formulated as follows. On the basis of mathe-

matical model (1.62), find the optimal parametric control law at the level of the

economic parameter b in the environment of the set of algorithms (1.64)

maximizing the performance criterion

Table 1.1 Statistical data on

endogenous variables of the

Richardson model

Year 1909 1910 1911 1912 1913

t 0 1 2 3 4

x* 115.3 123.4 132.8 144.4 167.4

y* 83.9 85.4 90.4 97.7 112.3


K ¼ 1

T

ZT0

yðtÞdt; (1.65)


yðtÞ � 1:1� xðtÞ (1.66)

Here the interval of control [0, T] corresponds to the years 1909–1913.

Numerical solution of the problem of choosing the optimal parametric control

law at the level of one economic parameter of the economic system shows that the

best result, K ¼ 111.51, can be obtained with use of the following law:

bðtÞ ¼ 0:3498þ 0:3208XðtÞ � Xð0Þ

Xð0Þ (1.67)

Note that the basic value of the criterion (without control) is equal toK¼ 96.8722.

The values of the endogenous variables of the model without the parametric

control, as well as with use of the parametric control, are presented in Figs. 1.6 and 1.7.

1.5.3.5 Analysis of Structural Stability of the Richardson MathematicalModel with Parametric Control

For carrying out this analysis, the expression for the optimal parametric control law

(1.67) is substituted into the right-hand side of the second equation of system (1.62)

instead of the parameter b. Then, by applying the numerical algorithm of the estima-

tion of the weak structural stability of the discrete-time dynamical system for the

chosen compact set N defined by the inequalities 100 � X � 150, 80 � Y � 120 in

the phase space of the variables (K, L), we find that the chain-recurrent set Rðf ;NÞ isempty. This means that the Richardson mathematical model with the optimal

parametric control law is estimated as weakly structurally stable in the compact setN.

Years

Fig. 1.6 Armament costs of the first group of countries without parametric control and with use of

the optimal law of parametric control. without parametric control, – law 0 is used


1.5.3.6 Analysis of the Dependence of the Optimal Value of CriterionK on the Parameter for the Variational Calculus ProblemBased on the Richardson Mathematical Model

Let’s analyze the dependence of the optimal value of the criterion K on the

exogenous parameter a, the threat level for the first group of countries for

parametric control laws (1.64) with the obtained optimal values of the adjusted

coefficientski . From computational experiments, the plots of dependencies of the

optimal value of the criterion K were obtained (see Fig. 1.8). Analysis of these plots

Law 1 is used

Opt

imal

val

ue o

f cr

iter

ion

Law 2 is usedLaw 3 is usedLaw 4 is used

Fig. 1.8 Plots of dependencies of the optimal value of criterion K on the exogenous parameter a

Years

Fig. 1.7 Armament costs of the second group of countries without parametric control and with use

of optimal law of parametric control. without parametric control, – law 0 is used


shows that there are no bifurcation points of the extremals of the problem for the

analyzed interval of the values of the exogenous parameter a. There are bifurcationpoints of the extremals in this case for the values a ¼ 0.315 and a ¼ 0.345.

1.5.4 Mathematical Model of a National Economic SystemSubject to the Influence of the Share of Public Expenseand the Interest Rate of Government Loans on EconomicGrowth


The mathematical model of a national economic system for analysis of the influ-

ence of the ratio of public expense to the gross domestic product and the influence

of interest rate on the rate of government loans on economic growth proposed in

[34], after appropriate transformation, is given by

dM

dt¼ FI

pb� mM; (1.68)

dQ

dt¼ Mf � F

p; (1.69)

dLG

dt¼ rGL

G þ FG � npF� nLsRL � nOðdP þ dBÞ; (1.70)

dp

dt¼ �a

Q

Mp; (1.71)

ds

dt¼ s

Dmax 0;

Rd � RS

RS

�;RL ¼ minfRd;RSg; (1.72)

Lp ¼ 1� xx

LG; (1.73)

dp ¼ 1� xx

br2LG; (1.74)

dB ¼ br2LG; (1.75)

x ¼ n1� d

1� snp

� �1�dd

!; (1.76)


Rd ¼ Mx; (1.77)

f ¼ 1� 1� 1� dn

x

� � 11�d

; (1.78)

F0 ¼ �0pMf ; (1.79)

FG ¼ ppMf ; (1.80)

FL ¼ ð1� nLÞsRd; (1.81)

FI ¼ 1� xxþ ð1� xÞnp

�ð1� npÞFG � �n0ðdB þ dPÞ þ npF0

� �nL � ð1� nLÞnp sRL�þ ðm þ rGÞLp;

(1.82)

F ¼ F0 þ FG þ FL þ FI; (1.83)

RS ¼ PA0 expðlptÞ

1

1þ no; o ¼ FL

pP0 expðlptÞ : (1.84)

Here М is the total productive capacity;

Q is the total stock-in-trade in the market with respect to some equilibrium state;

LG is the total public debt;

p is the level of prices;

s is the rate of wages;Lp is the indebtedness of production;dp and dB are the business and bank dividends, respectively;

RS and Rd are the supply and demand of the labor force;

D and v are the parameters of the function f(x),x is the solution to the equation f 0ðxÞ ¼ s

p ;

ФL and ФО are consumer expenditures of workers and owners, respectively;

ФI is the flow of investment;

ФG is the expenditure on consumers by the state;

x is the norm of reservation;

b is the ratio of the arithmetic mean return from business activity and the rate of

return of rentiers;

r2 is the deposit interest rate;rG is the interest rate of public bonds;

�0 is the coefficient of the propensity of owners to consume;

p is the share of consumer expenditure by the state in the gross domestic product;


np, nО, and nL are payment flow, dividends, and income taxes of workers,

respectively;

b is the norm of fund capacity of the unit of power;

m is the coefficient of the power unit retirement as a result of degradation;

m* is the depreciation rate;

a is the time constant;

D is the time constant defining the typical time scale of the wage relaxation process;

P0 and P0A are the initial number of workers and total number capable, respectively;

lp > 0 is the set rate of population growth;

o is per capita consumption in the group of workers.

The equation and relations from mathematical model (1.68), (1.69), (1.70),

(1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81),

(1.82), (1.83), and (1.84) correspond to the respective expressions from [33]

possibly after some simple transformations. Thus, the differential equation (1.68)

results from Ref. [33]’s (3.2.18), (3.2.6); (1.69) results from (3.2.19) and (3.2.8);

(1.70) is derived from (3.2.26) by substituting the expression for (FGК – НG) from

(3.2.25); (1.71) represents (3.2.9); (1.72) represents (3.2.30); expression (1.73)

represents the expression from page 150 [33]; expressions (1.74) and (1.75) repre-

sent expressions from (3.2.39); expression (1.76) represents the solution of (3.2.10)

f 0ðxÞ ¼ sp , where function (1.78) is defined on page 157 of [33]; expression (1.77)

represents one of expressions (3.2.10); relation (1.79) is derived from (3.2.15) and

(3.2.8); relation (1.80) is derived from (3.2.16) and (3.12); relation (1.81) is derived

from (3.2.22); expression (1.82) represents (3.2.36); expression (1.83) is (3.2.11);

expressions (1.84) are derived from (3.2.12), (3.2.13), and (3.2.14).

The model parameters and the initial conditions for differential equations (1.68),

(1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79),

(1.80), (1.81), (1.82), (1.83), and (1.84) are obtained on the basis of the economic

data of the Republic of Kazakhstan for the years 1996–2000 [37] (r2 ¼ 0.12; rG ¼0.12; b ¼ 2; np ¼ 0.08; nL ¼ 0.12; s ¼ 0.1; nО ¼ 0.5; m ¼ m* ¼ 0.012; D ¼ 1) or

estimated by solving the parametric control problem [x ¼ 0.1136; p ¼ 0.1348; d ¼0.3; n ¼ 34; �О ¼ 0.05; b ¼ 3.08; a ¼ 0.008; Q(0) ¼ � 125,000].

As illustrated in Table 1.2, presenting the results of parametric identification, the

relative value of the mean square deviation of the calculated values of variables

from the respective observed values is less than 5%.

In Table 1.2: М*, М**, p*, p** are the respective values of the total productive

capacity and the product price, both measured and model (calculated) ones.

Table 1.2 Parametric

identification resultsYear M* M** p* p**

1998 144,438 158,576 1.071 1.09

1999 168,037 183,162 1.160 1.20

2000 216,658 212,190 1.310 1.29


http://dx.doi.org/10.1007/978-1-4614-4460-2

http://dx.doi.org/10.1007/978-1-4614-4460-2

1.5.4.2 Analysis of the Structural Stability of the Mathematical Modelof the National Economic System Subject to the Influence of the Shareof Public Expenses and Interest Rate of Government Loans WithoutParametric Control

Let’s analyze the robustness (structural stability) of model (1.26), (1.27), (1.28),

(1.29), (1.30), (1.31), (1.32), (1.33), (1.34), (1.35), (1.36), (1.37), (1.38), (1.39),

(1.40), (1.41), and (1.42) on the basis of the theorem establishing the sufficient

conditions of structural stability [67] within a compact region of the phase space.

Assertion 1.1 Let N be a compact set lying in the region ðM>0; Q<0; p>0Þ orðM>0; Q>0; p>0Þ of the phase space of the system of differential equationsderived from (1.26), (1.27), (1.28), (1.29), (1.30), (1.31), (1.32), (1.33), (1.34),(1.35), (1.36), (1.37), (1.38), (1.39), (1.40), (1.41), and (1.42), that is, the four-dimensional space of variables ðM; Q; p; LGÞ . Let the closure of the interior ofN coincide with N. Then the flux f defined by (1.26), (1.27), (1.28), (1.29), (1.30),(1.31), (1.32), (1.33), (1.34), (1.35), (1.36), (1.37), (1.38), (1.39), (1.40), (1.41), and(1.42) is weakly structurally stable on N.

One can choose N as, for instance, the parallelepiped with boundariesM ¼ Mmin;M ¼ Mmax; Q ¼ Qmin; Q ¼ Qmax; p ¼ pmin; p¼pmax; LG ¼ LGmin; LG ¼ LGmax.Here 0<Mmin<Mmax , Qmin<Qmax<0 or 0<Qmin<Qmax , 0<pmin<pmax , LGmin<

LGmax.

Proof We’ll first prove that semitrajectory of the flux f starting from any point of

the set N with some value t (t > 0) leaves N.Consider any semitrajectory starting in N. With t>0, the following two cases are

possible; namely, all the points of the semitrajectory remain in N, or for some t the

point of the semitrajectory does not belong to N. In the first case, from (1.29) dpdt

¼ �a QM p of the system, it follows that for all t>0; the variable p(t) has a derivative

greater than some positive constant with Q<0 or less than some negative constant

withQ>0; that is, p(t) increases infinitely or converges to zero for increasing valuesof t. Therefore, the first case is impossible, and the orbit of any point in N leaves N.

Since any chain-recurrent set Rðf ;NÞ lying within N is an invariant set of this

flux, it follows that when it is nonempty, it consists of only whole orbits. Hence, in

the considered case Rðf ;NÞ is empty. The assertion follows from Theorem A [67].

1.5.4.3 Choosing the Optimal Laws of Parametric Control of MarketEconomic Development on the Basis of the Mathematical Modelof a Country Subject to the Influence of the Share of PublicExpenses and the Interest Rate of Government Loans

Let’s now consider the ability of the realization of efficient public policy by

choosing the optimal control laws using the following parameters: the share of

consumers’ public expenses in the gross domestic product p, the interest rate of thegovernment loans rG, and the norm of reservation x.


Evaluate the ability of choosing the optimal laws of parametric control in the

following order:

– Choosing the optimal control law at the level of one of the economic parameters

(x, p, rG)– Choosing the optimal pair of parametric control laws from the set of

combinations of two economic parameters out of three

– Choosing the optimal set of three parametric control laws for three economic

parameters


of the following relations:

1:U1jðtÞ ¼ þk1jDMðtÞMðt0Þ þ constj;

2:U2jðtÞ ¼ �k2jDMðtÞMðt0Þ þ constj;

3:U3jðtÞ ¼ þk3jDpðtÞpðt0Þ þ constj;

4:U4jðtÞ ¼ �k4jDpðtÞpðt0Þ þ constj;

5:U5jðtÞ ¼ þk5jDMðtÞMðt0Þ þ

DpðtÞpðt0Þ

� �þ const

j;

6:U6jðtÞ ¼ �k6jDMðtÞMðt0Þ þ

DpðtÞpðt0Þ

� �þ const

j j:

(1.85)

Here Uij is the ith control law of the jth parameter; the case i ¼ 1; :::; 6; j ¼ 1; 2; 3,j¼ 1 corresponds to the parameter x; the case j¼ 2 corresponds to the parameter p;the case j ¼ 3 corresponds to the parameter rG; kij is the nonnegative adjusted

coefficient of the ith control law of the jth parameter; constj is a constant equal to

the estimation of the values of the jth parameter as a result of parametric

identification.


of the economic parameters (x, p, rG) can be formulated as follows:

On the basis of the mathematical model (1.68), (1.69), (1.70), (1.71), (1.72),

(1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and

(1.84), find the optimal parametric control law Uij in the environment of the set of

algorithms (1.75) minimizing the criterion

K ¼ 1

T

Zt0þT

t0

pðtÞdt ! minfUij;kijg

(1.86)



MðtÞ �MðtÞj j � 0; 09MðtÞ; ðMðtÞ; QðtÞ; LGðtÞ; pðtÞ; sðtÞÞ 2 X ;

0 � Uij � aj; i ¼ 1; :::; 4; j ¼ 1; 2; t 2 ½t0; t0 þ T�; (1.87)

where М**(t) is the value of the total production capacity without parametric

control, aj is the maximum possible value of the jth parameter, and X is the compact

set of possible values of the system variables.

The stated problem is solved in two stages:

– First, the optimal values of the coefficients kij are determined for each law Uij by

enumerating their values on the intervals ½0; kmij Þ quantized with a step equal to

0.01, minimizing K under constraints (1.87). Here kmij is the first value of the

coefficient violating (1.87).

– Second, the law of the optimal control of the specific parameter (out of three) is

chosen on the basis of the results of the first stage by the minimum value of the

criterion K (1.86).

The results of the numerical solution of the first stage of the stated problem for

{Uij} are presented in Table 1.3.

The analysis of Table 1.3 in accordance with the requirements of the second

stage of the stated problem solution makes it possible to propose at the level of one-

parameter control of the market economy the following law for the parameter p:

p ¼ �0:84Dp1

þ 0:1348;

Table 1.3 First stage of

the numerical solution of

the stated problem of

choosing the optimal law

of parametric control

Notation for

parametric

control laws

Optimal value

of coefficient

of law

Value of

criterion K

U11 0.220 1.0980

U21 0.000 1.1734

U31 0.156 1.0370

U41 0.000 1.1734

U51 0.160 1.0900

U61 0.000 1.1734

U12 0.000 1.1734

U22 0.110 1.0900

U32 0.000 1.1734

U42 0.840 1.0230

U52 0.000 1.1734

U62 0.080 1.0840

U13 0.000 1.1734

U23 0.290 1.1700

U33 0.000 1.1734

U43 0.390 1.1701

U53 0.000 1.1734

U63 0.230 1.1702


which provides the minimum value of K ¼ 1.023 among all the laws Uij.

The problem of choosing the pair of optimal parametric control laws for simul-

taneous control of three parameters can be formulated as follows: Find the optimal

pair of parametric control laws (Uij, Uum) in the set of combinations of two

economic parameters out of three on the basis of the set of algorithms (1.85)

minimizing the criterion

K ¼ 1

T

Zt0þT

t0

pðtÞdt ! minðUij;kijÞ;ðUum;kumÞf g

;

i; v ¼ 1; :::; 6; j; m ¼ 1; 2; 3; j<m

(1.88)

under constraints (1.87).

The problem of choosing the optimal pair of laws is solved in two stages:

– In the first stage, the optimal values of the coefficients (kij, kum) are determined

for the chosen pair of control laws (Uij, Uum) by enumeration of their values from

the respective intervals quantized with the step equal to 0.01 minimizing K under

constraints (1.87).

– In the second stage, the optimal pair of parametric control laws is chosen

on the basis of the results of the first stage by the minimum value of the

criterion K.

The results of the numerical solution of the first stage of the stated problem of

choosing the optimal pair of parametric control laws are summarized in 18 tables

similar to Table 1.4, differing in the control law expression by at least one

parameter.

Choosing the optimal pair of parametric control laws according to the

requirements of the second stage, based on analysis of the content of 18 tables,

makes it possible to recommend the implementation of the control laws for the

Table 1.4 First-stage results of the numerical solution of the stated problem of choosing the

optimal pair of laws

Pairs of parametric control laws

Criterion

value

First law Second law

Law

denotation

Optimal

coefficient value

Law

denotation

Optimal

coefficient value

U21 0.000 U12 0.000 1.1734

U21 0.185 U22 0.123 0.9810

U21 0.000 U32 0.000 1.1734

U21 0.000 U42 0.840 1.0230

U21 0.000 U52 0.000 1.1734

U21 0.167 U62 0.167 0.9820


parameters (p, x) for the case of two-parameter control of the market economic

mechanism as follows:

x ¼ �0:185DMðtÞ139345

þ 0:1136; p ¼ �0:123DMðtÞ139345

þ 0:1348;

which provides the minimal value K ¼ 0.981 among all the pairs (Uij, Uum).

The problem of choosing the optimal set of three laws for simultaneous control

of the three parameters can be formulated as follows: Find the optimal set of

three-parameter control laws at the level of three parameters on the basis of the

set of algorithms (1.85) minimizing the criterion

K ¼ 1

T

Zt0þT

t0

pðtÞdt ! minðUi1;ki1Þ;ðUn2;kn2Þ;ðUg3;kg3Þf g;

i; n; g ¼ 1; :::; 6

(1.89)


This problem is solved in two stages:

– First, the optimal values of the coefficients are determined for the chosen set of

three control laws (Ui1;Un2;Ug3) by enumeration of their values from respective

intervals (quantized with the step equal to 0.01 for each coefficient) minimizing

K under constraints (1.87).

– Second, the optimal set of three parametric control laws is chosen on the basis of

the results of the first stage by the minimum value of the criterion K.

The results of the numerical solution of the first stage of the problem are featured

in 36 tables similar to Table 1.5 differing in the control law expression by at least

one parameter.

Table 1.5 First-stage results of the numerical solution of the stated problem of choosing optimal

set of three laws

Set of three parametric control laws

Criterion

value

First law of the set Second law of the set Third law of the set

Law

denotation

Optimal

coefficient

value

Law

denotation

Optimal

coefficient

value

Law

denotation

Optimal

coefficient

value

U21 0.185 U22 0.123 U13 0.00 0.981

U21 0.185 U22 0.123 U23 0.03 0.980

U21 0.185 U22 0.123 U33 0.00 0.981

U21 0.185 U22 0.123 U43 0.00 0.981

U21 0.185 U22 0.123 U53 0.00 0.981

U21 0.185 U22 0.123 U63 0.00 0.981


The choice of the optimal set of three laws according to the requirements of the

second stage makes possible a recommendation for implementing the control laws

for the parameters x, p, rG:

xðtÞ ¼ �0:185DMðtÞ139345

þ 0:1136; pðtÞ ¼ �0:123DMðtÞ139345

þ 0:1348;

rGðtÞ ¼ �0:03DMðtÞ139345

þ 0:01;

providing the minimum value K ¼ 0.980 among all combinations (Ui1;Un2;Ug3).

Thus, this work shows one possible way of choosing efficient laws of parametric

control of market economy.

In addition, alternative formulations and solutions of the problem of choosing

the optimal set of laws have been considered.

Choosing optimal parametric control laws on the basis of model (1.68), (1.69),

(1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80),

(1.81), (1.82), (1.83), and (1.84) at the level of one of two parameters x (j¼ 1) and p(j ¼ 2) was carried out under the following set of assumptions:

1:U1jðtÞ ¼ k1jM �M0

M0

þ constj; 2:U2jðtÞ ¼ �k2jM �M0

M0

þ constj;

3:U3jðtÞ ¼ k3jp� p0p0

þ constj; 4:U4jðtÞ ¼ �k4jp� p0p0

þ constj:(1.90)

Here Uij is the ith control law of the jth parameter (i ¼ 1; :::; 4; j ¼ 1; 2); the case

j¼ 1 corresponds to the parameter x; the case j¼ 2 corresponds to the parameter p;kij is the adjusted coefficient of the ith control law of the jth parameter, kij 0 ;

constj is a constant equal to the estimate of the value of the jth parameter as a result

of parametric identification;M0, p0 are the initial values of the respective variables;the mention of (1.68), (1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76),

(1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84) means that the

functions Uij from (1.90) should be substituted into (1.68), (1.69), (1.70), (1.71),

(1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82),

(1.83), and (1.84) instead of parameter x or p.The problem of choosing the optimal parametric control law at the level of one

out of two economic parameters (x, p) can be formulated as follows: On the basis of

the mathematical model (1.68), (1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75),

(1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84), find the

optimal parametric control law at the level of one of two economic parameters

(x, p) in the environment of the set of algorithms (1.90); that is, find the optimal

control law from the set {Uij} and its adjusted coefficient maximizing the criterion

K ¼ 1

T

Zt0þT

t0

YðtÞdt; (1.91)


where Y ¼ Mf is the gross domestic product, under constraints

pijðtÞ � pðtÞ�� 0:09pðtÞ; ðMðtÞ; QðtÞ; LGðtÞ; pðtÞ; sðtÞÞ 2 X ;

0 � uj � aj; i ¼ 1; :::; 4; j ¼ 1; 2; t 2 ½t0; t0 þ T�: (1.92)

Here aj is the maximum value of the jth parameter; pðtÞ is the model (calculated)

value of the price level without parametric control; pijðtÞ is the value of the price

level with the Uij th control law; X is the compact set of admissible values of the

given variables.

The problem formulated above is solved in two stages:

– First, the optimal values of the coefficients kij are determined for each law Uij by

enumerating their values on the intervals ½0; kmij Þ quantized with step size 0.01

minimizing K under constraints (1.92). Here kmij is the first value of the coefficient

violating (1.92).

– Second, the law of optimal control of the specific parameter (one of three) is

chosen on the basis of the results of the first stage by the maximum value of

criterion K (1.86).

Numerical solution of the problem of choosing the optimal law of parametric

control of a national economic system at the level of one economic parameter

shows that the best result K ¼ 177662 can be obtained by using the following

control law:

x ¼ �0:095M �M0

M0

þ 0:1136: (1.93)

Note that the criterion value without the use of parametric control isK ¼ 170784.

1.5.4.4 Parametric Control of Market Economic Development withVarying Objectives on the Basis of a Mathematical Model Subjectto the Influence of the Share of Public Expenses and the InterestRate of Governmental Loans

Let’s consider the parametric control of inflation processes in market economies on

the basis of themathematicalmodel (1.68), (1.69), (1.70), (1.71), (1.72), (1.73), (1.74),

(1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84). One can

accept the level of prices as a feasible characteristic of the development of economic

processes, taking into account that for the period 1996–2000, the years included in the

research, the economy of Kazakhstan was on the rise. The level of prices can be used

as some measure of the efficiency of the production of goods and services and can be

considered as characterizing the presence of inflationary or deflationary processes.

Within the context of price-level variation, one can conditionally distinguish two

regions, namely, admissible and inadmissible regions of the price-level variation.


The inadmissible region (В) of the price-level variation can be defined by the

inequalities pðtÞ � plðtÞ or pðtÞ puðtÞ , where plðtÞ is the admissible lower

bound of the price-level variation and puðtÞ is the admissible upper bound (pl(t)< pu(t), 0< t< T). Satisfying the inequalities pðtÞ � plðtÞ shows that there exists

some deflation process, whereas satisfying pðtÞ � puðtÞ indicates excessive infla-

tion. The admissible region (A) of the price-level variation can be defined by the

inequality plðtÞ<pðtÞ<puðtÞ; 0<t<T.Depending on the region, A or B, to which the price-level values belong, the

problem of choosing the optimal parametric control laws can be formulated as

the following problems:

– In region A, parametric control is not applied.

– In region B, we are interested in finding and realizing such parametric control

laws in the environment of some given set of algorithms that minimize the

criterion characterizing the transient performance under applied constraints on

the possible values of the respective indices of the economic state and control

parameters (block B).

The proposed approach is implemented as follows: First, the process of

simulating the economic system is begun based on the result of the parametric

identification problem. Regions A and B are determined as a preliminary to the

price-level values. The algorithm for computer simulation has a logical condition

determining the presence of the level of prices in one or another admissible region.

During this process, if it turns out that the value of p(t) is in region B, than block B is

switched on, solving the problem of taking the object out of inadmissible region Bto admissible region A. If the value of p(t) turns out to be in region A, the parametric

control is switched off.

Now consider the ability of implementing efficient public policy in the context

of block B by choosing the optimal control laws by the example of the following

economic parameters: the share of the state customers’ expenditure in the gross

domestic product ( p ); the interest rate of public bonds ( rG ); the norm of

reservation ( x ). These parameters are accepted for the research, taking into

consideration [40] and the analysis of the sensitivity matrix of the indices,

namely, the total production capacity (М), the volume of the public debt (LG),and the level of prices (p).

The algorithm for multiobjective control was tested for the model of an economy

of the Republic of Kazakhstan for the following bound of the price-level variation:

pl(t) ¼ 0.9 and pu(t) ¼ 1.1.

Let

dpðtÞ ¼0, if plðtÞ<pðtÞ<puðtÞ;pðtÞ � plðtÞ, if pðtÞ � plðtÞ;pðtÞ � puðtÞ; if pðtÞ puðtÞ:

8>><>>:


When the level of prices is in inadmissible region B, choosing the optimal

parametric control laws is carried out in the environment of the following relations

(control laws):

1: V1j ¼ �k1jdpðtÞpðt0Þ þ constj;

2: V2j ¼ � k2jt

Zt0þt

t0

dpðtÞpðt0Þ dtþ constj;

3: V3j ¼ �k3jdpðtÞpðt0Þ þ

1

t

Zt0þt

t0

dpðtÞpðt0Þdt

24

35þ constj:

(1.94)

Here the case j ¼ 1 corresponds to the parameter x; j ¼ 2 corresponds to the

parameter p; j ¼ 3 corresponds to the parameter rG; kij is the adjusted coefficient

of the ith control law of the j-parameter, kij 0; constj is a constant equal to the

estimate of the value of the jth parameter by the results of parametric identification.

Choosing the optimal laws of parametric control is carried out at the level of two

economic parameters from the set of three ðx; p; rGÞ.The problem of choosing the optimal pair of parametric control laws at the level

of two economic parameters from the triplet x; p; rGð Þ can be stated as follows:

Find the optimal pair of parametric control laws Vij;Vum� �

on the set of

combinations of two economic parameters out of three on the basis of the set of

algorithms (1.94) minimizing the performance criterion

K1 ¼Zt0þT

t0

dpðtÞ2dt ! minfVij;kijg

(1.95)


MðtÞ �MðtÞj j � 0:09MðtÞ; t 2 ½t0; t0 þ T�;0 � VijðtÞ � aj; 0 � VumðtÞ � am; i ¼ 1; 2; 3; m ¼ 1; 2; 3:

(1.96)

HereMðtÞ; pðtÞ are the values of the production capacity and level of prices with theuse of the parametric control, respectively; MðtÞ; pðtÞ are the values of the

production capacity and the level of prices without the parametric control, respec-

tively; aj, am are the maximum possible values of the respective control parameters.

Again, this problem is solved in two stages:

– First, the optimal values of the coefficients kij are determined for each pair of

laws Vij;Vum� �

by the enumeration of their values on the intervals from the

respective regions quantized with a sufficiently small step for each coefficient,

minimizing the value of the criterion K1 under constraints (1.96).


– Second, the optimal pair of parametric control laws is chosen based on an

analysis of the results of the first stage by the minimum value of criterion K1.

The results of the numerical solution of the first and second problems allow for a

recommendation to implement the following laws of parametric control of the

parameters (p, x) for the case of the two-parametric control of the market economy

mechanism:

x ¼ �k1jdpðtÞpðt0Þ þ 0:1136; p ¼ �k1j

dpðtÞpðt0Þ þ 0:1348:

The optimal value of criterion K1 is equal to 0.0086.

Analysis of the results of computational experiments shows that the chosen and

implemented laws of parametric control of the reservation norm x and the share of

public consumers’ expenditures in the gross domestic product p ensure that the

value of the price level is taken out of the inadmissible region and into the

admissible one.

The results of computer simulation on the parametric control of the market

economy mechanisms by means of one control law and a pair of laws of parametric

control are presented in Table 1.6 and Fig. 1.9.

1.5.4.5 Analysis of the Structural Stability of the Mathematical Model of theCountry Subject to the Influence of the Share of Public Expenses andthe Interest Rate of Governmental Loans with Parametric Control

Let’s analyze the robustness of system (1.68), (1.69), (1.70), (1.71), (1.72), (1.73),

(1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84),

where the parameters x, p, and rG are determined in accordance with the solution of

the parametric control problems as expressions

1:U1j ¼ þk1jMðtÞ �Mð0Þ

Mð0Þ þ constj;

2:U2j ¼ �k2jMðtÞ �Mð0Þ

Mð0Þ þ constj;

3:U3j ¼ þk3jpðtÞ � pð0Þ

pð0Þ þ constj;

4:U4j ¼ �k4jpðtÞ � pð0Þ

pð0Þ þ constj;

5:U5j ¼ þk5jMðtÞ �Mð0Þ

Mð0Þ þ pðtÞ � pð0Þpð0Þ

� �þ const

j;

6:U6j ¼ �k6jMðtÞ �Mð0Þ

Mð0Þ þ pðtÞ � pð0Þpð0Þ

� �þ const

j;

(1.97)


with any values of the adjusted coefficients kij 0. Here constj is a constant equal to

the estimate of the jth parameter based on the results of parametric identification.

The application of the parametric control laws Uij i ¼ 1; :::; 6; i ¼ 1; 2; 3means substituting the respective functions for the parameters x (j ¼ 1), p (j ¼ 2),

Table 1.6 Values of price level p(t) with applied control of economic parameters

Months

Value of price

level p(t)without control

Value of price

level p(t) withcontrol of

parameter p

Value of price

level p(t) withcontrol of

parameter x

Value of price

level p(t) withcontrol of pair of

parameters (p,x)

1 1.10 1.10 1.10 1.10

2 1.11 1.11 1.11 1.11

3 1.12 1.12 1.12 1.12

4 1.12 1.12 1.12 1.12

5 1.13 1.13 1.13 1.13

6 1.14 1.14 1.14 1.14

7 1.15 1.15 1.15 1.15

8 1.16 1.15 1.16 1.15

9 1.16 1.16 1.16 1.16

10 1.17 1.17 1.17 1.17

11 1.18 1.17 1.18 1.17

12 1.19 1.18 1.18 1.18

13 1.19 1.19 1.19 1.18

14 1.20 1.19 1.20 1.18

15 1.21 1.19 1.20 1.18

16 1.22 1.20 1.21 1.19

17 1.22 1.20 1.21 1.18

18 1.23 1.20 1.21 1.18

19 1.24 1.20 1.22 1.18

20 1.24 1.20 1.22 1.18

21 1.25 1.19 1.22 1.17

22 1.26 1.19 1.22 1.16

23 1.26 1.19 1.22 1.15

24 1.27 1.18 1.22 1.14

25 1.27 1.17 1.22 1.13

26 1.28 1.16 1.22 1.12

27 1.28 1.15 1.21 1.10

28 1.29 1.14 1.21 1.09

29 1.29 1.13 1.20 1.07

30 1.30 1.12 1.20 1.05

31 1.30 1.10 1.19 1.03

32 1.31 1.08 1.18 1.01

33 1.31 1.07 1.18 0.99

34 1.31 1.05 1.17 0.97

35 1.31 1.03 1.16 0.94

36 1.32 1.01 1.15 0.92


and rG (j ¼ 3) into the model equations (1.68), (1.69), (1.70), (1.71), (1.72), (1.73),

(1.74), (1.75), (1.76), (1.77), (1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84).

As a result of application of these laws, the following system is derived:

dM

dt¼ FI

pb� mM; (1.98)

dQ

dt¼ Mf � F

p; (1.99)

dLG

dt¼ Ui3L

G þ FG � npF� nLsRL � nOðdP þ dBÞ; (1.100)

dp

dt¼ �a

Q

Mp; (1.101)

ds

dt¼ s

Dmax 0;

Rd � RS

RS

�;RL ¼ minfRd;RSg; (1.102)

Lp ¼ 1� Ui1

Ui1LG; (1.103)

Months

Lev

el o

f pr

ices

Fig. 1.9 Values of price level p(t) with control of economic parameters. Notation: – values of

price level p(t) without control; – values of price level p(t) with control of parameter x; –

values of price level p(t) with control of pair of parameters ðp; xÞ


dp ¼ 1� Ui1

Ui1br2LG; (1.104)

dB ¼ br2LG; (1.105)

x ¼ n1� d

1� snp

� �1�dd

!; (1.106)

Rd ¼ Mx; (1.107)

f ¼ 1� 1� 1� dn

x

� � 11�d

; (1.108)

F0 ¼ �0pMf ; (1.109)

FG ¼ Ui2pMf ; (1.110)

FL ¼ ð1� nLÞsRd; (1.111)

FI ¼ 1� Ui1

Ui1 þ ð1� Ui1ðtÞÞnp�ð1� npÞFG � �n0ðdB þ dPÞ þ npF0

� �nL � ð1� nLÞnp

sRL�þ ðm þ Ui2ÞLp;

(1.112)

F ¼ F0 þ FG þ FL þ FI; (1.113)

RS ¼ PA0 expðlptÞ

1

1þ no;o ¼ FL

pP0 expðlptÞ : (1.114)

The proof of the weak structural stability of the mathematical model (1.68),

(1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79),

(1.80), (1.81), (1.82), (1.83), and (1.84), presented above and relying on (1.71),

indicates that the weak structural stability of the considered model will be preserved

with the use of each of the parametric control laws UijðtÞ in the form of the

following assertion.

Assertion 1.2 Let N be a compact set belonging to the region ðM>0; Q<0; p>0Þor ðM>0; Q>0; p>0Þ of the phase space of the system of differential equationsderived from (1.26), (1.27), (1.28), (1.29), (1.30), (1.31), (1.32), (1.33), (1.34),(1.35), (1.36), (1.37), (1.38), (1.39), (1.40), (1.41), and (1.42), that is, the four-dimensional space of variables ðM; Q; p; LGÞ. Let N coincide with the closure ofits interior. Then the flux f defined by (1.98), (1.99), (1.100), (1.101), (1.102),

(1.103), (1.104), (1.105), (1.106), (1.107), (1.108), (1.109), (1.110), (1.111),

(1.112), (1.113), and (1.114) is weakly structurally stable on N.


1.5.4.6 Finding the Bifurcation Points of the Extremals of the VariationalCalculus Problem on the Basis of the Mathematical Modelof the Country Subject to the Influence of the Share of PublicExpenses and the Interest Rate of Governmental Loans

Let’s consider the ability to find the bifurcation point for the extremals of

the variational calculus problem of choosing the law of parametric control of the

market economic mechanism at the level of one economic parameter in the environ-

ment of a fixed finite set of algorithms on the basis of mathematical model (1.68),

(1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77), (1.78), (1.79),

(1.80), (1.81), (1.82), (1.83), and (1.84) of the national economic system.

The ability to choose the optimal law of parametric control at the level of one of

two parametersx (j¼ 1) and p (j¼ 2) on the time interval ½t0; t0 þ T� is considered inthe environment of the following algorithms (1.90):

1:U1jðtÞ ¼ k1jM �M0

M0

þ constj;

2:U2jðtÞ ¼ �k2jM �M0

M0

þ constj

3:U3jðtÞ ¼ k3jp� p0p0

þ constj;

4:U4jðtÞ ¼ �k4jp� p0p0

þ constj:

In the considered problem, criterion (1.91) is used (mean value of the gross

domestic products for the period of 1997–1999):

K ¼ 1

T

Zt0þT

t0

YðtÞdt;

where Y ¼ Mf .The closed set in the space of continuous vector functions of the output variables

of system (1.68), (1.69), (1.70), (1.71), (1.72), (1.73), (1.74), (1.75), (1.76), (1.77),

(1.78), (1.79), (1.80), (1.81), (1.82), (1.83), and (1.84) and regulating parametric

actions are determined by relations (1.92):

pijðtÞ � pðtÞ�� 0:09pðtÞ; ðMðtÞ; QðtÞ; LGðtÞ; pðtÞ; sðtÞÞ 2 X ;

0 � uj � aj; i ¼ 1; :::; 4; j ¼ 1; 2; t 2 ½t0; t0 þ T�:

The following problems for finding the bifurcation points of the extremals of the

considered variational calculus problem were studied.

Problem 1 In this variational calculus problem, we consider its dependence on the

coefficient l ¼ r2 of the mathematical model with possible values on some interval

[a, b].


As a result of computer simulations, plots of the dependence of the optimal

values of criterion K on the deposit interest rate (in percentages) for the given set of

algorithms (Fig. 1.10) were obtained. As can be seen from Fig. 1.10, the conditions

of Theorem 1.8 are satisfied; for instance, for the interval [15.6; 21.6] since with

r2 ¼ 15:6 the optimal value of the criterion equal to 175,467 is attained with use of

the law U12. With r2 ¼ 21:6; the optimal value of the criterion equal to 171,309 is

attained with the use of another law U21. Using the proposed numerical algorithm

allows one to determine the bifurcation point of the extremal of the considered

problem r2 ¼ 18:0with an accuracy of up to 0.001. For this parameter, the lawsU21

and U12 are optimal, and the corresponding value of criterion K is 173,381

(monetary units per month).

Problem 2 Find the bifurcation point for the extremals of the variational calculus

problem of choosing the set of laws of parametric control of the market economic

mechanism subject to the influence of the public expenses at the level of two

economic parameters with one-parameter disturbance.

In this variational calculus problem, we consider its dependence on the coeffi-

cient l ¼ r2 of the mathematical model with possible values in some interval [a, b].As a result of computer simulations, plots a computer simulations, plots of

dependence of the optimal values of criterion K on the deposit interest rate (in

percentages) for all sets of algorithms (Fig. 1.11). As can be seen in Fig. 1.10, the

conditions of Theorem 1.8 are satisfied; for instance, for the interval [6; 9.6] since

with r2 ¼ 6 the optimal value of the criterion equal to 188803 is attained using laws

fU21;U32g. With r2 ¼ 9:6; the optimal value of the criterion equal to 190,831 is

attained with the use of other laws fU21;U12g . Using the proposed numerical

algorithm allows us to determine the bifurcation point of the extremal of the

considered problem r2 ¼ 0:075with an accuracy of up to 0.001. For this parameter,

two pairs of laws fU21;U32g and fU21;U12g are optimal, and the respective value of

criterion K is equal to 187,487 (monetary units per month).

Opt

imal

val

ues

of c

rite

rion

Fig. 1.10 Plots of dependencies of optimal criterion values on parameter of deposit interest rate

r2 . Notation: – U12 , – U32 , – U21 , – U41 , – without control


Problem 3 Find the bifurcation point for the extremals of the variational calculus

problem of choosing the set of laws of parametric control of the market economic

mechanism subject to the influence of public expenses at the level of one economic

parameter with two-parameter disturbance.

In this variational calculus problem, we consider its dependence on the two-

dimensional coefficient l ¼ ðr2; nOÞ of the mathematical model with possible

values in some region (rectangle) L of the plane.

As a result of a computer simulations, plots of the dependence of the optimal

values of criterion K on the values of the parameters ðr2; nOÞ for each of 12 possiblelawsUij; i ¼ 1; :::; 6; j ¼ 1; 2, were obtained. Figure 1.12 presents the plots for thetwo lawsU21 andU41, maximizing the criterion in region L, the intersection curve ofthe respective regions, and the projection of this intersection curve to the plane of

the values l consisting of the bifurcation points of these two-dimensional

parameters. This projection divides the rectangle L into two parts. The control

law U21 is optimal in one of these parts, whereas U41 is optimal in the other part.

Both laws are optimal on the curve projection.

Problem 4 As a result of a computer simulation experiment, the plots of the

dependence of the optimal values of criterion (1.95) K1 on the values of uncon-

trolled parameters ðr2; nOÞ for each of nine possible laws (1.94) Vij; i ¼ 1; 2; 3;j ¼ 1; 2; 3 were obtained. Figure 1.13 presents these plots for the four laws (V11;V12; V21; V22), minimizing criterion K1 in region L, the intersection curves of the

respective surfaces, and the projections of these intersection curves to the plane of

the values l . This projection consists of the bifurcation points of the two-

dimensional parameter l dividing the rectangle L into two parts, inside which

only one control law is optimal. Two or three different laws are optimal on the

projection curves.

187487

168000

173000

178000

183000

188000

193000

2,4 6 9,6 13,2 16,8 20,4r2

Opt

imal

val

ues

of c

rite

rion

Laws U41 and U32

Laws U21 and U12

Laws U41 and U12

Laws U21 and U32

Fig. 1.11 Plots of the dependencies of optimal criterion values on the parameter of deposit

interest rate r2


Opt

imal

val

ues

of c

rite

rion

Fig. 1.12 Plots of the dependencies of optimal criterion values on the parameters of deposit

interest rate r2 and dividend tax rate nO

Opt

imal

val

ues

of c

rite

rion

Fig. 1.13 Plot of optimal values of criterion K1


1.5.5 Mathematical Model of the National EconomicSystem Subject to the Influence of International Tradeand Currency Exchange on Economic Growth


The mathematical model proposed in [33] for researching the influence of

the international trade and currency exchange on economic growth after the respec-

tive transformations can be expressed as the following system of differential

and algebraic equations (where i ¼ 1 or 2 is the number of states, and t is the

time variable):

dMi

dt¼ FI

i

pibi� miMi; (1.115)

dQi

dt¼ Mifi � Fi

pi; (1.116)

dLGidt

¼ rG iLGi þ FG

i � np iFi � nL isiRLi � nO iðdPi þ dBi Þ; (1.117)

dpidt

¼ �aiQi

Mipi; (1.118)

dsidt

¼ siDi

max 0;Rdi � RS

i

RSi

�;RL

i ¼ minfRdi ;R

Si g; (1.119)

LPi ¼ 1� xixi

LGi ; (1.120)

dPi ¼ 1� xixi

bir2 iLGi ; (1.121)

dBi ¼ bir2 iLGi ; (1.122)

xi ¼ ni1� di

1� sinipi

� �1�didi

!; (1.123)

Rdi ¼ Mixi; (1.124)

fi ¼ 1� 1� 1� dini

xi

� � 11�di

; (1.125)

FOi ¼ �0 ipiMifi; (1.126)


FGi ¼ pipiMifi; (1.127)

FLi ¼ ð1� nL iÞsiRd

i ; (1.128)

FIi ¼

1

1þ np i

kqiMifixi

� ð1� npiÞFGi þ n0 iðdBi þ dPi Þ þ np iFO

i þ

þ nL i þ ð1� nL iÞnp i�

siRLi þ npiðFji � yiFijÞ þ miLpi � rG iL

Pi

�; (1.129)

RSi ¼ PA

0 i expðlp itÞ1

1þ nioi;oi ¼ FL

i

ð1þ CLi yi

pjpiÞP0 iðlpitÞ

; j ¼ 3� i; (1.130)

F12 ¼CL1p2p1

1þ CL1y

p2p1

FL1 þ

CO1

p2p1

1þ CO1 y

p2p1

FO1 ; (1.131)

F21 ¼CL2p1p2

1þ CL21yp1p2

FL2 þ

CO2

p1p2

1þ CO2

1yp1p2

FO2 ; (1.132)

F1 ¼ FI1 þ FL

1 þ FO1 þ FG

1 þ F21 � yF12; (1.133)

F2 ¼ FI2 þ FL

2 þ FO2 þ FG

2 þ F12 � 1

yF21: (1.134)

Here

Мi is the total productive capacity;

Qi is the total stock-in-trade in the market with respect to some equilibrium state;

LGi is the total public debt;pi is the level of prices;si is the rate of wages;LPi is the indebtedness of production;

dPi and dBi are the business and bank dividends, respectively;

Rdi and RS

i are the demand and supply of the labor force;

di, ni are the parameters of the function fi;xi is the solution to the equation f 0i ðxiÞ ¼ si

pi;

FLi and F

Oi are the consumer expenditures of the workers and owners, respectively;

FIi is the flow of investment;

FGi is the consumer expenditure of the state;

Fij are the consumer expenses of the ith country of the product imported from the

jth country;

y is the exchange rate of the currency of the first country with respect to the

currency of the second country, y1 ¼ y; y2 ¼ 1=y;


CLi ðCO

i Þ is the quantity of imported product items consumed by workers (owners) of

the ith country per domestic product item;

xi is the norm of reservation;

bi is the ratio of the arithmetic mean return from the business activity and the rate of

return of rentiers;

r2i is the deposit interest rate;rGi is the interest rate of public bonds;�Oi is the coefficient of the propensity of owners to consume;

pi is the share of consumer expenditure of the state in the gross domestic product;

nPi, nОi, nLi are the payments flow, dividends, andworkers’ income taxes, respectively;

bi is the norm of the fund capacity of the unit of power;

mi is the coefficient of the power unit retirement as a result of degradation;

m*i is the depreciation rate;

ai is the time constant;

Di is the time constant defining the typical time scale of the wage relaxation

process;

P0i;PA0iare, respectively, the initial number of workers and total number of those

capable;

oi is the per capita consumption in the group of workers;

lPi > 0 is rate of population growth;

kqi is the share of the gross domestic product of the country reserved in gold.

Among relations (1.115), (1.116), (1.117), (1.118), (1.119), (1.120), (1.121),

(1.122), (1.123), (1.124), (1.125), (1.126), (1.127), (1.128), (1.129), (1.130),

(1.131), (1.132), (1.133), and (1.134), Eqs. (1.131), (1.132), (1.133), and (1.134)

define the connection of the economic systems of two countries. Note that in the

case CL1 ¼ CL

2 ¼ CO1 ¼ CO

2 ¼ 0; there is no trade between these two countries, and

their economic systems are independent of one another.

For the purpose of analysis, the values of such parameters as bi, r2i, rGi, npi, nLi,bi, si, �0i, mi, mi

*, Di were taken from [37], [36]. Here we consider the case of

identical countries (i ¼ 1 and 2 correspond to statistical data of the Republic of

Kazakhstan) and the case of nonidentical countries (i ¼ 1 corresponds to the

Republic of Kazakhstan, i ¼ 2 corresponds to the Russian Federation).

For estimation of the remaining parameters of the model, xi, pi, di, ni, �Оi, bi, ai,Qi(0), the parametric identification problems were solved by the searching method

in the sense of the minimum of the squared discrepancies:

X2i¼1

XNj¼1

Mij �M

ij

Mij

!2

þ pij � pijpij

!224

35; (1.135)

where Mij*, Mij

**, pij*, pij

** are the respective values of the total product capacity

and product price of the ith country presented in [37], [36] and, calculated, N is the

number of observations, i ¼ 1; 2.


1.5.5.2 Analysis of the Structural Stability of the Mathematical Modelof the National Economic System Subject to the Influenceof International Trade and Currency Exchange WithoutParametric Control

Analysis of the robustness (structural stability) of model (1.115), (1.116), (1.117),

(1.118), (1.119), (1.120), (1.121), (1.122), (1.123), (1.124), (1.125), (1.126),

(1.127), (1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and (1.134) is based

on the theorem on sufficient conditions of weak structural stability in the compact

set of the phase space.

Assertion 1.3 Let N be a compact set residing within the region ðM1>0; Q1<0;p1>0Þ or ðM1>0; Q1>0; p1>0Þ of the phase space of the system of differentialequations of mathematical model (1.115), (1.116), (1.117), (1.118), (1.119),(1.120), (1.121), (1.122), (1.123), (1.124), (1.125), (1.126), (1.127), (1.128),(1.129), (1.130), (1.131), (1.132), (1.133), and (1.134), that is, the eight-dimensional space of the variables ðMi; Qi; pi; LGiÞ, i ¼ 1; 2. Let the closure ofthe interior of N coincide with N. Then the flux f defined by the system of modeldifferential equations is weakly structurally stable on N.

One can choose N such as, for instance, the parallelepiped with boundary Mi

¼ Mimin; Mi ¼ Mimax; Qi ¼ Qimin; Qi ¼ Qimax; pi ¼ pimin; pi ¼ pimax; LGi ¼ LGimin; LGi ¼ LGimax: Here 0<Mimin<Mimax, Qimin<Qimax<0 or 0<Qimin<Qimax, 0<pimin<pimax, LGimin<LGimax.

Proof First, let’s prove the semitrajectory of flux f starting from any point of the set

N for some value of t (t > 0) leaving N.Consider any semitrajectory starting in N. With t>0, the following two cases are

possible; namely, all the points of the semitrajectory remain in N, or for some t apoint of the semitrajectory does not belong to N. In the first case, from (1.118), dp1dt¼ �a1

Q1

M1p1 , of the system it follows that for all t>0 , the variable p1(t) has a

derivative greater than some positive constant withQ1<0or less than some negative

constant with Q1>0 ; that is, p1(t) increases infinitely or tends to zero with

unbounded increase of t. Therefore, the first case is impossible, and the orbit of

any point in N leaves N.Since any chain-recurrent set Rðf ;NÞ lying within N is the invariant set of this

flux, then if it is nonempty, it consists of only whole orbits. Hence, in the considered

case, Rðf ;NÞ is empty. The assertion follows from Theorem A [67].

1.5.5.3 Choosing Optimal Laws of Parametric Control of MarketEconomic Development on the Basis of the Mathematical Modelof the Country Subject to the Influence of International Tradeand Currency Exchange

Choosing the optimal laws of parametric control of the parametersxi; pi; y is carriedout in the environment of the following relations:


1: Ui1;b ¼ ki1;b

DMiðtÞMiðt0Þ þ constib;

2: Ui2;b ¼ �ki2;b


3: Ui3;b ¼ ki3;b

DpiðtÞpiðt0Þ þ constib;

4: Ui4;b ¼ �ki4;b

DpiðtÞpiðt0Þ þ constib:

(1.136)

HereUia;b is the ath control law of the bth parameter of the ith country, a ¼ 1; :::; 4;

b ¼ 1; :::; 3. The case b ¼ 1 corresponds to the parameterxi; b ¼ 2 corresponds

to the parameter pi; b ¼ 3, corresponds to the parameter yDMiðtÞ ¼ Ma;b;iðtÞ �Mi

ðt0Þ;DpiðtÞ ¼ pa;b;iðtÞ � piðt0Þ; t0 is the control starting time, t 2 t0; t0 þ T½ �. HereMa;b;iðtÞ, pa;b;iðtÞ are the values of the product capacity and the price level of the ithcountry, respectively, with theUi

a;bth control law; kia;b is the adjusted coefficient of

the respective law (kia;b 0i); constib is a constant number equal to the estimate of

the values of the b-parameter by the results of parametric identification.

The problem of choosing the optimal parametric control law for the economic

system of the ith country at the level of one of the economic parameters xi; pi; yð Þcan be formulated as follows: On the basis of mathematical model (1.115), (1.116),

(1.117), (1.118), (1.119), (1.120), (1.121), (1.122), (1.123), (1.124), (1.125),

(1.126), (1.127), (1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and (1.134),

find the optimal parametric control law in the environment of the set of algorithms

(1.136); that is, find the optimal law (and its coefficients kia;b) from the set {Uia;b}

minimizing the criterion

Ki ¼ 1

T

Zt0þT

t0

piðtÞdt ! minfkia;b;Ui

a;bg(1.137)


MiðtÞ�MiðtÞj j � 0:09Mi

ðtÞ;0�Ui

a;bðtÞ � aib; a¼ 1; :::;4;b¼ 1;2;3; piðtÞ 0; siðtÞ 0; where t 2 t0; t0 þ T½ � :

(1.138)

Here MiðtÞ are the values of the total production capacity of the ith country

without parametric control; aib is the bth parameter of the ith country.

The problem is solved in two stages:

1. In the first stage, the optimal values of the coefficients kia;b are determined for

each law Uia;b by enumerating their values on the respective intervals quantized

with step 0.01 minimizing K under constraints (1.138).

2. In the second stage, the law of optimal control of the specific parameter is chosen

on the basis of the results of the first stage by the minimum value of criterion Ki.


The problem of choosing the pair of optimal parametric control laws for the

simultaneous control of two parameters can be formulated as follows: Find the

optimal pair of parametric control laws (Uia;b,U

in;m) on the set of combinations of two

economic parameters from three parameters xi; pi; yð Þ on the basis of the set of

algorithms (1.136) minimizing the criterion

Ki ¼ 1

T

Zt0þT

t0

piðtÞdt ! minðUi

a;b; kia;bÞ;ðUi

n;m; kin;mÞ

� � ;a; n ¼ 1; :::; 4 ; b; m ¼ 1; 2; 3; b<m;

(1.139)


The problem of choosing the optimal pair of laws is solved in two stages:

1. In the first stage, the optimal values of the coefficients kia;b,kin;m are determined for

the chosen pair of the control laws (Uia;b ,U

in;m ) by enumeration of their values

from the respective intervals quantized with step equal to 0.01 minimizing Kt

under constraints (1.138);

2. In the second stage, the optimal pair of parametric control laws is chosen on the

basis of the results of the first stage by the minimum value of criterion Ki.

Here we present the results of numerical experiments on choosing efficient laws

of parametric control of the public consumers’ expenditures, the norm of reserva-

tion, and the currency exchange rate within the framework of the following part of

the research program:

– The estimation of the values of criterionKi on the basis of themathematical model

of the interaction between identical economic systems of two countries by foreign

trade (themodel coefficients are estimated by choosing and solving the parametric

identification problem with the data of one country, the Republic of Kazakhstan).

– On the basis of the mathematical model of the interaction between the identical

economic systems of two countries via foreign trade, choosing the optimal

parametric control law at the level of two of the economic parameters x1; p1; yð Þfor the economic system of the first country, and estimation of the values of

criterion K2 for the economic system of the second country.

– On the basis of the mathematical model of the interaction between identical

economic systems of two countries via foreign trade, choosing the optimal pair

of parametric control laws on the set of combinations of two economic parameters

from three parameters for the economic system of the first country and estimation

of the values of the criterion K2 for the economic system of the second country.

– The estimation of the values of criteria Ki i ¼ 1; 2ð Þ on the basis of the mathe-

matical model of the interaction between the nonidentical economic systems of

two countries (the Republic of Kazakhstan and the Russian Federation) via

foreign trade (the model coefficients are estimated by choosing and solving the

parametric identification problem for the data of two different countries).

– On the basis of the mathematical model of the interaction between the noniden-

tical economic systems of two countries via foreign trade, choosing the optimal


law of parametric control of the currency exchange rate y for the economic

system of the first country and estimating the values of criterion K2 for the

economic system of the second country.

– On the basis of the mathematical model of the interaction between nonidentical

economic system of two countries via foreign trade, choosing the optimal pair of

parametric control laws on the set x1; yð Þ, p1; yð Þ for the economic system of the

first country and estimate the values of criterion K2 for the economic system of

the second country.


economic systems of two countries via foreign trade, choosing the optimal law

of parametric control of the currency exchange rate y2 for the economic system

of the second country and estimating the values of criterion K1 for the economic

system of the first country.


economic systems of two countries via foreign trade, choosing the optimal pair

of the parametric control laws on the set (x2; y2), (p2; y2) for the economic system

of the second country and estimating of values of criterion K1 for the economic

system of the second country.


economic systems of two countries, the estimation of the influence of the control of

the economic system of one country on the economic indices of another country

with simultaneous application of the optimal control laws at the level of one

economic parameter of three (z1; p1; y) and (z2; p2; y) in two countries. Simulta-

neous control of the currency exchange rate y by two countries is not considered.

Within the framework of the first intended stage of research, we estimate the

coefficients of the mathematical model of the interaction between the two identical

economic systems of two countries via foreign trade on the basis of the data of one

country [40]. The results of parametric identification show that the value of the

standard deviation from the measured values of the respective variables is 5%. The

values of criteria Ki are equal and are given by K1 ¼ K2 ¼ 1:145 with CL1 ¼ CL

2

¼ CO1 ¼ CO

2 ¼ 0:1 and y ¼ 1.

The results of the numerical solution of the first stage of the stated problem of

choosing the optimal law of parametric control at the level of one of the economic

parameters (x1, p1, y) for the economic system of the first country are presented in

Table 1.7. Analysis of Table 1.7 shows that the best resultK1 ¼ 0:99 is attained withuse of the control law

p1 ¼ �0:8Dp1ðtÞ

1þ 0:1348:

With such a control law, the criterion of optimality of the economic system of

the second country is K2 ¼ 1.144, differing slightly from the case without control.

The results of numerical solution of the first stage of the stated problem of choosing

the optimal pair of parametric control laws are presented in the eight tables in the form

of Table 1.8, differing in the control law expression by at least one parameter.


The choice of the optimal pair of the parametric control laws according to the

requirements of the second stage on the basis of analysis of the data from these

tables allows the recommendation to implement the control laws for the parameters

p1 and y given as follows:

p1 ¼ �0:8DP1ðtÞ

1þ 0:1348; y ¼ �1:6

DM1ðtÞ139; 435

þ 0:2:

The value of the criterion of the economic system of the first country is equal to

K1 ¼ 0:97, and the value of the criterion for the economic system of the second

country differs slightly from the case without control and K1 ¼ 1.144.

Further, we estimate the coefficients of the mathematical model of interaction

between the nonidentical economic systems of two countries via foreign trade on

the basis of the data of two different countries [37], [36]. The parametric identifica-

tion results show the admissible precision of the description. The values of the

criterion Ki i ¼ 1; 2ð Þ are, respectively, K1 ¼ 1:137; K2 ¼ 1:775; with C1 ¼ 0:15;C2 ¼ 0:015; y ¼ 0:2:

Table 1.7 Results of the numerical solution of the first stage

of the problem of choosing the optimal parametric control law

at the level of one parameter

Notation of laws Law coefficient Value of criterion K1

U111 0.2 1.072

U121 0.0 1.145

U131 2.1 1.009

U141 0.0 1.145

U112 0.0 1.145

U122 0.1 1.068

U132 0.0 1.145

U142 0.8 0.990

U113 0.0 1.145

U123 1.8 1.070

U133 0.0 1.145

U143 1.9 1.100

Table 1.8 First-stage results of the numerical solution of the problem of choosing the optimal pair

of laws

Pairs of parametric control laws

Value of

criterion K1

First law Second law

Law denotation

Optimal

coefficient value

Law

denotation

Optimal

coefficient value

U142 0.8 U1

13 0.0 0.99

U142 0.8 U1

23 1.6 0.97

U142 0.8 U1

33 0.0 0.99

U142 0.8 U1

43 0.0 0.99


The solution of the problem of choosing the optimal law of parametric control of

the currency exchange y for the economic system of the first country on the basis of

the mathematical model of the interaction of the two nonidentical economic

systems of the two countries via foreign trade allows us to propose the law given by

y ¼ �1:2DM1ðtÞ139; 435

þ 0:2:

The application of this law to the control of the currency exchange rate of the

first country results in improving the criterion from 1.137 to 1.123. The criterion of

the second country goes down from 1.734 to 1.828.

The solution of the problem of choosing the optimal pair of parametric control

laws on the basis of the mathematical model of the interaction of the two noniden-

tical economic systems of the two countries via foreign trade allows us to propose

the following laws:

p1 ¼ 0:2DM1ðtÞ139; 435

þ 0:1136; y ¼ 1:5DM1ðtÞ139; 435

þ 0:2:

Criterion K2 is 1.83 for the economic system of the second country with

K1 ¼ 1.05.

In solving the problem of choosing the optimal parametric control law of the

second country from the given pair of countries, the following results are obtained:

The optimal control of the parameter y is realized by means of the law

y2 ¼ 1= �0:12Dp2ðtÞ þ 0:2ð Þ:

The value of criterion K2 improves from 1.775 to 1.73.

In solving the problem of choosing the optimal pair of parametric control laws

for the second country, the following pair of the laws is obtained:

y2 ¼ 1= �0:11Dp2ðtÞ þ 0:2ð Þ; p2 ¼ �0:01Dp2ðtÞ þ 0:1388:

With application of these control laws, the value of criterion K2 is equal to 1.66.

In both cases, the criterion of the first country K1 varies insignificantly (the increase

not exceeding 1%).

By carrying out the simultaneous control of the parameters of two countries, the

values of the criteria improve within the limits of 3% for each country in compari-

son with the control of each country separately. The optimal control of the first

country at the level of one parameter is implemented by means of lawU14;2; criterion

K1 is 0.99. The optimal control of the second country at the level of one parameter is

implemented by means of law U24;3 ; criterion K2 is 1.72. With the simultaneous

application of two control laws U14;2 and U2

4;3, for both countries the values of the

criteria turn out to be K1 ¼ 0.98 and K2 ¼ 1.66.


1.5.5.4 Analysis of the Structural Stability of the Mathematical Modelof the Country Subject to the Influence of International Tradeand Currency Exchange with Parametric Control

Let’s analyze the robustness of system (1.115), (1.116), (1.117), (1.118), (1.119),

(1.120), (1.121), (1.122), (1.123), (1.124), (1.125), (1.126), (1.127), (1.128),

(1.129), (1.130), (1.131), (1.132), (1.133), and (1.134), where the parameters xi;pi; y are defined in accordance with the solution of the parametric control

problems as the expressions

1: Ui1;b ¼ ki1;b


2: Ui2;b ¼ �ki2;b


3: Ui3;b ¼ ki3;b


4: Ui4;b ¼ �ki4;b

DpiðtÞpiðt0Þ þ constib

(1.140)

for any values of the adjusted coefficients kiab 0. Here constib is a constant number

equal to the estimate of the values of the bth parameter of the ith country by the

results of parametric identification, i ¼ 1; 2; a ¼ 1; :::; 4; b ¼ 1; 2.The application of parametric control law Ui

a;b means the substitution of the

respective functions into model equations (1.115), (1.116), (1.117), (1.118),

(1.119), (1.120), (1.121), (1.122), (1.123), (1.124), (1.125), (1.126), (1.127),

(1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and (1.134) for the parameters

xi (j ¼ 1), pi (j ¼ 2), and y (j ¼ 3).

As a result of the application of these laws to system (1.115), (1.116), (1.117),

(1.118), (1.119), (1.120), (1.121), (1.122), (1.123), (1.124), (1.125), (1.126),

(1.127), (1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and (1.134), the follow-

ing system is derived:

dMi

dt¼ FI

i

pibi� miMi; (1.141)

dQi

dt¼ Mifi � Fi

pi; (1.142)

dLGidt

¼ rG iLGi þ FG

i � np iFi � nL isiRLi � nO iðdPi þ dBi Þ; (1.143)

dpidt

¼ �aiQi

Mipi; (1.144)


dsidt

¼ siDi

max 0;Rdi � RS

i

RSi

�;RL

i ¼ minfRdi ;R

Si g; (1.145)

LPi ¼ 1� Uia;1

Uia;1

LGi ; (1.146)

dPi ¼ 1� Uia;1

Uia;1

bir2 iLGi ; (1.147)

dBi ¼ bir2 iLGi ; (1.148)

xi ¼ ni1� di

1� sinipi

� �1�didi

!; (1.149)

Rdi ¼ Mixi; (1.150)

fi ¼ 1� 1� 1� dini

xi

� � 11�di

; (1.151)

FOi ¼ �0 ipiMifi; (1.152)

FGi ¼ Ui

a;2piMifi; (1.153)

FLi ¼ ð1� nL iÞsiRd

i ; (1.154)

FIi ¼

1

1þ np i

kqiMifixi

� ð1� npiÞFGi þ n0 iðdBi þ dPi Þ þ np iFO

i þ

þ nL i þ ð1� nL iÞnp i�

siRLi þ npiðFji � Ui

a;3FijÞ þ miLpi � rG iLPi

�;

(1.155)

RSi ¼ PA

0 i expðlp itÞ1

1þ nioi; oi ¼ FL

i

ð1þ CLi U

ia;3

pjpiÞP0 iðlpitÞ

; j ¼ 3� i; (1.156)

F12 ¼CL1p2p1

1þ CL1U

ia;3

p2p1

FL1 þ

CO1

p2p1

1þ CO1U

ia;3

p2p1

FO1 ; (1.157)

F21 ¼CL2p1p2

1þ CL2

1Ui

a;3

p1p2

FL2 þ

CO2

p1p2

1þ CO2

1Ui

a;3

p1p2

FO2 ; (1.158)


F1 ¼ FI1 þ FL

1 þ FO1 þ FG

1 þ F21 � Uia;3F12; (1.159)

F2 ¼ FI2 þ FL

2 þ FO2 þ FG

2 þ F12 � 1

Uia;3

F21: (1.160)

The proof of the weak structural stability of the mathematical model indicates

that the weak structural stability of the considered model is maintained with the use

of each of the parametric control laws Uia;b in the form of the following assertion:

Assertion 1.4 Let N be a compact set belonging to region ðM1>0; Q1<0; p1>0Þor ðM1>0; Q1>0; p1>0Þ of the phase space of the model system of differentialequations (1.115), (1.116), (1.117), (1.118), (1.119), (1.120), (1.121), (1.122),(1.123), (1.124), (1.125), (1.126), (1.127), (1.128), (1.129), (1.130), (1.131),(1.132), (1.133), and (1.134), that is, the eight-dimensional space of variables ðMi

; Qi; pi; LGiÞ, i ¼ 1; 2. Let the closure of of the interior of N coincide with N. Thenthe flux f defined by system (1.141), (1.142), (1.143), (1.144), (1.145), (1.146),(1.147), (1.148), (1.149), (1.150), (1.151), (1.152), (1.153), (1.154),(1.155),(1.156), (1.157), (1.158), (1.159),and (1.160) is weakly structurally stable on N.

1.5.5.5 Finding the Bifurcation Points of the Extremals of the VariationalCalculus Problem on the Basis of the Mathematical Model of theCountry Subject to the Influence of International Trade andCurrency Exchanges

Besides the case considered above, the problem of choosing the optimal set of laws

was also solved in some other definition.

Choosing the optimal parametric control laws on the basis of model (1.115),

(1.116), (1.117), (1.118), (1.119), (1.120), (1.121), (1.122), (1.123), (1.124),

(1.125), (1.126), (1.127), (1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and

(1.134) at the level of one of the two parameters xi; pi is carried out in the

environment of the following relations:

1: Ui1;b ¼ ki1;b


2: Ui2;b ¼ �ki2;b


3: Ui3;b ¼ ki3;b


4: Ui4;b ¼ �ki4;b

DpiðtÞpiðt0Þ þ constib:

(1.161)


Here Uia;b is the ath control law of the bth parameter of the ith country, a ¼ 1; :::; 4;

b ¼ 1; 2. The case b ¼ 1 corresponds to parameter xi; b ¼ 2� pi; DMiðtÞ ¼ Ma;b;i

ðtÞ �Miðt0Þ;DpiðtÞ ¼ pa;b;iðtÞ � piðt0Þ; t0 is the control starting time, t 2 t0; t0 þ T½ �.HereMa;b;iðtÞ,pa;b;iðtÞ are the values of the product capacity and the level of prices ofthe ith country, respectively, with the Ui

a;b th control law; kia;b is the adjusted

coefficient of the respective law (kia;b 0i); constib is a constant equal to the estimate

of the values of the bth parameter by the results of parametric identification.

The problem of choosing the optimal parametric control law for the economic

system of the ith country at the level of one of the economic parameters (xi, pi, y)can be formulated as follows: On the basis of the mathematical model (1.115),

(1.116), (1.117), (1.118), (1.119), (1.120), (1.121), (1.122), (1.123), (1.124),

(1.125), (1.126), (1.127), (1.128), (1.129), (1.130), (1.131), (1.132), (1.133), and

(1.134), find the optimal parametric control law in the environment of the set of

algorithms (1.136); that is, find the optimal law (and its coefficients kia;b) from the

set {Uia;b} maximizing the criterion

Ki ¼ 1

T

Zt0þT

t0

YiðtÞdt; (1.162)

where Yi ¼ Mifi . In computational experiments, we research the influence of the

parametric control of the first country (i ¼ 1).

A closed set in the space of the continuous vector functions of the output

variables of system (1.115), (1.116), (1.117), (1.118), (1.119), (1.120), (1.121),

(1.122), (1.123), (1.124), (1.125), (1.126), (1.127), (1.128), (1.129), (1.130),

(1.131), (1.132), (1.133), and (1.134) and regulating parametric actions is defined

by the following relations:

p1ðtÞ � p1 ðtÞ�� 0:09p1 ðtÞ;ðMiðtÞ; QiðtÞ; LGiðtÞ; piðtÞ; siðtÞÞ 2 X ;

0 � Uiab � aib; a ¼ 1; :::; 4; b ¼ 1; 2; i ¼ 1; 2 t 2 ½t0; t0 þ T�:

(1.163)

Hereaib is the maximum possible value of the ath parameter of the ith country;pi ðtÞare the model (calculated) values of the price level of the ith country without

parametric control; X is the compact set of the admissible values of given variables.

In this variational calculus problem, we consider its dependence on the two-

dimensional coefficient l ¼ ðr2;1; yÞ of the mathematical model with possible

values in some region (rectangle) L in the plane.

As a result of a computer simulation experiment, the plots of the dependence of

the optimal value of criterion K on the values of the parameters ðr2;1; yÞ for each ofeight possible laws U1

a;b; a ¼ 1; :::; 4; b ¼ 1; 2 are established. Figure 1.14

presents the plots for the two laws U12;2 and U

14;2 maximizing the criterion in region


L , the intersection curve of the respective regions, and the projection of this

intersection curve to the plane of values l consisting of the bifurcation points of

this two-dimensional parameter. This projection divides rectangleL into two parts.

The control law U12;2 ¼ �k12;2

DM1ðtÞM1ðt0Þ þ const12 is optimal in one of these parts,

whereas U14;2 ¼ �k14;2

Dp1ðtÞp1ðt0Þ þ const12 is optimal in the other part. Both of the laws

are optimal on the curve projection.

1.5.6 Forrester’s Mathematical Model of Global Economy


Forrester’s mathematical model of the “world dynamics” [29] is given by the

following system of ordinary differential and algebraic equations (here t is time):

P0ðtÞ ¼ PðtÞðBnðtÞ � DðtÞÞ; (1.164)

V0ðtÞ ¼ CVGPðtÞVMðMÞ � CVDVðtÞ; (1.165)

Z0ðtÞ ¼ CZPðtÞZVðVRÞ � ZðtÞ=TZðZRÞ; (1.166)

Opt

imal

val

ues

of c

rite

rion

Fig. 1.14 Plots of the dependencies of optimal criterion values on the parameters of deposit

interest rate r2;1 and currency exchange rate y


R0ðtÞ ¼ �CRPðtÞRMðMÞ; (1.167)

S0ðtÞ ¼ ðCSSQQMðMÞSFðFÞ=QFðFÞ � SðtÞÞ=TS; (1.168)

MðtÞ ¼ VRðtÞð1� SðtÞÞERðRRÞ=½ð1� SNÞEN�; (1.169)

FðtÞ ¼ FSðSRÞFZðZRÞFPðPRÞFC=FN; (1.170)

BnðtÞ ¼ PðtÞCBBMðMÞBPðPRÞBFðFÞBZðZRÞ; (1.171)

DðtÞ ¼ PðtÞCDDMðMÞDPðPRÞDFðFÞDZðZRÞ; (1.172)

QðtÞ ¼ CQQMðMÞQPðPRÞQFðFÞQZðZRÞ; (1.173)

PRðtÞ ¼ PðtÞ=PN; (1.174)

VRðtÞ ¼ VðtÞ=PðtÞ; (1.175)

SRðtÞ ¼ VRðtÞSðtÞ=SN; (1.176)

RRðtÞ ¼ RðtÞ=R0; (1.177)

ZRðtÞ ¼ ZðtÞ=ZN: (1.178)

The model includes the following exogenous constants:

CQ is the standard quality of life;

CB is the normal rate of fertility;

CD is the normal rate of mortality;

FC is the nourishment coefficient;

CZ is normal pollution;

CR is the normal consumption of natural resources;

FN is the normal level of nourishment;

EN is the normal efficiency of the relative volume of funds;

CVD is the normal depreciation of funds;

CVG is the normal fund formation;

TS is the coefficient of pollution influence.

The exogenous functions of the model are as follows:

BM is the multiplier of the fertility dependence on the material standard of living;

BP is the coefficient of fertility dependence on the population density;

BF is the coefficient of fertility dependence on nourishment;

BZ is the coefficient of fertility dependence on the pollution;

DM is the coefficient of mortality dependence on the material standard of living;

DP is the coefficient of mortality dependence on the population density;


DF is the coefficient of mortality dependence on nourishment;

DZ is the coefficient of mortality dependence on pollution;

QM is the coefficient of life quality dependence on the material standard of living;

QP is the coefficient of life quality dependence on the population density;

QF is the coefficient of life quality dependence on nourishment;

QZ is the coefficient of life quality dependence on pollution;

FS is the food potential of the funds;

FZ is the coefficient of food production dependence on pollution;

FP is the coefficient of food production dependence on population density;

ER is the coefficient of dependence of the natural resources production cost;

ZV is the coefficient of pollution dependence on the specific volume of funds;

TZ is the time of the pollution decay (reflecting the difficulty of natural decay with

the growth of pollution);

RM is the coefficient of the natural resources’ production rate dependence on the

material standard of living;

SQ is the coefficient of the dependence of the fund share in agriculture on the relativequality of life;

SF is the coefficient of the dependence of the fund share in agriculture on the level ofnourishment;

RR is the share of the remaining resources;

PR is the relative population density;

VR is the specific capital;

ZR is the relative pollution;

SR is the relative volume of agriculture funds.

The endogenous variables of the model are as follows:

P is the world population;

V is the basic asset;

Z is the pollution level;

R is the remaining part of the natural resources;

S is the share of funds in agriculture (i.e., in the food supply industry);

M is the material standard of living;

F is the relative level of nourishment (quantity of food per head);

Q is the quality of life;

Bn is the rate of fertility;

D is the rate of mortality.

In [26], the following values of the coefficients and constants are used:

CB ¼ 0:04; CD ¼ 0:028; CZ ¼ 1; CR ¼ 1; CQ ¼ 1; FC ¼ 1; FN

¼ 1; EN ¼ 1; (1.179)

PN ¼ 3:6�109; ZN ¼ 3:6�109; SN ¼ 0:3; TS ¼ 15; TVD ¼ 40; CVG ¼ 0:05;


as well as the following initial conditions for the differential equations:

P0 ¼ 1:65�109; V0 ¼ 0:4�109; S0 ¼ 0:2; Z0 ¼ 0:2�109; R0 ¼ 9�1011;

corresponding to the time starting point t0 ¼ 1900. These data were obtained on the

basis of observations for the years 1900–1970.

Here we accepted the values of the parametersCD,CZ,CR,CQ,TS, andTVD equal tothe above data from (1.179). The values of the parameters CB , CVG , and FC are

estimated again on the basis of information about the global population for the years

1901–2009 [61] and the data calculated by the state functionsVðtÞ, SðtÞ,RðtÞ, andZðtÞ (accepted as the measured functions in solving the parametric identification

problem) based on the model (1.164), (1.165), (1.166), (1.167), (1.168), (1.169),

(1.170), (1.171), (1.172), (1.173), (1.174), (1.175), (1.176), (1.177), and (1.178).

These values are determined by solving the parametric identification problem by

the searching method in the sense of the minimum of the criterion

K ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

545

X2009t¼1901

PðtÞPðtÞ � 1

� �2þ SðtÞ

SðtÞ � 1

� �2þ RðtÞ

RðtÞ � 1

� �2þ ZðtÞ

ZðtÞ � 1

� �2þ VðtÞ

VðtÞ � 1

� �2 !vuut :

Here PðtÞ and PðtÞ are the measured and modeled (calculated) values of the

population, respectively; VðtÞ, SðtÞ, RðtÞ, and ZðtÞ are the calculated data of system

(1.164), (1.165), (1.166), (1.167), (1.168), (1.169), (1.170), (1.171), (1.172),

(1.173), (1.174), (1.175), (1.176), (1.177), and (1.178). As a result of the solution

of the stated problem of parametric identification, the following are estimates of

the values of the estimated parameters: CB ¼ 0.042095, CVG ¼ 0.049644, and FC

¼ 1.078077. The relative value of the mean square deviation of the calculated

values of the variables from the respective measured values is approximately

100K ¼ 4.27%.

1.5.6.2 Analysis of the Structural Stability of Forrester’s MathematicalModel Without Parametric Control

Assertion 1.5 Let N be a compact set residing in the region fP>0; V>0; S>0; Z>0; R>0 g of the phase space of the system of the differential derived from (1.164),(1.165), (1.166), (1.167), (1.168), (1.169), (1.170), (1.171), (1.172), (1.173),(1.174), (1.175), (1.176), (1.177), and (1.178), that is, the five-dimensional spaceof the variables fP; V; S; Z; Rg. Let the closure of the interior of N coincide withN. Then the flux f defined by system (1.164), (1.165), (1.166), (1.167), (1.168),(1.169), (1.170), (1.171), (1.172), (1.173), (1.174), (1.175), (1.176), (1.177), and(1.178) is weakly structurally stable on N.One can choose N such as, for instance, the parallelepiped with the boundary P¼ Pmin; P ¼ Pmax; V ¼ Vmin; V ¼ Vmax; S ¼ Smin; S ¼ Vmax; Z ¼ Zmin; Z ¼ Zmax


R ¼ Rmin;R ¼ Rmax . Here 0<Pmin<Pmax, 0<Vmin<Vmax, 0<Smin<Smax, 0<Zmin<Zmax, 0<Rmin<Rmax.

Proof First, let’s prove that the semitrajectory of the flux f starting from any point

of the set N with some value of t (t > 0) leaves N.Consider any semitrajectory starting in N. With t>0, the following two cases are

possible; namely, all the points of the semitrajectory remain in N, or for some

t the point of the semitrajectory does not belong to N. In the first case, from (1.167)

R0ðtÞ ¼ �CRPðtÞRMðMÞ of the system, it follows that for all t>0; the variable R(t)has a derivative less than some negative constant number. That is, R(t) tends to

converge to zero with increasing t. Therefore, the first case is impossible, and the

orbit of any point in N leaves N.Since any chain-recurrent set Rðf ;NÞ lying within N is the invariant set of this

flux, in the case when it is nonempty, it consists of only whole orbits. Hence, in the

considered case, Rðf ;NÞ is empty. The assertion follows from Theorem A [67].

1.5.6.3 Choosing Optimal Laws of Parametric Control on Basisof the Forrester Model

Let’s consider working out the recommendations on choosing a rational scenario of

world policy development (in terms of the objective to maximize the mean value of

quality of life for the years 1971–2100) by choosing the optimal control laws for the

example of economic parameters FC (coefficient of nourishment, j ¼ 1) and CB

(normal fertility rate, j ¼ 2).

The problem of choosing the optimal parametric control law at the level of the

parameter is solved in the environment of the following relations:

1: U1j ¼ constj þ k1jðPðtÞ=Pðt0Þ � 1Þ;2: U2j ¼ constj � k2jðPðtÞ=Pðt0Þ � 1Þ;3: U3j ¼ constj þ k3jðRðtÞ=Rðt0Þ � 1Þ;4: U4j ¼ constj � k4jðPðtÞ=Pðt0Þ � 1Þ;5: U5j ¼ constj þ k5jðZðtÞ=Zðt0Þ � 1Þ;6: U6j ¼ constj � k6jðZðtÞ=Zðt0Þ � 1Þ;7: U71j ¼ constj þ k7jðVðtÞ=Vðt0Þ � 1Þ;8: U8j ¼ constj � k8jðVðtÞ=Vðt0Þ � 1Þ;9: U9j ¼ constj þ k9jðSðtÞ=Sðt0Þ � 1Þ;

10: U10j ¼ constj � k10jðSðtÞ=Sðt0Þ � 1Þ;11: U11j ¼ constj þ k11jðQðtÞ=Qðt0Þ � 1Þ;12: U12j ¼ constj � k12jðQðtÞ=Qðt0Þ � 1Þ:

(1.180)


Here kij 0 is the adjusted coefficient of the respective law Uij i ¼ 1; :::; 12;ðj ¼ 1; 2Þ; constj is the base value (without parametric control) of the nourishment

coefficient FC(with j ¼ 1) or normal fertility rate C

B(with j ¼ 2), respectively.

The control starting time t0 corresponds to year 1971. Application of one of the

laws (1.180) means the substitution of the respective function for the right-hand

side of the corresponding relation (1.180) into (1.170) or (1.171) of system (1.164),

(1.165), (1.166), (1.167), (1.168), (1.169), (1.170), (1.171), (1.172), (1.173),

(1.174), (1.175), (1.176), (1.177), and (1.178) for the parameter FC or CB.

The problem of choosing the optimal parametric control law at the level of the

parameter FC in the environment of the algorithms (1.180) is stated as follows: On

the basis of mathematical model (1.164), (1.165), (1.166), (1.167), (1.168), (1.169),

(1.170), (1.171), (1.172), (1.173), (1.174), (1.175), (1.176), (1.177), and (1.178),

find the optimal parametric control law in the environment of algorithms (1.180);

that is, find the optimal law from this set of algorithms and its adjusted coefficient

that maximizes the criterion

K1 ¼ 1

130

X2100t¼1971

QðtÞ (1.181)

characterizing the mean the quality of life level on the interval of time 1971–2100


X2100t¼1971

ZðtÞ � �Z; FCðtÞ 2 ½0:9; 1:1�: (1.182)

Here �Z is the total value of the pollution levels for the years 1971–2100 without

parametric control.

The given problem is solved in two stages:

– In the first stage, the optimal values of the coefficients kij are determined for each

law (1.180) by the enumeration of their values on the intervals ½0; kmij Þ quantizedwith a sufficiently small step maximizing criterion K1 under constraints (1.182).

Here kmij is the first value of the coefficient violating (1.182).

– In the second stage, the law of optimal control of the specific parameter (of 12) is

chosen on the basis of the results of the first stage by the minimum value of

criterion K1.

The numerical solution of the problem of choosing the optimal parametric

control law of the economic system at the level of the stated economic parameter

shows that the best result K1 ¼ 0:70827 can be achieved with the application of thefollowing control law of type (8) from (1.180):

FC ¼ FC � 0:158ðVðtÞ=Vðt0Þ � 1Þ: (1.183)


Note that the value of criterion (1.180) without parametric control is equal to

K1 ¼ 0:6515: The increase of the criterion value with the given parametric control

in comparison with the base variant is equal to 5.025% (see Fig. 1.15).

The problem of choosing the optimal pair of parametric control laws at the level

of the parameters FC and CB in the environment of the set of algorithms (1.180) is

stated as follows: On the basis of mathematical model (1.164), (1.165), (1.166),

(1.167), (1.168), (1.169), (1.170), (1.171), (1.172), (1.173), (1.174), (1.175),

(1.176), (1.177), and (1.178), find the optimal pair of parametric control laws in

the environment of the set of algorithms (1.180); that is, find the optimal pair of

laws from this set of algorithms and its adjusted coefficients that maximize criterion

(1.181) under constraints (1.182).

The numerical solution of the problem of choosing the optimal pair of the

parametric control laws of the economic system at the level of two economic

parameters FC and CB shows that the best result K1 ¼ 0:703135 can be achieved

with the application of the following pair of control laws:

FC ¼ FC � 0:15ðVðtÞ=Vðt0Þ � 1Þ; CB ¼ C

B� 0:01ðPðtÞ=Pðt0Þ � 1Þ: (1.184)

In this case, the increase in the value of criterion K1 in comparison with the base

variant is equal to 7.93%.

Let’s compare the obtained results of the parametric control of the evolution of

dynamical system (1.164), (1.165), (1.166), (1.167), (1.168), (1.169), (1.170),

(1.171), (1.172), (1.173), (1.174), (1.175), (1.176), (1.177), and (1.178) with the

optimal laws found at the level of one (1.183) and two (1.184) parameters and the

results of the scenario consisting in the increase of the parameter FC by 25% in

Qua

lity

of life

Years

Tendency estimated by Forrester

Tendency obtained as a result of control of parameter of influence on agriculture production

Fig. 1.15 Trajectories characterizing the change in the quality of life Q


comparison with the base solution (obtained for the following values of constants

CB ¼ 0:042095; CD ¼ 0:028; CZ ¼ 1; CR ¼ 1; CQ ¼ 1; FC ¼ 1:078077; FN ¼ 1;

EN ¼ 1; PN ¼ 3:6�109; ZN ¼ 3:6�109; SN ¼ 0:3; TS ¼ 15; TVD ¼ 40; CVG ¼ 0:049644and initial conditions for the differential equationsP0 ¼ 1:65�109;V0 ¼ 0:4�109;S0 ¼ 0:2; Z0 ¼ 0:2�109; R0 ¼ 9�1011).

A comparison shows that for the stated scenario (the increase of parameterFC by

25%), the mean value of the quality of life (criterion K1) on the time interval

1971–2100 decreases by 9.77% in comparison with the base variant, and the mean

value of the pollution 1130

P2100t¼1971

ZðtÞ increases by 4.97% in comparison with the base

variant. With use of optimal law (1.183) alongFC , the index of the quality of life

improves by 5.025% and the mean value of the pollution decreases by 3.5% in

comparison with the base value. Furthermore, the value of the nourishment coeffi-

cient FC by optimal law (1.183) changes by no more than 10% in comparison with

the base value of this coefficientFC ¼ 1.078077. With the use of the optimal pair of

laws (1.183), the quality of life index improves by 7.93%, and the mean pollution

decreases by 1% in comparison with the base variants.

1.5.6.4 Analysis of the Structural Stability of Forrester’s MathematicalModel Subject to Parametric Control

Application of the optimal laws of parametric control (1.180) means substitution of

the respective functions into (1.170), (1.171) for the parametersFC andCB, while the

other model equations remain unchanged.

The proof of the weak structural stability of the mathematical model presented

above and relying on (1.167) allows us to derive the following assertion:

Assertion 1.6 Let N be a compact set belonging to the region fP>0; V>0; S>0;Z>0; R>0 g of the phase space of the system of differential equations derived from(1.164), (1.165), (1.166), (1.167), (1.168), (1.169), (1.170), (1.171), (1.172),

(1.173), (1.174), (1.175), (1.176), (1.177), and (1.178), that is, the five-dimensionalspace of variables fP; V; S; Z; Rg:

Let the closure of the interior of N coincide with N. Then the flux f defined by

(1.164), (1.165), (1.166), (1.167), (1.168), (1.169), (1.170), (1.171), (1.172),

(1.173), (1.174), (1.175), (1.176), (1.177), and (1.178) and (1.183) or (1.184) is

weakly structurally stable on N.

1.5.6.5 Finding Bifurcation Points of Extremals of the Variational CalculusProblem on the Basis of Forrester’s Mathematical Model

Let’s analyze the dependence of the solution of the problem considered above of

choosing the optimal parametric control law on the values of the two-dimensional

parameter (CVG,CVD)with possible values belonging to region (rectangle)Lin theplane.


As a result of computer simulations, we generated plots of the dependence of the

optimal values of criterionK on the values of the parameter (CVG,CVD) for each of the

24 possible lawsUij; i ¼ 1; :::; 12; j ¼ 1; 2. Figure 1.16 demonstrates such plots for

the four laws U2;1,U6;1,U11;1,U8;1maximizing the values of criterionK in regionL, aswell as the intersection curves of the corresponding surfaces. The projection of these

curves to the plane (CVG , CVD ) consists of the bifurcation points of this two-

dimensional parameter. This projection divides rectangle L into two parts; inside

each of them only one parametric control law out of (U2;1,U6;1,U11;1,U8;1) is optimal.

Two or three different laws are optimal on the projection curves.

1.5.7 Turnovsky’s Monetary Model


Turnovsky’s monetary model [72], after the corresponding transformations (for the

economic development scenario considered in [72], when the national deficit is

fully financed subject to invariable reserve of bonds per capita), is presented by the

system of the following differential and algebraic equations:

_p ¼ r½r� p� (1.185)

Fig. 1.16 Plot of dependencies of optimal values of criterion K on parameters (CVG , CVD ).

Here colorings correspond to parametric control laws as follows: –U2;1, –U6;1, –U11;1,

– U8;1


_m ¼ g� uyþ bðreð1� uÞ þ pÞ � ðmþ bÞðnþpÞ; (1.186)

_z ¼ gð1� uÞg� 1

½rez� Rk� þ gg� 1

z

k� 1

� �þ 1

� �gðk � kÞ þ n

z� k

g� 1; (1.187)

_k ¼ gðk � kÞ; (1.188)

y ¼ Aka; (1.189)

re ¼ Aaka�1; (1.190)

R ¼ Aaka�1; (1.191)

k� ¼y� c ðy� RkÞð1� uÞ þ bþz

l2½ð1� l4Þm� l4ðbþ zÞ � l1yþ l3p� þ zn� mp

h i� nk þ g

l czk � cþ 1� ;

(1.192)

i ¼ lðk � kÞ þ nk: (1.193)

Here the derivatives with respect to time t in years are denoted by an overdot.

The output (endogenous) variables of the model are as follows:

p is the instant expected rate of inflation (1/year);

m is the nominal reserve of the foreign money per capita (tenge per capita; tenge is

the monetary unit of Kazakhstan);

z is the real share of stock per capita (tenge per capita) (the real indicators are

defined by the prices of year 2000);

k is the real fixed capital per worker (tenge per capita);

y is the real product output per capita [(tenge per capita)/year];re is the real rate of stock income before tax (1/year);

R is the limiting real physical product of the capital [(tenge per capita)/year2];

k is the desired real fixed capital per capita (tenge per capita);

i is the real investment per capita [(tenge per capita)/year].

The input (exogenous) time-dependent parameters of the model are as follows:

p is the consumer price index (1/year);

g is the real public expense per capita [(tenge per capita)/year] (g>0);

n is the rate of population growth (1/year);

g is the coefficient of the equation for fixed capital per worker (1/year) ð0<g<1Þ;A and a are the coefficients of the production function ðA>0; 0<a<1Þ;c is the consumption share of the available income ð0<a<1Þ (dimensionless);

l1; l2; l3; and l4 are the coefficients of the equation for real demand for money per

capita (l1>0; l2<0; l3>0; 0<l4<1) (dimensionless);


l is the coefficient of the equation of investment per capita (1/year);

u is the income taxation rate, 0<u<1 (dimensionless).

The input model parameters are as follows:

b is the nominal reserve of public bonds per capita; b > 0 (tenge per capita).

The input model parameters also incorporate the initial values (t ¼ 0) of the

output variables of dynamic equations (1.1), (1.2), (1.3), and (1.4) of model p0, m0,

z0, and k0. The values of the input model functions for the integer moments of time

t are also considered to be the input model parameters. All input model functions

are considered to be piecewise linearly continuous functions defined by their values

for integer moments of t.

1.5.7.2 Estimation of Turnovsky’s Model Parametersand Retrospective Forecast

Solving the problem to estimate input parameters (parametric identification) of a

model, we obtain the values of input functions and parameters p(t), g(t), n(t), g(t),A(t), a(t), l(t), c(t), l1(t), l2(t), l3(t), and l4(t), where t ¼ 0; :::; 9; as well as b, u, p0,m0, z0, and k0 by the search method for minimizing the sum of squares of

discrepancies of the output variables based on statistical information on evolution

of the economy of the Republic of Kazakhstan from 2000 to 2009. The values of the

input functions and parameters were searched within small intervals from the

middle in measured values (if available) of the respective functions and parameters.

(Please check this)

The criterion of parametric identification is given by

KI ¼ 1Pvj¼1

Pnt¼0 Mjt

Xv

j¼1

Xn

t¼0Mjt

xjðtÞ � xj ðtÞxj ðtÞ

!2

! min : (1.194)

Here v is the number of the output variables used in the estimation of the para-

meters equal to 5, j is the variable number; n + 1 is the number of measurements,

t ¼ 0 corresponds to the beginning of year 2000; and xjðtÞ are the computed values

of the output variables ðyðtÞ; kðtÞ; zðtÞ;pðtÞ; iðtÞÞ and for the corresponding time

moments. The symbol (*) corresponds to the measured values’ respective variables.

MJt are the positive weights’ fitted reasoning from the significance of their respec-

tive values of the output variables while solving the problem of model parametric

identification. In Table 1.9, we present the weights of criterion KI. The weightMjt is

situated at the intersection of the row j with column t.For solving the problem of parametric estimation, we shall apply the the

Runge–Kutta and Nelder–Mead algorithms [66]. The stated parametric identifica-

tion problem is solved with the use of statistical information from 2000 to 2007

(n ¼ 7). As a result of solving the parametric identification problem, the values of


the mean square deviations of computed values of the model output variables from

their respective measured values ð100 ffiffiffiffiffiffiK1

p Þ do not exceed 1.2%.

For model verification, the problem of the retrospective forecast is solved. Using

values of input functions, parameters and initial values of output variables from

2000 to 2007 are determined by estimation (as well as the extrapolation of the

values of input functions for 2008–2009), between 2008 and 2009 and estimate

relative errors of the computed values of the model output variables relative to the

corresponding measured values. The results of the problem are presented in

Table 1.10. The symbol* corresponds to the measured values and “D” correspondsto the errors (percentage) of computed values of the corresponding measured

values.

The mean error of the variables indicated in Table 1.10 for the retro-forecasting

period is 3.7825%, hence showing the admissible accuracy of the description of

Kazakhstan’s economic evolution by the analyzed model.

Table 1.9 Weights Mjt of criterion KI

Year

Variable

2000,

t ¼ 0

2001,

t ¼ 1

2002,

t ¼ 2

2003,

t ¼ 3

2004,

t ¼ 4

2005,

t ¼ 5

2006,

t ¼ 6

2007,

t ¼ 7

yðtÞ; j ¼ 1 0.001 0.001 0.001 0.010 0.010 1.000 1.000 1.000

kðtÞ; j ¼ 2 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

zðtÞ; j ¼ 3 0.001 0.001 0.001 0.010 0.010 1.000 1.000 1.000

pðtÞ; j ¼ 4 0.001 0.001 0.001 0.010 0.010 0.100 0.100 0.100

iðtÞ; j ¼ 5 0.001 0.001 0.001 0.010 0.010 0.100 0.100 0.100

Table 1.10 Measured and computed values of the model output variables and respective errors

(in percentage)

Year 2008 2009

y 336,140 334,680

y 333,843 333,015

Dy 0.68322 0.49744

z 1,117,488 1,305,937

z 1,228,469 1,483,599

Dz 9.93126 13.60410

k 747,806 771,832

k 675,539 762,518

Dk 9.66383 1.20669

p 0.11828 0.07525

p 0.11881 0.07639

Dp 0.45321 1.51652

i 136,501 151,534

i 136,424 151,855

Di 0.05666 0.21187


1.5.7.3 Analysis of the Structural Stability of Turnovsky’s Monetary Model

This analysis is carried out on the basis of Robinson’s Theorem A [67] on sufficient

conditions of weak structural stability.

In this algorithm, the mapping f is defined as the shift along the trajectories

of dynamical system (1.185), (1.186), (1.187), (1.188), (1.189), (1.190), (1.191),

(1.192), and (1.193) corresponding to the variation of time t by a year. The input

functions of the model are considered to be the constants (same as values of

year 2007).

While realizing the previously mentioned algorithm, we consider a parallelepi-

ped, f0 � p � 0:2; 0 � m � 8; 000, 0 � z � 50; 000, 0 � k � 830; 000g in the statespace of models (1)–(9) as the initial compact N.

The partitioning of the initial parallelepiped (and other cells obtained as a result

of the algorithm) into 16 parts is realized by dividing each of its edges into two equal

parts. As a result of these computations according to the mentioned algorithm, after

four iterations we generated graphGwith an empty set of nodes. This means that the

Turnovsky’s monetary model with the considered values of input parameters is

considered to be weakly structurally stable in the indicated compact N.

1.5.7.4 Estimation of the Parametric Sensitivity of Turnovsky’s Model

Evaluating the influences of input parameters and model functions on the values of

its output variables, we form a matrix whose rows are enumerated by all input

parameters and functions, and whose columns are enumerated by the values of the

output variables for t¼ 9, which corresponds to year 2009. This matrix incorporates

the coefficients of sensitivity of the indicated output values of the model by its input

values computed by the formula

FpjðtÞ ¼ 100xnj ðtÞ � xjðtÞ

xjðtÞ : (1.195)

Here, p is the variable input parameter or value of the input function; xjðtÞ is the

value of the jth output variable at time t obtained via running the model with values

of the input parameters and functions derived from the estimate of the parameters or

taken from the statistical sources (the base computation); xnj ðtÞ is the value of the

respective output variable obtained by increasing variable parameter p by 1%; other

values of the parameters and functions remain invariable in comparison to the base

computation.

The results from solving the problem of constructing the parametric sensitivity

matrix are presented in Table 1.11. For instance, increasing the value of the

parameter c(9) by 1% while keeping the value of c(8) yields the respective increaseof the function c(t), which is linear in the interval [8, 9]. In turn, this yields a

variation of the values of the output variables (and coefficients FpjðtÞ for t ¼ 9.


The analysis of Table 1.11 shows that within the limits of the input parameters in

2009 presented in this table, the most influence on the values of the output variables

y(9), m(9), z(9), k(9), i(9) is exerted by the variation of coefficient a(9) of the

production function. The value of the output variable p(9) can be impacted only by

changing the price index p(9).

1.5.7.5 Finding Optimal Values of the Adjusted Parameters Basedon Turnovsky’s Model

Let’s now consider implementation of the effective public policy based on model

(1.185)–(1.193) via the synthesis of optimal values of economic parameters, public

expenses per capita g(t), and income taxation rate u(t) for 2010–2015.The problem of the synthesis of the optimal parametric control law at the level of

the previously mentioned parameters can be formulated as follows. Based on the

mathematical model (1.185)–(1.193), find such values g(t), u tð Þ; t ¼ 2010; . . . ;2015 that maximize criterion

K ¼ 1

6

X2015

t¼2010yðtÞ; (1.196)

which is the mean value of the real product output per capita in the years 2010–2015

under the following constraints on the model output variables and adjusted

parameters (here t 2 ½2010; 2015�):

mðtÞ>0; zðtÞ>0; kðtÞ>0; yðtÞ>0; reðtÞ>0;RðtÞ>0; k>0; iðtÞ>0; (1.197)

Table 1.11 Some elements of the model parametric sensitivity matrix for t ¼ 9

The output variableThe input

parameter y p M Z K I

C(9) �0.016370 0.0000 �0.01230 0.00309 �0.01650 1.693064

g(9) 0.280699 0.0000 0.26993 0.33196 0.28325 0.881013

n(9) �0.017260 0.0000 �0.02060 �0.02930 �0.01740 0.048407

p(9) �0.009930 0.5161 �0.05200 �0.01360 �0.01000 �0.397900

l(9) �0.038020 0.0000 0.02940 �0.08880 �0.03840 �2.592670

g(9) 0.136572 0.0000 �0.00200 �0.27080 0.13781 �0.455090

A(9) 0.790219 0.0000 �0.03340 �0.29850 �0.20960 �3.418850

a(9) 11.388490 0.0000 �0.31440 �2.22150 �2.60970 �53.240600

l1(9) �0.163710 0.0000 0.05780 �0.26970 �0.16520 �4.831100

l2(9) 0.009145 0.0000 0.01151 �0.00830 0.00923 �1.518530

l3(9) �0.043310 0.0000 �0.00520 �0.03870 �0.04370 0.849691

l4(9) �0.140340 0.0000 0.04360 �0.22460 �0.14160 �3.589840


gðtÞ>0; 0<uðtÞ<1: (1.198)

Note that for the base computation of the model until 2015 obtained from the

found values of the input parameters of the model and by the extrapolation of the

model input functions by the linear trend, the criterion value is K ¼ 437368 tenge

(in the prices of year 2000).

As a result of the numerical solution of the stated problem of finding optimal values

of parameters g(t) and u(t) of the economic system by the Nelder–Mead method [66],

we obtain the optimal valueK ¼ 511552. The increase in criterionKwith application

of the aforesaid parametric control in comparison with the base variant is 16.96%.

The diagrams of the computed values of the output variable of model y(t) (outputof real products per capita) without parametric control as well as with application of

the derived optimal parametric control law are presented in Fig. 1.17.

1.5.7.6 Analysis of the Dependence of Optimal Values of the ParametricControl Criterion on the Values of the Uncontrolled ParametersBased on Turnovsky’s Model

The optimization problem considered above was solved with fixed values of the

input parameters that do not participate in the control. Moreover, carrying out

the analysis, we realized the dependence of the optimal values of criterion K on

Fig. 1.17 Real product per capita


the values of the model’s uncontrolled parameters by the example of the two-

dimensional parameter a ¼ c 9ð Þ; l 9ð Þð Þ incorporating the share of real consump-

tion in the real income available and the coefficient of the investment equation for

2009. The range of variation of these parameters was determined based on the

estimated values of c(9) and l(9) in the form of a rectangle A ¼ ½0:0820; 0:1090��½0:708; 0:719�.

In Fig. 1.18, we present some results of the analysis, namely, the diagrams of the

dependence of criterion K on parameter a ( a 2 A ) for the parametric control

problem considered above. The diagrams in Fig. 1.18 describe the base and optimal

(for the problem of finding public expense per capita and the income tax rate) value

of criterion K.

1.5.8 Endogenous Jones’s Model


The mathematical model of technical progress and population [61], after some

transformations, is given by following difference and algebraic equations:

Nt ¼ Nt�1ð1þ ntÞ; (1.199)

LAt ¼ ltNtpt; (1.200)

LYt ¼ ltNtð1� ptÞ; (1.201)

Fig. 1.18 – the base variant, – adjustment of the public expense per capita and the income

taxation rate


At ¼ At�1 þ dLlAðt�1ÞA’t�1; (1.202)

Yt ¼ Ast L

bYtet (1.203)

wt ¼ YtptrLAt

(1.204)

Here time t is discrete, and the year number.

The endogenous variables of the model are as follows:

Nt is the population of the country (people);

LAt is the number of people involved in the production of ideas (people);

LYt is the number of people involved in the production of consumer goods (people);

At is the reserve of ideas in the economy (dimensionless);

Yt is the production of consumer goods expressed by the production function (tenge

per year);

wt is the wages per one worker in a unit of time (tenge/year).

The exogenous variables of the model are as follows:

nt is the rate of population growth (1/year);

lt is the share of people employed in the economy in the entire population

(dimensionless);

pt is the share of people involved in the production of ideas in the entire number of

people employed in the economy (dimensionless);

et is the exogenous production shock.

The exogenous parameters of the model are as follows:

d, l (dimensionless), and ’ (dimensionless) are parameters of the production

function of the production of ideas;

s and b are parameters of the production function of consumer goods production

(dimensionless);

r is the share of the wage fund in the entire production of consumer goods

(dimensionless).

It should be noted that this parameter is not present in [61]. It is assumed that the

entire output of consumer goods is intended for remuneration of workers’ labor.

The introduction of parameter r; 0 � r � 1; is made to obtain the model, which is

more adequate for real economic conditions.

The exogenous parameters of the model include initial values (with t ¼ 0) of the

endogenous variables of dynamical equations (1.199), (1.202) of models N0 and A0.

The values of the exogenous functions of the model with fixed time t are also

considered to be the exogenous parameters of this model.


1.5.8.2 Evaluation of the Exogenous Parameters of the Jones’s Modeland Retrospective Forecast

Solving the problem of estimation of input parameters (parametric identification) of

the model, we obtain values of the input functions and parameters nt, lt, pt, et, d, l,j, s, b, r, N0, and A0 by the search method for minimization of the sum of squares

of discrepancies of output variables based on statistical data from economic evolu-

tion of the Republic of Kazakhstan from 2000–2009. The values of the input

functions and parameters were searched within small intervals with centers in the

measured values (if available) of the corresponding functions and parameters.

The criterion of parametric identification is given by

KI ¼ 1

vðnþ 1ÞXv

j¼1

Xn

t¼0

xtj � xtjxtj

!2

! min (1.205)

Here v ¼ 5 is the number of the output variables used in the estimation of

parameters; n + 1 is the number of measurements; t ¼ 0 corresponds to year

2000; xtj are the computed values of the output variables (Nt, LAt, LYt, Yt, and wt).

The symbol * corresponds to the measured values of the respective variables.

To solve the problem of parametric estimation, we apply the Nelder–Mead

algorithm [66]. This parametric identification problem is solved with the use of

statistical information from 2000–2007 (n ¼ 7). As a result of solving the problem,

the relative values of the mean square deviations of computed values of the model

output variables from the respective measured values (100ffiffiffiffiffiKI

p) do not exceed 1%.

For model verification, the following problem of the retrospective forecast is

solved. Using the values of the input functions, parameters, and initial values of the

output variables from 2000–2007 determined by the estimation (as well as the

extrapolation of the values of the input functions for 2008–2009), within the period

of 2008–2009, estimate the relative errors of the computed values of the model

output variables relative to the corresponding measured values. The results of the

problem solution are presented in Table 1.12. Here the symbol * corresponds to the

measured values, and D corresponds to the errors (percentage) of the computed

values of the respective measured values.

The mean error of the variables indicated in Table 1.12 for the retro-forecasting

period is equal to 3.49%, thereby indicating admissible accuracy of the description

of Kazakhstan’s economic evolution by the analyzed model.

1.5.8.3 Analysis of the Structural Stability of the MathematicalJones’s Model

Let’s analyze the robustness (structural stability) of model (1.199), (1.200), (1.201),

(1.202), (1.203), and (1.204) on the basis of the mentioned theorem on sufficient

conditions of weak structural stability under the additional assumption on the


constant character of all the exogenous functions of the model. We’ll consider the

values of these functions as well as values of all exogenous parameters of the

analyzed model to be some constant positive numbers. In this case, (1.199), (1.200),

(1.201), (1.202), (1.203), and (1.204) define the homeomorphism f (and the respec-tive cascade) determined in the plane (N, A).

Assertion 1.7 Let M be a compact set in the region ðN>0;A>0Þ of state space ofthe cascade f defined by (1.199), (1.200), (1.201), (1.202), (1.203), and (1.204). Theclosure of the interior of M coincides with M. Then, cascade f is weakly structurallystable in M.

Choose M to be a rectangle with boundaries N ¼ Nmin ,N ¼ Nmax , A ¼ Amin;A ¼ Amax. Here 0<Nmin<Nmax, 0<Amin<Amax.

Proof Let’s first determine that the semitrajectory of cascade f,which begins at anypoint of setM with some value of t (t> 0), leavesM. It follows from (1.199) with nt> 0 since in that case the sequence of Nt is the increasing geometric progression.

As any chain-recurrent set R(f, M) lying within M is the invariant set of the

cascade, if it is nonempty, it consists only of the whole orbits. Hence, in our case,

set R(f, M) is empty. This statement follows from Robinson’s Theorem A [67].

1.5.8.4 Estimation of the Parametric Sensitivity of Jones’s Model

Solving the problem of evaluating the influences of input parameters and model

functions on the values of its output variables, we form a matrix whose rows are

enumerated by all the input parameters and functions and columns are enumerated

by the values of the output variables for t¼ 9, which corresponds to year 2009. This

matrix incorporates the coefficients of sensitivity of the indicated output values of

the model by its input values computed by the formula

Table 1.12 Measured

and computed values of

the model output variables

and the respective errors

(in percentage terms)

Year 2008 2009

Y:t 2.895 � 1013 3.450 � 1013

Yt 2.988 � 1013 3.660 � 1013

DYt 3.210 6.092

N:e 16.068 � 106 16.470 � 106

Nt 15.748 � 106 15.927 � 106

DNt 1.992 3.298

LAt 18,219 18,674

LAt 17,975 18,179

DLAt 1.335 2.650

Lt 7.844 � 106 8.041 � 106

Lyt 7.699 � 106 7.786 � 106

DLYt 1.847 3.155

wt 741.751 � 103 863.110 � 103

Wt 767.743 � 103 930.701 � 103

Dwt 3.504 7.831


Fpj ¼ 100xntj � xtj

xtj: (1.206)

Here p is the variable input parameter or value of the input function; xtj is the valueof the ith output variable at time t obtained via running the model with values of

input parameters and functions derived from the estimation of parameters or taken

from statistical sources (the base computation); xntj is the value of the respective

output variable obtained by increasing the variable parameter p by 1%, while

the other values of the parameters and functions remain invariable in comparison

to the base computation.

The results of the construction of the parametric sensitivity matrix are partially

presented in Table 1.13.

The analysis of Table 1.13 shows that the most influence on the values of

endogenous variables Y9 and W9 is exerted by the variation of the coefficient

of the production function s, and the value of the endogenous variable A9 is

exerted by the variation of the coefficient of the equation of the reserve of ideas

in economy j.

1.5.8.5 Finding the Optimal Values of the Adjusted ParametersBased on Jones’s Model

Let’s now consider implementation of the effective public policy based on model

(1.199), (1.200), (1.201), (1.202), (1.203), and (1.204) by via the synthesis of

optimal values of economic parameters, i.e., the shares of people involved in the

production of ideas in the number of people employed from 2010 to 2014. This

ability is in particular grounded by the analysis of the sensitivity matrix obtained in

the above subsection.

The problem of the synthesis of the optimal parametric control law at the level of

parameters pt can be formulated as follows. Based on mathematical models (1.199),

(1.200), (1.201), (1.202), (1.203), and (1.204) , find the optimal law of parametric

control of the parameters; i.e., find such values pt, t ¼ 10, . . ., 14, that maximize

criterion

K ¼ 1

5

X14t¼10

Yt; (1.207)

which is the mean value of the consumer goods output in the aforesaid period under

the following constraints:

Nt 0; LAt 0; LYt 0; Yt 0;wt 0; t ¼ 10; :::; 14; (1.208)

0:0012<pt � 0:0480; t ¼ 10; :::; 14: (1.209)


Note that for the base computation of the model until 2014 obtained for the

values found from the input parameters of the model and by extrapolating the model

input functions by a linear trend, the criterion value is K ¼ 8:128 � 1013 tenge.As the result of the numerical solution of the stated problem of finding the

optimal values of parameters pt of the economic system by the Nelder–Mead

method [66], we obtain the optimal value K ¼ 8:462 � 1013; which exceeds the

base value of the criterion by 4.1%.

Table 1.13 Some elements

of the model parametric

sensitivity matrix

Variable

Parameter Y9 W9 A9

A0 0.158903 0.158903 0.072459

N0 0.108772 0.007974 0.003775

s 4.075839 4.075839 0.000000

b 3.891727 3.891727 0.000000

d 0.198968 0.198968 0.089890

l 0.095282 0.095282 0.044117

j 2.071020 2.071020 0.702719

l0 0.000877 0.000877 0.000416

l1 0.000879 0.000879 0.000417

l2 0.000880 0.000880 0.000417

l3 0.000881 0.000881 0.000418

l4 0.000882 0.000882 0.000418

l5 0.000888 0.000888 0.000421

l6 0.000891 0.000891 0.000422

l7 0.000887 0.000887 0.000421

l8 0.000885 0.000885 0.000420

l9 0.100000 �9.53 � 10–10 0.000000

p0 0.000877 0.000877 0.000416

p1 0.000879 0.000879 4.17 � 10–4

p2 0.000880 0.000880 4.17 � 10–4

p3 0.000881 0.000881 4.18 � 10–4

p4 0.000882 0.000882 0.000418

p5 0.000888 0.000888 4.21 � 10–4

p6 0.000891 0.000891 4.22 � 10–4

p7 0.000887 0.000887 4.21 � 10–4

p8 0.000885 0.000885 0.000420

p9 –0.000210 –0.000210 0.000000

n0 –0.000260 �2.01 � 10–5 �9.56 � 10–6

n1 –0.00010 �7.23 � 10–6 �3.43 � 10–6

n2 0.000112 6.84 � 10–6 3.25 � 10–6

n3 0.000596 3.14 � 10–5 1.49 � 10–5

n4 0.000858 3.80 � 10–5 1.80 � 10–5

n5 0.000985 3.53 � 10–5 1.67 � 10–5

n6 0.001186 3.21 � 10–5 1.52 � 10–5

n7 0.001142 2.08 � 10–5 9.86 � 10–6

n8 0.001311 1.20 � 10–5 5.70 � 10–6

n9 0.001619 �1.62 � 10–11 0.000000


The obtained optimal values of the adjusted parameters pt are presented in

Table 1.14.

The diagrams of the computed values of the endogenous variable of the model,

namely, output of the consumer products Yt without parametric control as well as

with application of the determined parametric control law, are presented in

Fig. 1.19.

1.5.8.6 Analysis of the Dependence of the Optimal Values of theParametric Control Criterion on the Values of UncontrolledParameters Based on Jones’s Model

The optimization problem considered above was solved with fixed values of the

input parameters that do not participate in control. Moreover, in carrying out

the analysis, we discovered the dependence of the optimal values of criterion K(9) on the values of the model’s uncontrolled parameters via the example of the

two-dimensional parameter a ¼ d; sð Þ and by incorporating the parameters of

Table 1.14 Optimal values

of the adjusted parameters ptAdjusted parameter Optimal value

p10 0.0480

p11 0.0480

p12 0.0291

p13 0.0123

p14 0.0012

Fig. 1.19 Consumer products’ output (Y)


the equation from ideas of production and the production function of the consumer

goods output.

The range of variation for these parameters was determined based on the estimated

values of d and s in the form of rectangle A ¼ ½0:615; 0:769� � ½2:11; 3:16�.In Fig. 1.20, we present some results of the analysis, namely, the diagrams of the

dependence of criterion K on parameter a (where a 2 A) for the parametric control

problem considered before.

The diagrams in Fig. 1.20 describe the base and optimal (for the solved problem

of finding the shares of the whole wage fund, which is used for rewarding the

attempts of innovations) values of criterion K.

1.5.8.7 Appendix. Proof of Statements Given in Chapter 1

Proof of Theorem 1.1 From constraints (1.9), it follows that for any admissible

control signal u, the respective optimality criterion K(u) given by (1.7) makes sense

and there exists an upper bound of such values of function K(u). Therefore, thereexists a maximizing sequence such that sequence {uk} of the elements of set Wad

converge as

KðukÞ ! supu2Wad

KðuÞ: (1.210)

Obviously, set W is bounded in the space H1ðt0; t0þTÞ�q½ . Then, applying

Banach–Alaoglu’s theorem, one can choose a subsequence in {uk} such that

Fig. 1.20 – the base variant, – the adjustment of the shares of people involved in the

production of ideas in the number of people employed in the economy


(keeping the initial notation for brevity) the convergence uk ! u takes place in the

weak topology of H1ðt0; t0þTÞ�q½ ; Taking into account the closeness of setW, we’ll

determine the inclusion u 2 W . According to Rellikh–Kondrashov’s theorem, the

embedding H1ðt0; t0þTÞ�q � ðC½t0; t0þT�Þ½ :q is compact. Then, after a possible

choice of a subsequence, we determine the convergence

uk ! uin C t0; t0þT�½ Þq:ð (1.211)

Let’s denote x as the solution of problems (1.1) and (1.2) corresponding to the

limiting value of u. The following relations hold true:

_xkðtÞ�x: ðtÞ¼ f ðxkðtÞ; ukðtÞ; lÞ � f ðxðtÞ; uðtÞ; lÞ; xkðt0Þ � xðt0Þ¼ 0;

where xk denotes the solution of system (1.1), (1.2) corresponding to the control law

uk. As a result, we obtain the following inequality for t 2 ½0; T�:

jxkðtÞ � xðtÞj �Z t

t0

jf ðxkðtÞ; ukðtÞ; lÞ � f ðxðtÞ; uðtÞ; lÞjdt

� Lf

Z t

t0

jxkðtÞ � xðtÞjdtþ Lf

Z t

t0

jukðtÞ � uðtÞjdt

� Lf

Z t

t0

jxkðtÞ � xðtÞjdtþ Lf Tjjuk � ujj;

where Lf is the Lipschitz constant of function f. Hereinafter, the norm is understood

as the respective power (in general difference for the control and state function) of

space C½t0; t0 þ T� . Applying Gronwall’s lemma, we’ll show that there exists a

constant c > 0 independent of k such that the following estimate holds:

jjxk � xjj � cjjuk � ujj: (1.212)

Let’s proceed to the limit, taking into account closeness of set X(t). We deter-

mine that inclusion (1.8) holds true for the limiting function x; only then is control

law u admissible.

The following inequality holds true:

Fðxk�� ðtÞÞ � FðxðtÞÞj � LFjxkðtÞ � xðtÞj;

where LF is the Lipschitz constant of function F.Taking into account condition (1.212), we obtain the estimate

Z t0þT

t0

jFðxkðtÞÞ � FðxðtÞÞjdt � cTLf jjuk � ujj:


Hence, passing to the limit, we obtain

Z t0þT

t0

FðxkðtÞÞdt !Z t0þT

t0

FðxkðtÞÞdt:

Taking into account that the left-hand side of this expression is KðukÞ, and by

virtue of (1.210), we conclude that

Z t0þT

t0

FðxðtÞÞdt ¼ supu2Wad

KðuÞ:

Hence, control law u is optimal. And Theorem 1.1 is proved.

Proof of Theorem 1.2 Since the set of the control laws is finite, we suffice to prove

solvability of the problem of maximizing function Kj on set Vjad for the selected

number j ðj ¼ 1; :::; rÞ. By virtue of the constraint on the growth rate of function Fand condition (1.19) with the bounded set

SXðtÞ

t2ðt0;t0þT�, there exists an upper bound of

function Kj; hence, the corresponding maximizing subsequence {vk} of the

elements of set Vjad exists such that it converges as

KjðvkÞ ! supv2Vj

ad

KjðvÞ: (1.213)

Since set V is close and bounded, after choosing the subsequence, we obtain

convergence

vk ! v in Rl; (1.214)

and v 2 V.From (1.19) and (1.20), we obtain equalities

_xvkj ðtÞ � _xvj ðtÞ ¼ f ðxvkj ðtÞ;Gjðvk; xvkj ðtÞÞ; lÞ � f ðxvj ðtÞ;Gjðv; xvj ðtÞÞ; lÞ;

xvkj ðt0Þ � xvj ðt0Þ ¼ 0:

We also obtain the following inequalities for t 2 ðt0; t0 þ T� :

jxvkj ðtÞ � xvj ðtÞj �Z t

t0

jf ðxvkj ðtÞ;Gjðvk; xvkj ðtÞÞ; lÞ � f ðxvj ðtÞ;Gjðv; xvj ðtÞÞ; lÞjdt

� Lf

Z t

t0

jxvkj ðtÞ � xvj ðtÞjdtþ Lf

Z t

t0

jGjðvk; xvkj ðtÞÞ � Gjðv; xvj ðtÞÞjdt

� Lf ð1þ LjÞZ t

t0

jxvkj ðtÞ � xvj ðtÞjdtþ Lf LjTjvk � vj:


Here Lf and Lj are the Lipschitz constants for functions f and Gj, respectively. The

norm is understood as the space Rl. Applying Gronwall’s lemma and the last

inequality, we’ll show that there exists a constant c > 0 independent of k, j, l,and t; we thus have the estimate

jxvkj ðtÞ � xvj ðtÞj � cjvk � vj; (1.215)

from which it follows that

xvkj ðtÞ ! xvj ðtÞ: (1.216)

The condition vk 2 Vjad yields the inclusion

xvkj ðtÞ 2 XðtÞ; t 2 ðt0; t0 þ T�:

Hence, by virtue of (1.216) and the closeness of the set X(t), the following

condition holds true:

xvj ðtÞ 2 XðtÞ; t 2 ðt0; t0 þ T�: (1.217)

The Lipschitz condition for the function Gj yields the inequality

jGjðvk; xvkj ðtÞÞ ! Gjðvk; xvj ðtÞÞj � Ljðjvk � vj þ jxvkj ðtÞ � xvj ðtÞjÞ:

From the above inequality together with (1.215), we conclude that there exists a

positive constant c1 independent of any parameter such that the estimate

jGjðvk; xvkj ðtÞÞ ! Gjðv; xvj ðtÞÞj � c1jvk � vj

takes place. From the above estimate, the convergence

Gjðvk; xvkj ðtÞÞ ! Gjðv; xvj ðtÞÞ

follows. Taking into account the inclusionGjðvk; xvkj ðtÞÞ 2 UðtÞ and the closeness ofset U(t), we obtain Gjðv; xvj ðtÞÞ 2 UðtÞ; from which it follows that v 2 Vj

ad.

The proof of the convergence

ðt0þT

t0

F xvkj ðtÞ� �

dt !ðt0þT

t0

F xvkj ðtÞ� �

dt

based on the convergence in (1.216) can be made in the same way as the similar

property in Theorem 1.1. Then, from condition (1.213), it follows that the adjusted

coefficient v is optimal. Theorem 1.2 is proved.


Proof of Theorem 1.3 Obviously, the set of the admissible values of function K is

bounded from above (the boundaries of a continuous function in a compact set).

There exists an upper bound sup K of this set. Therefore, the sequence of the vector

functions ðxk; ukÞ exists such that the relations

xkðtþ 1Þ ¼ f ðxkðtÞ; ukðtÞ; lÞ; t ¼ 0; :::; n� 1; (1.218)

xkð0Þ ¼ x0; (1.219)

xkðtÞ 2 XðtÞ; t ¼ 1; :::; n; (1.220)

ukðtÞ 2 UðtÞ; t ¼ 0; :::; n� 1; (1.221)

hold and the convergence

KðxkÞ ! supK (1.222)

takes place. Since sets X(t) and U(t) are uniformly bounded, sequences {xk} and

{uk} are bounded as well. Then, by virtue of Bolzano–Weierstrass’s theorem, after

selecting a convergent subsequence, we obtain convergences uk ! u and xk ! x.Taking into account that sets X(t) and U(t) are closed, we conclude that the limiting

pair (u, x) satisfies inclusions (1.23) and (1.24). Proceeding to the limit in relations

(1.218) and (1.219), we determine that relations (1.20) and (1.21) hold and

hence the pair (u, x) is admissible. Since function K is continuous, the convergence

KðxkÞ ! supKðxÞ takes place. From this convergence and condition (1.222), it

follows that value K(x) coincides with the upper bound sup K in the set of admissi-

ble pairs of the system. Theorem 1.3 is proved.

Proof of Theorem 1.4 The boundaries of setsVjad follow from the boundaries of V.

The closeness of these sets follows from the closeness and boundaries of sets X(t)and U(t) and the continuity of functions f and Gj and the theorem about the

closeness of the complete preimage of a compact set with continuous mapping V! ðXðtÞ;Uðtþ 1ÞÞ given by the definition of set Vj

ad. The statement of the theorem

follows fromWeierstrass’s theorem about attaining the maximal value of a continu-

ous function in a compact set.

Proof of Theorem 1.5 By virtue of Weierstrass’s theorem, a continuous function

has a maximum on a nonempty closed bounded set. Thus, we suffice to show that

the multivariable function K ¼ K(u) defined by (1.34) is continuous and set Uad is

closed and bounded. Recall that set Uad is nonempty by the conditions of the

theorem.

Let’s show that there exist some expectations of the values entering state-space

constraint (1.36). Indeed, from (1.33), we have

E[xðtþ 1Þ] = E½f ðxðtÞ; uðtÞ; lÞ� þ E½xðtÞ�:


The second term on the right-hand side of this equation makes sense by virtue of

the theorem conditions; the first one is calculated by the formula

E[f ðxðtÞ; uðtÞ; lÞ� ¼ZRn

f ðo; uðtÞ; lÞpxðtÞðoÞdo:

The last integral converges absolutely [here px(t) denotes the probability density

function of a random variable x(t)]. This is really so because of the constraints on

the growth of function f and the existence of the expectation of x(t) for any t ¼ 1,

. . ., n (this fact is determined by induction).

The constraints on the growth of the function Ft and the existence of the

expectation of x(t) yield the existence of the expectation on the right-hand side of

(1.34). Let the convergence of the vectors uk ! u; uk 2 Uadtake place. From (1.33),

it follows that

jxkðtþ 1Þ � xðtþ 1Þj ¼ jf ðxkðtÞ; ukðtÞ; lÞ � f ðxðtÞ; uðtÞ; lÞj;

where xk and x are the solutions to problems (1.33) and (1.33) with control laws ukand u, respectively. Then the following relation holds:

jxkðtþ 1Þ � xðtþ 1Þj � Lf ½jxkðtÞ � xðtÞj þ jukðtÞ � uðtÞj�;

where Lf is the Lipschitz constant of function f. By the same reasoning and taking

into account that xkð0Þ ¼ xð0Þ and by (1.33), we have

jxkðtþ 1Þ � xðtþ 1Þj � ðLf Þ2jxkðt� 1Þ � xðt� 1Þj þ

þðLf Þ2jukðt� 1Þ � uðt� 1Þj þ Lf jukðtÞ � uðtÞ �

�Xts¼0

ðLf Þsþ1jukðt� sÞ � uðt� sÞj � ek;

where ek ! 0 as k ! 1.

Here LF is the maximum of the Lipschitz constants of functions Ft, for t¼ 1, . . .,n, and we obtain the estimate.

jFt½xkðtÞ� � Ft½xðtÞ�j � LFek

Taking the expectations of both parts of this inequality, we obtain

EfjFt½xkðtÞ� � Ft½xðtÞ�jg � LFek;

from which it follows that EfjFt½xkðtÞ� � Ft½xðtÞ�jg ! 0 and the convergence of the

considered sequence EfFt½xkðtÞg ! EfFt½xðtÞ�g takes place. By virtue of (1.34),

this yields the continuity of function K ¼ K(u).


The boundaries of the set Uad follow from the boundaries of U(t). The closenessof set Uad follows from the continuity of the mapping Uad ! X given by definition

of set Uad and the compactness of set X (the theorem about the closeness of a

complete preimage of a compact set under a continuous mapping). Then the

existence of the solution of the considered problem follows from Weierstrass’s

theorem. Theorem 1.5 is proved.

Proof of Theorem 1.6 We suffice to determine that all functions Kj are continu-

ous, and all sets Vjad are closed and bounded, where j ¼ 1; :::; r. The existence of all

expectations used below is proved similarly to the proof of Theorem 1.5.

Taking into account the additivity of the expectation, let’s find the values

Kj ¼ KjðvÞ ¼Xnt¼1

E Ft xvj ðtÞh in o

from which we obtain the inequalities for v;w 2 Vjad :

KjðvÞ � KjðwÞ�� ¼ Xn

t¼1

E Ft xvj ðtÞh in o

�Xnt¼1

E Ft xwj ðtÞh in o��

��Xnt¼1

jEf Ft½xvj ðtÞ�g � EfFt[xwj ðtÞ�gj:

(1.223)

From relations (1.42) and (1.43), it follows that

xvj ðtþ 1Þ � xwj ðtþ 1Þ�� ¼ f xvj ðtÞ;Gj v; xvj ðtÞ

� �; l

� �� f xwj ðtÞ;Gj w; xwj ðtÞ

� �; l

� �� Lf xvj ðtÞ � xwj ðtÞ

�� þ Gj v; xvj ðtÞ� �

� Gj w; xwj ðtÞ� �� h i

; t ¼ 0; . . . ; n� 1

(1.224)

where Lf is the Lipschitz constant of function f. Denoting by LG the maximum

Lipschitz constant of function Gj, we obtain the inequality

xvj ðtþ 1Þ � xwj ðtþ 1Þ�� Lf ð1þ LGÞ xvj ðtÞ � xwj ðtÞ

�� LfLGjv� wj; t ¼ 0; . . . ; ; n� 1

Taking into account the equality xvj ð0Þ � xwj ð0Þ�� ¼ 0; we obtain the estimate

xvj ðtþ 1Þ � xwj ðtþ 1Þ�� Lf LG

Xtl¼0

Lf ð1þ LG Þ� ljv� wj

� bjv� wj8v;w 2 Vjad; (1.225)


where

b= LfLG

Xtl¼0

Lf ð1þ LG Þ� l:

Denoting by LF, the maximum Lipschitz constant of function Ft, we have

Ft xvj ðtÞh i

� Ft xwj ðtÞh i�� LF xvj ðtÞ

�� xwj ðtÞ�� LF bjv� wj8v;w 2 Vj

ad: (1.226)

So, in the case of a sufficiently small difference in the adjusted coefficients v

and w, the values xvj ðtÞ and xwj ðtÞ (as well as Ft½xvj ðtÞ� and Ft xwj ðtÞh i

Þ are arbitrarilyclose to one another. Let’s define the convergent sequence w ¼ vk ! v. Then, aftertaking the expectations of the left- and right-hand sides of inequality (1.226), we

obtain the inequality

E Ft xvj ðtÞh i

� Ft xvkj ðtÞh i�� h i

� LFbjv� vkj;

which yields the convergence

E Ft xvkj ðtÞh in o

! E Ft xvj ðtÞh in o

from which it follows that function Kj(v) is continuous.Since Vj

ad 2 V , sets Vjad are bounded. The closeness of the sets Vj

ad follows

from the closeness of the sets U(t) and X(t+1), the continuity of the mappings

v ! E xvj ðtþ 1Þh i

; v ! E Gj v; xvj ðtÞ� �ih

proved above, and the definition of setVjad

as the complete preimage of the sets under continuous mappings. The theorem

statement follows from Weierstrass’s theorem about the attainment of the upper

bound by a continuous function in a compact set.

Proof of Lemma 1.1 Let the convergence of some sequence be ak ! a0, whereak 2 A: Let’s denote by uk the maximum point of function Kak in set Uak, and let u0denote the maximum point of function Ka0 in set Ua0 . Taking into account the

K-continuity of the family of sets {Ua} and the continuity of the function fa(x) on A� U, we conclude that for any e > 0, there exists a number k0 such that for k > k0there exists such a point u0k 2 Ua0 that satisfies неравенства Kakðu0kÞ � KakðukÞj j� e and, moreover, maxy2U KakðyÞ�j Ka0ðyÞj � e.

As a result, with k>k0; we obtain the following inequalities:

Ka0ðu0Þ Ka0ðu0kÞ Kakðu0kÞ � e KakðukÞ � 2e: (1.227)


By the same reasoning, one can verify the relation

KakðukÞ Ka0ðu0Þ � 2e: (1.228)

From (1.227) and (1.228), it follows that for sufficiently large k, the inequality

KakðuÞ � Ka0ðu0Þj j � 2e holds true. This inequality ensures the convergence of

sequence KakðuÞ ! Ka0ðu0Þ. The lemma is proved.

Proof of Theorem 1.7 Without loss of generality, by locality of the problem, set Lcan be considered to be compact. As noted above, the considered problems of the

synthesis of parametric control laws reduce to finding the maximal values of the

continuous functions (the optimization criterion K(u, l) in the compact sets of

admissible values of the adjusted parameters: maxu2Uad;l

Kðu; lÞ. HereUad; l is the

set of admissible values of the adjusted parameters of the corresponding problem for

somefixed values of the uncontrolled parameter l 2 L. The compactness of these sets

is already proved in Theorems 1.1, 1.3, and 1.5. From the continuity of mappings ðu;lÞ ! xu;lððu; lÞ ! E½xu;l�Þ for the stochastic cases), ðu; lÞ ! Kðu; lÞ, taking intoaccount the compactness of setsU andL, it follows that the set Uad;l : l 2 L

� �isK-

continuous. Then the required statement follows from Lemma 1.1.

Proof of Theorem 1.8 Just as in the previous theorem, the setL is considered to be

compact. For the given value of j, the problems of choice of the optimal control

laws reduce to finding the maximum values of continuous functions [the optimiza-

tion criterion Kjðu; lÞÞ in the compact sets of the admissible values of the adjusted

coefficients: maxv2Vj

ad; l

Kjðv; lÞ . Here Vjad; l is the set of the admissible values of the

adjusted coefficients of the respective problem for the chosen number j of the law and

value of the uncontrolled parameter l 2 L. The compactness of these sets is already

proved in Theorems 1.2, 1.4, and 1.6. From the continuity of mappings ðv; lÞ ! xv;lj

�ððu; lÞ ! E[xv;lj � for the stochastic cases), ðv; lÞ ! Gj v; xv;lj

� ��

ððu; lÞ ! E Gj v; xv;lj

� �h ifor the stochastic cases), and ðu; lÞ ! Kjðu; lÞ , taking

into account the compactness of setsV andA, it follows that the set Vjad;l : l 2 L

n oKj

is continuous. Then the required statement follows from Lemma 1.1.

Proof of Corollary 1.1 Since the optimal value of the criterion of the indicated

problems is defined as the maximal among the optimal values of the criteria Kj over

all possible parametric control laws from the given set, it can be written as

KðlÞ ¼ maxj¼1;...;r maxv2Vjad;l

Kjðv; lÞ:


As Kð lÞ is the maximum value from the finite set of the continuous function,

this function is also continuous in L.

Proof of Theorem 1.9 Let’s connect points l0 and l1 by a smooth curve S lying in

the region L : S ¼ lðsÞ; s 2 ½0; 1�f g; l(0) = l0; l(1) = l1 . We’ll denote by

Kj(s) the optimal value of criteria Kj of Problem 1.2, 1.4, 1.6 for the selected

parametric control law Gj and value l ¼ lðsÞ of the uncontrolled parameter.

From Theorem 1.9, it follows that the function y ¼ KjðsÞ is continuous in the

interval [0,1]. Hence, for the function y ¼ maxj¼1;:::;r

KjðsÞ ¼ KðsÞ, solving considered

Problem 1.2, 1.5, or 1.6 is also continuous in interval [0,1]. Denote by DðjÞ � ½0; 1�the set of all values of parameter s that satisfy KjðsÞ ¼ KðsÞ. This set is closed as a

preimage of a closed set for a continuous function y ¼ KjðsÞ � KðsÞ. The set DðjÞcan be empty. As a result, the interval [0, 1] is represented as the following finite

sum of sets consisting at a minimum of two nonempty closed sets (see the theorem

conditions):

½0; 1� ¼ [j¼1;:::;rDðjÞ:

Hence, by the theorem conditions 0 2 Dðj0Þ and 1=2Dðj0Þ; it follows that the point s0exists at the boundary of setDðj0Þ; which lies within the interval (0, 1) [let’s assume

that s0 is the lower bound of such boundary points of the set Dðj0Þ]. The point s0 isalso the boundary point of some other set Dðj2Þ and belongs to it. For the found

value of s0, pointl s0ð Þ is the desired bifurcation point, since forl ¼ l s0ð Þ there existat least two optimal laws (Gj0 andGj2), and for l ¼ lðsÞ, 0 � s � s0 there exists oneoptimal law Gj0 , Theorem 1.9 is proved.

Proof of Lemma 1.2 Setting the right-hand sides of the equations of system (1.49)

equal to zero, we obtain relations (1.50). Obviously, k>0; c>0. Let’s construct

the determinant of the Jacobian matrix for the right-hand sides of (1.50) at the point

ðkþ; cþÞ:

D ¼ a� 1

ð1� bÞa ð dþ pÞ ðnþ dÞð1� aÞ þ p� nð Þ:

Since D < 0 for all indicated values of parameters A, a, b, d, n, and r of the

mathematical model, the obtained singularity point ðk; cÞ is a saddle point of

system (1.49).

Proof of Theorem 1.10 Let’s show that system (1.51) does not have cyclic

trajectories in region O1. Assume the contrary: There exists a cyclic trajectory in

region O1. Then it must include at least one singularity point, and the sum of the

Poincare’s indices of the singularity points belonging to the interior of this cycle is

equal to zero [12, p. 117]. But the region O1 includes only one saddle point with

index �1. Thus, we have a contradiction.


Let’s show that the stable and unstable separatrices of the saddle point x; yð Þdonot form one trajectory in region O1. Assume the contrary: The stable and unstable

separatrices of the saddle point x; yð Þ form one singular trajectory g lying in O1.

Then this trajectory (or, if it exists, the second trajectory consists of some other

stable and unstable separatrices) together with the singularity point are the bound-

ary of the bounded cellO2, which lies in regionO1. Let’s consider the semitrajectory

L+ originating from some point ðx1; y1Þ, where ðx1; y1Þ is the interior point of O2.

Then, since there are no cyclic trajectories and the equilibrium state is unique, only

the boundary of cellO2 can be the limiting points of L+ [the point ðx1; y1Þ cannot bethe single limiting point of L+ since this is the saddle point] [12, p. 49]. Then, let’sconsider the semitrajectory L� originating from the point ðx1; y1Þ in the opposite

direction of L+. Obviously, the boundary of O2 cannot be the limiting points of L�.Since there are no other singularity points and singular trajectories in the regionO2,

we have a contradiction.

By virtue of [12, p. 146], this statement is proved.

Proof of Assertion 1.1 Let’s first show that the semitrajectory of flux f originatingfrom any point of the set N for some t (t > 0) leaves N.

Let’s consider any semitrajectory originating in N. With t>0; the following twocases are possible: All points of the semitrajectory remain in N or for some value of

t, a point of the semitrajectory does not belong to N. In the first case, from (1.71)dpdt ¼ �a Q

M p of the system, it follows that for all t>0; the variable p(t) has a

derivative that is greater than some positive constant with Q<0 or less than some

negative constant with Q>0; i.e., p(t) grows unboundedly or tends to converge to

zero with unbounded growth of t. Therefore, the first case is impossible and the

orbit of any point in N leaves N.Since any chain-recurrent set Rðf ;NÞ lying inside N is the invariant set of this

flux, then, if it is nonempty, it only consists of the whole orbits. Hence, in our case

Rðf ;NÞ is empty. The statement follows from Robinson’s Theorem A [67].


Chapter 2

Methods of Macroeconomic Analysisand Parametric Control of EquilibriumStates in a National Economy

Conducting a stabilization policy on the basis of the results of macroeconomic

analysis of a functioning market economy is an important economic function of the

state.

The AD–AS, IS, LM, IS-LM, IS–LM–BPmodels, as well as the Keynesian model

of common economic equilibrium for a closed economy and the model of a small

country for an open economy [39], include one of the efficient instruments of

macroeconomic analysis of the functioning of a national economy.

In previously published literature, we can see how these models are used for

carrying out a macroeconomic analysis of the conditions of equilibrium in national

economic markets. But there are no published results in the context of the estima-

tion of optimal values of the economic instruments on the basis of the Keynesian

model of common economic equilibrium and the model of an open economy of a

small country in the sense of some criteria, as well as an analysis of the dependence

of the optimal criterion value on exogenous parameters.

Based on the dependence of a solution to a system of algebraic equations on its

coefficients, we propose an approach to parametric control of the static equilibrium

of a national economy that reduces to the estimation of optimal values of economic

instruments as a solution of some respective mathematical program.

In this chapter, we construct IS, LM, and IS-LM models of the Keynesian all-

economic equilibrium and a small open national economy. We also present results

of macroeconomic analysis and parametric control of the static equilibrium of a

national economy.


117

2.1 Macroeconomic Analysis of a National Economic StateBased on IS, LM, and IS-LM Models, KeynesianAll-Economy Equilibrium. Analysis of the Influenceof Instruments on Equilibrium Solution

This section elaborates the construction of the IS, LM, and IS-LMmodels as well as

the Keynesian model of common economic equilibrium using an example of the

economy of the Republic of Kazakhstan, analysis of the influence of economic

instruments to the equilibrium conditions in the respective markets, as well as the

estimation of optimal values of economic instruments on the basis of the Keynesian

mathematical model of common economic equilibrium [39].

2.1.1 Construction of the IS Model and Analysisof the Influence of Economic Instruments

Let’s introduce the notation for economic indices used for model construction: Т is

the tax proceeds (to the state budget, in billions of tenge); S is the net savings,

billions of tenge; I is the investment in the capital asset, billions of tenge; G is the

public expenses, billions of tenge; Y is the gross national income, billions of tenge;

and C is the household consumption, billions of tenge.

Macroestimation of the equilibrium conditions in the wealth market can be done

on the basis of the IS model [39, p. 76] represented as

T þ S ¼ I þ G: (2.1)

The tax proceeds T to the state budget represented by the expressionT ¼ TyY hasthe following econometric estimation based on statistical information for the years

2000–2008:

T ¼ 0:2207 Y:

ð0:000Þ (2.2)

The statistical characteristics of model (2.2) are as follows: the determination

coefficient R2 ¼ 0.986; the standard error Se ¼ 209.5; the approximation coeffi-

cient A ¼ 10.47%; the Fisher statistics F ¼ 581.66. The statistical significance of

the coefficient of regression (2.2), as well as the regressions estimated below, is

given within parentheses under the respective coefficients of the regressions in the

form of p-values.

118 2 Methods of Macroeconomic Analysis and Parametric Control of Equilibrium. . .

The net savings S represented by the expression S ¼ aþ SyY has the following

econometric estimation:

S ¼ �366:055þ 0:222 Y:

ð0:000Þ 0:000ð Þ (2.3)

The statistical characteristics of model (2.3) are as follows: The determination

coefficient R2 ¼ 0.994; the standard error Se ¼ 69.2; the approximation coefficient

A ¼ 11.47%; the Fisher statistics F ¼ 1287.2; the Durbin–Watson statistics DW ¼1.96.

The investment in capital assets represented by the expression I ¼ aþ Iiiafter estimating the parameters of this model using statistical information becomes

the following:

I ¼ 1367:9� 81:3 iþ 0:2751 Ymean:

ð0:02Þ ð0:03Þ 0:00ð Þ (2.4)


coefficient R2 ¼ 0.99; the standard error Se ¼ 126.8; the approximation coefficient

A ¼ 4.2%; the Fisher statistics F ¼ 326.48; the Durbin–Watson statistics DW ¼1.72. Substituting into (2.4) the value of the mean nominal gross national income

for the years 2000–2008 in billions of tenge, Ymean ¼ 6662:7 , finally yields the

following model for the investment:

I ¼ 3202� 81:3 i: (2.5)

Substituting expressions (2.2), (2.3), and (2.5) into (2.1), we obtain the IS model

representation in the following form:

�366:055þ 0:222Y þ 0:2207Y ¼ 3202� 81:3 iþ G200X; (2.6)

which allows us to determine the equilibrium value of i for the given values of Y and

G200Х. From macroeconomic theory, a method [39, p. 77] of plotting the IS curve

exists, which is the set of combinations of the equilibrium values of Y and i(Fig. 2.1).

From the model IS2007 (Fig. 2.1), it follows that the equilibrium GNI2007 with

interest rate 13.6% equals 11,602.75 billion tenge, and the real GNI2007 with interest

rate 13.6% equals 11,371 billion tenge, which shows a lack of wealth in the

considered market. From the model IS2008 (Fig. 2.1), it follows that the equilibrium

GNI2008 with interest rate 15.3% equals 13,957.91 billion tenge, while the real

GNI2008 with the interest rate 15.3% equals 13,734 billion tenge, which also

shows a lack of wealth within that market.

2.1 Macroeconomic Analysis of a National Economic State Based on IS. . . 119

To estimate the multiplicative effects [41, p. 78] of the economic instruments Tyand G, we’ll construct an econometric model of the consumption of households C,which, on the basis of statistical information for the years 2000–2008, is given by

C ¼ 428:68þ 0:552Yv;

0:000ð Þ 0:000ð Þ

where Yv ¼ Y � TyY;CYv ¼ 0:552: The statistical characteristics of this model are

as follows: The determination coefficient R2 ¼ 0.999; the standard error

SE ¼ 68.92; the approximation coefficient A ¼ 1.78%; the Fisher statistics

F ¼ 5394; the Durbin–Watson statistics DW ¼ 1.53.

Table 2.1 presents the expressions and values of the multipliers [39, p. 83] of the

instruments Ty and G derived on the basis of the IS model (2.6).

Let’s estimate the multiplicative effects of the instruments Ty and G based on the

data for the year 2008. According to that data, we have G ¼ 3859.98,

Y ¼ 13,734.3, and Ty ¼ 0.2207. Now, let’s change G to DG ¼ 579. This change,

in accordance with the multiplier of DG, results in an increment of GNI by the

value DY ¼ 1308.54.

0

5

10

15

20

25

30

0,00 3000,00 6000,00 9000,00 12000,00 15000,00 18000,00

i (interest rate)

Y (Gross National Income)

actual point 2007: GNI = 11371.07; i=13.6 actual point 2007: GNI = 11374.29; i=15.3

IS 2007 IS 2008

Fig. 2.1 Plots of IS2007 and IS2008 models

Table 2.1 Consequences of changing public expenses and taxation

Action

Consequence Public expensesincrease by DG Taxes decrease by DТ

National income increases by1

Tyþ SyDG ¼ 2.26 DG

Cyv

Tyþ SyDТ ¼ 1.3 DТ

Budgeted deficit increases by 1� Ty

Tyþ Sy

� �DG ¼ 0.5 DG 1� TyCyv

Tyþ Sy

� �DТ ¼ 0.7 DТ


Also, from the data of year 2008, we have G ¼ 3859.98, Y ¼ 13,734.3, and

Ty ¼ 0.2207. Let’s replace Ty by DTy ¼ �0.01. This change, in accordance with

the multiplier ofDTy, results in an increment of GNI by the value DY ¼ 328.37. The

derived results agree with the macroeconomic theory that considers the influence of

the economic instruments on the changes in the domestic national income, which is

represented by Table 2.1, “Consequences of changing public expenses and taxa-

tion” [39, p. 83].

2.1.2 Macroeconomics of Equilibrium Conditionsin the Money Market

The macroestimation of equilibrium conditions in the money market can be realized

on the basis of the LM model represented as follows [41, p. 111]:

M ¼ lpr þ ltr; (2.7)

where М is the money supply, in billions of tenge; lpr is the volume of property

(deposits in deposit organizations by sectors and currencies), in billions of tenge; lpris the volume of transaction [the volume of credits given by second-level banks

(SLB) taking into account the money velocity], in billions of tenge.

To estimate the money velocity, let’s use the Fisher equation [39, p. 112]:

MV ¼ Y;

where V is the money velocity, Y is the nominal GNI, and the money aggregateM3is accepted in the Fisher equation as the active money volume M.

Estimation of the money velocity by the expression V ¼ Y/M on the basis of the

statistical information for the years 2007–2008 is presented in Table 2.2.

The value of the money supply represented in the Fisher equation by the

aggregate M3 can be checked again through its estimation determined by yearly

values of the money base and the money multiplier m.The money multiplier m is defined by the following relation [39, p. 99]:

m ¼ 1þ gð1� a� bÞaþ bþ gð1� a� bÞ ;

Table 2.2 Value of the money aggregate M3 and the velocity of money

Year GNI Value of money aggregate М3 V, velocity of money

2007 11,371 4,629.8 2.5

2008 13,734 6,266.4 2.2


where a ¼ RR/D is the normative of minimal reserve;

b ¼ ER/D is the coefficient of cash remainders of the commercial banks;

g ¼ CM/K is the share of money in cash in the total sum of credits of the

commercial banks;

RR the minimal reserves;

D is the check (current) deposits (we used the information about deposits in

the deposit organizations by sectors and currencies);

ER is the excess reserves;

K is the credits of the commercial banks accepted in accordance with the

expression K1/V;К1 is the statistical information about the given credits;

СM is the active money in cash.

Estimates of the money supply M by the money bases for the years 2007–2008

and values of m for the same period are respectively equal to the following: For the

year 2007,М ¼ mН ¼ 4519.9 billion tenge; for the year 2008,М ¼ mН ¼ 5343.6

billion tenge.

Table 2.4 presents the calculated values of the money supply and the values of

the money aggregate М3 by years. Table 2.4 shows that the calculated values ofMand values of the money aggregateM3 are of the same order and close to each other.

Taking into consideration this fact, together with the result on the money velocity

derived above, in this specific analysis we accept the calculated values as the money

supply, and the actual values of credits of the second-level banks are corrected

subject to the money velocity.

The property demand represented by the expression lpr ¼ eaþlii has the following

econometric estimate:

lpr ¼ 438883:3� 0:66i:

0:000ð Þ 0:01ð Þ (2.8)

The regression coefficients are statistically significant, although we have the

coefficient of determination R2 ¼ 0.33, the standard error Se ¼ 0.6, and the Fisher

Table 2.3 Values

of multipliersValues of multipliers

Year a b g Deposit Credit Money

2007 0.143 0.043 0.250 2.565 2.087 3.087

2008 0.045 0.069 0.252 2.969 2.632 3.632

Table 2.4 Calculated

values of money supply

and values of money

aggregate

Years

Calculated values

of money supply

Values of money

aggregate М3

2007 4,519.9 4,629.8

2008 5,343.6 6,266.4


statistics F ¼ 67. The demand for money for transactions represented by the

expression ltr ¼ a + bY describes the following econometric estimation:

ltr ¼ �1062:85þ 0:326 Y:

0:0005ð Þ 0:0000ð Þ (2.9)


coefficient R2 ¼ 0.965, the standard error SE ¼ 267, and the Fisher statistics

F ¼ 193.7.

Substituting expressions (2.8) and (2.9) into (2.7), we obtain the representation

of the LM model in the following form:

M200X ¼ 438883:3� 0:66i � 1062:85þ 0:326 Y; (2.10)

which allows us to determine the equilibrium value of i for the given values of Y and

M200X . In macroeconomic theory, a method exists [39, p. 113] to plot the LM curve,

which is the set of combinations of the equilibrium values of Y and i. Figure 2.2

presents the plots of the LM models for the years 2007 and 2008.

In accordance with the obtained results and plotted LM2007, LM2008, one can

conclude that the actual values of Y and i for the years 2007–2008 are situated abovethe respective curves LM2007, LM2008, which shows the relatively low demand for

the monetary assets.

The alarming aspect is that the actual state in which the money market found

itself in the year 2008 corresponds to a higher mean market interest rate than in the

year 2007, whereas the whole line LM for 2008 is situated below and to the right of

0

5

10

15

20

25

30

0,00 3000,00 6000,00 9000,00 12000,00 15000,00 18000,00 21000,00

i (interest rate)


actual point 2007: GNI = 11371.07; i=13.6 actual point 2007: GNI = 11374.29; i=15.3

LM 2007 LM 2008

Fig. 2.2 Plots of LM2007 and LM2008 models


the respective line for 2007; that is, the same volume of GNI corresponds to a lower

equilibrium interest rate than that of a year before. This is an indirect indicator that

the government has regulated the money market based on the necessity of making

money cheaper, but the second-level banks reacted to those signals in the opposite

way, raising the commercial rate. Exactly the same situation occurred in 2008 in

most developed countries on the threshold of the economic crisis.

2.1.3 Macroestimation of the Mutual Equilibrium Statein Wealth and Money Markets. Analysis of the Influenceof Economic Instruments

On the basis of the derived IS and LMmodels, the model for macroestimation of the

joint equilibrium state in the wealth and money markets can be represented by the

following system:

�366:055þ 0:222Y þ 0:2207Y ¼ 3202� 81:3iþ G200x;M200x ¼ 438833:3� 0:66i � 1062:85þ 0:326Y:

�(2.11)

The results of solving system (2.11) to estimate the joint equilibrium state in the

wealth and money market for the years 2007 and 2008 are presented in Table 2.5.

The plots of the IS and LM models in the same period are shown in Fig. 2.3.

From Fig. 2.3, it follows that the coordinates of the effective demand point for

years 2007 and 2008 are respectively represented by Y*2007 ¼ 11,670.89;

i*2007 ¼ 13.23, and Y*2008 ¼ 14,327.31; i*2008 ¼ 13.29. The points of the actual

state of the economy of the Republic of Kazakhstan in 2007 and 2008 are respec-

tively situated to the left of the corresponding IS2007 and IS2008 plots and above the

respective LM2007 and LM2008 plots. Such location of the points of the actual

economic state means a respective lack of wealth market and an excess of money

in the money market in 2007 and 2008.

Let’s estimate the influence of the instruments G andM on the joint equilibrium

conditions using the data for the year 2008.

By the results of the solution of system (2.11), on the basis of the data from 2008,

we have that G ¼ 3,859.98 andM ¼ 5,343.6. Let’s now increase G by DG ¼ 579.

With unchangedM, this fluctuation results in an increase of the Keynesian effective

demand – GNI up to 15,522 billion tenge and an increase of the interest rate up to

Table 2.5 Joint equilibrium and actual values of Y and i

Actual values Joint equilibrium conditions

i, interest rateof SLB, %

Y, gross domestic

income, billion tenge i*Y*, Keynesian effective

demand

2007 13.6 11,371.1 13.23 11,670.89

2008 15.3 13,734.3 13.29 14,327.31


13.9% due to the shift of IS to the right as a result of the multiplicative effect from

increasing the public expenses.

Let’s now increaseМ2008 by DМ ¼ 534. With unchanged G2008, this fluctuation

results in an increase of GNI up to 14,438.6 billion tenge and a decrease of the

interest rate to 12.7% due to the shift of IS to the right as a result of the multiplica-

tive effect from increasing the money supply.

The obtained results also agree with the macroeconomic theory on the influence

of the economic instruments in the wealth and money markets [39, pp. 78, 114].

2.1.4 Macroestimation of the Equilibrium State on the Basis ofthe Keynesian Model of Common Economic Equilibrium.Analysis of the Influence of Economic Instruments

The Keynesian mathematical model of common economic equilibrium on the basis

of the IS and LM models, as well as the econometric function of the labor supply

price and the econometric expression of the production function, is given by the

following [39, p. 223]:

TðYÞ þ SðYÞ ¼ IðiÞ þ G; ð2:12ÞM ¼ lðY; iÞ; ð2:13ÞWSðN;PÞ ¼ PYN; ð2:14ÞY ¼ YðNÞ; ð2:15Þ

ð2:12Þ

8>>>>><>>>>>:

0

5

10

15

20

25

30

0,00 5000,00 10000,00 15000,00 20000,00 25000,00

i (interest rate)


IS 2007LM 2008

actual point 2007: GNI = 11371.07; i=13.6 actual point 2007: GNI = 11374.29; i=15.3IS 2008 LM 2007

Fig. 2.3 Plots of IS2007, LM2008, LM2007, and LM2008 models


where Ws (N, P) is the function representing the labor supply price, YN is the

derivative of the production function, and Y(N) is the production function.

Equations (2.12) and (2.13) of the common economic equilibrium model are

given by the respective IS and LM equations (2.11).

The econometric representation of the labor supply price using the statistical

data for the years 2000–2008 is given by

Ws N;Pð Þ ¼ 60:12P� 0:007N;

ð0:000Þ ð0:000Þ (2.16)

where P is the level of prices for year 2000, and N is the busy population in

thousands per capita. The respective p-values (of t-statistics) in the equation in

Ws are presented in parentheses below the regression coefficients. The results of the

analysis of the statistical significance of the model for Ws are as follows: The

determination coefficient R2 ¼ 0.99, the standard error Se ¼ 3.37, the Fisher

statistics F ¼ 522.6, and the approximation coefficient A ¼ 7.4%.

The econometric representation of the production function Y(N) using the

statistical data for the years 2000–2008 is given by

Y ¼ �5:654 N þ 0:0009 N2:

0:000ð Þ 0:000ð Þ (2.17)

The results of analysis of statistical significance of the model for Ws are as

follows: The determination coefficient R2 ¼ 0.98, the standard error Se ¼ 1227,

and the Fisher statistics F ¼ 172.

The Keynesian model of common economic equilibrium on the basis of relations

(2.11), (2.16), and (2.17) is given by

�366:055þSyYþTyY¼ 3202� 81:30iþG200X;

M200X¼ 438883:3� 0:66i � 1062:85þ 0:326Y;

60:12P� 0:00698N¼�5:65Pþ0:0018NP;

Y¼�5:65Nþ0:0009N2:

8>>>><>>>>:

(2.18)

In this system describing the behavior of the macroeconomic subjects, the

exogenously given parameters include the value of public expenses G and the

nominal values of the money in cashM. The values of five endogenous parameters,

Y*, i*, P*, N*, and W*, that result in attaining equilibrium simultaneously in all

three mentioned markets are determined from the solution of this system of

equations.

Substituting the actual values of G200X and М200X of the respective year and

solving system (2.18), we obtain the values of variables that are in equilibrium

simultaneously in all three markets.


Table 2.6 presents the equilibrium values of the endogenous parameters by using

the solution of system (2.18) on the basis of the data for the years 2007 and 2008.

Let’s estimate the influence of instruments G and M on the Keynesian common

economic equilibrium from the data from 2008.

Increasing G by DG ¼ 579 while keeping the values of M results in an increase

of the GNI to 15,522.6 billion tenge and a decrease in the interest rate to 13.9%,

while at the same time unemployment drops by 1.6% and the level of prices

increases to 1.12.

Increasing М2008 by DМ ¼ 534.4 while keeping the values of G results in an

increase of the GNI to 14,438.56 billion tenge and a decrease of the interest rate to

12.68%, while unemployment is reduced by 0.15%, and the level of prices increases

insignificantly to 1.105.

Increasing G by DG ¼ 579 and increasing М2008 by DМ ¼ 534.4 result in an

increase of the GNI to 15,658.85 billion tenge and a decrease of the interest rate to

13.15%, while unemployment is reduced by 1.77% and the level of prices increases

to 1.13.

2.1.5 Parametric Control of an Open Economic State Basedon the Keynesian Model

Estimation of the optimal values of the instruments M and G for the given external

exogenous parameters Sy, Тy on the basis of model (2.18) for the year 2008 in the

sense of the GNI criterion gives

Y ! max : (2.19)

This estimate can be obtained by solving the following mathematical program-

ming problem.

Problem 1On the basis of mathematical model (2.18), find the values of (M, G) maximizing

criterion (2.19) under the constraints

Table 2.6 Comparative

analysis of actual and

equilibrium values of GNI,

interest rate, level of prices,

busy population

Y i P N

2007 Actual 11,371.10 13.60 1.789 7,631.10

Equilibrium 11,670.89 13.23 1.050 7,751.60

Deviation 2.64% �0.37 �0.740 1.58%

2008 Actual 13,734.30 15.30 1.959 7,857.20

Equilibrium 14,327.30 13.30 1.103 8,048.80

Deviation 4.32% �2.00 �0.900 2.44%


M �M�j j � 0:1M�;

G� G�j j � 0:1G�;

N � N�j j � 0:1N�;

P� P�j j � 0:1P�;

i� i�j j � 0:1i�;

Y � Y�j j � 0:1Y�:

8>>>>>>>>>>><>>>>>>>>>>>:

(2.20)

Here M* and G* are the respective actual values of the money and public expenses

supplies in 2008. The symbol (*) for the unknown variables of system (2.20)

corresponds to the equilibrium values of these variables with fixed values of

M* and G*.For Problem 1, the optimal values of the parameters are M ¼ 5877.96,

G ¼ 4245.98, which ensure attaining the maximum value of the criterion

Y ¼ 15,255.9. The value of this criterion without control is equal to 14,327.3. For

the optimal values of the instruments M and G that were obtained, the equilibrium

values of the other endogenous variables turn out to be N ¼ 8148.539, P ¼ 1.1210,

and i ¼ 12.986. Here we should also note that solving this optimization problem

results in an increase of the working segment of the population by approximately

100,000 people.

On the basis of Problem 1, we carry out the analysis of the dependence of the

optimal values of criterion Y on the pair of the exogenous parameters Ty; Sy� �

given

in their respective regions. The obtained plot of the optimal values of criterion

(2.19) is presented in Fig. 2.4.

Fig. 2.4 Plot of dependence of optimal values of criterion Y on parameters Ty, Sy


2.2 Macroeconomic Analysis and Parametric Controlof the National Economic State Based on the Modelof a Small Open Country

Ensuring a double equilibrium, that is, a common economic equilibrium in

conditions of full employment with a planned (assumed zero) balance of payments,

is an urgent problem in the conditions of an open economy, when the country is

engaged in the exchange of goods and capital with the outside world.

All of the remaining states in the national economy differing from the double

equilibrium represent various kinds of nonequilibrium states. Hence, unemployment

remains the same in spite of an excess in the balance of payments. Unemployment

can be accompanied by an excess in the balance of payments. The excess of

employment can be accompanied by both the excess and deficiency of the balance

of payments. Therefore, public economic policy aims at attainment of a double

equilibrium. The estimation of the equilibrium conditions for an open economy can

be partially considered on the basis of the model of a small country [39, p. 433].

This section is devoted to the construction of a mathematical model of an open

economy of a small country using the example of the Republic of Kazakhstan, to

the analysis of the influence of economic instruments on the conditions of common

economic equilibrium and state of the balance of payments, and to the estimation of

the optimal values of the economic instruments on the basis of the model of an open

economy of a small country, as well as an analysis of the dependencies of the

optimal values of the criteria on the values of one, two, and three parameters from

the set of the external economic parameters given in the respective regions.

2.2.1 Construction of the Model of an Open Economyof a Small Country and Estimationof Its Equilibrium Conditions

Let’s introduce the following notation for the economic indices used for the model

construction: Y is the gross national income (GNI); C is the household consump-

tion; I is the investment in capital assets; G is public expenses; NE is the net export

of wealth; P is the level of prices of RK; Pz is the level of prices abroad; l is the realcash remainder; I is the interest rate of second-level banks; N is the number of

employed; dY/dN is the derivative of the gross national income as a function of the

number of employed; WS is the level of wages; NKE is the net capital export; e isthe rate of exchange of the national currency; ее is the expected rate of exchange ofthe national currency; e

_eis the expected rate of increase of the exchange rate of the

national currency [39, p. 121];M is the money supply determined from [39, p. 412]

by the formula М ¼ mН, where Н is the money base of each year; m is the money

multiplier calculated from the balance equations of the banking system and defined

by the formula

2.2 Macroeconomic Analysis and Parametric Control of the National Economic. . . 129

m ¼ 1þ g 1� a� bð Þð Þ= aþ bþ g 1� a� bð Þð Þ; (2.21)

where a ¼ RR/D is the norm of the minimal reserve;

b ¼ ER/D is the coefficient of the cash remainder of the second-level banks;

g ¼ СМ/К is the share of cash in the whole sum of the credits of second-level

banks;

RR is the minimal reserve;

ER is the excessive reserve;

D is the check deposits;

CM is the active money in cash;

K is the credits of second-level banks corrected subject to the velocity of

money.

Let’s begin to construct a mathematical model of an open economy of a small

country by estimating the money multiplier, real cash remainders, and economic

functions characterizing the national economic state.

The estimations of values of the money multiplier calculated by formula (2.21)

using the statistical data for the period of years 2006–2008 are presented below:

Year 2006 2007 2008

m 2.372 3.087 3.632

The real cash remainder l is determined by the formula

l ¼ lpr þ ltr; (2.22)

where lpr is the property volume [deposits in the deposit organizations (by sectors

and kinds of currency)], billions of tenge, and ltr is the volume of the transaction (the

volume of the credits given by second-level banks subject to the money velocity),

billions of tenge.

The estimation of the money velocity is calculated by the Fisher equation [42]:

MV ¼ Y;

where V is the money velocity, and M is the quantity of the active money usually

represented by the money aggregate M3 in the Fisher equation.

From the latter formula, the estimates of the money velocity calculated by the

formula V ¼ Y=M on the basis of the statistical information for 2006–2008 [37] are

presented in Table 2.7.

In the macroeconomic theory, the behavior of the national economy is

characterized by the following functions constructed by econometric methods [1]

on the basis of official statistical information.

The consumption C represented by the expression C ¼ aþ CYY has the follow-

ing econometric estimation derived on the basis of the statistical information of the

Republic of Kazakhstan for the period 2000–2008:

C ¼ 555:8þ 0:4101Y:

0:00ð Þ 0:00ð Þ (2.23)


The statistical characteristics of the constructed model of the consumption C are

as follows: The determination coefficient R2 ¼ 0.994, and the approximation

coefficient A ¼ 1.8%. The statistical significance of the coefficients of regression

(2.23), as well as of the regressions estimated below, is presented in parentheses

under the respective regression coefficients as the p-values.The consumption of the imported wealth Qim is represented by the regression

equation

Qim ¼ a1Y þ b1ePZ=P

or, in estimated form,

Qim ¼ 0:4076Y � 2:6059ePZ=P

0;00ð Þ 0;17ð Þ (2.24)

with the determination coefficient R2 ¼ 0.975 and the approximation coefficient

A ¼ 10%.

The model of the demand of the real cash remainder is given by l ¼ a2 þ b2Yþ b3iþ b4e or, after estimating the parameters of this model using the statistical

information,

l ¼1:3320Y � 138:1i � 22:6e:

0;01ð Þ 0;08ð Þ 0;03ð Þ (2.25)

In constructing model (2.25), the values of l calculated in accordance with

formula (2.22) are accepted as the data for the left-hand side. The determination

coefficient is given by R2 ¼ 0.999, and the approximation coefficient is A ¼ 0.2%.

The statistical insignificance of the latter model concerns the fact that in the model

there are correlated factors.

The model of the labor supply price is given by WS ¼ b5N þ b6Pmean; where

Pmean ¼ 1� að ÞPþ aePZ=e0 has the following econometric estimation derived on

the basis of the statistical information:

WS ¼ �0:0219N þ 156:9 Pcp;

0;00ð Þ 0;00ð Þ (2.26)

Table 2.7 Values of GNI (billions of tenge), money aggregateM3 (billions of tenge), and money

velocity V

Year GNI М3 V

2007 11,371 4,629.8 2.5

2008 13,734 6,266.4 2.2


where Pmean ¼ 0:6Pþ 0:4ePz=e0; e0 is the currency exchange rate within the base

period (year 2000), and a is the share of the imported goods in their entire volume

accepted at the level of 0.4. We also have the determination coefficient R2 ¼ 0.98

and the approximation coefficient for 2007–2008 at the 5% level.

The model of the net capital export is given by NKE ¼ b7eðiZ þ e_e � iÞ or, after

estimating the parameters of this model by using the statistical information,

NKE ¼ � 0:3349eðiZ þ e_e � iÞ;

0;00ð Þ (2.27)

with the determination coefficient R2 ¼ 0.51.

The production function is represented in the regression pair Y ¼ a3 + b8 N or,

in the estimated form,

Y ¼ � 17;409:0þ 3:0866N;

0;00ð Þ 0;00ð Þ (2.28)

with the determination coefficient R2 ¼ 0.93 and the coefficient approximated

based on 2007–2008 data not exceeding A = 3.5%.

The model of investment in capital assets is given by

It ¼ a4 þ b9Yt�1 þ b10it;

where It and it are the values of the investments in the current period, and Yt-1 is thevalue of the gross national income in the preceding period.

After estimating the latter model parameters by the statistical data, the following

expression is derived:

It ¼ 862:8þ 0:3122 Yt�1 � 48:4it:

0:34ð Þ 0:00ð Þ 0:41ð Þ (2.29)

At that, the determination coefficient R2 ¼ 0.93 and the approximation coeffi-

cient A ¼ 5%.

Substituting the value Yt�1 ¼ Y2007 to (2.29), finally we obtain the following

model of investment in the year 2008:

I2008 ¼ 2;846:7� 48:4i: (2.30)

Similarly, substituting the value Yt�1 ¼ Y2006 into (2.29) for investment in 2007,

we obtain the following model:

I2007 ¼ 2;737:3� 48:4i: (2.31)


The wealth export model is a regression of the form Qex ¼ b11ePZ P= . After

estimating the parameters, this model becomes

Qex ¼ 17:87ePZ P= :

0; 01ð Þ (2.32)

The determination coefficient is R2 ¼ 0.65.

On the basis of derived econometric estimate (2.23), (2.24), (2.25), (2.26),

(2.27), (2.28), (2.29), (2.30), (2.31) and (2.32) characterizing the state of the

national economy, let’s proceed to the construction of a model of an open economy

of a small country for year 2008.

Within the framework of the IS curve, we constructed the function Y ¼ Cþ IþGþ Qex � Qim; which, subject to (2.23), (2.24), (2.29), (2.30), (2.31) and (2.32),becomes

Y ¼ 555:8þ 0:4101 Y þ 2;846:7� 48:4iþ Gþ 20:47e PZ=P� 0:4076 Y or

Y ¼ 3;410:9� 48:4iþ 20:52ePZ P= þ 1:0024G:

(2.33)

The equation of the LM line M/P ¼ l subject to the econometric model (2.25)

becomes

M=P ¼ 1:3320Y � 138:1iþ 22:6e;

from which one can derive the following relation:

i ¼ �0:1640eþ 0:0096Y � 0:0072M=P (2.34)

Substituting (2.34) into (2.33), we obtain the value of the aggregate demand YD:

YD ¼ 2;324:0þ 5:42eþ 13:98 ePZ P= þ 0:6830Gþ 0:2392M P= : (2.35)

Let’s substitute (2.33) into (2.34) and determine the function of the domestic

commercial interest rate:

i ¼ 22:42� 0:1117e� 0:0049M=Pþ 0:1349ePZ=Pþ 0:0065G (2.36)

The condition of equilibrium in the labor market is given by Р dY/dN ¼ WS [41,

p. 435], which, subject to the econometric functions (2.26) and (2.28), can be

represented by the expression

7; 5P ¼ �0:0219N þ 156:87ð0:6Pþ 0:4ePZ=e0Þ: (2.37)


From (2.37) we obtain the following relation for N:

N ¼ 4;165:8Pþ 20:20ePZ (2.38)

Substituting expression (2.38) into the production function (2.28), we obtain the

function of the aggregate supply:

YS ¼ �17; 408:6þ 12; 858:2Pþ 62:36ePZ (2.39)

The balance of payments has a zero balance if the net wealth export equals the

net capital export, i.e., NE ¼ NKE/P, is valid. The econometric representation of

the latter equality on the basis of (2.24), (2.27), and (2.32) is given by

17:87ePZ P= � 0:4076Y � 2:6059ePZ P=� � ¼ �0:3349eðiZ þ e

_e � iÞ=P:

Substituting the value of the domestic interest rate (2.36) into the latter equality,

after some transformation we obtain the following equation of the curve of the zero

balance of payments:

Y ZBO ¼ 50:23 ePZ P= � 0:8215 eiZ P= � 0:8215 ee P= þ 19:24 e P=

� 0:0918 e2 P= � 0:0041 eM P= 2 þ 0:111 e2PZ P=2 þ 0; 005 eG P= :

(2.40)

Thus, the model of an open economy of a small country in the year 2008 is given

by the following system of equations:

YD ¼ 2324:0þ 5:42eþ 13:98ePZ

Pþ 0:6830Gþ 0:2392

M

P;

YS ¼ �17408:6þ 12858:2Pþ 62:36ePZ;

YZBO ¼ 50:23ePZ

P� 0:8215

eiZ

P� 0:8215

ee

Pþ 19:24

e

P� 0:0918

e2

P�

�0:0041eM

P2þ 0:111

e2PZ

P2þ 0:005

eG

P;

YD ¼ YS ¼ YZBO:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(2.41)

Similarly to (2.41), a model of an open economy of a small country in 2007 can

be constructed.


Solving system (2.41) with prescribed values of the external economic indexes

PZ, iZ, ее and the economic instruments M and G, we’ll determine the equilibrium

conditions of the gross national income Y� ¼ YD ¼ YS ¼ YZBO, level of prices P*,

and exchange rate of the national currency е*. The equilibrium values of the credit

interest rate of the second-level banks i* and the number of employed are calculated

by formulas (2.36) and (2.38), respectively.

The following equilibrium values of the endogenous variables are obtained by

solving system (2.41) for the given external uncontrolled economic indices PZ, iZ, ее

and the controlled economic instruments M and G:

– by year 2007: Y� ¼ 6;383:1; P� ¼ 1:2054; e� ¼ 110:2; i� ¼ 16:4; N� ¼ 7;708:0– by year 2008: Y� ¼ 6;785:4; P� ¼ 1:2099; e� ¼ 114:7; i� ¼ 14:7; N� ¼ 7;838:4:

Figure 2.5 presents the double equilibrium state, where the point of intersection

of the IS–LM–ZBO curves corresponds to a simultaneous equilibrium in the wealth,

money, and labor markets with full employment and zero balance of payments in

2007. All combinations of the values of the national income and interest percent,

except i ¼ 14.7%, Y ¼ 6785.4, offer different types of non-equilibrium states.

In 2008 Kazakhstan, as per the presented figure, also has unemployment and deficit

of balance payments. In the graph of Fig. 2.2.1 this situation is represented by point

A (Y2008 ¼ 70098.0; i2008 ¼ 15.3%). However one can note that according to

official statistics in 2008 Kazakhstan had a surplus of balance payments.

Taking into account the obtained equilibrium values, the equilibrium values of

the economic indices C, I, and others calculated by econometric models are

constructed above. We present the results of comparison of the equilibrium

indices with the actual values of these indices in 2007. Table 2.8 shows similar

results for 2008.

−150

−100

−50

0

50

100

150

200

IS LM ZBO A

0 2000 4000 6000

A

8000 10000 12000 14000

Fig. 2.5 Double equilibrium by 2008


2.2.2 Influence of Economic Instruments on EquilibriumSolutions and Payment Balance States

Below, we’ll estimate the influence of economic instruments, namely, the money

supply and public expenses, on the conditions of common economic equilibrium

and the state of the balance of payments using the following algorithm:

1. Changing the value М2008 by DM ¼ 0.01 M2008 while keeping the values G2008

and iZ2008, PZ2008, ее2008 unchanged, define the values ðMDY�Þ ðY�DMÞ= ;

ðMDP�Þ ðP�DMÞ= ; ðMDe�Þ ðe�DMÞ= ; and ðMDi�Þ ði�DMÞ= that show the per-

centage by which the equilibrium values of the indicesY�; P�; e� i� change withvariation of М2008 by 1%.

2. Changing the value G2008 by DG ¼ 0:01G2008 while keeping the values M2008

and iZ2008, PZ2008, ее2008 unchanged, define the values ðGDY�Þ ðY�DGÞ= ;

ðGDP�Þ ðP�DGÞ= ; ðGDe�Þ ðe�DGÞ= ; and ðGDi�Þ ði�DGÞ= that show the percent-

age by which the equilibrium values of the indices Y�; P�; e�; i� change with

variation of G2008 by 1%.

3. Changing the value М2008 by DM ¼ 0.01M2008 and the value G2008 by DG ¼0:01G2008 while keeping the values iZ2008; PZ

2008; ee2008 unchanged, define thevalues 100DY� Y�= ; 100DP� P�= ; 100De� e�= ; and 100Di� i�= that show the

percentage by which the equilibrium values of the indices Y�; P�; e�; i� changewith simultaneous variation of М2008 and G2008 by 1%.

The results of computations carried out by the above algorithm are given in

Tables 2.9, 2.10, and 2.11.

According to the proposed algorithm, first we estimate the influence of the

economic instruments, namely, the money supply and public expenses, on the

conditions of the common economic equilibrium and the state of the balance of

payment individually. From Tables 2.9 and 2.10, it follows that increasing G2008 by

DG while keeping the value М2008 results in growth of the national income and an

Table 2.8 Equilibrium and actual values of indices in 2008

2008

Indices

Equilibrium

value of Y*Equilibrium

value of Yactual

Deviation Yactual � Y∗

Absolute %

Level of prices P 1.2099 1.96 0.7501 38.8

Currency exchange rate e 114.6 120.3 5.6687 4.7

Interest rate of SLB i 14.7 15.3 0.6246 4.1

National income Y 6785.4 7009.8 224.4 3.2

Consumption C 3338.2 3395.1 56.9 1.7

Import Qim 2467.5 2326.4 �141.2 �6.1

Investment I 2137.043 2149.2 12.2 0.6

Export Qex 2045.384 4370.6 2325.2 53.2


increase in the interest rate, whereas increasingМ2008 byDМwhile keeping the value

G2007 also results in growth of the common economic equilibrium of the GNI, but it

also results in a decrease in the interest rate. Also, from the tables it follows that the

growth in public expenses has a stronger influence on the national income growth,

whereas the money supply growth affects the currency exchange rate more strongly.

Here Y�; P�; e�; i� are the equilibrium solutions for the year 2008, DY� ¼ YM�

� Y�; DP� ¼ PM� � P�; De� ¼ eM

� � e�; Di� ¼ iM� � i�; where YM

*, PM*, eM

*,

iM* are the equilibrium solutions corresponding to M ¼ M2008 þ DM:According to the macroeconomic theory, the money supply growth shows the

following influence on the equilibrium solutions of system (2.41): The national

income, level of prices, and national currency exchange must increase, whereas the

interest rate must decrease. The results of the influence of the money supply

instrument on the equilibrium state of the national economy in 2008 presented in

Table 2.9 coincide with the theoretical assumptions except the price-level index,

which in this case decreases.

Here DY� ¼ YG� � Y�; DP� ¼ PG

� � P�; De� ¼ eG� � e�; Di� ¼ iG

� � i�;where YG

�; PG�; eG

�; iG� are the equilibrium solutions corresponding to

G ¼ G2008 þ DG:According to macroeconomic theory, the public expenses growth exerts the

following influence on the equilibrium solutions of system (2.41): The national

income, level of prices, national currency exchange rate, and interest rate must

grow. The results of the money supply instrument influence on the equilibrium state

of the national economy in 2008 presented in Table 2.9 completely coincide with

these theoretical assumptions.

Table 2.9 Influence of the money supply instrument on the equilibrium state of national economy

in 2007 for DM ¼ 0:01M2008 %ð ÞðMDY�Þ ðY�DMÞ= ðMDP�Þ ðP�DMÞ= ðMDe�Þ ðe�DMÞ= ðMDi�Þ ði�DMÞ=

0.359 �0.1128 0.4827 �1.4692

Table 2.10 Influence of the public expenses instrument on the equilibrium state of national

economy in 2007 for DG ¼ 0:01G2008 %ð ÞðGDY�Þ ðG�DMÞ= ðGDP�Þ ðP�DGÞ= ðGDe�Þ ðe�DGÞ= ðGDi�Þ ði�DGÞ=

0.1892 0.0345 0.0865 0.8017

Table 2.11 Influence of money supply and public expenses instruments on the equilibrium state

of the national economy in 2007 for DM ¼ 0:01M2008 and DG ¼ 0:01G2008 %ð Þ100DY� Y�= 100DP� P�= 100De� e�= 100Di� i�= :

0.5477 �0.0780 0.5675 �0.6434


Here DY� ¼ YMG� � Y�; DP� ¼ PMG

� � P�; De� ¼ eMG� � e�; Di� ¼ iMG

� � i�;where YMG

�; PMG�; eMG

�; iMG� are the equilibrium solutions corresponding to

M ¼ M2008 þ DM and G ¼ G2008 þ DG:Figures 2.7 and 2.8 present the plots of the IS, LM, and ZBO curves from the

derived econometric models for the actual statistical information for 2008.

As stated above (Fig. 2.5), the country has cyclical unemployment and a deficit

in balance of payments from the constructed models. In Fig. 2.6, such a situation is

represented by point Е0. According to the macroeconomic theory, the balance of

payments deficit can be eliminated by applying a restrictive monetary policy by

means of shifting the LM curve to the left up to its intersection with the IS curve at

point C, or the counteractive fiscal policy by means of the IS curve to the left up to

its intersection with the LM curve at point D.

EoC

D

−150

−100

−50

0

50

100

150

200

0 2000 4000 6000 8000 10000 12000 14000

IS LM ZBO Eo C D

Fig. 2.6 Plots IS–LM–ZBO by actual values of P, e for 2008

Fig. 2.7 Plot of the dependence of optimal values of criterion Qimp on pair PZ ; ee


2.2.3 Parametric Control of an Open Economy State Basedon a Small Country Model

Estimate the optimal values of instruments M and G given the external exogenous

parameters ee, iZ, PZ on the basis of model (2.41) for the year 2008 in the sense of

the criteria

Qex ¼ aePZ=P ! max (2.42)

and

Qimp ¼ bYS þ cePZ=P ! min: (2.43)

Such an estimate can be obtained by solving the following problems of mathe-

matical programming:

Problem 1On the basis of mathematical model (2.41), find the values (M, G) maximizing

criterion (2.42) under the constraints

Fig. 2.8 Plot of the dependence of optimal values of criterion Qex on pair PZ ; ee


M �M�j j � 0:1M�;

G� G�j j � 0:1G�;

P� P�j j � 0:1P�;

e� e�j j � 0:1e�;

i� i�j j � 0:1i�;

Y � Y�j j � 0:1Y�:

8>>>>>>>>>>><>>>>>>>>>>>:

(2.44)

Here M� and G� are the actual values of the money supply and public expenses in

the year 2008.

Problem 2On the basis of mathematical model (2.41), find the values (M, G) minimizing

criterion (2.42) under constraints (2.44).

Solving Problems 1 and 2 by the iterative technique [66] given the values

ee ¼ 120:3; iZ ¼ 1:32; PZ ¼ 1:2002; the following results are obtained:

For Problem 1, the optimal values of the parameters are M ¼ 5877.96,

G ¼ 4246, providing the attainment of the maximum value Qex ¼ 3122:74: Thevalue of this criterion without control is 3023.01.

For Problem 2, the optimal values of the parameters are M ¼ 4809.234,

G ¼ 3474, providing the attainment of the minimum value Qimp ¼ 4010:64:The value of this criterion without control is 4183.73.

On the basis of Problems 1 and 2, we carried out the analysis of the dependencies

of the optimal values of the criteriaQex andQimp on the one pair and one set of three

of the parameters from the set of the external parameters ee; iZ;PZf g given within

the respective regions. The plots of the dependencies of the optimal values of

criteria (2.42) and (2.43) for the single cases including that on the pair of the

parameters ðPZ; eeÞ and iZ; eeð Þ are shown in Figs. 2.7 and 2.8.


Chapter 3

Parametric Control of Cyclic Dynamicsof Economic Systems

The theory of market cycles is an important part of modern macroeconomic

dynamics. This theory is based on mathematical models [39] proposed for describ-

ing the evolution of business activity as an oscillatory processes. Readers can find a

number of mathematical models of market cycles in [20]. In this context, the main

factors causing oscillations in market tendencies are considered. Nevertheless,

issues of the structural stability of such models of parametric control of develop-

ment of the economic systems on the basis of mathematical models of business

cycles are not under consideration.

Developing a theory of business cycles is of great interest, including estimation

of the structural stability of mathematical models of business cycles and parametric

control of the evolution of economic systems based on the proposed mathematical

models.

This chapter is devoted to results in the theory of business cycles based on

mathematical models, namely, the Kondratiev cycle model [17] and the Goodwin

model [5, 39].

3.1 Mathematical Model of the Kondratiev Cycle

3.1.1 Model Description

This model [17] combines descriptions of nonequilibrium economic growth and

nonuniform scientific and technological advancement. The model is described by

the following system of equations, including two differential equations and one

algebraic equation:


141

nðtÞ ¼ AyðtÞa;dx=dt ¼ xðtÞðxðtÞ � 1Þðy0n0 � yðtÞnðtÞÞ;

dy=dt ¼ nðtÞð1� nðtÞÞyðtÞ2 xðtÞ � 2þ mþ l0n0y0

� �;

n0 ¼ Ay0a:

8>>>>>><>>>>>>:

(3.1)

Here t is the time (in months), x is the efficiency of innovations, y is the capital

productivity ratio, y0 is the capital productivity ratio corresponding to the equilib-

rium trajectory, n is the rate of savings, n0 is the rate of saving corresponding to theequilibrium trajectory, m is the coefficient of withdrawal of funds, l0 is the job

growth rate corresponding to the equilibrium trajectory, and A and a are some

model constants.

Preliminary estimation of the model parameters is carried out based on statistical

information from the Republic of Kazakhstan for the years 2001–2005 [24].

The deviations in the observed statistical data and the calculated data do not exceed

1.9% within the considered period.

As a result of solving the problem of the preliminary estimation of parametric

identification, the following values of the exogenous parameters are obtained:

a ¼ �0:0046235 , y0 ¼ 0:081173 , n0 ¼ 0:29317 , m ¼ 0:00070886 , l0 ¼ 0:00032161; andxð0Þ ¼ 1:91114.

A preliminary prediction for 2006 and 2007 is characterized by errors equal to

6.1% and 12.1%, respectively, for the capital productivity ratio, and 2.3% and 11%,

respectively, for the rate of savings.

The respective cyclic phase trajectory of the Kondratiev cycle model is

presented in Fig. 3.1.

The period of cyclic trajectory corresponding to the statistical information of the

Republic of Kazakhstan for the given years is estimated to be 232 months.

Phase trajectory

Fig. 3.1 Cyclic phase trajectory of the Kondratiev cycle model

142 3 Parametric Control of Cyclic Dynamics of Economic Systems

3.1.2 Estimating the Robustness of the Kondratiev CycleModel Without Parametric Control

The estimation of structural stability (robustness) of the mathematical model is

carried out according to Sect. 1.4 on parametric control theory (in Ch. 1) in the

chosen compact set of the model state space.

Figure 3.2 presents an estimate of the chain-recurrent set R f ;Nð Þ obtained by

the application of the chain-recurrent set estimation algorithm for the region N ¼½1:7; 2:3� � ½ 0:066; 0:098� of the phase plane Oxy of system (3.1). Since the set Rf ;Nð Þ is not empty, we can draw no conclusions about the weak structural stability

of the Kondratiev cycle model in N on the basis of Robinson’s theorem. However,

since there is a nonhyperbolic singular point in N, namely, the center

x0 ¼ 2� mþ l0n0y0

; y0

� �[16], then system (3.1) is not weakly structurally

stable in N.

Chain-recurrent set

Fig. 3.2 Chain-recurrent set for the Kondratiev cycle model

3.1 Mathematical Model of the Kondratiev Cycle 143

http://dx.doi.org/10.1007/978-1-4614-4460-2

3.1.3 Parametric Control of the Evolution of EconomicSystems Based on the Kondratiev Cycle Model

Choosing the optimal laws of parametric control is carried out in the environment of

the following four relations:

1. n0ðtÞ ¼ n0� þ k1

yðtÞ � yð0Þyð0Þ ;

2. n0ðtÞ ¼ n0� � k2

yðtÞ � yð0Þyð0Þ ; (3.2)

3. n0ðtÞ ¼ n0� þ k3

xðtÞ � xð0Þxð0Þ ;

4. n0ðtÞ ¼ n0� � k4

xðtÞ � xð0Þxð0Þ :

Here ki is the scenario coefficient, and n0� is the value of the exogenous parameter

n0 obtained as a result of the preliminary estimation of parameters.

The problem of choosing the optimal law of parametric control at the level of the

econometric parameter n0 can be formulated as follows.

On the basis of mathematical model (3.1), find the optimal parametric control

law in the environment of the set of algorithms (3.2) ensuring the attainment of

optimal values of the following criteria:

1. K1 ¼ 1

36

X36t¼1

yðtÞ ! max;

2. K2 ¼ 1

36

X36t¼1

xðtÞ ! max; (3.3)

3. K3 ¼ 1

36

P36t¼1

xðtÞxð0Þ þ

P36t¼1

yðtÞyð0Þ

0BBB@

1CCCA! max;

4. K4 ¼ 1

T

XTt¼1

xðtÞ � x0x0

� �2þ yðtÞ � y0

y0

� �2 !! min

(here T ¼ 232 is the period of one cycle) under the constraints

0 � yðtÞ � 1; 0 � nðtÞ � 1; 0 � xðtÞ: (3.4)

The base values of the criteria (without parametric control) are as follows:

K1 ¼ 0:06848;K2 ¼ 2:05489;K3 ¼ 2:08782;K4 ¼ 0:0307:


The values of all criteria for the control law, that is, optimal in the sense of the

criterion, from (3.2) represented before are obtained by solving the problems

formulated above through application of the parametric control approach to the

evolution of the economic system. The results are presented in Table 3.1.

The values of the model’s endogenous variables without applying parametric

control and with use of the optimal parametric control laws for each criterion are

presented in graphic form in Figs. 3.3, 3.4, 3.5, 3.6, and 3.7.

Table 3.1 Values of coefficients and criteria for optimal laws

Criterion Optimal law Coefficient value Criterion value

1 3 0.2404966 0.068890

2 3 0.4766800 2.230337

3 4 0.0718620 2.196740

4 4 0.3005190 0.007273

Months

no scenario scenario 3

Fig. 3.3 Capital productivity ratio without parametric control and with use of law 3, optimal in the

sense of criterion 1

Months




3.1 Mathematical Model of the Kondratiev Cycle 145

Months




Years

no control with optimal control law



Years

no control with optimal control law

Fig. 3.7 Efficiency of innovations without parametric control and with use of law 4, optimal in the



3.1.4 Estimating the Structural Stability of the KondratievCycle Mathematical Model with Parametric Control

To carry out this analysis, the expressions for optimal parametric control laws (3.2)

with the obtained values of the adjusted coefficients are substituted into the right-

hand side of the second and third equations of system (3.1) for the parametern0 .Then, by using a numerical algorithm to estimate the weak structural stability of the

discrete-time dynamical system for the chosen compact set N determined by the

inequalities 1:7 � x � 2:3, 0:066 � y � 0:098 in the state space of the variables

x; yð Þ , we obtain the estimation of the chain-recurrent set Rðf ;NÞ as the empty

(or one-point) set. This means that the Kondratiev cycle mathematical model with

optimal parametric control law is estimated as weakly structurally stable in the

compact set N.

3.1.5 Analysis of the Dependence of the Optimal Valueof Criterion K on the Parameter for the VariationalCalculus Problem Based on the Kondratiev CycleMathematical Model

Now we’ll analyze the dependence of the optimal value of criterion K on the

exogenous parameters m (share of withdrawal of capital production assets per

month) and a for parametric control laws (3.2) with the obtained optimal values

of the adjusted coefficients ki, where the values of the parameters m; að Þ belong to

the rectangle L ¼ ½0:00063; 0:00147� � ½�0:01; 0:71� in the plane.

Plots of dependencies of the optimal values of criterion K (for parametric control

laws 0 and 2, yielding the maximum criterion values) on the uncontrolled parameters

(see Fig. 3.8) were obtained by computational experimentation. The projection of the

intersection line of the two surfaces in the plane m; að Þ consists of the bifurcation

points of the extremals of the given variational calculus problem.

3.2 Goodwin Mathematical Model of MarketFluctuations of a Growing Economy

3.2.1 Model Description

The Goodwin model describing market fluctuations in a growing economy is

presented in [19, 41].

3.2 Goodwin Mathematical Model of Market Fluctuations. . . 147

The model is described by the following system of two differential equations:

d0ðtÞ ¼ ðalðtÞ � a0ÞdðtÞ;l0ðtÞ ¼ ð�bdðtÞ þ b0ÞlðtÞ:

((3.5)

Here d is the percentage of employed in the entire population; l is the percentage ofsupply for consumption in the GDP; a, a0, b, and b0 are constants in the model.

The estimation of the model parameters a, a0, b, b0 is carried out using the

statistical information of the Republic of Kazakhstan for the years 2001–2005 [40],

for which the deviations of the observed statistical data from the calculated results

do not exceed 4.93% during the period under consideration. In solving the

parametric identification preliminary estimation problem, we obtained the follow-

ing exogenous parameters:

a ¼ 0:1710; a0 ¼ 0:08; b ¼ 0:00211; b0 ¼ 0:001:

The calculated period was one cycle in this case, T ¼ 706:27 months.

The model relies on an assumption of invariability of the following economic

parameters:

Opt

imal

crite

rion

val

ues

Fig. 3.8 Plots of the dependencies of the optimal value of criterionK on exogenous parameters m, a


k is the capital output ratio, 0 < k < 1;

n is the population growth rate, n >�1;

g is the labor productivity growth rate, g >�1.

It is also assumed that the percentage of employed s depends linearly on the

wage growth rate o:

s ¼ s0 þ bo; 0 < s0 < 1; b > 0:

The constant parameters of model (3.5) are derived by the following relations:

a ¼ 1

bð1þ gÞ > 0; a0 ¼ s0bð1þ gÞ > 0; b ¼ 1

kð1þ gÞð1þ nÞ > 0;

b0 ¼ 1� kðgþ nþ ngÞkð1þ gÞð1þ nÞ :

Let’s also assume that gþ nþ ng < 1, in which case b0 > 0.

Let’s consider the solutions of system (3.5) in some closed, simply connected

region Owith boundary defined by a simple closed curve lying in the first quadrant

of the phase plane R2þ ¼ fd > 0; l > 0g: dð0Þ ¼ d0; lð0Þ ¼ l0; ðd0; l0Þ 2 O.

It is a well-known fact that in region R2þ , system (3.5) has only the following

state-space trajectories:

– The stationary singular point

l� ¼ a0=a; d� ¼ b0=b; 0 < l� < 1; 0 < d� < 1; (3.6)

– The nonstationary cyclic trajectories lying in R2þ and caused by the initial

conditions ðd0; l0Þ 6¼ ðd�; l�Þ . The singular point ðd�; l�Þ lies inside these

cycles.

3.2.2 Analysis of the Structural Stability of the GoodwinMathematical Model Without Parametric Control

Let’s estimate the structural stability of the Goodwin model without parametric

control in closed regionsO, being guided by the theorem on necessary and sufficient

conditions for structural stability [12]. First, let’s prove the following assertion.

Lemma 3.1The singular point ðd�; l�Þ of system (3.5) lies in the center.

ProofWith (3.6) in mind, let’s write down the Jacobian for the right-hand sides of system

(3.5) at the point ðd�; l�Þ:


A ¼ al� � a0 ad�

�bl� b0 � bd�

� �¼ 0 ab0=b

�ba0=a 0

� �:

It is obvious that this matrix has imaginary eigenvalues � iffiffiffiffiffiffiffiffiffia0b0

p; . Therefore,

the point ðd�; l�Þ is the structurally unstable center point (nonhyperbolic point).

Assertion 3.1System (3.5) is structurally unstable in the closed regionO ðO � R2

þÞwith boundarya simple closed curve containing the point ðd�; l�Þ of the form (3.6) for any fixedvalues of the parameters k; n; g; l0; b; each taken from their respective regionalvalues.

System (3.5) is structurally stable in the closed regionOðO � R2þÞwith boundary

a simple closed curve not containing the point ðd�; l�Þof the form (3.6) for any fixedvalues of parameters k; n; g; l0; b, each taken from their respective regionalvalues.

ProofLet the closed regionO � R2

þ contain the singular point ðd�; l�Þ. A neighborhood of

this system of points (3.5) is locally structurally unstable. Therefore, it is structur-

ally unstable in region O.Let the closed region O � R2

þ not contain the singular point ðd�; l�Þ. In this case,the region O does not contain any cycle, since at least one singular point must be

inside any cycle. Therefore, considering this case, system (3.5) is structurally stable

in region O.

3.2.3 Problem of Choosing Optimal Parametric Control Lawson the Basis of the Goodwin Mathematical Model

It should be noted that the estimates for parameters k; n; g; s0; b derived using

statistical information from the Republic of Kazakhstan for the period 2000–2008

do not describe the economy of the Republic of Kazakhstan with acceptable

accuracy. Therefore, choosing optimal parametric control laws is presented below

for conventional values of the given parameters.

Now, let’s consider implementing an efficient public policy by choosing optimal

control laws with the example of economic parameter k (capital output ratio). Thegoal of the economic policy is to reduce the magnitudes of fluctuations of the

indices ðd; lÞ of the evolution of the national economic system.

Choosing the optimal laws of parametric control is carried out using the follow-

ing sets of relations:

1:U1ðtÞ ¼ c1dðtÞ � d0

d0þ k0; 2:U2ðtÞ ¼ �c2

dðtÞ � d0d0

þ k0;

3:U3ðtÞ ¼ c3lðtÞ � l0

l0þ k0; 4:U4ðtÞ ¼ �c4

lðtÞ � l0l0

þ k0:

(3.7)


Here Ui is the ith control law of parameter k ( i ¼ 1; :::; 4 ); ci is the adjusted

coefficient of the ith control law, ci > 0; k0 is a constant equal to the base value

of parameter k. Application of control lawUi means the substitution of the function

from the right-hand side of (3.7) to system (3.5) for parameter k; t ¼ 0 is the control

system’s starting time, t 2 ½0; T�.The problem of choosing the optimal parametric control law at the level of the

economic parameter k can be stated as follows: On the basis of mathematical model

(3.5), find the optimal law of parametric control of the economic parameter k fromthe set of algorithms (3.7); that is, find the optimal law from the set Ui minimizing

the criterion characterizing the mean distance from the trajectory points to the

singular point ðd�; l�Þ of the system:

K ¼ 1

T

ðT0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdðtÞ � d�Þ2 þ ðlðtÞ � l�Þ2

qdt ! min

fUi; cig(3.8)


0 � k � 1; 0 � l � 1; 0 � d � 1; t 2 ½0; T�: (3.9)

Here T is the period of the controlled cyclic trajectory of system (3.5), and criterion

K characterizes the mean distance from the points of this trajectory to the stationary

point (3.6).

This problem is solved in two stages:

– In the first stage, the optimal values of the coefficients ci for each law Ui are

determined by enumerating their values in the respective intervals (quantized

with a small step size), minimizing K under constraints (3.9).

– In the second stage, the optimal law regulating parameter k is chosen based on

the outcome of the first stage using the minimum value of criterion K.

The problem is solved:

• with given values of parameters b ¼ 10=13, g ¼ 0:5, d0 ¼ 0:4, and l0 ¼ 0:5,• with a fixed value of the uncontrolled parameter n ¼ 0:3,• and with the base value of the controlled parameter k0 ¼ 10=19.

These values of the parameters yield the system stationary point with the

coordinates l� ¼ 0:5; d� ¼ 0:5.A numerical solution to the problem of choosing the optimal parametric control

law shows that the best result for K ¼ 0.03215307 can be achieved using the

following law:

k ¼ 4:28lðtÞ � 0:5

0:5þ 10=19: (3.10)

Note that the criterion value K without parametric control is 0.0918682.


Results of computational experiments allow us to determine the following facts:

– A decrease in the value of criterion K in comparison to the case without control

is obtained only with use of laws U1ðtÞ and U3ðtÞ from (3.7).

– Using laws of type U1ðtÞ , we can observe that cyclic character of the phase

trajectory of system (3.5) as seen in Fig. 3.9 is preserved.

– Using laws of type U3ðtÞ instead of a cyclic trajectory, we can observe

trajectories approaching the stable singular point of system (3.5) with parametric

control as t ! 1 approaches infinity (Fig. 3.10).

Fig. 3.9 Curve 1 corresponds

to the market cycle without

control; curve 2 corresponds

to the market cycle applying

control law U1ðtÞ

Fig. 3.10 Curve 1

corresponds to the market

cycle without control; curve

2 corresponds to the market

cycle applying control law

U3ðtÞ


3.2.4 Analysis of the Structural Stability of the GoodwinMathematical Model with Parametric Control

Let’s analyze the structural stability of system (3.5) using a parametric control law

of kindU3ðtÞ orU4ðtÞ from the set of algorithms (3.7) for any admissible fixed value

of the adjusted coefficient c 6¼ 0.

These laws are given by

k ¼ clðtÞ � l0

l0þ k0: (3.11)

Here k0 is a constant equal to the base value of parameter k. First, let’s find the

singular points of system (3.5) using parametric control. Substituting the respective

expression for k into the right-hand sides of the equations of system (3.5) and setting

them equal to zero, we obtain the following unknown variables ðd; lÞ (with the

remaining fixed admissible values of variables and constants):

ðal� a0Þd ¼ 0;

� 1

kðlÞð1þ gÞð1þ nÞ dþ1� kðlÞðgþ nþ ngÞkðlÞð1þ gÞð1þ nÞ

� �l ¼ 0:

8><>: (3.12)

Here kðlÞ ¼ c l�l0l0

þ k0: We only use values of c such that 0 < kðlÞ < 1 .

System (3.8) has a unique solution in R2þ:

l� ¼ a0=a;

d� ¼ 1� kðl�Þðgþ nþ ngÞ;

((3.13)

where 0 < l� < 1; 0 < d� < 1:Now, let’s write the Jacobian for the left-hand side of system (3.12) at point

(3.13):

A¼

al��a0 ad�

�bl�cd�

ðc1l��l0l0

þk0Þ2

l0ð1þgÞð1þnÞ� c

ðc1l��l0l0

þk0Þ2

l0ð1þgÞð1þnÞ

0BBB@

1CCCAl�þð�bd�þb0Þ

0BBBBB@

1CCCCCA

¼

0 ad�

�bl�cðd��1Þl�

ðcl��l0l0

þk0Þ2

l0ð1þgÞð1þnÞ

0BBB@

1CCCA:


The eigenvalues of matrix A are the roots of the equation

m2 þ cð1� d�Þl�

ðc l� � l0l0

þ k0Þ2

l0ð1þ gÞð1þ nÞmþ 1� kðl�Þðgþ nþ ngÞ

kðl�Þð1þ gÞð1þ nÞ a0 ¼ 0:

Denoting the coefficients of this equation by p and q, we obtain the quadratic

equation

m2 þ pmþ q ¼ 0; (3.14)

where q > 0, and the sign of p coincides with the sign of the coefficient c.The following cases are possible:

1. If the discriminant of Eq. (3.14) is

D ¼ cð1� d�Þl�

ðc l� � l0l0

þ k0Þ2

l0ð1þ gÞð1þ nÞ

0BBB@

1CCCA

2

� 41� kðl�Þðgþ nþ ngÞkðl�Þð1þ gÞð1þ nÞ a0 < 0;

then the singularity point ðd�; l�Þ of system (3.5) with parametric control (3.11)

is the focus, which is stable with c > 0 and unstable with c < 0.

2. If D 0, then singular point (3.13) of system (3.5) with parametric control

(3.11) is the node, and this node is stable with c>0 and unstable with c < 0.

Assertion 3.2System (3.5) with parametric control (3.11) is locally structurally stable in anysufficiently small closed region O ðO � R2

þÞ, small with the boundary as a simpleclosed curve containing the point ðd�; l�Þ of form (3.13) for any fixed values ofparameters c; n; g; l0; b from their respective regional values.

System (3.5) is structurally stable in any closed region OðO � R2þÞ with

boundary a simple closed curve not containing the point ðd�; l�Þ of form (3.13) forany fixed values of parameters c; n; g; l0; b from their respective regional values.

ProofLet the singular point ðd�; l�Þ not belong to the closed regionO � R2

þ. In this case,by the same reasoning as in the proof of Assertion 3.1, we obtain that system (3.5),

(3.11) is structurally stable in the region O.Now, let the singular point ðd�; l�Þ belong to the closed region O � R2

þ .

Since this point is hyperbolic (node or focus), then system (3.5), (3.11) is locally

structurally stable in its neighborhood.

Let’s analyze the structural stability of system (3.5) using a parametric control

law of type U1ðtÞ or U2ðtÞ from the set of algorithms (3.7) for any fixed admissible

value of the adjusted coefficient c 6¼ 0.


These laws are given by

k ¼ cdðtÞ � d0

d0þ k0: (3.15)

First, let’s find the singular points of system (3.5) with parametric control.

Substituting this expression for k into the right-hand sides of the equations of

system (3.5) and equating them to zero, we obtain the following system in the

unknown variables ðd; lÞ (with the remaining fixed admissible values of variables

and constants):

ðal� a0Þd ¼ 0;

� 1

kðdÞð1þ gÞð1þ nÞ dþ1� kðdÞðgþ nþ ngÞkðdÞð1þ gÞð1þ nÞ

� �l ¼ 0:

8><>: (3.16)

Here kðdÞ ¼ cd� d0d0

þ k0. System (3.16) has a unique solution

l� ¼ a0=a;

d� ¼ 1þ ðc� k0Þðgþ nþ ngÞ1þ cðgþ nþ ngÞ=d0 :

8><>: (3.17)

We use only the values of c such that 0< kðdÞ< 1; 0< l� < 1; 0< d� < 1:Now, the Jacobian for the left-hand side of the system (3.12) at point (3.13) is

A¼

0 ad�

cðd� �1Þ

ðcd� � d0d0

þ k0Þ2

d0ð1þ gÞð1þnÞ� 1

ðcd� � d0d0

þ k0Þd0ð1þ gÞð1þnÞ

0BBB@

1CCCAl� 0

0BBBB@

1CCCCA:

It is obvious that this matrix has imaginary eigenvalues. Therefore, singularity

point (3.17) is at the center. Applying methods from [11], it can be proved that

all phase trajectories of system (3.5) with parametric control (3.15) are the cycles in

R2þ except at point (3.13). The following assertion can be proved similarly to

Assertion 3.1.

Assertion 3.3System (3.5) with parametric control (3.15) is structurally unstable in the closedregion O ðO � R2

þÞ with boundary a simple closed curve containing the point ðd�;l�Þof the form (3.17) for any fixed values of parameters c; k; n; g; l0; b from theirrespective regional values.

System (3.5) with parametric control (3.15) is structurally stable in the closedregion OðO � R2

þÞ with the boundary as a simple closed curve not containing thepoint ðd�; l�Þof form (3.17) for any fixed values of the parameters c; k; n; g; l0; bfrom their respective regional values.


3.2.5 Analysis of the Dependence of the Optimal ParametricControl Law on Values of the Uncontrolled Parameterof the Goodwin Mathematical Model

Let’s consider the dependence of the results of choosing the optimal parametric

control law at the level of parameter k on the uncontrolled parameter n (population

growth rate) with values in the interval ½0; 0:4�.The results of computational experiments are presented in Table 3.2 and

Fig. 3.11. These results reflect the dependence of the optimal value of criterion

K on the values of parameter n for each of four possible laws (3.7).

An analysis of Table 3.2 shows that for all considered values of parameter n, thecontrol lawU3ðtÞ is optimal; that is, for the given interval of values of parameter n, abifurcation point of the extremals of the given variational calculus problem does not

exist.

Table 3.2

Parameter n 0 0.1 0.2 0.3 0.4

Control law Optimal value of the criterion for this law

U3ðtÞ 0.130000 0.093165 0.060932 0.032153 0.006379

U1ðtÞ 0.210856 0.167352 0.121062 0.069768 0.014642

U2ðtÞ, U4ðtÞ 0.336324 0.251121 0.151368 0.091868 0.018441

Opt

imal

crite

rion

val

ues

Fig. 3.11 Plots of the dependencies of the optimal values of criterionK on uncontrolled parameter n.Notation: – U3, – U1, – U2, U4


Chapter 4

Macroeconomic Analysis and ParametricControl of Economic Growth of a NationalEconomy Based on Computable Modelsof General Equilibrium

As is well known [39], in the context of implementing economic policy, one must

estimate values of economic instruments that will ensure uniform growth (dynamic

equilibrium) in order to provide economic development such that supply and

demand in macroeconomic markets increasing from one period to another are

always equal when labor and capital are fully employed. To a certain extent, this

is a requirement of the mathematical models used for estimating rational values of

economic instruments of public policy in the field of economic growth.

The problem of economic growth is covered at present by a large number of

phenomenological and econometric models [45].

Using the basic regression equation for estimating the determinants of economic

growth,

g ¼ a0 þXl

alxl þXp

bpzp þXr

crSLVr þ e

[where g is the rate of the economic growth of the main indexes of the gross national

product (GDP, GNP) in the country, a0 is a constant, al are the coefficients of the

economic variables, хl are the economic variables, bp are the coefficients of

additional variables, zp are additional variables (political, social, geographical,

etc.), cr are the coefficients of the slack variables, SLVr are the slack variables

reflecting the group effect, e is the random component], we are able to derive

various econometric models of dependencies of economic growth on various

types of determinants intended to estimate a wide spectrum of hypotheses and

assumptions about their influence on economic growth, econometric dynamic

interbranch models, as well as econometric macroeconomic model [57, 59, 68].

These models are mainly intended to give an estimate and do not meet the

aforementioned requirements. A wide range of phenomenological models [45],

starting with the mathematical model of neoclassical theory of Solow [69] and

Swan [71] [complemented by dynamical optimization models on the basis of the

Ramsey problem to such mathematical models of endogenous economic growth


157

that represent, for example, production of innovations as a product of a particular

economic sector (e.g., the Grossman and Helpman model [60]); activity aimed at

the development of people themselves (e.g., Robert Lucas model [65]); interna-

tional trade and dissemination of technologies (e.g., Lucas model [64]); and others],

answer the questions about economic growth sources, but do not meet the afore-

mentioned requirements of a mathematical model for estimating rational values of

the economic instruments for public policy in the field of economic growth.

In the context of the balance model [63], where the interbranch connections are

represented via a system of material balances for some set of products constituting

in aggregate the entire national economy, one can note that the system of material

balances expressing the interbranch connections is formed without market relations

between the agents. They also do not include descriptions of such prime agents as

the state, banking sector, and aggregate consumer. Therefore, the balance models

meet the aforementioned requirement to a lesser degree.

In [27], a number of computable models of general equilibrium are proposed.

These models to a greater degree meet the aforementioned requirement for mathe-

matical models applied for estimating rational values of the economic instruments

of the public policy in the field of economic growth.

In this chapter we present results of national economic growth control based on

computable models of general equilibrium subject to constraints on the level of

prices. To a certain extent, this allows us to take the requirements of an antiinflation

policy into consideration.

4.1 National Economic Evolution Control Basedon a Computable Model of General Equilibriumof Economic Branches

4.1.1 Model Description, Parametric Identification,and Retrospective Prediction

The considered CGE model [27] can be written using a system of relations

decomposed into the following subsystems:

1. A subsystem of difference equations connecting the values of the endogenous

variables for two consecutive years,

xtþ1 ¼ Fðxt; yt; zt; u; lÞ: (4.1)

Here t is a year number, in discrete time; t ¼ 0; :::; ext ¼ xt; yt; ztð Þ 2 Rm is

a vector of endogenous variables of the system; xt ¼ x1t ; x2t ; . . . ; x

m1t

� � 2 X1;yt ¼ y1t ; y

2t ; . . . ; y

m2t

� � 2 X2 , zt ¼ z1t ; z2t ; . . . ; z

m3t

� � 2 X3 m1 þ m2 þ m3 ¼ m:Here, variables xt include the values of the key assets, account balances of

158 4 Macroeconomic Analysis and Parametric Control of Economic Growth. . .

the agents, etc.; yt includes the values of demand and supply of the agents in

various markets; zt includes various types of market prices and shares

of budgets in markets with governmental prices for various economic agents;

u 2 U � Rq and l 2 ^ � Rs are the vectors of the exogenous parameters

(controlled and uncontrolled, respectively); X1;X2;X3; and U are compact

sets with nonempty interiors IntðXiÞ; i ¼ 1; 2; 3; and Int(WÞ, respectively; andF : X1 � X2 � X3 �W � ^ ! Rm1 is a continuous function.

2. A subsystem of algebraic equations describing the behavior and interaction of

the agents in various markets during a considered year. These equations allow us

to express the variables yt via the exogenous parameters and other endogenous

variables

yt ¼ Gðxt; zt; u; lÞ: (4.2)

Here G : X1 � X2 � X3 � U � ^ ! Rm2 is a continuous function.

3. A subsystem of recurrent relations for iterative computations of the equilibrium

values of market prices in various markets and shares of budget in the markets

with governmental prices for various economic agents

zt½Qþ 1� ¼ Zðyt½Q�; zt½Q�; L; u; lÞ: (4.3)

Here Q ¼ 0; 1; 2; . . . is the number of iterations. L is a set of positive numbers

(adjustable iteration constants). As L is decreased, the economic system goes into

the equilibrium state faster, although it increases the danger of the prices going to

the negative region. Here Z:X2 � X3 � ð0;þ1Þm3 � U � ^ ! Rm3 is a continuous

mapping (being contractive with fixed xt 2 X1; u 2 U , l 2 ^, and some fixed L.In this case, mapping Z has a unique stationary point, in which iterative process

(4.2), (4.3) converges.

For some fixed values of the exogenous parameters, for each moment of time

t, the CGE model of general equilibrium (4.1), (4.2) and (4.3) defines values of the

endogenous variables ext corresponding to the equilibrium of demand and supply

prices in markets with no governmental prices and shares of budget in markets

with governmental prices of the agents within the limits of the following

algorithm.

1. In the first step, we assume t ¼ 0 and give some initial values x0 to the variables.2. In the second step, for the current value of t, we give initial values zt½0� for the

variables in various markets and for different agents; then values yt½0� ¼ Gðxt;zt½0�; u; lÞ are computed (initial values of demand and supply of the agents in the

markets of goods and services) by (4.2).

3. In the third step, for the current value of time t, we start the iterative process

(2.12). For any Q, current values of supply and demand are found using (2.11):

yt½Q� ¼ Gðxt; zt½Q�; u; lÞ via adjustment of market prices and shares of the

budgets in the markets with the governmental prices of economic agents.

4.1 National Economic Evolution Control Based on a Computable Model of General. . . 159

http://dx.doi.org/10.1007/978-1-4614-4460-2_2

http://dx.doi.org/10.1007/978-1-4614-4460-2_2

Now, the iterative process is terminated if the values of supply and demand in

various markets are equal. As a result, we determine the equilibrium values of

market prices in every market and the shares of budget in the markets with

governmental prices for various economic agents. The index Q for such equilib-

rium values of the endogenous variables is omitted.

4. For the last step, using the derived equilibrium solution for moment t, wecompute the values of variables xtþ1 for the next moment of time via difference

equations (4.1). Then the value of t is incremented by 1. Then we proceed to

step 2.

A number of iterations for steps 2–4 are defined in accordance with the problems

of calibration, prediction, and control in the same time intervals as those chosen

before.

The CGE model can be represented in the form of a continuous mapping f : X�U � ^ ! Rm defining a transformation of values of the system’s endogenous

variables for year 0 to the respective values of the next year according to the

algorithm described above. Here the compact set X in the state space of the

endogenous variables is defined by the set of possible values of variables x(the compact set X1 with nonempty interior) and respective equilibrium values of

the variables y and z computed by relations (4.2) and (4.3).

Let’s suppose that for a chosen point x0 2 Int(X1Þ, the inclusion ext ¼ f tðex0Þ X1j

2 Int(X1Þ: is correct with fixed u 2 Int ðUÞ and l 2 ^ for t ¼ 0; . . . ; T (T is any fixed

natural number). This mapping f defines a discrete-time dynamic system (semi-

cascade) in set X with the following initial condition to its trajectories:

f t; t ¼ 0; 1; . . .f g; x t¼0j ¼ x0: (4.4)

4.1.1.1 Model Agents

Model [27] describes the behavior and interaction in 46 product markets and 16

labor markets of the following 20 economic agents (sectors).

Economic agent no. 1 represents agriculture, hunting, and forestry.

Economic agent no. 2 represents fishery and fish breeding.

Economic agent no. 3 is the mineral resource industry.

Economic agent no. 4 is the manufacturing and process industry.

Economic agent no. 5 is the production and distribution of electric power, gas, and

water.

Economic agent no. 6 is construction.

Economic agent no. 7 represents trade, repair services for cars, and goods for

household use.

Economic agent no. 8 represents hotels and restaurants.

Economic agent no. 9 represents transport and communications.

Economic agent no. 10 corresponds to financial activity.


Economic agent no. 11 represents real estate activities, rent, and services for

various businesses.

Economic agent no. 12 is public administration.

Economic agent no. 13 represents education.

Economic agent no. 14 is public health and social services.

Economic agent no. 15 represents other public utilities, social and personal

services.

Economic agent no. 16 represents housekeeping services.

A part of the products from economic agents producing goods and services

(economic agent nos. 1–16) are used in production, while the other part is spent to

invest and the remainder is sold to households. The producing agents deal in the

intermediate and investment products with each other.

Economic agent no. 17 is the aggregate consumer joining households.

The aggregate consumer purchases consumer goods produced by the producing

agents. Moreover, it purchases imported goods offered by the outer world.

Economic agent no. 18 is the government represented by the aggregate of the

central, regional, and local governments, as well as the off-budget funds. The

government establishes the taxation rates and defines the sum of the subsidies to

the producing agents, as well as the volumes of social transfers to the

households. Additionally, this sector includes the nonprofit organizations servic-

ing the households (the political parties, trade unions, public associations, etc.).

Economic agent no. 19 is the banking sector, including the central bank and

commercial banks.

Economic agent no. 20 is the outer world

The considered model includes 698 endogenous variables and 2045 exogenous

parameters to be estimated.

The following system of notations is used here for the constants and variables of

the CGE models:

<Type > <Parameter > _ < Price and its code > _ < Number of economic

agent and market code > [<instant time (t) or number of iteration (Q)>].

Here < Type > can take on two values; namely, С is the exogenous parameter,

and V is the endogenous variable.

<Parameter > corresponds to the action realized by the agent. The examples of

such actions can be given by S (product supply), D (product demand), O (determin-

ing the share of budget by the agent), and others.

For example, the notation CO p3 1z 0½ � corresponds to the exogenous parame-

ter, which is the share of budget of the first sector (agriculture, hunting, and

forestry) for purchasing the intermediate product produced by the third branch

(mineral resource industry) at the price of P 3z for the intermediate products of

the third branch in the year (2000).


4.1.1.2 Exogenous Variables of the Model

The exogenous parameters include the following:

– The coefficients of the production functions of the sectors;

– The various shares of the budgets of the sectors;

– The shares of the products for selling in the various markets;

– The depreciation rates of the capital assets and shares of the retired capital

assets;

– The deposit interest rates;

– The various taxation rates;

– The coefficients reflecting the level of nonpayments to the producing agents;

– The depreciation rates of the capital assets;

– The shares of the retired capital assets;

– The coefficient reflecting the level of wage liabilities for the employees in all

branches;

– The export prices and governmental prices of the goods, services, and labor

force, etc.

The list of the exogenous model parameters is given in Table 4.1.

Table 4.1 Exogenous variables of the computable model of economic branches

Producing agents nos. 1–16

CO pi il The share of the budget of the ith branch spent for paying the labor force at the

price of P il

CO pj iz The share of the budget of the ith branch spent for purchasing the intermediate

products produced by the branches j ¼ 1; :::; 16 at the price of P jz

CO p in The share of the budget of the ith branch spent for purchasing investment

products at the price of P n

CE pi iz The share of product produced by the ith branch for selling in the markets of the

intermediate products at the price of P iz

CE p ic The share of product produced by the ith branch for selling in the markets of final

products at the price of P ic

CE p in The share of product produced by the ith branch for selling in the markets of

investment products at the price of P in

CE pexi ic The share of product produced by the ith branch for selling in the markets of the

exported products at the price of P exi

CA r i The empirically determined coefficient of dimension of the production function

CA z ji The coefficients of the production function with the intermediate products

j ¼ 1; :::; 16 consumed by the ith branch

CA k i The coefficient of the capital in the production function

CA l i The coefficient of the labor in the production function

CO y i The coefficient reflecting the level of nonpayments to the producing agents

CA n The depreciation rate of the capital assets

CO w i The coefficient reflecting the level of arrears of wages to the employees in all

branches

CR i The share of the retired capital assets

(continued)


4.1.1.3 Endogenous Variables of the Model

The endogenous variables include the following:

– The budgets of the sectors and their various shares;

– The remainders of the agents’ budgets;

– The produced values-added of the producing sectors;

– Demand and supply of various products and services;

– The gains of the sectors;

– The capital assets of the producing sectors;

– The number of employees employed in Sectors 1–16;

– The wages of the employees;

– The various types of expenditures of the consolidated budget;

– The various types of prices of the products, services, and the labor force;

– The subsidies to the producing sectors;

– The social transfers to the citizens;

– The gross production of goods and services;

– The volume of production of the intermediate products;

– The volume of production of final products;

– The GDP of a country.

The list of the endogenous variables of the model is given in Table 4.2.

Table 4.1 (continued)

17 Households

CO p 17c The share of the budget of the aggregate consumer spent for purchasing final

products at the price of P c

CO b 17 The share of the budget deposited at the banks

CS pi 17l The number of employees employed in Sectors 1–16

18 Government

CT vad The VAT rate

CT pr The organization profit tax rate

CT pod The rate of physical body income tax

CT esn The rate of single social tax

CO s i 18 The shares of the consolidated budget for backing the producing agents

CO tr 18 The share of the consolidated budget for payment of social transfers to the

inhabitants

CO f 18 The share of off-budget funds for payment of pensions, welfare payments, etc.

CB other 18 The sum of the tax proceeds (not included into the considered ones), nontax

income, and other incomes of the consolidated budget

19 Banking sector

CP bpercent The deposit interest rate for enterprises

CP h bpercent The deposit interest rate for physical bodies

General part of the model

CP exi The price of the exported product produced by the ith branch

Technical parameters

Ceta 1 The iteration constant applied in the case of the exogenous price


Table 4.2 Endogenous variables of the computable model of economic branches

Producing agents nos. 1–16

VO tc i The share of the budget of the producing agent spent for discharging the taxes to the

consolidated budget

VO tf i The share of the budget of the producing agent spent for discharging the taxes to the

off-budget funds

VO s i The remainder of the agent’s budget

VD pi il The demand of labor power in the ith branch at the price of P il

VD pj iz The demand of the intermediate products produced by the branches j ¼ 1; :::; 16 in theith branch at the price of P jz

VD p in The demand of investment products in the ith branch at the price of P in

VY i Production of products and services in the prices of the base period

VY g i The value-added produced by the ith branch

VK i The capital assets of the producing agent

VS pi iz The supply of intermediate products

VS p ic The supply of final products

VS p in The supply of investment products

VS pex ic The supply of exported products

VY p i The gain of the producing agent

VY r i The profit of the producing agent

VB i The budget of the producing agent

VB b i The balance of banking accounts of the producing agent

17 Households

VO tc 17 The share of the budget of the aggregate consumer for discharging the taxes to the

consolidated budget

VO s 17 The remainder of the budget of Sector 17

VD p 17c The household demand for final products

VW i The wages of employees in Sectors 1–16

VB 17 The budget of households

VB b 17 The money of the households in banking accounts

18 Government

VO s 18 The share of the retained consolidated budget

VO s 18f The share of the retained off-budget funds

VG s i 18 Subsidies to the producing sectors

VG tr 18 Social transfers to the inhabitants

VG f 18 The off-budget funds allocated for the inhabitants

VB 18 The consolidated budget

VB b 18 The surplus (deficit) of the consolidated budget

VF 18 The monetary assets of the off-budget funds

VF b 18 The remainder of monetary assets of the off-budget funds

Integral indices of the model

VY The gross production of goods and services (in prices of the base period)

VS z The volume of production of the intermediate products (in prices of the base period)

VS c The volume of production of final products (in prices of the base period)

VY g The GDP

VP The consumer price index

(continued)


4.1.1.4 The Model Markets

The equilibrium prices are formed in 50 markets of the model as a result of leveling

the supply and demand of various types of products, services, and labor force.

The described model has

– 16 markets of the intermediate products and services produced and rendered by

the producing agents;

– A market of investment products; and

– A market of final products.

In addition, the model also includes the following:

– 16 foreign markets of exported products produced by the producing agents; and

– 16 markets of the labor force.

The total number of markets in this model is 46. The governmental and market

mechanisms of pricing are used in domestic markets. The prices of foreign markets

enter the model exogenously. We’ll now consider the formulas reflecting the

process of changes in prices in domestic markets (below, i is the agent number).

The price of labor force in the ith branch is given by

VP il Qþ 1½ � ¼ VP il Q½ � � Abs VD ps il t½ �=VS ps il t½ �ð Þ: (4.5)

The price of the intermediate product produced by the ith branch is as follows:

VP iz Qþ 1½ � ¼ VP iz Q½ � � Abs VD ps iz t½ �=VS ps iz t½ �ð Þ: (4.6)

The price of investment products is determined by

VP n Qþ 1½ � ¼ VP n Q½ � � Abs VD ps n t½ �=VS ps n t½ �ð Þ: (4.7)


General part of model

VP il The price of the labor force in the ith branch

VP iz The price of the intermediate product produced by the ith branch

VP n The price of investment products

VP c The price of consumer products


VD ps il The total demand for the labor force at the price of P il

VD ps iz The total demand for intermediate products at the price of P iz

VD ps n The total demand for investment products at the price of P n

VD ps c The total demand for consumer products at the price of P c

VS ps il The total supply of the labor force at the price of P il

VS ps iz The total supply of intermediate products at the price of P iz

VS ps n The total supply of investment products at the price of P n

VS ps c The total demand for consumer products at the price of P c


The price of consumer products is as follows:

VP c Qþ 1½ � ¼ VP c Q½ � � Abs VD ps c t½ �=VS ps c t½ �ð Þ: (4.8)

We now have 16þ 16þ 1 ¼ 33 prices of products sold in domestic markets for

the given model.

The notations of prices in foreign markets are given below.

The price of the exported product produced by the ith branch is

P exi: (4.9)

Thus, the total number of prices in this model is 33þ 16 ¼ 49.

Let’s now proceed to the formulas describing the mechanism of the formation of

the demand and supply of products produced by agent nos. 1–16 at governmental

and market prices.

The final formulas determining the demand and supply of each economic agent

in the product markets included in the model are presented below.

The total demand for the labor force at the price of VP il t½ � is given by

VD ps il t½ � ¼ VD pi il t½ �: (4.10)

For simplicity, we do not consider the demand for the labor force in the ithbranch from the other branches. In this connection, the total demand for the labor

force at the price of VP il t½ � is defined by the demand in the single ith branch.

The total supply of the labor force at the price of VP il t½ � is as follows:

VS ps il t½ � ¼ CS pi 17l: (4.11)

The total demand for the intermediate product at the price of VP jz t½ � producedby the jth branch is determined as

VD ps jz t½ � ¼ SUM VD pj iz t½ �ð ÞÞ: (4.12)

Hereafter, SUM X ið Þ corresponding toP16i¼1

X i, i ¼ 1; :::; 16; is the economic

agent number.

As can be seen, the total demand for the intermediate product at the price

of VP jz t½ � includes the demand for the intermediate products in the jth branch

j ¼ 1; :::; 16 from the direction of all 16 branches.

The total supply of the intermediate product at the price of VP iz t½ � is given by

VS ps iz t½ � ¼ VS pi iz t½ �: (4.13)

The total demand for investment products at the price of VP__n[t] is

VD ps n t½ � ¼ SUMðVD p in t½ �Þ: (4.14)


The total supply of investment products at the price of VP n t½ � is

VS ps n t½ � ¼ SUM VS p in t½ �ð Þ: (4.15)

The total demand for consumer products at the price of VP c t½ � is

VD ps c t½ � ¼ VD p 17c t½ �: (4.16)

The total supply of consumer products at the price of VP c t½ � is

VS ps c t½ � ¼ SUM VS p ic t½ �ð Þ: (4.17)

Thus, we have 32þ 32þ 2þ 2 ¼ 68 formulas for determining the total supply

of and demand for products in domestic markets.

Now we’ll present the notations defining the total supply of and demand for

exported products:

The total demand for exported products at the price of CP pex ic t½ � (given) is

VD pex ic t½ �: (4.18)

The total supply of exported products at the price of CP pex ic t½ � is

VS pex ic t½ �: (4.19)

Finally, we derive 68þ 32 ¼ 100 formulas for determining the total supply of

and demand for all products used in the model.

Next, we’ll describe the activities of the economic agents participating in this

model.

4.1.1.5 Economic Agent Nos. 1–16 Producing Products and Services

The equation of the production function of an economic producing agent is given by

VY i tþ 1½ � ¼ CA r i t½ � � Exp VD p1 iz t½ � � CA z 1i t½ �ð Þ � Exp VD p2 iz t½ � � CA z 2i t½ �ð Þ� Exp VD p3 iz t½ � � CA z 3i t½ �ð Þ � Exp VD p4 iz t½ � � CA z 4i t½ �ð Þ� Exp VD p5 iz t½ � � CA z 5i t½ �ð Þ � Exp VD p6 iz t½ � � CA z 6i t½ �ð Þ� Exp VD p7 iz t½ � � CA z 7i t½ �ð Þ � Exp VD p8 iz t½ � � CA z 8i t½ �ð Þ� Exp VD p9 iz t½ � � CA z 9i t½ �ð Þ � Exp VD p10 iz t½ � � CA z 10i t½ �ð Þ� Exp VD p11 iz t½ � � CA z 11i t½ �ð Þ � Exp VD p12 iz t½ � � CA z 12i t½ �ð Þ� Exp VD p13 iz t½ � � CA z 13i t½ �ð Þ � Exp VD p14 iz t½ � � CA z 14i t½ �ð Þ� Exp VD p15 iz t½ � � CA z 15i t½ �ð Þ � Exp VD p16 iz t½ � � CA z 16i t½ �ð Þ� Power VK i t½ �þVK i tþ1½ �ð Þ=2ð Þ;CA k i t½ �ð Þ�Power VD pi il t½ �;CA l i t½ �ð Þ:

(4.20)


Here CA r i;CA z ji, CA k i;CA l i are the parameters of the production function,

Power(X, Y) corresponds to XY, and Exp(X) corresponds to eX.The following formulas determine the demand for the production factors by the

ith agent.

The demand for the labor force is

VD pi il t½ � ¼ CO pi il� VB i t½ �ð Þ=VP il t½ �: (4.21)

The demand for intermediate products produced by all the producing agents is

VD p1 iz t½ � ¼ VO p1 iz t½ � � VB i t½ �ð Þ=CP 1z t½ �; (4.22)

VD p2 iz t½ � ¼ CO p2 iz� VB i t½ �ð Þ=VP 2z t½ �; (4.23)














VD p16 iz t½ � ¼ ðCO p16 iz� VB i t½ �Þ=VP 16z t½ �: (4.37)

The demand for investment products is

VD p in t½ � ¼ CO p in� VB i t½ �ð Þ=VP n t½ �: (4.38)


The following formulas determine the supply of products and services produced

by the producing agent.

The supply of the intermediate products is

VS pi iz t½ � ¼ CE pi iz� VY i t½ �: (4.39)

The supply of final products is

VS p ic t½ � ¼ CE p ic� VY i t½ �: (4.40)

The supply of investment products is

VS p in t½ � ¼ CE p in� VY i t½ �: (4.41)

The supply of the exported products is

VS pex ic t½ � ¼ CE pexi ic� VY i t½ �: (4.42)

The following formula calculates the gain of the producing agent:

VY p i t½ � ¼ VS pi iz t½ � � VP iz t½ � þ VS p ic t½ � � VP c t½ � þ VS p in t½ � � VP n t½ �þ VS pex ic t½ � � CP exi t½ �:

(4.43)

The profit of the producing agent is

VY r i t½ � ¼ CO y i� VY p i t½ � � ðVD p1 iz t½ � þ VD p2 iz t½ � þ VD p3 iz t½ �þ VD p4 iz t½ � þ VD p5 iz t½ � þ VD p6 iz t½ � þ VD p7 iz t½ � þ VD p8 iz t½ �þ VD p9 iz t½ � þ VD p10 iz t½ �þVD p11 iz t½ �þVD p12 iz t½ �þVD p13 iz t½ �þ VD p14 iz t½ � þ VD p15 iz t½ � þ VD p16 iz t½ �þ VW i t½ � � CO w ið Þ þ CA n t½ � � ðVK i t½ � � VP n t½ �ÞÞ:

(4.44)

Here CO y i is the coefficient reflecting the level of nonpayments; CA n is the

depreciation rate of capital assets. Here we calculate the profit of the sector

consisting of the gain corrected by the level of nonpayments. The assets spent for

the intermediate product, wages (without taking into account the debt, the coeffi-

cient CO w i), and amortization of capital assets are subtracted.

The value-added produced by the ith sector is given by

VY g i t½ � ¼ VY r i t½ � þ VW i t½ �: (4.45)


The value-added consists of the profit received in the current period and wages

paid to the employee sector.

The budget of the producing agent is as follows:

VB i t½ � ¼ VB b i t� 1½ � � 1þ CP bpercent t� 1½ �ð Þ þ CO y i� VY p i t½ � þ VG s i18 t� 1½ �:(4.46)

The agent budget consists of the following:

1. The funds in banking accounts (taking into consideration interests on deposits);

2. The gain received in the current period;

3. The subsidies received from the consolidated budget VG s i18.

The dynamics of the banking account balance of the producing agent is as

follows:

VB b i t½ � ¼ VO s i t½ � � VB i t½ �: (4.47)

The capital assets are determined by

VK i tþ 1½ � ¼ 1� CR i t½ �ð Þ � VK i t½ � þ VD p in t½ �: (4.48)

This formula calculates the volume of the capital assets, taking their retirement

into retirement. The asset put into operation enters the formula with the plus sign.

The share of the budget of the producing agent for discharging the taxes to the

consolidated budget is given by

VO tc i t½ � ¼ VY g i t½ � � CT vad t½ �ð Þ=VB i t½ �þ VY r i t½ � � CT pr t½ �ð Þ=VB i t½ �: (4.49)

This formula takes into consideration the value-added tax (VAT) and profit tax.

The share of the budget for discharging the single social tax to the off-budget

funds is described as

VO tf i t½ � ¼ VW i t½ � � CT esn t½ �ð Þ=VB i t½ �: (4.50)

The remainder of the budget of the producing agent is given by

VO s i t½ � ¼ 1� ðCO pi ilþ CO p inþ VO tc i t½ � þ VO tf i t½ � þ VO p1 iz t½ �þ CO p2 izþ CO p3 izþ CO p4 izþ CO p5 izþ CO p6 iz

þ CO p7 izþ CO p8 izþ CO p9 izþ CO p10 izþ CO p11 iz

þ CO p12 izþ CO p13 izþ CO p14 izþ CO p15 izþ CO p16 izÞ:(4.51)


4.1.1.6 Economic Agent No. 17: Aggregate Consumer

Let’s proceed to the formulas that determine the behavior of the aggregate

consumer.

The household demand for final products is given by

VD p 17c t½ � ¼ CO p 17c� VB 17 t½ �ð Þ=VP c t½ �: (4.52)

The wages of the employees of Sectors 1–16 are

VW i t½ � ¼ VD pi il t½ � � VP il t½ �: (4.53)

The budget of households is determined as follows:

VB 17 t½ � ¼ VB b 17 t� 1½ � � 1þ CP h bpercent t� 1½ �ð Þ þ VB 17 t� 1½ �� VO s 17 t� 1½ � þ VG tr 18 t� 1½ � þ VG f 18 t� 1½ � þ SUM VW i t½ �ð Þ:

(4.54)

The agent’s budget is formed using the following:

1. Funds in the banking accounts (subject to interests on deposits);

2. Retained money (in cash) kept from the preceding period;

3. Pensions, welfare payments, and subsidies received from the off-budget funds;

4. Wages received from the producing agent nos. 1–16.

The dynamics of the banking account balance of households is as follows:

VB b 17 t½ � ¼ CO b 17� VB 17 t½ �: (4.55)

The share of the budget for discharging income tax is given by

VO tc 17 t½ � ¼ tS� CT pod t½ �ð Þ=VB 17 t½ �: (4.56)

The remainder of the money in cash is calculated as follows:

VO s 17 t½ � ¼ 1� CO p 17c� VO tc 17 t½ � � CO b 17: (4.57)

4.1.1.7 Economic Agent No. 18 Government

As shown above, this economic agent is represented by the aggregate of the federal,

regional, and local governments, and as well as the off-budget funds. Additionally,

it includes the nonprofit organizations servicing the households.


Now we’ll move on to the formulas determining the behavior of economic agent

no. 18.

The consolidated budget is given by

VB 18 t½ � ¼ SUM VO tc i t½ � � VB i t½ �ð Þ þ VO tc 18 t½ � � VB 18 t½ �þ CB other 18 t½ � þ VB b 18 t½ � � 1þ CP bpercent t� 1½ �ð Þ:

(4.58)

This formula sums up money collected as taxes from the producing agents, as

well as from the inhabitants. The value CB_other_18 entering the model exoge-

nously is the sum of other taxes (not included in the list of taxes considered in

the model), nontaxable income, and other income of the consolidated budget. The

obtained sum is incremented by the funds in banking accounts (subject to the

deposit interests).

The dynamics of the banking account balance of the consolidated budget is

determined by

VB b 18 tþ 1½ � ¼ VO s 18 t½ � � VB 18 t½ �: (4.59)

The cash assets of off-budget funds are as follows:

VF 18 t½ � ¼ SUM VO tf i t½ � � VB i t½ �ð Þ þ VF b 18 t½ �� 1þ CP bpercent t� 1½ �ð Þ: (4.60)

This formula calculates the sum collected from the producing agents in the form

of the single social tax entering the accounts of the following off-budget funds: the

pension fund, the social insurance fund, and the federal and territorial funds of

obligatory medical insurance. The derived sum is added by the funds on the banking

accounts (subject to the deposit interests).

The dynamics of the banking account balance of the off-budget funds is deter-

mined by

VF b 18 tþ 1½ � ¼ VO s 18f t½ � � VF 18 t½ �: (4.61)

The subsidies to the producing sectors are as follows:

VG s i18 t½ � ¼ CO s i18� VB 18 t½ �: (4.62)

The social transfers to the inhabitants are

VG tr 18 t½ � ¼ CO tr 18� VB 18 t½ �: (4.63)


The assets of the off-budget funds made available for the inhabitants are

VG f 18 t½ � ¼ CO f 18� VF 18 t½ �: (4.64)

This includes the assets of the pension fund and social insurance fund for paying

pensions and welfare payments.

4.1.1.8 Integral Indexes of the Model

Let’s present the formulas for calculating some integral indexes of the economy of

Kazakhstan.

The gross production of goods and services (in prices of the base period) is

VY t½ � ¼ SUM VY i t½ �ð Þ: (4.65)

The total supply of the intermediate products (in prices of the base period) is

VS z t½ � ¼ SUM VS pi iz t½ �ð Þ: (4.66)

The total supply of final products (in prices of the base period) is

VS c t½ � ¼ SUM VS p ic t½ �ð Þ: (4.67)

The GDP of Kazakhstan is

VY g t½ � ¼ SUM VY g i t½ �ð Þ=VP c 0½ �: (4.68)

The consumer price index is

VP t½ � ¼ 100� VP c t½ �=VP c t� 1½ �ð Þ: (4.69)

The considered model is presented in the context of the common relations:

Relations (4.1) are presented by m1 ¼ 67 expressions.



4.1.1.9 Parametric Identification of the Model and Retrospective Prediction

The problem of parametric identification of the considered macroeconomic mathe-

matical model requires estimation of unknown parameters that minimize the value

of the objective function that characterizes the deviations of values of the output

model’s variables from the corresponding measured values (known statistical data).

This problem reduces to finding the minimal value of the function in several


variables (parameters) in some close region Ω of the Euclidean space under

constraints ex 2 eX imposed onto values of the model’s endogenous variables. In

the case of large dimensions of a region of possible parameter values, the standard

method for finding the extremum of a function is often inefficient due to the

objective function’s several local minimum points. Proposed here is an algorithm

taking into account peculiarities of the problem of parametric identification of

macroeconomic models that allows us to avoid “local extrema”.

The regionO ¼ Qqþsþm1

i¼1

½ai; bi�where ½ai; bi� is the interval of possible values of theparameteroi; i ¼ 1; . . . ; ðqþ sþ m1Þ, is considered the regionO � U � ^ � X1 for

estimating possible values of the exogenous parameters. The estimate of parameters

with their respective measured values available was searched within the intervals

½ai; bi�with centers in the respective measured values (in case of one such value) or

within some intervals covering the measured values (in case of several values).

Other intervals ½ai; bi� for searching the parameters were chosen by indirect

estimation of their possible values. For searching the minimal values of the contin-

uous function in severable variables F : O ! R with the additional constraints on

the endogenous variables, we applied the Nelder–Mead directed search algorithm.

Using this algorithm for an initial point, o1 can be interpreted as a sequence

o1;o2;o3; . . .f g [converging to the local minimum o0 ¼ arg minO;ex2ex F oð Þ offunction F], sequence {o1, o2, o3,…} where F ojþ1

� � � F oj

� �;oj 2 O; j ¼ 1; 2;

. . . . While describing the next algorithm, we consider that the point o0 can be

found with sufficient precision.

For parametric identification of the considered computable model based on the

obvious assumption about disagreement (in general case) of the minimum points of

two different functions, the following two criteria are proposed:

KAðoÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

naðt2 � t1 þ 1ÞXt2t¼t1

XnAi¼1

aiyit � yi�tyi�t

� �2

vuut ;

KBðoÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

nbðt2 � t1 þ 1ÞXt2t¼t1

XnBi¼1

biyit � yi�tyi�t

� �2

vuut :

(4.70)

Here ft1; . . . ; t2g is the identification time interval; yit; yi�t are computed and

measured values of the model output variables; KAðoÞ is the auxiliary criterion;

KBðoÞ is the main criterion; nB > nA ; ai > 0 and bi > 0 are some weighting

coefficients whose values are calculated during parametric identification of the

dynamic system; andPnA

i¼1 ai ¼ na;PnB

i¼1 bi ¼ nb:The algorithm for the parametric identification problem is as follows:

1. For some vector of initial values of parameter o1 2 O, solve problems A and Bsimultaneously. Then find the minimum pointsoA0 andoB0 of criteriaKA andKB,

respectively.

2. If KB oB0ð Þ<e for some sufficiently small number e, then the model parametric

identification problem is solved.


3. Otherwise, choose the point oB0 as the initial point o1, solve problem A, and,choosing the point oA0 as the initial point o1, solve problem B. Go to step 2.

After a sufficiently large number of iterations of stages 1, 2, and 3, searched

values of the parameters might leave neighborhoods of the nonglobal minima in one

criterion with the help of the other and thereby solve the parametric identification

problem.

As a result of a joint solution of problems A and B using the Nelder–Mead

algorithm, we obtain values KA ¼ 0:015 and KB ¼ 0:0063. The relative values ofdeviations in the computed variables used in the main criterion turn out to be less

than 0.63%.

The results of computation and retrospective prediction of the model for 2008

presented in Table 4.3 show computed VY;VY�g;VP� �

; measured values, and

deviations of the computed values of the main output variables from their respec-

tive measured values. Year 2000–2007 corresponds to the period of model

parametric identification; year 2008 is the retrospective prediction period; VY�gis the gross output ( �1012 tenge in prices of year 2000); VY�g is GDP ( �1012 in

prices of year 2000); VP is the consumer price index in percentage with respect to

the preceding year; the symbol «*» H= «D» corresponds to the measured values; the

symbol «*» corresponds to the deviations (in percentage) of the computed values

from their respective measured values.

The mean square error of the retrospective prediction of all 491 endogenous

variables of the model for the years 2008–2009 is 5.86%.

The results of verification show acceptable adequacy of the CGE model of

economic branches.

4.1.2 Macroeconomic Analysis on the Basis of the ComputableModel of General Equilibrium of Economic Branches

Rational implementation of the discretionary public policy determines carrying out

macroeconomic analysis on the basis of available instruments.

Table 4.3 Measured and computed values of the model output variables and respective deviations

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008

VY� 5.44 6.32 6.47 6.86 7.72 8.52 9.25 9.69 9.84

VY 5.38 6.32 6.47 6.86 7.72 8.52 9.27 9.64 9.82

DVY �1.22 �0.02 0.00 0.00 0.05 0.08 0.21 �0.51 �0.26

VY g� 2.45 2.78 3.05 3.36 3.72 4.09 4.55 5.01 5.18

VY g 2.47 2.78 3.05 3.35 3.72 4.09 4.55 5.01 5.20

DVY g 0.88 0.07 �0.04 �0.02 �0.02 �0.02 �0.04 �0.15 0.38

VP� 106.40 106.60 106.80 106.70 107.50 108.40 118.80 109.50

VP 107.60 106.80 106.90 106.70 107.30 108.20 118.60 109.40

DVP 1.13 0.18 0.08 �0.05 �0.23 �0.22 �0.24 �0.05


In this section we present some results of macroeconomic analysis on the basis

of the computable model of general equilibrium of economic branches after esti-

mation of its parameters using statistical data from the Republic of Kazakhstan for

the years 2000–2009.

We also analyzed the dynamics (and trends) of the exogenous and endogenous

indicators, the rates (and their trends) of the endogenous indicators of economic

agents, and the entire national economy from 2000 to 2009 and middle-term

prediction (perspective) from 2010 to 2015.

The list of the exogenous and endogenous indicators of the economic agents and

national economy forming the basis for retrospective and perspective macroeco-

nomic analysis is given in Tables 4.1 and 4.2.

4.1.2.1 Retrospective and Perspective Analysis of the Economic Agents’Indicators and National Economy on the Basis of the ComputableModel of General Equilibrium of Economic Branches

Below we present several results of the retrospective analysis at the level of the

branches of the national economy.

Results of the Retrospective Analysis of the Exogenous Variables

at the Level of Economic Branches

The retrospective analysis of the exogenous indicators by economic branches

for 2000–2009 shows that within this period

– The number employed in all branches except branch 16 (housekeeping services)

increased on average. The maximum average growth (computed by linear trend)

is observed for branch 6 (construction) and is equal to 4:2� 104 people per year.

The negative growth (decrease by 2:3� 103 people per year on average) was

observed in branch 16 (housekeeping services).

– The trend of the indicator “Share of intermediate products in branch’s output”

had a positive slope for branches 8, 11, 10, 2, 4, 1, and a negative slope for

branches 16, 9, 13, 14, 5, 12, 6, 7, 15, 3. The maximum growth of a trend that is

4.4% per year is observed in branch 8 (hotels and restaurants). The maximum

decrease by 3.4% per year is observed in branch 3 (mineral resource industry).

– The trend of the indicator “Share of final products in branch’s output” had a

positive slope for branches 5, 12, 16, 1, 4, 14, 11, 13 and a negative slope for

branches 3, 6, 15, 10, 9, 7, 2, 8. The maximum growth of a trend by 4.9% per

year is observed in branch 5 (production and distribution of electric power, gas,

and water). The maximum decrease by 3.8% per year is observed in branch

8 (hotels and restaurants).

– The trend of the indicator “Share of export in branch’s output” had a positive

slope for branches 7, 3, 4, 9, 15, 10, 2, 5 and a negative slope for branches 6, 13,


8, 11, 12, 1. In branches 14 and 16, the export component is approximately zero.

The maximum growth of trend by 3.6% per year is observed in branch 7 (trade,

repair services for cars, and goods for household use). The maximum decrease

by 1.8% per year is observed in branch 1.

– The trend of the indicator “Price of export supplies of branch’s products” had a

positive slope for all in branch 16. The maximum growth of the trend by 62% per

year is observed in branch 5 (production and distribution of electric power, gas,

and water). The minimum increase by 5.9% per year is observed in branch 13

(education).

– The trend of the indicator “Share of final products in branch’s output” had a

positive slope for branches 15, 13, 7, 4, 6, 5, 9, 18, 2 and a negative slope for

branches 14, 10, 11, 3, 16. The maximum growth of the trend by 1.9% per year is

observed in branch 15 (other public, social, and personal services). The maxi-

mum decrease by 1.4% per year is observed in branch 16 (housekeeping

services).

– The trend of the indicator “Share of budget of branch spent for purchasing

investment products” had a positive slope for branches 12, 6, 11, 5, 4, 1, 9, 14,

2, 7, 10 and a negative slope for branches 13, 15, 8, 3. This share was zero for

branch 16. The maximum growth of the trend by 10% per year is observed in

branch 12 (public administration). The maximum decrease by 2% per year is

observed in branch 3 (mineral resource industry).

– The indicator “Share of investment product in branch’s output” was nonzero in

8 branches out of 16. In branches 6, 7, 2, 1, 9 (hereinafter the branches are

ordered by descending trend angular coefficient), this indicator had a positive

slope of linear trend (the maximum value 2.9% per year was observed in branch

6, construction). This share was equal to 78% for branch 6 in 2008. Branches 15,

4, and 11 demonstrated negative slopes of linear trends. The maximum decrease

of the trend is observed in branch 11 (real estate activities, rent, and services for

business) and constitutes close to 0.94% per year.

In Fig. 4.1 we give a graphic illustration of the dynamics (and trend) of one of the

economic agents’ indicators for retrospective macroanalysis by the example of the

variable “Share of investment products in the output of economic agent 1 (Agricul-

ture, hunting, and forestry).”

The analysis of Fig. 4.1 shows a positive slope of linear trend of the considered

indicator, thereby demonstrating growth (on average) of the share of investment

products in the output of economic agent 1 (Agriculture, hunting, and forestry)

within the considered period.

In the context of the retrospective analysis of the indicators of economic

branches on the basis of the considered model, we also obtained the measured

and computed values (and linear trends) of such exogenous indicators for each of

the16 economic branches as CO pj iz (share of budget of the ith branch spent for

purchasing the intermediate products produced by the branch j; i; j ¼ 1; . . . ; 16).Particularly, retrospective analysis of the mentioned indicators of the branches for

2000–2009 shows


– The trend of the indicator CO pj 1z (share of budget of branch 1 that is

agriculture, hunting, and forestry) spent for purchasing intermediate products

produced by the jth branch had a positive slope for j ¼ 1, 8, 2, 13 and a negative

slope for j ¼ 11 , 3, 15, 14, 10, 6, 12, 5, 7, and 9. Branch 1 did not buy

intermediate goods produced by economic branch 16. The maximum growth

trend of 1.7% per year is observed for j ¼ 1. The maximum decrease of 0.63%

per year is observed for j ¼ 3 (mineral resource industry). The maximum value

of the indicator CO pj 1z of 36% was noted for j ¼ 1.

– The trend of the indicator CO pj 2z (share of budget of branch 2 that is fishery

and fish breeding) spent for purchasing intermediate products produced by the

jth branch had a positive slope for j ¼ 9, 11, 5, 10, 3, 6, 8 and a negative slope for

j ¼ 14, 15, 1, 12, 4, 7, and 2. Branch 2 did not buy intermediate goods produced

by economic branches 13 and 16. The maximum growth of the trend by 1.6% per

year is observed for j ¼ 9. The maximum decrease by 3.2% per year is observed

for j ¼ 2 (mineral resource industry). The maximum value of the indicator COpj 2z equal to 18% was noted for j ¼ 4 (manufacturing and process industry) in

2008.

– The trend of the indicator CO pj 3z (share of budget of branch 3 that is mineral

resource industry) spent for purchasing the intermediate products produced by

the jth branch had a positive slope for j ¼ 11, 10, 7, 12, 1, 2, 16 and a negative

slope for j ¼ 8, 15, 13, 4, 14, 6, 5, 3, and 9. Branch 3 did not buy intermediate

goods produced by economic branch 16. The maximum growth of the trend by

1.3% per year is observed for j ¼ 11 (real estate activities, rent, and services for

business). The maximum decrease by 1.1% per year is observed for j ¼ 9

(transport and communications). The maximum value of the indicator CO pj 3z equal to 19% was observed for j ¼ 11 (real estate activities, rent, and services

for business) in 2008.

Linear trend of parameter

Measured parameter values Estimated parameter values

Fig. 4.1 Measured and estimated values (and linear trend) of exogenous variable “Share of

investment products in the output of economic agent 1”



manufacturing and process industry) spent for purchasing intermediate products

produced by the jth branch had a positive slope for j ¼ 4, 3, 11, 1, 10, 6, 8, 2 and a

negative slope for j ¼ 14 , 12, 13, 15, 5, 9, and 7. Branch 4 did not buy

intermediate goods produced by economic branch 16. The maximum growth

of the trend by 0.46% per year is observed for j¼ 4. The maximum decrease by

0.46% per year is observed for j ¼ 7 (trade, repair services for cars, and goods for

household use). The maximum value of the indicatorCO pj 4z equal to 25% was

noted for j ¼ 4 (manufacturing and process industry) in 2008.

– The trend of the indicatorCO pj 5z (share of budget of branch 5 that is productionand distribution of electric power, gas, and water) spent for purchasing the

intermediate products produced by the jth branch had a positive slope for j ¼ 4,

5, 11, 3, 10, 8, 14, 1, 12, 2, 16 and a negative slope for j ¼ 15, 13, 6, 9, and 7.

The maximum growth of the trend by 2.0% per year is observed for j ¼ 4

(manufacturing and process industry). The maximum decrease by 1.0% per

year is observed for j ¼ 7 (trade, repair services for cars, and goods for household

use). The maximum value of the indicator CO pj 5z equal to 19% was noted for

j ¼ 4 (manufacturing and process industry) in 2008.


construction) spent for purchasing the intermediate products produced by the

jth branch had a positive slope for j ¼ 4, 6, 10, 8, 2, 14 and a negative slope for

j ¼ 13, 12, 1, 15, 11, 5, 7, 9, and 3. Branch 6 did not buy intermediate goods

produced by economic branch 16. The maximum growth of the trend by 2.3%

per year is observed for j ¼ 4 (manufacturing and process industry). The

maximum decrease by 0.79% per year is observed for j ¼ 3 (mineral resource

industry). The maximum value of the indicatorCO pj 6z equal to 37% was noted

for j ¼ 4 (manufacturing and process industry) in 2008.

– The trend of the indicator CO pj 7z (share of budget of branch 7 that is trade,

repair services for cars, and goods for household use) spent for purchasing

the intermediate products produced by the jth branch had a positive slope for

j ¼ 11, 3, 10, 8, 13 and a negative slope for j ¼ 2, 14, 12, 15, 1, 5, 6, 9, 4, and 7.

Branch 7 did not buy the intermediate goods produced by economic branch 16.



year is observed for j ¼ 7 (real estate activities, rent, and services for business).

Themaximumvalue of the indicatorCO pj 7zequal to 17%was noted for j ¼ 11

(real estate activities, rent, and services for business) in 2008.

– The trend of the indicator CO_pj_8z (share of budget of branch 8 that is hotels

and restaurants) spent for purchasing the intermediate products produced by the

jth branch had a positive slope for j ¼ 1, 9, 10, 6 and a negative slope for j ¼ 3,

14, 13, 12, 4, 2, 15, 7, 5, 8, and 11. Branch 8 did not buy intermediate goods


per year is observed for j ¼ 1 (agriculture, hunting, and forestry). The maximum

decrease by 0.62% per year is observed for j ¼ 11. The maximum value of the


indicator CO_pj_8z equal to 20% was noted for j ¼ 4 (manufacturing and

process industry) in 2008.

– The trend of the indicatorCO pj 9z (share of budget of branch 9 that is transportand communications) spent for purchasing the intermediate products produced

by the jth branch had a positive slope for j ¼ 9, 11, 10, 8, 2 and a negative slope

for j ¼ 14, 13, 15, 12, 1, 3, 6, 5, 4, and 7. Branch 8 did not buy intermediate

goods produced by economic branch 16. The maximum growth of the trend by

0.69% per year is observed for j ¼ 9. The maximum decrease by 1.0% per year

is observed for j ¼ 7 (trade, repair services for cars, and goods for household

use). The maximum value of the indicator CO pj 9z equal to 15% was noted for


– The trend of the indicator CO_pj_10z (share of budget of branch 10 that is

financial activity) spent for purchasing the intermediate products produced by

the jth branch had a positive slope for j ¼ 4, 9, 11, 8, 1, 6, 5, 15, 13, 2 and a

negative slope for branches j ¼ 16, 14, 12, 3, 7, and 10. The maximum growth of

the trend by 0.15% per year is observed for j ¼ 4 (manufacturing and process

industry). The maximum decrease by 0.98% per year is observed for j ¼ 10. The

maximum value of the indicator CO pj 10z equal to 16% was noted for j ¼ 4

(manufacturing and process industry) in 2008.

– The trend of the indicator CO pj 11z (share of budget of branch 11 that is real

estate activities, rent, and services for business) spent for purchasing the inter-

mediate products produced by the jth branch had a positive slope for j ¼ 4, 7, 9,5, 13, 14, 8, 2 and a negative slope for j ¼ 16, 12, 1, 15, 3, 6, and 11. The

maximum growth of the trend by 1.6% per year is observed for j ¼ 4


year is observed for j ¼ 11. The maximum value of the indicator CO pj 11zequal to 25% was noted for j ¼ 11 in 2008.

– The trend of the indicator CO pj 12z (share of budget of branch 12 that is real

estate activities, rent, and services for businesses) spent for purchasing the

intermediate products produced by the jth branch had a positive slope for

j ¼ 8, 10, 6, 11, 15, 1, 14, 13 and a negative slope for j ¼ 2, 9, 3, 5, 4, 7, and

12. Branch 12 did not buy intermediate goods produced by economic branch 16.


(hotels and restaurants). The maximum decrease by 0.76% per year is observed

for j ¼ 11. The maximum value of the indicator CO pj 12z equal to 18% was

noted for j ¼ 4 (manufacturing and process industry) in 2008.


education) spent for purchasing the intermediate products produced by the jthbranch had a positive slope for j ¼ 4, 6, 9, 15, 1, 8 and a negative slope for j ¼ 2,

10, 3, 12, 14, 7, 13, 5, and 11. Branch 13 did not buy intermediate goods


per year is observed for j ¼ 4 (manufacturing and process industry). The

maximum decrease by 0.66% per year is observed for j ¼ 11 (real estate

activities, rent, and services for business). The maximum value of the indicator


CO pj 13z equal to 24% was noted for j ¼ 4 (manufacturing and process

industry) in 2008.

– The trend of the indicator CO pj 14z (share of budget of branch 14 that is publichealth and social services) spent for purchasing the intermediate products pro-

duced by the jth branch had a positive slope for j ¼ 4, 6, 1, 7, 8 and a negative

slope for j ¼ 2, 3, 12, 11, 10, 15, 9, 13, 5, and 14. Branch 14 did not buy

intermediate goods produced by economic branch 16. The maximum growth of

the trend by 3.2% per year is observed for j ¼ 4 (manufacturing and process

industry). The maximum decrease by 1.1% per year is observed for j ¼ 14. The

maximum value of the indicator CO pj 14z equal to 37% was noted for j ¼ 4

(manufacturing and process industry) in 2008.

– The trend of the indicator CO pj 15z (share of budget of branch 15 that is other

public utilities, social and personal services) spent for purchasing the intermedi-

ate products produced by the jth branch had a positive slope for j ¼ 4, 8, 10, 5, 7,2, 16 and a negative slope for branches j ¼ 6, 1, 3, 13, 12, 9, 14, 15, and 11.

Branch 15 did not buy intermediate goods produced by economic branch 16. The

maximum growth of the trend by 0.22% per year is observed for j ¼ 4


year is observed for j ¼ 11 (real estate activities, rent, and services for business).

The maximum value of the indicator CO pj 15z equal to 21% was noted for



housekeeping services) spent for purchasing the intermediate products produced

by the jth branch had a positive slope for branches j ¼ 11, 4 and a negative slope

for j ¼ 1, 14, 16, 8, 15, 6, 9, and 13. Branch 16 did not buy intermediate goods

produced by economic branches 2, 3, 7, 10, 12. The maximum growth of the

trend by 1.2% per year is observed for j ¼ 11 (real estate activities, rent, and

services for business). The maximum decrease by 0.41% per year is observed for

j ¼ 13 (education). Branch 16 did not buy the intermediate goods in 2008.

In Fig. 4.2 we give a graphic illustration of the initial data for retrospective

macroanalysis at the level of economic branches within the framework of the

exogenous variable “Share of budget of branch 1 (agriculture, hunting, and forestry)

spent for purchasing the intermediate products produced by branch 4 (manu-

facturing and process industry).” The negative slope of the linear trend in Fig. 4.2

is evidence of a decrease (on average) of the mentioned share of expenses within the

considered time interval.

Also, we carried out the retrospective analysis of the endogenous variables at the

level of economic branches. In Figs. 4.3 and 4.4, we give a graphic illustration of

the dynamics, its rates, and trends of one of the economic agents’ indicators for

retrospective macroanalysis presented by the example of the indicator “Gross

values added in prices of year 2000” of the branch manufacturing and process

industry.

Note that the computed and measured values of the variable at Fig. 4.2 actually

coincide with each other. It is explained by the quality of parametric identification


of the considered model. An analysis of data presented in Figs. 4.3 and 4.4

(a positive slope of linear trends of both variable “Gross value-added of branch

4” and its rate) shows accelerated growth of the mentioned indicator within the

interval 2000–2009.

Retrospective Comparative Analysis

Within the framework of retrospective comparative analysis of economic branch

indicators for each year from 2000 to 2008, we also carried out the ranking of

branches with respect to the following indicators:

Linear trend of parameter

Measured parameter values

Estimated parameter values

Fig. 4.2 Exogenous variable “Share of budget of branch 1 (agriculture, hunting, and forestry)

spent for purchasing the intermediate products produced by branch 4 (manufacturing and process

industry)”, its trend, and measured values

Linear trend of variable

Measured variable values Computed variable values

Fig. 4.3 Gross value added of branch 4 (in prices of year 2000), its trend, and measured values


– Gross value-added (GVA) of a branch (in prices of year 2000);

– Fixed assets of a branch (in prices of year 2000);

– Capital productivity of a branch;

– Demand of labor force in a branch;

– Labor productivity in a branch.

An analysis of the results of ranking with respect to the mentioned indicators

allows us to conclude the following:

– The maximum GVA over all economic branches is observed within the branch

in 2000–2002, in branch 11 (real estate activities, rent, and services for business)

in 2003–2004, and in branch 3 (mineral resource industry) in 2005–2008.

– The maximum fixed assets were discovered in branch 3 (mineral resource

industry) for the entire period of observations.

– The maximum capital productivity was observed in branch 7 (trade, repair

services for cars, and goods for household use) in 2000–2005, and in branch

10 (financial activity) in 2006–2008.

– The maximum number of employees (about 30% of all employees) was noted in

branch 1 (agriculture, hunting, and forestry) for all periods of observations.

– Branch 1 demonstrated the minimum labor productivity in 2000–2008. The

maximum labor productivity was demonstrated by branch 3 (mineral resource

industry) for practically the entire considered period, with a growing gap with

respect to other branches.

The results of the Kazakhstan Republic’s economic branches’ ranking within

the framework of the given indicators in year 2008 are presented in Tables 4.4, 4.5,

4.6, 4.7, and 4.8.

Linear trend of variable rateVariable rate values

Fig. 4.4 Rate of the variable “Gross value added of branch 4 (in constant prices)” and its trend


Results of the Retrospective Analysis of the Main Integral Indicators of a Country’s

Development, Indicators of Public Administration and Households

Also, we present the results of retrospective analysis of the mentioned indicators for

2000–2009.

Table 4.4 Gross value-added

of branch

No. No. of branch

Value of indicator GVA

in year 2008 (tenge,

in prices of 2000)

1 3 9.92923 � 1011

2 11 7.94163 � 1011

3 7 6.49315 � 1011

4 4 6.24578 � 1011

5 9 5.83432 � 1011

6 6 4.29188 � 1011

7 10 2.79863 � 1011

8 1 2.78125 � 1011

9 13 1.49005 � 1011

10 12 9.00073 � 1010

11 5 8.88712 � 1010

12 15 8.88331 � 1010

13 14 7.84887 � 1010

14 8 4.36384 � 1010

15 16 4.40276 � 109

16 2 4.14457 � 109

Table 4.5 Fixed assets

of branch No. No. of branch

Value of indicator “Fixed assets of a branch”

in year 2008 (tenge, in prices of 2000)

1 3 1.55078 � 1012

2 11 1.15043 � 1012

3 4 6.43257 � 1011

4 9 6.41518 � 1011

5 12 3.21504 � 1011

6 6 2.94698 � 1011

7 5 2.75815 � 1011

8 1 2.33045 � 1011

9 7 2.21753 � 1011

10 13 9.67223 � 1010

11 14 8.86825 � 1010

12 10 7.66333 � 1010

13 15 5.98872 � 1010

14 8 3.36888 � 1010

15 2 1.44711 � 109

16 16 0


In Figs. 4.5 and 4.6, we present an example of initial data for the retrospective

macroanalysis of the national economic indicators in the context of the endogenous

variable “Volume of production of investment products.”

Analysis of the trends presented in Figs. 4.6 and 4.7 allows us to conclude that

from 2000 to 2009, one can observe an average growth of the real output of

investment products in a country with some average deceleration of its growth.

Table 4.6 Capital

productivity of branch No. No. of branch

Value of indicator “Capital productivity

of a branch” in year 2008

1 10 3.651970

2 7 2.928106

3 2 2.864025

4 13 1.540538

5 15 1.483340

6 6 1.456368

7 8 1.295338

8 1 1.193439

9 4 0.970963

10 9 0.909455

11 14 0.885053

12 11 0.690321

13 3 0.640272

14 5 0.322213

15 12 0.279957

16 16 0.000000

Table 4.7 Branch demand

for labor force No. No. of branch

Value of indicator “Branch demand

for labor force” in year 2008

1 1 2,349,730

2 7 1,150,307

3 13 754,299

4 9 588,847

5 4 572,886

6 6 548,895

7 11 378,129

8 12 352,489

9 14 347,329

10 15 205,425

11 3 200,283

12 5 164,816

13 8 103,048

14 10 96,165

15 16 24,360

16 2 19,971


Results of Prospective Analysis of the Endogenous Variables

at the Level of Economic Branches

The perspective analysis of the endogenous variables of the Kazakhstan Republic

was carried out on the basis of a computation of economic branches model until

year 2015 with use of extrapolation of the measured values of the exogenous

variables of the model until 2015.

Table 4.8 Labor

productivity in branch No. No. of branch

Value of indicator “Labor productivity

in branch” in year 2008 (in prices of 2000)

1 3 4,957,609.0

2 10 2,910,234.0

3 11 2,100,238.0

4 4 1,090,231.0

5 9 990,804.0

6 6 781,913.2

7 7 564,471.0

8 5 539,213.9

9 15 432,435.0

10 8 423,477.3

11 12 255,348.0

12 14 225,978.1

13 2 207,533.5

14 13 197,540.5

15 16 180,739.0

16 1 118,364.8

Linear trend of variableVariable values

Fig. 4.5 Endogenous variable “Volume of production of investment products” and its trend


The perspective analysis of the endogenous indicators for 2000–2015 by the

branches shows that

The trends of the indicator “Budget of a branch’s enterprises (in prices of year

2000)” increase for all branches. The maximum growth of 7.37 � 1010 tenge per

year is observed in branch 10, whereas the minimum growth of 3.3 � 108 tenge per

year is observed in branch 14. The trends of the rates of the considered indicator

have negative slopes for almost all branches except branches 12 and 4. This shows

that all branches except 12 and 4 are expected to demonstrate a drop in the growth

Linear trend of variable rate

Variable rate values

Fig. 4.6 Rate of endogenous variable “Volume of production of investment products” and its

trend

Linear trend of variable

Computed values of variableMeasured variable values

Fig. 4.7 Gross value added by branch 4 (in prices of year 2000), its trend, and measured values


rate of budgets (maximum drop by 3% per year by trend was observed in

branch 14).

The trends of the indicator “Gross value-added by branch (in prices of year

2000)” increased for almost all branches (except branches 12 and 14). The maxi-

mum growth of 7.4 � 1010 tenge per year is observed in branch 4, whereas the

maximum drop by 2.2 � 109 tenge per year is observed in branch 14. The trends of

the rates of the considered indicator have negative slopes for almost all branches

except 5, 15, 13, 2, 12, and 14. This shows that all branches except 5, 15, 13, 2, 12,

and 14 are expected to demonstrate a drop in the growth rate of the GVA (maximum

drop by 18% per year by trend was observed in branch 16; maximum growth of

1.6% was observed in branch 14).

The trends of the indicator “Output of branch (in prices of year 2000)” increased

for almost all branches (except branches 16, 2, 3, 14, 11, and 9). The maximum

growth of 1.6 � 1011 tenge per year is observed in branch 6, whereas the maximum

drop of 3.4 � 1010 tenge per year is observed in branch 9. The trends of the rates of

the considered indicator have negative slopes for almost all branches except 4, 5,

12, and 15. This shows that all branches except 4, 5, 12, and 15 are expected to

demonstrate a drop in the growth rate of the output of the branch (a maximum drop

by 19% per year by trend was observed in branch 16; a maximum growth of 1.1%

was observed in branch 4).

The trends of the indicator “Fixed assets of branch (in prices of year 2000)”

increased for almost all branches (except branches 2, 8, 14, 1, and 5). The maximum

growth of 5.5 � 1010 tenge per year is observed in branch 6, whereas the maximum

drop by 6.2 � 109 tenge per year is observed in branch 9. The trends of the rates of

the considered indicator have negative slopes for almost all branches except 10, 2,

8, 14, 1, and 5. This shows that all branches except 10, 2, 8, 14, 1, and 5 are expected

to demonstrate a drop in the growth rate of the fixed assets (a maximum drop by 1%

per year by trend was observed in branch 12; a maximum growth of 0.58% was

observed in branch 10).

The trends of the indicator “Supply of investment products of branch (in prices

of year 2000)” increased for branches 6, 4, 7, 1, 9, 15, and 2. The maximum growth

of 1.1 � 1011 tenge per year is observed in branch 6, whereas the maximum drop of

8.8 � 108 tenge per year is observed in branch 11. The trends of the rates of the

considered indicator have negative slopes for almost all branches or almost zero

slopes for branches 6, 4, 7, 1, 9, 15, and 2. This shows that all these branches except

branch 11 are expected to demonstrate a drop in the growth rate of investment

products. Investment products are not produced for all other branches that are not

mentioned in this paragraph.

The trends of the indicator “Supply of final products of branch (in prices of year

2000)” increased for almost all branches (except branches 11, 2, 14, and 9). The

maximum growth of 3.1 � 1010 tenge per year is observed in branch 4, whereas the

maximum drop of 6.9 � 109 tenge per year is observed in branch 9. The trends of

the rates of the considered indicator have negative slopes for almost all branches

except 15 and 7. This shows that all branches except 15 and 7 are expected to

demonstrate a drop in the growth rate of the supply of final products (the maximum


drop of 15% per year by trend was observed in branch 12; the maximum growth of

0.28% was observed in branch 15).

The trends of the indicator “Supply of intermediate products of branch (in prices

of year 2000)” increased for almost all branches (except branches 16, 2, 14, 3, and 9).

The maximum growth of 6.6 � 1010 tenge per year is observed in branch 4, whereas

the maximum drop of 1.0 � 1010 tenge per year is observed in branch 9. The trends

of the rates of the considered indicator have negative slopes for almost all branches

except 5 and 15. This shows that all branches except 5 and 15 are expected to

demonstrate a drop in the growth rate of the supply of intermediate products (the

maximum drop, with considerable jumps, was observed in branch 16; the maximum

growth of 2.9% was observed in branch 5).

The trends of the indicator “Supply of export products of branch (in prices of

year 2000)” increased for almost all branches (except branches 2, 13, 11, 1, and 9).

The maximum growth of 3.0 � 1010 tenge per year is observed in branch 4,

whereas the maximum drop of 1.6 � 109 tenge per year is observed in branch 9.

The trends of the rates of the considered indicator have negative slopes for almost

all branches except 12, 8, 13, and 1. This shows that all branches except 12, 8, 13,

and 1 are expected to demonstrate a drop in the growth rate of the supply of export

products (the maximum drop of 22% per year by trend was observed in branch 15;

the maximum growth of 3.3% was observed in branch 13). The export products are

not produced by branches 14 and 16.

The trends of the indicator “Profit of branch (in prices of year 2000)” increased

for almost all branches (except branches 16, 14, 12, and 13). The maximum growth

of 6.6 � 1010 tenge per year is observed in branch 4, whereas the maximum drop of

9.7 � 109 tenge per year is observed in branch 13. The trends of the rates of the

considered indicator have negative slopes for almost all branches except 4, 5, 1, 15,

2, and 14. This shows that all branches except 4, 5, 1, 15, 2, and 14 are expected to

demonstrate a drop in the growth rate of the profit of branch (the maximum drop of

15% per year by trend was observed in branch 12; the maximum growth of 4.0%

was observed in branch 15).

In Figs. 4.7 and 4.8, we present an example of the initial data for the retrospec-

tive macroanalysis at the level of economic branches on the basis of the endogenous

variables and in the context of the endogenous variable “Gross value-added (in

prices of year 2000) of economic agent 4 (manufacturing and process industry),” its

rate, and their trends.

Analysis of information presented in Figs. 4.7 and 4.8 and comparison of those

figures with Figs. 4.2 and 4.3 show that in perspective up until year 2015, one can

expect a growth (on average) in GVA of branch 4 with an increase (on average) in

the rate of that growth. That is, the tendencies similar to those of Figs. 4.2 and 4.3

noted before remain the same (according to the base prediction) for 2010–2015.

Nevertheless, the angular coefficient of the trend in Fig. 4.7 is somewhat higher

than the respective angular coefficient in Fig. 4.2. This is evidence to show an

increase (on average) in the rate of that growth in the perspective period. This is

also seen by a greater (on average) value of the trend in Fig. 4.8 in comparison with

its values for the retrospective period (Fig. 4.3).


Within the framework of perspective analysis of economic branch indicators on

the basis of the considered model, we also obtained the computed prediction values

(and their linear trend) of the indicators “Demand for intermediate products of jthbranch i (in constant prices of year 2000)” (i, j ¼ 1, . . ., 16) for each of the 16

economic branches.

The perspective analysis of the mentioned indicators of branches for 2000–2015

shows that

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 1 (agriculture, hunting, and forestry) (in constant prices of year 2000)”

had a positive slope for branches j ¼ 4, 5, 16, 10, 7, 12, 8, 16 and a negative

slope for branches j ¼ 13, 2, 15, 14, 3, 11, and 9. Branch 1 does not purchase

intermediate products of economic branch 16. The maximum increase of the

trend by 2.3 � 109 tenge per year is observed for j ¼ 4, whereas the maximum

decrease by 2.3 � 109 tenge per year is observed for j ¼ 9. The trend of the rate

of the considered indicator increases only for products of branch 5 and decreases

for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 2 (fishery and fish breeding) (in constant prices of year 2000)” had a

positive slope for branches j ¼ 4, 5, 9, 10, 11, 12, 6, 3, 7, and 8 and a negative

slope for branches j ¼ 15, 14, 1, and 2. Branch 2 did not purchase intermediate

products from economic branches 13 and 16. The maximum increase of the trend

by 5.1 � 107 tenge per year is observed for j ¼ 4, whereas the maximum

decrease by 7.7 � 107 tenge per year is observed for j ¼ 2. The trend of the

rate of the considered indicator increases only for products of branches 12, 6, and

8 and decreases for all other branches.

Linear trend of variable rate

Variable rate values

Fig. 4.8 Rate of variable “Gross value added by branch 4 (in fixed prices)” and its trend


– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 3 (mineral resource industry) (in constant prices of year 2000)” had a

positive slope for branches j ¼ 5, 4, 10, 7, 6, 8, 12, 15, and 11 and a negative

slope for branches j ¼ 2, 1, 16, 13, 14, 3, and 9. The maximum increase of the


decrease of 3.1 � 109 tenge per year is observed for j ¼ 9. The trend of the rate

of the considered indicator increases only for products of branches 3 and 16 and

decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of jth branch in

branch 4 (manufacturing and process industry) (in constant prices of year 2000)”

had a positive slope for branches j ¼ 4, 5, 10, 6, 11, 8, and 12 and a negative

slope for branches j ¼ 16, 15, 14, 13, 2, 1, 3, 7, and 9. The maximum increase of

the trend by 1.5 � 1010 tenge per year is observed for j ¼ 4, whereas the

maximum decrease by 2.3 � 109 tenge per year is observed for j ¼ 9. The

trend of the rate of the considered indicator increases only for products of

branches 4, 5, 6, 12, and 16 and decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 5 (manufacturing and process industry) (in constant prices of year 2000)”

had a positive slope for branches j ¼ 5, 4, 6, 10, 3, 7, 11, 12, 15, 8, 1, 2 and a

negative slope for branches j ¼ 16, 9, 13, and 14. The maximum increase of the



rate of the considered indicator increases only for products of branches 7, 2, and

16 and decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 6 (construction) (in constant prices of year 2000)” had a positive slope for

branches j ¼ 4, 6, 5, 7, 10, 11, 8, 12, 1, 15, 13 and a negative slope for branches

j ¼ 2, 14, 3, and 9. Branch 6 does not purchase intermediate products of

economic branch 6. The maximum increase of the trend by 1.0 � 1010 tenge

per year is observed for j ¼ 4, whereas the maximum decrease by 3.7 � 108

tenge per year is observed for j ¼ 9. The trend of the rate of the considered

indicator decreases for all 15 branches.

– The trend of the indicator “Demand for intermediate products of jth branch in

branch 7 (trade, repair services for cars, and goods for household use) (in

constant prices of year 2000)” had a positive slope for branches j ¼ 10, 4, 5,

11, 8, 7, 12 and a negative slope for branches j ¼ 13, 2, 14, 15, 1, 3, and 9.

Branch 7 does not purchase intermediate products of economic branch 16. The

maximum increase of the trend by 3.8 � 109 tenge per year is observed for

j ¼ 10, whereas the maximum decrease by 1.9 � 109 tenge per year is observed

for j ¼ 9. The trend of the rate of the considered indicator increases only for

products of branches 4, 5, 6, 15, and 1 and decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 8 (hotels and restaurants) (in constant prices of year 2000)” had a positive

slope for branches j ¼ 4, 5, 10, 1, 6, 7, 8, 9, 15, 11, 12, 3 and a negative slope for

branches j ¼ 14, 13, and 2. Branch 8 does not purchase intermediate products of


economic branch 16. The maximum increase of the trend by 1.0 � 109 tenge per

year is observed for j ¼ 4, whereas the maximum decrease by 1.2 � 107 tenge

per year is observed for j ¼ 2. The trend of the rate of the considered indicator

increases only for products of branches 4, 8, 15, and 12 and decreases for all

other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 9 (transportation and communications) (in constant prices of year 2000)”

had a positive slope for branches j ¼ 4, 5, 10, 6, 8, 12, 11 and a negative slope

for branches j ¼ 16, 2, 15, 14, 13, 1, 7, 3, and 9. The maximum increase of the



of the considered indicator increases only for products of branches 5, 16, 15, and

7 and decreases for other branches.

– The trend of the indicator “Demand for intermediate products of branch j inbranch 10 (financial activity) (in constant prices of year 2000)” had a positive

slope for all branches except 2 and 4. The maximum increase of the trend of

1.1 � 1010 tenge per year is observed for j ¼ 10, whereas the maximum


rate of the considered indicator increases only for products of branches 16 and

2 and decreases for other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 11 (real estate activities, rent, and services for business) (in constant

prices of year 2000)” had a positive slope for branches j ¼ 4, 10, 5, 6, 7, 8, 12, 9,

15, 13 and a negative slope for branches j ¼ 18, 2, 11, 1, 14, and 3. The

maximum increase of the trend of 5.3 � 109 tenge per year is observed for

j ¼ 4, whereas the maximum decrease by 3.0 � 108 tenge per year is observed

for j ¼ 3. The trend of the rate of the considered indicator increases only for

products of branch 15 and decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 12 (public administration) (in constant prices of year 2000)” had a

positive slope for branches j ¼ 5, 4, 10, 6, 8, 12, 15, 1, 11, 13 and a negative

slope for branches j ¼ 2, 14, 3, 7, and 9. Branch 12 does not purchase interme-

diate products of economic branch 16. The maximum increase of the trend of

3.2 � 109 tenge per year is observed for j ¼ 5, whereas the maximum decrease

by 2.8 � 108 tenge per year is observed for j ¼ 9. The trend of the rate of the

considered indicator increases only for products of branches 6, 12, and 7 and


– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 13 (public administration) (in constant prices of year 2000)” had a

positive slope for branches j ¼ 5, 4, 6, 15, 10, 13, 12, 8, 1 and a negative

slope for branches j ¼ 2, 3, 9, 7, 14, and 11. Branch 13 does not purchase


trend of 4.2 � 108 tenge per year is observed for j ¼ 5, whereas the maximum



rate of the considered indicator increases only for products of branch 13 and


– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 14 (public health and social services) (in constant prices of year 2000)”

had a positive slope for branches j ¼ 4, 5, 6, 7, 10, 15, 8, 12, 13 and a negative

slope for branches j ¼ 2, 3, 11, 1, 14, and 9. Branch 14 does not purchase


trend of 5.1 � 109 tenge per year is observed for j ¼ 4, whereas the maximum


of the considered indicator increases only for products of branch 13 and


– The trend of the indicator “Demand for intermediate products of branch j inbranch 15 (other public utilities, social and personal services) (in constant prices

of year 2000)” had a positive slope for branches j ¼ 4, 5, 10, 15, 7, 6, 8, 12, 11,

13, 1 and a negative slope for branches j ¼ 16, 2, 3, 9, and 14. The maximum

increase of the trend of 2.8 � 109 tenge per year is observed for j ¼ 4, whereas

the maximum decrease by 6.3 � 107 tenge per year is observed for j ¼ 14. The

trend of the rate of the considered indicator increases only for products of

branches 4 and 16 and decreases for all other branches.

– The trend of the indicator “Demand for intermediate products of the jth branch inbranch 16 (housekeeping services) (in constant prices of year 2000)” had a

positive slope for branches j ¼ 6, 4, 8 and a negative slope for branches j ¼ 1,

16, 14, 15, 9, 11, and 13. Branch 16 does not purchase intermediate products of

economic branches 10, 5, 12, 7, and 2. The maximum increase of the trend of

8.8 � 106 tenge per year is observed for j ¼ 6, whereas the maximum decrease

by 1.0 � 107 tenge per year is observed for j ¼ 13. The trend of the rate of the

considered indicator increases only for products of branches 4, 16, and 15 and


In Figs. 4.9 and 4.10, we present an example of initial data for the perspective

macroanalysis of the dynamics (and its trend) and rate (and its trend) of economic

branches in the context of the endogenous variable “Demand for intermediate

products of branch 4 (manufacturing and process industry) in branch 1 (agriculture,

hunting, and forestry) (in prices of year 2000)”.

A positive slope of the linear trend of the indicator (Fig. 4.9) argues for an

increase (on average) of the considered branch demand within the considered time

interval. At the same time, a negative slope of the trend of the rate of this indicator

argues for a decrease (on average within the period of 2000–2015) of the indicator

growth rate.

Perspective Comparative Analysis of the Indicators of Economic Branches

Within the framework of perspective comparative analysis of economic branches’

indicators for each year from 2009 to 2015, we also carried out a ranking of the


branches with respect to the same indicators just as in the case of retrospective

macroanalysis:

– Gross value-added (GVA) of a branch (in prices of year 2000);

– Fixed assets of a branch (in prices of year 2000);

– Capital productivity of a branch;

– Demand for labor force in a branch;

– Labor productivity in a branch.

Linear trend of variable rateComputed variable values

Fig. 4.9 Demand of intermediate products of branch 4 in branch 1 (in prices of year 2000) and

its trend


Fig. 4.10 Rate of the variable “Demand of intermediate products of branch 4 in branch 1 (in

prices of year 2000)” and its trend


Analysis of the results of ranking with respect to the mentioned indicators allows

us to conclude, e.g., the following:

– The maximum real GVA over economic branches is expected in branch 4

(manufacturing and process industry) in 2010–2015. The growth of the GVA

in this branch in 2015 will be equal to about 86% in comparison with 2009.

– The maximum fixed assets over economic branches is expected in branch 3

(mineral resource industry) for the entire period of retrospective and perspective

prediction. The growth of the fixed assets in this branch in 2015 will be equal to

about 9.2% in comparison with 2009.

– The maximum capital productivity over economic branches is expected in

branch 2 (fishery and fish-breeding) in 2010–2015. The growth of capital

productivity in this branch in 2015 will be more than 60 times the capital

productivity of branch 12 (public administration).

– The maximum number of employees is expected in branch 1 (agriculture,

hunting, and forestry) in 2000–2015. In 2015, the demand for the labor force

in this branch will constitute 2,581,607 people. The minimum number of people

starting from 2012 is expected in branch 16 (housekeeping services); in 2015, it

will constitute 24,361.

– The maximum labor productivity is expected in branch 10 (financial activity) in

2010–2015. In 2015, the branch labor of productivity will be 23 times greater

than branch 14 (public health and social services).

The results of the Kazakhstan Republic’s economic branches’ ranking within the

framework of the mentioned indicators in year 2015 are presented in Tables 4.9,

4.10, 4.11, 4.12, and 4.13.

Table 4.9 Gross value-added

of branch

No. No. of branch

Value of indicator GVA

in year 2015 (tenge,

in prices of 2000)

1 4 1.45122 � 1012

2 6 6.65949 � 1011

3 3 6.57340 � 1011

4 7 5.87018 � 1011

5 11 5.44444 � 1011

6 10 4.38485 � 1011

7 1 4.25092 � 1011

8 9 4.12240 � 1011

9 5 3.05231 � 1011

10 15 2.09079 � 1011

11 13 1.99452 � 1011

12 8 9.18216 � 1010

13 12 7.56724 � 1010

14 14 5.51708 � 1010

15 2 7.43652 � 109

16 16 5.15982 � 109


Perspective Analysis of the Main Integral Indicators of a Country’s Development,

Indicators of Public Administration and Households

The perspective analysis of the aforementioned indicators in 2000–2015 shows the

following:

– GVA of the country (in prices of year 2000) in the considered period grows with

an average rate of the trend equal to 2.5 � 1011 tenge per year. The rate of the

GVA indicator decreases (by its trend) by 0.84% per year.

Table 4.10 Fixed assets of branch

No. No. of branch

Value of indicator “Fixed assets of branch”

in year 2015 (tenge, in prices of 2000)

1 3 1.72577 � 1012

2 11 1.33253 � 1012

3 9 7.32356 � 1011

4 12 7.25682 � 1011

5 4 6.67990 � 1011

6 6 3.87501 � 1011

7 5 2.88265 � 1011

8 7 2.78329 � 1011

9 1 2.08751 � 1011

10 10 1.41096 � 1011

11 13 1.16788 � 1011

12 14 7.83038 � 1010

13 15 7.73435 � 1010

14 8 2.76785 � 1010

15 2 1.16724 � 1010

16 16 0

Table 4.11 Capital productivity of branch

No. No. of branch

Value of indicator “Capital productivity

of a branch” in year 2015

1 2 6.371005

2 8 3.317435

3 10 3.107695

4 15 2.703250

5 4 2.172512

6 7 2.109079

7 1 2.036358

8 6 1.718572

9 13 1.707822

10 5 1.058857

11 14 0.704573

12 9 0.562895

13 11 0.408581

14 3 0.380897

15 12 0.104278

16 16 0.000000


– Gross products’ output (in prices of year 2000) in the considered period grows

with average rate of the trend equal to 4.0 � 1011 tenge per year. The rate of this

indicator decreases (by its trend) by 0.45% per year.

– Investment products’ output (in prices of year 2000) in the considered period

grows with an average rate of the trend equal to 1.2 � 1011 tenge per year. The

rate of this indicator decreases (by its trend) by 2.0% per year.

– Final products’ output (in prices of year 2000) in the considered period grows

with an average rate of the trend equal to 9.9 � 1010 tenge per year. The rate of

this indicator decreases (by its trend) by 0.46% per year.

Table 4.12 Branch demand

for labor force No. No. of branch

Value of indicator “Branch demand

for labor force” in year 2015

1 1 2,581,608

2 7 1,235,941

3 13 963,120

4 6 841,841

5 9 604,098

6 4 595,420

7 11 516,757

8 12 406,500

9 14 401,955

10 3 247,131

11 15 225,833

12 5 184,281

13 8 150,238

14 10 139,326

15 2 27,907

16 16 24,361

Table 4.13 Labor

productivity in branch No.

No. of

branch

Value of indicator “Labor productivity

in branch” in year 2015 (in prices of 2000)

1 10 3,147,177.0

2 3 2,659,884.0

3 4 2,437,295.0

4 5 1,656,329.0

5 11 1,053,578.0

6 15 925,811.3

7 6 791,063.1

8 9 682,406.4

9 8 611,171.8

10 7 474,956.1

11 2 266,466.3

12 16 211,806.2

13 13 207,089.8

14 12 186,155.8

15 1 164,661.7

16 14 137,256.4


– Intermediate products’ output (in prices of year 2000) in the considered period



– Export products’ output (in prices of year 2000) in the considered period grows

with an average rate of the trend equal to 5.1 � 1010 tenge per year. The rate of

this indicator decreases (by its trend) by 1.4% per year.

– State budget (in prices of year 2000) in the considered period grows with an

average rate of the trend equal to 1.0 � 1011 tenge per year. The rate of this

indicator decreases (by its trend) by 1.5% per year.

– Social transfers to households (in prices of year 2000) in the considered period



– Fixed assets to households (in prices of year 2000) in the considered period



– Capital productivity in the considered period grows with an average rate of the

trend equal to 1.4 � 10�2 1/(year)2. The rate of this indicator decreases (by its

trend) by 0.69% per year.

– The total number employed within the mentioned period grows with the average

rate of trend equal to 1.9 � 105 people per year. The rate of this indicator

decreases (by its trend) by 0.14% per year.

– Labor productivity in the mentioned period grows with the average rate of trend

equal to 1.9 � 104 tenge/(people � year2). The rate of this indicator decreases

(by its trend) by 0.68% per year.

– The consumer price index in the mentioned period fluctuates within the limits of

6–8% per year along with an insignificant drop in the trend of this indicator with

the rate of 0.008% per year.

– Real prices of investment products in the considered period drop with the

average rate of trend equal to 1.2% per year. The rate of this indicator drops

(by its trend) by 0.16% per year.

– Money of households in banking accounts (in prices of year 2000) in the

mentioned period increases, with the average rate of the trend equal to

4.3 � 109 tenge per year. The rate of this indicator drops (by its trend) by

0.047% per year.

In Figs. 4.11 and 4.12, we carry out the analysis of initial data for the perspective

macroanalysis of the national economy indicators in the context of the endogenous

variable “Investment products output.”

Analysis of the trends presented in Figs. 4.11 and 4.12 allows us to conclude that

in 2000–2015, just as in the retro-analysis period (Figs. 4.6 and 4.7) , one can expect

average growth of the real output of investment products in the country with some

average deceleration of this growth. However, the decrease of a negative slope of

the rate trend (Fig. 4.12) for the period of 2000–2015 in comparison with the period

of retro-analysis (Fig. 4.6) argues for some increase of the rate of growth of

investment products by the end of the considered period.


4.1.2.2 Analysis of Elasticity Coefficients of Endogenous Variablesof Computable Model of General Equilibriumof Economic Branches

Within the limits of the retrospective analysis of economic branches, we obtained

the estimates of the elasticity coefficients of 18 endogenous variables for each of 16

economic branches, as well as 14 integral indicators of the country economy, public

Linear trend of variable rateComputed variable values

Fig. 4.11 Investment products output (in prices of year 2000) and its trend


Fig. 4.12 Rate of variable “Investment products output (in prices of year 2000)” and its trend


administration, and households in 2009 over 341 exogenous indicators of the model

for 2008–2014. These elasticity coefficients form a matrix of size 302 � 341 with

the elements defined by

FpjðtÞ ¼ 100xnj ðtÞ � xjðtÞ

xjðtÞ : (4.71)

Here p is the varied exogenous parameter; xjðtÞ is the value of the jth endogenousvariable for t ¼ 2009 obtained from base computation of the model; xnj ðtÞ is the

value of the respective endogenous variable obtained from the increase in the

values of the varied exogenous parameter p from 2000 to 2008; other values of

the exogenous parameters remain unchanged in comparison with the base

computation.

For example, the computed elasticity coefficient of the households’ budget by

the share of budget of branch 1 spent for paying the labor force appears to be

0.0229. This means that if the share of budget of branch 1 spent for paying the labor

force in each of 2000–2008 is increased by 1% in comparison with the base variant,

then in 2009, the households’ budget will increase by 0.0229 in comparison with the

base variant.

Within the limits of the perspective analysis of economic branches, we also

obtain the similar elasticity coefficients of endogenous indicators in 2015 by

exogenous indicators in 2009.

Analysis of influence of the exogenous parameters and, in particular, the

parameters CO_pj_iz, “Share of budget of the ith branch spent for purchasing

intermediate products produced by the jth branch,” on the country’s GVA shows

that

– The mean value of absolute values of the GVA elasticity coefficients by the

shares CO_pj_iz appears to be rather high and equal to 4.7450; and

– The mean value of absolute values of the GVA elasticity coefficients by shares

of budgets of other branches appears to be 2.379.

This remark allows recommendation of parameters CO_pj_iz for use in

parametric regulation of the national economic growth. We shall consider

experiments with the parametric regulation in 4.1.3.

4.1.2.3 Analysis of Economic Growth Sources on the Basisof the Computable Model of General Equilibriumof Economic Branches

Let’s now proceed to the analysis of economic growth sources by economic

branches on the basis of the retrospective data for the years 2000–2015. For this

purpose, using expressions (4.16) for the production functions, we’ll estimate the

influence of variation of these functions’ arguments on the rate of GVA growth of


branches VY__i[t + 1] under assumption about the constancy of the coefficients of

the intermediate products consumed by the branch CA_z_ji[t], coefficients of thecapital CA_k_i[t], and coefficients of labor CA_l_i[t]. Such an assumption is

applied in extrapolation of the mentioned functions within the prediction period

2010–2015 of the model computation.

Taking the logarithm of both parts of (4.20), finding the total increment of the

function ln(VY__i), and discarding the terms of higher infinitesimal order, we

obtain the following estimate of the growth rate yi of the real GVA of the ith branchin dependence on the growth rates of the production function arguments: CA_r_i,VD_pj_iz, Kiср ¼ (VK__i[t] + VK__i[t + 1])/2, and VD_pi_il[t]:

yi ¼DVY i

VY i¼ DCA r i

CA r iþX16j¼1

CA z ji� VD pj izð ÞDVD pj iz

VD pj iz

þ CA k iDKimKim

þ CA l iDVD pi il

VD pi il: (4.72)

Let ai ¼ DCA r iCA r i denote the rate of the technological progress in the ith branch; let

zij ¼ DVD pj izVD pj iz denote the rate of the intermediate products of the jth branch consumed

in the ith branch; letki ¼ DKimKim

denote the rate of capital accumulation in the ith branch;

let li ¼ DVD pi ilVD pi iz denote the rate of labor inputs growth in the ith branch, where the

sign “D” means the variable increment. The time in (4.72) is omitted for brevity.

The coefficients on the right-hand side of formula (4.72) with the rates indicated

above characterize the degree of influence of the considered factors on the eco-

nomic growth and allow us to compare their influence with that of the technological

progress with the coefficient equal to 1. Denoting these coefficients by

aij ¼ CA z ji� VD pj iz , bi ¼ CA k i , gi ¼ CA l i , from (4.72) we obtain its

abbreviated notation:

yi ¼ ai þX16j¼1

aijzij þ biki þ gili: (4.73)

Now we’ll present the values of the coefficients defining the contributions of the

sources of economic growth of the branches on the basis of the considered model in

2008 (Table 4.14). The coefficients in the table show the percentage of increase of

the GVA growth rate as a result of increase of growth factors (fixed assets, labor or

demand for intermediate products of the economic agents) by 1%.

Analysis of Table 4.15 shows that, except for the rate of technological progress

with equal influence on the rate of growth of all branches, which is the same in the

considered model, the maximum influence on the growth of GVA of economic

branches 1, 2, 5, 7, 12, 13, and 16 is exerted by the growth rate of labor inputs, GVA

of branches 4, 6, 8, 9, 10, 11, and 15 are exerted by the rate of the capital

accumulation, GVA of the branches 3 and 14 are exerted by the rate of intermediate

products of other branches consumed by the branch.


Table

4.14

Coefficientscharacterizinginfluence

offactorsofeconomic

growth

Number

ofbranch

ib i

g ia i1

a i2

a i3

a i4

a i5

a i6

a i7

10.3089

0.9051

1.345�10

�12

9.480�10

�02

2.897�10

�14

2.171�10

�13

1.602�10

�14

2.028�10

�14

1.345�10

�12

20.2426

2.4964

1.590�10

�16

7.308�10

�01

7.087�10

�16

9.884�10

�15

1.390�10

�15

1.120�10

�15

1.590�10

�16

30.9650

0.6886

2.886�10

�15

0.000

2.970�10

�12

8.269�10

�13

9.894�10

�14

1.478

2.886�10

�15

41.2900

0.0805

2.227�10

�13

1.343�10

�18.634�10

�13

9.989�10

�13

9.029�10

�03

2.641�10

�16

2.227�10

�13

51.0

�10�1

02.4083

3.199�10

�17

5.537�10

�17

8.913�10

�14

6.720�10

�45.406�10

�35.300�10

�43.199�10

�17

60.9343

0.7721

2.078�10

�16

9.186�10

�18

3.313�10

�14

3.940�10

�13

2.467�10

�15

1.324�10

�14

2.078�10

�16

71.0

�10�1

01.8792

8.133�10

�16

1.061�10

�15

3.115�10

�14

2.660�10

�13

4.124�10

�14

2.508�10

�13

8.133�10

�16

81.0691

0.4706

2.003�10

�02

1.498�10

�15

4.343�10

�17

6.171�10

�14

3.375�10

�15

7.288�10

�15

2.003�10

�02

90.8660

0.2153

4.289�10

�16

2.265

1.666�10

�13

8.753�10

�15

3.057�10

�14

4.005�10

�14

4.289�10

�16

10

0.6702

0.5492

0.000

0.000

0.000

4.397�10

�15

5.602�10

�15

8.811�10

�15

0.000

11

1.2022

0.1006

2.929�10

�05

2.267

5.064�10

�14

1.030�10

�12

1.664�10

�04

5.308�10

�13

2.929�10

�05

12

1.0

�10�1

02.5822

5.415�10

�14

0.000

0.000

4.255�10

�13

2.573�10

�14

0.000

5.415�10

�14

13

0.2635

1.7177

2.370�10

�14

0.000

0.000

8.847�10

�13

1.142�10

�13

5.739�10

�15

2.370�10

�14

14

0.0227

1.7814

6.736�10

�14

2.018�10

�16.280�10

�15

1.288�10

�12

1.919�10

�13

4.809�10

�13

6.736�10

�14

15

0.9304

0.2173

1.041�10

�15

1.408�10

�37.681�10

�15

3.712�10

�13

3.926�10

�14

1.579�10

�13

1.041�10

�15

16

01.9372

00

00

00

0


Table

4.15

Number

ofbranch

ia i8

a i9

a i10

a i11

a i12

a i13

a i14

a i15

a i16

14.325�10

�15

4.595�10

�21.807�10

�14

1.177�10

�14

01.681�10

�16

1.178�10

�15

9.304�10

�18

0

28.054�10

�17

1.087�10

�14

2.616�10

�17

2.280

00

00

0

34.633�10

�14

2.221�10

�21.436�10

�13

4.626�10

�12

01.581�10

�15

4.991�10

�4

1.571�10

�14

0

48.038�10

�15

1.444�10

�13

4.353�10

�14

6.338�10

�16

01.247�10

�16

3.379�10

�3

5.445�10

�16

0

51.642�10

�15

2.760�10

�16.132�10

�3

2.512�10

�14

03.273�10

�52.026�10

�17

5.494�10

�40

67.977�10

�15

1.117�10

�26.525�10

�16

2.014�10

�14

08.436�10

�17

1.554�10

�17

1.089�10

�15

0

76.067�10

�14

4.548�10

�13.407�10

�13

1.505�10

�12

05.394�10

�16

8.576�10

�5

4.404�10

�16

0

81.528�10

�15

8.267�10

�15

3.053�10

�14

1.507�10

�14

02.339�10

�17

1.338�10

�17

2.244�10

�16

0

93.346�10

�14

1.002�10

�12

3.203�10

�13

4.339�10

�13

03.657�10

�15

4.803�10

�16

1.091�10

�15

0

10

1.991�10

�14

3.270�10

�01

2.862�10

�13

5.408�10

�15

01.347�10

�14

08.805�10

�18

0

11

6.664�10

�14

6.596�10

�13

2.965�10

�13

2.646�10

�12

09.260�10

�14

1.661�10

�03

9.599�10

�06

0

12

4.185�10

�1

1.548�10

�19.506�10

�13

7.964�10

�30

2.126�10

�14

1.408�10

�15

2.886�10

�15

0

13

4.820�10

�14

5.807�10

�11.794�10

�14

1.331�10

�13

03.939�10

�15

7.738�10

�17

1.835�10

�14

0

14

1.891

2.118�10

�13

4.749�10

�17

2.743�10

�13

02.219�10

�16

1.782�10

�14

6.646�10

�14

0

15

6.285�10

�14

2.139�10

�19.739�10

�14

8.436�10

�14

09.538�10

�17

1.416�10

�17

1.222�10

�13

0

16

00

00

00

00

0


4.1.2.4 Estimation of the Macroeconomic Theory Provisions on the Basisof the Computable Model of General Equilibriumof Economic Branches

In this section we describe the results of computational experiments focused on the

estimation of some provisions of the macroeconomic theory on cyclic oscillations

of macroeconomic indicators as a result of shock changes of demands of final and

investment products.

Within the limits of the macroeconomic theory, cyclic oscillations of economic

processes can appear in the presence of [40]

– Linear dependence between the volume of consumer expenses and current

income;

– Linear dependence between investments and increase of income.

To verify these provisions, we carried out a number of computational

experiments aimed at computation of the following scenarios of variation of the

mentioned demands of final and investment products:

(a) O1½t� ¼ O1½t� 1� þ aðYg½t� � Yg½t� 1�Þ; (4.74)

(b) O2i½t� ¼ O2i½t� 1� þ bðYg½t� 3� � 2Yg½t� 2� þ Yg½t� 1�Þði ¼ 1; . . . ; 16Þ; (4.75)

(c) Joint application of scenarios (a) and (b);

(d) Increase of the shares O1ðtÞ by k times in comparison with the base variant;

(e) Increase of the shares O2iðtÞ ði ¼ 1; . . . ; 16Þ by l times in comparison with the

base variant;

(f) Joint application of scenarios (d) and (e).

Here t ¼ 2010; . . . ; 2015 is the time in years;O1½t� ¼ CO p 17c½t� is the share ofbudget of the households spent for purchasing final products (exogenous function);

O2iðtÞ is the share of the budget of the ith branch spent for purchasing investment

products ði ¼ 1; . . . ; 15Þ (exogenous function); Yg½t� ¼ VY g½t� is the sum of

branches’ GVA in constant prices (endogenous variable); a, b, l, and k are some

positive constants (l>1; k>1).

The application of scenarios (a) and (b) means the model computation with the

values of the parameters O1 and O2i defined by formulas (4.74) and (4.75),

respectively, starting from t ¼ 2010 . The values of all other exogenous model

parameters for scenarios (a)–(f) correspond to its base variant.

As a result of computational experiments on the base of the model aimed at

realization of scenarios (a)–(f), one can observe cyclic oscillations of the variable P(t) (consumer price index) (see Figs. 4.13, 4.14, 4.15, 4.16, 4.17, and 4.18). We did

not reveal the oscillation phenomenon for the values of the real indicators (in

particular, Yg[t]).The experimental results presented in Figs. 4.13, 4.14, 4.15, 4.16, 4.17, and 4.18

prove the respective provisions of the macroeconomic theory [40].


4.1.3 Finding Optimal Parametric Control Laws on the Basisof the CGE Model of Economic Branches

4.1.3.1 Attenuation of Cyclic Oscillations of Macroeconomic Indicatorsby Parametric Control Methods

In computational experiments we consider the following problem of attenuation of

cyclic oscillations of the consumer price level arising in application of scenario (c)

of the economic system development taking into account the linear dependence

Base Scenario

Fig. 4.13 Values of the consumer price index with application of scenario (a) with a ¼ 3 � 10�12

and for the base variant

Base Scenario

Fig. 4.14 Values of the consumer price index with application of scenario (b) with b ¼ 10�13 and

for the base variant


between the volume of consumer expenses and current income, as well as the linear

dependence between investments and income growth.

On the basis of the computable model of economic branches with application of

scenario (c) (see Sect. 4.1.2.4, where a ¼ 3 � 10�13, b ¼ 10�13), find the values of

the shares Oji½t� ¼ CO pj iz½t� of budgets of the j-th producing agent spent for

purchasing goods and services produced and rendered by the i-th producing agent

in 2010–2015, i; j ¼ 1; . . . ; 16, which provide the lower bound of the following

Base Scenario

Fig. 4.15 Values of the consumer price index with application of scenario (c) with a ¼ 3 � 10�13,

b ¼ 10�13 and for the base variant

Base Scenario

Fig. 4.16 Values of the consumer price index with application of scenario (d) with k ¼ 1.2 and



criterion KP which characterizes the deviations of the computed values of the

consumer price index VP½t� from the respective desired values of P½t�:

KP ¼X2015t¼2010

VP½t� � P½t�P½t�

� �2:

Here we use the computed base values of the model consumer price index

without parametric control as the desired values of P[t].

Base Scenario

Fig. 4.17 Values of the consumer price index with application of scenario (e) with l ¼ 1.4 and for

the base variant

Base Scenario

Fig. 4.18 Values of the consumer price index with application of scenario (f) with k ¼ 1.2,

l ¼ 1.4 and for the base variant


The constraints on the adjusted parameters are as follows:

0:5 � Oji½t� O

j

i

.� 1:5;

P16i¼1 O

jiðtÞ � 1; where i; j ¼ 1; . . . ; 16; t ¼ 2010; :::; 2015.

Here Oj

i are the base values of the mentioned shares obtained as a result of

solving the parametric identification problem by data of 2000–2008.

The constraints on the growth of the macroeconomic indicator:Yr½t� 0; 95YC½t�.Here YC½t� are the computed scenario values of the sum of GVA of the branches

without parametric control; Yr½t� are the computed values of the sum of GVA of the

branches with parametric control.

The value of the criterion KP without parametric control is KP ¼ 0; 424. Theoptimal values of the criterion KP with application of the parametric control law

appears to be equal to KP ¼ 0; 000844.The base values of macroeconomic indicators are YðtÞ and PðtÞ; values obtained

with application of scenario (c) and values obtained with application of the optimal

law of parametric control are presented in Figs. 4.19 and 4.20.

Analysis of the results of the computational experiments presented in Figs. 4.19

and 4.20 shows that with application of the derived optimal parametric control law,

the consumer price index within the controlled period practically coincides with the

desired values, while the values of the sum of economic branches’ GVAs appear to

be less than the respective base values except for 2015.

4.1.3.2 Finding Optimal Parametric Control Laws on the Basisof the Stochastic CGE Model of Economic Branches

The stochastic computable model of economic branches was derived from the

respective deterministic model (with estimates of values of the exogenous

Base Scenario Attenuation

Fig. 4.19 Computed values of sums of GVA of economic branches


parameters found from solving the parametric identification problem) via adding

the discrete Gaussian noise with independent constituents to the right-hand sides of

all of model dynamic equations (4.1). These equations include the following

equations for computing the following endogenous variables:

– Gross output of products and services in prices of the base period (VY i, i ¼ 1;. . . ; 16) of 16 producing sectors by means of the respective production functions;

– Fixed assets (VK i,i ¼ 1; . . . ; 15) of 15 producing sectors;

– Annual budgets (VB i,i ¼ 1; . . . ; 18) of 16 economic branches, households, and

consolidated budget.

The additive noises inserted into the expressions with respect toVB i can initiatethe respective cyclic oscillations caused by quick shifts (shocks) in development of

the technological progress and random changes of the population growth rate. The

additive noises inserted or added to the expressions with respect toVK icharacterizethe random changes of the shares of budgets of the producing agents spent for

purchasing investment goods and random character of the coefficients of the retire-

ment funds. The additive noises added to the expressions with respect to VB idescribe the random character of the income obtained by the sector within the

current period.

In this work, the estimates of mean square deviations of the generated Gaussian

random values defining the given noise were obtained on the basis of analysis of the

respective statistical data for economic development of the Kazakhstan Republic

from 2000 to 2008 as follows. For each time series of the measured values of the

aforementioned variables, we computed some selected mean square deviations of

differences between the measured values and trends of those values. The values

obtained in such a way were accepted to be the estimated mean square deviations of

the components of the generated discrete Gaussian noise xðtÞ added to the right-

hand sides of 50 dynamic equations indicated above.

Base Scenario Attenuation

Fig. 4.20 Computed values of the consumer price index


In computational experiments with the stochastic computable model of the

economic branch, we use the following optimization criterion:

Ks ¼ E1

6

X2015

t¼2010VY g½t�

� �! max : (4.76)

Here and below “E” denotes expectation. Here Ks is the expectation value of the

gross output of the country in the prices of year 2000 from years 2010–2015.

In computational experiments, we compute the criterion Ks as follows. We simulate

N realizations of random process xðtÞ by the Monte Carlo method and, after Ncomputations of the model for all these realizations used consecutively in equations

of type (1.33), we compute the criterion Ks as the arithmetic mean of the values of

the expressions 16

P2015t¼2010 YðtÞ over these N realizations. Similarly, we check the

condition of type (1.36) of the expectation values of the endogenous variables

belonging to given regions of the model state space.

The value of the criterion Ks for the base computation variant (with use of the

values of the exogenous parameters obtained as a result of the model parametric

identification) equals Ks ¼ 0:9891:1013.In experiments with the optimization criterion (4.76), we applied the constraints

on the growth of the consumer prices in the following form:

EðVPr½t�ÞÞ � 1:09E(VP[t]),.t = 2010, . . . ; 2015:

Here VP½t� is the computed level of the consumer prices in the model without

parametric control, and VPr½t� is the consumer price level with parametric control.

In the computational experiments, we realize the regulation of 1536 exogenous

parameters, shares of budgets of the jth producing agents spent for purchasing

goods and services produced by the ith producing agent from 2010 to 2015: OjiðtÞ;

t ¼ 2010; . . . ; 2015; i; j ¼ 1; . . . ; 16. HereP16

i¼1 OjiðtÞ � 1 for the mentioned values

of t. The base values of these shares obtained as a result of solving the model

parametric identification problem from data of 2000–2008 will be denoted by Oj

i;

i; j ¼ 1; . . . ; 16.We considered the problem of finding optimal values of the adjusted parameters’

vectors. On the basis of the stochastic computable model of economic branches,

find the mentioned values of the shares of the producing agents’ budgets OjiðtÞ ,

which provide the upper bound of the criterion Ks with additional constraints on

these shares of the following form:

0:5 � OjiðtÞ O

j

i

.� 2; i; j ¼ 1; . . . ; 16; t ¼ 2010; . . . ; 2015:

These optimization problemswere solved by applying theNelder–Mead algorithm.

After application of parametric control of the shares of the stochastic model budgets,


http://dx.doi.org/10.1007/978-1-4614-4460-2_1

http://dx.doi.org/10.1007/978-1-4614-4460-2_1

the value of the criterion appears to be equal to Ks ¼ 1:2453 � 1013 ; its value

increases by 25.89% in comparison with the base variant.

A similar parametric control problem with the respective constraints was also

solved on the basis of the deterministic CGE model of economic branches with the

use of the deterministic analog of criterion (4.76):

Kd ¼ 1

6

X2015t¼2010

Yt ! max:

After application of parametric control of the shares of budgets of producing

agents, the value of the criterion for the deterministic model appears to be equal to

Kd ¼ 1:6283 � 1013 . The criterion value increases by 33.14% in comparison with

the base variant.

The comparison of the results of solution of variational calculus problem on the

basis of stochastic and deterministic computable models of general equilibrium

shows that the computed value of the functional of the variational calculus problem

decreases while taking into account the disturbing violations in the deterministic

computable model of general equilibrium in the form of additive noise.

4.2 National Economic Evolution Control Basedon the Computable Model of General Equilibriumwith the Knowledge Sector


4.2.1.1 Model Agents

The considered model [27, 10] describes the behavior and interaction in nine

product markets and two labor markets for the following seven economic agents:

Economic agent no. 1 is the science and education (knowledge) sector rendering theservices on education of students and the production of knowledge. These include

educational institutions (public and private) rendering the services of higher educa-

tion, and as well as scientific (research) organizations.

This sector renders the services distributed among the following three areas:

1. The services for the innovation sector (mainly carrying out research and devel-

opment) and other sectors of the economy (mainly carrying out research and

development too), as well as services for economic agent no. 5, including, in

accordance with the methodology of National Economic Accounting (NEA),

4.2 National Economic Evolution Control Based on the Computable Model. . . 211

the services of nonmarket science. Additionally, a part of the services on

providing the knowledge is consumed by the sector itself.

2. The services for economic agent no. 5 (including, in accordance with methodol-

ogy of NEA, services of free education), services of paid education for the

innovation sector and other branches of the economy and households. Moreover,

a part of the educational services is consumed by the sector itself.

3. The services for the outer world, carrying out the works by scientific grants.

Economic agent no. 2 is the innovation sector, which is an aggregate of innovation-active enterprises and organizations. The sector produces the product distributed

between the following two areas:

1. The innovative products for the domestic market. The innovative products are

understood to be final products manufactured on the basis of various technologi-

cal and other innovations. This index corresponds to the volume of shipped

innovative products. The products manufactured by the sector are consumed by

all producing sectors (including this sector itself) as the costs of research and

development, as well as costs of the technological innovations, and by economic

agent no. 5 (this means government financing of the innovation activity).

2. The innovative products of the outer world.

Economic agent no. 3 is other branches of the economy.

The other branches of the economy produce the products distributed among the

following four areas:

1. Final products for households, including consumer goods of current consump-

tion (foodstuffs, etc.), durable products (home technical equipment, motor

vehicles, etc.), as well as services;

2. Final products for economic agent no. 5, including the following:

(a) Final products for public institutions (according to the NEA’s methodology,

expenditures of the public institutions on acquiring final products), including

free services for the inhabitants rendered by the enterprises and

organizations in the field of public health, culture (this does not include

the educational services, because they are rendered by economic agent no.

1); services satisfying the needs of the entire society, i.e., the general public

administration, protection of law and order, national defense, housing,

economy, etc.;

(b) Final products for nonprofit organizations servicing households, including

the free services of a social character;

3. Investment products, i.e., expenditures on improvement of produced and

nonproduced tangible assets (in other words, the expenditures on the creation

of the capital assets). In accordance with the NEA’s methodology, this type of

product is determined as the sum of gross saving in capital assets and change of

reserves of material circulating assets minus the cost of acquired new and

existing capital assets (with the deduction of withdrawal).


4. Export products. Since imported products are one of the constituent parts of the

products considered above, then, to avoid double counting, the exported

products include only the net export (i.e., export minus import).

To produce products and services, producing agents nos. 1–3 purchase the

following production factors:

1. The labor force (by governmental and market prices);

2. Investment products;

3. Innovative products;

4. Services for providing knowledge (e.g., R&D sector);

5. Educational services (paid education).

Economic agent no. 4 is the aggregate consumer joining households. The agent

purchases final products produced by other branches of the economy. Furthermore,

the households use the paid educational services as well. Also, this sector forms the

labor force.

Economic agent no. 5 is the government, establishing taxation rates, determining

the shares of budget for financing the producers and social transfers, and spending

its budget for purchasing final products produced by other branches of the economy.

Economic agent no. 6 is the banking sector determining the interest rate for the debt

deposits.

Economic agent no. 7 is the outer world.

4.2.1.2 Exogenous Parameters of Model

This model includes 86 exogenous parameters and 110 endogenous variables. The

exogenous parameters include the following:

– The coefficients of the production functions of the sectors;

– The various shares of the budgets of the sectors;

– The shares of the products for selling in the various markets;

– The depreciation rates of capital assets and shares of the retired capital assets;

– The deposit interest rates;

– The various taxation rates;

– The export prices and governmental prices of goods, services, and labor force,

etc.

The list of the exogenous model parameters is given in Table 4.16.




– The produced values-added;


Table 4.16 Exogenous variables of the computable model with sector of knowledge

Economic agent 1: Knowledge and education sector

CO_p1_1l The share of the budget for purchasing labor force at the price of P__1l

CO_p1_1z The share of the budget for purchasing knowledge provisional services at the

price of P__1z

CO_p1_1r The share of the budget for purchasing educational services at the price of P__1r

CO_p1_1n The share of the budget for purchasing innovative products at the price of P__1n

CO_p1_1i The share of the budget for purchasing investment products at the price of P__1i

CE_p1_1z The share of the produced product for selling in the markets of knowledge-

provisional services at the price of P__1z

CE_p2_1z The share of the produced product for selling in the markets of knowledge-

provisional services at the price of P__2z

CE_p1_1r The share of the produced product for selling in the markets of educational

services at the price of P__1r

CA_r_1 The dimension coefficient of the production function

CA_k_1 The coefficient of capital assets of the production function

CA_l_1 The coefficient of the labor of the production function

Calpha__1 The coefficient of the costs of the knowledge-provisional services of the

production function

Cbeta__1 The coefficient of the costs of the educational services of the production function

Cgamma__1 The coefficient of the costs of the innovative products of the production function

CA_0_1 The rate of depreciation for the capital assets

CR__1 The share of the retired capital assets

Economic agent 2: The innovation sector

CO_p1_2l The share of the budget for purchasing the labor force at the price of P__1l

CO_p1_2z The share of the budget for purchasing knowledge-provisional services at the

price of P__1z



CO_p1_2i The share of the budget for purchasing investment products at the price of P__1i

CE_p1_2n The share of the produced product for selling in the market of innovative

products at the price of P__1n

CE_p2_2n The share of the produced product for selling in the market of innovative

products at the price of P__2n

CA_r_2 The dimension coefficient of the production function

CA_k_2 The coefficient of the capital assets of the production function

CA_l_2 The coefficient of the labor of the production function

Calpha__2 The coefficient of the costs of knowledge-provisional services of the production

function

Cbeta__2 The coefficient of the costs of educational services of the production function

Cgamma__2 The coefficient of the costs of innovative products of the production function

CA_0_2 The rate of depreciation for capital assets

CR__2 The share of retired capital assets

Economic agent 3: Other branches of the economy

CO_p1_3l The share of the budget for purchasing the labor force at the price of P__1l

CO_p1_3z The share of the budget for purchasing knowledge-provisional services at the

price of P__1z


(continued)




CO_p1_3i The share of the budget for purchasing investment products at the price of P__1i.

CE_p1_3c The share of the produced product for selling in the markets of final products at

the price of P__1сCE_p1_3g The share of the produced product for selling in the markets of final products for

the economic agent no. 5 at the price of P__1g

CE_p1_3i The share of the produced product for selling in the markets of investment

products at the price of P__1i

CE_p2_3c The share of the produced product for selling in the markets of exported products

at the price of P__2сCA_r_3 The dimension coefficient of the production function

CA_k_3 The coefficient of capital assets of the production function

CA_l_3 The coefficient of labor of the production function

Calpha__3 The coefficient of the costs of knowledge-provisional services of the production

function

Cbeta__3 The coefficient of the costs of educational services of the production function

Cgamma__3 The coefficient of the costs of innovative products of the production function

CA_0_3 The rate of depreciation for capital assets

CR__3 The share of the retired capital assets

Economic agent 4: The aggregate consumer

CO_p1_4c The share of the budget for purchasing final products at the price of P__1c


CO_b_4 The share of the budget for saving in bank deposits

CS_p3_4l The supply of the labor force at the price of P__3l

CS_p1_4l The supply of the labor force at the price of P__1l

Economic agent 5: Government

CT_vad The VAT rate

CT_pr The organization profit tax rate

CT_pod The rate of physical body income tax

CT_esn The rate of single social tax

CO_p1_5g The share of the consolidated budget for purchasing the final goods at the price of

P__1g

CO_p1_5z The share of the consolidated budget for purchasing knowledge-provisional

services at the price of P__1z

CO_p1_5r The share of the consolidated budget for purchasing educational services at the

price of P__1r

CO_p1_5n The share of the consolidated budget for purchasing innovative products at the

price of P__1n

CO_s1_5 The share of the consolidated budget for backing Sector 1



CO_tr_5 The share of the consolidated budget for payment of social transfers to the

inhabitants

CO_f4_5 The share of off-budget funds for payment of pensions, welfare payments, etc.

CO_s_5b The share of the retained consolidated budget

CO_s_5f The share of the retained off-budget funds

(continued)


– The demand for and supply of various products and services;


– The capital assets of the sectors;

– The wages of employees;


– The various types of prices of the products, services, and the labor force.

The list of the endogenous model variables is given in Table 4.17.

4.2.1.4 Model Markets

As a result of leveling the demand for and supply of various types of products,

services, and labor force, the equilibrium prices are formed in the following

markets:

– The market for final products for households;

– The market for exported final products;

– The market for final products for economic agent no. 5;

– The market for investment products;

– The market for the labor force paid by privately owned enterprises;


CB_other_5 The sum of the tax proceeds (not included into the considered ones), nontax

income, and other incomes of the consolidated budget

Economic agent 6: The banking sector

CP__bpercent The deposit interest rate for enterprises

CP_h_bpercent The deposit interest rate for physical bodies

General parts of the model

CP__3l The governmental price of labor force

CP__2z The export price of knowledge-provisional services

CP__2n The export price of innovative products

CP__2c The export price of final products

CD_p2_sz The total demand for knowledge-provisional services at the export prices

CD_p2_sn The total demand for innovative products at the export prices

CD_p2_sc The total demand for final products at the export prices

Technical parameters

CC__1l The iteration constant applied in the case of the equilibrium price

CC__1c The iteration constant applied in the case of the equilibrium price

CC__1g The iteration constant applied in the case of the equilibrium price

CC__1n The iteration constant applied in the case of the equilibrium price

CC__1i The iteration constant applied in the case of the equilibrium price

CC__1r The iteration constant applied in the case of the equilibrium price

CC__1z The iteration constant applied in the case of the equilibrium price

Ceta__1 The iteration constant applied in the case of the exogenous price




Table 4.17 Endogenous variables of the computable model with sector of knowledge

Economic agent 1: Knowledge and education sector

VO_p3_1l The share of the budget for purchasing labor force at the price of P__3l

VO_t_1 The share of the budget for paying the taxes to the consolidated budget

VO_f_1 The share of the budget for paying the taxes to the off-budget funds

VO_s_1 The share of the retained budget

VY__1 The value-added produced by the sector

VS_p1_1z The supply of knowledge-provisional services at the price of P__1z

VS_p2_1z The supply of knowledge-provisional services at the price of P__2z

VS_p1_1r The supply of educational services at the price of P__1r

VD_p3_1l The demand for the labor force at the price of P__3l


VD_p1_1z The demand for knowledge-provisional services at the price of P__1z

VD_p1_1r The demand for educational services at the price of P__1r

VD_p1_1n The demand for innovative products at the price of P__1n

VD_p1_1i The demand for investment products at the price of P__1i

VY_p_1 The gain in current prices

VB__1 The budget of the sector

VB_b_1 The balance of banking accounts

VK__1 The capital assets of the sector

Economic agent 2: Innovation sector

VO_p3_2l The share of the budget for purchasing the labor force at the price of P__3l





VS_p1_2n The supply of innovative products at the price of P__1n

VS_p2_2n The supply of innovative products at the price of P__2n











Economic agent 3: Other branches of the economy

VO_p3_3l The share of the budget for purchasing the labor force at the price of P__3l





VS_p1_3c The supply of final products at the price of P__1c

VS_p1_3g The supply of final products at the price of P__1g

VS_p1_3i The supply of investment products at the price of P__1i

(continued)



VS_p2_3c The supply of final products at the price of P__2c












VO_tax_4 The share of the budget for discharging the income tax


VD_p1_4c The household demand for final products at the price of P__1c

VD_p1_4r The household demand for educational services at the price of P__1r

VW_3_1 The wages of the employees of Sector 1 (state-owned enterprises)

VW_1_1 The wages of the employees of Sector 1 (privately owned enterprises)





VB__4 The budget of the households



VD_p1_5g The demand for final products at the price of P__1g




VG_s_1 The expenditures of the consolidated budget aimed at backing Sector 1



VG_tr_4 The social transfers to the inhabitants from the consolidated budget

VG_f_4 The off-budget funds made available for the inhabitants

VB__5 The consolidated budget

VB_b_5 The remainder of the consolidated budget

VF__5 The monetary assets of the off-budget funds

VF_b_5 The remainder of the monetary assets of the off-budget funds

General parts of model

VP__1l The price of the labor force

VP__1c The price of final products for the households

VP__1g The price of final products for the economic agent no. 5

VP__1n The price of innovative products

VP__1i The price of investment products

VP__1r The price of educational services

VP__1z The price of knowledge-provisional services

(continued)


– The market for the labor force paid from the funds of the national state budget;

– The market for innovative products;

– The market for exported innovative products;

– The market for knowledge;

– The market for exported knowledge;

– The market for educational services.

The formula used in the model and determining the deficiency indicator for the

labor force market with governmental regulation of prices is given by

VI l t½ � ¼ VS p3 sl t½ �=VD p3 sl t½ �: (4.77)

The model formulas that describe the process of changing the prices for all these

markets are as follows:

The labor force price:

VP 1l Qþ 1½ � ¼ VP 1l Q½ � þ VD p1 sl t½ � � VS p1 sl t½ �ð Þ=CC 1l: (4.78)


VD_p3_sl The total demand for the labor force at the price of P__3l

VD_p1_sl The total demand for the labor force at the price of P__1l

VD_p1_sc The total demand for final products for the households at the price of P__1c

VD_p1_sg The total demand for final products for economic agent no. 5 at the price of P__1g

VD_p1_sn The total demand for innovative products at the price of P__1n

VD_p1_si The total demand for investment products at the price of P__1i

VD_p1_sr The total demand for educational services at the price of P__1r

VD_p1_sz The total demand for knowledge-provisional services at the price of P__1z

VS_p3_sl The total supply of the labor force at the price of P__3l

VS_p1_sl The total supply of the labor force at the price of P__1l

VS_p1_sc The total supply of final products the households at the price of P__1c

VS_p2_sc The total supply of final products for at the price of P__2c

VS_p1_sg The total supply of final products economic agent no. 5 at the price of P__1g

VS_p1_sn The total supply of innovative products for at the price of P__1n

VS_p2_sn The total supply of innovative products for at the price of P__2n

VS_p1_si The total supply of investment products for at the price of P__1i

VS_p1_sr The total supply of educational services for at the price of P__1r

VS_p1_sz The total supply of knowledge-provisional services for at the price of P__1z

VS_p2_sz The total supply of knowledge-provisional services for at the price of P__2z

Integral indices

VY GDP (in base period prices)

VY_p GDP (in current prices)


VK Capital assets

Technical variable

VI__l The deficiency indicator for the labor force market


The price of final products for households:

VP 1c Qþ 1½ � ¼ VP 1c Q½ � þ VD p1 sc t½ � � VS p1 sc t½ �ð Þ=CC 1c: (4.79)

The price of final products for economic agent no. 5:

VP 1g Qþ 1½ � ¼ VP 1g Q½ � þ VD p1 sg t½ � � VS p1 sg t½ �ð Þ=CC 1g: (4.80)

The price of innovative products:

VP 1n Qþ 1½ � ¼ VP 1n Q½ � þ VD p1 sn t½ � � VS p1 sn t½ �ð Þ=CC 1n: (4.81)

The price of investment products:

VP 1i Qþ 1½ � ¼ VP 1i Q½ � þ VD p1 si t½ � � VS p1 si t½ �ð Þ=CC 1i: (4.82)

The price of educational services:

VP 1r Qþ 1½ � ¼ VP 1r Q½ � þ VD p1 sr t½ � � VS p1 sr t½ �ð Þ=CC 1r: (4.83)

The price of knowledge-provisional services:

VP 1z Qþ 1½ � ¼ VP 1z Q½ � þ VD p1 sz t½ � � VS p1 sz t½ �ð Þ=CC 1z: (4.84)

We’ll now present the formulas determining the total demand for and supply of

the products for each of the prices used in the model. The final formulas determin-

ing the demand for and supply of a specific economic agent are given in the

respective items.

The total supply of and demand for the labor force at the governmental and market

prices:

VD p3 sl t½ � ¼ VD p3 1l t½ � þ VD p3 2l t½ � þ VD p3 3l t½ �; (4.85)

VD p1 sl t½ � ¼ VD p1 1l t½ � þ VD p1 2l t½ � þ VD p1 3l t½ �; (4.86)

VS p3 sl t½ � ¼ CS p3 4l t½ �; (4.87)

VS p1 sl t½ � ¼ CS p1 4l t½ �: (4.88)

The total supply of and demand for final products for households at market prices:

VD p1 sc t½ � ¼ VD p1 4c t½ �; (4.89)

VS p1 sc t½ � ¼ VS p1 3c t½ �: (4.90)


The total supply of and demand for final products for economic agent no. 5 at

market prices:

VD p1 sg t½ � ¼ VD p1 5g t½ �; (4.91)

VS p1 sg t½ � ¼ VS p1 3g t½ �: (4.92)

The total supply of and demand for innovative products at market prices:

VD p1 sn t½ � ¼ VD p1 1n t½ � þ VD p1 2n t½ � þ VD p1 3n t½ �þ VD p1 5n t½ �; (4.93)

VS p1 sn t½ � ¼ VS p1 2n t½ �: (4.94)

The total supply of and demand for investment products at market prices:

VD p1 si t½ � ¼ VD p1 1i t½ � þ VD p1 2i t½ � þ VD p1 3i t½ �; (4.95)

VS p1 si t½ � ¼ VS p1 3i t½ �: (4.96)

The total supply of and demand for educational services at market prices:

VD p1 sr t½ � ¼ VD p1 1r t½ � þ VD p1 2r t½ � þ VD p1 3r t½ �þ VD p1 4r t½ � þ VD p1 5r t½ �; (4.97)

VS p1 sr t½ � ¼ VS p1 1r t½ �: (4.98)

The total supply of and demand for knowledge-provisional services at market

prices:

VD p1 sz t½ � ¼ VD p1 1z t½ � þ VD p1 2z t½ � þ VD p1 3z t½ �þ VD p1 5z t½ �; (4.99)

VS p1 sz t½ � ¼ VS p1 1z t½ �: (4.100)

Thus, in total we have 16 formulas determining the total supply of and demand

for the products considered in this model.

Notations determining the total supply of and demand for exported products and

services are as follows:

The total supply of and demand for knowledge-provisional services (scientific

grants) at the export prices:

CD p2 sz t½ � is given; (4.101)

VS p2 sz t½ � ¼ VS p2 1z t½ �: (4.102)


The total supply of and demand for innovative products at the export prices:

CD p2 sn t½ � is given; (4.103)

VS p2 sn t½ � ¼ VS p2 2n t½ �: (4.104)

The total supply of and demand for final products at the export prices:

CD p2 sc t½ � is given; (4.105)

VS p2 sc t½ � ¼ VS p2 3c t½ �: (4.106)

Finally, we have 16 + 6 ¼ 22 formulas for determining the total supply of and

demand for all products used in this model.

Let’s now describe the activity of economic agents participating in this model.

4.2.1.5 Economic Agent No. 1: Science and Education Sector

As presented above, leveling of the total supply and demand in the markets with

governmental prices is realized by correcting the share of budget VO_p3_1. Thisprocess is described by the following formula:

VO p3 1l Qþ 1½ � ¼ VO p3 1l Q½ � � Ceta 1þ VO p3 1l Q½ �� VI l t½ � � 1� Ceta 1ð Þ: (4.107)

Here Q is the iteration step and 0<Ceta 1<1 is the model constant. With its

increase, the process of attaining equilibrium is slower. Nevertheless, the equation

system becomes more stable.

Let’s proceed to the formulas that determine the behavior of the science and

education sector.

The production function equation is given by

VY 1 tþ 1½ � ¼ CA r 1� Power VK 1 t½ � þ VK 1 tþ 1½ �ð Þ=2ð Þ;CA k 1ð Þ� PowerððVD p1 1l t½ � þ VD p3 1l t½ �Þ;CA l 1Þ� ExpðCalpha 1� VD p1 1z t½ � þ Cbeta 1� VD p1 1r t½ �þ Cgamma 1� VD p1 1n t½ �Þ:

(4.108)

Here Power(X, Y) corresponds to XY ; ExpðXÞ corresponds to eX.The following formulas determine the demand for production factors in the

science and education sector.


The demand for the labor force at the governmental prices:

VD p3 1l t½ � ¼ VO p3 1l t½ � � VB 1 t½ �ð Þ=CP 3l t½ �: (4.109)

The demand for the labor force at market prices:

VD p1 1l t½ � ¼ ðCO p1 1l t½ � � VB 1 t½ �Þ=VP 1l t½ �: (4.110)

The demand for knowledge-provisional services:

VD p1 1z t½ � ¼ CO p1 1z t½ � � VB 1 t½ �ð Þ=VP 1z t½ �: (4.111)

The demand for educational services:

VD p1 1r t½ � ¼ CO p1 1r t½ � � VB 1 t½ �ð Þ=VP 1r t½ �: (4.112)

The demand for innovative products:

VD p1 1n t½ � ¼ CO p1 1n t½ � � VB 1 t½ �ð Þ=VP 1n t½ �: (4.113)

The demand for investment products:

VD p1 1i t½ � ¼ CO p1 1i t½ � � VB 1 t½ �ð Þ=VP 1i t½ �: (4.114)

The following formulas determine the supply of the services rendered by the

science and education sector.

The supply of knowledge-provisional services at market prices:

VS p1 1z t½ � ¼ CE p1 1z� VY 1 t½ �: (4.115)

The supply of knowledge-provisional services at export prices:

VS p2 1z t½ � ¼ CE p2 1z� VY 1 t½ �: (4.116)

The supply of educational services:

VS p1 1r t½ � ¼ CE p1 1r � VY 1 t½ �: (4.117)

The following formula calculates the gain of the science and education sector

from the supplied services:

VY p 1 t½ � ¼ VS p1 1z t½ � � VP 1z t½ � þ VS p2 1z t½ � � CP 2z t½ �þ VS p1 1r t½ � � VP 1r t½ �: ð4:118Þ


The budget of the science and education sector is determined as follows:

VB 1 t½ � ¼ VB b 1 t½ � � 1þ CP bpercent t� 1½ �ð Þ þ VY p 1 t½ �þ VG s 1 t� 1½ �: (4.119)

The agent’s budget is formed from the following:

1. The funds in the banking accounts (subject to the interests on deposits);


3. Bounties received from the consolidated budget VG s 1 t� 1½ �:

The dynamics of the banking account balance in the science and education sector

is as follows:

VB b 1 tþ 1½ � ¼ VO s 1 t½ � � VB 1 t½ �: (4.120)


VK 1 tþ 1½ � ¼ 1� CR 1 t½ �ð Þ � VK 1 t½ � þ VD p1 1i t½ �: (4.121)


into account. The asset put into operation enters the formula with the plus sign.

The share of the budget of the science and education sector for discharging the

taxes to the consolidated budget is given by

VO t 1 t½ � ¼ VY p 1 t½ � � CT vad t½ �ð Þ=VB 1 t½ � þ ððVY p 1 t½ � � VW 3 1 t½ �� VW 1 1 t½ � � VK 1 t½ � � CA 0 1 t½ �Þ � CT pr t½ �Þ=VB 1 t½ �: ð4:122Þ

This formula takes into consideration the value-added tax (VAT) and profit tax.

While calculating the share of the budget for discharging the profit tax, the gain is

subtracted by the costs of the labor force of the state-owned VW 3 1 t½ �ð Þand privately owned VW 1 1 t½ �ð Þ enterprises, as well as the depreciation charges

VK 1 t½ � � CA 0 1 t½ �. The share of the budget for discharging the single social tax

to the off-budget funds is described as

VO f 1 t½ � ¼ VW 3 1 t½ � þ VW 1 1 t½ �ð Þ � CT esn t½ �ð Þ=VB 1 t½ �: (4.123)

The remainder of the budget of the science and education sector is given by

VO s 1 t½ � ¼ 1� CO p1 1l t½ � � VO p3 1l t½ � � CO p1 1z t½ � � CO p1 1r t½ �� CO p1 1n t½ � � CO p1 1i t½ � � VO t 1 t½ � � VO f 1 t½ �:

(4.124)


4.2.1.6 Economic Agent No. 2: Innovation Sector

As presented above, the leveling of the total supply and demand in the markets with

governmental prices is realized by correcting the share of budget VO_p3_2l. Thisprocess is described by the following formula:

VO p3 2l Qþ 1½ � ¼ VO p3 2l Q½ � � Ceta 2þ VO p3 2l Q½ �� VI l t½ � � 1� Ceta 2ð Þ: (4.125)

Here Q is the iteration step and 0 < Ceta__2 < 1 is the model constant. With its

increase, the process of attaining equilibrium is slower. Nevertheless, the equation

system becomes more stable. We’ll now examine the formulas determining the

behavior of the innovations sector.


VY 2 tþ 1½ � ¼ CA r 2� Power VK 2 t½ � þ VK 2 tþ 1½ �ð Þ=2ð Þ;CA k 2ð Þ� PowerððVD p1 2l t½ � þ VD p3 2l t½ �Þ;CA l 2Þ� ExpðCalpha 2� VD p1 2z t½ � þ Cbeta 2� VD p1 2r t½ �þ Cgamma 2� VD p1 2n t½ �Þ:

(4.126)

The following formulas determine the demand for the production factors in the

innovations sector:

The demand for the labor force at governmental prices:



VD p1 2l t½ � ¼ CO p1 2l t½ � � VB 2 t½ �ð Þ=VP 1l t½ �: (4.128)










The following formulas determine the supply of the products produced by the

innovations sector:

The supply of innovative products at market prices:

VS p1 2n t½ � ¼ CE p1 2n� VY 2 t½ �: (4.133)

The supply of innovative products at export prices:

VS p2 2n t½ � ¼ CE p2 2n� VY 2 t½ �: (4.134)

The following formula calculates the gain in the innovations sector:

VY p 2 t½ � ¼ VS p1 2n t½ � � VP 1n t½ � þ VS p2 2n t½ � � CP 2n t½ �: (4.135)

The budget of the innovations sector is determined as follows:



1. The funds on the banking accounts (subject to the interests on deposits);


3. Bounties received from the consolidated budget VG s 2:

The dynamics of the banking account balance of the innovations sector is as

follows:

VB b 2 tþ 1½ � ¼ VO s 2 t½ � � VB 2 t½ �: (4.137)




into account. The asset put into operation enters the formula with a plus sign.

The share of the budget of the innovations sector for discharging taxes to the



VO t 2 t½ � ¼ VY p 2 t½ � � CT vad t½ �ð Þ=VB 2 t½ � þ ððVY p 2 t½ � � VW 3 2 t½ �� VW 1 2 t½ � � VK 2 t½ � � CA 0 2 t½ �Þ � CT pr t½ �Þ=VB 2 t½ �: ð4:139Þ

This formula takes into consideration the VAT and profit tax. While calculating

the share of the budget for discharging the profit tax, the gain is subtracted by the

costs of the labor force of the state-owned VW 3 2ð Þ and privately owned VW 1 2ð Þenterprises, as well as the depreciation charges VK 2 t½ � � CA 0 2 t½ �:




The remainder of the budget of the innovations sector is given by

VO s 2 t½ � ¼ 1� CO p1 2l t½ � � VO p3 2l t½ � � CO p1 2z t½ � � CO p1 2r t½ �� CO p1 2n t½ � � CO p1 2i t½ � � VO t 2 t½ � � VO f 2 t½ �: ð4:141Þ

4.2.1.7 Economic Agent No. 3: Other Branches of the Economy

As presented above, the leveling of the total supply and demand in the markets with

governmental prices is realized by correcting the share of budget VO_p3_3l. Thisprocess is described by the following formula:

VO p3 3l Qþ 1½ � ¼ VO p3 3l Q½ � � Ceta 3þ VO p3 3l Q½ � � VI l t½ �� 1� Ceta 3ð Þ: (4.142)

Here Q is the iteration step and 0<Ceta 3<1 is the model constant.

We’ll now look at the formulas determining the behavior of the other branches of

economy.


VY 3 tþ 1½ � ¼ CA r 3� Power VK 3 t½ � þ VK 3 tþ 1½ �ð Þ=2ð Þ;CA k 3ð Þ� Power VD p1 3l t½ � þ VD p3 3l t½ �ð Þ;CA l 3ð Þ� ExpðCalpha 3� VD p1 3z t½ � þ Cbeta 3

� VD p1 3r t½ � þ Cgamma 3� VD p1 3n t½ �Þ: ð4:143Þ

The following formulas determine the demand of the production factors in other

branches of the economy:





VD p1 3l t½ � ¼ CO p1 3l t½ � � VB 3 t½ �ð Þ=VP 1l t½ �: (4.145)









The following formulas determine the supply of products produced by the other

branches of the economy:

The supply of final products for households:

VS p1 3c t½ � ¼ CE p1 3c� VY 3 t½ �: (4.150)

The supply of final products for economic agent no. 5:

VS p1 3g t½ � ¼ CE p1 3g� VY 3 t½ �: (4.151)

The supply of investment products:

VS p1 3i t½ � ¼ CE p1 3i� VY 3 t½ �: (4.152)

The supply of exported products:

VS p2 3c t½ � ¼ CE p2 3c� VY 3 t½ �: (4.153)

The following formula calculates the gain of the other branches of the economy:

VY p 3 t½ � ¼ VS p1 3c t½ � � VP 1c t½ � þ VS p1 3g t½ � � VP 1g t½ �þ VS p1 3i t½ � � VP 1i t½ � þ VS p2 3c t½ � � CP 2c t½ �: (4.154)


The budget of the other branches of the economy is determined as follows:



1. The funds in banking accounts (subject to the interests on deposits);


3. Bounties received from the consolidated budget VG s 3.

The dynamics of the banking account balance of the other branches of the

economy is as follows:

VB b 3 tþ 1½ � ¼ VO s 3 t½ � � VB 3 t½ �: (4.156)




into account. The asset put into operation enters the formula with the plus sign.

The share of the budget of the other branches of the economy for discharging the

taxes to the consolidated budget is given by

VO t 3 t½ � ¼ VY p 3 t½ � � CT vad t½ �ð Þ=VB 3 t½ � þ ððVY p 3 t½ � � VW 3 3 t½ �� VW 1 3 t½ � � VK 3 t½ � � CA 0 3 t½ �Þ � CT pr t½ �Þ=VB 3 t½ �:

(4.158)

This formula takes into consideration the VAT and profit tax. While calculating

the share of the budget for discharging the profit tax, the gain is subtracted by the

costs of the labor force of state-owned VW 3 3ð Þ and privately owned VW 1 3ð Þenterprises, as well as the depreciation charges.




The remainder of the budget of other branches of the economy is given by

VO s 3 t½ � ¼ 1� CO p1 3l t½ � � VO p3 3l t½ � � CO p1 3z t½ � � CO p1 3r t½ �� CO p1 3n t½ � � CO p1 3i t½ � � VO t 3 t½ � � VO f 3 t½ �: ð4:160Þ


4.2.1.8 Economic Agent No. 4: The Aggregate Consumer (Households)

Let’s now proceed to the formulas determining the behavior of the aggregate

consumer.

The household demand for final products is given by

VD p1 4c t½ � ¼ CO p1 4c t½ � � VB 4 t½ �ð Þ=VP 1c t½ �: (4.161)

The household demand for educational services:


The wages of the employees of state-owned enterprises in the science and education

sector:

VW 3 1 t½ � ¼ VD p3 1l t½ � � CP 3l t½ �: (4.163)

The wages of the employees of privately owned enterprises in the science and

education sector:

VW 1 1 t½ � ¼ VD p1 1l t½ � � VP 1l t½ �: (4.164)

The wages of the employees of state-owned enterprises in the innovations sector:

VW 3 2 t½ � ¼ VD p3 2l t½ � � CP 3l t½ �; (4.165)

The wages of the employees of privately owned enterprises in the innovations

sector:

VW 1 2 t½ � ¼ VD p1 2l t½ � � VP 1l t½ �: (4.166)

The wages of the employees of state-owned enterprises in the other branches of the

economy:

VW 3 3 t½ � ¼ VD p3 3l t½ � � CP 3l t½ �: (4.167)

The wages of the employees of privately owned enterprises in the other branches of

the economy:

VW 1 3 t½ � ¼ VD p1 3l t½ � � VP 1l t½ �: (4.168)



VB 4 t½ � ¼ VB b 4 t� 1½ � � 1þ CP h bpercent t� 1½ �ð Þþ VB 4 t� 1½ � � VO s 4 t� 1½ � þ VW 3 1 t½ � þ VW 1 1 t½ � þ VW 3 2 t½ �þ VW 1 2 t½ � þ VW 3 3 t½ � þ VW 1 3 t½ � þ VG f 4 t� 1½ �þ VG tr 4 t� 1½ �:

(4.169)


1. The funds in the banking accounts (subject to the interests on deposits);


3. Wages received from the three producing agents;

4. Pensions, welfare payments, and subsidies received from the off-budget funds.

The dynamics of the banking account balance of the households is as follows:

VB b 4 t½ � ¼ CO b 4 t½ � � VB 4 t½ �: (4.170)

The share of the budget for discharging income tax is given by

VO tax 4 t½ � ¼ ððVW 3 1 t½ � þ VW 1 1 t½ � þ VW 3 2 t½ � þ VW 1 2 t½ � þ VW 3 3 t½ �þ VW 1 3 t½ �Þ � CT pod t½ �Þ=VB 4 t½ �: ð4:171Þ

The remainder of the money in cash is as follows:

VO s 4 t½ � ¼ 1� CO p1 4c t½ � � CO p1 4r t½ � � VO tax 4 t½ � � CO b 4 t½ �: (4.172)

4.2.1.9 Economic Agent No. 5: Government

Let’s now review the formulas determining the behavior of economic agent no. 5.

The consolidated budget is given by

VB 5 t½ � ¼ VO t 1 t½ � � VB 1 t½ � þ VO t 2 t½ � � VB 2 t½ � þ VO t 3 t½ � � VB 3 t½ �þ VO tax 4 t½ � � VB 4 t½ � þ CB other 5þ VB b 5 t½ �� 1þ CP bpercent t� 1½ �ð Þ: ð4:173Þ

This formula sums up money collected as taxes from the producing agents, as

well as from inhabitants. The valueCB other 5 entered in the model exogenously is

the sum of other taxes (not included in the list of taxes considered in this model),


nontaxable income, and other income included in the consolidated budget. The

obtained sum is to be supplemented by the funds in banking accounts (subject to the

deposit interests).


determined by

VB b 5 tþ 1½ � ¼ CO s 5b t½ � � VB 5 t½ �: (4.174)

The cash assets of off-budget funds are as follows:

VF 5 t½ � ¼VO f 1 t½ � � VB 1 t½ � þ VO f 2 t½ � � VB 2 t½ � þ VO f 3 t½ � � VB 3 t½ �þ VF b 5 t½ � � 1þ CP bpercent t� 1½ �ð Þ:

(4.175)

This formula calculates the sum collected from the producing agents in the form

of the single social tax entering the accounts of the following off-budget funds:

– The pension fund;

– The social insurance fund;

– The federal and territorial funds of obligatory medical insurance.

The derived sum is supplemented by the funds on the banking accounts (subject

to the deposit interests).


mined by

VF b 5 tþ 1½ � ¼ CO s 5f t½ � � VF 5 t½ �: (4.176)

The demand for final products:

VD p1 5g t½ � ¼ CO p1 5g t½ � � VB 5 t½ �ð Þ=VP 1g t½ �: (4.177)

The knowledge-provisional service payment:


The educational service payment:





The subsidies to the producing sectors are as follows.

The science and education sector:

VG s 1 t½ � ¼ CO s1 5 t½ � � VB 5 t½ �: (4.181)

The innovations sector:

VG s 2 t½ � ¼ CO s2 5 t½ � � VB 5 t½ �: (4.182)

Other branches of the economy:

VG s 3 t½ � ¼ CO s3 5 t½ � � VB 5 t½ �: (4.183)

Social transfers to inhabitants:

VG tr 4 t½ � ¼ CO tr 5 t½ � � VB 5 t½ �: (4.184)

The assets of the off-budget funds made available for the inhabitants:

VG f 4 t½ � ¼ CO f4 5 t½ � � VF 5 t½ �: (4.185)

This includes assets of the pension and social insurance fund for paying out

pensions and welfare payments.

4.2.1.10 Integral Indices of Model

In this subsection we present the formulas for calculating some integral indices of

the economy of the Russian Federation.

The GDP (in prices of the base period):

VY t½ � ¼ VY 1 t½ � þ VY 2 t½ � þ VY 3 t½ �: (4.186)

The GDP (in current prices):

VY p t½ � ¼ VY p 1 t½ � þ VY p 2 t½ � þ VY p 3 t½ �: (4.187)

The consumer price index:

VP t½ � ¼ 100� VP 1c t½ �=VP 1c t� 1½ �ð Þ: (4.188)

Capital assets:

VK t½ � ¼ VK 1 t½ � þ VK 2 t½ � þ VK 3 t½ �: (4.189)


In this model:

– Relations (1.12) are represented by 12 expressions for finding the gross value-

added (GVA) of sectors by means of the production functions, capital assets of

the sectors, and balance of the banking accounts;

– Relations (1.13) are represented by 88 expressions for finding the supply and

demand for various products and services of the sectors, the budgets and shares

of budgets of the sectors, subsidies to the sectors from the consolidated budget,

etc.;

– Relations (1.14) are represented by 10 expressions for finding the equilibrium

market prices and shares of budgets of the sectors in the markets with the

exogenous prices.

The values of exogenous functions of the considered model were determined by

solving the problem of parametric identification of the model with the use of the

available statistical data at the Republic of Kazakhstan covering the period of 2000

to 2008. The validity of the model and identification process is ensured by the

following facts:

1. The identification criterion includes statistical information on basic macroeco-

nomic indices (the GDP and GVA of the sectors, the capital assets of the

sectors, etc.).

2. The estimations of the exogenous parameters and initial values of the difference

equations, which have measured values, are found in the intervals with their

centers in the respective measured values or those covering several measured

values.

3. The intervals for estimating other parameters are determined by indirect factors.

The values of parameters varying with years are found under assumptions on the

insignificance of their variations.

As a result of solving the parametric identification problem, the value of the

relative mean square deviation of the calculated values of the endogenous

variables from the respective measured values (statistical information) is less

than 1%.

4. The validity test of the model for the purpose of assessing its capability to

produce the precise prediction values was carried out via the retrospective

prediction. To do this, after solving the parametric identification problem

using the statistical information of the Republic of Kazakhstan [37] from

2000 to 2007, the values of all exogenous variables of the model were extended

to 2008, and model computation were performed for the test period without

performing additional parametric identification.

As a result of the joint solution of problems A and B (see Sect. 4.1.1), with

application of the Nelder–Mead algorithm [66], we obtain value KB ¼ 0.0073. The

relative value of deviations of the computed values of the variables used in the main

criterion from the respective measured values is less than 0.73%.

The results of computation and retrospective prediction of the model for

2008 partially presented in Table 4.18 demonstrate computed VY; VY 1; VY 2;ð


http://dx.doi.org/10.1007/978-1-4614-4460-2_1

http://dx.doi.org/10.1007/978-1-4614-4460-2_1

http://dx.doi.org/10.1007/978-1-4614-4460-2_1

Table

4.18

Measuredandcomputedvalues

ofthemodel

outputvariablesandrespectivedeviations

Year

2000

2001

2002

2003

2004

2005

2006

2007

2008

VY*

2.60

2.95

3.24

3.54

3.88

4.26

4.72

5.14

5.30

VY

2.60

2.95

3.24

3.54

3.88

4.26

4.72

5.15

5.26

DVY

�0.04

0.00

0.04

0.03

0.04

�0.01

0.07

0.26

�0.73

VY

1*

0.107

0.102

0.116

0.125

0.134

0.157

0.164

0.165

0.200

VY

10.109

0.102

0.116

0.126

0.134

0.158

0.164

0.164

0.199

DVY

11.80

�0.76

�0.01

1.32

0.46

0.49

�0.02

�0.59

�0.34

VY

2*

0.0330

0.0364

0.0410

0.0450

0.0500

0.0490

0.0680

0.0720

0.0610

VY

20.0410

0.0343

0.0390

0.0440

0.0500

0.0490

0.0670

0.0710

0.0610

DVY

225.00

�5.86

�4.73

�2.65

�0.57

�0.19

�0.13

�0.88

0.16

VY

3*

2.23

2.46

2.80

3.07

3.36

3.68

4.03

4.48

4.88

VY

31.900

2.463

2.800

3.070

3.360

3.680

4.030

4.480

4.890

VY

3�1

4.63

0.08

0.07

0.03

0.02

0.02

0.00

0.11

0.28

VP*

106.40

106.60

106.80

106.70

107.50

108.40

118.80

109.50

VP

106.57

106.81

106.95

106.83

107.64

108.50

118.90

109.50

DVP

0.00

0.16

0.20

0.14

0.12

0.13

0.09

0.09

0.00


V 3;VPÞ, measured values, and deviations of computed values of the main output

variables of the model from the respective measured values. Here years 2000–2007

correspond to the period of model parametric identification; 2008 is the retrospective

prediction period; VY is GDP (� 1012 tenge in prices of year 2000);VY i is the grossvalue-added by the ith sector (� 1012 tenge in prices of year 2000);

GDP (� 1012 tenge in prices of year 2000); VP is the consumer price index in

percentage with respect to a preceding year; the symbol * corresponds to the

measured values; and the symbol D corresponds to the deviations (in percentage)

of the computed values from the respective measured values.

The mean error of the computed values of endogenous variables of the model

relative to the respective measured values in the period of retrospective prediction

is 1.04%.

The results of verification show acceptable adequacy of the CGE model of

economic branches.

4.2.2 Estimation of the Macroeconomic Theory Provisionson the Basis of the Computable Model of GeneralEquilibrium with the Knowledge Sector

On the basis of the computable model with the knowledge sector, we carried out

experimental testing of the main provisions of the macroeconomic theory aimed at

detecting possible sources of cyclic oscillations of macroeconomic indicators as a

result of shocking changes of demands in final and investment goods. The

experiments are similar to the experiments described in Sect. 4.1.2.4 for a comput-

able model of general equilibrium of economic branches; they consist of

computations of the following scenarios of changes of demands of the final and

investment products:

(a) O1 t½ � ¼ O1 t� 1½ � þ a Y t½ � � Y t� 1½ �ð Þ; (4.190)

(b) O2i t½ � ¼ O2i t� 1½ � þ b Y t� 3½ � � 2Y t� 2½ � þ Y t� 1½ �ð Þ i ¼ 1; 2; 3ð Þ; (4.191)

(c) Joint application of scenarios (a) and (b);

(d) Increase in the shares O1ðtÞ k times in comparison with the base variant;

(e) Increase in the sharesO2iðtÞ i ¼ 1; :::; 16ð Þ l times in comparison with the base

variant;

(f) Joint application of scenarios (d) and (e).

Here t ¼ 2010, . . ., 2015 is the time in years; O1 t½ � ¼ CO p 4c t½ � is the share ofthe budget of households spent for purchasing final products (exogenous function);

O2iðtÞ is the share of the budget of the ith branch spent for purchasing investment

products (i ¼ 1, 2, 3) (exogenous function); Y[t] ¼ VY[t] is GDP in prices of year

2000 (endogenous variable); a, b, l, k are some positive constants (l > 1, k > 1).


As a result of computational experiments on the basis of the model aimed at

realization of scenarios (a)–(f), one can observe cyclic oscillations of variable P(t)(consumer price index) (see Figs. 4.21, 4.22, 4.23, and 4.24). We did not reveal the

oscillatory phenomena for values of real indicators (in particular, Y[t]). While

applying all proposed scenarios, one can observe a decrease in the GDP indicator

Y[t] in comparison to the base period.

The experimental results presented in Figs. 4.21, 4.22, 4.23, and 4.24 prove

respective provisions of the macroeconomic theory [40], just as for the case of the

computable model of economic branches considered above.

Base Scenario

Fig. 4.21 Values of the consumer price index with application of scenario (a) with a ¼ 3 � 10�13

and for the base variant

Base Scenario

Fig. 4.22 Values of the consumer price index with application of scenario (b) with b ¼ 10�13 and



4.2.3 Finding Optimal Parametric Control Laws Basedon the CGE Model with the Knowledge Sector

4.2.3.1 Attenuation of Cyclic Oscillations of Macroeconomic Indicatorsby Parametric Control Methods

In computational experiments, we consider the problem of attenuating cyclic

oscillations in the consumer price level arising from application of scenario (c) to

the economic system development while taking into account its linear dependence

between the volume of consumer expenses and its current income as well as the

linear dependence between investments and income growth. This problem is

Base Scenario

Fig. 4.23 Values of the consumer price index with application of scenario (c) with a ¼ 3 � 10�13;b ¼ 1013 and for the base variant

Base Scenario

Fig. 4.24 Values of the consumer price index with application of scenario (d) with and for the

base variant


similar to the problem considered in Sect. 4.1.3.1 for the case of a computable

model of economic branches.

On the basis of the computable model of economic branches and application of

scenario (c) (see Sect. 4.2.2, where a ¼ 3 � 10�13; b ¼ 10�13), find the values of

the taxation ratesCT vad t½ �; CT pr t½ �; CT pod t½ �; andCT esn t½ � from 2010 to 2015,

which provide the lower bound of the following criterion Kp; which characterizes

the deviations of computed values of the consumer price index VP½t� from their

respective desired values of P[t]:

Kp ¼X2015t¼2010

VP t½ � � P t½ �P t½ �

� �2:

Here we use the computed base values of the model consumer price index

without parametric control as the desired values of P[t].The constraints on the adjusted parameters are as follows: 0:104 � CT vad ½t�

� 0:156; 0:24 � CT pr½t� � 0: 36; 0:04 � CT pod½t� � 0:06; 0:16 � CT esn½t�� 0:24:

The constraints on the growth of the macroeconomic indicator:Yr t½ � 0:95Yc t:½ �Here Yc[t] are the computed scenario GDPs without parametric control; Yr[t] arethe computed values of the GDP with parametric control.

The value of criterion KP without parametric control is Kp ¼ 0.424. The optimal

values of the criterion KP with application of the parametric control law appears to

be Kp ¼ 0.000844.

The base values of macroeconomic indicators Y(t) and P(t), values obtained withapplication of scenario (c), and values obtained with application of the optimal law

of parametric control are presented in Figs. 4.25 and 4.26.

AttenuationBase Scenario

Fig. 4.25 Computed values of GDP


Analysis of the results of computation experiments presented in Figs. 4.25 and

4.26 shows that application of the derived optimal parametric control law to the

consumer price index in 2011 appears to be less than the scenario index and

practically coincides with the desired values starting from year 2012. Also, the

values of the GDP practically coincide with the initial values of the scenario

variant.

4.2.3.2 Finding Optimal Parametric Control Laws on the Basisof the Stochastic CGE Model with the Knowledge Sector

The stochastic computable model of economic branches was derived from the

respective deterministic model (with the estimates of values of the exogenous

parameters found from solving the parametric identification problem) via adding

discrete Gaussian noise with independent constituents to the right-hand sides of all

the model dynamic equations. These equations include the following for computing

the endogenous variables:

– Values added VY 1; VY 2; VY 3ð Þ in three producing sectors by means of the

respective production functions;

– Values added VK 1; VK 2; VK 3ð Þ in three producing sectors;

– Annual budgets VB 1; VB 2; . . . ; VB 5ð Þ of sectors 1–5.The noise added to the expressions with respect to VY__i can initiate respective

cyclic oscillations caused by quick shifts (shocks) in the development of techno-

logical progress and random changes of the population growth rate. The additive

noise in the expressions with respect to VK i also characterizes random changes to

the shares of budgets of the producing agents spent for purchasing investment

AttenuationBase Scenario

Fig. 4.26 Computed values of the consumer price index


goods and random character of the coefficients of retirement funds. The additive

noise in the expression with respect to VB i describes the random character of the

incomes obtained by the sector within the current period.

In this work the estimations of mean square deviations of the generated Gaussian

random values defining the mentioned noise were obtained by using statistical data

on economic development of the Kazakhstan Republic from 2000 to 2008 as

follows:

For each time series of measured values of the aforementioned variables, we

computed some selected mean square deviations of differences between the

measured values and trends of those values. The values obtained were accepted

to be the estimated mean square deviations of the components of the generated

discrete Gaussian noise xðtÞ added to the right-hand sides of the 11 dynamic

equations indicated above.

In computational experiments with the stochastic computable model of the

economic branch, we use the following optimization criterion:

Ks ¼ E1

6

X2015t¼2010

VY t½ �( )

! max (4.192)

Here Ks is the expectation value of the average GDP of the country in prices of

year 2000 in the period 2010–2015.

In experiments with optimization criterion (4.192), we applied additional

constraints on the growth of the consumer prices of the following form:

E VPr t½ �ð Þ � 1; 09E VP t½ �ð Þ; t ¼ 2010; . . . ; 2015 (4.193)

Here VP t½ � is the computed level of the consumer prices in the model without

parametric control, VPr t½ � is the consumer price level with parametric control.

In the computational experiments, we realize regulation of 18 adjusted

parameters Oi t½ � ¼ CO s1 i ½t� t ¼ 2010; . . . ; 2015; i ¼ 1; 2; 3ð Þ; shares of the

consolidated budget spent for the financing of three producing agents in

2010–2015. A natural limitation must hold; namely, the sum of all eight shares of

the consolidated budget used in the model (including the shares Oi t½ �; i ¼ 1; 2; 3,mentioned above) must not exceed 1:

X8

i¼1Oi t½ � � 1; t ¼ 2010; . . . ; 2015: (4.194)

We considered the following problem of finding optimal values of the adjusted

parameters. On the basis of the stochastic computable model with the knowledge

sector, find the values of the shares Oi ½t� of the consolidated budgets spent for

financing three producing agents, which provide the upper bound of the criterion Ks

with additional constraints (4.194) on these shares.

This optimization problem was solved using the Nelder–Mead algorithm.

After application of parametric control to the shares of the stochastic model


budgets, the value of the criterion appears to be KS ¼ 7:234 � 1012; its value

increases by 24,93% KS ¼ 5:79 � 1012: in comparison to the base variant.

A similar parametric control problem with the respective constraints was also

solved on the basis of the deterministic CGE model with the knowledge sector with

use of the deterministic analog of criterion (4.192):

Kd ¼ 1

6

X2015t¼2010

YðtÞ:

After application of parametric control to the shares of budgets of the producing

agents, the value of the criterion for the deterministic model appears to be equal to

Kd ¼ 9:23 � 1012: The criterion value increases by 33.14% in comparison with the

base variant.

4.3 National Economic Evolution Control Basedon the Computable Model of General Equilibriumwith the Shady Sector


As we know, the national economies of many countries [24] work with the shady

sector. Their activities negatively influence the national economy’s evolution.

Using a mathematical model of a national economy for the estimation of and search

for effective economic measures to decrease the influence of the shady sector

activity on the national economic system’s evolution seems to be proper direction.

4.3.1.1 Economic Agents of the Model

The considered model [27] describes the behavior and interaction of 10 product

markets and 3 labor markets in the following 7 economic agents: The first three are

producing agents.

Economic agent no. 1 is the state sector of the economy. This includes enterprises

with more than a 50% share of the state.

Economic agent no. 2 is the market sector consisting of the legally existing

enterprises and organizations with private and mixed ownership.


The state and market sectors produce products distributed among the following

four directions:

1. The final product for households including consumer nondurable products

(foodstuffs, etc.), durable products (house equipment, motor vehicles, etc.), as

well as services;

2. Final products for economic agent no. 5, including

(a) The final product for public institutions, including free services for

inhabitants rendered by the enterprises and organizations in the field of public

health, education, and culture; services satisfying the needs of the society as a

entirety, i.e., general public administration, maintaining law and order,

national defense, nonmarket science, housing, and communal servicing, etc.;

(b) The final product for nonprofit organizations servicing the households,

including the free services of a social character;

3. Investment products, namely, the costs of creation of capital assets. This type of

product does not include the state (or governmental) investment since it is taken

into account in the preceding type of products. The capital assets are considered

a separate type of product in this model;

4. The exported products. Since the imported products form one of the components

of the products considered above, only the net export is included in exported

products.

Besides produced products, the state and market sectors trade capital assets

represented by the capital products in the model.

Economic agent no. 3 is the shady sector.

There are various types of shady economics [27], namely,

– White-collar shady economics is the unofficial economic activity of employees

in the registered economy concerned with their official professional activity.

This includes the economics of informal ties (i.e., offstage performance of the

ordinary production programs), upward distortion economics (presenting ficti-

tious results as real), and bribe economics (abuse of official status of the public

officers for achieving private goals).

– So-called gray (informal) shady economics is the lawful economic activity that

is not accounted for by the official statistics. This sector of shady economics

produces mainly the ordinary products and services (just as in legal economics),

but the producers avoid official taxes, not wishing to pay additional costs

concerned with the discharge of taxes, etc.

– So-called black (underground) shady economics is the statute-prohibited eco-

nomic activity concerned with the production and selling of prohibited products

and services (selling drugs, racketeering, etc.).

As for the shady sector of the considered model, it includes “gray” shady

economics as well as white-collar shady economics represented by the production

of the final goods for households by the market sector of the economy.


The shady sector sells only one type of product, namely, the final product for

households. This economics agent does not pay taxes and receive subsidies. The

shady sector realizes the following actions:

– By distribution of its budget, it pays the services of the labor force and

determines the share of the retained budget;

– By distribution of produced products, it determines the share of final products for

selling in the market of final products for the households at the shady price.

Economic agent no. 4 is the aggregated consumer representing all households of the

country. Moreover, within the frameworks of this sector, the supplies of the labor

force for the state, market, and shady sectors are determined.

Economic agent no. 5 is the government represented by the aggregate of the central,

regional, and local governments as well as the off-budget funds. In addition, this

sector includes nonprofit organizations servicing households (political parties, trade

unions, public associations, etc.).

Economic agent no. 5 establishes taxation rates and defines the sum of the

subsidies to the producing agents and social transfer and spends its budget for

purchasing final products produced by the state and market sectors.

Economic agent no. 6 is the banking sector, including the central and commercial

banks.

The banking sector establishes interest rates for the attracted deposits and issues

money.

Economic agent no. 7 is the outer world.

4.3.1.2 Exogenous Parameters of the Model

The exogenous parameters of the model include the following:

– Coefficients of the production functions of the sectors;

– Various shares of the sectors’ budgets;

– Shares of production for selling in various markets;

– Depreciation rates for capital assets;

– Deposit interest rates;

– Issuance of money;

– Various taxation rates;

– Shares of the consolidated budget spent for purchasing final goods, backing the

state and market sectors, as well as for social transfers;

– Export prices of final goods for the outer world.

The list of the exogenous parameters of the model is given in Table 4.19.


Table 4.19 Exogenous variables of the computable model of economic branches with the shady

sector

Economic agent 1: State sector of the economy

O_1k_P2 The share of the state budget spent for purchasing the capital products at the price of

P_2k

O_1i_P2 The share of the state budget spent for purchasing investment products at the price of

P_2i

E_1c_P2 The share of the state sector products for selling in the markets of final products at the

price of P_2c

E_1g_P1 The share of the state sector products for selling in the markets of final products for

economic agent no. 5 at the price of P_1g

E_1g_P2 The share of the state sector products for selling in the markets of final products for


E_1i_P1 The share of the state sector products for selling in the markets of investment

products at the price of P_1i

E_1i_P2 The share of the state sector products for selling in the markets of investment


E_1k_P1 The share of the capital assets of the state sector for selling in the markets of capital

products at the price of P_1k

E_1k_P2 The share of the capital assets of the state sector for selling in the markets of capital

products at the price of P_2k

E_1ex_Pex The share of the capital assets of the state sector for selling in the markets of capital

products in the outer world countries at the price of P_ex

A_1_r The empirically determined coefficient of dimension of the state sector

A_1_k The coefficient of the state sector capital

A_1_l The coefficient of the state sector labor

A_1_n The depreciation rate for the capital assets of the state sector

Economic agent 2: Market sector

O_2l_P2 The share of the budget of the market sector spent for purchasing the labor force at the

price of P_2l

O_2k_P2 The share of the budget of the market sector spent for purchasing the capital products

at the price of P_2k

O_2i_P2 The share of the budget of the market sector spent for purchasing investment products

at the price of P_2i

E_2c_P2 The share of the market sector products for selling in the markets of final products at

the price of P_2c

E_2c_P3 The share of the market sector products for selling in the markets of final products at

the price of P_3c

E_2g_P2 The share of the market sector products for selling in the markets of final products for


E_2i_P2 The share of the market sector products for selling in the markets of investment


E_2k_P2 The share of the market sector products for selling in the markets of capital products

at the price of P_2k

E_2ex_Pex The share of the market sector products for selling in the markets of final products in

the outer world countries at the price of P_ex

A_2_r The empirically determined coefficient of the dimension of the market sector

A_2_k The coefficient of the market sector capital

A_2_l The coefficient of the market sector labor

A_2_n The depreciation rate for the capital assets of the market sector

(continued)



Economic agent 3: Shady sector

O_3l_P3 The share of the budget of the shady sector spent for purchasing the labor force at the

price of P_3l

E_3c_P3 The share of the shady sector products for selling in the markets of final products at

the price of P_3c

A_3_r The empirically determined coefficient of dimension of the shady sector

A_3_k The coefficient of the shady sector capital

A_3_l The coefficient of the shady sector labor


L_1_a The part of the employees entering the state sector (e.g., starting their working

activity in the state sector)

L_1_r The part of the employees withdrawing the state sector (for example, retired

employees)

L_2_a The part of the employees entering the market sector (e.g., starting their working

activity in the market sector)

L_2_r The part of the employees withdrawing from the market sector (for example, the

retired employees)

O_4c_P1 The share of the budget of the households for purchasing final products at the price of

P_1c


P_2c


P_3c

O_4_$ The share of the budget of the households for purchasing the foreign currency

O_4_b The share of the budget of the households for saving (as banking deposits)

L_1_2 The part of the state sector employees leaving for the market sector

L_2_1 The part of the market sector employees leaving for the state sector

L_12_3 The part of the state and market sector employees partially employed in the shady

market


T vad The value-added tax

T pr The profit tax for organizations

T prop The property tax

T pod The income tax for physical bodies

T esn The single social tax

O 5g P2 The share of the consolidated budget for purchasing final products at the price of

P_2g

O_5_s1 The share of the consolidated budget for backing the state sector

O_5_s2 The share of the consolidated budget for backing the market sector

O_5_tr The share of the consolidated budget for payment of the social transfers

O_5_f1 The share of expenditures of the off-budget funds spent for the state sector

O_5_f2 The share of expenditures of the off-budget funds spent for the market sector

O_5_f4 The share of expenditures of the off-budget funds spent for the households

B_5_Other The sum of tax proceeds (not included into consideration), nontaxable incomes, and

other incomes of the consolidated budget

(continued)





– The shares of the produced products for selling in various markets;

– The remainders of the agents’ budgets;

– The produced values-added of the producing sectors;

– Supply of and demand for various products and services;


– The capital assets of the producing sectors;

– The parts of the employees withdrawing from each of the producing sectors;

– The parts of the employees entering each of the producing sectors;

– The wages of employees;


– The various types of prices of products, services, and the labor force;

– The subsidies to the producing sectors;

– The social transfers to inhabitants;

– The gross production of goods and services;

– The GDP.

The list of the endogenous variables of the model is given in Table 4.20.


Economic agent 6: Banking sector

M_1 The issuance of money of the state sector

M_2 The issuance of money of the market sector

P_b The deposit interest rate for the enterprises

P_b_h The deposit interest rate for the physical bodies

General parts of the model

P_1l The state prices of labor force

P_1C The state prices of final products for households

P_1g The state prices of final products for economic agent no. 5

P_1i The state prices of investment products

P_1k The state prices of the capital products

P_1ex The state prices of investment products for the outer world

Model constants

etta The constant used for correction of the shares of budgets of the agents while leveling

the aggregate supply and demand in the markets with the state prices

Q The iteration step

C The iteration constant used for changing the velocity of computations of the

equilibrium state of the CGE model


Table 4.20 Exogenous variables of the computable model of the economic branches with the

shady sector

Economic agent 1: State sector of the economy

O_1l_P1 The share of the state sector budget for purchasing the labor force at the price of P_1l

O_1k_P1 The share of the state sector budget spent for purchasing capital products at the price

of P_1k

O_1i_P1 The share of the state sector budget spent for purchasing investment products at the

price of P_1i

O_1_t The share of the state sector budget for discharging taxes to the consolidated budget

O_1_f The share of the state sector budget for discharging taxes to off-budget funds

O_1_s The share of the retained budget of the state sector

E_1c_P1 The share of the state sector products for selling in the markets of final products at the

price of P_1c

Y_1 The value-added of the state sector (in the prices of the base period)

S_1c_P1 The supply of final products by the state sector at the price of P_1c

S_1c_P2 The supply of final products by the state sector at the price of P_2c

S_1g_P1 The supply of final products by the state sector for economic agent no. 5 at the price of

P_1g

S_1g_P2 The supply of final products by the state sector for economic agent no. 5 at the price of

P_2g

S_1i_P1 The supply of investment products by the state sector at the price of P_1i

S_1i_P2 The supply of investment products by the state sector at the price of P_2i

S_1k_P1 The supply of the capital products by the state sector at the price of P_1k

S_1k_P2 The supply of the capital products by the state sector at the price of P_2k

S_1ex_Pex The supply of the exported products by the state sector at the price of P_ex

D_1l_P1 The demand for the labor force in the state sector at the price of P_1l

D_1k_P1 The demand for the capital products in the state sector at the price of P_1k

D_1k_P2 The demand for the capital products in the state sector at the price of P_2k

D_1i_P1 The demand for investment products in the state sector at the price of P_1i

D_1i_P2 The demand for investment products in the state sector at the price of P_2i

Y_1_p The gain of the state sector in current prices

B_1 The state sector budget

B_1_b The balance of the banking accounts of the state sector

K_1 The capital assets of the state sector

Economic agent 2: Market sector of the economy

O_2k_P1 The share of the market sector budget spent for purchasing capital products at the

price of P_1k

O_2i_P1 The share of the market sector budget spent for purchasing capital products at the

price of P_1i

O_2_t The share of the market sector budget for discharging the taxes to the consolidated

budget

O_2_f The share of the market sector budget for discharging the taxes to the off-budget

funds

O_2_s The share of the retained budget of the market sector

Y_2 The value-added of the market sector (in the prices of the base period)

S_2c_P2 The supply of final products by the market sector at the price of P_2c

S_2c_P3 The supply of final products by the market sector at the price of P_3c

S_2g_P2 The supply of final products by the market sector for economic agent no. 5 at the price

of P_2g

(continued)



S_2i_P2 The supply of investment products by the market sector at the price of P_2i

S_2k_P2 The supply of the capital products by the market sector at the price of P_2k

S_2ex_Pex The supply of the exported products by the market sector at the price of P_ex

D_2l_P2 The demand for the labor force in the market sector at the price of P_2l

D_2k_P1 The demand for capital products in the market sector at the price of P_1k

D_2k_P2 The demand for capital products in the market sector at the price of P_2k

D_2i_P1 The demand for investment products in the market sector at the price of P_1i

D_2i_P2 The demand for investment products in the market sector at the price of P_2i

Y_p The gain of the market sector in the current prices

B_2 The market sector budget

B_2_b The balance of banking accounts in the market sector

K_2 The capital assets of the market sector

Economic agent 3: Shady sector

O_3_s The share of the retained budget of the shady sector

Y_3 The value-added of the shady sector (in the prices of the base period)

S_3c_P3 The supply of final products by the shady sector at the price of P_3c

D_3l_P3 The demand for the labor force in the shady sector at the price of P_3l

Y_3_p The gain of the shady sector in the current prices

B_3 The shady sector budget

B_3_b The balance of the banking accounts of the shady sector

K_3 The capital assets of the shady sector


M_4 The issuance of money to households

O_4_tax The share of the household budget for discharging taxes to the consolidated budget

O_4_s The share of the retained budget of the households

L_1 The supply of the labor force to the state sector

L_2 The supply of the labor force to the market sector

L_3 The supply of the labor force to the shady sector

D_4c_P1 The household demand for final products at the price of P_1c



W_1 The wages of employees of the state sector

W_2 The wages of employees of the market sector

W_3 The wages of employees of the shady sector

B_4 The budget of households

B_4_b The balance of banking accounts


O_5g_P1 The share of the consolidated budget spent for purchasing final products at the price

of P_1g

O_5_s The share of the retained consolidated budget

O_5f_s The share of the retained off-budget funds

D_5g_P1 The demand for final products at the price of P_1g

D_5g_P2 The demand for final products at the price of P_2g

G_1_s The expenditures of the consolidated budget aimed at backing the state sector

G_2_s The expenditures of the consolidated budget aimed at backing the market sector

G_4_tr

(continued)



The social transfers to the inhabitants formed from the funds of the consolidated

budget

G_1_f Off-budget funds assigned to the state sector

G_2_f Off-budget funds assigned to the market sector

G_4_f Off-budget funds assigned to the inhabitants

B_5 The consolidated budget

B_5_b The balance of banking accounts of the consolidated budget

F_5 The money assets of the off-budget funds

F_5_b The balance of banking accounts of the off-budget funds

General part of the model

P_2l The market prices of the labor force

P_2c The market prices of final products for households

P_2g The market prices of final products for economic agent no. 5

P_2i The market prices of investment products

P_2k The market prices of capital products

P_3l The market prices of the labor force

P_3c The market prices of final products for households

I_l The deficiency indicator for the labor force market

I_c The deficiency indicator for the market of final products for the households

I_g The deficiency indicator for the market of final products for economic agent no. 5

I_i The deficiency indicator for the market of investment products

I_k The deficiency indicator for the market of capital products

D_sl_P1 The total demand for the labor force at the price of P_1l



S_sl_P1 The total supply of the labor force at the price of P_1l



D_sc_P1 The total demand for final products at the price of P_1c



S_sc_P1 The total supply of final products at the price of P_1c.

S_sc_P2 The total supply of final products at the price of P_2c

S_sc_P3 The total supply of final products at the price of P_3c

D_sg_P1 The total demand for final products at the price of P_1g

D_sg_P2 The total demand for final products at the price of P_2g

S_sg_P1 The total supply of final products at the price of P_1g

S_sg_P2 The total supply of final products at the price of P_2g

D_si_P1 The total demand for investment products at the price of P_1i

D_si_P2 The total demand for investment products at the price of P_2i

S_si_P1 The total supply of investment products at the price of P_1i

S_si_P2 The total supply of investment products at the price of P_2i

D_sk_P1 The total demand for capital products at the price of P_1k

D_sk_P2 The total demand for capital products at the price of P_2k

S_sk_P1 The total supply of capital products at the price of P_1k

S_sk_P2 The total supply of capital products at the price of P_2k

(continued)


4.3.1.4 Model Markets

The equilibrium prices are formed in 13 markets of the model as a result of

equalization of the supply and demand of various types of products, services, and

labor force:

– The markets of final products for households with the governmental, market, and

shady prices;

– The markets of final products for economic agent no. 5 with the governmental

and market prices;

– The markets of the capital products with the governmental and market prices;

– The markets of investment products with the governmental and market prices;

– The markets of the labor force with the governmental, market, and shady prices;

– The markets of exported products.

For each market, we determine the total supply and demand equalized during

iterative recalculation. The formulas determining the deficiency indicators for the

markets with governmental prices used in the model are presented below.

The labor force market:

I l t½ � ¼ S sl P1 t½ �=D sl P1 t½ �; (4.195)

The market for final products for households:

I c t½ � ¼ S sc P1 t½ �=D sc P1 t½ �; (4.196)

The market for final products for economic agent no. 5:

I g t½ � ¼ S sg P1 t½ �=D sg P1 t½ �; (4.197)

The market for investment products:

I i t½ � ¼ S si P1 t½ �=D si P1 t½ �; (4.198)

The market for capital products:

I k t½ � ¼ S sk P1 t½ �=D sk P1 t½ �: (4.199)


Y The GDP (in the prices of the base period)

Y_p The GDP (in current prices)

P Inflation of consumer prices

L The number of people employed in the economy

K The capital assets


As is obvious, the deficiency indicator is the ratio of the product supply to its

demand.

Let’s now present the model formulas reflecting the market process of changing

the labor force prices:

P 2l Qþ 1½ � ¼ P 2l Q½ � � C 2l;C 2l ¼ Abs D sl P2 Q½ �=S sl P2 Q½ �ð Þ=C; (4.200)

The prices of final products for households:

P 2c Qþ 1½ � ¼ P 2c Q½ � � C 2c;C 2c¼ Abs D sc P2 Q½ �=S sc P2 Q½ �ð Þ=C; (4.201)

The prices of final products for economic agent no. 5:

P 2g Qþ 1½ � ¼ P 2g Q½ � � C 2g;C 2g¼ Abs D sg P2 Q½ �=S sg P2 Q½ �ð Þ=C; (4.202)

The prices of investment products:

P 2i Qþ 1½ � ¼ P 2i Q½ � � C 2i;C 2i ¼ Abs D si P2 Q½ �=S si P1 Q½ �ð Þ=C; (4.203)

The prices of capital products:

P 2k Qþ 1½ � ¼ P 2k Q½ � � C 2k;C 2k¼ Abs D sk P2 Q½ �=S sk P2 Q½ �ð Þ=C; (4.204)

The equilibrium price in the shady markets is the same as the market price. The

respective formulas are presented below:

The labor market price:

P 3l Qþ 1½ � ¼ P 3l t½ � � C 3l;C 3l ¼ Abs D sl P3 Q½ �=S sl P3 Q½ �ð Þ=C; (4.205)

The price of final products for households:

P 3c Qþ 1½ � ¼ P 3c t½ � � C 3c;C 3c ¼ Abs D sc P3 Q½ � S sc P3 Q½ �=ð Þ C= : (4.206)

The formulas describing the total supply of and demand for products for each

price used in this model are discussed below. The final formulas determining the

supply of and demand for each economic agent are also given ahead.

The total demand for the labor force at the governmental, market, and shady prices:

D sl P1 t½ � ¼ D 1l P1 t½ �; (4.207)


D sl P2 t½ � ¼ D 2l P2 t½ �; (4.208)

D sl P3 t½ � ¼ D 3l P3 t½ �: (4.209)

The total supply of the labor force at governmental, market, and shady prices:

S sl P1 t½ � ¼ L 1 t½ �; (4.210)

S sl P2 t½ � ¼ L 2 t½ �; (4.211)

S sl P3 t½ � ¼ L 3 t½ �: (4.212)

The total demand for final products at governmental, market, and shady prices:

D sc P1 t½ � ¼ D 4c P1 t½ �; (4.213)

D sc P2 t½ � ¼ D 4c P2 t½ �; (4.214)

D sc P3 t½ � ¼ D 4c P3 t½ �: (4.215)

The total supply of final products at governmental, market, and shady prices:

S sc P1 t½ � ¼ S 1c P1 t½ �; (4.216)

S sc P2 t½ � ¼ S 1c P2 t½ � þ S 2c P2 t½ �; (4.217)

S sc P3 t½ � ¼ S 2c P3 t½ � þ S 3c P3 t½ �: (4.218)

The total demand for final products for economic agent no. 5 at governmental and

market prices:

D sg P1 t½ � ¼ D 5g P1 t½ �; (4.219)

D sg P2 t½ � ¼ D 5g P2 t½ �: (4.220)

The total supply of final products for economic agent no. 5 at governmental and

market prices:

S sg P1 t½ � ¼ S 1g P1 t½ �; (4.221)

S sg P2 t½ � ¼ S 1g P2 t½ � þ S 2g P2: (4.222)

The total demand for investment products at governmental and market prices:

D si P1 t½ � ¼ D 1i P1 t½ � þ D 2i P1 t½ �; (4.223)


D si P2 t½ � ¼ D 1i P2 t½ � þ D 2i P2 t½ �: (4.224)

The total supply of investment products at governmental and market prices:

S si P1 t½ � ¼ S 1i P1 t½ �; (4.225)

S si P1 t½ � ¼ S 1i P2 t½ � þ S 2i P2 t½ �: (4.226)

The total demand for capital products at governmental and market prices:

D sk P1 t½ � ¼ D 1k P1 t½ � þ D 2k P1 t½ �; (4.227)

D sk P2 t½ � ¼ D 1k P2 t½ � þ D 2k P2 t½ �: (4.228)

The total supply of the capital products at governmental and market prices:

S sk P1 t½ � ¼ S 1k P1 t½ �; (4.229)

S sk P2 t½ � ¼ S 1k P2 t½ � þ S 2k P2 t½ �: (4.230)

So, we have 24 formulas for determining the total supply of and demand for

products considered in this model.

4.3.1.5 Economic Agent No. 1: The State Sector

In the markets with governmental pricing, equalization of the total supply and

demand occurs via the correction of budget shares and the share of the finished

product. This process is described by the formulas

O 1l P1 Qþ 1½ � ¼ O 1l P1 Q½ � � etta 1lþ O 1l P1 Q½ � � I l t½ �� 1� etta 1lð Þ; (4.231)

O 1i P1 Qþ 1½ � ¼ O 1i P1 Q½ � � etta 1iþ O 1i P1 Q½ � � I i t½ �� 1� etta 1ið Þ; (4.232)

O 1k P1 Qþ 1½ � ¼ O 1k P1 Q½ � � etta 1k þ O 1k P1 Q½ � � I k t½ �� 1� etta 1kð Þ; (4.233)

E 1c P1 Qþ 1½ � ¼ E 1c P1 Q½ � � etta 1cþ E 1c P1 Q½ � � I c t½ �� 1� etta 1cð Þ: (4.234)


HereQ is the iteration step and 0 < etta_1l, etta_1i, etta_1k, etta_1c < 1 are the

model constants. With its increase, the process of attaining equilibrium is slower.

Nevertheless, the equation system becomes more stable.

Let’s s now proceed to the formulas determining the behavior of the state sector.


Y 1 tþ 1½ � ¼A 1 r � power K 1 t½ � þ K 1 tþ 1½ �ð Þ=2ð Þ;A 1 kð Þ� powerðD 1l P1 t½ �;A 1 lÞ: ð4:235Þ

Here power(X, Y) corresponds to XY; A_1_r, A_1_k, and A_1_l are the

parameters of the production function.

The following formulas determine the demand for the production factors within

the state sector.


D 1l P1 t½ � ¼ O 1l P1 t½ � � B 1 t½ �ð Þ=P 1l: (4.236)

The demand for capital products:

At governmental prices:

D 1k P1 t½ � ¼ O 1k P1 t½ ��B 1 t½ �ð Þ=P 1; (4.237)

At market prices:

D 1k P2 t½ � ¼ O 1k P2� B 1 t½ �ð Þ=P 2k t½ �: (4.238)



D 1i P1 t½ � ¼ O 1i P1 t½ � � B 1 t½ �ð Þ=P 1i; (4.239)

At market prices:

D 1i P2 t½ � ¼ O 1i P2� B 1 t½ �ð Þ=P 2i t½ �: (4.240)

The following formulas determine the supply of products of the state sector.



S 1c P1 t½ � ¼ E 1c P1 t½ � � Y 1 t½ �; (4.241)


S 1c P2 t½ � ¼ E 1c P2� Y 1 t½ �: (4.242)

The supply of final products for economic agent no. 5:


S 1g P1 t½ � ¼ E 1g P1� Y 1 t½ �; (4.243)

At market prices:

S 1g P2 t½ � ¼ E 1g P2� Y 1 t½ �: (4.244)

The supply of investment products:


S 1i P1 t½ � ¼ E 1i P1� Y 1 t½ �; (4.245)

At market prices:

S 1i P2 t½ � ¼ E 1i P2� Y 1 t½ �: (4.246)

The supply of capital products:


S 1k P1 t½ � ¼ E 1k P1� K 1 t½ �; (4.247)

At market prices:

S 1k P2 t½ � ¼ E 1k P2� K 1 t½ �: (4.248)


S 1ex Pex t½ � ¼ E 1ex Pex� Y 1 t½ �: (4.249)

The following formula calculates the gain of the state sector:

Y 1 p t½ � ¼ S 1c P1 t½ � � P 1cþ S 1c P2 t½ � � P 2c t½ � þ S 1g P1 t½ � � P 1g

þ S 1g P2 t½ � � P 2g t½ � þ S 1i P1 t½ � � P 1iþ S 1i P2 t½ � � P 2i t½ �þ S 1k P1 t½ � � P 1k þ S 1k P2 t½ � � P 2k t½ � þ S 1ex Pex t½ �� P ex: ð4:250Þ


As is obvious, the gain consists of the gain from selling final products and

rendering services for households and economic agent no. 5, investment, capital,

as well as exported products.

The budget of the state sector is determined as follows:

B 1 t½ � ¼ B 1 b t½ � � 1þ CP b t� 1½ �ð Þ þ Y 1 p t½ � þ G 1 s t� 1½ � þ G 1 f� t� 1½ � þM 1: (4.251)


1. Funds in banking accounts (subject to the interests on deposits);

2. Gain received within the current period;

3. Bounties received from the consolidated budget G 1 s;4. Part of the off-budget funds G 1 f ;5. Emission of money M_1;

The dynamics of the banking account balance of the state sector is as follows:

B 1 b tþ 1½ � ¼ O 1 s t½ � � B 1 t½ �: (4.252)

The capital assets’ dynamics is determined by

K 1 tþ 1½ � ¼ K 1 t½ � � 1� E 1k P1� E 1k P2ð Þ � 1� A 1 nð Þ þ D 1k P1 t½ �þ D 1k P2 t½ � þ D 1i P1 t½ � þ D 1i P2 t½ �:

(4.253)

This formula calculates the volume of capital assets, taking into account their

selling and wear and tear. The purchased assets and investments to the capital assets

enter the formula with a plus sign.

The share of the budget of the state sector for discharging taxes to the


O 1 t t½ � ¼ Y 1 p t½ � � T vadð Þ=B 1 t½ � þ Y 1 p t½ � �W 1 t½ � � K 1 t½ � � P 1k � A 1 nð Þð� T prÞ=B 1 t½ � þ K 1 t½ � � P 1kð Þ � T propð Þ=B 1 t½ �:

(4.254)

This formula takes into consideration the value-added tax (VAT), profit tax, and

property tax. While calculating the share of the budget for discharging profit tax, the

gain is subtracted from the costs of labor forceW_1, as well as depreciation chargesK_1[t] � P_1k � A_1_n.



O 1 f t½ � ¼ W 1 t½ � � T esnð Þ B 1 t½ �= : (4.255)


The remainder of the budget of the state sector of the economy is given by

O 1 s t½ � ¼ 1� O 1l P1 t½ � � O 1k P1 t½ � � O 1k P2� O 1i P1 t½ �� O 1i P2� O 1 t t½ � � O 1 f t½ �: ð4:256Þ

4.3.1.6 Economic Agent No. 2: The Market Sector

Since the behavior of the market sector is similar to that of the state sector, we

shorten the description composed by analogy with agent no. 1 in some places.

The market sector corrects the shares of its budget O 2k P1 and O 2k P1 for

purchasing capital and investment products with governmental prices.

This process is described by the formulas

O 2k P1 Q½ � ¼ O 2k P1 Q½ � � etta 2k þ O 2k P1 Q½ � � I k t½ �� 1� etta 2kð Þ; (4.257)

O 2i P1 Q½ � ¼ O 2i P1 Q½ � � etta 2iþ O 2i P1 Q½ � � I i t½ �� 1� etta 2ið Þ; (4.258)

where Q is iteration step and 0 < etta_2k, etta_2i < 1 are model constants.

Let’s proceed to the formulas determining the behavior of the market sector.


Y 2 tþ 1½ � ¼ A 2 r � power K 2 t½ � þ K 2 tþ 1½ �ð Þ=2ð Þ;A 2 kð Þ� power D 2l P2 t½ �;A 2 lð Þ: ð4:259Þ

Here A_2_r, A_2_k, and A_2_l are the parameters of the production function.

The following formulas determine the demand for the production factors in the

market sector.


D 2l P2 t½ � ¼ O 2l P2� B 2 t½ �ð Þ P 2l t½ �= : (4.260)

The demand for capital products:


D 2k P1 t½ � ¼ O 2k P1 t½ � � B 2 t½ �ð Þ P 1k= ; (4.261)

At market prices:

D 2k P2 t½ � ¼ O 2k P2� B 2 t½ �ð Þ P 2k t½ �= : (4.262)




D 2i P1 t½ � ¼ O 2i P1 t½ � � B 2 t½ �ð Þ P 1i= ; (4.263)

At market prices:

D 2i P2 t½ � ¼ O 2i P2� B 2 t½ �ð Þ P 2i t½ �= : (4.264)

The following formulas determine the supply of products of the market sector.


At market prices:

S 2c P2 t½ � ¼ E 2c P2� Y 2 t½ �; (4.265)

At shady prices:

S 2c P3 t½ � ¼ E 2c P3� Y 2 t½ �: (4.266)

The supply of final products for economic agent no. 5 at market prices:

S 2g P2 t½ � ¼ E 2g P2� Y 2 t½ �: (4.267)

The supply of investment products at market prices:

S 2i P2 t½ � ¼ E 2i P2� Y 2 t½ �: (4.268)

The supply of capital products at market prices:

S 2k P2 t½ � ¼ E 2k P2� K 2 t½ �: (4.269)


S 2ex Pex t½ � ¼ E 2ex Pex� Y 2 t½ �: (4.270)

The following formula calculates the gain of the market sector:

Y 2 p ¼ S 2c P2 t½ � � P 2c t½ � þ S 2g P2� P 2g t½ � þ S 2i P2 t½ � � P 2i t½ �þ S 2k P2 t½ � � P 2k t½ � þ S 2ex Pex t½ � � P ex: ð4:271Þ


As is obvious, the gain consists of the gain from selling final products and

rendering services for households and economic agent no. 5, investment, capital,

as well as exported products. As presented above, the gain from selling final

products and services for households at the shady prices is not accounted for here.

The budget of the market sector is determined as follows:

B 2 t½ � ¼ B 2 b t½ � � 1þ CP b t� 1½ �ð Þ þ Y 2 pþ G 2 S t� 1½ � þ G 2 f� t� 1½ � þM 2: (4.272)



2. Gain received in the current period;

3. Subsidies received from the consolidated budget G 2 S2;4. Part of the off-budget funds G 2 f ;5. Emission of money M_2.

The dynamics of the banking account balance of the market sector is as follows:

B 2 b tþ 1½ � ¼ O 2 s t½ � � B 2 t½ �: (4.273)

The capital assets’ dynamics is determined by

K 2 tþ 1½ � ¼ K 2 t½ � � 1� E 2k P2ð Þ � 1� A 2 nð Þ þ D 2k P1 t½ � þ D 2k P2 t½ �þ D 2i P1 t½ � þ D 2i P2 t½ �: ð4:274Þ

This formula calculates the volume of the capital assets, taking into account their

selling and wear and tear. The purchased assets and investments to the capital assets

enter into the formula with a plus sign.

The share of the budget of the market sector for discharging taxes to the


O 2 t t½ � ¼ Y 2 p� T vadð Þ=B 2 t½ � þ Y 2 p�W 2 t½ � � K 2 t½ � � P 2k t½ � � A 2 nð Þð� T prÞ=B 2 t½ � þ K 2 t½ � � P 2k t½ �ð Þ � T propð Þ=B 2 t½ �:

(4.275)

This formula takes into consideration the VAT, the profit tax, and the property

tax. While calculating the share of budget for discharging the profit tax, the gain is

subtracted by the costs of the labor force W_2 as well as the depreciation charges

K_2[t] � P_2k[t] � A_2_n.The share of the budget for discharging the single social tax to the off-budget


O 2 f t½ � ¼ W 2 t½ � � T esnð Þ B 2 t½ �= : (4.276)


The remainder of the budget of the market sector of the economy is given by

O 2 s t½ � ¼1� O 2l P2� O 2k P1 t½ � � O 2k P2� O 2i P1 t½ � � O 2i P2

� O 2 t t½ � � O 2 f t½ �: ð4:277Þ

4.3.1.7 Economic Agent No. 3: The Shady Sector

Let’s compose the formulas determining the behavior of the shady sector.


Y 3 tþ 1½ � ¼ A 3 r � power K 3 t½ � þ K 3 tþ 1½ �ð Þ 2=ð Þ;A 3 kð Þ� power D 3l P3 t½ �;A 3 lð Þ: ð4:278Þ

Here A_3_r, A_3_k, and A_3_l are the parameters of the production function. The

production function equation is similar to that of the state and market sectors, but

one of its arguments (the capital assets) is calculated another way.

The shady sector does not have its own capital assets. The same can be seen in

real life, where the representatives of “white-collar” and “gray” economies use

capital assets of the state and market sectors. Therefore, the capital assets of the

shady sector are formed as follows:

K 3 t½ � ¼ gamma� K 1 t½ � þ K 2 t½ �ð Þ; (4.279)

where gamma is the share of capital assets of the state and market sectors used in

shady economics.

The demand for the labor force at shady prices is calculated similarly to that of

the other sectors:

D 3l P3 t½ � ¼ O 3l P3� B 3 t½ �ð Þ P 3l t½ �= : (4.280)

Now, we’ll calculate the supply of final products for households at shady prices:

S 3c P3 t½ � ¼ E 3c P3� Y 3 t½ �: (4.281)

The following formula calculates the gain of the shady sector:

Y 3 p t½ � ¼ S 2c P3 t½ � þ S 3c P3 t½ �ð Þ � P 3c t½ �: (4.282)

This formula takes into account the final goods produced by “white-collar” and

“gray” shady economics.

The budget of the shady sector is determined as follows:

B 3 t½ � ¼ B 3 b t½ � � 1þ CP b t� 1½ �ð Þ þ Y 3 p t½ �: (4.283)




2. Gain received in the current period.

The dynamics of the banking account balance of the shady sector is as follows:

B 3 b tþ 1½ � ¼ O 3 s t½ � � B 3 t½ �: (4.284)

The remainder of the budget of the shady sector of the economy is given by

O 3 s t½ � ¼ 1� O 3l P3 t½ � (4.285)

4.3.1.8 Economic Agent No. 4: The Aggregate Consumer (Households)

Let’s proceed to the formulas determining the behavior of the aggregate consumer.

The household demand for final products:


D 4c P1 t½ � ¼ O 4c P1� B 4 t½ �ð Þ P 1c= ; (4.286)

At market prices:

D 4c P2 t½ � ¼ O 4c P2� B 4 t½ �ð Þ P 2c t½ �= ; (4.287)

At shady prices:

D 4c P3 t½ � ¼ O 4c P3� B 4 t½ �ð Þ P 3c t½ �= : (4.288)

The movement of the labor force:

In the state sector:

L 1 t½ � ¼ L 1 t� 1½ � � ð1� L 1 2 t� 1½ � þ L 1 a t� 1½ �� L 1 r t� 1½ �Þ þ L 2 t� 1½ � � L 2 1 t� 1½ �; ð4:289Þ

In the market sector:

L 2 t½ � ¼ L 2 t� 1½ � � 1� L 2 1 t� 1½ � þ L 2 a t� 1½ � � L 2 r t� 1½ �ð Þþ L 1 t� 1½ � � L 1 2 t� 1½ �; ð4:290Þ

In the shady sector:

L 3 t½ � ¼ L 1 t½ � þ L 2 t½ �ð Þ � L 12 3: (4.291)


The number of employees in the shady sector is determined as the share of the

number of employees in the state and market sectors.

The wages of the employees:

In the state sector:

W 1 t½ � ¼ D 1l P1 t½ � � P 1l; (4.292)

In the market sector:

W 2 t½ � ¼ D 2l P2 t½ � � P 2l t½ �; (4.293)

In the shady sector:

W 3 t½ � ¼ D 3l P3 t½ � � P 3l t½ �: (4.294)


B 4 t½ � ¼ B 4 b t½ � � 1þ CP b h t� 1½ �ð Þ þ VB 4 t� 1½ � � VO 4 s t� 1½ � þW 1 t½ �þW 2 t½ � þW 3 t½ � þ G 4 tr t� 1½ � þ G 4 f t� 1½ � þM 4 t½ �: ð4:295Þ


1. Funds in banking accounts;

2. Retained money in cash remaining from the preceding period;

3. Wages received in the state, market, and shady sectors;

4. Pensions, welfare payments, and subsidies received from the off-budget funds;

part of the off-budget funds G_1_f;5. Emission of money M_4;6. Income from property, commercial activity, and other incomes. This constituent

part of the budget enters the model exogenously to complete the budget to the

values of official statistics.

The dynamics of the banking account balance of households is as follows:

B 4 b tþ 1½ � ¼ B 4 t½ � � O 4 b: (4.296)

The share of the budget for discharging income tax is as follows:

O 4 tax t½ � ¼ W 1 t½ � þW 2 t½ �ð Þ � T podð Þ B 4 t½ �= : (4.297)

The remainder of the money in cash is

O 4 s t½ � ¼ 1� O 4c P1� O 4c P2� O 4c P3� O 4 tax t½ �� O 4 b� O 4 buck: (4.298)


4.3.1.9 Economic Agent No. 5: Government

Economic agent no. 5 corrects the share of the budget for purchasing final products

at governmental prices. This process is described by the following formula:

O 5g P1 Q½ � ¼ O 5g P1 Q½ � � etta 5gþ O 5g P1 Q½ � � I g t½ �� 1� etta 5gð Þ; (4.299)

where Q is the iteration step, and 0 < etta_5g < 1 is the model constant.

Let’s now proceed to the formulas determining the behavior of economic

agent no. 5.

The consolidated budget obeys the relationship

B 5 t½ � ¼ O 1 t t½ � � B 1 t½ � þ O 2 t t½ � � B 2 t½ � þ O 4 tax t½ � � B 4 t½ � þ B 5 other

þ B 5 b t½ � � 1þ CP b t� 1½ �ð Þ: ð4:300Þ

This formula sums up the money collected as the taxes from the state and market

sectors as well as from inhabitants. The value B_5_other entering the model

exogenously is the sum of other taxes (not included in the list of taxes considered

in the model), nontaxable income, and other income of the consolidated budget.

The obtained sum is supplemented by the funds in banking accounts (subject to the

deposit interests).


described by

B 5 b tþ 1½ � ¼ O 5b s t½ � � B 5 t½ �ð Þ; (4.301)

f 5 t½ � ¼ O 1 f t½ � � B 1 t½ � þ O 2 f t½ � � B 2 t½ � þ F 5 b t½ �� 1þ CP b t� 1½ �ð Þ: (4.302)

This formula calculates the sum collected from the state and market sectors in

the form of the single social tax entering the accounts of the following off-budget

funds:

– Pension fund;

– Social insurance fund;

– Federal and territorial funds of obligatory medical insurance.

The derived sum is supplemented by the funds in banking accounts (subject to

the interest on deposits).


mined by

F 5 b tþ 1½ � ¼ O 5f s t½ � � F 5 t½ �: (4.303)


The demand for final products:


D 5g P1 t½ � ¼ O 5g P1 t½ � � B 5 t½ �ð Þ P 1g= ; (4.304)

At market prices:

D 5g P2 t½ � ¼ O 5g p2� B 5 t½ �ð Þ P 2g t½ �= : (4.305)

Subsidies to the producing sectors are as follows:

The state sector:

G 1 s t½ � ¼ O 5 s1� B 5 t½ �; (4.306)

The market sector:

G 2 s t½ � ¼ O 5 s2� B 5 t½ �: (4.307)

Social transfers to inhabitants:

G 4 tr t½ � ¼ O 5 tr � B 5 t½ �: (4.308)

The assets of the off-budget funds made available for

The state sector:

G 1 f t½ � ¼ O 5 f1� F 5 t½ �; (4.309)

The market sector:

G 2 f t½ � ¼ O 5 f2� F 5 t½ �: (4.310)

The assets of the off-budget funds made available to inhabitants:

G 4 f t½ � ¼ O 5 f4� F 5 t½ �: (4.311)

This includes the assets of the pension fund and social insurance fund for paying

out pensions and welfare payments.

4.3.1.10 Economic Agent No. 6: The Banking Sector

The banking sector of this model includes the central and commercial banks. This

economic agent implements the following functions:

1. It realizes the emission of money, M_1, M_2, M_4;

2. It establishes the deposit interest rate for enterprises and physical bodies.


4.3.1.11 Economic Agent No. 7: The Outer World

In this version of the model, all the economic indices of the outer world are

specified exogenously. This means that the domestic producers cannot export

more products than the outer world requires.

4.3.1.12 Integral Indices of the Model

In this subsection we present the formulas for calculating some integral indices of

the economy.

The GDP (in prices of the base period) is

Y t½ � ¼ Y 1 t½ � þ Y 2 t½ �: (4.312)

The GDP (in current prices) is

Y p t½ � ¼ Y 1 p t½ � þ Y 2 p t½ �: (4.313)

The inflation of consumer prices is

P t½ � ¼ 100� P 2c t½ � P 2c t� 1½ �=ð Þ: (4.314)

The number of people employed within the economy is

L t½ � ¼ L 1 t½ � þ L 2 t½ � þ L 3 t½ �: (4.315)

Capital assets are

K t½ � ¼ K 1 t½ � þ K 2 t½ �: (4.316)

The considered CGE model with the shady sector is presented in the context of

relation (1.12) by 11 expressions n1 ¼ 11ð Þ; in the context of relation (1.13) by 98

expressions n2 ¼ 98ð Þ; in the context of relation (1.14) by 14 expressions

n3 ¼ 14ð Þ:The analyzed model includes 144 exogenous parameters (whose values are

required to be estimated by solving the parametric identification problem) and

123 endogenous variables.

4.3.1.13 Parametric Identification of the CGE Model with the Shady Sector

The problem of the identification (calibration) of the exogenous model parameters

was solved by the methods applied to the parametric identification of the


http://dx.doi.org/10.1007/978-1-4614-4460-2_1

http://dx.doi.org/10.1007/978-1-4614-4460-2_1

http://dx.doi.org/10.1007/978-1-4614-4460-2_1

computable model of economic sectors and computable model with the knowledge

sector (see Sect. 4.1).

To estimate the quality of the retrospective prediction on the basis of economic

data from the Republic of Kazakhstan for the years 2000–2004, for some starting

point o1 2 O , we solve the problem (Problem A) of estimation of the model

parameters and initial conditions for the difference equations by searching for the

minimum of criterion KIA:

K2IA ¼ 1

10

X2004t¼2000

Y�½t� � Y½t�Y�½t�

� �2

þ P�½t� � P½t�P�½t�

� �2" #

: (4.317)

Here t is the number of years;

Y[t] is the calculated GDP in billions of tenge in the prices of year 2000;

P[t] is the calculated level of consumer prices.

Here and below, the * symbol corresponds to the measured values of their

respective variables. The problem of the model parametric identification is consid-

ered to be solved if such a pointo0KIA

2 O exists thatKIAðo0KIA

Þ< e for a sufficientlysmall e.

In parallel with Problem A for the point o1 , we also solve a similar problem

(problem B) with use of the extended criterion KIB instead of criterion KIA:

K2IB ¼ 1

12:15

( X2004t¼2000

Y�½t� � Y½t�Y�½t�

� �2

þ P�½t� � P½t�P�½t�

� �2" #

þ 0:1X2004

t¼2000

L� 1½t� � L 1½t�L� 1½t�

� �2

þ L� 2½t� � L 2½t�L� 2½t�

� �2" #

þ 0:1X2004

t¼2000

K� 1½t� � K 1½t�K� 1½t�

� �2

þ K� 2½t� � K 2½t�K� 2½t�

� �2" #

þ 0:01X2004

t¼2000

Y� 1½t� � Y 1½t�Y� 1½t�

� �2

þ Y� 2½t� � Y 2½t�Y� 2½t�

� �2

þ Y� 3½t� � Y 3½t�Y� 3½t�

� �2" #)

:

(4.318)

Here:

L_1[t] is the number of employees in the state sector;

L_2[t] is the number of employees in the market sector;

K_1[t] are capital assets of the state sector;K_2[t] are capital assets of the market sector;

Y_1[t] is the state sector GVA;Y_2[t] is the market sector GVA;

Y_3[t] is the shady sector GVA.


As a result of simultaneously solving problems A and B by the described

algorithm, we obtain the values KIA ¼ 0:0025 and KIB ¼ 0:12. The relative value

of the deviation of the calculated values of variables used in criterion (4.318) from

the respective measured values is less than 0.25%.

The results of the retrospective prediction of the model for the years 2005–2008

presented in Table 4.21 demonstrate the calculated values and measured values, as

well as the deviations of the calculated values of the output variables of the model

from the respective actual values.

4.3.2 Finding the Optimal Values of the Adjusted Parameterson the Basis of the CGE Model in the Shady Sector

In the context of analysis of the connection between some processes of shady

economics and the basic macroeconomic indices of a country (the GDP and

consumer price index), a number of computational experiments described below

(the simulation of the scenarios specifying some possible negative effects within

the country’s economy) are carried out. These simulations are similar to

experiments in [27].

We consider the following six scenarios:

1. Simulation of the process of cash withdrawals (10%, 20%, 30%) from a

consolidated budget of a country and reassigning this cash to households from

year 2005 (scenarios 1, 2, and 3). We also simulate the process of direct stealing

or the quite legal process of development of budgetary funds (the kickback

process).

2. Simulation of the process of cash withdrawal (10%, 20%, 30%) from producers

and reassigning this cash to households from year 2005 (scenarios 4, 5, and 6). In

this case we simulate the process of giving (from producers) and taking (by

households) bribes.

The results of applying the enumerated six scenarios of the economic develop-

ment of the country with a negative effect of shady economics in comparison with

the base variant of evolution are presented in Tables 4.22 and 4.23.

The analysis of Tables 4.22 and 4.23 also shows that the analyzed scenarios

insignificantly affect the country’s GDP. At the same time, the consumer price

Table 4.21 Results of

the model’s retrospective

prediction

Year 2005 2006 2007 2008

Y� t½ � 4258.03 4715.65 5136.54 5303.27

Y[t] 4221.69 4586.33 5004.12 5478.31

Error (%) �0.861 �2.820 �2.646 3.195

P� t½ � 107.6 108.4 118.8 109.5

P[t] 108.4 109.5 112.6 112.0

Error (%) 0.706 1.017 �5.528 2.240


index increases significantly during the first year of applying scenarios 1–6. In the

subsequent year, their effect on the price index becomes weaker.

Note that the considered aspects of shady economics, namely, stealing from

budget and bribes, results in pronounced negative consequences for the country’s

economy. In both cases, the demand for consumer products grows, which leads to a

natural rise in consumer prices. Additionally, the producer often includes the

expenditures on bribes in the price of its production, which also leads to a rise in

prices. In any case, finally, this hurts inhabitants who are not related to partitioning

the budgetary funds and not accepting bribes and kickbacks.

The next part of computational experiments is aimed at reducing the negative

effect from each of the considered scenarios to the main macroeconomic indexes,

namely, the level of prices by the methods of parametric control.

Within the framework of applying the parametric control approach, the problem

statement is as follows: Find the optimal values of 76 parameters (uli ; i ¼ 1; :::; 19 isthe parameter number, l ¼ 2005; :::; 2008 is the year number) regulated by the

government for 2005–2008 for each of the considered scenarios. The regulated

parameters are the following:

– Various taxation rates;

– Shares of the consolidated budget for financing the state and market sectors of

the economy as well as for purchasing final products;

– Shares of the state sector budgets for purchasing various types of products;

Table 4.22 Values of the

GDP (in billions of tenge

in the prices of the year 2000)

for the base variant and

for scenarios 1–6

GDP

Year 2005 2006 2007 2008

Base variant 4,300,103 4,618,653 4,963,707 5,337,048

Scenario 1 4,301,026 4,623,221 4,975,060 5,357,813

Scenario 2 4,301,887 4,627,487 4,985,442 5,376,495

Scenario 3 4,302,752 4,631,527 4,994,972 5,393,343

Scenario 4 4,298,244 4,612,732 4,953,878 5,324,870

Scenario 5 4,296,520 4,607,483 4,945,717 5,315,665

Scenario 6 4,294,935 4,602,927 4,939,176 5,309,146

Table 4.23 Values of the

consumer price index (in %

with respect to preceding

year) for the base variant

and for scenarios 1–6

Consumer price index

Year 2005 2006 2007 2008

Base variant 107.624 108.602 109.334 108.816

Scenario 1 115.575 109.706 109.986 108.989

Scenario 2 123.530 109.761 110.470 109.044

Scenario 3 131.481 108.962 111.001 109.006

Scenario 4 138.576 118.506 113.760 111.462

Scenario 5 171.450 123.439 115.029 111.904

Scenario 6 206.522 125.441 114.879 111.508


– Shares of various types of products produced by the state sector of the economy

for selling in the various markets.

The level of consumer prices of the country for 2008 in comparison to 2004 with

the use of the jth (j =1, . . . ,6) scenario is used as the minimized criterion K:

K ¼ P 2c½2008�=P 2c½2004�:

The following constraints on the GDP of the country are used among the

constraints of the solved variational problem:

Yj½t� �Yj½t�; j ¼ 1; :::; 6:

Here �Yj½t� is the value of the GDP with the use of the jth scenario without

parametric control; Yj½t� is the value of the GDP with the use of the jth scenario andthe values of the controlled parameters optimal in a sense of criterion K.

The constraints on the controlled parameters uli are presented in Table 4.24.

We consider a problem of finding optimal values of economic parameters uli. Onthe basis of model (4.195), (4.196), (4.197), (4.198), (4.199), (4.200), (4.201),

(4.202), (4.203), (4.204), (4.205), (4.206), (4.207), (4.208), (4.209), (4.210),

(4.211), (4.212), (4.213), (4.214), (4.215), (4.216), (4.217), (4.218), (4.219),

(4.220), (4.221), (4.222), (4.223), (4.224), (4.225), (4.226), (4.227), (4.228),

(4.229), (4.230), (4.231), (4.232), (4.233), (4.234), (4.235), (4.236), (4.237),

(4.238), (4.239), (4.240), (4.241), (4.242), (4.243), (4.244), (4.245), (4.246),

(4.247), (4.248), (4.249), (4.250), (4.251), (4.252), (4.253), (4.254), (4.255),

(4.256), (4.257), (4.258), (4.259), (4.260), (4.261), (4.262), (4.263), (4.264),

(4.265), (4.266), (4.267), (4.268), (4.269), (4.270), (4.271), (4.272), (4.273),

(4.274), (4.275), (4.276), (4.277), (4.278), (4.279), (4.280), (4.281), (4.282),

(4.283), (4.284), (4.285), (4.286), (4.287), (4.288), (4.289), (4.290), (4.291),

(4.292), (4.293), (4.294), (4.295), (4.296), (4.297), (4.298), (4.299), (4.300),

(4.301), (4.302), (4.303), (4.304), (4.305), (4.306), (4.307), (4.308), (4.309),

(4.310), (4.311), (4.312), (4.313), (4.314), (4.315), and (4.316), determine values

of the economic parameters uli ; i ¼ 1; :::; 19, l ¼ 2005; :::; 2008, which are optimal

in a sense of criterion K under the above constraints.

The stated problem of finding the minimum value of criterion K as the function

in 76 variables (and the respective values of the controlled parameters uli ¼ arg

minKÞ for each of the considered six scenarios under the aforementioned

constraints is solved using the Nelder–Mead algorithm. The results of the stated

problem solution are presented in Table 4.25.

Analysis of Table 4.23 shows that in the case of the considered scenarios, the

parametric control approach allows both reduction of the level of prices by

9.4–13.2% and increase of the GDP of the country by 1.3–2.44% for year 2008 in

comparison to the case without use of the parametric control approach.


Table 4.24 Controlled model parameters and the constraints imposed on them

No. Controlled parameter ui

Interval of admissible

values of controlled

parameter

1 Rate of the VAT [0.135; 0.165]

2 Income tax rate for organizations [0.27; 0.33]

3 Property tax rate [0.009; 0.011]

4 Income tax rate for physical bodies [0.135; 0.165]

5 Rate of the single social tax [0.099; 0.121]

6 Share of the consolidated budget for purchasing final products [0.117; 0.143]

7 Share of the consolidated budget for backing the state sector [0.325; 0.398]

8 Share of the consolidated budget for backing the market sector [0.028; 0.034]

9 Share of the consolidated budget for social transferring [0.320; 0.391]

10 Share of the consolidated budget for purchasing the capital products [0.129; 0.158]

11 Share of the consolidated budget for purchasing investment products [0.068; 0.083]

12 Share of products of the state sector for selling in the markets of final

products for the market sector

[0.101; 0.123]

13 Share of products of the state sector for selling in the markets of

final products for economic agent no. 5 at the exogenous prices

[0.039; 0.048]

14 Share of products of the state sector for selling in the markets

of final products for economic agent no. 5 at market prices

[0.039; 0.048]


of investment products at exogenous prices

[0.107; 0.131]


of investment products at market prices

[0.107; 0.131]


of the capital products at exogenous prices

[0.200; 0.244]


of the capital products at market prices

[0.200; 0.244]

19 Share of products of the state sector for selling in the markets of final

products in foreign countries

[0.230; 0.281]

Table 4.25 Results of the application of the parametric control approach

Year

Criterion K without

parametric control

Criterion Kcorresponding to

the obtained optimal

values of parameters

Yj � ½2008�in billions tenge

Yj½2008�in billions tenge

Scenario 1 1.52 1.32 5.35 5.47

Scenario 2 1.63 1.41 5.38 5.45

Scenario 3 1.73 1.50 5.40 5.50

Scenario 4 2.08 1.87 5.32 5.44

Scenario 5 2.72 2.47 5.31 5.44

Scenario 6 3.32 3.04 5.31 5.44


Chapter 5

Conclusion

This book describes the theory of parametric regulation of economic growth on the

basis of the deterministic continuous-time dynamical models, deterministic

discrete-time dynamical models, and stochastic discrete-time dynamical models

with additive noise. The presented theory consists of eight constituent parts. In the

context of these parts, we

• Proposed applied methods of estimation of structural stability and stability of the

mathematical models.

• Formulated and proved theorems on existence conditions for solutions of varia-

tional calculus problems of synthesis and choice of the optimal parametric

control laws.

• Proposed the method for analyzing dependence of the results of solving the

considered variational calculus problems on the values of uncontrolled

parameters. In particular, we gave a definition of the bifurcation points of

extremals of the variational calculus problem of choosing the optimal parametric

control laws and formulated and proved the theorem on a bifurcation point

existence conditions.

The efficiency of the parametric control theory is illustrated by a number of

examples, including the following: the mathematical model of the neoclassical

theory of optimal growth; the mathematical model of the national economic system

subject to influence of the share of public expenses and interest rate of governmen-

tal loans on economic growth; the mathematical model of the national economic

system that considers the influence of the international trade and currency exchange

on economic growth; Turnovsky’s monetary model; Jones’s endogenous model; the

computable model of general equilibrium of economic branches; Forrester’s math-

ematical model of global economy, etc.

Based on the dependence of a solution to a system of algebraic equations on their

coefficient, we proposed an approach to parametric control of the static equilibrium

of a national economy. The efficiency of this approach to parametric control

of static equilibrium is shown on the basis of the Keynes’s mathematical model


273

of general economic equilibrium and mathematical model of an open economy of a

small country.

This book illustrates the instrumental capabilities of a number of mathematical

models, such as the IS, LM, IS-LM models, Keynes’s model of general economic

equilibrium, model of an open economy of a small country, many models of market

cycles, a computable model of general equilibrium of economic branches, etc.,

respectively, in the following fields: macroeconomic analysis of the macroeconomic

markets’ conditions; estimation of characteristics of the market cycles in form of

amplitudes and periods of the respective business cycles; retrospective analysis

within the identification period; verification of the mathematical model and perspec-

tive analysis within the period of middle-term prediction of the indicators of the

economic agents and national economy, as well as their instrumental capabilities for

estimating the elasticity of the respective endogenous variables, sources of economic

growth, and estimation of several provisions of the macroeconomic theory on the

market cycles on the basis of the respective mathematical models.

The materials of this book have an important applied focus illustrating capabilities

of working out recommendations on economic policy. Hence, the diagrams of

dependencies of the optimal values of criterions of optimization problems formulated

on the basis of Keynes’s mathematical model of general economic equilibrium

and the mathematical model of the open economy in a small country (with the

coefficients estimated by the statistical data of the national economy) allow us to

consider the respective optimal values of economic instruments being a part of the

optimal solution of the formulated optimization problem as the recommendations on

economic policy.

The respective optimal parametric control laws can be considered the recom-

mendations on the economic policy for regulation of the business cycle within the

framework of the (selected by a decision maker) criteria for regulation of market

cycles’ evolution.

In the context of the computable models of general equilibrium, the respective

(agreed with a decision maker) optimal laws of parametric control of the criteria of

economic growth effectiveness can be considered the recommendations on economic

policy. Thus, the results of research presented in this book represent the elements

of the modern paradigm of macroanalysis and working out recommendations on

economic policy.

274 5 Conclusion

References

1. Aivazyan SA, Mhitaryan VS. Applied statistics and foundations of econometric: A course for

institutes of higher education. Moscow, UNITY, 1998; 1022 p. (in Russian).

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About the Authors

Ashimov Abdykappar Ashimovich is an academician at the National Academy of

Sciences of the Republic of Kazakhstan, doctor of technical sciences, and professor

at Kazakh National Technical University named after K.I. Satpaev, e-mail:

[email protected].

Borovskiy Yuriy Vyacheslavovich is a candidate of physical and mathematical

sciences and an assistant professor at Kazakh National Technical University

named after K.I. Satpaev, e-mail: [email protected].

Sultanov Bahyt Turlykhanovich is an adviser of the state scientific and technical

program at Kazakh National Technical University named after K.I. Satpaev, e-mail:

[email protected].

Abdilov Zheksenbek Makeevich is a doctor of economic sciences and a professor at

Kazakh National Technical University named after K.I. Satpaev, e-mail: [email protected].

NovikovDmitriy Alexandrovich is a correspondingmember of theRussianAcademy

of Sciences, doctor of technical sciences, and a professor at the Trapeznikov Institute

of Control Sciences of Russian Academy of Sciences, e-mail: [email protected].

Alshanov Rakhman Alshanovich is a doctor of economic sciences and a professor

at Kazakh National Technical University named after K.I. Satpaev, e-mail:

[email protected].

Ashimov Askar Abdykapparovich is a researcher of the state scientific

and technical program at Kazakh National Technical University named after

K.I. Satpaev, e-mail: [email protected].


279

Index

AAnalysis of national economy.

See Macroeconomic analysis and

parametric control

BBanach–Alaoglu’s theorem, 105–106

CChain-recurrent set, 7–8

Computable general equilibrium (CGE) model

economic branches

aggregate consumer, economic agent

17, 171

cyclic oscillations, 205–208

discretionary public policy

determination, 175

elasticity coefficients, 199–200

endogenous variables, 163–165

exogenous variables, 162–163

governmental prices, algorithm,

159–160

government, economic agent 18,

171–173

growth sources analysis, 200–203

integral indexes, 173

macroeconomic theory, 204–205

model agents, 160–161

model markets, 165–167

optimal parametric control laws,

208–211

parametric identification and

retrospective prediction, 173–175

producing products and services,

economic agent 1-16, 167–170

prospective analysis, endogenous

variables, 186–199

retrospective analysis, exogenous


statistical data, Republic of

Kazakhstan, 176

subsystems, 158–159

interbranch connections, 158

knowledge sector

aggregate consumer (households),

economic agent 4, 230–231

consumer price indexes,

237–238


economy branches, economic agent 3,

227–229

endogenous variables,

213–216

exogenous parameters, 213

general equilibrium, economic

branches, 236–237


231–233

innovation sector, economic agent 2,

225–227

integral indices, 233–236

macroeconomic indicators, cyclic

oscillations, 236




240–242

science and education sector,


mathematical models, 157

phenomenological models, 157

regression equation, 157


281

Computable general equilibrium (CGE) model

(cont.)shady sector



banking sector, economic agent 6, 265

economic agent no. 3, 261–262

economic agents, 242–244


exogenous parameters, 244–247


264–265

integral indices, 266

market sector, economic agent 2,

258–261

mathematical model, 242


optimal values, 268–271

outer world, economic agent 7, 266

parametric identification, 266–268

state sector, economic agent 1,

254–258

Continuous-time deterministic dynamical

system

parametric control laws for, 9–11

solvability conditions, 11–12

Cycle stability, Kondratiev cycle, 147

Cyclic market dynamics

Goodwin mathematical model

description, 147–149

optimal parametric control law,

150–152, 156

structural stability analysis, 149–150,

153–156

Kondratiev cycle, mathematical model

criterion K optimal value, variational

calculus problem, 147, 148


economic systems evolution, 144–146

estimation, robustness, 143

structural stability estimation, 147

DDecision makers (DM)

IDSS (see Information decision support

system (IDSS) and DM)

interaction vs. IDSS, 24Deterministic systems. See Discrete-time

deterministic dynamical system

Discrete-time deterministic dynamical system

parametric control laws, 13–14

solvability condition, 15

Discrete-time stochastic dynamical system



EEconomic branches, CGE model

aggregate consumer, economic

agent 17, 171

branch demand, labor force, 183, 185

budget, branch 1, 181, 182

capital productivity, 183, 185



determination, 175




fixed assets, branch, 183, 184


159–160


171–173

gross value added, branch 4, 181, 182

growth sources analysis

coefficients factors, 201–203

GVA, 200–201

retrospective data, 200

indicators, 178–183, 187–193


investment products, economic agent 1,

176, 177

macroanalysis, retrospective, 194–196


model agents

constants and variables, 161

description, 160

economic sectors, 160–161


optimal parametric control laws, 208–211

parametric identification and retrospective

prediction

algorithm, 174–175

macroeconomic mathematical

model, 173

measured and computed values, output

variables and deviations, 175

producing products and services, economic

agent 1-16, 167–170

production, investment products, 185–187

prospective analysis, 196–199

retrospective analysis, exogenous variables,

176–177

282 Index


Kazakhstan, 176


Economic equilibrium models, 125–127

Economic growth, one-sector Solow model

analysis, structural stability, 43, 45

choosing optimal laws, 43–45

dependence, optimal value, 45

estimation, model parameters, 42–43

model description, 42

Economic instruments

equilibrium solutions and payment balance

states, 136–139

equilibrium state macroestimation

equilibrium values, endogenous

parameters, 127

Keynesian mathematical model,

125–126

labor supply price and production

function, 126

IS model and analysis, 118–121

wealth and money markets

joint equilibrium and actual

values, 124

macroestimation, joint equilibrium

state, 124

plots, IS and LM models, 124, 125

Estimating defense cost, Richardson model

analysis, structural stability,

47, 48–49


dependence, optimal value, 49–50



FForrester’s mathematical model

analysis, structural stability, 86–87, 90


finding bifurcation points, 90–91

model description, 83–86

GGeneral equilibrium in economy. See National

economic evolution control based

CGE model

Global and national economy, Forrester’s

mathematical model


86–87, 90




Goodwin mathematical model


optimal parametric control laws

computational experiments, 16, 152

economic parameter k, 151

market cycle, 152

relations set, 150

solving problems, stages, 151

structural stability analysis

with parametric control, 153–156

without parametric control,

149–150

IInformation decision support system

(IDSS) and DM

application of parametric control theory,

23–24

effective public economic policy based

aggregate scheme, 25

“choice of one or several econometric

models”, 27–28

“economic instruments of the economic

policy implementation in the chosen

direction”, 34–35

“evaluation of the national economic

conditions”, 26–27

“statement and solution of the

problem(s) of estimation(s)”, 28–34

Invertibility test, 8

Investment-savings (IS) model. See IS model

and analysis, economic instruments

IS model and analysis, economic instruments

macroeconomic theory, 119, 120

macroestimation, 118

public expenses and taxation, 120

statistical characteristics, 118–119

JJones’s model

analysis, structural stability, 100–101

dependence of optimal values,

104–105

finding optimal values, 102–104


parameters evaluation and retrospective

forecast, 100

parametric sensitivity estimation,

101–102

Index 283

KKeynesian model

economic instruments

equilibrium values, endogenous

parameters, 127

labor supply price and production

function, 126

parametric control, 127–128

Knowledge sector, CGE model



consumer price indexes, 237–238



227–229

endogenous variables, 213, 216–218


general equilibrium, economic branches,

236–237

government, economic agent 5, 231–233


225–227

integral indices

economy, Russian Federation, 233–234

model and identification process, 234

Nelder-Mead algorithm, 234

output variables and respective

deviations, 234, 235

relations, 234


oscillations, 236

model agents


economic agent 2, 212


economic agent 4-7, 213

model markets, 216, 219–222

optimal parametric control laws

computational experiments, 241

endogenous variables, 240

Nelder-Mead algorithm, 241–242

parametric control problem, 242

science and education sector, economic

agent 1, 222–224

Kondratiev cycle, mathematical model

criterion K optimal value, variational

calculus problem, 147, 148


parametric control, economic system

evolution

capital productivity ratio, 145, 146

coefficients and criteria,

optimal laws, 145

efficiency, innovations, 146

optimal laws, relations, 144

optimal values, criteria, 144

robustness estimation, without parametric

control, 143

structural stability estimation, 147

MMacroeconomic analysis. See Computable

general equilibrium (CGE) model

Macroeconomic analysis and parametric

control

equilibrium solutions and payment balance

states

common economic equilibrium,

136–137

dependence, optimal values, 138, 139

IS-LM-ZBO, 138

money supply and public expenses, 236

national economy in 2007, 136, 137

open economy model, small country and

estimation, 129–136

small country model

dependencies, optimal values,

139, 140

external exogenous parameters, 139

unemployment, 129–130

Market economic development.

See Parametric control theory

Mathematical model, national economic

system

discrete-time dynamical system

(see Robinson approach)

international trade and currency exchange

analysis, structural stability, 73, 79–81




public expense and interest rate,

government loans


53, 62–65




parametric control, market economic

development, 59–62

stability indicators evaluation

absolute stability indicator, 5

CGE model, economic branches, 6

maximal absolute stability indicator, 5

Monte Carlo method, 4–5

284 Index

normalized input data vectors, 3–4

Orlov’s definition, 3

Mathematical models of cycles

Goodwin (see Goodwin mathematical

model)

Kondratiev cycle (see Kondratiev cycle,

mathematical model)

Money markets

equilibrium conditions

estimation, money velocity, 121

Fisher equation, 121

LM model, 121, 123

property demand, econometric

estimation, 122

statistical characteristics, 123

values, money supply and

aggregate, 122

values of multipliers, 122

and wealth, 124–125

NNational economic evolution control based

CGE model

economic branches

aggregate consumer, economic agent

17, 171



determination, 175





159–160


171–173

growth sources analysis, 200–203






208–211

parametric identification and

retrospective prediction,

173–175

producing products and services,

economic agent 1-16, 167–170

prospective analysis, endogenous


retrospective analysis, exogenous



Kazakhstan, 176


interbranch connections, 158

knowledge sector



consumer price indexes, 237–2238



227–229



general equilibrium, economic

branches, 236–237


231–233


225–227



oscillations, 236




240–242

science and education sector,


mathematical models, 157

phenomenological models, 157

regression equation, 157

shady sector



banking sector, economic

agent 6, 265

economic agent no. 3, 261–262





264–265


market sector, economic agent 2,

258–261



optimal values, 268–271

outer world, economic

agent 7, 266


state sector, economic agent 1,

254–258

Index 285

Neoclassical theory

analysis, structural stability, 37–38, 40–41


finding bifurcation points, 41


OOne-sector Solow model

analysis, structural stability, 43, 45


dependence, optimal value, 45



Open economy of small country

balance of payments, 134

consumption, 131

domestic commercial interest rate, 133

double balance in 2007, 135

economic indices, 129

equilibrium and actual values in 2007

and 2008, 135–136

investment model, 132

labor supply price, 131

money velocity, 130

net capital export model, 132

solving system, 134, 135

wealth export model, 133

Optimal control and uncontrolled

parameters, 156

Optimal control problems

cyclic oscillations

economic branches, 205–208

knowledge sector, 238–242

parameters, CGE model, 268–271

Optimal growth, neoclassical theory


37–38, 40–41


finding bifurcation points, 41


Optimal laws, parametric control theory

continuous-time deterministic

dynamical system

parametric control laws for, 9–11


discrete-time deterministic dynamical

system



discrete-time stochastic dynamical system



uncontrolled parameter variations

bifurcation points, 21–22

continuous dependence of optimal

values, 20–21

description, 19

Optimal parametric control

Goodwin mathematical model, 150–152


PParametric control laws

continuous-time deterministic dynamical

system, 9–11


system, 13–14

discrete-time stochastic dynamical system,

16–18

Parametric control theory

algorithm of application

definition and implementation, 23–24

DM with IDSS, 24

applications

endogenous Jones’s model, 98–105

Forrester’s mathematical model, global

economy, 83–91

mathematical model, national economic

system, 50–83

neoclassical theory, optimal growth,

36–41

one-sector Solow model, economic

growth, 42–45

Richardson model, 46–50

Turnovsky’s monetary model, 91–98

components, 2–3

linear constraints, 1–2

optimal laws (see Optimal laws, parametric

control theory)

proof of statements

Banach–Alaoglu’s theorem, 105–106

Jacobian matrix, 114

Lipschitz constant of function,

106–108, 111–112

Robinson’s Theorem A, 115

Weierstrass’s theorem, 109–110, 112

rules of interaction between DM and IDSS,

24–35

stability and structural stability,

mathematical model, 3–9

Parametric identification, 173–175, 211, 242,

266–268

Problem solutions, 270

Public economic policy, 24–35

286 Index

RRetrospective prediction

knowledge sector

economic agent 4, aggregate consumer

(households), 230–231

economic agent 3, economy branches,

227–229

economic agent 5, government,

231–233

economic agent 2, innovation sector,

225–227

economic agent 1, science and

education sector, 222–224






and parametric identification,

173–175

shady sector


economic agent 4, aggregate consumer

(households), 262–263

economic agent 6, banking

sector, 265

economic agent 5, government,

264–265

economic agent 2, market sector,

258–261

economic agent 7, outer world, 266


economic agent 1, state sector,

254–258





parametric identification,

266–268

Richardson model

analysis, structural stability, 47,

48–49


dependence, optimal value, 49–50



Robinson approach


chain-recurrent set, 7–8

construction, computational algorithm, 7

invertibility test, 8

weak structural stability, 9

Robinson’s theorem, 115, 143

SShady sector, CGE model



banking sector, economic agent 6, 265





government, economic agent 5, 264–265


market sector, economic agent 2, 258–261


model markets


equilibrium prices, 251

governmental prices, 251–252

labor prices, 252–253

optimal values

GDP, 270

parameters, 270

parametric control approach, 270, 271

simulation process, 268

outer world, economic agent 7, 266


state sector, economic agent 1

budgets, 256–257

capital assets’ dynamics, 257

economic agent 5, 256

exported products, 256

governmental prices and capital

products, 255

households, 255–256

investment products, 255

process, 254–255

VAT, profit tax and property tax,

257–258

Small country model

dependencies, optimal values, 139, 140

external exogenous parameters, 139

Solvability conditions

continuous-time deterministic dynamical

system, 11–12


system, 15

discrete-time stochastic dynamical system,

18–19

Stability indicators



maximal absolute stability indicator, 5




Index 287

Stability of mathematical models

discrete-time dynamical system

(see Robinson approach)

stability indicators evaluation



maximal absolute stability

indicator, 5




Stochastic systems, 18

TTheory of optimal growth, 36–41

Turnovsky’s monetary model

analysis, structural stability, 95

dependence of optimal values, 97–98

estimation, parametric sensitivity, 95–96

finding optimal values, 96–97


parameters estimation and retrospective

forecast, 93–94

UUncontrolled parameter variations


continuous dependence of optimal values,

20–21

description, 19

VVariational calculus problem

continuous-time deterministic

dynamical system

parametric control laws for,

9–11



system



discrete-time stochastic dynamical system

parametric control laws,

16–18


uncontrolled parameter variations


continuous dependence of optimal

values, 20–21

description, 19

WWeak structural stability, discrete-time

dynamical system


chain-recurrent set, 7–8

construction, computational

algorithm, 7

invertibility test, 8

Weierstrass’s theorem,

109–110, 112

288 Index

macroeconomic analysis and parametric control of a national economy

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