macroeconomic analysis and economic policy based on parametric control

Macroeconomic Analysis and Economic PolicyBased on Parametric Control

Abdykappar A. AshimovBahyt T. SultanovZheksenbek M. AdilovYuriy V. BorovskiyDmitriy A. NovikovRobert M. NizhegorodtsevAskar A. Ashimov

Macroeconomic Analysisand Economic Policy Basedon Parametric Control

Abdykappar A. AshimovKazakh National Technical UniversityNational Academy of Sciences of the RepuAlmaty City [email protected]

Bahyt T. SultanovKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City [email protected]

Zheksenbek M. AdilovKazakh National Technical UniversityAlmaty City [email protected]

Dmitriy A. NovikovInstitute of Control Sciences RASMoscow [email protected]

Askar A. AshimovKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City [email protected]

Yuriy V. BorovskiyKazakh National Technical UniversityState Scientific and Technical ProgramAlmaty City [email protected]

Robert M. NizhegorodtsevInstitute of Control Sciences RASMoscow [email protected]

ISBN 978-1-4614-1152-9 e-ISBN 978-1-4614-1153-6DOI 10.1007/978-1-4614-1153-6Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011936791

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Preface

Problems of macroeconomic analysis and the participation of the state in control

of market economic development were critically revealed by the latest global

economic crisis in 2007–2009.

This work presents the elements of parametric control theory, as well as some

results in the context of the aforementioned problems based on AD-AS, IS, LM,

IS–LM , IS–LM-BP mathematical models and the models of Keynes all-economic

equilibrium, open economy of a small country,market cycles, and computablemodels

of general equilibrium.

The materials of this book to a certain extent allow estimating the versions of

recommendations on stabilizing acyclic economic policy and choosing state policy

in the area of economic growth.

Chapter 1 is devoted to a presentation of parametric control theory. This chapter

includes the following:

l Components of parametric control theory.l Methods of analysis of the structural stability of mathematical models of a

national economic system.l Statements of variational calculus problems of choosing optimal sets of

parametric control laws for continuous- and discrete-time dynamical systems.

In these variational calculus problems, the objective functions express some

(global, intermediate, or tactical) goals of economic development. The phase

constraints and constraints in admissible form are presented by the mathematical

models of the economic systems. The considered variational calculus problems

of choosing optimal laws of parametric control in the environment of a given

finite set of algorithms differ from those considered earlier in the theory of

extremal problems [18] and are characterized by computationally acceptable

applications.l A solution existence theorem of the variational calculus problem of choosing the

optimal set of parametric control laws in the environment of a given finite set of

algorithms for continuous- and discrete-time systems.

v

l Defining the bifurcational points of extremals of the variational calculus

problem of choosing the optimal set of parametric control laws in the environ-

ment of a given finite set of algorithms.l A theorem establishing sufficient conditions for the existence of bifurcation

points. The presented results differ from similar well-known results of para-

metric disturbance analysis in the variational calculus problems considered in

[18], where parametric disturbance is used for obtaining sufficient extremum

conditions via construction of respective S-functions and using the constraint-

removing principle. The presented results also differ from the results of [42]

examining stability conditions for solutions of variational calculus problems

(Ulam problem). Research on this problem is reduced to finding the regularity

conditions under which the objective function of the disturbed problem has a

minimum point close to that of the objective function of the undisturbed

problem. Also, [13] offers a theorem stating existence conditions for the

bifurcation point of the variational calculus problem with the objective

function considered in the Sobolev space Wmp

0

ðOÞ ð2 � p<1Þ and depending

on some scalar parameter l∈[0,1].

The remainder of the chapter presents an algorithm for the application of

parametric control theory and examples of its application based on a number of

mathematical models of economic systems.

Chapter 2 presents economic estimates of functions obtained on the basis of

statistical information on the national economy of Kazakhstan that characterizes

the state of the national economy. A number of mathematical models including

AD–AS, IS, LM, IS-LM, IS–LM–BP, Keynesian general economic equilibrium

models (constructed on the basis of economic functions), as well as the model of

open economy of a small country are described. The results of analysis of influence

of economic instruments on the equilibrium solutions in the context of the afore-

mentioned mathematical models of economic equilibrium of the national economy

are presented.

Based on mathematical models of general economic equilibrium and open

economy, problems of estimation of optimal values of economic instruments in

the sense of certain criteria are stated and solved. Results on the dependence of the

optimal criteria values on the set of uncontrolled economic parameters given in the

respective ranges are described.

The main sources of inflation in the economy of Kazakhstan are revealed. It is

proved that prediction of inflation rates can be accomplished on the basis of

approaches of both the rational and adaptive expectation theories.

Chapter 3 is devoted to the development of the market cycle theory. It contains the

results of the analysis of structural stability of the Kondratiev and Goodwin

mathematical models of cycles and the solution of parametric control problems

on the basis of the aforementioned mathematical models.

vi Preface

Chapter 4 presents results on the parametric control of economic growth based on

computable models of general equilibrium. This chapter describes the proposed

algorithm of model parametric identification, taking into consideration the charac-

teristic features of the macroeconomic models with high dimensions and facilitating

the discovery of the global extremum of a function depending on a large number of

variables (more than a thousand). The algorithm uses two objective functions (two

criteria of identification, main and additional). This allows the withdrawal of the

values of identified parameters from neighborhoods of local (and nonglobal)

extreme points concurrently, maintaining the conditions of coordinated motion to

the global extreme point.

This chapter includes statements and solutions of parametric control problems of

economic growth on the basis of computable models with a sector of knowledge

of economic branches, as well as with the shady sector.

The authors are grateful to N.Yu. Borovskiy, D.T. Aidarkhanov, B.T.

Merkeshev, N.T. Sailaubekov, Zh.T. Dil’debayeva, O.V. Polyakova, and

M.V. Dzyuba for their help in carrying out computer simulation experiments.

Almaty City, Kazakhstan Abdykappar A. Ashimov

Almaty City, Kazakhstan Bahyt T. Sultanov

Almaty City, Kazakhstan Zheksenbek M. Adilov

Almaty City, Kazakhstan Yuriy V. Borovskiy

Moscow, Russia Dmitriy A. Novikov

Moscow, Russia Robert M. Nizhegorodtsev

Almaty City, Kazakhstan Askar A. Ashimov

Preface vii

Contents

1 Elements of Parametric Control Theory of Market EconomicDevelopment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Components of Parametric Control Theory of Market

Economic Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Methods of Analysis of Structural Stability of Mathematical

Models of National Economic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Approach to Choosing (Synthesis) Optimal Laws

of Parametric Control of the Development of National

Economic Systems and the Analysis of Existing

Conditions for the Solution of the Variational Calculus

Problem of Choosing (Synthesis) Optimal Laws

of Parametric Control in the Environment of the Given

Finite Set of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Statement of the Variational Calculus Problem

of Choosing an Optimal Set of Parametric

Control Laws for a Continuous-Time System. . . . . . . . . . . . . . . . 6

1.3.2 Analysis of Existing Conditions for the Solution

of the Variational Calculus Problem of Choosing

an Optimal Set of Parametric Control Laws

for a Continuous-Time Dynamical System. . . . . . . . . . . . . . . . . . . 8

1.3.3 Development of an Approach to Synthesis

of Optimal Parametric Control Laws for the

Development of National Economic Systems

and the Analysis of Existing Conditions for a Solution

to the Variational Calculus Problem of Choosing

(Synthesis) Optimal Parametric Control Laws

in the Environment of a Given Finite Set

of Algorithms for CGE Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ix

1.4 Analysis of the Influence of Uncontrolled Parametric

Disturbances on the Solution of the Variational Calculus

Problem of Synthesis of Optimal Parametric Control

Laws in the Environment of the Given Finite Set

of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Algorithm of the Application of Parametric

Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Examples of the Application of Parametric

Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.1 Mathematical Model of the Neoclassic

Theory of Optimal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.2 One-Sector Solow Model of Economic Growth. . . . . . . . . . . . 25

1.6.3 Richardson Model of the Estimation

of Defense Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6.4 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 . . . . . . . . . . . . . . . . . . 34

1.6.5 Choosing the Optimal Laws of Parametric Control

of Market Economic Development on the Basis

of the Mathematical Model of the Country Subject

to the Influence of the Share of Public Expenses

and the Interest Rate of Government Loans . . . . . . . . . . . . . . . . 37

1.6.6 Mathematical Model of the National Economic

System Subject to the Influence of International

Trade and Currency Exchange on Economic Growth. . . . . . 53

1.6.7 Forrester’s Mathematical Model of Global Economy. . . . . . 66

2 Macroeconomic Analysis and Parametric Control of EquilibriumStates in National Economic Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.1 Factor Modeling of the Aggregate Demand in a National

Economy: AD–AS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.1.2 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.1.3 Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2 Macroeconomic Analysis of the National Economic State

Based on IS, LM, IS–LM Models, Keynesian All-Economy

Equilibrium. Analysis of the Influence of Instruments

on Equilibrium Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.2.1 Construction of the IS Model and Analysis

of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . 92

2.2.2 Macroeconomics of Equilibrium Conditions

in the Money Market. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

x Contents

2.2.3 Macro-Estimation of the Mutual Equilibrium State

in Wealth and Money Markets. Analysis

of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . 97

2.2.4 Macro-Estimation of the Equilibrium State

on the Basis of the Keynesian Model of

Common Economic Equilibrium. Analysis of

the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . . . 99

2.2.5 Parametric Control of the Open Economy

State Based on the Keynesian Model. . . . . . . . . . . . . . . . . . . . . . 101

2.3 Long-Term IS–LM Model and Mundell–Flemming Model. . . . . . . 102

2.3.1 Problem Statement and Data Preparation . . . . . . . . . . . . . . . . . 102

2.3.2 Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.3.3 Final IS-LM-BP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

2.4 Macroeconomic Analysis and Parametric Control

of the National Economic State Based on the Model

of a Small Open Country . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.4.1 Construction of the Model of an Open Economy

of a Small Country and the Estimation of

Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.4.2 Influence of Economic Instruments on Equilibrium

Solutions and State of the Balance of Payments . . . . . . . . . . 121

2.4.3 Parametric Control of an Open Economy State

Based on a Small Country Model . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.5 Modeling of Inflationary Processes by Means of Regression

Analysis: Rational and Adaptive Expectations . . . . . . . . . . . . . . . . . . . . 125

2.5.1 Preparation of the Data for Factor Regression

Models of Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2.5.2 Construction of One-Factor Regression


2.5.3 Construction of Multifactor Regression


2.5.4 Construction of Autoregression Models

of Inflation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3 Parametric Control of Cyclic Dynamics of Economic Systems . . . . . 145

3.1 Mathematical Model of the Kondratiev Cycle . . . . . . . . . . . . . . . . . . . . 145

3.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.1.2 Estimating the Robustness of the Kondratiev Cycle

Model Without Parametric Control . . . . . . . . . . . . . . . . . . . . . . . . 147

3.1.3 Parametric Control of the Evolution

of the Economic System Based on the

Kondratiev Cycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.1.4 Estimating the Structural Stability of the Kondratiev

Cycle Mathematical Model with Parametric Control . . . . . 150

Contents xi

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.2 Goodwin Mathematical Model of Market Fluctuations

of Growing Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.2.2 Analysis of the Structural Stability of the Goodwin

Mathematical Model Without Parametric Control . . . . . . . . 153

3.2.3 Problem of Choosing Optimal Parametric

Control Laws on the Basis of Goodwin’s

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.2.4 Analysis of the Structural Stability of the Goodwin

Mathematical Model with Parametric Control . . . . . . . . . . . . 157

3.2.5 Analysis of the Dependence of the Optimal

Parametric Control Law on Values of the

Uncontrolled Parameter of the Goodwin

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4 Parametric Control of Economic Growth of a NationalEconomy Based on Computable Modelsof General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.1 National Economic Evolution Control Based

on a Computable Model of General Equilibrium

with the Knowledge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1.1 Model Description, Parametric Identification,

and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1.2 Finding Optimal Parametric Control Laws

on the Basis of the CGE Model with the

Knowledge Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189


Parametric Control Law on Values of

Uncontrolled Parameters Based on the CGE Model

with the Knowledge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193



of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195



4.2.2 Finding Optimal Parametric Control Laws

on the Basis of the CGE Model

of the Economic Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

xii Contents


Parametric Control Law on Values

of Uncontrolled Parameters on the Basis

of the CGE Model of Economic Sectors . . . . . . . . . . . . . . . . . . 214



with the Shady Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215



4.3.2 Finding the Optimal Values of the Adjusted

Parameters on the Basis of the CGE Model

with the Shady Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Contents xiii

Chapter 1

Elements of Parametric Control Theoryof Market Economic Development

1.1 Components of Parametric Control Theoryof Market Economic Development

As is well known, the state implements one of its prime economic functions,

namely, budgetary and fiscal policies, as well as monetary and credit policy, by

way of normatively establishing such economic parameters as various tax rates,

public expenses, discount rate, norm of reservation, credit rate, exchange rate, and

others.

The modern political economy [14, 21], within the framework of Keynesian

concepts, monetarism, and the theory of rational expectations, proposes various

sufficiently interesting views on the development of macroeconomic processes

depending on the values of one or another economic parameter (or a set of

economic parameters) mentioned above. Various conceptual (verbal) models of

economic regulation in the context of some (global, intermediate or tactical)

objective by means of choosing one or another economic parameter (parameters)

have been proposed.

Nevertheless, modern economic theory does not have a unified and clear

approach to determining optimal values of the aforementioned parameters, namely,

various tax rates, share of public expenses in the gross domestic product, discount

rate, exchange rate, and others.

In practice, the scale of governmental control in the fields of budgetary and fiscal

policies, as well as monetary and credit policies, its specific forms and methods,

essentially differ for various countries. They reflect the history, traditions, type, and

other factors of national culture, scale of a country, its geopolitical position, and

other factors.

In recent years, active research of the dynamics of economic parameters and

their influence on the evolution of economic processes has been carried out. Hence

in [45], econometric methods are applied for modeling dynamic series and statisti-

cal prediction of tax yields. In a number of papers [12], econometric methods have

A.A. Ashimov et al., Macroeconomic Analysis and Economic PolicyBased on Parametric Control, DOI 10.1007/978-1-4614-1153-6_1,# Springer Science+Business Media, LLC 2012

1

also been used for analysis of dependencies between the parameters of monetary

and credit politics (rate of refinancing, norm of reservation) and the indicators of

economic development (indicators of investment activity in the real sector and

others). In [34], on the basis of a mathematical model proposed by the authors, after

solving the parametric identification problem, the influence of the share of public

expense on the gross domestic product and the influence of the interest on govern-

mental loans on the mean income of the working population, mean public expenses,

and mean gross domestic product are analyzed.

In the mathematical economics, the so-called scenario approach is also proposed

for the estimation of a possible strategy of economic system development by means

of exploring various scenarios on the basis of the chosen mathematical model using

various sets of economic parameters and analysis of the respective solutions.

Thus, in the known literature and practice there are no scientific results in the

area of parametric control of the development of a market economy taking into

consideration the requirements of the optimality of the evolution of the economic

system of a country and recommendations on the development and implementation

of an efficient state economic policy developed on the basis of the aforementioned

scientific results.

Many dynamical systems [15], including national economic systems [27, 34],

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

ordinary differential equations:

dx

dt¼ f x; u; lð Þ; x t0ð Þ ¼ x0; (1.1)

where x ¼ x1; x2; :::; xnð Þ 2 X � Rn is the system state; u ¼ u1; u2; :::; ul� � 2 W �

Rl is the vector of controlled (regulated) parameters; W, X are compact sets with

nonempty interiors IntðWÞ and IntðXÞ, respectively; l ¼ l1; l2; :::; lm� � 2 L � Rm

is the vector of the uncontrolled parameters; L is an open connected set;

the mappings f x; u; lð Þ : X �W � L ! Rn and @f@x ,

@f@u ,

@f@l are continuous in

X �W � L; and t0; t0 þ T½ � is a fixed (time) interval.

As is well known, the solution (evolution) of the considered system of ordinary

differential equations depends on both the vector of initial values x0 2 IntðXÞ andthe values of vectors of controlled (u) and uncontrolled (l) 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 the vectors of both

controllable and uncontrollable parameters.

It is also known [3] that the process described by (1.1) can 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 of (1.1). In other words, system (1.1) must possess the property of robustness or

structural stability.

For this reason, a theory of parametric control of market economic development

is proposed in [7, 8, 54–56]. This theory consists of the following components:

2 1 Elements of Parametric Control Theory of Market Economic Development

1. Methods for forming a set (library) of macroeconomic mathematical models.

These methods are oriented toward a description of various specific socioeco-

nomic situations, taking into consideration environmental safety conditions.

2. Methods for estimating conditions for robustness (structural stability) of models

of national economic systems from the library without parametric control.

In this regard, conditions for the considered mathematical models to belong to

the Morse–Smale class of systems, or to the class of O-robust systems, or to the

class of uniformly robust systems, or to the class ofУ-systems, or to the class of

systems with weak structural stability are verified.

3. Methods of control or attenuation of the nonrobustness (structural instability)

of mathematical models of economic systems and the choice (synthesis) of

algorithms of control or attenuation of the structural instability of the respective

mathematical model of the national economic system.

4. Methods of choice and synthesis of the laws of parametric control of market

economic development based on mathematical models of the national economic

system.

5. Methods of estimating the robustness (structural stability) ofmathematical models

of national economic systems from the library with parametric control. In this

regard, conditions for the considered mathematical models with parametric

control to belong to the Morse–Smale class of systems, or to the class ofO-robustsystems, or to the class of uniformly robust systems, or to the class ofУ-systems,

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

6. Methods of adjustment of constraints on the parametric control of market

economic development in the case of structural instability of mathematical

models of national economic system with parametric control and adjustment

of the constraints on the parametric control of market economic development.

7. Methods of research and analysis of bifurcations of extremals of variational

calculus problems of choosing optimal laws of parametric control.

8. Development of recommendations on elaboration and implementation of effi-

cient governmental economic policy on the basis of the theory of parametric

control of market economic development taking into consideration specific

socioeconomic situations.

1.2 Methods of Analysis of Structural Stability of MathematicalModels of National Economic Systems

The methods of analysis of 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, Ω-robust systems,

У-systems, systems with weak structural stability).

1.2 Methods of Analysis of Structural Stability of Mathematical Models. . . 3

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

has available 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 by means of

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) of [69] on weak structuralstability.

Let N0 be some manifold, and N a compact subset in N0 such that the closure of

the interior of N is N. Let some vector field be given in a neighborhood of the set Nin N0: This field defines the C1-flux f in this neighborhood. Let R f ;Nð Þ denote thechain-recurrent set of the flux f on N.

Let R f ;Nð Þ be contained in the interior of N. Let it have a hyperbolic structure.Moreover, let the flux f upon R f ;Nð Þ also satisfy the transversability conditions of

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

particular, if Rð f ;NÞ is the empty set, then the flux f is weakly structurally stable onN. 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 by means of numerical estimation

of the chain-recurrent set R f ;Nð Þ for some compact region N of the phase state of

the considered dynamical system.

Let us 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 pro-

posed 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 set R f ;Nð Þ of 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 evolution for the considered time interval as 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 R f ;Nð Þ 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, 4, 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 R f ;Nð Þ.The method of estimating the chain-recurrent set for a compact subset of the

phase space of a dynamical system developed here allows one, in the case in which

the obtained chain-recurrent set R f ;Nð Þ is empty, to draw a conclusion about the

weak structural stability of the dynamical system.

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

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

Robinson’s theorem for estimating its weak structural stability.

Let us give a numerical algorithm for estimating the invertibility of the differen-

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

trajectory f tðx0Þ; t ¼ 0� Tf g in the phase space of the dynamical system is used

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

f tðx0Þ; t ¼ 0� Tf g. One can choose as such a curve a piecewise linear curve

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

semicascade.

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 the ascertainment of the fact that the

curve f ðLÞ does not have points of self-crossing, that is, x1 6¼ x2ð Þ )f x1ð Þ 6¼ f x2ð Þð Þ; x1; x2 2 L. For instance, one can determine the absence of self-

crossing points by means of testing monotonicity of the limitation of the mapping

f(L) onto L along any coordinate of the phase space of the semicascade f(L).2. An invertibility test for the mapping f(L) in neighborhoods of the points of the

curve L (local invertibility). Based on the inverse function theorem, such a test

1.2 Methods of Analysis of Structural Stability of Mathematical Models. . . 5

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

@x j ðxÞ� �

; 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 the 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 (semicascade defined by the mapping f ) with

phase space N0 2 Rn defined by the continuously differentiable mapping f canbe formulated as follows:

1. Find the discrete-time trajectory f tðx0Þ; t ¼ 0; T� �

and curve L in a closed

neighborhood N that is 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 R f ;Nð Þ. By virtue of the evident

inclusion R f ;N1ð Þ � R f ;N2ð Þ for N1 � N2 � N0, one can use any parallelepi-

ped belonging to N0 and containing L as the compact set N.4. In the case R f ;Nð Þ ¼ �, draw a conclusion about the weak structural stability of

the considered dynamical system in N.

This aggregate algorithm can be also applied for estimating the weak structural

stability of a continuous-time dynamical system (the flux f) if the trajectory L ¼f tðx0Þ; 0 � t � Tf g of the dynamical system is considered as the curve L. In this

case, item 2 of the aggregate algorithm is omitted.

1.3 Approach to Choosing (Synthesis) Optimal Laws ofParametric Control of the Development of NationalEconomic Systems and the Analysis of Existing Conditionsfor the Solution of the Variational Calculus Problem ofChoosing (Synthesis) Optimal Laws of Parametric Controlin the Environment of the Given Finite Set of Algorithms

1.3.1 Statement of the Variational Calculus Problemof Choosing an Optimal Set of Parametric ControlLaws for a Continuous-Time System

The statement of the variational calculus problem of choosing an optimal set of

parametric control laws from the set of combinations of p parameters taken r at atime in the environment of a given finite set of algorithms and the assertion of the


existence of a solution to the corresponding variational calculus problem in the

environment of the given finite set of algorithms can be formulated as follows:

Let xlðtÞ be the solution of system (1.1) defined in Sect. 1.1,

dx

dt¼ f x; u; lð Þ; x t0ð Þ ¼ x0;

on the interval t0; t0 þ T½ � with constant values of u 2 W and l 2 L. Let

xlðtÞ � IntðXÞ. Let us denote by xlðtÞ the solution to system (1.1) for chosen

u ¼ u1; u2; :::; u

l

� � 2 W. Further, u will be fixed.Denote by O the closed set in the space Cnþl t0; t0 þ T½ � consisting of all continu-

ous vector functions xðtÞ; uðtÞð Þ satisfying the following constraints:

x 2 X; u 2 U

2 W; x jðtÞ xjlðtÞj � axjlðtÞ; t 2 t0; t0 þ T½ �; j ¼ 1; n; a> 0:�� (1.2)

Let FiðxÞ : i ¼ 1� pf g and GðxÞ> 0 be the finite set of real-valued functions

that are continuous for x 2 X. All functions @Fi

@x j are also continuous in X. The abilityto chose an optimal set of parametric control laws from the set of combinations of

p parameters taken r at a time in the time interval t0; t0 þ T½ � is considered in the

environment of the following algorithms (control laws):

Uij ¼ kijFiðxÞ þ uj; i ¼ 1; p; j ¼ 1; l

� �: (1.3)

Here kij � 0 are the adjusted coefficients. Using a set of r (1� r� l, fixed here

and below) laws Uij from (1.3) with fixed kij in system (1.1) means the substitution

of the set of the functions ujs ¼ Uisjs

� �into the right-hand sides of the equations for

r different values of subscripts js 1� s� r; 1� js � l; 1� is � pð Þ. The other uj,where j is not included in the mentioned set of values of js, are considered to be

constant and equal to the values of uj.As can be seen from (1.3), each subset of this set enters mathematical model

(1.1) multiplicatively and gives an opportunity to obtain the multiplicative effect of

regulation owing to item kijFiðxÞ of the control algorithm.

The following functional (criterion) is considered for the solutions of system

(1.1) with the use of r control laws ujs ¼ Uisjs

� �:

K ¼ðt0þT

t0

G xlðtÞð Þdt: (1.4)

The statement of the problem of choosing a set of the parametric control laws

from the set of combinations of p parameters taken r at a time in the environment of

the given finite set of algorithms is as follows:

1.3 Approach to Choosing (Synthesis) Optimal Laws of Parametric Control. . . 7

With fixed l 2 L, find a set of r control laws

U ¼ Uis js ; s ¼ 1; r� �

from the set of algorithms (1.3) providing the supremum of the values of criterion

(1.4),

K ! supU

(1.5)

for the given time interval such that conditions (1.1, 1.2) hold.

1.3.2 Analysis of Existing Conditions for the Solution of theVariational Calculus Problem of Choosing an OptimalSet of Parametric Control Laws for a Continuous-TimeDynamical System

Let us prove the existence of a solution to problem (1.1, 1.2–1.5) by applying the

theorem about the continuous dependence of the solution to the Cauchy problem on

the parameters and the theorem about the continuous dependence of a definite

integral on a parameter.

Theorem 1.1 For any chosen set of laws U ¼ Uis js ; s ¼ 1� r� �

, where r� l,from the set (1.3) of algorithms under constraints (1.1) and (1.2), there exists asolution to the problem of finding the supremum of the criterion K (1.4):

ðt0þT

t0

G xlðtÞð Þdt ! supðki1 j1 ; ki2 j2 ; �� ; kir jr Þ

: (1.6)

If the set of possible values of the coefficients ki1j1 ; ki2j2 ; � � � ; kirjr� �

of the lawsfrom the considered problem is bounded, then the mentioned supremum for thechosen set of r laws is attained. For a finite set (1.3) of algorithms, problem(1.1–1.5) has a solution.

Proof Associating the respective output functions and regulating parametric

actions xlðtÞ; uðtÞð Þ of system (1.1) under control from the set of control laws U ¼Ui1j1 ; ki1j1� �

; Ui2j2 ; ki2j2� �

; � � � ; Uirjr ; kir jr� ��

with the set of the values of the

coefficients ki1j1 ; ki2j2 ; � � � ; kirjr� �

from this set of control laws defines a continuous

mapping H from some subset Rlþ ¼ ½0;þ1Þl to the space Cnþl t0; t0 þ T½ �.

The complete preimage H1ðOÞ of the set O for the mapping H is closed by

virtue of the theorem on the closure of the preimage of a closed set for a continuous

mapping. The set H1ðOÞ is nonempty, since it contains the coordinate origin Rlþ.


(With zero values of the coefficients of the function xðtÞ ¼ xlðtÞ; uðtÞ ¼ uð Þ,constraints (1.2) obviously hold.)

Associating the set of coefficients ~k 2: H1ðOÞ with the laws of criterion K (1.3)

for the solution of system (1.1) defines the continuous function

�K : H1ðOÞ ! ½0;1Þ:

Hence, with the chosen set U of laws, problem (1.1–1.5) is equivalent to the

problem of finding the supremum of the continuous bounded function

y ¼ �Kð~kÞ

on the closed set H1 Oð Þ. This function is continuous by virtue of the theorem on

the continuous dependence of the solution of a system of ordinary differential

equations on the parameters [21], the boundedness of this solution by virtue of

the inclusion x 2 X from (1.2), and continuous dependence of the definite integral

on the parameter. Therefore, problem (1.1–1.5) for a fixed set U of control laws

always has a solution, including the finite optimal values of criterion K. For thebounded set H1ðOÞ, this value of the criterion is attained with some values of

the coefficient ~k (the theorem on the existence of the maximum of a continuous

function on a compact set). For the unbounded set H1ðOÞ one can find a sequence

of values of the coefficients ~k from H1ðOÞ such that the values of the criterion Kcorresponding to the elements of this sequence approachK. Thus, we prove the factof existence of the solution to the variational calculus problem for the case of one

parametric control law. Finiteness of the set of possible control laws (1.3) yields the

correctness of the theorem, i.e., the fact of existence of a solution to problem

(1.1–1.5).

1.3.3 Development of an Approach to Synthesis of OptimalParametric Control Laws for the Development of NationalEconomic Systems and the Analysis of Existing Conditionsfor a Solution to the Variational Calculus Problem ofChoosing (Synthesis) Optimal Parametric Control Lawsin the Environment of a Given Finite Set of Algorithmsfor CGE Models

1.3.3.1 Description of Computable Models of General Equilibrium

In this section, the synthesis of optimal parametric control laws is extended to a new

class of models, namely, computable models of general equilibrium (CGE models).

The CGE model [24] can be generally defined by a system of relations that can

be decomposed into the following subsystems.


(1) The subsystem of difference equations connecting the values of the endogenous

variables for two consecutive years,

xtþ1 ¼ F xt; yt; zt; u; lð Þ: (1.7)

Here t is the year index (discrete-time index); t ¼ 0; 1; 2; :::;~xt ¼ xt; yt; ztð Þ 2 Rn is the vector of the endogenous variables of the system;

xt ¼ x1t ; x2t ; :::; x

n1t

� � 2 X1, yt ¼ y1t ; y2t ; :::; y

n2t

� � 2 X2, zt ¼ z1t ; z2t ; :::; z

n3t

� � 2 X3,

n1 þ n2 þ n3 ¼ n. Here variables xt include the values of capital assets, demand

balances of agents on banking accounts, and others; variables yt include the

values of supply and demand of agents in various markets, and others; variables

zt include various kinds of market prices and shares of the budget in the markets

with governmental prices for various economic agents; u and l are the vectors

of the exogenous parameters (controllable and uncontrollable, respectively);

X1, X2, X3, W are the compact nonempty sets IntðXiÞ; i ¼ 1; 2; 3, and IntðWÞ,respectively; F : X1 � X2 � X3 �W � L ! Rn1 is a continuous function.

(2) The subsystem of algebraic equations describing the behavior and interaction

of the agents in the various markets during the chosen year. These equations

allow one to express the variables yt via the exogenous parameters and other

endogenous variables,

yt ¼ G xt; zt; u; lð Þ: (1.8)

Here G : X1 � X3 �W � L ! Rn2 is a continuous function.

(3) The subsystem of recurrent relations for iterative computations of the equilib-

rium values of the market prices in various markets and shares of the budget in

the markets with governmental prices for various economic agents:

zt Qþ 1½ � ¼ Z zt Q½ �; yt Q½ �; L; u; lð Þ: (1.9)

Here Q ¼ 0; 1; 2; ::: is the iteration number; L is the set of the positive

numbers (adjusted constants of iterations). As these values decrease, the eco-

nomic system comes to an equilibrium state faster. However, the danger of

prices entering the negative range increases. Here Z : X2 � X3 � ð0;þ1Þn3 �W � L ! Rn3 is a continuous mapping that is contractive with fixed xt 2 X1;u 2 W, l 2 L and some fixed L. In this case, the mapping Z has a unique fixed

point, to which the iterative process (1.8, 1.9) converges.

For fixed values of the exogenous parameters, the CGE model of general

equilibrium (1.7–1.9) for each moment of time t defines the values of the endoge-nous variables ~xt corresponding to the equilibrium of prices of supply and demand

in markets with nongovernmental prices and the share of budget in the markets with

the governmental prices of the agents within the framework of the following

algorithm.


1. At the first step, it is assumed that t ¼ 0, and the initial values of the variables x0are set.

2. At the second step, for the current value of t, the initial values of the variables

zt½0� are set in the various markets and for the various agents. By means of (1.8),

the values of yt½0� ¼ G xt; zt½0�; u; lð Þ are computed. (These are the initial values

of supply and demand of the agents in the markets of goods and services.)

3. At the third step, for the current time t, the iterative process (1.9) starts. For eachQ, the current values of supply and demand are found from (1.8) as yt Q½ � ¼ G�xt; zt Q½ �; u; lð Þ via correction of the market prices and shares of the budgets in the

markets with the governmental prices of the economic agents.

Equality between the values of supply and demand in the various markets is a

condition for halting the iterative process. As a result, the equilibrium values of

the market prices in each market and the shares of the budget in the markets with

the governmental prices for various economic agents are determined. For such

equilibrium values of the endogenous variables, the number of iterations Q will

be omitted.

4. At the next step, by use of the obtained equilibrium solution for the time instance

t, the values of the variables xtþ1 for the next instant of time are computed by

means of difference equations (1.7). Then the value of t is increased by one.

Then go to step 2.

The number of iterations of steps 2, 3, 4 is determined in accordance with the

problems of calibration, prediction, and control at the time intervals chosen in

advance.

Extending the previously obtained results of parametric control theory in the

context of systems of ordinary differential equations to the class of CGE models

requires taking into account the fact that the models of such a class are the

semicascades. Therefore, it is necessary to extend the results of parametric control

theory for systems of nonlinear ordinary differential equations to the considered

class of CGE models.

All the reasoning of this section remains valid for other discrete-time systems,

for example, those obtained from continuous-time dynamical systems via

discretization.

1.3.3.2 Elements of Parametric Control Theory for the Classof Computable Models of General Equilibrium

The considered CGE model can be presented in the form of the continuous mapping

f : X �W � L ! Rn defining the transformation of the values of the system’s

endogenous variables for the year zero to the respective values of the next year

according to the algorithm presented above. Here the compact set X in the phase

space of the endogenous variables is determined by the set of possible values of the

variables x (the compact set X1 with nonempty interior) and the respective equilib-

rium values of the variables y and z calculated via relations (1.8) and (1.9).


Let us suppose that for the chosen point x0 2 Int X1ð Þ the inclusion

xt ¼ f tð~x0ÞjX12 Int X1ð Þ is correct with the fixed u 2 IntðWÞ and l 2 L for

t ¼ 0;N, where N is a fixed natural number. This mapping f defines a discrete-

time dynamical system (semicascade) on the set X:

f t; t ¼ 0; 1; :::f g: (1.10)

For the chosen u 2 IntðWÞ, let us denote by ~xt the points of the respective

trajectory ~xt ¼ f tð~x0Þ of the semicascade.

Let us denote by O the closed set in the space R nþlð Þ Nþ1ð Þ ([N + 1] sets of the

variables [~xt, ut] for t ¼ 0;N) defined by the constraints

~xt 2 X; ut 2 W; ~xjt ~x jt � aj ~x j

t; (1.11)

The latter inequalities in (1.11) are used for some values of j ¼ 1� n and with

positive values of xjt, aj > 0.

Let fHið~xÞ : i ¼ 1; pg and Ið~xÞ> 0 be a finite set of real-valued functions

continuous for ~x 2 X.The ability to choose an optimal set of parametric control laws from the set of

combinations of p parameters taken r at a time and for the finite trajectory ~xt,t ¼ 0;N, is analyzed in the environment of the following algorithms (control laws):

Uij ¼ kijHið~xÞ þ uj; i ¼ 1; p; j ¼ 1; l

� �: (1.12)

Here kij � 0 are the adjustable coefficients; u are the values of the regulated

parameter accepted or estimated by the results of calibration.

Using the set of r (1� r� l, r fixed here and below) laws Uij from (1.12) with

fixed kij for the semicascade defined by the mapping f means the substitution of the

set of functions ujs ¼ Uisjs

� �into the right-hand sides of the equations for r

different values of subscripts js 1� s� r; 1� js � l; 1� is � pð Þ. The other uj,where j is not included in the mentioned set of values of js, are considered to be

constant and equal to the values of uj. Let us denote by ut the values of the vectorsof parameters u obtained by means of control laws (1.12) for the time instant t. Thecoordinates of the vector ut are given by

u jt ¼ kijH

ið~xtÞ þ u j; j ¼ 1� l:

Let us consider the following objective function (criterion) for the trajectories of

semicascade (1.10) with use of a set of r control laws of the form ujs ¼ Uisjs

� �at the

time interval t ¼ 0; N (N is fixed):

K ¼ K ~x0; ~x1; :::; ~xNð Þ; (1.13)

where K is a function continuous in XN+1.


The statement of the problem of choosing a set of the parametric control laws

from the set of combinations of r parameters in the environment of the given finite

set (1.12) of algorithms for semicascade (1.10) is as follows: With fixed l 2 L, finda set of r control laws (and their coefficients) U ¼ Uisjs ; s ¼ 1; r

� �from the set

(1.12) of algorithms providing the supremum of the values of criterion (1.13):

K ! supU

(1.14)

under constraints (1.11).

The following theorem, similar to Theorem 1.1, can be formulated.

Theorem 1.2 For the semicascade (1.10) with use of any chosen set of laws

U ¼ Uis js ; s ¼ 1; r� �

, where r� l, from the set (1.12) of algorithms under

constraints (1.11), there exists a solution to the problem of finding the supremumof the criterion K:

K ! supðki1 j1 ; ki2 j2 ; �� ; kir jr Þ

: (1.15)

For the finite set (1.12) of algorithms and chosen 1� r� l, problem (1.10–1.15)has a solution.

Proof Associating the respective values of the endogenous variables and regulating

parametric actions ~xl;t; ut� �

, t ¼ 0;N, of the semicascade f tf g under control from

the set of control laws U ¼ Ui1j1 ; ki1j1� �

; Ui2j2 ; ki2j2� �

; � � � ;�Uirjr ; kir jr� �g with the

set of values of the coefficients ki1j1 ; ki2j2 ; � � � ; kir jr� �

from this set of control laws

defines a continuous mapping J from some subset Rrþ ¼ ½0;þ1Þl to the space

RðnþlÞðNþ1Þ.The complete preimage J1ðOÞ of the set O for the mapping J is closed by virtue

of the theorem on the closure of the complete preimage of a closed set under a

continuous mapping. The set J1ðOÞ is nonempty, since it contains the origin Rrþ.

(With zero values of the coefficients of the functions ~xt ¼ ~xl;t; ut ¼ u� �

,

constraints (1.11) obviously hold.)

Associating the laws of criterion (1.13) for the semicascade f tf g with the set of

coefficients �k 2 J1ðOÞ defines a continuous function

�K : J1ðOÞ ! ½0;1Þ:

Hence, with a chosen set U of laws, problem (1.10–1.15) is equivalent to the

problem of finding the supremum of the continuous bounded function

y ¼ �Kð�kÞ

on the closed set J1ðOÞ. This function is continuous by virtue of continuity of the

functions f,Hi, and I defined on a compact set. Therefore, problem (1.10–1.15) for a


fixed set U of control laws always has a solution including the finite optimal values

of the criterion K. Thus, we have proved the existence of a solution to the

variational calculus problem for the case of one parametric control law. Finiteness

of the set of possible control laws (1.12) yields correctness of the theorem, i.e., the

existence of a solution to problem (1.10–1.15).

For discrete-time dynamical systems, it is of practical interest to develop a

parametric control theory for the case in which the optimal (in the sense of some

criterion) values of the controlled parameters are estimated in some given set of

their values. Let us present the corresponding statement of the problem of finding

the optimal values of the criterion and the theorem on the existence of a solution to

this problem.

The statement of the problem of finding the optimal value of the controlled

vector of parameters (the problem of synthesis of the parametric control laws) for

semicascade (1.10) is as follows: For fixed l 2 L, find a set of N values of the

controlled parameters ut; t ¼ 1;N, that provides the supremum of the values of

criterion (1.13),

K ! suput; t¼1;N

; (1.16)


A similar problem can be stated for the case of minimization of the criterion K.The following theorem holds.

Theorem 1.3 For semicascade (1.10) under constraints (1.11), there exists asolution to problem (1.10, 1.11, 1.16) of finding the supremum of the criterion K.

The proof is based on the existence of the supremum of the values of a continu-

ous function defined on some compact set and reproduces the proof of the previous

theorem.

1.4 Analysis of the Influence of Uncontrolled ParametricDisturbances on the Solution of the Variational CalculusProblem of Synthesis of Optimal Parametric Control Lawsin the Environment of the Given Finite Set of Algorithms

Below we present the results of analysis of the influence of variations in the

uncontrolled parameters and bifurcation-point changes under the parametric

disturbances in the variational calculus problem of choosing the optimal parametric

control laws in the environment of a given finite set of algorithms with phase

constraints and constraints in the allowed form.


The functionals or phase constraints, as well as the constraints in the allowed

form of the considered problems, often depend on one or more parameters. Analysis

of similar problems requires defining the bifurcation point and the conditions for its

existence, and an analysis of the bifurcation value of the parameter. In applying

parametric control of the mechanisms of market economies, finding the extremal

solution of a given problem and its type can depend on the values of some

uncontrollable parameters, in which case the task of defining the bifurcation

value becomes practical.

We introduce the following definition characterizing such values of the parame-

ter l at which the substitution of one optimal control law instead of another one is

possible.

Definition A value l 2 L is called a bifurcation point of the extremal (1.1–1.5)

(or [1.10–1.15]) if for l ¼ l there exist at least two different optimal sets of r laws

from set (1.3) (or (1.12)) differing in at least one law Uij, and in each neighborhood

of the point l there exists a value l 2 L for which the problem as an immediate

corollary has a unique solution.

The following theorem establishes sufficient conditions for the existence of a

bifurcation point of the extremals of the given variational calculus problem of

choosing the parametric control law in the given environment of the algorithms

for the case of continuous- or discrete-time dynamical systems.

Theorem 1.4 (On the existence of the bifurcation point). With the parametervalues l1 and l2 (l1 6¼ l2; l1; l2 2 L), if problem (1.1–1.5) (or [1.10–1.15])has unique solutions for two different optimal sets of r laws from the set (1.3) (or(1.12)) differing in at least one law Uij, then there exists at least one bifurcationpoint l 2 L.

Proof Connect the points l1 and l2 by a smooth curve S lying in the region L:S ¼ lðsÞ; s 2 ½0; 1�f g; lð0Þ ¼ l1; lð1Þ ¼ l2. Denote by KUðsÞ the optimal value

of the criterion K of problem (1.1–1.5) (or [1.10–1.15]) for the chosen set of control

laws U ¼ Ui1j1 ; Ui2j2 ; � � � ;Uirjr

� �and the value lðsÞ. The function y ¼ KUðsÞ is

continuous at ½0; 1� by virtue of the theorem on the continuous dependence of the

solution to a system of ordinary differential equations, the continuous dependence

of the definite integral, and in general by virtue of the theorem proved above.

Consequently, the function y ¼ maxU KUðsÞ ¼ KðsÞ giving the solution to prob-

lem (1.1–1.5) (or [1.10–1.15]) is also continuous on the interval ½0; 1�. Denote by

DðUÞ � ½0; 1� the set of all values of the parameter s for which KUðsÞ ¼ KðsÞ. Thisset is closed as the preimage of the closed set f0g for the continuous function

y ¼ KUðsÞ KðsÞ. The set D U� �

can also be empty. As a result, the interval 0; 1½ �may be viewed as the following finite sum consisting of at least two closed sets (see

the conditions of the theorem):

0; 1½ � ¼[U

D U� �

:

1.4 Analysis of the Influence of Uncontrolled Parametric Disturbances. . . 15

Hence, since by the conditions of the theorem, 0 2 D U� �

for some set of laws Ucorresponding to l1 and 1=2DðUÞ, there exists a boundary point s of the set DðUÞbelonging to the interval 0; 1ð Þ (let us consider that s is the infimum of such

boundary points for the set D U� �

). The point s is also a boundary point of some

other set D U1

� �and belongs to it. For this value of s, the point l sð Þ is a bifurcation

point, since with l sð Þ there exist at least two sets of optimal laws, and with

0� s< s there exists the one optimal law U. Hence, the theorem is proved.

The following theorem is an immediate consequence of Theorem 1.4.

Theorem 1.5 Assume that with the value l ¼ l1 control by means of some set of rlaws from the set (1.3) (or [1.12]) results in the solution of problem (1.1–1.5)

(or [1.10–1.15]), but with l ¼ l2 (l1 6¼ l2; l1; l2 2 L), such a solution doesnot exist. Then there exists at least one bifurcation point l 2 L.

Let us present a numerical algorithm for finding the bifurcation value of the

parameter l when the conditions of Theorem 1.5 are satisfied. Connect the points

l1 and l2 by a smooth curve T � L. Partition this curve into n equal parts with a

sufficiently small step. For the obtained values lk 2 T; k ¼ 0� n; l0 ¼ l1; ln ¼l2 define the optimal sets r of the control laws Uk and find the first value of k at

which these sets of the laws differ from the set of laws U0 by at least one value of

the subscript. In this case the bifurcation point of the parameter l lies on the arc

lk1; lk� �

.

For the resultant section of the curve, the algorithm defining the bifurcation point

with given accuracy e implies an application of the bisection method. As a result,

one finds a point c 2 lk1; lk� �

from one side of which, within the limits of

deviation e from the value c, the set of laws U0 is optimal, but from the other

side, within the limits of deviation e from the value c, this set is not optimal. From

Theorem 1.4 it follows that the bifurcation point l exists on the given arc.

1.5 Algorithm of the Application of ParametricControl Theory

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, 54, 55]:

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

basis of estimation of 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 the mathematical models to the stated problems,

the calibration of the mathematical models (parametric identification and retro-

spective prediction by the current indexes of the evolution of the economic

system) and additional verification of the chosen mathematical models by means

of econometric analysis and political-economic interpretation of the sensitivity

matrices.

4. The analysis of the structural stability (robustness) of the mathematical models

without parametric control in accordance with the aforementioned methods of

estimation of the robustness conditions (see the second section on parametric

control theory and the preface). The robustness (structural stability) of the model

shows that the economic system is itself stable. In this case, the mathematical

model can be used, after econometric analysis and political-economic interpre-

tation of the results of the robustness analysis, for solving the problem of

choosing the optimal control laws for the economic parameters and prediction

of the macroeconomic indexes.

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

sary to choose algorithms and methods of stabilization of the economic system

in accordance with the methods of Sect. 1.3. After carrying out the economic

analysis and political-economic interpretation, the result can be accepted for

realization.

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

7. The estimation of structural stability (robustness) of mathematical models with

the chosen laws of parametric control according to the given methods of

estimation of the robustness conditions (Sect. 1.2). If the mathematical model

with the chosen laws of parametric control is structurally stable, then after the

econometric analysis and political-economic interpretation have been carried

out and the approval of the decision-makers obtained, the obtained results can be

put into practice. If the mathematical model with the chosen laws of parametric

control is structurally unstable, then the choice of parametric control laws 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 the uncontrolled parameters of the economic system. In this regard,

replacement of one optimal parametric control law by another one is possible.

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

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

economic parameters must be maintained by modern methods of analysis and

computer simulation. The aggregate scheme for making decisions is presented in

Fig. 1.1.

1.5 Algorithm of the Application of Parametric Control Theory 17

Yes No

Choice of the direction of the economic development based on the assessment of the economic conditions and the preferences of the decision maker

Selection of one or several mathematical models consistent with the problems of development direction andcoordination of the results with the preferences of the decision maker

Analysis of the robustness of the mathematical model. Econometric analysis, politico-economic interpretationof the results of the robustness analysis and coordination of the results with the preferences of the decision maker

Is the model robust?

Selection of the method and synthesis of the algorithm of control(attenuation) of structural instability of the mathematical model.

Econometric analysis, politico-economic interpretation of the resultsof control or the attenuation of structural uncertainty and

coordination of the results with the decision maker

Selection of the method and synthesis of the parametric control laws. Econometric analysis, politico-economicinterpretation of the results of the parametric control and coordination of their results with the decision maker

Robustness analysis of the mathematical models with the parametric control laws. Econometric analysis, politico-economic interpretation of the results of the robustness of the mathematical models with the parametric control laws

and coordination of the results with the decision maker

No YesIs the model robust?

Correction of the constraints on the parametric control in the case of structural instability of themathematical models of parametric control. Econometric analysis, politico-economic interpretation

of the results of correction of the constraints and coordination of the results with the decision maker

Analysis of the bifurcations of the extremals of the variational calculus problems of choosing the parametric control laws. Econometric analysis, politico-economic interpretation of the results of the analysis of the

extremals and coordination of the results with the decision maker

Formulation of recommendations on the application or replacement of the parametric control laws for the mechanisms of the market economy and coordination of the results with the decision makers

Particular decisions on the implementation of the parametric control laws for the mechanisms of the market economies

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

efficient public economic policy, part 1, part 2, part 3.


1.6 Examples of the Application of Parametric Control Theory

1.6.1 Mathematical Model of the Neoclassic Theoryof Optimal Growth

1.6.1.1 Model Description

A mathematical model of the economic growth [47] is given by the following

system of two ordinary differential equations containing the time derivatives (i.e.,

with respect to t):

dk

dt¼ Aka c ðnþ dÞk;

dc

dt¼ c

1 bðaAka1 ðdþ pÞÞ:

8>><>>: (1.17)

Here 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 the population: LðtÞ ¼ L0ent;

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

p is the discounting level;

ept is the discounting function (p> n);A and a are the parameters of the production function y ¼ fðkÞ ¼ Aka, where y is

the ratio of the gross domestic product to labor, that is, the mean labor produc-

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

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

population: UðcÞ ¼ Bcb (0< b< 1; B> 0).

The first equation of system (1.17) is the fundamental Solow equation from the

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

the maximum condition of the objective function

ð10

UðcÞLðtÞeptdt ¼ BL0

ð10

eb ln cðpnÞtdt

characterizing the total welfare of the whole population in the time interval

0� t<1. This functional is maximized under the constraints

kð0Þ ¼ k0; k0 ¼ Aka c nþ dð Þk; 0� cðtÞ�f kðtÞð Þ

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

1.6 Examples of the Application of Parametric Control Theory 19

The solutions of system (1.17) will be considered in some closed regionOwhose

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

R2þ ¼ k> 0; c> 0f g, kð0Þ ¼ k0; cð0Þ ¼ c0; ðk0; c0Þ 2 O:

1.6.1.2 Analysis of the Structural Stability of the Mathematical Modelof the Neoclassical Theory of Optimal Growth WithoutParametric Control

Let us carry out the estimation of robustness (structural stability) of the considered

model without parametric control in the aforementioned closed region O whose

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

R2þ ¼ k> 0; c> 0f g, kð0Þ ¼ k0; cð0Þ ¼ c0; ðk0; c0Þ 2 O, relying on the theorem

on necessary and sufficient conditions of robustness [11]. Let us prove the follow-

ing assertion:

Lemma 1.6 System (1.17) has the unique singular point

k ¼ aAdþ p

11a

;

c ¼ kðnþ dÞð1 aÞ þ p n

a

8>>><>>>:

(1.18)

in R2þ. This point is the saddle point of system (1.17).

Proof Setting the right-hand sides of the equations of system (1.17) to zero, we

obtain expressions (1.18). Obviously, k > 0; c > 0. Consider the determinant of

the Jacobian matrix for the right-hand sides of equations (1.18) at the point (k; c):

D ¼ a 1

1 bð Þa pþ dð Þ nþ dð Þ 1 að Þ þ p nð Þ:

Since for all stated values of the parameters A; a; b; p; n; d of the mathemati-

cal model we have D< 0, it follows that the singular point k; c is the saddle pointof system (1.17).

Theorem 1.7 Let the right-hand sides of the system

x0 ¼ f1 x; yð Þ;y0 ¼ f2 x; yð Þ

((1.19)

be smooth functions in some region O1 � R2, and suppose system (1.19) has aunique saddle singular point (x; y) in this region. Then system (1.19) is robust inthe closed region O (O � O1) containing the point (x; y).


Proof Let us make sure that system (1.19) does not have cyclic trajectories.

Assume the contrary. Let the region in 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 [11, p. 117]. But in the region O1

there is a unique saddle point with index equal to – 1. Thus, we have arrived at a

contradiction.

Let us make sure that the stable and unstable separatrices of the saddle point

(x; y) 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, are the boundary of the closed cell O2 lying in the region O1. Let

us consider the semitrajectory Lþ coming from some point (x1; y1), where (x1; y1)is the interior point of O2. Then, by virtue of the absence of cyclic trajectories and

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 of Lþ, since this point isa saddle [9, p. 49]). Now let us consider the semitrajectory L coming from the

point (x1; y1) in the direction opposite to Lþ. It is obvious that the boundary of O2

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 [11, p. 146, Theorem 12], the assertion is proved.

Corollary System (1.17) is robust in the closed region O (O � R2þ) containing the

point (k; c) for all fixed values of the parameters n; L0; d; p; A; a; B; b from therespective ranges of their definition.

In particular, it follows that there are no bifurcations of the phase-plane portrait

of system (1.17) in the region O under variation of the given parameters within their

range of definition.

1.6.1.3 Choosing Optimal Laws of Parametric Control of Market EconomicDevelopment Based on a Mathematical Model of the NeoclassicalTheory of Optimal Growth

Consider now the feasibility of the realization of an efficient public policy on the

basis of model (1.17) by choosing optimal control laws using the capital deprecia-

tion level (d) as an example of the economic parameter.

Choosing optimal parametric control laws is carried out in the environment of

the set of the following relations:

1Þ U1ðtÞ ¼ l1DkðtÞkð0Þ þ d; 2Þ U2ðtÞ ¼ l2

DkðtÞkð0Þ þ d;

3Þ U3ðtÞ ¼ l3DcðtÞcð0Þ þ d; 4Þ U4ðtÞ ¼ l4

DcðtÞcð0Þ þ d: (1.20)


Here Ui is the ith law of the control of the parameter d (i ¼ 1; 4); li is the

adjusted coefficient of the ith control law; li � 0; d is a constant equal to the basicvalue of the parameter d; DkðtÞ ¼ kiðtÞ kð0Þ;DcðtÞ ¼ ciðtÞ cð0Þ; (kiðtÞ, ciðtÞ) isthe solution of system (1.17) with initial conditions kið0Þ ¼ k0; cið0Þ ¼ c0 with useof the control law Ui. Use of the control law Ui means substitution of the function

from the right-hand side of (1.20) into system (1.17) instead of the parameterd;t ¼ 0 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 (1.17), find the optimal parametric control law at the level of the economic

parameter d in the environment of the set of algorithms (1.20), that is, find the

optimal law from the set {Ui} that maximizes the criterion

K ¼ BL0

ðT0

eb ln ciðtÞðpnÞtdt ! maxfUi; lig

(1.21)

under the constraints

kiðtÞ kðtÞj j � 0:09kðtÞ; kiðtÞ; ciðtÞð Þ 2 O; where t 2 ½0; T�: (1.22)

Here kðtÞ; cðtÞð Þ is the solution of system (1.17) without the parametric control.

The stated problem is solved in two stages:

– At 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 (1.22).

– At 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 a 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 use of the

following law:

d ¼ 0:19DkðtÞ4

þ 0:2 (1.23)

Note that the criterion value without use of the parametric control is equal to

K ¼ 1.901038.


1.6.1.4 Analysis of the Structural Stability of the Mathematical Model of theNeoclassical Theory of Optimal Growth with Parametric Control

Let us analyze the robustness of system (1.17), 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 (1.24)

with any fixed value of the adjusted coefficient l1 > 0. Here k0 > 0 and d0 > 0 are

some fixed numbers. Substitute (1.24) into the right-hand sides of the system (1.17)

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 baAka1 l1

k k0k0

þ d0

p

¼ 0:

8>>><>>>:

(1.25)

Since the function from the right-hand side of the second equation of system

(1.17) is strictly decreasing as a function of the variable k and takes on all values

with k > 0, it follows that the second equation has a unique solution k. For thissolution, there exists a unique solution c of the first equation in (1.25), that is,

system (1.25) has the unique solution ðk; cÞ. If ðk; cÞ=2R2þ, then obviously,

system (1.17) with the control law U1 is structurally stable in any closed region

O � R2þ.

Now, let ðk; cÞ 2 R2þ. Let us find the determinant of the Jacobian of the

functions f1, f2, which are the left-hand sides of the respective equations of system

(1.25) at this point. Since

@f1@c

k; cð Þ ¼ 1;@f2@k

k; cð Þ ¼ c

1 ba a 1ð ÞA kð Þa2 l1� �

< 0;

@f2@c

k; cð Þ ¼ 0;

the determinant of the matrix is negative: D< 0. Therefore, in this case, the point

ðk; cÞ is the saddle point of system (1.17) with control law U1. From Lemma 1.1

it follows that the system is structurally stable in the closed region O � R2þ

containing the point ðk; cÞ.In particular, with use of law (1.23), system (1.17) remains structurally stable.

The methods presented above allow one to analyze robustness conditions for

system (1.17) using the optimal control law d ¼ l1 cc0c0

þ d0 when the values of

the parameters ðn; pÞ are in a closed region in R2þ.


1.6.1.5 Finding Bifurcation Points for the Extremals of the VariationalCalculus Problem Based on the Mathematical Model of theNeoclassical Theory of Optimal Growth with Parametric Control

Let us analyze the dependence of the results of choosing the parametric control law

at the level of parameter d on the uncontrolled parameters ðn; pÞ with values in

some region (rectangle) L in the plane. In other words, let us find possible bifurca-

tion 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 of K in criterion (1.21) on the values of the parameters ðn; pÞ were obtainedfor each 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 regionL, the intersectioncurve for these surfaces, and the projection of the intersection curve onto the region

of the values of the parameters ðn; pÞ consisting of the bifurcation points of these

parameters. This projection divides the rectangle L into two parts, in one of which

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.

By 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

ðn; pÞ, one can approach choosing optimal parametric control laws in the following

way: If the values of the parameters ðn; pÞ lie to the left of the bifurcation curve in therectangle L (Fig. 1.2), then the law U1 is recommended as the optimal law. If the

values of the parameters ðn; pÞ lie to the right of the bifurcation curve in the rectangleL, then the lawU4 is recommended as the optimal law. If the values of the parameters

Fig. 1.2 Plots of optimal values of criterion K


ðn; pÞ lie on the bifurcation curve in the rectangle L, then any of the lawsU1, U4 can

be recommended as the optimal law.

1.6.2 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 (1.26), which includes one

differential equation and two algebraic equations:

LðtÞ ¼ Lð0Þent;dK

dt¼ mKðtÞ þ rXðtÞ;

XðtÞ ¼ AKðtÞaLðtÞ1a:

8>>><>>>:

(1.26)

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; n is the monthly

rate of increase of the population engaged in the economy; m is the share of basic

production 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; a is the

elasticity coefficient of the funds.

1.6.2.2 Estimation of the Model Parameters

In the context of the solution of the problem of preliminary estimation of the

parameters, it is required to estimate the values of the exogenous parameters n, m,r, A, a by the searching method in 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:

K¼ 1

9

Xð0ÞXð0ÞXð0Þ

2

þ Xð12ÞXð12ÞXð12Þ

2


2


2


2

þ Kð12ÞKð12ÞKð12Þ

2


2


2


2!!min

(1.27)


Here XðtÞ represent the 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 from 2001 to 2005, XðtÞ and KðtÞ are the calculatedvalues of the variables of system (1.27).

In computations, we use the value of L(0) equal to 6.698 and the value of K(0)equal to 4,004 (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%.

1.6.2.3 Analysis of the Structural Stability of the One-Sector SolowModel of Economic Growth Without Parametric Control

By applying a numerical algorithm of estimation of weak structural stability of the

discrete-time dynamical system for the chosen compact set N, defined by the

inequalities 3000�K� 10000, 5� L� 10 in the phase space of the variables (K,L), we discover that the chain-recurrent set Rðf ;NÞ is empty. This means that the

one-sector Solow model of economic growth for describing the interaction between

the benefit market and the money market is estimated as weakly structurally stable

in the compact set N.

1.6.2.4 Choosing Optimal Laws of Parametric Control of MarketEconomic Development Based on the Solow Mathematical Model

Let us consider now the feasibility of the realization of an efficient public policy on

the basis of model (1.26) by choosing the optimal control laws using the gross

investments to gross domestic product ratio (r) as an example of an economic

parameter.

The choice of optimal parametric control laws is made according to the follow-

ing scenarios:

#1 rðtÞ ¼ r þ k1KðtÞ Kð0Þ

Kð0Þ ; #2 rðtÞ ¼ r k2KðtÞ Kð0Þ

Kð0Þ ;

#3 rðtÞ ¼ r þ k5XðtÞ Xð0Þ

Xð0Þ ; #4 rðtÞ ¼ r k6XðtÞ Xð0Þ

Xð0Þ : (1.28)

Here ki is the adjusted coefficient of the ith control law, and ki � 0; r* is the valueof the 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 (1.26), find the optimal parametric control law at the level of

the economic parameter r in the environment of the set of algorithms (1.28)

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 : (1.29)

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.

1.6.2.5 Analysis of the Structural Stability of the One-Sector SolowModel of Economic Growth with Parametric Control

For carrying out this analysis, the expression for optimal parametric control law

(1.29) is substituted into the right-hand side of the second equation of system (1.26)

instead of parameter r. Then, by applying the numerical algorithm of estimation of

Fig. 1.3 Capital assets without optimal control and with use of law #3 optimal in the sense of

criterion K


weak structural stability of the discrete-time dynamical system for the chosen

compact set N, defined by the inequalities 3000�K� 10000, 5� L� 10 in the

phase space of the variables (K, L), we obtain that the chain-recurrent set Rðf ;NÞ isempty. 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.

1.6.2.6 Analysis of the Dependence of the Optimal Value of Criterion Kon the Parameter for the Variational Calculus Problem Basedon the Solow Mathematical Model

Let us analyze the dependence of the optimal value of criterion K on the exogenous

parameter m. Recall that this parameter represents the share of the basic production

assets retired for a month for parametric control laws (1.28) 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). Anal-

ysis 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 exoge-

nous parameter m.

1.6.3 Richardson Model of the Estimation of Defense Costs


The Richardson model of the estimation of defense costs is presented in [35],

Chap. 12.

Fig. 1.4 Gross domestic product without optimal control and with use of scenario law #3 optimal

in the sense of criterion K


The model is described by a system of two linear differential equations with

constant coefficients,

dx=dt ¼ ay mxþ r;

dy=dt ¼ bx nyþ s:

((1.30)

Here t is the time (in months); x(t) is the defense costs of the first country (groupof countries); y(t) is the defense costs of the second country (group of countries);

a is the scale of threat for the first country (group of countries); b is the scale of

threat for the second 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 is the scale of the past damage suffered by the first country

(group of countries); s is the scale of the past damage suffered by the second

country (group of countries).

1.6.3.2 Estimation of Model Parameters

In the context of the solution of the problem of preliminary estimation of the

parameters, it is required to estimate the values of the exogenous parameters a, b , m,n, r, 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).

Fig. 1.5 Plots of the dependencies of the optimal value of criterion K on the exogenous

parameter m


The parametric identification criterion is as follows:

K ¼ 1

8

xð1Þ xð1Þxð1Þ

2

þ xð2Þ xð2Þxð2Þ

2


2


2

þ yð1Þ yð1Þyð1Þ

2


2


2


2!

! min : (1.31)

Here xðtÞ represents statistical data on the armament costs of France and Russia

for the years 1910–1913 ; yðtÞ is statistical data about the armament costs

of Germany and the Dual Monarchy (Austria–Hungary) for the same years; xðtÞ,yðtÞ are the respective calculated values of the endogenous variables of the

system (1.30). 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 ¼ 0.2526; 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 ones (100ffiffiffiffiK

p) is

3.2819%.

1.6.3.3 Analysis of the Structural Stability of the RichardsonMathematical Model without Parametric Control

For the obtained values of the parameters of system (1.30), its stationary point has

the coordinates (x0 ¼ 0.2625; y0 ¼ 0.5273), and it does not lie in the first

quadrant of the phase plane R2þ ¼ x> 0; y> 0f g. Therefore, system (1.30) is robust

for any closed region O � R2þ.

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


1.6.3.4 Choosing Optimal Laws of Parametric Control of Market Economieson the Basis of the Richardson Mathematical Model

Let us consider now the feasibility of the realization of an efficient public policy on

the basis of model (1.30) by choosing optimal control laws using the threat level for

the second group of countries, b, as an example of the parameter.

Choosing optimal parametric control laws is carried out in the environment of

the following relations:

#0 bðtÞ ¼ b þ k1xðtÞ xð0Þ

xð0Þ ;

#1 bðtÞ ¼ b k2xðtÞ xð0Þ

xð0Þ ;

#2 bðtÞ ¼ b þ k3yðtÞ yð0Þ

yð0Þ ;

#3 bðtÞ ¼ b k4yðtÞ yð0Þ

yð0Þ : (1.32)

Here ki is the coefficient of the scenario; 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 (1.30), find the optimal parametric control law at the level of the

economic parameter b in the environment of the set of algorithms (1.32)

maximizing the performance criterion

K ¼ 1

T

ðT0

yðtÞdt; (1.33)


yðtÞ� 1:1 � xðtÞ: (1.34)

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 : (1.35)


Note that the basic value of the criterion (without control) is equal to

K ¼ 96.8722.

The values of the endogenous variables of the model without parametric control,

as well as with use of parametric control, are presented in Figs. 1.6 and 1.7.

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

Fig. 1.7 Armament costs of the second group of countries without parametric control and with use

of the optimal law of parametric control. without parametric control, law #0 is used


1.6.3.5 Analysis of Structural Stability of the RichardsonMathematical Model with Parametric Control

For carrying out this analysis, the expression for the optimal parametric control law

(1.35) is substituted into the right-hand side of the second equation of system (1.30)

instead of the parameter b. Then, by applying the numerical algorithm of the

estimation 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 (x, y), we find that the chain-

recurrent set Rðf ;NÞ is empty. This means that the Richardson mathematical model

with the optimal parametric control law is estimated as weakly structurally stable in

the compact set N.

1.6.3.6 Analysis of the Dependence of the Optimal Value of Criterion Kon the Parameter for the Variational Calculus ProblemBased on the Richardson Mathematical Model

Let us 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 (1.32) with the obtained optimal values of the adjusted coefficients ki.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 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 bifurcation points ofthe extremals in this case for the values a ¼ 0.315 and a ¼ 0.345.

120

100

80

60

40

20

00 0,04

Law #1 is used

Opt

imal

val

ue o

f cr

iter

ion

Law #2 is usedLaw #3 is usedLaw #4 is used

0,08 0,12 0,18 0,2 0,24 0,28 0,32 0,36 0,4 0,44 0,48a

Fig. 1.8 Plots of dependencies of the optimal value of criterion K on the exogenous parameter a


1.6.4 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 of government loans on economic growth proposed in [34], after

appropriate transformation, is given by

dM

dt¼ FI

pb mM; (1.36)

dQ

dt¼ Mf F

p; (1.37)

dLG

dt¼ rGL

G þ FG npF nLsRL nO dP þ dB

� �; (1.38)

dp

dt¼ a

Q

Mp; (1.39)

ds

dt¼ s

Dmax 0;

Rd RS

RS

� ;RL ¼ min Rd;RS

� �; (1.40)

Lp ¼ 1 xx

LG; (1.41)

dp ¼ 1 xx

br2LG; (1.42)

dB ¼ br2LG; (1.43)

x ¼ n1 d

1 snp

1dd

!; (1.44)

Rd ¼ Mx; (1.45)

f ¼ 1 1 1 dn

x

11d

; (1.46)


F0 ¼ �0pMf ; (1.47)

FG ¼ ppMf ; (1.48)

FL ¼ ð1 nLÞsRd; (1.49)

FI ¼ 1 xxþ ð1 xÞnp� ð1 npÞFG n0ðdB þ dPÞ þ npF0

� � nL ð1 nLÞnp� �

sRL� �

þ ðm þ rGÞLp; (1.50)

F ¼ F0 þ FG þ FL þ FI; (1.51)

RS ¼ PA0 expðlptÞ 1

1þno ; o ¼ FL

pP0 expðlptÞ : (1.52)

Here M 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, v are the parameters of the function f(x),x is the solution to the equation f 0ðxÞ ¼ s

p ;

ФL and ФO 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 lenders;

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 the expenditure on consumers by the state in the gross domestic

product;

np, nO, nL are payment flow, dividends, and income taxes of workers;

b is the norm of fund capacity of the unit of power;

m is the coefficient of the loss of manufacturing capacity due to equipment

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, P0A are the initial number of workers and the total available workforce reserve;

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 (1.36–1.52) correspond to

the respective expressions from [34], possibly after some simple transformations.

So, the differential equation (1.36) results from (3.2.6, 3.2.18); equation (1.37)

results from (3.2.19) and (3.2.8); equation (1.38) is derived from (3.2.6) by

substituting the expression for (FGК – HG) from (3.2.25); equation (1.39) represents

(3.2.9); equation (1.40) represents (3.2.30); expression (1.41) represents the expres-

sion from page 150 in [20]; expressions (1.42) and (1.43) represent expressions

from (3.2.39); expression (1.44) represents the solution of equation (3.2.10)

f 0ðxÞ ¼ sp , where the function (1.46) is defined on page 157 of [20]; expression

(1.45) represents one of expressions (3.2.10); relation (1.47) is derived from

(3.2.15) and (3.2.8); relation (1.48) is derived from (3.2.16) and (3.2.8); relation

(1.49) is derived from (3.2.22); expression (1.50) represents (3.2.36); expression

(1.51) is (3.2.11); expressions (1.52) are derived from (3.2.12, 3.2.13, 3.2.14).

The model parameters and the initial conditions for differential equations

(1.36–1.52) are obtained on the basis of the economic data of the Republic of

Kazakhstan for the years 1996–2000 [40] (r2 ¼ 0.12; rG ¼ 0.12; b ¼ 2; np ¼ 0.08;

nL ¼ 0.12; s ¼ 0.1; nO 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 ¼ 0.3; n ¼ 34;

�O 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%.

1.6.4.2 Analysis of the Structural Stability of the MathematicalModel of the National Economic System Subject to the Influenceof the Share of Public Expenses and Interest Rate of GovernmentLoans without Parametric Control

Let us analyze the robustness (structural stability) of model (1.36–1.52) on the basis

of the theorem establishing the sufficient conditions of structural stability [67]

within a compact region of the phase space.

Table 1.2 Parametric identification results

Years M* M** p* p**

1998 144,438 158,576 1.071 1.09

1999 168,037 183,162 1.16 1.20

2000 216,658 212,190 1.31 1.29

M*, M**, p*, p** are the respective values of the total productive capacity and the product price,

both measured and modeled (calculated) values


Assertion 1.8 Let N be a compact set lying in the region M> 0; Q< 0; p> 0ð Þ orM> 0; Q> 0; p> 0ð Þ of the phase space of the system of differential equationsderived from (1.36–1.52), that is, the four-dimensional space of variablesðM; Q; p; LGÞ. Let the closure of the interior of N coincide with N. Then the fluxf defined by (1.36–1.52) is weakly structurally stable on N.

One can choose N as, for instance, the parallelepiped with boundariesM¼Mmin;M¼Mmax;Q¼Qmin;Q¼Qmax; p¼pmin; p¼pmax;LG¼LGmin;LG¼LGmax.Here 0<Mmin<Mmax, Qmin<Qmax<0 or 0<Qmin<Qmax, 0<pmin<pmax,LGmin<LGmax.

Proof Let us first 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 equation

(1.39), dpdt ¼ a Q

M p, of the system it follows that for all t> 0, the variable p(t) has aderivative greater than some positive constant with Q< 0 or less than some

negative constant with Q> 0, that is, p(t) increases without bound or tends to

zero as t goes to infinity, and 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].

1.6.5 Choosing the Optimal Laws of Parametric Controlof Market Economic Development on the Basisof the Mathematical Model of the Country Subjectto the Influence of the Share of Public Expensesand the Interest Rate of Government Loans

Let us consider now the ability of the realization of efficient public policy by

choosing the optimal control laws using the following parameters: the share of

state expenses in the gross domestic product p, the interest rate of the government

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.


Choosing the optimal parametric control laws is carried out in the environment

of 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 constj;

#6 U6jðtÞ ¼ k6jDMðtÞMðt0Þ þ

DpðtÞpðt0Þ

þ constjj: (1.53)

HereUij is the ith control law of the jth parameter; the case i ¼ 1; 6; j ¼ 1; 3, j ¼ 1

corresponds to the parameter x; the case j ¼ 2 corresponds to the parameter p; thecase j ¼ 3 corresponds to the parameter rG; kij is the nonnegative adjusted coefficientof 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 (1.36–1.52), find the optimal parametric

control law Uij in the environment of the set of algorithms (1.43) minimizing the

criterion

K ¼ 1

T

ðt0þT

t0

pðtÞdt ! minfUij;kijg

(1.54)


MðtÞ MðtÞj j � 0; 09MðtÞ; MðtÞ; QðtÞ; LGðtÞ; pðtÞ; sðtÞ� � 2 X ;

0�Uij � aj; i ¼ 1; 4; j ¼ 1; 2; t 2 t0; t0 þ T½ �; (1.55)

where M**(t) is the value of the total production capacity without parametric

control; aj is the maximum possible value of the jth parameter; 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 (1.55). Here kmij is the first value of the

coefficient violating (1.55).


– 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 (1.54).

The results of the numerical solution of the first stage of the stated problem for

{Uij} are presented in Table 1.3.

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;

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 (1.53)

minimizing the criterion

K ¼ 1

T

ðt0þT

t0

pðtÞdt ! minðUij;kijÞ;ðUum;kumÞf g

;

i; u ¼ 1; 6; j; m ¼ 1; 3; j< m

(1.56)

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 values of

coefficients of laws

Values of

criterion K

U11 0.22 1.098

U21 0 1.1734

U31 0.156 1.037

U41 0 1.1734

U51 0.16 1.09

U61 0 1.1734

U12 0 1.1734

U22 0.11 1.09

U32 0 1.1734

U42 0.84 1.023

U52 0 1.1734

U62 0.08 1.084

U13 0 1.1734

U23 0.29 1.17

U33 0 1.1734

U43 0.39 1.1701

U53 0 1.1734

U63 0.23 1.1702



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 step equal to 0.01 minimizing K under

constraints (1.55).

– In the second stage, the optimal pair of parametric control laws are 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 could be summarized in 18

tables similar to Table 1.4 differing in the control law expression by at least one

parameter.

Choice of 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

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 of laws for simultaneous

control of the three parameters can be formulated as follows: Find the optimal set

of three parametric control laws at the level of three parameters on the basis of the

set of algorithms (1.53) minimizing the criterion

K ¼ 1

T

ðt0þT

t0

pðtÞdt ! minðUi1;ki1Þ;ðUn2;kn2Þ;ðUg3;kg3Þf g;

i; n; g ¼ 1; 6

(1.57)

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 U12 0 1.1734

U21 0.185 U22 0.123 0.981

U21 0 U32 0 1.1734

U21 0 U42 0.84 1.023

U21 0 U52 0 1.1734

U21 0.167 U62 0.167 0.982



This problem is solved in two stages:

– First, the optimal values of the coefficients are determined for the chosen set of

three control laws (Ki1;Kn2;Kg3) by enumeration of their values from respective

intervals (quantized with the step equal to 0.01 for each coefficient) minimizing

K under constraints (1.55).

– 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 could be

presented in 36 tables similar to Table 1.5 differing in the control law expression by

at least one parameter.

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 of the possible ways of choosing efficient laws of

parametric control of a 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 (1.36–1.52) at

the level of one of two parameters x ( j ¼ 1) and p ( j ¼ 2) was carried out under the

following set of assumptions:

Table 1.5 First-stage results of the numerical solution of the stated problem of choosing the

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 0.981

U21 0.185 U22 0.123 U23 0.03 0.980

U21 0.185 U22 0.123 U33 0 0.981

U21 0.185 U22 0.123 U43 0 0.981

U21 0.185 U22 0.123 U53 0 0.981

U21 0.185 U22 0.123 U63 0 0.981


#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:

(1.58)

Here Uij is the ith control law of the j-th parameter (i ¼ 1; 4; j ¼ 1; 2); the casej ¼ 1 corresponds to the parameter x; the case j ¼ 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.Utilizing the model (1.36–1.52) implies that the functions Uij from (1.58) should be

substituted into equations (1.36–1.52) instead of the 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 (1.36–1.52), find the optimal parametric control law at the

level of one out of two economic parameters (x, p) in the environment of the set of

algorithms (1.58), that is, find the optimal control law from the set {Uij} and its

adjusted coefficient maximizing the criterion

K ¼ 1

T

ðt0þT

t0

YðtÞdt; (1.59)

where Y ¼ Mf is the gross domestic product, under constraints

pijðtÞ pðtÞ�� 0; 09pðtÞ; MðtÞ; QðtÞ; LGðtÞ; pðtÞ; sðtÞ� � 2 X ;

0� uj � aj; i ¼ 1; 4; j ¼ 1; 2; t 2 t0; t0 þ T½ �: (1.60)

Here aj is the maximum value of the j-th 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 (1.60). Here kmij is the first value of the coeffi-

cient violating (1.60).

– Second, the law of optimal control of the specific parameter (one out of three) is

chosen on the basis of the results of the first stage by the maximum value of

criterion K (1.54).

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: (1.61)

Note that the criterion value without the use of parametric control is

K ¼ 170784.

1.6.5.1 Parametric Control of Market Economic Developmentwith Varying Objectives on the Basis of a Mathematical ModelSubject to the Influence of the Share of Public Expensesand the Interest Rate of Governmental Loans

Let us consider the parametric control of inflation processes in market economies

on the basis of the mathematical model (1.36–1.52). 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, and 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 (B) of the price-level variation can be defined by the

inequality pðtÞ� pHðtÞ or pðtÞ� pBðtÞ, where pHðtÞ is the admissible lower bound

of the price-level variation and pBðtÞ is the admissible upper bound (pH(t) < pH(t),0 < t < T). Satisfying the inequality pðtÞ� pHðtÞ shows that there exist some

deflation process, whereas satisfying pðtÞ� pBðtÞ indicates that there excessive

inflation exists. The admissible region (A) of the price-level variation can be definedby the inequality pн(t) < p(t) < pв(t), 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 two 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 indexes 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 the computer simulation has a logical condi-

tion determining the presence of the level of prices in one or another admissible

region. If during this process 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 B to admissible region A. If the value of p(t) turns out to be in region A, theparametric 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 expenditure in the gross domestic

product (p); the interest rate of public bonds (rG); norm of reservation (x). Theseparameters are accepted for the research, taking into consideration [40] and the

analysis of the sensitivity matrix of the indexes, namely, the total production

capacity (M), the volume of the public debt (LG), and the level of prices (p).The algorithm of multiobjective control was tested for a model of the economy

of the Republic of Kazakhstan for the following bounds of price-level variation:

pH(t) ¼ 0.9 and pB(t) ¼ 1.1.

Let dpðtÞ ¼0, if pHðtÞ< pðtÞ< pBðtÞ;pðtÞ pHðtÞ, if pðtÞ� pBðtÞ;pðtÞ pBðtÞ; if pðtÞ� pBðtÞ:

8>><>>: (1.61)

When the level of prices is in inadmissible region B, choosing of 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

ðt0þt

t0

dpðtÞpðt0Þ dtþ constj;

#3 V3j ¼ k3jdpðtÞpðt0Þ þ

1

t

ðt0þt

t0

dpðtÞpðt0Þdt

24

35þ constj:

(1.62)

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 ofthe ith control law of the jth 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 (x, p, rG).The problem of choosing the optimal pair of parametric control laws at the level

of two economic parameters from the triplet (x, p, rG) 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 (1.62) minimizing the performance criterion

K1 ¼ðt0þT

t0

dpðtÞ2dt ! minfVij;kijg

(1.63)



MðtÞ MðtÞj j � 0; 09MðtÞ; t 2 ½t0; t0 þ T�;0�VijðtÞ� aj; 0�VumðtÞ� am; i ¼ 1; 3;m ¼ 1; 3: (1.64)

Here M(t), p(t) are the values of the production capacity and level of prices with

the use of the parametric control, respectively; M**(t), p**(t) 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 (1.64).

– 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 allows 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

economic 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.

1.6.5.2 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 us analyze the robustness of system (1.36–1.52), where the parameters x, p, andrG are determined in accordance with the solution of the parametric control

problems as the expressions


Table 1.6 Values of price level p(t) with applied control of economic parameters

Months

Values of price

level p(t)without control

Values of price level

p(t) with control of

parameter p

Values of price level

p(t) with control of

parameter x

Values of price level p(t)with control of pair of

parameters (p,x)

1 1.1 1.1 1.1 1.1

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 1.2 1.19 1.2 1.18

15 1.21 1.19 1.2 1.18

16 1.22 1.2 1.21 1.19

17 1.22 1.2 1.21 1.18

18 1.23 1.2 1.21 1.18

19 1.24 1.2 1.22 1.18

20 1.24 1.2 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 1.1

28 1.29 1.14 1.21 1.09

29 1.29 1.13 1.2 1.07

30 1.3 1.12 1.2 1.05

31 1.3 1.1 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


1Þ U1j ¼ þk1jMðtÞ Mð0Þ

Mð0Þ þ constj;

2Þ U2j ¼ k2jMðtÞ Mð0Þ

Mð0Þ þ constj;

3Þ U3j ¼ þk3jpðtÞ pð0Þ

pð0Þ þ constj;

4Þ U4j ¼ k4jpðtÞ pð0Þ

pð0Þ þ constj;

5Þ U5j ¼ þk5jMðtÞ Mð0Þ

Mð0Þ þ pðtÞ pð0Þpð0Þ

þ const

j;

6Þ U6j ¼ k6jMðtÞ Mð0Þ

Mð0Þ þ pðtÞ pð0Þpð0Þ

þ const

j: (1.65)

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 (p,x)


with any values of the adjusted coefficients kij � 0. Here constj is a constant equal tothe 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; 3 means

substituting the respective functions for the parameters x ( j ¼ 1), p ( j ¼ 2), and

rG ( j ¼ 3) into the model equations (1.36–1.52). As a result of application of these

laws, the following system is derived:

dM

dt¼ FI

pb mM; (1.66)

dQ

dt¼ Mf F

p; (1.67)

dLG

dt¼ Ui3L

G þ FG npF nLsRL nOðdP þ dBÞ; (1.68)

dp

dt¼ a

Q

Mp; (1.69)

ds

dt¼ s

Dmax 0;

Rd RS

RS

� ;RL ¼ minfRd;RSg; (1.70)

Lp ¼ 1 Ui1

Ui1LG; (1.71)

d p ¼ 1 Ui1

Ui1br2LG; (1.72)

d B ¼ br2LG; (1.73)

x ¼ n1 d

1 snp

1dd

!; (1.74)

Rd ¼ Mx; (1.75)

f ¼ 1 1 1 dn

x

11d

; (1.76)

F0 ¼ �0pMf ; (1.77)

FG ¼ Ui2pMf ; (1.78)

F L ¼ ð1 nLÞsRd; (1.79)


FI ¼ 1 Ui1

Ui1 þ ð1 Ui1ðtÞÞnp� ð1 npÞFG n0ðdB þ dPÞ þ npF0

� � nL ð1 nLÞnp� �

sRL� �

þ ðm þ Ui2ÞLp; (1.80)

F ¼ F0 þ FG þ FL þ FI; (1.81)

RS ¼ PA0 expðlptÞ 1

1þno ; o ¼ FL

pP0 expðlptÞ : (1.82)

The proof of the weak structural stability of the mathematical model

(1.36–1.52), presented above and relying on equation (1.39) 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 1.9 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 (1.36–1.52), that is, the four-dimensional space of variablesðM; Q; p; LGÞ. Let N coincide with the closure of its interior. Then the flux f definedby (1.66–1.82) is weakly structurally stable on N.

1.6.5.3 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 the Share of Public Expensesand the Interest Rate of Governmental Loans

Let us consider the ability of finding 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

environment of a fixed finite set of algorithms on the basis of mathematical

model (1.36–1.52) of the national economic system.

The ability to choose the optimal law of parametric control at the level of one of

two parameters x ( j ¼ 1) and p ( j ¼ 2) on the time interval ½t0; t0 þ T� is consid-ered in the environment of the following algorithms (1.58):

1Þ U1jðtÞ ¼ k1jM M0

M0

þ constj;

2Þ U2jðtÞ ¼ k2jM M0

M0

þ constj

3Þ U3jðtÞ ¼ k3jp p0p0

þ constj;

4Þ U4jðtÞ ¼ k4jp p0p0

þ constj:


In the considered problem, criterion (1.59) is used (mean value of the gross

domestic products for the period of 1997–1999):

K ¼ 1

T

ðt0þT

t0

YðtÞdt;

where Y ¼ Mf .The closed set in the space of continuous vector functions of the output variables

of system (1.36–1.52) and regulating parametric actions are determined by relations

(1.60):

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 1.4 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 use of another law, U21. Using the

proposed numerical algorithm allows the determination of the bifurcation point

of the extremal of the considered problem r2 ¼ 18:0 with an accuracy of up to

0.001. For this parameter, the laws U21 and U12 are optimal, and the

corresponding value of the criterion K is 173,381 (monetary units per month).

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 public expenses at the level of two

economic parameters with one-parameter disturbance.

In this variational calculus problem, we consider its dependence on the

coefficient l ¼ r2 of the mathematical model with possible values in some

interval [a, b].As a result of a computer simulations, plots were obtained of the 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 1.4 are satisfied, for instance, for the interval [6, 9.6], since with

r2 ¼ 6 the optimal value of the criterion equal to 188,803 is attained using the

laws fU21;U32g. With r2 ¼ 9:6 the optimal value of the criterion equal to

190,831 is attained with use of other laws fU21;U12g. Using the proposed

numerical algorithm allows the determination of the bifurcation point of the

extremal of the considered problem r2 ¼ 0:075 with 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).

Fig. 1.10 Plots of dependencies of criterion optimal values on parameter of deposit interest rate

r2. Notation: U12, U32, U21, U41, without control

187487

168000

173000

178000

183000

188000

193000

2,4 6 9,6 13,2 16,8 20,4

r2

Opt

imal

val

ues

of c

rite

rion

Laws U41 and U32 Laws U21 and U12Laws U21 and U32Laws U41 and U12

Fig. 1.11 Plots of the dependencies of criterion optimal values on the parameter of deposit

interest rate r2


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 simulation experiment, plots of the dependence of

the optimal values of criterion K on the values of the parameters ðr2; nOÞ foreach of 12 possible laws Uij; i ¼ 1; 6; j ¼ 1; 2, were obtained. Figure 1.12

presents the plots for the two laws U21 and U41 maximizing the criterion in

region L, the intersection curve of the 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 rectan-

gle L into two parts. The control law U21 is optimal in one of these parts,

whereas U41 is optimal in the other part. Both of the laws are optimal on the

curve projection.

178000

176000

174000

172000

170000

168000

166000

164000

0.0050.0075

0.010.0125

0.0150.0175

0.020.0225

0.025 0.048

0.049

0.05

0.051

0.052

nor2

U41

U21

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


4. As a result of a computer simulation experiment, the plots of the dependence of

the optimal values of criterion (1.63), K1, on the values of uncontrolled

parameters ðr2; nOÞ for each of nine possible laws (1.62) Vij; i ¼ 1; 3;

j ¼ 1; 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 curvesof 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 several parts;

inside each of them only one control law is optimal. Two or three different laws

are optimal on the projection curves.

1.6.6 Mathematical Model of the National Economic SystemSubject to the Influence of International Trade andCurrency Exchange on Economic Growth


The mathematical model proposed in [34] for researching the influence of the

international trade and currency exchange on economic growth after the respective

transformations can be expressed as the following system of differential and

Fig. 1.13 Plot of optimal values of criterion K1


algebraic equations (where i ¼ 1, 2 is the number of states, and t is the time

variable):

dMi

dt¼ FI

i

pibi miMi; (1.83)

dQi

dt¼ Mifi Fi

pi; (1.84)

dLGidt

¼ rG iLGi þ FG

i np iFi nL isiRLi nO i d

Pi þ dBi

� �; (1.85)

dpidt

¼ aiQi

Mipi; (1.86)

dsidt

¼ siDi

max 0;Rdi RS

i

RSi

� ;RL

i ¼ min Rdi ;R

Si

� �; (1.87)

LPi ¼ 1 xixi

LGi ; (1.88)

dPi ¼ 1 xixi

bir2 iLGi ; (1.89)

dBi ¼ bir2 iLGi ; (1.90)

xi ¼ ni1 di

1 sinipi

1didi

!; (1.91)

Rdi ¼ Mixi; (1.92)

fi ¼ 1 1 1 dini

xi

11di

; (1.93)

FOi ¼ �0 ipiMi fi; (1.94)

FGi ¼ pipiMi fi; (1.95)

FLi ¼ ð1 nL iÞsiRd

i (1.96)

FIi ¼

1

1þ np i

�kqiMi fi

xi 1 npi� �

FGi þ n0 i d

Bi þ dPi

� �þ np iFOi þ

þ nL i þ 1 nL ið Þnp i� �

siRLi þ npi Fji yiFij

� �þ miLpi rG iLPi

8<:

9=;;

(1.97)


RSi ¼ PA

0 i exp lp it� �

11þnioi

; oi ¼ FLi

1þCLi yi

pjpi

� �P0 i lpitð Þ

; j ¼ 3 i;(1.98)

F12 ¼CL1p2p1

1þ CL1y

p2p1

FL1 þ

CO1

p2p1

1þ CO1 y

p2p1

FO1 ; (1.99)

F21 ¼CL2p1p2

1þ CL21yp1p2

FL2 þ

CO2

p1p2

1þ CO2

1yp1p2

FO2 ; (1.100)

F1 ¼ FI1 þ FL

1 þ FO1 þ FG

1 þ F21 yF12; (1.101)

F2 ¼ FI2 þ FL

2 þ FO2 þ FG

2 þ F12 1

yF21: (1.102)

Here:

Mi 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 FO

i are the consumers’ expenditures of workers and owners, respectively;

FIi is the flow of investment;

FGi is the expenditure of the state;

Fij are the expenses of consumers 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 expenditure of the state in the gross domestic product;


nPi, nOi, nLi are the payment flow, dividends, and workers’ income taxes,

respectively;

bi is the norm of fund capacity of the unit of power;

mi is the coefficient of the loss of manufacturing capacity due to equipment

degradation;

m*i is the depreciation rate;

ai is the time constant;

Di is the time constant defining the typical time scale of the wages adjustment

process;

P0i;PA0i are respectively the initial number of workers and total available workforce

reserve;

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 (1.83–1.102), equations (1.99–1.102) define the connection of

the economic systems of two countries. Note that in the case CL1 ¼ CL

2 ¼ CO1 ¼

CO2 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 [39, 40]. 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, �Oi, 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; (1.103)

where Mij*, Mij

**, pij*, pij

** are the respective values of the total product capacity

and product price of the ith country presented in [39, 40], and calculated N is the

number of observations, i ¼ 1; 2.

1.6.6.2 Analysis of the Structural Stability of the Mathematical Model of theNational Economic System Subject to the Influence of InternationalTrade and Currency Exchange without Parametric Control

Analysis of the robustness (structural stability) of model (1.83–1.102) is based on

the theorem on sufficient conditions of weak structural stability in the compact set

of the phase space.

Assertion 1.10 Let N be a compact set residing within the region M1 > 0; Q1 < 0;ðp1 > 0Þ or M1 > 0; Q1 > 0; p1 > 0ð Þ of the phase space of the system of differential


equations of mathematical model (1.83–1.102), that is, the eight-dimensional spaceof the variables ðMi; Qi; pi; LGiÞ, i ¼ 1; 2. Let the closure of the interior of Ncoincide with N. Then the flux f defined by the system of model differential equationsis weakly structurally stable on N.

One can choose N as, for instance, the parallelepiped with boundaryMi 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 or0<Qimin <Qimax, 0< pimin < pimax, LGimin < LGimax.


of the set N for 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 apoint of the semitrajectory does not belong to N. In the first case, from equation

(1.86), dp1dt ¼ a1

Q1

M1p1, of the system it follows that for all t> 0, the variable p1(t)

has derivative greater than some positive constant with Q1 < 0 or less than some

negative constant with Q1 > 0, that is, p1(t) increases infinitely or tends to zero withunbounded 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 withinN is the invariant set of this flux,

it follows that if it is nonempty, then it consists of only whole orbits. Hence, in the

considered case Rðf ;NÞ is empty. The assertion follows from Theorem A [67].

1.6.6.3 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 parameters xi; pi; y is carriedout in the environment of the following relations:

1Þ Ui1;b ¼ ki1;b

DMiðtÞMiðt0Þ þ constib;

2Þ Ui2;b ¼ ki2;b


3Þ Ui3;b ¼ ki3;b

DpiðtÞpiðt0Þ þ constib;

4Þ Ui4;b ¼ ki4;b

DpiðtÞpiðt0Þ þ constib: (1.104)

Here Uia;b is the a-th control law of the b-th parameter of the ith country,

a ¼ 1; 4; b ¼ 1; 3. The case b ¼ 1 corresponds to the parameter xi; b ¼ 2

corresponds to the parameter pi; b ¼ 3 corresponds to the parameter y, DMiðtÞ


¼ Ma;b;iðtÞ Miðt0Þ;DpiðtÞ ¼ pa;b;iðtÞ piðt0Þ; t0 is the control starting time,

t 2 t0; t0 þ T½ �. Here Ma;b;iðtÞ, pa;b;iðtÞ are the values of the product capacity and

the price level of the ith country, respectively, with theUia;bth control law; k

ia;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 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, y)can be formulated as follows: On the basis of mathematical model (1.83–1.102),

find the optimal parametric control law in the environment of the set of algorithms

(1.104), that is, find the optimal law (and its coefficients kia;b) from the set {Uia;b}

minimizing the criterion

Ki ¼ 1

T

ðt0þT

t0

piðtÞdt ! minfkia;b;Ui

a;bg(1.105)


MiðtÞ MiðtÞj j � 0:09Mi

ðtÞ;0�Ui

a;bðtÞ�aib; a¼ 1;4;b¼ 1;3; piðtÞ�0; siðtÞ�0; where t2 t0; t0þT½ �:(1.106)

Here MiðtÞ are the values of the total production capacity of the ith country

without parametric control; aib is the bth parameter of 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 size equal to 0.01 minimizing K under constraints (1.106).

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, y) on the basis of the set of

algorithms (1.104) minimizing the criterion

Ki ¼ 1

T

ðt0þT

t0

piðtÞdt ! minðUi

a;b; kia;bÞ;ðUi

n;m; kin;mÞ

� � ;a; n ¼ 1; 4 ; b; m ¼ 1; 3; b< m; (1.107)



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 Ki


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’ expenditure, 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 criterion Ki on the basis of the mathematical

model of the interaction between identical economic systems of two countries by

foreign trade (the model 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, y)for 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, 2) on the basis of the

mathematical model of the interaction between the nonidentical economic

systems of two countries (the Republic of Kazakhstan and the Russian Federa-

tion) 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 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 nonidentical

economic system of two countries via foreign trade, choosing the optimal pair of

parametric control laws on the set (x1, y), (p1, y) for the economic system of the

first country and estimating 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 system of two countries via foreign trade, choosing the optimal pair of

parametric control laws on the set (x2,y2), (p2, y2) for the economic system of the

second country and estimating the values of criterion K1 for the economic

system of the second country.


economic system of two countries, the estimation of the influence of the control

of the economic system of one country on the economic indexes 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.

Simultaneous 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 given by K1 ¼ K2 ¼ 1.145 with CL1 ¼

CL2 ¼ CO

1 ¼ CO2 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 result K1 ¼ 0.99 is attained

with use of the control law

Table 1.7 Results of the

numerical solution of the first

stage of the problem of

choosing optimal parametric

control laws at the level of

one parameter

Notations of laws Law coefficient Values of criterion K1

U111

0.2 1.072

U121

0 1.145

U131

2.1 1.009

U141

0 1.145

U112

0 1.145

U122

0.1 1.068

U132

0 1.145

U142

0.8 0.99

U113

0 1.145

U123

1.8 1.070

U133

0 1.145

U143

1.9 1.100


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 could be presented in eight

tables similar to 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

DM1ðtÞ139435

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 is equal to 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 [39, 40]. The parametric identifica-

tion results show the admissible precision of the description. The values of the

criterion Ki (i ¼ 1, 2) are respectively K1 ¼ 1.137, K2 ¼ 1.775 with C1 ¼ 0.15,

C2 ¼ 0.015, y ¼ 0.2.

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 proposing the law given by

y ¼ 1:2DM1ðtÞ139435

þ 0:2:

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 U113

0 0.99

U142

0.8 U123

1.6 0.97

U142

0.8 U133

0 0.99

U142

0.8 U143

0 0.99


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 proposing the

following laws:

p1 ¼ 0:2DM1ðtÞ139435

þ 0:1136; y ¼ 1:5DM1ðtÞ139435

þ 0:2:

Criterion K2 is equal to 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 the 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 equal to 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 equal to 1.72. With

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.

1.6.6.4 Analysis of the Structural Stability of the Mathematical Modelof the Country Subject to the Influence of International Tradeand Currency Exchange with Parametric Control

Let us analyze the robustness of system (1.83–1.102), where the parameters xi; pi; yare defined in accordance with the solution of the parametric control problems as

the expressions


1Þ Ui1;b ¼ ki1;b


2Þ Ui2;b ¼ ki2;b


3Þ Ui3;b ¼ ki3;b


4Þ Ui4;b ¼ ki4;b

DpiðtÞpiðt0Þ þ constib (1.108)

for any values of the adjusted coefficients kiab � 0. Here constib is a constant equal tothe 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; 3.The application of parametric control law Ui

a;b means the substitution of the

respective functions into model equations (1.83–1.102) 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 (1.83–1.102), the following

system is derived:

dMi

dt¼ FI

i

pibi miMi; (1.109)

dQi

dt¼ Mifi Fi

pi; (1.110)

dLGidt

¼ rG iLGi þ FG

i np iFi nL isiRLi nO i d

Pi þ dBi

� �; (1.111)

dpidt

¼ aiQi

Mipi; (1.112)

dsidt

¼ siDi

max 0;Rdi RS

i

RSi

� ;RL

i ¼ min Rdi ;R

Si

� �; (1.113)

LPi ¼ 1 Uia;1

Uia;1

LGi ; (1.114)

dPi ¼ 1 Uia;1

Uia;1

bir2 iLGi ; (1.115)

dBi ¼ bir2 iLGi ; (1.116)


xi ¼ ni1 di

1 sinipi

1didi

!; (1.117)

Rdi ¼ Mixi; (1.118)

fi ¼ 1 1 1 dini

xi

11di

; (1.119)

FOi ¼ �0 ipiMi fi; (1.120)

FGi ¼ Ui

a;2piMi fi; (1.121)

FLi ¼ ð1 nL iÞsiRd

i ; (1.122)

FIi ¼

1

1þ np i

�kqiMifixi

1 npi� �

FGi þ n0 i d

Bi þ dPi

� �þ np iFOi þ

þ nL i þ 1 nL ið Þnp i� �

siRLi þ npi Fji Ui

a;3Fij

� �þ miLpi rG iL

Pi

8><>:

9>=>;;

(1.123)

RSi ¼ PA

0 i exp lp it� �

11þnioi

; oi ¼ FLi

1þ CLi U

ia;3

pjpi

P0 i lpit� � ; j ¼ 3 i;

(1.124)

F12 ¼CL1P2

P1

1þ CL1U

ia;3

P2

P1

FL1 þ

CO1

P2

P1

1þ CO1U

ia;3

P2

P1

FO1 ; (1.125)

F21 ¼CL2P1

P2

1þ CL2

1Ui

a;3

P1

P2

FL2 þ

CO2

P1

P2

1þ CO2

1Ui

a;3

P1

P2

FO2 (1.126)

F1 ¼ FI1 þ FL

1 þ FO1 þ FG

1 þ F21 Uia;3F12; (1.127)

F2 ¼ FI2 þ FL

2 þ FO2 þ FG

2 þ F12 1

Uia;3

F21: (1.128)

The proof of weak structural stability of the mathematical model indicates that

the weak structural stability of the considered model is maintained with use of each

of the parametric control laws Uia;b in the form of the following assertion:


Assertion 1.11 Let N be a compact set belonging to region M1 > 0; Q1 < 0; p1 > 0ð Þor M1 > 0; Q1 > 0; p1 > 0ð Þ of the phase space of the model system of differentialequations (1.83–1.102), that is, the eight-dimensional space of variablesMi; Qi; pi; LGið Þ, i ¼ 1; 2. Let the closure of the interior of N coincide with N.Then the flux f defined by system (1.109–1.128) is weakly structurally stable on N.

1.6.6.5 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 Exchange

Besides the case considered above, some alternative statements of the problem of

choosing the optimal set of laws were considered.

Choosing the optimal parametric control laws on the basis of model

(1.83–1.102) at the level of one of the two parameters xi; pi is carried out in the

environment of the following relations:

1Þ Ui1;b ¼ ki1;b


2Þ Ui2;b ¼ ki2;b


3Þ Ui3;b ¼ ki3;b


4Þ Ui4;b ¼ ki4;b

DpiðtÞpiðt0Þ þ constib: (1.129)

Here Uia;b is the a-th control law of the b-th 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;iðtÞ Miðt0Þ;DpiðtÞ ¼ pa;b;iðtÞ piðt0Þ; t0 is the control starting

time, t 2 t0; t0 þ T½ �. Here Ma;b;iðtÞ, pa;b;iðtÞ are the values of the product capacity

and the level of prices of the ith country, respectively, with the Uia;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 (1.83–1.102),

find the optimal parametric control law in the environment of set of algorithms

(1.104); that is, find the optimal law (and its coefficients kia;b) from the set {Uia;b}

maximizing the criterion


Ki ¼ 1

T

ðt0þT

t0

YiðtÞdt; (1.130)

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 continuous vector functions of the output variables of

system (1.83–1.102) and regulating parametric actions are defined by the following

relations:

p1ðtÞ p1 ðtÞ�� 0:09p1 ðtÞ;MiðtÞ; QiðtÞ; LGiðtÞ; piðtÞ; siðtÞ� � 2 X ;

0�Uiab � aib; a ¼ 1; 4; b ¼ 1; 2; i ¼ 1; 2 t 2 t0; t0 þ T½ �: (1.131)

Here aib is the maximum possible value of the ath parameter of the ith country;

pi ðtÞ are the model (calculated) values of the price level of the ith country withoutparametric control; X is the compact set of the admissible values of the given

variables.

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


In this variational calculus problem, we consider the effect of 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 eachof eight 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 U1

4;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 ofthis two-dimensional parameter. This projection divides rectangle L 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.

1.6.7 Forrester’s Mathematical Model of Global Economy


Forrester’s mathematical model of “world dynamics” [26] is given by the following

system of ordinary differential and algebraic equations (here t is time):

P0ðtÞ ¼ PðtÞðBnðtÞ DðtÞÞ; (1.132)

V0ðtÞ ¼ CVGPðtÞVMðMÞ CVDVðtÞ; (1.133)

Z0ðtÞ ¼ CZPðtÞZVðVRÞ ZðtÞ=TZðZRÞ; (1.134)

R0ðtÞ ¼ CRPðtÞRMðMÞ; (1.135)

S0ðtÞ ¼ CSSQQMðMÞSFðFÞ=QFðFÞ SðtÞ� �=TS; (1.136)

MðtÞ ¼ VRðtÞð1 SðtÞÞERðRRÞ= ð1 SNÞEN½ �; (1.137)

FðtÞ ¼ FSðSRÞFZðZRÞFPðPRÞFC=FN; (1.138)

BnðtÞ ¼ PðtÞCBBMðMÞBPðPRÞBFðFÞBZðZRÞ; (1.139)

DðtÞ ¼ PðtÞCDDMðMÞDPðPRÞDFðFÞDZðZRÞ; (1.140)

QðtÞ ¼ CQQMðMÞQPðPRÞQFðFÞQZðZRÞ; (1.141)


PRðtÞ ¼ PðtÞ=PN; (1.142)

VRðtÞ ¼ VðtÞ=PðtÞ; (1.143)

SRðtÞ ¼ VRðtÞSðtÞ=SN; (1.144)

RRðtÞ ¼ RðtÞ=R0; (1.145)

ZRðtÞ ¼ ZðtÞ=ZN: (1.146)

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 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 relative quality of life;

SF is the coefficient of the dependence of the fund share in agriculture on the level

of nourishment;

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 capita);

Q is the level of 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;

PN ¼ 3:6�109; ZN ¼ 3:6�109; SN ¼ 0:3; TS ¼ 15;

TVD ¼ 40; CVG ¼ 0:05;

(1.147)

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 parameters CD, CZ, CR, CQ, TS, TVD equal to

the data from (1.147). 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 [59] and the data calculated by the state functions VðtÞ, SðtÞ, RðtÞ,ZðtÞ (accepted as the measured functions in solving the parametric identification

problem) based on the model (1.132–1.146). These values are determined by

solving the parametric identification problem by the searching method in the

sense of the minimum of the criterion


K¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

545

X2009t¼1901

PðtÞPðtÞ1

� �2þ SðtÞ

SðtÞ1

� �2þ RðtÞ

RðtÞ1

� �2þ ZðtÞ

ZðtÞ1

� �2þ VðtÞ

VðtÞ1

� �2 !vuut :

Here PðtÞ and PðtÞ are the measured and modeled (calculated) values of the

population, respectively; VðtÞ, SðtÞ, RðtÞ, ZðtÞ are the calculated data of system

(1.132–1.146). As a result of the solution of the given problem of parametric

identification, the following are estimates of the values of the estimated parameters:

CB ¼ 0.042095, CVG ¼ 0.049644, 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 100 K ¼ 4.27%.

1.6.7.2 Analysis of the Structural Stability of Forrester’s MathematicalModel without Parametric Control

Assertion 1.12 Let N be a compact set residing in the regionP> 0;V> 0; S> 0; Z> 0;R> 0f g of the phase space of the system of the differen-

tial equations derived from" (1.132–1.146), that is, the five-dimensional space ofthe variables fP; V; S; Z; Rg. Let the closure of the interior of N coincide with N.Then the flux f defined by system (1.132–1.146) is weakly structurally stable on N.

One can choose N as, for instance, the parallelepiped with the boundaryP ¼ Pmin; P ¼ Pmax; V ¼ Vmin; V ¼ Vmax; .S ¼ Smin; S ¼ Vmax; Z ¼ Zmin; Z ¼ZmaxR ¼ Rmin;R ¼ Rmax. Here 0<Pmin <Pmax, 0<Vmin <Vmax, 0< Smin < Smax,

0< Zmin < Zmax, 0<Rmin <Rmax.


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 apoint of the semitrajectory does not belong to N. In the first case, from equation

(1.135), R0ðtÞ ¼ CRPðtÞRMðMÞ, of the system it follows that for all t> 0, the

variable R(t) has derivative less than some negative constant number. That is, R(t)tends to zero with unbounded increase in 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, it follows that if it is nonempty, then it consists only of whole orbits. Hence, in

the given case Rðf ;NÞ is empty. The assertion follows from Theorem A [67].

1.6.7.3 Choosing Optimal Laws of Parametric Control on the Basisof Forrester’s Model

Let us 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:

1Þ U1j ¼ constj þ k1j PðtÞ=Pðt0Þ 1ð Þ;2Þ U2j ¼ constj k2j PðtÞ=Pðt0Þ 1ð Þ;3Þ U3j ¼ constj þ k3j RðtÞ=Rðt0Þ 1ð Þ;4Þ U4j ¼ constj k4j PðtÞ=Pðt0Þ 1ð Þ;5Þ U5j ¼ constj þ k5j ZðtÞ=Zðt0Þ 1ð Þ;6Þ U6j ¼ constj k6j ZðtÞ=Zðt0Þ 1ð Þ;7Þ U71j ¼ constj þ k7j VðtÞ=Vðt0Þ 1ð Þ;8Þ U8j ¼ constj k8j VðtÞ=Vðt0Þ 1ð Þ;9Þ U9j ¼ constj þ k9j SðtÞ=Sðt0Þ 1ð Þ;

10Þ U10j ¼ constj k10j SðtÞ=Sðt0Þ 1ð Þ;11Þ U11j ¼ constj þ k11j QðtÞ=Qðt0Þ 1ð Þ;12Þ U12j ¼ constj k12j QðtÞ=Qðt0Þ 1ð Þ: (1.148)

Here kij � 0 is the adjusted coefficient of the respective law Uij

i ¼ 1; 12; j ¼ 1; 2� �

; 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 the year 1971. Applicationof one of the laws (1.148) means the substitution of the respective function for the

right-hand side of corresponding relation (1.148) into equation (1.138) or (1.139) of

system (1.132–1.146) 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 (1.148) is stated as follows: On

the basis of mathematical model (1.132–1.146), find the optimal parametric control

law in the environment of algorithms (1.148); 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: (1.149)

characterizing the mean values of the quality of life level on the interval of time

from 1971 to 2100 under the constraints

P2100t¼1971

ZðtÞ� �Z; FCðtÞ 2 0:9; 1:1½ �: (1.150)

Here �Z is the total value of the pollution levels for the years 1971–2100 without

parametric control.



– In the first stage, the optimal values of the coefficients kij are determined for each

law (1.148) by the enumeration of their values on the intervals ½0; kmij Þ quantizedwith a sufficiently small step maximizing criterion K1 under constraints (1.150).

Here kmij is the first value of the coefficient violating (1.150).

– In the second stage, the law of optimal control of the specific parameter (of

twelve) 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 given 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 (1.148):

FC ¼ FC 0:158ðVðtÞ=Vðt0Þ 1Þ: (1.151)

Note that the value of criterion (1.149) 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 (1.148) is

stated as follows: On the basis of mathematical model (1.132–1.146), find the

optimal pair of parametric control laws in the environment of the set of algorithms

(1.148); that is, find the optimal pair of laws from this set of algorithms and its

adjusted coefficients that maximize criterion (1.149) under constraints (1.150).

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 0:15 VðtÞ=Vðt0Þ 1ð Þ; CB ¼ C

B 0:01 PðtÞ=Pðt0Þ 1ð Þ: (1.152)

In this case, the increase of the value of criterion K1 in comparison with the base

variant is equal to 7.93%.

Let us compare the obtained results of the parametric control of the evolution of

dynamical system (1.132–1.146) with the optimal laws found at the level of one

(1.151) and two (1.152) parameters and the results of the scenario consisting in the

increase of the parameter FC by 25% in 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 0:049644, and 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).


A comparison shows that for the given scenario (the increase of parameter FC by

25%), the mean value of the quality of life (criterion K1) on the time interval from

1971 to 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 (1.151) with respect to the parameter FC, the

index of the quality of life improves by 5.025% in comparison with the base value,

and the mean value of the pollution decreases by 3.5% in comparison with the base

value. Furthermore, the value of the nourishment coefficient FC by optimal law

(1.151) changes by no more than 10% in comparison with the base value of this

coefficient FC ¼ 1.078077. With use of the optimal pair of laws (1.51), the quality

of life index improves by 7.93%, and the mean pollution decreases by 1% in

comparison with the base variants.

1.6.7.4 Analysis of the Structural Stability of Forrester’s MathematicalModel Subject to Parametric Control

Application of the optimal laws of parametric control (1.148) determined above

means substitution of the corresponding functions into equations (1.138, 1.139) for

the parameters FC and CB, while the other model equations remain unchanged.

The proof of the weak structural stability of the mathematical model presented

above and relying on equation (1.135) allows us to derive the following assertion:

Fig. 1.15 Trajectories characterizing the change in the quality of life Q


Assertion 1.12 Let N be a compact set belonging to the region P> 0;V> 0; S> 0;fZ> 0;R> 0g of the phase space of the system of differential equations derived from(1.132–1.146), that is, the five-dimensional space of variables fP; V; S; Z; Rg. Letthe closure of the interior of N coincide with N. Then the flux f defined by(1.132–1.146) and (1.151) or (1.152) is weakly structurally stable on N.

1.6.7.5 Finding Bifurcation Points of Extremals of the Variational CalculusProblem on the Basis of Forrester’s Mathematical Model

Let us 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) L in the

plane. As a result of computer simulations, we have created plots of the dependence

of the optimal values of criterion K on the values of the parameter (CVG, CVD) for

each of 24 possible lawsUij; i ¼ 1; 12; j ¼ 1; 2. Figure 1.16 demonstrates plots for

the four laws U2;1, U6;1, U11;1, U8;1 maximizing the values of criterion K in region

L, as well as the intersection curves of the corresponding кsurfaces. The projectionof 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 control law is optimal. Two or three different laws are

optimal on the projection curves.

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


Chapter 2

Macroeconomic Analysis and ParametricControl of Equilibrium States in NationalEconomic Markets

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-BP models, 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 [41], are efficient instruments for the macroeconomic

analysis of the functioning of a national economy.

In the literature, one can find 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 estimation 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 certain criteria, as well as analysis of the dependence of

the optimal criterion value on exogenous parameters.

2.1 Factor Modeling of the Aggregate Demand in a NationalEconomy: AD–AS Model

2.1.1 Problem Statement

The problem consists in determining the relative position of the mean (aggregated)

curves expressing the values of aggregate demand and aggregate supply for the

Republic of Kazakhstan for the period of years from 2000 to 2008 [36]. The level of

the gross domestic product in comparable prices calculated by the manufacturing

method is used as the index of the aggregate supply. This is the manufacturing

method that is mainly used by the statistical services for calculation of the gross


75

domestic product. The aggregate demand is calculated reasoning from the basic

macroeconomic identity YAD ¼ C + I + G + NX. In other words, the level of the

gross domestic products calculated by the method of finite use is accepted as the

index of the aggregate demand.

2.1.2 Input Data

The official statistics of various state institutes (Statistical Agency of Kazakhstan

and National Bank of the Republic of Kazakhstan) are used for carrying out

computations. The data are presented in Table 2.1.

The problem consists in determining each term of the basic macroeconomic

identity YAD ¼ C + I + G + NX by the respective regressors.

As might be expected (Table 2.2), most of the macroeconomic parameters

closely correlate with the level of the gross domestic product. The basic macroeco-

nomic indexes show considerable correlation with the level of public expenses

(sometimes even more considerable than the correlation with the gross domestic

product), from which one can draw a conclusion about the significant role of the

state in the economy. The correlation of the exchange rate with the level of

consumption, investments, public expenses, and taxes is also considerable, from

which one can draw a conclusion about the considerable influence of the foreign

sector on the economy of Kazakhstan. This is also confirmed by the considerable

correlation between the oil price and basic macroeconomic indexes. Thus on the

one hand, the economy of Kazakhstan depends to a great extent on state interfer-

ence, and on the other hand, it depends on the actions of foreign countries.

2.1.3 Model Construction

As mentioned before, the problem consists in constructing the regression equations

for each component of YAD ¼ C + I + G + NX. The equations are constructed by

reasoning from the theoretically and empirically revealed connections between the

variables.

2.1.3.1 Finite Consumption

Let us consider the Keynesian model of consumption as the model for estimation of

the consumption level. According to this model, the consumption level depends

on the available income (the level of the gross domestic product minus taxes).

76 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .

Table

2.1

Statistical

dataofthemainindexes

ofmacroeconomic

dynam

icsoftheRepublicofKazakhstan

fortheyears

2000–2008

Year

YC

IG

PNX

ET

RPP

PP$

2000

2,599,901.6

1,589,061.6

519000

313,984.5

1195,126.8

142.13

174,530.5

8.68

24,874.62

175.0132

2001

3,055,068.891

1,739,472.838

729,323.3

409,808.647

1.064

�32,354.13534

139.4857

214,944.6

10.25

20,951.37

150.2044

2002

3,329,392.871

1,757,775.184

1,051,820

386,526.03

1.134224

�1,678.151758

140.2352

241,330.4

8.97

20,761.84

148.0502

2003

3,807,298.146

2,009,684.752

1,096,184

428,608.389

1.211351

204,479.0094

147.3835

261,403.6

8.46

24,249.48

164.5332

2004

4,541,648.642

2,362,576.484

1,318,119

527,489.899

1.292512

390,201.2452

114.6345

277,128.8

7.29

23,874.79

208.2689

2005

5,463,019.682

2,652,677.614

1,742,399

614,509.345

1.38945

481,646.4279

108.3157

341,560.2

5.98

31,602.33

291.7614

2006

6,781,287.746

3,019,149.723

1,875,309

690,393.561

1.506164

723,764.696

96.78848

447,985.3

4.61

34,471.47

356.1526

2007

7,181,488.578

3,152,724.009

1,895,757

793,823.402

1.789323

480,370.2316

82.57355

519,542

�4.02

32,179.73

389.7099

2008

8,133,751.486

3,395,129.647

1,957,906

850,104.741

1.959308

1,651,949.086

74.04733

581,246.1

5.77

40,700.53

549.6556

Here,Yistheyearlylevel

ofthereal

gross

domesticproduct

inmillionsoftenge(inpricesoftheyear2000)

Cistheconsumptionlevel


Iisthevolumeofinvestm

entto

thecapital

assetin

millionsoftenge(inpricesoftheyear2000)

Gisthelevel

ofpublicexpensesin


NXisthenet

exportin


Pisthelevel

ofprices(calculatedforthebaseoftheyear2000)

Tisthetaxationlevel


RistherealinterestratecalculatedbytheFisher

equationwithuse

ofthecurrentinflationlevel(theconsumer

price

index

isusedas

theinflationindex;the

meancreditinterestrate

isusedas

thenominal

interestrate)

Eisthereal

currency

exchangerate

ofthetengefortheUSdollar

(correctedfortheinflationofboth

thetengeandtheUSdollar)

PPistheprice

ofonetonofoilUralsin

tenge(inpricesoftheyear2000)

PP$istheprice

ofonetonofoilUralsin

USdollars(inpricesoftheyear2000)

2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 77

Tab

le2.2

Correlationmatrixoftheindexes

ofmacroeconomic

dynam


fortheyears

2000–2008

YC

IG

PNX

ET

RPP

Y1

C0.995568

1

I0.954742

0.964209

1

G0.992147

0.991159

0.946283

1

P0.976477

0.963525

0.902736

0.979412

1

NX

0.852656

0.835273

0.732093

0.823695

0.859411

1

E�0

.9683

�0.97038

�0.90791

�0.97677

�0.95128

�0.82786

1

T0.988341

0.971981

0.917414

0.98303

0.988739

0.843865

�0.95221

1

R�0

.71187

�0.72126

�0.69824

�0.73769

�0.72103

�0.3307

0.734325

�0.7284

1

PP

0.928747

0.919677

0.845586

0.901124

0.896504

0.924429

�0.8964

0.912055

�0.55308

1


In addition, let us also include the credit interest rate (taking into account the rather

significant correlation that is shown by the consumption level and credit interest

rate) and the currency exchange rate (also by virtue of strong correlation) as

explanatory variables. Thus, we estimate the following three models:

C ¼ a0 þ a1ðY � TÞ þ u1;

C ¼ a0 þ a1ðY � TÞ þ Rþ u2;

C ¼ a0 þ a1ðY � TÞ þ Rþ Eþ u3:

Estimation of the first model:

Dependent variable: C_

Method: least squares

Date: 06/22/09 Time: 03:20

Sample: 2000 2008

Included observations: 9

Variable Coefficient Std. error t-Statistic Prob

Y-T 0.363289 0.011641 31.20677 0.0000

C 720,078.5 57,818.27 12.45417 0.0000

R-squared 0.992863 Mean dependent var 2,408,695.

Adjusted R-squared 0.991844 S.D. dependent var 676,711.2

S.E. of regression 61,114.51 Akaike info criterion 25.07202

Sum squared resid 2.61E + 10 Schwarz criterion 25.11584

Log likelihood �110.8241 F-statistic 973.8625

Durbin–Watson stat 1.284489 Prob(F-statistic) 0.000000

Estimation of the second model:



Date: 06/22/09 Time: 03:21

Sample: 2000 2008


Variable Coefficient Std. error t-Statistic Prob.

Y-T 0.356045 0.017356 20.51473 0.0000

R �4,468.076 7,597.224 �0.588120 0.5779

C 781,551.9 120,884.5 6.465278 0.0006








Estimation of the third model:



Date: 06/22/09 Time: 03:22

Sample: 2000 2008



Y-T 0.292251 0.060955 4.794546 0.0049

R �1,261.369 8,037.910 �0.156928 0.8814

E �4,067.552 3,730.705 �1.090290 0.3253

C 1,496,067. 666,065.3 2.246127 0.0746






Durbin-Watson stat 1.335349 Prob(F-statistic) 0.000004

The first model is the best one, judging by the significance of the model

coefficients. Including the interest rate and exchange rate in the number of

regressors does not improve the model. Thus, we can assert that the credit market

insignificantly influences the consumption level in spite of the revealed correlation

between the interest rate and consumption level.

The resulting equation is given by

C ¼ 0:3632892184 Y � Tð Þ þ 720078:5098

It should be noted that the level of the limiting propensity to consumption is

rather low (0.36). It can be considered as the population uncertainty in the near

future, because two-thirds of income is not used for the purposes of current

consumption.

2.1.3.2 Investment

The theoretical approach implies that the investment level depends on the interest

rate. The high interest rate decreases the investment incentives of the economic

agents, since on the one hand, the credit resources rise in price, while on the other

hand, investments such as deposits become more attractive (in the view of both

profitability and risk). Taking into consideration the high degree of correlation of

the investment with the gross domestic product and currency exchange rate, let us

include these variables in the analysis. As a result, let us estimate the following

model:

I ¼ b0 þ b1 � Rþ b2 � Eþ b3 � Y þ u2


Dependent variable: I


Date: 06/22/09 Time: 03:26

Sample: 2000 2008



R �3,464.965 25,562.14 �0.135551 0.8975

E �1,717.959 11,622.83 �0.147809 0.8883

Y 0.225467 0.176924 1.274371 0.2585

C 435,858.2 2,076,119. 0.209939 0.8420







In spite of the significance of the model as a whole (F-test), each coefficient turns

out to be individually insignificant. Perhaps this is a natural result. Taking into

consideration the high cross correlation of the factors included into this model, it

turns out to be multicollinear.

Among the one-factor models (the estimation of the investment level depending

only on the interest rate, on the currency exchange rate, and on the level of the gross

domestic product), the model with the exchange rate as the regressor is the most

appropriate (by the results of the main tests), the next is the model with the gross

domestic product, and finally, the model with the interest rate. If one model includes

the gross domestic product and currency exchange rate, the model parameters

deteriorate severely. Therefore, the following model is used for further analysis:

I ¼ b0 þ b1 � Eþ u2



Date: 06/22/09 Time: 02:19

Sample: 2000 2008



E �15,783.44 2,159.018 �7.310470 0.0002

C 3,053,359. 241,485.3 12.64408 0.0000








I ¼ 3053359� 15783:44� E

The constructed model indicates that the “autonomous” volume of the invest-

ment, independent of the external factors in the economy, is equal to three billion

tenge (with respect to the gross domestic product of the year 2008, it constitutes

37.5%), which is 1.5 times the total volume of investment in comparable prices of

the year 2008. Thus, this equation shows an outflow instead of accumulation of the

investment because of the features of the external market situation. In this case, the

absolute term characterizes the investment potential of the country that theoreti-

cally should be realized inside the country under economic conditions more closed

and less dependent on external shocks.

Like any time series, the investment volume can show autoregression. Let us

examine this hypothesis. For this purpose, let us analyze the correlogram of this

series:

Autocorrelation Partial correlation AC PAC Q-Stat Prob

. |***** | . |***** | 1 0.686 0.686 5.8172 0.016

. |*** . | . **| . | 2 0.357 �0.214 7.6186 0.022

. |* . | . *| . | 3 0.086 �0.127 7.7417 0.052

. **| . | . ***| . | 4 �0.228 �0.346 8.7745 0.067

. ***| . | . *| . | 5 �0.417 �0.107 13.068 0.023

. ***| . | . | . | 6 �0.411 0.040 18.639 0.005

. ***| . | . *| . | 7 �0.356 �0.077 24.929 0.001

. **| . | . | . | 8 �0.217 0.026 29.576 0.000

The correlogram form shows that the investment is a first-order autoregression

AR(1). Constructing this dependence for the investment leads to the following

results:



Date: 10/25/09 Time: 23:14

Sample (adjusted): 2001 2008

Included observations: 8 after adjustments


I(�1) 1.105468 0.047283 23.37958 0.0000





Log likelihood �107.7591 Durbin–Watson stat 1.269912

The regression parameters are satisfactory thus allowing representing the invest-

ment series as follows:

I nð Þ ¼ 1:105 � I n� 1ð Þ þ u


Thus, the annual increase of investment in the Republic of Kazakhstan is equal

to 10.5%.

However, for the purpose of further analysis, the one-factor model remains the

priority. In this model, the level of investment depends on the currency exchange rate.

2.1.3.3 State Expenses

The value of the state expenses mainly depends on its preceding values (in part, it

can be explained by the budget procedure of planning these expenses). Therefore,

let us represent the consumption function as an autoregression function. To deter-

mine its order, let us look at the autocorrelation function of the series:


. |***** | . |***** | 1 0.667 0.667 5.5022 0.019

. |*** . | . *| . | 2 0.368 �0.138 7.4169 0.025

. | . | . **| . | 3 0.064 �0.226 7.4840 0.058

. **| . | . **| . | 4 �0.217 �0.242 8.4193 0.077

. ***| . | . *| . | 5 �0.385 �0.126 12.080 0.034

. ***| . | . | . | 6 �0.404 �0.010 17.467 0.008

. ***| . | . | . | 7 �0.347 �0.051 23.430 0.001

. **| . | . *| . | 8 �0.245 �0.071 29.395 0.000

From the function form we conclude that the consumption function is a first-

order autoregression function AR(1):

Gt ¼ a� Gt�1 þ et:

Let us estimate this model:

Dependent variable: G


Date: 06/22/09 Time: 02:32

Sample: 2000 2008



G-1 1.000002 1.85E-07 5,398,845. 0.0000

R-squared 1.000000 Mean dependent var 557,249.8


S.E. of regression 0.325331 Akaike info criterion 0.696492

Sum squared resid 0.846722 Schwarz criterion 0.718406


In spite of the outside perfection (the determination coefficient equal to one), this

model is unsatisfactory, since according to it, the level of public expenses varies

almost not at all, which is, of course, contrary to fact. Also, this model shows

heteroscedasticity in the residuals. Following the correlation matrix, let us estimate


another two models, whereby we consider the currency exchange rate and the level

of gross domestic product as the regressors:

Gt ¼ a� Eþ cþ et



Date: 06/22/09 Time: 02:45

Sample: 2000 2008



E �5,838.985 385.8181 �15.13404 0.0000

C 1,185,925. 43,153.60 27.48148 0.0000







Gt ¼ �5838:985� Eþ 1185925



Date: 06/22/09 Time: 03:30

Sample: 2000 2008



Y 0.094588 0.004507 20.98709 0.0000

C 85,435.24 24,030.68 3.555257 0.0093







Let us give preference to the second model:

G ¼ 0:0945881296� Y þ 85435:23751:


According to the results of this model, the level of state expenses in Kazakhstan

constitutes 9.4% of the gross domestic product of the current year plus the neces-

sary minimum of the autonomous state expenses.

Let us check the residuals on autocorrelation in this model too. The correlogram

looks as follows:


.****| . | .****| . | 1 �0.567 �0.567 3.9816 0.046

. *| . | *****| . | 2 �0.147 �0.691 4.2861 0.117

. |** . | .****| . | 3 0.328 �0.577 6.0595 0.109

. | . | . *| . | 4 0.038 �0.113 6.0882 0.193

. ***| . | . **| . | 5 �0.403 �0.282 10.102 0.072

. |*** . | . *| . | 6 0.381 �0.058 14.885 0.021

. *| . | . ***| . | 7 �0.148 �0.388 15.968 0.025

. | . | . **| . | 8 0.018 �0.273 16.000 0.042

Such a correlogram allows certification of the presence of autoregression in the

model residuals. Further analysis results in the clear conclusion that this is a second-

order autoregression.

Dependent variable: A


Date: 07/10/09 Time: 10:27




A(�1) �0.925561 0.222106 �4.167203 0.0088

A(�2) �0.796342 0.238031 �3.345539 0.0204

R-squared 0.782105 Mean dependent var �2,575.656





The situation in which the residuals of the regression model do not correlate with

any of the significant factors of the regression is said to be white noise. In this

model, there is not only white noise, but white wind, i.e., the values of the residualsshow autoregression, being in some way dependent on their preceding states. In

other words, there is some logic in the values of the derived residuals that cannot be

revealed by constructing the autoregression model.

Thus, one may speak of the presence of unrevealed factors affecting the state

expenses, whose effects are extended to two subsequent periods. Taking into

consideration the considerable institutional transformations that took place in the


Republic of Kazakhstan within the considered period, one may suppose that these

transformations affect public expenses.

Let us also estimate the regression of the public expenses depending on the

taxation level:



Date: 10/25/09 Time: 23:31

Sample: 2000 2008



T 1.303832 0.091964 14.17769 0.0000

C 113,994.6 33,652.48 3.387406 0.0116







G ¼ 1:3038 � T þ 113995:

If we compare this model with that derived above, G ¼ 0.0945881296*Y +

85,435.23751, we have to admit that it is inferior even in the parameter R2 (in

the latter model, it is equal to 0.98), and the sum of squared errors for the latter

model is twice as small (although taking into account the order of this index, this

discrepancy can be neglected). However, this regression can be useful in consider-

ing the Haavelmo alternative.

For this purpose, let us also estimate the dependence of the investment level on

the taxes:



Date: 10/26/09 Time: 00:00

Sample: 2000 2008



T 3.445681 0.564900 6.099630 0.0005

C 182,574.0 206,715.3 0.883214 0.4064








I ¼ 3:45� T þ 182573

Though this dependence is not as perfect as the previous, nevertheless it is

satisfactory and can be used for the purpose of analysis. From the derived equations

it can be seen that increasing the taxes by one unit results in an increment of public

expenses by 1.3 units and of investment by 3.45 units. That is, the main part of the

investment in Kazakhstan is not private (since it is supposed in this case that raising

taxes would result in a decrease of business economic activity and as a result,

a decrease in investment), but undertaken by the government. Therefore, in this

case, the Haavelmo alternative is out of the question. At this stage, the state is thesole institution that is able to have an effective influence on the economic situationin the country.

Let us give preference to the model

G ¼ 0:0945881296� Y þ 85435:23751:

Thus, we can assert that the level of public expense in Kazakhstan constitutes

9.4% of the gross domestic product of the current year plus the necessary minimum

of the autonomous public expenses.

2.1.3.4 Net Export

Theoretically, the net export must depend on the currency exchange rate; therefore,

let us test the following model:

NX ¼ e0 þ e1 � Eþ u4

Dependent variable: NX


Date: 06/22/09 Time: 02:51

Sample: 2000 2008



E �12,972.00 3,471.498 �3.736716 0.0073

C 1,851,510. 388,285.7 4.768423 0.0020








The coefficient of determination (index R2) of this model is not as expressive as

in the previous models, but it can be accepted as significant (almost 67%). We have

to acknowledge that we again observe a picture of confrontation between internal

potentials and external factors: when the dollar grows in value (E is greater), the net

export of Kazakhstan decreases:

NX ¼ 1851510� 12972� E:

To complete the analysis, let us test the net export on autoregression. The

correlogram for this index is as follows:

Date: 10/26/09 Time: 00:13

Sample: 2000 2008



. |** . | . |** . | 1 0.250 0.250 0.7752 0.379

. |** . | . |** . | 2 0.272 0.224 1.8251 0.402

. | . | . *| . | 3 0.023 �0.097 1.8336 0.608

. *| . | . *| . | 4 �0.097 �0.163 2.0206 0.732

. **| . | . *| . | 5 �0.215 �0.168 3.1681 0.674

. **| . | . **| . | 6 �0.301 �0.192 6.1641 0.405

. **| . | . *| . | 7 �0.283 �0.127 10.119 0.182

. *| . | . | . | 8 �0.149 0.027 12.316 0.138

The insignificance of the first values of the autocorrelation function gives reason

for not analyzing the autocorrelation of this time series. Thus, the hypothesis on

the autonomy of the net export and its dependence solely on foreign market

opportunities does not prove to be true.

Now let us estimate the dependence of the net export on the currency exchange

rate, tax, and oil price (in US dollars).

1. NX ¼ 3:001� T � 565470



Date: 10/26/09 Time: 00:16

Sample: 2000 2008



T 3.001217 0.721258 4.161085 0.0042

C �565,470.2 263,931.9 �2.142485 0.0694








2. NX ¼ �15295� Eþ 2231761



Date: 10/26/09 Time: 00:22

Sample: 2000 2008



E �15,294.99 3,917.023 �3.904749 0.0059

C 2,231,761. 466,379.4 4.785292 0.0020







Note that this regression is somewhat better in the main indexes in comparison

to the previous one.

3. NX ¼ 3455� PP$� 479230



Date: 10/26/09 Time: 00:24

Sample: 2000 2008



PP$ 3,454.733 485.5019 7.115796 0.0002

C �479,229.6 145,787.3 �3.287183 0.0134







The regression of net exports depending on oil price is the best in view of

parameter R2. This fact allows us to observe that net exports do, however, depend

to a greater extent on external factors than internal ones, i.e., their volume is

determined to a greater extent by external demand, but not the readiness of residents

to provide export supply. Moreover, as can be seen from the correlation matrix, the

taxation level depends (very considerably) on the world oil price. Thus, preference

is given to the last model,

NX ¼ 3455� PP$� 479230:


Now let us take the model values of the explanatory variables from each of the

equations derived above and add them to obtain the calculated value of the

aggregate demand. The initially given level of the gross domestic product appears

for the values of the aggregate supply. Let us trace their dynamics in Figs. 2.1, 2.2

and Table 2.3.

Based on the plots, one can ascertain that the economy of Kazakhstan is in a state

close to equilibrium.

First, our analysis confirms the initial hypotheses on the strong dependence of

the economy of the Republic of Kazakhstan on its governmental investment on

the one hand, and on the other, its dependence on the foreign sector. Second,

public policy appears to be efficient with respect to maintaining macroeconomic

equilibrium in the country. When in 2006 there was a tendency toward a

-5

0

5

10

15

0 2000000 4000000 6000000 8000000 10000000

Y AS

Y AD

Aggregate demand and aggregate supply

Million tenge

Inte

rest

Fig. 2.1 Values of aggregate demand and aggregate supply, million tenge in year-2000 prices and

in coordinates of real interest rate

0

0,5

1

1,5

2

2,5

0 2000000 4000000 6000000 8000000 10000000

Y ADY AS

leve

l of p

rices

Aggregate demand and aggregate supply

Million tenge

Fig. 2.2 Values of aggregate demand and aggregate supply, million tenge in year-2000 prices and

in coordinates of level of prices


recessionary gap (aggregate supply exceeding aggregate demand), the government

“bolstered” the demand, and in spite of increasing inflation (18.8% in 2007 against

8.4% in 2006), was able to return the system to a state of equilibrium (it should

also be recognized that inflation was successfully restrained to 9.5% in 2008).

However, equilibrium by itself cannot be a goal for a developing economy. Growth

unavoidably implies instability at certain stages for creating “reserves” for further

development. In this connection, the following question is still open: does the

maintenance of macroeconomic equilibrium restrain potential growth of the econ-

omy? And indeed, the fact of a restraining effect of external economic conditions on

the economy of Kazakhstan is unquestionable. The external factor does not allow the

realization of domestic investment potential. Thus, one can say that excessive

economic openness is not sufficiently strong to counteract external influences.

Within the current macroeconomic conditions, a strong state is an indispensable

condition of the economic stability of the Republic of Kazakhstan.

2.2 Macroeconomic Analysis of the National Economic StateBased on IS, LM, IS–LM Models, Keynesian All-EconomyEquilibrium. Analysis of the Influence of Instrumentson Equilibrium Solution

One of the main economic functions of the state is to carry out a stabilizing policy

based on the equilibrium conditions in various markets.

The IS, LM, IS-LM models, as well as the Keynesian model of common eco-

nomic equilibrium, are efficient instruments for macroeconomic analysis of market

states.

This section is devoted to the construction of the IS, LM, IS-LM models, as well

as the Keynesian model of common economic equilibrium by the example of the

economy of the Republic of Kazakhstan, analysis of the influence of the economic

instruments on equilibrium conditions in the respective markets, as well as the

estimation of the optimal values of the economic instruments on the basis of the

Keynesian mathematical model of common economic equilibrium.

Table 2.3 Values of

aggregate demand and

aggregate supply, million

tenge in year-2000 prices

Years Y AS Y AD

2000 2,599,902 2,782,329

2001 3,055,069 3,010,350

2002 3,329,393 3,171,492

2003 3,807,298 3,421,302

2004 4,541,649 4,557,192

2005 5,463,020 5,277,863

2006 6,781,288 6,140,684

2007 7,181,488 7,074,904

2008 8,133,751 8,254,390

2.2 Macroeconomic Analysis of the National Economic State Based. . . 91

2.2.1 Construction of the IS Model and Analysisof the Influence of Economic Instruments

Let us introduce the notation for the economic indexes used for model construction:

T is the tax proceeds (to the state budget, in billions of tenge); S is the net savings,

billions of tenge; I is the investment to the capital asset, billions of tenge; G is the

public expenses, billions of tenge; Y is the gross national income, billions of tenge;

C is household consumption, billions of tenge.

Macro-estimation of the equilibrium conditions in the wealth market can be

done on the basis of the IS model [41, p. 76] represented as

T þ S ¼ I þ G: (2.1)

The tax proceeds T to the state budget represented by the expression T ¼ TyYhas the following econometric estimation based on statistical information for the

years 2000–2008:

T ¼ 0:2207 Y:

0:000ð Þ (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.The net savings S represented by the expression S ¼ a + SyY has the following

econometric estimation:

S ¼ �366:055þ 0:222 Y

0:000ð Þ 0:000ð Þ (2.3)


coefficient R2 ¼ 0.994; the standard error Se ¼ 69.2; the approximation coefficient

A ¼ 11.47%; the Fisher statistics F ¼ 1,287.2; the Durbin–Watson statistics

DW ¼ 1.96.

The investment to capital assets represented by the expression I ¼ a + Ii i afterestimation of the parameters of this model using the statistical information becomes

the following:

I ¼ 1367:9� 81:3 iþ 0:2751Ymean:

0:02ð Þ 0:03ð Þ 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 ¼ 1.72. Substituting into (2.3) the value of the mean nominal gross national

income for the years 2000–2008 in billions of tenge Ym ¼ 6,662.7 finally yields the

following model for the investment:

I ¼ 3 202� 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 determining the equilibrium value of i for the given values of Y and

G200X. In macroeconomic theory, one has the method [41, p. 77] of plotting the IScurve, which is the set of combinations of the equilibrium values of Y and i(Fig. 2.3).

From the model IS2007 (Fig. 2.3) it follows that the equilibrium GNI2007 with

interest rate 13.6% equals 11,602.75 billion tenge, and that the real GNI2007 withinterest rate 13.6% equals 11,371 billion tenge, which shows a lack of wealth in the

considered market. From the model IS2008 (Fig. 2.3), it follows that the equilibriumGNI2008 with interest rate 15.3% equals 13,957.91 billion tenge, and that the real

GNI2008 with interest rate 15.3% equals 13,734 billion tenge, which also shows a

lack of wealth in that market.

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. 2.3 Plots of IS2007 and IS2008 models


To estimate the multiplicative effects [41, p. 78] of the economic instruments Tyand G, let us 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 0:552 Yv;

0:000ð Þ 0:000ð Þ

where Y v 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 5,394; the

Durbin–Watson statistics DW ¼ 1.53.

Table 2.4 presents the expressions and values of the multipliers [41, p. 83] of

instruments Ty and G derived on the basis of the IS model (2.6).

Let us estimate the multiplicative effects of the instruments Ty and G based on

the data for the year 2008. According to those data, we have G ¼ 3,859.98,

Y ¼ 13,734.3, Ty ¼ 0.2207. Now let us 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 1,308.54.

Also, from the data of the year 2008, we have G ¼ 3,859.98, Y ¼ 13,734.3,

Ty ¼ 0.2207. Let us change Ty by DTy ¼ �0.01. This change in accordance with

the multiplier of DTy 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

taxation” [41, p. 83].

2.2.2 Macroeconomics of Equilibrium Conditionsin the Money Market

The macro-estimation 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)

Table 2.4 Consequences of changing public expenses and taxation

Consequence

Action

Public expenses increase by DG Taxes decrease by DT

National income increases by 1TyþSy DG ¼ 2.26 DG Cyv

TyþSy DT ¼ 1.3 DT

Budgeted deficit increases by 1� TyTyþSy

� �DG ¼ 0.5 DG 1� TyCyv

TyþSy

� �DT ¼ 0.7 DT


where M is the money supply, in billions of tenge; lpr is the volume of property

(deposits in deposit organizations by sectors and currencies), billions of tenge; lpr isthe volume of transaction (the volume of credits given by second-level banks (SLB)

taking into account the money velocity), billions of tenge.

To estimate the money velocity, let us use the Fisher equation [41, 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 YM on the basis of the

statistical information for the years 2007–2008 is presented in Table 2.5.

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 [41, p. 99]:

m ¼ 1þ gð1� a� bÞaþ bþ gð1� a� bÞ ;

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;

CM 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,M ¼ mH ¼ 4,519.9 billion tenge; for the year 2008,M ¼ mH ¼ 5,343.6

billion tenge.

Table 2.6 presents the calculated values of the money supply and the values of

the money aggregate M3 by years. Table 2.7 shows that the calculated values of M

Table 2.5 Value of

the money aggregate M3and the velocity of money

Year GNIValue of money

aggregate M3

V, velocityof money

2007 11,371 4,629.8 2.5

2008 13,734 6,266.4 2.2


and 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 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; the Fisher

statistics F ¼ 67. The demand of money for transactions represented by the expres-

sion 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; the Fisher statistics

F ¼ 193.7.

Substituting expressions (2.8, 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 the determination of the equilibrium value of i for the given values ofY and M200X . In macroeconomic theory one has a method [41, p. 113] of plotting

the LM curve, which is the set of combinations of the equilibrium values of Y and i.Fig. 2.4 presents the plots of the LM models for the years 2007 and 2008.

Table 2.7 Calculated values of money supply and values of money aggregate

Years

Calculated values

of money supply

Values of money

aggregate M3

2007 4,519.9 4,629.8

2008 5,343.6 6,266.4

Table 2.6 Values of multipliers

Year a b g

Values of multipliers

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


In accordance with the obtained results and plotted LM2007, LM2008, one may

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 of

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

the respective line for 2007, i.e., 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.

2.2.3 Macro-Estimation of the Mutual Equilibrium Statein Wealth and Money Markets. Analysis of the Influenceof Economic Instruments

On the basis of the derived IS and LM models, the model for macro-estimation of

the joint equilibrium state in the wealth and money markets can be represented by

the following system:

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


LM 2007

LM 2008

Fig. 2.4 Plots of models LM2007 and LM2008


� 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 markets for the years 2007 and 2008 are presented in Table 2.8.

The plots of the IS and LM models in the same period are shown in Fig. 2.5.

From Fig. 2.5 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 in the wealth market and excess

of money in the money market in 2007 and 2008.

Let us estimate the influence of the instruments G andM on the joint equilibrium

conditions using the data for the year 2008.

0

5

10

15

20

25

30

0,00 5000,00 10000,00 15000,00 20000,00 25000,00

i (interest rate)



IS 2007

LM 2008


IS 2008

LM 2007

Fig. 2.5 Plots of models IS2007, LM2008, LM2007, and LM2008

Table 2.8 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*, Keynesianeffective demand

2007 13.6 11,371.1 13.23 11,670.89

2008 15.3 13,734.3 13.29 14,327.31


Based on 2008 data, solution of (2.1.11) result in G ¼ 3,859.98 and

M ¼ 5,343.6. Let us now increase G by DG ¼ 579. With unchanged M, 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 13.9% due to the

shift of IS to the right as a result of the multiplicative effect from increasing

the public expenses.

Let us now increaseM2008 by DM ¼ 534. With unchangedG2008, this fluctuation

results in an increase of GNI up to 15,438.6 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

multiplicative effect from increasing the money supply.

The obtained results also agree with the macroeconomic theory on the influence

of economic instruments in the wealth and money markets [41, p. 78; 114].

2.2.4 Macro-Estimation of the Equilibrium Stateon the Basis of the Keynesian Model of CommonEconomic Equilibrium. Analysis of the Influenceof Economic Instruments

The Keynesian mathematical model of common economic equilibrium on the basis

of the IS, 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

[41, 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

8>>>><>>>>:

where Ws (N,P) is the function representing the labor supply price; YN is the

derivative of the production function; Y(N) is the production function.

Equations (2.12, 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 60:12 P� 0:007 N;

0:000ð Þ 0:000ð Þ (2.16)

where P is the level of prices for the year 2000; N is the busy population in

thousand. 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; 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 � 5:654N þ 0:0009N2:

ð0:000Þ ð0:000Þ (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 122; 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

� 366:055þ SyY þ TyY ¼ 3202� 81:30iþ G200X;

M200X ¼ 438883:3� 0:66i � 1062:85þ 0:326 Y;

60:12 P� 0:00698N ¼ �5:65 Pþ 0:0018N P;

Y ¼ �5:65 N þ 0:0009 N2:

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*, W*, that result in attaining equilibrium simultaneously in all three

given markets are determined from the solution of this system of equations.

Substituting the actual values of G200X and M200X 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 2.9 presents the equilibrium values of the endogenous parameters using

the solution of system (2.18) on the basis of the data for the years 2007 and 2008.

Let us estimate the influence of instruments G andM on the Keynesian common

economic equilibrium from the data from 2008.

Table 2.9 Comparative analysis of actual and equilibrium values of GNI, interest rate, level of

prices, employed population

Y i P N

2007 Actual 11,371.1 13.6 1.789 7,631.1

Equilibrium 11,670.89 13.23 1.05 7,751.6

Deviation 2.64% �0.37 �0.74 1.58%

2008 Actual 13,734.3 15.3 1.959 7,857.2

Equilibrium 14,327.3 13.3 1.103 8,048.8

Deviation 4.32% �2 �0.9 2.44%


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 of the interest rate to 13.9%,

while at the same time, the unemployment drops by 1.6%, and the level of prices

increases to 1.12.

Increasing M2008 by DM 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 M2008 by DM ¼ 534.4 results in an

increase of 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.

2.2.5 Parametric Control of the Open Economy StateBased on the Keynesian Model

Let us estimate the optimal values of the instrumentsM andG for the given external

exogenous parameters Sy, Ty on the basis of model (2.18) for the year 2008 in the

sense of the GNI criterion

Y ! max : (2.19)

Such an estimate can be obtained by solving the following mathematical pro-

gramming problem:

Problem

1. On the base of mathematical model (2.18), find the values of (M, G) maximizing

criterion (2.19) under the constraints

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)

HereM* и 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 ofM*

and G*.For Problem 1, the optimal values of the parameters are M ¼ 5,877.96,

G ¼ 4,245.98, 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 8,148.539; P ¼ 1.1210; 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. 2.6.

2.3 Long-Term IS–LM Model and Mundell–Flemming Model

2.3.1 Problem Statement and Data Preparation

The problem consists in the construction of the long-term IS-LM-BP model for the

economy of Kazakhstan. The modeling is underlain by the constriction of the

regression equations for each of the curves included in the model, namely,

investment–savings (IS), liquidity–money (LM), and the balance of payments

(BP). The derived equation allows plotting the model in the traditional coordinates

“income–interest rate.”

Fig. 2.6 Plot of dependence of optimal values of criterion Y on parameters Ty, Sy


The statistical basis for constructing the model is the data of the official statistics

of the Republic of Kazakhstan, namely, the following indexes: the gross domestic

product (GDP), the interest rate, money aggregates M2 and M3, the investment

level, the volume of public expenses, the exchange rate of tenge to US dollars, and

the net export, as well as the price of Urals oil (the data are presented in Table 2.10).

Let us choose correlation analysis (Table 2.11) as the instrument of preliminary

analysis needed in further modeling.

2.3.2 Model Construction

The model is constructed under the assumption that the curves of IS, LM, and BPremain immovable over the considered period, an assumption that, strictly

speaking, does not correspond to reality. Therefore, as a result of computations,

we obtain the “averaged” Mundell–Fleming long-term model.

2.3.2.1 “Investment–Savings” (IS) Curve

The IS curve is the formalized reflection of all possible equilibrium states in the

market of goods and services. This equilibrium assumes equality between the

volume of national savings and the level of gross investment. The former depends

immediately on the national income (GDP), while the latter depends on the interestrate. So let us accept the regression dependence of GDP on the interest rate as the

initial dependence including the investment level or public expenses as the explan-

atory variables:

YIS ¼ Y R; Ið Þ; YIS ¼ Y R;Gð Þ:

Correlation analysis reflects the presence of a connection between the variables

entering the regression equation at which the level is significant (the correlation

coefficient between GDP and the interest rate is equal to �0.71, between GDP and

the investment is equal to 0.95, between GDP and the level of public expenses is

equal to 0.99). However, the correlation between the explanatory variables of the

model, namely the interest rate and investment as well as the interest rate and public

expenses, is also considerable (�0.698 and �0.737, respectively). This fact entails

the problem of multicollinearity in the model. Let us analyze this problem by the

basic econometric indexes:

YIS ¼ cþ a1Rþ a2Gþ e;

Y ¼ 20699:17234� Rþ 10:74660517� G� 1129241:822

2.3 Long-Term IS–LM Model and Mundell–Flemming Model 103

Tab

le2.10

Statistical

dataonbasic

indexes

ofmacroeconomic

dynam


fortheyears

2000–2008

Year

YR

IG

NX

EM2

M3

PP

2000

2,599,901.6

8.679417

519,000

313,984.5

195,126.8

142.13

249,142.9

322,312.7

24,874.62

2001

3,055,068.891

10.25376

729,323.3

409,808.6

�32,354.1

139.4857

304,050.5

443,335.1

20,951.37

2002

3,329,392.871

8.968105

1,051,820

386,526

�1,678.15

140.2352

338,320

554,087

20,761.84

2003

3,807,298.146

8.464419

1,096,184

428,608.4

204,479

147.3835

497,687.5

729,412.4

24,249.48

2004

4,541,648.642

7.291471

1,318,119

527,489.9

390,201.2

114.6345

702,109.3

960,779.1

23,874.79

2005

5,463,019.682

5.981395

1,742,399

614,509.3

481,646.4

108.3157

968,457.6

1,327,192

31,602.33

2006

6,781,287.746

4.612546

1,875,309

690,393.6

723,764.7

96.78848

1,394,648

1,835,797

34,471.47

2007

7,181,487.945

�4.02357

1,895,757

793,823.4

480,370.2

82.57355

1,901,954

2,384,855

32,179.73

2008

8,133,751.486

5.771689

1,957,906

850,104.7

1,651,949

74.04733

2,017,338

2,738,204

40,700.53

HereYistheannual

level

ofreal

GDPin


Iistheinvestm

entvolumein

milliontenge(inpricesoftheyear2000)

Gisthelevel

ofpublicexpensesin


NXisthenet

exportin


Risthereal

interestratecalculatedbytheFisher

equationwithuse


price

index

isusedas



isusedas

thenominal

interestrate)

Eisthereal

interestratecalculatedbytheFisher

equationwithuse


price

index

isusedas



isusedas

thenominal

interestrate)

M2,

M3aretherespectivemoney

aggregates

inmillionsoftenge(inpricesoftheyear2000)PPistheprice

ofonetonofUralsoilin

tenge(inpricesofthe

year2000)


Tab

le2.11

Correlationmatrixofindexes

ofthemacroeconomic

dynam


for2000–2008

YR

IG

NX

EM2

M3

PP

Y1

R�0

.711698533

1

I0.954742036

�0.69807

1

G0.992147203

�0.73752

0.946283

1

NX

0.852656102

�0.33054

0.732093

0.823695

1

E�0

.968299283

0.734224

�0.90791

�0.97677

�0.82786

1

M2

0.984931584

�0.78136

0.908705

0.985836

0.827694

�0.96861

1

M3

0.98991353

�0.74259

0.916602

0.988136

0.853583

�0.96661

0.997587

1

PP

0.928746735

�0.55291

0.845586

0.901124

0.924429

�0.8964

0.911159

0.92066

1


Dependent variable: Y


Date: 11/27/09 Time: 21:35

Sample: 2000 2008



R 20,699.17 34,610.14 0.598067 0.5717

G 10.74661 0.770466 13.94819 0.0000

C �1,129,242. 613,042.2 �1.842030 0.1151


Adjusted R-squared 0.980315 S.D. dependent var 1,997,879.





In spite of the high value of the index R2, the value of the coefficient of the

interest rate turns out to be insignificant (high p-level). This can be partially

explained by the sign of this coefficient. It is positive, though the theoretical

derivation of the model shows the presence of a negative connection between

GDP and the interest rate, and this connection is confirmed by the sign of the

correlation coefficient between them:

YIS ¼ cþ a1 � Rþ a2 � I þ e;

Y ¼ �41559:38539� Rþ 3:308409085� I þ 767164:6307



Date: 11/27/09 Time: 21:40

Sample: 2000 2008



R �41,559.39 78,083.25 �0.532245 0.6137

I 3.308409 0.613837 5.389718 0.0017

C 767,164.6 1,241,178. 0.618094 0.5592








In this case, we are faced with the same problem of statistical insignificance of

the coefficient of the interest rate, although its sign completely agrees with the

theory.

Let us also estimate the “lite” version of the equation:

YIS ¼ cþ a1 � Rþ e;

Y ¼ �335340:1444� Rþ 7074627:596



Date: 11/27/09 Time: 21:44

Sample: 2000 2008



R �335,340.1 125,105.8 �2.680452 0.0315

C 7,074,628. 925,242.8 7.646239 0.0001



S.E. of regression 1,500,383. Akaike info criterion 31.47347




Now, the coefficient of the interest rate is significant, but the common reliability

of the model suffers. It is expressed in the decrease of the coefficient R2. This

model is not worth considering for the characterization of the curve IS, sincethe connection “GDP–interest rate” is present in all three curves of the model

IS-LM-BP. Moreover, it would lead to an unacceptable situation when different

dependencies will be described by the same question.

In this situation, when none of the derived regression equations are fully

satisfactory, let us consider the other possible dependencies describing the curve IS.The insignificant coefficients of the interest rate in the regressions with two

explanatory variables lead to the idea of some “lag” of the reaction of the GDP to

the variation of the interest rate. From a logical point of view, this hypothesis is not

groundless. The macroeconomic indexes are rather resilient with respect to the

reactions, and the time lad with such reactions is a regular phenomenon. Therefore,

let us construct a model in which the GDP depends only on the value of the interest

rate in the preceding year. Also, let us keep the investment level in the model:

YIS ¼ cþ a1 � R�1 þ a2 � I þ e;

Y nð Þ ¼ �152650:914� R n� 1ð Þ þ 2:855488855� I nð Þ þ 2080721:113




Date: 11/27/09 Time: 22:40




R(�1) �152,650.9 38,816.77 �3.932602 0.0110

I 2.855489 0.374470 7.625414 0.0006

C 2,080,721. 746,195.3 2.788441 0.0385







This model seems to be perfect with respect to all the indexes (the significance of

the coefficients, the significance of the model as a whole, R2). The sign of the

coefficient of the interest rate also meets the theoretical requirements. Thus, we will

use just this regression equation as the curve describing the dependence

“investment–savings.”

While carrying out the analysis, the following regressions are also constructed:

YIS ¼ cþ a1 � R�1 þ a2 � Gþ e, YIS ¼ cþ a1 � R�1 þ e. However, the chosen

model turns out to be the best one for all the parameters.

2.3.2.2 “Liquidity–Money” (LM) Curve

This curve describes the money market equilibrium, as well as the stock market

equilibrium. According to the theory, the money demand depends on the income

level and the real interest rate,Ms ¼ MðY;RÞ. To preserve the model’s logic, let us

consider just this dependence, but not the implicit function Y ¼ Y(M;R):

Ms ¼ b1 � Y þ b2 � Rþ e;

M2 ¼ 0:2490654962� Y � 48136:28677� R

Dependent variable: M_2


Date: 11/28/09 Time: 01:38

Sample: 2000 2008



Y 0.249065 0.010246 24.30966 0.0000

R �48,136.29 7,386.492 �6.516799 0.0003

(continued)







M3 ¼ 0:3236175168� Y � 54199:73886� R

Dependent variable: M_3


Date: 11/28/09 Time: 01:40

Sample: 2000 2008



Y 0.323618 0.015388 21.03010 0.0000

R �54,199.74 11,094.16 �4.885432 0.0018

R-squared 0.960611 Mean dependent var 1255108.





The regression equation with the aggregateM2 as the index of the money supply

gives some better results in comparison with M3. This can be explained by the fact

that certificates of deposit, public bonds, exchequer savings stock, commercial

securities, by the value of which M3 differs from M2, are not so marketable to

represent the money demand as such. Therefore, for the further representation of

curve LM we use the first regression equation.

2.3.2.3 “Balance of Payment” (BP) Curve

The balance of payment curve characterizes the foreign market equilibrium (the

equilibrium in operations with the “foreign” sector). Therefore, the levels of the real

currency exchange rate, net export, as well as the oil price (besides the interest rate)

can appear for the regressors. Since from the correlation analysis it can be seen that

these additional regressors demonstrate a close connection, let us estimate the

regression equations with respect to each of them individually (of course, including

the interest rate):

YBP ¼ cþ d1 � Rþ d2 � Eþ e;

Y ¼ �767:0093069� R� 69893:29419� Eþ 13112868:16




Date: 11/28/09 Time: 01:54

Sample: 2000 2008



R �767.0093 70,774.86 �0.010837 0.9917

E �69,893.29 10,855.55 �6.438485 0.0007

C 13,112,868 1,002,904. 13.07490 0.0000







Here, the statistical insignificance of the interest rate coefficient can be explained

by multicollinearity, since the interest rate and the exchange rate are highly

correlated. Unfortunately, the interest rate cannot be removed from this model,

since by doing so, the required connection between theGDP and interest rate would

be corrupted:

YBP ¼ cþ d1 � Rþ d2 � NX þ e;

Y ¼ �227386:5054� Rþ 2:711387294� NX þ 5169694:357



Date: 11/28/09 Time: 01:56

Sample: 2000 2008



R �227,386.5 52,175.57 �4.358103 0.0048

NX 2.711387 0.433159 6.259562 0.0008

C 5,169,694. 474,598.0 10.89279 0.0000







YBP ¼ cþ d1 � Rþ d2 � PPþ e;

Y ¼ �134497:9641� Rþ 224:5877659 � PP� 505075:5724




Date: 11/28/09 Time: 01:57

Sample: 2000 2008



R �134,498.0 65,645.92 �2.048840 0.0864

PP 224.5878 40.58780 5.533381 0.0015

C �505,075.6 1,428,299. �0.353620 0.7357







As the p-level shows, in the third model, the confidence level with respect to the

value of the free term is low. Thus, the character of the dependence between Y and Ris revealed, but the precise position of the respective curve BP still remains unclear.

It is obvious that the second model most precisely reflects reality. In this model,

the interest rate and the net export appear for the regressors.

Let us draw our attention to the fact that in all three models of the curve BP, thedependence between Y and the market average interest rate becomes negative, and

moreover, the curve BP has a steeper slope than the curve IS situated similarly. This

is evidence of the relatively closed economic system of Kazakhstan for short-term

capital mobility and the presence of investment barriers to entering local markets.

2.3.3 Final IS-LM-BP Model

Now let us introduce the complete model, reasoning from the derived regression

equations:

IS : Y nð Þ ¼ �152650:914� R n� 1ð Þ þ 2:855488855� I nð Þ þ 2080721:113;

LM : M2 nð Þ ¼ 0:2490654962� Y nð Þ � 48136:28677� R nð Þ;

BP : Y nð Þ ¼ �227386:5054� R nð Þ þ 2:711387294� NX nð Þ þ 5169694:357:

Averaging the values of the “floating” variables I,M2, and NX for the considered

period, let us derive three equations expressing the dependence between the current

yearly GDP and the interest rate. The combination of these three equations gives

the Mundell-Flemming long-term model for the present-day Republic of

Kazakhstan.


Using these equations, let us find the respective model values for Y (Table 2.12).

The obtained data allow us to plot Fig. 2.7.

The curve IS is not a straight line, since it depends on the preceding values of theinterest rate. Furthermore, the “incorrect” slope of the line BP is very noticeable.

In theory, with an increase in the domestic interest rate, capital inflow to the country

must take place, i.e., the plot must be ascending. But the equilibrium of the balance

of payments (the line BP reflects just this) depends not so much on the value of the

domestic interest rate as on its deviation from the world average rate. Thus, we can

assume that during the considered period, investment abroad promised greater

profitability for the residents of Kazakhstan, that is, the difference between the

domestic and world interest rates was negative, which becomes apparent from

the sign of the coefficient of the variable R. Hence even the growth of the domestic

interest rate was accompanied by the outflow of capital from the country. Also, the

same may be said about the level of risk. As a rule, investment abroad is less risky

than that inside countries suffering from problems of reform and modernization.

Table 2.12 Model values of Y

Y IS Y LM Y BP

5,413,062 4,429,343

4,622,074 5,717,331 4,071,359

4,381,749 5,468,856 4,363,699

4,578,006 5,371,510 4,478,230

4,654,894 5,144,817 4,744,943

4,833,945 4,891,622 5,042,837

5,033,930 4,627,068 5,354,095

5,242,886 2,957,986 7,317,831

6,561,197 4,851,092 5,090,521

-6

-4

-2

0

2

4

6

8

10

12

200000 1200000 2200000 3200000 4200000 5200000 6200000 7200000 8200000

Y

R

Y IS Y LM Y BP

IS-LM-BP model

Fig. 2.7 IS-LM-BP model in coordinates “income–interest rate”


Let us draw attention to the fact that the current equilibrium of the money market

is achieved with little deficit of the balance of payments, and the current equilib-

rium of the real market is attained with an excess and deficit of the money supply (in

the points situated below curve LM, the money demand exceeds the money supply).

This means that the monetization level of the economy of Kazakhstan is insuffi-

cient, and as long as this drawback is not overcome, application of Keynesian

arrangements for stimulating economic growth is rather dangerous, because the

economy runs the risk of choking over the deficit of means of payment. However,

this does not reflect world problems concerned with a liquidity deficit. This is the

result of the rather tough monetary policy of the government of Kazakhstan.

As a whole, it should be noted that the year 2005 was closest to equilibrium.

However, in spite of two later years, the points on the curves are also rather close

one to another. It is evidence of the state’s effort to carry out the policy of

maintaining equilibrium in the real and money markets.

2.4 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 free exchange of goods and capital with the outside world.

All remaining states of the national economy differing from 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 [41, 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.

2.4 Macroeconomic Analysis and Parametric Control. . . 113

2.4.1 Construction of the Model of an Open Economyof a Small Country and the Estimationof Equilibrium Conditions

Let us introduce the following notation for the economic indexes used for the model

construction: Y is the gross national income (GNI); C is household consumption; I isthe investment in capital assets; G is public expenses; NE is the net export of

wealth; P is the level of prices of RK; l is the real cash remainder; i is the interestrate 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 is the rate of exchange of the national

currency; ee is the expected rate of exchange of the national currency; e_e

is the

expected rate of increase of the exchange rate of the national currency [41, p. 121];

M is the money supply determined from [41, p. 412] by the formulaM ¼ mH, whereH 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

m ¼ 1þ g 1� a� bð Þð Þaþ 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 CM/K is the share of cash in

the whole sum of the credits of second-level banks; RR is the minimal reserve; ERis 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 us 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 2,006–2008 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; 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 [43]:


MV ¼ Y;

where V is the money velocity; 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 [40] are

presented in Table 2.13.

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 consumptionC represented by the expressionC ¼ a + CYY has the following

econometric estimation derived on the basis of the statistical information of the

Republic of Kazakhstan for the period 2000–2008:

C ¼ 474:2þ 0:4531 Y:

0:00ð Þ 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 0.999, the approximation coefficient

A ¼ 1.9%. The statistical significance of the coefficients of regression (2.23),

as well as the regressions estimated below, are 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 b1 ePZ=P

or, in estimated form,

Qim ¼ 0:3946 Y � 2:6125 ePZ=P

0:00ð Þ 0:03ð Þ (2.24)

with the determination coefficient R2 ¼ 0.91 and the approximation coefficient

A ¼ 11%.

The model of the demand of the real cash remainder is given by l ¼ a2 + b2Y +b3 i + b4 e or, after estimating the parameters of this model using the statistical

information,

Table 2.13 Values of GNI(billions of tenge), money

aggregate M3 (billions of

tenge), and money velocity V

Year GNI M3 V

2007 11,371 4,629.8 2.5

2008 13,734 6,266.4 2.2


l ¼ �6758:3þ 0:9973 Y � 175:5 iþ 38:4 e:

0:3ð Þ 0:04ð Þ 0:7ð Þ 0:5ð 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 0.995, and the approximation coefficient is A ¼ 6%.

The statistical insignificance of the latter model concerns the fact that in the model

there are correlated factors. Thus, the gross national income has a strong correlation

with the exchange rate (R ¼ 0.92) and a direct connection with the interest rate

(R ¼ 0.65).

The model of the labor supply price is given by WS ¼ b5 N + b6 Pmean, where

Pmean ¼ (1�a)P + a ePZ/e0 has the following econometric estimation derived on

the basis of the statistical information:

Ws ¼ �0:025N þ 175:5Pmean

0:00ð Þ 0:00ð Þ (2.26)

where Pmean ¼ 0.6 P + 0.4 ePz/e0, e0 is the currency exchange rate within the baseperiod (year 2000); 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 A ¼ 0.07%.

The model of the net capital export is given by NKE ¼ b7e iZ þ e_e � i

� �or, after

estimating the parameters of this model using the statistical information,

NKE ¼ �0:47e iZ þ e_e � i

� �0:02ð Þ (2.27)


A ¼ 3.2%.

The production function is represented in the regression pair Y ¼ a3 + b8 Nor, in the estimated form,

Y ¼ �44477:9þ 7:5 N

0:00ð Þ 0:00ð Þ (2.28)


A ¼ 12%.

The model of investment in capital assets is given by

It ¼ a4 þ b9 Yt�1 þ b10 it;

where It and it are the values of the investments in the current period; Yt-1 is thevalue of the gross national income in the preceding period.


After estimating the latter model parameters using the statistical data, the

following expression is derived:

It ¼ 1367:9þ 0:2753 Yt�1 � 81:3 it

0:02ð Þ 0:03ð Þ 0:00ð Þ (2.29)

We have the determination coefficient R2 ¼ 0.98 and the approximation coeffi-

cient A ¼ 5%.

Substituting the value Yt-1 ¼ Y2007 into (2.29), finally we obtain the following

model of investment in the year 2008:

I2008 ¼ 5148:9� 81:3i: (2.30)

Similarly, substituting the value Yt-1 ¼ Y2006 into (2.29) for investment in 2007,

we obtain the following model:

I2007 ¼ 3857:6� 81:3i (2.31)

The wealth export model is a regression of the form Qex ¼ b11 ePZ/P. Afterestimating the parameters, this model becomes

Qex ¼ 25:68 ePZ

P:

0:02ð Þ (2.32)

The determination coefficient is R2 ¼ 0.50.

On the basis of derived econometric estimates (2.23–2.32) characterizing the

state of the national economy, let us proceed to the construction of a model of an

open economy of a small country for the year 2008.

Within the framework of the IS curve, we constructed the function Y ¼ C +I + G + Qex � Qim, which subject to (2.23, 2.24, 2.29–2.32), becomes

Y ¼ 474:2þ 0:4531 Y þ 5148:9� 81:3iþ Gþ 28:29 ePZ=P� 0:3946 Y or

Y ¼ 5985:2� 86:54 iþ 30:11 ePZ=Pþ 1:064 G

(2.33)

The equation of the LM line M/P ¼ l subject to the econometric model (2.25)

becomes

M

P¼ �6758:3þ 0:9973 Y � 175:5 iþ 38:4 e;


from which one can derive the following relation:

i ¼ �38:51þ 0:2190 eþ 0:0057 Y � 0:0057 M=P: (2.34)

Substituting (2.34) into (2.33), we obtain the value of the aggregate demand YD:

YD ¼ 6246:1� 12:70 eþ 20:18 ePZ Pþ 0:7135Gþ 0:3305M P= := (2.35)

Let us substitute (2.33) into (2.34) and determine the function of the domestic

commercial interest rate:

i ¼ �3:0147þ 0:1468 e� 0:0038 M P= þ 0:1147 ePZ Pþ 0:0041 G= : (2.36)

The condition of equilibrium in the labor market is given by P 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 �0:025N þ 175:5 0:6Pþ 0:4 ePZ=e0� �

: (2.37)

From (2.37) we obtain the following relation for N:

N ¼ 3915:9 Pþ 19:7758 ePZ: (2.38)

Substituting expression (2.38) into the production function (2.28), we obtain the

function of the aggregate supply:

YS ¼ �44477:9þ 29368:9 Pþ 148:3 ePZ: (2.39)

The balance of payments has a zero balance if the net wealth export equals the

net capital export, i.e., the following holds: NE ¼ NKE. The econometric represen-

tation of the latter equality on the basis of (2.24, 2.27, 2.32) is given by

25:68 ePZ=P� 0:3946 Y � 2:6125 ePZ=P� � ¼ �0:47eðiz þ e

_e� iÞ

Substituting the value of 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:

YZBO ¼ 72:0543 ePZ=P� 1:1971 eiZ=P� 1:1971 ee=P� 2:412 e=Pþ 0:1757 e2=

P� 0:0046 eM=P2 þ 0:1373 e2PZ=P2 þ 0:0049 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 6246; 1� 12:7eþ 2018 ePZ

P þ 0; 7135Gþ 0; 3305MP ;

YS ¼ �44477:9þ 29368:9Pþ 148:3ePZ;

YZBO ¼ 72:05 ePZ

P � 1:1971 eiZ

P � 1:1971 ee

P � 2:412 eP þ 0:1757 e2

P

�0:0046 eMP2 þ 0:1373 e2PZ

P2 þ 0:0049 eGP ;

YD ¼ YS ¼ YZBO:

8>>>>>>>>><>>>>>>>>>:

(2.41)

A model of an open economy of a small country in 2007 can be constructed

similar to (2.41).

Solving system (2.41) with prescribed values of the external economic indexes

PZ, iZ, ee and the economic instruments M and G, let us 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 e*. 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 indexes PZ, iZ,ee and the controlled economic instruments M and G:

– In the year 2007: Y* ¼ 9,398.1; P* ¼ 1.1699; e* ¼ 109.0; i* ¼ 16.8;

N* ¼ 7,183.5

– In the year 2008: Y* ¼ 11,383.0; P* ¼ 1.1924; e* ¼ 116.3; i* ¼ 26.1;

N* ¼ 7,448.1

Figure 2.8 presents the double equilibrium state, where the point of intersection

of the IS-LM-ZBO curves (i* ¼ 16.8%, Y* ¼ 9,398.1) corresponds to a simulta-

neous equilibrium in the wealth, money, and labor markets with full employment

and zero balance of payments in the year 2007. All combinations of the values of

the national income and interest rate besides this point represent various kinds of

nonequilibrium states. According to the plotted curves, Kazakhstan has cyclical

unemployment [41, p. 206] and a deficit in the balance of payments, which is

confirmed by the official statistics. In Fig. 2.8, such a situation is represented by the

point A (Y2007 ¼ 11,371.1; i2007 ¼ 13.6%).

Figure 2.9 presents the double equilibrium state, and the intersection point of the

IS-LM-ZBO curves corresponds to the simultaneous equilibrium in the wealth,

money, and labor markets with full employment and zero balance of payments in

the year 2008. All combinations of the values of the national income and interest

rate besides this point represent various kinds of nonequilibrium states. According

to the plotted curves, Kazakhstan also has cyclical unemployment and a deficit in

the balance of payments, which is confirmed by the official statistics. In Fig. 2.8,

such a situation is represented by the point B (Y2008 ¼ 13734, i2008 ¼ 15.3%).


However, it can be noted that in accordance with the official statistics, Kazakhstan

has an excessive balance of payments.

Taking into account the obtained equilibrium values, the equilibrium values of

the economic indexes C, I, and others calculated by the econometric models

constructed above, in Table 2.14 we present the results of comparison of the

equilibrium indexes with actual values of these indexes in 2007. Table 2.15

shows similar results for 2008.

A

-400,0

-300,0

-200,0

-100,0

0,0

100,0

200,0

300,0

400,0

500,0

0 5000 10000 15000 20000 25000 30000 35000 40000 45000

IS LM ZBO A Y* i*

Fig. 2.8 Double balance in the year 2007

-300,0

-200,0

-100,0

0,0

100,0

200,0

300,0

400,0

0 5000 10000 15000 20000 25000 30000 35000 40000 45000

IS LM ZBO Y* i*

B

B

Fig. 2.9 Double equilibrium in the year 2008


2.4.2 Influence of Economic Instruments on EquilibriumSolutions and State of the Balance of Payments

Below, let us 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 M2007 by DM ¼ 0.01 M2007 while keeping the values G2007

and iZ2007, PZ2007, ee2007 unchanged, define the values (MDY*)/(Y*DM),

(MDP*)/(P*DM), (MDe*)/(e*DM), and (MDi*)/(i*DM) that show the percent-

age by which the equilibrium values of the indexes Y*, P*, e*, i* change with

variation of M2007 by 1%.

2. Changing the value G2007 by DG ¼ 0.01 G2007 while keeping the values M2007

and iZ2007, PZ2007, e

e2007 unchanged, define the values (GDY*)/(Y*DG), (GDP*)/

(P*DG), (GDe*)/(e*DG), and (GDi*)/(i*DG) that show the percentage by which

Table 2.14 Equilibrium and actual values of indexes in 2007

Indexes

2007

Equilibrium value

of Y*Actual value

of Yactual

Deviation Yactual �Y*

Absolute %

Level of prices P 1.1699 1.7893 0.6194 34.6

Currency exchange rate e 109.0 122.6 13.6 11.1

Interest rate of SLB i 16.8 13.6 �3.2 �23.5

National income Y 9,398.1 11,371.1 1973 17.4

Consumption C 4,732.5 5,641.2 908.7 16.1

Import Qim 3,395.5 5,481.8 2,086.3 38.1

Investment I 2,495.2 3,392.1 896.9 26.4

Export Qex 2,891.1 6,360.5 3,469.4 54.5

Table 2.15 Equilibrium and actual values of indexes in 2008

Indexes

2008

Equilibrium value

of Y*Equilibrium value

of Yactual

Deviation Yactual �Y*

Absolute %

Level of prices P 1.1924 1.96 0.76 38.8

Currency exchange rate e 116.3 120.3 4 3.3

Interest rate of SLB i 26.1 15.3 �10.8 �70.0

National income Y 11,383.0 13,734.3 2,351.0 17.1

Consumption C 5,641.9 6,652.0 1,010.1 15.1

Import Qim 4,161.1 4,558.0 396.9 8.7

Investment I 3,026.2 3,836.0 809.8 21.0

Export Qex 3,026.1 8,563.4 5,618.4 65.6


the equilibrium values of the indexes Y*, P*, e*, i* change with variation of

G2007 by 1%.

3. Changing the value M2007 by DM ¼ 0.01 M2007 and the value G2007 by DG¼ 0.01 G2007 while keeping the values iZ2007, P

Z2007, e

e2007 unchanged, define

the values 100DY*/Y*, 100DP*/P*, 100De*/e*, and 100Di*/i* that show the

percentage by whichthe equilibrium values of the indexes Y*, P*, e*, i* change

with simultaneous variation of M2007 and G2007 by 1%.

The results of computations carried out by the above algorithm are given in

Tables 2.16–2.18.

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

payments individually. From Tables 2.16 and 2.17 it follows that increasingG2007 by

DG while keeping the value M2007 results in growth of the national income and an

increase in the interest rate, whereas increasingM2007 by DMwhile keeping the value

G2007 also results in growth of the common economic equilibrium of the GNI, butalso in a decrease in the interest rate. Moreover, from the tables it follows that the

growth in public expenses shows 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 2007, 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 M2007 + 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 2007 presented in

Table 2.16 coincide with the theoretical assumptions, except the price level index,

which in this case decreases.

Table 2.16 Influence of the money supply instrument on the equilibrium state of the national

economy in 2007 for DM ¼ 0.01 M2007 (%)

(MDY*)/(Y*DM) (MDP*)/(P*DM) (MDe*)/(e*DM) (MDi*)/(i*DM)

0.1829 �0.0709 0.2130 �0.5216

Table 2.17 Influence of the public expenses instrument on the equilibrium state of the national

economy in 2007 for DG ¼ 0.01 G2007 (%)

(GDY*)/(G*DM) (GDP*)/(P*DG) (GDe*)/(e*DG) (GDi*)/(i*DG)

0.2031% 0.0174 0.0672 0.7658

Table 2.18 Influence of money supply and public expenses instruments on the equilibrium state

of the national economy in 2007 for DM ¼ 0.01 M2007 and DG ¼ 0.01 G2007 (%)

100DY*/Y* 100DP*/P* 100De*/e* 100Di*/i*

0.3859 �0.0534 0.2799 0.2439


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 G2007 + 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 2007 presented in Table 2.16 completely coincide

with these theoretical assumptions.

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 M2007 + DM and G ¼ G2007 + DG.Figures 2.10 and 2.11 present the plots of the IS, LM, and ZBO curves from the

derived econometric models for the actual statistical information for 2007 and

2008.

DC

E0

-100,0

-50,0

0,0

50,0

100,0

150,0

200,0

0 5000 10000 15000 20000

IS LM ZBO DC E0

Fig. 2.10 Plots IS-LM-ZBO by actual values of P, e for 2007

DC

E0

-200,000-150,000-100,000-50,000

0,000

50,000100,000150,000200,000250,000300,000350,000

0 5000 10000 15000 20000 25000 30000

IS LM ZBO DC E0

Fig. 2.11 Plots IS-LM-ZBO by actual values of P, e for 2008


As stated above (Figs. 2.8 and 2.9), the country has cyclical unemployment and a

deficit in the balance of payments from the constructed models. In Figs. 2.10 and

2.11, such a situation is represented by point E0. According to macroeconomic

theory, the balance of payments deficit can be eliminated by applying a restrictive

monetary policy, namely by shifting the LM curve to the left up to its intersection

with the IS curve at the point C, or the counteractive fiscal policy by means of the IScurve to the left up to its intersection with the LM curve at the point D.

2.4.3 Parametric Control of an Open Economy State Basedon a Small Country Model

Let us estimate the optimal values of the 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 (2.44)

and

Qimp ¼ bYS þ cePZ=P ! min : (2.45)

Such an estimate can be obtained by solving the following problems of mathe-

matical programming:

Problem

1. On the basis of mathematical model (2.41), find the values (M, G) maximizing

criterion (2.44) under the constraints

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>>>>>>>>><>>>>>>>>>:

(2.46)

Here M* and G* are the actual values of the money supply and public

expenses in the year 2008.

2. On the basis of mathematical model (2.41), find the values (M, G) minimizing

criterion (2.44) under constraints (2.46).


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 5,877.96,

G ¼ 4,246, providing the attainment of the maximum value Qex ¼ 3,122.74.

The value of this criterion without control is 3,023.01.

For Problem 2, the optimal values of the parameters are M ¼ 4,809.234,

G ¼ 3,474, providing the attainment of the minimum value Qimp ¼ 4,010.64.

The value of this criterion without control is 4,183.73.

On the basis of Problems 1 and 2, we carried out the analysis of the

dependencies of the optimal values of the criteria Qex and Qimp on the one pair

and one set of three of the parameters from the set of the external parameters {ee,iZ, PZ} given within the respective regions. The plots of the dependencies of the

optimal values of criteria (2.44) and (2.45) for the single cases including that on

the pair of the parameters (PZ, ee) and (iZ, ee) are shown in Figs. 2.12–2.14.

From the plots in Figs. 2.12–2.14, one can see the general growth of Qimp and

Qex with increasing values of combinations PZ, ee and iZ, ee, in which the spikes

of values Qimp and Qex are observed for the pair PZ, ee.

2.5 Modeling of Inflationary Processes by Means of RegressionAnalysis: Rational and Adaptive Expectations

The goal of this section is to construct models of present-day inflation in the

Republic of Kazakhstan, reasoning from the concept of rational expectations (factor

regression models) and the concept of adaptive expectations (autoregression

models).

Qimp

PZ ee

4400

4300

4200

4100

4000

1.321.28

1.231.18

1.131.08 108.27

113.08117.89

127.7127.51

132.33

Fig. 2.12 Plot of the dependence of optimal values of criterion Qimp on pair PZ, ee

2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 125

Fig. 2.13 Plot of the dependence of optimal values of criterion Qex on pair PZ, ee

Fig. 2.14 Plot of the dependence of optimal values of criterion Qex on pair iZ, ee


2.5.1 Preparation of the Data for Factor RegressionModels of Inflation

First of all, let us distinguish a number of factors obviously wielding strong

influence on the rate of inflation in the Republic of Kazakhstan. These factors are

quite standard, and the level of effect for each of them on inflation processes is

subject to further analysis.

The initial data for the analysis are as follows (see Table 2.19):

The data expressed in value units are necessarily deflated to be put into compa-

rable prices. Then, to compare values (and put them in the same order), they are

converted from absolute values to relative ones (fractions of a unit). For data

expressed in percentages, the fraction of the yearly sum is calculated. As a result,

we obtain processed information suitable for carrying out regression analysis

(Table 2.20).

The matrix of the partial correlations of the derived dynamic series is presented

in Table 2.21.

In the regression formulas below, the following notation is used:

Infl Is the inflation rate

Nx Is the net export

Rd Is the wear and tear of capital assets

Rlr Is the ratio of the withdrawal coefficient to the renewal coefficient

Cipc Is the change in prices of consumer goods and services

Innov Is the cost of research and development (R&D) and innovation

Mh Is the income of the households used for consumption, mean yearly value per capita

Mm Is the monetary aggregate

Fv Is the index of the physical volume of industrial production

Invst Is the investment in capital assets

Cr Is the currency exchange rate (tenge per US dollar)

2.5.2 Construction of One-Factor Regression Models of Inflation

Now let us construct one-factor regression models of inflation, each time with one

of the distinguished factors.

2.5.2.1 Dependence of the Inflation Rate on the Net Export Volume

d Infl ¼ 0:00056� d Nxþ 0:1156


Table

2.19

Initialdataforregressionanalysis

Initialstatistics

2000

2001

2002

2003

2004

2005

2006

2007

2008

DeflatorofGDPof

USA(%

)

4.10

1.10

1.80

2.50

3.60

3.10

2.70

2.10

4.08

Monetaryaggregate

M3(m

illionsof

tenge)

397,015.00

576,023.00

764,954.00

971,213.00

1,650,115.00

2,065,348.00

3,677,561.00

4,629,829.00

6,266,395.00

Exportofwealthand

services

(in%

of

GDP)

56.60

45.90

46.99

48.42

52.23

53.54

51.14

49.43

60.71

GDP(incurrent

prices,USdollars)

18,291,

990,619

22,152,

689,130

24,636,

598,581

30,833,

692,831

43,151,

647,003

57,123,

671,734

81,003,

864,916

104,853,

480,212

132,228,

697,116

Importofwealthand

services

(in%

of

GDP)

49.10

46.95

47.04

43.05

43.49

44.73

40.43

42.60

40.66

Inflation,deflatorof

GDP(yearly,in%)

17.43

10.16

5.80

11.74

16.13

17.87

21.55

15.53

19.95

Exchangerate

(tenge/

USdollar)

142.13

147.93

153.28

168.79

131.40

132.88

126.09

122.55

120.30

Changeofpricesof

consumer

goods

andservices

(%,

increm

entto

the

precedingyear)

5.40

5.70

6.00

6.40

6.90

7.60

8.60

10.80

17.00

CostsofR&Dand

innovations($)

188,740,000

147,880,000

188,240,000

269,970,000

267,710,000

222,420,000

171,760,000

202,470,000

347,610,001

Coefficientofwearof

capital

asset,%

29.70

33.10

30.10

32.20

35.20

37.40

40.60

37.80


Index

ofphysical

volumeof

industrial

production,in%

w.r.t.preceding

year

115.50

113.80

110.50

109.10

110.40

104.80

107.20

105.00

102.10

Incomeofhouseholds

usedfor

consumption,

meanmonthly

valueper

capita,

tenge

4,731.00

5,398.00

6,478.00

7,569.00

8,387.00

9,751.00

13,723.00

16,935.00

20,037.08

Investm

entin

capital

asset,in%

w.r.t.

precedingyear

149.00

145.00

111.00

117.00

123.00

134.00

111.00

114.00

105.00

Ratio

ofwithdrawal

coefficientto

renew

al

coefficient

14.49

12.88

8.73

6.34

8.33

9.74

10.32

12.00

13.23


Tab

le2.20

Dataprepared

forcarryingoutregressionanalysis

Relativeincrem

ents

2001–2000

(%)

2002–2001

(%)

2003–2002

(%)

2004–2003

(%)

2005–2004

(%)

2006–2005

(%)

2007–2006

(%)

2008–2007

(%)

Net

export

�116.90

�94.80

�13,110.28

119.74

29.57

67.72

�19.10

255.42

Monetaryaggregate

37.88

25.90

12.48

110.66

20.05

82.72

26.87

32.47

CostsofR&Dandinnovations

�22.50

25.04

39.92

�4.28

�19.42

�24.81

15.46

64.95

Incomeofhouseholdsusedforconsumption,

meanyearlyvalueper

capita

8.43

13.77

3.52

37.39

11.51

44.41

24.36

15.81

Changeofpricesofconsumer

goodsand

services

7.26

7.66

8.06

8.60

9.27

10.22

11.56

14.52

Wearandtear

ofcapital

assets

10.76

11.99

10.90

11.66

12.75

13.55

14.70

13.69

Index

ofphysicalvolumeofindustrial

production

11.80

11.63

11.29

11.15

11.28

10.71

10.96

10.73

Investm

entto

capital

assets

13.44

13.07

10.01

10.55

11.09

12.08

10.01

10.28

Ratio

ofwithdrawal

coefficientto

renew

al

coefficient

�11.11

�32.22

�27.38

31.39

16.93

5.95

16.28

10.30

Currency

exchangerate

4.08

3.62

10.12

�22.15

1.13

�5.11

�2.81

�1.84

Inflationrate

12.80

7.46

4.26

8.62

11.85

13.13

15.83

11.40


Tab

le2.21

Partial

correlationmatrix

Net

export

Monetary

aggregate

Costsof

R&D

and

innovation

Incomeof

households

usedfor

consumption,

meanyearly

valueper

capita

Changeof

pricesof

consumer

goodsand

services

Wear

andtear

of

capital

assets

Index

of

physical

volumeof

industrial

production

Investm

ent

incapital

assets

Ratio

of

withdrawal

coefficient

torenew

al

coefficient

Currency

exchange

rate

Inflation

rate

Net

export

1

Monetary

aggregate

0.3731

1

CostsofR&D

and

innovation

�0.3660

�0.4233

1

Incomeof

households

usedfor

consumption,

meanyearly

valueper

capita

0.4675

0.8598

�0.3838

1

Changeofprices

ofconsumer

goodsand

services

0.2827

�0.0466

0.5125

0.2165

1

Wearandtear

of

capital

assets

0.4661

�0.0074

0.1445

0.4433

0.7967

1

Index

ofphysical

volumeof

industrial

production

�0.1210

�0.3065

�0.2458

�0.5998

�0.8160

�0.7611

1

(continued)


Tab

le2.21

(continued)

Net

export

Monetary

aggregate

Costsof

R&D

and

innovation

Incomeof

households

usedfor

consumption,

meanyearly

valueper

capita

Changeof

pricesof

consumer

goodsand

services

Wear

andtear

of

capital

assets

Index

of

physical

volumeof

industrial

production

Investm

ent

incapital

assets

Ratio

of

withdrawal

coefficient

torenew

al

coefficient

Currency

exchange

rate

Inflation

rate

Investm

entin

capital

assets

0.3665

0.0444

�0.4898

�0.0574

�0.5439

�0.3836

0.6220

1

Ratio

of

withdrawal

coefficientto

renew

al

coefficient

0.5255

0.5575

�0.2768

0.5955

0.4682

0.5003

�0.5607

�0.4570

1

Currency

exchangerate

�0.5067

�0.8876

0.2556

�0.7903

�0.1694

�0.2115

0.3844

0.2184

�0.7887

1

Inflationrate

0.7026

0.0909

�0.3935

0.3636

0.4485

0.6869

�0.3252

0.0825

0.5448

�0.2009

1


Regression statistics

Multiple R 0.702646212

Squared R 0.493711699

Normalized squared R 0.409330315

Standard error 0.028263362

Observations 8

Variance analysis

df SS MS F Significance of F

Regression 1 0.004673846 0.004673846 5.85095525 0.051942589

Residual 6 0.004792906 0.000798818

Total 7 0.009466752

Coefficients Standard error t-statistics P-Level

Y-intersection 0.115623264 0.010654327 10.85223578 3.6266E-05

Net export 0.000555822 0.000229785 2.418874789 0.05194259

The dependence of the inflation rate on a single factor with the rather high

determination coefficient (almost 50%) and low flexibility with respect to the

selected factor (0.00056) suggests that the inflation rate dynamics may weakly

depend on some dynamic factors and to a greater extent is determined by the

dynamics of its own preceding states. To test this hypothesis, one can apply

autoregression methods (see Sect. 2.5.4 below).

2.5.2.2 Dependence of the Inflation Rate on Monetary Aggregate Volume

d Infl ¼ 0:0097� d Mmþ 0:1024



Squared R 0.008262131

Normalized squared R �0.157027514


Observations 8

Variance analysis


Regression 1 7.82155E-05 7.82155E-05 0.04998577 0.830505924

Residual 6 0.009388536 0.001564756

Total 7 0.009466752


Y-intersection 0.102447184 0.023546608 4.350825545 0.00481825

Monetary aggregate 0.009707317 0.043418618 0.223574986 0.83050592


It can easily be seen that this model is not itself significant in any statistical

criteria. This unambiguously proves that there is no significant connection between

the monetary aggregate volume and the inflation rate in the Republic of

Kazakhstan. The absence of this connection is not a characteristic feature of

Kazakhstan; it is confirmed by theoretical reasoning [30, 31] and typical for a

large number of countries, in particular, for countries with transitional economies

during the last decade of the twentieth century [32].

2.5.2.3 Dependence of the Inflation Rate on the Costs of R&D and Innovation

d Infl ¼ �0:0442� d Innovþ 0:1108



Squared R 0.154846556



Observations 8

Variance analysis


Regression 1 0.001465894 0.001465894 1.09930255 0.334805228

Residual 6 0.008000858 0.001333476

Total 7 0.009466752


Y-intersection 0.110790788 0.013492163 8.211491911 0.00017599

Costs of R&D and innovation �0.044197082 0.04215363 �1.048476298 0.33480523

2.5.2.4 Dependence of the Inflation Rate on the Volume of Incomesof Households Used for Consumption per Capita

d Infl ¼ 0:0929� d Mhþ 0:0882



Squared R 0.132212044



Observations 8

(continued)


Variance analysis


Regression 1 0.001251619 0.001251619 0.91413145 0.375940254

Residual 6 0.008215133 0.001369189

Total 7 0.009466752

Coefficients Standard

error

t-statistics P-Level

Y-intersection 0.088202262 0.023339782 3.779052507 0.00919056

Yearly incomes of households

used for consumption per capita

0.092864177 0.097127875 0.956102218 0.37594025

2.5.2.5 Dependence of the Inflation Rate on Changes in the Pricesof Consumer Goods and Services

d Infl ¼ 0:6808� d Cipcþ 0:041



Squared R 0.201112654



Observations 8

Variance analysis


Regression 1 0.001903884 0.001903884 1.51044566 0.265081063

Residual 6 0.007562868 0.001260478

Total 7 0.009466752


error


Y-intersection 0.041023376 0.054879486 0.747517499 0.48300787

Changes of prices of consumer goods

and services

0.680840916 0.553978736 1.229001895 0.26508106

This model shows that the yearly inflation rate in this macro-system can be

explained by the dynamics of consumer prices by approximately 20%, although

the high values of p-level do not allow us to judge reliably the specific values of the

coefficients derived by the linear regression method.

2.5.2.6 Dependence of the Inflation Rate on Wear and Tear of Capital Assets

d Infl ¼ 1:7868 � d Rd � 0:1167




Squared R 0.471867645



Observations 8

Variance analysis


Regression 1 0.004467054 0.004467054 5.36078854 0.059829763

Residual 6 0.004999698 0.000833283

Total 7 0.009466752


Y-intersection �0.116679487 0.097008911 �1.202770814 0.27436974

Wear and tear of capital assets 1.786895333 0.771764462 2.315337673 0.05982976

This model confirms the fact that the technological backwardness and high wear

and tear of capital assets are significant inflation factors that can strongly contribute

to the development of inflationary processes [28, 29].

2.5.2.7 Dependence of the Inflation Rate on Index of PhysicalVolume of Industrial Production

d Infl ¼ �3:0344� d Fvþ 0:4464



Squared R 0.105753638



Observations 8

Variance analysis


Regression 1 0.001001143 0.001001143 0.70956042 0.431878752

Residual 6 0.008465608 0.001410935

Total 7 0.009466752


error


Y-intersection 0.446409333 0.403525118 1.106273969 0.31097982

Index of physical volume of

industrial production of the

Republic of Kazakhstan

�3.034474979 3.602374574 �0.842354096 0.43187875


2.5.2.8 Dependence of the Inflation Rate on the Volumeof Investment in Capital Asset

d Infl ¼ 0:22� d Invstþ 0:0817



Squared R 0.006801238



Observations 8

Variance analysis


Regression 1 6.43856E-05 6.43856E-05 0.04108687 0.846069149

Residual 6 0.009402366 0.001567061

Total 7 0.009466752


Y-intersection 0.081785455 0.123622153 0.661576044 0.53281154

Investment in capital assets 0.220005931 1.085382584 0.202698969 0.84606915

This model expresses a fact that is encouraging for the economy of Kazakhstan.

Investments in the main capital assets pose an inflationary danger of less than 1%. In

other macro-systems, this may not be so, the investment in capital assets in any case

stimulates aggregate demand, which unavoidably leads to growth of the common

level of prices.

For comparison, let us recall that the net export, which is also a part of aggregate

demand, explains the dynamics of the inflation rate in Kazakhstan by almost 50%

(model 2.5.2.1 above).

Under these conditions, it is short-sighted for a number of reasons to state the

problem as one of maximizing exports. The main reason is that this maximization

can make the national economy less controllable and more subject to the effect of

external shocks.

2.5.2.9 Dependence of the Inflation Rate on the Ratio of WithdrawalCoefficient to Renewal Coefficient

d Infl ¼ 0:0886� d Rlr þ 0:1055



Squared R 0.296841803



Observations 8

(continued)


Variance analysis


Regression 1 0.002810128 0.002810128 2.53293047 0.162598324

Residual 6 0.006656624 0.001109437

Total 7 0.009466752


Y-intersection 0.105558889 0.011797374 8.947659617 0.00010879

Ratio of withdrawal

coefficient to renewal

coefficient

0.088640382 0.055695484 1.59151829 0.16259832

2.5.2.10 Dependence of the Inflation Rate on the Currency Exchange Rate

d Infl ¼ �0:0772� d Cr þ 0:1054



Squared R 0.040347903



Observations 8

Variance analysis


Regression 1 0.000381964 0.000381964 0.25226581 0.633380857

Residual 6 0.009084788 0.001514131

Total 7 0.009466752


Y-intersection 0.10543109 0.013981186 7.540926073 0.00028201

Currency exchange rate �0.07722679 0.153758376 �0.502260702 0.63338086

This last model is not statistically significant. In addition to the low value of R2

(a little more 4%) and high p-level, this model also does not satisfy the Fisher

criterion, i.e., applying econometric methods gives grounds to conclude that the

form of dependence of a variable on the parameter explaining its dynamics has been

chosen incorrectly.

Model 2.5.2.1 is of the highest quality among the models constructed here;

model 2.5.2.6 is the next in quality. This reflects both the explanatory ability of

these models expressed by the determination coefficient and the quality (degree of

reliability) of the coefficients derived as a result of computations.


The next step consists in organizing the derived models in order of decreasing

R2. This gradation miraculously (but not randomly) coincides with increasing p-level of the coefficient of the explanatory parameter of the regression (Table 2.22).

Combining the factors that best explain the dynamics of the inflation rate and

avoiding multicollinearity, let us construct the multifactor regression inflation

models.

2.5.3 Construction of Multifactor Regression Models of Inflation

Combining three of the most significant factors, we derive model 2.5.3.1.

2.5.3.1 Dependence of the Inflation Rate on Net Export Volume, Wearand Tear of Capital Assets, and Ratio of WithdrawalCoefficient to Renewal Coefficient

d Infl ¼ 0:00038� d Nxþ 0:9021� d Rd þ 0:0192� d Rlr



Squared R 0.967909982



Observations 8

Table 2.22 Results of constructing one-factor linear regression models

Factor p-Level Squared R

2.1. Net export 0.052 0.494

2.6. Wear and tear of capital assets 0.060 0.472

2.9. Ratio of withdrawal coefficient to renewal coefficient 0.163 0.297

2.5. Change in prices of consumer goods and services 0.265 0.201

2.3. Costs of R&D and innovation 0.335 0.155

2.4. Income of households used for consumption,

yearly mean value per capita

0.376 0.132

2.7. Index of physical volume of industrial production 0.432 0.106

2.10. Currency exchange rate 0.633 0.040

2.2. Monetary aggregate volume 0.831 0.008

2.8. Investment in capital assets 0.846 0.007

(continued)


Variance analysis


Regression 3 0.097290321 0.032430107 50.2705633 0.001240905

Residual 5 0.003225556 0.000645111

Total 8 0.100515878


Net export 0.000380604 0.000239745 1.587538531 0.173254

Wear and tear of capital assets 0.902141768 0.078494697 11.49302828 8.7405E-05

Ratio of withdrawal

coefficient to renewal coefficient

0.019216022 0.050395611 0.381303485 0.71864334

With a very good determination coefficient (almost 97%), the p-level of thelatter regression parameter evidently exceeds its admissible value. Obviously, the

reason for this is because the ratio of the withdrawal coefficient to the renewal

coefficient is closely connected with the wear and tear of capital assets, so one of

these parameters should be considered unnecessary. Thus, the regression parameter

with the coefficient having the highest p-level in model 2.5.3.1 should be removed

from this model.

2.5.3.2 Dependence of the Inflation Rate on Net Export Volumeand Wear and Tear of Capital Assets

d Infl ¼ 0:00042� d Nxþ 0:9111� d Rd



Squared R 0.966976853



Observations 8

Variance analysis


Regression 2 0.097196527 0.048598263 87.8453696 0.000127375

Residual 6 0.003319351 0.000553225

Total 8 0.100515878


Net export volume 0.000429264 0.000187949 2.283935649 0.06246151

Wear and tear of capital assets 0.911137115 0.069329295 13.14216614 1.1977E-05


The correctness of the undertaken step is confirmed by the fact that the coeffi-

cient R2 of the model does not change significantly (the difference is one-tenth of a

percent). Nevertheless, the rest of the model’s statistical parameters demonstrate

the complete adequacy of the model.

Attempts to enter some additional factors into this model do not produce

good results. Indeed, at each attempt the p-level of the entered factor exceeds 0.5,

which of course is too high assuming normally distributed regression coefficients.

Construction of the models using another explaining factor leads to worse results

in comparison to model 2.5.3.2. Usually such models have a lower determination

coefficient and higher p-level of the coefficients of the explanatory parameters.

Let us consider model 2.5.3.3 as one of the typical (and quite successful)

attempts of this kind.

2.5.3.3 Dependence of the Inflation Rate on Costs of R&D and Innovation,Index of Physical Volume of Industrial Production, and MeanYearly Income of Households per Capita Used for Consumption

d Infl ¼ �0:0269� d Innovþ 0:8193� d Fvþ 0:0861� d Mh



Squared R 0.920975272



Observations 8

Variance analysis


Regression 3 0.092572638 0.030857546 19.4237782 0.007568807

Residual 5 0.00794324 0.001588648

Total 8 0.100515878


error


Costs of R&D and innovations �0.026926846 0.049077561 �0.548659005 0.60683873

Index of physical volume of

industrial production

0.819367611 0.242821599 3.374360495 0.0197954

Mean yearly income of households

per capita used for consumption

0.086165843 0.108645937 0.793088499 0.46368084

So, model 2.5.3.2 is most adequate among the constructed models. It allows us to

explain almost 97% of the dynamics of the inflation rate in the economy of the

Republic of Kazakhstan during the last ten years by means of two explaining factors.


2.5.4 Construction of Autoregression Models of Inflation Rate

2.5.4.1 Dependence of the Inflation Rate on Two Preceding Years

InflðtÞ ¼ 0:875� Infl t� 1ð Þ � 0:701� Infl t� 2ð Þ þ 0:087



Squared R 0.656784795



Observations 6

Variance analysis


Regression 2 0.005230478 0.002615239 2.870435745 0.201071158

Residual 3 0.002733284 0.000911095

Total 5 0.007963762


Y-intersection 0.087182255 0.042416364 2.055392011 0.132078479

Inflation (t-1) 0.875585255 0.372107188 2.353045795 0.100028831

Inflation (t-2) 0.701332319 0.444901675 �1.576375992 0.213029975

The derived model shows that 66% of variations of the yearly inflation rate in

Kazakhstan can be explained by the dynamics of the same rate for two preceding

years.

2.5.4.2 Dependence of Inflation Rate on Three Preceding Years

InflðtÞ ¼ 0:078� Infl t� 1ð Þ þ 0:349� Infl t� 2ð Þ � 0:524� Infl t� 3ð Þ þ 0:128



Squared R 0.660804348



Observations 5

(continued)


Variance analysis


Regression 3 0.001822794 0.000607598 0.649383666 0.697175376

Residual 1 0.000935653 0.000935653

Total 4 0.002758447


Y- intersection 0.128852949 0.066712285 1.931472569 0.304137731

Inflation (t-1) 0.078035028 0.707204219 0.11034299 0.930036504

Inflation (t-2) 0.349251506 0.907146362 0.385000173 0.766036509

Inflation (t-3) �0.524946165 0.628028501 �0.835863602 0.556766413

The constructed model is insignificant by all criteria. Such would be expected,

because it has four parameters, and no correct model can be constructed using five

observations.

2.5.4.3 Dependence of the Inflation Rate on the Values of Two and ThreePreceding Years

InflðtÞ ¼ 0:436� Infl t� 2ð Þ � 0:571� Infl t� 3ð Þ þ 0:133



Squared R 0.656674446



Observations 5

Variance analysis


Regression 2 0.001811402 0.000905701 1.912687353 0.343325554

Residual 2 0.000947045 0.000473523

Total 4 0.002758447


Y- intersection 0.133571731 0.036425493 3.666984809 0.066981695

Inflation (t-2) 0.436105273 0.320799038 1.359434479 0.306993601

Inflation (t-3) �0.571758565 0.329429306 �1.735603225 0.224769026


We derive a model of approximately the same quality as model 2.5.4.1. This

means that this model can be used for purposes of analysis and (in many cases)

prediction of the inflationary dynamics of this macro-system almost equally well.

Nevertheless, let us point out a difference between these two models. In the latter

(model 2.5.4.3), the free term characterizing the “autonomous” inflation rate inde-

pendent of the preceding dynamics of that parameter is the most reliable. In model

(2.5.4.1), the values of all three coefficients are reliable almost equally.

2.6 Conclusion

Finally, let us recall that the methodology of the construction of the factor regres-

sion models derives its strength and foundation from the concept of rational

expectations. In other words, to estimate approximately the inflation rate in

Kazakhstan in the future (for instance, the next year) by means of models of this

kind, one should ask the following question: What are the values of the parameters

on which the inflation rate function strongly depends? In this case, these are the net

export volume and wear and tear of capital assets (if one considers model 2.5.3.2 as

the basis, which is the best model in view of the econometric indexes).

The methodology of the construction of linear autoregression models originates

from the correctness of the concept of adaptive estimation. The inflation rate of the

next period strongly depends on the past history, and its subsequent dynamics can

be predicted on exactly this basis.

We have proved that methods based on both the concept of rational expectation

and the concept of adaptive expectation are suitable for estimation and analysis of

inflationary processes in the Republic of Kazakhstan.


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 [41] proposed for describ-

ing the evolution of business activity as oscillatory processes. In [19], one can find a

number of mathematical models of market cycles. In this context, the main factors

causing oscillations of market tendencies are considered. Nevertheless, issues of the

structural stability of such models parametric control of development of the eco-

nomic 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 [16] and the Goodwin

model [5, 41].

3.1 Mathematical Model of the Kondratiev Cycle

3.1.1 Model Description

This model [16] combines descriptions of nonequilibrium economic growth and

nonuniform scientific and technological advances. The model is described by the

following system of equations, including two differential equations and one

equation:


145

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 capitalproductivity ratio; y0 is the capital productivity ratio corresponding to the equilib-

rium trajectory; n is the rate of saving; n0 is the rate of saving corresponding to the

equilibrium trajectory; m is the coefficient of withdrawal of funds; l0 is the job

growth rate corresponding to the equilibrium trajectory; A and a are some model

constants.

Preliminary estimation of the model parameters is carried out based on the

statistical information of the Republic of Kazakhstan for the years 2001–2005

[24]. The deviations of the observed statistical data from the calculated data do

not exceed 1.9% within the period under consideration.

As a result of solving the problem of preliminary estimation of the

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, x(0) ¼ 1.91114.

A preliminary prediction for 2006 and 2007 is characterized by the errors 6.1%

and 12.1%, respectively, for the capital productivity ratio, and 2.3% and 11%,

respectively, for the rate of saving.

The respective cyclic phase trajectory of the Kondratiev cycle model is

presented in Fig. 3.1.

The period of the cyclic trajectory corresponding to the statistical information of

the Republic of Kazakhstan for the given years is estimated to be 232 months.

Fig. 3.1 Cyclic phase trajectory of the Kondratiev cycle model

146 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. 4 of Chap. 1 on parametric control theory in the

chosen compact set of the model state space.

Figure 3.2 presents the estimate of the chain-recurrent set R(f,N) obtained as by

the application of the chain-recurrent set estimation algorithm for the region N ¼½1:7; 2:3� of the phase plane Oxy of system (3.1). Since the set R(f,N) is not empty,

one can draw no conclusion 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� mþl0n0y0

; y0) [16], thensystem (3.1) is not weakly structurally stable in N.

Fig. 3.2 Chain-recurrent set for the Kondratiev cycle model

3.1 Mathematical Model of the Kondratiev Cycle 147

3.1.3 Parametric Control of the Evolution of the EconomicSystem Based on the Kondratiev Cycle Model

Choosing the optimal laws of parametric control is carried out in the environment of

the following four relations:

1Þ n0ðtÞ ¼ n0� þ k1

yðtÞ � yð0Þyð0Þ ;

2Þ n0ðtÞ ¼ n0� � k2

yðtÞ � yð0Þyð0Þ ;

3Þ n0ðtÞ ¼ n0� þ k3

xðtÞ � xð0Þxð0Þ ;

4Þ n0ðtÞ ¼ n0� � k4

xðtÞ � xð0Þxð0Þ : (3.2)

Here ki is the scenario coefficient; n0* is the value of the exogenous parameter n0

obtained as a result of the preliminary estimation of the 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 attainment of optimal

values of the following criteria:

1ÞK1 ¼ 1

36

X36t¼1

yðtÞ ! max;

2ÞK2 ¼ 1

36

X36t¼1

xðtÞ ! max;

3ÞK3 ¼ 1

36

P36t¼1

xðtÞxð0Þ þ

P36t¼1

yðtÞyð0Þ

0BBB@

1CCCA! max;

4ÞK4 ¼ 1

T

XTt¼1

xðtÞ � x0x0

� �2

þ yðtÞ � y0y0

� �2 !

! min (3.3)

(here T ¼232 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. 3.3–3.7.

Fig. 3.3 Capital productivity ratio without parametric control and with use of law 3, optimal in the

sense of criterion 1

Table 3.1 Values of coefficients and criteria for optimal laws

Criterion Optimal law Coefficient value Criterion value

1 3 0.2404966 0.06889

2 3 0.47668 2.230337

3 4 0.071862 2.19674

4 4 0.300519 0.007273




3.1.4 Estimating the Structural Stability of the Kondratiev CycleMathematical 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 parameter n0.



0,12

0,1

0,08

0,06

0,04

0,02

02001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021

y

y0

Yearsno control with optimal control law




Then, using a numerical algorithm for estimating 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, y),the estimation of the chain-recurrent set Rðf ;NÞ as the empty (or one-point) set is

obtained. This means that the Kondratiev cycle mathematical model with optimal

parametric control law is estimated asweakly structurally stable in the compact setN.

3.1.5 Analysis of the Dependence of the OptimalValue of Criterion K on the Parameterfor the Variational Calculus Problem Basedon the Kondratiev Cycle Mathematical Model

Let us 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 forparametric control laws (3.2) with the obtained optimal values of the adjusted

coefficients ki, where the values of the parameters (m,a) belong to the rectangle L ¼½0:00063; 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, a) consistsof the bifurcation points of the extremals of the given variational calculus problem.

2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021Years

no control with optimal control law

x

x0

2,5

2

1,5

1

0,5

0

Fig. 3.7 Efficiency of innovations without parametric control and with use of law 4, optimal in the



3.2 Goodwin Mathematical Model of Market Fluctuationsof Growing Economies

3.2.1 Model Description

The Goodwin model describing market fluctuations in a growing economy is

presented in [19, 41].

The model is described by the following system of two differential equations:

d0ðtÞ ¼ ðalðtÞ � a0Þ dðtÞ;l0ðtÞ ¼ ð�bdðtÞ þ b0Þ lðtÞ:

((3.5)

Here d is the percentage of employed in the entire population; l is the percentageof supply for consumption in the GDP; a, a0, b, b0 are constants of 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. Solving the problem of

Fig. 3.8 Plots of the dependencies of the optimal value of criterion K on exogenous parameters m, a


parametric identification, preliminary estimates of the following values of the

exogenous parameters were obtained:

a ¼ 0:1710; a0 ¼ 0:08; b ¼ 0:00211; b0 ¼ 0:001:

The calculated period of one cycle in this case is T ¼ 706.27 months.

The model relies on an assumption of invariability of the following economic

parameters:

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:

a¼ 1

bð1þgÞ>0; a0¼ s0bð1þgÞ>0; b¼ 1

kð1þgÞð1þnÞ>0; b0¼ 1�kðgþnþngÞkð1þgÞð1þnÞ :

Let us also assume that gþ nþ ng<1, in which case b0>0.

Let us consider the solutions of system (3.5) in some closed simply connected

region O with 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 the 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 inR2þ 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 us estimate the structural stability of the Goodwin model without parametric

control in closed regions O, being guided by the theorem on necessary and

sufficient conditions for structural stability [11]. First, let us prove the following

assertion.

3.2 Goodwin Mathematical Model of Market Fluctuations of Growing Economies 153

Lemma 3.1 The singular point ðd�; l�Þ of system (3.5) is the center.

Proof With (3.6) in mind, let us 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 the imaginary eigenvalues � iffiffiffiffiffiffiffiffiffia0b0

p; .

Therefore, the point ðd�; l�Þ is the structurally unstable center (nonhyperbolic

point).

Assertion 3.2 System (3.5) is structurally unstable in the closed region O(O � R2

þ) with boundary a simple closed curve containing the point ðd�; l�Þ ofthe form (3.6) for any fixed values of the parameters k; n; g; l0; b, each taken fromits domain of definition.

System (3.5) is structurally stable in the closed region O (O � R2þ) with bound-

ary a simple closed curve not containing the point ðd�; l�Þ of the form (3.6) for anyfixed values of the parameters k; n; g; l0; b, each taken from its domain ofdefinition.

Proof Let the closed region O � R2þ contain the singular point ðd�; l�Þ. A neigh-

borhood of this system of points (3.5) is locally structurally unstable. Therefore, it is

structurally unstable in the region O.Let the closed region O � R2

þ not contain the singular point ðd�; l�Þ. In this case,the regionO does not contain any cycle, since at least one singular point must be inside

any cycle. Therefore, in this case, system (3.5) is structurally stable in the region O.

3.2.3 Problem of Choosing Optimal Parametric Control Lawson the Basis of Goodwin’s Mathematical Model

It should be noted that the estimations of the parameters k, n, g, s0, b derived from

the statistical information of 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 us consider the implementation of an efficient public policy by choosing

optimal control laws with the example of the economic parameter k (capital outputratio). The goal of the economic policy is to reduce the magnitudes of fluctuations

of the indexes ðd; lÞ of the evolution of the national economic system.

Choosing optimal laws of parametric control is carried out in the environment of

the following set of relations:


1ÞU1ðtÞ ¼ c1dðtÞ � d0

d0þ k0; 2ÞU2ðtÞ ¼ �c2

dðtÞ � d0d0

þ k0;

3ÞU3ðtÞ ¼ c3lðtÞ � l0

l0þ k0; 4ÞU4ðtÞ ¼ �c4

lðtÞ � l0l0

þ k0: (3.7)

Here Ui is the ith control law of the parameter k ði ¼ 1; 4Þ; ci is the adjusted

coefficient of the ith control law, ci>0; k0 is a constant equal to the base value of theparameter k. Application of the control law Ui means substitution of the functions

on the right-hand sides of (3.7) into system (3.5) for the parameter k; t ¼ 0 is the

control 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), i.e., 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); criterion

K characterizes the mean distance from the points of this trajectory to the stationary

point (3.6).


– 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 results of the first stage using the minimum value of criterion K.

The problem is solved:

With given values of the parameters b ¼ 10=13, g ¼ 0:5, d0 ¼ 0:4, 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’s stationary point with

coordinatesl� ¼ 0:5; d� ¼ 0:5.


Numerical solution of the problem of choosing the optimal parametric control

law shows that the best result K ¼ 0.03215307 can be achieved with use of the

following law:

k ¼ 4:28lðtÞ � 0:5

0:5þ 10=19: (3.10)

Let us note that the criterion value without parametric control is K ¼ 0.0918682.

Computational experiments yield the following facts:

– A decrease in the value of criterion K in comparison with the case without

control is observed only with use of the laws U1ðtÞ and U3ðtÞ from (3.7);

– Using laws of type U1ðtÞ, one can observe that the cyclic character of the phase

trajectory of system (3.5) is preserved; see Fig. 3.9;

– With use of the laws of type U3ðtÞ, instead of a cyclic trajectory, one can observethe trajectories approaching the stable singular point of system (3.5) with

parametric control as t ! þ1 (see Fig. 3.10).

0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

0.420.35 0.4 0.45 0.55 0.6 0.65 0.70.5

δ

λ12

Fig. 3.9 Curve 1 corresponds to the market cycle without control; curve 2 corresponds to the

market cycle with control law U1ðtÞ


3.2.4 Analysis of the Structural Stability of the GoodwinMathematical Model with Parametric Control

Let us analyze the structural stability of system (3.5) using a parametric control law

of typeU3ðtÞ 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 the parameter k. First, let us findthe singular points of system (3.5) with parametric control. Substituting the expres-

sion for k into the right-hand sides of the equations of system (3.5) and setting them

equal to zero, we obtain the following system in the unknowns ðd; lÞ (along 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)

0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

0.42

λ

0.35 0.4 0.45 0.55 0.6 0.650.5δ

12

Fig. 3.10 Curve 1 corresponds to the market cycle without control; curve 2 corresponds to the

market cycle with control law U3ðtÞ


Here kðlÞ ¼ c l�l0l0

þ k0. We use only values 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 us write down the Jacobian for the left-hand

sides of system (3.12) at the point (3.13):

A¼al� �a0 ad�

�bl� cd�

c1l��l0l0

þk0

� �2

l0ð1þgÞð1þnÞ� c

c1l��l0l0

þk0

� �2

l0ð1þgÞð1þnÞ

0B@

1CAl� þð�bd� þb0Þ

0BBB@

1CCCA

¼0 ad�

�bl� cðd��1Þl�

cl� � l0

l0þ k0

� �2

l0ð1þ gÞð1þ nÞ

0B@

1CA:

The eigenvalues of matrix A are the roots of the equation

m2 þ cð1� d�Þl�

cl� � l0

l0þ k0

� �2

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 equation (3.14) is given by

D ¼ cð1� d�Þl�

c l��l0l0

þ k0

� �2l0ð1þ gÞð1þ nÞ

0B@

1CA

2

� 41� kðl�Þðgþ nþ ngÞkðl�Þð1þ gÞð1þ nÞ a0<0;

then the singular point ðd�; l�Þ of system (3.5) with parametric control (3.11)

is the focus, and this focus 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 3.3 System (3.5) with parametric control (3.11) is locally structurallystable in any sufficiently small closed region O (O � R2

þ) with boundary a simpleclosed curve containing the point ðd�; l�Þ of form (3.13) for any fixed values of theparameters k; n; g; l0; b from their respective domains of definition.

System (3.5) is structurally stable in any closed region O (O � R2þ) with

boundary a simple closed curve not containing the point ðd�; l�Þ of form (3.13)

for any fixed values of the parameters k; n; g; l0; b from their respective domainsof definition.

Proof Let the singular point ðd�; l�Þ not belong to the closed region O � R2þ. In

this case, by the same reasoning as in the proof of Assertion 3.2, 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. Sincethis point is hyperbolic (node or focus), then system (3.5, 3.11) is locally structur-

ally stable in its neighborhood.

Let us 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 us 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 setting them equal to zero, we obtain the following system in

the unknowns ðd; lÞ (along with the rest of the 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 c such that 0<kðdÞ<1; 0<l�<1; 0<d�<1:Now let us write down the Jacobian for the left-hand sides of system (3.12) at the

point (3.13):


A ¼0 ad�

cðd��1Þ

cd��d0d0

þk0

� �2

d0ð1þgÞð1þnÞ� 1

cd��d0d0

þk0

� �d0ð1þgÞð1þnÞ

0B@

1CAl� 0

0BB@

1CCA:

It is obvious that this matrix has imaginary eigenvalues. Therefore, the singular

point (3.17) is the center. Applying the methods from [11], it can be proved that all

phase trajectories of system (3.5) with parametric control (3.15) are cycles in R2þ

except point (3.13). The following assertion can be proved similarly to Assertion 3.2.

Assertion 3.4 System (3.5) with parametric control (3.15) is structurally unstablein the closed region O (O � R2

þ) with boundary a simple closed curve containingthe point ðd�; l�Þ of the form (3.17) for any fixed values of the parametersc; k; n; g; l0; b from their domains of definition.

System (3.5) with parametric control (3.15) is structurally stable in the closedregion O (O � R2

þ) with boundary a simple closed curve not containing the pointðd�; l�Þ of form (3.17) for any fixed values of the parameters c; k; n; g; l0; b fromtheir domains of definition.

Table 3.2 Values of criteria for different optimal control laws

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

Fig. 3.11 Plots of the dependencies of the optimal values of criterion K on uncontrolled parameter

n. Notation: U3, U1, U2, U4


3.2.5 Analysis of the Dependence of the Optimal ParametricControl Law on Values of the Uncontrolled Parameterof the Goodwin Mathematical Model

Let us 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 Kon the values of parameter n for each of four possible laws (3.7).

Analysis of Table 3.2 shows that for all considered values of parameter n, the controllaw U3ðtÞ is optimal, i.e., for the given interval of values of parameter n, a bifurcationpoint of the extremals of the given variational calculus problem does not exist.


Chapter 4

Parametric Control of Economic Growthof a National Economy Based on ComputableModels of General Equilibrium

As is well known [41], 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 such economic development 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 [46].

Using the basic regression equation for estimating the determinants of economic

growth,

g ¼ a0 þXl

alxl þXp

bpzp þXr

cr SLVr 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; xl 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), various econometric

models of dependencies of economic growth on various kinds 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 models, are derived [28, 60, 62]. These models are

mainly intended to provide estimates, and do not meet the aforementioned

requirements. A wide range of phenomenological models [46] starting from the

mathematical model of neoclassical theory of Solow [68] and Swan [70],

complemented by dynamic optimization models based on including the Ramsey

problem in mathematical models of endogenous economic growth that represent,


163

for example, production of innovation as a product of a particular economic sector

(e.g., the Grossman and Helpman model [61]); activity aimed at the development of

people themselves (e.g., Robert Lucas model [65]); international trade and dissem-

ination of technologies (e.g., Lucas model [64]); and others, answer questions about

economic growth sources, but do not meet the aforementioned requirements of a

mathematical model for estimating rational values of the economic instruments of

public policy in the field of economic growth.

In the context of the balance model [63, 15], 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 [24], a number of computable models of general equilibrium are proposed.

These models to a greater degree meet the aforesaid requirement for mathematical

models used to estimate rational values of the economic instruments of 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. This to a certain extent allows taking into account the requirements of

an anti-inflation policy.

4.1 National Economic Evolution Control Basedon a Computable Model of General Equilibriumwith the Knowledge Sector

4.1.1 Model Description, Parametric Identification,and Retrospective Prediction

4.1.1.1 Model Agents

The model under consideration [24, 10] describes behavior and interaction in nine

product markets and two labor markets for the following seven economic agents:

Economic agent 1 is the science and education sector (knowledge) providing

services of education and the production of knowledge. This includes educational

institutions (public and private) that provide higher education, as well as scientific

(research) organizations.

This sector renders services distributed among the following three areas:

1. Services for the innovation sector (mainly carrying out research and develop-

ment) and other sectors of the economy (mainly carrying out research and

164 4 Parametric Control of Economic Growth of a National Economy. . .

development, too), as well as services for economic agent 5 including, in

accordance with the methodology of National Economic Accounting (NEA),

the services of nonmarket science. Moreover, a portion of the services of

providing knowledge is consumed by the sector itself.

2. Services for economic agent 5 (including, in accordance with methodology of

NEA, services of free education), and services of paid education for the

innovation sector and other branches of the economy and households. Moreover,

a portion of these educational services is consumed by the sector itself.

3. The services for the outside world, carrying out work funded by scientific grants.

Economic agent 2 is the innovation sector, which is an aggregate of innovational

enterprises and organizations. This sector produces product distributed between the

following two areas:

1. Innovation products for the domestic market. Innovation products are under-

stood to be final products manufactured using various technological and other

innovations. This index corresponds to the volume of shipped innovation

products. The products manufactured by the sector are consumed by all produc-

ing sectors (including this sector itself) as the costs of research and development,

as well as the costs of the technological innovations, and by economic agent 5

(this means government financing of the innovation activity).

2. Innovation products for the outside world.

Economic agent 3 is the other branches of the economy.

It produces products distributed among the following four areas:

1. Final products for households including consumer goods for current consump-

tion (foodstuff, etc.), durable products (home technical equipment, motor

vehicles, etc.), as well as services.

2. Final products for economic agent 5 including the following:

(a) Final products for public institutions (according to NEA methodology,

expenditures by public institutions in acquiring final products), including

free services for the population rendered by enterprises and organizations in

the fields of public health, culture (this does not include educational services,

because they are rendered by economic agent 1); services satisfying the

needs of the society as a whole, e.g., general public administration, internal

security, national defense, and the housing economy.

(b) Final products for nonprofit organizations servicing households including

free services of a social character;

3. Investment products, i.e., expenditures on improvement of produced and

nonproduced tangible assets (in other words, expenditures for the creation of

capital assets). In accordance with NEA methodology, this kind of product is

determined as the sum of gross savings of capital assets and change in reserves of

material circulating assets minus the cost of acquiring new and existing capital

assets (with a deduction for withdrawal).

4.1 National Economic Evolution Control Based on a Computable. . . 165

4. Export products. Since imported products are a constituent part of the products

considered above, to avoid double counting, the exported products include only

net exports (i.e., exports minus imports).

To produce products and services, producing agents 1–3 purchase the following

production factors:

1. The labor force (based on governmental and market prices);

2. Investment products;

3. Innovation products;

4. Services for providing knowledge (e.g., R&D);

5. Educational services (paid education).

Economic agent 4 is the aggregate consumer combining all households. The agent

purchases the final products produced by other branches of the economy. Further-

more, the households use paid educational services. Also, this sector constitutes the

labor force.

Economic agent 5 is the government’s establishment of taxation rates, determin-

ing the portion of the budget used in financing producers and social transfers, and

spending its budget for purchasing final products produced by other branches of the

economy.

Economic agent 6 is the banking sector, determining interest rates for debt

deposits.

Economic agent 7 is the outside world.

The following system of notation 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 or number of iteration>].

Here < Type > can take on two values, namely, C is the exogenous parameter,

and V is the endogenous variable.

<Parameter > corresponds to the action realized by the agent. Examples of such

actions can be given by S (product supply), D (product demand), O (determining

the share of budget by the agent), and others.

For example, the notation CO_p3_1l[0] corresponds to the exogenous parameter

that represents the share of the budget of the first sector (knowledge sector) for

purchasing from the labor force at the price P3l (the governmental price of the labor

force in market 3) for year zero.

4.1.1.2 Exogenous Parameters of the Model

The 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 products for selling in various markets;

– The depreciation rates of capital assets and shares of 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 below.

4.1.1.3 Sector 1

CO_p1_1l The share of the budget for purchasing the labor force at the price P__1l.

CO_p1_1z The share of the budget for purchasing knowledge provision services at the price

P__1z.

CO_p1_1r The share of the budget for purchasing educational services at the price P__1r.

CO_p1_1n The share of the budget for purchasing innovation products at the price P__1n.

CO_p1_1i The share of the budget for purchasing investment products at the price P__1i.

CE_p1_1z The share of produced product for selling in the markets of knowledge-provision

services at the price P__1z.

CE_p2_1z The share of produced product for selling in the markets of knowledge-provision

services at the price P__2z.

CE_p1_1r The share of produced product for selling in the markets of educational services at

the price 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 labor of the production function.

Calpha__1 The coefficient of the costs of knowledge-provision services of the production

function.

Cbeta__1 The coefficient of the costs of educational services of the production function.

Cgamma__1 The coefficient of the costs of innovation products of the production function.

CA_0_1 The rate of depreciation for capital assets.

CR__1 The share of retired capital assets.

4.1.1.4 Sector 2


CO_p1_2z The share of the budget for purchasing knowledge-provision services at the price

P__1z.




CE_p1_2n The share of produced product for selling in the markets of innovation products at

the price P__1n.

CE_p2_2n The share of produced product for selling in the markets of innovation products at

the price P__2n.


(continued)





function.




CR__2 The share of retired capital assets.

4.1.1.5 Sector 3


CO_p1_3z The share of the budget for purchasing knowledge-provision services at the price

P__1z.




CE_p1_3c The share of produced product for selling in the markets of final products at the

price P__1c.

CE_p1_3g The share of produced product for selling in the markets of final products for

economic agent 5 at the price P__1g.

CE_p1_3i The share of produced product for selling in the markets of investment products at

the price P__1i.

CE_p2_3c The share of produced product for selling in the markets of exported products at the

price P__2c.





function.




CR__3 The share of the retired capital assets.

4.1.1.6 Sector 4

CO_p1_4c The share of the budget for purchasing final products at the price P__1c.


CO_b_4 The share of the budget for saving in bank deposits.

CS_p3_4l The supply of labor at the price P__3l.

CS_p1_4l The supply of labor at the price P__1l.


4.1.1.7 Sector 5

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 final products at the price

P__1g.

CO_p1_5z The share of the consolidated budget for purchasing knowledge-provision services

at the price P__1z.

CO_p1_5r The share of the consolidated budget for purchasing educational services at the price

P__1r.

CO_p1_5n The share of the consolidated budget for purchasing innovation products at the price

P__1n.

CO_s1_5 The share of the consolidated budget for backing Sector 1.

CO_s2_5 The share of the consolidated budget for backing Sector 2

CO_s3_5 The share of the consolidated budget for backing Sector 3.

CO_tr_5 The share of the consolidated budget for payment of social transfers to the

population.

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.

CB_other_5 The sum of tax proceeds (not included in those already considered), nontax income,

and other incomes of the consolidated budget.

4.1.1.8 Banking Sector

CP__bpercent The deposit interest rate for enterprises.

CP_h_bpercent The deposit interest rate for physical bodies.

4.1.1.9 General Part of the Model

CP__3l The governmental price of the labor force.

CP__2z The export price of knowledge-provision services.

CP__2n The export price of innovation products.

CP__2c The export price of final products.

CD_p2_sz The total demand for knowledge-provision services at export prices.

CD_p2_sn The total demand for innovation products at export prices.

CD_p2_sc The total demand for the final products at export prices.


4.1.1.10 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.



4.1.1.11 Endogenous Variables of the Model

The endogenous variables include the following:

– The budgets of the sectors and their various shares;

– The produced values added;

– The demands and supplies of the various products and services;

– The gains of the sectors;

– The capital assets of the sectors;

– The wages of the employees;

– The various kinds of expenditures of the consolidated budget;

– The various kinds of prices of the products, services, and the labor force.

A list of the model’s endogenous variables is given below:

4.1.1.12 Sector 1

VO_p3_1l The share of the budget for purchasing the labor force at the price P__3l.

VO_t_1 The share of the budget for paying taxes to the consolidated budget.

VO_f_1 The share of the budget for paying taxes to 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-provision services at the price P__1z.

VS_p2_1z The supply of knowledge-provision services at the price P__2z.

VS_p1_1r The supply of educational services at the price P__1r.

VD_p3_1l The demand of the labor force at the price P__3l.


VD_p1_1z The demand of knowledge-provision services at the price P__1z.

VD_p1_1r The demand of educational services at the price P__1r.

VD_p1_1n The demand of innovation products at the price P__1n.

(continued)


VD_p1_1i The demand of investment products at the price 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.

4.1.1.13 Sector 2



VO_f_2 The share of the budget for paying taxes to the off-budget funds.



VS_p1_2n The supply of innovation products at the price P__1n.

VS_p2_2n The supply of innovation products at the price P__2n.






VD_p1_2i The demand of investment products at the price P__1i.

VY_p_2 The gain in current prices.


VB_b_2 The balance of banking accounts.


4.1.1.14 Sector 3



VO_f_3 The share of the budget for paying taxes to the off-budget funds.



VS_p1_3c The supply of final products at the price P__1c.

VS_p1_3g The supply of final products at the price P__1g.

VS_p1_3i The supply of investment products at the price P__1i.

VS_p2_3c The supply of final products at the price P__2c.





(continued)


VD_p1_3n The demand of innovation products at the price P__1n.VD_p1_3i The demand of investment products at the price P__1i.

VY_p_3 The gain in current prices.


VB_b_3 The balance of bank accounts.


4.1.1.15 Sector 4

VO_tax_4 The share of the budget for discharging the income tax.


VD_p1_4c The households’ demand for products at the price P__1c

VD_p1_4r The households’ demand for educational services at the price P__1r

VW_3_1 The wages of the employees of Sector 1 (the state-owned enterprises).

VW_1_1 The wages of the employees of Sector 1 (the privately owned enterprises).





VB__4 The budget of the households.

VB_b_4 The balance of bank accounts.

4.1.1.16 Sector 5

VD_p1_5g The demand for final products at the price P__1g.




VG_s_1 The expenditures of the consolidated budget aimed at backing Sector 1.



VG_tr_4 The social transfers to the population from the consolidated budget.

VG_f_4 The off-budget funds made available for the population.

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.



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 5.

VP__1n The price of innovation products.

VP__1i The price of investment products.

VP__1r The price of educational services.

VP__1z The price of knowledge-provision services.

VD_p3_sl The total demand of the labor force at the price P__3l.

VD_p1_sl The total demand of the labor force at the price P__1l.

VD_p1_sc The total demand of final products for the households at the price P__1c.

VD_p1_sg The total demand of final products for economic agent 5 at the price P__1g.

VD_p1_sn The total demand of innovation products at the price P__1n.

VD_p1_si The total demand of investment products at the price P__1i.

VD_p1_sr The total demand of educational services at the price P__1r.

VD_p1_sz The total demand of knowledge-provision services at the price P__1z.

VS_p3_sl The total supply of the labor force at the price P__3l.

VS_p1_sl The total supply of the labor force at the price P__1l.

VS_p1_sc The total supply of final products for households at the price P__1c.

VS_p2_sc The total supply of final products for at the price P__2c.

VS_p1_sg The total supply of final products for economic agent 5 at the price P__1g.

VS_p1_sn The total supply of innovation products for at the price P__1n.

VS_p2_sn The total supply of innovation products for at the price P__2n.

VS_p1_si The total supply of investment products for at the price P__1i.

VS_p1_sr The total supply of educational services for at the price P__1r.

VS_p1_sz The total supply of knowledge-provision services for at the price P__1z.

VS_p2_sz The total supply of knowledge-provision services for at the price P__2z.

4.1.1.18 Integral Indexes

VY GDP (in base-period prices).

VY_p GDP (in current prices).

VP The consumer price index.

VK The capital assets.

4.1.1.19 Technical Variable

VI__l The deficiency indicator for the labor force market.


4.1.1.20 Model Markets

As a result of leveling demands and supplies of the various kinds of products,

services, and labor force, the equilibrium prices are formed in the following

markets:

– The market of final products for households;

– The market of exported final products;

– The market of final products for economic agent 5;

– The market of investment products;

– The market of the labor force paid by privately owned enterprises;

– The market of the labor force paid from funds of the national state budget;

– The market of innovation products;

– The market of exported innovation products;

– The market of knowledge;

– The market of exported knowledge;

– The market of educational services.

The formula used in the model for determining the deficiency indicator for the

labor force market with governmental regulation of prices is given by

VI��l½t� ¼ VS�p3�sl½t� VD�p3�sl½t�= : (4.1)

Let us now write down formulas describing the process of changing prices for all

these markets.

The labor force price:

VP��1l½t;Qþ 1� ¼ VP��1l½t;Q� þ VD�p1�sl½t;Q� � VS�p1�sl½t;Q�ð Þ CC��1l= ; (4.2)

The price of final products for households:

VP��1c t;Qþ 1½ � ¼ VP��1c t;Q½ � þ VD�p1�sc½t;Q� � VS�p1�sc½t;Q�ð Þ CC��1c= ; (4.3)

The price of final products for economic agent 5:

VP��1g½t;Qþ 1� ¼ VP��1g½t;Q� þ VD�p1�sg½t;Q� � VS�p1�sg½t;Q�ð Þ CC��1g= ; (4.4)

The price of innovation products:

VP��1n½t;Qþ 1� ¼ VP��1n½t;Q� þ VD�p1�sn½t;Q� � VS�p1�sn½t;Q�ð Þ CC��1n= ; (4.5)

The price of investment products:

VP��1i½t;Qþ 1� ¼ VP��1i½t;Q� þ VD�p1�si½t;Q� � VS�p1�si½t;Q�ð Þ CC��1i= ; (4.6)


The price of educational services:

VP��1r½t;Qþ 1� ¼ VP��1r½t;Q� þ VD�p1�sr½t;Q� � VS�p1�sr½t;Q�ð Þ CC��1r= ; (4.7)

The price of knowledge-provision services:

VP��1z½t;Qþ 1� ¼ VP��1z½t;Q� þ VD�p1�sz½t;Q� � VS�p1�sz½t;Q�ð Þ CC��1z= : (4.8)

Let us now present the formulas determining the total demand and supply of

products for each of the prices used in the model. The final formulas determining

the demand and supply of the specific economic agent are given in the respective

items.

The total supply and demand of the labor force at governmental and market

prices:

VD�p3�sl½t� ¼ VD�p3�1l½t� þ VD�p3�2l½t� þ VD�p3�3l½t�; (4.9)

VD�p1�sl½t� ¼ VD�p1�1l½t� þ VD�p1�2l½t� þ VD�p1�3l½t�; (4.10)

VS�p3�sl½t� ¼ CS�p3�4l½t�; (4.11)

VS�p1�sl½t� ¼ CS�p1�4l½t�: (4.12)

The total supply and demand of final products for households at market prices:

VD�p1�sc½t� ¼ VD�p1�4c½t�; (4.13)

VS�p1�sc½t� ¼ VS�p1�3c½t�: (4.14)

The total supply and demand of final products for economic agent 5 at market

prices:

VD�p1�sg½t� ¼ VD�p1�5g½t�; (4.15)

VS�p1�sg½t� ¼ VS�p1�3g½t�: (4.16)

The total supply and demand of innovation products at market prices:

VD�p1�sn½t� ¼ VD�p1�1n½t� þ VD�p1�2n½t� þ VD�p1�3n½t�þ VD�p1�5n½t�; (4.17)

VS�p1�sn½t� ¼ VS�p1�2n½t�: (4.18)


The total supply and demand of investment products at market prices:

VD�p1�si½t� ¼ VD�p1�1i½t� þ VD�p1�2i½t� þ VD�p1�3i½t�; (4.19)

VS�p1�si½t� ¼ VS�p1�3i½t�: (4.20)

The total supply and demand of educational services at market prices:

VD�p1�sr½t� ¼ VD�p1�1r½t� þ VD�p1�2r½t� þ VD�p1�3r½t�þ VD�p1�4r½t� þ VD�p1�5r½t�; (4.21)

VS�p1�sr½t� ¼ VS�p1�1r½t�: (4.22)

The total supply and demand of knowledge-provision services at market prices:

VD�p1�sz½t� ¼ VD�p1�1z½t� þ VD�p1�2z½t� þ VD�p1�3z½t�þ VD�p1�5z½t�; (4.23)

VS�p1�sz½t� ¼ VS�p1�1z½t�: (4.24)

Thus, we have 16 formulas determining the total supply and demand of the

products considered in the model.

Let us present the notation for determining the total supply and demand of

exported products and services.

The total supply and demand of knowledge-provision services (scientific grants)

at export prices:

CD�p2�sz½t� is given; (4.25)

VS�p2�sz½t� ¼ VS�p2�1z½t�: (4.26)

The total supply and demand of innovation products at export prices:

CD�p2�sn½t� is given; (4.27)

VS�p2�sn½t� ¼ VS�p2�2n½t�: (4.28)

The total supply and demand of final products at export prices:

CD�p2�sc½t� is given; (4.29)


VS�p2�sc½t� ¼ VS�p2�3c½t�: (4.30)

Finally, we have 16 + 6 ¼ 22 formulas for determining the total supply and

demand of all products used in the model.

Let us describe the activity of the economic agents participating in the model.

4.1.1.21 Economic Agent 1. Science and Education Sector

As presented above, leveling of the total supply and demand in the markets with

governmental prices is realized by means of the correction of the share of budget

VO_p3_1. This process is described by the following formula:

VO�p3�1l½t;Qþ 1� ¼ VO�p3�1l½t;Q� � Ceta��1þ VO�p3�1l½t;Q�� VI��l½t;Q� � ð1� Ceta��1Þ: (4.31)

Here Q is the iteration step; 0 < Ceta__1 < 1 is the model constant. When it

increases, equilibrium is attained more slowly. Nevertheless, the system of

equations becomes more stable.

Let us proceed to the formulas determining the behavior of the science and

education sector.

The production function equation is given by

VY��1½tþ 1� ¼ CA�r�1� Powerð VK��1½t� þ VK��1½tþ 1�ð Þ=2ð Þ;CA�k�1Þ� PowerððVD�p1�1l½t� þ VD�p3�1l½t�Þ;CA�l�1Þ� ExpðCalpha��1� VD�p1�1z½t� þ Cbeta��1� VD�p1�1r½t�þ Cgamma��1� VD�p1�1n½t�Þ:

(4.32)

Here Power(X, Y) corresponds to XY; Exp(X) corresponds to eX.The following formulas determine the demand of production factors in the

science and education sector.

The demand of the labor force at governmental prices:

VD�p3�1l½t� ¼ VO�p3�1l½t� � VB��1½t�ð Þ=CP��3l½t�: (4.33)

The demand of the labor force at market prices:

VD�p1�1l½t� ¼ ðCO�p1�1l½t� � VB��1½t�Þ=VP��1l½t�: (4.34)


The demand of knowledge-provision services:

VD�p1�1z½t� ¼ CO�p1�1z½t� � VB��1½t�ð Þ=VP��1z½t�: (4.35)

The demand of educational services:

VD�p1�1r½t� ¼ CO�p1�1r½t� � VB��1½t�ð Þ=VP��1r½t�: (4.36)

The demand of innovation products:

VD�p1�1n½t� ¼ CO�p1�1n½t� � VB��1½t�ð Þ=VP��1n½t�: (4.37)

The demand of investment products:

VD�p1�1i½t� ¼ CO�p1�1i½t� � VB��1½t�ð Þ=VP��1i½t�: (4.38)

The following formulas determine the supply of the services rendered by the

science and education sector.

The supply of knowledge-provision services at market prices:

VS�p1�1z½t� ¼ CE�p1�1z� VY��1½t�: (4.39)

The supply of knowledge-provision services at export prices:

VS�p2�1z½t� ¼ CE�p2�1z� VY��1½t�: (4.40)

The supply of educational services:

VS�p1�1r½t� ¼ CE�p1�1r � VY��1½t�: (4.41)

The following formula calculates the gain of the science and education sector

from the supplied services:

VY�p�1½t� ¼ VS�p1�1z½t� � VP��1z½t� þ VS�p2�1z½t� � CP��2z½t�þ VS�p1�1r½t� � VP��1r½t�: (4.42)

The budget of the science and education sector is determined as follows:

VB��1½t� ¼ VB�b�1½t� � 1þ CP��bpercent½t� 1�ð Þþ VY�p�1½t� þ VG�s�1½t� 1�: (4.43)

The agent’s budget is formed from the following:

1. The funds in bank accounts (subject to interest on deposits);

2. The gain received in the current period;

3. The bounties received from the consolidated budget VG_s_1[t � 1].


The dynamics of the banking account balance of the science and education

sector is as follows:

VB�b�1½tþ 1� ¼ VO�s�1½t� � VB��1½t�: (4.44)

The capital assets are determined by

VK��1½tþ 1� ¼ 1� CR��1½t�ð Þ � VK��1½t� þ VD�p1�1i½t�: (4.45)

This formula calculates the volume of capital assets, taking into account their

retirement. An asset put into operation enters the formula with a plus sign.

The share of budget of the science and education sector for discharging taxes to

the consolidated budget is given by

VO�t�1 t½ � ¼ VY�p�1½t� � CT�vad½t�ð Þ=VB��1½t� þ ððVY�p�1½t� � VW�3�1½t�� VW�1�1 t½ � � VK��1½t� � CA�0�1½t�Þ � CT�pr½t�Þ=VB��1½t�:

(4.46)

This formula takes into consideration the value added tax (VAT) and profit tax.

In calculating the share of budget for discharging the profit tax, the gain is

subtracted by the costs of the labor force of state-owned (VW_3_1[t]) and privatelyowned (VW_1_1[t]) enterprises, as well as the depreciation charges VK__1[t] � CA_0_1[t]. The share of budget for discharging the single social tax to the

off-budget funds is described as

VO�f�1½t� ¼ VW�3�1½t� þ VW�1�1½t�ð Þ � CT�esn½t�ð Þ=VB��1½t�: (4.47)

The remainder of the budget of the science and education sector is given by

VO�s�1½t� ¼ 1� CO�p1�1l½t� � VO�p3�1l½t� � CO�p1�1z½t� � CO�p1�1r½t�� CO�p1�1n½t� � CO�p1�1i½t� � VO�t�1½t� � VO�f�1½t�:

(4.48)

4.1.1.22 Economic Agent 2. Sector of Innovation

As presented above, the leveling of the total supply and demand in the markets with

governmental prices is realized by means of correction of the share of budget

VO_p3_2l. This process is described by the following formula:



Here Q is the iteration step; 0 < Ceta__2 < 1 is the model constant. When it

increases, equilibrium is attained more slowly. Nevertheless, the system of

equations becomes more stable. Let us proceed to the formulas determining the

behavior of the innovation sector.


VY��2½tþ 1� ¼ CA�r�2� Power VK��2½t� þ VK��2½tþ 1�ð Þ=2ð Þ;CA�k�2ð Þ� PowerððVD�p1�2l½t� þ VD�p3�2l½t�Þ;CA�l�2Þ� ExpðCalpha��2� VD�p1�2z½t� þ Cbeta��2� VD�p1�2r½t�þ Cgamma��2� VD�p1�2n½t�Þ:

(4.50)

The following formulas determine the demand of the production factors in the

innovation sector:




VD�p1�2l½t� ¼ CO�p1�2l½t� � VB��2½t�ð Þ=VP��1l½t�: (4.52)






VD�p1�2n½t� ¼ CO�p1�2n½t� � VB��2½t�ð Þ=VP��1n½t� (4.55)



The following formulas determine the supply of the products produced by the

innovation sector:

The supply of innovation products at market prices:

VS�p1�2n½t� ¼ CE�p1�2n� VY��2½t�: (4.57)


The supply of innovation products at export prices:

VS�p2�2n½t� ¼ CE�p2�2n� VY��2½t�: (4.58)

The following formula calculates the gain of the innovation sector:

VY�p�2½t� ¼ VS�p1�2n½t� � VP��1n½t� þ VS�p2�2n½t� � CP��2n½t�: (4.59)

The budget of the innovation sector is determined as follows:

VB��2½t� ¼ VB�b�2½t� � 1þ CP��bpercent½t� 1�ð Þ þ VY�p�2½t�þ VG�s�2½t� 1�: (4.60)




3. The bounties received from the consolidated budget VG_s_2.

The dynamics of the bank account balance of the innovation sector are as

follows:

VB�b�2½tþ 1� ¼ VO�s�2½t� � VB��2½t�: (4.61)



This formula calculates the volume of capital assets taking into account their


The share of budget of the innovation sector for discharging taxes to the

consolidated budget is given by

VO�t�2½t� ¼ VY�p�2½t� � CT�vad½t�ð Þ=VB��2½t�þ ððVY�p�2½t� � VW�3�2½t�� VW�1�2½t� � VK��2½t�� CA�0�2½t�Þ � CT�pr½t�Þ=VB��2½t�: (4.63)

This formula takes into consideration the VAT and profit tax. While calculating

the share of budget for discharging the profit tax, the gain is reduced by the costs of

the labor force of state-owned (VW_3_2) and privately owned (VW_1_2)enterprises, as well as the depreciation charges VK__2[t] � CA_0_2[t].


The share of budget for discharging the single social tax to off-budget funds

is described as


The remainder of the budget of the innovation sector is given by

VO�s�2½t� ¼ 1� CO�p1�2l½t� � VO�p3�2l½t� � CO�p1�2z½t� � CO�p1�2r½t�� CO�p1�2n½t� � CO�p1�2i½t� � VO�t�2½t� � VO�f�2½t�

(4.65)

4.1.1.23 Economic Agent 3. Other Branches of the Economy

As presented above, leveling of the total supply and demand in the markets with

governmental prices is realized by means of the correction of the share of budget

VO_p3_2l. This process is described by the following formula:


Here Q is the iteration step; 0 < Ceta__3 < 1 is the model constant.

Let us proceed to the formulas determining the behavior of the other branches of

the economy.


VY��3½tþ 1� ¼ CA�r�3� Power VK��3½t� þ VK��3½tþ 1�ð Þ=2ð Þ;CA�k�3ð Þ� PowerððVD�p1�3l½t� þ VD�p3�3l½t�Þ;CA�l�3Þ� ExpðCalpha��3� VD�p1�3z½t�þ Cbeta��3� VD�p1�3r½t� þ Cgamma��3� VD�p1�3n½t�Þ:

(4.67)


other branches of the economy:





VD�p1�3l½t� ¼ CO�p1�3l½t� � VB��3½t�ð Þ=VP��1l½t�: (4.69)









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�3c½t� ¼ CE�p1�3c� VY��3½t�: (4.74)

The supply of the final products for economic agent 5:

VS�p1�3g½t� ¼ CE�p1�3g� VY��3½t�: (4.75)

The supply of investment products:

VS�p1�3i½t� ¼ CE�p1�3i� VY��3½t�: (4.76)

The supply of exported products:

VS�p2�3c½t� ¼ CE�p2�3c� VY��3½t�: (4.77)

The following formula calculates the gain of the other branches of the economy:

VY�p�3½t� ¼ VS�p1�3c½t� � VP��1c½t� þ VS�p1�3g½t�� VP��1g½t� þ VS�p1�3i½t�� VP��1i½t� þ VS�p2�3c½t� � CP��2c½t�: (4.78)

The budget of the other branches of the economy is determined as follows:

VB��3½t� ¼ VB�b�3½t� � 1þ CP��bpercent½t� 1�ð Þ þ VY�p�3½t� þ VG�s�3½t� 1�:(4.79)





3. The bounties received from the consolidated budget VG_s_3.

The dynamics of the bank account balance of the other branches of the economy

are as follows:

VB�b�3½tþ 1� ¼ VO�s�3½t� � VB��3½t�: (4.80)



This formula calculates the volume of the capital assets taking into account their


The share of budget of the other branches of the economy for discharging taxes

to the consolidated budget is given by

VO�t�3½t� ¼ VY�p�3½t� � CT�vad½t�ð Þ=VB��3½t�þ ððVY�p�3½t� � VW�3�3½t� � VW�1�3½t� � VK��3½t�� CA�0�3½t�Þ � CT�pr½t�Þ=VB��3½t�: (4.82)

This formula takes into consideration the VAT and profit tax. In calculating the

share of budget for discharging the profit tax, the gain is reduced by the costs of the

labor force of state-owned (VW_3_3) and privately owned (VW_1_3) enterprises, aswell as by the depreciation charges.

The share of budget for discharging the single social tax to the off-budget funds

is described as


The remainder of the budget of the other branches of the economy is given by

VO�s�3½t� ¼ 1� CO�p1�3l½t� � VO�p3�3l½t� � CO�p1�3z½t� � CO�p1�3r½t�� CO�p1�3n½t� � CO�p1�3i½t� � VO�t�3½t� � VO�f�3½t�:

(4.84)

4.1.1.24 Economic Agent 4. Aggregate Consumer (Households)

Let us proceed to the formulas determining the behavior of the aggregate consumer.

The households’ demand for final products is given by


VD�p1�4c½t� ¼ CO�p1�4c½t� � VB��4½t�ð Þ=VP��1c½t�: (4.85)

The households’ demand for educational services:


The wages of the employees of state-owned enterprises in the science and

education sector:

VW�3�1½t� ¼ VD�p3�1l½t� � CP��3l½t�: (4.87)

The wages of the employees of privately owned enterprises in the science and

education sector:

VW�1�1½t� ¼ VD�p1�1l½t� � VP��1l½t�: (4.88)

The wages of the employees of state-owned enterprises in the innovation sector:

VW�3�2½t� ¼ VD�p3�2l½t� � CP��3l½t�; (4.89)

The wages of the employees of privately owned enterprises in the innovation

sector:

VW�1�2½t� ¼ VD�p1�2l½t� � VP��1l½t�: (4.90)

The wages of the employees of state-owned enterprises in the other branches of

the economy:

VW�3�3½t� ¼ VD�p3�3l½t� � CP��3l½t�: (4.91)

The wages of the employees of privately owned enterprises in the other branches

of the economy:

VW�1�3½t� ¼ VD�p1�3l½t� � VP��1l½t�: (4.92)

The budget of the households is determined as follows:

VB��4½t� ¼ VB�b�4½t� 1� � 1þ CP�h�bpercent½t� 1�ð Þ þ VB��4½t� 1�� VO�s�4½t� 1 þVW�3�1� ½t þVW�1�1� ½t þVW�3�2� ½t þVW�1�2� ½t�þVW�3�3½t þVW�1�3½t� þ VG�f�4½t� 1� þ VG�tr�4½t� 1�:�

(4.93)





3. The wages received from the three producing agents;

4. The 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�4½t� ¼ CO�b�4½t� � VB��4½t�: (4.94)

The share of the budget for discharging the income tax is given by

VO�tax�4½t� ¼ ððVW�3�1½t� þ VW�1�1½t� þ VW�3�2½t� þ VW�1�2½t�þ VW�3�3½t� þ VW�1�3½t�Þ � CT�pod½t�Þ=VB��4½t�: (4.95)

The remainder of the money in cash is as follows:

VO�s�4½t� ¼ 1� CO�p1�4c½t� � CO�p1�4r½t� � VO�tax�4½t� � CO�b�4½t�:(4.96)

4.1.1.25 Economic Agent 5. Government

Let us proceed to the formulas determining the behavior of economic agent 5.

The consolidated budget is given by

VB��5 t½ � ¼VO�t�1½t��VB��1½t�þVO�t�2½t��VB��2½t�þVO�t�3½t��VB��3½t�þVO�tax�4½t��VB��4½t�þCB�other�5þVB�b�5½t�� 1þCP��bpercent½t�1�ð Þ:

(4.97)

This formula sums the money collected as taxes from the producing agents as

well as from the population. The value CB_other_5 entered in the model exoge-

nously is the sum of the other taxes (not included in the list of taxes considered in

the model), nontaxable income, and other income included in the consolidated

budget. The obtained sum is to be supplemented by the funds in bank accounts

(subject to interest on deposits).

The dynamics of the banking account balance of the consolidated budget are

determined by

VB�b�5½tþ 1� ¼ CO�s�5b½t� � VB��5½t�: (4.98)


The cash assets of off-budget funds are as follows:

VF��5½t� ¼VO�f�1½t��VB��1½t�þVO�f�2½t��VB��2½t�þVO�f�3½t��VB��3½t�þVF�b�5½t�� 1þCP��bpercent½t�1�ð Þ:

(4.99)

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 in bank accounts (subject to

interest on deposits).

The dynamics of the banking account balance of the off-budget funds are

determined by

VF�b�5½tþ 1� ¼ CO�s�5f ½t� � VF��5½t�: (4.100)

The demand of the final products:

VD�p1�5g½t� ¼ CO�p1�5g½t� � VB��5½t�ð Þ=VP��1g½t�: (4.101)

The knowledge-provision service payment:


The educational service payment:


The demand of the innovation products:


The subsidies to the producing sectors are as follows.

The science and education sector:

VG�s�1½t� ¼ CO�s1�5½t� � VB��5½t�: (4.105)

The innovation sector:

VG�s�2½t� ¼ CO�s2�5½t� � VB��5½t�: (4.106)


The other branches of the economy:

VG�s�3½t� ¼ CO�s3�5½t� � VB��5½t�: (4.107)

The social transfers to the population:

VG�tr�4½t� ¼ CO�tr�5½t� � VB��5½t�: (4.108)

The assets of the off-budget funds made available to the population:

VG�f�4½t� ¼ CO�f4�5½t� � VF��5½t�: (4.109)

This includes the assets of the pension fund and social insurance fund for paying

out the pensions and welfare payments.

4.1.1.26 Integral Indexes of the Model

In this subsection we present the formulas for calculating some integral indexes of

the economy of the Russian Federation.

The GDP (in prices of the base period):

VY½t� ¼ VY��1½t� þ VY��2½t� þ VY��3½t�: (4.110)

The GDP (in current prices):

VY�p½t� ¼ VY�p�1½t� þ VY�p�2½t� þ VY�p�3½t�: (4.111)

The consumer price index:

VP½t� ¼ 100� VP��1c½t�=VP��1c½t� 1�ð Þ: (4.112)

Capital assets:

VK½t� ¼ VK��1½t� þ VK��2½t� þ VK��3½t�: (4.113)

In this model:

– Relations (1.7) are represented by 12 expressions for finding the gross value

added (GVA) of the sectors by means of the production functions, capital assets

of the sectors, balances of the banking accounts;

– Relations (1.8) are represented by 88 expressions for finding the supplies and

demands for the 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 (1.9) are represented by ten expressions serving to find the equilibrium

market prices and shares of budgets of the sectors in the markets with exogenous

prices.

The exogenous parameters of the considered model are determined by solving

the problem of parametric identification of the model using available statistical

information from the Russian Federation for the period 2000–2004. The validity of

the model and identification process is ensured by the following facts:

1. The identification criterion includes the statistical information on the basic

macroeconomic indexes (the GDP and GVA of the sectors, the capital assets

of the sectors, etc.).

2. The estimates of the exogenous parameters and the initial values of the differ-

ence equations, which have measured values, are found in the intervals with

centers in the respective measured values or those covering several measured

values.

3. The intervals for estimating the other parameters are determined by indirect

factors. The values of the parameters varying with years are found under the

assumption that their variations are insignificant.

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 revealing its availability to

produce precise prediction values is carried out by means of retrospective predic-

tion. To do this, after the parametric identification problem has been solved using

the statistical information of the Russian Federation [39] for the years 2000–2004,

the values of all exogenous variables of the model are extended to the period of

2005–2008, and the model computation for the tested period is carried out.

However, additional parametric identification is not carried out.

The mean error of the calculated values of the endogenous parameters with

respect to the corresponding measured values within the retrospective prediction

period is 1.04%.

4.1.2 Finding Optimal Parametric Control Laws on the Basisof the CGE Model with the Knowledge Sector

The results of applying the approach of parametric control theory are compared

with the results of applying the scenario approach [24] for control of market

economic development using the example of the Russian Federation. The scenario

of [24] implies an annual increase in financial investment from the state budget to

the innovation component of the economy, as well as to the science and education


sector (VG_s_1 and VG_s_2) by a factor of 2 over the 8 years (2007–2015) withouttaking into consideration changes in the level of consumer prices. According to

[24], as a result of the application of this approach, the value of the GDP (here

and below in constant prices of the year 2007) increases by 4.9% in the year

2015 (100% in 2007) in comparison to the base variant, which implies inertial

development of the economy without any additional financial investment. The

results of the GDP calculation in this experiment are presented in Fig. 4.1 (plot 1).

However, this scenario results in the rise of consumer prices by 22.19% by the

year 2015 in comparison with the base year (2015, 100%) variant. This is not

mentioned in the results of the scenario considered in [24].

In the context of the parametric control approach, the optimal laws are chosen

from the following set of algorithms:

Uij ¼ kij~xi � ~xi0~xi0

þ uj�; i ¼ 1; 11; j ¼ 1; l

� �: (4.114)

Here kij � 0 are the adjusted coefficients, uj� are the values of the controlled

parameter accepted or estimated by the results of calibration. These laws use such

state variables ~xi as

– The GDP of each of three sectors and the whole economic system;

– The levels of various kinds of prices encountered in the model;

– The production assets of three sectors of the economic system.

The mean value of the GDP of the country for the years 2008–2015 in constant

prices of the year 2007 is used as criterion K:

Fig. 4.1 Plots of changing GDP of the Russian Federation relative to the base variant of economic

development, as a percentage


K ¼ 1

8

X2015t¼2008

VY½t�:

The problems of choosing the set of the optimal parametric control laws at the

level of

– The economic parameter G+ (the coefficient determining the share of additional

investment from the state budget to the innovation component of the economy

(VG_s_2) and the science and education sector (VG_s_1)) and– The tax rates (CT_vad, CT_pr, CT_pod, CT_esn) can be formulated as follows.

On the basis of the mathematical model (4.1–4.113), find the optimal parametric

control law (the set of two laws) Uij in the environment of the set of algorithms

(4.114) maximizing criterion K under the chosen constraints on the values of the

endogenous variables and controlled parameters. The following constraints are

used in all computational experiments as the constraints on the endogenous

variables:

~x 2 X; Gþ � 0; 0 � CT�vad; CT�pr; CT�pod; CT�esn � 1: (4.115)


– In the first stage, we determine the optimal values of coefficients kij for each lawUij (or for the pair of laws) from (4.114) by enumeration of their values in their

respective ranges (quantized with a small step size) maximizing K under

constraints (4.115);

– In the second stage, the optimal law of control of the respective parameter(s) is

found using results of the first stage using the maximum value of criterion K.The application of the parametric control approach using analyzed model [24]

is carried out as the following sequence of computational experiments:

1. Determining the optimal law in the sense of criterion K from set (4.114)

by additional financial investment from the state budget in the innovation

component of the economy (VG_s_2), as well as the science and education

sector (VG_s_1) on the basis of the CGE model with the knowledge sector [24].

The optimal parametric control law of the coefficient of the given additional

investment G+ is given by

Gþ ¼ 1; 19VY½t� � VY½0�

VY½0� ;

where VY½t� is the GDP in year t, t ¼ 0 corresponds to year 2006. The final form

of the investment in the two sectors is as follows:

VG�s�1½t� ¼ ð1þ GþÞVG�s�1�½t�; VG�s�2 ½t� ¼ ð1þ GþÞVG�s�2� ½t�:


The sign (*) here and below corresponds to the base values of the variables

and controlled parameters.

With use of the law thus obtained, the GDP of the country increases by 7.38%

by the year 2015; the level of consumer prices increases by 24.67% by 2010 year

in comparison with the base variant. The results of the GDP computations from

this experiment are presented in Fig. 4.1 (plot 2).

2. Determining the optimal law in the sense of criterion K from set (4.114) to adjust

the coefficient of the additional investment G+ from the state budget in the

innovation component of the economy, as well as the science and education

sector (VG_s_1 and VG_s_2) on the basis of the CGE model with the knowledge

sector [24] with additional constraints on the growth of consumer prices.

For this model here and below, the additional constraint on the level of

consumer prices is given by

VP½t� � VP�½t� � aVP�½t�;

where VP is the level of consumer prices with parametric control; VP* is the

level of consumer prices of the base variant, a ¼ 0:09. This inequality means

that with parametric control, the increase in the level of consumer prices in

comparison with the base variant (without parametric control) is allowed by no

more than 9% over the whole control time interval.

The derived optimal parametric control law is given by

Gþ ¼ 0; 46VY½t� � VY½0�

VY½0� :

With use of this law, GDP of the country increases by 2.83% by 2015; the level

of consumer prices increases by 8.80% by 2015 year in comparison with the base

year.

3. Determining the optimal law in the sense of criterion K from set (4.114) for

adjusting the coefficient of the additional investment G+ from the state budget

in the innovation component of the economy, as well as in the science and

education sector by the rate of one of the taxes using the CGE model with the

knowledge sector [24] with constraints on the growth of consumer prices.

The derived optimal parametric control law by the organization profit tax

CT_pr is given by

CT�pr ½t� ¼ 2; 29VY��3½t� � VY��3½0�

VY��3½0� þ CT�pr�:

Here VY__3[t] is the GDP of the third sector (other branches of the economy)

in year t.Using the given law of parametric control of the organization profit tax (CT_pr),the GDP of the country increases by 17.79% by the year 2015; the level of


consumer prices decreases by 16.75% by 2015 in comparison with the base

variant.

4. Determining the pair of laws optimal in the sense of criterion K from set (4.114)

with constraints on the growth of consumer prices. Here the first law is applied

for adjusting the coefficient of the additional investment G+ from the state

budget in the innovation component of the economy, as well as in the science

and education sector. The second law is applied for adjusting one of the tax rates.

The optimal pair of parametric control laws is determined to be

Gþ ¼ 0; 46VY½t� � VY½0�

VY½0� ; CT�pr½t� ¼ 2; 16VY��3½t� � VY��3½0�

VY��3½0� þ CT�pr�:

Using this optimal pair of laws of parametric control of the parameters

(G+, CT_pr), the GDP of the country increases by 19.34% by 2015, and the

level of consumer prices decreases by 13.3% by 2015 in comparison with the

base variant. The results of the GDP calculation from the experiments 3 and 4 are

also presented in Fig. 4.1 (plots 3 and 4, respectively).

4.1.3 Analysis of the Dependence of the Optimal ParametricControl Law on Values of Uncontrolled ParametersBased on the CGE Model with the Knowledge Sector

Let us consider the dependence of the results of choosing the parametric control law

at the level of the parameter CT_pr on two uncontrolled parameters, namely,

l1 ¼ CP��bpercent(the deposit interest rate for enterprises) and l2 ¼ CO_p1_3n(the share of the third-sector budget for purchasing innovation products at price

P__1n), with values in some region (rectangle) L in the plane. In other words, let us

find the bifurcation points for the considered variational problem of choosing the

optimal parametric control law of the CGE model under consideration.

From computational experiments, we obtain the plots of dependence of the

optimal value of criterion K on the values of the parameters (l1; l2) for each of

11 possible laws Ui. Figure 4.2 presents the plots for the laws U1 and U2, ensuring

the maximum values of the criterion in the region L, the curve of intersection of

these surfaces, and the projection of the intersection curve onto the region of the

values of the parameters (l1; l2). This projection consists of the bifurcational

points of the parameters (l1; l2). It divides the rectangle L into two parts such

that the control law


U1 ¼ CT�pr½t� ¼ �k1VK��2½t� � VK��2½0�

VK��2½0� þ CT�pr�

is optimal in one part of the region, whereas the control law

U2 ¼ CT�pr½t� ¼ k2VY��3½t� � VY��3½0�

VY��3½0� þ CT�pr�

is optimal in the other. Both laws are optimal on the projection curve itself. Here

VK__2 denotes the capital assets of the innovation sector; k1, k2 are the adjusted

coefficients of the laws.

Depending on the result of this analysis of dependence of the solution of the

given variational calculus problem on the values of the uncontrolled parameters

(l1; l2), one can approach choosing the optimal parametric control law in the

following way. If the values of the parameters (l1; l2) are to the left of the

bifurcation curve in the rectangle L (Fig. 4.2), then law U1 is recommended as

the optimal law. If the values of the parameters (l1; l2) are to the right of the

bifurcation curve in the rectangle L, then law U2 is recommended as the optimal

law. When the values of the parameters (l1; l2) lie on the bifurcation curve in

rectangle L, either of the two laws U1, U2 can be recommended as the optimal law.

Fig. 4.2 Plots of dependencies of the criterion value for two optimal parametric control laws on

uncontrolled parameters (CP__bpercent, CO_p1_3n)


4.2 National Economic Evolution Control Basedon a Computable Model of General Equilibriumof Economic Branches


4.2.1.1 Model Agents

The considered model [24] describes the behavior and interaction in 46 product

markets and 22 labor markets of the following 26 economic agents.

Economic agent 1 is the power industry.

Economic agent 2 is the oil and gas industry.

Economic agent 3 is the coal-mining industry.

Economic agent 4 is other fuel industries.

Economic agent 5 is the iron industry.

Economic agent 6 is the nonferrous-metals industry.

Economic agent 7 is the chemical and petrochemical industry.

Economic agent 8 is the machine-building and metal-working industry.

Economic agent 9 is the timber, woodworking, and pulp and paper industry.

Economic agent 10 is the construction materials industry.

Economic agent 11 is the light industry.

Economic agent 12 is the food industry.

Economic agent 13 represents the remaining branches of the industry.

Economic agent 14 is the construction industry.

Economic agent 15 represents agriculture and forestry.

Economic agent 16 represents the transportation and communication industries.

Economic agent 17 represents trade, intermediate activity, and catering.

Economic agent 18 represents other kinds of activity in production of goods and

services.

Economic agent 19 represents housing and communal services, as well as

nonproduction types of consumer services rendered to the population.

Economic agent 20 represents the public health service, physical training, social

service, education, culture and art.

4.2 National Economic Evolution Control Based. . . 195

Economic agent 21 represents the science and scientific services, geology, and

exploration of the subsurface, as well as geodesic and hydrometeorological

services.

Economic agent 22 represents the finances, credit, insurance management, and

public associations.

A portion of the products of the economic agents producing goods and services

(economic agents 1–22) is used in production, another part is spent for investment,

and the remainder is sold to households. The producing agents deal in intermediate

and investment products with one another.

Economic agent 23 is the aggregate consumer uniting the households.

The aggregate consumer purchases the consumer goods produced by the pro-

ducing agents. Moreover, it purchases imported goods offered by the outside world.

Economic agent 24 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. Moreover, this

sector includes the nonprofit organizations servicing the households (the political

parties, trade unions, public associations, etc.).

Economic agent 25 is the banking sector including the central bank and commer-

cial banks.


This model also includes 1,722 exogenous parameters and 1,104 endogenous

variables.


The exogenous parameters of the model include the following:


– The various shares of the sectors’ budgets;

– The shares of the production for selling in the various markets;

– The depreciation rates for capital assets and shares of retired capital assets;



– The coefficients reflecting the level of nonpayments to the producing agents;

– The depreciation rate for capital assets;

– The share of retired capital assets;

– The coefficient reflecting the level of arrears of wages to employees in all

branches;

– The export prices and governmental prices of products, services, labor force,

and others.


A list of exogenous parameters of the model is given below:

4.2.1.3 Sectors 1–22

CO_pi_il The share of budget of the ith branch spent for paying the labor force at the price

P__il.

CO_pj_iz The share of budget of the ith branch spent for purchasing the intermediate products

produced by the branches j ¼ 2; 22; j 6¼ 19, at the price P__jz.

CO_p_in The share of budget of the ith branch spent for purchasing investment products at the

price P__n.

CE_pi_iz The share of product produced by the ith branch for selling in the markets of

intermediate products at the price 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 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 P__in.

CE_pexi_ic The share of product produced by the ith branch for selling in the markets of exported

products at the price P__exi.

CA_r_i The empirically determined coefficient of dimension.

CA_z_j_i The coefficients of the intermediate products j ¼ 1; 22 consumed by the ith branch.

CA_k_i The coefficient of capital.

CA_l_i The coefficient of labor.

CO_y_i The coefficient reflecting the level of nonpayments to the producing agents.

CA_n The depreciation rate of capital assets.

CO_w_i The coefficient reflecting the level of arrears of wages to employees in all branches.

CR__i The share of retired capital assets.

4.2.1.4 Sector 23

CO_p_23c The share of budget of the aggregate consumer spent for purchasing final products at

the price P__c.

CO_b_23 The share of budget deposited in banks.

CS_pi_23l The number of employees employed in Sectors 1.22.

4.2.1.5 Sector 24

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.

(continued)


CO_s_i_24 The shares of the consolidated budget for backing the producing agents.

CO_tr_24 The share of the consolidated budget for payment of social transfers to the

population.

CO_f_24 The share of off-budget funds for payment of pensions, welfare payments, etc.

CB_other_24 The sum of tax proceeds (not included in those already considered), nontax income,

and other incomes of the consolidated budget.


CP__bpercent The deposit interest rate for enterprises.

CP_h_bpercent The deposit interest rate for physical bodies.


CP__1z The price of electric power.

CP__19z The price of housing and communal services.

CP__exi The price of exported product produced by the ith branch.

4.2.1.8 Technical Parameters

Ceta__1 The iteration constant applied in the case of exogenous price.

Ceta__19 The iteration constant applied in the case of exogenous price.




– The remainders of the agents’ budgets;

– The added produced values of the producing sectors;

– Demands and supplies of the various products and services;


– The capital assets of the producing sectors;

– The number of employees employed in Sectors 1–22;

– The wages of the employees;


– The various kinds of prices of the products, services, and the labor force;

– The subsidies to the producing sectors;

– The social transfers to the population;

– The gross production of goods and services;


– The volume of production of intermediate products;

– The volume of production of final products;

– The GDP of the country.

A list of the endogenous variables of the model is given below:

4.2.1.10 Sectors 1–22

VO_p1_iz The share of the budget of the producing agent spent for purchasing intermediate

product from the branch producing electric power (for agent 1).

VO_p19_iz The share of the budget of the producing agent spent for purchasing intermediate

product from the branch rendering housing and communal services, as well as

nonproduction kinds of consumer services for the population (for agent 19).

VO_tc_i The share of the budget of the producing agent spent for discharging taxes to the

consolidated budget.

VO_tf_i The share of the budget of the producing agent spent for discharging taxes to the off-

budget funds.

VO_s_i The remainder of the agent budget.

VD_pi_il The demand of the labor power in the ith branch at the price P__il.

VD_pj_iz The demand of intermediate products produced by the branches j ¼ ð1; 22Þ in the ithbranch at the price P__jz.

VD_p_in The demand of investment products in the ith branch at the price 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 the bank accounts of the producing agent.

4.2.1.11 Sector 23

VO_tc_23 The share of the budget of the aggregate consumer for discharging taxes to the

consolidated budget.

VO_s_23 The remainder of the budget of Sector 23.

VD_p_23c The household demand for final products.

VW__i The wages of the employees in Sectors 1–22.

VB__23 The budget of the households.

VB_b_23 The money of the households in bank accounts.


4.2.1.12 Sector 24

VO_s_24 The share of the retained consolidated budget.

VO_s_24f The share of the retained off-budget funds.

VG_s_i_24 The subsidies to the producing sectors.

VG_tr_24 The social transfers to the population.

VG_f_24 The off-budget funds allocated for the population.

VB__24 The consolidated budget.

VB_b_24 The surplus (deficit) of the consolidated budget.

VF__24 The monetary assets of the off-budget funds.

VF_b_24 The remainder of the monetary assets of the off-budget funds.


VY The gross production of goods and services (in prices of the base period).

VS__z The volume of production of 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.


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, i 6¼ 1; i 6¼ 19:

VP__n The price of investment products.

VP__c The price of consumer products.

VD_ps_il The total demand of the labor force at the price P__il.

VD_ps_iz The total demand of intermediate products at the price P__iz.

VD_ps_n The total demand of investment products at the price P__n.

VD_ps_c The total demand of consumer products at the price P__c.

VS_ps_il The total supply of the labor force at the price P__il.

VS_ps_iz The total supply of intermediate products at the price P__iz.

VS_ps_n The total supply of investment products at the price P__n.

VS_ps_c The total demand of consumer products at the price P__c.

4.2.1.15 Technical Variables

VI_1_z The deficiency indicator for the electric power market.

VI_19_z The deficiency indicator for the market of housing and communal services.



The equilibrium prices are formed in 68 markets of the model as a result of leveling

the supplies and demands of the various kinds of products, services, and labor force.

The described model has:

– Twenty-two markets of intermediate products and services produced and ren-

dered by the producing agents;

– One market of investment products; and

– One market of final products.

In addition, the model includes the following:

– Twenty-two foreign markets of exported products produced by producing

agents; and

– Twenty-two markets of the labor force.

The total number of markets in the model is 46. The governmental and market

mechanisms of pricing are used in the domestic markets. The prices of the foreign

markets enter the model exogenously. Let us now consider the formulas reflecting

the process of changing prices in the domestic markets (below, i means the agent

number).

The price of the labor force in the ith branch is given by

VP��il½t;Qþ 1� ¼ VP��il½t;Q� � VD�ps�il½t;Q�=VS�ps�il½t;Q�: (4.116)

The price of the intermediate product produced by the ith branch is as follows:

VP��iz½t;Qþ 1� ¼ VP��iz½t;Q� � VD�ps�iz½t;Q�=VS�ps�iz½t;Q�; i 6¼ 1; i 6¼ 19:

(4.117)

The price of the electric power (the exogenous parameter) is

P��1z: (4.118)

The price of housing and communal services is

P��19z: (4.119)

Since these two prices enter the model exogenously, it is necessary to introduce

the deficiency indicators that assist in achieving the balance of demand and supply:

VI�1�z½t� ¼ VS�ps�1z½t�=VD�ps�1z½t�; (4.120)

VI�19�z t½ � ¼ VS�ps�19z t½ � VD�ps�19z t½ �= : (4.121)


The price of investment products is determined by

VP��n½t;Qþ 1� ¼ VP��n½t;Q� � VD�ps�n½t;Q�=VS�ps�n½t;Q� (4.122)

The price of consumer products is as follows:

VP��c½t;Qþ 1� ¼ VP��c½t;Q� � VD�ps�c½t;Q�=VS�ps�c½t;Q�: (4.123)

That is, we have 22 + 22 + 1 + 1 ¼ 46 prices of the products sold in the

domestic markets in the model.

The notation for the prices in foreign markets is given below.

The price of the exported product produced by the ith branch is

P�exi: (4.124)

Thus, the total number of prices in the model is 46 + 22 ¼ 68.

Let us now proceed to the formulas describing the mechanism of forming the

demand and supply of the products produced by agents 1–22 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 of the labor force at the price VP__il[t] is given by

VD�ps�il½t� ¼ VD�pi�il½t�: (4.125)

For simplicity, we do not consider the demand of the labor force in the ith branchfrom the other branches. In this connection, the total demand of the labor force at

the price VP__il[t] is defined by the demand in the ith branch alone.

The total supply of the labor force at the price VP__il[t] is as follows:

VS�ps�il½t� ¼ CS�pi�23l: (4.126)

The total demand of the intermediate product at the price VP__jz[t] produced bythe jth branch is determined as

VD�ps�jz½t� ¼ SUMi VD�pj�iz½t�ð ÞÞ: (4.127)

Here and below, SUM(X__i) corresponding toP22i¼1

X��i, i ¼ 1; 22, is the eco-

nomic agent number.

As can be seen, the total demand of the intermediate product at the price VP__jz[t] consists of the demands of the intermediate products in the jth branch j ¼ 1; 22from the direction of all 22 branches.


The total supply of the intermediate product at the price VP__iz[t] is given by

VS�ps�iz½t� ¼ VS�pi�iz½t�: (4.128)

The total demand of the investment products at the price VP__n[t]:

VD�ps�n½t� ¼ SUMðVD�p�in½t�Þ: (4.129)

The total supply of the investment products at the price VP__n[t]:

VS�ps�n½t� ¼ SUMðVS�p�in½t�Þ: (4.130)

The total demand of the consumer products at the price VP__c[t]:

VD�ps�c½t� ¼ VD�p�23c½t�: (4.131)

The total supply of the consumer products at the price VP__c[t]:

VS�ps�c½t� ¼ SUMðVS�p�ic½t�Þ: (4.132)

Thus, we have 44 + 44 + 2 + 2 ¼ 92 formulas for determining the total supply

and demand of the products in the domestic markets.

Let us present the notation defining the total supply and demand of the exported

products:

The total demand of the exported products at the price CP_pex_ic[t] (given) is

VD�pex�ic½t�: (4.133)

The total supply of the exported products at the price CP_pex_ic[t] is

VS�pex�ic½t�: (4.134)

Finally, we derive 92 + 44 ¼ 136 formulas for determining the total supply and

demand of all products used in the model.

Let us proceed to describing the activity of the economic agents participating in

the model.

4.2.1.17 Economic Agents 1–22 Producing Products and Services

Since the prices of the products of the producing agents 1 and 19 enter the model

exogenously, we introduce the following equations correcting the shares of budget

VO_p1_iz and VO_p19_iz:


VO�p1�iz½t;Qþ 1� ¼ VO�p1�iz½t;Q� � Ceta��1þ VO�p1�iz½t;Q� � VI�1�z½t;Q�� ð1� Ceta��1Þ;

(4.135)

VO�p19�iz½t;Qþ 1� ¼ VO�p19�iz½t;Q� � Ceta��19þ VO�p19�iz½t;Q�� VI�19�z½t;Q� � ð1� Ceta��19Þ (4.136)

Here Q is the step iteration; 0 < Ceta__1 < 1 and 0 < Ceta__19 < 1 are the

model constants. When they increase, equilibrium is attained more slowly. Never-

theless, the system of equations becomes more stable.

Let us proceed to the formulas determining the behavior of the producing agents.


VY��i½tþ1� ¼CA�r�i�PowerðVD�p1�iz½t�;CA�z�1iÞ�PowerðVD�p2�iz½t�;CA�z�2iÞ�PowerðVD�p3�iz½t�;CA�z�3iÞ�PowerðVD�p4�iz½t�;CA�z�4iÞ�PowerðVD�p5�iz½t�;CA�z�5iÞ�PowerðVD�p6�iz½t�;CA�z�6iÞ�PowerðVD�p7�iz½t�;CA�z�7iÞ�PowerðVD�p8�iz½t�;CA�z�8iÞ�PowerðVD�p9�iz½t�;CA�z�9iÞ�PowerðVD�p10�iz½t�;CA�z�10iÞ�PowerðVD�p11�iz½t�;CA�z�11iÞ�PowerðVD�p12�iz½t�;CA�z�12iÞ�PowerðVD�p13�iz½t�;CA�z�13iÞ�PowerðVD�p14�iz½t�;CA�z�14iÞ�PowerðVD�p15�iz½t�;CA�z�15iÞ�PowerðVD�p16�iz½t�;CA�z�16iÞ�PowerðVD�p17�iz½t�;CA�z�17iÞ�PowerðVD�p18�iz½t�;CA�z�18iÞ�PowerðVD�p19�iz½t�;CA�z�19iÞ�PowerðVD�p20�iz½t�;CA�z�20iÞ�PowerðVD�p21�iz½t�;CA�z�21iÞ�PowerðVD�p22�iz½t�;CA�z�22iÞ�Power VK��i½t�þVK��i½tþ1�ð Þ=2ð Þ;CA�k�ið Þ�PowerðVD�pi�il½t�;CA�l�iÞ:

(4.137)

Here CA_r_i, CA_z_ji (j ¼ ð1; 22Þ), 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 of the production factors by the

i-th agent.

The demand of the labor force:

VD�pi�il½t� ¼ ðCO�pi�il� VB��i½t�Þ=VP��il½t�: (4.138)

The demand of the intermediate products produced by all the producing agents:

VD�p1�iz½t� ¼ ðVO�p1�iz½t� � VB��i½t�Þ=CP��1z½t�; (4.139)

VD�p2�iz½t� ¼ ðCO�p2�iz� VB��i½t�Þ=VP��2z½t�; (4.140)











VD�p12�iz½t� ¼ CO�p12�iz� VB��i½t�ð Þ=VP��12z½t�; (4.150)







VD�p19�iz½t� ¼ VO�p19�iz� VB��i½t�ð Þ=CP��19z½t�; (4.157)



VD�p22�iz½t� ¼ CO�p22�iz� VB��i½t�ð Þ=VP��22z½t�: (4.160)

The demand of the investment products:

VD�p�in½t� ¼ CO�p�in� VB��i½t�ð Þ=VP��n½t�: (4.161)

The following formulas determine the supply of the products and services

produced by the producing agent.


The supply of the intermediate products:

VS�pi�iz½t� ¼ CE�pi�iz� VY��i½t�: (4.162)

The supply of the final products:

VS�p�ic½t� ¼ CE�p�ic� VY��i½t�: (4.163)

The supply of the investment products:

VS�p�in½t� ¼ CE�p�in� VY��i½t�: (4.164)

The supply of the exported products:

VS�pex�ic½t� ¼ CE�pexi�ic� VY��i½t�: (4.165)

The following formula calculates the gain of the producing agent:

VY�p�i½t� ¼ VS�pi�iz½t� � VP��iz½t� þ VS�p�ic½t� � VP��c½t� þ VS�p�in½t�� VP��n½t� þ VS�pex�ic½t� � CP��exi½t�:

(4.166)

The profit of the producing agent:

VY�r�i½t� ¼CO�y�i�VY�p�i½t�� ðVD�p1�iz½t�þVD�p2�iz½t�þVD�p3�iz½t�þVD�p4�iz½t�þVD�p5�iz½t�þVD�p6�iz½t�þVD�p7�iz½t�þVD�p8�iz½t�þVD�p9�iz½t�þVD�p10�iz½t�þVD�p11�iz½t�þVD�p12�iz½t�þVD�p13�iz½t�þVD�p14�iz½t�þVD�p15�iz½t�þVD�p16�iz½t�þVD�p17�iz½t�þVD�p18�iz½t�þVD�p19�iz½t�þVD�p20�iz½t�þVD�p21�iz½t�þVD�p22�iz½t�þ VW��i½t��CO�w�ið ÞþCA�n½t��ðVK��i½t��VP��n½t�ÞÞ:

(4.167)

Here CO_y_i is the coefficient reflecting the level of nonpayments; CA_n is the

depreciation rate of the 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 the capital assets are subtracted.

The value added produced by the ith sector is given by

VY�g�i½t� ¼ VY�r�i½t� þ VW��i½t�: (4.168)

The value added consists of the profit received in the current period and wages

actually paid to the sector employees.


The budget of the producing agent is as follows:

VB��i½t� ¼ VB�b�i½t� 1� � ð1þ CP��bpercent½t� 1�Þ þ CO�y�i� VY�p�i½t� þ VG�s�i24½t� 1�: (4.169)

The agent budget consists of the following:

1. The funds in bank accounts (taking into consideration interest on deposits);


3. The subsidies received from the consolidated budget VG_s_i24.

The dynamics of the bank account balance of the producing agent are as follows:

VB�b�i½t� ¼ VO�s�i½t� � VB��i½t�: (4.170)


VK��i½tþ 1� ¼ 1� CR��i½t�ð Þ � VK��i½t� þ VD�p�in½t�: (4.171)



The share of the budget of the producing agent for discharging the taxes to the


VO�tc�i½t� ¼ VY�g�i½t� � CT�vad½t�ð Þ=VB��i½t�þ VY�r�i½t� � CT�pr½t�ð Þ=VB��i½t�: (4.172)

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�i½t� ¼ VW��i½t� � CT�esn½t�ð Þ=VB��i½t�: (4.173)

The remainder of the budget of the producing agent is given by

VO�s�i½t� ¼ 1� ðCO�pi�ilþ CO�p�inþ VO�tc�i½t� þ VO�tf�i½t� þ VO�p1�iz½t�þ CO�p2�izþ CO�p3�izþ CO�p4�izþ CO�p5�izþ CO�p6�izþ CO�p7�izþ CO�p8�izþ CO�p9�izþ CO�p10�izþ CO�p11�izþ CO�p12�izþ CO�p13�izþ CO�p14�izþ CO�p15�izþ CO�p16�izþ CO�p17�izþ CO�p18�izþ CO�p19�iz½t� þ CO�p20�izþ CO�p21�izþ CO�p22�izÞ

(4.174)


4.2.1.18 Economic Agent 23. Aggregate Consumer (Households)

Let us proceed to the formulas determining the behavior of the aggregate consumer.

The household demand of the final products is given by

VD�p�23c½t� ¼ CO�p�23c� VB��23½t�ð Þ=VP��c½t�: (4.175)

The wages of the employees of Sectors 1–22:

VW��i½t� ¼ VD�pi�il½t� � VP��il½t�: (4.176)

The budget of the households is determined as follows:

VB��23½t� ¼ VB�b�23½t� 1� � 1þ CP�h�bpercent½t� 1�ð Þ þ VB��23½t� 1�� VO�s�23½t� 1� þ VG�tr�24½t� 1� þ VG�f�24½t� 1� þ SUMðVW��i½t�Þ:

(4.177)



2. The retained money in cash kept from the preceding period;

3. The pensions, welfare payments, and subsidies received from the off-budget

funds;

4. The wages received from the producing agents 1–22.

The dynamics of the banking account balance of the households are as follows:

VB�b�23½t� ¼ CO�b�23� VB��23½t�: (4.178)

The share of the budget for discharging the income tax is given by

VO�tc�23½t� ¼ ðSUMðVW��i½t� � CT�pod½t�Þ=VB��23½t�: (4.179)

The remainder of the money in cash is as follows:

VO�s�23½t� ¼ 1� CO�p�23c� VO�tc�23½t� � CO�b�23: (4.180)

4.2.1.19 Economic Agent 24. Government

As presented above, this economic agent is represented by the aggregate of the

federal, regional, and local governments, as well as the off-budget funds. Moreover,

it includes the nonprofit organizations servicing households.



The consolidated budget is given by

VB��24½t� ¼ SUM VO�tc�i½t� � VB��i½t�ð Þ þ VO�tc�23½t� � VB��23½t�þ CB�other�24½t� þ VB�b�24½t� � 1þ CP��bpercent½t� 1�ð Þ:

(4.181)

This formula sums up the money collected as taxes from the producing agents, as

well as from the population. The value CB_other_24 entering the model exoge-

nously is the sum of the 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 bank accounts (subject to the interest

on deposits).


determined by

VB�b�24½tþ 1� ¼ VO�s�24½t� � VB��24½t�: (4.182)

The cash assets of off-budget funds are as follows:

VF��24½t� ¼ SUM VO�tf�i½t� � VB��i½t�ð Þ þ VF�b�24½t�� 1þ CP��bpercent½t� 1�ð Þ: (4.183)

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 derived sum is added by the funds in bank accounts (subject to the interest

on deposits).

The dynamics of the bank account balance of the off-budget funds are deter-

mined by

VF�b�24½tþ 1� ¼ VO�s�24f ½t� � VF��24½t�: (4.184)

The subsidies to the producing sectors are as follows:

VG�s�i24½t� ¼ CO�s�i24� VB��24½t�: (4.185)


VG�tr�24½t� ¼ CO�tr�24� VB��24½t�: (4.186)


The assets of the off-budget funds made available for the population:

VG�f�24½t� ¼ CO�f�24� VF��24½t�: (4.187)




Let us present the formulas for calculating some integral indexes of the economy of

the Russian Federation.

The gross production of goods and services (in prices of the base period):

VY½t� ¼ SUM VY��i½t�ð Þ: (4.188)

The total supply of the intermediate products (in prices of the base period):

VS��z½t� ¼ SUM VS�pi�iz½t�ð Þ: (4.189)

The total supply of the final products (in prices of the base period):

VS��c½t� ¼ SUM VS�p�ic½t�ð Þ: (4.190)

The GDP of Russia:

VY�g½t� ¼ SUM VY�g�i½t�ð Þ=VP��c½0�: (4.191)

The consumer price index:

VP½t� ¼ 100� VP��c½t�=VP��c½t� 1�ð Þ: (4.192)

The model is presented in the context of the following common relations:

Relations (1.7) are represented by n1 ¼ 47 expressions;

Relations (1.8) are represented by n2 ¼ 945 expressions;

Relations (1.9) are represented by n3 ¼ 88 expressions.

The exogenous parameters of the model are determined by solving the problem

of parametric identification of the model using the available statistical information

from the Russian Federation for the period 2000–2008. The validity of the model

and identification process is checked just in the case of the model with the

knowledge sector.

As a result of solving the parametric identification problem, the relative mean

square deviation of the calculated values of the endogenous parameters from

the respective measured values (the statistical information) does not exceed


0.85%. The mean error of the retrospective prediction for the period of 2005–2008

also does not exceed 1%.

4.2.2 Finding Optimal Parametric Control Laws on the Basisof the CGE Model of the Economic Sectors

Some results of applying the parametric control theory approach and their compar-

ison with the results of applying the scenario approaches [24] are demonstrated

below.

The following two criteria were used in experiments as the optimization criteria:

– The mean of the value added of the country for the years 2004–2008 in prices

of the year 2000:

K1 ¼ 1

5

X2008t¼2004

VY�g½t� � VP½2000�=VP½t� ! max; (4.193)

– The mean of ratios of the value added to the country production for the years

2004–2008 in prices of the year 2000:

K2 ¼ 1

5

X2008t¼2004

VY�g½t� � VP½2000�=ðVP½t� � VY½t�Þ ! max : (4.194)

The values of these criteria for the base calculated variant (with the use of the

values of the exogenous parameters obtained as a result of the model identification)

are equal to K1* ¼ 112,005 � 108 (rubles) and K2* ¼ 0.57557, respectively.

The first part of the computational experiments with the model includes control

of the shares of the consolidated budget for backing the economic agents producing

the goods and services 1–22. The shares of the consolidated budget of the country

VG_s_j_24, j ¼ 1; 22, for backing Sectors 1–22 for the base variant of the model

computation were determined from [24] and by means of parametric identification

of the model of the economic branches.

The following problem finds the optimal value of the adjusted vector of

parameters.

Find the shares of the consolidated budget (VG_s_j_24[t], j ¼ 1; 22; t ¼2004; 2008) for backing the 22 sectors of the country’s economy that ensure the

supremum of criterion K1 (or K2) under the following additional constraints:

1. The subsidies from the consolidated budget of each branch of the economy

and in every considered year are to be not less than 50% of the respective

subsidies of the base variant, namely, VG�s�j�24 ½t� � aVG�s�j�24�½t�,j ¼ 1; 22; t ¼ 2004; 2008. In computations, a ¼ 0:5 is accepted.


2. For every considered year, the total subsidies to all 22 branches are not to exceed

the respective total subsidies of the base variant:

X22j¼1

VG�s�j�24½t� �X22j¼1

VG�s�j�24�½t�; t ¼ 2004; 2008:

The rest of the exogenous parameters of the model do not change in comparison

with the base variant.

The problem of maximizing the criteria K1 and K2 stated above is solved by

means of the Nelder–Mead algorithm [66], and the following results are obtained:

After applying the parametric control of the shares of the consolidated budget

Oj;t for backing Sectors 1–22, the values of the criteria (4.193, 4.194) increase

by c6.199% and 1.9179%, respectively, in comparison with the base variant.

The second part of the computational experiments with the model includes the

control of the supplementary subsidies allotted to five selected sectors.

In [24], the development scenario implies yearly (starting from the year 2004)

increase of the financial investments (in rubles) from the state budget to the selected

five sectors (22, 20, 16, 15, 19) in the following way:

Sector j Supplementary yearly investments

22 34 � 109

20 62 � 109

16 156 � 109

15 110 � 109

19 426 � 109

The total volume of the supplementary yearly financing is equal to 788 � 109.

Implementation of this scenario results in an increase of criteria (4.193, 4.194)

by 2.1205% and 1.2352% in comparison with the base result.

We consider the following problem of finding the optimal values of the yearly

increment of the financial investment to the given five sectors:

Find the optimal, in the sense of criteria (4.193, 4.194), supplementary yearly

financial investments to Sectors 22, 20, 16, 15, 19 under the following constraints:

the total yearly volume of these investments is not to exceed the respective value

from the scenario of [24], namely, 788 � 109.

Solving the stated parametric control problem by means of the Neldor–Mead

algorithm results in an increase of criteria (4.193, 4.194) by 11.8158%

and 7.4547%, respectively, in comparison with the base variant.

Also, we carried out the computational experiments with the model for

analyzing the abilities of the supplementary investments to other quintuples of

selected branches (in comparison with the scenario of [24]) subject to maintaining

the total volume of the supplementary investments equal to 788 � 109. Choosing

the groups of the sectors is carried out on the basis of a particular criterion, namely,


the mean ratio of the value added to the production of the jth sector for the years

2004–2008 in prices of the year 2000:

Kj;2 ¼ 1

5

X2008t¼2004

VY�g�j ½t� � VP½2000�=ðVP½t� � VY�g�j½t�Þ ! max; j ¼ 1; 22

(4.195)

On the basis of criterion (4.195), the following quintuple of sectors with

the highest values of criterion Kj;2 is determined:

Sector j Criterion Kj;2

4 0.75393

17 0.75050

18 0.62178

20 0.59417

3 0.57883

The following quintuple of sectors with the lowest values of criterion Kj;2 is

obtained:

Sector j Criterion Kj;2

5 0.34692

6 0.33229

7 0.32942

11 0.28573

12 0.27603

Applying the optimal values of the yearly increments of the financial

investments to the sectors of the first group (4, 17, 18, 20, 3) results in an increase

of criterion (4.193) by 7.8242% and an increase of criterion (4.194) by 4.1809%

in comparison with the base variant.

Applying the optimal values of the yearly increments of the financial

investments to the sectors of the second group (5, 6, 7, 11, 12) results in an increase

of criterion (4.193) by 6.9988%, whereas criterion (4.194) increases by 3.1795%

in comparison with the base variant.

Based on the analysis of this section, we can draw the following conclusion.

With fixed total volume of the supplementary investment, the optimal choice of

the quintuple of the sectors and volumes of investment to each sector can result in

an increase of the value added by up to 9% in comparison with the scenario

proposed in [24].


4.2.3 Analysis of the Dependence of the Optimal ParametricControl Law on Values of Uncontrolled Parameterson the Basis of the CGE Model of Economic Sectors

All optimization problems considered in Sect. 4.2.2 are solved with the fixed values

of the exogenous parameters not participating in the control. Moreover, while

carrying out the research, we determined the dependence of the values of criteria

(4.193) and (4.194) on the values of the uncontrolled model parameters by

the example of the bivariate parameter l ¼ ðl1; l2Þ, where l1 ¼ CO_p_23c is theshare of the budget of the aggregated consumer spent for purchasing final products;

l2 ¼ CO_b_23 is the share of the budget of the aggregated consumer deposited in

bank accounts.

The region of variation of these parameters is defined from the measured values

l1 and l2 : L ¼ ½0:759; 0:834� � ½0:070; 0:077�.Figures 4.3 and 4.4 present some results of the research, namely, the plots of

the dependence of criterion K1 on the parameter l (where l 2 L) for the parametric

control problems considered above.

The plots in Fig. 4.3 show the base and optimal values of criterion K1 for the

problem of finding the shares of the consolidated budget of the country

VG�s�j�24½t� for backing Sectors 1–22. The plots in Fig. 4.4 show the base and

1.2

1.18

1.16

1.14

1.12

1.1

1.08

1.000.759

0.7710.784

0.7960.809

0.8220.834

Op OB

K1

x 1013

0.0770.076

0.0750.074

0.070.071

0.072

Fig. 4.3 – base variant, – control of shares of consolidated budget of the country for backing

Sectors 1–22


optimal values of criterion K2 for the problem of finding the supplementary yearly

financial investments to Sectors 22, 20, 16, 15, and 19.

4.3 National Economic Evolution Control Basedon a Computable Model of General Equilibriumwith the Shady Sector


4.3.1.1 Economic Agents of the Model

The model discussed here [24] describes the behavior and interaction in ten product

markets and three labor markets of the following seven economic agents: The first

three of them are the producing agents.

0.64

0.62

0.6

0.58

0.759

0.771

0.784

0.796

0.809

0.822

0.834 0.077

0.076

0.075

0.074

0.0720.071

0.07

OBOp

K2

Fig. 4.4 – base variant, – control of supplementary yearly investments to Sectors 22, 20, 16,

15, 19

4.3 National Economic Evolution Control Based on a Computable Model. . . 215

Economic agent 1 is the state sector of the economy. This includes enterprises for

which the state owns more than a 50% share.

Economic agent 2 is the market sector consisting of legally existing enterprises and

organizations with private and mixed ownership.

The state and market sectors produce the products distributed among the following

four directions:

1. The final product for households including consumer nondurable goods

(foodstuffs, etc.), durable goods (house equipment, motor vehicles, etc.), as

well as services;

2. The final products for economic agent 5, including:

(a) The final product for public institutions including free services for the

population rendered by the enterprises and organizations in the field of

public health, education, and culture; services satisfying the needs of society

as a whole, 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 households including

free services of a social character;

3. The investment products, namely, the costs of creation of capital assets. This

kind of product does not include state (or governmental) investment, since it is

taken into account in the preceding kinds of products. Capital assets are consid-

ered a separate kind 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 the exported

products.

Besides the produced products, the state and market sectors trade the capital

assets represented by the capital products in the model.

Economic agent 3 is the shady sector.

There are various kinds of shady economics [24], namely:

– White-collar shady economics is the unofficial economic activity of the

employees of the registered economy concerned with their official professional

activity. This includes the economics of informal ties (i.e., offstage performance

of ordinary production programs); upward distortion economics (presenting

fictitious results as real); bribe economics (abuse by official status of public

officers for achieving private goals).

– So-called “gray” (informal) shady economics is lawful economic activity that is

not accounted for by official statistics. This sector of shady economics produces

mainly ordinary goods and services (just as in legal economics), but the

producers avoid official taxes, not wishing to pay the additional costs connected

with the discharge of taxes, etc.


– So-called “black” (underground) shady economics is statute-prohibited eco-

nomic activity concerned with the production and selling prohibited products

and services (selling drugs, racketeering, etc.).

As for the shady sector of the model under consideration here, it includes “gray”

shady economics, as well as white-collar shady economics represented by the

production of final goods for the households by the market sector of the economy.

The shady sector sells only one kind of product, namely, final products for

households. This economic agent does not pay taxes or receive subsidies. The

shady sector realizes the following actions:

– By the distribution of its budget, it pays for the services of the labor force and

determines the share of the retained budget;

– By the distribution of the produced products, it determines the share of the final

products for selling in the market of final products for the households at the

shady price.

Economic agent 4 is the aggregated consumer representing all the 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 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 5 establishes the taxation rates and defines the sum of the

subsidies to the producing agents and social transfer, and spends its budget for

purchasing the final products produced by the state and market sectors.

Economic agent 6 is the banking sector including the central bank and commercial

banks.

The banking sector establishes the interest rates for deposits and issues money.



The exogenous parameters of the model include the following:


– The various shares of the sectors’ budgets;

– The shares of the production for selling in the various markets;

– The depreciation rates for capital assets;


– The issue of money;


– The shares of the consolidated budget spent for purchasing final goods, backing

the state and market sectors, as well as for social transfers;

– The export prices of final goods for the outside world.


The list of the exogenous parameters of the model is given below.

4.3.1.3 Sector 1

O_1k_P2 The share of the state budget spent for purchasing capital products at the price P_2k.

O_1i_P2 The share of the state budget spent for purchasing investment products at the price

P_2i.

E_1c_P2 The share of the state sector products for selling in the markets of final products at the

price P_2c.

E_1g_P1 The share of the state sector products for selling in the markets of final products for

economic agent 5 at the price 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 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 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 P_2k.

E_1ex_Pex The share of the capital assets of the state sector for selling in the markets of capital

products in foreign countries at the price 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.

4.3.1.4 Sector 2

O_2l_P2 The share of the budget of the market sector spent for purchasing the labor force at the

price P_2l.

O_2k_P2 The share of the budget of the market sector spent for purchasing capital products at

the price P_2k.

O_2i_P2 The share of the budget of the market sector spent for purchasing investment products

at the price P_2i.

E_2c_P2 The share of the market sector products for selling in the markets of final products at

the price P_2c.

E_2c_P3 The share of the market sector products for selling in the markets of final products at

the price 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


(continued)


E_2k_P2 The share of the market sector products for selling in the markets of capital products

at the price P_2k.E_2ex_Pex The share of the market sector products for selling in the markets of final products in

foreign countries at the price 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.

4.3.1.5 Sector 3

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.

4.3.1.6 Aggregated Consumer

L_1_a The percentage of employees entering the state sector (e.g., starting their working

activity in the state sector).

L_1_r The percentage of employees withdrawing from the state sector (for example, retired

employees).

L_2_a The percentage of employees entering the market sector (e.g., starting their working

activity in the market sector).

L_2_r The percentage of employees withdrawing from the market sector (for example, retired

employees).

O_4c_P1 The share of household budgets for purchasing final products at the price P_1c.

O_4c_P2 The share of household budgets for purchasing final products at the price of P_2c.

O_4c_P3 The share of household budgets for purchasing final products at the price of P_3c.

O_4_$ The share of household budgets for purchasing foreign currency.

O_4_b The share of household budgets for saving (as bank deposits).

L_1_2 The percentage of state sector employees leaving for the market sector.

L_2_1 The percentage of market sector employees leaving for the state sector.

L_12_3 The percentage of state and market sector employees partially employed in the shady

market.


4.3.1.7 Government

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 social transfers.

O_5_f1 The share of expenditures of off-budget funds spent for the state sector.

O_5_f2 The share of expenditures of off-budget funds spent for the market sector.

O_5_f4 The share of expenditures of off-budget funds spent for households.

B_5_Other The sum of tax proceeds (not otherwise included elsewhere), nontax incomes, and

other incomes of the consolidated budget.


M_1 The issue of money of the state sector.

M_2 The issue of money of the market sector.

P_b The deposit interest rate for enterprises.

P_b_h The deposit interest rate for physical bodies.

4.3.1.9 General Part of Model

P_1l The state prices of the labor force.

P_1C The state prices of final products for households.

P_1g The state prices of final products for economic agent 5.

P_1i The state prices of investment products.

P_1k The state prices of capital products.

P_1ex The state prices of investment products for the outside world.


4.3.1.10 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 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.




– The shares of the produced products for selling in the various markets;

– The remainders of the agents’ budgets;

– The produced values added of the producing sectors;

– Supplies and demands of the various products and services;


– The capital assets of the producing sectors;

– The share of employees withdrawing from each of the producing sectors;

– The share of employees entering each of the producing sectors;

– The wages of employees;


– The various kinds of prices of products, services, and the labor force;

– The subsidies to the producing sectors;

– The social transfers to the population;

– The gross production of goods and services;

– The GDP.

A list of the endogenous variables of the model is given below.

4.3.1.12 Sector 1

O_1l_P1 The share of the state sector budget for purchasing the labor force at the price P_1l.

O_1k_P1 The share of the state sector budget spent for purchasing capital products at the price

P_1k.

O_1i_P1 The share of the state sector budget spent for purchasing investment products at the

price 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 the off-budget funds.

O_1_s The share of the retained budget of the state sector.

(continued)


E_1c_P1 The share of the state sector products for selling in the markets of final products at the

price P_1c.Y_1 The value added of the state sector (in prices of the base period).

S_1c_P1 The supply of final products by the state sector at the price P_1c.

S_1c_P2 The supply of final products by the state sector at the price P_2c.

S_1g_P1 The supply of final products by the state sector for economic agent 5 at the price P_1g.

S_1g_P2 The supply of the final products by the state sector for economic agent 5 at the price

P_2g.

S_1i_P1 The supply of investment products by the state sector at the price P_1i.

S_1i_P2 The supply of investment products by the state sector at the price P_2i.

S_1k_P1 The supply of capital products by the state sector at the price P_1k.

S_1k_P2 The supply of capital products by the state sector at the price P_2k.

S_1ex_Pex The supply of exported products by the state sector at the price P_ex.

D_1l_P1 The demand of the labor force in the state sector at the price P_1l.

D_1k_P1 The demand of capital products in the state sector at the price P_1k.

D_1k_P2 The demand of capital products in the state sector at the price P_2k.

D_1i_P1 The demand of investment products in the state sector at the price P_1i.

D_1i_P2 The demand of investment products in the state sector at the price 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 bank accounts of the state sector.

K_1 The capital assets of the state sector.

4.3.1.13 Sector 2

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 taxes to the consolidated

budget.

O_2_f The share of the market sector budget for discharging 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 prices of the base period).

S_2c_P2 The supply of final products by the market sector at the price P_2c.

S_2c_P3 The supply of final products by the market sector at the price P_3c.

S_2g_P2 The supply of final products by the market sector for the economic agent 5 at the

price P_2g.

S_2i_P2 The supply of investment products by the market sector at the price P_2i.

S_2k_P2 The supply of capital products by the market sector at the price P_2k.

S_2ex_Pex The supply of exported products by the market sector at the price P_ex.

D_2l_P2 The demand of the labor force in the market sector at the price P_2l.

D_2k_P1 The demand of capital products in the market sector at the price P_1k.

D_2k_P2 The demand of capital products in the market sector at the price P_2k.

D_2i_P1 The demand of investment products in the market sector at the price P_1i.

(continued)


D_2i_P2 The demand of investment products in the market sector at the price 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 bank accounts of the market sector.

K_2 The capital assets of the market sector.

4.3.1.14 Sector 3

O_3_s The share of the retained budget of the shady sector.

Y_3 The value added of the shady sector (in prices of the base period).

S_3c_P3 The supply of final products by the shady sector at the price P_3c.

D_3l_P3 The demand of the labor force in the shady sector at the price 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 bank accounts of the shady sector.

K_3 The capital assets of the shady sector.

4.3.1.15 Aggregated Consumer

M_4 The issue of money of the households.

O_4_tax The share of household budgets 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 households’ demand of final products at the price 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 bank accounts.


4.3.1.16 Government

O_5g_P1 The share of the consolidated budget spent for purchasing the final products at the price

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 of final products at the price P_1g.

D_5g_P2 The demand of final products at the price 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 The social transfers to the population formed from the funds of the consolidated

budget.

G_1_f The off-budget funds assigned to the state sector.

G_2_f The off-budget funds assigned to the market sector.

G_4_f The off-budget funds assigned to the population.

B_5 The consolidated budget.

B_5_b The balance of the banking accounts of the consolidated budget.

F_5 The money assets of the off-budget funds.

F_5_b The balance of the banking accounts of the off-budget funds.

4.3.1.17 General Part of Model

P_2l The market prices of the labor force.

P_2c The market prices of the final products for the households.

P_2g The market prices of the final products for economic agent 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 households.

I_g The deficiency indicator for the market of final products for economic agent 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 of the labor force at the price P_1l.



S_sl_P1 The total supply of the labor force at the price P_1l.



D_sc_P1 The total demand of final products at the price P_1c.



S_sc_P1 The total supply of final products at the price P_1c.

(continued)


S_sc_P2 The total supply of final products at the price P_2c.S_sc_P3 The total supply of final products at the price P_3c.

D_sg_P1 The total demand of final products at the price P_1g.

D_sg_P2 The total demand of final products at the price P_2g.

S_sg_P1 The total supply of final products at the price P_1g.

S_sg_P2 The total supply of final products at the price P_2g.

D_si_P1 The total demand of investment products at the price P_1i.

D_si_P2 The total demand of investment products at the price P_2i.

S_si_P1 The total supply of investment products at the price P_1i.

S_si_P2 The total supply of investment products at the price P_2i.

D_sk_P1 The total demand of capital products at the price P_1k.

D_sk_P2 The total demand of capital products at the price P_2k.

S_sk_P1 The total supply of capital products at the price P_1k.

S_sk_P2 The total supply of capital products at the price P_2k.

Y The GDP (in 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.


The equilibrium prices are formed in 13 markets of the model as a result of

equalization of the supplies and demands of the various kinds of products, services,

and labor force:

– The markets of final products for households with governmental, market, and

shady prices;

– The markets of final products for economic agent 5 with governmental and

market prices;

– The markets of capital products with governmental and market prices;

– The markets of investment products with governmental and market prices;

– The markets of the labor force with governmental, market, and shady prices;

– The markets of exported products.

For each market, we determine the total supply and demand equalized during the

iterative recalculation. The formulas determining the deficiency indicators for the

markets with the governmental prices used in the model are presented below.

The labor force market:

I�l½t� ¼ S�sl�P1½t�=D�sl�P1½t�: (4.196)

The market of the final products for the households:

I�c½t� ¼ S�sc�P1½t�=D�sc�P1½t�: (4.197)


The market of the final products for economics agent 5:

I�g½t� ¼ S�sg�P1½t�=D�sg�P1½t�: (4.198)

The market of investment products:

I�i½t� ¼ S�si�P1½t�=D�si�P1½t�: (4.199)

The market of capital products:

I�k½t� ¼ S�sk�P1½t�=D�sk�P1½t�: (4.200)

As is obvious, the deficiency indicator is the ratio of the product supply to its

demand.

Let us now present the model formulas reflecting the market process of changing

the labor force prices:

P�2l½t;Qþ 1� ¼ P�2l½t;Q� � C�2l; C�2l ¼ D�sl�P2½t;Q�=ðS�sl�P2½t;Q� � CÞ;(4.201)

The prices of the final products for households:

P�2c½t;Qþ 1� ¼ P�2c½t;Q� � C�2c; C�2c ¼ D�sc�P2½t;Q�=ðS�sc�P2½t;Q� � CÞ;(4.202)

The prices of the final products for economic agent 5:

P�2g½t;Qþ 1� ¼ P�2g½t;Q� � C�2g; C�2g ¼ D�sg�P2½t;Q�=ðS�sg�P2½t;Q� � CÞ;(4.203)

The prices of investment products:

P�2i½t;Qþ 1� ¼ P�2i½t;Q� � C�2i; C�2i ¼ D�si�P2½t;Q�=ðS�si�P1½t;Q� � CÞ;(4.204)

The prices of capital products:

P�2k½t;Qþ 1� ¼ P�2k½t;Q� � C�2k; C�2k ¼ D�sk�P2½t;Q�=ðS�sk�P2½t;Q� � CÞ;(4.205)

The equilibrium price in the shady markets forms similar to the market price.

The respective formulas are presented below.

The labor market price:

P�3l½t;Qþ 1� ¼ P�3l½t;Q� � C�3l; C�3l ¼ D�sl�P3½t;Q�=ðS�sl�P3½t;Q� � CÞ;(4.206)


The price of final products for households:

P�3c½t;Qþ 1� ¼ P�3c½t;Q� � C�3c; C�3c ¼ D�sc�P3½t;Q�=ðS�sc�P3½t;Q� � CÞ:(4.207)

Let us now present the formulas describing the total supply and demand of the

products for each of the prices used in this model. The final formulas determining

the supply and demand of each economic agent are given below.

The total demand of the labor force at governmental, market, and shady prices:

D�sl�P1½t� ¼ D�1l�P1½t�; (4.208)

D�sl�P2½t� ¼ D�2l�P2½t�; (4.209)

D�sl�P3½t� ¼ D�3l�P3½t�: (4.210)

The total supply of the labor force at governmental, market, and shady prices:

S�sl�P1½t� ¼ L�1½t�; (4.211)

S�sl�P2½t� ¼ L�2½t�; (4.212)

S�sl�P3½t� ¼ L�3½t�: (4.213)

The total demand of final products at governmental, market, and shady prices:

D�sc�P1½t� ¼ D�4c�P1½t�; (4.214)

D�sc�P2½t� ¼ D�4c�P2½t�; (4.215)

D�sc�P3½t� ¼ D�4c�P3½t�: (4.216)

The total supply of final products at governmental, market, and shady prices:

S�sc�P1½t� ¼ S�1c�P1½t�; (4.217)

S�sc�P2½t� ¼ S�1c�P2½t� þ S�2c�P2½t�; (4.218)

S�sc�P3½t� ¼ S�2c�P3½t� þ S�3c�P3½t�: (4.219)

The total demand of final products for economic agent 5 at governmental and

market prices:

D�sg�P1½t� ¼ D�5g�P1½t�; (4.220)

D�sg�P2½t� ¼ D�5g�P2½t�: (4.221)


The total supply of final products for economic agent 5 at governmental and

market prices:

S�sg�P1½t� ¼ S�1g�P1½t�; (4.222)

S�sg�P2½t� ¼ S�1g�P2½t� þ S�2g�P2½t� (4.223)

The total demand of investment products at governmental and market prices:

D�si�P1½t� ¼ D�1i�P1½t� þ D�2i�P1½t�; (4.224)

D�si�P2½t� ¼ D�1i�P2½t� þ D�2i�P2½t�: (4.225)

The total supply of investment products at governmental and market prices:

S�si�P1½t� ¼ S�1i�P1½t�; (4.226)

S�si�P1½t� ¼ S�1i�P2½t� þ S�2i�P2½t�: (4.227)

The total demand of capital products at governmental and market prices:

D�sk�P1½t� ¼ D�1k�P1½t� þ D�2k�P1½t�; (4.228)

D�sk�P2½t� ¼ D�1k�P2½t� þ D�2k�P2½t�: (4.229)

The total supply of capital products at governmental and market prices:

S�sk�P1½t� ¼ S�1k�P1½t�; (4.230)

S�sk�P2½t� ¼ S�1k�P2½t� þ S�2k�P2½t�: (4.231)

So, we have 24 formulas for determining the total supply and demand of the

products considered in the model.

4.3.1.19 Economic Agent 1. The State Sector

In the markets with governmental pricing, the equalization of the total supply and

demand proceeds by correction of the budget shares and the share of the finished

product. This process is described by the formulas

O�1l�P1½t;Qþ 1� ¼ O�1l�P1½t;Q� � etta�1lþ O�1l�P1½t;Q� � I�l½t;Q� � ð1� etta�1lÞ;(4.232)


O�1i�P1½t;Qþ 1� ¼ O�1i�P1½t;Q� � etta�1iþ O�1i�P1½t;Q�� I�i½t;Q� � ð1� etta�1iÞ; (4.233)

O�1k�P1½t;Qþ 1� ¼ O�1k�P1½t;Q� � etta�1k þ O�1k�P1½t;Q�� I�k½t;Q� � ð1� etta�1kÞ; (4.234)

E�1c�P1½t;Qþ 1� ¼ E�1c�P1½t;Q� � etta�1cþ E�1c�P1½t;Q�� I�c½t;Q� � ð1� etta�1cÞ: (4.235)

Here Q is the iteration step, and 0 < etta_1l, etta_1i, etta_1k, etta_1c < 1 are

the model constants. When they increase, equilibrium is attained more slowly.

Nevertheless, the system becomes more stable.

Let us proceed to the formulas determining the behavior of the state sector.


Y�1½tþ 1� ¼ A�1�r � power K�1½t� þ K�1½tþ 1�ð Þ=2ð Þ;A�1�kð Þ�powerðD�1l�P1½t�;A�1�lÞ

(4.236)

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 of the production factors within

the state sector.


D�1l�P1½t� ¼ ðO�1l�P1½t� � B�1½t�Þ=P�1l: (4.237)

The demand of the capital products:

At governmental prices:

D�1k�P1½t� ¼ ðO�1k�P1½t� � B�1½t�Þ=P�1; (4.238)

At market prices:

D�1k�P2½t� ¼ ðO�1k�P2� B�1½t�Þ=P�2k½t�: (4.239)



D�1i�P1½t� ¼ ðO�1i�P1½t� � B�1½t�Þ=P�1i; (4.240)

At market prices:

D�1i�P2½t� ¼ ðO�1i�P2� B�1½t�Þ=P�2i½t�: (4.241)


The following formulas determine the supply of products of the state sector.

The supply of final products for households:


S�1c�P1½t� ¼ E�1c�P1½t� � Y�1½t�; (4.242)

S�1c�P2½t� ¼ E�1c�P2� Y�1½t�: (4.243)

The supply of the final products for economic agent 5:


S�1g�P1½t� ¼ E�1g�P1� Y�1½t�; (4.244)

At market prices:

S�1g�P2½t� ¼ E�1g�P2� Y�1½t�: (4.245)

The supply of the investment products:


S�1i�P1½t� ¼ E�1i�P1� Y�1½t�; (4.246)

At market prices:

S�1i�P2½t� ¼ E�1i�P2� Y�1½t�: (4.247)

The supply of capital products:


S�1k�P1½t� ¼ E�1k�P1� K�1½t�; (4.248)

At market prices:

S�1k�P2½t� ¼ E�1k�P2� K�1½t�: (4.249)


S�1ex�Pex½t� ¼ E�1ex�Pex� Y�1½t�: (4.250)

The following formula calculates the gain of the state sector:

Y�1�p½t� ¼ S�1c�P1½t� � P�1cþ S�1c�P2½t� � P�2c½t� þ S�1g�P1½t� � P�1gþ S�1g�P2½t� � P�2g½t� þ S�1i�P1½t� � P�1iþ S�1i�P2½t� � P�2i½t�þ S�1k�P1½t� � P�1k þ S�1k�P2½t� � P�2k½t� þ S�1ex�Pex½t� � P�ex:

(4.251)


As is obvious, the gain consists of the gain from selling final products and

rendering services for households and economic agent 5, investment, capital, as

well as exported products.

The budget of the state sector is determined as follows:

B�1½t� ¼ B�1�b½t� � ð1þ CP�b½t� 1�Þ þ Y�1�p½t� þ G�1�s½t� 1�þ G�1�f ½t� 1� þM�1: (4.252)



2. The gain received within the current period;

3. The bounties received from the consolidated budget G_1_s;4. The part of the off-budget funds G_1_f;5. The emission of money M_1.

The dynamics of the banking account balance of the state sector are as follows:

B�1�b½tþ 1� ¼ O�1�s½t� � B�1½t�: (4.253)

The capital assets dynamics is determined by

K�1½tþ 1� ¼ K�1½t� � ð1� E�1k�P1� E�1k�P2Þ � ð1� A�1�nÞþ D�1k�P1½t� þ D�1k�P2½t� þ D�1i�P1½t� þ D�1i�P2½t�:

(4.254)

This formula calculates the volume of capital assets taking into account their

selling and wear and tear. Purchased assets and investments in 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�t½t� ¼ ðY�1�p½t� � T��vadÞ=B�1½t� þ ððY�1�p½t� �W�1½t� � K�1½t� � P�1k� A�1�nÞ � T��prÞ=B�1½t� þ ððK�1½t� � P�1kÞ � T��propÞ=B�1½t�:

(4.255)

This formula takes into consideration the value added tax (VAT), profit tax, and

property tax. While calculating the share of budget for discharging the profit tax, the

gain is reduced by the costs of the labor force W_1, as well as the depreciation

charges K_1[t] � P_1k � A_1_n.The share of the budget for discharging the single social tax to the off-budget


O�1�f ½t� ¼ ðW�1½t� � T��esnÞ=B�1½t�: (4.256)


The remainder of the budget of the state sector of the economy is given by

O�1�s½t� ¼ 1� O�1l�P1½t� � O�1k�P1½t� � O�1k�P2� O�1i�P1½t�� O�1i�P2� O�1�t½t� � O�1�f ½t�: (4.257)

4.3.1.20 Economic Agent 2. The Market Sector

The behavior of the market sector differs from that of the state sector

insignificantly. Therefore, we shorten the description of agent 1 in some places

where it is analogous to agent 1.

The market sector corrects the shares of its budget O_2k_P1 and O_2k_P1 for

purchasing capital and investment products at governmental prices. This process is

described by the formulas

O�2k�P1½t;Qþ 1� ¼ O�2k�P1½t;Q� � etta�2k þ O�2k�P1½t;Q�� I�k½t;Q� � ð1� etta�2kÞ; (4.258)

O�2i�P1½t;Qþ 1� ¼ O�2i�P1½t;Q� � etta�2iþ O�2i�P1½t;Q�� I�i½t;Q� � ð1� etta�2iÞ; (4.259)

where Q is the iteration step, and 0 < etta_2k, etta_2i < 1 are the model constants.

Let us proceed to the formulas for determining the behavior of the market sector.


Y�2½tþ 1� ¼ A�2�r � power ððK�2½t� þ K�2½tþ 1�Þ=2Þ;A�2�kð Þ� powerðD�2l�P2½t�;A�2�lÞ: (4.260)

Here A_2_r, A_2_k, A_2_l are the parameters of the production function.


market sector.


D�2l�P2½t� ¼ ðO�2l�P2� B�2½t�Þ=P�2l½t�: (4.261)

The demand of capital products:


D�2k�P1½t� ¼ ðO�2k�P1½t� � B�2½t�Þ=P�1k; (4.262)


At market prices:

D�2k�P2½t� ¼ ðO�2k�P2� B�2½t�Þ=P�2k½t�: (4.263)



D�2i�P1½t� ¼ ðO�2i�P1½t� � B�2½t�Þ=P�1i; (4.264)

At market prices:

D�2i�P2½t� ¼ ðO�2i�P2� B�2½t�Þ=P�2i½t�: (4.265)

The following formulas determine the supply of the products of the market

sector.

The supply of the products for households:

At market prices:

S�2c�P2½t� ¼ E�2c�P2� Y�2½t�; (4.266)

At shady prices:

S�2c�P3½t� ¼ E�2c�P3� Y�2½t�: (4.267)

The supply of final products for economic agent 5 at market prices:

S�2g�P2½t� ¼ E�2g�P2� Y�2½t�: (4.268)

The supply of investment products at market prices:

S�2i�P2½t� ¼ E�2i�P2� Y�2½t�: (4.269)

The supply of capital products at market prices:

S�2k�P2½t� ¼ E�2k�P2� K�2½t�: (4.270)


S�2ex�Pex½t� ¼ E�2ex�Pex� Y�2½t�: (4.271)

The following formula calculates the gain of the market sector:

Y�2�p ¼ S�2c�P2½t� � P�2c½t� þ S�2g�P2� P�2g½t� þ S�2i�P2½t�� P�2i½t� þ S�2k�P2½t� � P�2k½t� þ S�2ex�Pex½t� � P�ex: (4.272)


As is obvious, the gain consists of the gain from selling the final products and

rendering services for households and economic agent 5, investment, capital, as

well as exported products. As presented above, the gain from selling final products

and services for households at shady prices is not accounted for here.

The budget of the market sector is determined as follows:

B�2½t�¼B�2�b½t��ð1þCP�b½t�1�ÞþY�2�pþG�2�S½t�1�þG�2�f ½t�1�þM�2:

(4.273)




3. The subsidies received from the consolidated budget G_2_S2;4. The part of the off-budget funds G_2_f;5. The emission of money M_2.

The dynamics of the banking account balance of the market sector are as

follows:

B�2�b½tþ 1� ¼ O�2�s½t� � B�2½t�: (4.274)

The capital assets dynamics are determined by

K�2½tþ 1� ¼ K�2½t� � ð1� E�2k�P2Þ � ð1� A�2�nÞ þ D�2k�P1½t�þ D�2k�P2½t� þ D�2i�P1½t� þ D�2i�P2½t�: (4.275)


selling and wear and tear. Purchased assets and investments to capital assets enter

the formula with a plus sign.

The share of the budget of the market sector for discharging taxes to the


O�2�t½t� ¼ ðY�2�p� T��vadÞ=B�2½t� þ ððY�2�p�W�2½t� � K�2½t� � P�2k½t�� A�2�nÞ � T��prÞ=B�2½t� þ ðK�2½t� � P�2k½t�Þ � T��propð Þ=B�2½t�:

(4.276)

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 reduced by the costs of the labor force W_2, as well as the depreciation charges

K_2[t] � P_2k � A_2_n.


The share of the budget for discharging the single social tax to the off-budget


O�2�f ½t� ¼ ðW�2½t� � T��esnÞ=B�2½t�: (4.277)

The remainder of the budget of the market sector of the economy is given by

O�2�s½t� ¼ 1� O�2l�P2� O�2k�P1½t� � O�2k�P2� O�2i�P1½t�� O�2i�P2� O�2�t½t� � O�2�f ½t�: (4.278)

4.3.1.21 Economic Agent 3. The Shady Sector

Let us write down formulas for determining the behavior of the shady sector.


Y�3½tþ 1� ¼ A�3�r � power ðK�3½t� þ K�3½tþ 1�Þ=2ð Þ;A�3�kð Þ� powerðD�3l�P3½t�;A�3�lÞ: (4.279)

Here A_3_r, A_3_k, 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 in another way.

The shady sector does not have its own capital assets. The same can be seen in

real life, where the representatives of the “white-collar” and “gray” economies

use the capital assets of the state and market sectors. Therefore, the capital assets of

the shady sector are formed as follows:

K�3½t� ¼ gamma� ðK�1½t� þ K�2½t�Þ; (4.280)

where gamma is the share of the capital assets of the state and market sectors used

in shady economics.

The demand of the labor force at shady prices is calculated similarly to the other

sectors:

D�3l�P3½t� ¼ ðO�3l�P3� B�3½t�Þ=P�3l½t�: (4.281)

Then let us calculate the supply of final products for households at shady prices:

S�3c�P3½t� ¼ E�3c�P3� Y�3½t�: (4.282)


The following formula calculates the gain of the shady sector:

Y�3�p½t� ¼ ðs�2c�p3½t� þ s�3c�p3½t�Þ � p�3c½t�: (4.283)

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�3½t� ¼ B�3�b½t� � ð1þ CP�b½t� 1�Þ þ Y�3�p½t�: (4.284)


1. The funds in bank accounts (subject to the interest on deposits);

2. The gain received in the current period.

The dynamics of the banking account balance of the shady sector are as follows:

B�3�b½tþ 1� ¼ O�3�s½t� � B�3½t�: (4.285)

The remainder of the budget of the shady sector of the economy is given by

O�3�s½t� ¼ 1� O�3l�P3½t� (4.286)

4.3.1.22 Economic Agent 4. The Aggregated Consumer (Households)

Let us proceed to the formulas determining the behavior of the aggregated

consumer.

The households’ demand of final products:


D�4c�P1½t� ¼ ðO�4c�P1� B�4½t�Þ=P�1c; (4.287)

At market prices:

D�4c�P2½t� ¼ ðO�4c�P2� B�4½t�Þ=P�2c½t�; (4.288)

At shady prices:

D�4c�P3½t� ¼ ðO�4c�P3� B�4½t�Þ=P�3c½t�: (4.289)

The movement of the labor force:

In the state sector:


L�1½t�¼ L�1½t�1��ð1�L�1�2½t�1�þL�1�a½t�1��L�1�r½t�1�ÞþL�2½t�1��L�2�1½t�1�; (4.290)

In the market sector:

L�2½t� ¼ L�2½t�1��ð1�L�2�1½t�1�þL�2�a½t�1��L�2�r½t�1�ÞþL�1½t�1��L�1�2½t�1�; (4.291)

In the shady sector:

L�3½t� ¼ ðL�1½t� þ L�2½t�Þ � L�12�3: (4.292)

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�1½t� ¼ D�1l�P1½t� � P�1l; (4.293)

In the market sector:

W�2½t� ¼ D�2l�P2½t� � P�2l½t�; (4.294)

In the shady sector:

W�3½t� ¼ D�3l�P3½t� � P�3l½t�: (4.295)

The budget of households is determined as follows:

B�4½t� ¼ B�4�b½t��ð1þCP�b�h½t�1�ÞþVB�4½t�1��VO�4�s½t�1�þW�1½t�þW�2½t�þW�3½t�þG�4�tr½t�1�þG�4�f ½t�1�þM�4½t�:

(4.296)


1. The funds in bank accounts;

2. The retained money in cash remaining from the preceding period;

3. The wages received in the state, market, and shady sectors;

4. The pensions, welfare payments, and subsidies received from off-budget funds;

part of the off-budget funds G_1_f;5. The emission of money M_4;


6. The income fromproperty, commercial activity, and other income. This constituent

part of the budget enters the model exogenously to complete the budget to the

values of official statistics.

The dynamics of the bank account balance of households ape as follows:

B�4�b½tþ 1� ¼ B�4½t� � O�4�b: (4.297)

The share of the budget for discharging the income tax is as follows:

O�4�tax½t� ¼ ððW�1½t� þW�2½t�Þ � T��podÞ=B�4½t�: (4.298)

The remainder of the money in cash is

O�4�s½t� ¼ 1� O�4c�P1� O�4c�P2� O�4c�P3� O�4�tax½t�� O�4�b� O�4�buck: (4.299)

4.3.1.23 Economic Agent 5. The State

Economic agent 5 corrects the share of the budget for purchasing final products at

the governmental price. This process is described by the following formula:

O�5g�P1½t;Qþ 1� ¼ O�5g�P1½t;Q� � etta�5gþ O�5g�P1½t;Q�� I�g½t;Q� � ð1� etta�5gÞ; (4.300)

where Q is the iteration step, and 0 < etta_5g < 1 is the model constant.


The consolidated budget obeys the relation

B�5½t� ¼ O�1�t½t� � B�1½t� þ O�2�t½t� � B�2½t� þ O�4�tax t½ � � B�4 t½ �þ B�5�other þ B�5�b t½ � � ð1þ CP�b½t� 1�Þ: (4.301)

This formula sums the money collected as the taxes from the state and market

sectors, as well as from the population. The value B_5_other entering the model

exogenously is the sum of the 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 bank accounts (subject to

the interest on deposits).


described by

B�5�b½tþ 1� ¼ ðO�5b�s½t� � B�5½t�Þ; (4.302)


f�5½t� ¼ O�1�f ½t� � B�1½t� þ O�2�f ½t� � B�2½t� þ F�5�b½t�� 1þ CP�b½t� 1�ð Þ: (4.303)

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:




The derived sum is supplemented by the funds in bank accounts (subject to

interest on deposits).

The dynamics of the banking account balance of the off-budget funds are

determined by

F�5�b½tþ 1� ¼ O�5f�s½t� � F�5½t�: (4.304)

The demand of the final products:


D�5g�P1½t� ¼ ðO�5g�P1½t� � B�5½t�Þ=P�1g; (4.305)

At market prices:

D�5g�P2½t� ¼ ðO�5g�p2� B�5½t�Þ=P�2g½t�: (4.306)

The subsidies to the producing sectors are as follows:

The state sector:

G�1�s½t� ¼ O�5�s1� B�5½t�; (4.307)

The market sector:

G�2�s½t� ¼ O�5�s2� B�5½t�: (4.308)


G�4�tr½t� ¼ O�5�tr � B�5½t�: (4.309)

The assets of the off-budget funds made available for:

The state sector:

G�1�f ½t� ¼ O�5�f1� F�5½t�; (4.310)


The market sector:

G�2�f ½t� ¼ O�5�f2� F�5½t�: (4.311)

The assets of the off-budget funds made available for the population:

G�4�f ½t� ¼ O�5�f4� F�5½t�: (4.312)



4.3.1.24 Economic Agent 6. The Banking Sector

The banking sector of the model includes the central bank and the commercial

banks. This economic agent implements the following functions:

1. Realizes the emission of money, M_1, M_2, M_4;

2. Establishes the deposit interest rate for the enterprises and physical bodies.

4.3.1.25 Economic Agent 7. The Outside World

In this version of the model, all the economic indexes of the outside world are

specified exogenously. This means that the domestic producers cannot export more

products than the outside world needs.


In this subsection we present the formulas for calculating some integral indexes of

the economy.

The GDP (in prices of the base period):

Y½t� ¼ Y�1½t� þ Y�2½t�: (4.313)

The GDP (in current prices):

Y�p½t� ¼ Y�1�p½t� þ Y�2�p½t�: (4.314)

The inflation of consumer prices:

P½t� ¼ 100� ðP�2c½t�=P�2c½t� 1�Þ: (4.315)


The number of people employed within the economy:

L½t� ¼ L�1½t� þ L�2½t� þ L�3½t�: (4.316)

Capital assets:

K½t� ¼ K�1½t� þ K�2½t�: (4.317)

The considered CGE model with the shady sector is presented in the context of

relation (1.7) by 11 expressions (n1 ¼ 11); in the context of relation (1.8) by 98

expressions (n2 ¼ 98); in the context of relation (1.9) by 14 expressions (n3 ¼ 14).

The analyzed model includes 144 exogenous parameters (whose values are

required to be estimated by solving the parametric identification problem) and

123 endogenous variables.

4.3.1.27 Parametric Identification of the CGE Model with the Shady Sector

The problem of the identification (calibration) of the exogenous model parameters

in this case is reduced to finding the global minimum of some objective function

specified by means of the CGE model itself. The constraints on the optimization set

are also specified by means of the model. The problem of searching the global

extremum, generally of high dimension, is rather complex. The methods of random

search, parallel computational algorithms, and others are applied for solving

this problem [17, 20]. A review of the numerous publications devoted to global

extremum search is presented in [69]. For the model considered here, we use

an algorithm of parametric identification that is not presented in the literature.

This algorithm takes into account the characteristic features of the macroeconomic

models of high dimensionality and allows us in some cases to find the global

minimum of the objective function with a large number of variables (more than

100). The algorithm uses two objective functions (two criteria of identification,

namely the main and auxiliary criteria). This allows for the withdrawal of the values

of the identified parameters from the neighborhoods of the local (and nonglobal)

extremum points and to continue searching for the global extremum while simulta-

neously keeping conditions of consistency of the movement to the global

extremum.

Region O ¼ Qlþmþn1i¼1 ½ai; bi�, where ½ai; bi� is the interval of possible values of

the parameter � oi; i ¼ 1; ðlþ mþ n1Þ, is considered to satisfyO � W � L� X1

for estimating the possible values of the exogenous parameters. The estimates of

the parameters with available measured values are searched in the intervals ½ai; bi�with the centers in the respective measured values (in the case of a single value) or

in some intervals covering the measured values (in the case of multiple values).

The other intervals ½ai; bi� for searching the parameters are chosen by indirect

estimation of their possible values. To find the minimum values of the continuous


multivariable function F : O ! R under additional constraints on the endogenous

variables, we used the Neldor–Mead algorithm of directional search [66] in computa-

tional experiments. The application of this algorithm to the initial pointo1 2 O can be

interpreted as the sequence o1;o2;o3; :::� �

convergent to the local minimum o0F ¼

arg minO; ð2Þ

F of the function F, where Fðojþ1Þ � Fðo jÞ, o j 2 O; j ¼ 1; 2; :::. In the

description of the following algorithmwe suppose that the pointo0F can be found with

admissible accuracy.

To estimate the quality of the retrospective prediction on the basis of the

economic data of the Republic of Kazakhstan for the years 2000–2004, for some

starting point o1 2 O we solve the problem (Problem A) of estimating 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" #

: (4.318)

Here t is the number of years;

Y[t] is the calculated GDP in billions of tenge in the prices of the year 2000;

P[t] is the calculated level of consumer prices.

Here and below, the sign * corresponds to the measured values of the respective

variables. The problem of the model parametric identification is considered to be

solved if there exists a point o0KIA

2 O such that KIAðo0KIA

Þ< e for a sufficiently

small e.Parallel with Problem A for the point o1 we also solve a similar problem

(Problem B) using the extended criterion KIB instead of criterion KIA:

K2IB ¼

1

12:15

P2004t¼2000

Y�½t��Y½t�Y�½t�

� 2

þ P�½t��P½t�P�½t�

� 2 �

þ0:1P2004

t¼2000

L��1½t��L�1½t�L��1½t�

� 2

þ L��2½t��L�2½t�L��2½t�

� 2 �

þ0:1P2004

t¼2000

K��1½t��K�1½t�K��1½t�

� 2

þ K��2½t��K�2½t�K��2½t�

� 2 �

þ0:01P2004

t¼2000

Y��1½t��Y�1½t�Y��1½t�

� 2

þ Y��2½t��Y�2½t�Y��2½t�

� 2

þ Y��3½t��Y�3½t�Y��3½t�

� 2

#"

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

: (4.319)

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] is the capital assets of the state sector;K_2[t] is the 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.

The values of the reducing weights in criterion (4.319) are determined as a result

of the identification of the parameters of the specific dynamical system.

Because of the existence of several local minimum points of functions KIA and

KIB, it is rather hard to achieve the near-zero values of these criteria solving the

parametric identification problem for each of these criteria separately.

Therefore, the final algorithm for solving the problem of parametric identifica-

tion of the model is chosen in the form of the following stages:

1. Problems A and B are solved simultaneously for some vector of starting values of

the parameters o1 2 O. As a result, points o0KIA

and o0KIB

are found.

2. If KIA o0KIA

� <e or KIA o0

KIB

� <e, then the problem of parametric identification

of model (4.196–4.207) is solved.

3. Otherwise, using point o0KIB

as the starting point o1, solve Problem A and, using

the point o0KIA

as the starting point o1, solve Problem B. Go to stage 2.

A sufficiently large number of iterations of stages 1, 2, and 3 in some cases gives

an opportunity to the sought-for values of the parameters to withdraw from the

neighborhoods of the nonglobal minimum points of one criterion with the help of

another one and thereby solving the parametric identification problem.

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. Moreover, the

relative value of the deviation of the calculated values of the variables used in

criterion (4.319) 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 4.1 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.

Table 4.1 Results of the model retrospective prediction

Year 2005 2006 2007 2008

Y*[t] 4,258.03 4,715.65 5,136.54 5,303.27

Y[t] 4,221.69 4,586.33 5,004.12 5,478.31

Error (%) �0.861 �2.820 �2.646 3.195

P*[t] 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


4.3.2 Finding the Optimal Values of the Adjusted Parameterson the Basis of the CGE Model with the Shady Sector

In the context of analysis of the connection between some processes of shady

economics and the basic macroeconomic indexes of the country (the GDP and

consumer price index), a number of the computational experiments described

below (the simulation of the scenarios specifying some possible negative effects

within the country’s economy) were carried out. These simulations are similar to

the experiments in [24].

We consider the following six scenarios:

1. Simulation of the process of cash withdrawals (10%, 20%, 30%) from the

consolidated budget of the country and reassigning this cash to households

from the year 2005 (scenarios 1, 2, and 3). We simulate the process of direct

stealing or the quite legal process of the development of budgetary funds (the

kickback process).

2. Simulation of the process of cash withdrawal (10%, 20%, 30%) from the

producers and reassigning this cash to households from the year 2005 (scenarios

4, 5, and 6). In this case we simulate the process of giving (from the direction of

the producers) and taking (ultimately by households) bribes.

The results of applying the enumerated six scenarios of the economic develop-

ment of the country with the negative effect of shady economics in comparison with

the base variant of evolution are presented in Tables 4.2 and 4.3.

An analysis of Tables 4.2 and 4.3 also shows that the analyzed scenarios

insignificantly affect the country’s GDP. At the same time, the consumer price

index increases significantly during the first year of applying scenarios 1–6. In the

following year, their effect on the price index becomes weaker.

Note that the considered aspects of shady economics, namely stealing from the

budget and bribes, result in pronounced negative consequences for the country’s

economy. In both cases, the demand for consumer products grows, which leads to

the natural rise of consumer prices. Moreover, the producer often includes the

expenditures on the bribes in the price of its products, which also leads to a rise

Table 4.2 Values of the GDP (in billions of tenge in prices of the year 2000) for the base variant

and for scenarios 1–6

Year

GDP

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


in prices. In any case, this ultimately hurts the population who are not related to

partitioning the budgetary funds and not accepting bribes and kickbacks.

The next part of the computational experiments is aimed at reducing the negative

effect of each of the considered scenarios to one of 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 19 parameters (uli ; i ¼ 1; 19 is

the parameter number, l ¼ 2005; 2008 is the year number) regulated by the govern-

ment for the period 2005–2008 for each of the considered scenarios. The regulated

parameters are the following:


– The shares of the consolidated budget for financing the state and market sectors

of the economy, as well as for purchasing final products;

– The shares of the state sector budgets for purchasing various kinds of products;

– The shares of various kinds of products produced by the state sector of the

economy for selling in various markets.

The level of the consumer prices of the country for 2008 in relation to 2004 with

the jth scenario 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 4.4.

We consider the following problem of finding the optimal values of the eco-

nomic parameters uli . On the basis of model (4.196–4.317), determine the values of

Table 4.3 Values of the consumer price index (in% with respect to preceding year) for the base

variant and for scenarios 1–6

Year

Consumer Price Index

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


the economic parameters uli ; i ¼ 1:19, l ¼ 2005; 2008, which are optimal in the

sense of criterion K under the above constraints.

The stated problem of finding the minimum value of criterion K as a function in

76 variables (and the respective values of the controlled parameters

uli� � ¼ argminK) for each of the considered six scenarios under the given

Table 4.4 Controlled model parameters and the constraints imposed on them

No. Controlled parameter ui

Interval of admissible

values of controlled

parameter

1 The rate of the VAT [0.135; 0.165]

2 The income tax rate for organizations [0.27; 0.33]

3 The property tax rate [0.009; 0.011]

4 The income tax rate for physical bodies [0.135; 0.165]

5 The rate of the single social tax [0.099; 0.121]

6 The share of the consolidated budget for purchasing

final products

[0.117; 0.143]

7 The share of the consolidated budget for backing the

state sector

[0.325; 0.398]

8 The share of the consolidated budget for backing the

market sector

[0.028; 0.034]

9 The share of the consolidated budget for social

transferring

[0.320; 0.391]


capital products

[0.129; 0.158]


investment products

[0.068; 0.083]

12 The share of products of the state sector for selling in

the markets of final products for the market sector

[0.101; 0.123]


the markets of final products for economic agent

5 at exogenous prices

[0.039; 0.048]


the markets of final products for economic agent

5 at market prices

[0.039; 0.048]


the markets of investment products at exogenous

prices

[0.107; 0.131]


the markets of investment products at market

prices

[0.107; 0.131]


the markets of capital products at exogenous

prices

[0.200; 0.244]


the markets of capital products at market prices

[0.200; 0.244]


the markets of final products in foreign countries

[0.230; 0.281]


constraints is solved by the Nelder–Mead algorithm. The results are presented in

Table 4.5.

Analysis of Table 4.5 shows that in the case of the considered scenarios, the

parametric control approach allows reducing the level of prices of the year 2008 by

9.4–13.2% while increasing the GDP of the country in 2008 by 1.3–2.44% in

comparison with the case without control.

Table 4.5 Results of the application of the parametric control approach

Year

Criterion Kwithout

parametric

control

Criterion Kcorresponding to

found optimal

values of

parameters

Yj�½2008� inbillion tenge

Yj½2008� inbillion 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

The methodology of modern macroeconomic analysis and prediction includes

the achievements of parametric control theory and the methods of analysis and

prediction constructed on its basis.

One of the strong features of this theory consists in the fact that its development

and correct application to problems in macroeconomic analysis within a certain

range allow us to solve problems of the control of macroeconomic systems by

choosing macroeconomic parameters within admissible limits, which allows us

to direct the vector of economic dynamics to the course of an optimal scenario or

to explain why this cannot be successful. The scenario approach to the formation

and realization of macroeconomic politics based on the achievements of parametric

control theory allows making substantial conclusions about the modern state and

about the promises of development for the economy of the Republic of Kazakhstan.

The economic dynamics of the country are characterized by relatively high

stability of some internal parameters (the total level of prices, level of unemploy-

ment, refinancing rate). A level of lesser stability is observed for a number of

the parameters connecting the Kazakhstan economy with the world economy

(the exchange rate of the US dollar, world prices of energy, whose oscillations

result in “pulsation” of net exports). Preliminary estimates show that the maximum

sensitivity can be seen under the oscillation of prices of the exported products.

At the same time, the shocks of the aggregate demand caused by reducing

net exports can unbalance the economy. The unfavorable dynamics of the US dollar

exchange rate also shift the balance between the tendency to saving and the

tendency to consumption, which causes an outflow of investments from the

Kazakhstan economy.

Raising the volume of state investment and expenditures is the main instrument

for recovering the economy from crisis. A rational increase in the share of govern-

ment spending in the GDP in a short-term period results in increasing investments

and the volume of final consumption. The optimal volume of government spending

for the modern economy of Kazakhstan is equal to 40–45% of GDP, which is

significantly greater than the level that can be seen today. This also requires


249

increasing the share of gathered taxes in the GDP and carrying out relatively strict

monetary policy excluding the inflationary overheating of the economy.

The volume of aggregate demand determinatively depends on the value of

government spending and investments and, to a lesser degree, on the credit interest

rate and currency exchange rate. This is concerned with the fact that the level of

final consumption shows low correlation with the interest rate. Hence government

spending should be considered the main instrument of the short-term stimulation of

the GDP, which is consistent with the Keynesian logic of macroeconomic policy.

The rate of saving is significant for the long-term trajectory of economic growth.

Many of the equilibrium models (both Keynesian and neoclassical) show that the

rate of saving is not important in the long run, because the long-term rate of growth

will be close to the guaranteed rate of growth of the most critical recourse in

this macroeconomic system anyway. However, such a conclusion is correct for

the equilibrium economy, where investment is equal to saving. But this is not so in

real macroeconomic systems. Therefore, a decrease in the rate of saving that can be

seen during the last year gives reason for concern, because the threat of its further

decrease is quite realistic.

The net export is an important constituent part of the aggregate demand in

the Republic of Kazakhstan, and the equilibrium trajectory in the real sector of

the economy (the line “investment–saving”) is highly sensitive to its volume.

The macroeconomic system of the Republic of Kazakhstan is subject to the

effect of external shocks including financial ones. The sensitivity of the economy

to the level of the currency exchange rate is maximal. This is minimal with respect

to the volume of foreign loans. In other words, foreign loans (both governmental

and private) do not significantly influence the volume of the aggregate demand,

hence upon the volume of the produced GDP.

Aweakening of the national currency (the decrease in its exchange rate) causes the

outflow of investments from the economy and decreases the retirement coefficient and

the net export. This means that the domestic investment potential of the country is

actually higher than that currently realized in the conditions of shocks of the aggregate

demand conducted by the institutional (including financial) system of the country.

The restricting influence of foreign market trends on the development of the

Kazakhstan economy is beyond question. The “foreign sector” does not allow

the realization of the investment potential to exist internally. Thus, the economy

seems to be excessively open. It is not sufficiently strong to effectively withstand

external shocks. The Kazakhstan economy would be more stable with respect

to external shocks if the role of the government were stronger and the level of its

controllability higher.

First of all, there is a need for certain isolation of some individual elements

of credit and the banking system. The institutional division of “short-term” and

“long-term” risks, as well as strengthening the control of the market of “hot” short-

term liabilities having mainly a speculative nature, are required. Circulation of such

financial instruments in conditions of uncontrolled international capital mobility is

fraught with the formation of financial “bubbles” and collapse of individual

segments of the financial market of the country.

250 5 Conclusion

Econometric analysis also shows that the volume of the money aggregates,

besides the dynamics of the GDP and the common level of prices, is conditioned

by other factors of great significance. One can trace the statistically significant

dependence between the volume of the money aggregates and oscillations of the

currency exchange rate. This has a certain adverse effect on the economic dynamics

of the country.

Increasing the interest rate of government loans does not have any significant

influence on the inflow of investment resources and cannot weaken the inflationary

gap in the case of its appearance. The volume of the produced GDP is several times

more sensitive to the share of government spending in the GDP than to the volume

of foreign loans. This volume is almost insensitive (in a short-term period) to the

inflation rate.

In forming currency reserves, one should not figure on a speedy recovery of the

world means of payment. It is necessary to hold the currencies of stronger countries

(China and Russia) as currency reserves until they are officially declared to be

the regional reserve currencies.

The main directions of economic development must consist in the implementation

of moderate politics of protectionism with respect to domestic producers. At the same

time, there exists a danger of applying the methods of direct control of some local

markets (for instance, fixing the commercial interest rate). Such measures should not

be taken.

Principal attention should be given to stimulating aggregate demand and

forming a system of mutually guaranteeing the investments in the financial and

banking system.

In the modern economy of Kazakhstan, the line of the aggregate supply is more

stable, whereas the line of the aggregate demand is more mobile. At the same time,

the aggregate demand and aggregate supply notably diverge in structure.

This intensifies the country’s dependence on the import of high-tech products.

This requires optimizing the country’s participation in the international division

of labor.

Correlation analysis of the macroeconomic parameters shows that the Republic

of Kazakhstan remains on the threshold of large-scale technological modernization.

Emphasis on the development of its own high-tech industry with the aim of

reducing the national economy’s dependence on high-tech imports must become

one of the key directions of its economic policy.

A significant role in solving this problem will be played by the income policy.

In the absence of regulating measures, the unappreciated living labor (just as in

Russia) will become a serious obstacle along the path to the radical modernization

of the national economy. In this connection, special attention must be given to the

medium-term and long-term consequences of the measures taken.

The timely and correct solving of these complex problems is of crucial importance

for the development of a modern economy for the Republic of Kazakhstan.

5 Conclusion 251

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About the Authors

Abdykappar Ashimovich Ashimov is an academician at the National Academy

of Sciences of the Republic of Kazakhstan, doctor of technical sciences,

professor at Kazakh National Technical University named after K.I. Satpaev,

e-mail: [email protected].

Bahyt Turlykhanovich Sultanov is the adviser of the state scientific and technicalprogram of Kazakh National Technical University named after K.I. Satpaev,


Zheksenbek Makeevich Abdilov is a doctor of economic sciences, professor

at Kazakh National Technical University named after K.I. Satpaev, e-mail:

[email protected].

Yuriy Vyacheslavovich Borovskiy is a candidate in physical and mathematical

sciences, assistant professor at Kazakh National Technical University named after

K.I. Satpaev, e-mail: [email protected].

Dmitriy Alexandrovich Novikov is a corresponding member of the Russian

Academy of Sciences, doctor of technical sciences, professor, Trapeznikov Institute

of Control Sciences of the Russian Academy of Sciences, e-mail: [email protected].

Robert Mikhailovich Nizhagorodtsev is a doctor of economic sciences, principal

research officer of Trapeznikov Institute of Control Sciences of the Russian

Academy of Sciences, e-mail: [email protected].

Askar Abdykapparovich Ashimov is a researcher at the state scientific and tech-

nical program of Kazakh National Technical University named after K.I. Satpaev,


257

Index

AAD–AS model

finite consumption

credit interest rate, 79

interest and exchange rate, 80

Keynesian model, 76

model estimation, 79–80

investment

correlogram, 81

currency exchange rate, 80

economy, 81

interest rate, 80

macroeconomic dynamics, Republic of

Kazakhstan

correlation matrix, 78

statistical data, 77

net export

aggregate demand and supply,

90–91

autocorrelation function, 88


economy growth, 91

governmental investment, 90

internal potential and external

factors, 88

oil price, 89

problem statement

basic macroeconomic identity, 76

gross domestic product, 75

state expenses

autocorrelation function, 83

consumption function, 83

currency exchange rate, 83–84

gross domestic product, 87

investment level, 86

public expenses, 86

regression model, 85

Algorithms

numerical, 151

optimal parametric control law, 148, 155,

157, 159

BBalance of payment (BP) curve

economic system, Kazakhstan, 111

foreign market equilibrium, 109

GDP and interest rate, 110

regression equations, 109

CComputable general equilibrium (CGE) model

description, 9–11

economic branches

aggregate consumer (households),

economic agent 23, 208

endogenous variables, 198–200

exogenous parameters, 196–198

general part, 200

government, economic agent 24,

208–210

integral indexes, 200, 210–211

model agents, 195–196

model markets, 201–203

optimal parametric control laws,

211–215

producing products and services,

economic agents 1–22, 203–207

technical variables, 200

elements, parametric control theory, 11–14

with knowledge sector


economic agent 4, 184–186

259

banking sector, 169



general part, 169, 173


186–188




optimal parametric control laws, 189–194

other branches of the economy,


science and education sector, economic

agent 1, 177–179

sector of innovation, economic agent

2, 179–182

technical parameters, 170

technical variable, 173

shady sector

aggregated consumer (households),

economic agent 4, 219, 236–238

banking sector, economic agent 6, 220,

240

economic agents, 215–217



finding, optimal values, 244–247

general part, 220

government, 220

integral indexes, 240–241

market sector, economic agent 2,

232–235

model constants, 221


outside world, economic agent 7, 240

parametric identification, 241–244

state, economic agent 5, 238–240

state sector, economic agent 1, 228–232

Continuous-time dynamical system, 8–9

Continuous-time system, 6–8

Cycle stability, Kondratiev cycle, 150–151

Cyclic dynamics parametric control

Goodwin mathematical model

analysis, optimal parametric control

law, 161

description, 152–153

problem, optimal parametric control

laws, 154–157

structural stability analysis, 153–154,

157–160

Kondratiev Cycle, mathematical model

criterion K optimal value, variational

calculus problem, 151–152


economic system evolution, 148–150

robustness estimation, 147

structural stability estimation, 150–151

DDiscrete-time dynamic systems, 5, 6, 14, 28

EEconomic branches, CGE model



endogenous variables

Sectors 1–22, 199

Sectors 23, 199

Sectors 24, 200

exogenous parameters

banking sector, 198

Sectors 1–22, 197

Sectors 23, 197

Sectors 24, 197–198

general part, 200

government, economic agent 24, 208–210




optimal parametric control laws, 211–215

producing products and services, economic

agents 1–22, 203–207


Economic equilibrium models, 99

Economic growth control

general equilibrium (see Computable

general equilibrium (CGE) model)

national economic evolution control

based CGE model (see Nationaleconomic evolution control based

CGE model)

Economic instruments

equilibrium state macro-estimation

endogenous parameters, 100

GNI actual and equilibrium values, 100

Keynesian mathematical model, 97

level of prices, 100

regression coefficients, 99–100

wealth and money markets

actual economic state, 98

fluctuation results, 99

IS models, 98

joint equilibrium, 98

macro-estimation, 97–98

260 Index

FForrester’s mathematical model

bifurcation points, 74

choosing optimal laws, 70–73

model description, 66–70

structural stability, 70, 73–74

GGlobal extremum point, 241

Goodwin mathematical model

analysis, optimal parametric control law, 161


problem, optimal parametric control laws

computational experiments, 156

economic parameter k, 155

market cycle, 156, 157

relations set, 154–155

solving problems, stages, 155–156

structural stability analysis

with parametric control, 157–160

without parametric control, 153–154

IInflation rates prediction

autoregression models

three preceding years, 142–145

two preceding years, 142–144

factor regression models

capital assets, 135–136

consumer goods and services, 135

currency exchange rate, 138–139

initial data, 128–129

investment in capital asset, 137

monetary aggregate volume, 133–134

net export volume, dependence, 127

partial correlation matrix, 131–132

physical volume of industrial

production, 136

preparation, 127–139

R&D and innovation, 134

relative increments, 130

renewal coefficient, 137–138

volume of incomes, 134–135

multifactor regression models

net export volume, 139–140

R&D and innovation, 141

wear and tear of capital assets, 140–141

Republic of Kazakhstan, 125

Investment–savings (IS) Curve

correlation coefficient, 106

equilibrium states, 103

GDP, 103

investment level, 107


statistical insignificance, 107

theoretical requirements, 108

IS-LM-BP model

balance of payments, 112

income–interest rate, 112

model values, 112

money market, 113


IS–LM model and Mundell–Flemming model

“balance of payment” (BP) curve, 109–111

“investment–savings” (IS) curve, 103–107

IS-LM-BP model, 111–113

“liquidity–money” (LM) curve, 107–109

problem statement and data preparation,

102–103

IS model and analysis, economic instruments

macroeconomic theory, 94

macro-estimation, 92

plots, 93

public expenses and taxation, 94

statistical characteristics, 92

KKazakhstan, 242

Keynesian model

economic instruments

comparative analysis, 100

level of prices, 101

regression coefficients., 99–100

parametric control, 101–102

Knowledge sector, CGE model



banking sector, 169

endogenous variables

Sector 1, 170–171

Sector 2, 171

Sector 3, 171–172

Sector 4, 172

Sector 5, 172


Sector 1, 167

Sector 2, 167–168

Sector 3, 168

Sector 4, 168

Sector 5, 169


government, economic agent 5, 186–188


Index 261

Knowledge sector (cont.)model agents, 164–166



189–194

other branches of the economy, economic

agent 3, 182–184


agent 1, 177–179

sector of innovation, economic agent 2,

179–182



Kondratiev cycle, mathematical model

criterion K optimal value, variational

calculus problem, 151–152


parametric control, economic system

evolution

capital productivity ratio, 149, 150

coefficients and criteria values, 149

innovations efficiency, 149, 151

optimal laws, relations, 148

optimal values, criteria, 148

robustness estimation, without parametric

control, 147

structural stability estimation, 150–151

LLarge number of variables, 241

Liquidity–money (LM) curve

description, 108

regression equation, 109

MMacroeconomic analysis and parametric

control

equilibrium solutions and balance of

payments

common economic equilibrium, 122

IS-LM-ZBO, 122–123

money supply and public expense, 121

national currency exchange, 122

national economy in 2007, 122

open economy mathematical model,

small country


domestic commercial interest rate, 118

double balance in 2007, 119–120

double equilibrium in 2008, 119–120

econometric methods, 115

equilibrium and actual values in 2007

and 2008, 120–121

gross national income (GNI), 114

investment model, 116

money velocity, 114–115

second-level banks, 114

solving system, 119


wealth export model, 117

small country model

dependence of optimal values, 125–126

exogenous parameters external, 124

unemployment, 113

Macroeconomic models, 163

Market cycle, 145

Mathematical model, national economic

system

analysis methods, 3

chain-recurrent set

discrete-time dynamical system, 5

localization algorithm, 4–5

computational algorithm, 4

continuously differentiable mapping, 6

Forrester’s model (see Forrester’smathematical model)

international trade and currency exchange

bifurcation points, 65–66



structural stability, 56–57, 62–65

neoclassic theory, optimal growth




structural stability, 20–21, 23

public expense and interest rate of

government loans


choosing optimal law, 37–43


parametric control, 43–45

structural stability, 36–37, 45–49

stages, invertibility test, 5–6

structural stability, 4

Mathematical models of cycles

Goodwin (see Cyclic dynamics parametric

control)

Kondratiev cycle (see Cyclic dynamics

parametric control)

Money market equilibrium conditions

Fisher equation, 95

LM models, 94–97


262 Index

values of multipliers, 95–96

velocity of money, 95

NNational economic evolution control based

CGE model

economic branches





general part, 200


208–210





211–215

producing products and services,

economic agents 1–22, 203–207


knowledge sector



banking sector, 169





186–188





189–194

other branches of the economy,



agent 1, 177–179

sector of innovation, economic agent

2, 179–182



shady sector

aggregated consumer, 219



banking sector, economic agent 6, 240





general part, 220

government, 220


market sector, economic agent 2,

232–235







National economic markets, equilibrium states

AD–AS model

finite consumption, 76–80

input data, 76

investment, 80–83

net export, 87–91

problem statement, 75–76

state expenses, 83–87

inflationary process modeling

autoregression models, 142–144

factor regression models, 127–139

multifactor regression models, 139–141

IS–LM model and Mundell–Flemming

model

“balance of payment” (BP) curve,

109–111

“investment–savings” (IS) curve,

103–107

IS-LM-BP model, 111–113

“liquidity–money” (LM) curve,

107–109

problem statement and data

preparation, 102–103

macroeconomic analysis and parametric

control

equilibrium solutions and balance of

payments, 121–124

small country, equilibrium conditions

estimation, 114–121

small country model, 124–125

macroeconomic analysis economic

instruments

common economic equilibrium, 99–101

equilibrium conditions, money market,

94–97

IS model and analysis, 91–94

Keynesian model, parametric control,

101–102

mutual equilibrium state, 97–98

Neoclassic theory, optimal growth


Index 263

Neoclassic theory (cont.)choosing optimal laws, 21–22



OObjective functions, 12, 19

One-sector Solow model


dependence, 28

model description, 25

parameter estimation, 25–26


Optimal control

choosing optimal laws

CGE models, 9–14

continuous-time dynamical system, 8–9

continuous-time system, 6–8

Forrester’s mathematical model, 70–73

international trade and currency exchange

first-stage results, 61

numerical solution, 60

problem, choosing optimal pair, 62

research program, 59–60

stages, choosing optimal pair, 59

total production capacity, 58

neoclassic theory, optimal growth, 21–22

one-sector Solow model, 26–27


government loans

choice, optimal pair, 40

control law expression, 41

economic parameters, 42

numerical solution results, 39–40,

42–43

order and relations, choosing optimal

laws, 37–38

problem solving stages, 38

Richardson model, 31–33

PParametric control laws

CGE models, 9–14

Forrester’s mathematical model

bifurcation points, 74




influence, uncontrolled parametric

disturbances, 14–16

neoclassical theory, optimal growth




structural stability analysis, 20–21, 23

one-sector Solow model


dependence, 28




Richardson model, defense costs estimation


dependence, 33


parameters, 29–30

structural stability, 30, 33

variational calculus problem statement



Parametric control theory

analysis methods, structural stability, 3–6

application algorithm

aggregate scheme, decision-making,

17–18

definition and implementation, public

economic policy, 16–17

choosing optimal laws, variational calculus

problem

CGE models, 9–14



components, 2–3

econometric methods, 1–2



mathematical model, national economic

system (see Mathematical model,

national economic system)

neoclassic theory, optimal growth





one-sector Solow model


dependence, 28




Richardson model


dependence, 33

estimation, model parameters, 29–30

264 Index



scenario approach, 2

Parametric identification, 189, 195, 210,

215, 241–244

RRichardson model, defense costs estimation


dependence, 33


parameters, 29–30


SShady sector, CGE model


economic agent 4, 219, 236–238

banking sector, economic agent 6, 240




Sector 1, 218

Sector 2, 218–219

Sector 3, 219


general part, 220

government, 220


market sector, economic agent 2, 232–235







Solution existence theorem, 8, 14

Solution of parametric control problems

computational experiments, 156

economic parameter k, 155

market cycle, 156, 157

relations set, 154–155

solving problems, stages, 155–156

State of the national economy. See Nationaleconomic markets, equilibrium

states

Structural stability, mathematical model

Forrester’s mathematical model,

70, 73–74

international trade and currency exchange,

56–57, 62–65

neoclassic theory, optimal growth,

20–21, 23

one-sector Solow model, 26, 27–28


government loans, 36–37, 45–49

Richardson model, defense costs

estimation, 30, 33

VVariational calculus problem

bifurcation points

international trade and currency

exchange, 65–66

neoclassical theory, optimal growth

with parametric control, 24–25

CGE models, 9–14





Richardson mathematical model, 33

Solow mathematical model, 28

Index 265

macroeconomic analysis and economic policy based on parametric control

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